Abstract

A computer simulation technique useful for generating superposed coherent and chaotic radiation of arbitrary spectral shape is described. Its advantages over other techniques include flexibility and ease of implementation, as well as the capability of incorporating spectral characteristics that cannot be generated by other methods. We discuss the implementation of the technique and present results to demonstrate its validity. The technique can be used to obtain numerical solutions to photon statistics problems through computer simulation. We furthermore argue that experiments involving photon statistics can be carried out using a wideband source in place of an amplitude-stabilized source whenever the spectral characteristics of the source are not important. Experimental results that corroborate the argument are presented.

© 1980 Optical Society of America

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References

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  1. R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956).
    [CrossRef]
  2. R. Hanbury Brown, R. Q. Twiss, Nature 178, 1046 (1956).
    [CrossRef]
  3. B. Saleh, Photoelectron Statistics (Springer, Heidelberg, 1978), pp. 160ff.
  4. H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
    [CrossRef]
  5. H. Z. Cummins, in Photon Counting and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974).
  6. W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  7. F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
    [CrossRef]
  8. M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
    [CrossRef]
  9. B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), pp. 159–212.
  10. N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
    [CrossRef]
  11. G. Lachs, R. K. Koval, J. Appl. Phys. 43, 2918 (1972).
    [CrossRef]
  12. J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
    [CrossRef]
  13. B. Gold, C. M. Rader, Digital Processing of Signals (McGraw-Hill, New York, 1969), Chap. 6.
  14. M. Schwartz, L. Shaw, Discrete Spectral Analysis, Detection and Estimation (McGraw-Hill, New York, 1975), p. 33ff.
  15. L. Mandel, Proc. Phys. Soc. 74, 233 (1959).
    [CrossRef]
  16. G. J. Troup, J. Lyons, Phys. Lett. 29A, 705 (1969).
  17. B. L. Morgan, L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).
    [CrossRef]
  18. The first column of the histogram is seen to be slightly depressed below the fitted curve. This results from computer dead time; i.e., the computer software cannot resolve pulses occurring closer than ~20 μsec. The observed reduction of the column height is found to be in accord with the calculated effect of dead time.
  19. G. Vannucci, M. C. Teich, Opt. Commun. 25, 267 (1978).
    [CrossRef]

1978 (1)

G. Vannucci, M. C. Teich, Opt. Commun. 25, 267 (1978).
[CrossRef]

1972 (2)

N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
[CrossRef]

G. Lachs, R. K. Koval, J. Appl. Phys. 43, 2918 (1972).
[CrossRef]

1969 (1)

G. J. Troup, J. Lyons, Phys. Lett. 29A, 705 (1969).

1966 (2)

B. L. Morgan, L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).
[CrossRef]

M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
[CrossRef]

1965 (2)

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

1964 (2)

H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
[CrossRef]

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

1959 (1)

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

1956 (2)

R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

Bertolotti, M.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), pp. 159–212.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Crosignani, B.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), pp. 159–212.

Cummings, D. O.

N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
[CrossRef]

Cummins, H. Z.

H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
[CrossRef]

H. Z. Cummins, in Photon Counting and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974).

Di Porto, P.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), pp. 159–212.

Gold, B.

B. Gold, C. M. Rader, Digital Processing of Signals (McGraw-Hill, New York, 1969), Chap. 6.

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Keyes, R. J.

M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
[CrossRef]

Kingston, R. H.

M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
[CrossRef]

Knable, N.

H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
[CrossRef]

Koval, R. K.

G. Lachs, R. K. Koval, J. Appl. Phys. 43, 2918 (1972).
[CrossRef]

Lachs, G.

N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
[CrossRef]

G. Lachs, R. K. Koval, J. Appl. Phys. 43, 2918 (1972).
[CrossRef]

Lyons, J.

G. J. Troup, J. Lyons, Phys. Lett. 29A, 705 (1969).

Mandel, L.

B. L. Morgan, L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).
[CrossRef]

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

Martienssen, W.

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Morgan, B. L.

B. L. Morgan, L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).
[CrossRef]

Rader, C. M.

B. Gold, C. M. Rader, Digital Processing of Signals (McGraw-Hill, New York, 1969), Chap. 6.

Ruggieri, N. F.

N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
[CrossRef]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, Heidelberg, 1978), pp. 160ff.

Schwartz, M.

M. Schwartz, L. Shaw, Discrete Spectral Analysis, Detection and Estimation (McGraw-Hill, New York, 1975), p. 33ff.

Shaw, L.

M. Schwartz, L. Shaw, Discrete Spectral Analysis, Detection and Estimation (McGraw-Hill, New York, 1975), p. 33ff.

Spiller, E.

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Teich, M. C.

G. Vannucci, M. C. Teich, Opt. Commun. 25, 267 (1978).
[CrossRef]

M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
[CrossRef]

Troup, G. J.

G. J. Troup, J. Lyons, Phys. Lett. 29A, 705 (1969).

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Twiss, R. Q.

R. Hanbury Brown, R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

Vannucci, G.

G. Vannucci, M. C. Teich, Opt. Commun. 25, 267 (1978).
[CrossRef]

Yeh, Y.

H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
[CrossRef]

Am. J. Phys. (1)

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Appl. Phys. Lett. (1)

M. C. Teich, R. J. Keyes, R. H. Kingston, Appl. Phys. Lett. 9, 357 (1966).
[CrossRef]

J. Appl. Phys. (2)

N. F. Ruggieri, D. O. Cummings, G. Lachs, J. Appl. Phys. 43, 1118 (1972).
[CrossRef]

G. Lachs, R. K. Koval, J. Appl. Phys. 43, 2918 (1972).
[CrossRef]

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, Math. Comput. 19, 297 (1965).
[CrossRef]

Nature (2)

R. Hanbury Brown, R. Q. Twiss, Nature 177, 27 (1956).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, Nature 178, 1046 (1956).
[CrossRef]

Opt. Commun. (1)

G. Vannucci, M. C. Teich, Opt. Commun. 25, 267 (1978).
[CrossRef]

Phys. Lett. (1)

G. J. Troup, J. Lyons, Phys. Lett. 29A, 705 (1969).

Phys. Rev. Lett. (3)

B. L. Morgan, L. Mandel, Phys. Rev. Lett. 16, 1012 (1966).
[CrossRef]

H. Z. Cummins, N. Knable, Y. Yeh, Phys. Rev. Lett. 12, 150 (1964).
[CrossRef]

F. T. Arecchi, Phys. Rev. Lett. 15, 912 (1965).
[CrossRef]

Proc. Phys. Soc. (1)

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

Other (6)

The first column of the histogram is seen to be slightly depressed below the fitted curve. This results from computer dead time; i.e., the computer software cannot resolve pulses occurring closer than ~20 μsec. The observed reduction of the column height is found to be in accord with the calculated effect of dead time.

B. Gold, C. M. Rader, Digital Processing of Signals (McGraw-Hill, New York, 1969), Chap. 6.

M. Schwartz, L. Shaw, Discrete Spectral Analysis, Detection and Estimation (McGraw-Hill, New York, 1975), p. 33ff.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975), pp. 159–212.

H. Z. Cummins, in Photon Counting and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974).

B. Saleh, Photoelectron Statistics (Springer, Heidelberg, 1978), pp. 160ff.

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Figures (4)

Fig. 1
Fig. 1

Autocorrelation coefficients of the in-phase and quadrature components of computer-simulated Gaussian-Lorentzian radiation as a function of Nyquist sample separation. As expected, the two curves are very similar and appear as virtually straight lines on a log-linear plot, corresponding to an exponential falloff.

Fig. 2
Fig. 2

Cross-correlation coefficient between the in-phase and quadrature components of computer-simulated Gaussian-Lorentzian radiation as a function of Nyquist sample separation. It can be seen that the absolute value of the coefficient is always considerably less than unity, indicating that the two components are virtually uncorrelated.

Fig. 3
Fig. 3

Statistics of photons from a light-emitting diode (LED). Experimental histogram and theoretical distribution of the time interval between consecutive detected photons. The solid straight line represents the theoretical (exponential) probability density function for a Poisson point process with constant rate. The mean time interval is 2.25 msec.

Fig. 4
Fig. 4

Statistics of photons from a light-emitting diode (LED). Correlation between interphoton intervals for different interval separations. The first data point corresponds to consecutive intervals, whereas the last data point corresponds to intervals separated by eighteen intervening intervals. Observe that no correlation point exhibits an absolute value larger than 0.01.

Equations (20)

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A k ( t ) = a k cos ω k t + b k sin ω k t ,
A ( t ) = k = - a k cos ω k t + b k sin ω k t ,
ω k = ω 0 + k Δ ω .
E [ a k b l ] = 0 for any k , l E [ a k a l ] = E [ b k b l ] = { 0 for k l Δ ω σ 2 ( ω k ) for k = l .
A ( t ) = [ k = - ( a k cos k Δ ω t + b k sin k Δ ω t ) ] cos ω 0 t + [ k = - ( b k cos k Δ ω t - a k sin k Δ ω t ) ] sin ω 0 t
A ( t ) = x ( t ) cos ω 0 t + y ( t ) sin ω 0 t ,
{ x ( t ) = k = - ( a k cos k Δ ω t + b k sin k Δ ω t ) y ( t ) = k = - ( b k cos k Δ ω t - a k sin k Δ ω t )
R y y ( τ ) = R x x ( τ ) = E [ x ( t ) x ( t + τ ) = k = - { E [ a k 2 ] cos [ k Δ ω t ] cos [ k Δ ω ( t + τ ) ] + E [ b k 2 ] sin [ k Δ ω t ] sin [ k Δ ω ( t + τ ) ] } = k = - Δ ω σ 2 ( ω 0 + k Δ ω ) cos k Δ ω τ .
R y y ( τ ) = R x x ( τ ) = - σ 2 ( ω 0 + ω ) cos ω τ d ω .
R x y ( τ ) = E [ x ( t + τ ) y ( t ) ] = - σ 2 ( ω 0 + ω ) sin ω τ d ω .
I ( t ) = x 2 ( t ) + y 2 ( t ) .
R I I ( τ ) = E [ ( x 2 ( t ) + y 2 ( t ) ) ( x 2 ( t + τ ) + y 2 ( t + τ ) ) ] = 4 [ R x x 2 ( 0 ) + R x x 2 ( τ ) + R x y 2 ( τ ) ] .
σ L 2 ( ω ) = A ( ω - ω 0 ) 2 + Γ 2 + B δ ( ω - ω 0 ) ,
A l = k = - N / 2 N / 2 - 1 Z k W l k ,
A l = k = - N / 2 N / 2 - 1 Z k ( cos k 2 π N l + i sin k 2 π N l ) ,
{ Im A l = k = - N / 2 N / 2 - 1 ( Im Z k cos k 2 π N l + Re Z k sin k 2 π N l ) , Re A l = k = - N / 2 N / 2 - 1 ( Re Z k cos k 2 π N l - Im Z k sin k 2 π N l ) .
{ x ( t ) = k = - N / 2 N / 2 - 1 ( a k cos k Δ ω t + b k sin k Δ ω t ) y ( t ) = k = - N / 2 N / 2 - 1 ( b k cos k Δ ω t - a k sin k Δ ω t ) .
Im A l = x ( l Δ t )             Im Z k = a k             Δ ω = 2 π / N Δ t Re A l = y ( l Δ t )             Re Z k = b k             t = l Δ t .
Z k = { Z ( N + k + 1 ) for - N / 2 k < 0 Z ( k + 1 ) for 0 k < N / 2 ,
E [ ( Re Z k ) 2 ] = E [ ( Im Z k ) 2 ] = σ k 2 = A / [ 1 + ( B k ) 2 ] ,

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