Abstract

The laser power requirement for an adaptive-astronomical telescope using laser guide stars is determined largely by the effects of turbulence-induced anisoplanatism. Owing to the relatively low altitude of laser guide stars and the small size of the isoplanatic angle at visible wavelengths, multiple guide stars are required for correcting large telescope apertures. The laser power requirements are proportional to the required number of guide stars. Using an analysis technique that takes into account the realistic characteristics of the wave-front sensor and deformable mirror, as well as the spherical nature of the wave front from the laser guide star, we present computational results that show how the imaging performance of a laser-guided adaptive telescope varies as a function of the number and height of the guide stars. The results are presented as a function of the isoplanatic angle θIP as defined by Fried [ J. Opt. Soc. Am. 72, 52 ( 1982)]. A new parameter, the characteristic diameter of the largest telescope requiring a single guide star, is also introduced. This parameter is designated DIP and is related to the height of the guided star zg and the isoplanatic angle θIP by DIP = 2zgθIP. The effects of anisoplanatism on the design of a 2-m-diameter adaptive telescope using laser guide stars created in the mesospheric Na layer is considered. Using a Hufnagel Cn2 model, an isoplanatic angle of θIP = 1.64 arcsec (calculated for a value of ro = 20 cm), and zg = 92 km (the nominal height of the mesospheric Na layer), we find that three Na guide stars are required in order to achieve a rms wave-front error of approximately λ/10 across the telescope aperture.

© 1991 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. 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]
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    [CrossRef]
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  12. E. P. Wallner, “Optimal wave-front correction using slope measurements,”J. Opt. Soc. Am. 73, 1771–1776 (1983).
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. G. C. Valley, “Isoplanatic degradation of tilt correction and short-term imaging systems,” Appl. Opt. 19, 574–577 (1980).
    [CrossRef] [PubMed]
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    [CrossRef]
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  22. C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
    [CrossRef]

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 (2)

1988 (1)

1987 (2)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

J. H. Churnside, R. J. Lataitis, “Angle-of-arrival fluctuations of a reflected beam in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 1264–1272 (1987).
[CrossRef]

1986 (1)

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

1985 (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

1984 (1)

1983 (1)

1982 (1)

1980 (2)

1978 (1)

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

1975 (1)

1973 (1)

Buser, R. G.

Cho, K. H.

Churnside, J. H.

Cowie, L. L.

Dryden, G.

Foy, R.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Fried, D. L.

D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

D. L. Fried, “Varieties of isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 20–29 (1976).
[CrossRef]

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]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

T. J. Kane, C. S. Gardner, “Optimal design of an adaptive optic wavefront sensor,” EOSL report 89-002 (Department of Electrical and Computer Engineering, University of Illinois, Urbana, Ill., 1989).

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Adaptive Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6.

Hardy, J. W.

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

Hufnagel, R. E.

R. E. Hufnagel, in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Kane, T. J.

T. J. Kane, C. S. Gardner, “Optimal design of an adaptive optic wavefront sensor,” EOSL report 89-002 (Department of Electrical and Computer Engineering, University of Illinois, Urbana, Ill., 1989).

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Adaptive Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Korff, D.

Labeyrie, A.

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

Lataitis, R. J.

Leavitt, R. P.

Lutomirski, R. F.

Peterson, D. P.

Roddier, C.

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) infrared adaptive optics program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 298–304 (1986).
[CrossRef]

Roddier, F.

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) infrared adaptive optics program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 298–304 (1986).
[CrossRef]

Sechrist, C. F.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

Segal, A. C.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

Songaila, A.

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]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Adaptive Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Tyler, G. A.

Valley, G. C.

Voelz, D. G.

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

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. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Adaptive Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (1)

R. Foy, A. Labeyrie, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, 129–131 (1985).

J. Geophys. Res. (1)

C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime sodium layer over Urbana, Illinois: 1. Seasonal and nocturnal variations,”J. Geophys. Res. 91, 13659–13673 (1986).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Nature (1)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature 328, 229–231 (1987).
[CrossRef]

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 (6)

D. L. Fried, “Varieties of isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 20–29 (1976).
[CrossRef]

F. Roddier, C. Roddier, “National Optical Astronomy Observatories (NOAO) infrared adaptive optics program II: modeling atmospheric effects in adaptive optics systems for astronomical telescopes,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 298–304 (1986).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Adaptive Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 160–171 (1989).
[CrossRef]

T. J. Kane, C. S. Gardner, “Optimal design of an adaptive optic wavefront sensor,” EOSL report 89-002 (Department of Electrical and Computer Engineering, University of Illinois, Urbana, Ill., 1989).

R. E. Hufnagel, in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

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Figures (14)

Fig. 1
Fig. 1

Geometry between point sources and optical aperture.

Fig. 2
Fig. 2

Geometry of the isoplanatic angle superimposed upon a cross section of the telescope aperture.

Fig. 3
Fig. 3

Wave-front-sensor and mirror-actuator configuration.

Fig. 4
Fig. 4

Geometry of a single Hartmann tilt sensor (CCD, charge-coupled device).

Fig. 5
Fig. 5

Normalized plot of Cn2 versus altitude.

Fig. 6
Fig. 6

Residual rms wave-front error over the aperture versus DIP/D for the single-layer Cn2 model. Curves are shown for aperture dimensions ranging from D = 2L to D = 5L.

Fig. 7
Fig. 7

Residual rms wave-front error over the aperture versus DIP/D for the Hufnagel Cn2 model. Curves are shown for aperture dimensions ranging from D = 2L to D = 5L.

Fig. 8
Fig. 8

Strehl ratio versus DIP/D for both turbulence models.

Fig. 9
Fig. 9

OTF of the wave-front corrected aperture for DIP/D ranging from 1/3 to 3 and the single-layer Cn2 model.

Fig. 10
Fig. 10

OTF of the wave-front corrected aperture for DIP/D ranging from 2/3 to 3 and the Hufnagel Cn2 model.

Fig. 11
Fig. 11

PSF of the wave-front corrected aperture for DIP/D ranging from 1/3 to 3 and the single layer Cn2 model.

Fig. 12
Fig. 12

PSF of the wave-front corrected aperture for DIP/D ranging from 2/3 to 3 and the Hufnagel Cn2model.

Fig. 13
Fig. 13

Normalized angular resolution of the wave-front corrected aperture versus DIP/D. The normalizing resolution is 0.88λ/D and is computed from the unaberrated PSF in Fig. 11.

Fig. 14
Fig. 14

Residual rms wave-front error versus θx/θIP for DIP/D = 1, 2, and ∞ The angle θx is the offset (in the x direction) between the object and the axis of the telescope. The single-layer Cn2 model is assumed.

Equations (58)

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p o = x o + ( z z o ) { [ ( r o - x o ) · x ^ ] x ^ + [ ( r o - x o ) · y ^ ] y ^ } ,
Δ p o r = p o - p r .
Δ p o r ( x o , x r , z ) = x o - x r + z { [ ( r o - x o ) · x ^ ] z o - [ ( r r - x r ) · x ^ ] z r } x ^ + z { [ ( r o - x o ) · y ^ ] z o - [ ( r r - x r ) · y ^ ] z r } y ^ .
Δ p o g ( x o , x g , z ) = x o - x g [ 1 - ( z z g ) ] + z ( tan θ x x ^ + tan θ y y ^ ) ,
θ = tan - 1 ( tan 2 θ x + tan 2 θ y ) 1 / 2 .
Δ p o g ( x o , x g , z ) x o - x g [ 1 - ( z z g ) ] + z ( θ x x ^ + θ y y ^ ) .
Δ p g g ( x , x , z ) = ( x - x ) ( 1 - z z g ) ,
Δ p o o ( x , x , z ) = x - x .
d 2 x W A ( x ) = 1 ,
ϕ o , r ( x ) = ψ o , r ( x ) - d 2 x W A ( x ) ψ o , r ( x ) .
s n = d 2 x W n ( x ) [ ϕ r ( x ) · d ^ n ] + α n ,
s n = - d 2 x [ W n ( x ) · d ^ n ] ϕ r ( x ) + α n .
c j = n M j n s n ,
ϕ ^ ( x ) = j c j r j ( x ) ,
( x ) = ϕ ^ ( x ) - ϕ o ( x ) = j r j ( x ) n M j n s n - ϕ o ( x ) .
2 d 2 x W A ( x ) 2 ( x ) ,
2 = j j n n M j n M j n S n n R j j - 2 j n M j n A j n + o 2 ,
S n n = s n s n = d 2 x d 2 x W n s ( x ) W n s ( x ) ϕ r ( x ) ϕ r ( x ) + α n α n ,
R j j = d 2 x W A ( x ) r j ( x ) r j ( x ) ,
A j n = d 2 x W A ( x ) r j ( x ) s n ϕ o ( x ) = - d 2 x d 2 x W A ( x ) r j ( x ) W n s ( x ) ϕ o ( x ) ϕ r ( x ) ,
o 2 = d 2 x W A ( x ) ϕ o 2 ( x ) .
M j n * = R j j - 1 A j n S n n - 1 ,
2 min = o 2 j n ( R j j - 1 A j n S n n ) A j n .
H ( ρ ) = d 2 x W A ( x ) E ( x ) W A * ( x - ρ ) E * ( x - ρ ) d 2 x W A ( x ) E ( x ) 2 ,
H ( ρ ) = exp { - [ ϕ o ( x ) - ϕ o ( x - ρ ) ] 2 2 } d 2 x W A ( x ) 2 × d 2 x W A ( x ) W A * ( x - ρ ) ( exp { - 1 2 j i [ r j ( x ) - r j ( x - ρ ) ] [ r i ( x ) - r i ( x - ρ ) ] C j i + j [ r j ( x ) - r j ( x - ρ ) ] × c j [ ϕ o ( x ) - ϕ o ( x - ρ ) ] } ) ,
C j i = c j c i = n m M j n M i m S n m
c j [ ϕ o ( x ) - ϕ o ( x - ρ ) ] = - n M j n d 2 x × W n s ( x ) [ ϕ r ( x ) ϕ o ( x ) - ϕ r ( x ) ϕ o ( x - ρ ) ] .
D o r ( x , x ) = [ ψ o ( x ) - ψ r ( x ) ] 2 ,
ϕ o ( x ) ϕ r ( x ) = - ½ D o r ( x , x ) + g o r ( x ) + g r o ( x ) - f ,
g o r ( x ) = 1 2 d 2 x W A ( x ) D o r ( x , x )
f = 1 2 d x d x W A ( x ) W A ( x ) D o r ( x , x ) .
S n n = - 1 2 d 2 x d 2 x W n s ( x ) W n s ( x ) D r r ( x , x ) + α n α n
A j n = - d 2 x d 2 x W A ( x ) r j ( x ) W n s ( x ) × [ - 1 2 D o r ( x , x ) + g o r ( x ) ] ,
H ( ρ ) = exp [ - D o o ( x , x - ρ ) 2 ] d 2 x W A ( x ) 2 d 2 x W A ( x ) W A * ( x - ρ ) × ( exp { - 1 2 j i [ r j ( x ) - r j ( x - ρ ) ] [ r i ( x ) - r i ( x - ρ ) ] C j i + j [ r j ( x ) - r j ( x - ρ ) ] c j [ ϕ o ( x ) - ϕ o ( x - ρ ) ] } ) ,
c j [ ϕ o ( x ) - ϕ o ( x - ρ ) ] = 1 2 n M j n d 2 x W n s ( x ) × [ D o r ( x , x ) - D o r ( x - ρ , x ) ] .
D o r ( x , x ) = 2.91 ( 2 π λ ) 2 o z min C n 2 ( z ) Δ p o r ( x , x , z ) 5 / 3 d z ,
D o r ( x , x ) = 6.88 o z min C n 2 ( z ) Δ p o r ( x , x , z ) 5 / 3 d z r o 5 / 3 o z min C n 2 ( z ) d z ,
r o = [ 2.91 6.88 ( 2 π λ ) 2 o z min C n 2 ( z ) d z ] - 3 / 5 .
D o g ( x , x ) = 6.88 r o 5 / 3 o z min C n 2 ( z ) d z o z min C n 2 ( z ) × | x - x [ 1 - ( z z g ) ] + z ( θ x x ^ + θ y y ^ ) | 5 / 3 d z ,
D g g ( x , x ) = 6.88 x - x 5 / 3 o z min C n 2 ( z ) ( 1 - z z g ) 5 / 3 d z r o 5 / 3 o z min C n 2 ( z ) d z ,
D o o ( x , x ) = 6.88 x - x 5 / 3 r o 5 / 3 .
D I P = 2 z g θ I P ,
N g s D T 2 D I P 2 = D T 2 4 z g 2 θ I P 2 .
θ I P = 58.1 × 10 - 3 [ λ 2 0 z h C n 2 ( ξ ) ξ 5 / 3 d ξ ] 3 / 5 .
S ( 2 ) exp ( - 2 ) .
s ( γ / λ f D ) = F 2 - 1 [ H ( ρ ) ] ( λ f D ) 2 ,
s ( u / λ f D ) = d v s ( u / λ f D , v / λ f D ) = d x H ( x , 0 ) λ f D exp [ j 2 π x ( u λ f D ) ] = F 1 - 1 ( H ( x , 0 ) ) λ f D ,
r j ( x , y ) exp [ - ( x - x j ) 2 - ( y - y j ) 2 L a 2 ] ,
α n α n = σ n 2 k n n ,
σ n = { 0.86 π η N r o L > r o 0.74 π η N L L r o ,
C n 2 ( z ) = C n 2 δ ( z - h t ) ,
C n 2 ( z ) = A ( 2.2 × 10 - 23 z 10 e - z + 10 - 16 e - z / 1.5 ) ,
θ I P = r o / 3 h t             ( single - layer model ) ,
θ I P = 3.978 × 10 - 5 r o             ( Hufnagel model ) .
D D I P α = 2 z g θ I P α .
N g s D T 2 D 2 = α 2 D T 2 4 z g 2 θ I P 2 ,
N g s = 2.25 α 2 D T 2 h t 2 z g 2 r o 2             ( single - layer model ) ,
N g s = 1.58 × 10 8 α 2 D T 2 z g 2 r o 2             ( Hufnagel model ) .

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