Abstract

We are concerned with the derivation of the free-space form of the radiative transfer equation of traditional radiometry from statistical wave theory. It is shown that this equation governs the transport of all the generalized radiance functions of a wide class, for any field that is generated by a planar, secondary, quasi-homogeneous source, in the asymptotic limit as the wave number k = 2π/λ → ∞.

© 1992 Optical Society of America

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  1. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968).
    [CrossRef]
  2. A. Walther, J. Opt. Soc. Am. 63, 1622 (1973).
    [CrossRef]
  3. E. W. Marchand, E. Wolf, J. Opt. Soc. Am. 64, 1219 (1974).
    [CrossRef]
  4. E. Wolf, J. Opt. Soc. Am. 68, 6 (1978).
    [CrossRef]
  5. G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
    [CrossRef]
  6. W. H. Carter, E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
    [CrossRef]
  7. E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
    [CrossRef]
  8. Equation (2.5) is equivalent to requiring that B(r, s, ν) obey the free-space equation of radiative transfer: s · ∇B(r, s, ν) = 0.
  9. ℬW(0)(ρ,s,ν) was introduced in Ref. 1. ℬAS(0)(ρ,s,ν) is the complex version of the generalized radiance function introduced in Ref. 2; in that paper the real part of ℬAS(0)(ρ,s,ν) was used.
  10. A. T. Friberg, “Phase-space methods for partially coherent wavefields,” in Optics in Four Dimensions—1980, M. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313 (1981).
  11. A. T. Friberg, J. Opt. Soc. Am. 69, 192 (1979).
    [CrossRef]
  12. J. T. Foley, E. Wolf, Opt. Commun. 55, 236 (1985).
    [CrossRef]
  13. K. Kim, E. Wolf, J. Opt. Soc. Am. A 4, 1233 (1987).
    [CrossRef]
  14. G. S. Agarwal, E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
    [CrossRef]
  15. E. Wolf, J. Opt. Soc. Am. A 3, 76 (1986).
    [CrossRef]
  16. Lord Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II;Sec. 278 [with a modification appropriate to the time dependence exp(−2πiνt) used in the present paper].
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  18. C. J. Bouwkamp, Rep. Prog. Phys. (London Phys. Soc.)17, 35 (1954).
    [CrossRef]
  19. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1958), Sec. 21.

1987 (2)

G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
[CrossRef]

K. Kim, E. Wolf, J. Opt. Soc. Am. A 4, 1233 (1987).
[CrossRef]

1986 (1)

1985 (1)

J. T. Foley, E. Wolf, Opt. Commun. 55, 236 (1985).
[CrossRef]

1982 (1)

1979 (1)

1978 (1)

1977 (1)

1974 (1)

1973 (1)

1970 (1)

G. S. Agarwal, E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
[CrossRef]

1968 (1)

Agarwal, G. S.

G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
[CrossRef]

G. S. Agarwal, E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bouwkamp, C. J.

C. J. Bouwkamp, Rep. Prog. Phys. (London Phys. Soc.)17, 35 (1954).
[CrossRef]

Carter, W. H.

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1958), Sec. 21.

Foley, J. T.

G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
[CrossRef]

J. T. Foley, E. Wolf, Opt. Commun. 55, 236 (1985).
[CrossRef]

Friberg, A. T.

A. T. Friberg, J. Opt. Soc. Am. 69, 192 (1979).
[CrossRef]

A. T. Friberg, “Phase-space methods for partially coherent wavefields,” in Optics in Four Dimensions—1980, M. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313 (1981).

Kim, K.

Marchand, E. W.

Rayleigh, Lord

Lord Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II;Sec. 278 [with a modification appropriate to the time dependence exp(−2πiνt) used in the present paper].

Walther, A.

Wolf, E.

G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
[CrossRef]

K. Kim, E. Wolf, J. Opt. Soc. Am. A 4, 1233 (1987).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. A 3, 76 (1986).
[CrossRef]

J. T. Foley, E. Wolf, Opt. Commun. 55, 236 (1985).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. 72, 343 (1982).
[CrossRef]

E. Wolf, J. Opt. Soc. Am. 68, 6 (1978).
[CrossRef]

W. H. Carter, E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
[CrossRef]

E. W. Marchand, E. Wolf, J. Opt. Soc. Am. 64, 1219 (1974).
[CrossRef]

G. S. Agarwal, E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

J. T. Foley, E. Wolf, Opt. Commun. 55, 236 (1985).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, Opt. Commun. 62, 67 (1987).
[CrossRef]

Phys. Rev. D (1)

G. S. Agarwal, E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
[CrossRef]

Other (7)

Lord Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II;Sec. 278 [with a modification appropriate to the time dependence exp(−2πiνt) used in the present paper].

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

C. J. Bouwkamp, Rep. Prog. Phys. (London Phys. Soc.)17, 35 (1954).
[CrossRef]

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, Oxford, 1958), Sec. 21.

Equation (2.5) is equivalent to requiring that B(r, s, ν) obey the free-space equation of radiative transfer: s · ∇B(r, s, ν) = 0.

ℬW(0)(ρ,s,ν) was introduced in Ref. 1. ℬAS(0)(ρ,s,ν) is the complex version of the generalized radiance function introduced in Ref. 2; in that paper the real part of ℬAS(0)(ρ,s,ν) was used.

A. T. Friberg, “Phase-space methods for partially coherent wavefields,” in Optics in Four Dimensions—1980, M. Machado, L. M. Narducci, eds., AIP Conf. Proc.65, 313 (1981).

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Figures (1)

Fig. 1
Fig. 1

Illustrating the notation.

Equations (78)

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J ( s , ν ) = ( 2 π k s z ) 2 W ( 0 ) ( k s , k s , ν ) ,
W ( 0 ) ( f 1 , f 2 , ν ) = 1 ( 2 π ) 4 W ( 0 ) ( ρ 1 , ρ 2 , ν ) × exp [ i ( f 1 · ρ 1 + f 2 · ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 ,
k = 2 π ν / c ,
W ( 0 ) ( ρ 1 , ρ 2 , ν ) = U ( 0 ) ( ρ 1 , ν ) U ( 0 ) * ( ρ 2 , ν ) ,
J ( s , ν ) = s z σ B ( 0 ) ( ρ , s , ν ) d 2 ρ .
B ( r , s , ν ) = B ( 0 ) [ ρ ( z / s z ) s , s , ν ] .
W ( 0 ) ( ρ , s , ν ) = ( k 2 π ) 2 s z W ( 0 ) ( ρ + 1 2 ρ , ρ 1 2 ρ ; ν ) × exp ( i k s · ρ ) d 2 ρ ,
AS ( 0 ) ( ρ , s , ν ) = ( k 2 π ) 2 s z W ( 0 ) ( ρ , ρ ; ν ) × exp [ i k s · ( ρ ρ ) ] d 2 ρ .
ŝ = i ƛ ,
ƛ = λ / 2 π = 1 / k , = ( x , y ) .
[ x ̂ , ŝ x ] = i ƛ , [ ŷ , ŝ y ] = i ƛ ,
W ( 0 ) ( ρ 1 , ρ 2 , ν ) = ρ 1 | Ĝ | ρ 2 .
Ω ( 0 ) ( ρ , s , ν ) = s z ( 2 π ) 4 Ω ( u , v ) × exp [ i ( u · ρ + v · s + 1 2 ƛ u · v ) ] × exp ( i u · ρ 1 ) W ( 0 ) ( ρ 1 , ρ 1 ƛ v , ν ) d 2 u d 2 υ d 2 ρ 1 ,
Ω ( u , v ) = [ Ω ( u , v ) ] 1 ,
Ω W ( u , v ) = 1 ,
Ω AS ( u , v ) = exp ( i ƛ u · v / 2 )
W ( r 1 , r 2 , ν ) = U ( r 1 , ν ) U * ( r 2 , ν ) ,
Ω ( r , s , ν ) = s z ( 2 π ) 4 Ω ( u , v ) × exp [ i ( u · ρ + v · s + 1 2 ƛ u · v ) ] × exp ( i u · ρ 1 ) W ( r 1 , r 1 ƛ v , ν ) d 2 u d 2 υ d 2 ρ 1 .
ρ = ρ 1 ( 1 / 2 ) ƛ v ,
ρ = ƛ v .
Ω ( r , s , ν ) = k 2 ( 2 π ) 4 s z Ω ( u , k ρ ) × exp { i [ u · ( ρ ρ ) + k s · ρ ] } × W ( r + 1 2 ρ , r 1 2 ρ , ν ) d 2 u d 2 ρ d 2 ρ ,
W ( r 1 , r 2 , ν ) = U ( r 1 , ν ) U * ( r 2 , ν ) ,
U ( r , ν ) = G ( r ρ 0 , ν ) U ( 0 ) ( ρ 0 , ν ) d 2 ρ 0 .
G ( R , ν ) = 1 2 π z [ exp ( i k R ) R ] ,
W ( r 1 , r 2 , ν ) = G ( r 1 ρ 01 , ν ) × G * ( r 2 ρ 02 , ν ) W ( 0 ) ( ρ 01 , ρ 02 , ν ) d 2 ρ 01 d 2 ρ 02 .
ρ 1 = ρ + ( 1 / 2 ) ρ ,
ρ 2 = ρ ( 1 / 2 ) ρ ;
ρ 01 = ρ 0 + ( 1 / 2 ) ρ 0 ,
ρ 02 = ρ 0 ( 1 / 2 ) ρ 0 .
W [ r + ( 1 / 2 ) ρ , r ( 1 / 2 ) ρ , ν ] = G [ r ρ 0 + ( 1 / 2 ) ( ρ ρ 0 ) , ν ] × G * [ r ρ 0 ( 1 / 2 ) ( ρ ρ 0 ) , ν × W ( 0 ) ( ρ 0 + ( 1 / 2 ) ρ 0 , ρ 0 ( 1 / 2 ) ρ 0 , ν ) d 2 ρ 0 d 2 ρ 0 ,
Ω ( r , s , ν ) = k 2 ( 2 π ) 4 s z M ( u , r , ρ 0 , ν ) × exp { i [ u · ( ρ ρ ) + k s · ρ 0 ] } × d 2 u d 2 ρ d 2 ρ ,
M ( u , r , ρ 0 , ν ) = Ω [ u , k ( ρ b ρ a + ρ 0 ) ] × exp [ i k s · ( ρ b ρ a ) ] G ( r ρ a , ν ) G * ( r ρ b , ν ) × W ( 0 ) [ ½ ( ρ a + ρ b + ρ 0 ) , ( 1 / 2 ) ( ρ a + ρ b ρ 0 ) , ν ] d 2 ρ a d 2 ρ b .
W ( 0 ) ( ρ 01 , ρ 02 , ν ) = S ( 0 ) [ ( 1 / 2 ) ( ρ 01 + ρ 02 ) , ν ] g ( 0 ) ( ρ 01 ρ 02 , ν ) ,
W ( 0 ) [ ( 1 / 2 ) ( ρ a + ρ b + ρ 0 ) , ( 1 / 2 ) ( ρ a + ρ b ρ 0 ) , ν ] = S ( 0 ) [ ( 1 / 2 ) ( ρ a + ρ b ) , ν ] g ( 0 ) ( ρ 0 , ν ) ,
M ( u , r , ρ 0 , ν ) = g ( 0 ) ( ρ 0 , ν ) Ω [ u , k ( ρ b ρ a + ρ 0 ) ] × exp [ i k s · ( ρ b ρ a ) ] G ( r ρ a , ν ) × G * ( r ρ b , ν ) × S ( 0 ) [ ( 1 / 2 ) ( ρ a + ρ b ) , ν ] d 2 ρ a d 2 ρ b .
G ( R , ν ) = 1 2 π [ ( i k 1 R ) z R ] exp ( i k R ) R ,
G ( R , ν ) i k z 2 π R exp ( i k R ) R .
M ( u , r , ρ 0 , ν ) ( k z 2 π ) 2 g ( 0 ) ( ρ 0 , ν ) Ω [ u , k ( ρ b ρ a + ρ 0 ) ] × S ( 0 ) [ 1 2 ( ρ a + ρ b ) , ν ] × 1 | r ρ a | 2 exp [ i k ϕ ( r , ρ a ) ] × 1 | r ρ b | 2 exp [ i k ϕ ( r , ρ b ) ] d 2 ρ a d 2 ρ b ,
ϕ ( r , ρ j ) = | r ρ j | + s · ρ j ( j = a , b ) .
Ω ( u , v ) = f ( λ ¯ u , v ) .
Ω ( u , v ) = F ( ƛ u , v ) .
Ω ( u , k ρ ) = F ( ƛ u , k ρ ) .
Ω ( u , v ) = exp ( α u 2 + β u · v + γ υ 2 ) ,
Ω ( u , v ) = exp ( α ƛ 2 u 2 + β ƛ u · v ) ,
Ω ( u , k ρ ) = exp ( α ƛ 2 u 2 + β u · ρ ) ,
M ( u , r , ρ 0 , ν ) ( k z 2 π ) 2 g ( 0 ) ( ρ 0 , ν ) N ( u , r , ρ 0 , ν , ρ b ) × 1 | r ρ b | 2 exp [ i k ϕ ( r , ρ b ) ] d 2 ρ b ,
N ( u , r , ρ 0 , ν , ρ b ) = Ω [ u , k ( ρ b ρ a + ρ 0 ) ] × S ( 0 ) [ 1 2 ( ρ a + ρ b ) , ν ] × 1 | r ρ a | 2 exp [ i k ϕ ( r , ρ a ) ] d 2 ρ a .
ρ c ρ ( z / s z ) s ,
N ( u , r , ρ 0 , ν , ρ b ) 2 π i k z exp ( i k s · r ) Ω [ u , k ( ρ b ρ c + ρ 0 ) ] × S ( 0 ) [ 1 2 ( ρ c + ρ b ) , ν ] as k .
M ( u , r , ρ 0 , ν ) ( k z 2 π ) 2 g ( 0 ) ( ρ 0 , ν ) 2 π i k z exp ( i k s · r ) × Ω [ u , k ( ρ b ρ c + ρ 0 ) ] × S ( 0 ) [ 1 2 ( ρ c + ρ b ) , ν ] × 1 | r ρ b | 2 exp [ i k ϕ ( r , ρ b ) ] d 2 ρ b as k .
M ( u , r , ρ 0 , ν ) ( k z 2 π ) 2 g ( 0 ) ( ρ 0 , ν ) 2 π i k z × exp ( i k s · r ) [ 2 π i k z exp ( i k s · r ) ] × Ω [ u , k ( ρ c ρ c + ρ 0 ) ] × S ( 0 ) [ 1 2 ( ρ c + ρ c ) , ν ] = Ω ( u , k ρ 0 ) S ( 0 ) ( ρ c , ν ) g ( 0 ) ( ρ 0 , ν ) = Ω ( u , k ρ 0 ) W ( 0 ) ( ρ c + 1 2 ρ 0 , ρ c 1 2 ρ 0 , ν ) ,
Ω ( r , s , ν ) k 2 ( 2 π ) 4 s z Ω ( u , k ρ 0 ) × exp { i [ u · ( ρ ρ ) + k s · ρ 0 ] } × W ( 0 ) [ ρ ( z / s z ) s + 1 2 ρ 0 , ρ ( z / s z ) s 1 2 ρ 0 , ν ] d 2 u d 2 ρ d 2 ρ 0 .
ρ 0 = ρ ( z / s z ) s ,
ρ ρ = ρ ( z / s z ) s ρ 0 ,
Ω ( r , s , ν ) k 2 ( 2 π ) 4 s z Ω ( u , k ρ 0 ) × exp ( i { u · [ ρ ( z / s z ) s ρ 0 ] + k s ρ 0 } ) × W ( 0 ) ( ρ 0 + 1 2 ρ 0 , ρ 0 1 2 ρ 0 , ν ) × d 2 u d 2 ρ 0 d 2 ρ 0 .
Ω ( r , s , ν ) Ω ( 0 ) [ ρ ( z / s z ) s , s , ν ] , as k ,
( 2 + k 2 ) U ( r , ν ) = 0 ,
U ( r , ν ) = a ( s , ν ) exp ( i k s · r ) d 2 s ,
s = ( s x , s y , s z ) , s = ( s x , s y , 0 ) , s z = ( 1 s 2 ) 1 / 2 when s 1 = i ( s 2 1 ) 1 / 2 when s > 1 ,
exp ( i k s · r ) = exp { i k z [ 1 + ( 2 / k 2 ) ] 1 / 2 } exp ( i k s · ρ ) ,
U ( r , ν ) = exp { i k z [ 1 + ( 2 / k 2 ) ] 1 / 2 } a ( s · ν ) × exp ( i k s · ρ ) d 2 s = exp { i k z [ 1 + ( 2 / k 2 ) ] 1 / 2 } U ( 0 ) ( ρ , ν ) .
{ z i k [ 1 + ( 2 / k 2 ) ] 1 / 2 } U ( r , ν ) = 0 .
W ( ρ 1 , ρ 2 , z ) U ( ρ 1 , z ) U * ( ρ 2 , z ) = exp { i k z [ 1 + ( 1 2 / k 2 ) ] 1 / 2 } U ( ρ 1 , 0 ) U * ( ρ 2 , 0 ) × exp { i k z [ 1 + ( 2 2 / k 2 ) ] 1 / 2 } ,
Ĝ ( z ) = exp [ i k z ( 1 ŝ 2 ) 1 / 2 ] Ĝ ( 0 ) exp [ i k z ( 1 ŝ 2 ) 1 / 2 ] .
Ĝ ( z ) z = i λ ¯ [ ( 1 ŝ 2 ) 1 / 2 , Ĝ ( z ) ] .
i [ , ] { } PB .
i λ ¯ [ Â , B ̂ ] A ( ρ s s ρ ) B
B ( r , s , ν ) z = ( 1 s 2 ) 1 / 2 ( s ρ ) B ( r , s , ν ) = 1 ( 1 s 2 ) 1 / 2 s · B ( r , s , ν ) = 1 s z s · B ( r , s , ν ) ,
s · B ( r , s , ν ) = 0 .
Ω ( r , s , ν ) = k 2 ( 2 π ) 4 s z d 2 u d 2 ρ d 2 ρ Ω ( u , k ρ ) × exp { i [ u · ( ρ ρ ) + k s · ρ ] } × d 2 ρ 0 d 2 ρ 0 G [ r ρ 0 + ( 1 / 2 ) ( ρ ρ 0 ) , ν ] × G * [ r ρ 0 ( 1 / 2 ) ( ρ ρ 0 ) , ν ] × W ( 0 ) [ ρ 0 + ( 1 / 2 ) ρ 0 , ρ 0 ( 1 / 2 ) ρ 0 , ν ] .
δ = ρ ρ 0 ,
Ω ( r , s , ν ) = k 2 ( 2 π ) 4 s z M ( u , r , ρ 0 , ν ) × exp { i [ u · ( ρ ρ ) + k s · ρ 0 ] } × d 2 u d 2 ρ d 2 ρ 0 ,
M ( u , r , ρ 0 , ν ) = Ω [ u , k ( δ + ρ 0 ) ] exp ( i k s · δ ) × G [ r ρ 0 + ( 1 / 2 ) δ , ν ] × G * [ r ρ 0 , ( 1 / 2 ) δ , ν ] × W ( 0 ) [ ρ 0 + ( 1 / 2 ) ρ 0 , ρ 0 ( 1 / 2 ) ρ 0 , ν ] d 2 ρ 0 d 2 δ .
ρ a = ρ 0 ( 1 / 2 ) δ ,
ρ b = ρ 0 + ( 1 / 2 ) δ .
ρ 0 = ( 1 / 2 ) ( ρ a + ρ b ) ,
δ = ρ b ρ a .
M ( u , r , ρ 0 , ν ) = Ω [ u , k ( ρ b ρ a + ρ 0 ) ] × exp [ i k s · ( ρ b ρ a ) ] × G ( r ρ a , ν ) G * ( r ρ b , ν ) × W ( 0 ) [ ( 1 / 2 ) ( ρ a + ρ b + ρ 0 ) , ( 1 / 2 ) ( ρ a + ρ b ρ 0 ) , ν ] d 2 ρ a d 2 ρ b ,

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