Abstract

The variance σS2 of the Strehl ratio of a reasonably well-corrected adaptive optics system is derived as a power series in the log-amplitude variance σl2 and the residual phase error variance σδϕ2. It is shown that, to leading order, the variance of the Strehl ratio is dependent on the first power of the log-amplitude variance, (σl2)1, of the incident optical field but only on the second power of the residual phase variance, (σδϕ2)2, of that field after adaptive optics correction, and on the first power of the product of the log-amplitude variance times the phase variance, (σl2σδϕ2)1. As long as the adaptive optics correction is good enough to ensure that the variance of the residual phase, σδϕ2, is significantly less than unity, then even for fairly small values of the log-amplitude variance σl2, the value of the variance of the Strehl ratio, σS2, will be dominated by the value of the log-amplitude variance.

© 1998 Optical Society of America

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  1. In regard to the matter of an exact definition for resolution, it may be noted that if the concept of resolution is taken to be represented by the value of the optical transfer function integrated over the spatial frequency domain, then resolution as thus defined is directly proportional to the Strehl ratio.
  2. It is perhaps worth noting that in the matter of resolution the variations of intensity over the telescope’s aperture will affect the image of a point source, i.e., will affect the resolution. That this is indeed the case is perhaps made evident most easily by simply referring to the intensity variations as “random apodization” of the telescope.
  3. D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-3, 213–221 (1967).
    [CrossRef]
  4. M. E. Gracheva, A. S. Gurvich, “Averaging effect of the receiver aperture on fluctuations of light intensity,” Izv. Vyssh. Uchebn. Zaved. Radioviz. 12, 253–255 (1969).
  5. R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).
  6. A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

1969 (1)

M. E. Gracheva, A. S. Gurvich, “Averaging effect of the receiver aperture on fluctuations of light intensity,” Izv. Vyssh. Uchebn. Zaved. Radioviz. 12, 253–255 (1969).

1967 (1)

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-3, 213–221 (1967).
[CrossRef]

Fried, D. L.

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-3, 213–221 (1967).
[CrossRef]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, “Averaging effect of the receiver aperture on fluctuations of light intensity,” Izv. Vyssh. Uchebn. Zaved. Radioviz. 12, 253–255 (1969).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, “Averaging effect of the receiver aperture on fluctuations of light intensity,” Izv. Vyssh. Uchebn. Zaved. Radioviz. 12, 253–255 (1969).

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

Huschke, R. E.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).

Khmelesvtsov, S. S.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

Kon, A. I.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

Lutomirski, R. F.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).

Meecham, W. C.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).

Mironov, V. L.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

Yura, H. T.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).

IEEE J. Quantum Electron. (1)

D. L. Fried, “Atmospheric modulation noise in an optical heterodyne receiver,” IEEE J. Quantum Electron. QE-3, 213–221 (1967).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radioviz. (1)

M. E. Gracheva, A. S. Gurvich, “Averaging effect of the receiver aperture on fluctuations of light intensity,” Izv. Vyssh. Uchebn. Zaved. Radioviz. 12, 253–255 (1969).

Other (4)

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of laser systems by atmospheric turbulence,” (The Rand Corporation, Santa Monica, Calif., June1973).

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelesvtsov, Laser Radiation in a Turbulent Atmosphere (Nauka, Moscow, 1965); in Russian, quoted in S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarski, Principles of Statistical Radiophysics (Nauka, Moscow, 1965), Vol. IV, Fig. 2.12.

In regard to the matter of an exact definition for resolution, it may be noted that if the concept of resolution is taken to be represented by the value of the optical transfer function integrated over the spatial frequency domain, then resolution as thus defined is directly proportional to the Strehl ratio.

It is perhaps worth noting that in the matter of resolution the variations of intensity over the telescope’s aperture will affect the image of a point source, i.e., will affect the resolution. That this is indeed the case is perhaps made evident most easily by simply referring to the intensity variations as “random apodization” of the telescope.

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Equations (36)

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S
=dr1,dr2W(r1)W(r2)exp(F1)dr1,dr2W(r1)W(r2),
S2
=dr1,dr2dr3dr4W(r1)W(r2)W(r3)W(r4)exp(F2)dr1,dr2dr3dr4W(r1)W(r2)W(r3)W(r4).
W(r)=1if|r|D/20if|r|>D/2.
F1=-12[Dδϕ(r1-r2)+Dl(r1-r2)]
F2=-12[Dδϕ(r1-r2)-Dδϕ(r1-r3)+Dδϕ(r1-r4)+Dδϕ(r2-r3)-Dδϕ(r2-r4)+Dδϕ(r3-r4)+Dl(r1-r2)-Dl(r1-r3)+Dl(r1-r4)+Dl(r2-r3)-Dl(r2-r4)+Dl(r3-r4)]+2[Cl(r1-r3)+Cl(r2-r4)].
Dδϕ(r)=2σδϕ2[1-fδϕ(r)],
wherefδϕ(r)=Cδϕ(r)/σδϕ2,
Dl(r)=2σl2[1-fl(r)],wherefl(r)=Cl(r)/σl2.
F1=σδϕ2[-1+fδϕ(r1-r2)]+σl2[-1+fl(r1-r2)],
F2=σδϕ2[-2+fδϕ(r1-r2)-fδϕ(r1-r3)+fδϕ(r1-r4)+fδϕ(r2-r3)-fδϕ(r2-r4)+fδϕ(r3-r4)]+σl2[-2+fl(r1-r2)+fl(r1-r3)+fl(r1-r4)+fl(r2-r3)+fl(r2-r4)+fl(r3-r4)].
exp(F1)1+{σδϕ2[-1+fδϕ(r1-r2)]+σl2[-1+fl(r1-r2)]}+12((σδϕ2)2{1-2 fδϕ(r1-r2)+[fδϕ(r1-r2)]2}+(σl2)2{1-2 fl(r1-r2)+[fl(r1-r2)]2}+2σδϕ2σl2[1-fδϕ(r1-r2)-fl(r1-r2)+fδϕ(r1-r2)fl(r1-r2)])
exp(F2) 1+{σδϕ2[-2+2 fδϕ(r1-r2)]
+σl2[-2+6fl(r1-r2)]}+12{(σδϕ2)2[4-8fδϕ(r1-r2)+Kδϕ]+(σl2)2[4-24fl(r1-r2)+Kl]+2σδϕ2σl2[4-4fδϕ(r1-r2)-12 fl(r1-r2)+Kδϕ,l]},
Kδϕ=6fδϕ(r1-r2)fδϕ(r1-r2)-8fδϕ(r1-r2)fδϕ(r2-r3)+6fδϕ(r1-r2)fδϕ(r3-r4),
Kl=6fl(r1-r2)fl(r1-r2)+24fl(r1-r2)fl(r2-r3)+6fl(r1-r2)fl(r3-r4),
Kδϕ,l=2 fδϕ(r1-r2)fl(r1-r2)+8fδϕ(r1-r2)×fl(r2-r3)+2 fδϕ(r1-r2)fl(r3-r4).
S=1+σδϕ2(-1+αδϕ)+σl2(-1+αl)+12(σδϕ2)2(1-2αδϕ+βδϕ)+12(σl2)2(1-2αl+βl)+σδϕ2σl2(1-αδϕ-αl+βϕ,l),
S2=1+σδϕ2(-2+2αδϕ)+σl2(-2+6αl)+(σδϕ2)2(2-4αδϕ+3βδϕ-4γδϕ+3αδϕ2)+(σl2)2(2-12αl+3βl+12γl+3αl2)+σδϕ2σl2(4-4αδϕ-12αl+2βϕ,l+8γϕ,l+2αδϕαl),
αδϕ=dr1dr2W(r1)W(r2)fδϕ(r1-r2)dr1dr2W(r1)W(r2),
αl=dr1dr2W(r1)W(r2)fl(r1-r2)dr1dr2W(r1)W(r2),
βδϕ=dr1dr2W(r1)W(r2)[fδϕ(r1-r2)]2dr1dr2W(r1)W(r2),
βl=dr1dr2W(r1)W(r2)[fl(r1-r2)]2dr1dr2W(r1)W(r2),
βδϕ,l=dr1dr2W(r1)W(r2)fδϕ(r1-r2)fl(r1-r2)dr1dr2W(r1)W(r2),
γδϕ=dr1dr2dr3W(r1)W(r2)W(r3)fδϕ(r1-r2)fδϕ(r2-r3)dr1dr2dr3W(r1)W(r2)W(r3),
γl=dr1dr2dr3W(r1)W(r2)W(r3)fl(r1-r2)fl(r2-r3)dr1dr2dr3W(r1)W(r2)W(r3),
γδϕ,l=dr1dr2dr3W(r1)W(r2)W(r3)fδϕ(r1-r2)fl(r2-r3)dr1dr2dr3W(r1)W(r2)W(r3).
S2=1+σδϕ2(-2+2αδϕ)+σl2(-2+2αl)+(σδϕ2)2(2-4αδϕ+αδϕ2+βδϕ)+(σl2)2(2-4αl+αl2+βl)+σδϕ2σl2(4-4αδϕ-4αl+2αδϕαl+2βδϕ,l).
σS2=σl2(4αl)+(σδϕ2)2(2βδϕ+2αδϕ2-4γϕ)+(σl2)2(-8αl+2βl+2αl2+12γl)+σδϕ2σl2(-8αl+8γδϕ,l).
αδϕ(d/D)2,αl(ρ/D)7/3,
βδϕ(d/D)2,βl(ρ/D)7/3, βδϕ,l(d/D)2,
γδϕ(d/D)4,γl(ρ/D)14/3, γδϕ,l(d/D)2(ρ/D)7/3,
σS2σl2(ρ/D)7/3+(σδϕ2)2(d/D)2+(σl2)2(ρ/D)7/3+σδϕ2σl2(ρ/D)7/3.
σS2σl2(ρ/D)7/3ifσl2(σδϕ2)2(d/ρ)2(D/ρ)1/3(σδϕ2)2(d/D)2otherwise.
S1-σδϕ2-σl2,

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