Abstract

Interference fringes produced by the superposition of light beams from two independent lasers have been investigated under conditions where the light flux was so low that one photon was usually received at the detector before the next one was emitted by one or the other source. Because the average number of photons detected in each observation was only about 10, a photon correlation technique was used to demonstrate the presence of interference fringes. The measurement procedure was automated, and this led to much greater statistical accuracy than was previously reported. The effect of varying the observation time and the number of interference fringes sampled was investigated, and found to be in good agreement with theory.

© 1968 Optical Society of America

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References

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  1. G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).
    [Crossref]
  2. R. L. Pfleegor and L. Mandel, Phys. Letters 24A, 766 (1967); Phys. Rev. 159, 1084 (1967).
  3. The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
    [Crossref]
  4. An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
    [Crossref]
  5. L. Mandel, in Proceedings of the “Enrico Fermi” International School of Physics, Varenna, 1967 (Academic Press Inc., New York, 1968).
  6. A very different kind of interference experiment at very low light levels has recently been reported by Yu. P. Dontsov and A. I. Baz [Zh. Eksp. Teor. Fiz. 52, 3 (1967); English Trans.: Sov. Phys.—JETP 25, 1 (1967)]. These authors have investigated the visibility of interference fringes produced in a conventional Fabry–Perot interferometer by strongly attenuated light beams, and claim that interference fringes disappear at sufficiently low light levels. It is possible that the effect they observed is connected with lack of spatial coherence of the wavefront, since they made use of a large-aperture diaphragm at low light levels and a small-aperture diaphragm at high light levels. However, if there should be no elementary explanation of the observed effect, the results would seem to be at variance with the usual theory of photon detection, unlike the results of our experiment, which are shown to be consistent with theory. Interference effects produced in a Michelson interferometer at very low light levels have in recent years been investigated by L. Jánossy and Zs. Náray [Acta Phys. Hung. 7, 403 (1957)], who found no dependence on the light flux.
    [Crossref]
  7. We are indebted to Dr. N. Isenor for the suggestion of using a stack of suitably cut microscope cover slips for the interference detector.
  8. J. F. Kenney and E. S. Keeping, Mathematics of Statistics, Part 2, 2nd ed. (D. Van Nostrand Co., New York, 1951), Ch. 8.
  9. L. Mandel, in Progress in Optics2, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
    [Crossref]
  10. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [Crossref]
  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, New York, 1965), p. 63.
  14. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  15. E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
    [Crossref]
  16. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [Crossref]
  17. C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. 138, B274 (1965).
    [Crossref]
  18. J. R. Klauder, Phys. Rev. Letters 16, 534 (1966).
    [Crossref]
  19. L. Mandel, Phys. Rev. 138, B753 (1965).
    [Crossref]
  20. This factor was not included in the theory given previously, in Ref. 2, which was based on the assumption that ΔνT≪ 1. That the significance of the factor was overlooked in matching experimental data taken with ΔνT~ 0.6 is due to an error, for a factor 1/N2 was mistakenly included in Ref. 2 in the expression for F given by Eq. (15) above, and, as it happens, 1/N2 is comparable with [sin(πΔνT)/(πΔνT)]2 under the conditions of the experiment.
  21. L. Mandel, J. Opt. Soc. Am. 52, 1407 (1962).
    [Crossref]

1967 (2)

R. L. Pfleegor and L. Mandel, Phys. Letters 24A, 766 (1967); Phys. Rev. 159, 1084 (1967).

A very different kind of interference experiment at very low light levels has recently been reported by Yu. P. Dontsov and A. I. Baz [Zh. Eksp. Teor. Fiz. 52, 3 (1967); English Trans.: Sov. Phys.—JETP 25, 1 (1967)]. These authors have investigated the visibility of interference fringes produced in a conventional Fabry–Perot interferometer by strongly attenuated light beams, and claim that interference fringes disappear at sufficiently low light levels. It is possible that the effect they observed is connected with lack of spatial coherence of the wavefront, since they made use of a large-aperture diaphragm at low light levels and a small-aperture diaphragm at high light levels. However, if there should be no elementary explanation of the observed effect, the results would seem to be at variance with the usual theory of photon detection, unlike the results of our experiment, which are shown to be consistent with theory. Interference effects produced in a Michelson interferometer at very low light levels have in recent years been investigated by L. Jánossy and Zs. Náray [Acta Phys. Hung. 7, 403 (1957)], who found no dependence on the light flux.
[Crossref]

1966 (1)

J. R. Klauder, Phys. Rev. Letters 16, 534 (1966).
[Crossref]

1965 (3)

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

C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. 138, B274 (1965).
[Crossref]

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 (5)

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

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

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

The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
[Crossref]

G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).
[Crossref]

1962 (1)

1955 (1)

An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Baz, A. I.

A very different kind of interference experiment at very low light levels has recently been reported by Yu. P. Dontsov and A. I. Baz [Zh. Eksp. Teor. Fiz. 52, 3 (1967); English Trans.: Sov. Phys.—JETP 25, 1 (1967)]. These authors have investigated the visibility of interference fringes produced in a conventional Fabry–Perot interferometer by strongly attenuated light beams, and claim that interference fringes disappear at sufficiently low light levels. It is possible that the effect they observed is connected with lack of spatial coherence of the wavefront, since they made use of a large-aperture diaphragm at low light levels and a small-aperture diaphragm at high light levels. However, if there should be no elementary explanation of the observed effect, the results would seem to be at variance with the usual theory of photon detection, unlike the results of our experiment, which are shown to be consistent with theory. Interference effects produced in a Michelson interferometer at very low light levels have in recent years been investigated by L. Jánossy and Zs. Náray [Acta Phys. Hung. 7, 403 (1957)], who found no dependence on the light flux.
[Crossref]

Brunner, W.

The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
[Crossref]

Dontsov, Yu. P.

A very different kind of interference experiment at very low light levels has recently been reported by Yu. P. Dontsov and A. I. Baz [Zh. Eksp. Teor. Fiz. 52, 3 (1967); English Trans.: Sov. Phys.—JETP 25, 1 (1967)]. These authors have investigated the visibility of interference fringes produced in a conventional Fabry–Perot interferometer by strongly attenuated light beams, and claim that interference fringes disappear at sufficiently low light levels. It is possible that the effect they observed is connected with lack of spatial coherence of the wavefront, since they made use of a large-aperture diaphragm at low light levels and a small-aperture diaphragm at high light levels. However, if there should be no elementary explanation of the observed effect, the results would seem to be at variance with the usual theory of photon detection, unlike the results of our experiment, which are shown to be consistent with theory. Interference effects produced in a Michelson interferometer at very low light levels have in recent years been investigated by L. Jánossy and Zs. Náray [Acta Phys. Hung. 7, 403 (1957)], who found no dependence on the light flux.
[Crossref]

Forrester, A. T.

An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

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, New York, 1965), p. 63.

Gudmundsen, R. A.

An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Johnson, P. O.

An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

Keeping, E. S.

J. F. Kenney and E. S. Keeping, Mathematics of Statistics, Part 2, 2nd ed. (D. Van Nostrand Co., New York, 1951), Ch. 8.

Kelley, P. L.

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

Kenney, J. F.

J. F. Kenney and E. S. Keeping, Mathematics of Statistics, Part 2, 2nd ed. (D. Van Nostrand Co., New York, 1951), Ch. 8.

Klauder, J. R.

J. R. Klauder, Phys. Rev. Letters 16, 534 (1966).
[Crossref]

Kleiner, W. H.

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

Magyar, G.

G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).
[Crossref]

Mandel, L.

R. L. Pfleegor and L. Mandel, Phys. Letters 24A, 766 (1967); Phys. Rev. 159, 1084 (1967).

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]

G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).
[Crossref]

L. Mandel, J. Opt. Soc. Am. 52, 1407 (1962).
[Crossref]

L. Mandel, in Progress in Optics2, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
[Crossref]

L. Mandel, in Proceedings of the “Enrico Fermi” International School of Physics, Varenna, 1967 (Academic Press Inc., New York, 1968).

Mehta, C. L.

C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. 138, B274 (1965).
[Crossref]

Paul, H.

The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
[Crossref]

Pfleegor, R. L.

R. L. Pfleegor and L. Mandel, Phys. Letters 24A, 766 (1967); Phys. Rev. 159, 1084 (1967).

Richter, G.

The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
[Crossref]

Sudarshan, E. C. G.

C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. 138, B274 (1965).
[Crossref]

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]

Wolf, E.

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]

Ann. Physik (1)

The quantum theory of interference effects produced by the superposition of independent light beams has also been discussed by H. Paul, W. Brunner, and G. Richter, Ann. Physik 12, 325 (1963); L. Mandel, Phys. Rev. 134, A10 (1964); T. F. Jordan and F. Ghielmetti, Phys. Rev. Letters 12, 607 (1964); H. Haken, Phys. Rev. Letters 13, 329 (1964); H. Paul, Ann. Physik 19, 210 (1967).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (London) (1)

G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).
[Crossref]

Phys. Letters (1)

R. L. Pfleegor and L. Mandel, Phys. Letters 24A, 766 (1967); Phys. Rev. 159, 1084 (1967).

Phys. Rev. (6)

An experiment to demonstrate beats between two independent light beams by a statistical technique was performed some years ago by A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[Crossref]

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

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

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

C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. 138, B274 (1965).
[Crossref]

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

Phys. Rev. Letters (2)

J. R. Klauder, Phys. Rev. Letters 16, 534 (1966).
[Crossref]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (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)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

A very different kind of interference experiment at very low light levels has recently been reported by Yu. P. Dontsov and A. I. Baz [Zh. Eksp. Teor. Fiz. 52, 3 (1967); English Trans.: Sov. Phys.—JETP 25, 1 (1967)]. These authors have investigated the visibility of interference fringes produced in a conventional Fabry–Perot interferometer by strongly attenuated light beams, and claim that interference fringes disappear at sufficiently low light levels. It is possible that the effect they observed is connected with lack of spatial coherence of the wavefront, since they made use of a large-aperture diaphragm at low light levels and a small-aperture diaphragm at high light levels. However, if there should be no elementary explanation of the observed effect, the results would seem to be at variance with the usual theory of photon detection, unlike the results of our experiment, which are shown to be consistent with theory. Interference effects produced in a Michelson interferometer at very low light levels have in recent years been investigated by L. Jánossy and Zs. Náray [Acta Phys. Hung. 7, 403 (1957)], who found no dependence on the light flux.
[Crossref]

Other (6)

We are indebted to Dr. N. Isenor for the suggestion of using a stack of suitably cut microscope cover slips for the interference detector.

J. F. Kenney and E. S. Keeping, Mathematics of Statistics, Part 2, 2nd ed. (D. Van Nostrand Co., New York, 1951), Ch. 8.

L. Mandel, in Progress in Optics2, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963), p. 181.
[Crossref]

L. Mandel, in Proceedings of the “Enrico Fermi” International School of Physics, Varenna, 1967 (Academic Press Inc., New York, 1968).

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

This factor was not included in the theory given previously, in Ref. 2, which was based on the assumption that ΔνT≪ 1. That the significance of the factor was overlooked in matching experimental data taken with ΔνT~ 0.6 is due to an error, for a factor 1/N2 was mistakenly included in Ref. 2 in the expression for F given by Eq. (15) above, and, as it happens, 1/N2 is comparable with [sin(πΔνT)/(πΔνT)]2 under the conditions of the experiment.

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Figures (5)

Fig. 1
Fig. 1

Outline of the experimental setup. The insert shows details of the interference detector R.

Fig. 2
Fig. 2

Block diagram of the electronics used in the experiment.

Fig. 3
Fig. 3

Experimental values of the correlation coefficient r obtained for different ratios L/l, superimposed on the theoretical curve for N = 2, ΔνT ≈ 0.6, I1/I2 = 1, 〈n1〉 = 〈n2〉 = 8.1.

Fig. 4
Fig. 4

Experimental values of the correlation coefficient r obtained for different ratios L/l, superimposed on the theoretical curve for N = 3, ΔνT ≈ 0.72, I1/I2 = 1.1, 〈n1〉 = 〈n2〉 = 5.8

Fig. 5
Fig. 5

Experimental values of the correlation coefficient r obtained for different observation times T, superimposed on the theoretical curve for N = 2, L/l ≈ 1, I1/I2 = 2.8, 〈n1〉 = 9.61, 〈n2〉 = 9.76.

Equations (15)

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n 1 = α 1 c S 1 t t + T Â ( x , t ) · Â ( x , t ) d 2 x d t ,
n 2 = α 2 c S 2 t t + T Â ( x , t ) · Â ( x , t ) d 2 x d t ,
S 1 d 2 x b r = 0 N - 1 r L ( r + 1 2 ) L d x ,
S 2 d 2 x b r = 1 N ( r - 1 2 ) L r L d x .
n 1 n 2 = α 1 α 2 c 2 S 1 S 2 t t + T t t + T : Â ( x 1 , t 1 ) · Â ( x 1 , t 1 ) × Â ( x 2 , t 2 ) · Â ( x 2 , t 2 ) : d 2 x 1 d 2 x 2 d t 1 d t 2 ,
n 1 2 = n 1 + α 1 2 c 2 S 1 S 1 t t + T t t + T : Â ( x 1 , t 1 ) · Â ( x 1 , t 1 ) × Â ( x 2 , t 2 ) · Â ( x 2 , t 2 ) : d 2 x 1 d 2 x 2 d t 1 d t 2 ,
ρ ˆ = ϕ ( { v k , s } ) { v k , s } { v k , s } d 2 { v k , s } ,
ϕ ( { v k , s } ) = ϕ 1 ( { v k , s } ) ϕ 2 ( { v k , s } ) × k , s [ δ 2 ( v k , s - v k , s - v k , s ) d 2 v k , s d 2 v k , s ,
ϕ 1 ( { v k , s } ) = 1 2 π v ˜ k 1 , s 1 δ ( v k 1 , s 1 - v ˜ k 1 , s 1 ) k , s k 1 , s 1 δ 2 ( v k , s ) ,
ϕ 2 ( { v k , s } ) = 1 2 π v ˜ k 2 , s 2 δ ( v k 2 , s 2 - v ˜ k 2 , s 2 ) k , s k 2 , s 2 δ 2 ( v k , s ) .
n 1 = 1 2 α 1 c b L N ( I 1 + I 2 ) T ,
( Δ n 1 ) 2 = n 1 + 1 2 n 1 2 [ 2 ( I 1 / I 2 ) + ( I 2 / I 1 ) ] 2 × [ sin ( π N L / l ) sin ( π L / l ) ] 2 [ sin ( 1 2 π L / l ) 1 2 π N L / l ] 2 [ sin ( π Δ ν T ) π Δ ν T ] 2 ,
Δ n 1 Δ n 2 = 1 2 n 1 n 2 [ 2 ( I 1 / I 2 ) + ( I 2 / I 1 ) ] 2 × [ sin ( π N L / l ) sin ( π L / l ) ] 2 [ sin ( 1 2 π L / l ) 1 2 π N L / l ] 2 [ sin ( π Δ ν T ) π Δ ν T ] 2 .
r = Δ n 1 Δ n 2 / [ ( Δ n 1 ) 2 ( Δ n 2 ) 2 ] 1 2 = 1 2 n 1 n 2 F cos ( π L / l ) / { [ n 1 + 1 2 n 1 2 F ] 1 2 × [ n 2 + 1 2 n 2 2 F ] 1 2 } ,
F = [ 2 ( I 1 / I 2 ) + ( I 2 / I 1 ) ] 2 [ sin ( π N L / l ) sin ( π L / l ) ] 2 × [ sin ( 1 2 π L / l ) 1 2 π N L / l ] 2 [ sin ( π Δ ν T ) π Δ ν T ] 2 .