Abstract

The short-term average irradiance profile of a focused laser beam transmitted through a homogeneous-isotropic medium has been determined by using the extended Huygens-Fresnel principle and by modifying the phase structure function to remove tilt. In contrast to previous analysis, no assumption is made regarding the independence of the distribution of phase with tilt removed and the random vector β defining tilt. This analysis applies to the near field of the effective coherent transmitting aperture, where the beam wanders as a whole and does not break up into multiple patches or blobs. Central to the analysis is the short-term average mutual coherence function (MCF) of a spherical wave, which has been determined from the modified phase structure function. Assuming a Kolmogorov spectrum, the modified phase structure function has been determined for three specific aperture functions. These same aperture functions are then used to determine the short-term irradiance profiles. Numerical calculations have been performed, and the results are presented for uniform and Gaussian aperture functions for various values of aperture obscuration and for various strengths of turbulence values. Comparisons are made between the long-term average, short-term average, and Fried’s short-term average irradiance profiles. In particular, on-axis irradiance values and beam spread, as determined by the 1/e points in irradiance, are compared. It is found, in contrast to previous analysis, that the short-term beam spread remains relatively constant as the strength of turbulence becomes large and then increases slowly.

© 1976 Optical Society of America

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References

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  1. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
  2. R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).
  3. H. T. Yura, Appl. Opt. 10, 2771 (1971).
  4. A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).
  5. H. T. Yura, J. Opt. Soc. Am. 63, 567 (1973).
  6. C. Hogge, AFWL; private communication.
  7. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  8. R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 59, 99 (1969).
  9. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  10. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
  11. J. R. Kerr, AGARD Conference on Optical Propagation in the Atmosphere (October 1975), pp. 21–1.
  12. J. R. Kerr, Oregon Graduate Center, private communication. Dr. Kerr has recently modified his results to take this into account.
  13. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (U.S. Govt. Printing Office, Wash. D.C., 1965).

1973 (1)

1971 (3)

1970 (1)

A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).

1969 (1)

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 59, 99 (1969).

1966 (1)

1965 (1)

Brown, W. P.

Fried, D. L.

Hogge, C.

C. Hogge, AFWL; private communication.

Kerr, J. R.

J. R. Kerr, Oregon Graduate Center, private communication. Dr. Kerr has recently modified his results to take this into account.

J. R. Kerr, AGARD Conference on Optical Propagation in the Atmosphere (October 1975), pp. 21–1.

Kon, A. I.

A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).

Lutomirski, R. F.

R. F. Lutomirski, H. T. Yura, Appl. Opt. 10, 1652 (1971).

R. F. Lutomirski, H. T. Yura, J. Opt. Soc. Am. 59, 99 (1969).

Tatarskii, V. I.

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

Yura, H. T.

Appl. Opt. (2)

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, 61 (1970).

J. Opt. Soc. Am. (5)

Other (5)

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

J. R. Kerr, AGARD Conference on Optical Propagation in the Atmosphere (October 1975), pp. 21–1.

J. R. Kerr, Oregon Graduate Center, private communication. Dr. Kerr has recently modified his results to take this into account.

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (U.S. Govt. Printing Office, Wash. D.C., 1965).

C. Hogge, AFWL; private communication.

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Figures (9)

Fig. 1
Fig. 1

Normalized short-term irradiance profiles for various strength of turbulence values.

Fig. 2
Fig. 2

Normalized long-term irradiance profiles for various strength of turbulence values.

Fig. 3
Fig. 3

Normalized on-axis irradiance values using the short-term average, long-term average, and Fried’s approximation to short-term average for the MCF. A uniform aperture distribution with an obscuration of 0.5 is used.

Fig. 4
Fig. 4

Normalized on-axis irradiance values using the short-term average, long-term average, and Fried’s approximation to short-term average for the MCF. A Gaussian aperture distribution with βo equal to 1.0 and an obscuration of 0.5 is used.

Fig. 5
Fig. 5

Normalized on-axis irradiance values using the short-term average, long-term average, and Fried’s approximation to short-term average for the MCF. A Gaussian aperture distribution, with βo equal to 4.0 and no obscuration, is used.

Fig. 6
Fig. 6

Normalized beam spreads for the long-term and short-term irradiance profiles for βo = 1.0 and δ = 0.5 as determined from the 1/e point in intensity.

Fig. 7
Fig. 7

The improvement in on-axis irradiance due to tilt removal as a function of the effective strength of turbulence.

Fig. 8
Fig. 8

Geometric representation of aperture overlap area for the integration variable η.

Fig. 9
Fig. 9

Schematic of the integration area for the η variable in Figure 8(d).

Equations (62)

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I ( p ) = E ( k 2 π z ) 2 exp [ - i k p · ( r 1 - r 2 ) z ] × exp [ - i k 2 ( 1 f - 1 z ) ( r 1 2 - r 2 2 ) ] × exp [ ψ ( r 1 ) + ψ * ( r 2 ) ] U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
I ( p ) LT = E ( k 2 π z ) 2 exp [ - i k p · ( r 1 - r 2 ) z ] × M LT ( r 1 , r 2 , z ) × U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
M LT ( r 1 , r 2 , z ) = M LT ( ρ , z ) = exp [ - ½ D ψ ( ρ , z ) ] ,
D ψ = D χ ( ρ , z ) + D ϕ ( ρ , z ) .
D ψ = 2 ( ρ / ρ o ) 5 / 3 ,
D ϕ = α ( ρ ) D ψ ( ρ ) ,
γ ( r ) = ϕ ( r ) - β · r ,
β i d 2 r W ( r ) [ ϕ ( r ) - β · r ] 2 = 0.
β i = r i ϕ ( r ) W ( r ) d 2 r r i 2 W ( r ) d 2 r ,
I ( p ) ST = E ( k 2 π z ) 2 exp [ - i k p · ( r 1 - r 2 ) z ] × M ST ( r 1 , r 2 ) × U A ( r 1 ) U A * ( r 2 ) d 2 r 1 d 2 r 2 ,
M ST ( r 1 , r 2 ) = exp [ ψ ( r 1 ) + ψ * ( r 2 ) - i β · ( r 1 - r 2 ) ]
M ST ( r 1 , r 2 ) = exp { χ ( r 1 ) + χ ( r 2 ) + i [ ϕ ( r 1 ) - ϕ ( r 2 ) - β · ( r 1 - r 2 ) ] } .
M ST ( r , r 2 ) = exp ( - ½ [ D χ ( ρ ) + D γ ( r 1 , r 2 ) ] } ,
D γ ( r 1 , r 2 ) = [ γ ( r 1 ) - γ ( r 2 ) ] 2 = [ ϕ ( r 1 ) - ϕ ( r 2 ) ] 2 + ( β · ρ ) 2 - 2 ρ · β [ ϕ ( r 1 ) - ϕ ( r 2 ) ] .
D γ ( r 1 , r 2 ) = D ϕ ( ρ ) - A 2 2 ( ρ · r ) ( ρ · r ) W ( r ) W ( r ) D ϕ ( r - r ) d 2 r d 2 r + A ρ · [ r W ( r ) D ϕ ( r - r 1 ) d 2 r - r W ( r ) D ϕ ( r - r 2 ) d 2 r ] ,
A = [ π r 3 W ( r ) d r ] - 1 .
D γ ( ρ , R ) = D ϕ ( ρ ) - A 2 4 ρ 2 ( r · r ) W ( r ) W ( r ) D ϕ ( r - r ) d 2 r d 2 r + A [ ( r 1 · ρ ) r 1 2 r 1 · r W ( r ) D ϕ ( r - r 1 ) d 2 r - ( r 2 · ρ ) r 2 2 r 2 · r W ( r ) D ϕ ( r - r 2 ) d 2 r ] ,
U A = Q 1 exp ( - r 2 / 2 b 2 ) [ 1 - exp ( - r 2 / a 2 ) ] ,
M ST ν ( ρ , R ) = exp { - 5 / 3 ( ρ 5 / 3 + C ν ρ 2 + ( 2 ) 5 / 3 [ ρ · r 1 G ν ( r 1 ) - ρ · r 2 G ν ( r 2 ) ] } ,
I ν ( p ) ST = P π b 2 ( π b 2 λ z ) 2 ( 32 π S ν × { 0 1 B ν ( ρ ) ρ d ρ 0 cos - 1 ρ d θ ρ sec θ 1 H ν ( ρ , R - ρ ) R d R + 0 δ B ν ( ρ ) ρ d ρ 0 cos - 1 ρ / δ d θ ρ sec θ δ H ν ( ρ , R - ρ ) R d R - 0 1 + b / 2 B ν ( ρ ) ρ d ρ 0 π d θ 0 δ H ν ( ρ , R + ρ ) R d R + 1 - b / 2 1 + b / 2 B ν ( ρ ) ρ d ρ [ 0 α d θ 0 δ H ν ( ρ , R + ρ ) R d R - 0 θ d θ 2 ρ sin α csc φ 1 H ν ( ρ , R - ρ ) R d R ] } ) ,
B 1 ( ρ ) = J 0 ( 2 k p b ρ z ) exp [ - 5 / 3 ( ρ 5 / 3 + C 1 ρ 2 ) ] , B 2 ( ρ ) = J 0 ( 2 k p b ρ z ) exp [ - 5 / 3 ( ρ 5 / 3 + C 2 ρ 2 ) ] exp ( - β o 2 ρ 2 ) , H 1 ( ρ , R ) = exp { ( 2 ) 5 / 3 [ ρ · r 1 G 1 ( r 1 ) - ρ · r 2 G 1 ( r 2 ) ] } , H 2 ( ρ , R ) = exp { ( 2 ) 5 / 3 [ ρ · r 1 G 2 ( r 1 ) - ρ · r 2 G 2 ( r 2 ) ] } exp ( - β o 2 R 2 ) , S 1 = ( 1 - δ 2 ) - 1 , S 2 = β o 2 { exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) } - 1 , α = cos - 1 ( 1 - 4 ρ 2 - δ 2 4 δ ρ ) , θ ¯ = cos - 1 ( 1 + 4 ρ 2 - δ 2 4 ρ ) , φ = α - θ , δ = a / b ,
I 3 ( p ) ST = P π b 2 ( π b 2 λ z ) 2 { 32 π S 3 0 J 0 ( 2 k b p ρ z ) × exp [ - 5 / 3 ( ρ 5 / 3 + C 3 ρ 2 ) ] × [ exp ( - ρ 2 ) 0 exp ( - R 2 ) R d R 0 π / 2 H 3 ( ρ , R ) d θ - exp ( - ρ 2 ) 0 exp ( - R 2 ) R d R 0 π / 2 { exp [ - ( R + ρ ) 2 δ 2 ] + exp [ - ( R - ρ ) 2 δ 2 ] } H 2 ( ρ , R ) d θ + exp [ - ( ρ b e ) 2 ] 0 exp [ - ( R b e ) 2 ] R d R × δ / 2 H 3 ( ρ , R ) d θ ] ρ d ρ } ,
H 3 = exp { ( 2 ) 5 / 3 [ G 3 ( r 1 ) ρ · r 1 - G 3 ( r 2 ) ρ · r 2 ] } , S 3 = [ 1 - 2 ( c b ) 2 + ( e b ) 2 ] - 1 ,
D eff = 4 α o { 1 - exp ( β o / 2 ) [ 1 - exp ( β o ) ] 1 / 2 } .
D ϕ ( ρ ) = 2 ( ρ / ρ o ) 5 / 3 ,
D ϕ ( ρ ) = 2 5 / 3 ρ 5 / 3 .
W 1 ( r ) = 1 a r b = 0 0 r < a r > b ,
D y ( ρ , R ) = D ϕ ( ρ ) - X 1 ( ρ ) + Y 1 ( ρ , r 1 ) - Y 1 ( ρ , r 2 ) ,
A 1 = [ 4 / ( π b 4 ) ] ( 1 - δ 4 ) - 1 ,
δ = b / a .
Y 1 ( ρ , r 1 ) = 2 A 1 ρ · r 1 ρ o 5 / 3 r 1 2 0 2 II a b r · r 1 r - r 1 5 / 3 r d r d θ .
Y 1 ( ρ , r 1 ) = - 2 ( 2 ) 5 / 3 ρ · r 1 G 1 ( r 1 , δ ) ,
G 1 ( r 1 , δ ) = 40 3 ( 1 - δ 4 ) - 1 × δ 1 r 3 ( r 1 + r ) 1 / 3 F [ 1 6 , 3 2 , 3 , 4 r r 1 ( r 1 + r ) 2 ] d r ,
X 1 / ( ρ ) = A 1 2 ρ 2 2 ρ o 5 / 3 0 2 π a b 0 2 π a b r · r r - r 5 / 3 r d r r d r d θ d θ .
X 1 = 64 π b 8 A 1 2 ( ) 5 / 3 ( I 1 + I 2 + I 3 + I 4 ) ,
I 1 = δ 1 ξ 8 / 3 d ξ 0 cos - 1 ξ d θ ξ sec θ 1 ( η 2 - 2 η ξ cos θ ) η d η = - π Γ ( 4 / 3 ) Γ ( 11 / 6 ) 180 47311 ,
I 2 = - 0 ( 1 + b ) / 2 ξ 8 / 3 d ξ 0 π d θ 0 δ η d η ( η 2 + 2 ξ η cos θ ) = - 3 π 44 δ 4 ( 1 + δ 2 ) 11 / 3 ,
I 3 = 0 δ ξ 8 / 3 d ξ 0 cos - 1 ξ / δ d θ ξ cos θ δ ( η 2 - 2 η ξ cos θ ) η d η = δ 23 / 3 I 1 .
I 4 = ( 1 - b ) / 2 ( 1 + b ) / 2 ξ 8 / 3 d ξ [ 0 α d θ 0 δ ( η 2 + 2 ξ η cos θ ) η d η - 0 θ ¯ d θ 2 ξ sin α csc φ 1 ( η 2 - 2 ξ η cos θ ) η d η ] ,
α = cos - 1 ( 1 - 4 ξ 2 - δ 2 4 δ ξ ) , θ ¯ = cos - 1 ( 1 + 4 ξ 2 - δ 2 4 ξ ) , φ = α - θ .
I 4 = 1 4 ( 1 - b ) / 2 ( 1 + b ) / 2 ξ 8 / 3 d ξ T ( ξ , θ ¯ , α , φ ¯ ) ,
T ( ξ , θ ¯ , α , φ ) = δ 4 α + 8 3 δ 3 ξ sin α - θ ¯ + 8 3 ξ sin θ ¯ + 16 ξ 4 sin 4 α ( cot φ ¯ - cot α + cot 3 φ ¯ - cot 3 α 3 ) - 64 3 ξ 4 sin 3 α [ cos α 2 ( csc 2 φ ¯ - csc 2 α ) + sin α ( cot φ ¯ - cot α ) ] ,
X ( ρ ) = - 2 C 1 ɛ 5 / 3 ρ 2 ,
C 1 = 2 9 π ( 1 - δ 4 ) - 2 { [ π 1 / 2 Γ ( 4 / 3 ) 180 Γ ( 11 / 6 ) 47311 ] ( 1 + δ 23 / 3 + 3 π 44 ( 1 + δ 2 ) 11 / 3 δ 4 - ¼ ( 1 - b ) / 2 ( 1 + b ) / 2 ξ 8 / 3 T ( ξ , θ ¯ , α , φ ¯ ) d ξ } .
M ST = exp { - 5 / 3 ( ρ 5 / 3 + C 1 ρ 2 ) + ( 2 ) 5 / 3 [ ρ · r 1 G 1 ( r 1 ) - ρ · r 2 G 1 ( r 2 ) ] } .
W 2 = exp ( - r 2 / α o 2 ) a r b = 0 otherwise ,
A 2 = [ π a b r 3 exp ( - r 2 / α o 2 ) d r ] - 1 = 2 ( β o { δ 2 exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) } + exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) ) - 1 / ( π b 4 β o 4 ) ,
G 2 ( r 1 ) = 20 3 β o 4 [ β o 2 { δ 2 exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) + exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) ] } - 1 × δ l r 3 ( r + r 1 ) 1 / 3 exp ( - r 2 β o 2 ) × F [ 1 6 , 3 2 , 3 , 4 r r 1 ( r + r 1 ) 2 ] d r
C 2 = 32 π β o 6 ( β o 2 { δ 2 exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) } + exp [ - ( δ β o ) 2 ] - exp ( - β o 2 ) ) - 2 i = 1 5 I i .
I 1 = 0 1 ξ 8 / 3 d ξ exp ( - 2 β o 2 ξ 2 ) { [ 0 π / 2 exp ( - 2 β o 2 R 1 2 ) × ( R 1 2 + 1 2 β o 2 - ξ 2 ) d θ ] - π 2 ( 1 2 β o 2 - ξ 2 ) } ,
R 1 = ξ cos θ + ( 1 - ξ 2 sin 2 θ ) 1 / 2 .
I 2 = 0 δ ξ 8 / 3 d ξ exp ( - 2 β o 2 ξ 2 ) { [ 0 π / 2 exp ( - 2 β o 2 R 2 2 ) × ( R 2 2 + 1 2 β o 2 - ξ 2 ) d θ ] - π 2 ( 1 2 β o 2 - ξ 2 ) } ,
R 2 = - ξ cos θ + ( δ 2 - ξ 2 sin 2 θ ) 1 / 2 .
I 3 = 4 π β o 2 0 ( 1 + 3 δ 2 ) 1 / 2 / 2 ξ 8 / 3 d ξ exp ( - 2 β o 2 ξ 2 ) 0 δ η d η × exp [ - 2 β o 2 ( η - ξ ) 2 ] [ η 2 I ¯ 0 ( 4 β o 2 η ξ ) - 2 ξ η I ¯ 1 ( 4 β o 2 η ξ ) ] ,
I 4 = ( 1 - b ) / 2 ( 1 + 3 δ 2 ) 1 / 2 / 2 ξ 8 / 3 d ξ exp ( - 2 β o 2 ξ 2 ) 0 α d θ [ ( R 3 2 - ξ 2 + 1 2 β o 2 ) × exp ( - 2 β o 2 R 3 2 ) - ( R 1 2 - ξ 2 + 1 2 β o 2 ) exp ( - 2 β o 2 R 1 2 ) ] ,
R 3 = ξ cos θ + ( δ 2 - ξ 2 sin 2 θ ) 1 / 2 , α = sin - 1 ( sin γ Z ) , γ = cos - 1 ( 1 + 4 ξ 2 - δ 2 4 ξ ) , Z = ( 1 + ξ 2 - 2 ξ cos γ ) 1 / 2 .
I 5 = - ( 1 + 3 δ 2 ) 1 / 2 / 2 1 + δ / 2 ξ 8 / 3 d ξ exp ( - 2 β o 2 ξ 2 ) × 0 α d θ [ ( R 1 2 - ξ 2 + 1 2 β o 2 ) exp ( - 2 β o 2 R 1 2 ) - ( R 4 2 - ξ 2 + 1 2 β o 2 ) exp ( - 2 β o 2 R 4 2 ) ] ,
W 3 = exp ( - r 2 / b 2 ) [ 1 - exp ( - r 2 / a 2 ) ] 2 ,
A 3 = 2 [ 1 - 2 ( c b ) 2 + ( e b ) 4 ] - 1 / ( π b 4 ) ,
G 3 ( r 1 ) = 10 3 Γ ( 11 6 ) [ 1 - 2 ( c b ) 4 + ( e b ) 4 ] - 1 × { M ( 1 6 , 2 , - r 1 2 ) - 2 ( c b ) 11 / 6 M [ 1 6 , 2 , - ( b r 1 c ) 2 ] + ( e b ) 11 / 6 M [ 1 6 , 2 , - ( b r 1 e ) 2 ] } ,
C 3 = 5 6 2 1 / 6 Γ ( 11 6 ) [ 1 - 2 ( c b ) 4 + ( e b ) 4 ] - 2 × { 1 + 24 5 ( d b ) 4 ( g b ) 11 / 6 [ 1 - 11 6 ( g d ) 2 ( 1 - d 4 4 a 4 ) ] + 4 ( c b ) 23 / 3 + 24 5 ( f b ) 4 ( h b ) 11 / 3 [ 1 - 11 6 ( h f ) 2 ( 1 - f 4 4 a 4 ) ] + ( e b ) 23 / 3 } .
1 c 2 = 1 b 2 + 1 a 2 , 1 d 2 = 1 b 2 + 1 2 a 2 , 1 e 2 = 1 b 2 + 2 a 2 , 1 f 2 = 1 b 2 + 3 2 a 2 , 1 g 2 = 1 d 2 - d 2 4 a 4 , 1 h 2 = 1 f 2 - f 2 4 a 4 .

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