Abstract

The general solution presented here yields smooth, continuous, and monotonic refractive-index profiles for Luneburg–Morgan lenses with a boundary index of N > 1. The new formula incorporates an original apparent-immersing method as well as the continuous-deflection-function concept recently developed by one of the authors for the description of waveguide lenses.

© 1992 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; (U. Press, California. Berkeley, Calif., 1964), pp. 182–187.
  2. S. P. Morgan, “General solution of the Luneburg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
    [CrossRef]
  3. A similar concept of design was published slightly earlier by G. Toraldo di Francia, “Mathematical problem of the concentric stigmatic optical system,” Ann. Mat. (Italy) 44, 35–44 (1957) (in Italian). The formulation presented by Morgan is more general, however, and better known to the international optical community.
  4. Another possibility is to assume a certain particular form of the outer-shell, refractive-index distribution for A ≤ p ≤ N, but this coincides with the concept of Morgan and is associated with known disadvantages (see Section 1).
  5. S. Dorić, E. Munro, “General solution of the non-full-aperture Luneburg lens problem,” J. Opt. Soc. Am. 73, 1083–1086 (1983).
    [CrossRef]
  6. 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]
  7. J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” IEEE J. Lightwave Technol. LT-8, 667–672 (1990).
    [CrossRef]
  8. Before differentiating, integration by parts must be performed in Eq. (12a).
  9. S. Dorić, E. Munro, “Improvements of the ray trace through the generalized Luneburg lens,” Appl. Opt. 22, 443–445 (1983).
    [CrossRef]
  10. J. Sochacki, “Perfect geodesic lens designing,” Appl. Opt. 25, 235–243 (1986).
    [CrossRef] [PubMed]

1991 (1)

1990 (1)

J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” IEEE J. Lightwave Technol. LT-8, 667–672 (1990).
[CrossRef]

1986 (1)

1983 (2)

1958 (1)

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

1957 (1)

A similar concept of design was published slightly earlier by G. Toraldo di Francia, “Mathematical problem of the concentric stigmatic optical system,” Ann. Mat. (Italy) 44, 35–44 (1957) (in Italian). The formulation presented by Morgan is more general, however, and better known to the international optical community.

Doric, S.

Flores, J. R.

Gómez-Reino, C.

Luneburg, R. K.

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

Staronski, R.

Toraldo di Francia, G.

A similar concept of design was published slightly earlier by G. Toraldo di Francia, “Mathematical problem of the concentric stigmatic optical system,” Ann. Mat. (Italy) 44, 35–44 (1957) (in Italian). The formulation presented by Morgan is more general, however, and better known to the international optical community.

Ann. Mat. (Italy) (1)

A similar concept of design was published slightly earlier by G. Toraldo di Francia, “Mathematical problem of the concentric stigmatic optical system,” Ann. Mat. (Italy) 44, 35–44 (1957) (in Italian). The formulation presented by Morgan is more general, however, and better known to the international optical community.

Appl. Opt. (2)

IEEE J. Lightwave Technol. (1)

J. Sochacki, “New simplified method for designing the smooth-transition gradient-index and geodesic waveguide lenses of radial symmetry,” IEEE J. Lightwave Technol. LT-8, 667–672 (1990).
[CrossRef]

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 (3)

Before differentiating, integration by parts must be performed in Eq. (12a).

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

Another possibility is to assume a certain particular form of the outer-shell, refractive-index distribution for A ≤ p ≤ N, but this coincides with the concept of Morgan and is associated with known disadvantages (see Section 1).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the stigmatically imaging spherical gradient-index lens.

Fig. 2
Fig. 2

Cumulative deflection function ψ(k) [Eq. (3)], the modified deflection function ψ ˜ (k) [Eq. (9)], and the complementary deflection function ψ ˜ + (k) [Eq. (25)] for the lens where f1 = ∞, f2 = 2, N = 1.73, and A = 0.8.

Fig. 3
Fig. 3

Refractive-index profile resulting from Eqs. (27) for the lens with parameters as in Fig. 2.

Fig. 4
Fig. 4

Cumulative deflection function ψ(k) [Eq. (3)], modified deflection function ψ ˜ (k) [Eq. (9)], and complementary deflection function ψ ˜ + (k) [Eq. (25)] for the lens with f1 = ∞, f2 = 1.1, N = 1.46, and A = 0.8.

Fig. 5
Fig. 5

Refractive-index profile resulting from Eqs. (27) for the lens with parameters as in Fig. 4.

Equations (29)

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r * 1 k d r r ( p 2 - k 2 ) 1 / 2 = 1 2 ψ ( k ) + arccos ( k ) ,
p = p ( r ) n ( r ) r ,             0 r 1 ,
ψ ( k ) = arcsin ( k / f 1 ) + arcsin ( k / f 2 )
k = n ( r ) r sin φ = n ( r * ) r * = p ( r * ) ,
0 k A < 1.
n ( r = 1 ) N > 1 ,
d r / r = d p / p - d n / n = d p / p - { d [ ln n ( p ) ] / d p } d p , 0 p N .
- k N d [ ln n ( p ) ] d p k d p ( p 2 - k 2 ) 1 / 2 = 1 2 ψ ˜ ( k ) ,             0 k A ,
ψ ˜ ( k ) = arcsin ( k / f 1 ) + arcsin ( k / f 2 ) + 2 arcsin ( k / N ) - 2 arcsin ( k ) ,             0 k A ,
2 arcsin ( k / N ) - 2 arcsin ( k ) ,
- k N d [ ln n ( p ) ] d p k d p ( p 2 - k 2 ) 1 / 2 = { 1 2 ψ ˜ ( k ) , 0 k A , 1 2 ψ ˜ + ( k ) , A k N ,
n ( p ) = { N exp [ 1 π p A ψ ˜ ( k ) d k ( k 2 - p 2 ) 1 / 2 + 1 π A N ψ ˜ + ( k ) d k ( k 2 - p 2 ) 1 / 2 ] = N exp [ ω ( p , f 1 , A ) + ω ( p , f 2 , A ) + 2 ω ( p , N , A ) - 2 ω ( p , 1 , A ) + 1 π A N ψ ˜ + ( k ) d k ( k 2 - p 2 ) 1 / 2 ] ,             0 p A , N exp [ 1 π p N ψ ˜ + ( k ) d k ( k 2 - p 2 ) 1 / 2 ] ,             A p N ,
ω ( p , B , A ) 1 π p A arcsin ( k / B ) d k ( k 2 - p 2 ) 1 / 2 ω ( p A , B A )
ψ ˜ + ( k = N ) 0.
ψ ˜ + ( k = A ) = ψ ˜ ( k = A ) = arcsin ( A / f 1 ) + arcsin ( A / f 2 ) + 2 arcsin ( A / N ) - 2 arcsin ( A ) .
d n ( p ) d p = n ( p ) π p [ A N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 + F ( p ) ] ,             0 p A ,
F ( p ) arcsin ( A 2 - p 2 f 1 2 - p 2 ) 1 / 2 + arcsin ( A 2 - p 2 f 2 2 - p 2 ) 1 / 2 + 2 arcsin ( A 2 - p 2 N 2 - p 2 ) 1 / 2 - 2 arcsin ( A 2 - p 2 1 - p 2 ) 1 / 2 + A ( A 2 - p 2 ) 1 / 2 [ ψ ˜ + ( A ) - arcsin ( A / f 1 ) - arcsin ( A / f 2 ) - 2 arcsin ( A / N ) + 2 arcsin ( A ) ] ,
d n ( p ) d p = n ( p ) π p × [ p N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 ] ,             A p N .
F ( p ) = arcsin ( 1 - p 2 f 1 2 - p 2 ) 1 / 2 + arcsin ( 1 - p 2 f 2 2 - p 2 ) 1 / 2 + 2 arcsin ( 1 - p 2 N 2 - p 2 ) 1 / 2 - π + 1 ( 1 - p 2 ) 1 / 2 × [ ψ ˜ + ( 1 ) - arcsin ( 1 / f 1 ) - arcsin ( 1 / f 2 ) - 2 arcsin ( 1 / N ) + π ] ,             0 p 1 ,
d n / d r = n ( d n / d p ) 1 - r ( d n / d p ) .
A N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 + F ( p ) { < 0 > π             0 p A ,
p N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 { < 0 > π             A p N ,
F ( p ) = arcsin ( A 2 - p 2 f 1 2 - p 2 ) 1 / 2 + arcsin ( A 2 - p 2 f 2 2 - p 2 ) 1 / 2 + 2 arcsin ( A 2 - p 2 N 2 - p 2 ) 1 / 2 - 2 arcsin ( A 2 - p 2 1 - p 2 ) 1 / 2 .
0 < d n / d p < 1 / r = n / p ,
0 < A N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 + F ( p ) < π ,             0 p A ,
0 < p N d ψ ˜ + ( k ) d k k d k ( k 2 - p 2 ) 1 / 2 - ψ ˜ + ( N ) N ( N 2 - p 2 ) 1 / 2 < π ,             A p N .
ψ ˜ + ( k ) = c ( N 2 - k 2 ) 1 / 2 k ,
c = A ( N 2 - A 2 ) 1 / 2 [ arcsin ( A / f 1 ) + arcsin ( A / f 2 ) + 2 arcsin ( A / N ) - 2 arcsin ( A ) ] .
n ( p ) = { N exp { ω ( p , f 1 , A ) + ω ( p , f 2 , A )             + 2 ω ( p , N , A ) - 2 ω ( p , 1 , A ) - c π [ N p arccos ( 1 - p 2 / A 2 1 - p 2 / N 2 ) 1 / 2 - arccos ( A 2 - p 2 N 2 - p 2 ) 1 / 2 ] } , 0 p A , N exp [ c 2 ( N p - 1 ) ] , A p N ,

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