Abstract

A simple optical superposition experiment is analyzed in detail and a comparison is made between the photoelectric signal modulation which is characteristic of interference and the Alford and Gold effect. Some significant differences are pointed out. It is shown that the Alford and Gold effect is of the first order in the degeneracy parameter and of the second order in the degree of coherence, so that it resembles the intensity correlation effect discovered by Hanbury Brown and Twiss.

© 1962 Optical Society of America

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References

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  1. W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).
    [CrossRef]
  2. M. P. Givens, J. Opt. Soc. Am. 51, 1030, 1032 (1961).
    [CrossRef]
  3. L. Mandel [to be published].
  4. R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).
  5. Some similarities to a method of analyzing the frequency modulation spectrum of microwaves due to I. Berstein [Proc. Inst. Radio Engrs. 45, 94 (1957)] should also be pointed out.
  6. M. P. Givens [J. Opt. Soc. Am. 52, 225 (1962)] has more recently also drawn attention to this.
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).
  8. E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).
  9. L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
    [CrossRef]
  10. We are here using the Fourier transform conventionϕ11(ν)=∫-∞∞γ11(τ) exp-(2πiντ)dτ,which differs slightly from that used by Born and Wolf7 and makes the functions V1(t), γ11(τ), etc. analytic in the upper half—rather than the lower half—complex plane.
  11. E. M. Purcell, Nature 178, 1449 (1956).
    [CrossRef]
  12. L. Mandel, Proc. Phys. Soc. (London) 71, 1937 (1958).
  13. L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
    [CrossRef]
  14. L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).
    [CrossRef]
  15. L. Mandel (to be published in Progress in Optics, 1962).
  16. S. O. Rice, Bell System Tech. J. 23, 1, 282 (1944).
    [CrossRef]
  17. E. Parzen and N. Shiren, J. Maths and Phys. 35, 278 (1956).
  18. A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
    [CrossRef]
  19. A. W. Smith and G. W. Williams, J. Opt. Soc. Am. 52, 337 (1962).
    [CrossRef]
  20. D. Gabor, Prog. in Opt. 1, 111 (1961).
  21. L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
    [CrossRef]
  22. M. P. Givens [J. Opt. Soc. Am. 52, 225 (1962)] has recently drawn attention to this also.
    [CrossRef]
  23. A. Javan, W. R. Bennett, and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
    [CrossRef]
  24. B. J. McMurtry and A. E. Siegman, Appl. Opt. 1, 51 (1962).
    [CrossRef]
  25. A. Javan, E. A. Ballik, and W. L. Bond, J. Opt. Soc. Am. 52, 96 (1962).
    [CrossRef]
  26. V. V. Solodovnikov, Introduction to the Statistical Dynamics of Automatic Control Systems (Dover Publications, Inc., New York, 1960).
  27. This is true even if a separate filter is dispensed with. For the photodetector itself has a limited frequency response and must be regarded as a low-pass filter.
  28. L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
    [CrossRef]
  29. See L. Mandel and E. Wolf[Proc. Phys. Soc. (London) 80894, 1962] for a discussion of bandwidth and coherence time in this case.
    [CrossRef]

1962 (6)

1961 (8)

1959 (1)

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

1958 (2)

W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).
[CrossRef]

L. Mandel, Proc. Phys. Soc. (London) 71, 1937 (1958).

1957 (2)

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).

Some similarities to a method of analyzing the frequency modulation spectrum of microwaves due to I. Berstein [Proc. Inst. Radio Engrs. 45, 94 (1957)] should also be pointed out.

1956 (2)

E. M. Purcell, Nature 178, 1449 (1956).
[CrossRef]

E. Parzen and N. Shiren, J. Maths and Phys. 35, 278 (1956).

1955 (1)

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

1944 (1)

S. O. Rice, Bell System Tech. J. 23, 1, 282 (1944).
[CrossRef]

Alford, W. P.

W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).
[CrossRef]

Ballik, E. A.

Bennett, W. R.

A. Javan, W. R. Bennett, and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Berstein, I.

Some similarities to a method of analyzing the frequency modulation spectrum of microwaves due to I. Berstein [Proc. Inst. Radio Engrs. 45, 94 (1957)] should also be pointed out.

Bond, W. L.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Forrester, A. T.

Gabor, D.

D. Gabor, Prog. in Opt. 1, 111 (1961).

Givens, M. P.

Gold, A.

W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).
[CrossRef]

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).

Herriott, D. R.

A. Javan, W. R. Bennett, and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Javan, A.

A. Javan, E. A. Ballik, and W. L. Bond, J. Opt. Soc. Am. 52, 96 (1962).
[CrossRef]

A. Javan, W. R. Bennett, and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Mandel, L.

See L. Mandel and E. Wolf[Proc. Phys. Soc. (London) 80894, 1962] for a discussion of bandwidth and coherence time in this case.
[CrossRef]

L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
[CrossRef]

L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
[CrossRef]

L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
[CrossRef]

L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).
[CrossRef]

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

L. Mandel, Proc. Phys. Soc. (London) 71, 1937 (1958).

L. Mandel [to be published].

L. Mandel (to be published in Progress in Optics, 1962).

McMurtry, B. J.

Parzen, E.

E. Parzen and N. Shiren, J. Maths and Phys. 35, 278 (1956).

Purcell, E. M.

E. M. Purcell, Nature 178, 1449 (1956).
[CrossRef]

Rice, S. O.

S. O. Rice, Bell System Tech. J. 23, 1, 282 (1944).
[CrossRef]

Shiren, N.

E. Parzen and N. Shiren, J. Maths and Phys. 35, 278 (1956).

Siegman, A. E.

Smith, A. W.

Solodovnikov, V. V.

V. V. Solodovnikov, Introduction to the Statistical Dynamics of Automatic Control Systems (Dover Publications, Inc., New York, 1960).

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).

Williams, G. W.

Wolf, E.

See L. Mandel and E. Wolf[Proc. Phys. Soc. (London) 80894, 1962] for a discussion of bandwidth and coherence time in this case.
[CrossRef]

L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
[CrossRef]

L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).
[CrossRef]

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Am. J. Phys. (1)

W. P. Alford and A. Gold, Am. J. Phys. 26, 481 (1958).
[CrossRef]

Appl. Opt. (1)

Bell System Tech. J. (1)

S. O. Rice, Bell System Tech. J. 23, 1, 282 (1944).
[CrossRef]

J. Maths and Phys. (1)

E. Parzen and N. Shiren, J. Maths and Phys. 35, 278 (1956).

J. Opt. Soc. Am. (9)

Nature (1)

E. M. Purcell, Nature 178, 1449 (1956).
[CrossRef]

Phys. Rev. (1)

L. Mandel and E. Wolf, Phys. Rev. 124, 1696 (1961).
[CrossRef]

Phys. Rev. Letters (1)

A. Javan, W. R. Bennett, and D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Proc. Inst. Radio Engrs. (1)

Some similarities to a method of analyzing the frequency modulation spectrum of microwaves due to I. Berstein [Proc. Inst. Radio Engrs. 45, 94 (1957)] should also be pointed out.

Proc. Phys. Soc. (London) (3)

L. Mandel, Proc. Phys. Soc. (London) 71, 1937 (1958).

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[CrossRef]

See L. Mandel and E. Wolf[Proc. Phys. Soc. (London) 80894, 1962] for a discussion of bandwidth and coherence time in this case.
[CrossRef]

Proc. Roy. Soc. (London) (2)

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).

Prog. in Opt. (1)

D. Gabor, Prog. in Opt. 1, 111 (1961).

Other (6)

L. Mandel (to be published in Progress in Optics, 1962).

L. Mandel [to be published].

We are here using the Fourier transform conventionϕ11(ν)=∫-∞∞γ11(τ) exp-(2πiντ)dτ,which differs slightly from that used by Born and Wolf7 and makes the functions V1(t), γ11(τ), etc. analytic in the upper half—rather than the lower half—complex plane.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

V. V. Solodovnikov, Introduction to the Statistical Dynamics of Automatic Control Systems (Dover Publications, Inc., New York, 1960).

This is true even if a separate filter is dispensed with. For the photodetector itself has a limited frequency response and must be regarded as a low-pass filter.

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Figures (1)

Fig. 1
Fig. 1

Illustrating the superposition experiment.

Tables (1)

Tables Icon

Table I Comparison of the signal modulations.

Equations (32)

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V 3 ( t ) = V 1 ( t ) + V 2 ( t - T ) .
Ī 3 γ 33 ( τ ) = ( Ī 1 + Ī 2 ) γ 11 ( τ ) + ( Ī 1 Ī 2 ) 1 2 × [ γ 12 ( τ + T ) + γ 21 ( τ - T ) ] ,
γ 12 ( τ ) = γ 12 ( 0 ) γ 11 ( τ ) .
Ī 3 ϕ 33 ( ν ) = [ Ī 1 + Ī 2 + 2 ( Ī 1 Ī 2 ) 1 2 γ 12 ( 0 ) cos ( 2 π ν T ) ] ϕ 11 ( ν ) .
Ī 3 = Ī 1 + Ī 2 + 2 ( Ī 1 Ī 2 ) 1 2 γ 12 ( 0 ) cos ( 2 π ν 0 T )
ϕ 33 ( ν ) = ϕ 11 ( ν ) .
Ī 3 = Ī 1 + Ī 2
ϕ 33 ( ν ) = ϕ 11 ( ν ) × { 1 + [ 2 ( Ī 1 Ī 2 ) 1 2 / Ī 1 + Ī 2 ] γ 12 ( 0 ) cos ( 2 π ν T ) } .
S 1 ( t ) = I 4 ( t ) + N 4 ( t ) ,
S ¯ 1 = Ī 4 .
Ī 4 ϕ 44 ( ν ) = Ī 3 ϕ 33 ( ν ) α ( ν ) ,
ϕ 44 ( ν ) = ϕ 33 ( ν ) α ( ν ) / 0 ϕ 33 ( ν ) α ( ν ) d ν = ϕ 33 ( ν ) α ( ν ) / α ¯
Ī 4 = Ī 3 0 ϕ 33 ( ν ) α ( ν ) d ν = α ¯ Ī 3 ,
α ¯ = 0 ϕ 11 ( ν ) α ( ν ) d ν .
α ¯ = 0 ϕ 11 ( ν ) α ( ν ) d ν .
χ 11 ( ν ) = Δ 2 I ¯ 4 ψ 44 ( ν ) + Ī 4 ,
Δ 2 I 4 = Ī 4 2 ,
ψ 44 ( ν ) = 0 ϕ 44 ( ν + ν ) ϕ 44 ( ν ) d ν .
χ 11 ( ν ) = Ī 4 [ 1 + Ī 4 ψ 44 ( ν ) ] .
ψ 44 2 ( ν ) 0 ϕ 44 2 ( ν + ν ) d ν 0 ϕ 44 2 ( ν ) d ν ψ 44 2 ( 0 ) ,
ψ 44 ( ν ) ψ 44 ( 0 ) = 0 ϕ 44 2 ( ν ) d ν .
ψ 44 ( ν ) = 1 α ¯ 2 0 α ( ν + ν ) ϕ 11 ( ν + ν ) α ( ν ) ϕ 11 ( ν ) d ν ,
ψ 44 ( ν ) = ψ 44 I ( ν ) × { 1 + [ 2 Ī 1 Ī 2 / ( Ī 1 + Ī 2 ) 2 ] γ 12 2 ( 0 ) cos ( 2 π ν T ) } ,
S 2 ( t ) = 0 S 1 ( t - t ) b ( t ) d t ,
S ¯ 2 = S ¯ 1 0 b ( t ) d t = S ¯ 1 B ( 0 ) = α ¯ Ī 3 B ( 0 ) ,
χ 22 ( ν ) = B ( ν ) 2 χ 11 ( ν ) .
χ 22 ( ν ) = α ¯ Ī 3 B ( ν ) 2 [ 1 + α ¯ Ī 3 ψ 44 ( ν ) ] .
S ¯ 2 2 = S ¯ 2 2 + Δ 2 S 2 ¯ ( t ) = α ¯ 2 Ī 3 2 B 2 ( 0 ) + α ¯ Ī 3 - B ( ν ) 2 [ 1 + α ¯ Ī 3 ψ 44 ( ν ) ] d ν .
S 2 2 ¯ = α ¯ ( Ī 1 + Ī 2 ) × - B ( ν ) 2 d ν { 1 + 2 ( Ī 1 Ī 2 ) 1 2 Ī 1 + Ī 2 γ 12 ( 0 ) cos ( 2 π ν 0 T ) } × { 1 + α ¯ ( Ī 1 + Ī 2 ) ψ 44 I ( ν 1 ) × [ 1 + 2 ( Ī 1 Ī 2 ) 1 2 Ī 1 + Ī 2 γ 12 ( 0 ) ( cos 2 π ν 0 T ) ] }
S 2 2 ¯ = α ¯ ( Ī 1 + Ī 2 ) - B ( ν ) 2 d ν { 1 + α ¯ ( Ī 1 + Ī 2 ) ψ 44 I ( ν 1 ) × [ 1 + 2 Ī 1 Ī 2 ( Ī 1 + Ī 2 ) 2 γ 12 2 ( 0 ) cos ( 2 π ν 1 T ) ] } ,
ϕ 11 ( ν ) = 1 2 θ 11 ( ν + 1 2 ν 1 ) + 1 2 θ 11 ( ν - 1 2 ν 1 ) ,
ϕ11(ν)=-γ11(τ)exp-(2πiντ)dτ,