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Journals >Chinese Optics Letters >Volume 15 >Issue 2 >Page 021301 > Article

Chinese Optics Letters
Vol. 15, Issue 2, 021301 (2017)

Numerical evaluation of radiation and optical coupling occurring in optical coupler

Mansour Bacha^1,2,* and Abderrahmane Belghoraf¹

Author Affiliations

¹Electronics Department, University of Sciences and Technology of Oran Mohamed Boudiaf, Oran M’Naouer BP 1505, Algeria

²Centre of Satellites Development, Ibn Rochd USTO Oran BP 4065, Algeria

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DOI: 10.3788/COL201715.021301 Cite this Article Set citation alerts

Mansour Bacha, Abderrahmane Belghoraf, "Numerical evaluation of radiation and optical coupling occurring in optical coupler," Chin. Opt. Lett. 15, 021301 (2017) Copy Citation Text

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Abstract

We present in this work a new mathematical model to analyze and evaluate optical phenomena occurring in the nonuniform optical waveguide used in integrated optics as an optical coupler. By introducing some modifications to the intrinsic integral, we perfectly assess the radiation field present in the adjacent medium of the waveguide and, thus, follow the evolution of the optical coupling from the taper thin film to the substrate and cladding until there is a total energy transfer. The new model that is introduced can be used to evaluate electromagnetic field distribution in three mediums that constitute any nonuniform optical couplers presenting great or low wedge angles.

ϕ^{+ -} (θ^{+ -}) = 2 \arctan (\sqrt{\frac{n_{g}^{2} \cdot \cos^{2} (θ^{+ -}) - n_{c, s}^{2}}{n_{g}^{2} - n_{g}^{2} \cdot \cos^{2} (θ^{+ -})}}) .

(1)

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Φ_{e}^{+} (θ_{0}, θ_{n}) = \sum_{m = 1}^{n} [ϕ^{+} (θ_{m}^{+})] + \sum_{m = 1}^{n} [ϕ^{-} (θ_{m}^{-})],

(2)

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Φ_{o}^{+} (θ_{0}, θ_{n}) = Φ_{e}^{+} (θ_{0}, θ_{n}) - ϕ^{-} (θ_{n}^{-}),

(3)

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Φ_{e}^{-} (θ_{0}, θ_{n}) = \sum_{m = 1}^{n} [ϕ^{-} (θ_{m}^{-})] + \sum_{m = 1}^{n} [ϕ^{+} (θ_{m}^{+})],

(4)

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Φ_{o}^{-} (θ_{0}, θ_{n}) = Φ_{e}^{-} (θ_{0}, θ_{n}) - ϕ^{+} (θ_{n}^{+}),

(5)

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θ_{m}^{+ -} = θ_{0}^{+ -} + 2 \cdot a \cdot (m - 1) .

(6)

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m = 1 + \frac{θ_{m}^{+ -} - θ_{0}^{+ -}}{2 \cdot a} .

(7)

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ϕ_{e}^{+} (θ_{0}, θ_{n}) = \frac{1}{2} ϕ^{+} (θ_{0}^{+}) + \frac{1}{2} ϕ^{+} (θ_{n}^{+}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{+} (θ^{'}) \cdot d θ^{'} + \frac{1}{2} ϕ^{-} (θ_{0}^{-}) + \frac{1}{2} ϕ^{-} (θ_{n}^{-}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{-} (θ^{'}) \cdot d θ^{'} + E_{e}^{+} (θ_{n}^{+}, ϕ^{+} (θ_{n}^{+})) + E_{e}^{+} (θ_{n}^{-}, ϕ^{-} (θ_{n}^{-})),

(8)

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ϕ_{o}^{+} (θ_{0}, θ_{n}) = \frac{1}{2} ϕ^{+} (θ_{0}^{+}) + \frac{1}{2} ϕ^{+} (θ_{n}^{+}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{+} (θ^{'}) \cdot d θ^{'} + \frac{1}{2} ϕ^{-} (θ_{0}^{-}) - \frac{1}{2} ϕ^{-} (θ_{n}^{-}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{-} (θ^{'}) \cdot d θ^{'} + E_{e}^{+} (θ_{n}^{+}, ϕ^{+} (θ_{n}^{+})) + E_{e}^{+} (θ_{n}^{-}, ϕ^{-} (θ_{n}^{-})),

(9)

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ϕ_{e}^{-} (θ_{0}, θ_{n}) = \frac{1}{2} ϕ^{-} (θ_{0}^{-}) + \frac{1}{2} ϕ^{-} (θ_{n}^{-}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{-} (θ^{'}) \cdot d θ^{'} + \frac{1}{2} ϕ^{+} (θ_{0}^{+}) + \frac{1}{2} ϕ^{+} (θ_{n}^{+}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{+} (θ^{'}) \cdot d θ^{'} + E_{e}^{-} (θ_{n}^{-}, ϕ^{-} (θ_{n}^{-})) + E_{e}^{-} (θ_{n}^{+}, ϕ^{+} (θ_{n}^{+})),

(10)

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ϕ_{o}^{-} (θ_{0}, θ_{n}) = \frac{1}{2} ϕ^{-} (θ_{0}^{-}) + \frac{1}{2} ϕ^{-} (θ_{n}^{-}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{-} (θ^{'}) \cdot d θ^{'} + \frac{1}{2} ϕ^{+} (θ_{0}^{+}) - \frac{1}{2} ϕ^{+} (θ_{n}^{+}) + \frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{+} (θ^{'}) \cdot d θ^{'} + E_{e}^{-} (θ_{n}^{-}, ϕ^{-} (θ_{n}^{-})) + E_{e}^{-} (θ_{n}^{+}, ϕ^{+} (θ_{n}^{+})) .

(11)

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E (θ, ϕ^{\pm} (θ)) = 2 \cdot \sum_{q = - \infty}^{+ \infty} {\frac{1}{2 \cdot a} \int_{θ_{0}}^{θ} ϕ^{\pm} (θ^{'}) \cdot \cos [2 \cdot π \cdot p \cdot (1 + \frac{θ^{'} - θ_{0}}{2 a})] d θ^{'}} .

(12)

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W_{e}^{+} (X_{0}, X) = W_{e}^{+} (θ_{0}, θ) = Exp {j \cdot [ϕ_{e}^{+} (θ_{0}, θ) + k \cdot R_{e}^{+} (θ_{0}, θ)]},

(13)

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W_{o}^{+} (X_{0}, X) = W_{o}^{+} (θ_{0}, θ) = Exp {j \cdot [ϕ_{o}^{+} (θ_{0}, θ) + k \cdot R_{o}^{+} (θ_{0}, θ)]},

(14)

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W_{e}^{-} (X_{0}, X) = W_{e}^{-} (θ_{0}, θ) = Exp {j \cdot [ϕ_{e}^{-} (θ_{0}, θ) + k \cdot R_{e}^{-} (θ_{0}, θ)]},

(15)

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W_{o}^{-} (X_{0}, X) = W_{o}^{-} (θ_{0}, θ) = Exp {j \cdot [ϕ_{o}^{-} (θ_{0}, θ) + k \cdot R_{o}^{-} (θ_{0}, θ)]} .

(16)

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R_{p}^{+} (θ_{0}, θ_{m}) = r_{0} \cdot \cos (θ_{0} - a - x_{0}) - r \cdot \cos (θ + a - x),

(17)

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R_{i}^{+} (θ_{0}, θ_{m}) = r_{0} \cdot \cos (θ_{0} - a - x_{0}) - r \cdot \cos (θ - a + x),

(18)

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R_{p}^{-} (θ_{0}, θ_{m}) = r_{0} \cdot \cos (θ_{0} - a + x_{0}) - r \cdot \cos (θ - a + x),

(19)

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R_{i}^{-} (θ_{0}, θ_{m}) = r_{0} \cdot \cos (θ_{0} - a + x_{0}) - r \cdot \cos (θ + a - x) .

(20)

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W (θ_{0}, θ) = \frac{1}{2 a} \int_{c} \sum_{q = - \infty}^{+ \infty} {[W_{e}^{+} (θ_{0}, θ) + W_{o}^{+} (θ_{0}, θ) + W_{e}^{-} (θ_{0}, θ) + W_{o}^{-} (θ_{0}, θ)] \cdot Exp (- j \cdot 2 π \cdot q \cdot m)} d θ .

(21)

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W^{+} (X, θ) = \frac{1}{2 a} \int_{c} {[W_{e}^{+} (X, θ) + W_{o}^{+} (X, θ)] \cdot Exp (- j \cdot 2 π \cdot q \cdot m)} d θ,

(22)

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W^{-} (X, θ) = \frac{1}{2 a} \int_{c} {[W_{e}^{-} (X, θ) + W_{o}^{-} (X, θ)] \cdot \cdot Exp (- j \cdot 2 π \cdot q \cdot m)} d θ .

(23)

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W_{s} (X, θ) = \frac{1}{2 a} \int_{c} \sum_{q = - \infty}^{+ \infty} {[1 + Exp (j \cdot ϕ^{-})] \cdot W_{o}^{+} (X, θ) Exp (- j \cdot 2 π \cdot q \cdot m)} d θ,

(24)

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W_{s} (X, θ) = \frac{1}{2 a} \int_{c} \sum_{q = - \infty}^{+ \infty} {[1 + Exp (j \cdot ϕ^{+})] \cdot W_{o}^{-} (X, θ) Exp (- j \cdot 2 π \cdot q \cdot m)} d θ .

(25)

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Mansour Bacha, Abderrahmane Belghoraf, "Numerical evaluation of radiation and optical coupling occurring in optical coupler," Chin. Opt. Lett. 15, 021301 (2017)

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