Abstract

An integral expression is evaluated for the log-amplitude mean value of a gaussian or laser-like optical beam propagating along a horizontal path in a turbulent atmosphere. The mean-value integral expression has been taken from an analysis by Schmeltzer for a gaussian beam in the atmosphere. The expressions presented here for the log-amplitude mean value depend on the product of a strength factor, which is the log-amplitude variance for a point source, and a multiplying factor that depends on the source beam size. Numerical results for the beam-size-dependent factor are given for a wide range of beam sizes, wavelengths, and path lengths. Numerical results are also presented for the mean on-axis irradiance and mean beam size, for focused and collimated gaussian beams. These will find application in the design of systems for laser communication in the atmosphere. The data are obtained by combining our results for the log-amplitude mean with published data for the variance. The integral expression used here for the log-amplitude mean value is based on an assumption which restricts the range of validity of our results. The conditions necessary for its validity are discussed.

© 1969 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1961).
  2. L. A. Chernov, Wave Propagation in a Random Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1960).
  3. Reference 1, Chap. 9.
  4. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [CrossRef]
  5. Y. Kinoshita, T. Asakura, and M. Suzuki, J. Opt. Soc. Am. 58, 798 (1968).
    [CrossRef]
  6. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).
  7. D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [CrossRef]
  8. D. L. Fried, J. Opt. Soc. Am. 57, 980 (1967).
    [CrossRef]
  9. A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander and L. Topper, Eds. (Wiley–Interscience, Inc., New York, 1961), p. 151.
  10. As implied by our sign convention for R in Eq. (2.1), we are assuming e−iωt time dependence. It should be noted that this sign convention for R differs from that used by Schmeltzer [Ref. 6, Eq. (2.1)]. Also, it should be noted that the time convention in Ref. 6, p. 340 is stated to be eiωt but the mathematical formulation actually follows e−iωt time dependence, as can be readily determined from Eqs. (2.6) and (2.12) of the same reference.
  11. Equation (2.2) is obtained from Schmeltzer’s Eq. (7.6) by taking the real part of the right-hand side and setting s= z0 and q= s in the second term.
  12. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966), Eqs. (2.16)–(2.22).
    [CrossRef]
  13. The difference in sign of Im(1/α2) between our Eq. (2.6) and Schmeltzer’s Eq. (2.9) [also Eq. (2.6) in Ref. 7] is a result of the sign convention we are using for R.
  14. I. S. Gradshteyn and I. M. Ryshik, Table of Integrals, Series, and Products (Academic Press Inc., New York, 1965), Eqs. 3.381 (4), (5), p. 317.
  15. Reference 14, Eqs. 3.191(3), 3.194(1), p. 284.
  16. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Gov’t. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), Ch. 15.
  18. Reference 17, Eq. 15.1.1, p. 556.
  19. Equation (4.11) is obtained from Eq. (4.11) in Ref. 7 by using the relationships Γ(−5/6) = −(6/5)Γ(1/6) and Re(i5/6) = cos(5π/12).
  20. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966), Eq. (2.7), and Ref. 4, Eq. (1.13), or Ref. 7, Eq. (4.14).
    [CrossRef]
  21. Reference 1, p. 208.
  22. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967); J. Opt. Soc. Am. 58, 1164E (1968).
    [CrossRef]
  23. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967), Eq. (1.5).
    [CrossRef]
  24. Reference 17, Eq. 15.3.7, p. 559.
  25. The failure of this assumption to hold [see Ref. 6, Eq. (5.10)] in the spherical-wave limit was first pointed out to the authors by D. L. Fried, private communication (1967).
  26. M. E. Gracheva and A. S. Gurvich, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 8, 717 (1965).
  27. M. E. Gracheva, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 10, 775 (1967).
  28. D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968).
    [CrossRef]
  29. A gaussian mean beam shape is assumed here for the atmospherically perturbed laser beam because of the lack of available data for the log-amplitude statistics at off-axis points in the beam. For a randomly wandering gaussian-irradiance distribution, the average beam pattern can be shown to be gaussian when the orthogonal displacements Δx, Δy of the beam center are independent normal-random variables, with zero means and equal variances. Under more general conditions, however, (e.g., when the perturbed gaussian-beam pattern is also distorted) the average beam shape may not be gaussian.
  30. Reference 1, p. 140.

1968 (2)

1967 (6)

1966 (3)

1965 (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 8, 717 (1965).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Gov’t. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), Ch. 15.

Asakura, T.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1960).

Cloud, J. D.

deWolf, D. A.

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.

Fried, D. L.

Gracheva, M. E.

M. E. Gracheva, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 10, 775 (1967).

M. E. Gracheva and A. S. Gurvich, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 8, 717 (1965).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryshik, Table of Integrals, Series, and Products (Academic Press Inc., New York, 1965), Eqs. 3.381 (4), (5), p. 317.

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 8, 717 (1965).

Keister, M. P.

Kinoshita, Y.

Kolmogorov, A.

A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander and L. Topper, Eds. (Wiley–Interscience, Inc., New York, 1961), p. 151.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.

Mevers, G. E.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.

Ryshik, I. M.

I. S. Gradshteyn and I. M. Ryshik, Table of Integrals, Series, and Products (Academic Press Inc., New York, 1965), Eqs. 3.381 (4), (5), p. 317.

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).

Seidman, J. B.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Gov’t. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), Ch. 15.

Suzuki, M.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1961).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.

Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. (2)

M. E. Gracheva and A. S. Gurvich, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 8, 717 (1965).

M. E. Gracheva, Izv. Vysshikh Uchebn. Zavedenii-Radiofiz. 10, 775 (1967).

J. Opt. Soc. Am. (9)

Quart. Appl. Math. (1)

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).

Other (18)

V. I. Tatarski, Wave Propagation in a Turbulent Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1961).

L. A. Chernov, Wave Propagation in a Random Medium, transl. R. A. Silverman (McGraw–Hill Book Co., New York, 1960).

Reference 1, Chap. 9.

A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander and L. Topper, Eds. (Wiley–Interscience, Inc., New York, 1961), p. 151.

As implied by our sign convention for R in Eq. (2.1), we are assuming e−iωt time dependence. It should be noted that this sign convention for R differs from that used by Schmeltzer [Ref. 6, Eq. (2.1)]. Also, it should be noted that the time convention in Ref. 6, p. 340 is stated to be eiωt but the mathematical formulation actually follows e−iωt time dependence, as can be readily determined from Eqs. (2.6) and (2.12) of the same reference.

Equation (2.2) is obtained from Schmeltzer’s Eq. (7.6) by taking the real part of the right-hand side and setting s= z0 and q= s in the second term.

The difference in sign of Im(1/α2) between our Eq. (2.6) and Schmeltzer’s Eq. (2.9) [also Eq. (2.6) in Ref. 7] is a result of the sign convention we are using for R.

I. S. Gradshteyn and I. M. Ryshik, Table of Integrals, Series, and Products (Academic Press Inc., New York, 1965), Eqs. 3.381 (4), (5), p. 317.

Reference 14, Eqs. 3.191(3), 3.194(1), p. 284.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw–Hill Book Co., New York, 1953), Vol. I, Chapter 2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Gov’t. Printing Office, Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), Ch. 15.

Reference 17, Eq. 15.1.1, p. 556.

Equation (4.11) is obtained from Eq. (4.11) in Ref. 7 by using the relationships Γ(−5/6) = −(6/5)Γ(1/6) and Re(i5/6) = cos(5π/12).

Reference 1, p. 208.

Reference 17, Eq. 15.3.7, p. 559.

The failure of this assumption to hold [see Ref. 6, Eq. (5.10)] in the spherical-wave limit was first pointed out to the authors by D. L. Fried, private communication (1967).

A gaussian mean beam shape is assumed here for the atmospherically perturbed laser beam because of the lack of available data for the log-amplitude statistics at off-axis points in the beam. For a randomly wandering gaussian-irradiance distribution, the average beam pattern can be shown to be gaussian when the orthogonal displacements Δx, Δy of the beam center are independent normal-random variables, with zero means and equal variances. Under more general conditions, however, (e.g., when the perturbed gaussian-beam pattern is also distorted) the average beam shape may not be gaussian.

Reference 1, p. 140.

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Figures (5)

Fig. 1
Fig. 1

The dependence of the normalized log-amplitude mean 〈l〉/Cls(0) on the normalized source-beam size Ω (= 02/z). Collimated beam, solid curve; focused beam, dashed.

Fig. 2
Fig. 2

The collimated-beam mean forward antenna gain vs normalized source-beam size Ω (= 02/z) for various values of Cls(0), the spherical-wave variance. For no turbulence Cls(0) = 0. The values of α0 corresponding to the given Ω values are shown for ranges of z = 2 and 10 km; wavelength λ = 0.63 μ.

Fig. 3
Fig. 3

The focused-beam mean forward antenna gain vs normalized source-beam size Ω (= 02/z) for various values of spherical-wave variance Cls(0). For no turbulence Cls(0) = 0. The values of α0 corresponding to the given Ω values are shown for ranges z = 2 and 10 km; wave length λ = 0.63 μ.

Fig. 4
Fig. 4

Graph of the collimated-beam mean relative spot radius αt/α0 vs normalized source-beam size Ω (= 02/z) for various values of spherical-wave variance Cls(0). For no turbulence Cls(0) = 0. Values of α0 corresponding to the Ω values are shown for two ranges, z = 2 and 10 km; wavelength λ = 0.63 μ.

Fig. 5
Fig. 5

Graph of the relative focused-beam mean spot radius αt/α0 vs normalized beam size Ω (= 02/z) for various values of spherical-wave variance Cls(0). For no turbulence Cls(0) = 0. Values of α0 corresponding to the Ω values are shown for two ranges z = 2 and 10 km; wavelength λ = 0.63 μ.

Tables (2)

Tables Icon

Table I Collimated-beam propagation.

Tables Icon

Table II Focused-beam propagation.

Equations (65)

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u 0 ( r ) z = 0 = exp [ - ρ 2 / 2 α 0 2 - i k ρ 2 / 2 R ] ,
l = Re { - i k 8 π 0 z d z 0 0 z 0 d s 0 d σ σ Φ n ( σ 1 2 ; s ) × exp [ γ ( z 0 , s ) σ 2 ] } .
Φ n ( σ ; s ) = d r exp ( i σ · r ) C n ( r ; s ) ,
Φ n ( σ ; s ) = 8.16 C N 2 σ - 11 / 3 ,
γ ( z 0 , s ) = [ 2 ( z 0 - s ) / i k ] [ ( s - i k α 2 ) / ( z 0 - i k α 2 ) ] ,
1 / α 2 = 1 / α 0 2 + i k / R .
l = ( 8.16 / 8 π ) k C N 2 Re { - i 0 z d z 0 0 z 0 d s 0 d σ σ - 5 / 6 × exp γ ( z 0 , s ) σ 2 } .
0 x ν - 1 e - μ x d x = Γ ( ν ) μ - ν ,
ν = 1 - 5 / 6 = 1 / 6 ,
μ = - γ ( z 0 , s ) / 2 ,
Re [ μ ] = [ ( z 0 - s ) 2 / α 0 2 ] / [ ( z 0 / α 0 2 ) 2 + k 2 ( 1 - z 0 / R ) 2 ] 0 ,
l = ( 8.16 / 8 π ) k C N 2 Γ ( 1 / 6 ) × Re { - i 0 z d z 0 0 z 0 d s [ - γ ( z 0 , s ) 2 ] - 1 / 6 } .
- γ ( z 0 , s ) 2 = ( s - z 0 ) i k [ s - i k α 2 z 0 - i k α 2 ] ,
x = ( s - i k α 2 ) / ( z 0 - i k α 2 ) .
s = i k α 2 + x ( z 0 - i k α 2 ) ,
d s = ( z 0 - i k α 2 ) d x ,
- γ ( z 0 , s ) / 2 = - [ ( z 0 - i k α 2 ) / i k ] x ( 1 - x ) .
l = ( 8.16 / 8 π ) k 7 / 6 C N 2 Γ ( 1 / 6 ) × Re { ( - i ) 7 / 6 0 z d z 0 ( z 0 - i k α 2 ) 5 / 6 × U 0 1 x - 1 / 6 ( 1 - x ) - 1 / 6 d x } ,
U 0 = - i k α 2 / ( z 0 - i k α 2 ) .
U 0 1 x p - 1 ( 1 - x ) q - 1 d x = B ( p , q ) - ( U 0 p p ) F 2 1 ( p , 1 - q ; 1 + p ; U 0 ) ,
B ( p , q ) = Γ ( p ) Γ ( q ) / Γ ( p + q )
l = ( 8.16 / 8 π ) k 7 / 6 C N 2 Γ ( 1 / 6 ) × Re { ( - i ) 7 / 6 [ B ( 5 / 6 , 5 / 6 ) 0 z d z 0 ( z 0 - i k α 2 ) 5 / 6 - ( 6 5 ) ( - i k α 2 ) 5 / 6 0 z d z 02 F 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; U 0 ) ] } .
y = z 0 - i k α 2 ,
0 z d z 0 ( z 0 - i k α 2 ) 5 / 6 = - i k α 2 z - i k α 2 y 5 / 6 d y = ( 6 11 ) ( - i k α 2 ) 11 / 6 [ U - 11 / 6 - 1 ] ,
U = [ - i k α 2 / ( z - i k α 2 ) ] .
d x = ( x 2 / i k α 2 ) d z 0 .
0 z F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; U 0 ) d z 0 = ( i k α 2 ) 1 U x - 2 F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; x ) d x .
l = ( 8.16 / 8 π ) k 7 / 6 C N 2 Γ ( 1 / 6 ) Re ( ( - i ) 7 / 6 ( - i k α 2 ) 11 / 6 × { B ( 5 / 6 , 5 / 6 ) ( 6 / 11 ) [ U - 11 / 6 - 1 ] + ( 6 5 ) 1 U x - 2 F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; x ) d x } ) .
lim R k α 2 = k α 0 2 ,
U c = lim R U = Ω [ 1 + Ω 2 ] 1 2 exp [ i tan - 1 ( - 1 Ω ) ] ,
Ω = k α 0 2 / z ,
F 2 1 ( a , b ; c ; z ) = n = 0 ( a ) n ( b ) n ( c ) n n ! z n ,
( a ) n = Γ ( a + n ) / Γ ( a ) .
1 U c x - 2 F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; x ) d x = n = 0 ( 5 / 6 ) n ( 1 / 6 ) n ( 11 / 6 ) n n ! 1 U c x n - 2 d x = ( 5 / 6 ) ( 1 / 6 ) ( 11 / 6 ) ln U c + n = 0 , n 1 a n [ U c n - 1 - 1 ] ,
a n = ( 5 / 6 ) n ( 1 / 6 ) n ( 11 / 6 ) n n ! ( n - 1 ) .
Re ( c - a - b ) = 5 / 6 > 0.
U c 1
l = ( 8.16 / 8 π ) k 7 / 6 z 11 / 6 C N 2 Γ ( 1 / 6 ) Ω 11 / 6 × Re ( i { B ( 5 / 6 , 5 / 6 ) ( 6 / 11 ) [ U c - 11 / 6 - 1 ] + 1 11 ln U c + 6 5 · n = 0 , n 1 a n [ U c n - 1 - 1 ] } ) .
C l s ( 0 ) = ( 8.16 / 8 π ) C N 2 k 7 / 6 z 11 / 6 Γ ( 1 / 6 ) × B ( 11 / 6 , 11 / 6 ) cos ( 5 π / 12 ) ( 5 / 6 ) ,
l C l s ( 0 ) = - [ ( 5 / 6 ) / B ( 11 / 6 , 11 / 6 ) cos ( 5 π / 12 ) ] Ω 11 / 6 × { ( 6 / 11 ) B ( 5 / 6 , 5 / 6 ) Y - 11 / 6 sin ( 11 6 θ c ) - ( 6 / 5 ) Y - 1 sin θ c - ( 1 / 11 ) θ c - ( 6 / 5 ) × n = 2 a n Y n - 1 sin ( n - 1 ) θ c } ,
Y = U c = Ω / [ 1 + Ω 2 ] 1 2 ,
θ c = - arg ( U c ) = tan - 1 ( 1 / Ω ) .
lim α 0 γ ( z 0 , s ) = 2 ( z 0 - s ) / i k .
lim α 0 l / C l s ( 0 ) = - ( 6 / 11 ) B ( 11 / 6 , 11 / 6 ) = - 2.47332.
lim R z k α 2 = k α 0 2 / ( 1 + i Ω ) ,
U f = lim R z U = - i Ω .
l = ( 8.16 / 8 π ) C N 2 k 7 / 6 z 11 / 6 Γ ( 1 / 6 ) × Re ( i [ Ω 1 + i Ω ] 11 / 6 { B ( 5 / 6 , 5 / 6 ) ( 6 11 ) [ U f - 11 / 6 - 1 ] + 1 11 ln U f + 6 5 n = 0 ,             n 1 a n [ U f n - 1 - 1 ] } ) .
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( b - a ) Γ ( b ) Γ ( c - a ) ( - z ) - a × F 2 1 ( a , 1 - c + a ; 1 - b + a ; z - 1 ) + Γ ( c ) Γ ( a - b ) Γ ( a ) Γ ( c - b ) ( - z ) - b × F 2 1 ( b , 1 - c + b ; 1 - a + b ; z - 1 )
F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; x ) = A 1 ( - x ) - 5 / 6 F 2 1 ( 5 / 6 , 0 ; 5 / 3 ; x - 1 ) + A 2 ( - x ) - 1 / 6 F 2 1 ( 1 / 6 , - 2 / 3 ; 1 / 3 ; x - 1 ) ,
A 1 = Γ ( 11 / 6 ) Γ ( - 2 / 3 ) Γ ( 1 / 6 ) Γ ( 1 ) ,
A 2 = Γ ( 11 / 6 ) Γ ( 2 / 3 ) Γ ( 5 / 6 ) Γ ( 5 / 3 ) .
1 U f x - 2 F 2 1 ( 5 / 6 , 1 / 6 ; 11 / 6 ; x ) d x = - 6 11 A 1 ( - 1 ) - 5 / 6 [ U f - 11 / 6 - 1 ] - A 2 ( - 1 ) - 1 / 6 n = 0 b n [ U f - ( n + 7 / 6 ) - 1 ] ,
b n = ( 1 / 6 ) n ( - 2 / 3 ) n ( 1 / 3 ) n n ! ( n + 7 / 6 ) .
l = ( 8.16 / 8 π ) C N 2 k 7 / 6 z 11 / 6 Γ ( 1 / 6 ) × Re ( i ( Ω / 1 + i Ω ) 11 / 6 { B ( 5 / 6 , 5 / 6 ) ( 6 / 11 ) [ U f - 11 / 6 - 1 ] - ( 6 / 11 ) ( 6 / 5 ) A 1 ( - 1 ) - 5 / 6 [ U f - 11 / 6 - 1 ] - ( 6 / 5 ) A 2 ( - 1 ) - 1 / 6 n = 0 b n [ U f - ( n + 7 / 6 ) - 1 ] } ) ,
l C l s ( 0 ) = - ( 5 / 6 ) B ( 11 / 6 , 11 / 6 ) cos ( 5 π / 12 ) Y 11 / 6 { ( 6 11 ) B ( 5 / 6 , 5 / 6 ) [ Ω - 11 / 6 cos ( 5 π 12 - 11 6 ϕ f ) + sin ( 11 6 ϕ f ) ] - 6 5 [ sin ( 11 6 ϕ f ) + Ω - 1 cos ( 11 6 ϕ f ) ] - 1 11 [ ln Ω sin ( 11 6 ϕ f ) + π 2 cos ( 11 6 ϕ f ) ] + 6 5 n = 2 a n [ Ω n - 1 cos ( 11 6 ϕ f + n π 2 ) + sin ( 11 6 ϕ f ) ] } ,             0 Ω 1 ,
l C l s ( 0 ) = - ( 5 / 6 ) B ( 11 / 6 , 11 / 6 ) cos ( 5 π / 12 ) Y 11 / 6 { ( 6 11 ) B ( 5 / 6 , 5 / 6 ) [ Ω - 11 / 6 cos ( 5 π 12 - 11 6 ϕ f ) + sin ( 11 6 ϕ f ) ] - ( 6 11 ) ( 6 5 ) A 1 [ Ω - 11 / 6 sin ( π 12 - 11 6 ϕ f ) + cos ( π 3 + 11 6 ϕ f ) ] - 6 5 A 2 n = 0 b n [ Ω - ( n + 7 / 6 ) sin ( 5 π 12 - 11 6 ϕ f + n π 2 ) + cos ( π 3 - 11 6 ϕ f ) ] } ,             Ω 1 ,
ϕ f = tan - 1 Ω .
I = I 0 e 2 l ,
I = I 0 e 2 l = I 0 exp { 2 [ l + C l ( 0 ) ] } ,
G = 10 log 10 ( I I 0 ) ( dB ) .
G = ( 8.686 ) C l s ( 0 ) [ l / C l s ( 0 ) + C l ( 0 ) / C l s ( 0 ) ] ( dB ) .
α t / α 0 = { ( 1 + Ω 2 ) 1 2 Ω exp { - [ l + C l ( 0 ) ] } ; collimated beam , ( 1 Ω )             exp { - [ l + C l ( 0 ) ] } ; focused beam .
| μ ( r 0 ) μ ( r 0 ) | ψ 0 ( r ) - ψ 0 ( r 0 )
α z / l eq 1 ,
α 0 2 l eq 2 = k α 0 2 z = Ω 1.