Abstract

A critical component in a compensated imaging (CI) system is the wave-front sensor which measures the residual distortion of the wave front after reflecting off the active mirror. The sensor produces estimates of wave-front slopes or phase difference across the aperture. For many applications, the phase differences or slopes are not the most convenient form of data for processing or control, and they must be converted to absolute wave-front phases. This paper analyzes the conversion from phase differences to phases and derives the optimal linear estimator in terms of least noise propagation. Some remarks concerning hardware implementation are also made.

© 1977 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
    [Crossref]
  2. J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wave-front sensor,” (unpublished).
  3. H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
    [Crossref]
  4. “Restoration of Atmospherically Degraded Images,” Woods Hole Summer Study, July1966. .
  5. J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-time correction of optical imaging systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.
  6. J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
    [Crossref] [PubMed]
  7. Analog Data Processor, US Patent, 3, 921, 080November18, 1975, assigned to Itek Corporation, Lexington, Mass.
  8. J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
    [Crossref]
  9. J. W. Hardy, C. L. Koliopoulous, and J. K. Bowker, “Radial Grating A. C. Interferometer” (unpublished).
  10. Freeman J. Dyson, “Photon noise and atmospheric noise in active optical systems,” J. Opt. Soc. Am. 65, 551–558 (1975).
    [Crossref]
  11. When the wave front must be regarded as changing over the time interval of the sensor measurement, then ϕjk will be the average of the mean phase over the interval and Sjkl the average of the phase difference.
  12. We note that even for a Hartmann sensor (where the light from each subaperture is imaged onto a quad cell, and x-y tilts are measured using same photons), the probability density for photon arrival for a square subaperture is of the form P(x)-f(x)f(y) for an unresolved target. This means that for a single photon, the x coordinate of arrival is uncorrelated with the y coordinate of arrival, so the photon noises on the two components of tilt are uncorrelated even though using the same photons. Thus the analysis applies to this case. For a round subaperture the independence is based on a ± symmetric error distribution, but again the conclusion of uncorrelated tilt errors holds.

1977 (1)

1975 (1)

1974 (2)

J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
[Crossref] [PubMed]

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

1953 (1)

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[Crossref]

Babcock, H. W.

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[Crossref]

Bowker, J. K.

J. W. Hardy, C. L. Koliopoulous, and J. K. Bowker, “Radial Grating A. C. Interferometer” (unpublished).

Bowker, K.

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wave-front sensor,” (unpublished).

Cone, P. F.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Dyson, Freeman J.

Feinleib, J.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-time correction of optical imaging systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

Hardy, J.

J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
[Crossref]

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wave-front sensor,” (unpublished).

Hardy, J. W.

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-time correction of optical imaging systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

J. W. Hardy, C. L. Koliopoulous, and J. K. Bowker, “Radial Grating A. C. Interferometer” (unpublished).

Koliopoulos, C.

J. Hardy, J. Lefebvre, and C. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977) (this issue).
[Crossref]

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wave-front sensor,” (unpublished).

Koliopoulous, C. L.

J. W. Hardy, C. L. Koliopoulous, and J. K. Bowker, “Radial Grating A. C. Interferometer” (unpublished).

Lefebvre, J.

Lipson, S. G.

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Wyant, J. C.

J. C. Wyant, “White Light Extended Source Shearing Interferometer,” Appl. Opt. 13, 200 (1974).
[Crossref] [PubMed]

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-time correction of optical imaging systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Feinleib, S. G. Lipson, and P. F. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

J. Opt. Soc. Am. (2)

Pub Astr. Soc. Pac. (1)

H. W. Babcock, Pub Astr. Soc. Pac. 65, 229 (1953).
[Crossref]

Other (7)

“Restoration of Atmospherically Degraded Images,” Woods Hole Summer Study, July1966. .

J. W. Hardy, J. Feinleib, and J. C. Wyant, “Real-time correction of optical imaging systems,” presented at the OSA Meeting on Optical Propagation through Turbulence, Boulder, Colo.July 1974.

J. Hardy, C. Koliopoulos, and K. Bowker, “Real-time AC grating interferometer wave-front sensor,” (unpublished).

J. W. Hardy, C. L. Koliopoulous, and J. K. Bowker, “Radial Grating A. C. Interferometer” (unpublished).

Analog Data Processor, US Patent, 3, 921, 080November18, 1975, assigned to Itek Corporation, Lexington, Mass.

When the wave front must be regarded as changing over the time interval of the sensor measurement, then ϕjk will be the average of the mean phase over the interval and Sjkl the average of the phase difference.

We note that even for a Hartmann sensor (where the light from each subaperture is imaged onto a quad cell, and x-y tilts are measured using same photons), the probability density for photon arrival for a square subaperture is of the form P(x)-f(x)f(y) for an unresolved target. This means that for a single photon, the x coordinate of arrival is uncorrelated with the y coordinate of arrival, so the photon noises on the two components of tilt are uncorrelated even though using the same photons. Thus the analysis applies to this case. For a round subaperture the independence is based on a ± symmetric error distribution, but again the conclusion of uncorrelated tilt errors holds.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

FIG. 1
FIG. 1

Noise coefficient for various square arrays of size N × N.

FIG. 2
FIG. 2

Noise correlation function on the reconstructed phases.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

S j k 1 = ϕ j k - ϕ j + 1 , k ,
S j k 2 = ϕ j k - ϕ j , k + 1 .
n j k l n j k l = n 0 δ j j δ k k δ l l .
S ˜ j k l = S j k l + n j k l .
ϕ ˜ j k = l m n b j k m n * l S ˜ m n l ,
j k = ϕ ˜ j k - ϕ j k = ( l m n b j k m n * l S m n l - ϕ j k ) + l m n b j k m n * l n m n l .
ϕ j k = l m n b j k m n * l S m n l .
j k j k 2 = n 0 j k l m n ( b j k m n * l ) 2 ,
2 = j k j k 2 = n 0 l m n j k ( b j k m n * l ) 2
ϕ j k = l m n b j k m n * l S m n l
ϕ j k = l m n b j - m , k - n l S m n l ,
ϕ ˆ j k = l b ˆ j k l S ˆ j k l ,
ϕ ˆ j k = l m ϕ l m e ( 2 π i / N ) ( l j + m k ) ,
b ˆ j k n = l m b l m n e ( 2 π i / N ) ( l j + m k ) ,
S ˆ j k n = l m S l m n e ( 2 π i / N ) ( l j + m k ) .
δ 2 = j k n ( b ˆ j k n ) 2 .
S j k 1 = ϕ j k - ϕ j + 1 , k ,
S j k 2 = ϕ j k - ϕ j , k + 1 .
S ˆ j k 1 = ( 1 - e 2 π i j / N ) ϕ ˆ j k ,
S ˆ j k 2 = ( 1 - e 2 π i k / N ) ϕ ˆ j k .
ϕ ˆ j k = [ b ˆ j k 1 ( 1 - e 2 π i j / N ) + b ˆ j k 2 ( 1 - e 2 π i k / N ) ] ϕ ˆ j k .
b ˆ j k 1 ( 1 - e 2 π i j / N ) + b ˆ j k 2 ( 1 - e 2 π i k / N ) = 1.
δ j k 2 = ( b ˆ j k 1 ) 2 + ( b ˆ j k 2 ) 2 .
b ˆ j k 1 = ( 1 - e - 2 π i j / N ) 4 - e 2 π i j / N - e - 2 π i j / N - e 2 π i k / N - e - 2 π i k / N ,
b ˆ j k 2 = ( 1 - e - 2 π i k / N ) 4 - e 2 π i j / N - e - 2 π i j / N - e 2 π i k / N - e - 2 π i k / N .
ϕ ˆ j k = S ˆ j k 1 ( 1 - e - 2 π i j / N ) + S ˆ j k 2 ( 1 - e - 2 π i k / N ) 4 - e 2 π i j / N - e - 2 π i j / N - e 2 π i k / N - e - 2 π i k / N .
ϕ ˆ j k [ 4 - e 2 π i j / N - e - 2 π i j / N - e 2 π i k / N - e - 2 π i k / N ] = S ˆ j k 1 ( 1 - e - 2 π i j / N ) + S ˆ j k 2 ( 1 - e - 2 π i k / N ) ,
4 ϕ j k - ( ϕ j + 1 , k + ϕ j - 1 , k + ϕ j , k + 1 + ϕ j , k - 1 ) = S j k 1 - S j - 1 , k 1 + S j k 2 - S j , k - 1 2 .
ϕ j k = 1 4 ( ϕ j + 1 , k + ϕ j - 1 , k + ϕ j , k + 1 + ϕ j , k - 1 ) + 1 4 ( S j k 1 - S j - 1 , k 1 + S j k 2 - S j , k - 1 2 ) .
ϕ j k = 1 4 ( ϕ j + 1 , k + ϕ j - 1 , k + ϕ j , k + 1 + ϕ j , k - 1 ) + 1 4 ( S j k 1 - S j - 1 , k 1 + S j k 2 - S j , k - 1 2 ) .
2 = n 0 j k l m n b j k m n 2 l ,
j k l m n b j k m n 2 l 0.561 + 0.103 ln N ,
j k = l m n b j k m n * l n m n l l m n b j - m , k - n l n m n l ,
C r s = j + r , k + s j k
= n 0 b r + m , s + n l b m n l .
C r s = C 00 e - 5 ( r 2 + s 2 ) 1 / 2 / 14 .