Approximate formula for the propagation constant of the basic mode in slab waveguides of arbitrary-index profile

Shlomo Ruschin

doi:10.1364/AO.24.004189

Applied Optics
Vol. 24,
Issue 23,
pp. 4189-4191
(1985)
•https://doi.org/10.1364/AO.24.004189

Approximate formula for the propagation constant of the basic mode in slab waveguides of arbitrary-index profile

Shlomo Ruschin

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Shlomo Ruschin, "Approximate formula for the propagation constant of the basic mode in slab waveguides of arbitrary-index profile," Appl. Opt. 24, 4189-4191 (1985)

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Abstract

A simple approximate formula for the propagation constant of the basic mode in slab dielectric waveguides of arbitrary-symmetric-index profile is given. The formula is based on the correspondence between the given profile and an equivalent cosh⁻²(x) profile. The expression is compared to numerical calculations.

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Profile name	f(x/v)	M_o	M₂	n²_d	k
Cosecant hyperbolic	${Cosh}^{- 2} (\frac{x}{v})$	2v	$\frac{π^{2}}{6} v^{3}$	1	1
Exponential	$Exp (- \frac{x}{v})$	2v	4v³	$\frac{π}{4} \sqrt{\frac{2}{3}}$	$\frac{2}{π} \sqrt{6}$
Gaussian	$Exp (- \frac{x^{2}}{v^{2}})$	$\sqrt{π} v$	$\frac{\sqrt{π}}{2} v^{3}$	$\frac{\sqrt{π}}{4} \sqrt{\frac{2 π}{3}}$	$\frac{2}{π} \sqrt{\frac{3}{2}}$
Rectangular	$\begin{array}{l} 1 & \| \frac{x}{v} \| < 1 \\ 0 & \| \frac{x}{v} \| \geq 1 \end{array}$	2v	$\frac{2}{3} v^{3}$	$\frac{π}{2}$	$\frac{2}{π}$
Cladded parabolic	$\begin{array}{l} 1 - \frac{x^{2}}{v^{2}} & \| \frac{x}{v} \| < 1 \\ 0 & \| \frac{x}{v} \| \geq 1 \end{array}$	$\frac{4}{3} v$	$\frac{4}{15} v^{3}$	$π \sqrt{\frac{5}{27}}$	$\frac{2}{π} \sqrt{\frac{3}{5}}$

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Applied Optics