Abstract

The power loss due to propagation through a turbulent medium is considered for an optical-heterodyne detection system whose axis tracks perfectly the instantaneous direction for maximum signal power. Pertinent to both fixed- and tracking-axis cases, the general expression for power reduction is found to be given correctly by neglecting the angular diffraction spread of the local-oscillator field and using the signal field evaluated on the optical axis. A plane and broad incident wave is assumed. The result for the tracking aperture is indicated to be given correctly by ray optics if La2/λ, where L is the path length in the turbulent medium, a the aperture radius, and λ the signal wavelength, whereas for a fixed aperture the lateral homogeneity of the field is indicated to suffice without this condition. Refractive-index fluctuations are assumed to be described statistically by the usual Kolmogorov spectrum. For moderate aperture (a ae), the power reduction factor Γ is found to be given by

Γ1-s(a/ae)5/3

with s = 0.125 for a tracking axis and s = 0.955 for a fixed axis, where a. is ae certain effective radius for the fixed case. If the improvement due to tracking is extrapolated to arbitrary a/ae by conjecture of a fixed factor of increase in the effective radius, the factor of increase in maximum signal-to-noise ratio achievable by tracking is 11.5. To approach the maximum, the frequency response of the tracking system should extend beyond roughly 50 cps.

© 1966 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Company, Inc., New York, 1961).
  2. S. Gardner, 1964 IEEE International Convention Record,  Part 6, p. 337.
  3. J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959).
  4. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965). This work was called to my attention following completion of that reported here.
    [CrossRef]
  5. The degree of approach to perfect tracking as a function of frequency response of the control system is remarked in Sec. 7.
  6. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [CrossRef]
  7. The basic equations were given also in Ref. 2.
  8. E.g., see L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1951).
  9. This model becomes more realistic, but pertinent only for aperture sizes a≲l0, when le refers to the minimum eddy size (microscale l0) and δϕ0 is replaced by (3.44)12kCnL12l05/6, in which Cn is the coefficient referred to at Eq. (37) below [see Ref. 1, Eq. (6.64)].
  10. Some examination of the validity of this supposition could presumably be effected on the basis of the approximation for Φ1−Φ2 finally employed below [see Eq. (31)].
  11. The similar parameter r0 of Ref. 4 is related to ae by r0 = 2.1ae.
  12. More generally, Cn2 may not be constant in space, and Cn2L must then be replaced by the integral along the propagation path, ∫0LdsCn2(s).
  13. The stated rough upper limit on a for validity of (46) is established by requiring 〈(ϕ1−ϕ2)2〉 ≲ 1 for |ρ1−ρ2| = 2a, and the limit for (47) by increasing the former limit in accord with the reduction of the area average of 〈(Φ1−Φ2)2〉 relative to 〈(ϕ1−ϕ2)2〉. The error associated with (47) is smaller than that with (46) for equal a but roughly the same for equal power reduction.
  14. The sagittal condition1L≪l04/λ3 and the condition λ≪l0 are still imposed.
  15. Dr. R. E. Hufnagel has noted this statistical validity of ray optics in the perturbation and Rytov approximations on the basis of Tatarski’s work1 and pointed out the likelihood of more general validity (unpublished). General validity is not correctly proved in Ref. 6.16
  16. D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).
    [CrossRef]
  17. For the fixed-axis case, the asymptotic signal-to-noise ratio is equal to the ratio in the absence of propagation disturbance when the aperture radius in the latter instance is equal to (1.084)12ae. The effective radius so defined is thus nearly equal to ae, justifying the appellation for ae.
  18. The value 3.4 agrees with that obtained for the mathematically identical quantity in Ref. 4. In Ref. 4, as here, no computation is made that would yield the power-reduction factors outside the moderate-aperture approximation of Eq. (46) and (47).
  19. In the absence of propagation effects, for example,L(k1,k2)=(2π)2a2(k1k2)-1J1(k1a)J1(k2a).
  20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 1324.
  21. Bateman Manuscript Project, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. 2, p. 24, No. (7).
  22. Reference 21, Vol. 1, p. 331, No. (33).
  23. To obtain Eq. (A34), multiply Eq. (3) of Ref. 16 by exp[ik× (ζ1−ζ2)], take expectations, and perform the double integration over the aperture. The left member, by the second equation following (A20), is identified as the left member of Eq. (A34), and the right, since Φ = ϕ+kζ, as the right.

1965 (2)

1964 (1)

Chase, D. M.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 1324.

Fried, D. L.

Gardner, S.

S. Gardner, 1964 IEEE International Convention Record,  Part 6, p. 337.

Hinze, J. O.

J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959).

Hufnagel, R. E.

Landau, L.

E.g., see L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1951).

Lifshitz, E.

E.g., see L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1951).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 1324.

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Company, Inc., New York, 1961).

1964 IEEE International Convention Record (1)

S. Gardner, 1964 IEEE International Convention Record,  Part 6, p. 337.

J. Opt. Soc. Am. (3)

Other (19)

J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959).

For the fixed-axis case, the asymptotic signal-to-noise ratio is equal to the ratio in the absence of propagation disturbance when the aperture radius in the latter instance is equal to (1.084)12ae. The effective radius so defined is thus nearly equal to ae, justifying the appellation for ae.

The value 3.4 agrees with that obtained for the mathematically identical quantity in Ref. 4. In Ref. 4, as here, no computation is made that would yield the power-reduction factors outside the moderate-aperture approximation of Eq. (46) and (47).

In the absence of propagation effects, for example,L(k1,k2)=(2π)2a2(k1k2)-1J1(k1a)J1(k2a).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 1324.

Bateman Manuscript Project, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. 2, p. 24, No. (7).

Reference 21, Vol. 1, p. 331, No. (33).

To obtain Eq. (A34), multiply Eq. (3) of Ref. 16 by exp[ik× (ζ1−ζ2)], take expectations, and perform the double integration over the aperture. The left member, by the second equation following (A20), is identified as the left member of Eq. (A34), and the right, since Φ = ϕ+kζ, as the right.

The degree of approach to perfect tracking as a function of frequency response of the control system is remarked in Sec. 7.

The basic equations were given also in Ref. 2.

E.g., see L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1951).

This model becomes more realistic, but pertinent only for aperture sizes a≲l0, when le refers to the minimum eddy size (microscale l0) and δϕ0 is replaced by (3.44)12kCnL12l05/6, in which Cn is the coefficient referred to at Eq. (37) below [see Ref. 1, Eq. (6.64)].

Some examination of the validity of this supposition could presumably be effected on the basis of the approximation for Φ1−Φ2 finally employed below [see Eq. (31)].

The similar parameter r0 of Ref. 4 is related to ae by r0 = 2.1ae.

More generally, Cn2 may not be constant in space, and Cn2L must then be replaced by the integral along the propagation path, ∫0LdsCn2(s).

The stated rough upper limit on a for validity of (46) is established by requiring 〈(ϕ1−ϕ2)2〉 ≲ 1 for |ρ1−ρ2| = 2a, and the limit for (47) by increasing the former limit in accord with the reduction of the area average of 〈(Φ1−Φ2)2〉 relative to 〈(ϕ1−ϕ2)2〉. The error associated with (47) is smaller than that with (46) for equal a but roughly the same for equal power reduction.

The sagittal condition1L≪l04/λ3 and the condition λ≪l0 are still imposed.

Dr. R. E. Hufnagel has noted this statistical validity of ray optics in the perturbation and Rytov approximations on the basis of Tatarski’s work1 and pointed out the likelihood of more general validity (unpublished). General validity is not correctly proved in Ref. 6.16

V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Company, Inc., New York, 1961).

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Figures (1)

Fig. 1
Fig. 1

Geometry of perturbed wavefront and aperture planes. Without tracking, the aperture plane is fixed parallel to the mean wavefront. The actual wavefront at position ϱ in the aperture is advanced by a distance −ϕ(ϱ)/k, where ϕ is the phase increment due to index fluctuations. With perfect tracking, the aperture plane for each realization of the random medium is that which minimizes, for moderate aperture, the integrated phase dispersion ∫ d2ϱ1d2ϱ21−Φ2)2 over the aperture, where Φ is the phase increment relative to this rotated aperture. The wavefront is advanced with respect to the rotated aperture by a distance −Φ(ϱ)/k. Here the (small) angle α is analogous to the rotation vector (α1,α2) in three dimensions.

Equations (111)

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Γ 1 - s ( a / a e ) 5 / 3
P = c 0 i s 2 ,
i b D d 2 r 0 [ Re E ˜ ( r 0 ) ] 2 ,
E ˜ ( r 0 ) = E ˜ s ( r 0 ) + E ˜ l ( r 0 ) .
2 b D d 2 r 0 ( Re E ˜ s ) ( Re E ˜ l )
E ˜ s ( r 0 ) = A d 2 ϱ f s ( ϱ , r 0 ) E s ( ϱ ) , E ˜ l ( r 0 ) = A d 2 ϱ f l ( ϱ , r 0 ) E l ( ϱ )
f s ( ϱ , r 0 ) = B exp ( i k s r 0 - i k s · ϱ ) , f l ( ϱ , r 0 ) = B exp ( i k l r 0 - i k l · ϱ )
E l ( ϱ ) = E 0 .
E ˜ l ( θ ) E ˜ l ( r 0 ) = 2 π B E 0 a e i k l R sec θ J 1 ( k l a sin θ ) / k l sin θ ,
i s 2 N 0 π / 2 d θ 1 0 π / 2 d θ 2 J 1 ( k 1 a sin θ 1 ) J 1 ( k l a sin θ 2 ) × cos [ Δ k R ( sec θ 1 - sec θ 2 ) ] × A d 2 ϱ 1 A d 2 ϱ 2 J 0 ( k s ρ 1 sin θ 1 ) × J 0 ( k s ρ 2 sin θ 2 ) E s ( ϱ 1 ) E s * ( ϱ 2 )
N = ( 1 / 2 ) ( 2 π ) 4 b 2 B 4 E 0 2 k l - 2 R 4 a 2 ,             Δ k = k s - k l .
1 - cos [ Δ k R ( sec θ 1 - sec θ 2 ) ] ~ ( 1 / 2 ) ( Δ k R ) 2 ( θ 2 / 2 ) 2 [ ( 2 π ) 4 / 8 ] ( Δ k R ) 2 / ( k l a ) 4
E s ( ϱ 1 ) E s * ( ϱ 2 ) M ( ϱ 1 , ϱ 2 ) = M ( ρ ) ,
ρ = ϱ ,             ϱ = ϱ 1 - ϱ 2 .
i s 2 N 0 π / 2 d θ 1 0 π / 2 d θ 2 J 1 ( k s a sin θ 1 ) × J 1 ( k s a sin 2 ) L ( k s sin θ 1 , k s sin θ 2 ) ,
L ( k 1 , k 2 ) A d 2 ϱ 1 A d 2 ϱ 2 J 0 ( k 1 ρ 1 ) J 0 ( k 2 ρ 2 ) M ( ρ ) .
M ( ρ ) = ( 2 π ) - 1 d 2 K e i K · ρ m ( K ) = 0 d K K J 0 ( K ρ ) m ( K ) , m ( K ) = ( 2 π ) - 1 d 2 ϱ e - K · ρ M ( ρ ) = 0 d ρ ρ J 0 ( K ρ ) M ( ρ ) .
i s 2 ( 2 π ) 2 N k s - 2 0 d z z - 1 J 1 2 ( z ) m ( z / a ) N ( k s a ) - 2 A d 2 ϱ 1 A d 2 ϱ 2 M ( ϱ 1 - ϱ 2 ) . = N ( k s a ) 2 A d 2 ϱ 1 E s ( ϱ 1 ) A d 2 ϱ 2 E s * ( ϱ 2 ) .
i s 2 0 = π 2 N k s - 2 a 2 ,
γ i s 2 / i s 2 0 = 4 a - 2 0 d z z - 1 J 1 2 ( z ) m ( z / a ) = ( π a 2 ) - 2 A d 2 ϱ 1 A d 2 ϱ 2 M ( ϱ 1 - ϱ 2 ) .
M ( ϱ 1 - ϱ 2 ) = exp [ i { ϕ a ( ϱ 1 ) - ϕ a ( ϱ 2 ) } ] , ϕ a ( ϱ ) = k 0 L d s N ( ϱ , s ) + k ζ ( ϱ ) ;
δ ϕ ~ ( ρ / l e ) δ ϕ 0 .
δ Φ ~ ( ρ / l e ) 2 δ ϕ 0 .
Γ = ( π a 2 ) - 2 A d 2 ϱ 1 A d 2 ϱ 2 exp [ i ( Φ 1 - Φ 2 ) ] ,
Φ ( ϱ ) = ϕ ( ϱ ) + k ζ ( ϱ , α ) ,
ζ ( ϱ , α ) = x α 2 - y α 1 .
g / α 1 = 0 ,             g / α 2 = 0 ,
g ( α 1 , α 2 ) ( π a 2 ) - 2 A d 2 ϱ 1 A d 2 ϱ 2 exp [ i ( Φ 1 - Φ 2 ) ] = ( π a 2 ) - 2 A d 2 ϱ 1 A d 2 ρ 2 cos ( Φ 1 - Φ 2 )
Γ = g ( α 1 , α 2 ) .
d 2 ϱ 1 d 2 ϱ 2 ( y 1 - y 2 ) sin ( Φ 1 - Φ 2 ) = 0 , d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) sin ( Φ 1 - Φ 2 ) = 0 ,
Φ 1 - Φ 2 = ϕ 1 - ϕ 2 + k ( x 1 - x 2 ) α 2 - k ( y 1 - y 2 ) α 1
exp [ i ( Φ 1 - Φ 2 ) ] = exp ( - 1 2 ( Φ 1 - Φ 2 ) 2 ) .
Γ = ( π a 2 ) - 2 d 2 ϱ 1 d 2 ϱ 2 exp [ - 1 2 ( Φ 1 - Φ 2 ) 2 ] .
g ( α 1 , α 2 ) ( π a 2 ) - 2 d 2 ϱ 1 d 2 ϱ 2 [ 1 - 1 2 ( Φ 1 - Φ 2 ) 2 ] .
Γ ( π a 2 ) - 2 d 2 ϱ 1 d 2 ϱ 2 [ 1 - 1 2 ( Φ 1 - Φ 2 ) 2 ] ,
d 2 ϱ 1 = π a 2 ,             d 2 ϱ 1 x 1 = 0 , d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) ( y 1 - y 2 ) = 0 , d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) 2 = ( π 2 / 2 ) a 6 ,
( Φ 1 - Φ 2 ) 2 = { ϕ 1 - ϕ 2 + k α 2 ( x 1 - x 2 ) - k α 1 ( y 1 - y 2 ) } 2 , k α 1 ( 2 / π 2 ) a - 6 d 2 ϱ 1 d 2 ϱ 2 ( y 1 - y 2 ) ( ϕ 1 - ϕ 2 ) , k α 2 - ( 2 / π 2 ) a - 6 d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) ( ϕ 1 - ϕ 2 ) .
( Φ 1 - Φ 2 ) 2 = ( ϕ 1 - ϕ 2 ) 2 + [ ( π 2 / 2 ) a 6 ] - 2 ϱ 1 - ϱ 2 2 × ( 1 ) ( 2 ) ( 1 ) ( 2 ) ( x 1 - x 2 ) ( x 1 - x 2 ) × ( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) - 2 [ ( π 2 / 2 ) a 6 ] - 1 ( ϱ 1 - ϱ 2 ) · ( 1 ) ( 2 ) ( ϱ 1 - ϱ 2 ) ( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) ,
( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) = - 1 2 [ D ( 1 , 1 ) + D ( 2 , 2 ) - D ( 1 , 2 ) - D ( 2 , 1 ) ] ,
D ( 1 , 2 ) ( ϕ 1 - ϕ 2 ) 2 .
l 0 ρ L 0 ,
D ( ρ ) = 2 ( ρ / a e ) 5 / 3 .
a e = [ 1.45 ( 2 π ) 2 λ - 2 ( C n 2 L ) ] - 3 / 5
D ( ρ ) = 2 ( 2 π ) - 1 d 2 K [ 1 - exp ( i K · ϱ ) ] Δ ( K ) ,
Δ ( K ) = C a e - 5 / 3 K - 11 / 3             ( L 0 - 1 K l 0 - 1 )
C = - 2 8 / 3 Γ ( 11 / 6 ) / Γ ( - 5 / 6 ) 0.897
( Φ 1 - Φ 2 ) 2 = 2 a e - 5 / 3 ϱ 1 - ϱ 2 5 / 3 + 16 C a e - 5 / 3 a - 1 / 3 × { 2 E ϱ 1 - ϱ 2 2 - a ( ϱ 1 - ϱ 2 ) · [ ( ϱ 1 / ρ 1 ) × F ( ρ 1 / a ) - ( ϱ 2 / ρ 2 ) F ( ρ 2 / a ) ] } ,
F ( t ) = 0 d z z - 11 / 3 J 2 ( z ) J 1 ( t z ) ,
E = Γ ( 14 / 3 ) Γ ( 1 / 6 ) / 2 14 / 3 [ Γ ( 17 / 6 ) ] 2 Γ ( 29 / 6 ) 0.0579.
Γ 1 - ( a / a e ) 5 / 3 C ( U - 16 E ) ,
U = - Γ ( 14 / 3 ) Γ ( - 5 / 6 ) / 2 8 / 3 Γ ( 23 / 6 ) × [ Γ ( 17 / 6 ) ] 2 1.06.
( U - 16 E ) / U = 3 / 23 ,
γ 1 - 0.955 ( a / a e ) 5 / 3             ( a a e / 3 )
Γ 1 - 0.125 ( a / a e ) 5 / 3             ( a a e ) .
L l 0 2 / λ ,             λ l 0 .
L a 2 / λ .
γ = 2 b 0 ( a / a e ) - 2 1.084 ( a / a e ) - 2 ,
b 0 = 0 d z z exp ( - z 5 / 3 ) 0.542
γ 0 = 1 - s ( a / a e ) 5 / 3 ,
ω c v / a
ω c v / l 0 ,
2 π a 2 H ( k 1 a , k a ) A d 2 ϱ 1 J 0 ( k 1 ρ 1 ) exp ( i k · ϱ 1 ) = 2 π 0 a d ρ 1 ρ 1 J 0 ( k 1 ρ 1 ) J 0 ( k ρ 1 ) .
L ( z 1 / a , z 2 / a ) = ( 2 π ) 2 a 2 0 d z z H ( z 1 , z ) H ( z 2 , z ) m ( z / a ) .
i s 2 N 0 d θ 1 0 d θ 2 J 1 ( k s a θ 1 ) × J 1 ( k s a θ 2 ) L ( k s θ 1 , k s θ 2 ) .
i s 2 = ( 2 π ) 2 N k s - 2 0 d z z m ( z / a ) Q 2 ( z ) ,
Q ( z ) = 0 d z 1 J 1 ( z 2 ) H ( z 1 , z ) = 0 d z 1 J 1 ( z 1 ) 0 1 d z z J 0 ( z 1 z ) J 0 ( z z ) .
0 d z 1 J 1 ( z 1 ) J 0 ( z z 1 ) = 1             for             z 1 ,
Q ( z ) = 0 1 d z z J 0 ( z z ) = z - 1 J 1 ( z ) .
k - 1 J 1 ( k a ) = ( 2 π a ) - 1 A d 2 ϱ 1 exp ( i k · ϱ 1 ) .
ρ 5 / 3 = ( 2 π ) - 1 C d 2 K [ 1 - exp ( i K · ϱ ) ] K - 11 / 3 = C 0 d K K - 8 / 3 [ 1 - J 0 ( K ρ ) ] ,
C - 1 = 0 d z z - 8 / 3 [ 1 - J 0 ( z ) ] .
0 d z z - 8 / 3 J 0 ( y z ) .
I d 2 ϱ 1 d 2 ϱ 2 d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) × ( x 1 - x 2 ) ( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) .
I = - 2 ( π a 2 ) 2 I 1 ,
I 1 = d 2 ϱ 1 d 2 ϱ 1 x 1 x 1 D ( ϱ 1 - ϱ 1 ) = - 4 π 2 a 4 0 d K K - 1 J 2 2 ( K a ) Δ ( K ) = - ( 2 π ) 2 C a e - 5 / 3 a 23 / 3 E , E = 0 d z z - 14 / 3 J 2 2 ( z ) .
d 2 ϱ 1 d 2 ϱ 2 ( x 1 - x 2 ) ( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) = P ( ϱ 2 ) - P ( ϱ 1 ) ,
P ( ϱ ) = d 2 ϱ 1 d 2 ϱ 2 x 1 D ( ϱ 1 - ϱ ) = - ( 2 π ) 2 a 4 cos ψ 0 d K J 2 ( K a ) J 1 ( K ρ ) Δ ( K ) = - ( 2 π ) 2 C a s - 5 / 3 a 20 / 3 F ( ρ / a ) cos ψ ,
d 2 ϱ 1 d 2 ϱ 2 ( y 1 - y 2 ) ( ϕ 1 - ϕ 2 ) ( ϕ 1 - ϕ 2 ) = Q ( ϱ 2 ) - Q ( ϱ 1 ) ,
Q ( ϱ ) = d 2 ϱ 1 d 2 ϱ 2 y 1 D ( ϱ 1 - ϱ ) = - ( 2 π ) 2 a 4 sin ψ 0 d K J 2 ( K a ) J 1 ( K ρ ) Δ ( K ) = - ( 2 π ) 2 C a s - 5 / 3 a 20 / 3 F ( ρ / a ) sin ψ .
I 0 d 2 ϱ 1 d 2 ϱ 2 ϱ 1 - ϱ 2 5 / 3 .
I 0 = C ( π a 2 ) 2 0 d K K - 8 / 3 [ 1 - 4 ( k a ) - 2 J 1 2 ( K a ) ] = C ( π a 2 ) 2 a 5 / 3 U , U = 0 d z z - 8 / 3 [ 1 - 4 z - 2 J 1 2 ( z ) ] .
0 d x x s - 1 J 1 2 ( y x ) ,
d 2 ϱ 1 d 2 ϱ 2 ϱ 1 - ϱ 2 2 = π 2 a 6 .
I d 2 ϱ 1 d 2 ϱ 2 ( ϱ 1 - ϱ 2 ) · ( ϱ 1 / ρ 1 ) F ( ρ 1 / a ) = 0 a d ρ 2 ρ 2 0 2 π d ψ 1 0 2 π d ψ 2 × 0 a d ρ 1 ρ 1 ( ρ 1 - ρ 1 - 1 ϱ 2 · ϱ 1 ) F ( ρ 1 / a ) = 2 0 a d ρ 2 ρ 2 0 2 π d ν ( 2 π - ν ) 0 a d ρ 1 ρ 1 ( ρ 1 - ρ 2 cos ν ) × 0 d z z - 11 / 3 J 2 ( z ) J 1 ( z ρ 1 a ) = ( 2 π ) 2 ( a 2 / 2 ) 0 d z z - 11 / 3 J 2 ( z ) 0 a d ρ 1 ρ 1 2 J 1 ( z ρ 1 / a ) .
0 a d ρ 1 ρ 1 2 J 1 ( z ρ 1 / a ) = a 3 z - 1 J 2 ( z ) ,
I = ( 2 π ) 2 ( a 2 / 2 ) a 3 0 d z z - 14 / 3 J 2 2 ( z ) = 2 π 2 a 5 E .
2 C a e - 5 / 3 0 K e d K K - 8 / 3 [ 1 - J 0 ( K ρ ) ] 2 C a e - 5 / 3 ( 3 / 4 ) ( K e ρ ) 1 / 3 ρ 5 / 3 = ( 3 C / 4 ) ( K e ρ ) 1 / 3 D ( ρ )
M ( ϱ 1 - ϱ 2 ) = exp [ - 1 2 ( ϕ 1 - ϕ 2 ) 2 ]
m ( 0 ) = 0 d ρ ρ exp [ - ( ρ / a e ) 5 / 3 ] = b 0 a e 2 ,
ζ ( ϱ ) = x α 2 - y α 1 .
E ( ϱ , z , t ) = A ( ϱ , z , t ) exp [ i ( k z - ω t ) ] ,
E ( ϱ , L + ζ , t ) A ( ϱ , L ) exp { i [ k ( L + ζ ) - ω t ] } ,
A a ( ϱ , z ) = A ( ϱ , z ) exp [ i k ζ ( ϱ ) ] .
/ α i d 2 ϱ 1 d 2 ϱ 2 E * ( ϱ 2 , L + ζ 2 , t ) E ( ϱ 1 , L + ζ 1 , t ) = 0
/ α i d 2 ϱ 1 d 2 ϱ 2 A a 2 - A a 1 2 = 0 ,
A = exp ( χ + i β ) ,             whence             A a = exp [ χ + i ( β + k ζ ) ] ,
A a 2 - A a 1 2 ( χ 2 - χ 1 ) 2 + [ ( β 2 - β 1 ) 2 + k ( ζ 2 - ζ 1 ) ] 2 .
Γ = ( π a 2 ) - 2 d 2 ϱ 1 d 2 ϱ 2 A a 2 * A a 1
A a 2 * A a 1 = 1 - 1 2 ( χ 1 - χ 2 ) 2 - 1 2 ( β a 1 - β a 2 ) 2 , = 1 - 1 2 A a 1 - A a 2 2 ,
β a = β + k ζ .
A a 1 - A a 2 2 ( χ 1 - χ 2 ) 2 + ( β a 1 - β a 2 ) 2 = ( χ 1 - χ 2 ) 2 + ( β 1 - β 2 ) 2 + 2 k ( β 2 - β 2 ) ( ζ 2 - ζ 1 ) + k 2 ( ζ 2 - ζ 1 ) 2 .
( χ 1 - χ 2 ) 2 + ( β 1 - β 2 ) 2 = ( ϕ 1 - ϕ 2 ) 2 .
A a 1 - A a 2 2 ( ϕ 1 - ϕ 2 ) 2 + [ ( π 2 / 2 ) a 6 ] - 2 ϱ 1 - ϱ 2 2 × ( 1 ) ( 2 ) ( 1 ) ( 2 ) ( x 1 - x 2 ) ( x 1 - x 2 ) × ( β 1 - β 2 ) ( β 1 - β 2 ) - 2 [ ( π 2 / 2 ) a 6 ] - 1 ( ϱ 1 - ϱ 2 ) · ( 1 ) ( 2 ) ( ϱ 1 - ϱ 2 ) ( β 1 - β 2 ) ( β 1 - β 2 ) .
1 2 [ 1 + ( k / K 2 L ) sin 2 ( K 2 L / k ) ] K - 11 / 3 .
E ˜ 1 2 0 d z z - 14 / 3 [ 1 + ( a 2 k / z 2 L ) sin ( z 2 L / a 2 k ) ] J 2 2 ( z )
Ĩ ( 2 π ) 2 ( a 2 / 2 ) 1 2 0 d z z - 11 / 3 [ 1 + ( a 2 k / z 2 L ) sin ( z 2 L / a 2 k ) ] × J 2 ( z ) 0 a d ρ 1 ρ 1 2 J 1 ( z ρ 1 / a ) .
L / a 2 k 1 ,             or             L a 2 / λ .
L l 0 3 / λ 3 ,
Γ = ( π a 2 ) - 2 d 2 ϱ 1 d 2 ϱ 2 A a 2 * A a 1 = - ( π a 2 ) - 2 ( i 2 k ) - 1 d 2 ϱ 1 d 2 ϱ 2 0 L d z [ A 2 * ( z ) 1 2 A 1 ( z ) - A 1 ( z ) 2 2 A 2 * ( z ) ] exp { i [ Φ 1 ( z , L ) - Φ 2 ( z , L ) ] } + ( π a 2 ) - 2 d 2 ϱ 2 d 2 ϱ 2 × exp { i [ Φ 1 ( 0 , L ) - Φ 2 ( 0 , L ) ] } ,
Φ i ( z , z ) k z z d z N ( ϱ i , z ) + k ( x i α 2 - y i α 1 )
L(k1,k2)=(2π)2a2(k1k2)-1J1(k1a)J1(k2a).