Resonant Modes of Optic Interferometer Cavities. I. Plane-Parallel End Reflectors*

Leonard Bergstein; Harry Schachter

doi:10.1364/JOSA.54.000887

Journal of the Optical Society of America
Vol. 54,
Issue 7,
pp. 887-903
(1964)
•https://doi.org/10.1364/JOSA.54.000887

Resonant Modes of Optic Interferometer Cavities. I. Plane-Parallel End Reflectors

Leonard Bergstein and Harry Schachter

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Leonard Bergstein and Harry Schachter, "Resonant Modes of Optic Interferometer Cavities. I. Plane-Parallel End Reflectors*," J. Opt. Soc. Am. 54, 887-903 (1964)

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Abstract

A study is made of the resonant or normal modes of optic and quasioptic interferometer cavities with plane–parallel end reflectors. The solution of the integral equation governing the relation between the normal modes and the geometry of the cavity is found by means of a series expansion of orthogonal functions. The terms of the series for the normal modes can be interpreted as Fraunhofer diffraction patterns characteristic of the geometry of the end reflectors. Various geometries, such as the infinite-strip, rectangular, and circular end reflector cavities, are considered and the results plotted and interpreted.

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	$f_{n} (ξ) = exp {+ i (\frac{π}{2} H ξ^{2} - ψ_{n} (0))} (∑_{k = - \infty}^{+ \infty} α_{n, k} \frac{sin [π (H ξ - k)]}{π (H ξ - k)})$				${\begin{array}{l} α_{n, - k} = {(- 1)}^{n - 1} α_{n, k} \\ ψ_{n} (0) = arg {f_{n} (0)} + ψ_{n} (0) \end{array}$
	Mode order n	α_n_,0	α_n_{, 1}	α_n_{, 2}	α_n_{, 3}
H = 025	1	1.0	0.0282 − i0.1548
	2	0	1.0	−0.4934–i0.2006
H = 0.5	1	1.0	0.1187 − i0.2710
	2	0	1.0	−0.4414−i0.3525
H = 1.0	1	1.0	0.2605 − i0.4016	−0.0562−i0.0480
	2	0	1.0	−0.2525−i0.6499
H = 2.0	1	1.0	0.5715 − i0.5162	−0.2096−i0.2109	0.0479+i0.2220
	2	0	1.0	0.1280−i0.9260	−0.3448+i0.2542
	3	1.0	−0.3061+i0.3366	0.6102 +i0 9158	0.1824−i0 4526
	4	0	1.0	1.0220+i0.7896	1.5773−i1.3422
	5	1.0	−1.3862+i0.6413	0.7148−i0.5751	0.6101+i3.4693
	6	0	1.0	−1.04484+i0.4875	−1.3383+i0.1021

$\begin{array}{l} f_{m n} (ρ, ϕ) = R_{m n} (ρ) exp (- i m ϕ) & = exp {i [\frac{π}{2} H ρ^{2} - ψ_{m n} (0)]} (∑_{k = 0}^{\infty} α_{m n, k} \frac{2 (π H ρ) J_{m - 1} (π H ρ)}{J_{m} ({\tilde{p}}_{m k}) [{\tilde{p}}_{m k}^{2} - {(π H ρ)}^{2}]}) e^{- i m ϕ} \\ {\begin{array}{l} {\tilde{p}}_{m k} is the k th zero of the Bessel function J_{m - 1}, ie ., J_{m - 1} ({\tilde{p}}_{m k}) = 0 \\ ψ_{m n} (0) = arg {R_{m n} (0)} + ψ_{m n} (0) \end{array} \end{array}$
	Mode order mn	α_mn_{, 0}	α_mn_{, 1}	α_mn_{, 2}	α_mn_{, 3}
H = 0.5	01	1.0	0.0323−i0.1614	−0.0375+i0.0271
	11	0	1.0	0.1184−i0.2273	0.0162−i0.0870
H = 1.0	01	1.0	0.2295−i0.2691	−0.2238+0.0142
	11	0	1.0	0.8863−i0.6758	−1.22474+i1.4420
H = 2.0	01	1.0	0.5633−i0.2636	−0.2210−i0.0212	0.0775+i0.0597
	11	0	1.0	0.0010−i0.9742	−0.2931+i0.1466
	02	1.0	−0.0186−i0.2857	0.4098+i0.1196	−0.0369−i0.2422
	12	0	1.0	0.1333+i0.8000	1.2668−i0.1333
	03	1.0	−0.6484+i0.2853	−0.2740-i0.8674	−1.2944+i0.7026
	13	0	1.0	−1.1076+i0.5292	−0.6601−i2.9776

	$f_{n} (ξ) = exp {+ i (\frac{π}{2} H ξ^{2} - ψ_{n} (0))} (∑_{k = - \infty}^{+ \infty} α_{n, k} \frac{sin [π (H ξ - k)]}{π (H ξ - k)})$				${\begin{array}{l} α_{n, - k} = {(- 1)}^{n - 1} α_{n, k} \\ ψ_{n} (0) = arg {f_{n} (0)} + ψ_{n} (0) \end{array}$
	Mode order n	α_n_,0	α_n_{, 1}	α_n_{, 2}	α_n_{, 3}
H = 025	1	1.0	0.0282 − i0.1548
	2	0	1.0	−0.4934–i0.2006
H = 0.5	1	1.0	0.1187 − i0.2710
	2	0	1.0	−0.4414−i0.3525
H = 1.0	1	1.0	0.2605 − i0.4016	−0.0562−i0.0480
	2	0	1.0	−0.2525−i0.6499
H = 2.0	1	1.0	0.5715 − i0.5162	−0.2096−i0.2109	0.0479+i0.2220
	2	0	1.0	0.1280−i0.9260	−0.3448+i0.2542
	3	1.0	−0.3061+i0.3366	0.6102 +i0 9158	0.1824−i0 4526
	4	0	1.0	1.0220+i0.7896	1.5773−i1.3422
	5	1.0	−1.3862+i0.6413	0.7148−i0.5751	0.6101+i3.4693
	6	0	1.0	−1.04484+i0.4875	−1.3383+i0.1021

$\begin{array}{l} f_{m n} (ρ, ϕ) = R_{m n} (ρ) exp (- i m ϕ) & = exp {i [\frac{π}{2} H ρ^{2} - ψ_{m n} (0)]} (∑_{k = 0}^{\infty} α_{m n, k} \frac{2 (π H ρ) J_{m - 1} (π H ρ)}{J_{m} ({\tilde{p}}_{m k}) [{\tilde{p}}_{m k}^{2} - {(π H ρ)}^{2}]}) e^{- i m ϕ} \\ {\begin{array}{l} {\tilde{p}}_{m k} is the k th zero of the Bessel function J_{m - 1}, ie ., J_{m - 1} ({\tilde{p}}_{m k}) = 0 \\ ψ_{m n} (0) = arg {R_{m n} (0)} + ψ_{m n} (0) \end{array} \end{array}$
	Mode order mn	α_mn_{, 0}	α_mn_{, 1}	α_mn_{, 2}	α_mn_{, 3}
H = 0.5	01	1.0	0.0323−i0.1614	−0.0375+i0.0271
	11	0	1.0	0.1184−i0.2273	0.0162−i0.0870
H = 1.0	01	1.0	0.2295−i0.2691	−0.2238+0.0142
	11	0	1.0	0.8863−i0.6758	−1.22474+i1.4420
H = 2.0	01	1.0	0.5633−i0.2636	−0.2210−i0.0212	0.0775+i0.0597
	11	0	1.0	0.0010−i0.9742	−0.2931+i0.1466
	02	1.0	−0.0186−i0.2857	0.4098+i0.1196	−0.0369−i0.2422
	12	0	1.0	0.1333+i0.8000	1.2668−i0.1333
	03	1.0	−0.6484+i0.2853	−0.2740-i0.8674	−1.2944+i0.7026
	13	0	1.0	−1.1076+i0.5292	−0.6601−i2.9776

Abstract

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Equations (136)

Journal of the Optical Society of America