Abstract

The diffusion equation is derived from the ordinary space-time transport equation having a scattering cross section of the form σ(Ω·Ω), Ω and Ω being the azimuth vectors of incident and scattered waves. The condition of applicability is examined in detail. It is shown that the diffusion equation necessarily becomes first order in time in contrast to a recent paper on a similar subject in which it is given as second order in time. This is further confirmed for the case of isotropic scattering where an exact analytical solution is obtainable. The boundary conditions on a surface of medium discontinuity are also derived in the diffusion approximation and, in this connection, the pulse-width broadening in random media is investigated in terms of the pulse moments, which are obtained by solving boundary-value problems of a simple equation in space.

© 1980 Optical Society of America

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  1. C. H. Liu and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain, or fog,” J. Opt. Soc. Am. 67, 1261–1266 (1977).
    [Crossref]
  2. S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
    [Crossref]
  3. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (4978).
  4. V. I. Shishov, “Effect of refraction on scintillation characteristics and average pulse shape of pulsars,” Sov. Astron. 17, 598–602 (1974).
  5. L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics, II: A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
    [Crossref]
  6. L. Dolin, “Scattering of a light beam in a turbulent medium,” Radiophys. Quantum Electron. (USSR) 7, 380–391 (1964).
  7. K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
    [Crossref]
  8. K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. (N.Y.) 9, 1–23 (1960).
    [Crossref]
  9. N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
    [Crossref]
  10. U. M. Titulaer, “A systematic solution procedure for the Fokker–Planck equation of a Brownian particle in the high-friction case,” Physica 91A, 321–344 (1978).
  11. S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
    [Crossref]
  12. The second inequality does not always mean that |γ−1∂jJjkl′(ρ¯)|≪|Jkl′(ρ¯)|.
  13. It will be noted that Jr′(Ω,ρ¯)≠0even in a domain where Jc′(Ω,ρ¯)=0, giving J(r)=JJ(r)=0but JJk(r)≠0. However, the latter magnitude is of the order of (γ−1∂j)2γI, and the substitution of its Fourier transform (λj/γ)2γI˜(λ¯)with I˜(λ¯)given by Eq. (25) into the corresponding expression (23) for Jr″(ρ¯), given by Eq. (20) in terms of JJk(r)(ρ¯), shows that the contribution to the total intensity is smaller than I(ρ¯)by a factor of the order of (λj/γ)2.
  14. The boundary condition (39) differs in a few points from that given by Ref. 3, but agrees in the special case of time-independent isotropic scattering where a1= 0. In this case, the solution for a semi-infinite slab is known as the Milne solution and, in Ref. 3, it was pointed out that the condition is a good approximation to the exact one.
  15. At the end of Sec. III, a corrected flux vector Ij was introduced and was shown to give the same diffusion equation as (30) except for the source term J′(ρ¯), replaced byJ′(ρ¯)−(1−a1)−1γ−1∂jJj′(ρ¯). Here, Jj′is zero in free space and therefore ∂jJj′gives rise to a term of the surface δ function or surface source, which is not zero only on the boundary surface. This effect can be made explicit by applying Gauss’s theorem to both sides of the new diffusion equation over the infinitesimal space containing boundary surface, as used for Eq. (37), or simply by using njIj(ρ¯)|−+=0, to find the discontinuity3−1nj∂jI(ρ¯)|−+=njJj′(ρ¯)|+, while I(ρ¯)|=0since Is(Ω,ρ¯)|=0by Eq. (37). On the other hand, the boundary condition (39) is corrected to be{I(ρ¯)+2(1−a1)−1γ−1nj[−3−1∂jI(ρ¯)+Jj′(ρ¯)]}|+=0, which is also expressed, with the aid of the above relation, as[I(ρ¯)−(2/3)(1−a1)−1γ−1nj∂jI(ρ¯)]|−=0, being the same as the old condition except for |+replaced by |−. Thus it follows that the use of the corrected Ij does not change the boudary condition at all, as far as the source term is used over the entire space containing the boundary of medium discontinuity inside. The first corrected boundary condition mentioned above is similar to that given in Ref. 3 but is not quite the same, resulting from the second-order diffusion equation in time in the latter.
  16. K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980).
    [Crossref]

1980 (1)

K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980).
[Crossref]

1979 (1)

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[Crossref]

1978 (1)

U. M. Titulaer, “A systematic solution procedure for the Fokker–Planck equation of a Brownian particle in the high-friction case,” Physica 91A, 321–344 (1978).

1977 (2)

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
[Crossref]

C. H. Liu and K. C. Yeh, “Propagation of pulsed beam waves through turbulence, cloud, rain, or fog,” J. Opt. Soc. Am. 67, 1261–1266 (1977).
[Crossref]

1975 (1)

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics, II: A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[Crossref]

1974 (2)

V. I. Shishov, “Effect of refraction on scintillation characteristics and average pulse shape of pulsars,” Sov. Astron. 17, 598–602 (1974).

S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
[Crossref]

1966 (1)

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[Crossref]

1964 (1)

L. Dolin, “Scattering of a light beam in a turbulent medium,” Radiophys. Quantum Electron. (USSR) 7, 380–391 (1964).

1960 (1)

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. (N.Y.) 9, 1–23 (1960).
[Crossref]

Case, K. M.

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. (N.Y.) 9, 1–23 (1960).
[Crossref]

Dolin, L.

L. Dolin, “Scattering of a light beam in a turbulent medium,” Radiophys. Quantum Electron. (USSR) 7, 380–391 (1964).

Furutsu, K.

K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980).
[Crossref]

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[Crossref]

Gantsevich, S. V.

S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
[Crossref]

Gurevich, V. L.

S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
[Crossref]

Hong, S. T.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
[Crossref]

Ishimaru, A.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
[Crossref]

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (4978).

Jokipii, J. R.

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics, II: A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[Crossref]

Katilius, R.

S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
[Crossref]

Kušcer, I.

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[Crossref]

Lee, L. C.

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics, II: A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[Crossref]

Liu, C. H.

McCormick, N. J.

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[Crossref]

Shishov, V. I.

V. I. Shishov, “Effect of refraction on scintillation characteristics and average pulse shape of pulsars,” Sov. Astron. 17, 598–602 (1974).

Sreenivasiah, I.

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
[Crossref]

Titulaer, U. M.

U. M. Titulaer, “A systematic solution procedure for the Fokker–Planck equation of a Brownian particle in the high-friction case,” Physica 91A, 321–344 (1978).

Yeh, K. C.

Ann. Phys. (N.Y.) (1)

K. M. Case, “Elementary solutions of the transport equation and their applications,” Ann. Phys. (N.Y.) 9, 1–23 (1960).
[Crossref]

Astrophys. J. (1)

L. C. Lee and J. R. Jokipii, “Strong scintillations in astrophysics, II: A theory of temporal broadening of pulses,” Astrophys. J. 201, 532–543 (1975).
[Crossref]

IEEE Trans. Antennas Propag. (1)

S. T. Hong, I. Sreenivasiah, and A. Ishimaru, “Plane wave pulse propagation through random media,” IEEE Trans. Antennas Propag. AP-25, 822–827 (1977).
[Crossref]

J. Math. Phys. (3)

N. J. McCormick and I. Kuščer, “Bi-orthogonality relations for solving half-space transport problems,” J. Math. Phys. 7, 2036–2045 (1966).
[Crossref]

K. Furutsu, “An analytical theory of pulse wave propagation in turbulent media,” J. Math. Phys. 20, 617–628 (1979).
[Crossref]

K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropic random media,” J. Math. Phys. 21, 765–777 (1980).
[Crossref]

J. Opt. Soc. Am. (2)

Phys. Cond. Matter (1)

S. V. Gantsevich, V. L. Gurevich, and R. Katilius, “Diffusion near nonequilibrium steady state,” Phys. Cond. Matter 18, 165–178 (1974).
[Crossref]

Physica (1)

U. M. Titulaer, “A systematic solution procedure for the Fokker–Planck equation of a Brownian particle in the high-friction case,” Physica 91A, 321–344 (1978).

Radiophys. Quantum Electron. (USSR) (1)

L. Dolin, “Scattering of a light beam in a turbulent medium,” Radiophys. Quantum Electron. (USSR) 7, 380–391 (1964).

Sov. Astron. (1)

V. I. Shishov, “Effect of refraction on scintillation characteristics and average pulse shape of pulsars,” Sov. Astron. 17, 598–602 (1974).

Other (4)

The second inequality does not always mean that |γ−1∂jJjkl′(ρ¯)|≪|Jkl′(ρ¯)|.

It will be noted that Jr′(Ω,ρ¯)≠0even in a domain where Jc′(Ω,ρ¯)=0, giving J(r)=JJ(r)=0but JJk(r)≠0. However, the latter magnitude is of the order of (γ−1∂j)2γI, and the substitution of its Fourier transform (λj/γ)2γI˜(λ¯)with I˜(λ¯)given by Eq. (25) into the corresponding expression (23) for Jr″(ρ¯), given by Eq. (20) in terms of JJk(r)(ρ¯), shows that the contribution to the total intensity is smaller than I(ρ¯)by a factor of the order of (λj/γ)2.

The boundary condition (39) differs in a few points from that given by Ref. 3, but agrees in the special case of time-independent isotropic scattering where a1= 0. In this case, the solution for a semi-infinite slab is known as the Milne solution and, in Ref. 3, it was pointed out that the condition is a good approximation to the exact one.

At the end of Sec. III, a corrected flux vector Ij was introduced and was shown to give the same diffusion equation as (30) except for the source term J′(ρ¯), replaced byJ′(ρ¯)−(1−a1)−1γ−1∂jJj′(ρ¯). Here, Jj′is zero in free space and therefore ∂jJj′gives rise to a term of the surface δ function or surface source, which is not zero only on the boundary surface. This effect can be made explicit by applying Gauss’s theorem to both sides of the new diffusion equation over the infinitesimal space containing boundary surface, as used for Eq. (37), or simply by using njIj(ρ¯)|−+=0, to find the discontinuity3−1nj∂jI(ρ¯)|−+=njJj′(ρ¯)|+, while I(ρ¯)|=0since Is(Ω,ρ¯)|=0by Eq. (37). On the other hand, the boundary condition (39) is corrected to be{I(ρ¯)+2(1−a1)−1γ−1nj[−3−1∂jI(ρ¯)+Jj′(ρ¯)]}|+=0, which is also expressed, with the aid of the above relation, as[I(ρ¯)−(2/3)(1−a1)−1γ−1nj∂jI(ρ¯)]|−=0, being the same as the old condition except for |+replaced by |−. Thus it follows that the use of the corrected Ij does not change the boudary condition at all, as far as the source term is used over the entire space containing the boundary of medium discontinuity inside. The first corrected boundary condition mentioned above is similar to that given in Ref. 3 but is not quite the same, resulting from the second-order diffusion equation in time in the latter.

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Figures (2)

FIG. 1
FIG. 1

Boundary of medium discontinuity and notations for Eq. (37).

FIG. 2
FIG. 2

Branch cut and pole for the integral (A17) with (A16).

Equations (78)

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[ Ω j j + t + γ t ] I ( Ω , ρ ¯ ) = d Ω σ ( Ω | Ω ) I ( Ω , ρ ¯ ) + J c ( Ω , ρ ¯ ) .
γ t = γ + γ ab , γ = d Ω σ ( Ω · Ω ) ,
I ( Ω , ρ ¯ ) = I c ( Ω , ρ ¯ ) + I s ( Ω , ρ ¯ ) .
( Ω j j + t + γ t ) I c ( Ω , ρ ¯ ) = J c ( Ω , ρ ¯ ) ,
[ Ω j j + t + γ t ] I s ( Ω , ρ ¯ ) = d Ω σ ( Ω · Ω ) I s ( Ω , ρ ¯ ) + J c ( Ω , ρ ¯ ) ,
J c ( Ω , ρ ¯ ) = d Ω σ ( Ω · Ω ) I c ( Ω , ρ ¯ ) .
I ( ρ ¯ ) = d Ω I s ( Ω , ρ ¯ ) , I j ( ρ ¯ ) = d Ω Ω j I s ( Ω , ρ ¯ ) , I i j k ( ρ ¯ ) = d Ω Ω i Ω j Ω k I s ( Ω , ρ ¯ ) , i , j , k , = 1 , 2 , 3 ,
J ( ρ ¯ ) = d Ω J c ( Ω , ρ ¯ ) , J j ( ρ ¯ ) = d Ω Ω j J c ( Ω , ρ ¯ ) , J i j k ( ρ ¯ ) = d Ω Ω i Ω j Ω k J c ( Ω , ρ ¯ ) .
γ 1 d Ω Ω j σ ( Ω · Ω ) = a 1 Ω j ,
γ 1 d Ω Ω j Ω k σ ( Ω · Ω ) = b Ω j Ω k + ( 1 / 2 ) ( 1 b ) ( δ j k Ω j Ω k ) ,
γ 1 d Ω Ω i Ω j Ω k σ ( Ω · Ω ) = c 1 Ω i Ω j Ω k + c 2 [ ( δ i j Ω i Ω j ) Ω k + ( δ j k Ω j Ω k ) Ω i + ( δ k i Ω k Ω i ) Ω j ] , etc .
a n = γ 1 d Ω P n ( Ω · Ω ) σ ( Ω · Ω ) , a 0 = P n ( 0 ) = 1 ,
( 1 / 2 ) ( 1 b ) = ( 1 / 3 ) ( 1 a 2 ) , c 2 = ( 1 / 5 ) ( a 1 a 3 ) , c 1 3 c 2 = a 3 , etc .
j I j ( ρ ¯ ) + ( t + γ ab ) I ( ρ ¯ ) = J ( ρ ¯ ) ,
j I j k ( ρ ¯ ) + [ t + γ ab + ( 1 a 1 ) γ ] I k ( ρ ¯ ) = J k ( ρ ¯ ) ,
j I j k l ( ρ ¯ ) + [ t + γ ab + ( 3 / 2 ) ( 1 b ) γ ] I k l ( ρ ¯ ) = ( 1 / 2 ) ( 1 b ) γ I ( ρ ¯ ) δ k l + J k l ( ρ ¯ ) ,
j I j k l m ( ρ ¯ ) + [ t + γ ab + ( 1 + 3 c 2 c 1 ) γ ] I k l m ( ρ ¯ ) = c 2 γ [ I k ( ρ ¯ ) δ l m + I l ( ρ ¯ ) δ m k + I m ( ρ ¯ ) δ k l ] + J k l m ( ρ ¯ ) ,
| j I j k l | , | ( t + γ ab ) I k l | ( 3 / 2 ) ( 1 b ) γ | I k l | ,
I k l ( ρ ¯ ) ~ ( 1 / 3 ) δ k l I ( ρ ¯ ) + ( 1 a 2 ) 1 γ 1 J k l ( ρ ¯ ) ,
[ t + γ ab + ( 1 a 1 ) γ ] I k ( ρ ¯ ) ~ ( 1 / 3 ) k I ( ρ ¯ ) ( 1 a 2 ) 1 γ 1 j J j k ( ρ ¯ ) + J k ( ρ ¯ ) .
[ 1 + ( 1 a 1 ) 1 γ 1 ( t + γ ab ) ] ( t + γ ab ) I ( ρ ¯ ) 3 1 ( 1 a 1 ) 1 γ 1 j 2 I ( ρ ¯ ) = J c ( ρ ¯ ) ,
J c ( ρ ¯ ) = [ 1 + ( 1 a 1 ) 1 γ 1 ( t + γ ab ) ] J ( ρ ¯ ) + ( 1 a 1 ) 1 [ γ 1 j J j ( ρ ¯ ) + ( 1 a 2 ) 1 γ 2 j k J j k ( ρ ¯ ) ]
I ( ρ ¯ ) = ( 2 π ) 4 d λ ¯ exp [ i ( ν t λ · ρ ) ] I ˜ ( λ ¯ ) , λ = ( λ 1 , λ 2 , λ 3 ) , d λ ¯ = d λ d ν , d λ = d λ 1 , d λ 2 d λ 3 ,
{ [ 1 + ( 1 a 1 ) 1 γ 1 ( i ν / c + γ ab ) ] ( i ν / c + γ ab ) + 3 1 ( 1 a 1 ) 1 γ 1 λ j 2 } I ˜ ( λ ¯ ) = J ˜ c ( λ ¯ ) ,
J ˜ c ( λ ¯ ) = [ 1 + ( 1 a 1 ) 1 γ 1 ( i ν / c + γ ab ) ] J ˜ ( λ ¯ ) + ( 1 a 1 ) 1 [ i ( λ j / γ ) J ˜ j ( λ ¯ ) ( 1 a 2 ) 1 ( λ j λ k / γ 2 ) J j k ( λ ¯ ) ] ,
γ ab / γ , | ν / c γ | 1 a 2 ~ 1 ,
I ˜ ( λ ¯ ) ~ [ i ν / c + γ ab + 3 1 ( 1 a 1 ) 1 γ 1 λ j 2 ] 1 J ˜ c ( λ ¯ ) + δ [ i ν / c + γ ab + 3 1 ( 1 a 1 ) 1 γ 1 λ j 2 ] g ( λ ¯ ) ,
i ν / c γ = γ ab / γ 3 1 ( 1 a 1 ) 1 γ 2 λ j 2 , ( λ j / γ ) 2 3 ( 1 a 1 ) ( 1 a 2 ) ,
λ j / γ = i ( 3 / 2 ) ( 1 a 1 ) ( ρ j / c t ) , ( 3 / 4 ) ( 1 a 1 ) ( ρ / c t ) 2 1 a 2 .
I ( ρ ¯ ) = c [ 3 ( 1 a 1 ) γ / 4 π c t ] 3 / 2 × exp [ ( 3 / 4 ) ( 1 a 1 ) γ ρ 2 / c t γ ab c t ] J ˜ c ( λ ¯ s ) ,
[ i ν / c + γ ab + 3 1 ( 1 a 1 ) 1 γ 1 λ j 2 ] I ˜ ( λ ¯ ) = J ˜ ( λ ¯ ) ,
[ t + γ ab 3 1 ( 1 a 1 ) 1 γ 1 j 2 ] I ( ρ ¯ ) = J ( ρ ¯ ) .
I j ( ρ ¯ ) = 3 1 ( 1 a 1 ) 1 γ 1 j I ( ρ ¯ ) ,
I k l ( ρ ¯ ) = 3 1 δ k l I ( ρ ¯ ) ,
I k l m ( ρ ¯ ) ~ ( 1 a 3 ) 1 { γ 1 J k l m ( ρ ¯ ) + 5 1 ( a 1 a 3 ) [ I k ( ρ ¯ ) δ l m + I l ( ρ ¯ ) δ m k + I m ( ρ ¯ ) δ k l ] } ,
j I j k l ( ρ ¯ ) ~ ( 1 a 3 ) 1 [ γ 1 j J j k l ( ρ ¯ ) + 5 1 ( a 1 a 3 ) ( k I l + l I k + j I j δ k l ) ( ρ ¯ ) ] .
| J ˜ j k l | | J ˜ k l | , | γ 1 λ j J ˜ j k l | | J ˜ k l | , | λ j I ˜ k | = 3 1 ( 1 a 1 ) 1 γ 1 | λ j λ k I ˜ | ( 1 a 2 ) γ | I ˜ | ,
I s ( Ω , ρ ¯ ) = ( 4 π ) 1 [ I ( ρ ¯ ) ( 1 a 1 ) 1 γ 1 Ω j j I ( ρ ¯ ) ] + I r ( Ω , ρ ¯ ) ,
J r ( Ω , ρ ¯ ) = J c ( Ω , ρ ¯ ) ( 4 π ) 1 { J ( ρ ¯ ) + ( 1 a 1 ) 1 γ 1 × [ 3 1 j 2 ( Ω j j ) 2 ] I ( ρ ¯ ) + 3 Ω j ( γ ab + t ) I j ( ρ ¯ ) } ,
I j ( ρ ¯ ) = ( 1 a 1 ) 1 γ 1 [ 3 1 j I ( ρ ¯ ) + J j ( ρ ¯ ) ] ,
n · Ω I s ( Ω , ρ ¯ ) | ( n · Ω ) [ I s ( Ω , ρ ¯ ) | + I s ( Ω , ρ ¯ ) | ] = 0 ,
n · Ω > 0 d Ω ( n · Ω ) I s ( Ω , ρ ¯ ) | + = 0 ,
[ I ( ρ ¯ ) ( 2 3 ) ( 1 a 1 ) 1 γ 1 ( n j j ) I ( ρ ¯ ) ] | + = 0.
n · I ( ρ ¯ ) | + = ( 1 2 ) I ( ρ ¯ ) | + , I ( ρ ¯ ) = ( I 1 , I 2 , I 3 ) ( ρ ¯ ) ,
t n = d t t n I ( ρ , t ) / d t I ( ρ , t ) n = 1 , 2 , 3 , ,
I ( ρ , ν ) = d t exp ( i ν t ) I ( ρ , t ) ,
t n = ( i / ν ) n I ( ρ , ν ) | ν = 0 / I ( ρ , ν ) | ν = 0 .
I ( ρ , ν ) = exp [ κ 0 + κ 1 ( i ν ) + 1 2 ! κ 2 ( i ν ) 2 + 1 3 ! κ 3 ( i ν ) 3 + ] ,
κ n = ( i / ν ) n log I ( ρ , ν ) | ν = 0 ,
κ 1 = t , κ 2 = ( t t ) 2 , κ 3 = ( t t ) 3 ,
[ i ν / c + γ ab 3 1 ( 1 a 1 ) 1 γ 1 j 2 ] I ( ρ , ν ) = J ( ρ , ν ) ,
I ( Ω , λ ¯ ) = d ρ ¯ I ( Ω , ρ ¯ ) exp [ i ( λ · ρ ν t ) ] , d ρ ¯ = d ρ d t
[ i ( ν / c Ω · λ ) + γ t ] I ˜ ( Ω , λ ¯ ) = d Ω σ ( Ω · Ω ) I ˜ ( Ω , λ ¯ ) + J ˜ c ( Ω , λ ¯ ) ,
U ˜ ( Ω , λ ¯ ) = [ i ( ν / c Ω · λ ) + γ t ] 1 ,
I ˜ ( Ω , λ ¯ ) = U ˜ ( Ω , λ ¯ ) d Ω σ ( Ω · Ω ) I ˜ ( Ω , λ ¯ ) + I ˜ c ( Ω , λ ¯ ) ,
I ˜ c ( Ω , λ ¯ ) = U ˜ ( Ω , λ ¯ ) J ˜ c ( Ω , λ ¯ ) ,
δ [ i ( ν / c Ω · λ ) + γ t ] f ( Ω , λ ¯ ) ,
[ i ( ν / c Ω · λ ) + γ t ] I ˜ c ( Ω , λ ¯ ) = J ˜ c ( Ω , λ ¯ ) ,
[ c 1 t + γ t + Ω · ρ ] I c ( Ω , ρ ¯ ) = J c ( Ω , ρ ¯ ) .
I ˜ ( Ω , λ ¯ ) = I ˜ c ( Ω , λ ¯ ) + I ˜ s ( Ω , λ ¯ ) ,
I ˜ s ( Ω , λ ¯ ) = U ˜ ( Ω , λ ¯ ) d Ω σ ( Ω · Ω ) × [ I ˜ s ( Ω , λ ¯ ) + I ˜ c ( Ω , λ ¯ ) ] .
σ ( Ω · Ω ) = ( 4 π ) 1 γ
I ˜ ( λ ¯ ) d Ω I ˜ s ( Ω , λ ¯ )
[ 1 A ( λ ¯ ) ] I ˜ ( λ ¯ ) = A ( λ ¯ ) I ˜ ( λ ¯ ) .
A ( λ ¯ ) = ( 4 π ) 1 d Ω γ U ˜ ( Ω , λ ¯ ) , I ˜ ( λ ¯ ) = d Ω I ˜ c ( Ω , λ ¯ ) ,
A ( λ ¯ ) = i γ 2 λ log ( i ( ν / c λ ) + γ t i ( ν / c + λ ) + γ t ) , λ = | λ |
I ˜ ( λ ¯ ) = [ 1 A ( λ ¯ ) ] 1 A ( λ ¯ ) I ˜ ( λ ¯ ) + δ [ 1 A ( λ ¯ ) ] g ( λ ¯ ) ,
I ( ρ ¯ ) = ( 2 π ) 4 d λ ¯ I ˜ ( λ ¯ ) exp [ i ( ν t λ · ρ ) ] ,
ν / c = ± λ + i γ t
i ω / c = γ t + λ cot ( λ / γ ) = γ ab γ [ 1 3 ( λ γ ) 2 + 1 45 ( λ γ ) 4 + 2 945 ( λ γ ) 6 + ] , Im ( ω ) > 0
1 A ( λ ¯ ) ~ A ν | ν = ω [ ν ω ( λ ) ] , ν ~ ω ( λ ) , A ν | ν = ω = i ( c γ ) 1 w 1 ( λ ) , w ( λ ) = 1 + 1 3 ( λ γ ) 2 + O ( λ γ ) 4 ,
1 A ( λ ¯ ) ~ [ 1 γ ( i ν c + γ ab ) + 1 3 ( λ γ ) 2 + 1 45 ( λ γ ) 4 + ] w 1 ( λ ) .
[ 1 γ ( i ν c + γ ab ) + 1 3 ( λ γ ) 2 + 1 45 ( λ γ ) 4 + ] I ˜ ( λ ¯ ) ~ I ˜ ( λ ¯ ) .
[ c 1 t + γ ab ( 3 γ ) 1 ( ρ ) 2 ] I ( ρ ¯ ) = γ I ( ρ ¯ ) ,
J(ρ¯)(1a1)1γ1jJj(ρ¯).
31njjI(ρ¯)|+=njJj(ρ¯)|+,
{I(ρ¯)+2(1a1)1γ1nj[31jI(ρ¯)+Jj(ρ¯)]}|+=0,
[I(ρ¯)(2/3)(1a1)1γ1njjI(ρ¯)]|=0,