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 and 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, and E. Wolf, Opt. Commun. 62, 67 (1987).
    [Crossref]
  6. W. H. Carter and 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 and 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 and E. Wolf, Opt. Commun. 55, 236 (1985).
    [Crossref]
  13. K. Kim and E. Wolf, J. Opt. Soc. Am. A 4, 1233 (1987).
    [Crossref]
  14. G. S. Agarwal and 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 and 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, and E. Wolf, Opt. Commun. 62, 67 (1987).
[Crossref]

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

1986 (1)

1985 (1)

J. T. Foley and 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 and E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
[Crossref]

1968 (1)

Agarwal, G. S.

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

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

Born, M.

M. Born and 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, and E. Wolf, Opt. Commun. 62, 67 (1987).
[Crossref]

J. T. Foley and 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 and 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, and E. Wolf, Opt. Commun. 62, 67 (1987).
[Crossref]

K. Kim and 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 and 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 and E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
[Crossref]

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

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

M. Born and 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 and E. Wolf, Opt. Commun. 55, 236 (1985).
[Crossref]

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

Phys. Rev. D (1)

G. S. Agarwal and 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 and 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 and 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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