Universal coupled-mode theory formulation of quasi-normal modes in a 1D photonic crystal

Abstract

In terms of coupled-mode theory, only two real-valued normalized parameters are needed for describing the spectral properties of any 1D photonic crystal of finite size made of lossless materials: detuning (δL) with respect to the Bragg condition and coupling strength (κL). As far as its quasi-normal mode (QNM) spectrum is concerned, the usual “complex frequency” concept can be replaced by an effective “complex detuning” (δL + i αL), thus establishing a formal connection with the modal cartography, in the (κL, αL) plane, of purely index-coupled distributed-feedback lasers of normalized gain αL.

Appendices

Appendix 1: Couplonic equivalence

For the sake of completeness, we recall the correspondence rules enabling one to extract (κΛ, δΛ) from the elementary matrix [m] of any lossless unit cell, as established by (Matuschek et al. 1997). For the sake of simplicity, we also assume the unit cell to be symmetric. In all cases, det[m] = 1. Energy conservation ensures m₂₂ = m₁₁*, m₂₁ = m₁₂*. Because of symmetry, m₂₁ = − m₁₂. Introducing the half-trace V_m = (m₁₁ + m₂₂)/2, one can write the matrix elements as:

$$m_{11} = - \left( {\cosh \gamma \Lambda + i\;\frac{\delta \Lambda }{{\gamma \Lambda }}\;\sinh \gamma \Lambda } \right) = m_{22} *,$$

(14a)

$$m_{12} = - \left( { + i\;\frac{\kappa \Lambda }{{\gamma \Lambda }}\;{\text{e}}^{i\;\psi } \;\sinh \gamma \Lambda } \right) = m_{12} *,$$

(14b)

$${\text{with}}\quad \gamma \Lambda = {\text{arccosh}}( - V_{m} ) = \left( {(\kappa \Lambda )^{2} - (\delta \Lambda )^{2} } \right)^{1/2}$$

(14c)

By formal identification, we get eventually:

$$\kappa \Lambda \;{\text{e}}^{i\;\psi } = i\;\frac{\gamma \Lambda }{{\sqrt {V_{m}^{2} - 1} }}\;m_{12} ,$$

(15a)

$$\delta \Lambda = i\;\frac{\gamma \Lambda }{{\sqrt {V_{m}^{2} - 1} }}\;\left( {\frac{{m_{11} - m_{22} }}{2}} \right).$$

(15b)

These two quantities are real-valued. In order to keep κΛ > 0, phase ψ is put to either 0 or π, according to the sign of the right-hand side. We would like to point out that the procedure is easily extendable, at the cost of some extra degrees of freedom, to any asymmetric unit cell with loss or gain (Boucher 2000).

Appendix 2: Partial transfer matrices

In order to draw the internal mode profile, we use partial transfer matrices [L(z)] and [R(z)] such as [M] = [L][R]. Partial matrix [L], extending from z₀ = 0 to z, has exactly the same form than [M] except for its length (z instead of L):

$$L_{11} = \left( {\cosh (\gamma z) + i\;\frac{\delta }{\gamma }\;\sinh (\gamma z)} \right)\;{\text{e}}^{{ + i\,\beta_{B} z}} ,$$

(16a)

$$L_{12} = \left( { + i\;\frac{{\kappa \,{\text{e}}^{ + i\,\psi } }}{\gamma }\;\sinh (\gamma z)} \right)\;{\text{e}}^{{ - i\,\beta_{B} z}} ,$$

(16b)

$$L_{21} = \left( { - i\;\frac{{\kappa \,{\text{e}}^{ - i\,\psi } }}{\gamma }\;\sinh (\gamma z)} \right)\;{\text{e}}^{{ + i\,\beta_{B} z}} ,$$

(16c)

$$L_{22} = \left( {\cosh (\gamma z) - i\;\frac{\delta }{\gamma }\;\sinh (\gamma z)} \right)\;{\text{e}}^{{ - i\,\beta_{B} z}} .$$

(16d)

On the other hand, partial matrix [R], extending from z to z_S = L, exhibits phase terms that must be correctly accounted for, especially in its anti-diagonal elements:

$$R_{11} = \left( {\cosh \gamma (L - z) + i\;\frac{\delta }{\gamma }\;\sinh \gamma (L - z)} \right)\;{\text{e}}^{{ + i\,\beta_{B} (L - z)}} ,$$

(17a)

$$R_{12} = \left( { + i\;\frac{{\kappa \,{\text{e}}^{ + i\,\psi } }}{\gamma }\;\sinh \gamma (L - z)} \right)\;{\text{e}}^{{ - i\,\beta_{B} (L + z)}} ,$$

(17b)

$$R_{21} = \left( { - i\;\frac{{\kappa \,{\text{e}}^{ - i\,\psi } }}{\gamma }\;\sinh \gamma (L - z)} \right)\;{\text{e}}^{{ + i\,\beta_{B} (L + z)}} ,$$

(17c)

$$R_{22} = \left( {\cosh \gamma (L - z) - i\;\frac{\delta }{\gamma }\;\sinh \gamma (L - z)} \right)\;{\text{e}}^{{ - i\,\beta_{B} (L - z)}} .$$

(17d)

It is easily checked that [L(L)] = [R(0)] = [M], whereas [L(0)] = [R(L)] = Id, the (2 × 2) identity matrix.