Abstract

The first part of this paper is devoted to extending the Huygens-Fresnel principle to a medium that exhibits a spatial (but not temporal) variation in index of refraction. Utilizing a reciprocity theorem for a monochromatic disturbance in a weakly inhomogeneous medium, it is shown that the secondary wavefront will be determined by the envelope of spherical wavelets from the primary wavefront, as in the vacuum problem, but that each wavelet is now determined by the propagation of a spherical wave in the refractive medium. In the second part, the above development is applied to the case in which the index of refraction is a random variable; a further application of the reciprocity theorem results in a formula for the mean intensity distribution from a finite aperture in terms of the complex disturbance in the aperture and the modulation transfer function (MTF) for a spherical wave in the medium. The results are applicable for an arbitrary complex disturbance in the transmitting aperture in both the Fresnel and Fraunhofer regions of the aperture. Using a Kolmogorov spectrum for the index of refraction fluctuations and a second-order expression for the MTF, the formula is used to calculate the mean intensity distribution for a plane wave diffracting from a circular aperture and to give approximate expressions for the beam spreading at various ranges.

© 1971 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. 7.
  2. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1966).
  3. T. L. Ho, J. Opt. Soc. Amer. 60, 667 (1970).
    [CrossRef]
  4. D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
    [CrossRef]
  5. F. G. Gebhardt, S. A. Collins, J. Opt. Soc. Amer. 59, 1139 (1969).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), Chap. 8.
  7. R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Amer. 61, 482 (1971).
    [CrossRef]
  8. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  9. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 485.

1971 (1)

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Amer. 61, 482 (1971).
[CrossRef]

1970 (1)

T. L. Ho, J. Opt. Soc. Amer. 60, 667 (1970).
[CrossRef]

1969 (1)

F. G. Gebhardt, S. A. Collins, J. Opt. Soc. Amer. 59, 1139 (1969).
[CrossRef]

1967 (2)

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

1966 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), Chap. 8.

Collins, S. A.

F. G. Gebhardt, S. A. Collins, J. Opt. Soc. Amer. 59, 1139 (1969).
[CrossRef]

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

Gebhardt, F. G.

F. G. Gebhardt, S. A. Collins, J. Opt. Soc. Amer. 59, 1139 (1969).
[CrossRef]

Ho, T. L.

T. L. Ho, J. Opt. Soc. Amer. 60, 667 (1970).
[CrossRef]

Lutomirski, R. F.

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Amer. 61, 482 (1971).
[CrossRef]

Schmeltzer, R. A.

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

Seidman, J. B.

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 485.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), Chap. 8.

Yura, H. T.

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Amer. 61, 482 (1971).
[CrossRef]

J. Opt. Soc. Amer. (4)

T. L. Ho, J. Opt. Soc. Amer. 60, 667 (1970).
[CrossRef]

D. L. Fried, J. B. Seidman, J. Opt. Soc. Amer. 57, 181 (1967).
[CrossRef]

F. G. Gebhardt, S. A. Collins, J. Opt. Soc. Amer. 59, 1139 (1969).
[CrossRef]

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Amer. 61, 482 (1971).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Quart. Appl. Math. (1)

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

Other (3)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 7.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 485.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1965), Chap. 8.

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Figures (4)

Fig. 1
Fig. 1

The critical length zc for l0 = 0.1 cm, L0 = 100 cm. Curves A, B, and C correspond to C n 2 = 3 × 10 - 16 cm - 2 3 , 3 × 10 - 15 cm - 2 3, and 3 × 10 - 14 cm - 2 3 , respectively.

Fig. 2
Fig. 2

The overlap integral defined by Eq. (17). The quantity β = kD2/4z.

Fig. 3
Fig. 3

Normalized beam pattern as a function of α for D/L0 = 0.1 and various values of z/zc for β = 0 (far-field) and β = 8 (near-field).

Fig. 4
Fig. 4

The correction to the on-axis intensity F(z/zc) and the half-power angle, found by inverting G ( θ 1 2 / θ b , z / z c ) = 1 2 , as functions of normalized range z/zc.

Tables (1)

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Table I Summary of MTF’s and Beam Spreadsa

Equations (33)

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( 2 + k 2 n 2 ) U = 0 ,
( 2 + k 2 n 2 ) G ( r , P ) = - 4 π δ ( r - P ) ,
U ( P ) = ( 1 / 4 π ) S ( U G - G U ) · d A ,
U ( P ) = i k 4 π A U ( r ) exp ( i k s + ψ ) s × [ e s · e z - e n · e z + ψ i k · e z ] d 2 r ,
U ( P ) = A K ( χ ) G ( r , P ) U ( r ) d 2 r ,
U ( P ) = [ K ( χ 0 ) / s 0 ] A exp ( i k s + ψ ) U ( r ) d 2 r
I ( P ) = U ( P ) 2 = ( 1 + cos χ 0 2 λ s 0 ) 2 exp [ i k ( s 1 - s 2 ) ] × exp [ ψ ( r 1 ) + ψ * ( r 2 ) ] U ( r 1 ) U * ( r 2 ) d 2 r 1 d 2 r 2 ,
I = ( 2 π k z ) 2 exp [ i k ( s 1 - s 2 ) ] M s ( r 1 , r 2 , z ) U ( r 1 ) U * ( r 2 ) d 2 r 1 d 2 r 2
s 1 - s 2 = - [ ( p · ϱ + r · ϱ ) / z ] ,
I = ( k 2 π z ) 2 d 2 ϱ M s ( ϱ , z ) exp [ - ( i k / z ) p · ϱ ] + U ( r + 1 2 ϱ ) U * ( r - 1 2 ϱ ) exp [ ( i k / z ) ϱ · r ] d 2 r .
M p ( ρ , z ) = exp { - 2 z z c [ 1 - 0 J 0 ( K ρ ) Φ n ( K ) K d K 0 Φ n ( K ) K d K ] } ,
z c = [ 2 π 2 k 2 0 Φ n ( K ) K d K ] - 1
Φ n ( K ) = 0.033 C n 2 exp [ - ( K l 0 / 5.92 ) 2 ] ( K 2 + Ł - 2 ) ¹¹ / .
z c ( 0.4 k 2 C n 2 Ł 0 / ) - 1 .
M s ( ρ , z ) = exp ( - 2 z z c { 1 - ( 5 / 3 ) ( ρ / Ł 0 ) / 0 d u u J 0 ( u ) × 0 1 s / d s [ u 2 + ( s ρ / Ł 0 ) 2 ] 1 6 } ) .
U ( r 1 , 2 ) = 1 , r 1 , 2 D / 2 = 0 , r 1 , 2 > D / 2 ,
Γ β ( x ) = 0 cos - 1 ( x ) ( sin [ 2 β x ( cos θ - x ) ] ( 2 β x cos θ ) - { 1 - cos [ 2 β x ( cos θ - x ) ] } ( 2 β x cos θ ) 2 ) d θ             x 1 = 0 ,             x > 1 ,
I = ( 8 / π ) β 2 0 1 x J 0 ( 2 α x ) M s ( D x , z ) Γ β ( x ) d x ,
I N = 0 1 x J 0 ( 2 α x ) M s ( D x , z ) Γ β ( x ) d x 0 1 x Γ β ( x ) d x .
Γ 0 ( x ) = 1 2 [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 2 ] .
I ( θ , z ) = 2 β 2 D 2 0 ρ J 0 ( k ρ θ ) M s ( ρ , z ) d ρ .
I ( 0 , z ) β 2 ( ρ 0 / D ) 2 = ( D 2 / ( ρ 0 2 ) ( k ρ 0 2 / 4 z ) 2 ,
θ s ~ ( k ρ 0 ) - 1 ,
I ( 2 ) ( θ , z ) = 0.18 D 2 k 2 5 C n ¹² / z ¹⁶ / F ( z / z c ) G [ z / z c , θ / θ ( 2 ) ] ,
F ( z / z c ) = 0 ( z / z c ) 3 5 u exp { - u ¹⁵ / [ 1 - 0.67 ( z c / z ) 1 5 u 1 3 ] } d u , G [ z / z c , θ / θ ( 2 ) ] = 0 ( z / z c ) 3 5 u J 0 [ ( θ / θ ) ( 2 ) u ] exp { - u / [ 1 - 0.67 ( z c / z ) 1 5 u 1 3 ] } d u F ( z / z c )
θ ( 2 ) = 0.69 k 1 5 C n / z 3 5 .
I 3 ( θ , z ) = ( ρ 0 D ) 2 ( k D 2 4 z ) 2 exp ( - k 2 ρ 0 2 θ 2 / 4 ) = D 2 8 z 2 θ ( 3 ) exp [ - θ 2 / 2 θ ( 3 ) 2 ] = D 2 exp ( - p 2 / 4 q z 2 ) 16 q z 3 ,
D 2 Γ β ( ρ / D ) M s ( ρ , z ) = 2 π 0 J 0 ( k ρ z p ) I ( p , z ) p d p ,
( 2 + k 2 n 2 ) G 1 ( r , r 1 ) = - 4 π δ ( r - r 1 ) ,
( 2 + k 2 n 2 ) G 2 ( r , r 2 ) = - 4 π δ ( r - r 2 ) .
S [ G 1 ( r , r 1 ) G 2 ( r , r 2 ) - G 2 ( r , r 2 ) G 1 ( r , r 1 ) ] · d A = - 4 π [ G 2 ( r 1 , r 2 ) - G 1 ( r 2 , r 1 ) ] .
[ G 1 ( G 2 R - i k n G 2 ) - G 2 ( G 1 R - i k n G 1 ) ] R 2 d Ω .
lim R R ( G R - i k G ) = 0 ,

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