Abstract

An extremely simple procedure is given for estimating the mean intensity of each component of a two-component optical field from measurements of the number of photocounts recorded over counting intervals of T sec duration spaced τ sec apart. The only a priori knowledge required about each component is the functional form of the first- and second-order coherence functions and an order of magnitude estimate of their coherence times.

© 1974 Optical Society of America

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References

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  1. L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
    [CrossRef]
  2. B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
    [CrossRef]
  3. C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
    [CrossRef]
  4. D. L. Snyder, IEEE Trans. Inform. Theory IT-18, 91 (1972).
    [CrossRef]
  5. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), pp. 491–556.
  7. H. L. van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 52–74.
  8. S. Karp, J. R. Clark, IEEE Trans. Inform. Theory IT-16672 (1970).
    [CrossRef]
  9. P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
    [CrossRef]
  10. E. Jakeman, J. Phys. A: Gen. Phys. 3, 201 (1970).
    [CrossRef]
  11. H. L. van Trees, Ref. 7, p. 178. The difficulty in evaluating the expectations in Eqs. (1) and (2) lies in the inability to determine the statistics of the expansion coefficients for non-Gaussian processes.
  12. L. Mandel, Phys. Rev. 136, B1221 (1964).
    [CrossRef]
  13. L. Mandel, Phys. Rev. 138, B753 (1965).
    [CrossRef]
  14. R. L. Pfleegor, L. Mandel, J. Opt. Soc. Am. 58, 946 (1968).
    [CrossRef]
  15. B. W. Lindgren, Statistical Theory (Macmillan Co., New York, 1968), p. 142.
  16. W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  17. R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [CrossRef]
  18. V. Bluemel, L. M. Narducci, R. A. Tuft, J. Opt. Soc. Am. 62, 1309 (1972).
    [CrossRef]
  19. F. Davidson, F. Tittel, IEEE J. Quant. Electron. 10, 409April (1974).
    [CrossRef]

1974 (1)

F. Davidson, F. Tittel, IEEE J. Quant. Electron. 10, 409April (1974).
[CrossRef]

1972 (2)

1970 (5)

S. Karp, J. R. Clark, IEEE Trans. Inform. Theory IT-16672 (1970).
[CrossRef]

P. Diament, M. C. Teich, J. Opt. Soc. Am. 60, 682 (1970).
[CrossRef]

E. Jakeman, J. Phys. A: Gen. Phys. 3, 201 (1970).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
[CrossRef]

1968 (1)

1965 (1)

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

1964 (2)

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

L. Mandel, Phys. Rev. 136, B1221 (1964).
[CrossRef]

1963 (2)

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

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

1958 (1)

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Bluemel, V.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), pp. 491–556.

Clark, J. R.

S. Karp, J. R. Clark, IEEE Trans. Inform. Theory IT-16672 (1970).
[CrossRef]

Davidson, F.

F. Davidson, F. Tittel, IEEE J. Quant. Electron. 10, 409April (1974).
[CrossRef]

Diament, P.

Glauber, R. J.

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

Gordon, J. P.

C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
[CrossRef]

Jakeman, E.

E. Jakeman, J. Phys. A: Gen. Phys. 3, 201 (1970).
[CrossRef]

Karp, S.

S. Karp, J. R. Clark, IEEE Trans. Inform. Theory IT-16672 (1970).
[CrossRef]

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Lindgren, B. W.

B. W. Lindgren, Statistical Theory (Macmillan Co., New York, 1968), p. 142.

Liu, J. W. S.

C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
[CrossRef]

Mandel, L.

R. L. Pfleegor, L. Mandel, J. Opt. Soc. Am. 58, 946 (1968).
[CrossRef]

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

L. Mandel, Phys. Rev. 136, B1221 (1964).
[CrossRef]

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Martienssen, W.

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

Narducci, L. M.

Pfleegor, R. L.

Reiffen, B.

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

Sherman, H.

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

Snyder, D. L.

D. L. Snyder, IEEE Trans. Inform. Theory IT-18, 91 (1972).
[CrossRef]

Spiller, E.

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

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Teich, M. C.

Tittel, F.

F. Davidson, F. Tittel, IEEE J. Quant. Electron. 10, 409April (1974).
[CrossRef]

Tuft, R. A.

van Trees, H. L.

H. L. van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 52–74.

H. L. van Trees, Ref. 7, p. 178. The difficulty in evaluating the expectations in Eqs. (1) and (2) lies in the inability to determine the statistics of the expansion coefficients for non-Gaussian processes.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), pp. 491–556.

Am. J. Phys. (1)

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

IEEE J. Quant. Electron (1)

F. Davidson, F. Tittel, IEEE J. Quant. Electron. 10, 409April (1974).
[CrossRef]

IEEE Trans. Inform. Theory (2)

D. L. Snyder, IEEE Trans. Inform. Theory IT-18, 91 (1972).
[CrossRef]

S. Karp, J. R. Clark, IEEE Trans. Inform. Theory IT-16672 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A: Gen. Phys. (1)

E. Jakeman, J. Phys. A: Gen. Phys. 3, 201 (1970).
[CrossRef]

Phys. Rev. (3)

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

L. Mandel, Phys. Rev. 136, B1221 (1964).
[CrossRef]

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

Proc. IEEE (3)

B. Reiffen, H. Sherman, Proc. IEEE 51, 1316 (1963).
[CrossRef]

C. W. Helstrom, J. W. S. Liu, J. P. Gordon, Proc. IEEE 58, 1578 (1970).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[CrossRef]

Proc. Phys. Soc. (1)

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Other (4)

B. W. Lindgren, Statistical Theory (Macmillan Co., New York, 1968), p. 142.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1964), pp. 491–556.

H. L. van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), pp. 52–74.

H. L. van Trees, Ref. 7, p. 178. The difficulty in evaluating the expectations in Eqs. (1) and (2) lies in the inability to determine the statistics of the expansion coefficients for non-Gaussian processes.

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Figures (5)

Fig. 1
Fig. 1

The probability distribution function P(Xx), x = nm/E{nm} is plotted as a function of x for pure pseudothermal light. T was set at 0.1, 2.0 and 6.0 msec, E{n} = 12.8, and the coherence time of the light was measured to be 2.6 msec.

Fig. 2
Fig. 2

The parameter μ is plotted as a function of E{n}. H2 is the reduced, normalized second factional moment, [E{n(n − 1)}/E2{n} − 1], which is unity for pure pseudothermal or Gaussian light and zero for pure laser light.

Fig. 3
Fig. 3

The parameter ν is plotted as a function of E{n} for various values of H2.

Fig. 4
Fig. 4

The parameter p is plotted as a function of E{n} for various values of H2.

Fig. 5
Fig. 5

The mean number of counts n ¯1, as computed from Eq. (8) with E{nm} and E{n} replaced by measured sample means is plotted as a function of the number of samples N for various values of H2. n ¯1 is the average total number of counts E{n}.

Tables (1)

Tables Icon

Table I Variation of σnm/E{nm} with E{nm}

Equations (19)

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p ( n , T | n ¯ ) = E { 1 n ! ( α c S t t + T I ( t ) d t ) n exp ( α c S t t + T I ( t ) d t ) } .
p ( n , m , T | n ¯ ) = E { 1 n ! m ! ( α c S t 1 t 1 + T I ( t ) d t ) n × ( α c S t 2 t 2 + T I ( t ) d t ) m × exp ( α c S t 1 t 1 + T I ( t ) d t α c S t 2 t 2 + T I ( t ) d t ) } .
E { n } = n = 0 n p ( n , T | n ¯ ) = α c S t 1 t 1 + T G ( 1,1 ) ( r , t ; r , t ) d t E { n m } = n = 0 m = 0 n m p ( n , m , T | n ¯ ) = ( α c S ) 2 t 1 t 1 + T t 2 t 2 + T G ( 2,2 ) ( r 1 , t 1 , r 2 t 2 ; r 1 , t 1 , r 2 , t 2 ) d t 1 d t 2 .
G ( n , n ) ( r 1 , t 1 , r n 1 , t n ; r 1 , t 1 r n , t n ) T r P ( { α k s } ) | { α k s } > < { α k s } | A ˆ ( ) ( r 1 , t 1 ) A ˆ ( ) ( r n , t n ) A ˆ ( + ) ( r 1 , t 1 ) A ˆ ( + ) ( r n , t n ) d 2 { α k s } ,
G ( n , n ) ( r 1 , t 1 r n 1 t n ; r 1 , t 1 , r n t n ) = T r P 1 ( { α k s } ) P 2 ( { α k s } ) all k s δ ( 2 ) ( α k s α k s α k s ) × | { α k s } > < { α k s } | A ˆ ( ) ( r 1 , t 1 ) × A ˆ ( ) ( r n , t n ) A ˆ ( + ) ( r 1 , t 1 ) A ˆ ( + ) ( r n , t n ) × d 2 { α k s } d 2 { α k s } d 2 { α k s } .
G ( n , n ) ( r 1 , t I , r n , t n ; r 1 , t 1 , r n , t n ) = P 1 ( { α k s } ) P 2 ( { α k s } ) i = 1 n ( V 1 * ( r i , t i ) + V 2 * ( r i , t i ) ) ( V 1 ( r i t i ) + V 2 ( r i , t i ) ) d 2 { α k s } d 2 { α k s } .
E { n } = α c S t t + T [ G 1 ( 1,1 ) ( r , t ; r t ) + G 2 ( 1,1 ) ( r , t ; r , t ) ] d t E { n m } = ( α c S ) 2 t 1 t 1 + T t 2 t 2 + T [ G 1 ( 2,2 ) ( r 1 , t 1 , r 2 , t 2 ; r 1 , t 1 , r 2 , t 2 ) + G 1 ( 1,1 ) ( r 1 , t 1 ; r 1 , t 1 ) G 2 ( 1,1 ) ( r 2 , t 2 ; r 2 , t 2 ) + G 1 ( 1,1 ) ( r 2 , t 2 ; r 2 , t 2 ) G 2 ( 1,1 ) ( r 1 , t 1 ; r 1 , t 1 ) + G 1 ( 1,1 ) * ( r 1 , t 1 ; r 2 , t 2 ) G 1 ( 1,1 ) ( r 1 , t 1 ; r 2 , t 2 ) + G 1 ( 1,1 ) ( r 1 , t 1 ; r 2 , t 2 ) G 2 ( 1,1 ) * ( r 1 , t 1 : r 2 , t 2 ) + G 2 ( 2,2 ) ( r 1 , t 1 , r 2 , t 2 ; r 1 , t 1 , r 2 , t 2 ) ] d t 1 d t 2 ,
g ( n , n ) ( r 1 , t 1 r n , t n ; r 1 , t 1 r n , t n ) = G ( n , n ) ( r 1 , t 1 r n , t n ; r 1 , t 1 r n , t n ) / i = 1 2 n × { G ( 1,1 ) ( r i , t i ; r i , t i ) } 1 / 2
G i ( 2,2 ) ( r 1 , t 1 , r 2 , t 2 ; r 1 , t 1 , r 2 , t 2 ) = E { I i ( r 1 ) } × E { I i ( r 2 ) } ( 1 + { E [ Δ I i ( r 1 , t 1 ) Δ I i ( r 2 , t 2 ) ] / E [ I i ( r 1 ) ] × E [ I i ( r 2 ) ] } ) , i = 1,2.
E { n } = α c S I 1 ¯ T + α c S I 2 ¯ T = n ¯ 1 + n ¯ 2 E { n m } = ( n ¯ 1 + n ¯ 2 ) 2 + β n ¯ 1 2 + δ n ¯ 2 2 + γ n ¯ 1 n ¯ 2 ,
β 1 T 2 0 T τ τ + T λ 1 ( r , r , τ ) d t d τ , δ 1 T 2 0 T τ τ + T λ 2 ( r , r , τ ) d t d τ , γ 1 T 2 2 Re 0 T τ τ + T g 1 ( 1,1 ) * ( r , r , τ ) g 2 ( 1,1 ) ( r , r , τ ) d τ d t ,
λ i ( r , r , τ ) E [ Δ I i ( r , t ) Δ I i ( r , t + τ ) ] / E [ I i ( r ) ] 2 .
n 1 ¯ = E { n } ( 2 δ γ ) ± { E 2 { n } ( 2 δ γ ) 2 + 4 [ E { n m } E 2 { n } ( 1 + δ ) ] ( β + δ γ ) } 1 / 2 / 2 ( β + δ γ ) , n 2 ¯ = E { n } ( 2 β γ ) ± [ E 2 { n } ( 2 β γ ) 2 + 4 [ E { n m } E 2 { n } ( 1 + β ) ] ( β + δ γ ) } 1 / 2 / 2 ( β + δ γ ) .
N 1 i = 1 N n i m i and N 1 i = 1 N n i
σ sample mean of n m 2 = 1 N σ n m 2 σ sample mean of n 2 = 1 N σ n 2
P ( X x ) = 1 μ exp ( ν x p ) ,
p ( x ) = μ ν p x p 1 exp ( ν x p ) , x 0 = δ ( x ) [ 1 μ ] . x = 0
P ( X = O ) = 1 μ , P ( X 1 ) = 1 μ exp ( ν ) , P ( X 2 ) = 1 μ exp ( v 2 p ) .
σ x 2 = ( μ / v 2 / p ) Γ ( 1 + 2 / p ) 1.

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