Field penetrations in photonic crystal Fano reflectors

Transcription

1 Field penetrations in potonic crystal Fano reflectors Deyin Zao, Zenqiang Ma, and Weidong Zou,* Department of Electrical Engineering, NanoFAB Center, University of Texas at Arlington, Texas 769, USA Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Wisconsin 576, USA Abstract: We report ere te field and modal caracteristics in potonic crystal (PC) Fano reflectors. Due to te tigt field confinement and te compact reflector size, te cavity modes are igly localized and confined inside te single layer Fano reflectors, wit te energy penetration dept of only nm for a 4 nm tick Fano reflector wit a design wavelengt of 55 nm. On te oter and, te pase penetration depts, associated wit te pase discontinuity and dispersion properties of te reflectors, vary from nm to 4 nm, over te spectral range of 5 nm to 58 nm. Tis unique feature offers us anoter design freedom of te dispersion engineering for te cavity resonant mode tuning. Additionally, te field distributions are also investigated and compared for te Fabry-Perot cavities formed wit PC Fano reflectors, as well conventional DBR reflectors and D sub-wavelengt grating reflectors. All tese caracteristics associated wit te PC Fano reflectors enable a new type of resonant cavity design for a large range of potonic applications. Optical Society of America OCIS codes: (4.948) Fabry-Perot cavity; ( ) Fano resonance; (.) Optical devices; (5.58) Pase sift; ( ) Potonic crystal. References and links. S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitc, Ultimate vertical Fabry- Perot cavity based on single-layer potonic crystal mirrors, Opt. Express 5(9), (7).. M. Sagawa, S. Goto, K. Hosomi, T. Sugawara, T. Katsuyama, and Y. Arakawa, 4-Gbit/s Operation of Ultracompact Potodetector-Integrated Dispersion Compensator Based on One-Dimensional Potonic Crystals, Jpn. J. Appl. Pys. 47(8), (8).. A. Cutinan, N. P. Kerani, and S. Zukotynski, Hig-efficiency potonic crystal solar cell arcitecture, Opt. Express 7(), (9). 4. O. Kilic, M. Digonnet, G. Kino, and O. Solgaard, External fibre Fabry-Perot acoustic sensor based on a potonic-crystal mirror, Meas. Sci. Tecnol. 8(), (7). 5. J. D. Joannopoulos, S. G. Jonson, J. N. Winn, and R. D. Meade, Potonic Crystals: Molding te Flow of Ligt, nd ed. (Princeton University Press, 8). 6. U. Fano, Effects of Configuration Interaction on Intensities and Pase Sifts, Pys. Rev. 4(6), (96). 7. R. Magnusson, and S. S. Wang, New principle for optical filters, Appl. Pys. Lett. 6(9), (99). 8. S. Fan, and J. D. Joannopoulos, Analysis of guided resonances in potonic crystal slabs, Pys. Rev. B 65(), 5 (). 9. D. K. Jacob, S. C. Dunn, and M. G. Moaram, Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters, Appl. Opt. 4(7), 4 45 ().. S. T. Turman, and G. M. Morris, Controlling te spectral response in guided-mode resonance filter design, Appl. Opt. 4(6), 5 ().. C. F. R. Mateus, M. C. Y. Huang, L. Cen, C. J. Cang-Hasnain, and Y. Suzuki, Broadband mirror (.-.6 µm) using single-layer sub-wavelengt grating, IEEE Poton. Tecnol. Lett. 6(7), (4).. W. Su, and S. Fan, All-pass transmission or flattop reflection filters using a single potonic crystal slab, Appl. Pys. Lett. 84(4), 495 (4).. S. Boutami, B. B. Bakir, H. Hattori, X. Letartre, J.-L. Leclercq, P. Rojo-Rome, M. Garrigues, C. Seassal, and P. Viktorovitc, Broadband and compact -D potonic crystal reflectors wit controllable polarization dependence, IEEE Poton. Tecnol. Lett. 8(7), (6). #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 45

2 4. Y. Ding, and R. Magnusson, Resonant leaky-mode spectral-band engineering and device applications, Opt. Express (), (4). 5. M. L. Wu, Y. C. Lee, C. L. Hsu, Y. C. Liu, and J. Y. Cang, Experimental and Teoretical demonstration of resonant leaky-mode in grating waveguide structure wit a flattened passband, Jpn. J. Appl. Pys. 46(No. 8B), (7). 6. R. Magnusson, and M. Sokoo-Saremi, Pysical basis for wideband resonant reflectors, Opt. Express 6(5), (8). 7. W. Zou, Z. Ma, H. Yang, Z. Qiang, G. Qin, H. Pang, L. Cen, W. Yang, S. Cuwongin, and D. Zao, Flexible potonic-crystal Fano filters based on transferred semiconductor nanomembranes, J. Pys. D. 4(), 47 (9). 8. L. Coldren, and S. Corzine, Diode lasers and potonic integrated circuits, (Wiley New York, 995). 9. J. H. Kim, L. Crostowski, E. Bisaillon, and D. V. Plant, DBR, Sub-wavelengt grating, and Potonic crystal slab Fabry-Perot cavity design using pase analysis by FDTD, Opt. Express 5(6), 9 (7).. D. Babic, and S. Corzine, Analytic expressions for te reflection delay, penetration dept, and absorptance of quarter-wave dielectric mirrors, IEEE J. Quantum Electron. 8(), (99).. C. Sauvan, J. Hugonin, and P. Lalanne, Difference between penetration and damping lengts in potonic crystal mirrors, Appl. Pys. Lett. 95(), (9).. M. Born, E. Wolf, and A. Batia, Principles of optics: electromagnetic teory of propagation, interference and diffraction of ligt: Cambridge Univ Pr, Introduction Ultra-compact dielectric broadband reflectors are essential elements in te design of laser cavities, ligt trapping microcavities, cavity QEDs, nonlinear optics, and quantum computing systems [ 4]. Traditionally, tey can be realized by using te metal films or te stacked dielectric tin films (i.e., distributed Bragg reflectors, DBRs). Metal films can offer larger reflection bandwidt but are limited by te intrinsic absorption losses. Stacked dielectric tin films can acieve very low losses. But tey typically require many individual layers wit te stringent refractive and tickness tolerances for eac layer. Two dimensional potonic crystal slab (D PCS) broadband reflectors can be realized for in-plane directions based on te potonic bandgap principle [5]. Recently more attention as been paid to te single layer patterned broadband reflectors for surface-normal incident direction operation, based on te guide mode resonance, or Fano resonance [6 8] principles, for single layer one-dimensional sub-wavelengt grating (D SWG) structures, or D PCS structures [8 7]. Under surface-normal incidence, DBR, D SWG, and D PCS reflectors can all exibit similar reflection properties wit extremely ig reflection and broad reflection spectral band. However, reflection mecanisms are different. For D SWG and D PC mirrors, te incident wave couples to te in-plane guided-mode based on pase matcing conditions. Te wave ten reradiates at one edge wit a zero pase difference and at anoter edge wit a π pase difference. Consequently, tese constructive and destructive interferences result in ig reflection and low transmission, respectively [6]. Wile for DBR, te ig reflectivity arises from te multiple reflections wit constructive interference among tese reflected waves. Due to te large difference, Bragg mirrors possess a broad reflection spectral band [8]. For D SWG and D PCS mirrors, te broad reflection spectral band most likely originates from te cooperating of te several adjacent guided mode resonances [6]. In addition to te reflector spectral amplitude properties, suc as ig reflectivity R and broad reflection band, it is equally important to understand te pase discontinuity and dispersion beavior (reflection pase sift, Φ r ), te field/energy penetration depts (L p, L e ), and te field/cavity modal caracteristics in cavities formed by tese types of patterned single layer dielectric reflectors. However, most attentions so far ave been paid to te spectral reflection amplitude properties. Very little work as been reported on pase/dispersion and mode/energy caracteristics. In Ref [9], te autors reported excellent work on te pase discontinuity Φ r and te energy penetration dept L e, wic were estimated from te mode spacing in te Fabry-Perot (FP) cavity. In tis work, we investigate te pase discontinuity and te dispersion properties in D PCS Fano reflectors, for applications in multi-wavelengt cavity design. We will also discuss te distinctively different beavior for pase and energy penetration depts. Te pase penetration dept L p is related to te reflection delay. A large #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 45

3 pase penetration dept will lead to longer poton lifetime and longer cavity resonance. Te energy penetration dept L e is related to te energy decay. A smaller energy penetration dept will lead to more modal confinement [8,,]. In addition, te different reflection mecanisms, guided mode resonance in PC mirrors and constructive interferences in DBRs, result in distinctively different field distribution profiles inside te cavity, wic is anoter important factor to be considered in te laser cavity design. In wat follows, we first introduce te cavity configurations of tree types of surface-normal dielectric reflectors. Secondly, we investigate and compare te two penetration depts, L p and L e according to te calculated Φ r and R values, based on te D finite difference time domain (FDTD) tecnique. We ten compare te field distributions of te resonant modes in FP cavity formed by tese tree types of dielectric reflectors. Finally, a conclusion is given.. Dielectric mirrors configuration and corresponding FP cavities Here, we consider tree types of dielectric mirrors and teir corresponding FP cavities, as sown in Fig.. Tey all consist of two kinds of materials, Si and SiO. Here Si and SiO are assumed to be lossless and dispersion-free, wit te refractive es are of.48 and.48, respectively, over te spectral range of interest around 55 nm. In Fig. (a), te top and te bottom mirrors of te FP cavity (denoted as Cavity I ) are D Si SWG structures wit te same lattice parameters, wit te Si layer tickness of.46µm, te grating period Λ of.7µm, and te air slit widt w of.5λ. Te bottom SWG mirror is formed on a silicon-oninsulator (SOI) substrate, wit te buffered oxide (BOX) layer tickness of.8µm. Tese two D SWG reflectors exibit a ig reflection over.-.6µm wavelengt band for TM polarization only (H field is parallel to te air slit, y direction) [6]. Λ Si L c w z yλ L c r t x SiNM PC r b DBR (Si/SiO ) L c Si SiO SiNM PC SiO BOX DBR (Si/SiO ) Si substrate Si Substrate SiO substrate (a) (b) (c) Fig.. Sketces of different dielectric reflectors and te corresponding Fabry-Perot cavities: (a) cross section (xz plane) of Cavity I consisting of top and bottom reflectors based on D single Si-layer sub-wavelengt grating (SWG) wit te same pattern parameters; (b) overview of Cavity II consisting of top and bottom mirrors based on D PCS patterns wit a square air oles lattice; (c) cross section (xz plane) of Cavity III consisting of top and bottom mirrors based on 4-pairs of Si/SiO (./.7µm) DBR stacked layers. Sown in Fig. (b) is te second FP cavity ( Cavity II ) formed wit top and bottom D PCS Fano reflectors, were te Si slab layers wit a tickness =.4um are patterned wit D square lattice air ole arrays. Te PC lattice constant Λ is equal to.98µm. Again, te bottom reflector is processed on a SOI wafer, wit te BOX layer tickness of µm. To enable te reflection bands of te top and bottom mirrors wit large spectral overlap, teir radius of air ole are set to r t =.6Λ and r b =.8Λ, respectively. Te resulting overlapping reflection range of tese two D PC mirrors is over µm wavelengt band for bot TE and TM polarizations (Broader and more flat band could be found troug carefully optimizing te structure parameters). For comparison, we also consider te tird FP cavity ( Cavity III ), based on two classical ig contract Si/SiO DBRs, as sown in Fig. (c). Te DBRs consist of 4-pairs of Si/SiO stacked layers, were te tickness of Si and SiO are cosen to be. and.7µm, respectively. Suc a DBR wit a very large difference possesses a #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 454

4 wide reflection band over.-.7µm. Note all te parameters cosen ere are optimized for broadband reflectors wit peak reflection around 55nm.. Pase penetration dept and energy penetration dept In dielectric mirrors, te reflection is not an instantaneous process. It includes a reflection time delay (τ) and energy storage in te mirrors. Te reflection delay increases te laser cavity round-trip time or te poton lifetime. Te energy storage in mirrors decreases te modal volume and te confinement factor in small cavity in wic te cavity round-trip time and te cavity volume ave a comparable magnitude as te reflection delay and te mirror storage [,]. Te reflection delay is directly related to te slope of te reflection pase sift Φ r. Te relation can be expressed as τ = Φ r / ω. Te pase penetration dept, L p, is defined as te alf-distance tat ligt propagates in te incident medium during tis delay time, L p vg = vgτ = Φr, ω were v g is te group velocity of te incident wave. Te energy storage is always associated to te parameter of energy penetration dept, L e. It is te lengt tat te field intensity decays to /e of its maximum from te edge of cavity into te mirrors. However, tis metod is not suitable to calculate L e of te PC mirrors we discuss ere, because te guided modes are excited inside te mirrors. But L e can be estimated from te mirror transmission or reflection based on te following equation, () T = R= exp( ), () L were T is te transmission and is te mirror tickness [8,]. Wile, for DBRs, L e can be obtained [8]: e meff λ λ Le = ( + ), () 4n 4n were m eff = tan(mr)/(r) is te effective period number seen by te incident ligt. r = (n - n )/(n + n ) and m is te actual period number in DBRs. n and n are te refractive of two materials in DBR, respectively. In te following we will numerically investigate te pase and energy penetration dept according to te above definitions.. Reflection and te pase sift First, we utilize FDTD simulation metod to get R and Φ r of tese dielectric mirrors. A Gaussian temporal pulse excitation is used to simulate te reflectivity R. In order to calculate Φ r, a continuous plane wave of a single wavelengt (λ) is vertically incident on te dielectric mirrors and Φ r is extracted from te stable reflected field. Here te calculated Φ r is set in te range of [, π]. In order to validate our numerical simulation metod, we compare te simulated R and Φ r of te top DBR based on FDTD wit te teoretical calculated values according to te multiple tin film matrix teory []. Te calculated results completely overlap wit te teoretical ones. Plotted in Fig. (a) are te simulated R and Φ r values for bot top and bottom DBRs. In te ig reflection (R>.95) spectral band, covering from. to.µm, Φ r slowly increases from.85π to.9π. Sown in Fig. (b) are te simulated R and Φ r spectra of top and bottom D SWG reflectors. Te ig reflection TM (R>.95) spectral band spans from. to.6µm. Altoug bot DBR and D SWG reflector ave similar broad ig reflection spectra bands, teir Φ r canges are very different. Φ r of D SWG reflector rapidly varies in te range of [, π], muc faster tan tose of DBRs. Te calculated R and Φ r spectra of for te top and #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 455

5 bottom D PCS reflectors are plotted in Fig. (c), were te overlapping ig reflection spectra band covers from.49 to.58µm. Owing to te different air fill factors (r t =.6Λ < r b =.8Λ), ig reflection spectral band for te bottom mirror is narrower tan tat of te top mirror. In tis ig reflection band range, te pase sift of top mirror Φ rt varies in range of (.9π,.7π), and te pase sift of bottom mirror Φ rb canges in range of (.7π,.4π). For all tese tree types of mirrors, Φ r varies over te ig reflection spectral band at drastically different cange rates, wic can be found by comparing te pase penetration dept L p. R & Φ r (π ).5.5 R t R b Φ r,t Φ r,b DBR R & Φ r (π ).5.5 D SWG R t R b Φ r,t Φ r,b R & Φ r (π ).5.5 R t R b Φ r,t Φ r,b D PCS.5.5 Wavelengt (µm) Wavelengt (µm) Wavelengt (µm) (a) (b) (c) Fig.. Calculated reflection R (blue solid and black das lines) and pase sift Φ r (red das-dot and green dot lines) spectra of top and bottom (a) DBR reflectors, (b) D SWG reflectors and (c) D PCS reflectors.. Pase penetration dept and energy penetration dept Te pase penetration dept L p can ten be calculated based on Eq. () and te simulated Φ r sown in Fig. Te results are plotted in Fig. (a), for te bottom DBR, D SWG, and D PCS reflectors in te wavelengt range of.5-.6µm, denoted as L p, DBR, L p, DSWG, L p, DPCS, respectively. It can be seen tat L p, DPCS (~.µm) and L p, DSWG (~.µm) are one order larger tan L p, DBR (~.µm). Additionally, wile L p, DBR and L p, DSWG remains largely uncanged over te spectral range, L p, DPCS does cange significantly over different spectral locations. Tis not only results in a muc longer reflection delay time in PC mirrors, it also leads to different resonance cavity locations. Suc a long pase delay may be due to te guided mode excitation, even if D PCS reflectors are very tin. To verify tis point, we record te dynamic process of te reflected field wit λ =.54µm at one monitor above te P PCS reflector and te DBR, as sown in Fig. (b). One can find, for te PCS reflector based on guided mode Fano resonance, te reflected field reaces stable condition only after a long fs period, wile it only takes about 6fs for DBR to reac te stable condition. So, it is very clear tese two different reflection mecanisms result in very different L p values in PC reflectors and DBRs. Lp (µm) 5 4 L p,dbr L p,dswg L p,dpcs Wavelengt (µm) (a) H DSWG E DPCS E DBR D SWG D PCS DBR 4 time (fs) (b) Fig.. (a) Te pase penetration depts for tree types of bottom reflectors; (b) Reflected field at λ =.54µm canges as function of time, for D SWG reflectors (top), D PCS reflectors (middle), and DBR reflectors (bottom). #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 456

6 Te energy penetration dept L e can be obtained based on T = -R, and Eqs. () and (). Te results are sown in Fig. 4 for te bottom DBR, D SWG, and D PCS reflectors, respectively. Different from te pase penetration caracteristics discussed earlier, bot L e, DPCS (~ nm) and L e, DSWG (~6 nm) are muc lower tan L e, DBR (~ nm). And all energy penetration depts ave muc less wavelengt dependence, as compared to tat of pase penetration depts. It is wort noting tat muc iger energy penetration depts are expected in DBRs wit smaller contrast (e.g. GaAs/AlGaAs, InGaAsP DBRs). Suc a small energy penetration lengt in PCS mirrors can attribute from te tigt mode confinement. Tis is also very favorable in acieving better modal confinement inside te cavity. For classical DBRs, te pase penetration dept is typically smaller or similar to te energy penetration dept, due to smaller contract and larger energy penetration dept [8,]. However, for D SWG and D PCS reflectors discussed ere, L p is muc larger tan L e, consistent wit inplane D PCS reflectors, as reported in Ref. 8. Tis could be due to te large pase discontinuities originated from te modal interaction between in-plane guided modes and vertical radiation modes. On te oter and, te energy penetration dept can be very small and localized wit te tin PCS layer..4 L e,dbr L e,dswg L e,dpcs. Le (µm) Wavelengt (µm) Fig. 4. Te energy penetration depts for tree types of bottom reflectors. 4. FP cavities and field distributions Finally, for te resonant FP cavity modes, we can investigate te field distribution properties in tese FP cavities sown in Fig.. To ave resonance cavity modes wit similar spectral locations, te cavity lengts are cosen as L c = 5.4µm, L c = 5. and 5.4µm, L c = 5.4µm in tree different cavities. We cose te two resonant modes in eac cavity: λ =.69µm and.5µm in Cavity I, λ =.54µm and.54µm in Cavity II, λ =.76µm and.54µm in Cavity III. Figure 5(a) represents te field intensity of two resonant modes in DBR Cavity III, λ =.76µm and.54µm, were te dielectric profile is also plotted wit a black tin line. For a classical DBR cavity, te field intensity inside te cavity is always larger tan tat in mirrors, and te field gradually decays into te mirrors. However, it is not te case for Cavity I and II based on Fano or guided mode resonances. In Cavity I at L c = 5.4µm, for te two resonant modes (λ =.69µm and.5µm) sown in Fig. 5(b), te magnetic field intensity inside cavity is muc smaller tan tat in mirrors. It is because te guide modes are excited into te mirrors. Outside te cavity, te field intensity decays rapidly, wic is in consistent wit te very small L e obtained earlier. Figure 5(c) and Fig. 5(d) correspond to te case for Cavity II. It can be seen tat te electric field intensity of te two resonant modes (λ =.54µm and.54µm) inside cavity is muc larger tan tat in reflector slabs and te field outside cavity also decays very fast. Cases sown in Figs. 5(b) 5(d) can exist in bot D SWG and D PCS based FP cavities. However, te ratio of te intensity inside te reflector slabs to te intensity inside te cavity (between two reflector slabs) is still larger tan tat of Cavity III, mostly due to nature of Fano or guide mode resonance excitation inside reflector slabs. Neverteless, it is expected a very ig modal confinement inside te cavity wit ig field intensity. #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 457

7 E λ=.76µm λ=.54µm Position along te cavity (µm) (a) H λ=.69µm λ=.5µm Position along te cavity (µm) (b).5.5 E λ=.54µm Position along te cavity (µm) (c).5.5 E λ=.54µm Position along te cavity (µm) (d) 5. Conclusion Fig. 5. Field intensity distributions of te resonant modes inside te cavities (a) Cavity III wit L c = 5.4µm, (b) Cavity I wit L c = 5.4µm, and (c) Cavity II wit L c = 5.µm, and (d) Cavity II wit L c = 5.4µm. In conclusion, we ave numerically investigated pase and energy penetration depts, and field distributions of D SWG and D PCS reflectors based on Fano or guided mode resonances. Comparing to te DBR reflectors, tese new types of single layer ultra-compact broadband reflectors can ave more complicated larger pase delays and smaller energy penetration properties, wic can be engineered via dispersion engineering for large spectral dependent pase delays, and ultra-small energy penetration depts. Te work reported ere is mostly based on one set of optimized design parameters for 55nm band reflectors. Following similar procedures, oter design parameters can be used for reflectors wit different reflection requirements, as well as different pase delays, energy penetrations, and field distributions. All te results and conclusions can be very elpful for te design of resonant cavities for a wide range of potonic applications. Acknowledgments DZ appreciates te elp from Dr. Zexuan Qiang. Tis work is supported in part by US Air Force Office of Scientific Researc (AFOSR) MURI program under Grant FA , by AFOSR under grant FA955-9-C-, and in part by US Army Researc Office (ARO) under Grant W9NF #87 - $5. USD Received 6 May ; revised 9 Jun ; accepted Jun ; publised 6 Jun (C) OSA June / Vol. 8, No. / OPTICS EXPRESS 458