Abstract

Zernike polynomials are often used to analyze turbulence-induced optical phase aberrations. Previous investigations have examined the spatial and temporal characteristics of the expansion coefficients of the turbulence-induced optical phase with respect to these polynomials. The results of these investigations are valid only for certain geometries and atmospheric models and do not take into account the effects of relative motion between the sensor and the object of interest. We introduce a generalized analysis geometry and use this aperture-and-source geometry with conventional methods to arrive at a general expression for the inter- aperture cross correlation of the Zernike coefficients. Aperture-and-source motion considerations are introduced to derive an expression for the temporal cross correlation and cross power spectra of these expansion coefficients. Temporal correlation and spectrum results are presented for several low-order Zernike modes, given certain aperture-and-source motions.

© 1998 Optical Society of America

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References

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    [CrossRef]
  13. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  14. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1981), Vol. XIX, pp. 283–376.
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  16. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  19. V. I. Tatarski, V. U. Zavorotny, “Atmospheric turbulence and the resolution limits of large ground-based telescopes: comment,” J. Opt. Soc. Am. A 10, 2410–2414 (1993).
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    [CrossRef]
  21. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, Washington, D.C., 1964), p. 360.

1997 (1)

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

1995 (2)

1993 (2)

1992 (1)

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for an adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).
[CrossRef]

1991 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. (Bellingham) 29, 1174–1180 (1990).
[CrossRef]

1989 (1)

1978 (1)

1976 (2)

1965 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, Washington, D.C., 1964), p. 360.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Butts, R. R.

Dai, G.

Fried, D. L.

Fugate, R. Q.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

Gardner, C. S.

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Graves, J. E.

Hogge, C. B.

Hu, P. H.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Markey, J. K.

McKenna, D. L.

Noll, R. J.

Northcott, M. J.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Paschall, R. N.

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for an adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).
[CrossRef]

Roddier, D.

Roddier, F.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. (Bellingham) 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC, Boca Raton, Fla., 1996).

Stanley, T.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, Washington, D.C., 1964), p. 360.

Stone, J.

Takato, N.

Tatarski, V. I.

Von Bokern, M. A.

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for an adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).
[CrossRef]

Wang, J. Y.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC, Boca Raton, Fla., 1996).

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for an adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yamaguchi, I.

Zavorotny, V. U.

Comput. Elect. Eng. (1)

M. A. Von Bokern, R. N. Paschall, B. M. Welsh, “Modal control for an adaptive optics system using LQG compensation,” Comput. Elect. Eng. 18, 421–433 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Eng. (Bellingham) (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. (Bellingham) 29, 1174–1180 (1990).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

Other (8)

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, Washington, D.C., 1964), p. 360.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

M. C. Roggemann, B. Welsh, Imaging Through Turbulence (CRC, Boca Raton, Fla., 1996).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1981), Vol. XIX, pp. 283–376.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

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Figures (7)

Fig. 1
Fig. 1

Analysis geometry used for computing the interaperture cross correlation of Zernike expansion coefficients.

Fig. 2
Fig. 2

Illustration of the particular aperture and source positions and velocities used to obtain temporal correlation and spectrum results. The position of the source is specified relative to the position of the aperture by the parameter d, which represents the distance between source and aperture.

Fig. 3
Fig. 3

(a) Autocorrelation and (b) frequency times normalized power spectrum results for Zernike coefficient a2 (tilt), calculated for several values of aperture–source separation d.

Fig. 4
Fig. 4

(a) Autocorrelation and (b) frequency times normalized power spectrum results for Zernike coefficient a3 (tilt), calculated for several values of aperture–source separation d.

Fig. 5
Fig. 5

(a) Autocorrelation and (b) frequency times power spectrum results for Zernike coefficient a4 (defocus) and the orthogonal components of astigmatism, a5 and a6, calculated for fixed aperture–source separation distance d=100 km.

Fig. 6
Fig. 6

(a) Autocorrelation and (b) frequency times normalized power spectrum results for the orthogonal components of coma, a7 and a8, and for Zernike coefficient a11 (spherical aberration), calculated for fixed aperture–source separation distance d=100 km.

Fig. 7
Fig. 7

(a) Cross correlation for combinations of the orthogonal components of tilt (i=2, j=3), astigmatism (i=5, j=6), and coma (i=7, j=8). (b) Cross correlation for combinations of tilt and defocus (i=2, j=4), defocus and astigmatism (i=4, j=6), astigmatism and coma (i=6, j=8), and coma and spherical aberration (i=8, j=11). All results are for aperture–source separation d=100 km.

Equations (81)

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q1l=(1-A1l)R1ρ1,
q2l=(1-A2l)R2ρ2,
A1l=zl-(ra1·zˆ)(rs1-ra1)·zˆ,
A2l=zl-(ra2·zˆ)(rs2-ra2)·zˆ.
sl=ra2-ra1+A2l(rs2-ra2)-A1l(rs1-ra1).
Δsl=q2l+sl-q1l.
ϕ1(R1ρ1)=ia1iZi(ρ1),
ϕ2(R2ρ2)=ja2jZj(ρ2),
a1i=dρ1ϕ1(R1ρ1)Zi(ρ1)W(ρ1),
a2j=dρ2ϕ2(R2ρ2)Zj(ρ2)W(ρ2),
W(ρ)=1πif|ρ|10otherwise.
Ba1ia2j=E{a1ia2j},
Ba1ia2j=Edρ1dρ2ϕ1(R1ρ1)ϕ2(R2ρ2)×Zi(ρ1)Zj(ρ2)W(ρ1)W(ρ2).
Ba1ia2j=dρ1dρ2E{ϕ1(R1ρ1)ϕ2(R2ρ2)}×Zi(ρ1)Zj(ρ2)W(ρ1)W(ρ2).
ϕ1(R1ρ1)=2πλl1Δzl1·nl1(q1l1),
ϕ2(R2ρ2)=2πλl2Δzl2·nl2(q2l2),
E{nl1(q1l1)nl2(q2l2)}δl1l2,
E{ϕ1(R1ρ1)ϕ2(R2ρ2)}=2πλ2lΔzl2E{nl(q1l)nl(q2l)}.
E{ϕ1(R1ρ1)ϕ2(R2ρ2)}=l2πλ2Δzl2Bnl(q1l, q2l).
E{ϕ1(R1ρ1)ϕ2(R2ρ2)}=lBϕl(q1l, q2l).
Bϕl(Δsl)=Bϕl(q2l+sl-q1l).
E{ϕ1(R1ρ1)ϕ2(R2ρ2)}=lBϕl(q2l+sl-q1l).
Ba1ia2j=dρ1dρ2lBϕl(q2l+sl-q1l)×Zi(ρ1)Zj(ρ2)W(ρ1)W(ρ2).
Ba1ia2j(ra1, rs1; ra2, rs2)=lBa1ia2jl(ra1, rs1; ra2, rs2; zl),
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)
[R1R2(1-A1l)(1-A2l)]-1
×dq1ldq2lBϕl(q2l+sl-q1l)
×Ziq1lR1(1-A1l)Zjq2lR2(1-A2l)
×Wq1lR1(1-A1l)Wq2lR2(1-A2l).
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=dkWϕl(k)exp(-j2πk·sl)Qi[R1(1-A1l)k]×Qj*[R2(1-A2l)k],
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)
=[πR1R2(1-A1l)(1-A2l)]-1[(ni+1)(nj+1)]1/2(-1)12(ni+nj)2[1-12(δmi0+δmj0)](-1)mj
×(-1)32(mi+mj) cos(mi+mj)θsl+π4{(1-δmi0)[(-1)i-1]+(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ(mi+mj)[slx]J(ni+1)[R1(1-A1l)x]J(nj+1)[R2(1-A2l)x]+(-1)32|mi-mj| cos(mi-mj)θsl+π4{(1-δmi0)[(-1)i-1]-(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ|mi-mj|[slx]J(ni+1)[R1(1-A1l)x]J(nj+1)[R2(1-A2l)x],
ra2=ra1+vaτ,
rs2=rs1+vsτ,
A2l=(zl-ra1·zˆ)-(va·zˆ)τ(rs1-ra1)·zˆ+[(vs-va)·zˆ]τ.
Al(τ)=(zl-ra·zˆ)-(va·zˆ)τ(rs-ra)·zˆ+[(vs-va)·zˆ]τ.
sl(τ)=vaτ+[Al(τ)-Al(0)](rs-ra)+Al(τ)(vs-va)τ.
sl(τ)=vaτ+[Al(τ)-Al(0)](rs-ra)+Al(τ)(vs-va)τ-vlτ.
Raiaj(τ; ra, rs; va, vs; vl)=lRaiajl[sl(τ)].
Raiajl[sl(τ)]=0dkWϕl(k)exp[-j2πk·sl(τ)]×Qi{R[1-Al(0)]k}Qj*{R[1-Al(τ)]k}.
Raiajl[sl(τ)]={πR2[1-Al(0)][1-Al(τ)]}-1[(ni+1)(nj+1)]1/2(-1)12(ni+nj)2[1-12(δmi0+δmj0)](-1)mj×(-1)32(mi+mj) cos(mi+mj)θsl(τ)+π4{(1-δmi0)[(-1)i-1]+(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ(mi+mj)[sl(τ)x]J(ni+1){R[1-Al(0)]x}J(nj+1){R[1-Al(τ)]x}+(-1)32|mi-mj| cos(mi-mj)θsl(τ)+π4{(1-δmi0)[(-1)i-1]-(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ|mi-mj|[sl(τ)x]J(ni+1){R[1-Al(0)]x}J(nj+1){R[1-Al(τ)]x},
Saiaj(ν; ra, rs;va; vs; vl)=-dτRaiaj(τ; ra; rs; va, vs;vl)exp(j2πντ).
Saiaj(ν; ra, rs; va, vs; vl)
=l0dkWϕl(k)Qi{R[1-Al(0)]k}
×-dτ exp(-j2πντ)exp[j2πk·sl(τ)]
×Qj*{R[1-Al(τ)]k}.
rs·xˆ=(d2-(3000m)2)1/2.
Wϕl(k)=Δzl·0.033Cn2(zl)2πλ2·exp-k2km2(k2+k02)11/6,
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=[R1R2(1-A1l)(1-A2l)]-1×dq1ldq2lBϕl(q2l+sl-q1l)×Ziq1lR1(1-A1l)Zjq2lR2(1-A2l)×Wq1lR1(1-A1l)Wq2lR2(1-A2l).
drf(r)g*(r)=dfF(f)G*(f),
FTfrb=|b|F(bf),
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=[R1(1-A1l)]-1dq1ldkFT{Bϕl(q2l+sl-q1l)}×Ziq1lR1(1-A1l)Wq1lR1(1-A1l)×Qj*[R2(1-A2l)k],
FT{Bϕl(q2l+sl-q1l)}=Wϕl(k)exp[j2πk·(q1l-sl)],
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)
=[R1(1-A1l)]-1
×dq1ldkWϕl(k)exp[j2πk·(q1l-sl)]
×Ziq1lR1(1-A1l)Wq1lR1(1-A1l)×Qj*[R2(1-A2l)k].
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)
=[R1(1-A1l)]-1dkWϕl(k)Qj*[R2(1-A2l)k]
×exp(-j2πk·sl)dq1lZiq1lR1(1-A1l)×Wq1lR1(1-A1l)exp(j2πk·q1).
FTZiq1lR1(1-A1l)Wq1lR1(1-A1l)=R1(1-A1l)Qi[R1(1-A1l)k],
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=dkWϕl(k)exp(-j2πk·sl)×Qi[R1(1-A1l)k]Qj*[R2(1-A2l)k],
Qi(f, θ)=ni+1(-1)(ni-mi)/2-1mi21-δmi0·Jni+1(2πf)πfcosmiθ+π4(1-δmi0)×[(-1)i-1],
k·sl=ksl cos(θ-θsl).
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=0dk·k·Wϕl(k)θslθsl+2πdθ×exp[-j2πksl cos(θ-θsl)]×Qi[R1(1-A1l)k,θ]Qj*[R2(1-A2l)k,θ].
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)=[π2R1R2(1-A1l)(1-A2l)]-1×[(ni+1)(nj+1)]1/2(-1)12(ni+nj)2[1-12(δmi0+δmj0)](-1)mj×0dk·k-1·Wϕl(k)·Jni+1[2πR1(1-A1l)k]Jnj+1[2πR2(1-A2l)k]×θslθsl+2πdθ exp[-j2πksl cos(θ-θsl)]·cosmiθ+π4(1-δmi0)[(-1)i-1]×cosmjθ+π4(1-δmj0)[(-1)j-1].
cosmiθ+π4(1-δmi0)[(-1)i-1]
×cosmjθ+π4(1-δmj0)[(-1)j-1]
=12cos[(mi+mj)θ+g]+12cos[(mi-mj)θ+h],
gπ4{(1-δmi0)[(-1)i-1]+(1-δmj0)[(-1)j-1]},
hπ4{(1-δmi0)[(-1)i-1]-(1-δmj0)[(-1)j-1]}.
12cos[(mi+mj)θ+g]+12cos[(mi-mj)θ+h]=12cos[(mi+mj)ϕ]cos[(mi+mj)θsl+g]-12sin[(mi+mj)ϕ]sin[(mi+mj)θsl+g]+12cos[(mi-mj)ϕ]cos[(mi-mj)θsl+h]-12sin[(mi-mj)ϕ]sin[(mi-mj)θsl+h].
Iθ=02πdϕ exp[-j2πksl cos ϕ]·12cos[(mi+mj)ϕ]cos[(mi+mj)θsl+g]-12sin[(mi+mj)ϕ]sin[(mi+mj)θsl+g]+12cos[(mi-mj)ϕ]cos[(mi-mj)θsl+h]-12sin[(mi-mj)ϕ]sin[(mi-mj)θsl+h].
Jλ(z)=1-1λπ0πdϕ exp(jz cos ϕ)cos(λϕ),
02πdϕ exp(-jz cos ϕ)cos(λϕ)=2π(-1)32λJλ(z),
02πdϕ exp(-jz cos ϕ)sin(λϕ)=0.
Iθ=-π{(-1)3/2(mi+mj)J(mi+mj)(2πksl)×cos[(mi+mj)θsl+g]+(-1)3/2|mi-mj|J|mi-mj|(2πksl)×cos[(mi-mj)θsl+h]}.
Ba1ia2jl(ra1, rs1; ra2, rs2; zl)
=[πR1R2(1-A1l)(1-A2l)]-1[(ni+1)(nj+1)]1/2(-1)12(ni+nj)2[1-12(δmi0+δmj0)](-1)mj
×(-1)32(mi+mj) cos(mi+mj)θsl+π4{(1-δmi0)[(-1)i-1]+(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ(mi+mj)[slx]J(ni+1)[R1(1-A1l)x]J(nj+1)[R2(1-A2l)x]+(-1)32|mi-mj| cos(mi-mj)θsl+π4{(1-δmi0)[(-1)i-1]-(1-δmj0)[(-1)j-1]}×0 dxxWϕlx2πJ|mi-mj|[slx]J(ni+1)[R1(1-A1l)x]J(nj+1)[R2(1-A2l)x],

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