Abstract

The Luneburg integral has many applications in optics and optoelectronics, among which is determination of the refractive-index profile of a Luneburg lens with a full or nonfull aperture. Consequently, computationally efficient and accurate methods for evaluating this integral represent an important challenge. An alternative approach to numerical evaluation of the Luneburg integral that is five times faster than existing methods is described. Several improvements in the ray-tracing procedure in gradient-index media are also presented. A combination of these methods increases the speed of ray tracing through the generalized Luneburg lens by as many as 2 orders of magnitude compared with earlier algorithms. The precision of our method can be easily controlled.

© 1996 Optical Society of America

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References

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  1. S. Doric, E. Munro, “General solution of the nonfullaperture Luneburg lens problem,” J. Opt. Soc. Am. 73, 1083–1086 (1983).
  2. J. Sochacki, C. Gomez-Reino, “Nonfull-aperture Luneburg lenses: a novel solution,” Appl. Opt. 24, 1371–1373 (1985).
  3. W. H. Southwell, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 72, 908–911 (1982).
  4. G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).
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  8. S. Doric, E. Munro, “Improvements of the ray trace through the generalized Luneburg lens,” Appl. Opt. 22, 443–445 (1983).
  9. J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).
  10. J. Sochacki, “Functional approach to the Luneburg's integral for the planar Luneburg lenses design,” IEEE J. Lightwave Technol. LT-3, 684–687 (1985).
  11. J. Sochacki, J. R. Flores, R. Staronski, C. Gomez-Reino, “Improvements in the computation of refractive-index profiles for the generalized Luneburg lens,” J. Opt. Soc. Am. A 8, 1248–1255 (1991).
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  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of the Appl. Math. Ser. (Dover, New York, 1972), p. 916.

1994 (1)

1991 (1)

1989 (1)

J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).

1985 (2)

J. Sochacki, “Functional approach to the Luneburg's integral for the planar Luneburg lenses design,” IEEE J. Lightwave Technol. LT-3, 684–687 (1985).

J. Sochacki, C. Gomez-Reino, “Nonfull-aperture Luneburg lenses: a novel solution,” Appl. Opt. 24, 1371–1373 (1985).

1983 (2)

1982 (2)

1981 (1)

1977 (1)

1968 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of the Appl. Math. Ser. (Dover, New York, 1972), p. 916.

Beliakov, G.

G. Beliakov, “Reconstruction of optical characteristics of waveguide lenses by the use of ray tracing,” Appl. Opt. 33, 3401–3404 (1994).

G. Beliakov, “Study of mathematical model of the refractive index reconstruction of a smooth thin film waveguide by the use of ray tracing,” Ph.D. dissertation (Russian Peoples’ Friendship University, Moscow, 1992), in Russian.

Colombini, E.

Doric, S.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.

Flores, J. R.

Ghatak, A. K.

Gomez-Reino, C.

Kumar, D. V.

Montagnino, L.

Munro, E.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.

Rogus, D.

J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).

Sharma, A.

Sochacki, J.

J. Sochacki, J. R. Flores, R. Staronski, C. Gomez-Reino, “Improvements in the computation of refractive-index profiles for the generalized Luneburg lens,” J. Opt. Soc. Am. A 8, 1248–1255 (1991).

J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).

J. Sochacki, C. Gomez-Reino, “Nonfull-aperture Luneburg lenses: a novel solution,” Appl. Opt. 24, 1371–1373 (1985).

J. Sochacki, “Functional approach to the Luneburg's integral for the planar Luneburg lenses design,” IEEE J. Lightwave Technol. LT-3, 684–687 (1985).

Southwell, W. H.

Staronski, R.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of the Appl. Math. Ser. (Dover, New York, 1972), p. 916.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.

Appl. Opt. (4)

Fiber Integr. Opt. (1)

J. Sochacki, D. Rogus, C. Gomez-Reino, “Paraxial designing of planar waveguide. Variable-index focusing elements: Part 1. Lenses of circular symmetry,” Fiber Integr. Opt. 8, 121–127 (1989).

IEEE J. Lightwave Technol. (1)

J. Sochacki, “Functional approach to the Luneburg's integral for the planar Luneburg lenses design,” IEEE J. Lightwave Technol. LT-3, 684–687 (1985).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Other (3)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989), p. 122.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Vol. 55 of the Appl. Math. Ser. (Dover, New York, 1972), p. 916.

G. Beliakov, “Study of mathematical model of the refractive index reconstruction of a smooth thin film waveguide by the use of ray tracing,” Ph.D. dissertation (Russian Peoples’ Friendship University, Moscow, 1992), in Russian.

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Equations (16)

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n = exp [ w ( p , s 0 ) + w ( p , s 1 ) ] ,        p = n r ,
w ( p , s ) = 1 π p 1 arcsin ( t s ) ( t 2 p 2 ) 1 / 2 d t ,
w ( p , s ) p = 1 π p [ π 4 1 2 arcsin ( p 2 + s 2 2 s 2 p 2 ) arcsin ( 1 / s ) ( 1 p 2 ) 1 / 2 ] ,
f ( p ) = f ( a ) { 1 + [ 1 ( p / a ) 2 ] 1 / 2 } exp ( w ( p / a , v / a ) 2 w ( p / a , 1 / a ) q C 2 + ( 2 q + 1 ) ( a p ) + 2 q π [ C A + ( 1 p 2 ) 1 / 2 arctan ( C / A ) ] 2 ( 2 q + 1 ) π × { C B + arctan ( C / A ) p    arcsin [ C a ( 1 p 2 ) 1 / 2 ] } ) ,
A = ( 1 a 2 ) 1 / 2 ,     B = In ( 1 + A a ) , q = a B 1 2    arcsin ( a / v ) arccos ( a ) 2 a ( A B )
n ( r ) = 1 r + 1 [ 1 4 q  In ( r ) ] 1 / 2 2 q r ,       b r 1.
w ( p , s ) ( 1 p 2 ) 1 / 2 π m = 0 M 1 ( 2 m + 1 ) 2 s 2 m + 1 × n = 0 m ( 2 n ) ! p 2 ( m n ) 4 n ( n ! ) 2 .
max ( M ) 1 2 ( 2 M 1 ) ( 1 p 2 ) 1 / 2 [ 1 + In ( s ) ] s 2 M + 1 .
max ( M ) ( s 2 1 ) M + 1 2 π ( M + 1 ) ( 2 M + 1 ) 1 / 2 ( 2 M + 3 ) 1 / 2 .
w ( p , s ) = 2 π 0 ( 1 p ) * 1 / 2 arcsin ( x 2 + p s ) ( x 2 + 2 p ) 1 / 2 d x .
0 ( 1 p ) 1 / 2 arcsin ( x 2 + p s ) ( x 2 + 2 p ) 1 / 2 d x i = 1 N w i arcsin ( x i 2 + p s ) ( x i 2 + 2 p ) 1 / 2 ,
max ( N ) ( 1 p ) N + 1 / 2 ( N ! ) 4 ( 2 N + 1 ) [ ( 2 N ) ! ] 3 M 2 N 2 ( 1 p ) 1 / 2 5 N [ ( 1 p ) 1 / 2 3 N ] 2 N M 2 N ,
M 2 N = max f ( 2 N ) ( x )    [ 0 , ( 1 r ) 1 / 2 ] ,          f ( x ) = arcsin ( x 2 + r s ) ( x 2 + 2 r ) 1 / 2 .
d d s ( n d r d s ) = n ,
d 2 r d t 2 = n n ,     d t = d s / n ,
n y 1 + ( y ) 2 + n x y n y = 0 ,

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