Abstract

A new method of ray tracing through an inhomogeneous lens with radial symmetry of the refractive-index distribution is proposed that consists of a parametric determination of the polar coordinates of the current ray position.

© 1992 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (Brown University, Providence, R.I., 1944), pp. 189–213, or (University of California, Berkeley, 1964), pp. 182–187.
  2. W. H. Southwell, “Inhomogeneous optical waveguide lens analysis,” J. Opt. Soc. Am. 67, 1004–1009 (1977).
    [Crossref]
  3. S. Dorić, E. Munro, “Improvements of the ray trace through the generalized Luneburg lens,” Appl. Opt. 22, 443–445 (1983).
    [Crossref]
  4. S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
    [Crossref]
  5. J. Sochacki, J. R. Flores, R. Staroński, C. Gómez-Reino, “Improvements in computation of the refractive index profiles for generalized Luneburg lenses,” J. Opt. Soc. Am. A 8, 1248–1255 (1991).
    [Crossref]

1991 (1)

1983 (1)

1977 (1)

1958 (1)

S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[Crossref]

Doric, S.

Flores, J. R.

Gómez-Reino, C.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown University, Providence, R.I., 1944), pp. 189–213, or (University of California, Berkeley, 1964), pp. 182–187.

Morgan, S. P.

S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[Crossref]

Munro, E.

Sochacki, J.

Southwell, W. H.

Staronski, R.

Appl. Opt. (1)

J. Appl. Phys. (1)

S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[Crossref]

J. Opt. Soc. Am. (1)

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

Other (1)

R. K. Luneburg, Mathematical Theory of Optics (Brown University, Providence, R.I., 1944), pp. 189–213, or (University of California, Berkeley, 1964), pp. 182–187.

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Figures (2)

Fig. 1
Fig. 1

Ray trajectory of the Luneburg lens.

Fig. 2
Fig. 2

Ray trace through the perfectly focusing Luneburg lens of the focal length, which is twice as large as the lens radius.

Equations (18)

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k = p sin φ ,
p = p ( r ) n r             ( 0 p 1 for 0 r 1 ) ,
k = n ( r * ) r * = p ( r * ) .
x c = r c cos θ c , y c = r c sin θ c .
d θ = ± k d r r ( p 2 - k 2 ) 1 / 2 .
θ c ( ± ) = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± r * r c k d r r ( p 2 - k 2 ) 1 / 2 ,             0 k p c 1.
d r r = d p p - d n n = d p p - d [ ln n ( p ) ] d p d p ,             0 p 1 ;
θ c ( ± ) = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± k p c k d p p ( p 2 - k 2 ) 1 / 2 k p c d [ ln n ( p ) ] d p × k d p ( p 2 - k 2 ) 1 / 2 = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± arccos k p c k p c d [ ln n ( p ) ] d p × k d p ( p 2 - k 2 ) 1 / 2 ,             0 k p c 1 ,
ln n ( p ) = 1 π p 1 arcsin ν f 1 d ν ( ν 2 - p 2 ) 1 / 2 + 1 π p 1 arcsin ν f 2 d ν ( ν 2 - p 2 ) 1 / 2 ω ( p , f 1 ) + ω ( p , f 2 ) ,             0 p 1 ,
d ω ( p , f m ) d p = 1 π p [ arctan ( 1 - p 2 f m 2 - 1 ) 1 / 2 - arcsin 1 f m ( 1 - p 2 ) 1 / 2 ] ,             0 p 1 ,             m = 1 , 2.
θ c ( ± ) = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± arccos k p c k π k p c arctan ( 1 - p 2 f 1 2 - 1 ) 1 / 2 d p p ( p 2 - k 2 ) 1 / 2 k π k p c arctan ( 1 - p 2 f 2 2 - 1 ) 1 / 2 d p p ( p 2 - k 2 ) 1 / 2 ± 1 π ( arcsin 1 f 1 + arcsin 1 f 2 ) × k p c k d p p ( 1 - p 2 ) 1 / 2 ( p 2 - k 2 ) 1 / 2 .
θ c ( ± ) ( p c ) = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± arccos k p c ± arcsin 1 f 1 + arcsin 1 f 2 π × arcsin ( 1 - k 2 / p c 2 1 - k 2 ) 1 / 2 [ I ( p c , k , f 1 ) + I ( p c , k , f 2 ) ] ,             0 k p c 1 ,
I ( p c , k , f m ) k π k p c arctan ( 1 - p 2 f m 2 - 1 ) 1 / 2 p ( p 2 - k 2 ) 1 / 2 d p ,             0 k p c 1 ,             m = 1 , 2.
r c = r c ( p c ) = p c / n ( p c ) = p c / exp [ ω ( p c , f 1 ) + ω ( p c , f 2 ) ] .
I ( 1 , k , f m ) = 1 2 ( arcsin 1 f m - arcsin k f m ) ,             m = 1 , 2 ,
I ( p c , k , 1 ) = 1 2 arccos k p c .
I ( p c , k , f m ) = k π 0 ( p c 2 - k 2 ) 1 / 2 arctan ( 1 - k 2 - z 2 f m 2 - 1 ) 1 / 2 k 2 + z 2 d z ,             0 k p c 1 ,             m = 1 , 2.
{ θ c ( ± ) ( p c ) = 1 2 ( π + arcsin k f 1 - arcsin k f 2 ) ± arccos k p c k p c d [ ln n ( p ) ] d p k d p ( p 2 - k 2 ) 1 / 2 , r c ( p c ) = p c / n ( p c ) ,             0 k p c 1 ,

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