Abstract

The mean square of the beating component of the photodetector output current of a coherent optical or infrared receiver, illuminated by partially coherent quasi-monochromatic radiation, is expressed in terms of the mutual intensity functions of the signal and local reference fields on the detector’s active surface. The result is of interest for heterodyne detection problems in which partially coherent sources occur. The use of the result is illustrated and is compared with a known special case by treating the heterodyne detection of a spatially incoherent source in the far-field zone of the receiver aperture, a problem previously discussed by H. T. Yura [Appl. Opt. 13, 150 (1974)] from a different viewpoint.

© 1980 Optical Society of America

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References

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  1. H. T. Yura, Appl. Opt. 13, 150 (1974).
    [CrossRef] [PubMed]
  2. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, pp. 516–518.
  3. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  4. Ref. 2, pp. 519–520.
  5. S. C. Cohen, Appl. Opt. 14, 1953 (1975).
    [CrossRef] [PubMed]
  6. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), pp. 276–278, 285–287.
  7. Ref. 2, pp. 494–500.

1975

1974

1967

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, pp. 516–518.

Cohen, S. C.

Fried, D. L.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, pp. 516–518.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), pp. 276–278, 285–287.

Yura, H. T.

Appl. Opt.

Proc. IEEE

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

Other

Ref. 2, pp. 519–520.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, pp. 516–518.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, New York, 1971), pp. 276–278, 285–287.

Ref. 2, pp. 494–500.

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Figures (1)

Fig. 1
Fig. 1

Source and detector geometry.

Equations (22)

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J S ( r 1 , r 2 ) = Γ S ( r 1 , r 2 ; 0 ) , J L ( r 1 , r 2 ) = Γ L ( r 1 , r 2 ; 0 ) ,
Γ S ( r 1 , r 2 ; τ ) = J S ( r 1 , r 2 ) exp ( - i ω τ ) , Γ L ( r 1 , r 2 ; τ ) = J L ( r 1 , r 2 ) exp ( - i ω L τ ) ,
V S ( r , t ) = E S ( r , t ) exp ( - i ω t ) , V L ( r , t ) = E L ( r , t ) exp ( - i ω L t ) ,
i b A D d A η Re ( V S V L * ) ,
i b 2 = | A D d A η Re ( V S V L * ) | 2 .
i b 2 = | A D d A η Re ( V S V L * ) | 2 ¯ ,
i b 2 = A D d 2 r 1 A D d 2 r 2 η ( r 1 ) η ( r 2 ) × Re [ V S ( r 1 , t ) V L * ( r 1 , t ) ] Re [ V S ( r 2 , t ) V L * ( r 2 , t ) ] ¯ .
Re ( V S V L * ) = 1 2 ( V S V L * + V S * V L ) ,
L = 1 2 Re [ V S ( r 1 , t ) V S ( r 2 , t ) V L * ( r 1 , t ) V L * ( r 2 , t ) + V S ( r 2 , t ) V S * ( r 1 , t ) V L ( r 1 , t ) V L * ( r 2 , t ) ] .
f = 1 2 T - T T d t E S ( r 1 , t ) E S ( r 2 , t ) × E L * ( r 1 , t ) E L * ( r 2 , t ) exp ( 2 i ω 0 t ) ,
s = E S ( r 2 , t ) E S * ( r 1 , t ) E L ( r 1 , t ) E L * ( r 2 , t ) ,
V S ( r 2 , t ) V S * ( r 1 , t ) = E S ( r 2 , t ) E S * ( r 1 , t )
V L ( r 1 , t ) V L * ( r 2 , t ) = E L ( r 1 , t ) E L * ( r 2 , t ) .
i b 2 = 1 2 A D d 2 r 1 A D × d 2 r 2 η ( r 1 ) η ( r 2 ) Re [ J S ( r 2 , r 1 ) J L ( r 1 , r 2 ) ] ,
J S ( r 2 , r 1 ) = π a 2 I S R 2 2 J 1 ( k a R | r 2 - r 1 | ) k a R | r 2 - r 1 | ,
Re [ J S ( r 2 , r 1 ) J L ( r 1 , r 2 ) ] = π a 2 I S I L R 2 2 J 1 ( k a R | r 2 - r 1 | ) k a R | r 2 - r 1 | .
i b 2 = 1 2 ( η a R ) 2 π I S I L A D d 2 r 1 A D × d 2 r 2 2 J 1 ( k a R | r 2 - r 1 | ) k a R | r 2 - r 1 | .
2 J 1 ( k a R | r 2 - r 1 | ) k a R | r 2 - r 1 | = 1 π a 2 σ × d 2 x exp [ i k R x · ( r 2 - r 1 ) ] ,
i b 2 = 1 2 ( η R ) 2 I S I L σ d 2 x × | A D d 2 r exp ( i k R x · r ) | 2 ,
i b 2 = 1 2 ( 2 π η b k ) 2 π I S I L × [ 1 - J 0 2 ( k a b R ) - J 1 2 ( k a b R ) ] ,
lim t 0 1 2 ( 2 π η b k ) 2 π a 2 I S I L × 1 t 2 [ 1 - J 0 2 ( k t b R ) - J 1 2 ( k t b R ) ] ,
1 2 ( 2 π η b k ) 2 π a 2 I S I L ( k b 2 R ) 2 .

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