Abstract

An analysis is made of the problem of detecting a weak. light beam from a distant source in the presence of a background of much greater intensity, by the photoelectric heterodyne technique. In this method the incident light is superposed on the light beam from a local laser, whose frequency can be adjusted by a feedback arrangement so as to maximize a certain “beat note” in the output of the detector. With the aid of plausible assumptions it is shown that the effectiveness of the method is largely independent of the intensity of the background light, and of the fluctuation properties of the incident light. The key parameter is the number of photoelectrons released by the signal beam in a time comparable with its coherence time.

© 1966 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
    [Crossref]
  2. S. F. Jacobs and P. J. Rubinowitz in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 481.
  3. G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.
  4. L. Mandel, Phys. Rev. 138, B753 (1965).
    [Crossref]
  5. R. J. Glauber, Phys. Rev,  130, 2529 (1963).
    [Crossref]
  6. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [Crossref]
  7. E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
    [Crossref]
  8. See for example the review by L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  9. L. Mandel and E. Wolf, Phys. Rev. 149 (1966).
    [Crossref]
  10. For an introduction to coherence theory, see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., p. 491, or Ref. 8.
  11. L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
    [Crossref]
  12. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
    [Crossref]
  13. R. J. Glauber in Quantum Optics and Electronics, C. deWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach Science Publishers, New York, 1965), p. 63.
  14. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., Inc., New York, 1958).
  15. V. I. Tatarski, Wave Propagation in a Turbulent Medium, [translated] (McGraw-Hill Book Co., Inc., New York, 1961).
  16. Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
    [Crossref]

1966 (1)

L. Mandel and E. Wolf, Phys. Rev. 149 (1966).
[Crossref]

1965 (2)

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

See for example the review by L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

1964 (2)

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

1963 (4)

Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
[Crossref]

R. J. Glauber, Phys. Rev,  130, 2529 (1963).
[Crossref]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

1961 (1)

Born, M.

For an introduction to coherence theory, see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., p. 491, or Ref. 8.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., Inc., New York, 1958).

Emmons, R. B.

G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.

Forrester, A. T.

Glauber, R. J.

R. J. Glauber, Phys. Rev,  130, 2529 (1963).
[Crossref]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

R. J. Glauber in Quantum Optics and Electronics, C. deWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach Science Publishers, New York, 1965), p. 63.

Gould, G.

Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
[Crossref]

Harned, B.

G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.

Jacobs, S. F.

S. F. Jacobs and P. J. Rubinowitz in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 481.

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

LaTourrette, J.

Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
[Crossref]

Lucovsky, G.

G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.

Mandel, L.

L. Mandel and E. Wolf, Phys. Rev. 149 (1966).
[Crossref]

See for example the review by L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

Powers, J. K.

G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.

Rabinowitz, P.

Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
[Crossref]

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., Inc., New York, 1958).

Rubinowitz, P. J.

S. F. Jacobs and P. J. Rubinowitz in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 481.

Sudarshan, E. C. G.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium, [translated] (McGraw-Hill Book Co., Inc., New York, 1961).

Wolf, E.

L. Mandel and E. Wolf, Phys. Rev. 149 (1966).
[Crossref]

See for example the review by L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

For an introduction to coherence theory, see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., p. 491, or Ref. 8.

J. Opt. Soc. Am. (1)

Phys. Rev (1)

R. J. Glauber, Phys. Rev,  130, 2529 (1963).
[Crossref]

Phys. Rev. (4)

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[Crossref]

L. Mandel and E. Wolf, Phys. Rev. 149 (1966).
[Crossref]

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

Phys. Rev. Letters (1)

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

Proc. IEEE (1)

Compare the arrangement of P. Rabinowitz, J. LaTourrette, and G. Gould, Proc. IEEE 51, 857 (1963).
[Crossref]

Proc. Phys. Soc. (London) (1)

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

Rev. Mod. Phys. (1)

See for example the review by L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Other (6)

For an introduction to coherence theory, see M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., p. 491, or Ref. 8.

S. F. Jacobs and P. J. Rubinowitz in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 481.

G. Lucovsky, R. B. Emmons, B. Harned, and J. K. Powers in Quantum Electronics III, edited by P. Grivet and N. Bloembergen (Dunod Cie., Paris; Columbia University Press, New York, 1964), p. 1731.

R. J. Glauber in Quantum Optics and Electronics, C. deWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach Science Publishers, New York, 1965), p. 63.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., Inc., New York, 1958).

V. I. Tatarski, Wave Propagation in a Turbulent Medium, [translated] (McGraw-Hill Book Co., Inc., New York, 1961).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

F. 1
F. 1

Outline of the setup for heterodyne detection.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

ρ ̂ = ϕ ( { ϑ λ } ) | { ϑ λ } { ϑ λ } | d 2 { ϑ λ } ,
V ( t ) = V 0 ( t ) + V 1 ( t ) + V 2 ( t ) ,
V * ( t ) · V ( t ) I = V 0 * ( t ) · V 0 ( t ) + V 1 * ( t ) · V 1 ( t ) + V 2 * ( t ) · V 2 ( t ) I 0 + I 1 + I 2 ,
p 1 ( t ) δ t = α I ( t ) δ t ,
p N ( t 1 , , t N ) δ t 1 δ t N = α N I ( t 1 ) I ( t N ) δ t 1 δ t N .
S ( t ) = K j δ ( t t j ) ,
R ( τ ) = S ( t ) S ( t + τ ) = K 2 Σ Σ i j δ ( t t j ) δ ( t t j + τ ) + K 2 Σ i δ ( t t j ) δ ( t t i + τ ) ,
R ( τ ) = K 2 α 2 δ ( t t i ) δ ( t t j + τ ) I ( t i ) I ( t j ) d t i d t j + K 2 α δ ( t t i ) δ ( t t i + τ ) I d t i = K 2 [ α 2 I ( t ) I ( t + τ ) α I δ ( τ ) ] .
R ( τ ) = K 2 [ 2 α 2 ( I 0 I 1 + I 0 I 2 + I 1 I 2 ) + α 2 I 0 ( t ) I 0 ( t + τ ) + α 2 I 1 ( t ) I 1 ( t + τ ) + α 2 I 2 ( t ) I 2 ( t + τ ) + α ( I 0 + I 1 + I 2 ) δ ( τ ) + α 2 { I 0 I 1 γ 00 ( τ ) γ 11 * ( τ ) + I 0 I 2 γ 00 ( τ ) γ 22 * ( τ ) + I 1 I 2 γ 11 ( τ ) γ 22 * ( τ ) + c . c . } ] ,
I 0 γ 00 ( τ ) = V 0 * ( t ) · V 0 ( t + τ ) , I 1 γ 11 ( τ ) = V 1 * ( t ) · V 1 ( t + τ ) , I 2 γ 22 ( τ ) = V 2 * ( t ) · V 2 ( t + τ ) . }
I 0 ( t ) I 0 ( t + τ ) = I 0 2 .
I 2 ( t ) I 2 ( t + τ ) = I 2 2 [ 1 + | γ 22 ( τ ) | 2 ] .
I 1 ( t ) I 1 ( t + τ ) = I 1 2 [ 1 + f 11 ( τ ) ] ,
R ( τ ) = K 2 [ α 2 ( I 0 + I 1 + I 2 ) 2 + α 2 I 1 2 f 11 ( τ ) + α 2 I 2 2 | γ 22 ( τ ) | 2 + α 2 ( I 0 I 1 γ 00 ( τ ) γ 11 * ( τ ) + I 0 I 2 γ 00 ( τ ) γ 22 * ( τ ) + I 1 I 2 γ 11 ( τ ) γ 22 * ( τ ) + c . c . ) + α ( I 0 + I 1 + I 2 ) δ ( τ ) ] .
γ 00 ( τ ) = 0 ϕ 00 ( ν ) exp ( 2 π i ν τ ) d ν , γ 11 ( τ ) = 0 ϕ 11 ( ν ) exp ( 2 π i ν τ ) d ν , γ 22 ( τ ) = 0 ϕ 22 ( ν ) exp ( 2 π i ν τ ) d ν , }
f 11 ( τ ) = g 11 ( ν ) exp ( 2 π i ν τ ) d ν ,
| γ 22 ( τ ) | 2 exp ( 2 π i ν τ ) d τ = 0 0 ϕ 22 ( ν ) ϕ 22 ( ν ) exp [ 2 π i ( ν + ν ν ) τ ] d ν d ν d τ , = 0 0 ϕ 22 ( ν ) ϕ 22 ( ν ) δ ( ν + ν + ν ) d ν d ν = 0 ϕ 22 ( ν ) ϕ 22 ( ν + ν ) d ν ,
[ γ 00 ( τ ) γ 11 * ( τ ) + c . c . ] exp ( 2 π i ν τ ) d τ = 0 3 2 [ ϕ 00 ( ν ) ϕ 11 ( ν + ν ) + ϕ 11 ( ν ) ϕ 00 ( ν + ν ) ] d ν ,
0 ϕ 22 ( ν ) ϕ 22 ( ν + ν ) d ν ψ 22 ( ν ) , 0 ϕ 00 ( ν ) ϕ 11 ( ν + ν ) d ν ψ 01 ( ν ) , 0 ϕ 11 ( ν ) ϕ 00 ( ν + ν ) d ν ψ 10 ( ν ) , }
χ ( ν ) = K 2 α 2 { ( I 0 + I 1 + I 2 ) 2 δ ( ν ) + I 1 2 g 11 ( ν ) + I 2 2 ψ 22 ( ν ) + I 0 I 1 [ ψ 01 ( ν ) + ψ 10 ( ν ) ] + I 0 I 2 [ ψ 02 ( ν ) + ψ 20 ( ν ) ] + I 1 I 2 [ ψ 12 ( ν ) + ψ 21 ( ν ) ] } + K 2 α ( I 0 + I 1 + I 2 ) .
Q 2 ( t ) = χ ( ν ) | B ( ν ) | 2 d ν .
[ ψ s r ( ν ) + ψ r s ( ν ) ] | B ( ν ) | 2 d ν 2 [ ψ s r ( ν ) + ψ r s ( ν ) ] 0 | B ( ν ) | 2 d ν , r , s = 0 , 1 , 2
g 11 ( ν ) | B ( ν ) | 2 δ ν 2 g 11 ( ν ) 0 | B ( ν ) | 2 d ν ,
Q 2 ( t ) = { α 2 I 1 2 g 11 ( ν ) + α 2 I 2 2 ψ 22 ( ν ) + α 2 I 0 I 1 [ ψ 01 ( ν ) + ψ 10 ( ν ) ] + α 2 I 0 I 2 [ ψ 02 ( ν ) + ψ 20 ( ν ) ] + α 2 I 1 I 2 [ ψ 12 ( ν ) + ψ 21 ( ν ) ] + α ( I 0 + I 1 + I 2 ) } 2 K 2 0 | B ( ν ) | 2 d ν .
δ ν ν Δ ν 2 ν 0 , ν 1 , ν 2 Δ ν 0 < Δ ν 1 Δ ν 2 ν 0 , ν 1 , ν 2 }
ψ 22 ( ν ) ψ 22 ( 0 ) 1 / Δ ν 2 ,
ψ 02 ( ν ) ϕ 22 ( ν + ν 0 ) ϕ 22 ( ν 0 ) ~ 1 / Δ ν 2 , ψ 20 ( ν ) ϕ 22 ( ν + ν 0 ) ϕ 22 ( ν 0 ) ~ 1 / Δ ν 2 , ψ 12 ( ν ) ϕ 22 ( ν + ν 1 ) ϕ 22 ( ν 1 ) ~ 1 / Δ ν 2 , ψ 21 ( ν ) ϕ 22 ( ν + ν 1 ) ϕ 22 ( ν 1 ) ~ 1 / Δ ν 2 , }
g 11 ( ν ) = β / Δ ν 1 ,
1 / Δ f max ψ 10 ( ν ) = 0 ϕ 00 ( ν ) ϕ 11 ( ν + ν 1 ν 0 ) d ν = ϕ 00 ( ν 0 + ν ) ϕ 11 ( ν 1 + ν ) d ν = max ψ 10 ( ν ) .
ϕ 00 2 ( ν 0 + ν ) d ν = 1 / Δ ν 0
ϕ 11 2 ( ν 1 + ν ) d ν = 1 / Δ ν 1 ,
1 / Δ f 1 / ( Δ ν 0 Δ ν 1 ) 1 2 .
Δ f Δ ν 1 .
max Q 2 ( t ) [ α 2 I 1 2 β / Δ ν 1 + α 2 I 2 2 / Δ ν 2 + α 2 I 0 I 1 / Δ ν 1 + 2 α 2 I 2 ( I 0 + I 1 ) / Δ ν 2 + α ( I 0 + I 1 + I 2 ) ] 2 K 2 0 | B ( ν ) | 2 d ν .
I 1 I 0 I 2 I 0 , }
max Q 2 ( t ) α I 0 [ α I 1 / Δ ν 1 + 2 α I 2 / Δ ν 2 + 1 ] 2 K 2 0 | B ( ν ) | 2 d ν .
max Q 2 ( t ) on max Q 2 ( t ) off max Q 2 ( t ) off α I 1 / Δ ν 1 1 + 2 α I 2 / Δ ν 2 ,
α I 1 / Δ ν 1 δ 1 , α I 2 / Δ ν 2 δ 2 ,
δ 2 1 ,
= δ 1 .
V 2 * ( r ; t ) · V 2 * ( r ; t ) = I 2 ( r ) δ r · r ,
V ( r ; t ) = ( 1 / n ) V 0 ( t ) + ( 1 / n ) V 1 ( t ) + V 2 ( r ; t ) ,
I ( r ; t ) = ( 1 / n ) I 0 ( t ) + ( 1 / n ) I 1 ( t ) + I 2 ( r ; t ) + [ ( 1 / n ) V 0 * ( t ) · V 1 ( t ) + c . c . ] + [ ( 1 / n ) V 0 * ( t ) · V 2 ( r ; t ) + c . c . ] + [ ( 1 / n ) V 1 * ( t ) · V 2 * ( r ; t ) + c . c . ] ,
R ( τ ) S ( t ) S ( t + τ ) = K 2 [ α 2 τ τ I ( r ; t ) I ( r ; t + τ ) + α τ I ( r ) δ ( τ ) ] ,
Q 2 ( t ) = { α 2 I 1 2 g 11 ( ν ) + ( α 2 / n ) I 2 2 ψ 22 ( ν ) + α 2 I 0 I 1 [ ψ 01 ( ν ) + ψ 10 ( ν ) ] + ( α 2 / n ) I 0 I 2 [ ψ 02 ( ν ) + ψ 20 ( ν ) ] + ( α 2 / n ) I 1 I 2 [ ψ 12 ( ν ) + ψ 21 ( ν ) ] + α ( I 0 + I 1 + I 2 ) } 2 K 2 0 | B ( ν ) | 2 d ν ,
max Q 2 ( t ) = α I 0 [ δ 1 + 2 δ 2 + 1 ] 2 K 2 0 3 2 | B ( ν ) | 2 d ν ,
δ 1 / ( 1 + 2 δ 2 ) δ 1 ,