Analysis of Photonic Crystal Waveguide Discontinuities. Using the Mode Matching Method and Application to Device. Performance Evaluation

Transcription

1 Analysis of Photonic Crystal Waveguide Discontinuities Using the Mode Matching Method and Application to Device Performance Evaluation Athanasios Theocharidis, Thomas Kamalakis and Thomas Sphicopoulos The authors are with the Optical Communication Laboratory of the National and Kapodistrian University of Athens, Panepistimiopolis Ilyssia GR15784, Athens Greece This work was supported by the PENED2003 program of the Greek Secretariat for Research and Technology. In this paper, the application of the Mode Matching (MM) method in the case of photonic crystal waveguide discontinuities is presented. The structure under consideration is divided into a number of cells and the modes of each cell are calculated by an alternative formulation of the Plane Wave Expansion (PWE) method. This formulation allows the calculation of both guided and evanescent modes at a given frequency. A matrix equation is then formed relating the modal amplitudes at the beginning and the end of the structure. The accuracy of the MM method is compared to the Finite Difference Frequency Domain (FDFD) and the Finite Difference Time Domain (FDTD) and good agreement is observed. The MM method requires much fewer resources than the FDFD and the FDTD methods while providing a useful physical insight to the calculation of the frequency response of waveguide discontinuities. The method is also applied to the calculation of the power loss, due to structural fabrication-induced variations. 1

2 Remark 1 Rev. #1. Copyright OCIS codes : Optical communications, Optical devices, Photonic crystal Waveguides 1. INTRODUCTION Photonic Crystals (PCs) [1],[2] are constantly attracting increased attention as a potential solution for the realization of ultra-compact integrated optical circuits. The strong confinement of light in a PC waveguide (PCW) allows the design of sharp waveguide bends in which light can change direction 90 o without significant power losses [3]. This is in contrast to conventional low index-contrast integrated optical components, in which the bending radii must be kept rather large (in order to limit the bending losses). Large bending radii may increase the overall size of the integrated circuit such as the Arrayed Waveguide Grating [4]. PC-based devices can perform many photonic functionalities such as light generation [5] and processing [6]. Various designs have also been demonstrated for optical filtering [7]-[11]. Many of the aforementioned designs are based on the introduction of discontinuities (defects) inside a PCW. Coupled optical cavities (COCs) [12] are also receiving attention for optical telecommunication applications. Systems consisting of a few coupled cavities have been proposed both for filtering and modulation applications [13],[14]. In order to implement compact COCs one can use coupled PC defect cavities. A large chain of COCs can be thought of as a novel type of waveguide where light propagates through coupling by hopping from cavity to cavity. This new type of waveguide is called the Coupled Resonator Optical Waveguide (CROW) [15] and has many interesting properties. Another type of device based on COCs, the Side Coupled Integrated Spaced Sequence of Resonators (SCISSOR) [16], is also receiving 2

3 attention. The SCISSOR consists of a number of COCs, side-coupled to a waveguide. Light propagates with small group velocity inside a SCISSOR and nonlinear effects are enhanced. By cascading a CROW and a SCISSOR it is possible to compensate third-order dispersion effects occurring in each separate device and significantly increase the bandwidth on which slow light propagation can take place [17]. Finite Difference Time Domain (FDTD) [18] has proved successful for the electromagnetic simulation of many PC-based devices. However as the size of the device increases, FDTD may require increased memory resources and computational time which can become prohibitive in large structures. Especially in the case of a PCW, long Perfectly Matched Layer (PML) sections must be used in the input and output of the device in order to prevent reflections [19]. In addition, the size of the grid can pose restrictions in modelling small dimension fluctuations due to fabrication imperfections. On the other hand the Finite Difference Frequency Domain (FDFD) method [20] could be modified in order to account for small geometry perturbations [21] but requires prohibitively large memory resources in order to solve a practical PC problem. In this paper, we demonstrate the effectiveness of a method based on Plane Wave Expansion (PWE) [2] and Mode Matching (MM) [22],[23] in the analysis of PCW discontinuities, Remark2 Rev. #1. such as the ones encountered in PC-based filters and SCISSORs. In order to apply the MM technique, the modes corresponding to a given frequency ω must be calculated including the evanescent modes with complex propagation constants β. By applying the PWE to the wave equation [2], one may determine the various values of ω corresponding to a given β. However, in contrast to conventional, constant cross-section waveguides, where β for the evanescent modes Remark 1 Rev. #1. lie on the imaginary axis, in PCWs β may lie on the entire complex plane. To avoid sweeping the entire complex plane, an alternative formulation of the PWE is used for the first time, allowing 3

4 the determination of the propagation constant and the distribution of the guided and the evanescent modes at a given frequency. It is shown that the MM method can provide accurate results without requiring significant memory resources and computational time. In the framework of the MM method, whenever a discontinuity is encountered inside a waveguide, we attempt to match the field expressed in terms of the waveguide modes to the modal fields of the discontinuity. This allows the computation of the reflection and transmission coefficients of each guided waveguide mode. In this way, the MM method provides a useful physical insight to the problem. Furthermore, since in most large device designs distinct cell types of discontinuities are encountered, one needs to calculate the modal fields only once for each type of cell. This can significantly speed up the computation process. The method is also applied to the study of fabrication induced disorder by calculating the performance degradation of a PCW in terms of the scattering loss and it is shown that MM can handle small perturbations without excessive Remark 1,3 Rev. #1. computational time requirements. The rest of the paper is organized as follows: in section 2, a method for computing the guided and evanescent modal fields of both the PCW and the discontinuity cell at a given frequency ω, is presented. The method is based on the formulation of Maxwell s equations at a given ω as a generalized Hermitian eigenproblem, as in Ref.[24]. In section 3, the MM method is formulated in the case of PCW discontinuities. A simplified version based on 2 2 Transfer Matrices (TrM) is presented in section 4 for the case where the consecutive discontinuities are spaced far apart. In section 5, the results of the MM method results are compared with the FDFD and FDTD methods and applied to study of SCISSORs and the performance degradation in a PCW due to structural perturbations. 2. CALCULATION OF THE MODES 4

5 In order to implement the MM method, one first needs to estimate the propagation constants β and the modal fields of the various cells of the structure under consideration. This can be achieved by solving the wave equation numerically. For example, the field can be written as a sum of plane waves [2] and then the wave equation is transformed to an eigenproblem allowing the estimation of the frequencies ω=ω(β) that correspond to a given propagation constant β. Given the fact that β will lie on the entire complex plane, it is preferable to solve the inverse problem: given a frequency ω one must determine the values of β=β(ω) corresponding to this frequency. In this section, a novel plane wave expansion formulation for determining the evanescent and guided mode properties of a periodic structure at a given ω is outlined based on the formulation of the source-free Maxwell s equations in terms of a generalized Hermitian eigenproblem as in Ref. [24],[25]. This method will be used in order to calculate the modal fields required in order to implement the MM method illustrated in the next section. Using Bloch s theorem, the modes of a periodic dielectric structure along the z-direction can be written (Ref. [2]) Er () = ur () e jβ z (1) Hr () = vr () e jβ z (2) where β is the propagation constant of the mode and u,v are periodic functions along the z direction. Defining Ψ β > to be a four component vector comprising of the tangential parts u t and Remark 4 Rev. #1. v t of u and v respectively, i.e. T ( t t) ( ux uy vx vy) T Ψ >= u, v =,,, (3) β one can write Maxwell s equations in the following form (Ref. [26]) ˆ ˆ A j B Bˆ + Ψ β >= β Ψ β > z (4) 5

6 where the operators  and ˆB are defined by 1 1 ωε t t 0 Aˆ ω μ = ωμ t t ω ε (5) and ˆ 0 z B = z 0 (6) In (5), ε and μ are the dielectric constant and the magnetic permeability of the structure. The eigenvalues of the eigenproblem in (4) can be used to determine the propagation constants of both evanescent and guided modes of the structures while the eigenvectors determine their modal fields. In order to solve (5) one can expand the periodic four component vector in terms of plane and standing waves as Gmxx Gny y jglz z Ψ β >= BG ( mnl )sin sin e (7) mnl,, 2 2 where B(G mnl ) are the Fourier coefficients of Ψ β > and we have assumed that the periodic cell is rectangular. In this case the reciprocal lattice vectors G mnl =[G mx,g ny,g lz ] have components G mx =2πm/b, G ny =2πn/d, G lz =2πl/a where m,n,l are integers with 0 m Nx, 0 n Ny, Remark 5 Rev # 1. N l N and b,d,a are the sizes of the cell along the x,y and z direction, respectively. The z z total number of terms in the expansion of (7) is therefore (N x +1)(N y +1)(2N z +1). Remark 8 Rev # 1. Note that the fields in (7) vanish at the edges of the cell and hence the structure can be thought as being enclosed by perfectly conducting walls. As in the case of the dielectric slab waveguide (Ref. [27]), b and d must be taken infinite but as the walls move further and further apart from the waveguide center, the guided modes of the waveguide remain practically the same 6

7 while more evanescent modes tend to appear having their field primarily outside the core of the PCW. This means that for discontinuities near the core these extra evanescent modes will not be significantly excited and hence will not affect the transmission and reflection of the guided Remark 2 Rev # 1. modes. In practice b and d are assumed finite and their value must be taken such that the guided modes of the structure decay significantly near the perfectly conducting walls of the cell. Substituting (7) in (4), the operator eigenproblem is transformed to a matrix eigenproblem which can be solved using standard techniques. For a 2D PCW, where ε does not change with y, the eigenproblem is further simplified in the Transverse Magnetic (TM y ) case, since one needs to consider only one y-directed electric and one x-directed magnetic field tangential components which we will designate as u y and v x. In this case the fields do not depend on y and hence the reciprocal lattice vectors G mnl are such that G mnl = G ml =[G mx,0,g lz ] and the matrix eigenproblem is written as where the vector MV = β V (8) ( V,..., V, U,..., U ) 1 N 1 N T V = (9) comprises of all the spectral components V 1,,V N and U 1,,U N of v x and u y respectively (note that a finite number N of spectral components must be assumed for computational purposes). The (2N) (2N) square matrix M is given by M MG Mε = Mω M G (10) where M G is a N N diagonal matrix whose diagonal [M G ] ll elements are given by [M G ] ll =G lz while M ω is another N N diagonal matrix with [M ω ] ll =ωμ. The elements [M ε ] pq of the N N matrix M ε are given by 7

8 4 2 1 k G ( k ) qx M ε = ω 1 ( pq) pq % ε G δ pq (11) 2 ωμ [ ] ( ) In equation (11), % ε ( G) is the Fourier transform of the dielectric constant, k = 1 1 d e j Gr S ( G) rε( r) % ε = S (12) S being the surface (in the 2D case) or volume (in the 3D case) of the basic cell of the structure. In (11), p and q are integers with 0 p N x, Nz q Nz, k is also an integer with 1 k 4, Remark 7 Rev. 1. δ pq is Kronecker's delta and the vectors G, are defined by ( k ) pq G = ( G + G, G G ) T (13) (1) pq px qx pz qz G = ( G G, G G ) T (14) (2) pq px qx pz qz G = ( G G, G G ) T (15) (3) pq px qx pz qz G = ( G + G, G G ) T (16) (4) pq px qx pz qz In many cases the Fourier transform of the dielectric constant can be calculated in closed form. For example, if as shown in figure 1 the cell of the structure is comprised of a set of N r rods, each centered at p n =(x n,z n ) and each with radius r n, then generalizing the result of Ref. [2], % ε ( G) is given by J ( ) ( 1 Gr ) n jgp n 2 εa εb fn e, G 0 % ε n ( G) = Grn (17) ( εa εb) fn + εb, G = 0 n In (17), f n =πr n 2 /S is the filling factor for each rod, G= G, ε a and ε b is the dielectric constant of the rods and the background medium, respectively. Figure 2, illustrates examples of the modal field intensities u y 2 calculated by solving the eigenproblem in (8) for a PCW assuming ε a =9ε 0, ε b =ε 0, a=0.6μm, b=9a, while the radii r n of the 8

9 all the rods are taken equal r n =r a =0.12μm. Figure 2(a) corresponds to β=2.58μm -1, figure 2(b) to β=( j)μm -1 and figure 2(c) to β=3.12j μm -1. Note that figure 2(b) corresponds to an evanescent mode whose propagation constant has both imaginary and real parts and is the mode with the smallest dumping constant Im{β}. The third field in figure 2(c), corresponds to the mode with a purely imaginary β that has the smallest dumping constant of all purely imaginary β modes. 3. MODE MATCHING EQUATIONS Since the modes of the structure can be calculated, one can proceed to apply the MM technique. In this section the equations related to the above technique are derived. Figure 3 depicts the general situation where a sequence of N cells containing dielectric rods is considered. The field must satisfy the continuity equations, i.e. the tangential fields at the left of a boundary must equal the tangential fields at the right of the boundary. At the i th cell the tangential electric and magnetic fields are written as: i ( ) β ( ) ( i) ( ) i () i () i jβm z zi 1 () i () i j m z zi t = am tme + b m tm e m m E e e (18) where e, () i tm a and () i m β are the tangential electric Bloch functions, the coefficients and the () i m propagation constants of the m th forward mode of the i th cell respectively, while e and () i tm b are () i m the tangential electric Bloch functions propagation constants of the m th backward mode of the i th cell respectively and i ( ) β ( ) ( i) ( ) i () i () i jβm z zi 1 () i () i j m z zi t = am tme + b m tm e m m H h h (19) where h () i tm, and h are the tangential magnetic Bloch functions propagation constants of the m th () i tm forward mode and the m th backward mode of the i th cell respectively. 9

10 At each interface between two cells, the tangential fields must be continuous at the boundary z=z i (Ref. [28]). This implies: E = E (20) i i+ 1 t( zi) t ( zi) i i+ 1 t( zi) = t ( zi) H H (21) Using (20) and (21), one can derive a linear system of equations relating the modal amplitudes of cell i to the modal amplitudes of cell (i+1). Defining the product f, g = f g dv (22) V and projecting (20) along h and (21) along e () i, for 1 n M, one obtains a matrix equation ( i+ 1) tn tn relating the mode coefficients in cells i and i+1: i+ 1 i A A Z i+ 1 = i i B B (23) where vectors A i =[a 1 i,,a M i ] T, B i =[b 1 i,b M i ] T contain the coefficients of the M forward and M backward modes of the i th cell. The matrix Z i is given by Z = Y X (24) 1 i i i where the element of the matrices Y i and X i are given by [ X ] ( i ) () i ( i+ 1) jβm a etm, htn e 1 mn, M () i ( i+ 1) e tm, htn 1 m M, n M = 1 mn, M M () i () i M + 1 mn, 2M htm, etn i nm ( i ) () i () i jβm a htm, etn e (25) and 10

11 [ Y ] ( i+ 1) ( i+ 1) etm, htn 1 mn, M ( i+ 1) ( i+ 1) ( i+ 1) jβm a e tm, htn e 1 m M, n M = 1 mn, M M ( i+ 1) ( i+ 1) ( i) jβ 1, 2 m a M + mn M htm, etn e i nm ( i+ 1) ( i) htm, etn (26) If the structure consists of many cells, one can relate the modal amplitudes at its input to the modal amplitudes of its output using the transfer matrix properties leading to the following equation: N 1 A A = Z N 1 B B (27) and the matrix Z is given by Z = Z... Z (28) N 1 1 To examine the transmission and reflection properties of the structure one can set all the output backward modes equal to zero and assume that only the guided modes are excited at the input. In this case one obtains B = Z Z A (29) N 1 1 A = Z11A + Z12B (30) where the M M submatrices of Z are determined by Z Z Z = Z21 Z 22 (31) To summarize, once the propagation constants and the mode distribution are calculated by (8) the transmission and reflection properties of a structure can be calculated by dividing the structure into N sections and calculating the matrices Z i at each boundary. One can then obtain 11

12 the Z matrix using (28) and calculate the amplitudes of the coefficients of the backward modes at the input using (29). The modal amplitudes of the forward modes at the device output are given by (30). If a single forward guided mode exists (say m=1) in the 1 st cell, one sets A 1 =(1,0,,0) T and the power transmission coefficients T and R of the guided mode are given by T= a N 1 2 / a and R= b / a Referring to figure 3, note that at the beginning of the device at z=z 0, one can assume that the PCW cells extend infinitely from z=z 0 to z=-, and hence no mode conversion takes place before the first cell (i=1). Similarly and since the backward PCW modes at the last cell (i=n) equal to zero, no reflection will occur at the end of the structure. Hence no absorber cells are required at the edge of the structure unlike the FDTD and the FDFD method TRANSFER MATRIX FORMULATION A further simplification is possible whenever the discontinuities inside the PCW are spaced far apart as in figure 4. If there are many waveguide cells between the discontinuities, then the evanescent modes excited at the first discontinuity will decay significantly before they reach the second discontinuity and will not play an important role in the results, while the guided mode will simply undergo a phase shift. This means that one can calculate the transmission and reflection properties of the entire structure by only considering the two smaller structures A and B, shown in the figure. Applying the mode matching method to each of the structures one can calculate the 2 2 matrices Z A and Z B that relate the amplitudes of the forward and backward modes at the input and output of each structure. Using these matrices one can calculate a 2 2 matrix corresponding to the entire structure using Z = Z D Z (32) A B where D is given by 12

13 jβg L e 0 D = jβg L 0 e (33) and accounts for the phase shift experienced by the forward and backward modes at the waveguide layers (having total length equal to L) between the two structures. In (33), β g is the propagation constant of the guided mode of the waveguide. This procedure can be easily generalized in the case where the waveguide supports two or more guided modes. As will be shown in the next section, this simplification is quite accurate and can be used as an alternative in cases where the computation of the inverse matrix of Z 22 in (29) requires increased numerical accuracy. This can occur in large devices where the evanescent modes with large dumping constants may result in large fluctuations of the elements of Z and hence the computation of Z must be carried out with increased precision. 5. RESULTS AND DISCUSSION In this section the mode matching method is illustrated by applying it to some example structures. Its accuracy is verified by comparing it both with the FDFD and the FDTD methods. A. COMPARISON WITH FDFD To compare the results of the MM method with the FDFD method, a sequence of 1,2 and 3 defect rods with radius r d is placed inside a PC waveguide. Figures 5(a)-5(c) depict the power reflection coefficients calculated with the FDFD (dots) and the MM method (solid lines). The radius of the rods of the PCW was taken r a =0.12μm, while the lattice constant was a=0.6μm. The wavelength in free space was taken λ=1.55μm. The dielectric constant of the rods was assumed ε a =9ε 0 and that of the surrounding medium was ε b =ε 0. The radius of the defects r d varied from 0.3r a to 2.0r a. For the calculation of the modes the number of plane waves used was 15 for the Remark 9 Rev #1. 13

14 propagation direction (z-direction) while 19 standing waves were used for the transverse direction (x-direction). The grid of the FDFD was taken Δ G =r a /8 in order to account for the small variations in the size of the defect rods and 10 PML rods were used along the z-direction in both sides, necessary in order to minimize reflections from the edge of the computational window [19]. Note that the FDFD required more than 1GB of RAM in order to solve its system of equations. On the other hand no serious memory requirements were imposed by the MM method. Both FDFD and the combination of the PWE and MM methods required roughly the same amount of time to produce their results. In the application of the MM method, the time required is primary Remark 3 Rev #1. determined by the PWE calculations for each cell. As observed in figure 5, there is a very good agreement between the two methods in terms of the power reflection coefficient and this verifies the accuracy of the MM method. A similar agreement is obtained when the position of the defect rods is changed. Table I, shows the values for the power reflection coefficient, calculated with both methods, assuming a single defect rod (as in fig. 5(a)) with r d =r a whose position changes ±r a in either the x or the z direction. Note that the MM method computes practically the same values for R when the rods are displaced ±r a along the x-direction and this is not surprising since the structure is symmetric along this direction. The same is true for the z-direction as well. B. CONVERGENCE OF THE MM METHOD The accuracy of the MM method greatly depends on the number of evanescent modes taken into account and on the accuracy of the computed modal fields and propagation constants. If the plane wave expansion method is used for the modal calculations, then there are three parameters that primarily determine the accuracy of the method: the number of plane and standing waves determined by N z, N x along the z-direction and x-directions and the size of the cell b. Since the structure is periodic, the propagation constants of the modes can be grouped into 14

15 a number of zones [2π(p-½)/a, 2π(p+½)/a] where p is an integer. According to Bloch s theorem [2] one can consider the modes lying in the first zone (p=0). However, the mode solver may compute values for β that may lie on other zones as well and the number of these zones is primarily dependent on N z. On the other hand the value of N x determines how many modes are actually computed inside each zone. Figure 6(a) depicts the values of the power reflection coefficient computed for the structure of figure 5(a) for N z =5 and various values of N x. The structure supports a single guided mode and the number of evanescent modes in the first zone is N e N x. As N e is increased, R begins to converge to 0.76 and for N e >14, R is practically constant. Note that for small values of N e the value of R changes whenever N e is increased by 2 modes. This is not surprising because as N e increases, new evanescent modes appear possessing either odd or even symmetry. Since the guided mode has even symmetry, only the even evanescent modes are excited so the result will change only if even modes are included. Moving from N e =7 to N e =8 produces one extra odd mode that does not affect the result. This explains the step-like behavior observed in the figure. On the other hand, as seen in figure 6(b) the value of N z has a less critical role since the guided modal fields and propagation constants are accurately estimated even with N z =3 (implying 2N z +1=7 plane waves in the z-direction). In the figure legend N rods refers to the number of rods positioned both above and below the waveguide core. Figure 6(b) indicates that the size of the cell in the x-direction does not alter the results significantly since for N rods =3,4,5 the values of R are not very different. This is because the incident guided mode of the waveguide is tightly confined inside the core as seen in figure 2(a). C. SCISSOR ANALYSIS 15

16 It is also interesting to demonstrate the accuracy of the MM method in more complex structures as well. Figure 7(a) depicts a SCISSOR comprising of four PC defect cavities coupled with a PC waveguide. The cavities are spaced one rod apart and there is one rod spacing between the cavities and the waveguide. The rest of the PC parameters (lattice constant a, rod radius r a, etc) are the same as in section 5.A. As shown in figure 7(a), in the context of the MM method, the SCISSOR is broken down to 9 cells, 5 of which are ordinary single mode PCW and the rest are two-mode PCWs. This device required prohibitively large memory in order to be solved by FDFD ( 2GB) and therefore FDTD was employed to verify the results of the MM method. The FDTD simulation required about 2 days to complete (on a 3GHz Pentium desktop computer) for a grid Δ G =r a /8. The parameters for the MM method simulation were N x =59 and N z =7. In figure 7(b), a comparison between the results of the MM (solid line) and the FDTD methods (dots) is being presented. It is shown that both methods agree very well. The FDTD required the use of 10 additional single PCW PML cells on each side to prevent reflections. The computational time was significantly less for the MM method (0.5h for each frequency value). Due to the device periodicity the modal fields need to be calculated only once for each type of cell. D. APPLICATION IN THE STUDY OF FABRICATION IMPERFECTIONS In this sub-section the MM method will be applied in the calculation of optical losses due to scattering at fabrication imperfections in a PCW. Towards this end a number of PCW cells will Remark 1 Rev #1. be considered having the centres ( z, x ) ( z z, x x ) respect to the centres (, ) = +Δ +Δ of the rods slightly displaced with i i i i i i zi x i of the rods of the ideal PCW and their radius r i r a r i = +Δ perturbed as shown in figure 3. For simplicity, the perturbations Δz i, Δx i and Δr i are independently selected 16

17 from the samples of a uniform distribution inside [-Δ, +Δ]. Table 2 quotes the mean power loss (expressed in db/mm) due to scattering obtained assuming Δ=1nm, 3nm and 5nm considering 100 perturbed PCWs of 10 cells length for each case. For these simulations, the values of N z and N x where taken 9 and 19 respectively and about 9 hours were required for the completion of 100 runs (including both PWE calculations for the cells and the application of the MM method). Note that this computation time is short compared to FDTD which requires a very fine mesh for the treatment of such small perturbations (for Δ=1nm, Δ/r a is less than 1%). It is deduced that although small deviations of 1nm do not introduce significant losses, the losses increase significantly for Δ=5nm exceeding 1dB/mm in this case. Table 2 also quotes the standard deviation (StD) of the power loss in each case, which is comparable to the mean value implying a spread of the loss values of the samples. This is illustrated in figure 8 where a bar plot of the power losses of the samples is given and it is deduced that although for the majority of the samples the loss is close to the mean value, there are some samples with significantly higher loss. A similar behavior was observed in the study of fabrication induced imperfections in a PC coupler using couple mode theory [29] and may be attributed to the fact that for these samples, the larger rod perturbations happen to occur near the waveguide core and hence their influence in the propagation of the guided mode is more pronounced. E. ANALYSIS USING 2 2 TRANSFER MATRICES In this section, the results obtained with the MM and the 2 2 transfer matrices are compared for the structure depicted in figure 9(a). This structure is formed by placing single rod and double rod type discontinuities spaced 6a apart inside a PCW. One can analyze the structure as explained in section 4 by assuming only the propagation of the guided modes between the two discontinuities. In figure 9(b) the results obtained by conventional MM and the 2 2 Transfer 17

18 Matrices (TrM) are compared and it is shown that the latter method is quite accurate in predicting the shape of the power transfer function T of the entire structure. The parameters used in this simulation were N x =11 and N z =5, while the rest of the PC lattice parameters are identical to those used in the previous sections. It is therefore deduced that the simplified 2 2 TrM can be used in order to determine the filtering characteristics of a large device, provided the discontinuities are spaced sufficiently far apart. 6. CONCLUSIONS The mode matching method has been applied in the study of PC-based waveguide discontinuities. The method is based in the expansion of the field in terms of the eigenmodes of the cells of the structure and their matching at the boundary interfaces. At a given frequency the modes are calculated by an alternative formulation of the plane wave expansion method. The MM method was verified by comparing it to FDFD and FDTD simulations for various structures. Compared to FDFD the MM method requires much less memory while compared to the FDTD it requires less computational time. Finally a simplification of the method was presented in the case where the discontinuities are spaced far apart and was shown to provide accurate results. The MM method can provide significant physical insight and can be useful in the study of performance degradation due to fabrication induced imperfections or the design of PC devices based on waveguide discontinuities. REFERENCES [1]. J.D Joannopoulos, R.D. Meade and J.N. Winn, Photonic Crystals, Molding the flow of Light, Princeton University Press, [2]. K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag Berlin,

19 [3]. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve and J. D. Joannopoulos, High Transmission through Sharp Bends in Photonic Crystal Waveguides, Phys. Rev. Lett. 77, , [4]. Y. Hibino, Recent Advances in High-Density and Large Scale AWG Multi/Demultiplexers With Higher Index Contrast-Based PLCs, IEEE J. Selected Topics in Quant. Elec. Vol. 8, No. 6, November 2002, pp [5]. I. Vurgaftman and J. R. Meyer Photonic-Crystal Distributed-Feedback Quantum Cascade Lasers, IEEE J. Quantum Electronics, Vol. 38, No. 6, June 2002, pp [6]. M. F. Yanik and S. Fan, M. Soljaˇcic and J. D. Joannopoulos All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry, OSA Optics Letters Vol. 28, No. 24 December 2003, pp [7]. M. Koshiba, Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Couplers, Vol. 19, No. 12, December 2001, pp [8]. T. Matsumoto and T. Baba, Photonic Crystal k-vector Superprism, IEEE Journal of Lightwave Technlogy, Vol. 22, No. 3, March 2004, pp [9]. M. Imada, S. Noda A. Chutinan, M. Mochizuki and T. Tanaka Channel Drop Filter Using a Single Defect in a 2-D Photonic Crystal Slab Waveguide, IEEE J. Lightwave Technology, Vol. 20, No. 5, May 2002, pp [10]. R. Costa, A. Melloni and M. Martinelli, Bandpass Resonant Filters in Photonic-Crystal Waveguides, IEEE Photon. Techn. Letters, Vol. 15, No. 3, March 2003, pp [11]. D. Park, S. Kim, I. Park and H. Lim, Higher Order Optical Resonant Filters Based on Coupled Defect Resonators in Photonic Crystals, IEEE J. Lightwave Technology Vol. 23, May 2005, No. 5 pp

20 [12]. N. Stefanou and A. Modinos, Impurity bands in photonic insulators, Physical Review B, Vol. 57, No. 19, pp , [13]. C.K Madsen, General IIR optical filter design for WDM applications using all-pass filters, IEEE Journal of Lightwave Technology, Vol. 18, pp , 2000 [14]. B.E. Little, S.T. Chu, W.Pan, D. Ripin, T. Kaneko, Y. Kokubun and E. Ippen, Vertically coupled glass microring resonator channel dropping filters, IEEE Photonics Technology Letters, Vol. 11, pp , [15]. A. Yariv, Y. Xu, R.K. Lee and A. Scherer, Coupled-resonator optical waveguide: a proposal and analysis, OSA Optics Letters, Vol. 24, pp , [16]. E. Heebner, R. W. Boyd, Q-Han Park, SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides, OSA J. Opt. Soc. Am. B/Vol. 19, No. 4/ April 2002, pp [17]. J. B. Khurgin, Expanding the bandwidth of slow-light photonic devices based on coupled resonators, OSA Optics Letters, Vol. 30, No. 5, 2005 pp [18]. A. Tafflove and S. Hagness Computational Electrodynamics: the finite difference timedomain method, Artech House Publishers,2000. [19]. M. Koshiba, Y. Tsuji, S. Sasaki High-Performance Absorbing Boundary Conditions for Photonic Crystal Waveguide Simulations, IEEE Microwave and Wireless Components Letters,Vol. 11,No.4,April 2001, pp [20]. S. D. Wu and E. N. Glytsis, Finite-number-of-periods holographic gratings with finitewidth incident beams: analysis using the finite-difference frequency-domain method, J. Opt. Soc. Am. A, Vol. 19, No. 10, October 2002, pp

21 [21]. G. Veronis, R.W. Dutton, S. Fan, Method for sensitivity analysis of photonic crystal devices, OSA Optics Letters, Vol. 29, No. 19, October 2004, pp [22]. H. Shigesawa, M. Tsuji, A new equivalent network method for the analyzing discontinuity properties of open dielectric waveguides, IEEE Transactions On Remark 1 Rev #1. Microwave Theory and Techniques, Vol. 37, No. 1, January 1989, pp [23]. R.E. Collin, Field Theory of Guided Waves, McGraw Hill, 1992 [24]. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs and Y. Fink, Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarizarion-mode dispersion and group-velocity dispersion, OSA J. Optical Soc. Am. B, Vol. 19, No. 12, Dec 2002, pp [25]. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, J. D. Joannopoulos, Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals, Phys. Rev. E 66, (2002). [26]. M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, Effect of a photonic band gap on scattering from waveguide disorder, Applied Physics Letters, Vol. 84, No. 12, May 2004, pp [27]. D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press Inc, Second Edition [28]. G. A. Gesell, I. R. Ciric, Recurrence model analysis for multiple waveguide discontinuities and its application to circular structures, IEEE Transactions on Microwave Theory and Techniques, Vol. 41, No. 3, March 1993, pp

22 [29]. T. Kamalakis and T. Sphicopoulos, Numerical study of the implications of size nonuniformities in the performance of photonic crystal couplers using couple mode theory, IEEE J. Quantum Electronics, Vol. 41, No. 6, June 2005, pp

23 Figure 1: A cell of a periodic waveguide comprising of dielectric rods having arbitrary centers and radius. Figure 2: Examples of guided and evanescent modes of a 2D photonic crystal waveguide: a) the guided mode, b) the evanescent mode with the smallest dumping constant and c) the evanescent mode with the smallest purely imaginary β. Figure 3: A structure comprising of discontinuities with arbitrarily positioned dielectric rods between two PCW cells. Figure 4: Widely spaced waveguide discontinuities. Figure 5: Comparison of power reflection coefficient of the MM and the FDFD methods for a) single, b) double, c) triple defect rods inside a PCW. Figure 6: Convergence of the MM method with a) increasing N x and b) increasing N z for various cell sizes. Figure 7: a) A SCISSOR comprising of four PC defect cavities side coupled to a PCW, b) Comparison of the power transmission obtained by the FDTD and the MM method. Figure 8: Power loss (expressed in db/mm) due to scattering obtained considering 100 perturbed PCWs assuming a) Δ=1nm b) 3nm and c) 5nm. Figure 9: a) Structure used in order to compare the conventional MM method and its 2x2 TM simplification, b) Power transmissions obtained by the two methods. 23

24 Figure 1 24

25 Figure 2 25

26 Figure 3 26

27 Figure 4 27

28 Figure 5 28

29 Figure 6 29

30 Figure 7 30

31 Figure 8 31

32 Figure 9 32

33 Table 1. Comparison between the FDFD and MM method for various defect rod positions Position Power Reflection R MM method FDFD method +r a (x-direction) 0,7781 0,7685 -r a (x-direction) 0,7727 0,7977 +r a (z-direction) 0,8286 0,8426 -r a (z-direction) 0,8286 0,

34 Table 2. Mean value and standard deviation of the power loss due to scattering Δ (nm) Mean Power Loss StD of Power Loss (db/mm) (db/mm)