Abstract

In optical propagation through atmospheric turbulence, the performance of compensation with adaptive optics depends on a beacon’s spatial distribution. With distributed beacons, the inefficiency of the modal correction, which is defined as the ratio of the anisoplanatic error of the jth mode and the Zernike-coefficient variance, is derived by use of the wave-front expansion on the Zernike polynomials for non-Kolmogorov turbulence. Numerical results are presented for laser beam propagation through constant turbulence with an offset point beacon and an on-axis uniform circular beacon. The results show that compensation for an on-axis uniform circular beacon is much more effective than that for an offset point beacon. The low-order modes are much more correlated than the higher-order modes. The larger the power-law exponent of the refractive-index power spectrum β, the smaller the propagation path length L and the larger the diameter D of the telescope aperture, the more effective the compensation is. For a specific extended degree of beacon for which there are a maximum number of modes N max to be corrected, only low-order-correction systems are useful.

© 2001 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam (1981), Vol. 19, pp. 281–376.
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    [CrossRef]
  3. C. Rao, W. Jiang, “Error and performance analysis for an infrared adaptive optics system at 2.16 m telescope,” Acta Astrophys. Sin. 16, 428–437 (1996).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2000 (1)

C. Rao, W. Jiang, N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

1999 (1)

1997 (1)

1996 (3)

G. D. Boreman, C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13, 517–522 (1996).
[CrossRef]

C. Rao, W. Jiang, “Error and performance analysis for an infrared adaptive optics system at 2.16 m telescope,” Acta Astrophys. Sin. 16, 428–437 (1996).

P. D. Stroud, “Anisoplanatism in adaptive optics compensation of a focused beam with use of distributed beacons,” J. Opt. Soc. Am A 13, 868–874 (1996).
[CrossRef]

1995 (2)

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 2, 1359–1361 (1995).
[CrossRef]

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

1994 (2)

1992 (1)

1982 (1)

1976 (1)

Belen’kii, M. S.

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 2, 1359–1361 (1995).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 113–123 (1997).
[CrossRef]

Belsher, J. F.

Boreman, G. D.

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 113–123 (1997).
[CrossRef]

Dainty, C.

Fried, D. L.

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 113–123 (1997).
[CrossRef]

Hardy, J. W.

J. W. Hardy, “Adaptive optics—a progress review,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 2–17 (1991).
[CrossRef]

Jiang, W.

C. Rao, W. Jiang, N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

C. Rao, W. Jiang, N. Ling, “Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack–Hartmann wave-front sensor,” Opt. Lett. 24, 1008–1010 (1999).
[CrossRef]

C. Rao, W. Jiang, “Error and performance analysis for an infrared adaptive optics system at 2.16 m telescope,” Acta Astrophys. Sin. 16, 428–437 (1996).

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

C. Rao, W. Jiang, N. Ling, “Atmospheric parameter measurements for non-Kolmogorov turbulence with Shack–Hartmann wavefront sensor,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 84–91 (1999).
[CrossRef]

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 113–123 (1997).
[CrossRef]

Li, M.

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

Ling, N.

C. Rao, W. Jiang, N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

C. Rao, W. Jiang, N. Ling, “Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack–Hartmann wave-front sensor,” Opt. Lett. 24, 1008–1010 (1999).
[CrossRef]

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

C. Rao, W. Jiang, N. Ling, “Atmospheric parameter measurements for non-Kolmogorov turbulence with Shack–Hartmann wavefront sensor,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 84–91 (1999).
[CrossRef]

Molodij, G.

Noll, R. J.

Rao, C.

C. Rao, W. Jiang, N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

C. Rao, W. Jiang, N. Ling, “Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack–Hartmann wave-front sensor,” Opt. Lett. 24, 1008–1010 (1999).
[CrossRef]

C. Rao, W. Jiang, “Error and performance analysis for an infrared adaptive optics system at 2.16 m telescope,” Acta Astrophys. Sin. 16, 428–437 (1996).

C. Rao, W. Jiang, N. Ling, “Atmospheric parameter measurements for non-Kolmogorov turbulence with Shack–Hartmann wavefront sensor,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 84–91 (1999).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam (1981), Vol. 19, pp. 281–376.

Rousset, G.

Sasiela, R. J.

Stroud, P. D.

P. D. Stroud, “Anisoplanatism in adaptive optics compensation of a focused beam with use of distributed beacons,” J. Opt. Soc. Am A 13, 868–874 (1996).
[CrossRef]

Tang, G.

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

Tyler, G. A.

Wolf, E.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam (1981), Vol. 19, pp. 281–376.

Zheng, D.

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

Acta Astrophys. Sin. (1)

C. Rao, W. Jiang, “Error and performance analysis for an infrared adaptive optics system at 2.16 m telescope,” Acta Astrophys. Sin. 16, 428–437 (1996).

J. Mod. Opt. (1)

C. Rao, W. Jiang, N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

J. Opt. Soc. Am A (1)

P. D. Stroud, “Anisoplanatism in adaptive optics compensation of a focused beam with use of distributed beacons,” J. Opt. Soc. Am A 13, 868–874 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

W. Jiang, M. Li, G. Tang, N. Ling, M. Li, D. Zheng, “Adaptive optics image compensation experiments on stellar objects,” Opt. Eng. 34, 15–20 (1995).
[CrossRef]

Opt. Lett. (2)

Other (4)

C. Rao, W. Jiang, N. Ling, “Atmospheric parameter measurements for non-Kolmogorov turbulence with Shack–Hartmann wavefront sensor,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann, L. R. Bissonnette, eds., Proc. SPIE3763, 84–91 (1999).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE3126, 113–123 (1997).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam (1981), Vol. 19, pp. 281–376.

J. W. Hardy, “Adaptive optics—a progress review,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 2–17 (1991).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Geometry of the telescope aperture and the distributed beacon.

Fig. 2
Fig. 2

Modal coefficient versus beacon extended degree for several radial degrees n. β = 44/12, D = 600 mm, and L = 1 km.

Fig. 3
Fig. 3

Modal coefficient versus beacon extended degree for various turbulence power-law exponents β. D = 600 mm, L = 1 km.

Fig. 4
Fig. 4

Modal coefficient versus beacon extended degree for several propagation path lengths L. β = 44/12, D = 600 mm.

Fig. 5
Fig. 5

Modal coefficient versus beacon extended degree for several diameters D of the telescope aperture. β = 44/12, L = 1 km.

Equations (45)

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ϕPRr=k 0LdznχzRr, z,
ϕBRr=k  d2bBb0LdznχzRr+χˆzb, z,
ϕPRr=j=2 ajPzjr,
ϕBRr=j=2 ajBzjr,
ajP= d2rWrϕPRrZjr,
ajB= d2rWrϕBRrZjr,
Wr=1/π|r|10elsewhere.
Zevenj=n+1Rnmr2 cosmθ,  m0, Zoddj=n+1Rnmr2 sinmθ, m0, Zj=n+1Rn0r,  m=0,
Rnmr=s=0n-m/2-1sn-s!s!n+m/2-s!n-m/2-s!×rn-2s,
ε2J= d2rWrϕPRr-ϕBRr, J2,
ε2J=j=J+1 ajP2+j=2JajP2+ajB2-2ajPajB.
εj2=ajP2+ajB2-2ajPajB.
ηj=εj2ajP2=ajP2+ajB2-2ajPajBajP2.
ajPajB= d2x1  d2x2Wx1Zjx1Wx2Zjx2×ϕPRx1ϕBRx2,
WxZjρ, θ= d2KQjK, φexp-2iπK·x,
QjK, φ=n+1Jn+12πKπK-1n-m/2im2 cosmφj=even,m0-1n-m/2im2 sinmφj=odd,m0-1n/2m=0,
ajPajB=k2  d2bBb  d2K1× d2K2QjK1, φ1Qj*K2, φ2× d2x1  d2x2 exp-2iπK1·x1× exp2iπK2·x20Ldz×0LdznχzRx1, znχzRx2+χˆzb, z,
Wnf, z=2π3Φn2πf, z=2π3-βBβΩn2zf-β,
nx1z, znx2z, z=2π3-βBβΩn2z+z2× d2f  dfzf2+fz2-β/2exp2iπf·x1z-x2z+2iπfzz-z,
0Ldznx1z, znx2z, z=2π3-βBβ×0LdzΩn2z× d2ff-β×exp2iπfx1z-x2z,
 d2x exp2iπz-zx=δz-z
ajPajB=k22π3-βBβ  d2bBb0LdzΩn2z× d2K1  d2K2QjK1, φ1Qj*K2, φ2× d2x1  d2x2 exp-2iπK1·x1×exp2iπK2x2 d2ff-β×exp2iπRχzf·x1-x2+2iπχˆzf·b.
δK2-K1= d2x2 exp2iπK2-K1·x2;
ajPajB=k22π3-βBβ  d2bBb0LdzΩn2z× d2KQjK, φQj*K, φ× d2x exp2iπK·x  d2ff-β×exp-2iπRχzf·x+2iπχˆzf·b.
ajPajB=k22π3-βπ-2n+1BβRβ-2× d2bBb0LdzΩn2zχzβ-2×0dKK-β+1Jn+122πK02πdφ×exp-2iπ χˆzχzRK·bqj,
qj=-1n-m2 cos2mφ,  j=even, m0-1n-m2 sin2mφ,  j=odd, m0-1n,           m=0,
ρ0=Dβk20L Ωn2zχzβ-2dz-1/β-2,
ajPajB=π226-βn+1BβDβ-1Dρ0β-2× dbBb0LdzΩn2zχzβ-2In,mχˆzb/χzR, β0LdzΩn2zχzβ-2,
In,mx, β=sn,m0dtt-β+1Jn+12t×J0xt+rj,mJ2mxt,
sn,m=1,m=0-1n-m,m0, rj,m=0m=0-1j+m,m0
02πdθ expjz cos θcosnθ=2π-13n/2Jnz,
02πdθ expjz cos θsinnθ=0
ajB2=π226-βn+1BβDβ-1Dρ0β-2× dbBb  dbBb0LdzΩn2zχzβ-2In,mχˆz|b-b|/χˆzR, β0LdzΩn2zχzβ-2.
ajP2=π226-βn+1BβDβ-1D/ρ0β-2In,00, β,
ηj=1+ dbBb  dbBb0LdzΩn2zχzβ-2In,mχˆz|b-b|/χzR, βIn,00, β0LdzΩn2zχzβ-2-2  dbBb0LdzΩn2zχzβ-2In,mχˆzb/χzR, βIn,00, β0LdzΩn2zχzβ-2.
ηj=1+β-1  dbBb  dbBb01dz1-zβ-2In,mz|b-b|/1-zR, βIn,00, β-2β-1  dbBb01dz1-zβ-2In,mzb/1-zR, βIn,00, β.
Bb=δb-b0δθ-θ0.
Bb=1πb02circb/b0,
circx=1|x|10otherwise.
Λ=b01.22 λL2R.
ηj=2-2β-101dz1-zβ-2In,mzb0/1-zR, βIn,00, β.
In,mx, β=sn,mH1x, n+1, 0, β+1+rj,mH1x, n+1, m, β+1,
H1x, n+1, m, β+1=0dKK-β+1Jn+12K×J2mxK.
H1x, n+1, m, β+1=p=012π-1pp!x22p+β+1Γn+p+3/2Γm-p-β/2-1/2Γn-p+3/2Γ1/2-pΓm+p+β/2+3/2+p=012π-1pp!x22p+2mΓn+m+p-β/2+1Γ-m-p+β/2+1/2Γ2m+p+1Γn-p-m+β/2+2Γ-m-p+β/2+1,
H1x, n+1, m, β+1=p=012π-1pp!×x2-2p-2n+β-2×Γn+p+3/2Γm+n+p-β/2+1Γ2n+p+3Γp+n+2Γm-p-n+β/2.

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