Abstract

Inversion of measurements of optical pathlength through strongly refracting, radially symmetric phase objects, such as plasmas, is discussed. An exact inversion scheme, based upon methods originally applied in seismology is developed and applied to interferometry. It is shown that Abel inversion, which assumes that the probing rays are straight lines, yields rather accurate results if the interferogram is formed with appropriate imaging.

© 1975 Optical Society of America

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References

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  1. R. A. Jeffries, Phys. Fluids 13, 210 (1970).
    [CrossRef]
  2. J. L. Seftor, J. Appl. Phys. 14, 4965 (1973).
    [CrossRef]
  3. F. Keilmann, Plasma Phys. 14, 111 (1972).
    [CrossRef]
  4. W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, Eds. (Academic Press, New York1970), Vol. 6.
    [CrossRef]
  5. W. L. Howes, D. R. Buchelle, J. Opt. Soc. Am. 56, 1517 (1966).
    [CrossRef]
  6. K. Bockasten, J. Opt. Soc. Am. 51, 943 (1961).
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  7. W. L. Barr, J. Opt. Soc. Am. 52, 885 (1962).
    [CrossRef]
  8. S. I. Herlitz, Ark. Fys. 23, 571 (1963).
  9. R. Baraket, J. Maths. Phys. 43, 325 (1964).
  10. D. W. Sweeney, (Abstract) J. Opt. Soc. Am. 64, 559 (1974).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 122–123.
  12. K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).
  13. P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
    [CrossRef]
  14. R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
    [CrossRef]

1974

D. W. Sweeney, (Abstract) J. Opt. Soc. Am. 64, 559 (1974).

1973

P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
[CrossRef]

J. L. Seftor, J. Appl. Phys. 14, 4965 (1973).
[CrossRef]

1972

F. Keilmann, Plasma Phys. 14, 111 (1972).
[CrossRef]

1970

R. A. Jeffries, Phys. Fluids 13, 210 (1970).
[CrossRef]

1968

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

1966

1964

R. Baraket, J. Maths. Phys. 43, 325 (1964).

1963

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

1962

1961

Anderson, D. L.

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

Baraket, R.

R. Baraket, J. Maths. Phys. 43, 325 (1964).

Barr, W. L.

Bockasten, K.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 122–123.

Buchelle, D. R.

Bullen, K. E.

K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).

Grigull, U.

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, Eds. (Academic Press, New York1970), Vol. 6.
[CrossRef]

Hauf, W.

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, Eds. (Academic Press, New York1970), Vol. 6.
[CrossRef]

Herlitz, S. I.

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

Howes, W. L.

Hunter, A. M.

P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
[CrossRef]

Jeffries, R. A.

R. A. Jeffries, Phys. Fluids 13, 210 (1970).
[CrossRef]

Keilmann, F.

F. Keilmann, Plasma Phys. 14, 111 (1972).
[CrossRef]

Phinney, R. A.

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

Schreiber, P. W.

P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
[CrossRef]

Seftor, J. L.

J. L. Seftor, J. Appl. Phys. 14, 4965 (1973).
[CrossRef]

Smith, D. R.

P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
[CrossRef]

Sweeney, D. W.

D. W. Sweeney, (Abstract) J. Opt. Soc. Am. 64, 559 (1974).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 122–123.

Ark. Fys.

S. I. Herlitz, Ark. Fys. 23, 571 (1963).

J. Appl. Phys.

J. L. Seftor, J. Appl. Phys. 14, 4965 (1973).
[CrossRef]

J. Geophys. Res. Space Phys.

R. A. Phinney, D. L. Anderson, J. Geophys. Res. Space Phys. 73, 1819 (1968).
[CrossRef]

J. Maths. Phys.

R. Baraket, J. Maths. Phys. 43, 325 (1964).

J. Opt. Soc. Am.

Phys. Fluids

R. A. Jeffries, Phys. Fluids 13, 210 (1970).
[CrossRef]

Plasma Phys.

P. W. Schreiber, A. M. Hunter, D. R. Smith, Plasma Phys. 15, 635 (1973).
[CrossRef]

F. Keilmann, Plasma Phys. 14, 111 (1972).
[CrossRef]

Other

W. Hauf, U. Grigull, in Advances in Heat Transfer, J. P. Hartnett, T. F. Irvine, Eds. (Academic Press, New York1970), Vol. 6.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), pp. 122–123.

K. E. Bullen, An Introduction to the Theory of Seismology (Cambridge U. P., New York, 1963).

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Figures (6)

Fig. 1
Fig. 1

Optical rays traversing a radially symmetric refractive index field. (a) A straight ray (no refraction). (b) A curved ray (strong refraction).

Fig. 2
Fig. 2

Two optical rays that interfere in the image of the center plane of the phase object.

Fig. 3
Fig. 3

Ray segments contributing to the apparent optical pathlength change measured interferometrically. Φ m = B A n ( s ) d s + n 0 B C n 0 ( D E + E F ).

Fig. 4
Fig. 4

Results of computer simulations for two typical refractive index fields. (a) This field has a total decrease in refractive index of 5.2%; the direct inversion of Φ(p) does not deviate measurably from the solid curve. (b) This field has a total decrease in refractive index of 68.3%.

Fig. 5
Fig. 5

Comparison of the apparent measured pathlength changes through phase objects with those along straight lines at the same positions x. (a) This corresponds to the field in Fig. 4(a). (b) This corresponds to the field in Fig. 4(b).

Fig. 6
Fig. 6

Paths of optical rays through refractive index fields. (a) This corresponds to the field in Fig. 4(a). (b) This corresponds to the field in Fig. 4(b).

Tables (1)

Tables Icon

Table I Reconstruction of Refractive Index Fields from Simulated Interferometric Data

Equations (24)

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n = ( 1 N e / N c ) 1 / 2 ,
N c = 1.113 × 10 13 λ 0 2
Φ ( x ) = 2 x r 0 n ( r ) rdr ( r 2 x 2 ) 1 / 2 .
n ( r ) = 1 π r r 0 ( d Φ / d x ) d x ( x 2 r 2 ) 1 / 2 .
r n ( r ) sin i = p .
d s 2 = d r 2 + ( r d θ ) 2 = d r 2 + ( d r tan i ) 2 ,
η r n ( r ) ,
d s = d r [ η 2 / ( η 2 p 2 ) ] 1 / 2 .
Φ ( p ) = 2 r p r 0 η 2 r d r ( η 2 p 2 ) 1 / 2 ,
Φ ( p ) = 2 p η 0 η 2 r ( d r / d η ) d η ( η 2 p 2 ) 1 / 2 ,
θ ( p ) = 2 p η 0 p r ( d r / d η ) d η ( η 2 p 2 ) 1 / 2 .
Φ ( p ) = 2 p η 0 ( η d ln r / d η ) η d η ( η 2 p 2 ) 1 / 2 ,
η d ln r d η = 1 π η η 0 ( d Φ / d p ) d p ( p 2 η 2 ) 1 / 2 .
ln ( r 0 r ) = 1 π r η 0 d η η η η 0 ( d Φ / d p ) d p ( p 2 η 2 ) 1 / 2 .
ln ( r 0 r ) = 1 π η η 0 d Φ d p d p η r d η η ( p 2 η 2 ) 1 / 2 .
r r 0 = exp [ 1 π η η 0 cosh 1 ( p η ) 1 p d Φ d p d p ] .
r r 0 = exp [ 1 π η η 0 cosh 1 ( p η ) d θ d p d p ] .
Φ m ( x ) = A B n ( s ) d s + n 0 ( B C ¯ D E ¯ E F ¯ ) ,
Φ m ( x ) = Φ ( p ) n 0 { ( r 0 2 p 2 / n 0 2 ) 1 / 2 x cos [ θ ( p ) + i 0 ] / ( p / n 0 r 0 ) } ,
Φ ( p ) = 2 r p r 0 n ( r ) rdr [ r 2 ( p / n ( r ) ) 2 ] 1 / 2 ,
Φ s ( x ) = 2 x r 0 n ( r ) rdr ( r 2 x 2 ) 1 / 2 .
I ( p ) = f ( r ) d s
I ( p ) = 2 p η 0 [ f ( r ) n ( r ) d r d η ] η d η ( η 2 p 2 ) 1 / 2 .
f ( r ) n ( r ) ( d r / d η ) = 2 p η 0 ( d I / d p ) d p ( η 2 p 2 ) .

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