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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 25346–25360
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Coupled waveguide modes in hexagonal photonic crystals

J. Scott Brownless, Sahand Mahmoodian, Kokou B. Dossou, Felix J. Lawrence, Lindsay C. Botten, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 25346-25360 (2010)
http://dx.doi.org/10.1364/OE.18.025346


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Abstract

We investigate the modes of coupled waveguides in a hexagonal photonic crystal. We find that for a substantial parameter range the coupled waveguide modes have dispersion relations exhibiting multiple intersections, which we explain both intuitively and using a rigorous tight-binding argument.

© 2010 Optical Society of America

1. Introduction

Coupled photonic crystal waveguides (CPCWs) have received substantial attention due to their ability to guide slow light with significant control over dispersion [1

1. S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. 32, 2981–2983 (2007). [CrossRef] [PubMed]

3

3. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25, C65–C74 (2008). [CrossRef]

]. The unique properties of CPCWs have led to high bandwidth delay lines [4

4. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

] and directional couplers with extremely short coupling lengths [5

5. A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694 –696 (2003). [CrossRef]

] needed to create ultra compact devices [6

6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

]. Analysis of the underlying coupled waveguide modes (CWMs) is critical to understanding how these properties are achieved. Coupled waveguides in uniform media are well understood: the fundamental mode is always even, the second mode is odd [7

7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.

, 8

8. K. Iizuka, Elements of Photonics, Volume II (Wiley, 2002). Chap. 9.

] and the dispersion curves of the two modes do not cross. de Sterke et al. [9

9. C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004). [CrossRef]

] showed that the fundamental CWM of square lattice PCWs can be either even or odd, and that this depends on the number of rows between the waveguides. Since the coupling coefficient is given by C = (βevenβodd)/2, the existence of an odd fundamental CWM in square photonic crystals (PCs) led to the realisation of structures with negative coupling coefficients exhibiting discrete negative refraction [10

10. A. Locatelli, M. Conforti, D. Modotto, and C. D. Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343–1345 (2006). [CrossRef] [PubMed]

].

There are key differences between square and hexagonal lattice CPCWs which make the hexagonal case a more interesting, and ultimately more challenging, problem to study. First, there exist two distinct geometries for coupled waveguides, the inline case with an odd number of rows between PCWs [inset Fig. 1(b)], and the staggered case with an even number of rows between PCWs [inset in Fig. 1(a)]. The inline case has reflection symmetry since the centers of the cylinder defects in the two waveguides line up. No such symmetry exists for the staggered case as the cylinder centers do not line up. The two waveguide configurations exhibit different behaviour at the Brillouin zone (BZ) edge. Figure 1 shows that in the staggered arrangement, the even and odd dispersion curves intersect at the BZ edge, while they are well separated in the inline configuration. This behaviour was reported previously [11

11. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008). [CrossRef] [PubMed]

, 12

12. A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A, Pure Appl. Opt 11, 094016 (2009). [CrossRef]

], and has resulted in staggered CPCWs being proposed for use as slow light couplers [11

11. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008). [CrossRef] [PubMed]

].

Fig. 1 Dispersion curves of CWMs for (a) PCWs separated by four rows (staggered geometry) and (b) PCWs separated by five rows (inline geometry). The purple dashed curve and the solid green curves are for a single waveguide and for CPCWs, respectively. The red shaded area indicates the projected bands and the green area is the braided region, separated from the typical region by the blue dashed curve.

Here, we take a more general interest in CWMs in hexagonal lattices. We consider a PC with a background index of nb = 3, air holes of radius rc = 0.3d, where d is the period, and use Hz polarisation. We create the PCWs by altering the refractive index of rows of holes to nd = 1.5. Figure 1 shows that for both the inline and staggered geometries, the dispersion curves, which were computed using the generalised fictitious source superposition method [13

13. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method,” Int. J. Microwave Opt. Technol. 1, 133–145 (2006).

], differ on either side of the dashed blue curve. We refer to the area to the right of this curve (green background) as the braided region as the dispersion curves of the coupled modes are interwoven around the single waveguide mode leading to multiple degeneracies. At these degeneracies the PCW modes do not couple. Such a point in the dispersion curve can be used to design compact demultiplexers [14

14. F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express 12, 1119–1125 (2004). [CrossRef] [PubMed]

]. The presence of the braiding means that the coupling coefficient depends not only on the geometry of the system but also depends strongly on the Bloch wavevector kx. The degeneracies here are not accidental, but are associated with how the mode of the PCW decays in the bulk PC separating the waveguides. We refer to the region to the left of this curve as typical, since the modes display similar properties to those in a square lattice.

Unlike square lattices, where the symmetry of the fundamental mode depends on the spacing between the waveguides, the symmetry of the fundamental mode in hexagonal lattice CWMs changes both with the spacing and with the Bloch wavevector, i.e. the symmetry of the fundamental mode varies across the BZ. As the spacing between the waveguides increases by two rows an extra crossing appears within the braided region. In this paper we analyse the intricacies of CWMs in hexagonal CPCWs. We do this in Section 2 by providing a physical argument as to why such degeneracies should exist inside the BZ, and explain how they depend on the parameters of the CPCWs and the underlying bulk PC. In Section 3 we provide a rigorous analysis of the CWMs using a perturbative method based on the modes of the single uncoupled PCW. We discuss our results in Section 4. The Appendix provides some proofs for Section 2.

2. Physical Argument

Deep inside the band gap, the interaction between waveguides is weak, allowing us to use a tight binding approximation. In this regime, the splitting of the coupled modes occurs symmetrically around the single waveguide mode. This splitting is proportional to the J overlap integral [7

7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.

],
J=δɛE1(r)E2*(r)d2r=δɛ(Ex1(r)Ex2*(r)+Ey1(r)Ey2*(r))d2r.
(1)
Here δε is the perturbation formed by one of the waveguides. For the PCW mode that has a Hz field that is even (odd) with respect to its center, there is a nodal line at the center of the PCW for the Ex (Ey) component, so the dominant contribution to the splitting is from the Ey (Ex) component. This means that the magnitude of the coupling depends on only one field component of the single PCW mode.

Figure 2 shows the relevant electric field for the even mode of a PCW, situated at y = 0. When moving away from the waveguide in the braided region, the field has nodal lines (at four rows away in Fig. 2(b) and at one, three and five rows away in 2(c)). If the second waveguide is situated on such a nodal line the waveguide coupling is small and their dispersion curves intersect. In the typical region [Fig. 2(a)] the envelope of the field simply decays exponentially in y/d and intersections do not occur. In the braided region [Fig. 2(b)] the mode decays exponentially but with an underlying periodic feature, the novel element considered here.

Fig. 2 |Ey| fields of the even single PCW mode for a PCW centered at y = 0. The scale has been chosen so as to emphasize the weak fields which are involved in coupling the PCWs. The value of kxd for Fig. 2(b) corresponds to a crossing at spacing 4.

The nature of this decay is best explained by considering the complex bands of the bulk PC. The complex bands arise by finding the k values, real or complex, associated with a real frequency. Since complex bands are continuous when the frequency is varied, a bandgap can be considered to be a frequency interval with only complex bands but no real ones. PCWs are periodic in x therefore their modes propagate with a fixed value of a (real) kx. When describing how the modes decay in the bulk we thus choose to make ky complex while keeping kx real. The modes of a bulk PC are Bloch modes which acquire a Bloch factor μ after translation along a lattice vector. For a propagating Bloch mode |μ| = 1, while for an evanescent solution |μ| < 1. For this lattice we define the lattice vector, e2=[d/2,d3/2], shown in Fig. 5(a). When translating along the lattice vector −e2, the Bloch factor is given by μ = eik·e2, and thus the imaginary part of the Bloch vector k denotes the decay rate.

Fig. 5 (a): Geometry of the PC unit cell, defined by the basis vectors e1 and e2. Points P1 and P2 are the phase origins of the plane wave expansion respectively at the upper (Π1) and lower (Π2) interfaces of a grating layer. (b): Schematic of a PC with a double waveguide, a composite of 5 elements characterized by their scattering matrices: RW, TW for the waveguides, RB, TB for the barrier between the waveguides and R for the semi-infinite PCs.

The complex band diagrams shown in Fig. 3 are for three different values of kxd, with the color representing decay rate of the Bloch mode: Fig. 3(a) is for kxd = 0.31 in the typical region, (b) is for kxd = 2.51, in the typical region for d/λ < 0.2531 and in the braided region for d/λ > 0.2531, while (c) is at the BZ edge and completely within the braided region. Figure 3(b) and 3(c) show that in the braided region there are two equally dominant evanescent Bloch modes. The presence of either one or two dominant evanescent Bloch modes defines the typical and braided regions respectively. The line separating these regions in Fig. 1 corresponds to the bifurcation point in the complex band-structure. This extends the work of Mahmoodian et al [15

15. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009). [CrossRef] [PubMed]

] to complex bands.

Fig. 3 Complex band diagrams at different slices of kx for the bulk hexagonal lattice. Dark blue indicates real bands, while other colours indicate complex bands. (a) kxd is in the typical region; (b) kxd intersects the blue dashed curve in Fig. 1; (c) kxd is at the BZ edge.

To illustrate the behaviour of the bands within the braided region, we consider how the field decays along lattice vectors in the bulk PC. As shown in Fig. 3 in the braided region, for a given value of kx there are two complex Bloch modes with the same decay rate, κ, with opposite signs of Re(ky) (which we refer to as ky). Thus, when translating m times by the bulk lattice vector −e2, we acquire a Bloch factor
μ±m=eimke2=eikxmd/2e±ikymd3/2eκmd3/2,
(2)
where ky refers to the real part of the Bloch vector. Since we are interested in coupled PCW modes we assume the single PCW mode has been computed and analyze how the mode decays in the bulk. As described in the Appendix, we can write the single PCW mode in the bulk region as a superposition of the decaying bulk Bloch modes, φi. In the braided region there are two leading order Bloch modes which decay at the same rate. We ignore all but these modes. Taking their amplitudes as c1 and c2, after decaying along m lattice vectors the field of the PCW is
ψ(r0+me2)=c1φ1(r0)eikxmd/2eikymd3/2eκmd3/2+c2φ2(r0)eikxmd/2eikymd3/2eκmd3/2.
(3)
We show in the Appendix that in the braided region, φ2(x,y)=φ1*(x,y) and that the modes have equal magnitudes with c2=eiϑc1*, thus we choose the origin such that r0 = (0, y0) and get
ψ(r0+me2)=2eκmd3/2eikxmd/2Re[c1φ1(r0)ei(kymd3/2)+iϑ/2].
(4)
Therefore in the barrier region the PCW mode consists of a decaying envelope modulating a periodic oscillation. When a second waveguide is situated on a zero of this oscillation, the PCWs do not couple and their modes are degenerate. Though the relative phase ϑ is known only after computing the single PCW mode, Eq. (4) shows that the serpentine nature of the coupled PCW bands is due to the interference of the two evanescent Bloch modes in the barrier.

We now examine the symmetry of the CWMs. Dossou et al [16

16. K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, “Modes of symmetric composite defects in two-dimensional photonic crystals,” Phys. Rev. A 80, 013826 (2009). [CrossRef]

] showed that the fundamental mode of coupled point defects has the same symmetry as the underlying bulk Bloch mode. We have observed the same behaviour here for CWMs. Two CPCWs separated by rows have cylinder centers which are a distance ( + 1)e2 apart. The fundamental mode is a superposition of the two individual PCW modes such that the phase difference between the two PCWs is that of the underlying bulk Bloch mode. Since the underlying bulk PC has two Bloch modes, we combine these as in Eq. (4) and write
|Ψ=|ψ1±eikx(+1)d/2|ψ2
(5)
where |ψ1〉 is the mode of the single waveguide, ψ2(r) = ψ1(r +( + 1)e2), shown in Fig. 5(b), and the sign is given by sgn[Re(c1φ1(r0)ei(ky(+1)d3/2+iϑ/2)]. The effect of the real part of the exponent is to flip the sign of the fundamental mode as kx moves along the BZ. A special case occurs when kxd = π, i.e. at the BZ edge. As shown in the Appendix, the single PCW mode can be written as a real valued function at the BZ-edge and thus we have c1 = c2. Figure 3(c) shows that at kxd = π there are two dominant evanescent Bloch modes (the third is separated by v a reciprocal lattice vector) where Re(ky)=±π/(d3) along the entire bandgap. Thus Eq. (4) becomes
ψ(r0+me2)=2eκmd3/2eikxmd/2c1φ1(r0)cos(mπ/2).
(6)
We find the coupling goes to zero for all odd m causing the degeneracy seen in Fig. 2(c). For even m the coupling is locally maximized. Using Eq. (5), the modes at the BZ-edge are |Ψ〉 = |ψ1〉 ± i|ψ2〉. This is illustrated by the Hz field densities in Fig. 4(a)4(d). Here, both modes have fields that are completely imaginary in one PCW, but are real in the other.

Fig. 4 Hz fields for two degenerate coupled waveguide modes separated by two rows at kxd = π. (a) The real part of both modes is identical. (b) The imaginary part of mode 1. (c) Imaginary part of mode 2. (d) Both modes have the same absolute value.

3. Formulation of the perturbation theory

3.1. Computation of the modes of the unperturbed photonic crystals

The transfer matrix 𝒯 relates the fields at upper and lower interfaces of the grating. If we denote by f1f1+, f2 and f2+ the column vectors whose elements are respectively the plane wave expansion coefficients fp,1, fp,1+, fp,2 and fp,2+ in Eq. (7), then the Bloch modes are given by the condition
𝒯[f1f1+]=μ[f1f1+]
(8)
where μ is the phase factor μ = eik·e2. Thus the Bloch modes are the eigenvectors of the transfer matrix 𝒯 of a single grating layer; 𝒯 can be diagonalized as 𝒯 = ℱℒℱ−1 where
=[FFF+F+]and=[Λ00Λ].
(9)
In Eq. (9) the columns of ℱ represent the eigenvectors which constitute the Bloch modes [17

17. 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 left partition F and F+ contain the downward propagating modes, whereas the right partition contains the upward propagating modes. The matrix ℒ is diagonal and comprises the eigenvalues μ, partitioned into downward (Λ) and upward propagating (Λ′) modes. The grating layer in Fig. 5(a) has up-down symmetry, i.e., it is invariant by the transformation (x,y)  (x, −y), assuming without loss of generality that the coordinate origin is the midpoint between P1 and P2. This transformation changes a downward propagating mode into an upward propagating mode and vice-versa. It also permutes the fields [and their plane wave expansions (7)] at the lower interface and upper interfaces. To obtain the new plane wave expansion, for instance, at the upper interface, we must take into account (xx2) = (xx1) – d/2, i.e., P1 and P2 are shifted horizontally by a half period, together with (−yy2) = −(yy1). It follows from these properties that the downward and upward propagating modes can be chosen such that they satisfy the symmetry relations
ΛΛ=eikxdI,[FF+]=[Q0F+Q0F]andQ0=diag(eipπ)p=diag((1)p)p.
(10)

3.2. Photonic crystal waveguides

To derive a dispersion equation for the double waveguide in Fig. 5(b), we model the structure as a composite of 5 elements: the upper and lower waveguides, a barrier consisting of periodic PC layers, and upper and lower semi-infinite PCs. Each element is characterised by its reflection and transmission matrices under plane wave incidence. Since the phase origins P1 and P2 are shifted horizontally, incidence by downward propagating plane waves (on the upper interface) and by upward propagating plane waves (on the lower interface) have different scattering matrices; primed symbols apply to the matrices of the latter case. Let RW, TW, RW and TW denote the plane wave scattering matrices of a single waveguide. The scattering matrices of the barrier are denoted RB, TB, RB and TB. The Fresnel reflection matrices of semi-infinite PCs are represented by R and R. From Botten et al. [17

17. 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 scattering matrices of the barrier and the semi-infinite PCs can be computed using the Bloch modes of the unperturbed PCs
R=F+F1andR=FF+1
(11a)
RB=F+(R21+ΛR21Λ)(IR21ΛR21Λ)1F1
(11b)
TB=F(IR21R21)Λ(IR21ΛR21Λ)1F1
(11c)
RB=F(R21+ΛR21Λ)(IR21ΛR21Λ)1F+1
(11d)
TB=F+(IR21R21)Λ(IR21ΛR21Λ)1F+1
(11e)
with R21 = −F+−1 F+ and R21 = −F−1 F; R21 and R21 are the Fresnel reflection matrices for Bloch mode incidence on a semi-infinite homogeneous background material. The scattering matrices RW, TW, RW and TW can be computed by solving a grating diffraction problem. However these scattering matrices can be obtained analytically if a waveguide is created by removing a row of cylinders: since the waveguide is homogeneous and has the same refractive index as the background, RW = RW = 0, TW = exp(ikx d/2) Q0 P and TW = exp(−ikx d/2) Q0 P with P = diag (exp(p h)) and h=d3/2. In the general case of waveguide gratings, RW0 and, at the upper waveguide, we have the plane wave scattering relations
{f^1+=RWf^1+TWf1+f1=TWf^1+RWf1+f^1=Rf^1+{f^1+=(IRWR)1TWf1+f^1=R(IRWR)1TWf1+f1=(RW+TWR(IRWR)1TW)f1+
(12)

Similarly for the lower waveguide we obtain
f2+=(RW+TWR(IRWR)1TW)f2.
(13)

We have the relations across the barrier between the two waveguides
f1+=RBf1+TBf2+andf2=TBf1+RBf2+.
(14)
We now use the symmetry relations (10) to halve the size of the eigenproblem and so get some insight about the symmetry of the modes. It follows from Eq. (10) that the scattering matrices in Eqs (11a)(11e) and Eq. (13) satisfy the symmetry properties (note that Q01=Q0)
RB=Q0RBQ0,TB=eikxdQ0TBQ0andR21=R21.
(15)
Next, we translate horizontally the bottom phase origin P2 of the barrier by Δx = ℓd/2; the new phase origin is aligned vertically with the top phase origin P1 which is useful for analyzing the field symmetry. With the new phase origin, the expression for the vector of plane wave coefficients f2 is
f˜2=diag(eiαpd2)f2=eikxd2Q0f2.
(16)

Substituting the relations (15) into Eq. (14), together with Eq. (16) gives
(f1+±Q0+1f˜2)=(RB±eikxd2Q0TB)(f1±Q0+1f˜2+)
(17)
with both relations holding simultaneously. A similar treatment for the Fresnel-type equations (13) and its counterpart in Eq. (12), together with the symmetry relations
R=Q0RQ0,
(18)
RW+TWR(IRWR)1TW=Q0(RW+TWR(IRWR)1TW)Q0.
(19)
yield
f1=R¯¯f1+andQ0+1f˜2+=R¯¯Q0+1f˜2.
(20)
Hence
(f1±Q0+1f˜2+)=R¯¯(f1+±Q0+1f˜2+)
(21)
where
R¯¯=RW+TWR(IRWR)1TW.
(22)

Substituting expression (21) into Eq. (17) leads to
A(σ)(f1++σQ0+1f˜2+)=0,withA(σ)=I(RB+σeikxd2Q0TB)R¯¯
(23)
where σ = 1 gives one equation and σ = −1 gives the other, both of which are simultaneously true. For a pair of coupled waveguide modes, f1+ or f˜2+ are not zero, so if A(σ) is singular then, in general, A(−σ) is not. Thus the dispersion equation is given by det A(σ) = 0 while the mode symmetry follows from f1+σQ0+1f˜2+=0. In particular when is odd we have Q0+1=I so that f1+=σf˜2 and, from Eq. (20), f1=σf˜2+, i.e., the waveguide mode has even symmetry when σ = 1 and odd symmetry when σ = −1. In practice, the nonlinear eigenvalue problems can be solved by searching for the roots of the determinant of the matrix A(σ).

As the thickness parameter increases, RB tends to R while TB tends to zero since the bulk PC is in a band gap. We obtain the single waveguide dispersion equation as → ∞
A0f1+=0withA0=IRR¯¯.
(24)

3.3. Perturbation theory

We assume that the propagation constant kx is fixed while the normalized frequency ν = d/λ is unknown. The dispersion equation (23) for the double waveguide problem is equivalent to finding a frequency ν and a mode x such that A(σ,ν) x = 0. Similarly, for a single waveguide, we write the dispersion equation as A0(ν0) x0 = 0. To find a solution through a perturbation analysis, we consider the problem (23) as a perturbation to Eq. (24) and introduce the notation
ν=ν0+δν,x=x0+δxandA(σ,ν)=A0(ν0+δν)+δA(σ,ν0+δν).
(25)

The term δA, which accounts for the perturbation due to the finite width of the barrier between the waveguides, is defined below in Eq. (33). The equation A(σ,ν) x = 0 is thus that for
A0(ν0+δν)+δA(σ,ν0+δν))(x0+δx)=0.
(26)

As a first order analysis, we can derive the leading order equation
(A0(ν0)+A0νδν+δA(σ,ν0))(x0+δx)=0.
(27)

The size of the matrix A0 = IRR̿ is the number of plane wave orders included in our calculations. When the plane wave orders are truncated to just the propagating plane wave orders, then the matrices R and R̿ are unitary when the background PC is in a bandgap (as a consequence, RR̿ is also unitary). For the PCs considered here, there are either one or two propagating plane wave orders. The most interesting cases, corresponding to the braided region, have two propagating plane wave orders. Then the unitary matrix RR̿ has two orthogonal eigenvectors x0(1) and x0(2) associated respectively with eigenvalues γ1 and γ2. The single waveguide equation A0x = (IRR̿)x = 0 has a solution if 1 is an eigenvalue of the matrix RR̿. Let assume, for instance, that the eigenvector x0(1) is associated with the eigenvalue γ1 = 1 and construct the solution of the double waveguide as the perturbation
x=x0+δx=x0(1)+c(1)x0(1)+c(2)x0(2)with|c(1)|1and|c(2)|1.
(28)

By substituting this expression in Eq. (27) and using the fact that A0(ν0)x0(1)=0, we are led to the first order perturbation equation
δνA0νx0(1)=δA(σ,ν0)x0(1)c(2)A0(ν0)x0(2).
(29)
and by taking the inner product with x0(1), we find
δν=x0(1)HδA(σ,ν0)x0(1)x0(1)HA0νx0(1)
(30)
since x0(1)H(A0(ν0)x0(2))=(1γ2)x0(1)Hx0(2)=0. The superscript H denotes the Hermitian transpose, i.e., the conjugate transpose. A matrix norm is used in the analysis below and any type of norm can be considered since, in a finite dimensional space, all matrix norms are equivalent. From Eqs. (11a)(11c) and (15), we can derive the leading order estimates with respect to the small parameter ||Λ||
RB=R+O(||Λ||2)
(31)
TB=F(IR212)ΛF1+O(||Λ||2)=(I(Q0R)2)FΛF1+O(||Λ||2)
(32)
and it follows that
δA(σ,ν)=A(σ,ν)A0(ν)=σeikxd2Q0(I(Q0R)2)FΛF1R¯¯+O(||Λ||2)
(33)
so that, to leading order, the numerator x0(1)HδA(σ,ν0)x0(1) is given by
x0(1)HδA(σ,ν0)x0(1)=σx0(1)HQ0(I(Q0R)2)F(eikxd2Λ)F1R¯¯x0(1)
(34)
or
x0(1)HδA(σ,ν0)x0(1)=σx0(1)HQ0(I(Q0R)2)F(eikxd2Λ)F1R1x0(1)
(35)
since, from Eq. (24), RR¯¯x0(1)=x0(1). The denominator term x0(1)HA0νx0(1) is also derived from Eq. (24)
x0(1)HA0νx0(1)=x0(1)H(R(ν)R¯¯(ν))νx0(1).
(36)
From unitarity and mode orthogonality RHT′ + TH R′ = 0, for propagating plane orders only (see Ref. [18

18. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory,” Phys. Rev. E 70, 056606 (2004). [CrossRef]

, Eq. (24b)]), it follows that the matrix exp(ikxℓd/2) Q0TR−1 is skew-hermitian, and as a consequence its leading term is also skew-hermitian. Thus x0(1)HδA(ν0)x0(1) is a pure imaginary number. The denominator x0(1)HA0νx0(1) in Eq. (36) is also pure imaginary at ν = ν0. We prove this by using RR¯¯x0(1)=(RR¯¯)Hx0(1)=x0(1) and the fact that the derivative of a parameterized family of unitary matrices U(ν) is skew-hermitian at ν = ν0 if U(ν0) = I since, from U(ν)UH(ν) = I, we have 0 = (U(ν)UH(ν))′ = U′(ν)UH(ν)+U(ν)UH(ν) and in particular U′(ν0) + UH(ν0) = 0. The parameterized family U(ν) = (R(ν0)P(ν0))HR(ν)P(ν) satisfies such a property. Since the denominator does not depend on the length , to study the impact of the barrier thickness on the frequency shift δν in Eq. (30), we just have to analyze the numerator (35). Since R is unitary, we can show that Eq. (35) can be written as
x0(1)HδA(σ,ν0)x0(1)=σx0(1)H(Q0RHQ0R)Q0F+(eikxd2Λ)F+1x0(1)
(37)
and when only one propagating plane wave order exists, Eq. (30) simplifies to a scalar problem:
δν=σIm(R)(R(ν)R¯¯(ν))ν(eikxd2μ1)
(38)
which is a generalisation of Eq. (20) in Ref. [19

19. L. C. Botten, R. A. Hansen, and C. M. de Sterke, “Supermodes in multiple coupled photonic crystal waveguides,” Opt. Express 14, 387–396 (2006). [CrossRef] [PubMed]

].

In the cases where two propagating plane wave orders are considered, there are two evanescent modes with Bloch factors μ1 and μ2, and Eq. (37) takes the form
x0(1)HδA(ν0)x0(1)=σ(a1(eikxd2μ1)+a2(eikxd2μ2)).
(39)
If one of the evanescent Bloch modes is dominant, i.e., for instance, |μ1| > |μ2|, when is large enough |μ2| becomes negligible with respect to |μ1| and we get
δνσa1(eikxd2μ1).
(40)
When μ1 is associated with a dominant evanescent mode (as in Eqs (38) and (40)) the quantity (exp(ikx d/2) μ1) must be real. Otherwise, as shown below, we can find μ2μ1 such that |μ2| = |μ1|. If (exp(ikxd/2) μ) is real and negative, we get an oscillatory dependence since the sign of δν depends on the parity of .

Indeed when (exp(ikxd/2) μ1) has a nonzero imaginary part, if φ1(x, y) is an associated Bloch mode, from the invariance of the PC lattice with the geometric transformation (x,y)  (−x,y), it follows that for lossless PC, φ2=φ1*(x,y) is a Bloch mode associated with kx and
eikxd2μ2=(eikxd2μ1)*eikxd2μ1.
(41)
This also means that |μ1| = |μ2|, i.e., φ1(x, y) and φ2(x, y) form a pair of linearly independent dominant evanescent Bloch modes. If f1 and f1+ are vectors of plane wave components of the mode φ1(x, y) in the plane wave expansion (7), then φ2=φ1*(x,y) has as components f2=Q0f1+* and f2+=Q0f1*; matrix Q accounts for the fact that the transformation (x,y)  (−x,y) shifts the phase origins x0 = d/4 and x0 = −d/4. Furthermore, when (exp(ikx d/2)μ1) and (exp(ikx d/2)μ2) form a conjugate pair, to satisfy the requirement that x0(1)HδA(ν0)x0(1) be purely imaginary, it is sufficient that the prefactors a1 and a2 in Eq. (39) satisfy a2=a1*. Varying shows that this condition is also necessary. Thus for a pair of dominant evanescent modes, Eq. (39) becomes
x0(1)HδA(ν0)x0(1)=σ(a1(eikxd2μ1)a1*(eikxd2μ1*))=2σi|a1||μ1|sin((arg(μ1)+kxd2)+arg(a1))
(42)
which shows that x0(1)HδA(ν0)x0(1)/|μ1| is periodic with respect to ; in particular for some values of x0(1)HδA(ν0)x0(1)=0, i.e., δν = 0, although this root is physically meaningful only when is an integer; if that is the case we have a crossing between the even (σ = 1) and odd (σ = −1) dispersion curves of the double waveguides.

Figure 6 shows the root of δν versus kx d ∈ [0,π] when (kx, ν0) varies along the dispersion curve of a single waveguide; the crossing points correspond to integer values of the root . The results in Table 1 show that the first order perturbation theory agrees well with full numerical calculations. The perturbation theory is not accurate near kx d = 1.8555 where the dispersion relation of a single waveguide is degenerate. For the PCs studied here, such degeneracy occurs outside the braided region so that the perturbation theory is valid inside this region.

Fig. 6 Roots of the correction term δν(kx, d/λ, ) when (kx d, d/λ) varies along the dispersion curve of a single waveguide (purple dashed curve in Fig. 1). The roots occurring at = 4 and = 5 are indicated by green and cyan dots respectively (see Table 1 for their coordinates). The blue and red curves correspond, respectively, to the upper and lower frequency modes in the braided region in Fig. 1. The former enters the braided region at kx d = 1.998, the latter at kx d = 2.048).

Table 1. Crossing points (kx d, d/λ) of the even and odd CWM dispersion curves in Fig. 1, according to the dispersion results in Fig. 1 (columns “converged results”) and the roots of δν(kx, d/λ, ) (columns “perturbation theory”)

table-icon
View This Table

For kx d = π, all even values of > 0 are a root. Our theory explains this property. Since kx d = ±π are equivalent wave vector components (same quasi-periodicity with respect to e1), in addition to φ2=φ1*(x,y), φ˜2=φ1*(x,y) is also a permissible Bloch mode and since exp(iπd/2) = i, it follows that {iμ1,iμ2}={iμ1,iμ2}*={iμ1*,iμ2*}; from the assumption that 1 and 2 form a conjugate pair, this implies that μ1=μ1*, i.e., μ1 is real and μ2 = −μ1. Here Eq. (42) is imaginary if a1 is real. Thus when kx d = π, Eq. (42) becomes
x0(1)HδA(ν0)x0(1)=σa1((iμ1)(iμ1))=2σia1|μ1|sin(π2)
(43)
so that x0(1)HδA(ν0)x0(1)=0 when is even.

4. Discussion and conclusion

We have given a detailed description of coupling of hexagonal lattice PCW modes, showing that their dispersion curves intertwine due to the beating of two equally dominant evanescent Bloch modes in the barrier regions. This work highlights that there is a hierarchy in the understanding of coupled waveguides. In the simplest case involving two conventional waveguides the fundamental mode is always even and the second mode is odd. In the tight-binding limit these modes can be understood as even and odd superpositions of the modes of the individual waveguides. In square lattices, the modes are similar, but the fundamental mode can be odd and the second mode can be even. In the hexagonal lattices we have considered here the coupled modes can be considered complex superpositions of the modes of the individual waveguides, with coefficients which depend on the wavenumber, leading to the braiding effect. While the rigorous perturbation theory from Section 3 explains the observed behaviour very well (see Table 1), considerable insight may be obtained from intuitive description using the complex bands of the barrier region in Section 2.

Though all results described here were obtained for one particular structure, in which only the parameter , defining the thickness of the barrier separating the waveguide, was varied, the behaviour is generic and applies to coupled waveguides in any type of hexagonal lattice. Similarly, though the treatment here approximates the PCs as being two-dimensional, our results are generic and apply equally well to slab geometries.

Finally, the braiding leads to complicated dispersion relations, which may have implications for the study of slow light or for the creation of geometry induced, frequency selective index media. In practice the braiding may be somewhat difficult to observe since increasing the spacing between the waveguides, increases the number of intersection points, but decreases the amplitude of the oscillations.

A. Appendix

A.1. Dominant evanescent modes

We number the evanescent modes such that 1>|μ1||μ2||μ3| and 1<|μ1+||μ2+||μ3+|, i.e., they are numbered from the least evanescent to the most evanescent. If |μ1|>|μ2| and |μ1+|>|μ2+| the contribution from the evanescent modes φ1(x,y) and φ1+(x,y) dominate the series in Eq. (45) for large values of |y| so that the waveguide field |Ψ(x,y)| decays by a factor |μ1|=|μ1+| across each row of PC cylinders [as in Fig. 3(a)].

There are two dominant evanescent modes if |μ1|=|μ2|>|μ3| and |μ1+|=|μ2+|<|μ3+|. As we now show, for lossless PCs, a pair of dominant evanescent modes occurs when (β1d3/2) is not a multiple of π. The hexagonal lattice is invariant by the geometric transformation T : (x,y)  (−x,y), since T e1 = −e1 and T e2 = e2e1 are both lattice vectors. It follows that if φ1(x, y) is a Bloch mode associated with a wave vector k = [kx,β1 + 1], then, for a lossless PC, φ1*(x,y) is also a Bloch mode associated k = [−kx, – β1 + 1] thus, because of the symmetry, φ1*(x,y) is a Bloch mode which is associated with a permissible wave vector: k = [kx, – β1 + 1]. If (β1d3/2) is not a multiple of π, we can derive from the definition (44) that φ1(x, y) and φ1*(x,y) have the same decay rate κ but different phase factors; thus we can take φ2 as φ2=φ1*(x,y) and conclude that both φ1 and φ2 are dominant modes.

A.2. Field at the band edge

A special case occurs when kx is at the edge of the (one-dimensional) BZ edge. For lossless PCs, kx d = π and kx d = −π are equivalent. The field Ψ*(x,y) is then also a waveguide mode associated with kx and d/λ and if Ψ(x,y) is non-degenerate then there exist γ ∈ 𝒞 such that |γ| = 1 and Ψ*(x,y) = γΨ(x,y). Let γ̂ be a square root of γ, then γ̂Ψ(x,y) is a real since
Ψ*(x,y)=γ^2Ψ(x,y)Ψ*(x,y)=γ^γ^*Ψ(x,y)(γ^Ψ(x,y))*=γ^Ψ(x,y).
(46)
We can then consider the chosen mode Ψ(x,y) to be real. As a consequence, the dominant term (c1φ1(x,y)+c2φ1*(x,y)) must have real values. This implies that c1 and c2 are real since, as discussed in Sect. A.3, φ1(x, y) can be chosen to be real if it is non-degenerate. Since |c1| = |c2| we then have c2 = ±c1. The choice c2 = −c1 cancels the field (c1φ1(x,y)+c2φ1*(x,y)) along x = 0, but this would contradict the fact that in Fig. 2, the component Ey is strong around the cylinder center (x,y) = (0,0). Thus we must have c2 = c1. When translating the bulk lattice vector e2, times, we acquire a Bloch factor
μ1=eikxd/2eiβ1d3/2eκ1d3/2=(i)(i)eκ1d3/2=(1)eκ1d3/2
(47)
μ2=eikxd/2eiβ1d3/2eκ1d3/2=(i)(i)eκ1d3/2=eκ1d3/2
(48)
and the corresponding dominant field is
c1(φ1((x,y)e2)+φ2((x,y)e2))=c1(μ1φ1(x,y)+μ2φ2(x,y))
(49)
=c1eκ1d3/2((1)φ1(x,y)+φ1(x,y))
(50)

For all odd , the field cancels at the cylinder centers on the row y=3/2 and the coupling goes to zero for all odd causing the degeneracy seen in Fig. 2(c).

A.3. Properties of the phase factor at the band edge

When kx d = π, φ(x,y), φ(−x,y), φ*(x,y) and φ*(−x,y) are all permissible Bloch modes, i.e., they satisfy the same quasi-periodicity condition with respect to the lattice vector e1. They are associated with the wave vectors k = [±kx, ±β + ]. Since exp (±ikx d/2) = ±i, the wave vectors k = [±kx, ±β + ] correspond to at least two different phase factors μ [see Eq. (44)]. Indeed we have the three situations
  1. If exp(iβd3/2)=exp(iβd3/2), i.e., if (βd3/2) is a multiple of π, then we have a pair of opposite pure complex phase factors. We have φ*(−x,y) = γφ(x,y), with γ ∈ 𝒞, if there is no degeneracy.
  2. If exp(iβd3/2)=exp(iβd3/2), i.e., if (βd3/2)±π/2, then we have a pair of opposite real phase factors. We have φ*(x,y) = γφ(x,y), with γ ∈ 𝒞, if there is no degeneracy; furthermore, by following the arguments used for Eq. (46), it turn out that φ(x,y) can be chosen as a real valued function.
  3. If exp(iβd3/2)±exp(iβd3/2), then we have quadruple Bloch factors μ, −μ, μ* and −μ*.

Our numerical calculations confirm the occurrence of the first two cases; in the first gap of the hexagonal lattice, the dominant eigenvalues μ1 and μ2 are a pair of opposite real numbers while some higher order eigenvalues form a pair of opposite imaginary complex numbers. However we have not observed the last case, with quadruple evanescent modes having the same decay.

Acknowledgments

The authors thank P.Y. Chen and Dr A.A. Sukhorukov for useful discussions. Support of the Australian Research Council through its Centres of Excellence Program is acknowledged.

References and links

1.

S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. 32, 2981–2983 (2007). [CrossRef] [PubMed]

2.

A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). [CrossRef]

3.

A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion in coupled periodic waveguides,” J. Opt. Soc. Am. B 25, C65–C74 (2008). [CrossRef]

4.

D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

5.

A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,” IEEE Photon. Technol. Lett. 15, 694 –696 (2003). [CrossRef]

6.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

7.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.

8.

K. Iizuka, Elements of Photonics, Volume II (Wiley, 2002). Chap. 9.

9.

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004). [CrossRef]

10.

A. Locatelli, M. Conforti, D. Modotto, and C. D. Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343–1345 (2006). [CrossRef] [PubMed]

11.

S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16, 1104–1114 (2008). [CrossRef] [PubMed]

12.

A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, “Slow-light vortices in periodic waveguides,” J. Opt. A, Pure Appl. Opt 11, 094016 (2009). [CrossRef]

13.

L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, “Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method,” Int. J. Microwave Opt. Technol. 1, 133–145 (2006).

14.

F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, “Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides,” Opt. Express 12, 1119–1125 (2004). [CrossRef] [PubMed]

15.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009). [CrossRef] [PubMed]

16.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, “Modes of symmetric composite defects in two-dimensional photonic crystals,” Phys. Rev. A 80, 013826 (2009). [CrossRef]

17.

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]

18.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory,” Phys. Rev. E 70, 056606 (2004). [CrossRef]

19.

L. C. Botten, R. A. Hansen, and C. M. de Sterke, “Supermodes in multiple coupled photonic crystal waveguides,” Opt. Express 14, 387–396 (2006). [CrossRef] [PubMed]

OCIS Codes
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Integrated Optics

History
Original Manuscript: September 23, 2010
Revised Manuscript: November 12, 2010
Manuscript Accepted: November 13, 2010
Published: November 19, 2010

Citation
J. Scott Brownless, Sahand Mahmoodian, Kokou B. Dossou, Felix J. Lawrence, Lindsay C. Botten, and C. Martijn de Sterke, "Coupled waveguide modes in hexagonal photonic crystals," Opt. Express 18, 25346-25360 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25346


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References

  1. S. Kubo, D. Mori, and T. Baba, "Low-group-velocity and low-dispersion slow light in photonic crystal waveguides," Opt. Lett. 32, 2981-2983 (2007). [CrossRef] [PubMed]
  2. A. Y. Petrov, and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004). [CrossRef]
  3. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, "Slow-light dispersion in coupled periodic waveguides," J. Opt. Soc. Am. B 25, C65-C74 (2008). [CrossRef]
  4. D. Mori, and T. Baba, "Dispersion-controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004). [CrossRef]
  5. A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photon. Technol. Lett. 15, 694-696 (2003). [CrossRef]
  6. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005). [CrossRef] [PubMed]
  7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). Chap. 11.
  8. K. Iizuka, Elements of Photonics, Volume II (Wiley, 2002). Chap. 9.
  9. C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, "Modes of coupled photonic crystal waveguides," Opt. Lett. 29, 1384-1386 (2004). [CrossRef]
  10. A. Locatelli, M. Conforti, D. Modotto, and C. D. Angelis, "Discrete negative refraction in photonic crystal waveguide arrays," Opt. Lett. 31, 1343-1345 (2006). [CrossRef] [PubMed]
  11. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar, "Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers," Opt. Express 16, 1104-1114 (2008). [CrossRef] [PubMed]
  12. A. A. Sukhorukov, S. Ha, A. S. Desyatnikov, A. V. Lavrinenko, and Y. S. Kivshar, "Slow-light vortices in periodic waveguides," J. Opt. A, Pure Appl. Opt. 11, 094016 (2009). [CrossRef]
  13. L. C. Botten, K. B. Dossou, S. Wilcox, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, and A. A. Asatryan, "Highly accurate modelling of generalized defect modes in photonic crystals using the FSS method," Int. J. Microwave Opt. Technol. 1, 133-145 (2006).
  14. F. S. Chien, Y. Hsu, W. Hsieh, and S. Cheng, "Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides," Opt. Express 12, 1119-1125 (2004). [CrossRef] [PubMed]
  15. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009). [CrossRef] [PubMed]
  16. K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. Martijn de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009). [CrossRef]
  17. 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]
  18. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. theory," Phys. Rev. E 70, 056606 (2004). [CrossRef]
  19. L. C. Botten, R. A. Hansen, and C. M. de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express 14, 387-396 (2006). [CrossRef] [PubMed]

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