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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 8 — Apr. 19, 2004
  • pp: 1592–1604
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Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics

L. C. Botten, T. P. White, C. Martijn de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry  »View Author Affiliations


Optics Express, Vol. 12, Issue 8, pp. 1592-1604 (2004)
http://dx.doi.org/10.1364/OPEX.12.001592


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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

Photonic crystals (PC), which are arguably amongst the most exciting of optical structures, are at the forefront of theoretical and experimental research [1

1. C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht2001).

]. Their significance is based on properties which resemble those of semiconductors, and which enable them to tailor the propagation of light on the scale of optical wavelengths with minimal losses. Their ability to guide light on the wavelength scale [2

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

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]

] and to control the radiation dynamics of embedded sources [5

5. T. D. Happ, I.I. Tartakovskii, V.D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, “Enhanced light emission of InxGa1-xAs quantum dots in a two-dimensional photonic-crystal defect microcavity,” Phys. Rev. B 66, 041303(R) (2002). [CrossRef]

] make them ideally suited to the miniaturization of optical components and to the development of compact devices that incorporate a large number of elements. Already demonstrated are the bending of light around corners [6

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]

], interconnections such as T- and Yjunctions [1

1. C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht2001).

,7

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]

], channel-drop filters [8

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

], superprisms [9

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]

] and Mach-Zehnder interferometers [10

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]

]. Attention is now being turned to the modeling and fabrication of ultra-compact optical devices (such as the coupler in Fig. 1) which comprise various elements connected through complex waveguide structures embedded in photonic crystals [11

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]

].

The exploitation of the technological potential of PC based structures requires a thorough understanding of the mechanisms by which light is coupled into, and guided through, these devices. In order to understand the performance of such structures it is important to develop rigorous analytical techniques that can provide physical insight into the scattering and diffraction processes, facilitating the conceptualization of the device using generalizations of familiar ideas derived from other areas of optics (e.g. thin film optics). Until now, much of the modeling has been undertaken using finite difference time domain (FDTD) methods or other computational schemes. Though these methods produce accurate results, they do not easily reveal insight since they neither exploit the underlying physics to any significant extent, nor take advantage of the structure of the problem or its geometry in order to accelerate calculations. For a device such as that in Fig. 1, comprising a sequence of components and waveguides, there exists the real possibility that semi-analytic techniques, which exploit the geometry and the mode structure of the individual devices, can be superior to exclusively numerical simulations.

2. Theoretical formulation

We consider a two-dimensional (2D) structure, as in Fig. 1, that consists of three segments, M1, M2, and M3, each of which comprises a set of identical layers which may be conceptualized as diffraction gratings with a transverse period Dx . 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 Dx 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 Dx , 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 Rpq 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.

Fig. 1. A typical three segment photonic crystal device showing the component regions M1, M2 and M3, three lateral supercells of the model and a constituent grating bounded by dashed lines.

The modelling of the composite device comprises two distinct elements: (a) Bloch mode propagation in a given region and (b) the scattering/diffraction of modes that occurs at interfaces, and which is modeled with generalized Fresnel (matrix) coefficients that satisfy various energy conservation, reciprocity and transitivity relations.

Each of the structural regions may be homogeneous (e.g., free space or dielectric) or a periodically modulated structure such as a photonic crystal device (e.g., with waveguides as shown), with its Bloch modes generated via a transfer matrix technique [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]

] that is based on supercell methods and diffraction grating theories [17

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]

]. Between the grating layers (e.g., on the dashed lines of Fig. 1) in any given region, the modes are characterized by plane wave expansions represented by vectors of plane wave amplitudes 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 𝒻2 =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]

]). The eigenvalue equation to be solved follows from the imposition of the Bloch condition (with Bloch factor µ), i.e., 𝒻2 µ𝒻 1. That is,

𝓣𝓯=μ𝓯where𝓣=(TRT1RRT1T1RT1),𝓯=(ff+),
(1)

where the complex eigenvalues, 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

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]

]. While the scattering matrices can be computed in a variety of ways (e.g., differential methods, integral methods etc.), we use a multipole analysis [17

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]

]. The multipole formulation analytically preserves energy conservation and reciprocity within the scattering matrices [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]

] and leads to desirable corresponding properties within the Bloch mode analysis.

It is important to ensure that each supercell is sufficiently isolated from its neighbours. To achieve this, an isolating barrier of between 7–10 cylinders of bulk crystal is needed between one defect and its nearest neighbour in an adjacent supercell. In turn, this determines the period Dx of the grating supercells. Once effective isolation is achieved, the calculations are independent of the lateral component of the Bloch vector, k0x . For analytic convenience in what follows, we set kox =0 and thus work with periodic boundary conditions.

𝓣T𝓠𝓣=𝓠,where𝓠=[0QQ0].
(2)

In Eq. (2),Q is the reversing permutation (derived by reversing the rows of the unit matrix) and arises through the reciprocity-based derivation [22

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.

]. 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

𝓣=𝓕𝓕1with𝓕=[FF+F+F],=[Λ00Λ1],Λ=diag(μi).
(3)

𝓕T𝓠𝓕=where=(0II0)
(4)

𝓕H𝓣p𝓕=𝓣mwhere𝓣p=(IriIeiIeIr)and𝓣m=(ImiIm¯iIm¯Im),
(5)

arising as a consequence of the flux conservation relation 𝒯H 𝒯p 𝒯=𝒯p that is satisfied by the transfer matrix [22

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.

], which in turn follows from the representation 𝒻H 𝒯p 𝒻 of the energy flux carried by a plane wave field 𝒻=[f T 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]

]. In Eq. (5), 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.

Rij=(Fi)1(IRjRi)1(RjRi)Fi,
Tij=(Fj)1(IRiRj)1(IRi2)Fi,
(6)

where R i =F +i(Fi )-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 TijT =T ji and, and RijT =R ij energy conservation relations which take the form RijH R ij +TijH T ij =1—a simplified form with R ij and T ij trimmed to contain only the propagating states.

We can now formulate the propagation problem of the structure in Fig. 1, expanding the field in M 2 in terms of Bloch modes of amplitudes c - and c +, with phase origins respectively located at the upper and lower boundaries of M2. The mode matching equations are then

r=R12δ+T21ΛLc+,c=T12δ+R21ΛLc+,
(7)
c+=R23ΛLc,t=T23ΛLc,
(8)

Eqs. (7) and (8) respectively derive from field matching at the upper and lower interfaces of 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

23. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

]

R=R13=R12+T21ΛLR23ΛL(IR21ΛLR23ΛL)1T12,
T=T13=T23ΛL(IR21ΛLR23ΛL)1T12.
(9)

The form of the reflection matrix in Eq. (9) can be simplified further with the introduction of 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

S122=I,whereS12=(R12T21T12R21)
(10)

denotes the Bloch mode S-matrix for the 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,

R13=T121(R21+ΛLR23ΛL)(IR21ΛLR23ΛL)1T12,
=T211(IΛLR23ΛLR21)1(R21+ΛLR23ΛL)T21.
(11)

With the modes in their normalized form (i.e. satisfying Eqs. (4) and (5)), the reflected and transmitted fluxes may be computed directly by taking the square magnitude of the relevant elements in the reflection and transmission scattering matrices. In fact, the generalized form of the energy conservation relations [22

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.

] are best expressed in terms of the matrix. In the case of the propagating modes, we may trim the Bloch mode scattering matrices so they contain only entries for the propagating states and show, using an approach analogous to that in Ref. [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]

], that

S13HS13=I13,whereS13=(R13T31T13R31),I13=(I100I3),
(12)

where I 1and 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. R13H R 13+T13H T 13=l), while the off-diagonal partitions contain a host of phase relationships (e.g. R13H T 13+T13H 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

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.

].

Finally, we observe that the forms in Eqs. (9) and (11) are precise generalizations of the Airy formulae for a Fabry-Perot interferometer [23

23. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1970).

], with the Bloch mode theory having enabled the formulation of the propagation problem in a stratified photonic crystal device to be cast into a form that is structurally identical to that of thin film optics in uniform media.

3. Applications

3.1 Asymptotic expansions and the folded directional coupler

Recently, we reported upon the design of the folded directional coupler (FDC) [11

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]

], a novel compact resonant filter which combines the characteristics of both a directional coupler and a Fabry-Perot interferometer, and which is the basis of a high-Q notch rejection filter. This device (Fig. 1) is simply two semi-infinite waveguides with a common coupling region of length L where the guides run parallel to each other, separated by Nc columns of cylinders. As is discussed in Ref. [11

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]

], the properties of the FDC are governed by only the propagating modes, since the coupling region is sufficiently long to ensure substantial decay of the evanescent states. The behaviour of the coupler is thus described by its two propagating Bloch modes—the supermodes which, for well separated guides, are well approximated by the symmetric and anti-symmetric superpositions of the modes of a single guide. We now form asymptotic approximations of the reflection and transmission 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

w1T=w3T=[1000],w2T=[10000100].
(13)

Here, 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 Lw 2Λ L ̃w2T 2, where Λ˜ is a 2×2 diagonal matrix containing the eigenvalues of the two supermodes of the coupler. Some manipulation of Eq. (9) then reveals

T˜13=T˜23Λ˜L(IR˜21Λ˜LR˜23Λ˜L)1T˜12,
(14)

where ij =wjT T ij wi and ij =wiT R ij wi , together with an analogous expression for 13. The derivation of these follows from the Woodbury formula [25

25. W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

] for the inversion of a rank-p matrix perturbation. In the case of the FDC, both 13 and 13 are scalars, the value of which can be obtained in a variety of ways including direct computation. In our earlier paper [11

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]

], we derived, in a heuristic manner,

R13=ρf=cos2(ΔβL)exp(2iβ¯L)1+sin2(ΔβL)exp(2iβ¯L),T13=τf=iexp(iβ¯L)sin(ΔβL)(1+exp(2iβ¯L))1+sin2(ΔβL)exp(2iβ¯L),
(15)

3.2 Serpentine waveguide

The unit cell of the serpentine waveguide is a pair of coupled FDC devices joined input-to- output by a single waveguide of length 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.

Fig. 2. Two equivalent serpentine waveguide geometries (assuming no tunneling through the guide ends). Both are characterized by the period Dy and the double and single guide lengths, L 1 and L 2.

Analysis of the periodic structure begins with the transfer matrix 𝒯s through a complete period of the serpentine

𝓣s=(τsρs2τsρsτsρsτs1τs),
(16)

where ρs and τs are the appropriately padded Fresnel coefficients for a full period, double FDC structure of length DY =(L 1+L 2). As observed above, the minimum segment of the serpentine is the single FDC, which leads to

𝓣s=𝓣f2,where𝓣f=(τfρf2τfρfτfρfτf1τf)
(17)

is the transfer matrix through a single FDC with input and output guides of length L 2/2, and the superscripted prime denotes the padded Fresnel coefficients described above.

The dispersion relation for the periodic serpentine structure is then obtained by solving the eigenvalue problem 𝒯s𝒻=µ𝒻 where the eigenvalues are µ=exp(iqDy ) and q is the Bloch factor in the longitudinal direction of the serpentine. Since 𝒯s =tf2, a consequence of the serpentine period being regarded as a pair of FDC devices in series, the eigenvalue equation can be written as 𝒯f𝒻µ 1/2 𝒻, the solution of which leads to the dispersion relation

μ+1μ12=τf2ρf2+1τf,i.e.cos(qDy2)=(1τf),
(18)

where the simplification exploits the energy conservation properties satisfied by the Fresnel coefficients Eq. (12), guaranteeing that |ρ′f |2+|τf |2=1 and arg (τ′f )-arg (ρf )=±π/2.

The right hand side of Eq. (18) can be calculated using the full Bloch mode method, or alternatively, from the analytic expressions Eq. (15) obtained above, with the appropriate padding terms included. The latter method results in an analytic form of the right hand side, which can provide a better understanding of the serpentine waveguide properties. With τ′=τf exp(iβL 2), we can express Eq. (18) in the form

cos(qD2)=sin(ΔβL1)sin(β¯L1+βL2)+cos2(ΔβL1)sin(2β¯L1+βL2)2sin(ΔβL1)cos(β¯L1).
(19)

Transmission bands exist for frequencies where the right hand side of Eq. (19), ℜ(1/τ′f ), has magnitude less than unity. Correspondingly, band gaps, which lie between the transmission bands, occur when |ℜ(1/τf )|>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 τf may vanish. The first occurs when sin(ΔβL 1)=0, i.e., for cavity lengths L 1=nLB /2=(2Δβ), for an odd integer n, where LB is the beat length of the double waveguide. The second condition occurs when, cos(β¯ L 1)=0, L 1=/(2β¯), also for odd 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]

] appears in the serpentine band diagram. We label these two types of resonator band gap RG1 and RG2 respectively. The third class of band gap that appears in the serpentine band diagram is a Bragg 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]

] that occurs as a result of the overall periodicity of the serpentine structure, rather than a single feature of the FDC.

Fig. 3. (a) Band diagram for a serpentine tine waveguide with L 1=L 2=5d, where q is the Bloch coefficient along the waveguide, and Dy =L 1+L 2. (b) Transmission through an FDC with same parameters (dashed), one period of the serpentine waveguide (dotted) and two periods (solid). Note that for frequencies below ωd/(2πc)=0.3064, the double guide cavity only supports a single, odd mode, and thus the analytic result of (19) does not apply.

Figure 3(a) shows the band diagram of a serpentine waveguide with L 1=L 2=5d, 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 Nc =1. The FDC structures making up this device have the same parameters as those in Ref. [11

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]

], and indeed, the very narrow RG2-type band gap at ωd/(2πc)=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 τf 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 β̄β, it is apparent that resonator gaps of type RG1 occur less frequently than RG2-type gaps.

Another feature of both types of resonator band gap is the positioning of the bands above and below the gaps. Recall that in a one-dimensional Kronig-Penney model [28

28. P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.

], the right hand side (RHS) of the dispersion relation is continuous, and hence only direct gaps exist, the RHS of the relation being unable to change sign without passing through zero. In the case of the resonator gaps however, the RHS of the dispersion relation (19) diverges when either of the resonance conditions is satisfied. When this happens, the RHS can change sign without passing back through zero, thus resulting in an indirect band gap. Although band gaps with this property occur commonly in two-dimensional photonic crystals, such features have only recently been observed in other coupled resonator waveguides [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]

] consisting of a linear periodic array of resonant elements.

A number of interesting band structures can be created with appropriate choices of the lengths and and waveguide separation. In Ref. [11

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]

] it was shown that the width of a resonance of type RG2 is largely a function of Δβ, 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 Δβ, 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.

Fig. 4. (a) Band diagram for a serpentine waveguide with L 1=L 2=7d. The solid curve is calculated with the full numerical simulation while the dashed curve is calculated using the approximation (19) L 1=7.5d, L 2=6.7d (b) Transmission through a FDC with L=7d (dashed), 2 periods of the serpentine guide (dotted) and 3 periods of the serpentine guide.

In Fig. 4(a), the band structure is shown for a serpentine waveguide with L 1=L 2=7d. The constituent waveguides are identical to those used previously. The band gaps at centre frequencies ωd/(2πc)=0.3135 and ωd/(2πc)=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=7d and a 2-period long serpentine waveguide with L 1=L 2=7d. 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

Acknowledgments

The authors wish to thank Sergei Migaleev for providing numerical data with which to compare results from the Bloch mode method. This work was produced with the assistance of the Australian Research Council under its ARC Centres of Excellence Program.

References and links

1.

C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht2001).

2.

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

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. 79, 2690–2692 (2001). [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 InxGa1-xAs 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]

14.

S.F. Mingaleev and K. Busch, “Scattering matrix approach to large-scale photonic crystal circuits,” Opt. Lett. 28, 619–621 (2003). [CrossRef] [PubMed]

15.

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. E 68, 155101 (2003).

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. 17, 2165–2176 (2000). [CrossRef]

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]

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]

21.

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, 1962).

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, Principles of Optics (Pergamon Press, Oxford, 1970).

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]

25.

W. H. Press, B. P. Flannery, S. A. Teulolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).

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]

28.

P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.

29.

S. Wolfram, The Mathematica Book, 3rd Ed., (Wolfram Media/Cambridge University Press, 1996).

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

  1. C. M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, (Kluwer, Dordrecht 2001).
  2. A. Chutinan, and S. Noda, �??Waveguides and waveguide bends in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 62, 4488-4492 (2000). [CrossRef]
  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. 79, 2690-2692 (2001). [CrossRef]
  4. Z. Wang, 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 InxGa1-xAs 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]
  14. S.F. Mingaleev, and K. Busch, �??Scattering matrix approach to large-scale photonic crystal circuits,�?? Opt. Lett. 28, 619-621 (2003). [CrossRef] [PubMed]
  15. Z.-Y. Li, and K.-M. Ho, �??Light propagation in semi-infinite photonic crystals and related waveguide structures,�?? Phys. Rev. E 68, 155101 (2003).
  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. 17, 2165-2176 (2000). [CrossRef]
  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]
  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]
  21. M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, 1962).
  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, Principles of Optics (Pergamon Press, Oxford, 1970).
  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]
  25. W. H. Press, B. P. Flannery, S. A. Teulolsky and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1988).
  26. M. Bayindir and E. Ozbay, �??Band-dropping via coupled photonic crystal waveguides,�?? Opt. Express 10, 1279�??1284 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1279">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1279</a> [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]
  28. P. Yeh, Optical waves in layered media, (Wiley, New York, 1988), Ch. 6.
  29. S. Wolfram, The Mathematica Book, 3rd Ed., (Wolfram Media / Cambridge University Press, 1996).

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