Abstract

Blackbody radiation is a gaussian stochastic process and a knowledge of the moments of its energy spectrum (Planck radiation law) is sufficient to enable the determination of the expected number of zero crossings of the electric field vector as well as the expected number of peaks and expected number of maxima. Explicit formulas are presented for these quantities in terms of the Einstein function and transport integrals. In addition, the spectrum of the intensity fluctuations is obtained and similar calculations performed. Typical numerical results are shown.

© 1966 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). See Sec. 4.5 for discussion along with additional references.
    [CrossRef]
  2. R. Bourret, Nuovo Cimento 18, 347 (1960).
    [CrossRef]
  3. C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964).
    [CrossRef]
  4. L. Mandel, in Progress in Optics, II edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1963), Ch. 5.
  5. D. Slepian, Bell System Tech. J. 37, 163 (1958).
    [CrossRef]
  6. S. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover Publications Inc., New York, 1954), Ch. 4.
  7. W. M. Rogers and R. L. Powell, Tables of Transport Integrals, Natl. Bur. Std. Circ. 595 (U. S. Government Printing Office, Washington, 1958).
  8. C. L. Mehta, Nuovo Cimento 28, 401 (1963).
    [CrossRef]
  9. Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). See Sec. 4.5 for discussion along with additional references.
[CrossRef]

1964 (1)

C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964).
[CrossRef]

1963 (1)

C. L. Mehta, Nuovo Cimento 28, 401 (1963).
[CrossRef]

1960 (1)

R. Bourret, Nuovo Cimento 18, 347 (1960).
[CrossRef]

1958 (1)

D. Slepian, Bell System Tech. J. 37, 163 (1958).
[CrossRef]

Bourret, R.

R. Bourret, Nuovo Cimento 18, 347 (1960).
[CrossRef]

Kopal, Z.

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). See Sec. 4.5 for discussion along with additional references.
[CrossRef]

L. Mandel, in Progress in Optics, II edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1963), Ch. 5.

Mehta, C. L.

C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964).
[CrossRef]

C. L. Mehta, Nuovo Cimento 28, 401 (1963).
[CrossRef]

Powell, R. L.

W. M. Rogers and R. L. Powell, Tables of Transport Integrals, Natl. Bur. Std. Circ. 595 (U. S. Government Printing Office, Washington, 1958).

Rice, S.

S. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover Publications Inc., New York, 1954), Ch. 4.

Rogers, W. M.

W. M. Rogers and R. L. Powell, Tables of Transport Integrals, Natl. Bur. Std. Circ. 595 (U. S. Government Printing Office, Washington, 1958).

Slepian, D.

D. Slepian, Bell System Tech. J. 37, 163 (1958).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). See Sec. 4.5 for discussion along with additional references.
[CrossRef]

C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964).
[CrossRef]

Bell System Tech. J. (1)

D. Slepian, Bell System Tech. J. 37, 163 (1958).
[CrossRef]

Nuovo Cimento (2)

R. Bourret, Nuovo Cimento 18, 347 (1960).
[CrossRef]

C. L. Mehta, Nuovo Cimento 28, 401 (1963).
[CrossRef]

Phys. Rev. (1)

C. L. Mehta and E. Wolf, Phys. Rev. 134, A1143 (1964).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965). See Sec. 4.5 for discussion along with additional references.
[CrossRef]

Other (4)

Z. Kopal, Numerical Analysis (John Wiley & Sons, Inc., New York, 1955).

L. Mandel, in Progress in Optics, II edited by E. Wolf (North Holland Publishing Co., Amsterdam, 1963), Ch. 5.

S. Rice, in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover Publications Inc., New York, 1954), Ch. 4.

W. M. Rogers and R. L. Powell, Tables of Transport Integrals, Natl. Bur. Std. Circ. 595 (U. S. Government Printing Office, Washington, 1958).

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Figures (3)

F. 1
F. 1

Zero-crossing rate E0≡(hN0/kT) versus the upper bandpass frequency x2 for different lower-bandpass frequencies x1: (A) x1=0, (B) x1=3, (C) x1=4, (D) x1=5, (E) x1=6.

F. 2
F. 2

Expected number of maxima Emax≡(hM/kT) versus the upper bandpass frequency x2 for different lower bandpass frequencies x1: (A) x1=0, (B) x1=4, (C) x1=5.

F. 3
F. 3

Spectral density of intensity fluctuations Ψ(x).

Tables (1)

Tables Icon

Table I Values of the normalized spectral density of the intensity fluctuations Ψ(x).

Equations (31)

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ϕ ( ν ) = ( 8 π h / c 3 ) ν 3 [ e h ν / k t 1 ] 1 .
1 2 T T T | V ( t ) | 2 d t ,
N 0 = 2 [ m 2 / m 0 ] 1 2 .
m n = ν 1 ν 2 ϕ ( ν ) ν n d ν , = [ k T h ] n + 4 x 1 x 2 z n + 3 d z ( e z 1 ) ,
F l ( ) 0 z l d z e z 1 = Γ ( l ) ζ ( l ) .
J n ( x ) 0 x e z z n d z ( e z 1 ) 2 ,
F n 1 ( x ) 0 x t n 1 d t ( e t 1 ) = x n 1 n [ J n ( x ) x n 1 + x e x 1 ] .
E 0 h N 0 k T = 2 [ F 5 ( x 2 ) F 5 ( x 1 ) F 3 ( x 2 ) F 3 ( x 1 ) ] 1 2 .
N 0 = 1.867 10 11 T ( zero / sec ) ,
N 0 = 2 ν 0 ,
ν 0 = 0.934 10 11 T
ν ¯ = 3.32 k T / h = 0.692 10 11 T .
N A ( x 2 , x 1 ) = 1 2 N 0 ( x 2 , x 1 ) e n 2 / 2 ,
n 2 = V 2 A ( t ) / 2 ν 1 ν 2 ϕ ( ν ) d ν = V 2 A ( t ) 2 m 0 ( x 2 , x 1 ) .
M = [ m 4 / m 2 ] 1 2 ,
E max h M k T = [ F 7 ( x 2 ) F 7 ( x 1 ) F 5 ( x 2 ) F 5 ( x 1 ) ] 1 2 ,
M = 1.332 10 11 T ( max / sec ) .
λ 11 ( τ ) = | γ 11 ( τ ) | 2 ,
ψ ( ν ) = | γ 11 ( τ ) | 2 e 2 π i ν τ d τ ,
= 0 ϕ ( f + ν ) ϕ ( f ) d f .
ψ ( ν ) = [ 8 π h c 3 ] 2 0 f 3 [ e h f / k T 1 ] ( ν + f ) 3 [ e h ( ν + f ) k t 1 ] d f .
ψ ( x ) = [ 8 π h c 3 ] 2 [ k T h ] 7 0 e z z 3 ( z + x ) 3 × [ 1 e z ] 1 [ e z + x 1 ] 1 d z .
0 e z f ( z ) d z ,
D 2 0 z 6 d z [ e z 1 ] 2 = 6 ! [ ξ ( 6 ) ξ ( 7 ) ] = 6.477521 ,
ψ ( x ) = [ 8 π D h / c 3 ] 2 [ k T / h ] 7 Ψ ( x ) ,
0 Ψ ( x ) Ψ ( 0 ) = 1 .
N 0 ( I ) = 11.42 ( k T / h ) sec 1 = 2.38 10 11 T sec 1 .
N A ( I ) = 1 2 N 0 ( I ) e n 2 / 2 ,
n 2 = I 2 A ( t ) / 2 0 ψ ( ν ) d ν = I 2 A ( t ) 2 m 0 .
ν 0 = 1.19 10 11 T sec 1 ,
M ( I ) = 5.44 ( k T / h ) sec 1 = 1.13 10 11 T sec 1 .