Intracavity adaptive optics. 4: Comparison of theory and experiment

Kurt E. Oughstun; J. M. Spinhirne; D. Anafi

doi:10.1364/AO.23.001529

Applied Optics
Vol. 23,
Issue 10,
pp. 1529-1541
(1984)
•https://doi.org/10.1364/AO.23.001529

Intracavity adaptive optics. 4: Comparison of theory and experiment

Kurt E. Oughstun, J. M. Spinhirne, and D. Anafi

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Kurt E. Oughstun, J. M. Spinhirne, and D. Anafi, "Intracavity adaptive optics. 4: Comparison of theory and experiment," Appl. Opt. 23, 1529-1541 (1984)

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Abstract

A numerical analysis of the system performance of the intracavity adaptive optics experiment is presented. The analysis begins with a detailed comparison of the calculated and experimentally measured sensitivities of the unstable laser to both intracavity phase-tilt and astigmatism aberrations. The appropriate sensitivity measures considered here will be the outcoupled power and the far-field beam quality and power-in-the-bucket due to the field distribution outcoupled from the cavity. Of primary importance here is the correction obtained by the multidither zonal COAT system when a known intracavity phase aberration is introduced. The experimental performance with either beam quality or power in the bucket optimization is numerically assessed along with a comparison of intracavity and extracavity adaptive optic control.

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Distance from the convex mirror plane to the intracavity phase perturbation plane	Z₁ = 47.3 cm
Distance from the intracavity phase perturbation plane to the intracavity deformable mirror plane	Z₂ = 223.75 cm
Distance from the intracavity deformable mirror plane to the concave mirror plane	Z₃ = 27.95 cm
Total cavity length	Z_T = 299.0 cm
Geometric magnification of the cavity	M = 1.44
Transverse radial extent of the convex feedback mirror	a₁ = 3.06 cm
Transverse radial extent of the geometric mode	a₂ = Ma₁ = 4.406 cm
Radius of curvature of the convex feedback mirror	$R_{1} = - \frac{2 Z_{T}}{M - 1} = - 1359.09 cm$
Radius of curvature of the concave mirror	R₂ = 2Z_T − R₁ = 1957.09 cm
Wavelength of the cavity radiation field	λ = 10.6 μm
Equivalent Fresnel number of the cavity	$N_{eq} = \frac{(M - 1) a_{1}^{2}}{2 λ Z_{T}} = 6.50$
Collimated Fresnel number of the cavity	$N_{c} = \frac{a_{2}^{2}}{(M + 1) λ Z_{T}} = 25.10$
Geometric eigenvalue	γ_G = 0.694
Critical angle at the phase-perturbation plane	$θ_{c} (Z_{1}) = \frac{M - 1}{M + 1 + (M - 1) \frac{Z_{1}}{Z_{T}}} \cdot \frac{a_{1}}{∣ R_{1} ∣} = 394.75 μ rad$
Number of wavelengths of phase tilt (center to edge) necessary to attain the critical angle at the phase-perturbation plane	$m_{c} (Z_{1}) = \frac{a_{2}}{λ} θ_{c} (Z_{1}) = 1.6408$
Clear aperture diameter of the gain discharge tube	D = 10.2 cm
Distance from the phase perturbation plane to the first gain tube window	Z′ = 42.0 cm
Distance from the second gain tube window to the intracavity deformable mirror plane	Z″ = 31.0 cm

		Beam quality optimization			Power-in-the-bucket optimization			ΔΦ DM1-DM3
Aberration type	Zernike polynomial	Open loop DM1	Closed loop DM2	ΔΦ DM2-DM1	Open loop DM3	Closed loop DM4	ΔΦ DM4-DM3	ΔΦ DM1-DM3
x-Tilt	ρ cosθ	0.0196	0	−0.0196	−0.0902	0	0.0902	0.1098
y-Tilt	ρ sinθ	0.0258	0	−0.0258	−0.0159	0	0.0159	0.0417
Defocus	2ρ² − 1	0.0434	0.0525	0.0091	0	0.0506	0.0506	0.0433
Third order 0° astigmatism	ρ² cos2θ	0.0189	0.0439	0.0250	0	0.0704	0.0704	0.0189
Third order 45° astigmatism	ρ² sin2θ	0	0	0	0.0100	0	−0.0100	−0.0100
Third order x-coma	(3ρ³ − 2ρ) cosθ	−0.0292	0	0.0292	0	0	0	−0.0292
Third order y-coma	(3ρ³ − 2ρ) sinθ	0	0	0	0	0.0151	0.0151	0
Third order x-clover	ρ³ cos3θ	−0.0141	−0.0169	−0.0028	0	0.0445	0.0445	−0.0141
Third order y-clover	ρ³ sin3θ	−0.0234	−0.0210	0.0024	−0.0278	−0.0519	−0.0241	0.0044
Third order spherical	6ρ⁴ − 6ρ² + 1	−0.0406	−0.0436	−0.0030	−0.0232	−0.0327	−0.0095	−0.0174
Fifth order 0° astigmatism	(4ρ⁴ − 3ρ²) cos2θ	−0.0129	−0.0220	−0.0091	0	−0.0232	−0.0232	−0.0129
Fifth order 45° astigmatism	(4ρ⁴ − 3ρ²) sin2θ	0.0323	0.0287	−0.0036	0.0287	0.0413	0.0126	0.0036
Four leaf x-clover	ρ⁴ cos4θ	0	0	0	0	0	0	0
Four leaf y-clover	ρ⁴ sin4θ	0.0235	0.0169	−0.0066	0.0314	0	−0.0314	−0.0079
Fifth order x-coma	(10ρ⁵ − 12ρ³ + 3ρ) cosθ	0.0366	0.0274	−0.0092	0.0303	0.0226	−0.0077	0.0063
Fifth order y-coma	(10ρ⁵ − 12ρ³ + 3ρ) sinθ	0	0	0	0.0171	0	−0.0171	−0.0171
Fifth order x-clover	ρ⁵ cos5θ	0.0120	0.0135	0.0015	0.0260	0.0308	0.0048	−0.0140
Fifth order y-clover	ρ⁵ sin5θ	0.0134	0.0100	−0.0034	0.0223	0.0481	0.0258	−0.0089
Fifth order spherical	20ρ⁶ − 30ρ⁴ + 12ρ² − 1	0	0	0	0	0.0100	0.0100	0
Seventh order 0° astigmatism	(15ρ⁶ − 20ρ⁴ + 6ρ²) cos2θ	0.0238	0.0281	0.0043	0.0109	0.0191	0.0082	0.0129
Seventh order 45° astigmatism	(15ρ⁶ − 20ρ⁴ + 6ρ²) sin2θ	−0.0201	−0.0158	0.0043	−0.0223	−0.0213	0.0010	0.0022

Distance from the convex mirror plane to the intracavity phase perturbation plane	Z₁ = 47.3 cm
Distance from the intracavity phase perturbation plane to the intracavity deformable mirror plane	Z₂ = 223.75 cm
Distance from the intracavity deformable mirror plane to the concave mirror plane	Z₃ = 27.95 cm
Total cavity length	Z_T = 299.0 cm
Geometric magnification of the cavity	M = 1.44
Transverse radial extent of the convex feedback mirror	a₁ = 3.06 cm
Transverse radial extent of the geometric mode	a₂ = Ma₁ = 4.406 cm
Radius of curvature of the convex feedback mirror	$R_{1} = - \frac{2 Z_{T}}{M - 1} = - 1359.09 cm$
Radius of curvature of the concave mirror	R₂ = 2Z_T − R₁ = 1957.09 cm
Wavelength of the cavity radiation field	λ = 10.6 μm
Equivalent Fresnel number of the cavity	$N_{eq} = \frac{(M - 1) a_{1}^{2}}{2 λ Z_{T}} = 6.50$
Collimated Fresnel number of the cavity	$N_{c} = \frac{a_{2}^{2}}{(M + 1) λ Z_{T}} = 25.10$
Geometric eigenvalue	γ_G = 0.694
Critical angle at the phase-perturbation plane	$θ_{c} (Z_{1}) = \frac{M - 1}{M + 1 + (M - 1) \frac{Z_{1}}{Z_{T}}} \cdot \frac{a_{1}}{∣ R_{1} ∣} = 394.75 μ rad$
Number of wavelengths of phase tilt (center to edge) necessary to attain the critical angle at the phase-perturbation plane	$m_{c} (Z_{1}) = \frac{a_{2}}{λ} θ_{c} (Z_{1}) = 1.6408$
Clear aperture diameter of the gain discharge tube	D = 10.2 cm
Distance from the phase perturbation plane to the first gain tube window	Z′ = 42.0 cm
Distance from the second gain tube window to the intracavity deformable mirror plane	Z″ = 31.0 cm

		Beam quality optimization			Power-in-the-bucket optimization			ΔΦ DM1-DM3
Aberration type	Zernike polynomial	Open loop DM1	Closed loop DM2	ΔΦ DM2-DM1	Open loop DM3	Closed loop DM4	ΔΦ DM4-DM3	ΔΦ DM1-DM3
x-Tilt	ρ cosθ	0.0196	0	−0.0196	−0.0902	0	0.0902	0.1098
y-Tilt	ρ sinθ	0.0258	0	−0.0258	−0.0159	0	0.0159	0.0417
Defocus	2ρ² − 1	0.0434	0.0525	0.0091	0	0.0506	0.0506	0.0433
Third order 0° astigmatism	ρ² cos2θ	0.0189	0.0439	0.0250	0	0.0704	0.0704	0.0189
Third order 45° astigmatism	ρ² sin2θ	0	0	0	0.0100	0	−0.0100	−0.0100
Third order x-coma	(3ρ³ − 2ρ) cosθ	−0.0292	0	0.0292	0	0	0	−0.0292
Third order y-coma	(3ρ³ − 2ρ) sinθ	0	0	0	0	0.0151	0.0151	0
Third order x-clover	ρ³ cos3θ	−0.0141	−0.0169	−0.0028	0	0.0445	0.0445	−0.0141
Third order y-clover	ρ³ sin3θ	−0.0234	−0.0210	0.0024	−0.0278	−0.0519	−0.0241	0.0044
Third order spherical	6ρ⁴ − 6ρ² + 1	−0.0406	−0.0436	−0.0030	−0.0232	−0.0327	−0.0095	−0.0174
Fifth order 0° astigmatism	(4ρ⁴ − 3ρ²) cos2θ	−0.0129	−0.0220	−0.0091	0	−0.0232	−0.0232	−0.0129
Fifth order 45° astigmatism	(4ρ⁴ − 3ρ²) sin2θ	0.0323	0.0287	−0.0036	0.0287	0.0413	0.0126	0.0036
Four leaf x-clover	ρ⁴ cos4θ	0	0	0	0	0	0	0
Four leaf y-clover	ρ⁴ sin4θ	0.0235	0.0169	−0.0066	0.0314	0	−0.0314	−0.0079
Fifth order x-coma	(10ρ⁵ − 12ρ³ + 3ρ) cosθ	0.0366	0.0274	−0.0092	0.0303	0.0226	−0.0077	0.0063
Fifth order y-coma	(10ρ⁵ − 12ρ³ + 3ρ) sinθ	0	0	0	0.0171	0	−0.0171	−0.0171
Fifth order x-clover	ρ⁵ cos5θ	0.0120	0.0135	0.0015	0.0260	0.0308	0.0048	−0.0140
Fifth order y-clover	ρ⁵ sin5θ	0.0134	0.0100	−0.0034	0.0223	0.0481	0.0258	−0.0089
Fifth order spherical	20ρ⁶ − 30ρ⁴ + 12ρ² − 1	0	0	0	0	0.0100	0.0100	0
Seventh order 0° astigmatism	(15ρ⁶ − 20ρ⁴ + 6ρ²) cos2θ	0.0238	0.0281	0.0043	0.0109	0.0191	0.0082	0.0129
Seventh order 45° astigmatism	(15ρ⁶ − 20ρ⁴ + 6ρ²) sin2θ	−0.0201	−0.0158	0.0043	−0.0223	−0.0213	0.0010	0.0022

Abstract

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Figures (18)

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Applied Optics