Optimization of triangular lattice defect in dynamic photonic crystal structures for optical storage and processing

Transcription

1 Optmzaton of trangular lattce defect n dynamc photonc crystal structures for optcal storage and processng Mostafa Shalaby a, A. K. AboulSeoud a,b, Moustafa H. Aly a, Amr Marzouk c a Arab Academy for Scence, Technology and Martme Transport, Alexandra, Egypt b Unversty of Alexandra, Alexandra, Egypt c Smon Fraser Unversty, Surrey, Canada ABSTRACT A trangular lattce GaAs photonc crystal structure was proposed n a prevous work [] for optcal storage n a dynamc modulaton process. Ths work presents a defect optmzaton of ths tunable coupled resonator array. Preservng translatonal nvarance and adabatcty, ths structure exhbts an optcal analogue to electromagnetc nduced transparency. Ths trangular lattce structure shows an advantage over the prevously proposed square one [, 3] n compressng hgher bandwdth pulses. The man problem of ths structure s the ntroducton of hgher group-velocty dsperson. In the present work, the structure s redesgned so as to change the operatng range of frequency for the propagatng pulse. In ths way, the group-velocty dsperson s elmnated to values close to that of the square lattce structure. The fnal desgn, therefore, combnes both hgher compressble bandwdth and lower group-velocty dsperson n addton to a fabrcaton advantage. Keywords: Dynamc structures, temporal coupled mode theory, optcal storage, group-velocty dsperson, trangular lattce, refractve ndex modulaton.. INTRODUCTION Development of dynamc structures for optcal storage represents one of the recent advances n the area of photonc crystals. These structures make good use of the fascnatng propertes of photonc crystals n controllng lght. A tunable coupled resonator array structure was proposed for dynamc tunng of the bandwdth of optcal pulses [3]. In ths structure, the bandwdth s adjusted to be large enough to accommodate the optcal pulse. When t s completely nsde the structure, the transmsson propertes of the structure are tuned dynamcally. Under approprate condtons, the bandwdth of the pulse can be reduced to zero. By cascadng such tunable band pass flters, t s possble to reduce the velocty of a propagatng pulse to zero and even to tme reverse t. The pulse can then be kept n the structure ndefntely. Ths process can then be reversed and the pulse s released. The system therefore behaves as both a tunable band pass flter and a tunable delay element [3]. Ths stop-freeze-release process allows for new possbltes for optcal storage and processng. The orgnal structure used to valdate the theory s based on a two dmensonal square lattce structure [3, 4]. In a prevous work [], a trangular structure was suggested nstead. Ths trangular structure offers an advantage over the square one. It represents a better approxmaton to the deal case of sphercal Brlloun zone n two dmensons. Ths structure shows hgher capabltes for bandwdth compresson [] especally when used wth GaAs. However, ths structure has two man dsadvantages; t requres a longer wavegude length and t shows hgher group-velocty dsperson. Whle the former can be compromsed wth the requred system specfcatons [], the latter needs more consderaton. In the present work, the trangular lattce dynamc structure s revsted. The orgnal model s redesgned so as to reduce the group-velocty dsperson whle mantanng hgher bandwdth compresson capabltes. Fgure shows two unt cells of the basc system for lght manpulaton. Each unt cell conssts of a wavegude sde coupled to two cavtes [5]. The two suffcent condtons for dynamc refractve ndex modulaton are the translatonal nvarance and adabatcty [5-7]. It has been shown [] that usng a trangular lattce nstead of a square one wll not volate any of them. The sngle mode wavegude s created here by removng a sngle row of delectrc rods. In prevous works [-3], the sngle mode cavty was obtaned by reducng both refractve ndex and radus of a sngle rod. By modulatng the refractve ndex of ths system, the resonance frequences of the unt cell cavtes can be vared creatng Photoncs North 009, edted by Réal Vallée, Proc. of SPIE Vol. 7386, 7386D 009 SPIE CCC code: X/09/$8 do: 0.7/ Proc. of SPIE Vol D-

2 two cavtes (n each unt cell) wth resonance frequences of ω, =ω o ±δω/ [6], δω s the sgnal bandwdth. In both square and trangular lattce structures, both cavtes have the same symmetry and the system supports nonorthogonal modes [8]. Ths system exhbts an electromagnetc nduced transparency. Fg.. Schematc of two unt cells of the coupled-cavty structure used to stop lght. The cavtes couple to the wavegude wth a couplng rate of /τ. Length of the unt cell s l = l +l [3].. TRIANGULAR GAAS LATTICES The orgnal system used to valdate the theory s based on a square lattce wth a delectrc constant of.5 [3]. The trangular lattce GaAs structure, suggested n a prevous work [], shows some advantages over the orgnal one. These advantages are summarzed below:. Effect of the trangular lattce At the same background delectrc constant and center frequency, the trangular lattce shows an advantage of compressng larger bandwdths than that of the square one. In addton, the rato of the fnal bandwdth (after modulaton) to the orgnal bandwdth (before modulaton) s to the advantage of the trangular lattce. The reason for ths behavor s two-fold []: frst, because the photonc bandgap n the case of a trangular lattce s wder than that n the case of a square lattce, an ntroduced defect mode at the mddle of each gap s more localzed n the case of a trangular lattce. The evanescent modes n the gap of the structure have exponental nature and they share the same fnal value of leakng nto the extended bands. Therefore, any devaton, represented by the refractve ndex modulaton, s more effectve n the case of the trangular lattce. Ths results n a wder pulse bandwdth to compress. The other factor arses from the fact that the mode s better localzed n the photonc bandgap of the trangular lattce because t s wder than the square one. A better qualty factor s, thus, obtaned n the case of the trangular lattce. Both the precedng factors can be ntroduced n the bandwdth Q ω ω /ω 0 leadng to a hgher compressble bandwdth.. Effect of usng GaAs Usng GaAs (ε=.4 at λ=.5µm [4]) leads to a larger compressble bandwdth than that obtaned f the orgnal materal [] wth ε=.5 s used. Because GaAs has a lower delectrc constant, t offers narrower photonc bandgap and s, thus, weak at the localzaton of cavty modes. Ths weakness can result nto a hgher bandwdth under the followng crcumstances. For a materal wth ε=.5, a large value for delectrc shft ε s requred for a defect mode to cross half the bandgap (whch s large) and be ntroduced at the mddle of the gap. On the other hand, n the case of GaAs, no such bg effort s requred because the bandgap s smaller. If the dfference n the orgnal reference value s small ( ε=.5-.4) and t wll be further reduced when the refractve ndex modulaton s consdered ( n= ), the requred delectrc constant for the defect cavty mode can be larger for GaAs (n cavty =.4) than n the case of ε=.5 (n cavty =.4) at the same defect frequency. Once the defect mode s establshed, the perturbaton and varatonal theores can be appled. Ths s done n [] and usng GaAs shows a hgher compressble bandwdth. Proc. of SPIE Vol D-

3 3. PARAMETERS OPTIMIZATION The prevously dscussed model based on GaAs trangular lattces shows two man problems []; the frst s the requrement of longer wavegude lengths for the compresson of the extra band. In an attempt to solve ths problem, there are two suggested scenaros []. The second problem s the hgher group-velocty dsperson shown n the new desgn. The objectve of ths work s to address ths problem. In an attempt to solve ths problem, the lnear defect ntroduced n the forbdden bandgap s spanned searchng for an approprate range of lnear group-velocty. At the same tme, t should be as close as possble to the mddle of the forbdden regon for maxmum localzaton of the defect mode. Zoomng n Fg. shows that a shft from the old center frequency ω o =0.357 (Fg. 3 (a)) to the new one, ω o =0.375 (Fg. 3 (b)) satsfes both condtons, especally when used wth GaAs. The new range of operaton shows a close approxmaton to the (almost) lnear curve of the square lattce structure. All curves are calculated at the same refractve ndex modulaton rate. Frequency ωa/πc Wave vector ka/π Fg. The band structure of the lnear defect (due to wavegude only) for a background delectrc constant of.4. In order to produce no change n the center frequency of operaton, the orgnal model parameters should be redesgned; more specfcally, the defect modes of the cavtes. Moreover, by changng the center frequency, the condton for maxmum transmsson s volated and the transmsson through the structure s reduced. 3. Defect optmzaton In the prevous desgns, square lattce [3] and trangular lattce [] structures, the center frequency s kept the same at a value of Ths defect frequency s obtaned by reducng both the radus to 0.a for both structures and delectrc constant of a sngle rod to and for square and trangular lattces respectvely. These calculatons are done at a background delectrc of GaAs. In order to obtan a new center frequency of 0.375, only the delectrc constant s changed. The radus of the defect rod s kept untouched. Ths desgn, therefore, shows a fabrcaton advantage over the past two desgns. In a GaAs background of trangular rods, reducng the delectrc constant of a sngle rod to.55 creates a sngle mode cavty of a resonance frequency ω o = Proc. of SPIE Vol D-3

4 Frequency ωa/πc Frequency ωa/πc Wave vector ka/π Fg. 3. (a) The orgnal frequency range of operaton (ω=0.357) s zoomed n and shown n relaton to the perfect lnear mode (dashed lne) Wave vector ka/π Fg. 3. (b) The new frequency regon of operaton (ω=0.375) s, agan, zoomed n for a trangular lattce structure. 3. Cavtes separaton The desgn of ths coupled resonator array s dependent on the adjustment of the system components n such a way that the system shows an optcal analogue to electromagnetcally nduced transparency. For maxmum transmsson through such a structure, the separaton between the cavtes should be adjusted such that the followng equaton s satsfed: nπ L = () B( ω) where B(ω) s the wavegude dsperson relaton, and n s an nteger. Equaton () represents a condton for maxmum transmsson n the common problem of smple parallel mrrors resonator (lke Fabry-Perot etalon). An mportant characterstc of a photonc crystal based wavegude s that lght s manly drected n the low-ε regons whch are usually ar. As a result, the dsperson relaton β(ω) can be approxmated to (ω/c). Ths result and equaton () at the operatng frequency found above gves cavtes separaton of 4a for maxmum transmsson. 4. ANALYSIS OF THE OPTIMIZED MODEL In order to gve a comprehensve analyss of the dscussed system, the transmsson characterstcs and band structure should be found. The ntensty transmsson coeffcent, T, of ths system s calculated as follows [5] [9] [0] t t T = () r r r r + 4 sn θ ( ) r r where θ s the round trp phase angle accumulated n the wavegude sectons: θ=0.5arg(r r e -jβ(ω)l ), β(ω) s the wavegude dsperson. r and t are the reflecton and transmsson coeffcents of the th cavty gven [5] by γ r = (3) j( ω ω ) + γ j( ω ω ) t = (4) j ω ω + ( ) γ Proc. of SPIE Vol D-4

5 In order to obtan the band structure of such a system, the transmsson matrx should be found a pror. If T c and T l are the transmsson matrces for the th wavegude sde coupled to a sngle resonator and wavegude secton, respectvely, (=, ), the transmsson matrx through an entre unt cell s found [] to be Τ = Τc Τl Τc Τl (5) where Τ c + j /( ω ω = j /( ω ω j /( ω ω j /( ω ω where ω, and /τ are the resonance frequency and couplng rate to the wavegude for the th resonator and jβ ( ω ) l Τ l e = 0 e 0 ( ) ω l jβ Usng the above model, the band dagram for equal couplng rate cavtes and gnorng any drect couplng between sde cavtes s found [] to be where Tr ( Τ ) = cos( kl) = f ( ω) cos( βl ) + C + + C (8) sn( βl) sn( βl ± = ± τ ( ω ω ( ω ω ) ( ω ω ) )sn( βl) C (9) Ths analyss s based on the representaton of the egenvalues of T as e -jk(ω)l, and e jk(ω)l (because det (T) =); where k s the Bloch wavevector of the entre system. 5. SIMULATION RESULTS The transmsson characterstcs for the system shown n Fg. wth the new parameters are shown n Fg. 4 (a). The transmsson characterstcs are calculated usng the parameters obtaned from smulatng these systems and usng equatons () (4) at a center frequency of ω o = Fgure 4 (b) shows the band structure calculated usng equatons (5) (9). Ths system exhbts three photonc bands. The wdth of the mddle band depends strongly on the cavtes resonant frequences. By modulatng the cavtes frequency spacng, the system bandwdth s compressed. Fgure 4 (b) shows the possblty of nearly flat band n the entre Brlloun zone of the structure. In ths way, by cascadng the bandwdth flters shown n Fg. 4 (a), t becomes possble to reduce the group velocty of the propagatng pulse to zero. After the general behavor of the system s assessed, t s very mportant to compare these systems wth the other systems prevously dscussed. Beng an optmzaton process, the new system should combne the low group-velocty dsperson of the orgnal square lattce at ε=.5 and the hgher compressble bandwdth of the trangular lattce structure of GaAs. The former has been taken nto consderaton n the system desgn. As shown n Fg. 5, the performance of the new desgn s comparable to that of the trangular lattce at ε=.4 and r defect =0.a at ω o = CONCLUSION Ths work represents an attempt to overcome the hgh group-velocty dsperson brought about by the ntroducton of trangular lattce n a prevous work [] to the dynamc lght stoppage structure. Shftng the center frequency of operaton from to represents a more lnear regon of operaton. In ths way, the problem of group-velocty dsperson s overcome and the performance of the system becomes close to that of the orgnal model of a square lattce. As the center frequency s modfed, the necessary condton for maxmum transmsson s volated and the transmsson dropped below maxmum. Ths problem s handled by redesgnng the cavtes separatons so that the round trp phase angle accumulated n the resonator s an ntegral multple of π. In ths way, the transmsson returned back to maxmum (6) (7) Proc. of SPIE Vol D-5

6 and the compressble bandwdth of the fnal structure s comparable to that of the trangular lattce at the orgnal center frequency of Intensty Transmsson Coeffcent (ω-ω0)/γ U Before Compresson - After Compresson I I I I I I 5 I I I I U US (ω-ω 0 )/γ Fg. 4. (a) Transmsson characterstcs for a trangular lattce structures wth ε=.4 and ω o =0.375 Wave vector kl/π Fg. 4. (b) Band structure for a trangular lattce structures wth ε=.4 and ω o =0.375 Intensty Transmsson Coeffcent (ω-ω0)/γ (ω-ω 0 )/γ Fg. 5. (a) Transmsson characterstcs for a square lattce (ε=.5, r defect =0.a), a trangular lattce (ε=.4, r defect =0.a), and a trangular lattce (ε=.4, r defect =0.a). Wave vector kl/π Fg. 5. (b) Band structure for a square lattce (ε=.5, r defect =0.a), a trangular lattce (ε=.4, r defect =0.a), and a trangular lattce (ε=.4, r defect =0.a). REFERENCES [] [] Shalaby, M., AboulSeoud, A. K., and Aly, M. H., Dynamc Photonc Crystal Structures for Optcal Storage and Processng Based on GaAs and Trangular Lattce, To be Publshed. Fan, S., Yank, M. F., Povnell, M. L., and Sandhu, S., "Dynamc Photonc Crystals," Optcs and Photoncs News, 8, 4-45 (007). Proc. of SPIE Vol D-6

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