Abstract

An analytical expression for the gradient of the generalized Luneburg lens is found and applied to the computation of the profile and the gradient of the refractive index.

© 1983 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 182–187.
  2. S. P. Morgan, J. Appl. Phys. 29, 1358 (1958).
    [CrossRef]
  3. W. H. Southwell, J. Opt. Soc. Am. 67, 1010 (1977).
    [CrossRef]
  4. E. Colombini, J. Opt. Soc. Am. 71, 1403 (1981).
  5. A. Sharma, A. K. Ghatak, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA3-1.
  6. A. Sharma, D. V. Kumar, A. K. Ghatak, Appl. Opt. 21, 984 (1982).
    [CrossRef] [PubMed]

1982 (1)

1981 (1)

1977 (1)

1958 (1)

S. P. Morgan, J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

Colombini, E.

Ghatak, A. K.

A. Sharma, D. V. Kumar, A. K. Ghatak, Appl. Opt. 21, 984 (1982).
[CrossRef] [PubMed]

A. Sharma, A. K. Ghatak, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA3-1.

Kumar, D. V.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 182–187.

Morgan, S. P.

S. P. Morgan, J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

Sharma, A.

A. Sharma, D. V. Kumar, A. K. Ghatak, Appl. Opt. 21, 984 (1982).
[CrossRef] [PubMed]

A. Sharma, A. K. Ghatak, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA3-1.

Southwell, W. H.

Appl. Opt. (1)

J. Appl. Phys. (1)

S. P. Morgan, J. Appl. Phys. 29, 1358 (1958).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (2)

A. Sharma, A. K. Ghatak, in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1981), paper MA3-1.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), pp. 182–187.

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Figures (1)

Fig. 1
Fig. 1

Ray trace through the Luneburg lens for different conjugates: (a) s0 is infinite, s1 = 5; (b) s0 = s1 = 3; (c) s0 = 4.5, s1 = 1.5.

Equations (18)

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ρ = n r ,
n = exp [ ω ( ρ , s 0 ) + ω ( ρ , s 1 ) ] 0 r 1 , n 1 , 0 ρ 1 ,
ω ( ρ , s ) = 1 π ρ 1 arcsin ( x / s ) ( x 2 ρ 2 ) 1 / 2 d x .
d d s ( n d r d s ) = n ,
d 2 r d t 2 = n n ,
F ( n ) = exp [ ω ( ρ , s 0 ) + ω ( ρ , s 1 ) ] n .
ω ( ρ , s ) = ( 1 ρ ) 1 / 2 π m = 0 a m s ( 2 m + 1 ) r = 0 m b r ρ 2 ( m r ) ,
n i + 1 = n i F ( n i ) F ( n i ) ,
F ( n i ) = exp [ ω ( ρ , s 0 ) + ω ( ρ , s 1 ) ] · [ d ω ( ρ , s 0 ) d ρ + d ω ( ρ , s 1 ) d ρ ] · r 1.
d n d r = n [ d ω ( ρ , s 0 ) d ρ + d ω ( ρ , s 1 ) d ρ ] ( d n d r r + n ) ,
d n d r = n 2 [ d ω ( ρ , s 0 ) d ρ + d ω ( ρ , s 1 ) d ρ ] 1 ρ [ d ω ( ρ , s 0 ) d ρ + d ω ( ρ , s 1 ) d ρ ] .
d ω d ρ = 1 π ρ [ arcsin ( 1 ρ 2 s 2 ρ 2 ) 1 / 2 1 ( 1 ρ 2 ) 1 / 2 arcsin ( 1 s ) ] .
GPL = n d t , OPL = n 2 d t .
ω ( ρ , s ) = 1 π ρ 1 arcsin ( x / s ) ( x 2 ρ 2 ) 1 / 2 d x
ω ( θ 1 , k ) = 1 π 0 θ 1 arcsin ( k cosh θ ) d θ .
d ω d ρ = ω θ 1 d θ 1 d ρ + ω k d k d ρ .
ω θ 1 = 1 π arcsin ( 1 s ) , d θ 1 d ρ = 1 ρ ( 1 ρ ) 1 / 2 , ω k = s π ρ arcsin ( 1 ρ 2 s 2 ρ 2 ) 1 / 2 , d k d ρ = 1 s .
d ω d ρ = 1 π ρ [ arcsin ( 1 ρ 2 s 2 ρ 2 ) 1 / 2 1 ( 1 ρ 2 ) 1 / 2 arcsin ( 1 s ) ] .

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