Abstract

Optical-field correction with deformable mirrors can be accomplished by correction of both amplitude and phase. As a result of developments over the past 20 years, phase correction with deformable mirrors has become a mature technology. We discuss simply the phase correction when it is concerned with field correction. The basic principle of amplitude correction with deformable mirrors is that if a certain phase distribution is constructed at the deformable mirror, after a vacuum diffraction, a certain amplitude distribution can be obtained. Some algorithms for implementing the principle have been put forward by several researchers [ T. T. Karr, Proc. Soc. Photo-Opt. Instrum. Eng. 1221, 26 ( 1990); Wang Kai-yun et al., Proc. Soc. Photo-Opt. Instrum. Eng. 1628, 244 ( 1992)]. But there are two problems that need to be solved. The first is that the vacuum path is too long. The second is that the precisions of these algorithms are relatively low. We describe a new algorithm, which not only yields a 1–2 order-of-magnitude reduction in the vacuum distance but also improves the amplitude correction precision.

© 1994 Optical Society of America

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References

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  1. T. J. Karr, “Instabilities of atmospheric laser propagation,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 26–55 (1990)
    [Crossref]
  2. Wang Kai-yun, Sun Jin-wen, Zhang Wei, “Paraxial theory of amplitude correction,” in Intense Laser Beams, P. B. Ulrich, R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244–252 (1992).
    [Crossref]
  3. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-VerlagBerlin, 1978).
    [Crossref]
  4. D. F. Elliot, K. Ramamohan Rao, Fast Transforms: Algorithms, Analysis, Applications (Academic, New York, 1982), p. 24.

Elliot, D. F.

D. F. Elliot, K. Ramamohan Rao, Fast Transforms: Algorithms, Analysis, Applications (Academic, New York, 1982), p. 24.

Jin-wen, Sun

Wang Kai-yun, Sun Jin-wen, Zhang Wei, “Paraxial theory of amplitude correction,” in Intense Laser Beams, P. B. Ulrich, R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244–252 (1992).
[Crossref]

Kai-yun, Wang

Wang Kai-yun, Sun Jin-wen, Zhang Wei, “Paraxial theory of amplitude correction,” in Intense Laser Beams, P. B. Ulrich, R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244–252 (1992).
[Crossref]

Karr, T. J.

T. J. Karr, “Instabilities of atmospheric laser propagation,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 26–55 (1990)
[Crossref]

Ramamohan Rao, K.

D. F. Elliot, K. Ramamohan Rao, Fast Transforms: Algorithms, Analysis, Applications (Academic, New York, 1982), p. 24.

Strohbehn, J. W.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-VerlagBerlin, 1978).
[Crossref]

Wei, Zhang

Wang Kai-yun, Sun Jin-wen, Zhang Wei, “Paraxial theory of amplitude correction,” in Intense Laser Beams, P. B. Ulrich, R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244–252 (1992).
[Crossref]

Other (4)

T. J. Karr, “Instabilities of atmospheric laser propagation,” in Propagation of High-Energy Laser Beams Through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1221, 26–55 (1990)
[Crossref]

Wang Kai-yun, Sun Jin-wen, Zhang Wei, “Paraxial theory of amplitude correction,” in Intense Laser Beams, P. B. Ulrich, R. C. Wade, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1628, 244–252 (1992).
[Crossref]

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-VerlagBerlin, 1978).
[Crossref]

D. F. Elliot, K. Ramamohan Rao, Fast Transforms: Algorithms, Analysis, Applications (Academic, New York, 1982), p. 24.

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Figures (5)

Fig. 1
Fig. 1

Field correction with two deformable mirrors.

Fig. 2
Fig. 2

Compressing-beam principle of field correction.

Fig. 3
Fig. 3

One-dimensional simulations of intensity (or amplitude) correction using the system shown in Fig. 1: (a) small-signal algorithm; (b) paraxial theory. The correction precisions of (a) and (b) are 1.2215 and 1.1255, respectively, and z = 1 km, D = 0.4 m, N = 12,800, and DF = 1.2 m. 1, |E1|2; 2, |E2|2; 3, |E2|2; 4, 0.018ϕ1; 5, 0.018ϕ2.

Fig. 4
Fig. 4

One-dimensional simulations of intensity (or amplitude) correction using the system shown in Fig. 2 with an iterative algorithm. The number of iterations is 1. The correction precision is 0.7028, z2 = 50 m, z1 = 6 m, D0 = 0.4 m, D1 = 0.08 m, N = 80,000, and DF = 0.6 m. (a) E1 and E2: 1, |E1|2; 2, |E2|2; 3, 0.05ϕ1; 4,0.05ϕ2. (b) E3: 1, |E3|2; 2, 0.03ϕ3. (c) E1 and E4: 1, |E1|2; 2, |E4|2; 3, |E4|2; 4, 0.02ϕ1; 5, 0.02ϕ4.

Fig. 5
Fig. 5

Two-dimensional simulations of intensity (or amplitude) correction using the system shown in Fig. 1 with an iterative algorithm. The number of iterations n is 40. The correction precision is 0.3240, and z = 1000 m, D = 0.4 m, N = 256 × 256, and the Fourier transform area = 0.5 × 0.5 m. (a) Intensity pattern (|E2|2) that is needed, (b) Intensity pattern (|E2|2) after correction, (c) |E1|2 and |E2|2 on the X axis: 1, |E1|2; 2, |E2|2; 3, |E2|2. (d) Variation of Pa with the number of iterations.

Equations (44)

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E z = i 2 K 0 2 E ,
2 = 2 / x 2 + 2 / y 2
E = e ψ , ψ = A + i ϕ ,
ψ z = i 2 K 0 [ 2 ψ + ( ψ ) 2 ] .
ψ z = i 2 K 0 2 ψ .
F ( ψ 2 ) = F ( ψ 1 ) exp ( i a k z ) ,
F ( A 2 ) = F ( A 1 ) cos ( a k z ) + F ( ϕ 1 ) sin ( a k z ) ,
F ( ϕ 2 ) = F ( ϕ 1 ) cos ( a k z ) F ( A 1 ) sin ( a k z ) .
F ( ϕ 1 ) = [ F ( A 2 ) F ( A 1 ) cos ( a k z ) ] sin ( a k z ) .
F ( ϕ 1 ) = F ( A 2 ) / sin ( a k z ) .
A 2 = A 1 z 2 K 0 ( 2 ϕ 1 + 2 A 1 x ϕ 1 x + 2 A 1 y ϕ 1 y ) ,
ϕ 2 = ϕ 1 + z 2 K 0 [ 2 A 1 + ( A 1 x ) 2 + ( A 1 y ) 2 ( ϕ 1 x ) 2 ( ϕ 1 y ) 2 ] .
F ( A 2 ) = F ( A 1 ) + F ( ϕ 1 ) a k z ,
F ( ϕ 2 ) = F ( ϕ 1 ) F ( A 1 ) a k z ,
F ( ϕ 1 ) = [ F ( A 2 ) F ( A 1 ) ] a k z .
ϕ 2 ( x , y ) = ϕ 1 ( α x , α y ) ,
A 2 ( x , y ) = A 1 ( α x , α y ) + ln α ,
A 3 ( x , y ) = A 4 ( α x , α y ) + ln α ,
F ( ϕ 2 ) = [ F ( A 3 ) F ( A 2 ) ] / a k z 2 .
F [ ϕ 2 ( x , y ) ] = ϕ 2 F ( k 1 , k 2 ) = ϕ 1 F ( k 1 / α , k 2 / α ) α 2 ,
A 3 F ( k 1 , k 2 ) = [ A 4 F ( k 1 / α , k 2 / α ) + F ( ln α ) ] α 2 ,
A 2 F ( k 1 , k 2 ) = [ A 1 F ( k 1 / α , k 2 / α ) + F ( ln α ) ] α 2 ,
ϕ 1 F ( k 1 , k 2 ) = [ A 4 F ( k 1 , k 2 ) A 1 F ( k 1 , k 2 ) ] / α 2 α k z 2 .
F ( E 2 ) = F ( E 1 ) exp ( i a k z ) ;
F [ exp ( A 2 + i ϕ 2 ) ] = F [ exp ( A 1 + i ϕ 1 ) ] exp ( i a k z ) .
ψ = ψ p + δ ψ ;
E = E p ( 1 + δ E ) ,
| δ E | 2 = | exp ( δ ψ ) 1 | 2 = 1 2 exp ( δ A ) cos ( δ ϕ ) + exp ( 2 δ A ) .
δ A = 0 δ ϕ = 0.
F ( E p 2 ) = F ( E p 1 ) exp ( i a k z ) .
F ( δ E 2 E p 2 ) = F ( δ E 1 E p 1 ) exp ( i a k z ) .
| δ E 2 | 2 | E p 2 | 2 ¯ = | δ E 1 | 2 | E p 1 | 2 ¯ ,
ϕ 21 = ϕ p 2 + δ ϕ 21 = F 1 [ F ( A p 2 ) F ( A p 1 ) s c a k z ] ,
ϕ 11 = ϕ p 1 + δ ϕ 11 = Im ( clg { F 1 [ F ( E 21 ) exp ( i a k z ) ] } ) ,
| δ E 2 ( 0 , δ ϕ 21 ) | 2 | E p 2 | 2 ¯ = | δ E 1 ( δ A 11 , δ ϕ 11 ) | 2 | E p 1 | 2 ¯ .
ϕ 22 = ϕ p 2 + δ ϕ 22 = Im ( clg { F 1 [ F ( E 11 ) exp ( i a k z ) ] } ) ,
| δ E 2 ( δ A 22 , δ ϕ 22 ) | 2 | E p 2 | 2 ¯ = | δ E 1 ( 0 , δ ϕ 11 ) | 2 | E p 1 | 2 ¯ .
ϕ 12 = ϕ p 1 + δ ϕ 12 = Im ( clg { F 1 [ F ( E 22 ) exp ( i a k z ) ] } ) ,
| δ E 2 ( 0 , δ ϕ 22 ) | 2 | E p 2 | 2 ¯ = | δ E 1 ( δ A 12 , δ ϕ 12 ) | 2 | E p 1 | 2 ¯ .
| δ E 1 ( δ A 11 , δ ϕ 11 ) | 2 ¯ > | δ E 1 ( 0 , δ ϕ 12 ) | 2 ¯ , | δ E 2 ( δ A 22 , δ ϕ 22 ) | 2 ¯ > | δ E 2 ( 0 , δ ϕ 22 ) | 2 ¯ .
ϕ p 1 = lim n ϕ 1 n .
α 2 ( z 2 + z 1 / α + z 3 / α ) a k .
ϕ 2 = Im ( clg { F 1 [ F ( A 0 + ϕ 1 x ) exp ( i a k z ) ] } ) .
P a = [ ( | E 2 | | E 2 | ¯ ) 2 d s / ( | E 2 | | E 2 | ¯ ) 2 d s ] 1 / 2 ,

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