Abstract

The two-path method (utilizing a mirror) is shown to provide sufficient information to solve for the true radial brightness distribution of a luminous medium in the presence of self-absorption. In the case where self-absorption is not large, an approximate solution to the problem is shown to be no more difficult than a solution for the “optically thin” approximation.

© 1960 Optical Society of America

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References

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  1. J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Oxford University Press, New York, 1953), p. 400.
  2. William J. Pearce, Conference on Extremely High Temperatures (John Wiley & Sons, Inc., New York, 1958), p. 123.
  3. F. S. Simmons and A. G. DeBell, J. Opt. Soc. Am. 48, 717 (1958).
    [Crossref]
  4. S. S. Penner, Am. J. Phys. 17, 422, 491 (1949); S. S. Penner and E. K. Bjovnerud, J. Chem. Phys. 23, 143 (1955).
    [Crossref]
  5. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948), 2nd ed., p. 331.

1958 (1)

1949 (1)

S. S. Penner, Am. J. Phys. 17, 422, 491 (1949); S. S. Penner and E. K. Bjovnerud, J. Chem. Phys. 23, 143 (1955).
[Crossref]

Craggs, J. D.

J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Oxford University Press, New York, 1953), p. 400.

DeBell, A. G.

Meek, J. M.

J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Oxford University Press, New York, 1953), p. 400.

Pearce, William J.

William J. Pearce, Conference on Extremely High Temperatures (John Wiley & Sons, Inc., New York, 1958), p. 123.

Penner, S. S.

S. S. Penner, Am. J. Phys. 17, 422, 491 (1949); S. S. Penner and E. K. Bjovnerud, J. Chem. Phys. 23, 143 (1955).
[Crossref]

Simmons, F. S.

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948), 2nd ed., p. 331.

Am. J. Phys. (1)

S. S. Penner, Am. J. Phys. 17, 422, 491 (1949); S. S. Penner and E. K. Bjovnerud, J. Chem. Phys. 23, 143 (1955).
[Crossref]

J. Opt. Soc. Am. (1)

Other (3)

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, New York, 1948), 2nd ed., p. 331.

J. M. Meek and J. D. Craggs, Electrical Breakdown of Gases (Oxford University Press, New York, 1953), p. 400.

William J. Pearce, Conference on Extremely High Temperatures (John Wiley & Sons, Inc., New York, 1958), p. 123.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of apparatus suitable for the two-path method experiment.

Fig. 2
Fig. 2

Typical Jx(——) and log(Kx/Jx) data are shown for a plasma jet with the corresponding radial brightness distribution function Br(– – – –) calculated from Eq. (14).

Fig. 3
Fig. 3

(a) The points represent the function Br calculated by direct solution of Eq. (16), while the curves represent the analytic solution (18). (b) Densitometer traces taken across a stigmatic image (negative) of an optically thin plasma jet at approximately 2-mm intervals.

Fig. 4
Fig. 4

(——) Optical density of photographic negatives of a cross section of an optically thin blowpipe flam with nonluminous center. (a) Schlieren photograph; (b) direct photograph. The points represent the direct solution of Eq. (16), while the dashed line represents the analytic solution (18). Vertical dashed lines have been drawn through the extrema, while the horizontal dashed line represents the zero gradient exposure for the schlieren photograph.

Equations (25)

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d y · B r · exp { - y ( R 2 - x 2 ) 1 2 d η · α ρ } ;             ρ 2 = x 2 + η 2 r 2 = x 2 + y 2 ,
J x = - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · B r exp { - y ( R 2 - x 2 ) 1 2 d η · α ρ } ;             ρ 2 = x 2 + η 2 r 2 = x 2 + y 2 .
K x = J x + μ J x · exp { - - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · α r } ;             r 2 = x 2 + y 2 ,
ln 1 1 + ω x = - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · α r ;             r 2 = x 2 + y 2 ,
ω x = ( 1 / μ ) [ ( K x / J x ) - 1 ] - 1.
F ( x ) = 2 0 ( R 2 - x 2 ) 1 2 d y · G ( r ) ;             r 2 = x 2 + y 2 ,
G ( r ) = α r
F ( x ) = ln [ 1 / ( 1 + ω x ) ] .
G ( r ) = - 1 π r d d r r R F ( x ) x d x ( x 2 - r 2 ) 1 2 .
d y · B r · exp { - - ( R 2 - x 2 ) 1 2 y d η · α ρ } ;             ρ 2 = x 2 + η 2 r 2 = x 2 + y 2 ,
J ˜ x = - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · B r · exp { - - ( R 2 - x 2 ) 1 2 y d η · α ρ } .
J ˜ x = J x ,
J x = 1 2 ( J x + J ˜ x ) .
J x - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · B r · { 1 - y ( R 2 - x 2 ) 1 2 d η · α ρ } ;
J ˜ x - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · B r · { 1 - - ( R 2 - x 2 ) 1 2 y d η · α ρ } ;
ω x - - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d y · α r
ω x - - ( R 2 - x 2 ) 1 2 ( R 2 - x 2 ) 1 2 d η · α ρ .
J x 1 + 1 2 ω x = 2 ( R 2 - x 2 ) 1 2 d y · B r ,
G ( r ) = B r
F ( x ) = J x / ( 1 + 1 2 ω x ) .
J x = 2 0 ( R 2 - x 2 ) 1 2 d y · B r .
F ( x ) = i = 0 m F i ( R 2 - x 2 ) i ,
G ( r ) = i = 0 m G i ( R 2 - r 2 ) i - 1 2 ,
G i = [ Γ ( i + 1 ) / Γ ( 1 2 ) Γ ( i + 1 2 ) ] F i
G i = ( 2 2 i / π ) [ ( i ! ) 2 / ( 2 i ) ! ] F i .