Abstract

We present a new technique for the correction of atmospheric distortion in telescope images. Most of this distortion arises from a random phase variation of the incoming light across the telescope aperture. This variation limits the resolving power of even large telescopes to about one arc second. If the sharpness of the images is defined in a suitable way, this sharpness is maximized only when the phase distortion of the incoming light is zero. We present computer simulations of a simple feedback system in which active optical elements, set to maximize the sharpness, correct most of the atmospheric distortion. Photon statistics set the limiting magnitude of the object for which a practical feedback system can work. Details in a sixth magnitude object smaller than 0.1 sec of arc should be resolvable. The system can be conveniently employed within existing telescopes.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Isaac Newton, Opticks, Book I, Part I, Prop VIII, Prob. II (1730) (Dover, New York, 1952).
  2. D. Kelsall, J. Opt. Soc. Am. 63, 1472 (1973).
    [CrossRef]
  3. R. E. Danielson, High Resolution Imagery with the Large Space Telescope, Astronomy from a Space Platform, Vol. 28 in the Science and Technology series (American Astronautical Society, Tarzana, Calif., 1972), p. 197.
  4. A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
    [CrossRef]
  5. R. Brown Hanbury and R. Q. Twiss, Nature 178, 1046 (1956); Proc. R. Soc. (Lond.) A248, 199 (1958); Proc. R. Soc. (Lond.) A248, 222 (1958).
    [CrossRef]
  6. A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Gezari, A. Labeyrie, and R. Stacknik, Astrophys. J. 173, L1 (1972).
    [CrossRef]
  7. W. T. Rhodes and J. W. Goodman, J. Opt. Soc. Am. 63, 647 (1973).
    [CrossRef]
  8. Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (National Academy of Science, Washington D. C., 1966).
  9. A. D. Code, Ann. Rev. Astron. Astrophys. 11, 239 (1973).
    [CrossRef]
  10. W. A. Baum, D. M. Busby, and T. V. Pettauer, Photo-Electronic Image Devices, Vol. 33B, Advances in Electronics and Electron Physics (Academic, New York, 1972), p. 781; R. B. Leighton, Sci. Am. 194, (6), 156 (1956).
  11. H. W. Babcock, Publ. Astron. Sci. Pac. 65, 229 (1953); H. W. Babcock, J. Opt. Soc. Am. 48, 500 (1958).
    [CrossRef]
  12. I. G. Kolchinskii, in Optical Instability of the Earth’s Atmosphere, edited by N. I. Kucherov (Moscow, 1965), p. 7. Translated by Z. Lerman (1966), Israel Program for Scientific Translation (IPST), Cat. No. 1822.
  13. P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).
  14. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964); R. E. Hufnagel, Appendix 3, Vol. 2 of Ref. 8.
    [CrossRef]
  15. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966). Also D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  16. J. L. Bufton, P. O. Minott, and M. W. Fitzmaurice, J. Opt. Soc. Am. 62, 1068 (1972); also A. Labeyrie, private communication.
    [CrossRef]
  17. W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, and W. P. Brown, Appl. Opt. 13, 291 (1974).
    [CrossRef] [PubMed]
  18. H. F. Wischina, in Ref. 3, p. 347.
  19. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
    [CrossRef]
  20. See, for example, Optical Spectra, p. 46 (October1973).
  21. Can be calculated from the quantities given on p. 191, C. W. Allen, Astrophysical Quantities, 2nd ed. (Oxford U. P., New York, 1964).
  22. F. J. Dyson, private communication.
  23. Multiple-image systems have been suggested to ns by Jack Franck and Freeman Dyson (private communication).
  24. H. A. Hill and C. A. Zanoni, J. Opt. Soc. Am. 56, 1655 (1966).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 380.

1974 (1)

1973 (4)

See, for example, Optical Spectra, p. 46 (October1973).

D. Kelsall, J. Opt. Soc. Am. 63, 1472 (1973).
[CrossRef]

W. T. Rhodes and J. W. Goodman, J. Opt. Soc. Am. 63, 647 (1973).
[CrossRef]

A. D. Code, Ann. Rev. Astron. Astrophys. 11, 239 (1973).
[CrossRef]

1972 (1)

1970 (2)

P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Gezari, A. Labeyrie, and R. Stacknik, Astrophys. J. 173, L1 (1972).
[CrossRef]

1966 (2)

1964 (1)

1956 (2)

R. Brown Hanbury and R. Q. Twiss, Nature 178, 1046 (1956); Proc. R. Soc. (Lond.) A248, 199 (1958); Proc. R. Soc. (Lond.) A248, 222 (1958).
[CrossRef]

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
[CrossRef]

1953 (1)

H. W. Babcock, Publ. Astron. Sci. Pac. 65, 229 (1953); H. W. Babcock, J. Opt. Soc. Am. 48, 500 (1958).
[CrossRef]

1921 (1)

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[CrossRef]

Allen, C. W.

Can be calculated from the quantities given on p. 191, C. W. Allen, Astrophysical Quantities, 2nd ed. (Oxford U. P., New York, 1964).

Babcock, H. W.

H. W. Babcock, Publ. Astron. Sci. Pac. 65, 229 (1953); H. W. Babcock, J. Opt. Soc. Am. 48, 500 (1958).
[CrossRef]

Baum, W. A.

W. A. Baum, D. M. Busby, and T. V. Pettauer, Photo-Electronic Image Devices, Vol. 33B, Advances in Electronics and Electron Physics (Academic, New York, 1972), p. 781; R. B. Leighton, Sci. Am. 194, (6), 156 (1956).

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 380.

Bridges, W. B.

Brown, W. P.

Brown Hanbury, R.

R. Brown Hanbury and R. Q. Twiss, Nature 178, 1046 (1956); Proc. R. Soc. (Lond.) A248, 199 (1958); Proc. R. Soc. (Lond.) A248, 222 (1958).
[CrossRef]

Brunner, P. T.

Bufton, J. L.

Busby, D. M.

W. A. Baum, D. M. Busby, and T. V. Pettauer, Photo-Electronic Image Devices, Vol. 33B, Advances in Electronics and Electron Physics (Academic, New York, 1972), p. 781; R. B. Leighton, Sci. Am. 194, (6), 156 (1956).

Button, P. A.

P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).

Code, A. D.

A. D. Code, Ann. Rev. Astron. Astrophys. 11, 239 (1973).
[CrossRef]

Danielson, R. E.

R. E. Danielson, High Resolution Imagery with the Large Space Telescope, Astronomy from a Space Platform, Vol. 28 in the Science and Technology series (American Astronautical Society, Tarzana, Calif., 1972), p. 197.

Duke, D.

P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).

Dyson, F. J.

F. J. Dyson, private communication.

Dyson, Freeman

Multiple-image systems have been suggested to ns by Jack Franck and Freeman Dyson (private communication).

Fitzmaurice, M. W.

Franck, Jack

Multiple-image systems have been suggested to ns by Jack Franck and Freeman Dyson (private communication).

Fried, D. L.

Goodman, J. W.

Hill, H. A.

Hufnagel, R. E.

Kelsall, D.

Kolchinskii, I. G.

I. G. Kolchinskii, in Optical Instability of the Earth’s Atmosphere, edited by N. I. Kucherov (Moscow, 1965), p. 7. Translated by Z. Lerman (1966), Israel Program for Scientific Translation (IPST), Cat. No. 1822.

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Gezari, A. Labeyrie, and R. Stacknik, Astrophys. J. 173, L1 (1972).
[CrossRef]

Lazzara, S. P.

Linfoot, E. H.

Michelson, A. A.

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[CrossRef]

Minott, P. O.

Newton, Isaac

Isaac Newton, Opticks, Book I, Part I, Prop VIII, Prob. II (1730) (Dover, New York, 1952).

Nussmeier, T. A.

O’Meara, T. R.

Pease, F. G.

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[CrossRef]

Pettauer, T. V.

W. A. Baum, D. M. Busby, and T. V. Pettauer, Photo-Electronic Image Devices, Vol. 33B, Advances in Electronics and Electron Physics (Academic, New York, 1972), p. 781; R. B. Leighton, Sci. Am. 194, (6), 156 (1956).

Reusch, M. F.

P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).

Rhodes, W. T.

Sage, B.

P. A. Button, M. F. Reusch, B. Sage, and D. Duke, J. Opt. Soc. Am. 60, 1550A (1970).

Sanguinet, J. A.

Stanley, N. R.

Twiss, R. Q.

R. Brown Hanbury and R. Q. Twiss, Nature 178, 1046 (1956); Proc. R. Soc. (Lond.) A248, 199 (1958); Proc. R. Soc. (Lond.) A248, 222 (1958).
[CrossRef]

Wischina, H. F.

H. F. Wischina, in Ref. 3, p. 347.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 380.

Zanoni, C. A.

Ann. Rev. Astron. Astrophys. (1)

A. D. Code, Ann. Rev. Astron. Astrophys. 11, 239 (1973).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Gezari, A. Labeyrie, and R. Stacknik, Astrophys. J. 173, L1 (1972).
[CrossRef]

Astrophys. J. (1)

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[CrossRef]

J. Opt. Soc. Am. (8)

Nature (1)

R. Brown Hanbury and R. Q. Twiss, Nature 178, 1046 (1956); Proc. R. Soc. (Lond.) A248, 199 (1958); Proc. R. Soc. (Lond.) A248, 222 (1958).
[CrossRef]

Optical Spectra (1)

See, for example, Optical Spectra, p. 46 (October1973).

Publ. Astron. Sci. Pac. (1)

H. W. Babcock, Publ. Astron. Sci. Pac. 65, 229 (1953); H. W. Babcock, J. Opt. Soc. Am. 48, 500 (1958).
[CrossRef]

Other (10)

I. G. Kolchinskii, in Optical Instability of the Earth’s Atmosphere, edited by N. I. Kucherov (Moscow, 1965), p. 7. Translated by Z. Lerman (1966), Israel Program for Scientific Translation (IPST), Cat. No. 1822.

H. F. Wischina, in Ref. 3, p. 347.

R. E. Danielson, High Resolution Imagery with the Large Space Telescope, Astronomy from a Space Platform, Vol. 28 in the Science and Technology series (American Astronautical Society, Tarzana, Calif., 1972), p. 197.

Restoration of Atmospherically Degraded Images, Woods Hole Summer Study (National Academy of Science, Washington D. C., 1966).

W. A. Baum, D. M. Busby, and T. V. Pettauer, Photo-Electronic Image Devices, Vol. 33B, Advances in Electronics and Electron Physics (Academic, New York, 1972), p. 781; R. B. Leighton, Sci. Am. 194, (6), 156 (1956).

Can be calculated from the quantities given on p. 191, C. W. Allen, Astrophysical Quantities, 2nd ed. (Oxford U. P., New York, 1964).

F. J. Dyson, private communication.

Multiple-image systems have been suggested to ns by Jack Franck and Freeman Dyson (private communication).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 380.

Isaac Newton, Opticks, Book I, Part I, Prop VIII, Prob. II (1730) (Dover, New York, 1952).

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 (11)

Fig. 1
Fig. 1

Schematic diagram of image-restoring system. Arrow heads indicate the relative phase of the wave. The adjustable phase shifter (perhaps an extra flat mirror with movable segment incorporated into the optical path) corrects the phase of the incoming wave.

Fig. 2
Fig. 2

Computer-simulated diffraction patterns, (a) No disturbing atmosphere, (b) random amplitude variation, (c) random phase variation. The telescope aperture for this figure was a 2.5 m strip. All of the displays in this paper are normalized so that 100% irradiance is the peak for an undisturbed diffraction pattern. The smooth curve in (a) is a [sin (x)/x]2 diffraction-pattern shape, whereas the histograms indicate the results of our discrete computer calculations.

Fig. 3
Fig. 3

Image of a point star as each phase-shifter segment is adjusted in sequence. The initial image (shaded) is the same as the speckle pattern shown in Fig. 2(c). The image improves as each of the 25 segments is adjusted in turn to compensate for the atmospheric distortion. The final diffraction-limited image (image No. 25 in the figure) is also shaded. This final image was formed at the position of what was originally the brightest speckle. The sharpness function used to correct the image was S1 = ∫I2.

Fig. 4
Fig. 4

Two-dimensional image from a computer simulation of a ring-shaped telescope aperture. The ring had an outer diameter of 75 cm and an inner diameter of 55 cm. No disturbing atmosphere is present. The contour plot (but not the sectional histogram) has been smoothed to average out unphysical features due to the discrete nature of the computer calculations. Peak= 100.

Fig. 5
Fig. 5

Speckle pattern. Same geometry as Fig. 4, but random atmospheric phase distortion present at the aperture plane. Peak ≈ 25.

Fig. 6
Fig. 6

Restored image of Fig. 5 after a single iteration cycle of the 25 phase shifter segments, with the use of S3 = ∫MI, with M = a round hole the size of the central diffraction maximum.

Fig. 7
Fig. 7

Images of a triple star. The 2.5 m strip telescope is used to view three stars of equal intensity and separated by 0.04 and 0.20 arc sec. The sharpness definition S1 = ∫I2 was used, (a) Original speckle pattern, (b) image after 1 iteration cycle of the 25 segments.

Fig. 8
Fig. 8

Phase angle vs position for the ring-telescope geometry Position is indicated by numbering the 25 correction segments around the ring. The original disturbed phase of the incident wave (smooth curve) and segment setting resulting for a Single iteration (histogram) are shown. The image resulting from this wavefront was shown in Figs. 5 and 6. Segments 1 and 2 were misplaced by 2π and have been redrawn (dotted) to show another equivalent position.

Fig. 9
Fig. 9

Image of an unresolved star corrected by use of the moment-of-interia sharpness function S6 = ∫r2I. Although some improvement results, it is not comparable to that of Fig. 3. (a) Original speckle pattern, (b) image after 1 iteration of the 25 segments, (c) after 2 iterations.

Fig. 10
Fig. 10

Image of a triple star in the presence of photoelectron fluctuations. The distorted wavefront was the same as that used in Fig. 7, but a random-number generator varied the number of photoelectrons in each image bin according to Poisson statistics. Roughly 1000 photoelectrons were available for each measurement of the sharpness function S1. For the strip mirror simulated here (25 elements of 100 cm2 each, photon detection efficiency η = 0.25, and time available for the correction τ = 0.02 s), each star would be ninth magnitude. Statistical fluctuations evident in the figure result from the very short effective exposure time used (0.4 × 10−3 s); a better image would result from the superposition of many such fast images at an observatory.

Fig. 11
Fig. 11

Fractional image quality after one iteration vs photoelectrons per segment setting. Image quality Q = S/S0, where S is the image sharpness after one iteration, and S0 is the image sharpness for the simulation when no statistical fluctuations were applied. The telescope geometry for this figure was the 0.75 m diam annular mirror with area A = 2500 cm2. The hatched region indicates the typical range of variations of Q due to photoelectron fluctuations. The sharpness function used was S3 = ∫MI, where M = a round hole the same size as the central diffraction pattern of a point star. The number of photoelectrons was defined to be 0.25 times the number of photons incident on the entire telescope in one setting time t. Very similar results were found for S1 and for the strip-telescope geometry. ● Typical Monte Carlo points.

Tables (1)

Tables Icon

Table I Satisfactory definitions of image-plane sharpness.a

Equations (25)

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

S 1 = d x d y I 2 ( x , y ) ,
W ( u , v ) = W ( u , v ) · Δ ( u , v )
S 1 = d x d y I 2 ( x , y ) ,
S 3 = d x d y M ( x , y ) · I ( x , y ) ,
t = τ / n N = τ a / n A .
p = η B a t = η B a τ / n ,
1 / N d p / p = ( η B a τ / n ) - 1 2 .
B n A 2 / a 3 η τ .
p N 2 .
B = ( 4 × 10 6 ) × 10 - m / 2.5 ,
tan ϕ = [ S ( - ) - S ( + ) ] / [ S ( + ) + S ( - ) - 2 S ( 0 ) ] .
B n A 2 a 3 τ η ,
θ = 10 - 3 λ 10 m / 5 a ( η τ / n ) 1 2 ,
I ( x , y ) = d θ d ϕ O ( θ , ϕ ) × | d u d v e i k [ δ ( u , v ) + ( u x + v y ) / f + ( u θ + v ϕ ) ] | 2 ,
S 4 = d x d y | l + m I ( x , y ) x l y m | 2 .
I ( x , y ) = d θ d ϕ O ( θ , ϕ ) d u d u d v d v · e i k [ δ ( u , v ) - δ ( u , v ) + ( u - u ) x / f + ( v - v ) y / f ] · e i k [ ( u - u ) θ + ( v - v ) ϕ ] .
S 4 = d x d y d θ 1 d θ 2 d ϕ 1 d ϕ 2 O ( θ 1 , ϕ 1 ) · O ( θ 2 , ϕ 2 ) d u 1 d u 1 d u 2 d u 2 d v 1 d v 1 d v 2 d v 2 · ( k / f ) 2 ( l + m ) ( u 1 - u 1 ) l ( v 1 - v 1 ) m ( u 2 - u 2 ) l ( v 2 - v 2 ) m · e i k [ ( u 1 - u 1 ) ( x / f + θ 1 ) + ( v 1 - v 1 ) ( y / f + ϕ 1 ) ] · e i k [ ( u 2 - u 2 ) ( x / f + θ 2 ) + ( v 2 - v 2 ) ( y / f + ϕ 2 ) ] · e i k { δ ( u 1 , v 1 ) - δ ( u 1 , v 1 ) + δ ( u 2 , v 2 ) - δ ( u 2 , v 2 ) } .
S 4 = ( 2 π ) 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 O ( θ 1 , ϕ 1 ) · O ( θ 2 , ϕ 2 ) d u 1 d z d u 2 d v 1 d w d v 2 · ( k / f ) 2 ( l + m - 1 ) z 2 l w 2 m e i k [ ( θ 1 - θ 2 ) z + ( ϕ 1 - ϕ 2 ) w ] · e i k { δ ( u 1 , v 1 ) - δ ( u 1 - z , v 1 - w ) + δ ( u 2 , v 2 ) - δ ( u 2 + z , v 2 + w ) } .
S 4 = ( 2 π ) 3 d u 1 d z d u 2 d v 1 d w d v 2 · ( k / f ) 2 ( l + m - 1 ) z 2 l w 2 m ( Õ k z , k w ) 2 e i k { } ,
δ ( u , v ) = a + b u + c v ,
S 5 = d x d y I n ( x , y ) ,             n 2 , integer .
S 5 = ( 2 π f k ) 2 O n ( 0 , 0 ) d n u d n v d n - 1 z d n - 1 w e i k { } ,
j = 1 n - 1 [ δ ( u j , v j ) - δ ( u j - z j , v j - w j ) ] + δ ( u n , v n ) - δ ( u n + j = 1 n - 1 z j , v n + j = 1 n - 1 w j )
S 8 = - d x d y I ( x , y ) - I 0 ( x , y ) 2 .
S 8 = - d x d y I 2 ( x , y ) + 2 d x d y I 0 ( x , y ) I ( x , y ) - d x d y I 0 2 ( x , y ) .