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 (atheoh@di.uoa.gr).

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.

2

Copyright

OCIS codes : 060.4510 Optical communications , 230.0230 Optical devices , 999.9999

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 90o 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

Remark 1 Rev. #1.

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,

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

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

Remark2 Rev. #1.

Remark 1 Rev. #1.

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

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

Remark 1,3 Rev. #1.

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])

( ) ( ) j ze β=E r u r (1)

( ) ( ) j ze β=H r v r (2)

where β is the propagation constant of the mode and u,v are periodic functions along the z

direction. Defining |Ψβ> to be a four component vector comprising of the tangential parts ut and

vt of u and v respectively, i.e.

( ) ( )| , , , ,TT

t t x y x yu u v vβΨ >= =u v (3)

one can write Maxwell’s equations in the following form (Ref. [26])

ˆ ˆ ˆ| |A j B Bz β ββ∂⎛ ⎞+ Ψ >= Ψ >⎜ ⎟∂⎝ ⎠

(4)

Remark 4 Rev. #1.

6

where the operators A and B are defined by

1 1 0ˆ

1 10

t t

t t

Aωε

ω μ

ωμω ε

⎛ ⎞− ∇ × ∇ ×⎜ ⎟⎜ ⎟=⎜ ⎟− ∇ × ∇ ×⎜ ⎟⎝ ⎠

(5)

and

0ˆ0

B− ×⎛ ⎞

= ⎜ ⎟×⎝ ⎠

zz

(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

, ,| ( ) sin sin

2 2lzny jG zmx

mnlm n l

G yG x eβ⎛ ⎞⎛ ⎞Ψ >= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ B G (7)

where B(Gmnl) are the Fourier coefficients of |Ψβ> and we have assumed that the periodic cell is

rectangular. In this case the reciprocal lattice vectors Gmnl=[Gmx,Gny,Glz] have components

Gmx=2πm/b, Gny=2πn/d, Glz=2πl/a where m,n,l are integers with 0 xm N≤ ≤ , 0 yn N≤ ≤ ,

z zN l N− ≤ ≤ and b,d,a are the sizes of the cell along the x,y and z direction, respectively. The

total number of terms in the expansion of (7) is therefore (Nx+1)(Ny+1)(2Nz+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

Remark 5 Rev # 1.

Remark 8 Rev # 1.

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

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 (TMy) case, since one needs to

consider only one y-directed electric and one x-directed magnetic field tangential components

which we will designate as uy and vx. In this case the fields do not depend on y and hence the

reciprocal lattice vectors Gmnl are such that Gmnl= Gml=[Gmx,0,Glz] and the matrix eigenproblem

is written as

M β=V V (8)

where the vector

( )1 1,..., , ,..., TN NV V U U=V (9)

comprises of all the spectral components V1,…,VN and U1,…,UN of vx and uy 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 MM

M Mε

ω

⎡ ⎤= − ⎢ ⎥

⎣ ⎦G

G

(10)

where MG is a N×N diagonal matrix whose diagonal [MG]ll elements are given by [MG]ll=Glz

while Mω is another N×N diagonal matrix with [Mω]ll=ωμ. The elements [Mε]pq of the N×N

matrix Mε are given by

Remark 2 Rev # 1.

8

[ ] ( )24

( )

1

1 1 ( )2

k qxkpq pqpq

k

GM ε ω ε δ

ωμ=

= − −∑ G% (11)

In equation (11), ( )ε G% is the Fourier transform of the dielectric constant,

( ) ( )1 j

Sd e

Sε ε ⋅= ∫ G rG r r% (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 xp N≤ ≤ , z zN q N− ≤ ≤ , k is also an integer with 1 4k≤ ≤ ,

δpq is Kronecker's delta and the vectors ( )kpqG , are defined by

(1) ( , )Tpq px qx pz qzG G G G= + −G (13)

(2) ( , )Tpq px qx pz qzG G G G= − −G (14)

(3) ( , )Tpq px qx pz qzG G G G= − − −G (15)

(4) ( , )Tpq px qx pz qzG G G G= − + −G (16)

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 Nr

rods, each centered at pn=(xn,zn) and each with radius rn, then generalizing the result of Ref. [2],

( )ε G% is given by

( )( )

( )

1( )2 , 0

, 0

njna b n

n n

a b n bn

J Grf eGrf

ε εε

ε ε ε

⋅⎧ − ≠⎪= ⎨⎪ − + =⎩

∑

∑

G p GG

G% (17)

In (17), fn=πrn2/S is the filling factor for each rod, G=|G|, εa and εb is the dielectric

constant of the rods and the background medium, respectively.

Figure 2, illustrates examples of the modal field intensities |uy|2 calculated by solving the

eigenproblem in (8) for a PCW assuming εa=9ε0, εb=ε0, a=0.6μm, b=9a, while the radii rn of the

Remark 7 Rev. 1.

9

all the rods are taken equal rn=ra=0.12μm. Figure 2(a) corresponds to β=2.58μm-1, figure 2(b) to

β=(5.24+1.19j)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 ith cell the tangential electric

and magnetic fields are written as:

( ) ( )( ) ( )1( ) ( ) ( ) ( )i i

m i m ij z z j z zi i i i it m tm m tm

m m

a e b eβ β−− − −′= +∑ ∑E e e (18)

where ( )itme , ( )i

ma and ( )imβ are the tangential electric Bloch functions, the coefficients and the

propagation constants of the mth forward mode of the ith cell respectively , while ( )itm′e and ( )i

mb are

the tangential electric Bloch functions propagation constants of the mth backward mode of the ith

cell respectively and

( ) ( )( ) ( )1( ) ( ) ( ) ( )i i

m i m ij z z j z zi i i i it m tm m tm

m ma e b eβ β−− − −′= +∑ ∑H h h (19)

where ( )itmh , and ( )i

tm′h are the tangential magnetic Bloch functions propagation constants of the mth

forward mode and the mth backward mode of the ith cell respectively.

10

At each interface between two cells, the tangential fields must be continuous at the boundary z=zi

(Ref. [28]). This implies:

1( ) ( )i it i t iz z+=E E (20)

1( ) ( )i it i t iz z+=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

,V

dV= ⋅∫f g f g (22)

and projecting (20) along ( 1)itn+h and (21) along ( )i

tne , for 1≤n≤M, one obtains a matrix equation

relating the mode coefficients in cells i and i+1:

1

1

i i

ii iZ

+

+

⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

A A=

B B (23)

where vectors Ai=[a1i,…,aM

i]T, Bi=[b1i…,bM

i]T contain the coefficients of the M forward and M

backward modes of the ith cell. The matrix Zi is given by

1i i iZ Y X−= (24)

where the element of the matrices Yi and Xi are given by

[ ]

( )

( )

( ) ( 1)

( ) ( 1)

( ) ( )

( ) ( )

, 1 ,

, 1 ,1 ,,

1 , 2,

im

im

j ai itm tn

i itm tn

i nm j ai itm tn

i itm tn

em n M

m M n MX

m n M MeM m n M

β

β

+

+

⎧≤ ≤⎪

⎪ ′ ≤ − ≤⎪= ⎨ ≤ − ≤⎪⎪ + ≤ ≤

′⎪⎩

e h

e h

h e

h e

(25)

and

11

[ ]( 1)

( 1)

( 1) ( 1)

( 1) ( 1)

( 1) ( )

( 1) ( )

, 1 ,

, 1 ,1 ,,

1 , 2,

im

im

i itm tn

j ai itm tn

i nm i itm tn

j ai itm tn

m n Me m M n M

Ym n M M

M m n Me

β

β

+

+

+ +

−+ +

+

−+

⎧≤ ≤⎪

⎪ ′ ≤ − ≤⎪= ⎨ ≤ − ≤⎪⎪ + ≤ ≤

′⎪⎩

e h

e h

h e

h e

(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:

Z⎛ ⎞ ⎛ ⎞

⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

N 1

N 1

A A=

B B (27)

and the matrix Z is given by

1 1...NZ Z Z−= ⋅ ⋅ (28)

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

1 1 122 21Z Z−= −B A (29)

1 111 12

N Z Z= +A A B (30)

where the M×M submatrices of Z are determined by

11 12

21 22

Z ZZ

Z Z⎡ ⎤

= ⎢ ⎥⎣ ⎦

(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 Zi at each boundary. One can then obtain

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 1st cell, one sets A1=(1,0,…,0)T

and the power transmission coefficients T and R of the guided mode are given by T=|a1N|2/|a1

1|2

and R=|b11|2/|a1

1|2.

Referring to figure 3, note that at the beginning of the device at z=z0, one can assume that

the PCW cells extend infinitely from z=z0 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.

4. 2×2 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 ZA and ZB 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

A BZ Z D Z′ = ⋅ ⋅ (32)

where D is given by

13

0

0

g

g

j L

j L

eD

e

β

β−

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

(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), βg 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 Z22 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 Z22-1 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 rd 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 ra=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

εa=9ε0 and that of the surrounding medium was εb=ε0. The radius of the defects rd varied from

0.3ra to 2.0ra. For the calculation of the modes the number of plane waves used was 15 for the

Remark 9 Rev #1.

14

propagation direction (z-direction) while 19 standing waves were used for the transverse

direction (x-direction). The grid of the FDFD was taken ΔG=ra/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

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 rd=ra whose position changes

±ra 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 ±ra 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 Nz, Nx 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

Remark 3 Rev #1.

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 Nz. On the other hand the value of Nx 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 Nz=5 and various values of Nx. The structure supports a single guided

mode and the number of evanescent modes in the first zone is Ne≅Nx. As Ne is increased, R

begins to converge to 0.76 and for Ne>14, R is practically constant. Note that for small values of

Ne the value of R changes whenever Ne is increased by 2 modes. This is not surprising because as

Ne 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 Ne=7 to Ne=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 Nz has a less critical role since the guided

modal fields and propagation constants are accurately estimated even with Nz=3 (implying

2Nz+1=7 plane waves in the z-direction). In the figure legend Nrods 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 Nrods=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

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 ra,

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 ΔG=ra/8. The parameters for the MM method simulation were Nx=59 and Nz=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

be considered having the centres ( ) ( ), ,i i i i i iz x z z x x′ ′ = + Δ + Δ of the rods slightly displaced with

respect to the centres ( ),i iz x of the rods of the ideal PCW and their radius i a ir r r′= + Δ perturbed

as shown in figure 3. For simplicity, the perturbations Δzi, Δxi and Δri are independently selected

Remark 1 Rev #1.

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 Nz and

Nx 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/ra 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

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 Nx=11 and Nz=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.

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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 Nx and b) increasing Nz 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.

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Figure 1

25

Figure 2

26

Figure 3

27

Figure 4

28

Figure 5

29

Figure 6

30

Figure 7

31

Figure 8

32

Figure 9

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Table 1. Comparison between the FDFD and MM method for various defect rod positions

Power Reflection R Position MM method FDFD method

+ra (x-direction) 0,7781 0,7685 -ra (x-direction) 0,7727 0,7977 +ra (z-direction) 0,8286 0,8426 -ra (z-direction) 0,8286 0,8638

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Table 2. Mean value and standard deviation of the power loss due to scattering

Δ (nm) Mean Power Loss (dB/mm)

StD of Power Loss (dB/mm)

1 -0.0412 0.0511 3 -0.3526 0.4150 5 -1.1498 1.2775