Diffractive effect of a scraper mirror in an unstable resonator

W. P. Latham; Martin E. Smithers

doi:10.1364/JOSA.72.001321

Journal of the Optical Society of America
Vol. 72,
Issue 10,
pp. 1321-1327
(1982)
•https://doi.org/10.1364/JOSA.72.001321

Diffractive effect of a scraper mirror in an unstable resonator

W. P. Latham and Martin E. Smithers

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W. P. Latham and Martin E. Smithers, "Diffractive effect of a scraper mirror in an unstable resonator," J. Opt. Soc. Am. 72, 1321-1327 (1982)

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Abstract

In an unstable optical resonator, a scraper mirror located in front of and displaced from the feedback mirror clips the feedback beam twice on each round trip through the resonator. The increased diffractive scattering produced by this double application of the scraper aperture enhances the peaking of the intracavity intensity and raises the feedback ratio. This increased diffractive scattering is proportional to the square root of the scraper displacement and to the equivalent Fresnel number of the resonator.

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k	$Γ_{k} = \int_{0}^{γ} d r r^{k} exp (- i b r^{2})$
0	$\frac{1}{2} \frac{π}{i b} {1 - erfc [{(i b)}^{1 / 2} γ]}$
1	$\frac{1}{2 i b} [1 - exp (- i b γ^{2})]$
2	$\frac{1}{2 i b} (- γ exp (- i b γ^{2}) + \frac{1}{2} \frac{π}{i b} {1 - erfc [{(i b)}^{1 / 2} γ]})$
n	$\frac{1}{2 i b} [- γ^{n - 1} exp (- i b γ^{2}) + (n - 1) Γ_{n - 2}]$

Constant	Exact Expression	Approximate Form	Limiting Value
C₁	$\frac{(1 - \frac{d}{R_{0}})}{R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})}$	$\frac{(1 - \frac{d}{R_{0}})}{R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})}$	$\frac{1}{R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})} = 2 f_{g}$
c₂	$- \frac{C_{1}}{2} {[\frac{(1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}{(1 - \frac{d}{R_{0}})}]}^{1 / 2}$	$- \frac{C_{1}}{2} [1 + \frac{1}{2} \frac{N_{eq}}{N}]$	$- \frac{C_{1}}{2} = - f_{g}$
c₃	$i^{1 / 2} \frac{C_{1}}{2} {(\frac{N_{eq}}{2 N})}^{1 / 2} {[\frac{(1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}{(1 - \frac{d}{R_{f}}) (1 - \frac{d}{R_{0}})}]}^{1 / 2}$	$i^{1 / 2} \frac{C_{1}}{2} {(\frac{N_{eq}}{2 N})}^{1 / 2} [1 + d (\frac{1}{R_{0}} - \frac{1}{R_{f}})]$	0
c₄	$- i^{1 / 2} C_{1} \frac{N_{eq}}{{(N)}^{1 / 2}} \frac{1}{(1 - \frac{d}{R_{0}}) {(1 - \frac{d}{R_{f}})}^{1 / 2}}$	$- i^{1 / 2} C_{1} \frac{N_{eq}}{{(N)}^{1 / 2}} [1 + d (\frac{1}{R_{0}} + \frac{1}{2 R_{f}})]$	0
C₅	$\frac{- 1}{2 π i^{1 / 2}} \frac{C_{1}}{2} \frac{1}{{(N)}^{1 / 2}} \frac{(1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}{{(1 - \frac{d}{R_{f}})}^{1 / 2}}$	$\frac{- 1}{2 π i^{1 / 2}} \frac{C_{1}}{2} \frac{1}{{(N)}^{1 / 2}} [1 + d (\frac{1}{R_{0}} - \frac{3}{2 R_{f}})]$	0
C₆	$- i \frac{C_{1}}{2} \frac{N_{eq}}{2 N} \frac{1}{{[(1 - \frac{d}{R_{f}}) (1 - \frac{d}{R_{0}})]}^{1 / 2}}$	$- i \frac{C_{1}}{2} \frac{N_{eq}}{2 N} [1 + \frac{d}{2} (\frac{1}{R_{0}} + \frac{1}{R_{f}})]$	0
c₇	$\frac{i^{1 / 2}}{4 π} \frac{C_{1}}{2} \frac{N_{eq}}{N^{1 / 2} N} \frac{1}{{(1 - \frac{d}{R_{f}})}^{1 / 2}}$	$\frac{i^{1 / 2}}{4 π} \frac{C_{1}}{2} \frac{N_{eq}}{N^{1 / 2} N} (1 + \frac{d}{2 R_{f}})$	0
P₁	$\frac{2 π N_{eq}}{(1 - \frac{d}{R_{0}}) (1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}$	$2 π N_{eq} (1 + \frac{2 d}{R_{f}})$	2πN_eq
P₂	$\frac{4 π N}{(1 - \frac{d}{R_{0}})}$	$4 π N (1 + \frac{d}{R_{0}})$	∞

Constant	Geometric Expression
A	$\frac{(1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}{2 d (1 - \frac{d}{R_{f}}) (1 - \frac{d}{R_{0}})}$
B	$2 d (1 - \frac{d}{R_{f}})$
R_d	$R_{0} (1 - \frac{d}{R_{0}})$
$A + \frac{1}{B}$	$\frac{1}{d (1 - \frac{d}{R_{0}})}$
$A - \frac{1}{B}$	$\frac{(\frac{1}{R_{0}} + \frac{1}{R_{f}})}{(1 - \frac{d}{R_{f}}) (1 - \frac{d}{R_{0}})}$
$A^{2} - \frac{1}{B^{2}}$	$\frac{(\frac{1}{R_{0}} - \frac{1}{R_{f}})}{d (1 - \frac{d}{R_{f}}) {(1 - \frac{d}{R_{0}})}^{2}}$
${[B R_{d} (A^{2} - \frac{1}{B^{2}})]}^{- 1}$	$\frac{(1 - \frac{d}{R_{0}})}{2 R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})}$
N_eq	$\frac{α^{2}}{λ} (\frac{1}{R_{0}} - \frac{1}{R_{f}})$
N	$\frac{α^{2}}{2 d λ}$

Constant	Exact Expression	Limiting Value
$\frac{C_{1}}{2}$	$\frac{(1 - \frac{d}{R_{0}})}{2 R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})}$	$\frac{1}{2 R_{0} (\frac{1}{R_{0}} - \frac{1}{R_{f}})} = f_{g}$
C₈	$\frac{1}{π i^{1 / 2}} \frac{C_{1}}{2} \frac{N^{1 / 2}}{N_{eq}} {(1 - \frac{d}{R_{f}})}^{3 / 2}$	0
C₉	$\frac{i^{1 / 2}}{4 π} \frac{C_{1}}{2} \frac{N_{eq}}{N^{1 / 2} N} {(1 - \frac{d}{R_{f}})}^{- 1 / 2}$	0
P₁	$\frac{2 π N_{eq}}{(1 - \frac{d}{R_{0}}) (1 - \frac{2 d}{R_{f}} + \frac{d}{R_{0}})}$	2πN_eq
P₂	$\frac{4 π N}{(1 - \frac{d}{R_{0}})}$	4πN
P₃	$\frac{2 π N_{eq}}{(1 - \frac{d}{R_{f}}) (1 - \frac{d}{R_{0}})}$	2πN_eq

k	$Γ_{k} = \int_{0}^{γ} d r r^{k} exp (- i b r^{2})$
0	$\frac{1}{2} \frac{π}{i b} {1 - erfc [{(i b)}^{1 / 2} γ]}$
1	$\frac{1}{2 i b} [1 - exp (- i b γ^{2})]$
2	$\frac{1}{2 i b} (- γ exp (- i b γ^{2}) + \frac{1}{2} \frac{π}{i b} {1 - erfc [{(i b)}^{1 / 2} γ]})$
n	$\frac{1}{2 i b} [- γ^{n - 1} exp (- i b γ^{2}) + (n - 1) Γ_{n - 2}]$

Abstract

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Figures (6)

Tables (5)

Equations (83)

Journal of the Optical Society of America