## Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics

Optics Express, Vol. 12, Issue 8, pp. 1592-1604 (2004)

http://dx.doi.org/10.1364/OPEX.12.001592

Acrobat PDF (378 KB)

### Abstract

A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions and generalized Fresnel coefficients, is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

© 2004 Optical Society of America

## 1. Introduction

2. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B **62**, 4488–4492 (2000). [CrossRef]

4. Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E **68**, 066616 (2003). [CrossRef]

5. T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of In_{x}Ga_{1-x}As quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B **66**, 041303(R) (2002). [CrossRef]

6. 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**, 3787–3790 (1996). [CrossRef] [PubMed]

7. J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. **17**, 1500–1508 (1999). [CrossRef]

8. S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. **80**, 960–963 (1998). [CrossRef]

9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. **74**, 1370–1372 (1999). [CrossRef]

10. A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer employing coupled-resonator optical waveguides,” Opt. Lett. **28**, 405–407 (2003). [CrossRef] [PubMed]

11. T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. **28**, 2452–2454 (2003). [CrossRef] [PubMed]

12. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E **64**, 046603 (2001). [CrossRef]

13. L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. **28**, 854–856 (2003). [CrossRef] [PubMed]

11. T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. **28**, 2452–2454 (2003). [CrossRef] [PubMed]

## 2. Theoretical formulation

_{1}, M

_{2}, and M

_{3}, each of which comprises a set of identical layers which may be conceptualized as

*diffraction gratings*with a transverse period

*D*

_{x}. While this derivation is restricted to a three segment device, the approach extends naturally to

*N*-segment structures via recursion. Because of the gratings’ periodicity, the functional elements of the structure are contained within a supercell (see Fig. 1), the dimension

*D*

_{x}of which is chosen large enough to ensure effective isolation from neighbouring supercells when operated within a band gap of the bulk crystal. The periodicity imposed by the diffraction grating model means that individual layers are coupled together by plane wave diffraction orders, the directions of which are given by the grating equation. Between each layer, the field can be represented in the plane wave basis, by an expansion over all plane wave orders. The action of the individual gratings, all of which must have a common transverse period

*D*

_{x}, is handled by plane wave scattering matrices (

**R**and

**T**) that characterize the reflection and transmission of plane waves incident on a grating layer. For example, the element

*R*

_{pq}specifies the reflected amplitude in order

*p*due to unit amplitude incidence in order

*q*. For simplicity here, we assume gratings are up-down symmetric and arranged in a square or rectangular lattice.

12. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E **64**, 046603 (2001). [CrossRef]

17. L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.1, Method,” J. Opt. Soc. Am. A. **17**, 2165–2176 (2000). [CrossRef]

**f**- and

**f**

_{+}(which include entries for both propagating and evanescent waves), respectively denoting the downward and upward propagating plane wave field components. The Bloch modes are represented by the solutions of the eigenvalue problem for the transfer matrix

*𝒯*that relates fields on either side of a grating layer according to

*𝒻*

_{2}=

*𝒯𝒻*

_{1}and which is derived from relations between outgoing and incoming fields expressed in terms of the reflection and transmission matrices

**R**and (e.g. [12

12. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E **64**, 046603 (2001). [CrossRef]

*µ*), i.e.,

*𝒻*

_{2}

*µ𝒻*

_{1}. That is,

*µ*, corresponding to propagating Bloch modes have unit amplitude, and |

*µ*|≠1 for non-propagating modes. The transfer matrix form in Eq. (1) is analogous to transfer matrices used in conventional thin-film optics. We observe that while the eigenvalue problem is stated formally in terms of the transfer matrix, numerical instabilities require the use of alternative methods for solving the eigenvalue problem [18

18. G.H. Smith, L.C. Botten, R.C. McPhedran, and N.A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E **67**, 056620 (2003). [CrossRef]

19. B. Gralak, S. Enoch, and G. Tayeb, “From scattering or impedance matrices to Bloch modes of photonic crystals,” J. Opt. Soc. Am. A **19**, 1547–1554 (2002). [CrossRef]

17. L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.1, Method,” J. Opt. Soc. Am. A. **17**, 2165–2176 (2000). [CrossRef]

20. L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.2, Method,” J. Opt. Soc. Am. A. **17**, 2177–2190 (2000). [CrossRef]

*D*

_{x}of the grating supercells. Once effective isolation is achieved, the calculations are independent of the lateral component of the Bloch vector,

*k*

_{0x}. For analytic convenience in what follows, we set

*k*

_{ox}=0 and thus work with periodic boundary conditions.

**Q**is the reversing permutation (derived by reversing the rows of the unit matrix) and arises through the reciprocity-based derivation [22]. The symplectic nature of

*𝒯*holds even in a system with loss, and ensures that the eigenstates are arranged into forward and backward propagating pairs, respectively associated with eigenvalues

*µ*and 1/

*µ*. For the structure in Fig. 1, in which each layer is up-down symmetric and the lattice is rectangular, the diagonalized form of

*𝒯*is then

*𝓕*of Eq. (3), the left and right partitions represent the forward and backward propagating modes, with the matrices

**F**

_{∓}comprising the column vector components of the eigenvectors of

*𝒯*. In turn, the partitioned matrix contains

*ℒ*the eigenvalues for the forward and backward propagating states respectively in the diagonal matrices Λ and Λ

^{-1}, where Λ=diag{

*µ*

_{j}} and {

*µ*

_{j}} denotes the set of eigenvalues for the forward states. The columns of the matrix

*𝓟*are then scaled so that physical properties such as reciprocity and energy conservation, and also modal orthogonality, are represented in a physically normalized form. For completeness, we state these results without proof, referring the reader to Ref. [22] for their derivation. Modal reciprocity, which follows from the symplectic nature of

*𝒯*is characterized by

*𝒯*

^{H}

*𝒯*

_{p}

*𝒯*=

*𝒯*

_{p}that is satisfied by the transfer matrix [22], which in turn follows from the representation

*𝒻*

^{H}

*𝒯*

_{p}

*𝒻*of the energy flux carried by a plane wave field

*𝒻*=[

**f**

^{T}

_{−}

^{T}[20

20. L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.2, Method,” J. Opt. Soc. Am. A. **17**, 2177–2190 (2000). [CrossRef]

**I**

_{r}and

**I**

_{e}are diagonal matrices with unit entries on the diagonal respectively for real (propagating) and evanescent plane waves. Correspondingly,

**I**

_{m}and

**I**

_{m}

*̄*are diagonal matrices with unit entries on the diagonal respectively for propagating and non-propagating Bloch modes.

*M*

_{1}to

*M*

_{3}, through the

*L*layers of

*M*

_{2}, is characterized by an incident field

**δ**, a vector of forward propagating Bloch mode amplitudes in

*M*

_{1},

**r**=

**Rδ**, a vector of backward propagating modal amplitudes in

*M*

_{1}, and

**t**=

**Tδ**, a vector of forward propagating modes in

*M*

_{3}. At the interface between each stack (e.g. the

*ij*interface from

*M*

_{i}to

*M*

_{j}), the reflection and transmission of Bloch modes can be characterized by Fresnel matrices

**R**

_{ij}and

**T**

_{ij}, which are expressed in terms of the Bloch modes of media

*M*

_{i}and

*M*

_{j}. These matrices are essentially generalizations of planar interface Fresnel coefficients. The Fresnel matrices are derived by considering a Bloch mode expansion represented by the amplitude vector

*ij*interface, giving rise to reflected (

**R**

_{ij}

**T**

_{ij}

*M*

_{i}and

*M*

_{j}. Field matching at the

*ij*interface then leads to

**R**

_{i}=

**F**

^{+}

_{i}(

^{-1}denotes the plane wave reflection scattering matrix of a semi-infinite region of material

*i*, with incidence from free space. With the modes suitably normalized, these coefficients satisfy reciprocity relations which take the form

**T**

_{ji}and, and

**R**

_{ij}energy conservation relations which take the form

**R**

_{ij}+

**T**

_{ij}=

**1**—a simplified form with

**R**

_{ij}and

**T**

_{ij}trimmed to contain only the propagating states.

*M*

_{2}in terms of Bloch modes of amplitudes

**c**

_{-}and

**c**

_{+}, with phase origins respectively located at the upper and lower boundaries of M

_{2}. The mode matching equations are then

*M*

_{2}and express the reflection and transmission of the Bloch modes at these interfaces. The Λ

^{L}terms in Eqs. (7) and (8) correspond to mode propagation through the layers between the upper and lower interfaces of medium

*M*

_{2}. Solving Eqs. (7) and (8), we derive the reflection (

**R**) and transmission (

**T**) scattering matrices of Eq. (9) and see that these are generalizations of the Airy formulation for a Fabry-Perot (FP) interferometer [23]

*transitivity relations*which are implicit in this formulation. These relations evolve from (9) by setting the width of

*M*

_{2}to

*L*=0. In doing so, we derive a pair of relations that are analytically satisfied by the Fresnel matrices of Eq. (6). We now consider the special case when the media

*M*

_{1}and

*M*

_{3}are identical, for which

**T**

_{11}=

**I**and

**R**

_{11}=

**0**, and the corresponding problem of the transition from

*M*

_{2}to

*M*

_{2}via a

*L*=0 length layer of

*M*

_{1}. From these we derive the four transitivity relations that are summarized in the notable result

*M*

_{1}-

*M*

_{2}interface. With these relations, which are independent of the energy and reciprocity relations, we arrive at a simplified form for

**R**,

20. L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.2, Method,” J. Opt. Soc. Am. A. **17**, 2177–2190 (2000). [CrossRef]

**I**

_{1}and

**I**

_{3}are identity matrices, the dimensions of which are given by the number of propagating states in each of regions

*M*

_{1}and

*M*

_{3}respectively. When Eq. (12) is expanded, the diagonal partitions contain the energy conservation relations (e.g.

**R**

_{13}+

**T**

_{13}=

**l**), while the off-diagonal partitions contain a host of phase relationships (e.g.

**T**

_{13}+

**R**

_{13}=

**0**), the origins of which lie in the application of time reversibility for lossless systems. The generalization of Eq. (12) to include both propagating and evanescent modes is given in Ref. [22].

*S*matrix all hold analytically within the multipole formulation and are verified numerically to within effectively machine procession (14 significant figures), a consequence of the use of the multipole formulation for the computation of the scattering matrices. This therefore rules out the use of energy conservation and reciprocity as valid physical tests of the accuracy of the Bloch mode formulation in the case of an implementation in which the grating scattering matrices already preserve these properties analytically. In the case of the multipole formulation that we use, such test are merely tests of the efficacy of the coding of the algorithm. We note, however, that for alternative implementations, in which the grating scattering matrices are not imbued with these analytic properties, energy conservation and reciprocity are indeed valid physical tests of the scattering matrix calculation and have flowon effects in the Bloch mode method. In our treatment, however, convergence of the method, which is dependent on the truncation dimensions of plane wave and modal fields (i.e. the number of evanescent terms included), is the only real means of validating results and comparing them against those obtained by entirely different means. We have confirmed the accuracy of this method using results obtained from a recently developed Wannier function method [24

24. K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter **15**, 1233–1254 (2003). [CrossRef]

## 3. Applications

### 3.1 Asymptotic expansions and the folded directional coupler

11. T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. **28**, 2452–2454 (2003). [CrossRef] [PubMed]

*L*where the guides run parallel to each other, separated by

*N*

_{c}columns of cylinders. As is discussed in Ref. [11

**28**, 2452–2454 (2003). [CrossRef] [PubMed]

**R**

_{13}matrices

**T**

_{13}and in these limits. First, we introduce notation that will enable us to take projections that extract the relevant rows and columns of matrices. The projection matrices comprise columns of the identity matrix, the number of which is identical to the number of propagating modes in the given region. In the case of the FDC, we have

**w**

_{1}and

**w**

_{3}, are used to handle projections in regions

*M*

_{1}and

*M*

_{3}in which there is only a single propagating mode, while

**w**

_{2}is used in the coupler (region 2 w

*M*

_{2}) which has two propagating modes. We then observe that for

*L*sufficiently large, Λ

^{L}≈

**w**

_{2}Λ

^{L}

*̃*

**T̃**

_{ij}=

**T**

_{ij}

*w*

_{i}and

**R̃**

_{ij}=

**R**

_{ij}

*w*

_{i}, together with an analogous expression for

**R̃**

_{13}. The derivation of these follows from the Woodbury formula [25] for the inversion of a rank-p matrix perturbation. In the case of the FDC, both

**R̃**

_{13}and

**T̃**

_{13}are scalars, the value of which can be obtained in a variety of ways including direct computation. In our earlier paper [11

**28**, 2452–2454 (2003). [CrossRef] [PubMed]

*β̄*=(

*β*

_{1}+

*β*

_{2})/2,Δ

*β*=(

*β*

_{1}-

*β*

_{2})/2, with

*β*

_{j}denoting the modal propagation constants of the propagating supermodes in

*M*

_{2}, for which

*µ*

_{j}=exp(

*iβ*

_{j}). However, these results Eq. (15) may also be derived [22] as a limiting case of the general theory by assigning particular, idealized values to the Fresnel matrices

**R̃**

_{ij}and

**T̃**

_{ij}in Eq. (14) and its reflection matrix equivalent.

### 3.2 Serpentine waveguide

26. M. Bayindir and E. Ozbay, “Band-dropping via coupled photonic crystal waveguides,” Opt. Express **10**, 1279–1284 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1279. [CrossRef] [PubMed]

27. S. Pereira, P. Chak, and J.E. Sipe, “Gap-soliton switching in short microresonator structures,” J. Opt. Soc. Am. B **19**, 2191–2202 (2002) [CrossRef]

*L*

_{2}as shown in Fig 2(a). Each of the individual components of this structure is essentially the same as those of the isolated FDC device. In the sections where there are two parallel waveguides (length

*L*

_{1}), a pair of propagating modes is supported, and light is coupled from one guide to the other. The sections of single guide (of length

*L*

_{2}) act as the input/output guides to the next/previous coupling region. If

*L*

_{2}is sufficiently long that evanescent tunnelling between the blocked waveguide ends is negligible, the only connection between the coupling regions is via the single (even-symmetry) mode of the single guide. In this limit, the structures shown in Figs. 2(a) and 2(b) are equivalent. This property means that a single FDC, rather than a complete unit cell, is the minimum segment of the serpentine required to determine the band structure.

*L*

_{2}provides a few lattice periods of separation between each cavity region. The complex Fresnel reflection and transmission coefficients,

*ρ*

_{f}and

*τ*

_{f}of Eq. (15), for the single propagating mode through the double guide cavity of the FDC, are calculated using either the full Bloch method or the approximate analytic form given in Eq. (15). Since

*ρ*

_{f}and

*τ*

_{f}are the reflection and transmission amplitudes of the cavity itself, they must be padded to provide for propagation through the input and output guides to give the properties of the minimum serpentine segment. Padding is included as an extra phase term due to propagation through the single guide, exp(

*iβL*

_{2}), where

*β*is the modal propagation constant in the single mode guide that connects the coupler regions of length.

*L*

_{2}

*𝒯*

_{s}through a complete period of the serpentine

*ρ*

_{s}and

*τ*

_{s}are the appropriately padded Fresnel coefficients for a full period, double FDC structure of length

*D*

_{Y}=(

*L*

_{1}+

*L*

_{2}). As observed above, the minimum segment of the serpentine is the single FDC, which leads to

*L*

_{2}/2, and the superscripted prime denotes the padded Fresnel coefficients described above.

*𝒯*

_{s}

*𝒻*=

*µ𝒻*where the eigenvalues are

*µ*=exp(

*iqD*

_{y}) and

*q*is the Bloch factor in the longitudinal direction of the serpentine. Since

*𝒯*

_{s}=

*𝒯*

_{f}

*𝒻*=±

*µ*

^{1/2}

*𝒻*, the solution of which leads to the dispersion relation

*ρ′*

_{f}|2+|τ

*′*

_{f}|

^{2}=1 and arg (

*τ′*

_{f})-arg (

*ρ*′

_{f})=±

*π*/2.

*τ*′=

*τ*

_{f}exp(

*iβL*

_{2}), we can express Eq. (18) in the form

*τ′*

_{f}), has magnitude less than unity. Correspondingly, band gaps, which lie between the transmission bands, occur when |ℜ(1/τ

*′*

_{f})|>1. There are three distinct types of band gap, each originating through a distinct physical effect. Following Eq. (19), or from the analysis of a single FDC, there are two conditions for which

*τ*

_{f}may vanish. The first occurs when sin(Δ

*βL*

_{1})=0, i.e., for cavity lengths

*L*

_{1}=

*nL*

_{B}/2=

*nπ*(2Δ

*β*), for an odd integer

*n*, where

*L*

_{B}is the beat length of the double waveguide. The second condition occurs when, cos(

β ¯
L

_{1})=0,

*L*

_{1}=

*nπ*/(2

*), also for odd*β ¯

*n*. When either of these conditions is satisfied, the right hand side of Eq. (19) diverges, and a resonator band gap [27

27. S. Pereira, P. Chak, and J.E. Sipe, “Gap-soliton switching in short microresonator structures,” J. Opt. Soc. Am. B **19**, 2191–2202 (2002) [CrossRef]

27. S. Pereira, P. Chak, and J.E. Sipe, “Gap-soliton switching in short microresonator structures,” J. Opt. Soc. Am. B **19**, 2191–2202 (2002) [CrossRef]

*L*

_{1}=

*L*

_{2}=5

*d*, formed in a square symmetric photonic crystal, with cylinders of normalized radius 0.3d and refractive index 3.0, operated in TM polarized light with

*N*

_{c}=1. The FDC structures making up this device have the same parameters as those in Ref. [11

**28**, 2452–2454 (2003). [CrossRef] [PubMed]

*ωd*/(2

*πc*)=0.33107 corresponds to the sharp resonance of the FDC filter. In Fig. 3(b), the calculated intensity transmission is shown for the single FDC (half a period), a single whole period and two whole periods of the serpentine waveguide. The band diagram was calculated using

*τ*

_{f}obtained from the full Bloch mode method and the transmission spectra were also calculated using the full numerical approach. Results for the approximation Eq. (19) do not agree exactly because of the choice of and the phase change on reflection from the guide ends, which is not included in the FDC model. Counting from the top of Fig 3(a), we see that top two resonator gaps are both of type RG2. Given the resonance conditions for RG1 and RG2, and since

*β̄*>Δ

*β*, it is apparent that resonator gaps of type RG1 occur less frequently than RG2-type gaps.

**19**, 2191–2202 (2002) [CrossRef]

**28**, 2452–2454 (2003). [CrossRef] [PubMed]

*β*, being only weakly dependent on

*β̄*. As can be seen in Fig. 3(a), very flat bands with low group velocities occur for sharp resonances, and hence there is the potential to tune the group velocity and group delay with Δ

*β*, by changing either the guide separation or the properties of the cylinders between the guides. Another band structure of interest occurs when two or more consecutive band gaps are resonator gaps.

*L*

_{1}=

*L*

_{2}=7

*d*. The constituent waveguides are identical to those used previously. The band gaps at centre frequencies

*ω*d/(2

*πc*)=0.3135 and

*ω*d/(2

*πc*)=0.3183 are resonator gaps of type RG1 and RG2 respectively. Since these are consecutive gaps, and each is indirect, the three bands above, below and between the gaps are almost parallel to one other and all have a positive slope. With careful optimization, it may be possible to design a band structure with several equally-spaced bands. Fig. 4(b) shows the transmission spectrum of a single FDC with

*L*=7

*d*and a 2-period long serpentine waveguide with

*L*

_{1}=

*L*

_{2}=7

*d*. Observe that, for these parameters, the resonator gaps are relatively strong even for such a small number of periods, whereas the Bragg gaps only appear as very weak perturbations in the transmission spectrum for two periods.

## 4. Conclusions

*Mathematica*[29] and Fortran—with the applications suite being implemented in the former, exploiting the convenient programming language and numerical linear algebra library, and the grating scattering matrices implemented in the latter, with the two linked together using the

*MathLink*toolkit. The method demonstrates computational advantages when handling extended structures which have components with many identical gratings, thus allowing the propagation problem in a long segment to be handled by a single set of modes. However, there is no performance advantage when dealing with varying structures such as tapers, in which each layer can differ from its neighbours.

## Acknowledgments

## References and links

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2. | A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B |

3. | A. Chutinan, M. Mochizuki, M.. Imada, and S. Noda, “Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

4. | Z. Wang and S. Fan, “Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines,” Phys. Rev. E |

5. | T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of In |

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7. | J. Yonekura, M. Ikeda, and T. Baba, “Analysis of Finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. |

8. | S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunnelling through Localized States,” Phys. Rev. Lett. |

9. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Photonic crystals for micro lightwave circuits using wavelength dependent angular beam steering,” Appl. Phys. Lett. |

10. | A. Martinez, A. Griol, P. Sanchis, and J. Marti, “Mach-Zehnder interferometer employing coupled-resonator optical waveguides,” Opt. Lett. |

11. | T.P. White, L.C. Botten, R.C. McPhedran, and C.M. de Sterke, “Ultracompact resonant filters in photonic crystals,” Opt. Lett. |

12. | L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E |

13. | L.C. Botten, A.A. Asatryan, T.N. Langtry, T.P. White, C.M. de Sterke, and R.C. McPhedran, “Semi-analytic treatment for propagation in finite photonic crystal waveguides,” Opt. Lett. |

14. | S.F. Mingaleev and K. Busch, “Scattering matrix approach to large-scale photonic crystal circuits,” Opt. Lett. |

15. | Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E |

16. | S. Chen, R. C. McPhedran, C. M. De Sterke, and L. C. Botten, “Optimal tapers in photonic crystal waveguides,” submitted to Appl. Phys. Lett. |

17. | L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.1, Method,” J. Opt. Soc. Am. A. |

18. | G.H. Smith, L.C. Botten, R.C. McPhedran, and N.A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E |

19. | B. Gralak, S. Enoch, and G. Tayeb, “From scattering or impedance matrices to Bloch modes of photonic crystals,” J. Opt. Soc. Am. A |

20. | L.C. Botten, N.A. Nicorovici, A.A. Asatryan, R.C. McPhedran, C.M. de Sterke, and P.A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations.Part.2, Method,” J. Opt. Soc. Am. A. |

21. | M. Hamermesh, |

22. | L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures,” in preparation. |

23. | M. Born and E. Wolf, |

24. | K. Busch, S.F. . Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys.: Condens. Matter |

25. | W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, |

26. | M. Bayindir and E. Ozbay, “Band-dropping via coupled photonic crystal waveguides,” Opt. Express |

27. | S. Pereira, P. Chak, and J.E. Sipe, “Gap-soliton switching in short microresonator structures,” J. Opt. Soc. Am. B |

28. | P. Yeh, |

29. | S. Wolfram, |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(290.4210) Scattering : Multiple scattering

**ToC Category:**

Focus Issue: Photonic crystals and holey fibers

**History**

Original Manuscript: March 1, 2004

Revised Manuscript: April 5, 2004

Published: April 19, 2004

**Citation**

L. Botten, T. White, C. de Sterke, R. McPhedran, A. Asatryan, and T. Langtry, "Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics," Opt. Express **12**, 1592-1604 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-8-1592

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### References

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