Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals

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1 2192 J. Opt. Soc. A. A/ Vol. 23, No. 9/ Septeber 2006 W. Jiang and R. T. Chen Rigorou analyi of diffraction grating of arbitrary profile uing virtual photonic crytal Wei Jiang and Ray T. Chen Microelectronic Reearch Center and Departent of Electrical and Coputer Engineering, The Univerity of Texa at Autin, Autin, Texa Received February 10, 2006; revied April 4, 2006; accepted April 4, 2006; poted April 7, 2006 (Doc. ID 67976) A new approach i developed to calculate diffraction efficiency for a dielectric grating with an arbitrary refractive index profile. By treating a one-dienional grating a a egent of a virtual two-dienional (2D) photonic crytal, we exploit a rigorou theory of photonic crytal refraction and calculate the diffraction efficiencie. We expand, analytically in any cae, the dielectric function of the grating into 2D Fourier erie. We find the eigenode for the virtual photonic crytal, and then ue thee eigenode to atch the boundary condition by olving a et of linear equation. In two uch iple tep, the diffraction efficiencie can be coputed rigorouly without licing the grating into thin layer Optical Society of Aerica OCIS code: , INTRODUCTION Dielectric grating have been ubject to extenive tudie in the pat due to their wide application in holography, pectrocopy, laer, and optoelectronic Nuerou device and yte have been conceived and built with dielectric grating a eential functional part. Aong the, ditributed-feedback laer, optical pectru analyzer, and wavelength diviion deultiplexer are wellknown exaple. In any application, one of the ot eential perforance characteritic of grating i diffraction efficiency, which ha been analyzed by nuerou theoretical technique. During the pat two decade, layering approache 2 6 have gained popularity in nuerical odeling of dielectric grating with arbitrary profile. Generally, thee approache lice a urface-relief grating into a large nuber of thin layer, each of which can be conidered unifor in the thickne direction. The electroagnetic field on the front and back urface of each layer can be eaily related by olving Maxwell equation. The relation between the field on the front and back urface i coonly expreed in certain atrix for, often called tranfer atrice or T atrice. 6 Once the T atrice are calculated for each layer, one can, in principle, ultiply the T atrice equentially to obtain the overall tranfer atrix that relate the front and back urface of the entire grating. Becaue it turned out that uch ultiplication reulted in nuerical intability for a large nuber of layer, a nuber of alternative forulation, 3 6 including the R-atrix or S-atrix approache a reviewed in Ref. 6, were later devied to overcoe the intability iue. Thee advanceent have greatly iproved the accuracy and efficiency of calculating the diffraction efficiencie for dielectric grating with arbitrary profile. In thi paper, we preent a new approach to calculating diffraction efficiency by treating an arbitrary grating a one layer of a photonic crytal, which retain the tructural ignature of the original grating. Thi rigorou approach i atheatically iple and allow u to exploit the analytic Fourier expanion of any coon grating profile. 2. VIRTUAL PHOTONIC CRYSTAL In recent year, photonic crytal reearch ha further broadened our view of dielectric grating by introducing the concept of photonic band and bandgap. The tudy of light refraction on a photonic crytal urface, or the uperpri effect, 11 propted u to reconider certain iue that have not received uch attention in grating diffraction. Thee include the characteritic of light refraction on a ingle photonic crytal urface, which oetie exhibit abnorality or high enitivity to wavelength and incident angle. 11 The high enitivity andate nuerical accuracy beyond popular coputation technique uch a the finite-difference tie-doain ethod. Recently, we developed a rigorou theoretic fraework to odel the refraction, traniion, and propagation inide a dielectric photonic crytal that ha an arbitrary lattice type and arbitrary urface orientation. 12 Our theory doe not need to dicretize a photonic crytal or lice it into any thin layer. Motivated by that work, we have developed a rigorou theory of odeling grating diffraction without licing a grating into thin layer. A dicued in our previou work, 12 our approach i applicable to both TE and TM polarization, although for iplicity we will focu on one polarization here. The key idea of thi work i to conider a grating with one-dienional periodicity a part of a two-dienional (2D) photonic crytal. In Fig. 1 we illutrate thi concept. For the grating hown in Fig. 1(a), the correponding virtual 2D photonic crytal i hown in Fig. 1(b). A unit cell of the photonic crytal i indicated by the dotted line in Fig. 1(b). The grating can be een a a egent of the 2D photonic crytal, having only one period along the y axi. In fact, covering region I and III and looking at region II /06/ /$ Optical Society of Aerica

2 W. Jiang and R. T. Chen Vol. 23, No. 9/Septeber 2006/J. Opt. Soc. A. A 2193 chee that can approxiate the Fourier integral of Eq. (1) with fater convergence, but they generally reort to technique beyond the taircae approxiation that i characteritic of the layering approache. Furtherore, to achieve tability, the tranfer-atrix approach require relatively coplicated atrix operation uch a atrix inverion for iteration through each layer. 6 The overall tranfer atrix of the entire grating depend on the original tranfer atrix of each layer in a coplicated way, which ake evaluation of the error bound of the overall calculation a difficult tak. The ethod to be preented here how a proiing way to avoid thee iue. It eparate the calculation of the 2D Fourier coefficient fro the electroagnetic field calculation; therefore the 2D Fourier coefficient can be evaluated with fater and ore accurate nuerical algorith. For any regularly haped urface-relief grating, it i often poible to analytically calculate the Fourier coefficient. Fig. 1. (Color online) Concept of a virtual photonic crytal, where we treat a grating a a ingle layer of a 2D photonic crytal. (a) A urface-relief grating with an arbitrary profile. The origin of the coordinate yte i located on the farthet extruion of the urface. Three region are arked by I, II, and III. (b) The correponding virtual 2D photonic crytal, where a unit cell i indicated by dotted line. For a rectangular lattice a 1 =, a 2 =d. The layer actually preent in the grating proble i encloed by dahed line. only in Fig. 1(a), one could not tell whether the dielectric tructure between y=0 and y=d i a grating or i a part of a photonic crytal. Either view of the tructure in region II i valid. Therefore, we can convert the grating diffraction proble into a traniion/reflection proble for a onolayer of photonic crytal. Note that except for one onolayer encloed by the dahed line in Fig. 1(b), the other part of the 2D photonic crytal doe not phyically exit. For thi reaon, we call it a virtual photonic crytal. In principle, the unit cell could be an arbitrary parallelogra with it bae parallel to the grating urface (or x axi in the diagra). For iplicity, we chooe a rectangular unit cell throughout thi work. The 2D Fourier coefficient of the pace-dependent dielectric function x of uch a photonic crytal can then be calculated a G = 1 A dxdy exp igx x, where A i the area of a unit cell. The patially varying dielectric contant i given by x = G G exp ig x, where G i a reciprocal lattice vector, and G l, = lb 1 + b 2. Here b 1 and b 2 are the bai vector of the reciprocal lattice of the virtual photonic crytal. For a finite Fourier erie, we aue L l L, M M, and the total nuber of Fourier coponent i N= 2L+1 2M+1. Fro the Fourier analyi viewpoint, licing a grating into thin layer ean that the Fourier coefficient of the grating are eentially calculated by the taircae approxiation of Fourier integral along the y direction. Such approxiated Fourier coefficient converge relatively lowly with the nuber of layer. There are other 1 3. THEORY FOR DIFFRACTION EFFICIENCY Conider the TE polarization (called TM in photonic crytal reearch), for which the agnetic field lie in plane. Generally, Maxwell equation can be converted into a econd-order partial differential equation, which ha a handy for for the 2D TE polarization: 2 E x + 2 x E x =0, where i the circular frequency (aue the peed of light c=1), and E x = exp ik x E G exp ig x, 3 G according to the Bloch theore. Equation (1) can be written a k x + G x 2 + k y + G y 2 E G + 2 G G E G =0. G 4 For the grating diffraction proble, we know the wavelength and the incident angle, fro which the circular frequency and tangential incident wave k x can be eaily obtained. With and k x known, we need to olve Eq. (4) to find the eigenvalue k y, =1,2,...,2N, and the correponding eigenvector E G. The eigenvalue proble can be reforulated into a atrix for: k y 2 I +2k y B + C E =0, where I, B, and C are N-by-N atrice, particularly I i the identity atrix; and E i a N-by-1 colun vector. Such an eigenvalue proble can be eaily converted into an ordinary eigenvalue proble by aigning Z = k y I + B E. The eigenvalue proble now take the expanded for B I 2 5 D B E Z = k y E Z, 6 where D = B 2 C. It i traightforward to how that B i a diagonal atrix in thi proble; therefore the calculation of B 2 i traightforward. Note that one ay a-

3 2194 J. Opt. Soc. A. A/ Vol. 23, No. 9/ Septeber 2006 W. Jiang and R. T. Chen ign a different Z, for exaple, Z =k y E, and obtain a different for of an expanded eigenvalue proble; neverthele the eigenvalue k y and the correponding eigenvector E reain the ae. Note that if I i replaced by an arbitrary atrix A, then an expanded 2N-by-2N eigenvalue proble iilar to Eq. (6) can till be obtained through a tandard approach. 13 However, one ay again define Z in convenient way without following the tandard approach. We chooe the current for of Eq. (6) priarily for atheatical iplicity. For the current proble, one can how that B = G y, C = G y 2 + k x + G x 2 2 G, D = 2 G k x + G x 2, where, =1,2,...,N are linear indice for l, pair. That i, for each pair l, where l= L, L+1,...L 1, L, and = M, M+1,...,M 1,M, a unique nuber (or ) between 1 and N i aigned to it. Moreover, we define G G G. Apparently, there are 2N eigenvalue. Typically, N i around However, a proved in Ref. 12, the eigenvalue are highly degenerate. Only the 2 2L+1 eigenvalue in the firt Brillouin zone (BZ), defined by b 2 /2 Re k y b 2 /2, hould be conidered. A dicued in Ref. 12, thoe eigenvalue outide the firt BZ differ fro thoe inide by b 2, where = M, M+1,...,M. Yet the real-pace eigenfunction E x for thoe eigenvalue outide are identical to thoe in the firt BZ. Thi reveal that the forer are plainly the replica of the latter with the apparent difference of k y attributed to the periodic BZ chee. 12 Thi degeneracy enure that we only need to calculate a all ubet of eigenvalue k y atifying inequality (7), which contribute the nuerical efficiency of thi approach for a large value of M. Thoe eigenode outide the firt BZ have no ue in the ubequent calculation jut like thoe replica eigenode have no ue in the photonic crytal refraction calculation. 12 Furtherore, for ufficiently large L,M (e.g., L,M 4 in ot cae), we oberved that the ajority of the 2 2L+1 eigenvalue in the firt BZ could be approxiated by k y ± i k x + G l0 x 2 0 2, l =±L 0,± L 0 +1,..., ±L, where typically L 0 2. To prove thi, conider an eigenvector whoe larget coponent i E G l, where G l i large enough. Note that for a ufficiently large G l, the firt part of Eq. (4), which i proportional to k+g 2, doinate over the econd part that contain G G. Keeping only the leading ter of the u in Eq. (4) and retricting k y to the firt BZ, one readily obtain approxiation (8). It i traightforward to derive the correponding approxiated eigenvector. With uch good approxiated eigenvalue (uually le than 2% error) a a tarting point, one can nuerically find the accurate 7 8 value of each correponding eigenvalue very fat. For a large value of L, all but a few eigenvalue can be obtained through thi highly expedited technique. It i not abolutely neceary to ue thi technique for the grating calculation preented here that involve only 2D photonic crytal. We developed thi technique priarily for iulating the uperpri effect in a three-dienional photonic crytal, where the coputational workload i ore deanding. The detail of three-dienional photonic crytal coputation will be preented elewhere. Unlike for a ei-infinite photonic crytal, in principle, there i no need to eparate the up and down ode 12 for a grating, which eentially i a photonic crytal with a finite dienion along the y axi. Neither up ode nor down ode diverge under uch circutance and both are phyically preent. Neverthele, for nuerical tability, certain partition of the evanecent ode i helpful a we hall ee. With eigenode of the virtual photonic crytal, the boundary condition can now be olved through a et of linear equation. Conider the grating hown in Fig. 1(a). The incident ediu, the grating, and the half-pace behind the grating are called region I, II, and III, repectively. A planar wave with unity aplitude ipinge upon the grating urface at an angle. Firt, we write down the electric field in three region: where E I x = exp iq 0 x + l r l exp ip l x, E II x = c E G l exp i k x + lb 1 x l, + i k y + b 2 y, E III x = l t l exp iv l x, p l = q 0x + lb 1 e x I 2 q 0x + lb 1 2 e y, v l = q 0x + lb 1 e x + III 2 q 0x + lb 1 2 e y ; and c, r l, and t l repreent the coplex aplitude of the eigenode E x, reflected wave, and tranitted wave, repectively. Note that q 0x k x. Again, we ephaize that only 2 2L+1 eigenode E G whoe eigenvalue k y are in the firt BZ are included in Eq. (9). By cobining the boundary condition for entering and exiting a photonic crytal, one can now derive the coplete boundary condition at the front and back urface of the grating: l,0 + r l = q 0y l,0 + p l,y r l = exp iv l,y d t l = c E G l, c k y + b 2 E G l, c exp ik y d E G l, 9

4 W. Jiang and R. T. Chen Vol. 23, No. 9/Septeber 2006/J. Opt. Soc. A. A 2195 v l,y exp iv l,y d t l = c exp ik y d k y + b 2 E G l. 10 reflection: r l 2 p l,y q 0y, The firt two equation are for the front urface and the lat two are for the back urface. The firt and third equation are baed on the continuity of the electric field, and the econd and fourth are baed on the continuity of the tangential coponent of the agnetic field. Note that if we had not choen a rectangular unit cell (i.e., a 2 i not parallel to the y axi, but a 1 ut alway be parallel to the grating urface or the x axi), b 1 ay not be parallel to the x axi. Then it can be eaily hown that the ter b 2 in Eq. (9) and (10) would becoe lb 1y +b 2. There hould be a factor exp i lb 1y +b 2 d for each ter on the right-hand ide of the lat two equation of Eq. (10). However, for a rectangular cell, it i traightforward to how that exp i lb 1y +b 2 d =1 becaue b 2 =2 /d for a grating and b 1y =0 by chooing a rectangular cell. Becaue coplex k y eigenvalue appear in conjugate pair, the coefficient containing exp ik y d for I k y 0 for a large d ay doinate over other ter in Eq. (10). 14 To avoid thi proble, we define c = c exp ik y d I k y 0 c otherwie. 11 Alo, we define t l=t l exp i l,y d. Now Eq. (10) becoe l,0 + r l = + q 0y l,0 + p l,y r l = + c + E + G l + c exp ik y d + E G l, c + k y + + b 2 E + G l c exp ik y d k y + b 2 E G l, t l 2 v l,y traniion:. q 0y A entioned above, for coon grating profile, the Fourier coponent G can be calculated analytically. In fact, the Fourier coponent for a inuoidal profile are found to have the for 2 G l = 1 I + III l, III I l,1 + l, 1 =0 III I 2 i l,0 + 1 i l 1 J l 0, 13 where J l x i the lth-order Beel function of the firt kind. The diffraction efficiencie for a inuoidal grating are hown in Fig. 2 for varying groove depth. The croe repreent the publihed reult fro Fig. 4 of Ref. 2 (data obtained by coputer oftware fro the canned iage of the original plot). Our reult are in excellent agreeent with thoe of Ref. 2. The Fourier coponent of a awtooth profile are found to have the for 2 G l = 1 I + III l = =0 III I / 2 il l 0, =0. III I 2 i l,0 l+, The diffraction efficiencie for a awtooth grating are hown in Fig. 3. Again, we copare our reult with thoe obtained by reading the canned plot of Fig. 7 in Ref. 2, and the agreeent i excellent. t l = + c + exp ik y + d E + G l + c E G l, v l,y t l = + c + exp ik y + d k y + + b 2 E + G l + c k y + b 2 E G l, 12 where refer to thoe ode with I k y 0, wherea + refer to other ode. Thi new for enure the nuerical tability when olving the linear equation. For each diffraction order with real p l, v l, the diffraction efficiency i given by noralized y coponent of Poynting vector that eaure the power-plitting ratio Fig. 2. (Color online) Diffraction efficiencie for a inuoidal grating, where I =1, III =2.5, =, and =30. One reflection order R 0 and three traniion order T 1,T 0,T 1 are hown. The croe repreent the publihed reult fro Fig. 4 of Ref. 2. Our reult are in excellent agreeent with thoe of Ref. 2.

5 2196 J. Opt. Soc. A. A/ Vol. 23, No. 9/ Septeber 2006 W. Jiang and R. T. Chen Fig. 3. (Color online) Diffraction efficiencie for a awtooth grating, where I =1, III =2.5, =, and =30. One reflection order and three traniion order are hown. The croe repreent the data fro Fig. 7 of Ref. 2. The agreeent i excellent. 4. DISCUSSION Generally, the R-or S-atrix approache require a erie of coplicated atrix operation. Our ethod require only two atrix operation: olving eigenvalue and then olving a linear equation. In thi way, a long a we have a good eigenvalue olver and a good linear equation olver, there i no need for pecialized, coplicated algorith for nuerical tability. A uch, reearcher could focu ore on the optic than on coputational detail. Note that good eigenvalue olver and linear equation olver are available fro open ource uch a ARPACK and LAPACK. Our approach eparate the calculation of Fourier coefficient of the grating fro the diffraction efficiency calculation. Therefore, the accuracy of the Fourier integral in Eq. (1) doe not depend on the ethod we ued to calculate the diffraction efficiencie. In the layering approache, the nuber of layer required for the calculation of the diffraction efficiency eentially deterine the accuracy of the iplicit Fourier integral. Furtherore, the evaluation of error bound for our ethod would be uch ipler than for the conventional layering approache becaue our ethod ha only two tep. In contrat, it i highly coplicated and therefore ipractical to evaluate the overall error bound of the R-and S-atrix approache that involve a long erie of repeated atrix operation, including atrix ultiplication and inverion. Developing a detailed procedure of rigorouly etiating the error bound for thi ethod could be cloely related to the perturbation technique to be dicued below, reulting in an intereting direction for further tudy. We would like to point out one intereting correpondence between our ethod and the layering approach. In our ethod, one eek the eigenvalue k y of a atrix V, wherea the layering involve a atrix exp i V d that connect the electroagnetic field on the front and back urface. If the original eigenvalue of V have coparable agnitude, the eigenvalue of exp i V d would generally ditribute over a uch larger nuerical range a ot eigenvalue of V are coplex. Furtherore, it could be conductive to incorporate our ethod into the R- or S-atrix approach. With our ethod, one can now divide a deep grating into a all nuber (e.g., three or four) of relatively thick layer. Then one ay ue the R- or S-atrix approach recurively through thee few layer to obtain the overall atrix for the entire grating. The traniion through each thick layer i rigorouly coputed with our new ethod. Thi differ fro the conventional layering ethod in that each layer doe not have to be very thin (thi wa required for the taircae approxiation in the conventional ethod). Thi certainly offer flexibility and advantage unavailable prior to thi work. For extreely deep grating, uch a ethod of cobining the current ethod with an R-atrix or S-atrix approach ay be epecially ueful. It would be an intereting proble to tudy how to chooe the axiu layer thickne or the iniu nuber of layer to optiize the nuerical efficiency againt the accuracy for an extreely deep grating. Meanwhile, follow-up work could help to evaluate whether perturbation technique can becoe ore effective uing thi theory a a tarting point. In oe cae, with the analytic forula of G, it ight be poible to analytically carry out the proble olution to a deeper level with coon atrix perturbation theorie 15 or,ina for ore failiar to ot phyicit, quantu perturbation theory. 16 Thee effort oetie ay give conducive, intuitive analytic forula of diffraction efficiencie that can be ued for the evaluation of trend or further analytical tudie. In thi regard, our approach ay have a unique value a well. A we entioned earlier, the unit cell could be an arbitrary parallelogra with it bae parallel to the grating urface. If the grating profile ha it intrinic geoetric boundarie following the lanting ide of a parallelogra, then a nonrectangular unit cell ay be beneficial. Note that our ethod can handle thi type of proble a well, but the right-hand ide of Eq. (10) or (12) ut be odified with b 1y explicitly preent a dicued earlier. It i poible to chooe an alternative tructure for the virtual photonic crytal a hown in Fig. 4(b). Correpondingly, one need to olve the boundary condition at Fig. 4. (Color online) Alternative choice of the virtual photonic crytal. (a) Region II ha to be redefined (now between y=0 and y=a 2 ). And the backide boundary i now located at y=a 2. (b) The correponding 2D photonic crytal. A unit cell i indicated by dotted line. The layer actually preent in the grating proble i encloed by dahed line.

6 W. Jiang and R. T. Chen Vol. 23, No. 9/Septeber 2006/ J. Opt. Soc. A. A 2197 Table 1. Coparion of Diffraction Efficiencie with Thoe in Ref. 2 Diffraction Order a / Ref. 2 y=a 2, rather than at y=d, a hown in Fig. 4(a). Any proper choice of virtual photonic crytal will not affect the calculated value of diffraction efficiency jut a any proper choice of contour doe not affect the value of a coplex contour integral. However, any choice with a 2 d would generally require extra Fourier coponent to repreent the enlarged unit cell. Thi unnecearily increae the coputational workload and coplicate the calculation, and therefore it i not ued. In Table 1 we copare our nuerical reult with thoe of Ref. 2 for a inuoidal urface-relief grating with I =1.0, III =2.3104, and d/ = The incident angle i 36. Our reult are in excellent agreeent with the reult lited in Table 3 of the widely cited work by Mohara and Gaylord. 2 All reult in the preent work are obtained with L, M in the range of CONCLUSION Preent Work DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, DE 3, a DE 3,i i the ith-order diffraction efficiency in ediu III. In uary, a rigorou theory of grating diffraction i developed baed on the concept of virtual photonic crytal. Our ethod eparate the calculation of the Fourier coefficient of a grating fro the diffraction efficiency calculation, and analytical for of Fourier coefficient can be ued for any coon urface-relief profile. In two iple tep, diffraction efficiencie can be calculated fro the 2D Fourier erie of the dielectric function without licing the grating profile into thin layer. Direction for further tudy have been dicued. ACKNOWLEDGMENTS The author are indebted to Lifeng Li of Tinghua Univerity, Beijing, China, for extenive helpful dicuion. We thank J. Chen for reading the paper and providing coent. Thi work i upported in part by the U.S. Air Force Reearch Laboratory. The author alo acknowledge partial upport fro the U.S. Air Force Office of Scientific Reearch. Author W. Jiang can be reached by e-ail at jiang@ece.utexa.edu. REFERENCES 1. H. Kogelnik, Coupled wave theory for thick hologra grating, Bell Syt. Tech. J. 48, (1969). 2. M. G. Mohara and T. K. Gaylord, Diffraction analyi of dielectric urface-relief grating, J. Opt. Soc. A. 72, (1982). 3. L. Li, Multilayer odal ethod for diffraction grating of arbitrary profile, depth, and perittivity, J. Opt. Soc. A. A 10, (1993). 4. N. Chateau and J. P. Hugonin, Algorith for the rigorou coupled-wave analyi of grating diffraction, J. Opt. Soc. A. A 11, (1994). 5. M. G. Mohara, D. A. Poet, E. B. Grann, and T. K. Gaylord, Stable ipleentation of the rigorou coupledwave analyi for urface-relief grating: enhanced tranittance atrix approach, J. Opt. Soc. A. A 12, (1995). 6. L. Li, Forulation and coparion of two recurive atrix algorith for odeling layered diffraction grating, J. Opt. Soc. A. A 13, (1996). 7. R. T. Chen, H. Lu, D. Robinon, and T. Jannon, Highly ultiplexed graded-index polyer waveguide hologra for near-infrared eight-channel wavelength diviion deultiplexing, Appl. Phy. Lett. 59, (1991). 8. M. R. Wang, G. J. Sonek, R. T. Chen, and T. Jannon, Large fanout optical interconnect uing thick holographic grating and ubtrate wave propagation, Appl. Opt. 31, (1992). 9. L. Gu, X. Chen, Z. Shi, B. Howley, J. Liu, and R. T. Chen, Bandwidth-enhanced volue grating for dene wavelength-diviion ultiplexer uing a phaecopenation chee, Appl. Phy. Lett. 86, (2005). 10. L. Gu, X. Chen, W. Jiang, and R. T. Chen, A olution to the fringing-field effect in liquid crytal baed high-reolution witchable grating, Appl. Phy. Lett. 87, (2005). 11. H. Koaka, T. Kawahia, A. Toita, M. Notoi, T. Taaura, T. Sato, and S. Kawakai, Superpri phenoena in photonic crytal, Phy. Rev. B 58, R10096 (1998). 12. W. Jiang, R. T. Chen, and X. Lu, Theory of light refraction at the urface of a photonic crytal, Phy. Rev. B 71, (2005). 13. P. Lancater, Labda-Matrice and Vibrating Syte (Pergaon, 1966). 14. W. Jiang, Wavelength-elective icro- and nano-photonic device for wavelength diviion ultiplexing network, Ph.D. diertation (Univerity of Texa at Autin, 2005). 15. J. H. Wilkinon, The Algebraic Eigenvalue Proble (Clarendon, 1965). 16. L. I. Schiff, Quantu Mechanic, 3rd ed. (McGraw-Hill, 1968).