Abstract

A generalized representation of the radiation field incident on the photosensitive surface of a photoelectric device accommodates the consideration of many laser beams superposed or the consideration of radiation due to multimodes or multiple filaments of a single laser. The magnitudes and frequencies of the fluctuations in electron emission from the photosurface associated with photon noise and shot noise and the parameters affecting these magnitudes are determined. Also, the influence of these fluctuations on the probability of detecting low-level laser radiation is investigated.

© 1962 Optical Society of America

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  1. A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
    [CrossRef]
  2. R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A242, 300 (1957).
  3. R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).
  4. Referred to by Brown and Twiss as wave-interaction noise and/or excess photon noise.
  5. See, for example, M. Born and E. Wolf, Principle of Optics, (Pergamon Press, New York, 1959), p. 509.
  6. R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 (1956).
    [CrossRef]
  7. This representation is convenient since A can be interpreted classically or quantum mechanically in a direct manner. Insofar as energy and momentum properties are concerned, a plane wave corresponds to a beam of particles (photons) each of energy ħω and momentum k(k=ħω/c). The photon flux is n=(2ω/ħ) (A2/4πc2) photons/cm2/sec.
  8. The rate of these changes are orders of magnitude less than those corresponding to frequencies ωn.
  9. Lower case Greek letters are used to label atomic states. Lower case Latin letters are used to label the frequencies and phases of the Fourier components of the radiation field.
  10. As usual, expi(knj · r) is not integrated with respect to electronic coordinates as it occurs very slowly with respect to these in the region of interest.
  11. See Appendix.
  12. L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
    [CrossRef]
  13. A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
    [CrossRef]

1961 (2)

1957 (2)

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

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

1956 (1)

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

1955 (1)

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[CrossRef]

Born, M.

See, for example, M. Born and E. Wolf, Principle of Optics, (Pergamon Press, New York, 1959), p. 509.

Forrester, A. T.

A. T. Forrester, J. Opt. Soc. Am. 51, 253 (1961).
[CrossRef]

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[CrossRef]

Gudmundsen, R. A.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[CrossRef]

Hanbury Brown, R.

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

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

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

Johnson, P. O.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[CrossRef]

Mandel, L.

Twiss, R. Q.

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

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

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

Wolf, E.

See, for example, M. Born and E. Wolf, Principle of Optics, (Pergamon Press, New York, 1959), p. 509.

J. Opt. Soc. Am. (2)

Nature (1)

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

Phys. Rev. (1)

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
[CrossRef]

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

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

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

Other (7)

Referred to by Brown and Twiss as wave-interaction noise and/or excess photon noise.

See, for example, M. Born and E. Wolf, Principle of Optics, (Pergamon Press, New York, 1959), p. 509.

This representation is convenient since A can be interpreted classically or quantum mechanically in a direct manner. Insofar as energy and momentum properties are concerned, a plane wave corresponds to a beam of particles (photons) each of energy ħω and momentum k(k=ħω/c). The photon flux is n=(2ω/ħ) (A2/4πc2) photons/cm2/sec.

The rate of these changes are orders of magnitude less than those corresponding to frequencies ωn.

Lower case Greek letters are used to label atomic states. Lower case Latin letters are used to label the frequencies and phases of the Fourier components of the radiation field.

As usual, expi(knj · r) is not integrated with respect to electronic coordinates as it occurs very slowly with respect to these in the region of interest.

See Appendix.

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Figures (5)

Fig. 1
Fig. 1

Envelope of probability-density function of electron emission vs number of electrons emitted ( ν ¯T=1).

Fig. 2
Fig. 2

Envelope of probability density function of electron emission vs number of electrons emitted ( ν ¯T=10).

Fig. 3
Fig. 3

Normalized variance of electron emission vs average number of electrons emitted per unit time per unit bandwidth.

Fig. 4
Fig. 4

Probability of detection vs average number of electrons emitted.

Fig. 5
Fig. 5

Integrand of Eq. (A3) vs ω1.

Equations (66)

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J ¯ 2 = J ¯ n 2 + J ¯ c 2 ,
ω 1 < ω < ω 2 ,
ω 1 > ω 2 - ω 1 .
A = n A n ɛ n exp i ( k n · r - ω n t + ϕ n ) + c . c . ,
A = n j A n j ɛ n j exp i ( k n j · r - ω n t + ϕ n j ) + c . c .
ψ = ν a ν ( t ) u ν exp i ( E ν t ) ,
H = ( i e / m c ) A · grad .
a λ ( t ) = ( i ) - 1 0 t H λ ν a ν ( 0 ) exp i ( ω λ ν ξ ) d ξ ,
a λ ( t ) = ( i ) - 1 0 t electron coordinates i e m c n j ɛ n j × u λ * { A n j exp i ( k n j · r - ω n ξ + ϕ n j + ω λ ν ξ ) + A n j exp - i ( k n j · r - ω n ξ + ϕ n j - ω λ ν ξ ) } · grad u ν d τ d ξ .
ɛ n j · u λ * grad u ν d τ - m ω λ ν ( r λ ν ) n j ,
a λ ( t ) = - ( i ) - 1 0 t i e c n j A n j ω λ ν ( r ν λ ) n j × { exp i [ k n j r + ( ω λ ν - ω n ) ξ + ϕ n j ] + exp - i [ k n j · r - ( ω λ ν + ω n ) ξ + ϕ n j ] d ξ } .
a λ ( t ) = - e c ω λ ν n j ( r λ ν ) n j A n j × ( exp i { k n j · r + ϕ n j + ( ω λ ν - ω n ) ξ } i ( ω λ ν - ω n ) ] 0 t + exp - i { k n j · r + ϕ n j - ( ω λ ν + ω n ) ξ } i ( ω λ ν + ω n ) ] 0 t ) .
a λ ( t ) = - e i c ω λ ν n j ( r λ ν ) n j A n j exp i ( k n j · r + ϕ n j ) × [ exp i ( ω λ ν - ω n ) t - 1 ω λ ν - ω n ] .
a λ a λ * ( t ) = e 2 2 c 2 ω 2 { n j r n j A n j exp ( - i α n j ) × exp [ - i ( ω - ω n ) t ] - 1 ( ω - ω n ) } × { m k r m k A m k exp i α m k exp [ i ( ω - ω m ) t ] - 1 ( ω - ω m ) } ,
α n j ( k n j · r + ϕ n j ) ,             r n j r λ ν n j ,             ω ω λ ν .
w = 1 t a λ a λ * ρ ( λ ) d ω λ ν = 1 t 2 e 2 2 c 2 ω 2 n j N J ( r n j A n j ) 2 - 2 sin 2 1 2 ( ω - ω n ) t ( ω - ω n ) 2 d ω + 1 t 4 e 3 2 c 2 ω 2 n j k j > k N J J ( r n j A n j ) ( r n k A n k ) - cos ( α n j - α n k ) × 2 sin 2 1 2 ( ω - ω n ) t ( ω - ω n ) 2 d ω + 1 t 8 e 2 2 c 2 ω 2 n m j k n > m N N J J ( r n j A n j ) ( r m k A m k ) × - cos [ ( ω m - ω n ) t 2 + ( α n j - α m k ) ] × sin 1 2 ( ω - ω n ) t sin 1 2 ( ω - ω m ) t ( ω - ω n ) ( ω - ω m ) d ω .
w ( t ) = 2 π e 2 2 c 2 ω 2 ρ ( λ ) n j N J ( r n j A n j ) 2 + 4 π e 2 2 c 2 ω 2 ρ ( λ ) n j k j > k N J J ( r n j A n j ) ( r n k A n k ) cos ( α n j - α n k ) + 8 π e 2 2 c 2 ω 2 ρ ( λ ) n m j k n > m N N J J ( r n j A n j ) ( r m k A m k ) × cos [ ω m - ω n 2 t + ( α n j - α m k ) ] sin 1 2 ( ω m - ω n ) t ( ω n - ω m ) t .
w ( t ) = 2 π e 2 2 c 2 ω 2 r 2 ρ ( λ ) n j N J A n j 2 + 4 π e 2 2 c 2 ω 2 r 2 ρ ( λ ) n j k j > k N J J A n j A n k cos ( ϕ n j - ϕ n k ) + 8 π e 2 2 c 2 ω 2 r 2 ρ ( λ ) n m j k n > m N N J J A n j A m k × cos [ ω m - ω n 2 + t ( α n j - α m k ) ] sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ,
( t ) = 4 π 2 e 2 ω ρ ( λ ) r 2 n j N J n n j + 8 π 2 e 2 ω ρ ( λ ) r 2 n j k j > k N J J ( n m j ) 1 / 2 ( n n k ) 1 / 2 cos ( ϕ n j - ϕ n k ) + 16 π 2 e 2 ω ρ ( λ ) r 2 n m j k n > m N N J J ( n n j ) 1 / 2 ( n m k ) 1 / 2 × cos [ ω m - ω n 2 t + ( α n j - α m k ) ] sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t .
μ = M w ( t ) d t .
N J n = n j N J n n j ,
w ¯ i = N J [ ( 4 π 2 e 2 / ) ω r 2 ρ ( λ ) ] = α N J n ,
w ¯ c = α N J 2 n ,
w i ( t ) = w ¯ i + w i 0 + w i t = α N J n + 2 α n n l 1 2 N ( J 2 - J ) cos ϕ n l + 4 α n n m l n > m N N J 2 sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t × cos [ ω m - ω n 2 t + ( k n - k m ) · r + ϕ n m l ] ,
w c ( t ) = w ¯ c + w c t = α N J 2 n + 4 α J 2 n n m n > m N N sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) × [ cos ω m - ω n 2 t + ( k n - k m ) · r + ϕ n m ] ,
p n ( x ) = [ 1 / σ ( 2 π ) 1 2 ] exp ( - x 2 / 2 σ 2 )
σ 2 = 1 2 ( A 1 2 + A 2 2 + + A n 2 ) .
σ i 0 2 = α 2 N ( J 2 - J ) n 2 ,             σ i 0 < α N 1 2 J n .
4 α n [ sin 1 2 ( ω m - ω n ) t / ( ω m - ω n ) t ] ,
ρ ( w n m ) = ( 1 / σ n m 2 π ) exp [ - ( w n m ) 2 / 2 σ n m ] ,
σ n m 2 = 1 2 J 2 { 4 α n [ sin 1 2 ( ω m - ω n ) t / ( ω m - ω n ) t ] } 2 .
σ i t 2 = n m σ n m 2 = 1 2 J 2 n m n > m N N ( 4 α n ) 2 [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 .
p ( w i ) = [ 1 / σ i ( 2 π ) 1 2 ] exp [ - ( w i - w ¯ i ) 2 / 2 σ i 2 ] ,
w ¯ i = α N J n
σ i 2 = 8 N 2 w ¯ i n m n > m N N [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 .
4 α J 2 n [ sin 1 2 ( ω m - ω n ) t / ( ω m - ω n ) t ] ,
σ c t 2 = 1 2 n m n > m N N ( 4 α J 2 n ) [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 ,
p ( w c ) = [ 1 / σ c ( 2 π ) 1 2 ] exp [ - ( w c - w ¯ c ) 2 / 2 σ c 2 ] ,
w ¯ c = α N J 2 n
σ c 2 = 8 N 2 w ¯ c 2 n m n > m N N [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 .
ν = M w ( t ) .
p 1 ( K ) = [ ( ν ¯ T ) K / K ! ] / exp ( - ν ¯ T ) ,
ν ¯ = M w ¯ .
m 1 = ν ¯ T ,             σ 1 2 = m 1 = ν ¯ T .
p ( K ) = 0 p 1 ( K ) p 2 ( ν T ) d ν T .
p 2 ( ν T ) = 1 σ 2 ( 2 π ) 1 2 exp [ - ( ν T - ν ¯ T ) 2 2 σ 2 2 ] ,
σ 2 2 = M 2 T 2 σ c 2 = ( ν ¯ T ) 2 8 N 2 n m n > m N N [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) ] 2 .
p 2 ( ν T ) = A ( ν ¯ T , σ 2 ) σ 2 ( 2 π ) 1 2 exp [ - ( ν T - ν ¯ T ) 2 2 σ 2 2 ]
A ( ν ¯ T , σ 2 ) σ 2 ( 2 π ) 1 2 0 exp [ - ( ν T - ν ¯ T ) 2 2 σ 2 2 ] d ν T = 1 .
p ( K ) = A σ 2 ( 2 π ) 1 2 0 ( ν T ) K K ! exp ( - ν T ) × exp [ - ( ν T - ν ¯ T ) 2 2 σ 2 2 ] d ν T .
m = K = 0 K p ( K ) = 0 K = 0 K x K K ! p 2 ( x ) e - x d x = ν ¯ T
σ 2 + m 2 = K = 0 K 2 p ( K ) = 0 K = 1 x K 1 ( K - 2 ) ! + 1 ( K - 1 ) ! p 2 ( x ) e - x d x = σ 2 2 + m 2 + m ,
σ 2 = ( ν ¯ T ) 2 8 N 2 n m n > m N N [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 + ν ¯ T .
σ 2 = μ ¯ 2 T ( b - a ) + μ ¯ T ( b - a ) ,
P D = 1 - p ( 0 ) .
p ( K ) = - p K ¯ ( K ) p ( K ¯ ) d K ¯ ,
p ( K ¯ ) = ( K ¯ ) K K ! e - K ¯ .
p ( K ¯ ) = ( 1 / 2 3 σ 2 ) ( ν ¯ T - 3 σ 2 ) < K ¯ < ν ¯ T + 3 σ 2 = 0             elsewhere .
p ( 0 ) = 1 2 γ ν ¯ T - γ ν ¯ T + γ e - x d x
= ( 1 / γ ) exp ( - ν ¯ T ) sinh γ ,
P D = 1 - [ exp ( - ν ¯ T ) sinh γ / γ ] .
σ c 2 = σ i 2 = w ¯ 2 8 N 2 n m n > m N N [ sin 1 2 ( ω m - ω n ) t ( ω m - ω n ) t ] 2 .
σ 2 = w ¯ 2 [ ( N 2 - N ) / N 2 ] w ¯ 2 .
σ 2 = w ¯ 2 4 ( b - a ) 2 a b a b sin 2 1 2 ( ω 1 - ω 2 ) t ( ω 1 - ω 2 ) 2 t 2 d f 1 d f 2 .
I = 1 4             ω 2 - π / t < ω 1 < ω 2 + π / t = 0 elsewhere .
σ 2 = [ w ¯ 2 / ( b - a ) 2 ] [ ( b - a ) / t ] .