Abstract

A decomposition of the field at the aperture of an optical system in terms of the eigenfunctions of a certain integral equation is useful in analyzing the detectability of incoherent objects. The kernel of the integral equation is the mutual coherence function of the light from the object. The decomposition permits specification of the number of degrees of freedom in the aperture field contributing to detection of the object. Quantum mechanically the coefficients of the modal decomposition become operators similar to the usual creation and annihilation operators for field modes. The optimum detector of the object is derived in terms of these operators. Specific detection probabilities are calculated for a uniform circular object whose light is observed at a circular aperture. The modal decomposition is also applied to estimating the radiance distribution of the object plane.

© 1970 Optical Society of America

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References

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  1. C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969), herein referred to as I.
    [CrossRef]
  2. C. W. Helstrom, J. Opt. Soc. Am. 59, 331 (1969) (II).
    [CrossRef]
  3. D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.
  4. C. W. Helstrom, Statistical Theory of Signal Detection, 2nd. ed. (Pergamon Press, Ltd., Oxford, 1968), Chs. 3, 8.
  5. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969). This paper provides references to previous work on image degrees of freedom.
    [CrossRef] [PubMed]
  6. W. Louisell, Radiation and Noise in Quantum Electronics (McGraw–Hill Book Co., New York, 1964), Ch. 4.
  7. See Ref. 4, pp. 69–72; Ref. 1, Sec. I.
  8. C. W. Helstrom, J. Opt. Soc. Am. 59, 924 (1969), herein referred to as III.
  9. See Ref. 4, Ch. 11, Sec. 1 (iii), p. 383.
  10. See Ref. 8, Sec. V and Appendix.
  11. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958), Sec. 8.6, pp. 136–139.
  12. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
    [CrossRef]
  13. See Ref. 1, Eqs. (1.8), (1.9), (A2).
  14. C. W. Helstrom, J. Opt. Soc. Am. 60, 233 (1970), herein referred to as IV. See Sec. IV.
    [CrossRef]
  15. D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 138.
  16. A. Kuriksha, Radio Engr. Electron. Phys. 13, 1567 (1968).
  17. G. Wentzel, Quantum Theory of Fields (Interscience Publishers, John Wiley & Sons, Inc., New York, 1949), Ch. I.
  18. R. J. Glauber, Phys. Rev. 131, 2766 (1963), Sec. VIII.
    [CrossRef]
  19. See Ref. 6, Sec. 6.10, pp. 242–245.
  20. C. W. Helstrom, Trans. IEEE IT-10, 275 (1964).
  21. See Ref. 4, p. 152.
  22. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
    [CrossRef]
  23. See Ref. 14, Sec. 5.
  24. See Ref. 4, p. 219.
  25. H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1946), pp. 473 ff.
  26. See also Ref. 1, Sec. VI, and Ref. 14.
  27. See Ref. 4, pp. 260–261.

1970 (1)

1969 (4)

1968 (1)

A. Kuriksha, Radio Engr. Electron. Phys. 13, 1567 (1968).

1964 (2)

C. W. Helstrom, Trans. IEEE IT-10, 275 (1964).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[CrossRef]

1963 (1)

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

1961 (1)

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[CrossRef]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1946), pp. 473 ff.

Gabor, D.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 138.

Glauber, R. J.

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

Grenander, U.

U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958), Sec. 8.6, pp. 136–139.

Helstrom, C. W.

Kuriksha, A.

A. Kuriksha, Radio Engr. Electron. Phys. 13, 1567 (1968).

Louisell, W.

W. Louisell, Radiation and Noise in Quantum Electronics (McGraw–Hill Book Co., New York, 1964), Ch. 4.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.

Pollak, H. O.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[CrossRef]

Slepian, D.

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[CrossRef]

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[CrossRef]

Szegö, G.

U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958), Sec. 8.6, pp. 136–139.

Toraldo di Francia, G.

Wentzel, G.

G. Wentzel, Quantum Theory of Fields (Interscience Publishers, John Wiley & Sons, Inc., New York, 1949), Ch. I.

Bell System Tech. J. (2)

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
[CrossRef]

D. Slepian, Bell System Tech. J. 43, 3009 (1964).
[CrossRef]

J. Opt. Soc. Am. (5)

Phys. Rev. (1)

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

Radio Engr. Electron. Phys. (1)

A. Kuriksha, Radio Engr. Electron. Phys. 13, 1567 (1968).

Trans. IEEE (1)

C. W. Helstrom, Trans. IEEE IT-10, 275 (1964).

Other (17)

See Ref. 4, p. 152.

See Ref. 6, Sec. 6.10, pp. 242–245.

See Ref. 14, Sec. 5.

See Ref. 4, p. 219.

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J., 1946), pp. 473 ff.

See also Ref. 1, Sec. VI, and Ref. 14.

See Ref. 4, pp. 260–261.

G. Wentzel, Quantum Theory of Fields (Interscience Publishers, John Wiley & Sons, Inc., New York, 1949), Ch. I.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 138.

W. Louisell, Radiation and Noise in Quantum Electronics (McGraw–Hill Book Co., New York, 1964), Ch. 4.

See Ref. 4, pp. 69–72; Ref. 1, Sec. I.

See Ref. 1, Eqs. (1.8), (1.9), (A2).

See Ref. 4, Ch. 11, Sec. 1 (iii), p. 383.

See Ref. 8, Sec. V and Appendix.

U. Grenander and G. Szegö, Toeplitz Forms and Their Applications (University of California Press, Berkeley, 1958), Sec. 8.6, pp. 136–139.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.

C. W. Helstrom, Statistical Theory of Signal Detection, 2nd. ed. (Pergamon Press, Ltd., Oxford, 1968), Chs. 3, 8.

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Figures (1)

Fig. 1
Fig. 1

Probability Qd of detecting a uniform circular object of radius ao by observations at an aperture of radius a, vs the average number Ns of photons received from the object. The average number of background photons is N0= NWT=1.0; the false-alarm probability is Q0=0.01. The curves are indexed by the parameter α=2πaaoR.

Tables (1)

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Table I Parameters in probability computation.

Equations (75)

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ψ + ( r , t ) = ψ s ( r , t ) + ψ n ( r , t ) .
1 2 E [ ψ n ( r 1 , t 1 ) ψ n * ( r 2 , t 2 ) ] = φ n ( r 1 , t 1 ; r 2 , t 2 ) = N δ ( r 1 - r 2 ) δ ( t 1 - t 2 ) ,
1 2 E [ ψ s ( r 1 , t 1 ) ψ s * ( r 2 , t 2 ) ] = φ s ( r 1 , t 1 ; r 2 , t 2 ) ,
φ s ( r 1 , t 1 ; r 2 , t 2 ) = φ s ( r 1 , r 2 ) χ ( t 1 - t 2 ) exp i Ω ( t 1 - t 2 ) ,
E s = ( 2 Ω 2 c ) - 1 T A φ s ( r , r ) d 2 r .
X ( ω ) = - χ ( τ ) e i ω τ d τ ,
W = [ - X ( ω ) d ω / 2 π ] 2 / - [ X ( ω ) ] 2 d ω / 2 π = χ ( 0 ) 2 / - χ ( τ ) 2 d τ .
ψ + ( r , t ) = p m a p m f p m ( r , t ) ,
A 0 T f p m * ( r , t ) f q n ( r , t ) d 2 r d t = δ p q δ m n .
λ p m f p m ( r , t ) = C A d 2 s 0 T d u φ s ( r , t ; s , u ) f p m ( s , u )
f p m ( r , t ) = η p ( r ) γ m ( t ) e - i Ω t ,
λ p m = h p g m ,
g m γ m ( t ) = T - 1 0 T χ ( t - s ) γ m ( s ) d s
h p η p ( r ) = ( 2 Ω 2 c T / E s ) A φ s ( r , s ) η p ( s ) d 2 s .
m g m = 1 ,             p h p = 1 ,
g m T - 1 X ( 2 π m / T ) , m = - 2 , - 1 , 0 , 1 , 2 , .
γ m ( t ) T - 1 2 exp ( - 2 π i m t / T ) .
φ s ( r 1 , r 2 ) = ( 8 π R 2 Ω 2 c ) - 1 × exp [ i k 2 R ( r 1 2 - r 2 2 ) ] β ( r 2 - r 1 ) ,
β ( r ) = o B ( u ) exp ( i k u · r / R ) d 2 u .
η p ( r ) = η p ( r ) exp ( - i k r 2 / 2 R ) ,
h p η p ( r 1 ) = [ A β ( 0 ) ] - 1 A β ( r 2 - r 1 ) η p ( r 2 ) d 2 r 2 ,
h p A δ B ( p x δ x , p y δ y ) / B T ,
B T = β ( 0 ) = O B ( u ) d 2 u
η p ( r ) A - 1 2 exp [ 2 π i ( p x x a - 1 + p y y b - 1 ) ] .
sinc [ ( x - ξ p x ) / δ x ] sinc [ ( y - ξ p y ) / δ y ] , δ x = λ R / a ,             δ y = λ R / b ,
( ξ p x , ξ p y ) = ( p x δ x , p y δ y )
M = B T 2 [ O B 2 ( v ) I A ( v ) 2 d 2 v ] - 1 ,
B 2 ( v ) = O B ( u ) B ( u - v ) d 2 u
I A ( v ) = A - 1 A exp ( i k v · r / R ) d 2 r
M A A o , e / λ 2 R 2
A o , e = B T 2 { O [ B ( u ) ] 2 d 2 u } - 1
ψ - ( r , t ) = [ ψ + ( r , t ) ] + .
φ ( r 1 , t 1 ; r 2 , t 2 ) = 1 2 Tr     [ ρ ψ - ( r 2 , t 2 ) ψ + ( r 1 , t 1 ) ] ,
b p m = ( Ω c / ) 1 2 A 0 T e i Ω t η p * ( r ) γ m * ( t ) ψ + ( r , t ) d 2 r d t .
b q n + = ( Ω c / ) 1 2 A 0 T e - i Ω t η q ( r ) γ n ( t ) ψ - ( r , t ) d 2 r d t
[ b p m , b q n + ] = b p m b q n + - b q n + b p m = δ pq δ m n , [ b p m , b q n ] = [ b p m + , b q n + ] = 0.
[ ψ + ( r 1 , t 1 ) , ψ - ( r 2 , t 2 ) ] = 1 2 ( 2 π ) - 3 ω - 1 exp [ - i ω ( t 1 - t 2 ) ] + i k · ( r 1 - r 2 ) ] d 3 k , ω 2 = c 2 k 2 = c 2 ( k x 2 + k y 2 + k z 2 ) .
[ b p m , b q n + ] = ( Ω / 8 π 3 c ) A 0 T A 0 T 0 k z - 1 η p * ( r 1 ) γ m * ( t 1 ) η q ( r 2 ) γ n ( t 2 ) × exp [ i ( Ω - ω ) ( t 1 - t 2 ) + i k · ( r 1 - r 2 ) ] d 2 r 1 d 2 r 2 d t 1 d t 2 d k x d k y d ω .
k · ( r 1 - r 2 ) = k x ( x 1 - x 2 ) + k y ( y 1 - y 2 ) ,
k z = ( ω 2 c - 2 - k x 2 - k y 2 )
[ ψ + ( r 1 , t 1 ) , ψ + ( r 2 , t 2 ) ] = 0
p i ( n p m ) = ( 1 - v p m ( i ) ) exp ( n p m ln v p m ( i ) ) , v p m ( i ) = N p m ( i ) / ( N p m ( i ) + 1 ) ,             i = 0 , 1 ,
N p m ( 0 ) = N = [ exp ( Ω / K T ) - 1 ] - 1
N p m ( 1 ) = N + h p g m N s .
U = ln p , m [ p 1 ( n p m ) / p 0 ( n p m ) ] = p , m { n p m ln ( v p m ( 1 ) / v p m ( 0 ) ) + ln [ ( 1 - v p m ( 1 ) ) / ( 1 - v p m ( 0 ) ) ] } .
Q 0 = Pr ( U > U 0 H 0 ) .
U = p , m [ n p m ln ( 1 + h p g m N s / N ) - h p g m N s ] .
U = p [ n p ln ( 1 + h p N s / N W T ) - h p N s ] ,
n p = n p m
U = p , m h p g m n p m = p , m h p g m b p m + b p m .
U = p h p n p
h N , k = ( 4 / α 2 ) λ N , k ( α ) ,             α = k a a 0 / R = 2 π a a 0 / λ R ,
M = α 2 / 4 = A A o / λ 2 R 2 ,
Q d = Pr { U > U 0 H 1 } ,
Q 0 = Pr { U > U 0 H 0 }
U = p σ h p n p
Q d = 1 - p σ ( n p ! ) - 1 exp ( n p ln n ¯ p - n ¯ p ) ,
p σ h p n p < U 0 .
n ¯ p = N 0 + N s h p
u ( s ; N s ) = E [ exp ( s U ) H 1 ] = exp { p σ ( N 0 + N s h p ) [ exp ( h p s ) - 1 ] } ,
Q 0 = [ 2 π Φ ( t ) ] - 1 2 × exp [ Φ ( t ) ] { 1 + 1 8 Φ ( t ) [ Φ ( t ) ] - 2 } ,
Φ ( s ) = ln μ ( s ; N s ) - s U 0 - ln s ,
κ n = d n d s n [ ln μ ( s ; N s ) ] s = 0 = p σ h p n ( N 0 + N s h p ) .
ln μ ( s ) = ln E [ exp ( s U ) H 1 ] = - p , m ln { 1 - N p m ( 1 ) [ exp ( h p g m s ) - 1 ] } ,
ln μ ( s ) = p , m ( N + h p g m N s ) [ exp ( h p g m s ) - 1 ] .
ln μ ( s ) = N M T O ( d 2 u / A o ) - ( d ω / 2 π ) × [ 1 + N s B ( u ) X ( ω ) / B ¯ N M T ] × { exp [ B ( u ) X ( ω ) s / B ¯ M T ] - 1 } ,
p ( { n p m } ; θ ) = m ( 1 - v p m ) exp ( n p m ln v p m ) , v p m = N p m / ( 1 + N p m ) ,
N p m = N + g m θ ,
E ( θ ˆ - θ ) 2 > { E [ θ ln p ( { n p m } ; θ ) ] 2 } - 1 = { m [ g m 2 / N p m ( 1 + N p m ) ] } - 1 .
W T 1 ,             N p m 1 ,
E ( B ˆ p - B p ) 2 / B p 2 N p s - 1 [ f 1 ( D p ) ] - 1 , D p = N p s / N W T ,
f 1 ( D ) = D W - [ X ( ω ) ] 2 [ 1 + D W X ( ω ) ] - 1 d ω / 2 π ,
n p = m n p m
f 1 ( D ) = D / ( D + 1 ) .
E ( B ˆ p - B p ) 2 / B p 2 = ( N / E p s ) 2 W T ,