Abstract

For non-Kolmogorov turbulence we develop a differential angle-of-arrival fluctuation coefficient, which is the ratio between the transverse and longitudinal differential angle-of-arrival variances, and a slope structure-correlation coefficient, which is the ratio between the transverse and longitudinal differences of the slope correlation function and the slope structure function, to measure the power-law exponent of a phase power spectrum with a Shack–Hartmann wave-front sensor: The differential arrival-of-angle fluctuation coefficient and the slope structure-correlation coefficient are both related to power-law exponent β and are independent of strength parameter ρ0 of the turbulence. We compare the methods developed and use them to evaluate β in recently completed horizontal atmospheric experiments for 1000-m laser beam propagation.

© 1999 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. M. S. Belen’kii, Opt. Lett. 20, 1359 (1995).
    [CrossRef] [PubMed]
  3. M. S. Belen’kii and A. S. Gurvich, Proc. SPIE 2471, 260 (1995).
    [CrossRef]
  4. G. D. Boremann and J. C. Dainty, J. Opt. Soc. Am. A 13, 517 (1996).
    [CrossRef]
  5. T. W. Nicholls, G. D. Boremann, and J. C. Dainty, Opt. Lett. 20, 2460 (1995).
    [CrossRef]
  6. E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, J. Opt. Soc. Am. A 13, 2453 (1996).
    [CrossRef]
  7. C.-H. Rao, W.-H. Jiang, and N. Ling, “Compensation effectiveness analysis for adaptive optical system used as low-order correction,” Acta Opt. Sin. (to be published).

1996 (2)

1995 (3)

Belen’kii, M. S.

M. S. Belen’kii, Opt. Lett. 20, 1359 (1995).
[CrossRef] [PubMed]

M. S. Belen’kii and A. S. Gurvich, Proc. SPIE 2471, 260 (1995).
[CrossRef]

Boremann, G. D.

Dainty, J. C.

Gurvich, A. S.

M. S. Belen’kii and A. S. Gurvich, Proc. SPIE 2471, 260 (1995).
[CrossRef]

Jiang, W.-H.

C.-H. Rao, W.-H. Jiang, and N. Ling, “Compensation effectiveness analysis for adaptive optical system used as low-order correction,” Acta Opt. Sin. (to be published).

Ling, N.

C.-H. Rao, W.-H. Jiang, and N. Ling, “Compensation effectiveness analysis for adaptive optical system used as low-order correction,” Acta Opt. Sin. (to be published).

Nicholls, T. W.

Rao, C.-H.

C.-H. Rao, W.-H. Jiang, and N. Ling, “Compensation effectiveness analysis for adaptive optical system used as low-order correction,” Acta Opt. Sin. (to be published).

Roggemann, M. C.

Silbaugh, E. E.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Welsh, B. M.

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Proc. SPIE (1)

M. S. Belen’kii and A. S. Gurvich, Proc. SPIE 2471, 260 (1995).
[CrossRef]

Other (2)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

C.-H. Rao, W.-H. Jiang, and N. Ling, “Compensation effectiveness analysis for adaptive optical system used as low-order correction,” Acta Opt. Sin. (to be published).

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Figures (3)

Fig. 1
Fig. 1

Dependence of the ratio of transverse to longitudinal differential angle-of-arrival variance on the power-law exponent β of the phase power spectrum. The results calculated from Eq. (12) are plotted as dotted curves; those from Ref. 5 are shown by dashed curves.

Fig. 2
Fig. 2

Ratio of differential variances found in experimental data (symbols) compared with theoretical curves for several values of β.

Fig. 3
Fig. 3

Slope structure-correlation coefficient as a function of the subaperture separation for several values of β. The measured results (symbols) from the experimental data and the theoretical curves from Eq. (16) are shown.

Equations (16)

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Wϕk=0.023k-11/3/r05/3,
Wϕk=Aβk-β/ρ0β-2 2<β<4,
Saˆx=1/k0drWr-xϕr·aˆ=-1/k0drWr-x·aˆϕr,
Wr=1d2rectrd=d-2rxd/2,ryd/20elsewhere.
Wr=xˆrxWrx,ry+yˆryWrx,ry=xˆd2δrx+d2-δrx-d2rectryd+yˆd2δry+d2-δry-d2rectrxd,
DSaˆx,x=Saˆx-Saˆx2,
CSaˆx,x=Saˆx+Saˆx2.
DSxˆx,x=1/k02dr1dr2Wr1-x·xˆ×Wr2-x·xˆDϕr1,r2-1/k02dr1dr2Wr1·xˆ×Wr2·xˆDϕr1,r2,
CSxˆx,x=-1/k02dr1dr2Wr1-x·xˆ×Wr2-x·xˆDϕr1,r2-1/k02dr1dr2Wr1·xˆ×Wr2·xˆDϕr1,r2.
Dϕr1,r2=ϕr1-ϕr22=γβr1-r2/ρ0β-2,
τR=σt2/σl2=DSxˆ0,R/DSxˆR,0,
τR=dutriu20,u+Rβ-2-21,u+Rβ-2-20,uβ-2-21,uβ-2dutriu2R,uβ-2-R-1,uβ-2-R+1,uβ-2-20,uβ-2-21,uβ-2,
triu=1-uu10elsewhere,
τR=σt2+2σn2σl2+2σn2.
μSxˆR=CSxˆ0,R-DSxˆ0,RCSxˆR,0-DSxˆR,0.
μSxˆR=dutriu20,u+Rβ-2-21,u+Rβ-2dutriu2R,uβ-2-R-1,uβ-2-R+1,uβ-2.

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