Abstract

This paper defines the time-invariant detection scheme which yields the largest average signal-to-noise ratio in the heterodyne detection of a randomly distorted optical signal. It is shown that the detection scheme may be realized by properly shaping both the isophase surface and the irradiance distribution of the local-oscillator beam. Applied to the case of an atmospherically distorted optical plane wave it is shown that although the optimum local-oscillator beam differs significantly from a plane wave, the increase of average signal-to-noise ratio, over the plane-wave case as reported by Fried, is negligible (less than 10%).

© 1969 Optical Society of America

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References

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  1. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [Crossref]
  2. J. Gardner, IEEE International Convention Record 12, 337 (1964).
  3. C. W. Helstrom, J. Opt. Soc. Am. 57, 353 (1967).
    [Crossref]
  4. R. F. Lucy, Proc. IEEE 51, 162 (1963).
    [Crossref]
  5. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1953), Ch. 3.
  6. James P. Moreland, Ph.D. dissertation, Ohio State University (1967).

1967 (2)

1964 (1)

J. Gardner, IEEE International Convention Record 12, 337 (1964).

1963 (1)

R. F. Lucy, Proc. IEEE 51, 162 (1963).
[Crossref]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1953), Ch. 3.

Fried, D. L.

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

Gardner, J.

J. Gardner, IEEE International Convention Record 12, 337 (1964).

Helstrom, C. W.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1953), Ch. 3.

Lucy, R. F.

R. F. Lucy, Proc. IEEE 51, 162 (1963).
[Crossref]

Moreland, James P.

James P. Moreland, Ph.D. dissertation, Ohio State University (1967).

IEEE International Convention Record (1)

J. Gardner, IEEE International Convention Record 12, 337 (1964).

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

R. F. Lucy, Proc. IEEE 51, 162 (1963).
[Crossref]

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

Other (2)

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1953), Ch. 3.

James P. Moreland, Ph.D. dissertation, Ohio State University (1967).

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Figures (3)

Fig. 1
Fig. 1

Normalized irradiance patterns I/I0 for circular apertures of diams D = 1 4 r 0 , 1 2 r 0, r0, 2r0, and 4r0. The patterns are symmetric and only half is shown.

Fig. 2
Fig. 2

Normalized signal-to-noise ratio vs normalized aperture diameter for plane-wave local oscillator (dashed curve) and optimized local oscillator (solid curve).

Fig. 3
Fig. 3

Equipment arrangement for discussion of improvement using optical spatial filtering. The aperture a multiplies the signal field by W( x ˜), filter b multiplies the signal by G1( x ˜), and filter d multiplies the signal by G2( y ˜). The signals at various points are A: U( x ˜,t); B: U ( x ˜ , t ) W ( x ˜ ); C: U ( x ˜ , t ) W ( x ˜ ) G 1 ( x ˜ ); D: Z( y ˜,t); E: Z ( y ˜ , t ) G 2 ( y ˜ ).

Tables (1)

Tables Icon

Table I Normalized local-oscillator amplitude, B(ρ)/B(0), vs normalized distance from aperture center ρ; and normalized signal-to-noise ratio γ, for various aperture sizes D, from D = 0.25r0 to D = 4.0r0. [B(ρ)/B(0)].

Equations (20)

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E S = Re A ( x ˜ , t ) exp { j [ ω S t + ϕ ( x ˜ , t ) ] } × W ( x ˜ ) = Re U ( x ˜ , t ) W ( x ˜ ) e j ω S t ,
E L = Re B ( x ˜ ) exp { j [ ω L t + θ ( x ˜ ) ] } = Re V ( x ˜ ) e j ω L t .
P 0 = 1 2 η 2 M u ( x ˜ , x ˜ ) V * ( x ˜ ) V ( x ˜ ) d x ˜ d x ˜ ,
M u ( x ˜ , x ˜ ) = U ( x ˜ , t ) U * ( x ˜ , t ) W ( x ˜ ) W ( x ˜ ) ,
S / N = ( 1 / 2 e ) ( P 0 / P L ) ,
P L = 1 2 η V ( x ˜ ) V * ( x ˜ ) d x ˜
M u ( x ˜ , x ˜ ) V ( x ˜ ) d x ˜ = λ V ( x ˜ )
( S / N ) max = ( η / 2 e ) λ max .
tan [ θ ( x ˜ ) - θ ( x ˜ ) ] = Im M u ( x ˜ , x ˜ ) / Re M u ( x ˜ , x ˜ ) ,
B ( x ˜ ) { [ Re M u ( x ˜ , x ˜ ) ] 2 + [ Im M u ( x ˜ , x ˜ ) ] 2 } 1 2 d x ˜ = λ B ( x ˜ ) ,
M u ( x ˜ , x ˜ ) = A ¯ 2 exp [ - 3.44 ( r / r 0 ) 5 / 3 ] W ( x ˜ ) W ( x ˜ ) ,
θ ( x ˜ ) - θ ( 0 ) = 0 ,
W ( x ˜ ) W ( x ˜ ) A ¯ 2 exp [ - 3.44 ( r / r 0 ) 5 / 3 ] × B ( x ˜ ) d x ˜ = λ B ( x ˜ ) ,
γ = 8 π r 0 2 ( η / e ) A ¯ 2 ( S / N ) ,
Z ( y ˜ , t ) = { U ( x ˜ , t ) W ( x ˜ ) G 1 ( x ˜ ) } L ( x ˜ , y ˜ ) d x ˜ .
S N = 1 2 e 1 2 η 2 R * ( y ˜ ) R ( y ˜ ) G 2 ( y ˜ ) G 2 * ( y ˜ ) M Z ( y ˜ , y ˜ ) d y ˜ d y ˜ 1 2 η R ( y ˜ ) R * ( y ˜ ) d y ˜ ,
M Z ( y ˜ , y ˜ ) = Z ( y ˜ , t ) Z * ( y ˜ , t )
R * ( y ˜ ) R ( y ˜ ) = R ( y ˜ ) R ( y ˜ ) exp { - j [ θ R ( y ˜ ) - θ R ( y ˜ ) ] } = R R e - j Δ θ R G 2 ( y ˜ ) G 2 * ( y ˜ ) = G 2 G 2 e j Δ θ G 2 Z ( y ˜ , t ) Z * ( y ˜ , t ) = Z Z e j Δ ϕ Z
S / N = η 2 e R R G 2 G 2 [ Z Z cos Δ ϕ Z 2 + Z Z sin Δ ϕ Z 2 ] 1 2 cos β d y ˜ d y ˜ R 2 d y ˜ ,
tan β = cos ( Δ θ R - Δ θ G 2 ) Z Z sin Δ ϕ Z - sin ( Δ θ R - Δ θ G 2 ) Z Z cos Δ ϕ Z cos ( Δ θ R - Δ θ G 2 ) Z Z cos Δ ϕ Z 2 + sin ( Δ θ R - Δ θ G 2 ) Z Z sin Δ ϕ Z .