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Journals >Chinese Optics Letters >Volume 15 >Issue 8 >Page 081901 > Article

Chinese Optics Letters
Vol. 15, Issue 8, 081901 (2017)

Matrix formalism for radiating polarization sheets in multilayer structures of arbitrary composition

Hui Shi, Yu Zhang, Hongqing Wang, and Weitao Liu^*

Author Affiliations

Physics Department, State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nano Photonic Structures (MOE), Collaborative Innovation Center of Advanced Microstructures (Nanjing), Fudan University, Shanghai 200433, China

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DOI: 10.3788/COL201715.081901 Cite this Article Set citation alerts

Hui Shi, Yu Zhang, Hongqing Wang, Weitao Liu. Matrix formalism for radiating polarization sheets in multilayer structures of arbitrary composition[J]. Chinese Optics Letters, 2017, 15(8): 081901 Copy Citation Text

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Abstract

In optical studies on layered structures, quantitative analysis of radiating interfaces is often challenging due to multiple interferences. We present here a general and analytical method for computing the radiation from two-dimensional polarization sheets in multilayer structures of arbitrary compositions. It is based on the standard characteristic matrix formalism of thin films, and incorporates boundary conditions of interfacial polarization sheets. We use the method to evaluate the second harmonic generation from a nonlinear thin film, and the sum-frequency generation from a water/oxide interface, showing that the signal of interest can be strongly enhanced with optimal structural parameters.

Δ E_{x} = σ_{z} = - \frac{4 π}{ϵ^{'}} i k_{x} P_{s z}, Δ E_{y} = 0, Δ H_{x} = σ_{y} = - \frac{4 π i}{c} ω P_{s y}, Δ H_{y} = σ_{x} = \frac{4 π i}{c} ω P_{s x},

(1)

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E_{I, 1} + σ_{I, E} = E_{I, 2}, H_{I, 1} + σ_{I, H} = H_{I, 2},

(2)

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E_{I, 2} = E_{I, 2}^{R} + E_{I, 2}^{T} .

(3a)

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H_{I, 2} = (E_{I, 2}^{T} - E_{I, 2}^{R}) {\bar{n}}_{2},

(3b)

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[\begin{matrix} E_{I, 1} + σ_{I, E} \\ H_{I, 1} + σ_{I, H} \end{matrix}] = [\begin{matrix} E_{I, 2} \\ H_{I, 2} \end{matrix}] = [\begin{matrix} 1 & 1 \\ {\bar{n}}_{2} & - {\bar{n}}_{2} \end{matrix}] [\begin{matrix} E_{I, 2}^{T} \\ E_{I, 2}^{R} \end{matrix}] .

(4a)

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[\begin{matrix} E_{I, 2}^{T} \\ E_{I, 2}^{R} \end{matrix}] = [\begin{matrix} \exp (- i δ_{2}) & 0 \\ 0 & \exp (i δ_{2}) \end{matrix}] [\begin{matrix} E_{I I, 2}^{T} \\ E_{I I, 2}^{R} \end{matrix}] .

(4b)

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[\begin{matrix} E_{I I, 3} \\ H_{I I, 3} \end{matrix}] = [\begin{matrix} E_{I I, 2} \\ H_{I I, 2} \end{matrix}] = [\begin{matrix} 1 & 1 \\ {\bar{n}}_{2} & - {\bar{n}}_{2} \end{matrix}] [\begin{matrix} E_{I I, 2}^{T} \\ E_{I I, 2}^{R} \end{matrix}] .

(4c)

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[\begin{matrix} E_{I, 1} + σ_{I, E} \\ H_{I, 1} + σ_{I, H} \end{matrix}] = (\begin{matrix} \cos δ_{2} & - \frac{i \sin δ_{2}}{{\bar{n}}_{2}} \\ - i {\bar{n}}_{2} \sin δ_{2} & \cos δ_{2} \end{matrix}) [\begin{matrix} E_{I I, 3} \\ H_{I I, 3} \end{matrix}] = M_{2} [\begin{matrix} E_{I I, 3} \\ H_{I I, 3} \end{matrix}],

(5a)

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[\begin{matrix} E_{I, 1}^{R} + σ_{I, E} \\ - E_{I, 1}^{R} {\bar{n}}_{1} + σ_{I, H} \end{matrix}] = M_{2} [\begin{matrix} E_{I I, 3}^{T} \\ - E_{I I, 3}^{T} {\bar{n}}_{3} \end{matrix}] .

(5b)

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[\begin{matrix} E_{I I, 2} \\ H_{I I, 2} \end{matrix}] = [\begin{matrix} E_{I I, 3} \\ H_{I I, 3} \end{matrix}] = M_{3} [\begin{matrix} E_{I I I, 4} \\ H_{I I I, 4} \end{matrix}],

(5c)

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[\begin{matrix} E_{I, 1}^{R} + σ_{I, E} \\ H_{I, 1}^{R} + σ_{I, H} \end{matrix}] = M_{2} M_{3} [\begin{matrix} E_{I I I, 4}^{T} \\ H_{I I I, 4}^{T} \end{matrix}] .

(6a)

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[\begin{matrix} E_{I, 1}^{R} \\ H_{I, 1}^{R} \end{matrix}] = M_{2} [\begin{matrix} E_{I I, 2} \\ H_{I I, 2} \end{matrix}], [\begin{matrix} E_{I I, 2} + σ_{I I, E} \\ H_{I I, 2} + σ_{I I, H} \end{matrix}] = [\begin{matrix} E_{I I, 3} \\ H_{I I, 3} \end{matrix}] = M_{3} [\begin{matrix} E_{I I I, 4}^{T} \\ H_{I I I, 4}^{T} \end{matrix}] .

(6b)

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[\begin{matrix} E_{I, 1}^{R} \\ - E_{I, 1}^{R} {\bar{n}}_{1} \end{matrix}] = (\prod_{j = 2}^{i} M_{j}) [\begin{matrix} E_{i, i} \\ H_{i, i} \end{matrix}], [\begin{matrix} E_{i, i} + σ_{i, E} \\ H_{i, i} + σ_{i, H} \end{matrix}] = (\prod_{j = i + 1}^{N} M_{j}) [\begin{matrix} E_{N, N + 1}^{T} \\ - E_{N, N + 1}^{T} {\bar{n}}_{N + 1} \end{matrix}] .

(7)

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[\begin{matrix} E_{I, 1}^{R} \\ H_{I, 1}^{R} \end{matrix}] = M_{2} \dots M_{i} [\begin{matrix} E_{i, i} \\ H_{i, i} \end{matrix}], [\begin{matrix} E_{i, i} + σ_{i, E} \\ H_{i, i} + σ_{i, H} \end{matrix}] = M_{i + 1} \dots M_{j} [\begin{matrix} E_{j, j} \\ H_{j, j} \end{matrix}], [\begin{matrix} E_{j, j} + σ_{j, E} \\ H_{j, j} + σ_{j, H} \end{matrix}] = M_{j + 1} \dots M_{N} [\begin{matrix} E_{N, N + 1}^{T} \\ H_{N, N + 1}^{T} \end{matrix}] .

(8a)

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[\begin{matrix} E_{I, 1}^{'} \\ H_{I, 1}^{'} \end{matrix}] = M_{2} \dots M_{i} [\begin{matrix} E_{i, i}^{'} \\ H_{i, i}^{'} \end{matrix}], [\begin{matrix} E_{i, i}^{'} + σ_{i, E} \\ H_{i, i}^{'} + σ_{i, H} \end{matrix}] = M_{i + 1} \dots M_{j} [\begin{matrix} E_{j, j}^{'} \\ H_{j, j}^{'} \end{matrix}], [\begin{matrix} E_{j, j}^{'} \\ H_{j, j}^{'} \end{matrix}] = M_{j + 1} \dots M_{N} [\begin{matrix} E_{N, N + 1}^{'} \\ H_{N, N + 1}^{'} \end{matrix}] .

(8b)

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[\begin{matrix} E_{I, 1}^{''} \\ H_{I, 1}^{''} \end{matrix}] = M_{2} \dots M_{i} [\begin{matrix} E_{i, i}^{''} \\ H_{i, i}^{''} \end{matrix}], [\begin{matrix} E_{i, i}^{''} \\ H_{i, i}^{''} \end{matrix}] = M_{i + 1} \dots M_{j} [\begin{matrix} E_{j, j}^{''} \\ H_{j, j}^{''} \end{matrix}], [\begin{matrix} E_{j, j}^{''} + σ_{j, E} \\ H_{j, j}^{''} + σ_{j, H} \end{matrix}] = M_{j + 1} \dots M_{N} [\begin{matrix} E_{N, N + 1}^{''} \\ H_{N, N + 1}^{''} \end{matrix}],

(8c)

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[\begin{matrix} E_{I, 1}^{'} + E_{I, 1}^{''} \\ H_{I, 1}^{'} + H_{I, 1}^{''} \end{matrix}] = M_{2} \dots M_{i} [\begin{matrix} E_{i, i}^{'} + E_{i, i}^{''} \\ H_{i, i}^{'} + H_{i, i}^{''} \end{matrix}], [\begin{matrix} E_{i, i}^{'} + E_{i, i}^{''} + σ_{i, E} \\ H_{i, i}^{'} + H_{i, i}^{''} + σ_{i, E} \end{matrix}] = M_{i + 1} \dots M_{j} [\begin{matrix} E_{j, j}^{'} + E_{j, j}^{''} \\ H_{j, j}^{'} + H_{j, j}^{''} \end{matrix}], [\begin{matrix} E_{j, j}^{'} + E_{j, j}^{''} + σ_{j, E} \\ H_{j, j}^{'} + H_{j, j}^{''} + σ_{j, E} \end{matrix}] = M_{j + 1} \dots M_{N} [\begin{matrix} E_{N, N + 1}^{'} + E_{N, N + 1}^{''} \\ H_{N, N + 1}^{'} + H_{N, N + 1}^{''} \end{matrix}] .

(8d)

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d P_{S}^{(2)} (2 ω, z^{'}) = P_{B}^{(2)} (2 ω, z^{'}) d z^{'} = {\overset{\leftrightarrow}{χ}}_{B}^{(2)} (2 ω, z^{'}) : E (ω, z^{'}) E (ω, z^{'}) d z^{'},

(9)

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[\begin{matrix} d E_{a}^{R} (z^{'}) \\ - d E_{a}^{R} (z^{'}) {\bar{n}}_{a} \end{matrix}] = M_{f} (z^{'}) [\begin{matrix} d E_{f} (z^{'}) \\ d H_{f} (z^{'}) \end{matrix}], [\begin{matrix} d E_{f} (z^{'}) + d σ_{E} (z^{'}) \\ d H_{f} (z^{'}) + d σ_{H} (z^{'}) \end{matrix}] = M_{f} (d - z^{'}) [\begin{matrix} d E_{s}^{T} (z^{'}) \\ - d E_{s}^{T} (z^{'}) {\bar{n}}_{s} \end{matrix}],

(10)

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Hui Shi, Yu Zhang, Hongqing Wang, Weitao Liu. Matrix formalism for radiating polarization sheets in multilayer structures of arbitrary composition[J]. Chinese Optics Letters, 2017, 15(8): 081901

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