Abstract

Certain functions useful for representing axisymmetric refractive-index distributions are shown to have exact solutions for Abel transformation of the resulting angular deflection data.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
    [CrossRef]
  2. C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods,” Appl. Opt. 31, 1146–1152 (1992).
    [CrossRef] [PubMed]
  3. C. M. Vest, “Interferometry of strongly refracting axisymmetric phase objects,” Appl. Opt. 14, 1601–1606 (1975).
    [CrossRef] [PubMed]
  4. A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.
  5. B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
    [CrossRef]

1992

1982

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

1975

1826

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Abel, N. H.

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Dasch, C. J.

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Hughey, B. J.

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Santavicca, D. A.

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

Tricomi, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Vest, C. M.

Appl. Opt.

Combust. Sci. Technol.

B. J. Hughey, D. A. Santavicca, “A comparison of techniques for reconstructing axisymmetric reacting flow fields from absorption measurements,” Combust. Sci. Technol. 29, 167–190 (1982).
[CrossRef]

J. Reine Angew. Math.

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Other

A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Geometry of ray deflections in an axisymmetric refractive-index distribution n(r): y i , entrance coordinates of the initial rays; ds, differential length along any given ray; r, radial coordinate of the refractive-index distribution; ∊(y 2), exit angle of the ray originating at the coordinate y 1 = y 2.

Fig. 2
Fig. 2

Error in seven-term reconstruction of a model refractive-index field.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

L | d δ d r | d s 1 ,
δ ( r ) 0             for r ,
( y ) = 2 y y d δ d r d r ( r 2 - y 2 ) 1 / 2 ,
δ ( r ) = - 1 π r ( y ) d y ( y 2 - r 2 ) 1 / 2 .
x ν exp ( - ν 2 ) d ν ( ν 2 - x 2 ) 1 / 2 = 1 2 ( π ) 1 / 2 exp ( - r 2 ) ,
δ ( r ) = A exp ( - r 2 ) ,
( y ) = B y exp ( - y 2 ) .
x ν 2 n + 1 exp ( - ν 2 ) d ν ( ν 2 - x 2 ) 1 / 2 = 1 2 ( π ) 1 / 2 x 2 n + 1 exp ( - x 2 ) U ( 1 2 , n + 3 2 , x 2 ) ,
( c - 3 2 ) U ( 1 2 , c - 1 , z ) + ( 1 - c - z ) U ( 1 2 , c , z ) + z U ( 1 2 , c + 1 , z ) = 0 , U ( 1 2 c + 1 , z ) = U ( 1 2 , c , z ) - U ( 1 2 , c , z ) ,
U ( 1 2 , 1 2 , z 2 ) = ( π ) 1 / 2 exp ( z 2 ) erfc ( z ) .
U ( 1 2 , 3 2 , z ) = 1 2 z - 1 / 2 , U ( 1 2 , 5 2 , z ) = 1 2 z - 1 / 2 + 1 4 z - 3 / 2 .
δ ( r ) = ( a 0 + a 2 r 2 + ) exp ( - r 2 ) ,
( y ) i odd a i H i ( y ) exp ( - y 2 )
a i = - ( y ) H i ( y ) ( π ) 1 / 2 i ! 2 i .
x 2 n + 1 U ( 1 2 , n + 3 2 , x 2 ) = V ( n , x 2 ) .
V ( n , x 2 ) = ( x 2 + n - 1 2 ) V ( n - 1 , x 2 ) + x 2 ( 1 - n ) V ( n - 2 , x 2 ) .
H n ( x ) = m 0 H n , m x m .
H 0 , 0 = 1 , H 0 , m = 0             for m 1 , H 1 , 0 = 0 , H 1 , 1 = 2 , H 1 , m = 0             for m 2 , H n , 0 = - 2 ( n - 1 ) H n - 2 , 0             for n 2 , H n , m = - 2 ( n - 1 ) H n - 2 , m + 2 H n - 1 , m - 1             for n 2 , m 1.
( y ) = n o d d a n H n ( y ) exp ( - y 2 ) = n odd , p 0 a n H n , 2 p + 1 y 2 p + 1 exp ( - y 2 ) ,
δ ( r ) = - 1 π r d y ( y 2 - r 2 ) n odd , p 0 a n H n , 2 p + 1 y 2 p + 1 exp ( - y 2 ) = - 1 2 ( π ) 1 / 2 n odd , p 0 ν 2 p + 1 × exp ( - ν 2 ) U ( 1 2 p + 3 2 , ν 2 ) H n , 2 p + 1 = - ( π ) 1 / 2 2 n odd , p 0 a n H n , 2 p + 1 exp ( - ν 2 ) V ( p , ν 2 ) ,
δ ( r ) = exp [ - ( r - 1 ) 2 ] + exp [ - ( r + 1 ) 2 ] ,
0 V ( n , x 2 ) exp ( - x 2 ) d x C 0 ν 2 n + 1 exp ( - ν 2 ) d ν
0 ν 2 n + 1 exp ( - ν 2 ) d ν = 1 2 n !
0 ν 2 n exp ( - ν 2 ) d ν n ! .
V ( n , x 2 ) = a 0 x 2 n + a 1 x 2 n - 2 + + a 2 n .
a p 3 ( 2 p - 1 ) ! ! / 2 p + 1 3 p !             for p 1 ,
0 V ( n , x 2 ) exp ( - x 2 ) d x / 0 ν 2 n + 1 exp ( - ν 2 ) d ν 6 π 1 / 2 [ p 1 ( 2 p - 1 ) ! ! 2 p + 1 ( 2 n - 2 p - 1 ) ! ! n ! 2 2 n - 2 p + 1 ] 6 π 1 / 2 [ p 1 p ! ( n - p ) ! / n ! + 1 ] .
0 p n p ! ( n - p ) ! / n ! 8 3

Metrics