Modes of optical waveguides

Allan W. Snyder; William R. Young

doi:10.1364/JOSA.68.000297

Journal of the Optical Society of America
Vol. 68,
Issue 3,
pp. 297-309
(1978)
•https://doi.org/10.1364/JOSA.68.000297

Modes of optical waveguides

Allan W. Snyder and William R. Young

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Allan W. Snyder and William R. Young, "Modes of optical waveguides," J. Opt. Soc. Am. 68, 297-309 (1978)

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Abstract

A simple method is presented for finding the modes on those optical waveguides with a cladding refractive index that differs only slightly from the refractive index of the core. The method applies to waveguides of arbitrary refractive index profile, arbitrary number of propagating modes, and arbitrary cross section. The resulting modal fields and their progagation constants display the polarization properties of the waveguide contained within the ∇ ∊ term of the vector wave equation. Examples include modes on waveguides with circular symmetry and waveguides with two preferred axes of symmetry, e.g., an elliptical core. Only a minute amount of eccentricity is necessary for the well-known LP modes to be stable on an elliptical core, while the circle modes couple power among themselves.

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	Mode 1 Even EH_l_−1,_m (even EH_0,_m = TM_0,_m)	Mode 2 Even HE_l_+1,_m	Mode 3 Odd HE_l_+1,_m	Mode 4 Odd EH_l_−1,_m (odd EH_0,_m = TE_0,_m)

Tranverse fields e_t
$f_{l} (r) = \begin{matrix} J_{l} (Ũ r / ρ) / J_{l} (Ũ), r \leq ρ \\ K_{l} (\tilde{W} r / ρ) / K_{l} (\tilde{W} r / ρ), r \geq ρ \end{matrix}$	$\begin{array}{l} e_{t 1} = {\tilde{e}}_{e x} + {\tilde{e}}_{o y} \\ = {\hat{x} cos l ϕ + ŷ sin l ϕ} f_{l} (r) \end{array}$	$\begin{array}{l} e_{t 2} = {\tilde{e}}_{e x} - {\tilde{e}}_{o y} \\ = {\hat{x} cos l ϕ + ŷ sin l ϕ} f_{l} (r) \end{array}$	$\begin{array}{l} e_{t 3} = {\tilde{e}}_{o x} + {\tilde{e}}_{e y} \\ = {\hat{x} sin l ϕ + ŷ cos l ϕ} f_{l} (r) \end{array}$	$\begin{array}{l} e_{t 4} = {\tilde{e}}_{o x} - {\tilde{e}}_{e y} \\ = {\hat{x} sin l ϕ + ŷ cos l ϕ} f_{l} (r) \end{array}$

Longitudinal fields e_z
$\begin{array}{l} e_{z} = - i ρ θ_{c} (\nabla_{t} \cdot e_{t}) / V \\ g_{l}^{\pm} (r) = \begin{matrix} Ũ J_{l \pm 1} (Ũ r / ρ) / J_{l} (Ũ), r < ρ \\ \tilde{W} K_{l \pm 1} (\tilde{W} r / ρ) / K_{l} (\tilde{W}), r > ρ \end{matrix} \end{array}$	$(i / V) θ_{c} g_{l}^{-} (r) cos (l - 1) ϕ$	$- (i / V) θ_{c} g_{l}^{+} (r) cos (l + 1) ϕ$	$- (i / V) θ_{c} g_{l}^{+} (r) sin (l + 1) ϕ$	$(i / V) θ_{c} g_{l}^{-} (r) sin (l - 1) ϕ$

Propagation constants β
$\begin{array}{l} β - \tilde{β} = \frac{ρ θ_{c} \int_{A_{\infty}} {\tilde{e}}_{t} \cdot \nabla_{t} (e_{t} \cdot \nabla_{t} ln ∊) d A}{2 V \int_{A_{\infty}} {\tilde{e}}_{t} \cdot e_{t} d A} \\ {\tilde{β}}^{2} = k_{co}^{2} - Ũ^{2} / ρ^{2} \end{array}$	$\begin{array}{l} β_{1} - \tilde{β} = - \frac{θ_{c}^{3} Ũ^{2} \tilde{W} K_{1} (\tilde{W})}{ρ V^{3} K_{2} (\tilde{W})} \\ β_{1} - \tilde{β} = - \frac{\begin{matrix} (l = 1) \\ θ_{c}^{3} Ũ^{2} \tilde{W} K_{l} (\tilde{W}) \end{matrix}}{\begin{matrix} 2 ρ V^{3} K_{l + 1} (\tilde{W}) \\ (l \neq 1) \end{matrix}} \end{array}$	$β_{2} - \tilde{β} = β_{3} - \tilde{β} = - \frac{θ_{c}^{3} Ũ^{2} K_{l} (\tilde{W})}{2 ρ V^{3} K_{l - 1} (\tilde{W})}$		$\begin{array}{l} \begin{matrix} β_{4} - \tilde{β} = 0 \\ (l = 1) \end{matrix} \\ β_{4} - \tilde{β} = - \frac{θ_{c}^{3} Ũ^{2} \tilde{W} K_{l} (\tilde{W})}{\begin{matrix} 2 ρ V^{3} K_{l + 1 (\tilde{W})} \\ (l \neq 1) \end{matrix}} \end{array}$

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Journal of the Optical Society of America