Designing a novel microlaser on a hybrid III-V/silicon platform using resonant photonic crystal cavity reflectors

Transcription

1 Designing a novel microlaser on a hybrid III-V/silicon platform using resonant photonic crystal cavity reflectors Andreas De Groote Promotoren: prof. dr. ir. Günther Roelkens, prof. dr. ir. Roel Baets Begeleider: Yannick De Koninck Masterproef ingediend tot het behalen van de academische graad van Master in de ingenieurswetenschappen: fotonica Vakgroep Informatietechnologie Voorzitter: prof. dr. ir. Daniël De Zutter Faculteit Ingenieurswetenschappen en Architectuur Academiejaar

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3 Designing a novel microlaser on a hybrid III-V/silicon platform using resonant photonic crystal cavity reflectors Andreas De Groote Promotoren: prof. dr. ir. Günther Roelkens, prof. dr. ir. Roel Baets Begeleider: Yannick De Koninck Masterproef ingediend tot het behalen van de academische graad van Master in de ingenieurswetenschappen: fotonica Vakgroep Informatietechnologie Voorzitter: prof. dr. ir. Daniël De Zutter Faculteit Ingenieurswetenschappen en Architectuur Academiejaar

4 Preface In the spring of 200, I was in my third and last year of Bachelor in Electrical Engineering. I had started this studies with lots of expectations when it came to designing possibilities. A computer, or more in general electronics, could do everything and I wanted to be part of the making of one that could do even more. After three years I had learned a lot and I enjoyed combining the different principles to understand and even create new devices. Although my bachelor thesis had earned a lot of my time and attention, I had this hollow feeling deep inside me working his way up. The improvement in electronics relied too much on little tricks and the major leap forward I wanted to be part of, had already occurred. At the same time, I was enrolled in this new course called photonics. It introduced a new and exciting technology platform with a lot of possibilities ahead. It managed to combine the infinite design space of electronics and the promise of a technological revolution. Week after week, photonics nibbled from the lead of electronics and from time to time even a big chunk at once. At the end of my Bachelor, the choice was made. The adventurous path of photonics would be mine. Photonics has a lot of possibilities, but since it is young also a lot of problems. The main problem, and therefore also the bottleneck holding back the technology, is a silicon laser and, ambitious as I am, I wanted immediately to tackle the big one. The master thesis subjects were listed and I had to force myself to read through the list carefully not to rush into something I would regret. I asked extra information to the professors and went to see the supervisors. When I met Yannick, there was a match in personalities. The subject was promising and, after our talk, ingeniously simple at the same time. The execution of this master thesis was frustrating at times. The long simulation times make the setback of a bad result even bigger and require you to be available to start a new one at any moment of the day as time is valuable. Of course this works in two ways. The harder you need to work for a result, the more satisfying it is. To illustrate this, I will try to describe the feeling I experienced when I saw my first reflection peak. For weeks, I had been trying to match the III-V waveguide to the triplerail cavity, which I had created only shortly before the mask deadline. I had encountered some decent spectra in my simulations, but not near good enough for the laser. After some dead-ends I had tried a new approach and all of a sudden I saw this reflection spectrum. It stayed low for almost the entire wavelength range, but at the right wavelength, at the resonance, there was a peak. The reflection was only 35% but it was a reflection nonetheless. If I would have been a great dancer, I would have put up quite a performance, but luckily I am not and I only did a modest dance on my seat. One does not live trough a year of hard work without the good guidance of supervisors and the unconditional support of friends and family. In view of this, there are a lot of people to thank, but some deserve special attention. First of all, I would like to thank prof. Baets and prof. Roelkens for the opportunity. They were very involved with my master thesis and always open for questions, even when they had to rush to a more important meeting. They did not rule with iron hand, but also my opinion mattered in what I was doing. They have created a very innovative atmosphere in which it is pleasant to work and think for yourself. iv

5 v All this also applies to Yannick, my supervisor, but he had the extra trouble of solving my practical problems like getting the software to work and interpreting strange results. He always had time, no matter how insignificant a problem might be. He even had time to discuss who would win the next cycling game. I am very happy for Yannick that Tom Boonen won as much as he did. The professors and Yannick were also keen to learn more about the subject. Nowadays, scientists do not stay on their island, but cooperate with other groups with more, different experience. We invited Nicolas Le Thomas from the École Polytechnique Fédérale de Lausanne to discuss the photonic crystal cavity of our design. His input was very helpful and triggered the start of a long line of simulations and results. In the technicum, the building I worked in, I shared a room with the other master thesis students. I had a great time with Daan, Jelle, Christopher, Agnes and the others. They were always in for a discussion or a joke and eased the burden when it was getting heavy. In the evening, I was able to tell what I did that day to Sarah, my girlfriend. She listened with honest interest and often asked simple but important questions. Sarah was the sounding board I needed. Last but not least, I want to thank my family. Not for the important questions they asked or the scientific input, as they have a totally different background, but just because they were there and tried to grasp a fraction what I have been doing for an entire year. Andreas De Groote June 4 th, 202

6 Permissions The author(s) gives (give) permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Andreas De Groote, June 202 vi

7 Designing a novel microlaser on a hybrid III-V/silicon platform using resonant photonic crystal cavity reflectors by Andreas De Groote Master thesis dissertation for achieving the academic degree of Master of Science in Engineering: Photonics Academic year Promotors: prof. dr. ir. Günther Roelkens, prof. dr. ir. Roel Baets Supervisor: ir. Yannick De Koninck Department of Information Technology President: prof. dr. ir. Daniël De Zutter Faculty of Engineering and Architecture Ghent University Summary This master thesis dissertation deals with the problem of creating a hybrid microlaser using deep UV techniques. To ensure reflection, the mechanism of resonant cavities is used. This already proved to work using gratings, but the option of photonic crystals was not yet investigated. Three mirrors were built: a passive resonant monorail mirror, an active resonant monorail mirror and an active resonant triplerail mirror. A monorail is a waveguide with a D photonic crystal, a row of holes, written in it. The triplerail is the multirow analogon. Both monorails showed a nearly perfect reflection of 97% at the resonance wavelength, the triplerail achieved 90% reflection. The different parameters are discussed: the number of holes on the left and the right of the cavity, the waveguide width and the bonding thickness. Also the misalignment error was taken a look at. It was explained that the triplerail suffers from fundamental problems, preventing perfect reflection. Keywords photonic crystals, resonant cavity, microlaser, hybrid silicon/iii-v, monorail vii

8 Designing a novel microlaser on a hybrid III-V/silicon platform using resonant photonic crystal cavity reflectors Andreas De Groote Supervisor(s): Yannick De Koninck, Günther Roelkens, Roel Baets Abstract This article discusses the use of photonic crystals in a resonant cavity system in the hybrid silicon/iii-v platform. The most important parameters are discussed and three highly reflective structures are proposed (maximal reflection of 97%, 96% and 90%), based on monorail and triplerail cavities. These mirrors can be used to form a microlaser cavity. Keywords photonic crystals, resonant cavity, microlaser, hybrid silicon/iii-v, monorail I. Introduction AN integrated laser source is the bottleneck of silicon photonics. Hybrid integration seems the most promising platform, using the gain characteristics of a III-V die bonded on the silicon chip. Depending on the location of the mode, mainly in the die or in the silicon, various problems occur. The device, e.g. a microdisk laser, can be very small when the mode lives in the III-V die, but often it is hard to process it. A silicon mode device, e.g. an evanescent laser, can make use of CMOS compatible deep UV technology, but is a long device due to the low gain. Therefore, we propose a laser design making use of both, separating a gain and a mirror section based on resonant cavities. II. Working principle of resonant cavities The resonant cavity design consists of a III-V waveguide and a silicon cavity buried beneath it. The working principle is illustrated in figure, where we show one of the two mirrors forming the laser cavity. due to the resonant behaviour of the cavity a power buildup can occur. At the other side of this cavity, part of the light couples back to the III-V waveguide. Under the right conditions, this light interferes destructively with the light that never coupled to the silicon and the transmission will be zero. At the same time the light coupling in the other direction to the III-V waveguide will ensure reflection. Note that the output to a silicon passive waveguide can be done very elegantly by making the mirror on one of the sides of the silicon cavity shorter. [] There are two conditions to be fulfilled to realize the destructive interference. Firstly, we need phase matching. The effective refractive index of the cavity and of the waveguide should be equal, so that the relative phase shift is only due to the double coupling. Since every coupling introduces a π 2 phase shift, the total shift is π. Secondly, the coupling should be at the critical level compared to the distributed reflection. We come back to this in the discussion on the bonding thickness in section III-A. The silicon cavity can be made using gratings, but we opted for photonic crystals. Since the III-V medium is shaped as a waveguide, the effective index in the vertical direction will be completely different at the center and at the cladding. This causes the photonic band gap of the crystal to change substantially, making the design more difficult. Therefore, we opted for a monorail design, which makes use of index guiding in this direction. Only in the direction of propagation, confinement is ensured by the band gap. [2] III. Discussion of design parameters A resonant monorail cavity mirror structure is illustrated on figure 2. The structure shown is a passive device for ease Fig.. Working principle of a resonant cavity design. [] Let us say light is coming from the left, propagating to the right in the III-V waveguide. At the position of the silicon cavity a small fraction of the light will couple to the silicon. This is only a very small fraction as the spacing is typically 300nm. Normally we would neglect this, but Source Detector Detector Fig. 2. Waveguide gap Waveguide width Taper out N left Taper in Top view of a passive resonant monorail cavity mirror. of illustration, but the goal is to make an active one. In that case, the silicon input waveguide will be a III-V waveguide lying on top of the structure. The different parameters are Taper in Nright Taper out

9 shown. The inner tapers serve the purpose of increasing the cavity quality factor, the outer one is to suppress band edge modes. However, the main parameters in a resonant cavity design are N left, N right, the waveguide width and the waveguide gap. In an active design, the bonding thickness will play the role of the waveguide gap. We will now discuss the influence of each of these parameters. A. Waveguide gap - bonding thickness If the bonding layer or gap is too thick, the cavity mode will need to be very strong in order to make sure the destructive interference results in 0% transmission. In this case, the scattering losses of the cavity will increase to the order of the coupling losses. A larger fraction of the light is scattered away, thus lost for reflection, and the cavity mode is not strong enough to compensate for the light in the III-V waveguide, thus leading to transmission. If the bonding layer or gap is too thin, the structure will be overcoupled. The reflected light will couple an extra time from the waveguide to the silicon layer, where it is normally scattered away. B. Waveguide width: phase matching condition If the structure is not phase matched, the destructive interference will not be ideal. A fraction of the light will be transmitted and thus, can not contribute to reflection. The III-V waveguide we used was very broad (.85µm), meaning that the influence of the width on the effective refractive index is limited. For smaller waveguide widths, the change in index can be larger, and the reflection will drop substantially. C. Waveguide misalignment Due to misalignment of the III-V waveguide in an active device, the coupling strength might drop. Therefore, a misalignment of the waveguide with respect to the cavity has a similar influence as a too thick bonding layer. Due to a large offset, coupling to higher order modes might also become more favorable. As these do not interfere with the ground mode of the waveguide, this light is lost. D. The number of holes on the left The left hand side of the cavity (i.e. the input side) has a double function. On one hand, it is one of the mirrors forming the silicon cavity, on the other hand it defines the distance over which coupling is possible. Making N left very small results in a very lossy cavity. Most of the light will be scattered away at the edge meaning that it can not contribute to reflection anymore. Making N left very large has the same effect as making the bonding thickness too small. Light will couple back to the silicon and lower the total reflection. Ideally, one should make sure the amplitude of the mode at the edge of the silicon cavity is zero. This way, no scattering can occur at the edge and all this light can contribute to reflection. Power flux (a.u.) Reflection Transmission Wavelength (µm) Fig. 3. Transmission and reflection spectrum of a passive monorail mirror. E. The number of holes on the right The right hand side of the cavity is less critical than the left hand side. As long as this side is long enough, the cavity will not be lossy. Ideally, no light is present in the waveguide (at the right of the cavity) meaning that nothing can couple back to the silicon. This takes away the limit on too many holes we encountered for N left. We can use this parameter N right to suppress other modes. We see that in the monorail design, the band edge modes mainly reside at the right of the cavity. By shortening this side, we can make these unwanted modes lossy and thus prevent reflection at unwanted wavelengths. IV. Simulation results We have investigated three different devices: a passive monorail mirror, an active monorail mirror and an active triplerail mirror. The passive resonant monorail cavity mirror is the device shown in figure 2, where a silicon waveguide couples sideways to the monorail cavity. The transmission and reflection spectrum is shown in figure 3. The maximal reflection situated at the cavity mode, at a wavelength of 527nm, is 96% with a Q of 30. If we use a III-V waveguide on top of the cavity in stead of a sideways coupled silicon one, we can build a laser cavity. The spectrum, shown in figure 4, is very similar to the one in figure 3. The maximal reflection is quasi the same (97%), the Q is 240. As said before, it is important to phase match the structures. Since the effective refractive index of the cavity is relatively low (n eff = 2.29), phase matching implicates the III-V waveguide needs to be narrow. In the active monorail mirror, the waveguide was only 585nm wide. These dimensions are possible for advanced III-V processing, but we would like to be able to use more rudimentary techniques. In order to increase the index we proposed the multirail design, being a broader silicon waveguide with several rows of holes in it, resulting in an index of The spectrum of a triplerail mirror with a.85 micron wide III-V overlay waveguide is shown in figure 5. The maximal reflection is 90% at a wavelength of 563nm (with a Q of 220). The reason for the lower reflection in comparison with the ones

10 Power flux (a.u.) Reflection Transmission Wavelength (µm) Fig. 4. Transmission and reflection spectrum of an active monorail mirror. was 90%. Acknowledgments We would like to thank Nicolas Le Thomas of the École Polytechnique Fédérale de Lausanne for his input to this research. References [] Y. De Koninck, G. Roelkens, and R. Baets, Hybrid III-V/silicon micro-lasers based on resonant cavity reflectors, 5th Annual Meeting of the IEEE Photonics Society Benelux chapter, 200. [2] A.R.M. Zain, N.P. Johnson, M. Sorel, R.M. De La Rue, et al., Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI), Opt. Express, vol. 6, no. 6, pp , Reflection Transmission Power flux (a.u.) Wavelength (µm) Fig. 5. Transmission and reflection spectrum of an active triplerail mirror. above is to be found in the coupling to higher order waveguide modes. The Bloch mode of the triplerail resembles the higher order modes better and thus gives rise to more efficient coupling. As the modes of a waveguide are orthogonal, no destructive interference can occur. V. From mirror to cavity We can use two of these mirrors and create a laser cavity with it. When doing this, it is important to synchronize the three resonances: two resonance wavelengths of the silicon cavity of the mirror and a resonance wavelength of the longitudinal mode between the mirrors. A laser cavity with a total length of 20µm, the gain section is then 5µm, results in a quality factor of the order of VI. Conclusions We have successfully used photonic crystal cavities to build a hybrid mirror based on the resonant cavity principle. After shortly introducing this principle, we discussed the different parameters of our cavities. By choosing them appropriately we were able to build three highly reflecting mirrors. Both the passive and active monorail design achieved nearly a perfect reflection (96-97%). In order to create a device that can be fabricated with the simple contact lithography technique, we proposed a triplerail cavity. Because of the high effective refractive index, broader III-V waveguides can be used. This cavity s maximal reflection

11 Designen van een nieuwe microlaser op het hybride III-V/silicium platform met behulp van resonante caviteiten op basis van fotonische kristallen Andreas De Groote Begeleider(s): Yannick De Koninck, Günther Roelkens, Roel Baets Abstract Dit artikel handelt over het gebruik van fotonische kristallen in het mechanisme van resonante caviteiten in het hybride silicium/iii-v platform. De belangrijkste parameters worden besproken en drie hoogreflectieve structuren worden voorgesteld (maximale reflectie van 97%, 96% en 90%), gebaseerd op monorail- en multirailcaviteiten. Deze spiegels kunnen vervolgens gebruikt worden om een microlasercaviteit te vormen. Trefwoorden fotonische kristallen, resonante caviteiten, microlaser, hybride silicium/iii-v, monorail I. Inleiding EEN geïntegreerde laser is de flessenhals van silicon photonics. Hybride integratie lijkt de beste oplossing, aangezien het gebruik maakt van de winstkarakteristieken van III-V-materialen die op een silicium chip gehecht worden. Afhankelijk van de locatie van de mode, vooral in de III-Vlagen of in het silicium, moeten verschillende problemen aangepakt worden. De laser, bijvoorbeeld een microdisk laser, kan zeer klein zijn wanneer de mode in het III-V-deel leeft, maar vaak is het dan moeilijk om te fabriceren. Wanneer de mode zich vooral in het silicium situeert, kunnen we gebruik maken van de CMOS compatibele deep UV lithografie, maar dan is de laser vaak zeer lang vanwege de lage winst. Daarom stellen we een laserdesign voor dat gebruik maakt van beide, met een winst- en spiegelsectie gebaseerd op resonante caviteiten. II. Werking van resonante caviteiten De volledige resonante caviteit bestaat uit een III-Vgolfgeleider en een siliciumcaviteit eronder. De werking wordt geïllustreerd in figuur, waar we een van de twee spiegels afbeelden die de lasercaviteit zal vormen. zouden verwaarlozen. Hier echter, zal er door de resonantie van de caviteit energie opgestapeld worden. Aan de andere kant van de caviteit koppelt een deel van het licht terug naar de III-V-golfgeleider. Onder de juiste voorwaarden zal dit licht destructief interfereren met het licht dat nooit naar de caviteit koppelde. Het licht dat in de andere richting koppelt van de caviteit naar de waveguide, zorgt voor reflectie. Bemerk dat we ook heel elegant het licht kunnen uitkoppelen naar een passieve siliciumgolfgeleider door een kant van de siliciumcaviteit korter maken. [] Er zijn twee voorwaarden om destructieve interferentie te bekomen. In de eerste plaats hebben we phasematching nodig. De effectieve brekingsindex van de caviteit en de golfgeleider moeten gelijk zijn, zodat de relatieve fasedraaiing enkel bepaald wordt door de dubbele koppeling. Aangezien elke koppeling een π 2 fasedraaiing met zich mee brengt, is de totale draaiing π. Naast phasematching moeten de structuren ook kritisch gekoppeld zijn. Hier komen we op terug in paragraaf III-A. We kunnen de caviteit met behulp van gratings maken, maar wij opteerden voor fotonische kristallen. Aangezien de III-V-lagen eigenlijk een golfgeleider zijn, zal de effectieve index in de verticale richting in het centrum sterk verschillen van die ter hoogte van de cladding. Dit zorgt ervoor dat de optische verboden zone van het kristal verschilt naargelang de III-V-bedekking, wat het designing sterk bemoeilijkt. Daarom stellen we een monorail voor, die gebruik maakt van index geleiding in de laterale richting. Enkel in de propagatierichting wordt het licht opgesloten door de bandkloof. [2] III. Bespreking van de design parameters Fig.. Werking van een resonante caviteit. [] Stel dat het licht van links komt en naar rechts propageert in de III-V-golfgeleider. Ter hoogte van de siliciumcaviteit zal een klein deel van het licht naar de caviteit koppelen. Dit is slechts een zeer klein deel dat we normaal Bron Detector Detector Een spiegel gebaseerd op een resonante monorailcaviteit, is geïllustreerd in figuur 2. Om niet naar een 3D afbeel- Golfgleiderkloof Golfgeleiderbreedte Taper uit N links Fig. 2. Bovenaanzicht van een passieve resonante monorailcaviteitsspiegel. ding te moeten grijpen, hebben we een passieve structuur Taper in Taper in Nrechts Taper uit

12 afgebeeld, maar het doel is een actieve. In dat geval zal de silicium golfgeleider een III-V-golfgeleider zijn die boven de caviteit ligt. We hebben de verschillende parameters aangeduid. De binnenste taper verhoogt de kwaliteitsfactor Q van de caviteit, de externe tracht de band edge modes te onderdrukken. De belangrijkste parameters zijn echter N links, N rechts, de golfgeleiderbreedte en de golfgeleiderkloof. We zullen nu de invloed van elk van deze parameters onderzoeken. Vermogensflux (a.u.) Reflectie Transmissie A. Golfgeleiderkloof - hechtingsdikte Wanneer de hechtingslaag te dik is, zal de caviteitsmode zeer sterk moeten zijn om met de destructieve interferentie een nultransmissie te bereiken. De verstrooiingsverliezen van de caviteit zullen verhogen tot ze van dezelfde orde als de koppelingsverliezen zijn. Een groter deel wordt verstrooid, en is dus verloren voor reflectie, en de caviteitsmode is niet sterk genoeg om het licht in de III-Vgolfgeleider te compenseren. Wanneer de hechtingslaag te dun is, zal de structuur overgekoppeld zijn. Het gereflecteerd licht zal terug naar de siliciumlaag koppelen, waar het meeestal verloren gaat. B. Golfgeleiderbreedte: phasematching voorwaarde Wanneer de structuur niet gephasematched is, zal de destructieve interferentie niet perfect zijn. Een deel van het licht zal doorgelaten worden en kan dus niet bijdragen aan de reflectie. Onze III-V-golfgeleider was zeer breed (.85µm), dus de invloed van de breedte op de brekingsindex is beperkt. Voor smallere golfgeleiders zal deze invloed groter zijn en de reflectie dus sterker dalen. C. Alignatiefout van de golfgeleider Een alignatiefout van de golfgeleider in een actief toestel zal de koppelingssterkte doen dalen. Daarom heeft een foute alignatie een soortgelijke invloed dan een te dikke hechtingslaag. Een grote offset kan er ook toe leiden dat koppeling naar hogere orde modes sterker wordt. Aangezien deze niet interfereren met de grondmode van de golfgeleider, is dit licht verloren. D. Het aantal gaten aan de linkerzijde De linkerzijde van de caviteit (i.e. de input zijde) heeft een dubbele functie. Enerzijds is het een van de spiegels die de siliciumcaviteit vormt, anderzijds bepaalt het de afstand waarover koppeling mogelijk is. Een zeer kleine N links zorgt voor hoge verliezen in de caviteit. Veel van het licht wordt verstrooid en zal dus niet bijdragen aan de reflectie. Een zeer grote N links heeft het zelfde effect als een te dunne hechtingslaag. Het reeds gereflecteerde licht zal terug naar de caviteit koppelen en de totale reflectie verlagen. In het ideale geval is de amplitude van de mode nul aan de rand van de siliciumcaviteit. Op deze manier zal er geen verstrooiing kunnen gebeuren en zal alle licht bijdragen aan de reflectie Golflengte (µm) Fig. 3. Transmissie- en reflectiespectrum van een passieve monorailspiegel. E. Het aantal gaten aan de rechterzijde De rechterzijde is niet zo cruciaal als de linker. Zolang deze zijde lang genoeg is, zullen de caviteitsverliezen binnen de perken blijven. In het ideale geval is er geen licht aanwezig in de golfgeleider (rechts van de caviteit), wat betekent dat er ook niets naar het silicium kan koppelen. Het probleem van de dubbele koppeling zoals aan de linkerzijde bestaat hier niet en het aantal gaten moet dus enkel naar beneden beperkt worden. We kunnen deze parameter wel gebruiken om band edge modes te onderdrukken. We zien dat in de monorailstructuur de band edge mode zich vooral aan de rechterkant bevindt. Wanneer we deze kant korter maken, kunnen we de verliezen van die modes vergroten en dus reflectie voorkomen op ongewilde golflengtes. IV. Simulatieresultaten We hebben drie verschillende structuren onderzocht: de passieve monorailspiegel, de actieve monorailspiegel en de actieve triplerailspiegel. De passieve resonante monorailcaviteitsspiegel is de structuur die in figuur 2 werd afgebeeld, waar een silicium golfgeleider zijdelings naar een monorailcaviteit koppelt. Het transmissie- en reflectiespectrum wordt in figuur 3 getoond. De maximale reflectie bevindt zich op de caviteitsmode, met een golflengte van 527nm, en is 96% met een Q van 30. Als we een III-V-golfgeleider boven de caviteit in plaats van een van silicium erlangs gebruiken, kunnen we een lasercaviteit maken. Het spectrum, zie figuur 4, lijkt zeer goed op dat van figuur 3. De maximale reflectie is quasi dezelfde (97%) en de Q is 240. Zoals we eerder al beschreven, is de phasematchingvoorwaarde belangrijk. Aangezien de effectieve brekingsindex van de monorailcaviteit relatief laag is (n eff = 2.29), mag de III-V-golfgeleider niet breed zijn. In de actieve monorailspiegel was de III-V-golfgeleider slechts 585nm breed zijn. Deze dimensies zijn haalbaar met geavanceerde III-Vtechnieken, maar wij willen gebruik maken van de simpelere. Om de brekingsindex te verhogen stellen we het multiraildesign voor. Dit is een bredere silicium golfgeleider met

13 Vermogensflux (a.u.) Reflectie Transmissie Golflengte (µm) Fig. 4. Transmissie- en reflectiespectrum van een actieve monorailspiegel. We hebben op een succesvolle manier gebruik gemaakt van fotonische kristallen om een hybride spiegel te maken met het resonante caviteitsprincipe. Na een korte introductie van dit principe, hebben we de verschillende parameters van de caviteiten belicht. Door ze goed te kiezen, waren we er toe in staat om drie hoogreflectieve spiegels te maken. Zowel het passieve als het actieve monoraildesign bereiken een quasi perfecte reflectie van 96-97%. Opdat we een lasercaviteit zouden kunnen maken met simpele III-V-fabricatietechnieken, hebben we de triplerailcaviteit gecreëerd. Nu kunnen we bredere golfgeleiders gebruiken dankzij de hoge brekingsindex. Dit design leidt tot een maximale reflectie van 90%. Erkentelijkheid We willen Nicolas Le Thomas van de École Polytechnique Fédérale de Lausanne bedankt voor zijn bijdrage aan dit onderzoek. Vermogensflux (a.u.) Reflectie Transmissie Referenties [] Y. De Koninck, G. Roelkens, and R. Baets, Hybrid III-V/silicon micro-lasers based on resonant cavity reflectors, 5th Annual Meeting of the IEEE Photonics Society Benelux chapter, 200. [2] A.R.M. Zain, N.P. Johnson, M. Sorel, R.M. De La Rue, et al., Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI), Opt. Express, vol. 6, no. 6, pp , Golflengte (µm) Fig. 5. Transmissie- en reflectiespectrum van een actieve triplerailspiegel. verschillende rijen van gaten. Ons tripleraildesign heeft een index van 2.59, waar we dus een.85µm brede III-Vgolfgeleider kunnen aan matchen. Het spectrum wordt in figuur 5 getoond. De maximale reflectie is 90% bij een golflengte van 563nm (met een Q van 220). Deze reden voor deze lagere reflectie in vergelijking met de monorailspiegels moet bij de koppeling naar hogere orde modes gezocht worden. De Bloch mode van de triplerail lijkt beter op deze hogere orde modes dan die van een monorail. Dit leidt tot een efficiëntere koppeling en aangezien de modes van een golfgeleider orthogonaal zijn, kunnen ze niet destructief interfereren. V. Van spiegel naar caviteit We kunnen twee van deze spiegels gebruiken om een lasercaviteit te maken. Wanneer we dit doen, is het belangrijk om drie resonanties te synchroniseren: de twee spiegelresonanties en de resonanties van de longitudinale mode tussen de spiegels. Een lasercaviteit met een totale lengte van 20µm, de winstsectie is dan 5µm, leidt tot een Q van orde VI. Conclusies

14 Contents Motivation of the proposed design. Introduction Current status of silicon integrated laser sources The hybrid silicon/iii-v platform Microdisk lasers Evanescent lasers DVS-BCB bonded Fabry-Perot laser diode Other platforms General structure of the proposed design Overview of this master thesis dissertation Photonic Crystals 9 2. A brief recapitulation on Maxwell s equations Photonic crystals as a discrete translational symmetric device Brillouin zone and photonic band gap Real life photonic crystals Defects in photonic crystals Methods of increasing the quality factor of the cavity Delocalization Cancellation Conclusion Resonant cavities 9 3. Analytical model Parameters Conclusion Monorail structure The choice of a monorail Designing the crystal Designing the cavity Designing the mirror The influence of the waveguide gap The influence of the number of holes on the left N left The influence of the number of holes on the right N right III-V-like cavity using a resonant monorail cavity Estimation of the III-V-like cavity Simulation of the III-V-like cavity Conclusion xiv

15 Contents xv 5 Multirail structure The choice of a multirail Designing the crystal Designing the cavity Designing the mirror The influence of the III-V - to - silicon thickness The influence of the waveguide width: phase matching condition Using the band edge modes Influence of alignment errors III-V cavity using a resonant triplerail cavity Estimation of the quality factor Simulation of the quality factor Estimation of the lasing threshold Conclusion Full optimization and comparison of monorail and multirail structures A resonant monorail cavity mirror using III-V overlay waveguide The monorail cavity Optimizing the III-V parameters Optimizing the silicon parameters Optimal design Tuning the silicon parameters of the resonant multirail mirror Discussion and comparison of the optimized structures Conclusion Masks The IMEC mask Mask parameters Passive structures Active structures Contact mask The Glasgow mask Mask parameters Straight waveguides Monorail structures Multirail structures: the doublerail and triplerail Material stack and fabrication The silicon chip The III-V die Die to die bonding Fabrication process Conclusion Conclusion and future prospects Conclusion Future prospects

16 Contents xvi A Tools 77 A. MEEP A.. Theory A..2 Practice A.2 Harminv A.3 MPB A.3. Theory A.3.2 Practice Bibliography 82 List of Figures 84 List of Tables 87

17 Chapter Motivation of the proposed design. Introduction Since 970, Moore s law is dominating the world of electronics. Every 8 months the number of transistors on a chip doubles, providing more and more calculation power to computer processors. Because of this incredible innovation speed, we can today use pocket devices capable of doing millions of instructions in a second. Moore s law is a direct result of the constant shrinking of the transistor size. Since the transistors become smaller and smaller, more and more transistors can be put on the same chip area. [] Up to now, electricity was the data carrier on a chip. Since recent years, this is changing. The electrical transistor is reaching dimensions at which quantum effects come to mess around. The oxide layer beneath the gate electrode is only a few atoms thick. making diminishing the transistor size a lot more difficult. Next to this, and more importantly, the available bandwidth is limited on a copper wire. Tens of gigahertz seems to be the maximum one can reach. In photonics however, bandwidth is less of an issue. The optical bandwidth exceeds the electrical one by multiple orders of magnitude. Hence, lots of effort is being made to replace the old electrical technology with a new, light-driven one. Microelectronics is dominated by the silicon platform. Because of the very good isolating properties of the oxide, silicon was chosen as substrate and after decades of working with this material, it is the best known semiconductor material by far. The designing platform, called complementary metal-oxidesemiconductor or in short CMOS, is a very reliable, low-cost platform by now. It is because of these properties, a chip only costs as much as it does. Silicon photonics tries to use this platform as a starting point to engineer light. At first glance, silicon does not have much in his toolbox for optoelectronics. It is an indirect band gap semiconductor leading to inefficient band to band light emission. On top of this, the band gap is.2ev corresponding to absorption of wavelengths of ±.µm and shorter, making it poor for light detection at the telecommunication wavelength windows at.3µm and.5µm. The inversion symmetry of the crystal results in the absence of the linear electro-optic effect. These disadvantages are countered by the high refractive index contrast of silicon and its oxide, making it very suitable to build nanoscale photonic devices. The potential to leverage silicon s high volume, low cost manufacturing infrastructure as means to break the cost barrier, is so promising that we will learn to live with the problems. The most important passive devices have been proven to work. Low loss waveguides have been demonstrated with losses on the order of 0.2dB/cm, modulators have broken the 40GHz bandwidth barrier and high speed silicon germanium photodetectors have been found to work in the 550nm and 300nm regime with data rates of 40Gb/s. The most important missing piece in the toolbox for photonic integrated circuits, is an electrically pumped laser source on silicon. Several solutions have been proposed to solve this. There are the more exotic

18 Chapter. Motivation of the proposed design 2 solutions like quantum dots and strained silicon who are still in research phase, but we focus on the silicon/iii-v solution. III-V compounds have been used to make lasers for the last decades. Therefore, the electro- and photoluminescent properties are well known and found to be very satisfying. Implementing these III-V device onto silicon is not straightforward though. Thin crystalline films are transferred to the silicon. These films are first bonded, and only then processed ensuring that alignment losses are defined by the quality of lithography rather than bonding. This method is called the hybrid silicon/iii-v platform. Next to gain, we also need feedback to make a laser. By defining the feedback structures in silicon, we can make use of the precision of the CMOS compatible deep UV (DUV) lithography. As will be explained in section.3, resonant cavities offer a high reflectivity in the III-V waveguide and an elegant solution for the output on the silicon chip. The resonant cavity will be designed using photonic crystals, mainly because of their potential for high distributed reflection coefficients and low mode volumes. More in particular, we will focus on monorails who use index guiding in two dimensions and photonic band gap confinement in the other.

19 Chapter. Motivation of the proposed design 3.2 Current status of silicon integrated laser sources As stated in the introductory section, an electrically pumped integrated laser source is the bottleneck of silicon photonics. Therefore, a large variety of designs and design platforms is being explored to find the optimal solution. In this section, we give an overview of what has been achieved so far..2. The hybrid silicon/iii-v platform When working in the hybrid Si/III-V platform, the location of the optical mode is a very important choice to be made. If the optical mode is predominantly in the silicon region, providing feedback is fairly easy. The reflecting structure, e.g. a grating, will work on the maximum of the field profile and thus will be very efficient. Wavelength selective features to control the lasing wavelength can easily be defined using DUV and the output to a passive silicon waveguide is straightforward. The big drawback lies in the fact that not a lot light is interacting with the gain material. This gives rise to very long laser cavities and higher power consumption. On the other side of the design space, if the optical mode is completely defined in the III-V waveguide, gain can be very effective. The optical cavity is often defined in the III-V semiconductor, losing the benefits of DUV to create the laser source. Coupling to the passive silicon waveguide can be problematic..2.. Microdisk lasers Figure. shows the structure and the modes of a microdisk laser. The cavity sustains several whispering gallery modes confined at the periphery of the disk. The confinement in the quantum wells in the III-V material is engineered such that only the TE00 mode will laser, because of a supreme gain to losses ratio. (a) The structure (b) Four lowest modes being sustained by the microdisk Figure.: A microdisk laser. [2] Microdisk lasers have the advantage of being very small devices, with typical radii of a few micrometers, and a very low threshold current. The latter is of course a direct consequence of the confinement in the III-V layers rather than the silicon. In [3], the threshold current was ±500µA. The laser is single mode, with a side mode suppression ratio of 22dB. Because the mode has only very little overlap with the SOI waveguide, the coupling to the rest of the chip is very bad. Next to this, the lasing wavelength tends to vary from design to design. The resonance condition, and thus resonance wavelength, is defined by the radius of the disk, which is defined through III-V processing. This technology lacks unique combination of precision and low cost of the silicon

20 Chapter. Motivation of the proposed design 4 technology. The processing becomes even more important when one takes into account that the mode lives very close to the interface, meaning that surface roughness will dominate the losses Evanescent lasers In evanescent laser devices, the mode is primarily confined in silicon. Gratings-based lasers are an attractive option for heterogeneous integration as they offer facet-less single-wavelength light sources. Figure.2 shows the schematic of an evanescent DFB laser as described in [4]. There are three regions: a gain section and two taper sections. In the gain section, the modal overlap with the III-V layers is maximized in order to have enough gain. The taper section modifies this mode to fit the passive SOI waveguide mode to maximize the coupling. The taper section also increases the influence of the grating, thus enhancing the reflection. Figure.2: Schematic of an evanescent DFB laser design. [4] As the mode is predominantly confined in the silicon, the output can be very elegant. The most important structures such as the grating and the tapers are defined in silicon and thus benefit from DUV. The size is the main disadvantage. The laser cavity is of the order of hundreds of micrometers, as the gain per unit length is low. If one wants to increase the output power, even longer cavities are needed. A DBR design is then used rather than the DFB shown above DVS-BCB bonded Fabry-Perot laser diode As a last example, we take a look at the DVS-BCB bonded Fabry-Perot laser diode, a schematic of which is shown in figure.3. This laser makes use of an adiabatic taper, indicating that in one device it is possible to have modes predominantly living in the III-V in one section and in the silicon in another. By cleaving the facets of the III-V die, a Fabry-Perot laser is created. After bonding to a silicon waveguide, a polymer waveguide is added that has the same mode profile as the laser. By using an inverted taper in the silicon waveguide, this mode is transformed to a SOI waveguide one. This is a way of very efficient and broadband coupling. Unfortunately, there are a lot of issues with Fabry-Perot lasers. They are not very stable and often required a high current density. The quality of the surface recombination in the III-V stack and the etched facets are of paramount importance. The laser being used in the experiments reviewed in [2], was also very long (700µm)..2.2 Other platforms Of course the hybrid Si/III-V platform is not the only option. alternatives. Here, we will just touch upon other

21 Chapter. Motivation of the proposed design 5 Figure.3: A Fabry-Perot laser using a III-V/SOI coupling scheme. [2] The most straightforward method of getting light into a silicon photonic integrated circuit is injecting the light of an off-chip laser. Note that this method is not only used in measurement setups, but also in a so-called optical power supply. This approach has some advantages in thermal separation, but alignment is a major issue here, as light is normally injected trough a fiber. The dimensions of the fiber are much larger than the waveguide (a core of 9µm versus a waveguide width of ±µm). V-grooves and other passive alignment techniques are used to reduce packaging costs. The approach is illustrated in figure.4a. (a) Fiber coupling an off chip laser (b) die attaching prefabricated III-V lasers (c) Hybrid integration Figure.4: Three methods to supply III-V laser light to a photonic integrated circuit on silicon. [4] A slightly more elegant solution is die-attaching of prefabricated compound semiconductors, shown in figure.4b. Also here, alignment is very difficult as the III-V waveguide and the passive silicon waveguide need to be aligned during the bonding step. Quantum dots have shown electro- and photoluminescent properties. There are several routes and one has to choose between silicon quantum dots (often called silicon nanocrystals) or quantum dots from other materials. This domain is still very much in research phase and fabrication remains very unreliable. Of these different fabrication techniques, we like to draw your attention to the route of ion implantation, making use of CMOS technology to fabricate the nanocrystals. [5], [6], [7] Silicon is also being strained to change the fundamental properties. E.g. strained SiC has shown photoluminescence. As was the case for quantum dots, this is still in research phase and no practical devices are to be expected soon. [8]

22 Chapter. Motivation of the proposed design 6.3 General structure of the proposed design As was made clear in the previous section, the hybrid silicon/iii-v platform is very promising but the ideal integrated laser design has not been found yet. The microdisk laser is very small, but has important fabrication steps in the III-V layers. The evanescent laser avoids these low yield steps, but is a lot larger due to the low gain per unit length. We think that the resonant cavity design, proposed by De Koninck et.al. [9], might be able to break through this trade-off barrier. As will be explained below, the light will be predominantly confined in the III-V waveguide at the gain section, providing high gain possibilities. At the reflecting section, the bigger part of the light is situated in the silicon. This allows us to build small, highly reflecting mirrors. Due to the clever design no taper section is needed! Previously, this cavity was made using surface gratings. In this master thesis, we will use photonic crystals, whose high refractive index contrast results in a high distributed reflection coefficient. A resonant cavity design consists of a III-V waveguide and a cavity buried beneath it. Suppose light is traveling in the III-V waveguide towards the cavity. Because of the (relative) vicinity of the silicon cavity, a small fraction of the light will couple to the silicon cavity. Normally, one would neglect this light, but because it is coupled to a resonant cavity a power buildup may occur. Eventually the light in the cavity will couple back to the III-V waveguide. If the III-V waveguide and the silicon cavity are phase matched and critically coupled, the light that couples back co-directionally to the incoming signal will interfere destructively with the latter. As a result, there will be 0% transmission and 00% reflection at the resonance wavelength of the silicon cavity. Off resonance, the reflection will be a lot lower, making sure we have built a monochromatic laser source. The output to a passive silicon waveguide can be done very elegantly by making the far side of the resonant cavity shorter. The entire process is being illustrated in figure.5. Figure.5: Schematic view of a resonant cavity reflector in a hybrid silicon laser. [9] As a result, the light in the III-V waveguide is reflected by a structure defined in the silicon layer. The last step is to use two of these mirrors and create the laser cavity. An abstract schematic is shown in figure.6. In the gain section, the III-V waveguide is completely surrounded by low index materials (we got rid of the silicon since it has no purpose here). Therefore, the III-V confinement is maximal offering high gain possibilities, giving rise to short gain sections. In the mirror section, the power buildup in the silicon cavity ensures that the mode is predominantly confined in the silicon. The cavity mode volume can be kept to a minimum and after the destructive interference the light in the III-V waveguide destroyed. The length of the mirror section can thus be limited to approximately the cavity mode volume. The most important process steps occur in the silicon layer, meaning that we can fully use the leverage possibilities of the CMOS compatible deep UV

23 Chapter. Motivation of the proposed design 7 Mirror section Gain section Mirror section III - V Si cavity Si cavity Figure.6: Abstract schematic of the laser cavity created using two resonant cavities. The cavity contains a gain section and two mirror sections, the mode profiles of which are shown in red. technology. Using photonic crystals, we think we can reduce the mirror section size to a minimum. On top of this, the small mode volume inherent to photonic crystals should lower the influence of the phase mismatch of the silicon cavity and the III-V waveguide. This should increase the tolerance to fabrication errors (e.g. in III-V processing) even further.

24 Chapter. Motivation of the proposed design 8.4 Overview of this master thesis dissertation In this chapter, we have illustrated the need for the research on silicon microlasers. The advantages and disadvantages of current laser source designs were taken a look at. The microdisk laser is small but has fabrication issues, whereas the evanescent laser is easy to make but often very long. We are convinced that we can get rid of the shortcomings and combine the merits of these devices using the proposed mechanism of resonant cavities, the working principle of which we just explained in the previous section. The hands-on creation of these structures will of course happen in phases. First, two chapters provide us with a theoretical frame to work in. Chapter 2 describes the theory of photonic crystals and what features to look at when using them in designs. In chapter 3, a simple analytical model on resonant cavities is derived, showing perfect reflection is possible under the right conditions. This model is being used as a guideline throughout the designing chapters 4, 5 and 6. In chapter 4, we optimize a resonant monorail cavity mirror. Because of practical reasons, we only look at passive devices, implying that we will only vary the silicon cavity parameters. In order to increase the effective refractive index of the cavity, we introduce the multirail - the multiple row of holes analogon - cavity in chapter 5. It is possible to phase match a III-V waveguide to this cavity and investigate the effect of the III-V parameters. Chapter 6 uses the information of the previous chapters to design two optimal mirror structures. The first one is a monorail with III-V overlay waveguide (however this can not be fabricated with simple III-V processing techniques), the other one is a triplerail with III-V overlay waveguide. These two optimum designs are compared. We have ordered two masks en did the III-V processing on one of them. These are discussed in chapters 7 and 8 respectively. In chapter 9, we summarize the obtained results one last time and draw the conclusions from this work. We also take a look at the future challenges that lie ahead.

25 Chapter 2 Photonic Crystals 2. A brief recapitulation on Maxwell s equations Electromagnetic waves, among which optical waves, are dominated by Maxwell s equations:.b(r, t) = 0 (2.).D(r, t) = ρ(r, t) (2.2) B(r, t) E(r, t) + t = 0 (2.3) D(r, t) H(r, t) t = J(r, t) (2.4) Where E and H are the macroscopic electric and magnetic fields, D and B are the displacement and magnetic induction fields, and ρ and J are the free charge and current densities. We make the usual approximations: the dielectric structure does not change in time, the material behaves linearly, we neglect material dispersion and the material is mainly transparent at the frequency region of interest. We also put the source terms to zero. We can now state that D = ɛ 0 ɛe and B = µ 0 H. The equations become: To make life even easier we express the fields in the phasor form: Remember that in this H(r) and E(r) are complex numbers. The two curl equations 2.7 and 2.8 simplify to:.h(r, t) = 0 (2.5). [ɛ(r)e(r, t)] = 0 (2.6) H(r, t) E(r, t) + µ 0 δt = 0 (2.7) E(r, t) H(r, t) ɛ 0 ɛ(r) t = 0 (2.8) H(r, t) = Re { H(r)e jωt} (2.9) E(r, t) = Re { E(r)e jωt} (2.0) E(r) jωµ 0 H(r) = 0 (2.) H(r) jωɛ 0 ɛ(r)e(r) = 0 (2.2) This is an approximation which is commonly made, but not really necessary if we allow complex refractive indices. 9

26 Chapter 2. Photonic Crystals 0 These equations can be decoupled by dividing equation 2.2 and take the curl. After this, we can use equation 2. to eliminate E(r). Note that ɛ0µ 0 is of course the vacuum speed of light c. We find: ( ) ( ω ) 2 ɛ(r) H(r) = H(r) (2.3) c Equation 2.3 will be the master equation. Together with equation 2.5 we can solve the problem to find H, E then follows immediately from equation Photonic crystals as a discrete translational symmetric device Unfortunately, Maxwell s equations can only be solved in extreme simple cases. If there is symmetry in the setup, we can however deduce a lot of information without solving the equations. Photonic crystals are systems that exhibit discrete translational symmetry. In order to describe a photonic crystal we have to take into account that ɛ(r) = ɛ(r + a) in the master equation 2.3, with a the period, also called pitch or lattice constant. For reasons of simplicity we focus on a D case, the structure has discrete translational symmetry in the z-direction: ɛ(z + la) = ɛ(z) (2.4) Note that the master equation 2.3 is an eigenvalue problem. In order not to overload the notation we rewrite the master equation and introduce the operator T la : ΘH = λh (2.5) T la ɛ(z) = ɛ(z + la) (2.6) Because of the symmetry present, Θ must commute with T la : [Θ, T la ]. Some magic mathematics provide us with an interesting result: T la+l ah(z) = λ(l a + la)h(z) (2.7) = T l at la H(z) = T l aλ(la)h(z) = λ(l a)λ(la)h(z) (2.8) λ(l a + la) = λ(l a)λ(la) (2.9) From equation 2.9 we can deduce λ(la) = e jkzla. This leads to the first form of Bloch s theorem: H(z + a) = e jkza H(z) (2.20) A more practical formulation is Bloch s second form: H(z) = e jkzz u(z) (2.2) in which u(z) is a periodic function with the same periodicity as the lattice. This can be shown very easily, using Bloch s first form, equation u(z + la) = e jkz(z+la) H(z + la) = e jkzz e jkzla e jkzla H(z) (2.22) = e jkzz H(z) = u(z) (2.23) The interpretation of the second form of Bloch s theorem, equation 2.2 is one of a plane wave, as it should be in free space, but modulated by a periodic function due to the periodic lattice. [0]

27 Chapter 2. Photonic Crystals 2.3 Brillouin zone and photonic band gap We start from the second form of Bloch s theorem, equation 2.2. Since the function u(z) is a periodic function with the same period as the lattice, we can write this as a Fourier series: u(z) = u m e jm 2π a z (2.24) m= This means that the eigenfunction can be written as: H(z) = u m e j(kz+m 2π a )z m= (2.25) From equation 2.25 we see that we can define a set of wave vectors k z in the interval [ π a, π a [. To each of the wave vectors k z from this interval, there will be a contribution added due to the periodicity. Indeed, suppose there is a need for a k z from outside of this interval, we can write: H(z) = = u m e j(kz+m 2π a )z m= u m e j(k z +m 2π a )z m = (2.26) (2.27) By definition, we can say k z is in the interval. Since this interval is so important, it has its own name: the Brillouin zone. We can now represent all wave vectors of the dispersion diagram in the Brillouin zone. Let s do this for the simple case of a uniform material. To this material, we dictate the artificial period a. The dispersion diagram is a straight line, but due to the periodicity we can shift this straight line over a distance 2π a, as is being shown in figure 2.. As was stated before, only the band structures in the Brillouin zone are needed for a full description. This leads to so-called band folding, as is being shown on the left panel of figure 2.2. Figure 2.: Dispersion diagram for a uniform medium with an artificial periodicity a. [0] We now move to a more realistic example. We have a multilayer structure of two layers of different refractive index. Both layers have the same optical thickness and are repeated infinitely. If the refractive index contrast is low, the band structures will very much look like the uniform case. Only at the edge of the Brillouin zone, the curve will flatten so the first derivative is zero. This is necessary because we have to remember that at the other side of the edge of the Brillouin zone π a, we have the same line (which is actually shown at π a ). Thus, because of the continuity of the first derivative, the curve should be flat

28 Chapter 2. Photonic Crystals 2 Figure 2.2: Dispersion diagrams for normal incidence for three different multilayer films, where each layer is a/2 thick. Left: all layers have ɛ = 3. Middle: layers alternate betweens ɛ = 2 and ɛ = 3. Right: layers alternate betweens ɛ = and ɛ = 3. [0] at the edge. This is the case for both the upper and the lower band, meaning that they will split. Hence, there exist certain energy levels for which there are no wave vectors, i.e. we have created a photonic band gap in the multilayer structure. The larger the refractive index contrast, the larger the band gap, as is shown in figure Real life photonic crystals In the theory of photonic crystals, set forth above, we assumed a one dimensional world. We have several options to move to three dimensional structures. The most straight-forward way of going to more dimensions is doing the same discretization in the extra dimensions. In figure 2.3 we see a two-dimensional photonic crystal, a square lattice of rods of infinite length. Figure 2.3: A square lattice of rods of infinite length. [] On the left panel of figure 2.3 the unit cell is indicated. This unit cell is then repeated along the two base vectors, in this case along x and along y. There is no need however for these base vectors to be perpendicular to each other, as long as they are no linear combination of other base vectors. We can now write the eigenmodes as: 2 H (n,k )(ρ) = e jk ρ u (n,k )(ρ) (2.28) 2 Note that this is only the in-plane component. To be exact we should also describe the out-of-plane component. In

29 Chapter 2. Photonic Crystals 3 We can again make the same dispersion diagrams as we did in the one dimensional case. In stead of a simple one dimensional Brillouin zone we have a 2 dimensional one. On the inset of figure 2.4, the irreducible Brillouin zone is shown. In this case, the Brillouin zone was further reduced because of spatial symmetries of the unit cell (e.g. point symmetry around the origin). To know all the information of the structure one should plot the energies of all the different k-vectors of the irreducible Brillouin zone. This is still a three dimensional plot, so this does not really fit in a textbook. Because most of the extremes are on the edge, we only plot those. The result is shown in figure 2.4. From this figure we can see that the structure has a band gap for TM waves (transverse magnetic), but not for TE waves (transverse electric). Figure 2.4: Band structures of a square lattice of rods of infinite height. The rods have a relative radius r a = 0.2a and ɛ = 8.9. The surrounding material is air (ɛ = ). [] Figure 2.5: Band structures of a diamond lattice of air spheres in a high dielectric material (ɛ = 3). [] In a three dimensional world we can go up one more dimension. We can try to build a diamond lattice of spheres. If everything is designed properly, it is possible to get a full photonic band gap, as is being shown in figure 2.5. Today s technology is mostly planar though, e.g. DUV technology. In the vertical direction, it is very hard to define structures other than grown layers. Because of this, we will not use this option throughout this master thesis. In order to confine light in a three dimensional world, we can also make use of index guiding, next to confinement by band gaps. Index guiding is used extensively in dielectric waveguides, where modes are confined because of high index materials. We can now start from a 2D photonic crystal and make it finite in the third dimension, an example is shown in the inset of figure 2.6. In this example, we see that light is confined by the band gap of the crystal in the (x, y) direction and by index guiding in the vertical z direction. The band diagram, shown in figure 2.6, is similar to the one of a 2D crystal (i.e. of infinite height) but there is a light cone to be taken into account. In the light cone (the purple region in the figure), the mode is not being confined in the z direction. The mode is radiative, as we know from simple waveguides. Outside the light cone, the mode is confined in the dielectric slab. Strictly speaking, we can not speak of TE and TM modes anymore. If the slab is thick enough, the modes will still be very much alike. This is why we speak of TE-like and TM-like modes. As can be seen in figure 2.6, the structure proposed has a TE-like band gap. This band gap is incomplete though, there are certain wave vectors k for which there is a solution at energy levels in the gap. This is a direct consequence of the limitations of index guiding. [] that case we can write the Bloch states as: H (n,kz,k )(ρ) = e jkzz e jk ρ u (n,kz,k )(ρ)

30 Chapter 2. Photonic Crystals 4 Figure 2.6: A photonic crystal slab: a triangular lattice in a dielectric slab. A light cone is limiting the band gap. [] There are now a lot of parameters to play with, both in the z direction as in the (x, y) plane. ˆ ˆ ˆ ˆ The shape of the structure in the unit cell does not have to be a hole or a rod. However, due to fabrication simplicity and mathematical elegance, this is very often the case. The radius of the hole is a very important parameter. If we look at the extreme case of a very small radius, the structure is almost a homogeneous slab. In the other extreme case, a very large radius, the structure is also almost homogeneous. In both cases, a band gap is out of the question. Somewhere in between there is an optimum of the band gap size. To find this, one makes use of gap maps which are plots of the band gap versus the radius of the holes of a certain lattice. The lattice constant a is often used as a scaling parameter of the simulations. If we take a closer look at figure 2.6, we indeed see that the frequency is relative to the lattice constant a. By choosing an appropriate value of a, we can center the band gap around a wavelength of 550nm, or at the visible spectrum if needed. The slab thickness also has an optimum value. In a very thin slab, the light is not confined anymore. Also the guided modes are only very weakly guided, with very large exponential tails in the cladding. The mode is so delocalized that the periodic structure is not sensed anymore. For a very thick slab, one could reason that we can apply the theory of 2D photonic crystals of infinite height. However, higher order modes (with more vertical nodes) will be pulled down and populate the band gap. From this reasoning, we can intuitively feel that the optimum thickness will be around half a wavelength (since the ground mode has a wavelength of approximately twice the thickness). The big question is what refractive index to use when calculating this vertical wavelength, the one from the cladding or the one from the slab. The answer is something in between, we should use an effective index which is a weighted average of both. It turns out that the index for TE-like modes is closer to the high index material and TM-like modes closer to the low index material. This is also illustrated in figure Defects in photonic crystals With a photonic crystal slab, we can generate a material structure in which certain energies are not allowed. However to get things interesting, we need to introduce defects.

31 Chapter 2. Photonic Crystals 5 Figure 2.7: The size of the band gap in the guided modes of a photonic crystal slab of holes and rods. [] Let s start with a point defect, shown in figure 2.8a. To create a point defect, we can change the dielectric constant of one of the rods in an otherwise perfect lattice, change the radius of one of the rods (even remove it) or change the shape. If we now suppose that there is light in the cavity with an energy level in the band gap. This light is forbidden in the crystal and, thus, is surrounded by a mirror. The light starts to resonate and, in the case of constructive interference, a mode exists. (a) Point defect containing a monopole mode in a square lattice of rods. (b) Evolution of localized modes depending on refractive index contrast of lattice rods with defect rod. n = 0 corresponds to the perfect crystal, n = 2 corresponds to the complete removal of the defect rod. Figure 2.8: The behaviour of point defects in photonic crystals. [] In figure 2.8b we see how different defects are pulled into the band gap by enlarging the radius of the defect rod. It is interesting to make a simple reasoning of perturbation to understand why the mode is pulled down out of the air band. Suppose that you have a material with mirrors at the edges. There exist Fabry-Perot modes with certain wavelengths (e.g. the fundamental mode with λ mode = 2L). If we increase the index of the material, the vacuum wavelength will have to go up because of the resonance condition (i.e. λ mode = λ0 n = 2L). If we increase the radius of the defect rod, the effective index will go up, the vacuum wavelength will rise, so the frequency will go down. It is as if we are pulling modes out of the air band.

32 Chapter 2. Photonic Crystals 6 Next to point defects, also line defects are very popular. In these, an entire row is manipulated in a perfect crystal. They can be used as waveguides or hollow core fibers. Surface defects are not very widely spread. By clinging a homogeneous low index material and a photonic crystal together, we can make a surface state. These are important to keep in mind as in reality we almost always have such an interface and very rarely desire a mode. Since we will not be using these defects in the rest of the thesis, we will not go into further detail Methods of increasing the quality factor of the cavity Because of the incomplete band gap, very high Q s seem impossible in photonic crystal slabs. When designing high Q cavities, one has to pay a lot of attention to reducing the radiation losses. No clearly superior design is currently known, although several high-performance devices have been achieved. Through the literature study we found two basic mechanisms who are being exploited. [], [2], [3], [4] Delocalization By increasing the mode volume, we may decrease the losses and increase the quality factor Q. The simplest example of a cavity is of course the Fabry-Perot cavity. If we increase the distance separating the two mirrors, the total losses will not change (they are always the sum of the transmission of the mirrors), assuming a non-absorbing medium. The quality factor will increase, because it is a number expressing the losses per time unit. If a pulse is resonating in a Fabry-Perot cavity of two times as big as before, the distributed losses will divided by four (the propagation length is two times the length of the cavity). This is a linear increase of Q with the length. An other example is given by a conventional dielectric ring resonator. Because in a ring resonator, the waveguide is not straight, there will be radiation losses. By making the radius larger, the bend curvature will be smaller and the losses will shrink. The quality factor Q increases exponentially, rather than linear. [5] In a photonic crystal one can increase the mode volume by diminishing the effect of the defect. In a delocalized mode, more Fourier components will be in the k-space of the photonic crystal reducing the scattering. Delocalizing will push the mode away from the light line in the dispersion diagram, i.e. the mode is more guided. If we take the example of a point defect, shown in figure 2.8a, we can make the defect weaker by reducing the refractive index contrast relative to the rest of the slab. We can also change the refractive index of the neighbours of the defect. A smoother transition from defect to crystal will increase the exponential tail. This larger mode volume benefits in turn the Q factor. Of course we can also use the radius of the rods or even the pitch. In engineering, every advantage brings disadvantages. By increasing the mode volume we will of course limit the degree of miniaturization. These larger designs often require larger driving powers. Structures with large mode volumes tend to become multimode and the spectral separation between these mode diminishes, i.e. the free spectral range shrinks. Specifically to our proposed design of phase matched resonant cavities, a larger mode volume will increase the importance of the phase matching condition Cancellation In order to increase the quality factor dramatically one has to take care of the out of plane scattering. This is possible by cancellation. If we calculate the multipole expansion of the radiated field, we decompose it into a sum of dipole, quadrupole, hexapole, etc. radiation patters. It turns out that the radiated power is simply the sum of

33 Chapter 2. Photonic Crystals 7 the contributions of each term, i.e. there is no inter-term interference. Also, it is very often the case that one term is dominant (usually the dipole term). By arranging this term to vanish, by destructive interference with other scattering centers, we can greatly enhance the Q. Figure 2.9: Increase in the quality factor Q due to cancellation of the lowest order multipole moment in the two-dimensional cavity shown in the upper right. [] In figure 2.9 we see a two dimensional example. The cavity is formed by a single line of dielectric rods, with a defect formed by a rod with increased radius. This structure has a TM band gap. In the lower right corner of the figure, the Q is shown for different frequencies. It is striking how there is a giant peak of at ωa 2πc 0.3. On the left hand side of the figure, the far field patterns are shown at different frequencies (and thus also different quality factors). Note the extra node in figure b, indicating that the major component of the other patterns was canceled. The scattering is much weaker, the field in the cavity remains untouched. This example shows the power of cancellation. The use of cancellation is not straight-forward though and is normally only used for very high Q s. Building a cavity using this method would be a good topic for a master thesis, but with the extras of this topic not realistic. The method of cancellation was discussed nonetheless, for completeness.

34 Chapter 2. Photonic Crystals Conclusion This chapter provided us with the theoretical background of photonic crystals and the ways of characterizing them. We started with Maxwell s equations as one always should when studying photonics. Using the discrete translational symmetry, we were able to derive an expression for the typical waves in a photonic crystal: the Bloch wave H(z) = e jkzz u(z) This equation states that we can express the magnetic field as a plane wave modulated by a periodic function with the same period as the crystal. From this theory, we derived the tools to work with. The introduction of Brillouin zone led to a photonic band gap, a frequency range for which there are no modes available. The introduction of a photonic crystal slab proved necessary in a three dimensional world. In this slab, we have band gap confinement in two dimensions and index guiding in the third. Because of this, the band gap became incomplete. On top of the old band gap we needed to lay a light cone overlay. Within this light cone, the mode is not confined in the slab but scatters away. A perfect crystal may exhibit band gaps, but in order to use it, we need defects. A point defect may create a cavity in which light can resonate. Delocalization and cancellation are the two basic mechanisms to increase the quality factor. Delocalization makes sure more Fourier components are in the crystal plane enhancing the influence of band gap. In cancellation, two scattering centers cancel out to take care of out of plane scattering.

35 Chapter 3 Resonant cavities 3. Analytical model Using coupled wave theory, we can make a very simple model of a resonant cavity. Both structures, the III-V waveguide and the silicon cavity, can be said to have a forward and backward propagating wave, as shown in figure 3.. We will describe three phenomena: propagation through a waveguide, coupling from one waveguide to another and reflection by a discrete translational symmetric structure (such a photonic crystal or a grating). Figure 3.: A simple model of a resonant cavity using coupled wave theory. Propagation The propagation in a waveguide is described by a propagation factor: A + (x) = A 0 exp( jβ A x) (3.) B + (x) = B 0 exp( jβ B x) (3.2) If we now define a new framework in which we take the phase of B + (x) as a reference, we can rewrite this to: A + (x) = A 0 exp(j βx) (3.3) B + (x) = B 0 (3.4) 9

36 Chapter 3. Resonant cavities 20 with β = β B β A = 2π λ 0 n. Because of what comes next, it is better to use the differential equation format: da + (x) = j βa + (x) (3.5) dx Coupling For the coupling between the waveguides, we turn to [6]: da + (x) dx = jκ AA A + (x) + jκ AB B + (x) (3.6) db + (x) dx = jκ BA A + (x) + jκ BB B + (x) (3.7) In most cases the coupling factor from waveguide A to B will be equal to the one from B to A, κ AB = κ BA = κ., and the self coupling factors κ AA and κ BB can often be neglected compared to the propagation factors. The equations simplify to: da + (x) dx db + (x) dx = jκb + (x) (3.8) = jκa + (x) (3.9) Distributed reflection Distributed reflections using coupled wave theory are described in [7]. First we write down the one dimensional wave equation and the Fourier series of the photonic crystal: d 2 ψ(x) dx 2 + k 2 0n 2 (x)ψ(x) = 0 (3.0) n 2 (x) n 2 0 = m 0 a m e jmkx with K = 2π Λ (3.) In this, ψ is a linear combination between the forward and backward propagating wave, Λ is the length of the unit cell, the so-called pitch, n 0 is the average refractive index. After some mathematics we find the following differential equation: db (x) dx = jr exp( j2 Bragg )B + (x) (3.2) with the distributed reflection coefficient r and the Bragg deviation Bragg = 2β B mk 2. This Bragg deviation is nothing else than the Bragg condition, i.e. at the Bragg wavelength Bragg will be zero. All three phenomena in one model In our full analytical model, we have to take these 3 equations into account, equation 3.5, 3.8 and 3.2. This gives us a differential equation in A + (x), A (x), B + (x) and B (x). Remember that the propagation is relative to B and there is no distributed reflection in waveguide A. We assume to be working at the Bragg wavelength. da + (x) dx da (x) dx db + (x) dx db (x) dx = j βa + (x) + jκb + (x) (3.3) = j βa (x) jκb (x) (3.4) = jκa + (x) + jrb (x) (3.5) = jκa (x) jrb + (x) (3.6)

37 Chapter 3. Resonant cavities 2 These equations can be solved numerically. Typical boundary conditions are: A + (0) = A 0 (3.7) A (L tot ) = B + (0) = B (L tot ) = 0 (3.8) The transmission coefficient T and reflection coefficient R are defined as: T = A + 2 (L tot ) A + (0) R = A 2 (0) A + (0) (3.9) (3.20) Figure 3.2 shows a map of the reflection coefficient R as a function of the distributed reflection coefficient r and the coupling coefficient κ, in the case of perfect phase matching. We see that, contrary to our first intuitive thoughts of the more coupling the better, there is a critical coupling for every distributed reflection coefficient at which the total reflection coefficient will be maximal. If we couple too much, the structure is overcoupled and reflected light will couple back to the silicon and scatter away n = kl rl Figure 3.2: Map of the reflection coefficient when the waveguide and the resonant cavity are phase matched. Figure 3.3 shows the same as figure 3.2 but in the phase mismatched case, the refractive index contrast is 0.5. It is obvious that the phase matching condition is of great importance. When there is a phase mismatch, the light coupling back to the waveguide will not interfere destructively with the light already in the waveguide. It is not possible anymore to achieve a perfect reflection R = (note that the R=0.990 contour has vanished). When comparing figure 3.2 and 3.3, we see that there is a major change in the regions of higher coupling. In perfect phase matched conditions, there is perfect reflection when critically coupled, but also a small local maximum when the structure is heavily overcoupled (the small green stripe at the bottom of the figure). This maximum is enhanced by the phase mismatch. Plotting the fields at the different maxima clarifies the nature of the modes, see figure 3.4 (note that the scale on the y-axis differs in both graphs). The amount of power in the cavity is limited in figure 3.4b, an important fraction is to be found in the grating itself. We also see that the power drop in the III-V waveguide is mainly over the mode in the grating. Figure 3.5b illustrates how the phase mismatch can give rise to this additional mode. Waveguide A and waveguide B are shown together with the phasors of the light. Remember that the angle of the phasor

38 Chapter 3. Resonant cavities 22 n = kl rl Figure 3.3: Map of the reflection coefficient when the refractive index contrast between the waveguide and the resonant cavity is 0.5. (a) Mode profile of the mode in a resonant cavity with parameters: n = 0.5, rl = 6 and κl = 0.4 (b) Mode profile of the mode in a resonant cavity with parameters: n = 0.5, rl = 6 and κl = 3.6 Figure 3.4: The spatial distribution of the intensity of two modes from figure 3.3 with the horizontal axis (real axis) is proportional to the phase of the wave. The phase of the waveguide A was used as a reference, meaning that the angle stays zero under propagation. At the left of the figure, there is no light in waveguide B other than the light coupled from the waveguide A. This means the phase of the light in waveguide B is π 2 at the left of the figure. Since there is a phase mismatch, the phase of the lower waveguide will change with respect to the upper one. The waves coupling to the waveguide have a different phase than the ones already living there. This is illustrated in figure 3.5a. The sum of the phasors, shown is red, is not in phase with the light of waveguide A. It is this red phasor that is shown in figure 3.5b. The same reasoning can be done throughout the entire waveguide. We see that the photons at the right of the figure are in antiphase with the ones on the left. In this story, we neglected the fact that waveguide B is mirroring part of the light. If the distance is long enough, the mirroring can be sufficient to sustain a mode. The mechanism from figure 3.5b ensures that the phase in the end is a multiple of 2π.

39 Chapter 3. Resonant cavities 23 Imag A Real (a) Addition of two phasors with different phase (b) Phasor evolution under propagation through waveguide A and B when they are not phase matched B Figure 3.5: Explanation of modes living in the grating when the waveguides are not phase matched. The arrows indicate the phasors of the light. 3.2 Parameters A resonant cavity is a complex structure, but thanks to our rough analytical model we can make some estimations. ˆ ˆ ˆ The distributed reflection coefficient r should be as large as possible. This is one of main reasons to turn to photonic crystals in stead of gratings. The high refractive index contrast will lead to high values of r. For every structure, there is a critical coupling coefficient. When the coupling coefficient is lower than this value, not enough light is exchanged between the waveguide and the cavity. When the coupling coefficient is higher than this value, the reflected light is coupled back to the edge of the cavity structure where it scatters away. Phase matching is critical. If there is a phase mismatch, light coupling back to the waveguide will not interfere destructively with the light already in the waveguide. The reflection coefficient drops heavily because of this. It does not follow from the model, but we can intuitively feel that the mode volume is also important when there is a phase mismatch. If the mode volume is very low, let us say that the mode is a delta peak, the coupling back to the waveguide will be instantaneously. There is no distance traveled in the cavity, so the phase matching is always zero. In section on delocalization, we saw we could increase the quality factor Q by increasing the mode volume. Specifically to this design we have to keep in mind that increasing the mode volume will not only decrease the level of miniaturization but also increase the influence of phase mismatch. It is of course not a coincidence that photonic crystals have a very good reputation of small mode volumes, while retaining a high Q. 3.3 Conclusion In this chapter, we created an analytical model to describe resonant cavities based on coupled wave theory. After the theoretical background we studied a map of relating the total reflection R to the coupling factor κ and the distributed reflection coefficient r. For every r there is a critical coupling for which the reflection is maximal. In phase matched conditions and for highly reflecting crystals we can have a perfect reflection. A phase mismatch reduces the reflection and gives rise to new modes, living outside of the cavity. At the end of the chapter, we set our goals for the rest of the master thesis being a high distributed reflection coefficient r, a critical coupling thickness, a phase matched structure and a low mode volume.

40 Chapter 4 Monorail structure After the theoretical models, it is time to design a laser cavity. In this chapter, we go through the design process of the monorail structure. A monorail is a waveguide with a D photonic crystal of holes in it, shown on figure 4.. In a first section, we will motivate our choice for the monorail rather than a two dimensional photonic crystal slab. The designing follows the logical steps: crystal - silicon cavity - mirror - III-V cavity. The choice of the crystal is dictated by the photonic band gap, which should be very wide and should contain the target wavelength 550nm. The main parameters are the pitch and the hole diameter. With this crystal, we can make a cavity. The design will be based on cavities found in literature. By varying the cavity length, we can tune the resonance wavelength to the right range. If we bring the right waveguide close to the cavity, the principle of resonating cavities will ensure reflection. In this chapter we will use a silicon waveguide rather than a III-V one. Thus, this is the passive analogon of the proposed design from chapter.3. To emphasize this, we will often refer to this waveguide as the III-V-like waveguide. The influence of the silicon parameters is investigated, showing the design flexibility this structure has to offer and the limitations one has to cope with. By ending a waveguide with two of these mirrors, we have created a cavity in the III-V-like waveguide. This will be our lasing cavity in the active design. The reason we used a silicon waveguide in stead of a III-V one is a practical one. Because of the limited time, we ordered a set of chips fabricated with e-beam. The delivery time is limited this way, but only a few chips are fabricated and thus, ruining structures because of bad III-V processing is highly unwelcome. Therefore we only discuss the silicon parameters in this chapter. The influence of the III-V waveguide is discussed in the next chapter. In chapter 6, a monorail with III-V overlay is optimized. As we are not able to fabricate it (but one could use more advanced processing tools to do it), we kept it separate. 4. The choice of a monorail Throughout chapter 2, we used band gap confinement in the (x, y) plane and total internal reflection in the z direction. There is no fundamental reason to use this configuration however. In the rest of this master thesis, we opted to use a monorail design in stead of this photonic crystal slab. A monorail is a waveguide with a pattern written into it, as shown in figure 4.. We still have index guiding in the z direction and band gap confinement in the x direction. In the y direction, light is now confined by index guiding. Although all the structures in this chapter are passive, the motivation of our choice was also inspired by the hybrid Si/III-V platform. The reason for the use of a D photonics crystal is fourfold: 24

41 Chapter 4. Monorail structure 25 z x y Figure 4.: A monorail design. First of all, it was already proven that very high Q cavities can be achieved with monorails [2]. Lots of these devices are fabricated in III-V materials, but some are also designed in silicon-on-insulator (SOI). Since we want to define a cavity in SOI (with a III-V overlay), there are some very good starting points, especially [2] was very useful. Photonic crystals are severely subject to fabrication errors. The fundamental reason for this is the discrete translational symmetry. All the unit cells have to be the same and it is because of the repetitive behaviour that the photonic band gap arises. Index guiding does not suffer from this specific problem (although surface roughness may be a greater problem here). On top of this, the asymmetry in the z direction is causing problems. Typically, we will use photonic crystal structures in a very symmetrical environment, e.g. the typical layer stack of a silicon-on-insulator device with oxide cladding SiO 2 / Si / SiO 2. In our stack structure we will use SiO 2 / Si / BCB / SiO 2 / III-V as described in chapter 8. In this we also have to add the fact that the III-V medium is a waveguide rather than a layer. The problem is illustrated in figure 4.2, where we see a cross section of a 2D photonic crystal lattice (e.g. a square lattice) with a III-V waveguide bonded on top. We can use the effective refractive index approximation along the z direction, making clear that the situation under the center of the waveguide and at the left of the figure (so with air cladding) will be very different. The effective index in the middle of the figure will be substantially higher, breaking the translational symmetry of the photonic crystal and therefore ruining the band gap. A monorail obviously does not suffer from this asymmetry. The last reason is a more practical one. We can think of monorail designs in a more intuitive way, especially because we are more used to index guiding. Simpler structures are a big advantage in any project, especially in the limited time frame of a master thesis. 4.2 Designing the crystal The band gap The most important property of a photonic band gap material is of course the photonic band gap. To simulate this, we used the frequency approach of MPB, see also the appendix A.3. Based on the monorail cavity proposed by Zain et.al. [2], we choose a pitch of 350nm and a hole diameter of 82nm. The band diagram of this photonic crystal is shown in figure 4.3. Note that we express on the y-axis the dimensionless frequency a λ. This means that, for a pitch of 350nm, the lowest order TE band gap (between the ground mode and the first order mode) lies from 408nm to

42 Chapter 4. Monorail structure 26 z x y Figure 4.2: Illustration of how the III-V waveguide poses a problem for photonic crystals a/λ 0.5 a/λ k*a/2π k*a/2π (a) TE modes (b) TM modes Figure 4.3: The band diagram of a monorail with a height of 220nm, a width of 500nm, a pitch of 350nm and a hole diameter of 82nm in a SOI system. The red dashed line shows the location of our target wavelength 550nm. 637nm and thus contains the target wavelength of 550nm (the red dashed line). It is only in the light cone that there are modes available at the target wavelength. If we plot the band gap as a function of the relative radius, we call the diagram a gap map. In figure 4.4, the gap map of the crystal described above is shown. When we increase the radius, we can further increase the band gap. However, we already have a very wide band gap, meaning that an increase would only result in a marginal improvement of our cavity. On top of this, an increased radius pushes the band gap to higher frequencies. This means that the light cone will be more prominent and the incompleteness of the gap will manifest itself more. This proximity of the light cone is not displayed in the gap map, but plays of course an important role. It is always useful to check the results of a program with an other simulation program. Figure 4.5b shows the spectrum of a transmission simulation in MEEP, the FDTD solver we used (see appendix A.). We have excited the structure from the left, two detectors measure the transmission and reflection spectrum.

43 Chapter 4. Monorail structure 27 Figure 4.4: The TE gap map of a monorail crystal with a height of 220nm, a width of 500nm and a pitch of 350nm in a SOI system. The chosen crystal with a relative radius of 0.26 is highlighted in green. Source Detector Detector Power flux (a.u.) (a) The structure Reflection Transmission Wavelength (µm) (b) The transmission and reflection spectrum of a crystal containing 8 holes. Reflection coefficient Simulated data Theoretical fit Number of holes N (c) The reflection coefficient as a function of the crystal length Figure 4.5: MEEP simulation of a monorail crystal with a height of 220nm, a width of 500nm, a pitch of 350nm and a hole diameter of 82nm in a SOI system. The region of high reflection indicates the band gap and confirms the results found earlier. The distributed reflection coefficient The distributed reflection coefficient r, as defined in section 3., of the crystal can be estimated by simulating the transmitted and reflected power of a monorail grating. According to [7], we can describe the reflection of a finite discrete translational symmetric device (the theory was actually developed for

44 Chapter 4. Monorail structure 28 gratings) using coupled wave theory. The total reflection coefficient can be estimated to be: R max = tanh 2 (rl) (4.) with L the total length of the crystal and r the distributed reflection coefficient. R max is the maximal reflection from the entire structure. Figure 4.5c shows a sweep over the number of holes in the crystal. The fitted curve is found to be: R max = tanh 2 (.2282 [ ] µm an ) (4.2) with a the pitch 0.350µm. The last term is an error term, needed because of insertion losses. The insertion losses are such that the simulations of very long crystal do not result in perfect reflection R max =. However, this is not important as we are more interested in what happens in short crystals in order to deduce the distributed reflection coefficient r. This data can be mapped onto the figure 3.2 of the analytical model. As we will see later, we will use 0 holes at the left of the cavity (including the tapers), meaning that we have to choose rl = on the x-axis of the figure. We can now compare this value to the ones found in a similar way using gratings. A typical distributed reflection coefficient r for a shallow etched grating is 0.45, thus only a bit more that a third of the value of the monorail. If we use the same parameters, a pitch of 350nm and 0 periods, rl will only be.575. In this case, figure 3.2 tells us that the maximal reflection one can achieve is only ±60%. We would need much longer structures (almost four times as long) for perfect reflection when using surface gratings. With the monorail, this is already possible using this length. 4.3 Designing the cavity We will now play with the other parameters in order to design a cavity with a high Q (order of 0 3 ). If the Q is too low, a considerable fraction of the light in the cavity will be scattered away. Once this light is scattered (e.g. from the sides of the cavity) there is no way of using the light to our benefit. It might be possible to have a low transmission through the waveguide, but also a low reflection because the light is just lost. Since it is reflection we are considered about, this is not what we want. As we will see in section 4.4, the quality factor of the mirror will be of the order 0 2. Since we can calculate this Q using Q mirror = Q coupling + Q intrinsic, a Q of order 0 3 is high enough. One or two extra orders will not make a big difference. For the design of the monorail cavity, we first neglect the III-V-like waveguide. Again, we start from the SOI monorail cavity found in [2], from which the crystal pitch of 350nm and the hole diameter of 82nm were already discussed in the previous section. The diameters and center-to-center distances of the four holes closest to the cavity are varied to make a taper (and this on both sides of the cavity). From inside to outside, the different diameters are 3nm, 66nm, 80nm and 70nm; the different center-tocenter distances are 290nm, 30nm, 304nm and 342nm. The different center-to-center distances should smoothen the refractive index change between the cavity and the crystal, whereas the different diameters should make sure the mode can change its spatial profile gradually from waveguide-like to Bloch mode. This internal taper is discussed more in detail in section 5.3 of the chapter on multirails, where we create a taper of our own. The cavity length in [2] was 425nm, resulting in a resonance wavelength of 484nm. However, our slab is a little bit thinner (220nm i.s.o. 260nm), meaning that we have to account for the loss of high index material (i.e. silicon). In a simple approximation, we could compare the vertical effective refractive Note the taper holes will actually have a different distributed reflection coefficient. However, as a first approximation, we can use the crystal value for all holes.

45 Chapter 4. Monorail structure 29 indices n eff and scale the vacuum resonance wavelength λ 0. Remember that the resonance conditions applies to the material wavelength: λ material = 2L = λ 0 n eff (4.3) with L the length of the cavity and λ material the material wavelength. A mode solver (e.g. CAMFR) presents us the effective index of a slab of 260nm thickness and 220nm: and respectively. Thus, we estimate the resonance wavelength using our slab and the dimensions from [2] to be λ 0 [220nm] = λ 0[260nm] n eff [260nm] n eff [220nm] = 426nm. The target wavelength is 550nm, meaning that we have to increase the cavity length, thereby increasing the effective refractive index and changing the resonance condition (equation 4.3). After some simulations, we changed the cavity length to 465nm, resulting in a resonance wavelength of 525nm. There are also tapers at the end of the photonic crystal, consisting of only two holes. The center-to-center distances are 30nm and 290nm, the hole diameters are 60nm and 30nm. Zain et.al. added this taper to increase the transmission through the cavity. Although we are going to use the cavity in a different way, we did not get rid of the tapers. They proved to be useful to suppress some arising band edge modes. We have simulated this cavity using MEEP, a three dimensional finite difference time domain solver. We performed a transmission simulation exciting the waveguide on the left of figure 4.6a with a gaussian pulse. This light travels to the cavity and part of it is reflected, part of it is transmitted. The spectra of the light is captured at fluxplanes left and right of the cavity and shown in figure 4.6c. To estimate the refractive index, we took a field plot of the cavity mode (figure 4.6b) and counted the number of pixels to calculate the material wavelength. Since we know the vacuum wavelength from the spectra, we can derive the effective refractive index. Reflection Transmission Source Detector (a) The structure Detector Power flux (a.u.) Wavelength (µm) (b) The cavity mode (c) The transmission and reflection spectrum Figure 4.6: A monorail cavity with the following parameters: a height of 220nm, a width of 500nm, a pitch of 350nm, a hole diameter of 82nm and a cavity length of 465nm. Both inner and outer tapers were added. The crystal itself consists of 4 holes on the left and 0 holes on the right This configuration of a cavity yields a quality factor Q = 955, using harminv (see the appendix A.2), and the resonance wavelength is.525µm. By counting the number of pixels we can estimate the material wavelength. From this, we can deduce the effective index of this cavity to be There is also a secondary resonance at a wavelength of.60µm with a quality factor Q = 76. This is not a traditional cavity mode, but a band edge mode.

46 Chapter 4. Monorail structure Designing the mirror In the previous section, we have designed a monorail cavity. As already made clear, we will add a waveguide to show the resonant cavity effect as is shown in figure 4.7. Depending on the phase matching and coupling condition, this effect might be very strong or non-detectable. Source Detector Detector Waveguide gap Waveguide width Taper out N left Taper in Taper in Nright Taper out Figure 4.7: A resonant monorail cavity through sideways coupling. In a first approximation, we can say that the cavity will not be influenced by adding the waveguide. This approximation is valid as long as the distance to these other structures is large enough. This first approximation serves as a starting point. If necessary, we can deviate the parameters a little bit and see whether the simulated result is better. A silicon waveguide is added so that sideways coupling is possible. First we have to predict the waveguide width needed. In order to obey the phase matching condition, the effective refractive index of the waveguide should be 2.29, the same as the one from the cavity. Under the same approximation as the paragraph above, we can look at the waveguide alone. A SOI rib waveguide with a width of 390nm and a height of 220nm results in an effective refractive index of 2.29, according to the simple effective index method. As we will see, adding this waveguide leads to a high reflection (96%) under the right conditions. The mirror then has N left equal to 4 and N right to 6. The entire structure has a length of only 7.4µm. Below, we will discuss the most important parameters The influence of the waveguide gap The coupling is determined by the waveguide gap and the length over which coupling is possible. We opted to work with a small gap of 00nm. Because of this, one might doubt the correctness of the approximation we made. Therefore we varied the waveguide width with steps of 30nm for almost all the simulations. It turned out that the results are optimal as we predicted them. We now estimate the effective refractive index of the cavity mode to be This is a little bit higher than the 2.29 of the stand-alone cavity found earlier but also the index waveguide has increased due to the presence of a high index material. Because of this, the difference in index has not changed that much. Also, note that our method of estimating the refractive index is not very accurate, exactly because it changes anyway when adding the waveguide. Under a larger waveguide gap, the structures are decoupled. Therefore, it would be most interesting to lower the coupling width and see what happens. However, 00nm is already a very low value and going lower would bring fabrication difficulties. It seems that the passive design with sideways coupling is not the best to investigate this parameter. In section 5.4. we do discuss the coupling condition in further detail.

47 Chapter 4. Monorail structure The influence of the number of holes on the left N left The number of holes on the left N left impacts on two processes. On one hand, it is the left side of the silicon cavity and thus it is important to have enough holes in order to limit the scattering. On the other hand, it defines the length over which coupling is possible from the III-V-like waveguide to the cavity. A sweep of this parameter N left is shown in figure Power flux (a.u.) Power flux (a.u.) Power flux (a.u.) Reflection Transmission Wavelength (µm) 0.2 Reflection Transmission Wavelength (µm) 0.2 Reflection Transmission Wavelength (µm) (a) N left = 2 (b) N left = 4 (c) N left = 2 Figure 4.8: A resonant monorail cavity: variation of the coupling length. A schematic of the structure is shown, followed by a field plot at the cavity resonance wavelength and the transmission and reflection spectrum. The reflection from the middle structure (N left = 4) reaches levels up to 96% with a Q factor of 30 at the resonance wavelength 527nm. The coupling here is (very close to) critical. In the mode profile, we see a nicely confined cavity mode resonating with the source on the left. Note that we did not chose a plot when the cavity mode is maximal since the field in waveguide is too weak compared to this maximum. Thus, in reality, the cavity mode is stronger than may be expected from this plot. On figure 4.8a, the coupling length is too short. As predicted with the analytical model, the reflection drops (to 78% with a Q of 04 at the cavity mode). From the field plot, it is clear what too short means. A significant fraction of the light scatters away at the left end of the monorail (the red circle in the field plot of figure 4.8a), because the cavity mode becomes lossy. Ideally, as in figure 4.8b, the cavity mode profile is zero at the edge of the monorail, so no scattering can occur. It can be seen that also the transmission at the resonance is not zero on figure 4.8a (note that for the other graphs, the transmission is zero). The extra losses in the cavity, make sure the mode can not become strong enough to compensate the light in the III-V-like waveguide. On figure 4.8c, the opposite case is shown. At the cavity mode, the reflection has now dropped to 82% with a Q of 26. In the field plot we see that the mode has a node, but because the monorail is too long on the left hand side, power is again built up (the red circle in the field plot). This power is mainly coming from reflected light in the III-V-like waveguide coupling back to the monorail. The influence of the coupling length, expressed in number of holes N left, is shown in a graph on figure 4.9. We have a broad maximum of both the power reflection and the quality factor ranging from 4 to 0 holes (note that this number does not include the tapers). This range corresponds with the width of the node visible on the field plot of figure 4.8c. As long as the edge of the monorail coincides with the node of the cavity mode, there no scattering to be expected, meaning that the reflection is high (provided that all the other conditions such as phase matching are preserved of course).

48 Chapter 4. Monorail structure 32 Power reflection coefficient R Nleft (a) Influence on the reflection coefficient Quality factor Q Nleft (b) Influence on the quality factor Figure 4.9: The influence of the coupling length on the reflection coefficient and the quality factor of a resonant monorail cavity. From the structures of figure 4.9a, the ones with N left equal to 4 until 0 are good choices for a cavity. The reflection coefficient is astonishingly high: 96% and more. The mirror has a quality factor of 29 or more and there is no reasonable reflection other than the desired one below a wavelength of ±.580µm. The quality factor seems to be dominated by the coupling, as apparent from equation 4.4. Since the total quality factor of long cavities is of the order 0 3, it has to be the coupling defining the Q factor. This relation is illustrated in figure 4.0. Q total = Q coupling + Q intrinsic (4.4) Q coupling Q intrinsic Q total N left Figure 4.0: The relation between Q total and the coupling and intrinsic cavity Q For small coupling lengths, the coupling is negligible if it comes to losses, and thus quality factor Q coupling will be very large. Therefore, we can conclude that the cavity itself is very lossy for small N left and thus the Q drops tremendously. This result confirms the interpretation of figure 4.8a The influence of the number of holes on the right N right At longer wavelengths, we again have a reflection coefficient of 80% up to 92% (see figure 4.8b). The mode responsible for this high reflection is a band edge mode. Contrary to the number of holes on the left

49 Chapter 4. Monorail structure 33 hand side of the cavity, those of the right hand side have nothing to do with the coupling to the cavity. Light that coupled from the cavity to the waveguide will interfere destructively with the light already present, making sure that nothing can couple to the monorail at the right hand side of the cavity. Thus, we can manipulate this side without influencing the cavity mode coupling to the monorail. A sweep of this parameter N right is shown in figure Reflection Transmission 0.8 Reflection Transmission Power flux (a.u.) Power flux (a.u.) Power flux (a.u.) Reflection Transmission Wavelength (µm) Wavelength (µm) Wavelength (µm) (a) N right = 0 (b) N right = 6 (c) N right = 4 Figure 4.: A resonant monorail cavity: variation of the number of holes at the right hand side of the cavity. A schematic of the structure is shown, followed by a field plot at the cavity resonance wavelength and a field plot at the band edge mode wavelength. At the bottom, the transmission and reflection spectrum is shown. Figure 4.a shows the same structure as figure 4.8b. We see that the right hand side is long enough for the cavity mode to be nicely confined in the monorail. There is virtually no scattering at the right end of the waveguide in the upper field plot (at the cavity mode resonance). Also in the field plot at the band edge resonance, there is very little scattering, making sure the power reflection is high. This structure reflects a lot of light at both resonances: 96% at the cavity mode (Q = 29) and up to 92% at the band edge (Q = 23). In figure 4.b, the cavity mode resonance is as high as for larger N right (95% reflection with a Q = 26). Indeed, on the upper field there is still no scattering at right visible. There is some scattering for the band edge mode though (encircled in red). This makes sure that the reflection coefficient is not as high as before, R = 69%. This trend is more explicit in figure 4.c, where the reflection coefficient has dropped to 52%. Now the right hand side of the monorail cavity has become too short for the cavity mode to be confined. At the edge of the monorail of the upper field plot, we clearly see some scattering. This scattering causes the reflection coefficient at the cavity mode to drop to 8%. Figure 4.2 shows the reflection of several simulations as a function of the number of holes at the right N right (also here, N right does not include the holes forming the tapers). Above a certain threshold value (N right = 6) the power reflection of the cavity mode is virtually constant. Below this value, it lowers dramatically. The band edge mode reflection drops steadily by decreasing the number of holes. This mode is a very broad mode, so there is always a significant fraction extra loss by reducing N right. Of course this fraction will be bigger and bigger when we approach the maximum of the mode. For very long crystals we expect this behaviour to be more complex due to the existence of spatial maxima and minima. If the edge of the monorail coincides with a node, the scattering will be very low and therefore the reflection coefficient will be high.

50 Chapter 4. Monorail structure 34 Power reflection coefficient R Cavity mode Band edge mode Nright Figure 4.2: The reflection coefficient as a function of the number of holes at the right side N right. It is interesting to note that N left =4 still leads to good reflections, meaning that the scattering is low, while N right =4 leads to significant scattering. The reason that N right has to be at least 6, is the double coupling of the light. A fraction of the light that coupled from the cavity to the waveguide (going to the left) can couple back to the monorail and interfere destructively with the light already propagating in the monorail. Because of this additional destruction of light, the cavity mode will be asymmetrical, decaying faster at the left than at the right. 4.5 III-V-like cavity using a resonant monorail cavity In the section above, we designed a mirror based on the idea of resonant cavities. If we now put two of these mirrors facing each other, we can make a cavity in the III-V-like waveguide. In this paragraph, we will first try to estimate the quality factor of the cavity we have built. Afterwards a simulation will check our estimation Estimation of the III-V-like cavity We will use the structures from figures 4.8b and 4.a as a mirror. This means that the maximal reflection is 96% at a wavelength of.527µm. A small calculation estimates the quality factor of the III-V like cavity. We start with the intensity degradation of a pulse in a typical Fabry-Perot cavity as a function of time telling us that after one round trip the intensity in the cavity has lowered because of losses. I(2L) = I(0)R R 2 (4.5) with R and R 2 the reflection coefficient of the two mirrors. The mathematics become much easier if we use continuous losses in stead of the discrete losses of a pulse traveling in the cavity: This allows us to calculate the loss factor α: α = ( ) 2L ln R R 2 I(x) = I 0 exp( αx) (4.6) I(2L) = I 0 exp( α2l) = I 0 R R 2 (4.7) (4.8) We can also look at the intensity as a fuction of time t rather than location of the pulse x: ( ) t I(t) = I 0 exp τ p (4.9)

51 Chapter 4. Monorail structure 35 with the photon lifetime τ p. In order for equation 4.9 to correspond with equation 4.6, the exponents need to be linked through the group velocity v g : with c the vacuum speed of light and n g the group index. Using equations 4.8 and 4.0, we can calculate the photon lifetime: x t = ατ p = v g = c n g (4.0) τ p = n g c α = n g c The quality factor and photon lifetime are connected through Q = ωτ p : Q = ωτ p = 2πn g λ 0 2L ( ) (4.) ln R R 2 2L ( ) (4.2) ln R R 2 With a mode solver, we can calculate the group index through the definition: n g = n λ 0 dn dλ 0 (4.3) Our silicon waveguide has a group index n g = 4.2. We can choose the length to be a little bit over 5.µm, as we will space the mirrors 5.µm apart. We assume that a longitudinal mode can exist at the wavelength of maximal reflection of the mirror: R = R 2 = These values lead to an estimation of Q = Simulation of the III-V-like cavity In order to create a cavity with reasonable losses (meaning a low threshold current), we need to synchronize three resonances: two resonances from the mirrors and one from the III-V-like cavity itself. From the spectra shown before, it is clear that the reflection coefficient of the mirror is very wavelength dependent. In simulations it is of course no problem to match the spectra of the mirrors, but in real life this can be a problem due to fabrication errors. In this we would like to point out that fabrication errors do not necessarily lead to a mismatch. If the errors are the same for both mirrors (e.g. due to overetching the monorail is too thin), both spectra will change in the same way and they will still be matched with respect to each other. Of course, it may be possible that the maximal reflection coefficient diminishes, but the wavelength of this maximum will be the same for both. In the III-V stack we can say to have a Fabry-Perot cavity, although the mirrors are very wavelength dependent. One has to make sure that one of the resonances of this Fabry-Perot cavity coincides with the maximum of both mirror reflection coefficients. We opted to use the structure from figures 4.8b and 4.a as a mirror. The spacing between these two mirrors (5.µm) was chosen such that there is a Fabry-Perot resonance at the cavity mode resonance wavelength of the monorail. A different spacing between the mirrors will result in a different resonance condition in the III-V-like waveguide, leading to a different resonance wavelength. This III-V-like resonance wavelength might not coincide with the mirror resonance. The resulting spectrum is shown in figure 4.3, together with the spectrum of the mirrors as a reference. This spectrum is quite complicated, since there are a lot of conditions to be fulfilled to have a resonance. We will go over the different areas. Under the wavelength of 55nm, the mirror is not reflecting and quasi everything is transmitted. The light just propagates through the III-V waveguide without interfering too much with the resonant cavity mirrors.

52 Chapter 4. Monorail structure 36 Source Detector Detector Full cavity reflection Full cavity transmission Mirror reflection Mirror transmission Power flux (a.u.) Wavelength (µm) Figure 4.3: Simulation of a III-V cavity using a resonant monorail cavity. The reflection and transmission spectrum of the entire structure is shown in full lines. The dotted line is the spectrum of the structure used as a mirror (this is the one from figures 4.8b and 4.a). From 55nm to 535nm, the mirrors are reflecting. It is only at 526nm that also the Fabry-Perot is sustaining a mode. Therefore, the larger part of this wavelength range is reflected, primarily by the first mirror. Due to the lack of a longitudinal resonance, a power buildup between the mirrors is not allowed. At the wavelength of 526nm all the conditions are fulfilled. Both mirrors are reflecting and a longitudinal mode is sustained. There will be a large power buildup between the mirrors. It is the leakage of this mode that is ensuring the transmission. The wavelength is very close the maximum of the mirror reflection in order to optimally make use of it. The quality factor Q is 958, which is very similar to the estimated one. In the wavelength range of 535nm to 570nm, the mirror reflection is again very low. However, since this time the mirror transmission is also quite low, the transmission output is not as high as in the range below 55nm. Most of the light is scattered away. Above 570nm the transmission becomes very low. The mirror reflection is very high, but the total reflection does not follow it very well anymore. Again, it is mainly the first mirror who will be the reflector, but we have to pay attention to the fact that the mirror is excited backwards. As was stated before, one end is longer than the other. At these wavelengths, the reflection is triggered by the band edge mode living in the far end of the structure. Thus, now this band edge mode is very lossy due to the lack of space. Most of the light is being scattered. In the end, the only mode important for lasing structures is the cavity mode near the maximum of the mirror reflection. It has a Q of 958 at a wavelength of.526µm. 4.6 Conclusion We started this chapter with explaining why we choose for monorails. One of the main reasons, and specifically to this design, was the robustness against the effect of the III-V waveguides. The fact that

53 Chapter 4. Monorail structure 37 this is a waveguide rather than a layer makes sure that we have different effective refractive indices in the crystal buried under or next to the waveguide. Designing photonic crystal devices starts with the band gap. Literature provided us with a crystal having a large TE band gap and a high distributed reflection coefficient. Using this crystal, we could build a cavity resonating close to the target wavelength of 550nm (525nm) with a quality factor of the order 0 3. We used photonic crystal tapers to enhance the Q, making use of the delocalization mechanism. By bringing a silicon waveguide close to the cavity, we were able to build a mirror reflecting 96% of the incident light. This mirror structure has a few parameters, which we took a look to. The design platform was especially useful to check the influence of the number of holes to the left and to the right of the cavity. N left has a major influence on both the coupling and the intrinsic cavity losses, making it a critical parameter. Making this parameter too short or too long can degrade the reflection substantially. We have more freedom with N right, with which we can suppress the band edge modes. Two of these mirrors create a cavity. The distance between them has to be chosen carefully as the resonance should coincide with the reflection maximum. A spacing of 5.µm resulted in a mode with a Q of 2000.

54 Chapter 5 Multirail structure 5. The choice of a multirail In the previous chapter we saw some highly reflective resonant cavity devices. They were based on a monorail and sideways coupling. Unfortunately the effective refractive index of the cavity is only 2.29, meaning that phase matching this cavity to a III-V waveguide is impossible. The effective refractive index of the III-V waveguide will be around There are various ways of tuning the index: ˆ ˆ By making the hole size smaller, we effectively add more silicon to the structure. Since this is a high index material, the effect will be as desired. Of course, we have to keep an eye at the band gap. As can be seen in the gap maps, the band gap eventually closes when the holes are shrunk too much. In a monorail, the D photonic crystal is embedded in a waveguide. If we increase the width of the waveguide, the amount of silicon will rise and thus the index will increase. However, increasing the waveguide width also broadens the mode in the y direction of figure 5., i.e. the lateral direction. Because of this delocalization, less light will feel the photonic crystal and the distributed reflection coefficient will drop. This means that we need much longer photonic crystals and the mode volume will be enlarged substantially. As stated in the section on delocalization, this larger mode volume undermines the miniaturization of the laser and the phase matching condition of the resonant cavity. From analyical model, figure 3.2, it is clear that a large distributed reflection coefficient r is desirable. The solution we propose to this problem is the multirail. This multirail is the multiple row analogon of a monorail. The idea is that we have some discrete translational symmetry in the y direction, more specifically in the waveguide. The D photonic crystal is being transformed to a one-and-alittle-bit dimensional crystal. The different rows have a spacing equal to the pitch. The light, more spread out in the y direction due to the broad waveguide, now feels the crystal a lot more. This ensures the distributed reflection coefficient to be higher. These tuning techniques were incorporated in our new cavity. Of course, the influence of one is bigger than that of another, but every little helps. 38

55 Chapter 5. Multirail structure Designing the crystal The band gap When creating the new lattice, we have to pay attention to the fact that the smallest dimension of a structure has to be larger than 00nm to be feasible for fabrication. Figure 5.2 shows the band diagram of a triplerail (meaning three rows of holes in a waveguide, illustrated in figure 5.) with a pitch of 300nm. z x y Figure 5.: A multirail design a/λ 0.5 a/λ k*a/2π k*a/2π (a) TE modes (b) TM modes Figure 5.2: The band diagram of a triplerail with a height of 220nm, a width of µm, a pitch of 300nm and a hole diameter of 50nm in a SOI system. The red dashed line shows the location of our target wavelength 550nm. We can see that the gap is not as large as before and there are modes at the target wavelength of 550nm (shown in red). However, this should not be a large problem. In the laser structure, we will assume that the III-V waveguide is lasing in the ground mode of the waveguide. The coupling to the first higher order

56 Chapter 5. Multirail structure 40 mode will be zero due to the fact that it is an odd function whereas the III-V waveguide ground mode is even. This means that the light will couple to the ground mode and the second order mode. The latter is very close to the light cone and the phase matching condition only enhances the preference to couple to the ground mode. Thus, we can say that for our structure, the band gap covers the relative frequencies between 0.9 and When using a pitch of 300nm, this corresponds to wavelengths of.565µm and.337µm respectively. This band gap is confirmed by the FDTD simulation performed using MEEP, shown in figure 5.3b. The distributed reflection coefficient As in the case of the monorail, the distributed reflection coefficient of the crystal can be deduced from the transmitted and reflected power. The total reflection coefficient can be estimated to be:[7] R max = tanh 2 (rl) (5.) with L the total length of the crystal and r the distributed reflection coefficient. R max is the maximal reflection of the entire structure. Source Power flux (a.u.) Detector (a) The structure Reflection Transmission Detector Wavelength (µm) (b) The transmission and reflection spectrum Reflection coefficient Simulated data Theoretical fit Number of holes N (c) The reflection coefficient as a function of the crystal length. Figure 5.3: FDTD transmission and reflection simulation of a finite triplerail crystal with the following parameters: a height of 220nm, a width of µm, a pitch of 300nm and a hole diameter of 50nm. Figure 5.3c shows the obtained data and the fitted curve with the form: [ ] ) R max (N) = tanh ( a N 0.03 (5.2) µm with a the pitch of the crystal 300nm. The last term is an error term due to the insertion losses of the structure. We map this data on the analytical model. As we will see later, we will use 25 holes at the left of the cavity (N left = 25). Together with the four taper holes, this makes a length of 29a = 8.67µm, meaning that we have to choose rl =.6 in the analytical model (the x-axis of figure 3.2). This is a very high level and very high reflection (>99%) should be obtainable with this resonant cavity design. Note that this value of r is a little bit higher than the one from the monorail and much higher than that of surface gratings. Also here, the taper holes will actually have a different distributed reflection coefficient. In a first approximation, we can use the crystal value for all holes.

57 Chapter 5. Multirail structure Designing the cavity Starting from the crystal above, we create a cavity. The pitch is 300nm, the hole diameter 50nm. The cavity length is 380nm and the waveguide width is µm. There is also a taper to enhance the intrinsic quality factor with distances of 292nm, 30nm, 3nm and 322nm (from inside to outside). The diameter of the holes in the taper is constant (at 50nm) because the cavity becomes multimode otherwise. For the creation of the taper, we were inspired by the following design rule, given earlier by Schriever et.al. [3] and Sauvan et.al. [4]. The effective refractive indices of the cavity waveguide mode (n eff ) and the Bloch mode of the crystal (n B ) should match (n eff = n B ). This way, the modal mismatch between both should be minimized. We use a four hole taper to smoothen this transition. center n eff taper taper2 taper3 taper4 crystal n B Design rule n a(nm) Monorail cavity a(nm) n Triplerail cavity a(nm) n Table 5.: Comparison of the effective index design rule and the final taper structures. Table 5. shows an implementation of this design rule. First, we calculate the effective index of the cavity. Assuming the index to be the one from a SOI waveguide of µm wide and 220nm high, so without any influence of the photonic crystal, we find n eff = Second, we calculate the refractive index of the crystal, which can be expressed as n B = λ0 2a, with a the pitch, leading to an index of (using λ 0 =.55µm). The refractive index mismatch is significant: Therefore we want to create four intermediate regions, bridging the gap in a linear way. If we assume the mode in the taper resembles the Bloch mode, we can use the same formula to calculate the different pitches : a = λ0 2n. In literature, it is however also emphasized that the proper parameters can only be obtained by successive variation of all taper parameters. This of course results in extensive calculations. The design rule mentioned above can be used as a starting point, but the final result can differ a lot. The second row of table 5. applies the design rule to the monorail cavity from chapter 4, adopted from Zain et.al. [2]. The discrepancy between the design rule and the monorail cavity is striking and one wonders whether this design rule was even used as starting point. The last row of the table shows the final taper structure of the triplerail cavity we created. The values are different from the ones of the design rule, but the trends are preserved (a monotonically decaying refractive index in the taper). In tapers one normally also changes the hole diameter. This smoothens the transition of the spatial mode profiles. In the cavity mode the profile will be much like a waveguide, so with a maximum in the center and exponentially decaying tails. The Bloch mode will be more spread out and fills the places of high index materials. However, in our multirail a change in hole size lead to a multimode cavity. To avoid this, we did not change the hole size. As a result of the extensive simulation work, we created a cavity with a quality factor Q = 4684 and a resonance wavelength.558µm. The effective refractive index was estimated to be 2.59, using the same technique described in chapter 4, i.e. we count the number of pixels and the index is found using resolution periods n eff = λ 0 pixels. An index of 2.59 should be high enough to phase match the III-V waveguide to the cavity. The profile and spectrum of the cavity mode are shown in figure One may note that the sum of the transmission and reflection on the resonance is not one. We believe this is not due to scattering, but because of the finite simulation time there is still a lot of light in the cavity. It is not worth waiting for as we can derive all the information about losses through the estimation of the Q of Harminv.

58 Chapter 5. Multirail structure 42 Reflection Transmission Source Detector (a) The structure Detector Power flux (a.u.) Wavelength (µm) (b) The cavity mode (c) The transmission and reflection spectrum Figure 5.4: A triplerail cavity with the following parameters: a height of 220nm, a width of µm, a pitch of 300nm, a hole diameter of 50nm and a cavity length of 380nm. An inner taper of four holes was added. The crystal itself consists of 25 holes on the left and 25 holes on the right. 5.4 Designing the mirror In the previous section, we have created a cavity resonating at telecom wavelengths and an acceptable Q. We have made sure that the effective refractive index of the cavity is high (2.59), so phase matching with our III-V stack is possible. For details on the structure of the III-V stack, we refer to section 8.2. In this section we will review the influence of the different parameters we can play with in the III-V design. The silicon parameters were discussed in section 4.4. When choosing the III-V parameters at or close to the optimum, we get a spectrum as is shown in figure 5.5d. The silicon part of this design consists of a triplerail cavity with the parameters described in the previous section. The number of holes N left and N right were chosen 25 each. This high number was chosen because of practical reasons as we wanted to make sure the intrinsic losses of the cavity would not limit our mirror design. At the time of ordering the chips, too little was simulated on the mirror design to risk fewer holes. As we see from the mode profile in figure 5.5b, also a shorter triplerail cavity would not lead to a lot of scattering. The bonding thickness was chosen 30nm and the III-V waveguide width.85µm. These parameters are further discussed below. As a result we get a reflection coefficient of 88% at the cavity mode (λ =.568µm), with a quality factor Q of 247. The band edge mode reflection - at a wavelength of.585µm - is even higher, reaching levels of 95% reflection and a Q of 275. The mode profiles are shown in figures 5.5b and 5.5c. Note that this mirror only has a length of 7µm The influence of the III-V - to - silicon thickness To make a mirror, we have to bring a III-V waveguide close to the silicon cavity. But how close is close? The analytical model predicted that there is an optimum as a function of the coupling. It is possible to calculate the coupling factor κ using the different field profiles and use an overlap integral as in equation 5.3, in which k 0 is the vacuum wave vector, n AB is the refractive index of the supermode of the two structures and n A and n B those of the individual structures and ψ A and ψ B are the field profiles. κ = 2 k2 0 (n 2 AB n 2 A )ψ Aψ B dx (n 2 AB n 2 B )ψ Bψ A dx = 2 k2 0 (5.3)

59 Chapter 5. Multirail structure 43 Source Detector (a) The structure: cross section perpendicular to y Detector (b) Mode profile of the cavity mode (cross section as in (a)) (c) Mode profile of the band edge mode (cross section as in (a)) Reflection Transmission Power flux (a.u.) Wavelength (µm) (d) The transmission and reflection spectrum Figure 5.5: Characteristics of a resonant triplerail cavity mirror. However, the calculation of this overlap integral is not that straight-forward and we thought it easier and even faster to try with different bonding thicknesses. Figure 5.6 gives an overview of how the different cases (bonding thickness too small, too large or good) look like. The top figure indicates the cross section is taken perpendicular to the y direction. This allows us to see the two different structures propagating along each other. Figure 5.6a shows the mode profile under too tight coupling. The first thing that pops up is that all the colors, especially in the III-V waveguide, are much brighter. This is because of normalization. Due to the lack of a very strong cavity mode, the power differences are smaller and therefore, low powers can be expressed better. We see that light couples multiple times from and to the III-V waveguide. Not one the lobes is explicitly stronger, but the biggest reflection occurs over the first one (this can be derived from the power drops in the III-V waveguide occurring at the different lobes). As was predicted from the analytical model in figures 3.4 and 3.5, most of the reflection occurs before the cavity itself. From the scattering at the end of the silicon waveguide, we conclude that there will be a lot of losses which will lower the reflectivity considerably. When the coupling is too weak, we get a mode profile as in figure 5.6c. The light in the silicon behaves nicely, it is very well confined in the cavity without too much extras. However, we can also see that a significant part of the light in the III-V waveguide propagates past the cavity.

60 Chapter 5. Multirail structure 44 (a) too small bonding thickness (20nm) (b) critical bonding thickness (30nm) (c) too large bonding thickness (400nm) Figure 5.6: Mode profiles of the resonant triplerail cavity of under different coupling strengths. Let us say that there is a continuous flow of light from the left. This flow couples, be it weakly, to the cavity where the resonant mode is building up power. This power buildup is counteracted by the coupling back to the waveguide. Because the reflection is too large compared to the coupling, the mode volume will not be large enough for coupling back enough light. Also this can be deduced from the analytical mode. Figure 5.7a shows the power of the different modes as a function of the position x in the undercoupled case. (a) Weak coupling (κ = 0.66) (b) Critical coupling (κ = 0.55) Figure 5.7: Analytical mode powers under phase matching conditions. The distributed reflection r was chosen 3.5. There is a very strong cavity mode and this mode ensures a considerable drop in transmission. However, the transmission stays 5% because the length to couple back to III-V waveguide (waveguide A in the model) is too short, or equivalently, the reflection is too strong. Note that sum of reflected power (B (0)) and transmitted power A + (L tot ) is not equal to the input power (A + (0)) because the strong cavity mode results in extra losses. Comparing this mode strength to the critically coupled one of figure 5.7b, the difference in scales is remarkable. Without these extra losses, the mode would be even stronger and the transmission would be lower. Also note that even under critically coupled conditions, the reflection is

61 Chapter 5. Multirail structure 45 not neccesarily 00% (see figure 3.2) Figure 5.6b shows the case of critical coupling. There is a strong cavity mode that does not suffer from scattering from the edges. The light in the III-V waveguide clearly loses most of its power when passing past the cavity. Still, if one looks very closely there is still a small fraction of the light passing. This can have various reasons. First of all, the analytical model s main result was that there is an optimal coupling, however this optimum needs not to result in a perfect reflection. Secondly, although we did extensive simulations the parameters might not be exactly correct and there might be a small error on the phase matching or on the coupling thickness. Of course, it makes no sense to simulate further and further, since the fabrication technology will not be able to follow our simulation resolution. A last possible reason is coupling to higher order modes. In the analytical model this was neglected, but in reality it can of course occur. The III-V waveguide has a width of.85µm and thus sustains higher order modes. A cross section of the III-V waveguide perpendicular to the z-direction is shown in figure 5.8. Figure 5.8: A cross section of the III-V waveguide perpendicular to the z direction under critical coupling conditions. The source is the bright spot on the left (it is so bright because half of the light is immediately lost propagating to the left) and excites only the ground mode. Near the middle of the figure, we can clearly detect a sudden power drop in the mode, indicating the location of the silicon cavity buried beneath it. After this cavity, we see that an important fraction of the transmitted light is located in higher order modes (mainly second order). Note that it is impossible to couple to the first higher order mode as this is a function with odd parity. Since the resonator is oscillating in the ground mode, the coupling overlap integral is zero. Although an important fraction of the light is contained in higher order modes, we also see regions resembling the ground mode. The non-zero transmission is probably due to a combination of the reasons explained above. Figure 5.9 shows the reflection coefficient when sweeping the coupling thickness, figure 5.0 the corresponding spectra. Both for the cavity mode and the band edge mode, there is an optimal coupling condition. It appears so that this occurs at the same bonding thickness. At large thicknesses the modes are coupled too weakly and the mode profiles resemble figure 5.6a. Shorter thicknesses lead to mode profiles as in 5.6c. When looking at the spectra, we see that for small bonding thicknesses strange effects start to take place. This is already a bit visible in figure 5.0b, but manifests itself fully in figure 5.0c. The III-V waveguide and the cavity influence each other very heavily and therefore have to be considered as one at all times. This makes designing very difficult and therefore we will not work in this area The influence of the waveguide width: phase matching condition Throughout this master thesis, we tried to emphasize the importance of phase matching. Without phase matching, the light in the III-V waveguide and the light coupling from the cavity to the waveguide are not in antiphase and will therefore not cancel each other out. We used a mode solver to estimate the width of the III-V waveguide. When we only use one pair of layers of InP-InGaAsP(Q=.22), the effective refractive index is 2.59 when the thickness is.85µm. This is a very broad III-V waveguide meaning that an error on this width does not result in a large error in terms

62 Chapter 5. Multirail structure 46 Reflection coefficient cavity mode band edge mode III V to silicon distance (nm) Figure 5.9: The reflection coefficient of the cavity mode and band edge mode as a function of the coupling thickness. Note that the coupling thickness is determined by the sum of the BCB thickness and the oxide thickness (see figure 8.)..2 Reflection Transmission.2 Reflection Transmission.2 Reflection Transmission Power flux (a.u.) Power flux (a.u.) Power flux (a.u.) Wavelength (µm) Wavelength (µm) Wavelength (µm) (a) Si to III-V thickness: 30nm (b) Si to III-V thickness: 280nm (c) Si to III-V thickness: 260nm Figure 5.0: Spectra of the resonant multirail cavity mirror at different silicon to III-V thicknesses effective index waveguide width (µm) Figure 5.: The effective refractive index versus the III-V waveguide width. of effective refractive index. Figure 5. shows the effective index of the fundamental mode versus the waveguide width. Note that we have swept the width over a very large range, as the typical error is less than 00nm. This small dependence on waveguide width is confirmed by the performed simulations in the same range

63 Chapter 5. Multirail structure 47 of widths. In figure 5.2, the cavity reflection coefficient is given as a function of waveguide width. We see that there is a broad region, much broader than the error margin of the technology, for which there is a high reflection. Especially for broad waveguides, the reflection coefficient of the cavity is stable. Reflection coefficient cavity mode band edge mode Waveguide width (µm) Figure 5.2: The reflection coefficient of the cavity mode and band edge mode as a function of the III-V waveguide width. We also see that there is a discrepancy between the cavity mode and the band edge mode. Whereas the cavity mode prefers broad waveguides and thus high refractive indices, the band edge mode has an optimum near thinner waveguides. Reflection Transmission Reflection Transmission Power flux (a.u.) Power flux (a.u.) Wavelength (µm) Wavelength (µm) (a) Width = 2.µm (b) Width =.55µm (c) Width = 2.µm, cavity mode λ =.568 (d) Width =.55µm, cavity mode λ =.568 Figure 5.3: Comparison of different phase matching conditions for the cavity mode. Let us first check the phase matching condition of the cavity mode using broad waveguide on figure 5.3c. Between the two vertical lines, we count 9.5 wavelengths in the cavity. Also in the III-V waveguide we count 9.5 wavelengths. Note the π 2 phase shift, clearly visible at the left drawn line and therefore also at the right line. In figure 5.3d, we count the same 9.5 wavelengths in the cavity, but in the III-V waveguide we have almost half a wavelength short. The refractive index in the waveguide is lower, making the material

64 Chapter 5. Multirail structure 48 wavelength λ0 n eff longer. We see that at the line at the right hand side, the cavity and the waveguide are in phase rather than shifted π 2. As we already stated before, the band edge mode will mainly reside in the crystal itself. This mode does not necessarily need a cavity, as it is a consequence of the weak reflections on the borders of the finite crystal. It is also easy to see that the effective refractive index in the crystal will be lower than the one in the cavity, as we can look at the cavity as a local excess of silicon. Therefore, the effective index of the band edge mode will be lower than the one of the cavity mode. We need narrower waveguides to phase match to this lower index Using the band edge modes Up to now, we coped with band edge modes as a problem. In section 4.4.3, we saw that by reducing the number of holes on the right N right we can suppress these modes. However, it is also possible to suppress the cavity mode and use the band edge mode reflection to our benefit. Suppressing the cavity mode is very simple, we just have to remove the cavity. The result are structures as in figure 5.4a. In the spectrum on figure 5.4c, we see it is possible to create highly reflecting structures. Source Detector (a) Structure Detector Power flux (a.u.) Reflection Transmission Wavelength (µm) (b) Mode profile (c) The transmission and reflection spectrum Figure 5.4: Using the band edge mode reflection. The number of holes is 54 and the coupling width 00nm. The upper silicon waveguide is 630nm wide. The reflection goes up to 80% and the transmission is quasi zero. The remaining 20% is scattered away, mainly at the left side. The mode goes through a zero, but power has again built up towards the edge. Previously, we reduced this scattering by making N left smaller or larger. Naturally, in these structure we can not speak of N left and N right, but only of N. If we decrease the number of holes N, the mode starts to scatter at the right hand side and it seems not to influence the left hand side of the cavity. A band edge mode is of course determined by the boundary conditions and apparently it is the boundary at the left is dominant. Therefore, it was not possible to reduce the scattering at the left. This limits the maximum reflection fundamentally making the structure not attractive for designing. Perhaps it is possible to tune this by making tapers at the edge of the monorail, but this was not further investigated Influence of alignment errors Up to now, we considered the system under ideal conditions. In this section, we will discuss the influence of alignment errors on our design. Even though the alignment is defined in a lithographic step rather than the bonding step, errors of the order of 200nm are not uncommon. On the influence of an offset in the x direction, one can be very brief. The III-V layer consists of a simple waveguide that is not required to stop when reflection has occurred. Thus, in this direction, we can not even talk of alignment errors since there is no aligning necessary.

65 Chapter 5. Multirail structure 49 When there is an offset in the y direction, the coupling overlap integral of equation 5.3 will change and therefore also the coupling factor κ. The reflection coefficient as a function of the offset is shown in figure 5.5. Reflection coefficient cavity mode band edge mode Alignment offset (nm) Figure 5.5: The reflection coefficient as a function of the alignment error. Below an alignment error of ±200nm, the change in reflection coefficient is very small, not to say negligible. For larger offsets, the reflection drops due to lack of coupling. The fact that this error works with a threshold is a direct result of the overlap integral. For small offsets, the maximums of the modes stay aligned, for larger offsets this is not the case and the maximum on one mode will have to be multiplied with the dying tail of the other. Returning to the band gap discussion, we have to review the statement that no coupling to the first order Bloch mode can occur. Since we now have broken the symmetry, even and odd modes will no longer cancel out. The phase mismatch still limits the coupling to this first order mode. For small errors, the overlap integral will of course also remain quasi zero. We would also like to point out the similarities between the variation in the y-offset and the variation in the coupling thickness. Both impact on the coupling factor and therefore have similar effect. 5.5 III-V cavity using a resonant triplerail cavity With the mirror structures designed in the previous section, we can now build a laser cavity. For the bigger part of this cavity, the mode is primarily confined in the III-V waveguide, ensuring high gain possibilities. At the edge of the cavity, close to the mirrors, the silicon confinement is dominant making the use of small size mirror structures possible, in this case of the order of 0µm Estimation of the quality factor We can make use of the same method of estimation as in section Equation 4.2 is here repeated for convenience. Q = ωτ p = 2πn g 2L ( ) (5.4) λ 0 ln R R 2 The group index of our III-V waveguide is 3.48 and we aim at a distance between the mirrors of 5µm. The cavity mode resonates at a wavelength of 568nm and has a maximal reflection of 88%. This results in a quality factor of approximately 550.

66 Chapter 5. Multirail structure 50 For the same structure, the band edge reflection is stronger though. At a wavelength of 585, the reflection is 95%, making the Q equal to Simulation of the quality factor A simulation with the mirrors as described in section 5.4 and a mirror spacing of 5.0µm was done using 3D FDTD. The cavity was excited between the mirrors, as shown in figure 5.6. It is best not to excite exactly in the middle as this might be a node of the longitudinal mode. Excitation (a) The structure and mode profile perpendicular to y. (b) The structure and mode profile perpendicular to z, in the silicon layer. Excitation (c) The structure and mode profile perpendicular to z, in the III-V layer. Figure 5.6: Simulation of a III-V cavity using a resonant monorail cavity. The field in the III-V waveguide was recorded and using Harminv (see chapter A.2) we deduce the quality factor. The main mode resonates at a wavelength of.588µm with a Q of 525. This is higher than was estimated, but the mode profile shows the length is larger a lot larger than the distance between the mirror. This explains the additional factor 4. If we use a spacing of 5.µm between the mirrors, the resonances will shift. In the same way as above, we simulated that there is a longitudinal mode near the cavity mode, with a resonance wavelength of 566nm and a quality factor of 940. There is however still a stronger resonance due to band edge mode reflections with a Q of 590 at the wavelength of 592nm Estimation of the lasing threshold We start from the typical rate equations used in almost all works on lasers, e.g. [8]. dn dt ds dt = I qv R spont(n) ΓGS V (5.5) = ΓGS + βr spont V S τ p (5.6) with N the carrier density and S the number of photons. I is the injected current, q is the charge of an electron and V is the gain volume. The term R spont (N) accounts for carrier recombinations due to both non-radiative and radiative processes. One often states R spont (N) = AN + BN 2 + CN 3, were we have

67 Chapter 5. Multirail structure 5 terms for non-radiative processes, spontaneous radiative recombination and non-radiative Auger processes respectively. We will assume that only the spontaneous radiative recombination has to be accounted for. The last term of equation 5.5 describes the carrier recombination due to stimulated emission. In this, Γ is the mode confinement and G is the gain. In equation 5.6 the first term describes stimulated emission, the second spontaneous emission in the same direction as the stimulated (hence the coupling β is very small) and the third describes the mirror losses (τ p is the photon lifetime). In steady state lasing conditions, we have to put the derivatives dn dt and ds dt to zero. When neglecting the spontaneous emission term, equation 5.6 results in the fact that lasing can only occur when the gain compensates for the mirror losses (as these are the only losses we take into account). This gain can be expressed as a function of the carrier density N using the differential gain G 0 and transparency carrier density N 0. The photon lifetime τ p relates to the quality factor Q of the cavity Q = ωτ p. ΓG = ΓG 0 (N N 0 ) = τ p = ω Q (5.7) This is in particular true for the lasing threshold, providing us with an expression for the threshold carrier density N th. N th = N 0 + ω (5.8) G 0 ΓQ With this threshold carrier density N th we go to equation 5.5. At threshold, there are no photons in the cavity yet. The equation reduces to: 0 = I qv BN th 2 (5.9) We are planning to optically pump the laser in our measurements. Of course, not every incident photon will create a carrier suitable for lasing. Mathematically, we have to take an efficiency factor along: = I q, in which P pump is the power of the pump laser and ω pump is the pump light frequency. η Ppump ω pump Note that Ppump ω pump and I q are the number of photons incident per time unit, number of electrons creating free carriers per time unit respectively. P pump = V B ω pump η N th 2 (5.0) Using equations 5.8 and 5.0, we can calculate our threshold pump power. We use the following estimation of numbers: G 0 = cm 2 N 0 = cm 3 Γ = cm3 B = 2 0 s η = 0.0 V =.85 µm 5.0 µm 80nm = 0.74 µm 3 Note that G 0 is expressed in cm 2, the unit used in the derivation is cm3 s. Therefore, we have to multiply c with the group velocity: G 0,derivation = G 0,given n g. The active volume is defined by the III-V waveguide width, the distance between the mirrors and the thickness of the InGaAsP(Q=.55) layer (it is only this layer that is providing gain at this wavelength). With a Q of 525 and a reference wavelength of.588µm, we find: N th = cm 3 P pump 5 photons = ω pump s

68 Chapter 5. Multirail structure 52 If we pump with a laser with a wavelength of 800nm, corresponding to an energy of µm =.55eV, the pump power becomes: P pump =.6 mw ev µm If we use a distance between the mirrors of 5.µm, we also have a longitudinal mode at the cavity mode. We do the same calculation using Q = 940 and λ 0 = 566. In this configuration, the band edge mode still has a higher Q however, Q = 590 at λ 0 = 592. We find: N th, cavity mode = cm 3 P pump, cavity mode 6 photons =.5 0 ω pump s P pump, cavity mode = 3.6 mw N th, band edge mode = cm 3 P pump, band edge mode 6 photons =.0 0 ω pump s P pump, band edge mode = 2.5 mw Thus, the laser will still work in the band edge mode regime, unless fabrication errors change the Q factors relative to one another. 5.6 Conclusion This chapter started with the introduction of a new type of crystal: the multirail crystal. The incentive for this creation was the low effective refractive index of the monorail cavity. This prevented to phase match the structure to the III-V waveguides we can make using our simple III-V processing techniques. As in the previous chapter, we started with the design of the crystal. Because of the trade-off between band gap and refractive index, the gap is more narrow than in the monorail case, but this should not prove to be a problem. The distributed reflection coefficient r is still high, even higher than the monorails. The cavity we created, resonates at the wavelength of 558nm with a quality factor of The effective refractive index is now 2.59, which is a lot larger than the 2.29 from the monorail cavity. Because of the high refractive index, we can phase match a III-V waveguide to the cavity and thus we can build a mirror suitable for a laser cavity. The different III-V parameters are discussed (a discussion of the silicon parameters was given in chapter 4). The III-V - to - silicon thickness mainly impacts on the coupling factor κ between the waveguide and the cavity. It is the sum of the bonding and oxide thicknesses. Too thin results in an overcoupled structure, meaning that the light that has been reflected by the cavity, couples back to the silicon layer. There it is scattered away and lost for reflection. If the thickness is too large, the system is undercoupled. In that case, not enough light is able to couple from the cavity to the right of the waveguide. The destructive interference will not result in a field equal to zero and hence, the transmission will not be zero. In between, there is the critical coupled case, avoiding the problems of both. In the same type of errors, we also find alignment errors. A misalignment of the waveguide results in a reduced mode overlap, reducing the coupling factor κ. The effects of misaligning are thus very similar to a too large III-V - to - silicon thickness. The waveguide width determines the phase matching condition. The effective refractive indices of the cavity and the waveguide need to be equal, otherwise the phase difference will not be π. The waveguide we use is very wide (.85µm) and therefore a dimension error does not result in a large refractive index error. It was also shown that the cavity mode and the band edge modes have different phase matching optima, due to the different refractive indices. Finally, we also took a quick look of how we could exploit the band edge mode reflection. It turned out that removing the cavity introduced a lot of scattering for the band edge mode. This might be solved using tapers, but this was not further investigated.

69 Chapter 5. Multirail structure 53 As said, we can build a laser cavity because we can phase match the cavity to a III-V waveguide. As before, we need to synchronize the different resonances. When synchronizing them for the band edge modes, we find a total quality factor of 525. From this, we estimate the pump threshold power to be.6mw. When we synchronize them for the cavity mode, the quality factor is 940 resulting a pump threshold power of 3.6mW. However, even then, the band edge mode is still stronger with a Q of 590 and thus a threshold pump power of 2.5mW.

70 Chapter 6 Full optimization and comparison of monorail and multirail structures In the two previous chapters we discussed resonant monorail and multirail cavity structures. Because of practical implications of the limited number of chips through e-beam fabrication, we focused on the silicon parameters (number of holes on the left N left or on the right N right of the cavity) in chapter 4 on monorails. Because of timing, it were mainly the III-V parameters that were discussed in chapter 5 on multirails. We had to order the chips very early and thus, our main design space consisted of the bonding thickness and the III-V waveguide width. In this chapter we will try to compare the two structures. In order to do this properly, we need to optimize both using all parameters. We will first create a monorail cavity with a III-V overlay waveguide, then optimize the multirail mirror using the silicon parameters. Finally we will compare the two. 6. A resonant monorail cavity mirror using III-V overlay waveguide The design of this mirror again follows the same procedure as the previous chapters. Since we already did two of these designs, we can now go quicker thereby referring to the chapters 4 and The monorail cavity We use the same cavity as in section 4.3, figure 4.6. The quality factor is 955 at a wavelength of 525nm and the effective refractive index is Optimizing the III-V parameters It is important to phase match the waveguide and the cavity, meaning that we need the same refractive indices. We used a mode solver on different waveguide widths and a refractive index of 2.29 was found using a 585nm wide waveguide with only one pair of InP - InGaAsP(Q=.22) layers (see material stack on figure 8.). For the bonding thickness, we use the information from the multirail. The analytical model of chapter 3 predicted that the coupling coefficient κ and the distributed reflection coefficient r should relate to one another. Since the distributed reflection coefficient r, i.e. the reflection of one period in the crystal, of the monorail and the multirail crystal are very similar (.2282µm - and.2868µm - respectively), also 54

71 Chapter 6. Full optimization and comparison of monorail and multirail structures 55 the coupling coefficients κ will be similar and therefore the bonding thicknesses. The optimal III-V - to - silicon distance was 30nm for the multirail. Starting from this value, we vary this parameter and find the optimal value of 335nm. The spectrum and mode profiles of this resonant mirror are shown in figure 6.. Excitation Detector Detector (a) The structure and mode profile, perpendicular to y Reflection Transmission (b) The structure and mode profile, perpendicular to z in the silicon layer Power flux (a.u.) Excitation Detector Detector 0.2 (c) The structure and mode profile, perpendicular to z in the III-V layer Wavelength (µm) (d) The transmission and reflection spectrum Figure 6.: A resonant monorail mirror with a pitch of 350nm and a hole diameter of 82nm. A cavity length of 465nm and a four hole taper were used. The monorail was 500nm wide and consisted of 5 holes left and right. The III-V waveguide width 585nm and the bonding thickness 335nm. Already without optimizing the silicon parameters we reach a reflection of 90%. To optimize this further, we will use the number of holes on the left N left to reduce the scattering from the left edge of the monorail Optimizing the silicon parameters If we change the normalization of the mode profile massively (remember that we are only looking for a few percent of the light), as shown in figure 6.2a, we can see that most of the lost light is scattered at the left side of the monorail. This is reflected light that couples again to the silicon since the coupling (a) N left = 5 (b) N left = 8 Figure 6.2: A zoom in on the mode profile perpendicular to z in the silicon layer. Note the difference in scattering from the left side of the cavity. distance is too long. We reduce N left such that the monorail stops at a node of the mode profile. After a few simulations, we find that we have to choose N left equal to 8 (although one hole less or extra does not make much difference as we saw in section 4.4.2). This is illustrated in figure 6.2b. The resulting spectra and mode profiles are shown in figure 6.3. The maximum reflection, at the cavity mode, is 97% with a Q of 220.

72 Chapter 6. Full optimization and comparison of monorail and multirail structures 56 Excitation Detector Detector (a) The structure and mode profile, perpendicular to y Reflection Transmission (b) The structure and mode profile, perpendicular to z in the silicon layer Power flux (a.u.) Excitation Detector Detector 0.2 (c) The structure and mode profile, perpendicular to z in the III-V layer Wavelength (µm) (d) The transmission and reflection spectrum Figure 6.3: A resonant monorail mirror with a pitch of 350nm and a hole diameter of 82nm. A cavity length of 465nm and a four hole taper were used. The monorail was 500nm wide and consisted of 8 holes on the left and 5 holes on the right. The III-V waveguide width 585nm and the bonding thickness 335nm. If we want, we can vary N right in order to limit the band edge reflection. As the spectral distance between the cavity mode and the band edge mode is large enough, this is not very important Optimal design We started from a monorail with a pitch of 350nm, hole diameter 82nm and a waveguide width of 500nm. In order to have a resonance around 550nm, the cavity length was 465 and a four hole taper was used to increase the Q. In the optimal design the III-V waveguide is 585nm wide and the bonding thickness is 335nm. The number of holes on the left of the cavity N left was chosen 8 to prevent scattering. The ones on the right are less critical and are chosen 5. This results in an almost perfect reflection of 97% at the resonance wavelength of.530µm. The quality factor of the mirror is 240. The length is only 2 microns 6.2 Tuning the silicon parameters of the resonant multirail mirror In chapter 5 we have optimized a resonant multirail mirror using the III-V parameters. We have kept the silicon parameters of this design constant up to now, as we had already ordered the chips. In this section, we will look for the optimal design using both parameters. In the section on monorails above, we obtained very good results by first optimizing the III-V parameters and only then the silicon ones. Hence, we can start from the results of chapter 5.

73 Chapter 6. Full optimization and comparison of monorail and multirail structures 57 Figure 5.5 summarizes the resonant multirail mirror of chapter 5. The multirail has a pitch of 300nm and a hole size of 50nm. The width is µm, the height 220nm. A four hole taper was added to increase the Q and the cavity length was chosen 380nm. We used 25 holes on either side. The III-V waveguide has a width of.85µm and the bonding thickness is 30nm. This results in a reflection coefficient of 88% at the cavity mode with a Q of 247. If we zoom in on the mode profile, as we do in figure 6.4a, we see that there is scattering from the left edge of the cavity. As before, we can tune the number of holes on the left N left to fix this. (a) N left = 25 (b) N left = 7 Figure 6.4: A zoom in on the mode profile of a resonant multirail mirror perpendicular to z in the silicon layer. We have decreased N left to 7, in order to reduce the scattering on the left of figure 6.4b. The resulting spectrum is shown in figure 6.5. The reflection has risen up to 90% at the cavity mode, at a resonance wavelength of.563µm. The quality factor of the mirror is 220. Note that the band edge reflection is lower, as it normally lives in the left side of the cavity. We have now made this side lossy. 6.3 Discussion and comparison of the optimized structures We have now optimized both structures and we can compare them properly. They both offer high reflection of at least 90% and can therefore both be used in laser sources. Also the quality factor is very similar. It is mainly the coupling determining the Q mirror through the formula Q mirror = Q coupling + Q cavity. Since the bonding thickness is almost equal, the Q factor must also be approximately the same. The monorail mirror offers a very high reflection of 97%. The lost 3% is scattered away, as figure 6.3d shows a quasi zero transmission. Probably scattering in other directions than the direction of propagation, becomes imporant. The multirail mirror on the other hand offers a reflection of 90%, meaning that 0% is lost. It might be the case, as for the monorail, that a small fraction is lost due to scattering. On figure 6.4b we see that there is scattering from the right edge. As the profile does not resemble the cavity mode, it is improbable that this scattering is due to leakage of the silicon cavity. As we see on figure 6.5c, there is still considerable transmission past the cavity. This is confirmed by the transmission spectrum telling us that at least 3% is captured by the detector in the III-V waveguide. It is this light that couples to the right of the silicon cavity and scatters away. Thus this 3% is a lower limit and the real value will be higher. In figure 6.5c, we also see that most of this light is situated in the second order mode of the waveguide rather than the first. The different modes of a waveguide are orthogonal and therefore can not cancel each other out. In the monorail structure, we do not have this problem because the III-V waveguide only supports the ground mode It seems not straight-forward to come up with a solution for this problem. The problem is that the Bloch mode (visible at the right hand side of the cavity on figure 6.4b) and the second order mode of the

74 Chapter 6. Full optimization and comparison of monorail and multirail structures 58 Excitation Detector Detector (a) The structure and mode profile, perpendicular to y Reflection Transmission (b) The structure and mode profile, perpendicular to z in the silicon layer Power flux (a.u.) Excitation Detector Detector 0.2 (c) The structure and mode profile, perpendicular to z in the III-V layer Wavelength (µm) (d) The transmission and reflection spectrum Figure 6.5: A resonant monorail mirror with a pitch of 350nm and a hole diameter of 82nm. A cavity length of 465nm and a four hole taper were used. The monorail was 500nm wide and consisted of 8 holes on the left and 5 holes on the right. The III-V waveguide width 585nm and the bonding thickness 335nm. waveguide (visible at the right of figure 6.5c) spatially resemble each other very well. Because of that, the overlap integral defining the coupling is large, making sure that substantial coupling occurs. This is a clear disadvantage of the triplerail design. Normal surface gratings suffer much less from this problem. Their profile stays very waveguide-like and will always have the maximum in the middle of the waveguide. When creating a triplerail, we actually create several railways of high index material (between the rows of holes) giving rise to a more spatially distributed mode, which is more suitable for coupling to higher order modes. 6.4 Conclusion In this chapter, we have optimized the monorail and triplerail resonant cavity mirrors using both the silicon and III-V parameters. We have done this in a very methodological way. Firstly, we estimate the needed III-V waveguide width based on the effective refractive index. Secondly, we try a few bonding thicknesses to find the critical coupling. Thirdly, we tune N left in order not to scatter away the light at the left of the cavity. If N left was large enough, this light was reflected by the cavity, but has couples back to the silicon layer. Otherwise, the cavity is leaky, meaning that the scattered light comes directly from the cavity itself. N right does not have a double function as N left has. It has to be large enough, so the cavity does not become lossy at this side. If we want to suppress band edge modes by limiting N right, but that was not necessary in our case. The result was 97% reflection for the monorail and 90% for the triplerail. The difference is due to coupling to higher order modes. This is a clear disadvantage of multirails.

75 Chapter 7 Masks In this thesis, we made two masks to order chips. The first mask was sent to IMEC, where chips are being made using CMOS compatible deep UV (DUV) technology. Because of the long delivery time, we also made a second mask of structures to be made in Glasgow with e-beam technology. When using e-beam, one writes the structure every chip, meaning that the cost of making a hundred chips is approximately a hundred times the cost of one chip. In the wafer-based DUV technology, this problem does not arise. Therefore, we ordered structures on December 2 th, 202 through the route of IMEC suitable for bonding. With the in-house contact mask technology, we then defined the III-V structures. These chips were fabricated and delivered in Ghent at April 27 th, 202. Next to this, we also ordered chips on April 24 th, 202 through the route of Glasgow. The limitation on the number of chips through e-beam forces us to work with the passive analogon of the proposed design. Too many chips might be ruined because of bad bonding. These chips were fabricated and delivered in Ghent at May 30 th, The IMEC mask 7.. Mask parameters The structures are defined in the 220nm thick silicon layer. This designs are completely etched through, making sure we have a degree of symmetry for the photonic crystal. The structures are separated from the substrate by a 2 micron thick oxide layer. After the chips have been fabricated, we will bond the III-V die on top and do the processing with the in-house technology. A laser does not require an input signal, but only a pump laser which we do not have to bring into the silicon layers. It is important that we have a mechanism to couple light out of the chip though. As the output of the laser will be in a passive silicon waveguide, we need a way of bringing the light to a detector in order to measure it. For this we used shallow etched grating couplers. The output of the laser (i.e. one end of a mirror) tapers to a grating coupling that brings it out of the plane of the chip. There, we can use a detector to measure the laser working. Figure 7.: grating couplers for in and output on the IMEC mask 59

76 Chapter 7. Masks 60 We also want to make one additional remark regarding the cylindrical holes. Since circles are not supported by the used type of gds-files, the holes are shaped octagonal. As the corners will be rounded due to fabrication, the approximation will be very good Passive structures To check the quality of our fabricated cavity designs, we put some passive structures on the mask. These structures are for reference use only and thus, only a limited sweep of the parameters is done. All structures are connected to two grating couplers, so we can always perform reflection and transmission measurements. Straight waveguides The straight waveguides can be used as a reference for most measurements. Certainly the spectrum of the grating couplers is essential to know as they are being used to couple light in and out throughout the entire chip. Since straight waveguides are the simplest structures possible, it is a good way to check the quality of the chip. Crystal structures To check the quality of the photonic crystals we also put some crystal structures on the mask. This will allow us to check the band gap and the distributed reflection coefficient r using equation 5.: R max = tanh 2 (rl). The sweep for the number of holes contained in the crystal is shown in table 7.. We expect to measure a resemblance of figure 5.3c. number of holes Table 7.: Passive crystal structures on the IMEC mask Cavity structures The silicon cavity makes or breaks the resonant cavity principle. It is very important to know the resonance wavelength and the quality factor. The sweeps we applied are shown in table 7.2. pitch (nm) hole diameter (nm) cavity length (nm) internal taper yes no Table 7.2: Passive cavity structures on the IMEC mask Figure 7.2: Image of one of the passive cavity structure on the IMEC mask The sweep over the different pitches and hole diameters makes sure we have different band gaps. This variation is primarily a safety net. If for some reason the hole diameters turn out to be smaller than expected, we still have decent structures. Next to the pitches and hole diameters, we also vary the cavity lengths. This should make sure that we definitely have a cavity resonance with wavelength near 550nm.

77 Chapter 7. Masks 6 To measure the influence of a taper, there was also a cavity without tapers put on the mask Active structures This is of course the main content of the mask and therefore contains a vast majority of the number of structures. They are all laser cavities, with an output to a passive silicon waveguide on one side. The number of holes at this side is reduced for efficient coupling. The other end of the passive waveguide is connected to a fiber coupler, so we can capture the light with a fiber. Table 7.3 describes the sweeps we made. The sweep of the upper four parameters was explained in the previous section. cavity length (nm) pitch (nm) hole diameter (nm) internal taper yes no mirror separation (µm) Table 7.3: Laser cavity structures on the IMEC mask The mirror separation consists of two times three values. This parameter is the distance between the two mirror structures of the laser cavity. It is measured between the edges of the multirails. The small variation is being made in order to align the different resonances in the laser cavity. Remember that we have to align three resonances: the resonances of the silicon cavity of the two mirrors and the resonance of the III-V cavity formed by these mirrors. We can assume that the first two are aligned as fabrication errors are the same for the two mirrors. By sweeping the length of the III-V cavity, the chance that we do not have a match is quasi zero. The large variation offers a degree of freedom in the amount of gain. In a longer cavity, the light will be amplified more before part of the power is lost due to imperfect reflection. A smaller cavity has the obvious advantage of being small and is also less power hungry. For lasing, we need a threshold current density. Thus, a longer cavity consumes more current. Figure 7.3 shows an image of the laser structures on the mask. Note that the left side of the left mirror is shorter to make sure most of the laser light is outputted at this side. Figure 7.3: The active laser structure in the IMEC mask.

78 Chapter 7. Masks Contact mask After the chips are fabricated at IMEC and delivered in our clean room, we can start the III-V processing. We will bond a III-V die on top of our structures and etch away all the superficial III-V materials. The aligning of the III-V waveguides and the silicon cavities is done in a lithographic step. For this we of course need a mask telling us where the waveguides have to come. As said before, we had not simulated a lot mirror structures when we ordered the chips. Because of this we had to sweep the parameters as much as possible and we did not have the space to put the structures on the mask multiple times. Therefore, we can not sweep in the III-V waveguide width, all of them are.85µm wide. Figure 7.4 shows the overlay on one of the devices. Note that we have ended the III-V waveguide with a triangle. This avoids reflections that may disturb the resonances of the laser cavity. Figure 7.4: Illustration the contact mask for the III-V processing on the IMEC mask. 7.2 The Glasgow mask The Glasgow chips were fabricated using the e-beam technology. This is not a wafer based technology, but in every chip the structure is written and therefore it is very time consuming (and thus also costly) to make a lot of chips. The big advantage is the fabrication time which is in days or weeks. When ordering chips using deep UV at IMEC, we have to wait for other partners to sign in on the multiproject wafer. On top of this, other - often industrial - companies order wafers with higher priority. The short waiting time for e-beam patterning makes that researchers like to use this technology, even if they can only give proof of principle. Since the number of chips returned is limited and it is always possible that things go wrong during post processing, we only put passive structures on this mask. The vertical coupling of III-V to silicon made room for sideways silicon-to-silicon coupling Mask parameters Also on this mask the structures are defined in the 220nm thick silicon layer. There is only one mask step, meaning that there can be only one etch depth. This etch step etches through the entire silicon layer making sure we have a degree of symmetry for the photonic crystal. The structures are separated from the substrate by a 2 micron thick oxide layer. Only one etch step limits the costs but has the big disadvantage that we can not use shallow etched grating couplers. The alternative of fully etched ones is not very desirable as these suffer from very high reflections. Nevertheless, in these passive structures, it is important to be able to get light in and out as good as possible. We will have to use horizontal coupling for this. The chip will be cleaved in order to separate the different columns of structures. At the location of cleaving, the waveguides are very wide

79 Chapter 7. Masks 63 (3µm) in order to have a flat mode profile. We then bring a lensed fiber close and the flat mode profiles will match. This is an inefficient way of coupling, but usable nonetheless. Also here, the circles were approximated by octagons. One might reason that with the higher resolution, we could better use more sides. However, we thought it better to use the same type of holes over the two masks Straight waveguides Also on this chip, we put some straight waveguides. This time we did not use grating couplers, but horizontal coupling. Using these straight waveguides, we can measure the input losses and wavelength dependence of the coupling. Since we will be using different widths of waveguides on this chip, we made a sweep over the waveguide width, shown in table 7.4. Note that this width is only used in the center part over a length of 50µm, the rest is a long taper to larger width for coupling efficiency. waveguide width (nm) Table 7.4: Straight waveguides on the Glasgow mask Monorail structures Monorail crystal structures The photonic crystal quality is of course of paramount importance in a photonic crystal cavity. We put several photonic crystals on the chip with different lengths. A transmission measurement allows us to check the position of the band gap. Using coupled wave theory, we can also find the distributed reflection coefficient r and see whether it is really as high as we hoped for. We expect to be able to fit it to figure 4.5c. Remember that it followed from the analytical mode that it was advantageous to have a high distributed reflection. The parameters of the sweep are shown in table 7.5 number of holes Table 7.5: Monorail crystals on the Glasgow mask Contrary to the IMEC mask, we did not sweep the pitch and hole diameter since the resolution of e-beam is far better than of deep UV (order of a nanometer i.s.o. a hundred nanometer). On top of this, we ordered several chips of different exposure times. This will already sweep the hole diameter as more or less silicon is etched. Monorail cavity structures In order to know the resonance wavelength and the quality factor, all the cavities that are going to be used as a resonant cavity mirror were put on the mask. This way we know where to look for reflections in our mirror designs. The sweeps are shown in table 7.6. We chose not to sweep the cavity length as we did in the IMEC order, but opted to sweep the number of holes on the left N left and on the right N right of the cavity (this number does not include the taper holes). As we saw in chapter 4, the influence on the reflection of these parameters can be very big. To fully interpret the results from those measurements, we need the data from the stand-alone cavities. If

80 Chapter 7. Masks 64 N left N right Table 7.6: Monorail cavities on the Glasgow mask Figure 7.5: A monorail cavity structure on the Glasgow mask we see a low reflection from the mirror, but also a low quality factor from the cavity measurement, we know it is the cavity mode that is lossy and scatters away most of the light leaving the option open that the phase matching might still be good. Monorail mirror structures The focus of the thesis lies of course on the mirror structures. Have we built a good mirror? We expect to find spectra similar to the ones in section 4.4. The sweeps are itemized in table 7.7. III-V like waveguide width (nm) Coupling width (nm) N left N right Table 7.7: Monorail mirror structures on the Glasgow mask Figure 7.6: A resonant monorail cavity mirror structure on the Glasgow mask If there is a phase mismatch, perfect reflection is impossible. Therefore, a sweep of the III-V-like waveguide width is made. Note that we do not expect surprises when it comes to the effective index of the III-V like waveguide (this is a simple silicon waveguide), but there can be a certain variance on the effective index of the cavity mode. The sweep of the widths implies a sweep of the refractive index of 2.03, 2.2, The best simulation results were obtained with a coupling width of 00nm. However, it is very interesting to see what happens at larger coupling widths, since this is one of the only parameters that intervenes in only one process (being the coupling). We opted not to go below the 00nm barrier as this is already below the resolution of lots of technologies. Sweeping the N left and N right allows us to experimentally check the simulations and explanations given in paragraph and

81 Chapter 7. Masks 65 Cavity using resonant monorail cavity mirrors The design of the mirror was done with the goal of laser cavity in mind. Thus, a lot of these cavities were put on the mask, allowing us to find the resonances. When we perform a transmission measurement over the III-V-like waveguide, we can distinct several cases depending on the wavelength, also discussed in section If the wavelength is outside the reflecting wavelength range, the output may be high or low. If the reason for the lack of reflection is scattering, the light is lost and thus the output will be low. If it is phase matching that is causing problems, the light is still confined in the waveguide and thus the output will be high. If the light is in the reflecting wavelength range, but no resonance between the mirrors is sustained, most of the light will be reflected. If the light is in the reflecting wavelength range and a longitudinal resonance is sustained, a mode will build up between the mirrors. The output will be high due to leakage of this strong resonance mode. The sweep used for these complex structures is listed in table 7.8. mirror spacing µm III-V like waveguide width (nm) Coupling width (nm) N left N right Table 7.8: Monorail mirror structures forming a cavity on the Glasgow mask Figure 7.7: Monorail mirror structures forming a cavity on the Glasgow mask. (The box between the mirrors makes sure all the silicon is certainly gone in between the mirrors.) The mirror spacing is varied over four different values in order to have a longitudinal resonance near the maximum reflection of the resonant monorail cavity mirror. The other parameters are varied because of the same reasons as described above Multirail structures: the doublerail and triplerail A multirail structure is a monorail with more than one row of holes in the waveguide. In this section, we will consider the doublerail and triplerail containing two and three rows of holes respectively. These structures were created to account for the high refractive index of the III-V waveguide. Although this chip contains only passive devices, we also put multirails on the mask. They should give us more insight on how they work and how we can relate them to monorails. Multirail crystal structures In the doublerail simulations, we used two designs. One design was based on a waveguide of 750nm wide, the other on a waveguide of µm. Although they both use the same crystal constants, they may exhibit different properties due to the difference in index confinement. The sweep in crystal structures is shown in table 7.9. The sweep in number of holes again allows us to derive the band gap and the distributed reflection.

82 Chapter 7. Masks 66 For the triplerail, we opted not to use the same pitch as in the monorail and doublerail designs. Instead, we used a pitch of 300nm as we did on the IMEC chips. The width is µm. The sweep is shown in table 7.0. waveguide width µm 0.75 number of holes number of holes Table 7.9: Doublerail crystal structures on the Glasgow mask Table 7.0: Triplerail crystal structures on the Glasgow mask Multirail cavity structures All designs aimed at a resonance wavelength around 550nm. Because of the considerable waveguide width difference, the doublerail cavity lengths had to be chosen differently. The 0.75µm doublerail has a cavity length of 45nm, the µm doublerail of 380nm. The triplerail cavity has the same width as the wide doublerail and, as a consequence, an equally long cavity (380nm). As with the monorail, we vary the number of holes on either side of the cavity, although we had to nibble on the number of doublerails due to the two designs. The parameters are shown in table 7. and table 7.2 for the doublerail and triplerail respectively. N left N right 0 doublerail width (µm) 0.75 N left 7 9 N right Table 7.: Doublerail cavity structures on the Glasgow mask Table 7.2: Triplerail cavity structures on the Glasgow mask Multirail mirror structures A famous quote during the master thesis research was In the end, it is all about reflection. Thus it is logical that also for the multirail this section is one of the most important ones. Tables 7.3 and 7.4 show the sweeps for these mirrors. The reasoning behind these sweeps is completely analogous to the monorail mirrors. III-V like waveguide width (nm) coupling width (nm) N left N right 0 doublerail width (µm) 0.75 III-V like waveguide width (nm) coupling width (nm) N left 7 9 N right Table 7.3: Doublerail mirror structures on the Glasgow mask Table 7.4: Triplerail mirror structures on the Glasgow mask

83 Chapter 7. Masks 67 Cavity using resonant multirail cavity mirrors As one might guess, also with the multirail mirror structures we created a cavity. The sweep is given in tables 7.5 and 7.6. mirror spacing (µm) III-V like waveguide width (nm) coupling width (nm) N left N right 0 doublerail width (µm) 0.75 mirror spacing (µm) III-V like waveguide width (nm) coupling width (nm) N left 7 9 N right Table 7.5: Doublerail mirror structures forming a cavity on the Glasgow mask Table 7.6: Triplerail mirror structures forming a cavity on the Glasgow mask

84 Chapter 8 Material stack and fabrication As stated in the introduction, we opted to work on a hybrid silicon/iii-v platform. In this chapter, we will go into further detail of the various aspects of the material platform. We start briefly on the silicon chip and will then introduce the III-V die being used. To bring this together we need a bonding step, discussed in section 8.3. We have fabricated the devices using the process flow explained in the last section. 8. The silicon chip Every chip starts with a substrate, providing strength and a foundation to build on. In silicon photonics this substrate is of course silicon. It is usually 750µm thick. Since silicon is a high refractive index material (3.47), it is important to have thick layer of oxide separating the structures and the substrate. SiO 2 has a refractive index of.45 and a layer of 2 micron thick should be a good buffer to limit the substrate losses. All structures, except the III-V waveguide, are defined in a 220nm thick silicon layer. By etching on the right places, very small structures can be formed. There are several ways of fabrication such as deep UV lithography (resolution of tens of nanometers) and e-beam patterning (resolution of nanometers). 8.2 The III-V die In order to have a gain medium in the microlaser, we bond a III-V die on top. We opted for a stack of InGaAsP materials. The exact stack of materials is shown in figure 8.. Note that this figure shows the layers as we see the stack when it is bonded, not as it was grown. This means that the bottom layer on the figure was the top layer before the bonding occurred. The stack consists of a bottom layer of SiO 2 that will be used in the definition of the bonding thickness. Since it is easier to control the thickness of a SiO 2 layer than that of the BCB layer, we opted for a constant BCB thickness of 50nm and a variable oxide thickness. Above this we have an InP - InGaAsP(Q=.55) - InP waveguide where the ground mode will mainly reside. The top of the stack consists of four times an InP - InGaAsP(Q=.22) layer pair. These thin layers are an additional degree of freedom to satisfy the phase matching condition. The designs were simulated using one or two of these layers, but once fabricated, we might want to measure with an extra layer or a layer less. It is important to remember that reality and simulations can be quite different. This structure has an air cladding. 68

85 Chapter 8. Material stack and fabrication 69 Figure 8.: A schematic of the III-V material stack used as gain medium. When we are fabricating the III-V waveguide, we etch away all the superfluous III-V material. This means that the waveguide cladding will only have the SiO 2 layer (and everything beneath it). Everything else consists of air. Of course one wants to the test the photoluminescent properties of this gain medium before using it. The red curve in figure 8.2a shows the photoluminescence of a measurement of the stack prior to bonding. This stack has an InP substrate and a thick InGaAs layer. As we see, it is mainly this layer that is emitting light and on top of that absorbs a fraction of the generated light. After bonding we measure the correct spectrum. In figure 8.2b, the properties of the bonded stack are shown in the wavelength region Photoluminescence (a.u.) stand alone stack bonded stack Photoluminescence (a.u.) Wavelength (nm) (a) The entire measured spectrum Wavelength (nm) (b) Zoom at the region of interest: 500nm to 600nm Figure 8.2: Measurement of the photoluminescence of the III-V material stack. of interest, being from 500nm to 600nm. We see that after 550nm, there is a sharp decrease. As the triplerail oscillates in that region rather than below 550nm, we may have to heat the chip to get enough gain.

86 Chapter 8. Material stack and fabrication Die to die bonding An important but easily overlooked part of the hybrid silicon/iii-v platform is the glue in between, the bonding layer. It is our goal to utilize die-to-die bonding. In order to do this, the III-V structures are only processed after the bonding step, which should be possible in our proposed design as the III-V structure is a simple waveguide. There are two mainstream bonding technologies: direct bonding and adhesive bonding. In direct bonding, we bring the III-V die very close to the silicon surface. Intermolecular reactions will come into play due to Van Der Waals forces, hydrogen bonds and strong covalent bonds. As the name suggests, there is direct contact meaning that no intermediate layer is formed. The problem is that the intermolecular reactions often produce gas byproducts. These will be trapped in the bonding layer, giving rise to large voids. To avoid these, we can make use of gas absorbing layers or vertical outgassing channels. These complicate the bonding significantly and sometimes limit the design freedom. [2] In adhesive bonding we spincoat a BCB layer (or an other adhesive) on the surfaces. When we bring these close to each other and heat it, a polymerization reaction will occur and two polymers will be clung to one another. This is the so-called Diels-Adler reaction. In a simple way of speaking, the adhesive is a glue. As no gases are created in the Diels-Adler reaction, no special measures have to be taken to limit the number of voids. [2] From an industrial point of view, it is very hard which bonding technology is supreme. Both have advantages and disadvantages, both have produced comparable devices. Further research on this topic is under way at IMEC and INTEL. From the point of view of this master thesis, BCB is preferable simply because the Photonics Research Group has more experience in the adhesive bonding. 8.4 Fabrication process To fabricate the chips we follow the process flow shown in figure 8.3. We will shortly discuss the different steps. SOI chip III-V die SiO2 deposition Bonding Substrate removal Litho + etch Figure 8.3: Process flow of the structures. The silicon-on-insulator chip was fabricated in IMEC using DUV. The protective layer of photoresist is removed and the chip is cleaned using an SC solution. This results in a very clean surface, needed for the bonding.

87 Chapter 8. Material stack and fabrication 7 The III-V chip was fabricated at ITME. After the removal of a sacrificial pair of InP and InGaAsP layers, the surface is very clean and we can deposit the oxide. Remember that the oxide thickness will (together with the bonding thickness) define the distance between the III-V waveguide and the cavity. Because of the use of sacrificial layers, also this results in a very clean surface. For the bonding step, we spincoat the promotor AP3000 and the adhesive BCB on the silicon chip. We then bring the silicon chip and III-V die in contact and bake them. The polymer will be B-staged and the Diels-Adler reaction starts. Now we have a silicon chip, with our desired structures on it, and the III-V die on top. The die still has its substrate and in order to pump it, we need to remove this first. We grind the substrate away until we have only ±70µm left. The rest of the InP substrate is removed in a wet etch step using HCl. After the removal of the ears (sides of InP still standing up because of anisotropic etching direction) we remove the thick InGaAs layer that served as an etch stop for the substrate. The remaining III-V material stack is now as shown in figure 8.. In order to do define the III-V waveguide, we also do a lithographic step. We will use a hard mask approach, so first SiO2 is deposited. A photoresist is spincoated, illuminated trough a mask and developed. The oxide layer is dry-etched and the hard mask is created. We can now etch using the SiO2 as a hard mask and the waveguide has been created. Figure 8.4 shows a few images taken from a bonded III-V die. The substrate has been removed, but the waveguide has not yet been defined yet. We see on figure 8.4a that near the edge there are a lot of 5 mm (a) Lots of defects at the edge of the die 5 mm (b) Defects at various places that may cause disfunctionalities and stripes of good bonding giving rise to good devices Figure 8.4: An image of a III-V die bonded on a SOI chip, taken with contrast microscopy. defects. The reason for this is the blanking of the structures next to ours, resulting in a large unbalance of the heights. When the III-V die is brought into contact and some pressure is applied, the die will bend down on the blanked structures. This results in tension on the part of the die lying over our structures. In figure 8.4b, there are some cracks shown as a consequence of this tension (the stripe at the left and below the center of the figure). Still, we would like to point out that we do not need the whole die to be bonded perfectly to the silicon. Most of it will be etched away, it is only the III-V stack on top of the device itself that needs to be of high quality.

88 Chapter 8. Material stack and fabrication Conclusion In this chapter, we discussed the material structure and the fabrication process flow. We started with the silicon chip, which is a typical SOI chip. There is a thick oxide layer separating the substrate from the structures, that are defined in the 220nm thick silicon layer. The other side of the III-V/silicon platform is of course the III-V stack. We have explained how this looks like and at which wavelengths the photoluminescence is maximal. These two parts are glued together using BCB bonding. This type of adhesive bonding is not capable of making very thin bonding layers, but that is not necessary for our design. The design freedom is a big advantage, as is the experience of the photonics research group in BCB bonding. After introducing the different parts of the platform, we discussed the fabrication process flow. The two chips are cleaned and bonded together. After the removal of the substrate, we can use the hard mask approach to etch the III-V waveguide. We showed some results of the fabrication and highlighted the origin of the defects.

89 Chapter 9 Conclusion and future prospects To finish this master thesis dissertation, we would like to summarize the most important results and insights we have come to and how we see the future on this subject. 9. Conclusion In this work, we have designed a few laser cavities in the hybrid silicon platform. Since we wanted to overcome the fabrication problems of microdisk lasers and the too large size of the evanescent lasers, we spatially separated the gain and mirror section. Resonant cavities are a very attractive mechanism to do this. We have used photonic crystals to achieve small cavities leading to high reflections. We first properly introduced photonic crystals and resonant cavities, learning how to use photonic band gaps on the one hand and the interplay between coupling and distributed reflection on the other hand. The three designing chapters (4, 5 and 6) form the core of this report. We have created three types of resonant cavity mirrors based on photonic crystals: a passive resonant monorail mirror, an active resonant monorail mirror and an active resonant triplerail mirror. We have discussed the influence of the most important parameters: ˆ ˆ ˆ ˆ The number of holes on the left of the cavity N left is a parameter impacting on two processes, being the coupling from and to the waveguide and the distributed reflection ensuring the left mirror of the silicon cavity. If N left is too large, the reflected light will couple again to the silicon. Very often it will encounter the edge of the monorail and be scattered away. If N left is too short, the silicon cavity itself becomes lossy, and again light will be scattered away. Contrary to N left, the number of holes on the right of the cavity N right is less critical. It should be large enough to prevent the silicon cavity to become lossy. Ideally, there is no light in the III-V waveguide after the cavity, meaning we do not have the same problem of extra coupling to the silicon layer. Still it can be useful not to choose N right too large to suppress band edge modes living there. The phase matching condition is taken care of by the waveguide width. If the effective refractive indices of the silicon cavity and the III-V waveguide are different, the destructive interference will not be complete. The waves are not exactly in antiphase and thus, part of the light will be transmitted. One of the results of the analytical model was that there is a critical coupling factor κ. Therefore, there is also a critical bonding thickness or waveguide gap (in an active or passive device respectively). If this thickness or gap is too small, we will again have the problem of multiple coupling 73

90 Chapter 9. Conclusion and future prospects 74 ˆ at the left hand side of the cavity. If it is too large, the coupling will be too weak compared to the distributed reflection. The extremely strong cavity mode suffers from extra losses. Because of that, the mode will have decayed before enough light has coupled to the III-V waveguide to interfere destructively with the light already present. In the ideal design, the center of the waveguide and of the cavity are on top of each other. In a real device, there will always be an alignment error. We proved that, in a first approximation, we can consider these errors as a drop in the coupling factor. Altogether, the influence was limited in our triplerail design. By optimizing these parameters, we achieved very high reflections in the monorail design, shown in figures 9.a and 9.b. The resonance wavelength of the cavity mode is 527nm. This renders a maximal reflection of 96% and a Q of 30, for the passive mirror. The active mirror has values in the same range, reflecting 97% of the incoming light with a Q of 240. Excitation Detector Detector Excitation Detector Detector Reflection Transmission Reflection Transmission Power flux (a.u.) Power flux (a.u.) Wavelength (µm) Wavelength (µm) (a) A passive resonant monorail mirror (b) An active resonant monorail mirror Excitation Detector Detector Reflection Transmission Power flux (a.u.) Wavelength (µm) (c) An active resonant triplerail mirror Figure 9.: The transmission and reflection spectrum of the various designs. Because of the low effective refractive index of the monorail cavity, only small III-V waveguides can be used. To counter this, we proposed the triplerail cavity. The reflection is still very high but a few percent lower than the monorail reflection. The drop from 97% to 90% can be explained by the coupling to higher

91 Chapter 9. Conclusion and future prospects 75 order modes of the waveguide. These modes are orthogonal to the ground mode, preventing destructive interference. After one year, one should be able to answer the question Is it desirable to use resonant cavities for a laser and what type of cavities should one?. My answer to the first question would be a convincing yes. Resonant cavities are one of the most elegant solutions of separating a gain and reflection section in the laser. Without any taper section whatsoever, they transfer a mode confined in the III-V waveguide to one primarily confined in the silicon, giving rise to very small and easy to fabricate laser sources. The second part of the question would be a yes, but not as convincing. Multirails show very high distributed reflection coefficients r. These do not only miniaturize the design substantially, but also make sure that we are able to achieve theoretical perfect reflection using our resonant cavities. The downside is coupling to higher order modes, meaning the reflection will never be perfect. Surface gratings do not suffer as much from this but r is smaller. Thus, it is a trade-off between miniaturization and reflection. Note that is we use more advance III-V processing, the monorail can be used. This also does not suffer from the higher order mode coupling and reflection of 97% was shown. 9.2 Future prospects To conclude, we would like to shed our light on the future. There are several topics one should investigate further. ˆ ˆ ˆ ˆ This master thesis s emphasis lay on the designing part. Because of the limited time frame, we were not able to do real measurements. Although we did pay attention to the impact of certain fabrication errors, simulations and reality can be very different. There is no reason why we would think the laser will not work, but we can only say it works after it has been measured to work. The ultimate goal of this research topic is to be able to fabricate an electrically pumped silicon integrated laser source using only DUV and simple III-V processing. In this work, we did not bother about the creation of population inversion as we assumed an optical pump beam would do the trick. In an electrical design, lots of attention has to be paid to the metals and where the current will flow. While playing with the parameters, we noticed that the ambiguity of the number of holes on the left N left can be problematic. On one hand it plays the role of left mirror of the silicon cavity, on the other hand it defines the length over which coupling is possible. Maybe a different design can be able to deduplicate this parameter. A simple implementation of doing this, could be possibly achieved done by turning the side of the left side of the monorail away from the III-V waveguide. In this case, we risk extra scattering centers due to the bend and as we saw in the designing chapters scattering can reduce the reflection substantially. Throughout this thesis, we made use of DFB designs. We could evolve more to a DBR design, where we separate the mirror sections from the cavity. This results in larger mirrors, but it might be possible to decouple the parameter in this way. The reflection would be defined in the mirror section, the coupling in the combination of mirror and cavity section. The coupling of the multirail to higher order modes poses a problem. We think that this is inherent to the multirail design, meaning one will have to think of a creative solution to solve this while still clinging to the multirail. Surface gratings do not suffer from this problem because their mode profile resembles the waveguide modes very well. Thus, if one can live with the much lower distributed reflection coefficient of

92 Chapter 9. Conclusion and future prospects 76 the surface grating, these might by a more attractive solution. We also would like to highlight the intermediate structure called high contrast grating. These are fully etched gratings and thus probably combine the waveguide-like mode profiles and a higher distributed reflection coefficient. Of course, as with all solutions, other problems might show up.

93 Appendix A Tools Throughout this thesis, we made use of different tools. The main simulation tool was MEEP, a FDTD solver, but also MPB and Harminv were used. In this section we want to give a brief overview of what these tools can do and what not. We will briefly go into theoretical detail, but mainly focus on how to use them. A. MEEP MEEP, or in its full name MIT Electromagnetic Equation Propagation, is a free finite-difference timedomain solver (FDTD) developed by MIT. A.. Theory The equations to be solved are the simplest in a 2D system, since then Maxwell s equations decouple in a transverse electric (TE) and a transverse magnetic (TM) set of equations. The wave equation for TM modes (i.e. with the electric field along the third dimension z) in a 2D system is: 2 E(x, y) x E(x, y) y 2 = ɛ(x, y) 2 E(x, y) t 2 (A.) This equation is now discretized using a simple square lattice. The space-time points are separated by fixed basic units of time t and space s. Using the short notation where E(i s, j s, n t) = E n i,j and ɛ(i s, j s) = ɛ i,j, we can approximate the derivatives in the wave equation by a centered difference. This results in a finite-difference equation: E n i+,j 2En i,j + En i,j ( s) 2 + En i,j+ 2En i,j + En i,j E n+ i,j ( s) 2 = ɛ i,j 2E n i,j + En i,j ( t) 2 (A.2) The idea is to solve this equation for the future time component E n+ i,j and to use this to update the field in the simulation domain. There exist similar, more complicated equations for 3D problems where TE and TM become TE-like and TM-like mode. [9] A..2 Practice The version we used was Python-MEEP, a python wrap around the original C version. This made sure we could interface it with the Ipkiss-Picazzo framework, the in-house mask design software. The simulation steps can be summarized as follows:. We generate the component we wish to simulate using Ipkiss-Picazzo. 77

94 Appendix A. Tools MEEP will discretize the structure using a uniform mesh. Within each cell of the mesh, we can make an average of the epsilons or just take the center value. Here we have to make an important choice between accuracy and time efficiency. 3. A perfectly matching layer (PML) is grown at the edges of the simulation window. A thick layer absorbs a lot of the light incident but increases the simulation window considerably. A thin layer may reflect light back into the structure. 4. Sources are added at the desired positions. If needed, we can use a mode solver (e.g. CAMFR), to excite a mode of the cross-section. We have the option to use a pulsed source (what we will do for spectra) or a continuous, monochromatic source (what we will do to grasp the details of a certain minimum/maximum in the spectrum). 5. Fluxplanes are added. These will integrate the power flux of a certain wavelength span over time. All the spectra have to be simulated in this manner and it is very important to be sure the simulation time is long enough. Otherwise, there is too much light still trapped in some cavity without contributing to a fluxplane. If needed, one can also load some data from previous simulations (see below). 6. Probers are added, if indicated. These are positions (so only one point, not a plane) at which a desired field (e.g. E y ) is recorded as a function of time. The result of these probers is then used as input for Harminv, section A Finally, the actual simulation starts. Maxwell s equations are solved in time domain. Other then the discretizations, there are no approximations meaning that it is mostly brute force calculating. During this step, the data for the probers and fluxplanes is filled in. At regular times, a field plot is made, so we can follow the propagation of the fields. A problem might arise because of superposition of forward and backward propagating waves when working with fluxplanes. Since the direction of propagation through the plane defines the sign (plus for forward, minus for backward), a fluxplane can only be trusted if we can account for this. For example, if we want to measure the transmission and reflectivity of a grating, we put a flux plane in front of the grating and one after the grating. If we now look at the light propagation, we see that the light coming from the source first propagates through the fluxplane in front of the grating, before it interacts with the grating. After the interaction, a part is transmitted and propagates through the second fluxplane, the other part is reflected and propagates backwards through the first grating. From this we see that in case of perfect reflection, the resulting spectra will both be a flat zero. In order to solve this, we first do a reference simulation, where we don t take the grating into account. When doing the real simulation (with the grating), we first load the fluxes from the first fluxplane from this reference measurement and make it negative. Because of this, the light coming from the source propagating through the fluxplane will result in a flux of value zero. After this, the reflection can be measured. There is also a second, smaller problem with fluxplanes. The source has a gaussian spectrum. In order to see which wavelength is transmitted most, we also should take this into account. This is simply done in the analysis section by dividing the spectra by the source spectrum. The plotting of fluxplanes During our research, we plotted the fluxplanes in a clever way. In stead of only plotting the final result, we made use of a time dimension to check convergence of the simulation

95 Appendix A. Tools 79 Every now and then the current values of the fluxplanes are written down in a textfile. This means that at the end of the simulation, we can see the pulse pass through the fluxplane simply because the values are rising, an example is shown in figure A., and thus we have a sense of what happens in time..5 Power flux 0.5 Power flux (a.u.) Wavelength (µm) Wavelength (µm) Figure A.: A pulse propagating through a fluxplane. Figure A.2: The reflection fluxplane of a monorail mirror (see figure 4.8b). Note how certain wavelength are already converged whereas others are not. We use this knowledge to see whether the simulation has converged or too much light is still in the cavity. In a fully converged simulation, we would only see one line on the spectrum, even though we have plotted fifty of them. They are all on top of each other. To illustrate this, we take a look at the spectra from figure 4.8b repeated in figure A.2. We see that the simulation is converged outside of the resonances as light at this wavelengths can not be trapped at certain locations in the structure. We also see that the band edge modes (wavelength of.6µm) are more or less converged but the cavity mode (at.527µm) is still changing very rapidly. This is because the Q factor of the cavity mode is much bigger, meaning that light is trapped much longer. A.2 Harminv Harminv stands for Harmonic inversion and resolves the decomposition of a field into the harmonics. Mathematically, we say: f(t) = a n e jωnt (A.3) n If the input field is long enough, Harminv is able to calculate the amplitude factor a n and the complex frequencies ω n very accurately. From this complex frequency, it is straightforward to calculate the quality factor Q of the cavity: Q = Re(ω) (A.4) 2Im(ω) We have to keep in mind that the amplitude factor should be considerable in order to be sure of the result. The fields were captured by a prober which is a single point. Because of this, it is a good habbit to put at least two probers. If the prober is positionned coincidentally at a zero of the harmonic mode, the amplitude factor will of course be very low. This also accounts for the excitation. In order to have a high amplitude, it is also favourable to put the prober inside the cavity. The extinction factor does not depend on the amplitude however. So provided that there is an acceptable amplitude factor (so the mistakes are not important), the Q should be correct.

96 Appendix A. Tools 80 The big advantage of Harminv lies in the simulation time that is needed. In principle, we can also calculate the Q from the transmission spectrum. To do this, it is very important that there is no light left in the cavity meaning that the simulation time should be very long. Harminv uses the decaying constant rather than the absolute value of the spectrum. This trend is clear after a much shorter time, it is as if a extrapolation is being made. Note that we have to make sure, we cut off the start of the field. In the beginning, the field is defined by the source. It is only after the source has died out, we can measure the decay constants. A.3 MPB MPB is also a free package developed by MIT. It is an acronym for MIT Photonic Bands and is an eigensolver for periodic structures. Because of the limited combination possibilities with other software available in our research group, we did not opt for a python wrap as we did with MEEP, but used the programming language scheme. A.3. Theory MPB makes use of the frequency domain version of Maxwell s equations. It uses a plane wave strategy to find the bloch modes of the fully discrete translational symmetrical structure. The starting point is the master equation 2.3, here repeated for convenience: ( ) ( ω ) 2 ɛ(r) H = H (A.5) c Using a basis of transverse plane waves e λ exp[j(k + G)] r where k is the wave vector in the Brillouin zone, G is a reciprocal lattice vector and e λ is perpendicular to k + G, we can rewrite this equation into a matrix eigenvalue equation: (Gλ) Θ k (Gλ),(Gλ) h (Gλ) = ω2 h (Gλ) (A.6) h (Gλ) is the coefficient of the corresponding plane wave, the matrix can be shown to be equal to (with ɛ (G, G ) the inverse fourier transform of ɛ(r): Θ k (Gλ),(Gλ) = [(k + G) e λ] [(k + G ) e λ ]ɛ (G, G ) (A.7) Once this matrix is generated, it is only a matter of diagonalizing and obtaining the eigenvalues and eigenmodes. [9] A.3.2 Practice When using MPB it is very important to know that it makes use of the supercell approximation. The eigensolver requires a perfect periodic structure, so the first action that is being done is to copy-paste the cell we defined. This means, that if we define one hole in a block, MPB turns it into a lattice. By default this is a square lattice, but this can be redefined. If we now want to calculate the modes of a monorail lattice, we have to define a supercell. This means that the unit distance is kept constant (normally the lattice constant a), but the size of cell that is being copied is enlarged. For the monorail, we change the supercell size in the y and z direction, since the intended lattice is only along the x direction. After the copy-paste, we have an infinite set of monorails of infinite length, spaced some distance apart, see also figure A.3

97 Appendix A. Tools 8 Figure A.3: Due to the copy-paste behaviour of MPB, lots of monorails are generated in stead of one. As was explained in chapter 2.4, light is confined by photonic bandgap confinement in the x direction and index guiding in the other dimensions. If the distance between two monorails is large enough, field of a mode will be approximately zero when reaching his neighbour because of the exponentially decaying tail. If this is not the case, increasing the distance should do the trick. The story is totally different if there is no index guiding, i.e. we are in the light cone. In that case, there is no exponentially decaying tail, meaning that the field will never be zero, no matter what distance we choose. Light will start to resonate between the different monorails. The solutions being found this way are pure nonsense. Because of this, we have to generate a light cone overlay during the data analysis. This overlay covers all the nonsense solutions.