Abstract

A method for determining the spatial distribution of the emission coefficient in the limit of small self-absorption is based on solving the transfer equation for the flux which is observed at the detector when a mirror is present. From this solution and that corresponding to an interchange of the detector and mirror, integral equations for the absorption and emission coefficients are obtained. In the limit of small self-absorption, when the optical density over the entire path is much less than one, the integral equation for emission coefficient is approximated by an integral equation which is mathematically identical to that for the absorption coefficient. The method which was recently presented by Maldonado et al. for inverting such integral equations is then used to obtain series representations of the absorption and emission coefficients. The numerical procedure of Maldonado et al. for summing these series representations is applied to two hypothetical examples. The numerical results obtained for the absorption and emission coefficients are within 1% of exact theoretical results.

© 1967 Optical Society of America

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References

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  1. C. D. Maldonado, A. P. Caron, and H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965).
    [Crossref]
  2. C. D. Maldonado and H. N. Olsen, J. Opt. Soc. Am. 56, 1305 (1966).
    [Crossref]
  3. M. P. Freeman and S. Katz, J. Opt. Soc. Am. 50, 826 (1960).
    [Crossref]
  4. R. W. Porter, S. I. A. M. Rev. 6, 228 (1964).
  5. S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).
  6. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.
  7. C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
    [Crossref]
  8. W. Gröbner and N. Hofreiter, Integraltafel erster Teil Unbestimmte Integrals (Springer-Verlag, Vienna, 1961).

1966 (1)

1965 (2)

1964 (1)

R. W. Porter, S. I. A. M. Rev. 6, 228 (1964).

1960 (1)

Caron, A. P.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.

Freeman, M. P.

Gröbner, W.

W. Gröbner and N. Hofreiter, Integraltafel erster Teil Unbestimmte Integrals (Springer-Verlag, Vienna, 1961).

Hofreiter, N.

W. Gröbner and N. Hofreiter, Integraltafel erster Teil Unbestimmte Integrals (Springer-Verlag, Vienna, 1961).

Katz, S.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.

Maldonado, C. D.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.

Olsen, H. N.

Porter, R. W.

R. W. Porter, S. I. A. M. Rev. 6, 228 (1964).

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.

J. Math. Phys. (1)

C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
[Crossref]

J. Opt. Soc. Am. (3)

S. I. A. M. Rev. (1)

R. W. Porter, S. I. A. M. Rev. 6, 228 (1964).

Other (3)

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Company, New York, 1963), Vols. I and II.

W. Gröbner and N. Hofreiter, Integraltafel erster Teil Unbestimmte Integrals (Springer-Verlag, Vienna, 1961).

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Figures (10)

Fig. 1
Fig. 1

Diagram of source coordinate system (x′,y′) rotated by an angle ξ relative to the fixed laboratory coordinate system (x,y) together with two arrangements of a plane reflecting mirror and detector for obtaining the integrated flux distributions of asymmetrical light sources.

Fig. 2
Fig. 2

Qualitative sketch of a typical step approximation to the surface of h(ξ,x) at the point (ξ=ξjξj, x=xixi) together with the symbols and nomenclature that are pertinent to the numerical procedure for evaluating the double integrals of Eqs. (20) and (21).

Fig. 3
Fig. 3

Theoretical (solid curves) and numerical results for the spatial distribution of the absorption coefficient, κ(x′,y′), along the three directions x′=0, x′=y′, and y′=0, for the first hypothetical example.

Fig. 4
Fig. 4

Theoretical (solid curve) and numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the y′=0 direction for the first hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin assumption).

Fig. 5
Fig. 5

Theoretical (solid curve) and numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the x′=y′ direction for the first hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin assumption).

Fig. 6
Fig. 6

Theoretical (solid curve) and numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the x′=0 direction for the first hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin absorption).

Fig. 7
Fig. 7

Theoretical (solid curves) and numerical results for the spatial distribution of the absorption coefficient, κ(x′,y′), along the three directions x′=0, x′=y′, and y′=0, for the second hypothetical example.

Fig. 8
Fig. 8

Numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the y′=0 direction for the second hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin assumption).

Fig. 9
Fig. 9

Numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the x′=y′ direction for the second hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin assumption).

Fig. 10
Fig. 10

Numerical results for the spatial distribution of the emission coefficient, f(x′,y′), along the x′=0 direction for the second hypothetical example with absorption present. Also plotted is the theoretical curve for the emission coefficient, f0(x′,y′), when no absorption is present (optically thin assumption).

Equations (55)

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( d / d y ) g ( x , y , ξ ) = f ( x , y ) - κ ( x , y ) g ( x , y , ξ ) ,
x = x cos ξ + y sin ξ y = - x sin ξ + y cos ξ ,
g d + ( ξ , x ) = g m - ( ξ , x ) exp [ - - + d y κ ( x , y ) ] + - + d y f ( x , y ) exp [ - y + d y κ ( x , y ) ] .
g m - ( ξ , x ) = R - + d y f ( x , y ) exp [ - - y d y κ ( x , y ) ] ,
g d - ( ξ , x ) = g m + ( ξ , x ) exp [ - - + d y κ ( x , y ) ] + - + d y f ( x , y ) exp [ - - y d y κ ( x , y ) ] ,
g m + ( ξ , x ) = R - + d y f ( x , y ) exp [ - y + d y κ ( x , y ) ]
g d + ( ξ , x ) + g d - ( ξ , x ) = [ g m + ( ξ , x ) + g m - ( ξ , x ) ] exp [ - - + d y κ ( x , y ) ] + - + d y f ( x , y ) { exp [ - y + d y κ ( x , y ) ] + exp [ - - y d y κ ( x , y ) ] } ,
g ˜ d + ( ξ , x ) + g ˜ d - ξ , ( x ) = - + d y f ( x , y ) { exp [ - y + d y κ ( x , y ) ] + exp [ - - y d y κ ( x , y ) ] }
h ( ξ , x ) = - + d y κ ( x , y ) ,
h ( ξ , x ) ln { R [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] / ( [ g d + ( ξ , x ) + g d - ( ξ , x ) ] - [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] ) } ,
g ˜ d + ( ξ , x ) = - + d y f ( x , y ) .
h ( ξ , x ) = - + d y κ ( x , y ) 1 ,
h ( ξ , x ) ( 1 + R ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] - [ g d + ( ξ , x ) + g d - ( ξ , x ) ] [ g d + ( ξ , x ) + g d - ( ξ , x ) ] - [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] 1.
0 - y d y κ ( x , y ) - + d y κ ( x , y ) 1
0 y + d y κ ( x , y ) - + d y κ ( x , y ) 1 ,
exp [ - y + d y κ ( x , y ) ] + exp [ - - y d y κ ( x , y ) ] 2 - - + d y κ ( x , y ) .
[ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] / [ 2 - h ( ξ , x ) ] - + d y f ( x , y ) ,
( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] - + d y f ( x , y )
κ ( x , y ) = ( α / π 3 2 ) k = 0 m = 0 m [ ( k ! ( m + k ) ! ) 1 2 / ( m + 2 k ) ! ] × Re { [ - π π d ξ exp ( - i m ξ ) - + d x h ( ξ , x ) H m + 2 k ( α x ) ] U m + 2 k m ( α x , α y ) } exp [ - α 2 ( x 2 + y 2 ) ]
f ( x , y ) = ( α / π 3 2 ) k = 0 m = 0 m [ ( k ! ( m + k ) ! ) 1 2 / ( m + 2 k ) ! ] × Re { [ - π π d ξ exp ( - i m ξ ) - + d x { ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] × [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } H m + 2 k ( α x ) ] U m + 2 k m ( α x , α y ) } exp [ - α 2 ( x 2 + y 2 ) ] .
U m + 2 k m ( α x , α y ) = ( - 1 ) k ( α / π 1 2 ) [ k ! / ( m + k ) ! ] 1 2 [ α 2 ( x 2 + y 2 ) ] m / 2 exp ( i m φ ) L k m [ α 2 ( x 2 + y 2 ) ] ,
κ ( x , y ) ( α / π ) 2 k = 0 K m = 0 M m ( - 1 ) k [ k ! / ( m + 2 k ) ! ] [ A m + 2 k m ( α ) cos ( m φ ) + B m + 2 k m ( α ) sin ( m φ ) ] [ α 2 ( x 2 + y 2 ) ] m / 2 L k m [ α 2 ( x 2 + y 2 ) ] exp [ - α 2 ( x 2 + y 2 ) ]
f ( x , y ) ( α / π ) 2 k = 0 K m = 0 M m ( - 1 ) k [ k ! / ( m + 2 k ) ! ] [ C m + 2 k m ( α ) cos ( m φ ) + D m + 2 k m ( α ) sin ( m φ ) ] [ α 2 ( x 2 + y 2 ) ] m / 2 L k m [ α 2 ( x 2 + y 2 ) ] exp [ - α 2 ( x 2 + y 2 ) ] ,
{ A m + 2 k m ( α ) C m + 2 k m ( α ) } = - π π d ξ cos ( m ξ ) - + d x { h ( ξ , x ) ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } H m + 2 k ( α x )
{ B m + 2 k m ( α ) D m + 2 k m ( α ) } = - π π d ξ sin ( m ξ ) - + d x { h ( ξ , x ) ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } H m + 2 k ( α x )
{ A m + 2 k m ( α ) C m + 2 k m ( α ) } [ 1 / 2 α m ( m + 2 k + 1 ) ] i = 0 I - 1 j = 0 J - 1 { h ( ξ , x ) ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } ξ = ξ j + Δ ξ j , x = x i + Δ x i × [ sin ( m ξ j + 1 ) - sin ( m ξ j ) ] [ H m + 2 k + 1 ( α x i + 1 ) - H m + 2 k + 1 ( α x i ) ]
{ B m + 2 k m ( α ) D m + 2 k m ( α ) } [ 1 / 2 α m ( m + 2 k + 1 ) ] i = 0 I - 1 j = 0 J - 1 { h ( ξ , x ) ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } ξ = ξ j + Δ ξ j , x = x i + Δ x i × [ cos ( m ξ j + 1 ) - cos ( m ξ j ) ] [ H m + 2 k + 1 ( α x i + 1 ) - H m , 2 k + 1 ( α x i ) ]
( ξ = ξ j + Δ ξ j , x = x i + Δ x i ) .
( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] = A ( π 1 2 / β ) exp [ - β 2 ( x - d cos ξ ) 2 ]
h ( ξ , x ) = B ( π 1 2 / λ ) exp [ - λ 2 ( x - d cos ξ ) 2 ] ,
{ ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] × [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } ξ = ξ j + Δ ξ j ,
( 1 / 2 ) [ g d + ( ξ , x ) + g d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] = A ( π 1 2 / β ) exp [ - β 2 ( x - d cos ξ ) 2 ] + A B ( π / 2 β λ ) exp [ - ( β 2 + λ 2 ) ( x - d cos ξ ) 2 ] ,
κ ( x , y ) = B exp { - λ 2 [ ( x - d ) 2 + y 2 ] }
f ( x , y ) = f 0 ( x , y ) × { 1 + ( π 1 2 / 2 λ ) [ 1 + ( λ / β ) 2 ] 1 2 κ ( x , y ) } ,
f 0 ( x , y ) = A exp { - β 2 [ ( x - d ) 2 + y 2 ] }
( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] = A [ π / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] 1 2 × exp [ - β 2 λ 2 x 2 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ]
h ( ξ , x ) = B [ π / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] 1 2 × exp [ - β 2 λ 2 x 2 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] ,
{ ( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] × [ 1 + ( 1 / 2 ) h ( ξ , x ) ] } ξ = ξ j + Δ ξ j ,
( 1 / 2 ) [ g ˜ d + ( ξ , x ) + g ˜ d - ( ξ , x ) ] [ 1 + ( 1 / 2 ) h ( ξ , x ) ] = A [ π / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] 1 2 × exp [ - β 2 λ 2 x 2 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] + A B ( π / 2 ) [ 1 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) × ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] 1 2 × exp [ - x 2 { [ β 2 λ 2 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] + [ β 2 λ 2 / ( λ 2 sin 2 ξ + β 2 cos 2 ξ ) ] } ] ,
κ ( x , y ) = B exp [ - ( λ 2 x 2 + β 2 y 2 ) ] .
f 0 ( x , y ) = A exp [ - ( λ 2 x 2 + β 2 y 2 ) ] .
κ ( x , y ) = B ( α 2 / π 3 2 λ ) k = 0 m = 0 m ( - 1 ) k [ k ! / ( m + 2 k ) ! ] × [ - π π d ξ { cos ( m ξ ) / [ sin 2 ξ + ( β / λ ) cos 2 ξ ] 1 2 } × - + d x exp { - β 2 x 2 / [ sin 2 ξ + ( β / λ ) cos 2 ξ ] } × H m + 2 k ( α x ) ] cos ( m φ ) [ α 2 ( x 2 + y 2 ) ] m / 2 × L k m [ α 2 ( x 2 + y 2 ) ] exp [ - α 2 ( x 2 + y 2 ) ] ,
J m + 2 k m ( α , β , λ ) = - π π d ξ { cos ( m ξ ) / [ sin 2 ξ + ( β / λ ) 2 cos 2 ξ ] 1 2 } × - + d x exp { - β 2 x 2 / [ sin 2 ξ + ( β / λ ) 2 × cos 2 ξ ] } H m + 2 k ( α x ) ,
- + d x exp { - β 2 x 2 / [ sin 2 ξ + ( β / λ ) 2 cos 2 ξ ] } H 2 ( p + k ) ( β x ) = ( π 1 2 / β ) { [ 2 ( p + k ) ] ! / ( p + k ) ! } [ ( β / λ ) 2 - 1 ] p + k × [ sin 2 ξ + ( β / λ ) 2 cos 2 ξ ] 1 2 cos 2 ( p + k ) ( ξ ) ,
J 2 ( p + k ) 2 p ( α = β , β , λ ) = ( π 1 2 / β ) { [ 2 ( p + k ) ] ! / ( p + k ) ! } [ ( β / λ ) 2 - 1 ] p + k × - π π d ξ cos ( 2 p ξ ) cos 2 ( p + k ) ( ξ )
J 2 ( p + k 2 p ( α = β , β , λ ) = ( 2 π 3 2 / β ) { [ 2 ( p + k ) ] ! [ 2 ( p + k ) ] ! / × k ! ( p + k ) ! ( k + 2 p ) ! 2 2 ( p + k ) } [ ( β / λ ) 2 - 1 ] p + k
- π π d ξ cos ( 2 p ξ ) cos 2 ( p + k ) ( ξ ) = 2 π { [ 2 ( p + k ) ] ! / k ! ( k + 2 p ) ! 2 2 ( p + k ) } .
κ ( x , y ) = 2 B ( β / λ ) × p = 0 p ( β / 2 λ ) 2 p [ ( λ 2 - β 2 ) ( x 2 + y 2 ) ] p cos ( 2 p φ ) × k = 0 [ ( 2 p + 2 k ) ! / ( 2 p + k ) ! ( p + k ) ! 2 2 k ] [ 1 - ( β / λ ) 2 ] k × L k 2 p [ β 2 ( x 2 + y 2 ) ] exp [ - β 2 ( x 2 + y 2 ) ]
M [ a , c ; z t / ( t - 1 ) ] = ( 1 - t ) a [ Γ ( c ) / Γ ( a ) ] × k = 0 [ Γ ( a + k ) / Γ ( c + k ) ] t k L k c - 1 ( z )
k = 0 [ ( 2 p + 2 k ) ! / ( 2 p + k ) ! 2 2 k ] [ 1 - ( β / λ ) 2 ] k × L k 2 p [ β 2 ( x 2 + y 2 ) ] = [ ( λ / β ) 2 p + 1 / p ! ] × M [ p + 1 2 , 2 p + 1 ; ( β 2 - λ 2 ) ( x 2 + y 2 ) ] ,
M [ p + 1 2 , 2 p + 1 ; ( β 2 - λ 2 ) ( x 2 + y 2 ) ] = [ p ! 2 2 p / ( β 2 - λ 2 ) ( x 2 + y 2 ) p ] × I p [ ( β 2 - λ 2 ) ( x 2 + y 2 ) / 2 ] × exp [ ( β 2 - λ 2 ) ( x 2 + y 2 ) / 2 ] .
κ ( x , y ) = 2 B p = 0 p cos ( 2 p φ ) I p [ ( β 2 - λ 2 ) ( x 2 + y 2 ) / 2 ] × exp [ - ( β 2 + λ 2 ) ( x 2 + y 2 ) / 2 ] ,
κ ( x , y ) = B exp { - ( β 2 [ 1 - cos ( 2 φ ) ] + λ 2 [ 1 + cos ( 2 φ ) ] ) ( x 2 + y 2 ) / 2 } ,
2 p = 0 p cos ( p ψ ) I p ( z ) = exp [ z cos ψ ]
κ ( x , y ) = B exp [ - ( λ 2 x 2 + β 2 y 2 ) ] .