Abstract

An analytical solution of the Luneburg integral equation is derived that provides a transcendental relationship for the refractive-index distribution. Using this expression, the index profiles for particular generalized Luneburg lenses with one and two finite image distances are calculated.

© 1983 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 189–213.
  2. R. Stettler, “Über die optische Abbildung von Flächen und Räumen,” Optik 12, 529–543 (1955).
  3. S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
    [Crossref]
  4. W. H. Southwell, “Index profiles for generalized Luneburg lenses and their use in planar optical waveguides,” J. Opt. Soc. Am. 67, 1010–1014 (1977).
    [Crossref]
  5. The lens called the Maxwell fisheye is described, among other places, in M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 146–148.

1977 (1)

1958 (1)

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

1955 (1)

R. Stettler, “Über die optische Abbildung von Flächen und Räumen,” Optik 12, 529–543 (1955).

Born, M.

The lens called the Maxwell fisheye is described, among other places, in M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 146–148.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 189–213.

Morgan, S. P.

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

Southwell, W. H.

Stettler, R.

R. Stettler, “Über die optische Abbildung von Flächen und Räumen,” Optik 12, 529–543 (1955).

Wolf, E.

The lens called the Maxwell fisheye is described, among other places, in M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 146–148.

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)

Optik (1)

R. Stettler, “Über die optische Abbildung von Flächen und Räumen,” Optik 12, 529–543 (1955).

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 189–213.

The lens called the Maxwell fisheye is described, among other places, in M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 146–148.

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Figures (2)

Fig. 1
Fig. 1

Refractive-index profiles for generalized Luneburg lenses. From top to bottom, the curves correspond to lenses with focal lengths f = 4, 6, 7, 8, 10, respectively. n is the refractive index and r is the radial position, both normalized to unity at the edge of the lens.

Fig. 2
Fig. 2

Refractive-index profiles for generalized Luneburg lenses with two finite image distances. Respectively, from top to bottom, the curves correspond to lenses with the following image distances: f1 = 2 and f2 = 3, f1 = 3 and f2 = 4, f1 = 4 and f2 = 10. n is the refractive index and r is the radial position, both normalized to unity at the edge of the lens.

Tables (2)

Tables Icon

Table 1 Refractive-Index Profiles n(ρ) and Radial Distances r(ρ) Using Eqs. (24)(27) for Lenses with f = 4, 6, 7, 8, 10a

Tables Icon

Table 2 Refractive-Index Profiles n(ρ) and Radial Distances r(ρ) Using Eqs. (26), (27), and (33) for Three Lenses with Two Finite Image Distancesa

Equations (44)

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n = exp [ ω ( ρ , f 1 ) + ω ( ρ , f 2 ) ] , ρ 1 , n 1 , ρ > 1 ,
ρ n r
ω ( ρ , f i ) = 1 π ρ 1 arcsin x / f i ( x 2 - ρ 2 ) 1 / 2 d x ,             i = 1 , 2.
n 1 ,         r 0.
n ( ρ ) = [ 1 + ( 1 - ρ 2 ) 1 / 2 ] 1 / 2 ,
n ( r ) = ( 2 - r 2 ) 1 / 2 ,     0 r 1 , n = 1 , r > 1.
n ( ρ ) = 1 + ( 1 - ρ 2 ) 1 / 2 ,
n ( r ) = 2 1 + r 2 , 0 r 1 , n = 1 , r > 1.
ω ( ρ , f ) = i = 1 5 p i ( 1 - ρ ) ( 2 i - 1 ) / 2 ,
ω ( ρ , f ) = 1 π ρ 1 arcsin x / f d x ( x 2 - ρ 2 ) 1 / 2 = 1 π ρ 1 ( x f + 1 2 × 3 x 3 f 3 + 1 × 3 2 × 4 × 5 x 5 f 5 + 1 × 3 × 5 2 × 4 × 6 × 7 x 7 f 7 + ) x ( 1 - ρ 2 / x 2 ) 1 / 2 d x = 1 π [ 1 f ρ 1 d x ( 1 - ρ 2 / x 2 ) 1 / 2 + 1 2 × 3 × f 3 ρ 1 x 2 d x ( 1 - ρ 2 / x 2 ) 1 / 2 + 1 × 3 2 × 4 × 5 × f 5 ρ 1 x 4 d x ( 1 - ρ 2 / x 2 ) 1 / 2 + ] .
ω ( ρ , f ) = 1 π [ ρ f ρ 1 d z z 2 ( 1 - z 2 ) 1 / 2 + ρ 3 2 × 3 × f 3 × ρ 1 d z z 4 ( 1 - z 2 ) 1 / 2 + 3 × ρ 5 2 × 4 × 5 × f 5 × ρ 1 d z z 6 ( 1 - z 2 ) 1 / 2 + 3 × 5 × ρ 7 2 × 4 × 6 × 7 × f 7 × ρ 1 d z z 8 ( 1 - z 2 ) 1 / 2 + ] .
f k ( z ) = 1 z 2 k ( 1 - z 2 ) 1 / 2 ,             k = 1 , 2 , .
d z z 2 ( 1 - z 2 ) 1 / 2 = - ( 1 - z 2 ) 1 / 2 z φ 1 ( z ) .
φ 2 ( z ) = - ( 1 - z 2 ) 3 / 2 z 3 .
φ 2 ( z ) = [ - ( 1 - z 2 ) 3 / 2 z 3 ] = 3 z 4 ( 1 - z 2 ) 1 / 2 - 3 z 2 ( 1 - z 2 ) 1 / 2 ,
d z z 4 ( 1 - z 2 ) 1 / 2 = φ 2 ( z ) 3 + d z z 2 ( 1 - z 2 ) 1 / 2 = φ 2 ( z ) 3 + φ 1 ( z ) = - 1 3 ( 1 - z 2 ) 3 / 2 z 3 - ( 1 - z 2 ) 1 / 2 z ,
φ 3 ( z ) = - ( 1 - z 2 ) 5 / 2 z 5
d z z 6 ( 1 - z 2 ) 1 / 2 = - 1 5 ( 1 - z 2 ) 5 / 2 z 5 - 2 3 ( 1 - z 2 ) 3 / 2 z 3 - ( 1 - z 2 ) 1 / 2 z .
d z z 8 ( 1 - z 2 ) 1 / 2 = - 1 7 ( 1 - z 2 ) 7 / 2 z 7 - 3 5 ( 1 - z 2 ) 5 / 2 z 5 - 3 3 ( 1 - z 2 ) 3 / 2 z 3 - ( 1 - z 2 ) 1 / 2 z ,
d z z 10 ( 1 - z 2 ) 1 / 2 = - 1 9 ( 1 - z 2 ) 9 / 2 z 9 - 4 7 ( 1 - z 2 ) 7 / 2 z 7 - 6 5 ( 1 - z 2 ) 5 / 2 z 5 - 4 3 ( 1 - z 2 ) 3 / 2 z 3 - ( 1 - z 2 ) 1 / 2 z .
d z z 2 k ( 1 - z 2 ) 1 / 2 = - l = 1 k ( k - 1 l - 1 ) 2 ( k - l ) + 1 ( 1 - z 2 ) [ 2 ( k - l ) + 1 ] / 2 z 2 ( k - l ) + 1 ,             k = 1 , 2 , .
C ρ 2 k - 1 ρ 1 d z z 2 k ( 1 - z 2 ) 1 / 2 ,             k = 1 , 2 , ,
C ρ 2 k - 1 ρ 1 d z z 2 k ( 1 - z 2 ) 1 / 2 = C ρ 2 k - 1 lim 0 ρ 1 - d z z 2 k ( 1 - z 2 ) 1 / 2 = C { ρ 2 k - 1 lim 0 [ - l = 1 k ( k - 1 l - 1 ) 2 ( k - l ) + 1 [ 1 - ( 1 - ) 2 ] [ 2 ( k - l ) + 1 ] / 2 ( 1 - ) 2 ( k - l ) + 1 ] + l = 1 k ( k - 1 l - 1 ) 2 ( k - 1 ) + 1 ρ 2 ( l - 1 ) ( 1 - ρ 2 ) [ 2 ( k - l ) + 1 ] / 2 } = 0 + C l = 1 k ( k - 1 l - 1 ) 2 ( k - l ) + 1 ρ 2 ( l - 1 ) ( 1 - ρ 2 ) [ 2 ( k - l ) + 1 ] / 2 .
ω ( ρ , f ) = 1 π { 1 f ( 1 - ρ 2 ) 1 / 2 + 1 2 × 3 × 3 × f 3 × [ ( 1 - ρ 2 ) 3 / 2 + 3 ρ 2 ( 1 - ρ 2 ) 1 / 2 ] + 1 × 3 2 × 4 × 5 × 5 × f 5 [ ( 1 - ρ 2 ) 5 / 2 + 5 × 2 3 × ρ 2 ( 1 - ρ 2 ) 3 / 2 + 5 ρ 4 ( 1 - ρ 2 ) 1 / 2 ] + + 1 × 3 × 5 ( 2 k - 5 ) ( 2 k - 3 ) 2 × 4 × 6 × ( 2 k - 4 ) ( 2 k - 2 ) ( 2 k - 1 ) 2 f 2 k - 1 l = 1 k × 2 k - 1 2 ( k - l ) + 1 ( k - 1 l - 1 ) ρ 2 ( l - 1 ) ( 1 - ρ 2 ) [ 2 ( k - l ) + 1 ] / 2 + } .
ω ( 0 , f ) = 1 π ( 1 f + 1 2 × 3 × 3 × f 3 + 1 × 3 2 × 4 × 5 × 5 × f 5 + 1 × 3 × 5 2 × 4 × 6 × 7 × 7 × f 7 + ) .
n ( ρ ) = exp [ ω ( ρ , f ) ]
r ( ρ ) = ρ exp - [ ω ( ρ , f ) ] .
ω ( ρ , f ) = ( 1 - ρ 2 ) 1 / 2 π { 1 f + 1 2 × 3 × 3 × f 3 [ ( 1 - ρ 2 ) + 3 ρ 2 ] + 1 × 3 2 × 4 × 5 × 5 × f 5 [ ( 1 - ρ 2 ) 2 + 5 × 2 3 ρ 2 ( 1 - ρ 2 ) + 5 ρ 4 ] + 1 × 3 × 5 2 × 4 × 6 × 7 × 7 × f 7 [ ( 1 - ρ 2 ) 3 + 7 × 3 5 ρ 2 ( 1 - ρ 2 ) 2 + 7 × 3 3 ρ 4 ( 1 - ρ 2 ) + 7 ρ 6 ] + } = ( 1 - ρ 2 ) 1 / 2 π ( 1 ρ [ ρ f + ρ 3 2 × 3 × f 3 + 3 × ρ 5 2 × 4 × 5 × f 5 + ] + ( 1 - ρ 2 ) { 1 2 × 3 × 3 × f 3 + 1 × 3 2 × 4 × 5 × 5 × f 5 [ ( 1 - ρ 2 ) + 5 × 2 3 ρ 2 ] + 1 × 3 × 5 2 × 4 × 6 × 7 × 7 × f 7 [ ( 1 - ρ 2 ) 2 + 7 × 3 5 ρ 2 ( 1 - ρ 2 ) + 7 × 3 3 ρ 4 ] + } ) = ( 1 - ρ 2 ) 1 / 2 π ( arcsin ρ / f ρ + ( 1 - ρ 2 ) { 1 2 × 3 × 3 × f 3 + 1 × 3 2 × 4 × 5 × 5 × f 5 [ ( 1 - ρ 2 ) + 5 × 2 3 ρ 2 ] + 1 × 3 × 5 2 × 4 × 6 × 7 × 7 × f 7 × [ ( 1 - ρ 2 ) 2 + 7 × 3 5 ρ 2 ( 1 - ρ 2 ) + 7 × 3 3 ρ 4 ] + 1 × 3 × 5 × × ( 2 k - 5 ) ( 2 k - 3 ) 2 × 4 × 6 × × ( 2 k - 4 ) ( 2 k - 2 ) ( 2 k - 1 ) 2 f 2 k - 1 × l = 1 k - 1 2 k - 1 2 ( k - l ) + 1 ( k - 1 l - 1 ) ρ 2 ( l - 1 ) ( 1 - ρ 2 ) k - l - 1 + } ) .
ω ( ρ , f ) = ( 1 - ρ 2 ) 1 / 2 π [ s 0 ( f ) + s 1 ( f ) ρ 2 + s 2 ( f ) ρ 4 + ] ,
s 0 ( f ) = 1 f ( 1 1 × 1 + 1 2 × 3 × 3 × f 2 + 1 × 3 2 × 4 × 5 × 5 × f 4 + 1 × 3 × 5 2 × 4 × 6 × 7 × 7 × f 6 + ) , s 1 ( f ) = 1 f 3 ( 1 3 × 3 + 1 2 × 5 × 5 × f 2 + 1 × 3 2 × 4 × 7 × 7 × f 4 + 1 × 3 × 5 2 × 4 × 6 × 9 × 9 × f 6 + ) , s 2 ( f ) = 1 f 5 ( 1 5 × 5 + 1 2 × 7 × 7 × f 2 + 1 × 3 2 × 4 × 9 × 9 × f 4 + 1 × 3 × 5 2 × 4 × 6 × 11 × 11 × f 6 + ) ,
ω ( ρ , f ) = ( 1 - ρ 2 ) 1 / 2 π k = 0 s k ( f ) ρ 2 k
s k ( f ) = 1 f 2 k + 1 l = 0 1 2 × 3 2 × 5 5 × × ( 2 l - 3 ) 2 ( 2 l - 1 ) 2 ( 2 l ) ! [ 2 ( k + l ) + 1 ] 2 f 2 l ,             k = 0 , 1 , .
n ( ρ ) = exp 1 π { ( 1 - ρ 2 ) 1 / 2 k = 0 [ s k ( f 1 ) + s k ( f 2 ) ] ρ 2 k } ,             ρ 1 ,             n 1 ,             ρ > 1 ,
d z z 2 k ( 1 - z 2 ) 1 / 2 = - l = 1 k ( k - 1 l - 1 ) 2 ( k - l ) + 1 ( 1 - z 2 ) [ 2 ( k - l ) + 1 ] / 2 z 2 ( k - l ) + 1 ,             k = 1 , 2 , ,
d z z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = - l = 1 k + 1 ( k + 1 - 1 l - 1 ) 2 [ ( k + 1 ) - l ] + 1 × ( 1 - z 2 ) { 2 [ ( k + 1 ) - l ] + 1 } / 2 z 2 [ ( k + 1 ) - l ] + 1 ,
φ k + 1 ( z ) - ( 1 - z 2 ) [ 2 ( k + 1 ) - 1 ] / 2 z 2 ( k + 1 ) - 1 .
[ - ( 1 - z 2 ) [ 2 ( k + 1 ) - 1 ] / 2 z 2 ( k + 1 ) - 1 ] = [ 2 ( k + 1 ) - 1 z 2 ( 1 - z 2 ) k + ( 1 - z 2 ) k + 1 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = [ 2 ( k + 1 ) - 1 ] [ 1 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 + l = 0 k ( - 1 ) l ( k l ) z 2 l + 2 + l = 0 k ( - 1 ) l + 1 ( k + 1 l + 1 ) z 2 l + 2 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 ] = [ 2 ( k + 1 ) - 1 ] × { 1 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 + l = 0 k - 1 ( - 1 ) l [ ( k l ) - ( k + 1 l + 1 ) ] z 2 l + 2 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 } ,
( k l ) - ( k + 1 l + 1 ) = ( k l + 1 ) ,
1 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = 1 2 ( k + 1 ) - 1 [ - ( 1 - z 2 ) [ 2 ( k + 1 ) - 1 ] / 2 z 2 ( k + 1 ) - 1 ] - l = 0 k - 1 ( - 1 ) l ( k l + 1 ) z 2 l + 2 z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2
d z z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = - 1 2 ( k + 1 ) - 1 ( 1 - z 2 ) [ 2 ( k + 1 ) - 1 ] / 2 z 2 ( k + 1 ) - 1 + ( k 1 ) d z z 2 k ( 1 - z 2 ) 1 / 2 - ( k 2 ) d z z 2 ( k - 1 ) ( 1 - z 2 ) 1 / 2 + ( k 3 ) d z z 2 ( k - 2 ) ( 1 - z 2 ) 1 / 2 - - ( - 1 ) k ( k k ) d z z 2 ( 1 - z 2 ) 1 / 2 .
d z z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = - 1 2 [ ( k + 1 ) - 1 ] + 1 ( 1 - z 2 ) { 2 [ ( k + 1 ) - 1 ] + 1 } / 2 z 2 [ ( k + 1 ) - 1 ] + 1 - ( k 1 ) ( k - 1 0 ) 2 [ ( k + 1 ) - 2 ] + 1 ( 1 - z 2 ) { 2 [ ( k + 1 ) - 2 ] + 1 } / 2 z 2 [ ( k + 1 ) - 2 ] + 1 - [ ( k 1 ) ( k - 1 1 ) - ( k 2 ) ( k - 2 0 ) ] 2 [ ( k + 1 ) - 3 ] + 1 ( 1 - z 2 ) { 2 [ ( k + 1 ) - 3 ] + 1 } / 2 z 2 [ ( k + 1 ) - 3 ] + 1 - [ ( k 1 ) ( k - 1 2 ) - ( k 2 ) ( k - 2 1 ) + ( k 3 ) ( k - 3 0 ) ] 2 [ ( k + 1 ) - 4 ] + 1 ( 1 - z 2 ) { 2 [ ( k + 1 ) - 4 ] + 1 } / 2 z 2 [ ( k + 1 ) - 4 ] + 1 - - [ ( k 1 ) ( k - 1 k - 1 ) - ( k 2 ) ( k - 2 k - 2 ) + ( k 3 ) ( k - 3 k - 3 ) - + ( - 1 ) k + 1 ( k k ) ( 0 0 ) ] 2 [ ( k + 1 ) - ( k + 1 ) ] + 1 ( 1 - z 2 ) 1 / 2 z .
n = 1 m k ( - 1 ) n + 1 ( k n ) ( k - n m - n )
n = 1 m k ( - 1 ) n + 1 ( k n ) ( k - n m - n ) = n = 1 m k ( - 1 ) n + 1 k ! ( k - n ) ! n ! ( k - n ) ! ( m - n ) ! ( k - m ) ! = k ! ( k - m ) ! [ - n = 1 m ( - 1 ) n 1 n ! ( m - n ) ! ] = k ! ( k - m ) ! m ! [ - n = 1 m ( - 1 ) n m ! ( m - n ) ! n ! ] = ( k m ) [ 1 - 1 - n = 1 m ( - 1 ) n ( m n ) ] = ( k m ) [ 1 - n = 0 m ( - 1 ) n ( m n ) ] = ( k m ) [ 1 - ( 1 - 1 ) m ] = ( k m ) .
d z z 2 ( k + 1 ) ( 1 - z 2 ) 1 / 2 = - l = 1 k + 1 ( k + 1 - 1 l - 1 ) 2 [ ( k + 1 ) - l ] + 1 × ( 1 - z 2 ) { 2 [ ( k + 1 ) - l ] + 1 } / 2 z 2 [ ( k + 1 ) - l ] + 1 .