Abstract

An iterative algorithm for tomographic reconstruction of refractive-index fields from measured values of path integrals along rays which have been bent by refraction is presented. The behavior of the algorithm is studied by applying it to path length data obtained by computer simulation of experiments in which holographic or Mach-Zehnder interferograms of the field are recorded for several different viewing directions. A special form of the algorithm is also used to measure concentration profiles in the boundary layer formed at the cathode of an electrolytic cell containing ZnCl2. The Appendix contains a discussion of series expansion techniques for reconstructing object fields from measured values of line integrals through the field.

© 1981 Optical Society of America

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  1. D. W. Sweeney, C. M. Vest, Appl. Opt. 12, 2649 (1973).
    [CrossRef] [PubMed]
  2. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 6.
  3. B. P. Hildebrand, D. E. Huffered, in Acoustical Holography, Vol. 7, L. W. Kessler, Ed. (Plenum, New York, 1976), pp. 245–262.
  4. A. C. Kak, Proc. IEEE 67, 1245 (1979).
    [CrossRef]
  5. J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.
  6. J. A. Greenleaf, S. A. Johnson, Ultrasound Med. Biol. 3, 327 (1978).
    [CrossRef] [PubMed]
  7. G. H. Glover, J. C. Sharp, IEEE Trans. Sonics Ultrason. SU-24, 229 (1977).
    [CrossRef]
  8. H. Schomberg, J. Phys. D: 11, L181 (1978).
    [CrossRef]
  9. K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
    [CrossRef]
  10. R. Gordon, G. T. Herman, Int. Rev. Cytol. 38, 111 (1974).
    [CrossRef] [PubMed]
  11. S. Cha, C. M. Vest, Opt. Lett. 4, 311 (1979).
    [CrossRef] [PubMed]
  12. G. C. McKinnon, R. H. T. Bates, Ultrason. Imaging 2, 48 (1980).
    [PubMed]
  13. H. Krauss, “Ein Auswerteverfahren fur allgemeine dreidimensionale Dichtefelder mit Hilfe der Interferenzenmethode,” Doctoral Dissertation, U. Stuttgart (1977).
  14. C. M. Vest, Appl. Opt. 14, 1601 (1975).
    [CrossRef] [PubMed]
  15. S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).
  16. IBM Application Program: System 1360 Scientific Subroutine Package (360 A-CM-03X), Version III, Programmers Manual (IBM, White Plains, N.Y., 1970).
  17. G. Golub, Numer. Math. 7, 206 (1965).
    [CrossRef]
  18. J. L. Synge, Hamilton's Method in Geometrical Optics (Institute for Fluid Dynamics and Applied Mathematics, U. Maryland, 1951, Lecture Series 9.
  19. B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  20. A. C. Hindmarsh, Gear: Ordinary Differential Equation System Solver, Report UCID-30001, Rev. 3, Lawrence Livermore Laboratory (1974).
  21. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).
  22. W. L. Howes, D. L. Buchele, J. Opt. Soc. Am. 56, 1517 (1966).
    [CrossRef]
  23. D. Ludwig, Commun. Pure Appl. Math. 19, 49 (1966).
    [CrossRef]
  24. M. Ein-Gal, Tech. Report 6851-1, Information Systems Laboratory, Stanford U. (1974).
  25. A. M. Cormack, J. Appl. Phys. 35, 2908 (1964).
    [CrossRef]
  26. C. D. Maldonado, H. N. Olsen, J. Opt. Soc. Am. 56, 1305 (1966).
    [CrossRef]

1980 (1)

G. C. McKinnon, R. H. T. Bates, Ultrason. Imaging 2, 48 (1980).
[PubMed]

1979 (2)

1978 (2)

J. A. Greenleaf, S. A. Johnson, Ultrasound Med. Biol. 3, 327 (1978).
[CrossRef] [PubMed]

H. Schomberg, J. Phys. D: 11, L181 (1978).
[CrossRef]

1977 (2)

K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
[CrossRef]

G. H. Glover, J. C. Sharp, IEEE Trans. Sonics Ultrason. SU-24, 229 (1977).
[CrossRef]

1975 (1)

1974 (1)

R. Gordon, G. T. Herman, Int. Rev. Cytol. 38, 111 (1974).
[CrossRef] [PubMed]

1973 (1)

1966 (3)

1965 (1)

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

1964 (1)

A. M. Cormack, J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

Bates, R. H. T.

G. C. McKinnon, R. H. T. Bates, Ultrason. Imaging 2, 48 (1980).
[PubMed]

Buchele, D. L.

Carnahan, B.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Cha, S.

S. Cha, C. M. Vest, Opt. Lett. 4, 311 (1979).
[CrossRef] [PubMed]

S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).

Cormack, A. M.

A. M. Cormack, J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

Duck, F. A.

J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.

Ein-Gal, M.

M. Ein-Gal, Tech. Report 6851-1, Information Systems Laboratory, Stanford U. (1974).

Gear, C. W.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Glover, G. H.

G. H. Glover, J. C. Sharp, IEEE Trans. Sonics Ultrason. SU-24, 229 (1977).
[CrossRef]

Golub, G.

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Gordon, R.

R. Gordon, G. T. Herman, Int. Rev. Cytol. 38, 111 (1974).
[CrossRef] [PubMed]

Greenleaf, J. A.

J. A. Greenleaf, S. A. Johnson, Ultrasound Med. Biol. 3, 327 (1978).
[CrossRef] [PubMed]

Greenleaf, J. F.

J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.

Herman, G. T.

R. Gordon, G. T. Herman, Int. Rev. Cytol. 38, 111 (1974).
[CrossRef] [PubMed]

Hildebrand, B. P.

B. P. Hildebrand, D. E. Huffered, in Acoustical Holography, Vol. 7, L. W. Kessler, Ed. (Plenum, New York, 1976), pp. 245–262.

Hindmarsh, A. C.

A. C. Hindmarsh, Gear: Ordinary Differential Equation System Solver, Report UCID-30001, Rev. 3, Lawrence Livermore Laboratory (1974).

Howes, W. L.

Huffered, D. E.

B. P. Hildebrand, D. E. Huffered, in Acoustical Holography, Vol. 7, L. W. Kessler, Ed. (Plenum, New York, 1976), pp. 245–262.

Johnson, S. A.

J. A. Greenleaf, S. A. Johnson, Ultrasound Med. Biol. 3, 327 (1978).
[CrossRef] [PubMed]

J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.

Kak, A. C.

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Krauss, H.

H. Krauss, “Ein Auswerteverfahren fur allgemeine dreidimensionale Dichtefelder mit Hilfe der Interferenzenmethode,” Doctoral Dissertation, U. Stuttgart (1977).

Ludwig, D.

D. Ludwig, Commun. Pure Appl. Math. 19, 49 (1966).
[CrossRef]

Luther, H. A.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Maldonado, C. D.

McKinnon, G. C.

G. C. McKinnon, R. H. T. Bates, Ultrason. Imaging 2, 48 (1980).
[PubMed]

Olsen, H. N.

Samayo, W. F.

J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.

Schomberg, H.

H. Schomberg, J. Phys. D: 11, L181 (1978).
[CrossRef]

Sharp, J. C.

G. H. Glover, J. C. Sharp, IEEE Trans. Sonics Ultrason. SU-24, 229 (1977).
[CrossRef]

Smith, K. T.

K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
[CrossRef]

Solomon, D. C.

K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
[CrossRef]

Sweeney, D. W.

Synge, J. L.

J. L. Synge, Hamilton's Method in Geometrical Optics (Institute for Fluid Dynamics and Applied Mathematics, U. Maryland, 1951, Lecture Series 9.

Vest, C. M.

Wagner, S. L.

K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
[CrossRef]

Wilkes, J. O.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Appl. Opt. (2)

Bull. Am. Math. Soc. (1)

K. T. Smith, D. C. Solomon, S. L. Wagner, Bull. Am. Math. Soc. 83, 1227 (1977).
[CrossRef]

Commun. Pure Appl. Math. (1)

D. Ludwig, Commun. Pure Appl. Math. 19, 49 (1966).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

G. H. Glover, J. C. Sharp, IEEE Trans. Sonics Ultrason. SU-24, 229 (1977).
[CrossRef]

Int. Rev. Cytol. (1)

R. Gordon, G. T. Herman, Int. Rev. Cytol. 38, 111 (1974).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

A. M. Cormack, J. Appl. Phys. 35, 2908 (1964).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Phys. D (1)

H. Schomberg, J. Phys. D: 11, L181 (1978).
[CrossRef]

Numer. Math. (1)

G. Golub, Numer. Math. 7, 206 (1965).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Ultrason. Imaging (1)

G. C. McKinnon, R. H. T. Bates, Ultrason. Imaging 2, 48 (1980).
[PubMed]

Ultrasound Med. Biol. (1)

J. A. Greenleaf, S. A. Johnson, Ultrasound Med. Biol. 3, 327 (1978).
[CrossRef] [PubMed]

Other (11)

M. Ein-Gal, Tech. Report 6851-1, Information Systems Laboratory, Stanford U. (1974).

H. Krauss, “Ein Auswerteverfahren fur allgemeine dreidimensionale Dichtefelder mit Hilfe der Interferenzenmethode,” Doctoral Dissertation, U. Stuttgart (1977).

J. F. Greenleaf, S. A. Johnson, W. F. Samayo, F. A. Duck, in Acoustical Holography, Vol. 6, N. Booth, Ed. (Plenum, New York, 1975), pp. 71–90.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 6.

B. P. Hildebrand, D. E. Huffered, in Acoustical Holography, Vol. 7, L. W. Kessler, Ed. (Plenum, New York, 1976), pp. 245–262.

J. L. Synge, Hamilton's Method in Geometrical Optics (Institute for Fluid Dynamics and Applied Mathematics, U. Maryland, 1951, Lecture Series 9.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

A. C. Hindmarsh, Gear: Ordinary Differential Equation System Solver, Report UCID-30001, Rev. 3, Lawrence Livermore Laboratory (1974).

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, N.J., 1971).

S. Cha, “Reconstruction of Strongly Refracting Asymmetric Fields from Interferometric Measurements,” Doctoral Dissertation, U. Michigan, Ann Arbor (1980).

IBM Application Program: System 1360 Scientific Subroutine Package (360 A-CM-03X), Version III, Programmers Manual (IBM, White Plains, N.Y., 1970).

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Figures (9)

Fig. 1
Fig. 1

Formation of an interferogram.

Fig. 2
Fig. 2

Reconstruction of an asymmetric object field: (a) plot of the absolute value of the refractive-index distribution described by Eq. (18); (b) reduction of error by iterative reconstruction of this field.

Fig. 3
Fig. 3

Reconstruction of the axisymmetric refractive-index distribution described by Eq. (19): (a) field and its reconstruction; (b) reduction of error by iterative reconstruction.

Fig. 4
Fig. 4

Reconstruction of the axisymmetric refractive-index distribution described by Eq. (20): (a) field and its reconstruction; (b) reduction of error by iterative reconstruction.

Fig. 5
Fig. 5

Refraction of an optical ray by a boundary layer and test section window. Apparent displacement of the object surface and an appropriate distortion of the boundary layer result.

Fig. 6
Fig. 6

Reconstruction of a boundary layer with refractive index n(x) − n0 = −0.003 erfc(2.5x) imaged so that p = 2.05: (a) distribution and its reconstruction; (b) reduction of errors by iterative reconstruction.

Fig. 7
Fig. 7

Reconstruction of the same boundary layer as in Fig. 6 imaged so that p = 1.0: (a) distribution and its reconstruction; (b) reduction of errors by iterative reconstruction.

Fig. 8
Fig. 8

Interferogram of a portion of the cathodic boundary layer: (a) current density 3 mA/cm2; (b) current density 15 mA/cm2.

Fig. 9
Fig. 9

Reconstructed boundary layer concentration profiles.

Equations (44)

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Δ Φ ( ρ , θ ) = A B n ( r , ϕ ) d s + n 0 ( B C ¯ D E ¯ E F ¯ ) .
d d s ( n d r d s ) = n .
Δ Φ ( ρ , θ ) = P [ n ( r , ϕ ) ; n 0 ] .
Δ Φ ( ρ , θ ) = λ 0 N f ( ρ , θ ) ,
Δ Φ ¯ ( ρ , θ ) = E F [ n ( r , ϕ ) n 0 ] d l ,
P ¯ [ n ( r , ϕ ) ; n 0 ] = R [ n ( r , ϕ ) n 0 ] = Δ Φ ¯ ( ρ , θ ) .
D ( ρ , θ ) = Δ Φ ( ρ , θ ) Δ Φ ¯ ( ρ , θ ) .
Δ Φ ¯ i ( ρ , θ ) = Δ Φ ( ρ , θ ) D i ( ρ , θ ) .
n i ( r , ϕ ) n 0 = P ¯ 1 ( Δ Φ ¯ i ) .
Δ Φ i ( ρ , θ ) = P [ n i ( r , ϕ ) ; n 0 ] .
D i ( ρ , θ ) = Δ Φ i Δ Φ ¯ i .
Δ ¯ Φ ( ρ , θ ) = m n A m n g m n ( ρ ) exp ( i m θ ) ,
Δ Φ ¯ ( ρ i , θ i ) = m n A m n g m n ( ρ i ) exp ( i m θ i )
f ( r , ϕ ) = m n A m n f m n ( r ) exp ( i m ϕ ) .
m = 0 , ± 1 , ± 2 , ± 3 , , n = | m | + 2 i , i = 0 , 1 , 2 , 3 , .
d y d x = tan γ ,
d γ d x = n y n x tan γ n ,
d ( Δ E ) d x = n cos γ ,
d ϕ d r = 1 r tan ϕ ,
d χ d r = 1 r n [ n ϕ ( r n r + n ) tan χ ] ,
d ( Δ E ) d r = n cos χ ,
n n 0 = 0.06 [ 1 ( r / R ) 2 ] [ ( x / R ) + 1.5 ] 2 .
n ( r ) n 0 = { 1 0.9 exp [ 4 ( r / R ) 2 ] } 1 / 2 .
n ( r ) n 0 = 0.05 [ 1 ( r / R ) 2 ] + 0.025 [ cos ( 3 π r / R ) + 1 ] .
P [ n ( x ) ; n 0 ] = Δ Φ ( x ) = A B n ( x ) d s + n 0 ( B B 0 ¯ w M w S ) + n w ( B 0 C 0 ¯ w M ) n a C C 0 ¯ ,
P ¯ [ n ( x ) ; n 0 ] = Δ Φ ¯ ( x ) = w M [ n ( x ) n 0 ] ,
T [ Δ Φ ( x ) ] = p Δ Φ ( x / p ) ,
Δ Φ ¯ ( x ) = e ( x ) n = 0 N A n x n .
n ( x ) n 0 = 0.003 erfc ( 2.5 x ) n 0 = 1.3311 ; n w = 1.5231 n a = 1.0002674 ; w w = 12.7 mm w s = 0 ; w m = 20.00 mm δ b = 1.0 mm .
n a = 1.002674 , n 0 = 1.3695 , n w = 1.4571 , w m = 5 mm , w s = 3 mm , w w = 6.35 mm .
C C 0 = 52.63 ( n n 0 ) ,
g ( ρ , θ ) = e ( ρ R ) [ 1 ( ρ R ) 2 ] 1 / 2 m = 0 n = m , 2 A ± m , n q n ( ρ R ) exp ( ± i m θ ) .
R 1 [ g n ( ρ ) exp ( i m θ ) ] = f m ( r ) exp ( i m θ ) ,
H n [ f ( x ) ] = 2 π 0 x f ( x ) J n ( 2 π x t ) d x ,
F [ f ( x ) ] = f ( x ) exp ( i 2 π x t ) d x ,
g ( ρ , θ ) = m = 0 n = m , 2 A ± m , n P n ( ρ R ) exp ( ± i m θ ) ,
f ( r , ϕ ) = 1 π R [ 1 ( r / R ) 2 ] 1 / 2 m = 0 n = m , 2 A ± m , n ( 1 ) ( m n ) / 2 ( 1 ) n m 2 ½ ( n m ) ( r R ) m P ( m , 1 2 ) n m 2 × [ 1 2 ( r R ) 2 ] exp ( ± i m ϕ ) .
g ( ρ , θ ) = [ 1 ( ρ R ) 2 ] 1 / 2 m = 0 n = m , 2 A ± m , n U n ( ρ R ) exp ( ± i m θ ) ,
f ( r , ϕ ) = 1 2 R m = 0 n = m , 2 A ± m , n ( n + 1 ) P ( 0 , m ) m n 2 × [ 2 ( r R ) 2 1 ] exp ( ± i m ϕ ) ,
g ( ρ , θ ) = exp ( α 2 ρ 2 ) m = 0 n = m , 2 A ± m , n H n ( α ρ ) exp ( ± i m θ ) ,
f ( r , ϕ ) = α π exp ( α 2 ρ 2 ) m = 0 n = m , 2 A ± m , n ( 1 ) n 2 m ( 1 ) n m 2 2 n ( α r ) m × L m n m 2 ( α 2 r 2 ) exp ( ± i m ϕ ) ,
( a ) 0 = 1 , ( a ) n = a ( a + 1 ) ( a + 2 ) ( a + n 2 ) ( a + n 1 ) ,
g ( ρ , θ ) = m = 0 n = m + 2 , 2 A ± m , n [ P n ( ρ R ) P m ( ρ R ) ] exp ( ± i m θ ) .
f ( r , ϕ ) = 1 π R [ 1 ( r / R ) 2 ] 1 / 2 m = 0 n = m , 2 A ± m , n exp ( ± i m ϕ ) ( r R ) m { ( 1 ) n m 2 ( m + 1 ) n m 2 ( ½ ) n m 2 k = 0 n m 2 1 ( m n 2 ) k ( m n + 1 2 ) k ( 1 ) k ( m + 1 ) k ( r R ) 2 k [ 1 ( r R ) 2 ] ( n m 2 1 k ) k = 0 n m 2 1 ( r R ) 2 k } .

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