## Supermodes in multiple coupled photonic crystal waveguides

Optics Express, Vol. 14, Issue 1, pp. 387-396 (2006)

http://dx.doi.org/10.1364/OPEX.14.000387

Acrobat PDF (1070 KB)

### Abstract

We analyze the supermodes in multiple coupled photonic crystal waveguides for long-wavelengths. In the tight-binding limit we obtain analytic results that agree with fully numerical calculations. We find that when the field flips sign after a single photonic crystal period, and there is an odd number of periods between adjacent waveguides, the supermode order is reversed, compared to that in conventional coupled waveguides, generalizing earlier results obtained for two coupled waveguides.

© 2006 Optical Society of America

## 1. Introduction

1. See, e.g., Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in
GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up
to 1 cm in length” Opt. Express **12**, 1090–1096
(2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090 [CrossRef] [PubMed]

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

3. 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]

3. 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]

3. 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]

*all*modes, both bound and unbound [3

**29**, 1384–1386
(2004). [CrossRef]

**29**, 1384–1386
(2004). [CrossRef]

*tight-binding limit*, analytic results can be obtained that are the PC equivalent of well-known expressions in the literature [4]. We develop this theory in Section 2 and present the results, both in terms of the propagation constants and in terms of the associated fields, in Section 3. Section 4 contains a discussion of our results and concludes.

## 2. Theory

*β*=

_{p}*β*+ 2

*πp*/

*d*, χ

_{p}= (

*k*

^{2}-

*β*

_{p}

^{2})

^{1/2},

*k*= 2π/λ, with λ denoting the wavelength in the background medium, and

*y*=

*y*

_{j}denoting the center line of guide

*j*. The factor χ

_{p}

^{-1/2}is included to normalize energy calculations conveniently [6

6. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic bandstructure calculations using
scattering matrices,” Phys. Rev. E **64**, 046603:1–18
(2001). [CrossRef]

**= [**

*f*

*f*_{-}

^{T}

**f**_{+}

^{T}]

^{T}, we apply Bloch’s theorem to derive a transfer matrix equation ℑ

*= μ*

**f***. Here, the transfer matrix ℑ may be calculated using the reflection and transmission scattering matrices of a single grating layer (*

**f***c.f*. Eq. (6) below). The set of Bloch modes that arise from the solution of the eigenvalue problem may be partitioned into sets of forward and backward propagating states, with the forward modes associated with a matrix of eigenvalues

**Λ**= diag μ

_{j}(with |μ

_{j}| = 1 for propagating modes and |μ

_{i}| < 1 for evanescent modes) and eigenvectors [

**f**_{j}

^{-T}

**f**_{j}

^{+T}. For a square lattice, the corresponding set of backward modes have Bloch factors

**Λ**

^{-1}and eigenvectors [

**f**_{j}

^{+T}

**f**_{j}

^{-T}]

^{T}. The two key quantities to emerge from this eigenproblem are thus the matrix of Bloch factors

**Λ**, which determines the propagation of modes in the bulk crystal, and the scattering matrix

**R**_{∞}=

*+ (*

**F****F**

_{-})

^{-1}which characterizes the reflection of a field incident on a semi-infinite bulk crystal from free space. Here,

**F**

_{±}are matrices whose columns are the eigenvector components

**f**_{±}, with the matrices

**F**

_{-}and

**F**

^{-1}

_{-}respectively performing changes of basis from plane waves to Bloch modes and

*vice versa*[6

6. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic bandstructure calculations using
scattering matrices,” Phys. Rev. E **64**, 046603:1–18
(2001). [CrossRef]

*l*-layer barriers [6

6. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, “Photonic bandstructure calculations using
scattering matrices,” Phys. Rev. E **64**, 046603:1–18
(2001). [CrossRef]

**Q**^{l}=

*F*_{-}

**Λ**

^{l}

*F*^{-1}quantifies the propagation of the Bloch modes within the

*l*-layer barriers.

*m*- 1) barriers that are sandwiched between two semi-infinite PCs. The presence of each of the guides of width

*h*is taken into account, by “padding,” relative to the field origins located in the center of the guides, the relevant scattering matrices to allow for propagation to and from the phase origin. The fields in the uppermost and lowermost guides thus satisfy

*m*-1) barrier structure. In Equations (4) and (5),

*R̃*_{∞}=

*P*^{1/2}

*R*_{∞}

*P*^{1/2}, with

**= diag[exp(**

*P**iχ*)] representing for each plane wave order, the phase accumulation when traversing a waveguide of width

_{p}h*h*.

*R̃*_{n}and

**T̃**_{n}, for the

*n*=

*m*- 1 “padded” barriers can be derived by recursion [7

7. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Electromagnetic scattering and propagation
through grating stacks of metallic and dielectric cylinders for photonic crystal
calculation. Part 1: Formulation”, J. Opt. Soc.
Am. **17**, 2165–76
(2000). [CrossRef]

**R̃**_{1}=

*P*^{1/2}

*R*_{l}

*P*^{1/2}and

*T̃*_{1}=

*P*^{1/2}

*T*_{l}

*P*^{1/2}, we will follow an alternative route that involves the calculation of the Bloch modes of an infinite structure composed of these padded barrier sections. This leads us to the dispersion equation in a form that is suitable for computational use, and which is also amenable to an asymptotic analysis in the long wavelength and tight binding limits. To do so, we introduce the transfer matrix

*for crossing a padded layer. Proceeding as before, we may solve the eigenvalue problem for this transfer matrix, deduce the corresponding Bloch factors*ℑ ˜

**and eigenvector matrices**Λ ˜

**F̃**_{±}, and compute the reflection and transmission matrices

*R*_{n}and

*T*_{n}in a form analogous to Equations (2) and (3). We next introduce the transfer matrix to cross the

*n*=

*m*-1 padded barriers, namely

*satisfies the symmetry relationship*ℑ ˜

ℑ ˜

^{-1}=

*U*

ℑ ˜

*U*, where

*U*is the matrix that reverses the partition order. Accordingly, some straightforward manipulation reveals that

*g*^{±}

_{m}, = σ

*g*^{∓}

_{1}.

*g*^{-}

_{1}=

*R̃*_{∞}

*g*^{+}

_{m}and

*g*^{+}=

*R̃*_{∞}

*g*^{-}

_{m}, and the symmetry relations

*g*^{±}

_{m}= σ

*g*^{∓}

_{1}into the transfer matrix equation (7). This leads to

*T̃*_{n}being invertible. The modes of the structure may now be found by solving det

**(**

*M**k*,

*β*) = 0 for the propagation constant

*β*, after which the mode may be reconstructed from the null vector

**.**

*g**n*

^{th}power, namely

*u*(

_{n}*t*) = sin[(

*n*+1)ϕ]/sinϕ denotes the Chebyshev polynomial of the second kind of degree

*n*in the variable

*t*= cos ϕ, and

*R*and

_{l}*T*are given respectively by the scalar versions of Equations (2) and (3),

_{l}

*R*_{∞}by its specular (0,0)

^{th}order component ρ, and the matrix

**by the single, dominant Bloch factor μ for the bulk crystal, a number which is real and has a magnitude |μ| <**

*Q*^{1}(since we are operating in a band gap).

*l*is sufficiently large that ξ = μ

^{l}is small. Guided by the scalar form of the dispersion equation for a single waveguide,

*P*

_{ρ}= ± 1, in which both

*P*= exp(

*iχh*) and ρ have unit magnitude, the latter because we are operating in a band gap, we seek a solution of the form

*v*is a small argument that represents the effect on the phase upon reflection off a waveguide-cladding interface due to presence of all other waveguides. While we may expand

*v*(ξ) in a power series and seek a self-consistent perturbation solution to arbitrary order, we need to work only to first order with

*v*(ξ) = αξ to obtain the results that we are after. In what follows, we work with the for

*P*

_{ρ}= exp[

*iv*(ξ)] corresponding to the fundamental symmetric mode for each of the single waveguides. The ansatz

*P*

_{ρ}= - exp[

*iv*(ξ)] gives essentially the same result and it is therefore not considered further here.

*u*(

_{n}*t*)}, namely

*u*(

_{n}*t*) = 2

*tu*-1(

_{n}*t*) -

*u*-2(

_{n}*t*), we derive the tight binding form of the dispersion relation which, correct to first order, is

*T*_{n}is invertible, translates in the long wavelength limit to

*T*≠ 0 which, physically, requires the guides be not completely isolated from one another. To first order, in the tight binding limit, this requires that

_{n}*u*

_{n-1}(

*t*) ≠ 0. Subject to this constraint we solve (18) to deduce the solutions

*t*= cosϑ

_{s}_{s}, with ϑ

_{s}=

*sπ*/(

*m*+ 1), for

*s*= 1,2,…,

*m*, with odd values of

*s*corresponding to the symmetric, or even, modes (σ = 1) and with even values of

*s*corresponding to the antisymmetric, or odd, modes (σ = - 1). Here, the integer

*s*enumerates the different supermodes of the structure.

*χh*+ arg(ρ) = αξ, into which we substitute the possible values of α = α

_{s}= 2ℑ(ρ) cos ϑ

_{s}. Thus,

*β*is the propagation constant of mode

_{s}*s*and

*j*is an integer. Since the corresponding form for a single waveguide is

*h*+ arg[ρ(

*β*

_{0})] =

*jπ*, we deduce from a simple power series expansion of the left hand side of Eq. (19) that

*β*, this leads to an implicit equation that needs to be solved self consistently. However, in the spirit of the tight-binding approximation we evaluate it at

_{s}*β*

_{0}leading to an explicit expression for the

*β*

_{s}*β*

_{0})/∂

*β*corresponds to the lateral beam displacement δ

*x*due to the Goos-Hänchen shift [4]. The displacement can be associated with a barrier penetration

*t*= -χ

_{0}δ

*x*/ (2β

_{0}); note that in the special case of a planar waveguide the parameter

*t*corresponds to the 1/

*e*decay length of the evanescent field in the cladding. With this, the magnitude of the denominator of Eq. (20) can be written as (

*h*+ 2

*t*)/χ

_{0}=

*h*

_{eff}/χ

_{0}, where the effective width

*h*

_{eff}, a standard parameter in the theory of planar waveguides, can be considered to be the width of the mode that includes the effect of the evanescent tails. Equation (20) can thus be simplified as

*j*, the plane wave field (1) originates from multiple reflections and transmissions through the structure and we write

*g*^{±}

_{j}in terms of

*g*^{-}

_{1}and

*g*^{+}

_{m}, from which we calculate the fields in each of the guides, using the null vectors of the dispersion equation (10) for both the symmetric (σ = 1) and antisymmetric (σ = -1) driving fields

*g*^{+}

_{1}= σ

*g*_{m}

^{-}=

**. Of particular interest in the long wavelength limit is the total field which is represented by the amplitude of the specular order plane wave**

*g**s*, we assign

*g*

^{+}

_{1}= σ

*g*

_{m}

^{-}= 1 according to its symmetry, and perform an asymptotic analysis of the total field (23) in the tight binding limit (|ξ| ≪ 1), leading to

## 3. Results

*a*and refractive index

*n*= 3 are arranged in a square lattice with period

*d*, where

*a*= 03

*d*. We consider W1 waveguides,

*i.e.*waveguides that are formed by the removal of one row of inclusions, so that

*h*=

*d*, and solutions in which the electric field points in the direction parallel to the inclusions. At the normalized frequency

*d*/χ = 0.32, the single mode of a single, isolated waveguide has a propagation constant

*β*

_{0}

*d*= 1.16223. The dominant Bloch function here has μ = -0.468476, and ρ = exp(-0.52224

*πi*). Since μ < 0, this is the situation in which the fundamental mode of two coupled waveguides is odd when

*l*is odd [3

**29**, 1384–1386
(2004). [CrossRef]

*β*

_{2}

_{s}-

*β*

_{0}

^{2}with increasing thickness of the separating layer

*l*for

*m*= 5 (Fig. 2(a)) and for

*m*= 6 (Fig. 2(b)). The propagation constants are calculated using both an independent numerical method [9

9. 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
modelling extended photonic crystal structures. Part I:
Theory”, Phys. Rev. E **70**, 056606, 1–13
(2004). [CrossRef]

**64**, 046603:1–18
(2001). [CrossRef]

*l*and that, as expected from the μ

^{l}factor in Eq. (20), the

*β*values approach

_{s}*β*

_{0}with increasing

*l*. The color of the dots indicates whether the super mode is even (red dots) or odd (blue dots). Note that for

*m*= 5 the fundamental mode, the mode with the highest propagation constant, is always even, whereas for

*m*= 6 the fundamental mode is alternatingly even (when

*l*is even), or odd (when

*l*is odd). We return to this below.

*β*

_{s}of the supermodes is that the μ

^{l}in (20) causes them all to converge exponentially to

*β*

_{0}with large

*l*, thus making the verification of the key asymptotic form (20) difficult. Accordingly, we scale out the exponential term by plotting values of

^{l}, rather than μ

^{l}, so that the multiplying factor, and therefore the order in which the modes appear, does not change sign with

*l*. The results are depicted by the dots in Figures 3, with

*m*= 5 in Fig. 3(a) and

*m*= 6in Fig. 3(b). The dots again indicate exact numerical results, with the red dots corresponding to even supermodes and blue dots corresponding to odd super-modes. Note that the color patterns are the same as in Figures 2. According to Equations (20) and (21), and plotted in this way, the normalized propagation constants should not depend on

*l*and should be given by

*t*= (- 1)

^{l}cosϑ

_{s}where ϑ

_{s}=

*sπ*/(

*m*+ 1) (10). The horizontal lines in Figures 3 indicate the values of cos ϑ

_{s}.

*l*≥ 6, since the dots converge to the the results given by Eq. (25). In both figures, the dots converge to the asymptotic lines. The quality of the agreement is not surprising since |μ|

^{l}< |μ|

^{6}≪ 0.01, indicating that ξ ≪ 1 as required for the tight-binding approximation to apply. However, even for

*l*= 3, for which |μ|

^{3}≈ 0.1, the tight-binding result (20) is fairly accurate, though the need for higher order corrections can be discerned. Fig. 3(b) is similar to Fig. 3(a), but is for a set of

*m*= 6 coupled waveguides and similar conclusions can be drawn as for Fig. 3(a). Note also that the colors of the dots in Figures 3 are consistent with the earlier observation that modes associated with cos ϑ

_{s}for odd

*s*are even while those for even s are odd.

*m*= 5 the fundamental mode is always even, whereas for

*m*= 6 the fundamental mode alternates between even and odd symmetry according to whether

*l*is odd or even. We find that this applies to all even and odd

*m*, respectively, and is consistent with the result for

*m*= 2 [3

**29**, 1384–1386
(2004). [CrossRef]

*j*are given by Eq. (24), which are consistent with results for conventional waveguides. Figures 4 and 5 show the supermodes for

*m*= 5 and

*m*= 6, respectively, both for

*l*= 6 (Figures 4(a) and 5(a)), and for

*l*= 7 (Figures 4(b) and 5(b)). For

*l*= 6 (Figures 4(a) and 5(a)) the order of the modes is identical to those for conventional structures: in the fundamental supermodes the fields in the waveguides are mutually in phase, whereas in the highest order supermode the fields in adjacent waveguides are out of phase. The key difference with conventional structures occurs when

*l*is odd (and μ < 0, the case we consider here). In this case the order of the supermodes is reversed with respect to that for conventional coupled waveguides: for example, in the highest order supermode the fields in all the waveguides are in phase, whereas in conventional waveguides the field of the fundamental supermode has this property. This is illustrated in Figures 4(b) and 5(b). This shows that when

*m*is even, the fundamental mode is odd (Figures 5(b)), consistent with the earlier result for

*m*= 2 [3

**29**, 1384–1386
(2004). [CrossRef]

*m*= 2.

## 4. Discussion and Conclusions

11. C.M. de Sterke, “Superstructure gratings in the tight-binding
approximation,” Phys. Rev. E **57**3502–3509 (1998). [CrossRef]

12. M. Bayindir, B. Temelkuran, and E. Özbay, “Tight-binding description of the coupled
defect modes in three-dimensional photonic crystals,”
Phys. Rev. Lett. **84**, 2140 (2000). [CrossRef] [PubMed]

13. S. Mookherjea and A. Yariv, “Optical pulse propagation in the
tight-binding approximation,” Opt.
Express **9**, 91–96
(2001). [CrossRef] [PubMed]

11. C.M. de Sterke, “Superstructure gratings in the tight-binding
approximation,” Phys. Rev. E **57**3502–3509 (1998). [CrossRef]

12. M. Bayindir, B. Temelkuran, and E. Özbay, “Tight-binding description of the coupled
defect modes in three-dimensional photonic crystals,”
Phys. Rev. Lett. **84**, 2140 (2000). [CrossRef] [PubMed]

13. S. Mookherjea and A. Yariv, “Optical pulse propagation in the
tight-binding approximation,” Opt.
Express **9**, 91–96
(2001). [CrossRef] [PubMed]

^{l}, which is associated with exponential decay of the field between adjacent wells or waveguides. However, here we are interested in the multiplying prefactor. The presence of

*h*

_{eff}in the denominator of (21) is consistent with the perturbative results for planar waveguides, where this parameter often appears.

*m*= 2 guides, we represent the fields by plane wave expansions (1) and write down relationships analogous to Equations (4) and (5) in which, in scalar form,

*R̃*

_{∞}= ρP,

*R̃*

_{1}=

*R*

_{l}

*P*, and

*T̃*

_{1}=

*T*. In the tight binding limit, in which we neglect terms of

_{l}P*O*(ξ

^{2}) or higher, we approximate

*R*≈ ρ and

_{l}*T*≈ (1 - ρ

_{l}^{2})ξ. Appropriate to the case of a symmetric single guide mode, we construct the following total field with

*v*=

_{j}*g*

_{j}^{-}+

*g*

_{j}^{+}by adding the pairs of Equations (4) and (5) for

*g*

_{j}^{±}, while for an antisymmetric mode, we would construct a quantity proportional to the field derivative,

*v*=

_{j}*g*-

_{j}*g*

_{j}^{+}. For the symmetric case, then, with

*v*=

_{j}*g*

_{j}^{-}+

*g*

_{j}^{+}, we have

*g*

^{-}

_{1}and

*g*

^{+}

_{1}to the total field according to

*g*

^{-}

_{1}= ρ

*P*/(1 + ρ

*P*)

*v*

_{1}≈ ρ

*P*

*v*

_{1}/2 and similarly

*g*

^{+}

_{1}≈ ρ

*P*

*v*

_{2}/2 for the symmetric single mode case, for which

*P*ρ ≈ 1. The final form of the dispersion relation is then derived from the coupled system of equations

*v*= ρ

_{j}*Pv*for

_{j}*j*= 1,2 are perturbed by the presence of the neighboring guide through the term -

*i*ℑ(ρ)ξ

*v*for

_{l}*l*= 2,1. It is thus clear that the factor ℑ(ρ) arises directly from the transmission coefficient for a sequence of

*l*layers, and we observe that if ρ = ± 1 (as for an electric or magnetic mirror) the perturbation vanishes, in keeping with the guides being completely isolated.

*l*is odd. As a consequence of this, we find that the fundamental supermode is then odd when

*m*is even, consistent with earlier findings [3

**29**, 1384–1386
(2004). [CrossRef]

## Acknowledgments

## References and links

1. | See, e.g., Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in
GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up
to 1 cm in length” Opt. Express |

2. | A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional
couplers,” IEEE Photonics Technol. Lett. |

3. | 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. |

4. | P. Yeh, |

5. | L. D. Landau and E.M. Lifshitz, |

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

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

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

9. | 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
modelling extended photonic crystal structures. Part I:
Theory”, Phys. Rev. E |

10. | D. Felbacq, A. Moreau, and Rafik Smaâli, “Goos-Hänchen effect in the gaps of
photonic crystals,” Opt. Lett. |

11. | C.M. de Sterke, “Superstructure gratings in the tight-binding
approximation,” Phys. Rev. E |

12. | M. Bayindir, B. Temelkuran, and E. Özbay, “Tight-binding description of the coupled
defect modes in three-dimensional photonic crystals,”
Phys. Rev. Lett. |

13. | S. Mookherjea and A. Yariv, “Optical pulse propagation in the
tight-binding approximation,” Opt.
Express |

14. | N.W. Ashcroft and N.D. Mermin |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(130.2790) Integrated optics : Guided waves

(230.7370) Optical devices : Waveguides

**ToC Category:**

Photonic Crystals

**Citation**

L. C. Botten, R. A. Hansen, and C. Martijn de Sterke, "Supermodes in multiple coupled photonic crystal waveguides," Opt. Express **14**, 387-396 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-387

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### References

- See, e.g., Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length" Opt. Express 12, 1090-1096 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090</a>. [CrossRef] [PubMed]
- A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photonics Technol. Lett. 15, 694-696 (2003). [CrossRef]
- 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]
- P. Yeh, Optical Waves in Layered Media (Wiley, Hoboken, 1988).
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 3rd Ed. (Pergamon, Oxford, 1977).
- L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, "Photonic bandstructure calculations using scattering matrices," Phys. Rev. E 64, 046603:1-18 (2001). [CrossRef]
- L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculation. Part 1: Formulation", J. Opt. Soc. Am. 17, 2165-76 (2000). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 6th Edition (Pergamon, Oxford, 1980), p. 66.
- 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 modelling extended photonic crystal structures. Part I: Theory," Phys. Rev. E 70, 056606, 1-13 (2004). [CrossRef]
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