Abstract

A method for decomposition of phase difference error between measurements of atmospheric turbulence-induced phase distortion at two different wavelengths is described. Calculations are made of the phase difference errors in the first five Zernike radial modes for both ground-to-ground and ground-to-space transmission of laser radiation. It is found that the phase difference error compared with the uncorrected wavefront phase error is relatively insignificant in the first (tilt) Zernike mode but increases in significance with the order of the Zernike mode. Relative phase difference error is also found to depend on transmitted and received wavelengths, aperture diameter, propagation path, and strength of turbulence.

© 1983 Optical Society of America

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References

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  1. C. B. Hogge, R. R. Butts, J. Opt. Soc. Am. 72, 606 (1982).
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    [CrossRef] [PubMed]
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1979 (1)

1978 (1)

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1971 (1)

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Figures (13)

Fig. 1
Fig. 1

Power spectra of phase error and phase difference error of radial mode n = 1. Nf = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 2
Fig. 2

Power spectra of phase error and phase difference error of radial mode n = 2. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 3
Fig. 3

Power spectra of phase error and phase difference error of radial mode n = 3. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 4
Fig. 4

Power spectra of phase error and phase difference error of radial mode n = 5. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 5
Fig. 5

Normalized residual phase error vs control loop bandwidth of single- and dual-wavelength control systems for radial mode n = 1. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 6
Fig. 6

Normalized residual phase error vs control loop bandwidth of single- and dual-wavelength control systems for radial mode n = 2. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 7
Fig. 7

Normalized residual phase error vs control loop bandwidth of single- and dual-wavelength control systems for radial mode n = 3. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 8
Fig. 8

Normalized residual phase error vs control loop bandwidth of single- and dual-wavelength control systems for radial mode n = 5. NF = 16, NT = 4, λ1 = 1 μm, λ2 = 10 μm.

Fig. 9
Fig. 9

Normalized phase difference error vs Fresnel number for the first five Zernike radial modes. NT = 4, λ1 = 1 μm, λ2 = 10 μm infinite bandwidth.

Fig. 10
Fig. 10

Normalized phase difference error vs received wavelength. NF = 16, NT = 4, λ1 = 1 μm, infinite bandwidth.

Fig. 11
Fig. 11

Comparison of single- and dual-wavelength control showing anisoplanatic effects of lead-angle compensation on ground-to-space propagation. Diameter = 0.2 m, λ1 = 1 μm, λ2 = 10 μm, infinite bandwidth.

Fig. 12
Fig. 12

Comparison of single- and dual-wavelength control showing anisoplanatic effects of lead-angle compensation on ground-to-space propagation. Diameter = 0.5 m, λ1 = 1 μm, λ2 = 10 μm, infinite bandwidth.

Fig. 13
Fig. 13

Bandwidth requirements for compensation of Zernike radial modes n = 1 and n = 5 due to ground-to-space laser propagation with no lead-angle compensation. Diameter = 0.5 m, λ1 = 1 μm, λ2 = 10 μm, Vwind = 5 m/sec.

Equations (7)

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Δ σ ϕ 2 ( λ 1 ) = ( 4.08 k 1 2 / π ) d z C n 2 ( z ) d K K 8 / 3 [ 1 ( KRz / 2 ) 2 J 1 2 ( KRz ) ] × { cos [ K 2 L ( 1 z ) z / ( 2 k 1 ) ] cos [ K 2 L ( 1 z ) z / ( 2 k 2 ) ] } 2 ,
Δ Ω n ( f ) = ( 16 π 2 k 1 2 L / f ) ( n + 1 ) 2 0 1 d z z 2 × 0 π / 2 d θ Φ Δ n ( Q , z ) J n + 1 2 ( QRz ) × [ cos Q 2 L z ( 1 z ) 2 k 1 cos Q 2 L z ( 1 z ) 2 k 2 ] 2 ,
Φ Δ n ( Q , z ) = 0.033 C n 2 ( z ) L 0 11 / 3 exp ( Q 2 ρ m 2 ) ( 1 + Q 2 L 0 2 ) 11 / 6 ,
Δ Ω n ( f ) = Ω n 1 ( f ) + Ω n 2 ( f )
Ω n i ( f ) = ( 16 π 2 k 1 2 L / f ) ( n + 1 ) 2 0 1 d z z 2 × 0 π / 2 d θ Φ Δ n ( Q , z ) J n + 1 2 ( QRz ) × [ cos 2 Q 2 L z ( 1 z ) 2 k i ]
ϕ 2 ( f 0 ) = Ω n ( f ) H ( f , f 0 ) H * ( f , f 0 ) d f + Δ Ω n ( f ) [ 1 H ( f , f 0 ) ] [ 1 H * ( f , f 0 ) ] d f ,
C n 2 ( h ) = 2.7 [ 2.2 × 10 23 h 10 exp ( h ) + 10 16 exp ( h / 1.5 ) ] + [ 10 14 exp ( 10 h ) ]

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