Modeling and optimization of complex photonic resonant cavity circuits

M. Sumetsky; B. J. Eggleton

doi:10.1364/OE.11.000381

Optics Express
Vol. 11,
Issue 4,
pp. 381-391
(2003)
•https://doi.org/10.1364/OE.11.000381

Modeling and optimization of complex photonic resonant cavity circuits

M. Sumetsky and B. J. Eggleton

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M. Sumetsky and B. J. Eggleton, "Modeling and optimization of complex photonic resonant cavity circuits," Opt. Express 11, 381-391 (2003)

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Abstract

The simple method for modeling of circuits of weakly coupled lossy resonant cavities, previously developed in quantum mechanics, is generalized to enable calculation of the transmission and reflection amplitudes and group delay of light. Our result is the generalized Breit-Wigner formula, which has a clear physical meaning and is convenient for fast modeling and optimization of complex resonant cavity circuits and, in particular, superstructure gratings in a way similar to modeling and optimization of electric circuits. As examples, we find the conditions when a finite linear chain of cavities and a linear chain with adjacent cavities act as bandpass and double bandpass filters, and the condition for a Y-shaped structure to act as a bandpass 50/50 light splitter. The group delay dependencies of the considered structures are also investigated.

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Fig. 1. Resonant cavity structure.

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Fig. 2. Single cavity structures: a – a cavity between two ports, b and c – transmission and group delay spectrum of this structure for γ₁₁=γ₁₂=γ₁/2.

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Fig. 3. A structure of three resonant cavities, a – not optimized, b – optimized, λ₀=1500 nm.

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Fig. 4. Transmission and group delay spectrum for the apodized 50-cavity structure, λ₀=1500 nm; a – transmission spectrum for the lossless cavities, b – transmission spectrum for the cavities having internal loss γ _int = 0.01δ₀.

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Fig. 5. Transmission and group delay spectrum for the double bandpass structure, λ₀=1500 nm.

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Fig. 6. a – Y-splitter assembled of resonant cavities; b – transmission and group delay spectrum for the coupling parameters shown in Fig. 6a, λ₀=1500 nm.

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Fig. 7. a – all-reflecting single port device; b – model of a device consisting of cavities having hexagonal internal symmetry; c – generalized model of a port.

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Equations (14)

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u (r) = \sum_{n = 1}^{N} C_{n} u_{n} (r) + u_{m}^{(in)} (r)

H (u) = λ^{- 2} u,

H_{m}^{(port)} (u_{m}^{(in)}) = λ^{- 2} u_{m}^{(in)}, H_{n}^{(cav)} (u_{n}) = {(λ_{n}^{(0)} + \frac{i}{2} γ_{n})}^{- 2} u_{n},

ΛC = χ^{(m)}, C = (\begin{matrix} C_{1} \\ C_{2} \\ \dots \\ C_{N} \end{matrix}), χ^{(m)} = (\begin{matrix} χ_{1 m} \\ χ_{2 m} \\ \dots \\ χ_{Nm} \end{matrix}) .

Λ (λ) = (\begin{matrix} λ - λ_{1} + \frac{i}{2} γ_{1} & δ_{12} & \dots & δ_{1 N} \\ δ_{21} & λ - λ_{2} + \frac{i}{2} γ_{2} & \dots & δ_{2 N} \\ \dots & \dots & \dots & \dots \\ δ_{N 1} & δ_{N 2} & \dots & λ - λ_{N} + \frac{i}{2} γ_{N} \end{matrix})

γ_{jm} = {∣ χ_{jm} ∣}^{2} .

A_{lm} (λ) = χ_{il} χ_{jm}^{*} t_{ij}, R_{ll} (λ) = 1 - i {∣ χ_{ll} ∣}^{2} t_{ii}, l \neq m

τ_{lm} = - \frac{λ^{2}}{2 πc} Im [\frac{d}{dλ} \ln [A_{lm} (λ)]], τ_{ll} = - \frac{λ^{2}}{2 πc} Im [\frac{d}{dλ} \ln [R_{ll} (λ)]] .

A_{12} (λ) = \frac{χ_{11} χ_{12}^{*}}{λ - λ_{1} + \frac{i}{2} γ_{1}}, R_{11} (λ) = 1 - \frac{i {∣ χ_{11} ∣}^{2}}{λ - λ_{1} + \frac{i}{2} γ_{1}},

τ_{12} (λ) = \frac{λ^{2}}{πc} \frac{γ_{1}}{4 {(λ - λ_{1})}^{2} + γ_{1}^{2}}, τ_{11} (λ) = τ_{12} (λ) .

λ_{n} = λ_{n}^{(0)} + Δ λ_{n}

Δ λ_{n} = \frac{1}{2} {(λ_{n}^{(0)})}^{3} \int d r u_{n} (r) (H - H_{n}) u_{n} (r)

δ_{ij} = \frac{1}{2} {(λ_{i}^{(0)})}^{3} \int d r u_{i} (r) (H - H_{j}) u_{j} (r)

χ_{im} = \frac{1}{2} {(λ_{m}^{(0)})}^{3} \int d r u_{i} (r) (H - H_{m}^{(port)}) u_{m}^{(in)} (r) .

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Equations (14)

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