Variational principle in whispering gallery mode sensor responses

Iwao Teraoka; Stephen Arnold

doi:10.1364/JOSAB.25.001038

Journal of the Optical Society of America B
Vol. 25,
Issue 6,
pp. 1038-1045
(2008)
•https://doi.org/10.1364/JOSAB.25.001038

Variational principle in whispering gallery mode sensor responses

Iwao Teraoka and Stephen Arnold

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Iwao Teraoka and Stephen Arnold, "Variational principle in whispering gallery mode sensor responses," J. Opt. Soc. Am. B 25, 1038-1045 (2008)

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Abstract

We present a variational method for whispering gallery modes (WGMs) and apply it to sensor responses of WGMs in a spherical resonator to two types of perturbation for which we know the exact answers. The perturbations are adsorption of a thin dielectric layer and a uniform change in the surroundings’ refractive index. The variational method gives the perturbed wave function and, if a suitable trial function is chosen, the resonance wavelength shift up to the second order in the perturbation. The linear part is identical to the result of the first-order perturbation theory.

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$f (r)$	$\int_{0}^{\infty} f (r) d r$
$n_{o}^{2} S_{l}^{2}$	$(\frac{a}{2}) (n_{1}^{2} - n_{2}^{2}) [S_{l} (a)]^{2}$
$n_{o}^{2} 2 r S_{l} S_{l}^{'}$	$(\frac{a}{2}) (n_{1}^{2} - n_{2}^{2}) [S_{l} (a)]^{2}$
$n_{o}^{2} r^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$(n_{1}^{2} - n_{2}^{2}) {a^{2} S_{l} (a) S_{l}^{'} (a) - (\frac{a}{2}) [S_{l} (a)]^{2}}$
$n_{o}^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$(n_{1}^{2} - n_{2}^{2}) S_{l} (a) S_{l}^{'} (a)$
${S_{l}^{'}}^{2} + S_{l} S_{l}^{″}$	0
$(\frac{d n_{o}^{2}}{d r}) r S_{l}^{2}$	$- (n_{1}^{2} - n_{2}^{2}) a [S_{l} (a)]^{2}$
$(\frac{d n_{o}^{2}}{d r}) r^{2} S_{l} S_{l}^{'}$	$- (n_{1}^{2} - n_{2}^{2}) a^{2} S_{l} (a) S_{l}^{'} (a)$
$(\frac{d^{2} n_{o}^{2}}{d r^{2}}) r^{2} S_{l}^{2}$	$2 (n_{1}^{2} - n_{2}^{2}) {a [S_{l} (a)]^{2} + a^{2} S_{l} (a) S_{l}^{'} (a)}$

$f (r)$	$\int_{a}^{\infty} f (r) d r$
$S_{l}^{2}$	$(\frac{a}{2}) {- \frac{[S_{l}^{'} (a)]^{2}}{n_{2}^{2} k^{2}} + [\frac{l (l + 1)}{(n_{2} k a)^{2}} - 1] [S_{l} (a)]^{2} + \frac{S_{l} (a) S_{l} (a)^{'}}{n_{2}^{2} k^{2} a}} \equiv I$
$2 r S_{l} S_{l}^{'}$	$- a [S_{l} (a)]^{2} - I$
$r^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$- a^{2} S_{l} (a) S_{l}^{'} (a) + a [S_{l} (a)]^{2} + I$

$f (r)$	$\int_{0}^{\infty} f (r) d r$
$n_{o}^{2} S_{l}^{2}$	$(\frac{a}{2}) (n_{1}^{2} - n_{2}^{2}) [S_{l} (a)]^{2}$
$n_{o}^{2} 2 r S_{l} S_{l}^{'}$	$(\frac{a}{2}) (n_{1}^{2} - n_{2}^{2}) [S_{l} (a)]^{2}$
$n_{o}^{2} r^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$(n_{1}^{2} - n_{2}^{2}) {a^{2} S_{l} (a) S_{l}^{'} (a) - (\frac{a}{2}) [S_{l} (a)]^{2}}$
$n_{o}^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$(n_{1}^{2} - n_{2}^{2}) S_{l} (a) S_{l}^{'} (a)$
${S_{l}^{'}}^{2} + S_{l} S_{l}^{″}$	0
$(\frac{d n_{o}^{2}}{d r}) r S_{l}^{2}$	$- (n_{1}^{2} - n_{2}^{2}) a [S_{l} (a)]^{2}$
$(\frac{d n_{o}^{2}}{d r}) r^{2} S_{l} S_{l}^{'}$	$- (n_{1}^{2} - n_{2}^{2}) a^{2} S_{l} (a) S_{l}^{'} (a)$
$(\frac{d^{2} n_{o}^{2}}{d r^{2}}) r^{2} S_{l}^{2}$	$2 (n_{1}^{2} - n_{2}^{2}) {a [S_{l} (a)]^{2} + a^{2} S_{l} (a) S_{l}^{'} (a)}$

$f (r)$	$\int_{a}^{\infty} f (r) d r$
$S_{l}^{2}$	$(\frac{a}{2}) {- \frac{[S_{l}^{'} (a)]^{2}}{n_{2}^{2} k^{2}} + [\frac{l (l + 1)}{(n_{2} k a)^{2}} - 1] [S_{l} (a)]^{2} + \frac{S_{l} (a) S_{l} (a)^{'}}{n_{2}^{2} k^{2} a}} \equiv I$
$2 r S_{l} S_{l}^{'}$	$- a [S_{l} (a)]^{2} - I$
$r^{2} ({S_{l}^{'}}^{2} + S_{l} S_{l}^{″})$	$- a^{2} S_{l} (a) S_{l}^{'} (a) + a [S_{l} (a)]^{2} + I$

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Equations (53)

Journal of the Optical Society of America B