Abstract

An analysis is performed to determine the accuracy with which an ac heterodyne lateral shear interferometer can measure wavefront aberrations if a white light extended source is used with the interferometer, and shot noise is the predominate noise source. The analysis shows that for uniform circular or square sources larger than a derived minimum size, the wavefront measurement accuracy depends only upon the radiance of the source and not upon the angular subtense of the source. For a 1-msec integration time, a 25-cm2 collecting area, and a source radiance of 10 W/m2-sr the rms wavefront error is approximately 1/30 wave, assuming the signal is shot noise limited. It is shown that for both uniform circular and square sources an optimum shear distance is approximately ½ the aperture diameter required to resolve the light source. Comments are made on the optimum shear for nonuniform radiance distributions.

© 1975 Optical Society of America

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References

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  1. H. W. Babcock, Publ. Astron. Soc. Pac. 65, 229 (1953).
    [CrossRef]
  2. R. A. Muller, A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
    [CrossRef]
  3. J. Hardy, J. Feinleib, J. C. Wyant, in Digest of OSA Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).
  4. W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
    [CrossRef] [PubMed]
  5. W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
    [CrossRef] [PubMed]
  6. L. J. Golden, R. V. Shack, P. N. Slater, NASA Final Report for contract NAS8-27863 (1974).
  7. R. Crane, Appl. Opt. 8, 538 (1969).
  8. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, Appl. Opt. 13, 2693 (1974).
    [CrossRef] [PubMed]
  9. J. C. Wyant, Appl. Opt. 13, 200 (1974).
    [CrossRef] [PubMed]
  10. J. B. Saunders, R. J. Bruening, Astron. J. 73, 415 (1968).
    [CrossRef]
  11. M. P. Rimmer, Appl. Opt. 13, 623 (1974).
    [CrossRef] [PubMed]
  12. R. V. Shack, J. Opt. Soc. Am. 64, 1362A (1974).
  13. E. D. Sunde, Communication Systems Engineering Theory (Wiley, New York, 1969), p. 120
  14. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 463.
  15. The author acknowledges a beneficial conversation with V. P. Bennett on the derivation.
  16. A. G. Worthing, J. Geffner, Treatment of Experimental Data (Wiley, New York, 1943), p. 208.
  17. S. P. Wyatt, Principles of Astronomy (Allyn and Bacon, Boston, 1964), p. 205.
  18. RCA Electro-Optics Handbook Technical Series EOH-11 (Harrison, N.J., 1974), p. 68.

1974

W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
[CrossRef] [PubMed]

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

R. V. Shack, J. Opt. Soc. Am. 64, 1362A (1974).

J. C. Wyant, Appl. Opt. 13, 200 (1974).
[CrossRef] [PubMed]

M. P. Rimmer, Appl. Opt. 13, 623 (1974).
[CrossRef] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, Appl. Opt. 13, 2693 (1974).
[CrossRef] [PubMed]

R. A. Muller, A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
[CrossRef]

1969

R. Crane, Appl. Opt. 8, 538 (1969).

1968

J. B. Saunders, R. J. Bruening, Astron. J. 73, 415 (1968).
[CrossRef]

1953

H. W. Babcock, Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Arnold, R. A.

W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
[CrossRef] [PubMed]

Babcock, H. W.

H. W. Babcock, Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Brangaccio, D. J.

Bridges, W. B.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Brown, W. P.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Bruening, R. J.

J. B. Saunders, R. J. Bruening, Astron. J. 73, 415 (1968).
[CrossRef]

Bruning, J. H.

Brunner, P. T.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Buffington, A.

Crane, R.

R. Crane, Appl. Opt. 8, 538 (1969).

Feinleib, J.

J. Hardy, J. Feinleib, J. C. Wyant, in Digest of OSA Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Gallagher, J. E.

Geffner, J.

A. G. Worthing, J. Geffner, Treatment of Experimental Data (Wiley, New York, 1943), p. 208.

Golden, L. J.

L. J. Golden, R. V. Shack, P. N. Slater, NASA Final Report for contract NAS8-27863 (1974).

Gowrinathan, S.

W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
[CrossRef] [PubMed]

Hardy, J.

J. Hardy, J. Feinleib, J. C. Wyant, in Digest of OSA Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Herriott, D. R.

Lassera, S. P.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Muller, R. A.

Nussmeier, T. A.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

O'Meara, T. R.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Peters, W. N.

W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
[CrossRef] [PubMed]

Rimmer, M. P.

Rosenfeld, D. P.

Sanguinet, J. A.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Saunders, J. B.

J. B. Saunders, R. J. Bruening, Astron. J. 73, 415 (1968).
[CrossRef]

Shack, R. V.

R. V. Shack, J. Opt. Soc. Am. 64, 1362A (1974).

L. J. Golden, R. V. Shack, P. N. Slater, NASA Final Report for contract NAS8-27863 (1974).

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 463.

Slater, P. N.

L. J. Golden, R. V. Shack, P. N. Slater, NASA Final Report for contract NAS8-27863 (1974).

Sunde, E. D.

E. D. Sunde, Communication Systems Engineering Theory (Wiley, New York, 1969), p. 120

White, A. D.

Worthing, A. G.

A. G. Worthing, J. Geffner, Treatment of Experimental Data (Wiley, New York, 1943), p. 208.

Wyant, J. C.

J. C. Wyant, Appl. Opt. 13, 200 (1974).
[CrossRef] [PubMed]

J. Hardy, J. Feinleib, J. C. Wyant, in Digest of OSA Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Wyatt, S. P.

S. P. Wyatt, Principles of Astronomy (Allyn and Bacon, Boston, 1964), p. 205.

Appl Opt.

W. B. Bridges, P. T. Brunner, S. P. Lassera, T. A. Nussmeier, T. R. O'Meara, J. A. Sanguinet, W. P. Brown, Appl Opt. 13, 291 (1974).
[CrossRef] [PubMed]

Appl. Opt

W. N. Peters, R. A. Arnold, S. Gowrinathan, Appl. Opt 13, 1785 (1974).
[CrossRef] [PubMed]

Appl. Opt.

Astron. J.

J. B. Saunders, R. J. Bruening, Astron. J. 73, 415 (1968).
[CrossRef]

J. Opt. Soc. Am.

R. V. Shack, J. Opt. Soc. Am. 64, 1362A (1974).

R. A. Muller, A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
[CrossRef]

Publ. Astron. Soc. Pac.

H. W. Babcock, Publ. Astron. Soc. Pac. 65, 229 (1953).
[CrossRef]

Other

J. Hardy, J. Feinleib, J. C. Wyant, in Digest of OSA Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

L. J. Golden, R. V. Shack, P. N. Slater, NASA Final Report for contract NAS8-27863 (1974).

E. D. Sunde, Communication Systems Engineering Theory (Wiley, New York, 1969), p. 120

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 463.

The author acknowledges a beneficial conversation with V. P. Bennett on the derivation.

A. G. Worthing, J. Geffner, Treatment of Experimental Data (Wiley, New York, 1943), p. 208.

S. P. Wyatt, Principles of Astronomy (Allyn and Bacon, Boston, 1964), p. 205.

RCA Electro-Optics Handbook Technical Series EOH-11 (Harrison, N.J., 1974), p. 68.

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Figures (1)

Fig. 1
Fig. 1

Sample wavefront grid. × denotes points at which the wavefront difference function is measured, and ○ denotes points at which the wavefront is calculated.

Equations (39)

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W ( x , y ) = W ( x d , y ) + a 1 ,
W ( x , y ) = W ( x + d , y ) a 2 ,
W ( x , y ) = W ( x , y + d ) + b 1 ,
W ( x , y ) = W ( x , y d ) b 2 .
a 1 = V i ( d / S x ) ,
a 2 = V j ( d / S x ) ,
b 1 = V k ( d / S y ) ,
b 2 = V l ( d / S y ) .
W ( x , y ) = ¼ [ W ( x d , y ) + W ( x + d , y ) + W ( x , y d ) + W ( x , y + d ) + d s x { V i V j } + d s y { V k V l } ] .
Δ θ = ( M Δ ϕ ) / 3 ,
M = d 2 ( 1 S x 2 + 1 S y 2 ) 1 / 2 .
M = d / S .
i = i s [ 1 + γ sin ( ω t + ϕ ) ] ,
Δ θ = 1 / ( S / N ) ,
P = T / 8 T / 8 i d t e + T / 8 3 T / 8 i d t e + 3 T / 8 5 T / 8 i d t e + 5 T / 8 7 T / 8 i d t e
= A + B + C + D ,
A = P ( 1 4 + 2 γ 2 π sin ϕ ) ,
B = P ( 1 4 + 2 γ 2 π cos ϕ ) ,
C = P ( 1 4 2 γ 2 π sin ϕ ) ,
D = P ( 1 4 2 γ 2 π cos ϕ ) .
tan ϕ = A C B D .
ϕ = tan 1 ( A C B D ) .
( Δ ϕ ) 2 = [ B D ( B D ) 2 + ( A C ) 2 ] 2 [ ( Δ A ) 2 + ( Δ C ) 2 ] + [ A C ( B D ) 2 + ( A C ) 2 ] 2 [ ( Δ B ) 2 + ( Δ D ) 2 ] .
( Δ A ) 2 + ( Δ C ) 2 = A + C = P / 2
( Δ B ) 2 + ( Δ D ) 2 + B + D = P / 2 ,
Δ ϕ = π / ( 2 γ P ) .
Δ θ = M π 2 3 γ P = ( 0.9 ) M γ P
P = N Ω T A 0 τ ρ / e ,
Δ θ = 0.9 M γ ( e N Ω T A 0 τ ρ ) 1 / 2
γ = 2 J 1 ( π S α / λ ) π S α / λ ,
Δ θ = 0.9 d λ J 1 ( π S α / λ ) ( e π N T A 0 ρ τ ) 1 / 2 ,
S = 1.8 λ / π α .
Δ θ = ( 1.1 × 10 9 ) d λ ( N T A 0 ρ τ ) 1 / 2 .
γ = sin ( π S x α x / λ ) π S α x / λ ,
Δ θ = ( α x α y + α y α x ) 1 / 2 0.9 d π λ sin ( π S x α x / λ ) ( e 2 N T A 0 ρ τ ) 1 / 2 .
S x = ( n + ½ ) λ / α x , n = 0 , 1 , 2 , . . . ,
Δ θ = ( 8 × 10 10 ) ( α x α y + α y α x ) 1 / 2 d λ ( N T A 0 ρ τ ) 1 / 2 .
Δ θ = ( 1.1 × 10 9 ) d λ ( N T A 0 ρ τ ) 1 / 2 .
Δ θ = 0.9 d S γ P .

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