Abstract

We investigate the expression of non-Kolmogorov turbulence in terms of Zernike polynomials. Increasing the power-law exponent of the three-dimensional phase power spectrum from 2 to 4 results in a higher proportion of wave-front energy being contained in the tilt components. Closed-form expressions are given for the variances of the Zernike coefficients in this range. For exponents greater than 4 a von Kármán spectrum is used to compute the variances numerically as a function of exponent for different outer-scale lengths. We find in this range that the Zernike-coefficient variances depend more strongly on outer scale than on exponent and that longer outer-scale lengths lead to more energy in the tilt terms. The scaling of Zernike-coefficient variances with pupil diameter is an explicit function of the exponent.

© 1996 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), Chap. 2.
  2. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
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  3. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
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  4. F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
    [CrossRef]
  5. M. S. Belen’kii, A. S. Gurvich, “Influence of the stratospheric turbulence on infrared imaging,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 260–271 (1995).
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  6. R. G. Buser, “Interferometric determination of the distance dependence of the phase structure function for near-ground horizontal propagation at 6326 Å,” J. Opt. Soc. Am. 61, 488–491 (1971).
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  7. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
    [CrossRef]
  8. D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
    [CrossRef] [PubMed]
  9. A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Media 4, 297–306 (1994).
    [CrossRef]
  10. B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 181–196 (1995).
    [CrossRef]
  11. B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” M. S. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1994).
  12. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
    [CrossRef]
  13. T. W. Nicholls, G. D. Boreman, J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
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  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
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  15. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  16. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994), formula 6.574.2.
  17. V. V. Voitsekhovich, “Outer scale of turbulence: comparison of different models,” J. Opt. Soc. Am. A 12, 1346–1353 (1995).
    [CrossRef]

1995 (2)

1994 (2)

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Media 4, 297–306 (1994).
[CrossRef]

1992 (2)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, “Atmospheric structure function measurements with a Shack–Hartmann wave-front sensor,” Opt. Lett. 17, 1737–1739 (1992).
[CrossRef] [PubMed]

1991 (1)

1978 (1)

1976 (1)

1971 (1)

1966 (1)

1965 (1)

Belen’kii, M. S.

M. S. Belen’kii, A. S. Gurvich, “Influence of the stratospheric turbulence on infrared imaging,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 260–271 (1995).
[CrossRef]

Bester, M.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Boreman, G. D.

Buser, R. G.

Dainty, J. C.

Dalaudier, F.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

Danchi, W. C.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Dayton, D.

Degiacomi, C. G.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Fried, D. L.

Gonglewski, J.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994), formula 6.574.2.

Greenhill, L. J.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Gurvich, A. S.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

M. S. Belen’kii, A. S. Gurvich, “Influence of the stratospheric turbulence on infrared imaging,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 260–271 (1995).
[CrossRef]

Kan, V.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

Kon, A. I.

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Media 4, 297–306 (1994).
[CrossRef]

Markey, J. K.

Nicholls, T. W.

Noll, R. J.

Pierson, B.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 181–196 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994), formula 6.574.2.

Sidi, C.

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” M. S. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1994).

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 181–196 (1995).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw Hill, New York, 1961), Chap. 2.

Townes, C. H.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Voitsekhovich, V. V.

Wang, J. Y.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 181–196 (1995).
[CrossRef]

Winker, D. M.

Adv. Space Res. (1)

F. Dalaudier, A. S. Gurvich, V. Kan, C. Sidi, “Middle stratosphere temperature spectra observed with stellar scintillation and in situtechniques,” Adv. Space Res. 14, 61–64 (1994).
[CrossRef]

Astrophys. J. (1)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations—empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

J. Opt. Soc. Am. (5)

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

Opt. Lett. (2)

Waves Random Media (1)

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Media 4, 297–306 (1994).
[CrossRef]

Other (5)

B. E. Stribling, B. M. Welsh, M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 181–196 (1995).
[CrossRef]

B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” M. S. thesis (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1994).

M. S. Belen’kii, A. S. Gurvich, “Influence of the stratospheric turbulence on infrared imaging,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2471, 260–271 (1995).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw Hill, New York, 1961), Chap. 2.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994), formula 6.574.2.

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Figures (6)

Fig. 1
Fig. 1

Zernike-tilt-coefficient variances (for the case of D = r 0) 〈|a2–3|2〉 as a function of β, for 2 < β < 4.

Fig. 2
Fig. 2

Higher-order Zernike-coefficient variances (for the case of D = r 0) 〈|a4–6|2〉, 〈|a7–10|2〉 and 〈|a11|2〉, as functions of β, for 2 < β < 4.

Fig. 3
Fig. 3

Zernike-coefficient variances peak at lower values of β, as n increases. Plots are for 〈|aj|2〉 (for the case of D = r 0) corresponding to n = 20, 24, 30, and 40.

Fig. 4
Fig. 4

Zernike-tilt-coefficient variances (for the case of D = r 0) 〈|a2–3|2〉 as a function of β, for β > 4, with L0/D = 10, 100, and 1000.

Fig. 5
Fig. 5

Zernike-coefficient variances (for the case of D = r 0) 〈|a4–6|2〉 as a function of β, for β > 4, with L0/D = 10, 100, and 1000.

Fig. 6
Fig. 6

Zernike-coefficient variances (for the case of D = r 0) 〈|a7–10|2〉 as a function of β, for β > 4, with L0/D = 10, 100, and 1000.

Tables (2)

Tables Icon

Table 1 Correspondence between j, n, m, and the Lowest-Order Aberrations

Tables Icon

Table 2 Comparison of 〈|aj|2〉 for β = 11/3 and D = r 0 Calculated with Eq. (17) to Those of Ref. 12

Equations (38)

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Φ φ ( k ) = 0.023 k - 11 / 3 r 0 5 / 3 ,
Φ φ ( k ) = A β k - β r 0 β - 2 ,
D φ ( r ) 2 k ¯ - β { 1 - exp [ i 2 π ( k ¯ · r ¯ ) ] } d k ¯ .
d k ¯ 0 2 π d ϕ 0 k d k ,
Q j ( k , ϕ ) = n + 1 J n + 1 ( 2 π k ) π k × { ( - 1 ) n / 2 2 cos m ϕ , j even ( - 1 ) n / 2 2 sin m ϕ , j odd ( - 1 ) n / 2 , m = 0 ,
Δ J = φ 2 - j = 1 J a j 2 ,
φ 2 = j = 1 a j 2 ,
Δ 1 = φ 2 - a 1 2 = j = 2 a j 2 = 1 ,
a j 2 = - - Q j * ( k ) Q j ( k ) Φ φ ( k / R , k / R ) d k ¯ d k ¯ ,
Φ φ ( k / R , k / R ) = A β ( R r 0 ) β - 2 k - β δ ( k - k ) .
a j 2 = A β ( R r 0 ) β - 2 - - Q j * Q j k - β δ ( k - k ) d k ¯ d k ¯ .
a j 2 = 2 A β π ( R r 0 ) β - 2 ( n + 1 ) 0 k - ( β + 1 ) J n + 1 2 ( 2 π k ) d k ,
a j 2 = 2 A β π ( R r 0 ) β - 2 ( n + 1 ) ( 2 π ) β 0 k - ( β + 1 ) J n + 1 2 ( k ) d k .
a j 2 = 8 A β ( D r 0 ) β - 2 ( n + 1 ) π β - 1 0 k - ( β + 1 ) J n + 1 2 ( k ) d k .
0 k - ( β + 1 ) J n + 1 2 ( k ) d k = Γ ( β + 1 ) Γ ( 2 n + 2 - β 2 ) 2 β + 1 [ Γ ( β + 2 2 ) ] 2 Γ ( 2 n + 4 + β 2 )
a j 2 = 8 A β ( D r 0 ) β - 2 ( n + 1 ) π β - 1 × Γ ( β + 1 ) Γ ( 2 n + 2 - β 2 ) 2 β + 1 [ Γ ( β + 2 2 ) ] 2 Γ ( 2 n + 4 + β 2 ) .
a j 2 = ( D r 0 ) β - 2 ( n + 1 ) π × Γ ( 2 n + 2 - β 2 ) Γ ( β + 4 2 ) Γ ( β 2 ) sin ( π β - 2 2 ) Γ ( 2 n + 4 + β 2 ) .
a 2 - 3 2 a 2 2 = a 3 2 ,
a 4 - 6 2 a 4 2 = a 5 2 = a 6 2 ,
a 7 - 10 2 a 7 2 = a 8 2 = a 9 2 = a 10 2 .
a j 2 = 4 ( n + 1 ) ( D r 0 ) β - 2 0 k - ( β + 1 ) J n + 1 2 ( k ) d k 0 k - ( β - 1 ) { 1 - 4 J 1 2 ( k ) k 2 } d k .
Φ φ ( k ) = A β [ k 2 + ( R L 0 ) 2 ] - β / 2 r 0 β - 2 .
Φ φ ( k ) = A β [ k 2 + ( 2 π R L 0 ) 2 ] - β / 2 r 0 β - 2 .
Φ φ ( k ) = A β [ k 2 + ( π D L 0 ) 2 ] - β / 2 r 0 β - 2 .
a j 2 = 4 ( n + 1 ) ( D r 0 ) β - 2 × 0 [ k 2 + ( π D L 0 ) 2 ] - β / 2 k - 1 J n + 1 2 ( k ) d k 0 [ k 2 + ( π D L 0 ) 2 ] - β / 2 k ( 1 - 4 J 1 2 ( k ) k 2 ) d k .
φ 2 = 2 π 0 k Φ φ ( k / R ) d k ,
φ 2 = 2 π A β ( R r 0 ) β - 2 0 k - ( β - 1 ) d k ,
φ 2 = π A β ( D r 0 ) β - 2 2 - ( β - 3 ) 0 k - ( β - 1 ) d k .
φ 2 = π A β ( D r 0 ) β - 2 2 - ( β - 3 ) 0 ( k 2 π ) - ( β - 1 ) d k 2 π ,
φ 2 = π A β ( D r 0 ) β - 2 2 - ( β - 3 ) ( 2 π ) ( β - 2 ) 0 k - ( β - 1 ) d k ,
φ 2 = A β ( D r 0 ) β - 2 2 π β - 1 0 k - ( β - 1 ) d k .
Δ 1 = φ 2 - a 1 2 = 2 A β ( D r 0 ) β - 2 π β - 1 0 k - ( β - 1 ) d k - 8 A β ( D r 0 ) β - 2 π β - 1 0 k - ( β + 1 ) J 1 2 ( k ) d k ,
Δ 1 = π β - 1 2 A β ( D r 0 ) β - 2 { 0 k - ( β - 1 ) - 4 [ k - ( β + 1 ) J 1 2 ( k ) ] d k } ,
Δ 1 = π β - 1 2 A β ( D r 0 ) β - 2 ( 0 k - ( β - 1 ) { 1 - 4 [ J 1 2 ( k ) k 2 ] } d k ) .
A β = 1 / { 2 π β - 1 0 k - ( β - 1 ) [ 1 - 4 J 1 2 ( k ) k 2 ] d k } .
0 k - ( β - 1 ) { 1 - 4 J 1 2 ( k ) k 2 } d k = π Γ ( β + 1 ) 2 β - 1 [ Γ ( β + 2 2 ) ] 2 Γ ( β + 4 2 ) Γ ( β 2 ) sin ( π β - 2 2 ) ,
A β = 1 2 π β - 1 π Γ ( β + 1 ) 2 β - 1 [ Γ ( β + 2 2 ) ] 2 Γ ( β + 4 2 ) Γ ( β 2 ) sin ( π β - 2 2 ) ,
A β = 2 β - 2 [ Γ ( β + 2 2 ) ] 2 Γ ( β + 4 2 ) Γ ( β 2 ) sin ( π β - 2 2 ) π β Γ ( β + 1 ) .

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