An Analysis of Optical Effects Caused by Thermally Induced Mirror Deformations

R. F. Ogrodnik

doi:10.1364/AO.9.002028

Applied Optics
Vol. 9,
Issue 9,
pp. 2028-2034
(1970)
•https://doi.org/10.1364/AO.9.002028

An Analysis of Optical Effects Caused by Thermally Induced Mirror Deformations

R. F. Ogrodnik

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R. F. Ogrodnik, "An Analysis of Optical Effects Caused by Thermally Induced Mirror Deformations," Appl. Opt. 9, 2028-2034 (1970)

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Abstract

This paper analyzes thermally induced mirror deformations and their resulting wavefront distortions which occur under the conditions of radially nonuniform mirror heating. The analysis is adaptable to heating produced by any radially nonuniform incident radiation. Specific examples of radiation distributions which are considered are the cosine squared and the gaussian and TEM(0, 1) laser distributions. Deformation effects are examined from two aspects, the first of which is the reflected wavefront radial phase distortion profile caused by the thermally induced surface irregularities at the mirror face. These phase distortion effects appear as aberrations in noncoherent optical applications and as the loss of spatial coherence in coherent applications. The second aspect is the gross wavefront bending due to mirror curvature effects. The analysis considers substrate material, geometry, and cooling in order to determine potential deformation controlling factors. Substrate materials are compared, and performance indicators are suggested to aid in selecting an optimum material for a given heating condition. Deformation examples are given for materials of interest and specific absorbed power levels.

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Distribution	Power density, p(R) W cm⁻²	Enclosed power distribution, P(R) in %
		If $b / w < π / 2, \frac{2 γ^{2} R^{2} + 2 γ R sin 2 γ R + cos 2 γ R - 1}{2 γ^{2} + 2 γ sin 2 γ + cos 2 γ - 1}$
Cosine squared	$\frac{4 γ^{2} {cos}^{2} (γ R)}{π b^{2} [2 β^{2} + 2 β sin 2 β + cos 2 β - 1]}$	If $b / w > π / 2, 2 (\frac{2 γ^{2} R^{2} + 2 γ R sin 2 γ R + cos 2 γ R - 1}{π^{2} - 4})$
TEM(0, 0)	$\frac{a exp (- a R^{2})}{π b^{2} [1 - exp (- 2)]}$	$\frac{exp (a) [exp (a R^{2}) - 1]}{exp (a R^{2}) [exp (a) - 1]}$
TEM(0, 1)	$\frac{a^{2} R^{2} exp (- a R^{2})}{π b^{2} [1 - 3 exp (- 2)]}$	$\frac{exp (a) [exp (a R^{2}) - a R^{2} - 1]}{exp (a R^{2}) [exp (a) - a - 1]}$
where a = 2(b/w)², β = arc cos(1/e), γ = βb/w, and b/w = mirror-to-1/e field diameter ratio.

(a) κ⁻¹ in tens of kilometers for b/w = 1
k	α × 10⁻⁷	F × 10⁻⁴	M	cos²	TEM (0, 0)	TEM (0, 1)
				κ⁻¹
0.01	1	10	10	659	762	770
			100	371	416	417
0.1	1	100	10	688	788	790
			100	379	425	426
1	100	10	10	6.9	7.3	7.3
			100	3.8	4.0	4.0

(b) κ⁻¹ in tens of kilometers for b/w = 5

k	α × 10⁻⁷	F × 10⁻⁴	M	cos²	TEM (0, 0)	TEM (0, 1)
				κ⁻¹
0.01	1	10	10	794	667	495
			100	422	370	275
0.1	1	100	10	797	695	505
			100	488	380	277
1	100	10	10	7.0	6.5	4.8
			100	3.8	3.6	2.6

Material	Distribution b/w = 1 cosine square	TEM (0, 1)	Distribution b/w = 5 cosine square	TEM (0, 1)
Super Invar	667	764	772	492
Cer-vit	622	672	642	441
Copper	4.1	4.4	4.2	2.9

Distribution	Power density, p(R) W cm⁻²	Enclosed power distribution, P(R) in %
		If $b / w < π / 2, \frac{2 γ^{2} R^{2} + 2 γ R sin 2 γ R + cos 2 γ R - 1}{2 γ^{2} + 2 γ sin 2 γ + cos 2 γ - 1}$
Cosine squared	$\frac{4 γ^{2} {cos}^{2} (γ R)}{π b^{2} [2 β^{2} + 2 β sin 2 β + cos 2 β - 1]}$	If $b / w > π / 2, 2 (\frac{2 γ^{2} R^{2} + 2 γ R sin 2 γ R + cos 2 γ R - 1}{π^{2} - 4})$
TEM(0, 0)	$\frac{a exp (- a R^{2})}{π b^{2} [1 - exp (- 2)]}$	$\frac{exp (a) [exp (a R^{2}) - 1]}{exp (a R^{2}) [exp (a) - 1]}$
TEM(0, 1)	$\frac{a^{2} R^{2} exp (- a R^{2})}{π b^{2} [1 - 3 exp (- 2)]}$	$\frac{exp (a) [exp (a R^{2}) - a R^{2} - 1]}{exp (a R^{2}) [exp (a) - a - 1]}$
where a = 2(b/w)², β = arc cos(1/e), γ = βb/w, and b/w = mirror-to-1/e field diameter ratio.

(a) κ⁻¹ in tens of kilometers for b/w = 1
k	α × 10⁻⁷	F × 10⁻⁴	M	cos²	TEM (0, 0)	TEM (0, 1)
				κ⁻¹
0.01	1	10	10	659	762	770
			100	371	416	417
0.1	1	100	10	688	788	790
			100	379	425	426
1	100	10	10	6.9	7.3	7.3
			100	3.8	4.0	4.0

(b) κ⁻¹ in tens of kilometers for b/w = 5

k	α × 10⁻⁷	F × 10⁻⁴	M	cos²	TEM (0, 0)	TEM (0, 1)
				κ⁻¹
0.01	1	10	10	794	667	495
			100	422	370	275
0.1	1	100	10	797	695	505
			100	488	380	277
1	100	10	10	7.0	6.5	4.8
			100	3.8	3.6	2.6

Abstract

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Applied Optics