Complex coupled-mode theory for optical waveguides

Wei-Ping Huang; Jianwei Mu

doi:10.1364/OE.17.019134

Optics Express
Vol. 17,
Issue 21,
pp. 19134-19152
(2009)
•https://doi.org/10.1364/OE.17.019134

Complex coupled-mode theory for optical waveguides

Wei-Ping Huang and Jianwei Mu

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Wei-Ping Huang and Jianwei Mu, "Complex coupled-mode theory for optical waveguides," Opt. Express 17, 19134-19152 (2009)

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Abstract

A coupled-mode formulation is described in which the radiation fields are represented in terms of discrete complex modes. The complex modes are obtained from a waveguide model facilitated by the combination of perfectly matched boundary (PML) and perfectly reflecting boundary (PRB) condition. By proper choice of the PML parameters, the guided modes of the structure remain unchanged, whereas the continuous radiation modes are discretized into orthogonal and normalizable complex quasi-leaky and PML modes. The complex coupled-mode formulation is identical to that for waveguides with loss and/or gain and can be solved by similar analytical and numerical techniques. By identifying the phase-matching conditions between the complex modes, the coupled mode formulation may be further simplified to yield analytical solutions. The complex coupled-mode theory is applied to Bragg grating in slab waveguides and validated by rigorous mode-matching method. It is for the first time that we can treat guided and radiation field in a unified and straightforward fashion without having to resort to cumbersome radiation modes. Highly accurate and insightful results are obtained with consideration of only the nearly phase-matched modes.

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Fig. 1 Ideal waveguide models

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Fig. 2 (a) Real and imaginary parts of the effective indices for the first twenty (20) modes (in descent order based on the values of the real part for the mode effective indices) supported in the waveguide for the different outer cladding indices. (b) A blow-up view for the portion of graph in (a) enclosed by the dashed circle.

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Fig. 3 Mode leakage loss for the quasi-leaky modes.

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Fig. 4 Mode field patterns for the three waveguide structures. (a) the fundamental guided mode; (b) the quasi-leaky cladding modes; (c) the PML modes.

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Fig. 5 Field confinement factor as a function of PML parameters.

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Fig. 6 Field confinement factors in core and cladding region as functions of computation window (

n_{s}

= 1.455).

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Fig. 7 Mode field overlap integrals for different modes. (a)Non-conjugate overlapping integral using Eqs. (13); (b) Conjugate overlapping integral using Eqs. (12). (

n_{s} = 1.45

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Fig. 8 Volume Bragg grating structure based on slab waveguides

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Fig. 9 The transmission spectrum for Case A with lower outer cladding index n_s = 1.0. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the reduced CMT involving only two phase matching modes; (b) The transmission spectrum calculated by the reduced CMT (dash lines), the full CMT (dotted lines) and the rigorous MMM (solid lines).

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Fig. 10 The transmission spectrum for Case B with equal outer cladding index n_s = 1.450. (a) Phase matching wavelengths, corresponding coupling strengths, and the transmission spectrum predicted by the full CMT involving from 2 up to 11 modes; (b) The transmission spectrum calculated by the full CMT (dotted lines) and the rigorous MMM (solid lines).

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Fig. 12 Transmission spectra with n_s lager than n_cl. (a) n_s = 1.60; (b) n_s = 1.90.

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Fig. 11 Transmission spectra with index of the outer cladding n_s slightly larger than the index of the inner cladding n_cl (n_s = 1.455).

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Equations (81)

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\nabla_{t} \times e_{n} - j γ_{n} e_{n} = - j ω μ_{o} [Λ] h_{n}

\nabla_{t} \times h_{n} - j γ_{n} h_{n} = + j ω ε [Λ] e_{n}

[Λ] = [\begin{array}{c} S_{y} / S_{x} & 0 & 0 \\ 0 & S_{x} / S_{y} & 0 \\ 0 & 0 & S_{x} S_{y} \end{array}]

S_{x} = κ_{x} - j \frac{σ_{x}}{ω ε}

S_{y} = κ_{y} - j \frac{σ_{y}}{ω ε}

σ = σ_{\max} {(\frac{ρ}{T_{P M L}})}^{m}

R_{P M L} = \exp {- \frac{2 σ_{\max}}{n \sqrt{ε_{o} / μ_{o}}} \int_{0}^{T_{P M L}} {(\frac{ρ}{T_{P M L}})}^{m} d ρ}

S = κ - j \frac{λ}{4 π n T_{P M L}} [(m + 1) \ln (\frac{1}{R_{P M L}})] {(\frac{ρ}{T_{P M L}})}^{m}

γ_{n} = β_{n} - j α_{n}

\nabla_{t} \times e_{- n} - j γ_{- n} e_{- n} = - j ω μ_{o} [Λ] h_{- n}

\nabla_{t} \times h_{- n} - j γ_{- n} h_{- n} = + j ω ε [Λ] e_{- n}

γ_{n} = - γ_{n}

e_{t n} = + e_{t, - n}

e_{_{z n}} = - e_{_{z, - n}}

h_{t n} = - h_{t, - n}

h_{z n} = + h_{z, - n}

\iint_{} (e_{t m} \times h_{t n}) \cdot \hat{z} d a = 0

\frac{1}{2} ℜ \iint_{} (e_{t n}^{} \times h_{t n}^{*}) \cdot \hat{z} d a = 1

\iint_{} (e_{t m}^{} \times h_{t n}^{*}) \cdot \hat{z} d a \neq 0

\tilde{ε} (x, y, z) = ε (x, y) + Δ ε (x, y, z)

\nabla \times E (x, y, z) = - j ω μ_{o} H (x, y, z)

\nabla \times H (x, y, z) = + j ω \tilde{ε} (x, y, z) H (x, y, z)

E_{t} (x, y, z) = \sum_{n = 1} [a_{n} (z) + b_{n} (z)] e_{t n} (x, y)

H_{t} (x, y, z) = \sum_{n = 1} [a_{n} (z) - b_{n} (z)] h_{t n} (x, y)

E_{z} (x, y, z) = \sum_{n = 1} [a_{n} (z) - b_{n} (z)] \frac{ε}{\tilde{ε}} e_{z n} (x, y)

H_{z} (x, y, z) = \sum_{n = 1} [a_{n} (z) + b_{n} (z)] h_{z n} (x, y)

N_{m} \frac{d a_{m}}{d z} + j γ_{m} a_{m} = - j \sum_{n = 1} κ_{m n} a_{n} - j \sum_{n = 1} χ_{m n} b_{n}

N_{m} \frac{d b_{m}}{d z} - j γ_{m} b_{m} = + j \sum_{n = 1} κ_{m n} b_{n} + j \sum_{n = 1} χ_{m n} a_{n}

κ_{m n} = \frac{ω ε_{o}}{4} \int_{A} ({\tilde{n}}^{2} - n^{2}) (e_{t n} \cdot e_{t m} - \frac{n^{2}}{{\tilde{n}}^{2}} e_{z n} \cdot e_{z m}) d a

χ_{m n} = \frac{ω ε_{o}}{4} \int_{A} ({\tilde{n}}^{2} - n^{2}) (e_{t n} \cdot e_{t m} + \frac{n^{2}}{{\tilde{n}}^{2}} e_{z n} \cdot e_{z m}) d a

N_{m} = \frac{1}{2} \iint_{} (e_{t m} \times h_{t m}) \cdot \hat{z} d a

κ_{m n} = κ_{n m}

χ_{m n} = χ_{n m}

a_{n} = A_{n} \exp (- j γ_{n} z)

b_{n} = B_{n} \exp (+ j γ_{n} z)

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} κ_{m n} A_{n} \exp [- j (γ_{n} - γ_{m}) z] - j \sum_{n = 1} χ_{m n} B_{n} \exp [+ j (γ_{n} + γ_{m}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} κ_{m n} B_{n} \exp [+ j (γ_{n} - γ_{m}) z] + j \sum_{n = 1} χ_{m n} A_{n} \exp [- j (γ_{n} + γ_{m}) z]

κ_{m n} = \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp (j l \frac{2 π}{Λ} z)

χ_{m n} = \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp (j l \frac{2 π}{Λ} z)

\begin{array}{l} N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [- j (γ_{n} - γ_{m} - l \frac{2 π}{Λ}) z] \\ - j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [+ j (γ_{n} + γ_{m} + l \frac{2 π}{Λ}) z] \end{array}

\begin{array}{c} N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [+ j (γ_{n} - γ_{m} + l \frac{2 π}{Λ}) z] \\ + j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [- j (γ_{n} + γ_{m} - l \frac{2 π}{Λ}) z] \end{array}

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [+ j (γ_{n} + γ_{m} + l \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} C_{m n}^{(l)} \exp [- j (γ_{n} + γ_{m} - l \frac{2 π}{Λ}) z]

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [- j (γ_{n} - γ_{m} - l \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} \sum_{l = - \infty}^{+ \infty} D_{m n}^{(l)} \exp [+ j (γ_{n} - γ_{m} + l \frac{2 π}{Λ}) z]

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} B_{n} C_{m n}^{(- 1)} \exp [+ j (γ_{n} + γ_{m} - \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} A_{n} C_{m n}^{(+ 1)} \exp [- j (γ_{n} + γ_{m} - \frac{2 π}{Λ}) z]

N_{m} \frac{d A_{m}}{d z} = - j \sum_{n = 1} A_{n} D_{m n}^{(- 1)} \exp [- j (γ_{n} - γ_{m} + \frac{2 π}{Λ}) z]

N_{m} \frac{d B_{m}}{d z} = + j \sum_{n = 1} B_{n} D_{m n}^{(+ 1)} \exp [+ j (γ_{n} - γ_{m} + \frac{2 π}{Λ}) z]

N_{1} \frac{d A_{1}}{d z} = - j C_{1 n}^{(- 1)} B_{n} \exp [+ j (γ_{n} + β_{1} - \frac{2 π}{Λ}) z]

N_{n} \frac{d B_{n}}{d z} = + j C_{n 1}^{(+ 1)} A_{1} \exp [- j (γ_{n} + β_{1} - \frac{2 π}{Λ}) z]

N_{1} \frac{d A_{1}}{d z} = - j D_{1 n}^{(- 1)} A_{n} \exp [- j (γ_{n} - β_{1} + \frac{2 π}{Λ}) z]

N_{n} \frac{d A_{n}}{d z} = - j D_{n 1}^{(+ 1)} A_{1} \exp [+ j (γ_{n} - β_{1} + \frac{2 π}{Λ}) z]

Δ β_{n} = \frac{1}{2} (γ_{1} + γ_{n} - \frac{2 π}{Λ})

Δ β_{n} = \frac{1}{2} (γ_{1} - γ_{n} - \frac{2 π}{Λ})

ℜ (Δ β_{n}) = 0

Λ = \frac{2 π}{ℜ (γ_{1} + γ_{n})}

Λ = \frac{2 π}{ℜ (γ_{1} - γ_{n})}

N_{1} \frac{d A_{1}}{d z} = - j C_{1 n}^{(- 1)} B_{n} \exp [+ j (2 Δ β_{n}) z]

N_{n} \frac{d B_{n}}{d z} = + j C_{n 1}^{(+ 1)} A_{1} \exp [- j (2 Δ β_{n}) z]

N_{1} \frac{d A_{1}}{d z} = - j D_{1 n}^{(- 1)} A_{n} \exp [- j (2 Δ β_{n}) z]

N_{n} \frac{d A_{n}}{d z} = - j D_{n 1}^{(+ 1)} A_{1} \exp [+ j (2 Δ β_{n}) z]

a_{1} (z) = a_{1} (0) \frac{Δ β_{n} \sin h S (z - L) - j S \cos h S (z - L)}{- Δ β_{n} \sin h S L - j S \cos h S L} e^{- j (β_{1} - Δ β_{n}) z}

b_{n} (z) = - a_{1} (0) \frac{C_{n 1}^{(+ 1)}}{N_{n}} \frac{\sin h S (z - L)}{Δ β_{n} \sin h S L + j S \cos h S L} e^{j (γ_{n} - Δ β_{n}^{}) z}

S = \sqrt{κ_{n}^{2} - {(Δ β_{n})}^{2}}

κ_{n} = \sqrt{\frac{C_{1 n}^{(- 1)} C_{n 1}^{(+ 1)}}{N_{1} N_{n}}}

a_{1} (z) = a_{1} (0) \frac{j Δ β \sin (Q_{n} z) + Q_{n} \cos (Q_{n} z)}{Q_{n}} e^{- j (β_{1} - Δ β_{n}^{}) z}

a_{n} (z) = - j e^{j Δ β_{n}^{} z} \frac{D_{n 1}^{(+ 1)}}{N_{n}} \frac{\sin (Q_{n} z)}{Q_{n}} a_{1} (0) e^{- j (γ_{n} + Δ β_{n}^{}) z}

Q_{n} = \sqrt{χ_{n}^{2} + {(Δ β_{n})}^{2}}

χ_{n} = \sqrt{\frac{D_{n 1}^{(+ 1)} D_{1 n}^{(- 1)}}{N_{n} N_{1}}}

P (z) = \frac{1}{2} ℜ \int_{A} [E_{t} (x, y, z) \times H_{t}^{*} (x, y, z)] \cdot \hat{z} d a

P (z) = \sum_{m = 1} \sum_{n = 1} M_{m n} [a_{m} a_{n}^{*} - b_{m} b_{n}^{*}] - \sum_{m = 1} \sum_{n = 1} N_{m n} [a_{m} b_{n}^{*} - b_{m} a_{n}^{*}]

M_{m n} = \frac{1}{4} \iint_{A} (e_{t m} \times h_{t n}^{*} + e_{t n}^{*} \times h_{t m}^{}) \cdot \hat{z} d a

N_{m n} = \frac{1}{4} \iint_{A} (e_{t m} \times h_{t n}^{*} - e_{t n}^{*} \times h_{t m}^{}) \cdot \hat{z} d a

P (z) ≅ \sum_{m = 1} \sum_{n = 1} M_{m n} [a_{m} a_{n}^{*} - b_{m} b_{n}^{*}]

P (z) ≅ \sum_{n = 1} [{| a_{n} |}^{2} - {| b_{n} |}^{2}]

P (0) = 1 - R (λ)

R (λ) ≅ \sum_{m = 1} \sum_{n = 1} M_{m n} b_{m} (0) b_{n}^{*} (0)

R (λ) ≅ \sum_{n = 1} {| b_{m} (0) |}^{2}

P (L) = {| a_{1} (L) |}^{2} = T (λ)

T (λ) = 1 - R (λ)

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Equations (81)

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