Abstract

Modal estimation of wave-front phase from phase derivatives is discussed. It is shown that it is desirable to minimize the number of modes estimated and the number of measurements used to maintain the quality of the estimates of low-order modes. It is also shown that mode cross coupling occurs when one tries to estimate modes higher than astigmatism.

© 1979 Optical Society of America

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References

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  1. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [Crossref] [PubMed]
  2. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [Crossref]
  3. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [Crossref] [PubMed]
  4. P. B. Liebelt, An Introduction to Optimal Estimation (Addison-Wesley, Reading, Massachusetts, 1967).
  5. W. P. Brown (private communication).
  6. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  7. J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [Crossref]
  8. R. D. Hudson, Infrared System Engineering (Wiley, New York, 1969).

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TABLE I Zernike polynomials and directional derivatives

Equations (62)

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ϕ ( X , Y ) = K a K Z K ( X , Y ) .
P ( X , Y ) i = ϕ ( X , Y ) i X + ν i
Q ( X , Y ) i = ϕ ( X , Y ) i Y + μ i ,
ϕ ( X , Y ) X = K 1 [ ϕ ( X + S , Y ) ϕ ( X , Y ) ]
ϕ ( X , Y ) Y = K 2 [ ϕ ( X , Y + S ) ϕ ( S , Y ) ]
P = { P ( X , Y ) 1 , P ( X , Y ) 2 , , P ( X , Y ) K , Q ( X , Y ) 1 , Q ( X , Y ) 2 , , Q ( X , Y ) L } T .
 = { â 2 , â 3 , , â I } .
Φ ̂ ( X , Y ) = i = 2 I â i Z i ( X , Y ) .
Φ ̂ X = i = 2 I â i Z i ( X , Y ) X ,
Φ ̂ Y = i = 2 I â i Z i ( X , Y ) Y .
F = k = 1 K ( P ( X , Y ) k i = 2 I â i Z i ( X , Y ) k X ) 2 + l = 1 L ( Q ( X , Y ) l i = 2 I â i Z i ( X , Y ) l Y ) 2 .
k = 1 K ( P ( X , Y ) k Z j ( X , Y ) k X ) + l = 1 L ( Q ( X , Y ) l Z i ( X , Y ) l Y ) = i = 2 I [ â i ( k = 1 K Z j ( X , Y ) k X Z i ( X , Y ) k X + l = 1 L Z j ( X , Y ) l Y Z i ( X , Y ) l Y ) ]
D P = E Â ,
D = ( Z 2 ( X , Y ) 1 X Z 2 ( X , Y ) 2 X Z 2 ( X , Y ) K X Z 2 ( X , Y ) 1 Y Z 2 ( X , Y ) L Y Z 3 ( X , Y ) 1 X Z 3 ( X , Y ) 2 X Z 3 ( X , Y ) K X Z 3 ( X , Y ) 1 Y Z 3 ( X , Y ) L Y Z I ( X , Y ) 1 X Z I ( X , Y ) 2 X Z I ( X , Y ) K X Z I ( X , Y ) 1 Y Z I ( X , Y ) L Y )
E = D D T .
 = E 1 D P = ( D D T ) 1 D P .
C A = ( D D T ) 1 D C P D T ( D D T ) 1 ,
C A = ( σ â 2 â 2 2 σ â 2 â 3 2 σ â 2 â I 2 σ â 3 â 2 2 σ â 3 â 3 2 σ â 2 â I 2 σ â I â 2 2 σ â I â 3 2 σ â I â I 2 )
C P = ( σ P 2 ( X , Y ) 1 P ( X , Y ) 1 σ P 2 ( X , Y ) 1 P ( X , Y ) 2 σ P 2 ( X , Y ) 1 Q ( X , Y ) L σ P 2 ( X , Y ) 2 P ( X , Y ) 1 σ P 2 ( X , Y ) 2 P ( X , Y ) 2 σ P 2 ( X , Y ) 2 Q ( X , Y ) L σ Q 2 ( X , Y ) L P ( X , Y ) 1 σ Q 2 ( X , Y ) L P ( X , Y ) 2 σ Q 2 ( X , Y ) L Q ( X , Y ) L )
C A = ( D D T ) 1 σ P 2 = E 1 σ P 2 ,
σ P 2 = σ P ( X , Y ) 1 , P ( X , Y ) 1 2 = σ P ( X , Y ) 2 ( P ( X , Y ) 2 2 = = σ Q ( X , Y ) L Q ( X , Y ) L 2 .
E = [ 1 0 0 1 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 .
E = [ 2 0 0 2 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 2
σ P 2 = N 2 σ TOTAL 2 ,
σ P 2 = N σ TOTAL 2 .
σ â 2 â 2 2 = σ â 3 â 3 2 = 2 N σ TOTAL 2 .
σ â 2 â 2 2 = σ â 3 â 3 2 = 2 σ TOTAL 2 .
E = [ 2 0 0 0 2 0 0 0 16 ]
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 2 ,
σ â 4 â 4 2 = σ P 2 / 16 .
E = [ 4 0 0 0 0 0 4 0 0 0 0 0 32 0 0 0 0 0 8 0 0 0 0 0 8 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 4 ,
σ â 4 â 4 2 = σ P 2 / 32 ,
σ â 5 â 5 2 = σ â 6 â 6 2 = σ 2 / 8 ,
E = [ 6 0 0 0 0 0 6 0 0 0 0 0 512 9 0 0 0 0 0 128 9 0 0 0 0 0 12 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 6 ,
σ â 4 â 4 2 = 9 σ P 2 / 512 σ P 2 / 56.89 ,
σ â 5 â 5 2 = 9 σ P 2 / 128 σ P 2 / 14.22 ,
σ â 6 â 6 2 = σ P 2 / 12
E = [ 6 0 0 0 0 0 6 0 0 0 0 0 24 0 0 0 0 0 6 0 0 0 0 0 6 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 6 ,
σ â 4 â 4 2 = σ P 2 / 24 ,
σ â 5 â 5 2 = σ â 6 â 6 2 = σ P 2 / 6 .
E = [ 6 0 0 0 0 0 6 0 0 0 0 0 384 9 0 0 0 0 0 32 3 0 0 0 0 0 32 3 ] ,
σ â 2 â 2 2 = σ â 3 â 3 2 = σ P 2 / 6 ,
σ â 4 â 4 2 = ( 9 σ P 2 / 384 ) ( σ P 2 / 42.67 ) ,
σ â 5 â 5 2 = σ â 6 â 6 2 = 3 σ P 2 / 32 σ P 2 / 10.67 .
E = [ 6 0 0 0 0 62 4 0 5 6 0 0 6 0 0 0 0 62 4 0 5 6 0 0 512 9 0 0 0 0 0 0 0 0 0 128 9 0 0 0 0 0 0 0 0 0 12 0 0 0 0 62 4 0 0 0 0 619 9 0 265 24 0 0 62 4 0 0 0 0 619 8 0 265 24 5 6 0 0 0 0 265 24 0 1331 72 0 0 5 6 0 0 0 0 265 24 0 1331 72 ]
E 1 = [ 0.359 0 0 0 0 0.076 0 0.029 0 0 0.426 0 0 0 0 0.096 0 0.077 0 0 0.018 0 0 0 0 0 0 0 0 0 0.070 0 0 0 0 0 0 0 0 0 0.083 0 0 0 0 0.076 0 0 0 0 0.030 0 0.015 0 0 0.096 0 0 0 0 0.036 0 0.026 0.029 0 0 0 0 0.015 0 0.062 0 0 0.077 0 0 0 0 0.026 0 0.073 ]
σ â 2 â 2 2 = σ P 2 / 2.79 , σ â 3 â 3 2 = σ P 2 / 2.35 , σ â 4 â 4 2 = σ P 2 / 55.56 , σ â 5 â 5 2 = σ P 2 / 14.29 , σ â 6 â 6 2 = σ P 2 / 12 , σ â 7 â 7 2 = σ P 2 / 33.33 , σ â 8 â 8 2 = σ P 2 / 27.78 , σ â 9 â 9 2 = σ P 2 / 16.13 , σ â 10 â 10 2 = σ P 2 / 13.70 .
A S = A / N .
Q S = ( H A / h ν ) ( A / N ) τ 0 η q t d ,
Q N = [ ( ω d N b / h ν ) ( A / N ) η q t d ] 1 / 2 ,
SNR = Q S Q N = H A τ 0 ( η q t d h ν N b ) 1 / 2 ( A ω d N ) .
β = H A τ 0 ( η q t d / h ν N b ) 1 / 2
SNR = β ( A / ω d N ) 1 / 2 .
σ P 2 = K / SNR 2 = K ω d N / β 2 A ,
ω d = K 1 N λ 2 / A ,
K 1 = ( 1.22 ) 2 π .
σ P 2 = K K 1 λ 2 N 2 / β 2 A 2 .