Abstract

The performance of ground-based optical imaging systems is severely degraded from the diffraction limit by the random effects of the atmosphere. Adaptive-optics techniques have been used to compensate for atmospheric-turbulence effects. A critical component in the adaptive-optics system is the wave-front sensor. At present, two types of sensors are common: the Hartmann–Shack wave-front sensor and the shearing interferometer wave-front sensor. In this paper we make a direct performance comparison of these two sensors. The performance calculations are restricted to common configurations of these two sensors and the fundamental limits imposed by shot noise and atmospheric effects. These two effects encompass the effects of extended reference beacons and sensor subaperture spacings larger than the Fried parameter r 0. Our results indicate comparable performance for good seeing conditions and small beacons. However, for poor seeing conditions and extended beacons, the Hartmann sensor has lower error levels than the shearing interferometer.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  2. R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winkler, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, “Two generations of laser-guide-star adaptive optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
    [CrossRef]
  3. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  4. K. A. Winnick, “Cramer–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  5. G. A. Tyler, D. L. Fried, “Image position error associated with a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
    [CrossRef]
  6. D. Sandler, L. Cuellar, M. Lefebvre, T. Barrett, R. Arnold, P. Johnson, A. Rego, G. Smith, G. Taylor, B. Spivey, “Shearing interferometry for laser-guide-star atmospheric correction at large D/ro,” J. Opt. Soc. Am. A 11, 858–873 (1994).
    [CrossRef]
  7. T. J. Kane, B. M. Welsh, C. S. Garnder, “Wavefront detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1114, 160–171 (1989).
  8. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [CrossRef]
  9. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [CrossRef]
  10. T. L. Pennington, B. M. Welsh, M. C. Roggemann, “Performance comparison of the shearing interferometer and the Hartmann wave front sensor,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 508–518 (1994).
  11. R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994).
    [CrossRef]
  12. H. Yura, M. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
    [CrossRef]
  13. E. P. Wallner, “Optimal wave front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  14. 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]
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1985) Chap. 8, p. 436.
  16. G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
    [CrossRef]
  17. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  18. J. Walkup, J. Goodman, “Limitations of fringe-parameter estimation at low light levels,” J. Opt. Soc. Am. 63, 399–407 (1973).
    [CrossRef]
  19. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]

1994 (3)

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

1989 (1)

1988 (1)

1986 (1)

1985 (1)

1983 (1)

1982 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1976 (1)

1975 (1)

1973 (1)

1967 (1)

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

1966 (1)

Arnold, R.

Barrett, T.

Boeke, B. R.

Cuellar, L.

Ellerbroek, B. L.

Fried, D. L.

Fugate, R. Q.

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

Garnder, C. S.

T. J. Kane, B. M. Welsh, C. S. Garnder, “Wavefront detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1114, 160–171 (1989).

Goodman, J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985) Chap. 8, p. 436.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Heidbreder, G. R.

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

Higgins, C. H.

Jelonek, M. P.

Johnson, P.

Kane, T. J.

T. J. Kane, B. M. Welsh, C. S. Garnder, “Wavefront detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1114, 160–171 (1989).

Lange, W. J.

Lefebvre, M.

Moroney, J. F.

Noll, R. J.

Oliker, M. D.

Parenti, R. R.

Pennington, T. L.

T. L. Pennington, B. M. Welsh, M. C. Roggemann, “Performance comparison of the shearing interferometer and the Hartmann wave front sensor,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 508–518 (1994).

Rego, A.

Roddier, F.

Roggemann, M. C.

T. L. Pennington, B. M. Welsh, M. C. Roggemann, “Performance comparison of the shearing interferometer and the Hartmann wave front sensor,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 508–518 (1994).

Ruane, R. E.

Sandler, D.

Sasiela, R. J.

Slavin, A. C.

Smith, G.

Spinhirne, J. M.

Spivey, B.

Swindle, D. W.

Tavis, M.

Taylor, G.

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Tyler, G. A.

Walkup, J.

Wallner, E. P.

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

T. J. Kane, B. M. Welsh, C. S. Garnder, “Wavefront detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1114, 160–171 (1989).

T. L. Pennington, B. M. Welsh, M. C. Roggemann, “Performance comparison of the shearing interferometer and the Hartmann wave front sensor,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 508–518 (1994).

Wild, W. J.

Winkler, D. M.

Winnick, K. A.

Wyant, J. C.

Wynia, J. M.

Yura, H.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

G. R. Heidbreder, “Image degradation with random wavefront tilt compensation,” IEEE Trans. Antennas Propag. AP-15, 90–98 (1967).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (6)

Proc. IEEE (2)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (3)

T. L. Pennington, B. M. Welsh, M. C. Roggemann, “Performance comparison of the shearing interferometer and the Hartmann wave front sensor,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2201, 508–518 (1994).

T. J. Kane, B. M. Welsh, C. S. Garnder, “Wavefront detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1114, 160–171 (1989).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985) Chap. 8, p. 436.

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 (7)

Fig. 1
Fig. 1

Diagram of the Hartmann WFS.

Fig. 2
Fig. 2

Diagram of the shearing interferometer.

Fig. 3
Fig. 3

Tilt-removed optical transfer function (OTF) H tr(f x , f y ) plotted versus the normalized spatial frequency λf l f x /d. These results were calculated for a square subaperture. Also shown is an OTF result predicted by the tilt-removed structure function derived by Fried.17

Fig. 4
Fig. 4

Plot of visibility μ a versus the normalized subaperture size d/r o . These results were calculated for a square subaperture.

Fig. 5
Fig. 5

Normalized WFS rms error versus d/r o for a point-source beacon (β = 0). The SI-WFS error curves are for the shears s ranging from 0.1 to 2.0.

Fig. 6
Fig. 6

Normalized WFS error versus d/r o for three beacon sizes (β = 0.5, 1, and 2). The optimum shear is used for each of the SI-WFS results.

Fig. 7
Fig. 7

Optimum shear s plotted versus beacon size β for d/r o = 1. Also shown is the previously calculated optimum shear result.10

Equations (38)

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

tan ( θ h ) = x c f l θ h = tan - 1 ( x c f l ) ,
I d = d 2 x W ( x ) | u ( x + Δ x 2 ) exp [ j ϕ ( x + Δ x 2 ) ] + u ( x - Δ x 2 ) exp [ j ϕ ( x - Δ x 2 ) ] | 2 ,
I d = d 2 x W ( x ) { | u ( x + Δ x 2 ) | 2 + | u ( x - Δ x 2 ) | 2 + 2 u ( x + Δ x 2 ) u * ( x - Δ x 2 ) × cos [ ϕ ( x + Δ x 2 ) - ϕ ( x - Δ x 2 ) ] } ,
ϕ ( x - Δ x 2 ) - ϕ ( x + Δ x 2 ) = Δ ϕ ( x , Δ x ) ,
u ( x ) u * ( x ) = μ b ( x - x ) I b ,
| u ( x + Δ x 2 ) | 2 = | u ( x - Δ x 2 ) | 2 = I b .
I b = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x ) cos [ Δ ϕ ( x , Δ x ) ] } .
I d ( t ) = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x ) cos [ ω t + Δ ϕ ( x , Δ x ) ] } ,
I d ( i ) = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x ) cos [ Φ i + Δ ϕ ( x , Δ x ) ] } ,
σ c = 2 f l λ / d 2 N d f x I ˜ b ( f x , 0 ) H tr ( f x , 0 ) ,
I ˜ b ( f ) = F J [ I b ( α λ f l d ) ] F J [ I b ( α λ f l d ) ] | f = 0 ,
E H = ( k d f l ) σ c = 2 π N d f x I ˜ b ( f x , 0 ) H tr ( f x , 0 ) × ( radians subaperture ) ,
W ( x , y ) = 1 d 2 rect ( x d ) rect ( y d ) = { 1 d 2 - d 2 x d 2 and - d 2 y d 2 0 otherwise .
I b ( x ) exp [ - x 2 ( f l σ b z b ) 2 ] ,
I ˜ b ( f ) = exp [ - π 2 f 2 ( σ b d λ z b ) 2 ] .
I ˜ b ( f ) = exp [ - π 2 f 2 β 2 ( d r o ) 2 ] ,
σ Δ ϕ = 2 2 N μ b ( s ) ,
μ b ( s ) = exp ( - π 2 β 2 s 2 ) ,
E SI = d r o S σ Δ ϕ = 2 2 N s ( d r o ) exp ( π 2 β 2 s 2 ) ( radians subaperture ) .
Δ ϕ ¯ ( Δ x ) = d 2 x W ( x ) Δ ϕ ( x , Δ x ) .
I d = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x ) × cos [ Δ ϕ ( x , Δ x ) - Δ ϕ ¯ ( Δ x ) + Δ ϕ ¯ ( Δ x ) ] } .
I d = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x ) × cos [ Δ ϕ ˜ ( x , Δ x ) + Δ ϕ ¯ ( Δ x ) ] } .
I d = 2 I b d 2 x W ( x ) [ 1 + μ b ( Δ x ) exp { j [ Δ ϕ ˜ ( x , Δ x ) + Δ ϕ ¯ ( Δ x ) ] } + exp { - j [ Δ ϕ ˜ ( x , Δ x ) + Δ ϕ ¯ ( Δ x ) ] } 2 ] = 2 I b d 2 x W ( x ) [ 1 + μ b ( Δ x ) ( exp [ - ½ Δ ϕ ˜ 2 ( x , Δ x ) ] exp [ j Δ ϕ ¯ ( Δ x ) ] + exp [ - ½ Δ ϕ ˜ 2 ( x , Δ x ) ] exp [ - j Δ ϕ ¯ ( Δ x ) ] 2 ) ] = 2 I b d 2 x W ( x ) { 1 + μ b ( Δ x exp [ - ½ Δ ϕ ˜ 2 ( x , Δ x ) ] cos [ Δ ϕ ¯ ( Δ x ) ] } ,
μ a ( Δ x ) = d 2 x W ( x ) exp [ - ½ Δ ϕ ˜ 2 ( x , Δ x ) ] .
I d = 2 I b { 1 + μ b ( Δ x ) μ a ( Δ x ) cos [ Δ ϕ ¯ ( Δ x ) ] } .
E SI = 2 2 N s μ a ( s ) μ b ( s ) ( d r o ) = 2 2 N s μ a ( s ) ( d r o ) exp ( π 2 β 2 s 2 ) ( radians subaperture ) .
E H = 2 π N d f x I ˜ b ( f x , 0 ) H tr ( f x , 0 ) ( radians subaperture ) ,
E SI = 2 2 N s μ a ( s ) ( d r o ) exp ( π 2 β 2 s 2 ) ( radians subaperture ) .
E H = 2 π N ( radians subaperture ) ,
E SI = 2 2 N ( d Δ x ) ( radians subaperture ) .
μ a ( Δ x ) = d 2 x W ( x ) exp [ - ½ Δ ϕ ˜ 2 ( x , Δ x ) ] .
Δ ϕ ˜ 2 ( x , Δ x ) = [ Δ ϕ ( x , Δ x ) - Δ ϕ ¯ ( Δ x ) ] 2 = [ Δ ϕ ( x , Δ x ) - d 2 x W ( x ) Δ ϕ ( x , Δ x ) ] 2 .
Δ ϕ ˜ 2 ( x , Δ x ) = [ ϕ ( x ) - ϕ ( x + Δ x ) ] 2 - 2 d 2 x W ( x ) × [ ϕ ( x ) - ϕ ( x + Δ x ) ] [ ϕ ( x ) - ϕ ( x + Δ x ) ] + d 2 x d 2 x W ( x ) W ( x ) [ ϕ ( x ) - ϕ ( x + Δ x ) ] × [ ϕ ( x ) - ϕ ( x + Δ x ) ] .
ϕ ( x ) ϕ ( x ) = ϕ 2 ( x ) 2 + ϕ 2 ( x ) 2 - 1 2 D ϕ ( x , x ) ,
D ϕ ( x , x ) = [ ϕ ( x ) - ϕ ( x ) ] 2 .
Δ ϕ ˜ 2 ( x , Δ x ) = D ϕ ( Δ x ) - d 2 x W ( x ) [ D ϕ ( x - x + Δ x ) + D ϕ ( x - x - Δ x ) - 2 D ϕ ( x - x ) ] + 1 2 d 2 x d 2 x W ( x ) W ( x ) D ϕ ( x - x + Δ x ) + D ϕ ( x - x - Δ x ) - 2 D ϕ ( x - x ) ] .
Δ ϕ ˜ 2 ( x , Δ x ) = D ϕ ( Δ x ) - d 2 x W ( x ) [ D ϕ ( x - x + Δ x ) + D ϕ ( x - x - Δ x ) - 2 D ϕ ( x - x ) ] + 1 2 d 2 x [ D ϕ ( x + Δ x ) + D ϕ ( x - Δ x ) - 2 D ϕ ( x ) ] [ W ( x ) W ( x ) ] ,
D ϕ ( x - x ) = 6.88 ( x - x r o ) 5 / 3 ,

Metrics