Abstract

The wave structure function associated with atmospheric turbulence is important in the effective design of telescopes for coherent optical receivers. This paper describes a method of measurement of the wave structure function by employing an already existing optical receiver. The aperture of this receiver’s telescope is masked by an opaque screen with two small holes. It is then shown that when the incident beam obeys Gaussian statistics, the receiver’s A.M. output can be simply related to the wave structure function of the incident beam.

© 1973 Optical Society of America

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References

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  1. S. Gardner, IEEE Int. Conv. Rec. 12, 337 (1964).
  2. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  3. N. McAvoy, NASA, Goddard Space Flight Center, Greenbelt, Maryland, Private Communication.
  4. It has been assumed that polarization effects are small so that a scalar analysis can be used; see Ref. 5.
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.
  6. V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 12.

1967 (1)

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

1964 (1)

S. Gardner, IEEE Int. Conv. Rec. 12, 337 (1964).

Fried, D. L.

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

Gardner, S.

S. Gardner, IEEE Int. Conv. Rec. 12, 337 (1964).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

McAvoy, N.

N. McAvoy, NASA, Goddard Space Flight Center, Greenbelt, Maryland, Private Communication.

Tartarski, V. I.

V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 12.

IEEE Int. Conv. Rec. (1)

S. Gardner, IEEE Int. Conv. Rec. 12, 337 (1964).

Proc. IEEE (1)

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

Other (4)

N. McAvoy, NASA, Goddard Space Flight Center, Greenbelt, Maryland, Private Communication.

It has been assumed that polarization effects are small so that a scalar analysis can be used; see Ref. 5.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 12.

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Figures (2)

Fig. 1
Fig. 1

Optical heterodyne receiver with masked telescope aperture.

Fig. 2
Fig. 2

View of aperture and photomixer planes.

Equations (44)

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i p = γ P E p 2 d A . γ constant ,
i IF ( t ) = A ( t ) cos [ Δ ω t + ϕ ( t ) ] ,
i d ( t ) = σ A ( t ) , σ constant ,
E s ( ρ , t ) = A 1 + A 2 h ( ρ , ρ ) E inc ( ρ t ) d ρ ,
ρ = x a x + y a y ,             ρ = x a x + y a y ,
E inc ( ρ , t ) = A s exp [ j ( ω s t + ϕ s ) ] ,
E inc ( ρ , t ) = A s ( ρ , t ) exp { j [ ω s t + ϕ s ( ρ , t ) ] } ,
A s ( ρ , t ) = A ¯ s exp [ l ( ρ , t ) ] .
E s ( ρ , t ) = M 1 ( ρ ) exp { l 1 ( t ) + j [ ω s t + ϕ 1 ( t ) ] } + M 2 ( ρ ) exp { l 2 ( t ) + j [ ω s t + ϕ 2 ( t ) ] }
M 1 2 ( ρ ) = A ˜ s A 1 h ( ρ , ρ ) d ρ .
l 1 2 ( t ) = l s ( ρ 1 2 , t )
ϕ 1 2 ( t ) = ϕ s ( ρ 1 2 , t )
E p = E 0 + E s .
E 0 ( ρ , t ) = A 0 ( ρ ) exp { j [ ω 0 t + ϕ 0 ( ρ ) ] } .
i p ( t ) = γ D ( E 0 + E s ) ( E ¯ 0 + E ¯ s ) d ρ ,
i p ( t ) = γ D E 0 E ¯ 0 d ρ + Re 2 γ D E s E ¯ 0 d ρ + γ D E s E s d ρ ,
i p ( t ) = γ D E 0 E ¯ 0 d ρ + 2 γ Re D E s E ¯ 0 d ρ .
i p ( t ) = γ D A 0 2 ( ρ ) d ρ + Re ( B 1 ( t ) exp { l 1 ( t ) + j [ Δ ω t + ϕ 1 ( t ) ] } + B 2 ( t ) exp { l 2 ( t ) + j [ Δ ω t + ϕ 2 ( t ) ] } ) ,
B 1 2 ( t ) = 2 γ D M 1 2 ( ρ ) A 0 ( ρ ) exp [ j ϕ 0 ( ρ ) ] d ρ
Δ ω = ω s - ω 0 .
B 1 2 ( t ) = B 1 2 ( t ) exp [ j ψ 1 2 ( t ) ]
i p ( t ) = γ D A 0 2 ( ρ ) d ρ + B 1 ( t ) exp [ l 1 ( t ) ] cos [ Δ ω t + ϕ 1 ( t ) + ψ 1 ( t ) ] + B 2 ( t ) exp [ l 2 ( t ) ] cos [ Δ ω t + ϕ 2 ( t ) + ψ 2 ( t ) ] .
i IF ( t ) = B 1 ( t ) exp [ l 1 ( t ) ] cos [ Δ ω t + ϕ 1 ( t ) + ψ 1 ( t ) ] + B 2 ( t ) exp [ l 2 ( t ) ] cos [ Δ ω t + ϕ 2 ( t ) + ψ 2 ( t ) ] .
i IF ( t ) = A ( t ) cos [ Δ ω t + θ ( t ) ] ,
A ( t ) = [ B 1 2 exp ( 2 l 1 ) + B 2 exp ( 2 l 2 ) + 2 B 1 B 2 exp ( l 1 + l 2 ) ] cos ( ϕ 1 - ϕ 2 + ψ 1 - ψ 2 ) ] 1 / 2 .
i d ( t ) = A ( t ) .
D l ( r ) = [ l ( ρ 1 ) - l ( ρ 2 ) ] 2
D ϕ ( r ) = [ ϕ ( ρ 1 ) - ϕ ( ρ 2 ) ] 2 .
D ( r ) = D l ( r ) + D ϕ ( r ) .
A 2 = B 1 2 exp ( 2 l 1 ) + B 2 2 exp ( 2 l 2 ) + 2 B 1 B 2 exp ( l 1 + l 2 ) cos ( ϕ 1 - ϕ 2 + ψ 1 - ψ 2 ) .
exp [ 2 l ( ρ 1 2 ) = 1
exp { l ( ρ 1 ) + l ( ρ 2 ) ± [ ϕ ( ρ 1 ) - ϕ ( ρ 2 ) ] } = exp [ - ½ D ( r ) ] ,
A 2 = B 1 2 + B 2 2 + 2 B 1 B 2 cos ( ψ 1 - ψ 2 ) exp [ - ½ D ( r ) ] .
A 2 = 2 B 2 { 1 + exp [ - ½ D ( r ) ] } .
A 1 2 = B 1 2 = B 2 .
A ( t ) = ( 2 ) 1 / 2 B [ 1 + cos Δ ϕ ( t ) ] 1 / 2 ,
Δ ϕ ( t ) = ϕ 1 ( t ) - ϕ 2 ( t ) .
D M 1 ( ρ ) A 0 ( ρ ) exp [ j ϕ ( ρ , t ) ] d ρ = D M 2 ( ρ ) A 0 ( ρ ) exp [ j ϕ ( ρ , t ) ] d ρ .
D M 1 ( ρ ) d ρ = D M 2 ( ρ ) d ρ .
h ( ρ , ρ ) = σ exp [ - j ( k / 2 f ) ρ · ( ρ - 2 ρ ) ] σ = - [ exp ( - j k f ) / j λ f ] .
M 1 2 ( ρ ) = f ( ρ ) exp [ + j ( k / f ) · ρ · ρ 1 2 ] ,
f ( ρ ) = ( k a f σ / π ρ ) J 1 ( k a ρ / 2 f ) exp [ - j ( k / 2 f ) ρ 2 ] , ρ = ρ .
x = ρ cos θ , x 1 2 = ρ 1 2 cos θ 1 2 , y = ρ sin θ , y 1 2 = ρ 1 2 sin θ 1 2 ,
0 J 0 ( k f ρ ρ 1 ) f ( ρ ) ρ d ρ = 0 J 0 ( k f ρ ρ 2 ) f ( ρ ) ρ d ρ .

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