Abstract

It is shown that the equations relating the radial profiles of the volume emission and absorption coefficients to the transmission and emitted intensity profiles in self-absorbing cylindrically symmetric sources, can be written in such a way that the problem of spatially resolving the volume emission coefficient gives rise to a Volterra integral equation of the second kind in a standard form. The theory of equations of this type is invoked to show the formal convergence of an iterative solution to the problem, subject only to a finite transmission and bounded slope to the absorption coefficient. A prescription for applying this iterative procedure is given that involves a series of numerical integrations and Abel inversions, and the convergence of some numerical solutions is demonstrated.

© 1968 Optical Society of America

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References

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  1. F. G. Tricomi, Integral Equations (Interscience Publishers, New York, 1957).
  2. O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
    [CrossRef]
  3. M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
    [CrossRef]
  4. K. Bockasten, J. Opt. Soc. Amer. 51, 943 (1961).
    [CrossRef]
  5. W. L. Barr, J. Opt. Soc. Amer. 52, 885 (1962).
    [CrossRef]
  6. H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).
  7. D. R. Paquette, W. L. Wiese, Appl. Opt. 3, 291 (1964).
    [CrossRef]
  8. M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 50, 826 (1960).
    [CrossRef]
  9. P. Elder, J. Jerrick, J. W. Birkeland, Appl. Opt. 4, 589 (1965).
    [CrossRef]
  10. V. W. Hochstrasser, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegren, Eds. National Bureau of Standards Applied Mathematics Series 55 (IT. S. Govt. Printing Office, Wash., D. C., 1964).

1965 (1)

1964 (1)

1963 (1)

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

1962 (2)

W. L. Barr, J. Opt. Soc. Amer. 52, 885 (1962).
[CrossRef]

H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).

1961 (1)

K. Bockasten, J. Opt. Soc. Amer. 51, 943 (1961).
[CrossRef]

1960 (2)

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 50, 826 (1960).
[CrossRef]

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Barr, W. L.

W. L. Barr, J. Opt. Soc. Amer. 52, 885 (1962).
[CrossRef]

Birkeland, J. W.

Bockasten, K.

K. Bockasten, J. Opt. Soc. Amer. 51, 943 (1961).
[CrossRef]

Edels, H.

H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).

Elder, P.

Freeman, M. P.

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 50, 826 (1960).
[CrossRef]

Hearns, K.

H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).

Hochstrasser, V. W.

V. W. Hochstrasser, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegren, Eds. National Bureau of Standards Applied Mathematics Series 55 (IT. S. Govt. Printing Office, Wash., D. C., 1964).

Jerrick, J.

Katz, S.

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 50, 826 (1960).
[CrossRef]

Nestor, O. H.

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Olsen, H. N.

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Paquette, D. R.

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Interscience Publishers, New York, 1957).

Wiese, W. L.

Young, A.

H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).

Appl. Opt. (2)

J. Math. Phys. (1)

H. Edels, K. Hearns, A. Young, J. Math. Phys. 41, 62 (1962).

J. Opt. Soc. Amer. (4)

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 50, 826 (1960).
[CrossRef]

M. P. Freeman, S. Katz, J. Opt. Soc. Amer. 53, 1172 (1963).
[CrossRef]

K. Bockasten, J. Opt. Soc. Amer. 51, 943 (1961).
[CrossRef]

W. L. Barr, J. Opt. Soc. Amer. 52, 885 (1962).
[CrossRef]

SIAM Rev. (1)

O. H. Nestor, H. N. Olsen, SIAM Rev. 2, 200 (1960).
[CrossRef]

Other (2)

F. G. Tricomi, Integral Equations (Interscience Publishers, New York, 1957).

V. W. Hochstrasser, Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegren, Eds. National Bureau of Standards Applied Mathematics Series 55 (IT. S. Govt. Printing Office, Wash., D. C., 1964).

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of the geometry. The emitted intensity and transmission profiles are obtained from observations along chords ( x , - y ) ( x , y ) ¯ as shown.

Fig. 2
Fig. 2

Emitted intensity (arbitrary units) and transmission profiles. The circles are observed profiles and the solid curves are obtained from the calculated emission and absorption coefficients.

Fig. 3
Fig. 3

Error in reproducing the observed data [see Eq. (28)] as a function of N for the profiles given in Fig. 2.

Fig. 4
Fig. 4

Radial profiles of the volume emission and absorption coefficients (arbitrary units) calculated from the data shown in Fig. 2.

Fig. 5
Fig. 5

Emitted intensity and transmission profiles obtained from the observed profiles given in Fig. 2 after increasing the self-absorption as described in the text. The circles are the data, I0 and I1 are the profiles obtained from the calculated coefficients after 0 and 1 iteration, respectively. In both cases, the error E in the intensity profiles was less than 6 × 10−4 after 1 iteration.

Fig. 6
Fig. 6

Volume emission coefficient profiles calculated from the data given in Fig. 5; j0 and j1 are the results from 0 and 1 iteration, respectively; j* is the result obtained if the self-absorption is completely ignored

Equations (40)

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- ln T ( x ) = 2 r = x r = R k ( r ) r ( r 2 - x 2 ) 1 2 d r
I ( x ) = r = x r = R j ( r ) [ 1 + exp - 2 r = x r = r k ( r ) r ( r 2 - x 2 ) 1 2 d r ] × exp [ - r = r r = R k ( r ) r ( r 2 - x 2 ) 1 2 d r ] r ( r 2 - x 2 ) 1 2 d r .
T - 1 2 ( x ) = exp r = x r = R k ( r ) r ( r 2 - x 2 ) 1 2 d r
I ( x ) / T ( x ) 1 2 = 2 r = x r = R j ( r ) cosh [ r = r r = r k ( r ) r ( r 2 - x 2 ) 1 2 d r ] × r ( r 2 - x 2 ) 1 2 d r .
- ln T ( v ) = R u = 0 u = r k ( u ) ( v - u ) 1 2 d u
I ( v ) / T 1 2 ( v ) = R u = 0 u = r j ( u ) cosh [ R 2 u = 0 u = u k ( u ) ( v - u ) 1 2 d u ] × 1 ( v - u ) 1 2 d u .
k ( u ) = 1 π R d d u v = 0 v = u - ln T ( v ) ( u - v ) 1 2 d v .
1 π d d u v = 0 v = u d v 1 ( u - v ) 1 2 ,
1 π d d u v = 0 v = u I ( v ) T 1 2 ( v ) ( u - v ) 1 2 d v = R π d d u v = 0 v = u u = 0 u = r j ( u ) H ( u v ) ( u - v ) 1 2 ( v = u ) d u d v ,
H ( u , v ) = cosh R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u .
H ( v , v ) = 1 ,
v = u v = u 1 ( u - v ) 1 2 ( v - u ) 1 2 d v = π ,
1 π d d u v = 0 v = u I ( v ) T 1 2 ( v ) 1 ( u - v ) 1 2 d v = R π u = 0 u = u j ( u ) u [ v = u v = u H ( u , v ) ( u - v ) 1 2 ( v - u ) 1 2 ] d u + R j ( u ) .
g ( u ) = 1 π R d d u v = 0 v = u I ( v ) T 1 2 ( v ) 1 ( u - v ) 1 2 d v ,
K ( u , u ) = - 1 π u u = v v = u H ( u , v ) ( u - v ) 1 2 ( v - u ) 1 2 d v ,
j ( u ) - u = 0 u = u j ( u ) K ( u , u ) d u = g ( u ) .
j ( u ) = n = 0 j n ( u ) .
j 0 ( u ) = g ( u )
j n ( u ) = u = 0 u = u j n - 1 ( u ) K ( u , u ) d u .
I n ( v ) = R u = 0 u = v j n - 1 ( u ) 1 - H ( u , v ) ( v - u ) 1 2 d u ,
j n ( u ) = 1 π R d d u v = 0 v = u I n ( v ) ( u - v ) 1 2 d v .
k ( r ) = - 1 π r d d r x = r x = R - ln T ( x ) x ( x 2 - r 2 ) 1 2 d x ,
H ( r , x ) = cosh r = x r = r k ( r ) r ( r 2 - x 2 ) 1 2 d r ,
j ( r ) = n = 0 j n ( r ) ,
j n ( r ) = - 1 π r d d r x = r x = R I n ( x ) x ( x 2 - r 2 ) 1 2 d x ,
I n ( x ) = 2 r = x r = R j n - 1 ( r ) 1 - H ( r , x ) ( r 2 - x 2 ) 1 2 d r ,
I 0 ( x ) = I ( x ) / T 1 2 ( x ) .
1 π d d u 0 u v i + 1 2 ( u - v ) 1 2 d v = ( i + 1 ) ( 1 ) ( 3 ) ( 2 i - 1 ) ( 2 i + 1 ) ( 2 ) ( 4 ) ( 2 i ) ( 2 i + 2 ) u i ,
F ( x ) = v 5 2 P N ( v ) | v = 1 - x 2 / R 2 ,
g n ( x ) = ( 2 R h n ) 1 2 v 5 2 G n ( ¹¹ / , 6 , v ) | v = 1 - x 2 / R 2 ,
f ( r ) = x = 0 x = R A N ( x , r ) F ( x ) d x ,
A N ( x , r ) = [ n = 0 N g n ( v ) ( 1 π R d d u v = 0 v = u g n ( v ) ( u - v ) 1 2 d v ) ] v = 1 - x 2 / R 2 u = 1 - r 2 / R 2
E = 0 R [ ( F ( x ) - F c ( x ) ] 2 d x / 0 R F 2 ( x ) d x ,
K ( u , u ) = - 1 π u v = u v = u H ( u , v ) ( u - v ) 1 2 ( v - u ) 1 2 d v = - 1 π ( u - u ) v = u v = u ( v - u ) 1 2 ( u - v ) 1 2 v H ( u , v ) d v
( / v ) H ( u , v ) < M ,
K ( u , u ) M π ( u - u ) v = u v = u ( v - u ) 1 2 ( u - v ) 1 2 d v < M π ( u - u ) 1 2 v = u v = u 1 ( u - v ) 1 2 d v = 2 M π .
v cosh R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u = [ sinh R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u ] v R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u = [ sinh R 2 u = u u = r k ( u ) ( v - u ) 1 2 d u ] × { R 2 ( v - u ) u = u u = v [ ( u - u ) k ( u ) u - 1 2 k ( u ) ] 1 ( v - u ) 1 2 d u } .
[ ( u - u ) [ k ( u ) / u ] - 1 2 k ( u ) ] < B .
| v H ( u , v ) | B R ( v - u ) 1 2 sinh R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u .
lim v u 1 ( v - u ) 1 2 sinh R 2 u = u u = v k ( u ) ( v - u ) 1 2 d u = R k ( u ) ,

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