Abstract

Gradient-index lenses can be viewed from the perspectives of both imaging and nonimaging optics, that is, in terms of both image fidelity and achievable flux concentration. The simple class of gradient-index lenses with spherical symmetry, often referred to as modified Luneburg lenses, is revisited. An alternative derivation for established solutions is offered; the method of Fermat’s strings and the principle of skewness conservation are invoked. Then these nominally perfect imaging devices are examined from the additional vantage point of power transfer, and the degree to which they realize the thermodynamic limit to flux concentration is determined. Finally, the spherical gradient-index lens of the fish eye is considered as a modified Luneburg lens optimized subject to material constraints.

© 2000 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 39–47, 231–242.
  2. J. C. Maxwell, “On the general laws of optical instruments,” Q. J. Pure Appl. Math. 2, 233–247 (1854).
  3. A. L. Mikaelian, “Self-focusing media with variable index of refraction,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Chap. V, pp. 281–345.
  4. R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 208–213.
  5. S. Doric, E. Munro, “General solution of the non-full-aperture Luneburg lens problem,” J. Opt. Soc. Am. 73, 1083–1086 (1983).
    [CrossRef]
  6. S. Doric, “Generalized non-full-aperture Luneburg lens: a new solution,” Opt. Eng. 32, 2118–2121 (1993).
    [CrossRef]
  7. L. Matthiesen, “Über die Beziehungen welche zwischen dem Brechungsindex des Kernzentrums der Krystallinse und den Dimensionen des Auges bestehen,” Pfluegers Arch. 27, 510–528 (1886).
    [CrossRef]
  8. M. F. Land, “Optics and vision in inverterbrates,” in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, New York, 1981), Vol. 7/6B, pp. 471–594.
    [CrossRef]
  9. R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
    [CrossRef] [PubMed]
  10. W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
    [CrossRef] [PubMed]
  11. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
    [CrossRef]
  12. A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
    [CrossRef]
  13. A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
    [CrossRef]

1999 (1)

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

1993 (1)

S. Doric, “Generalized non-full-aperture Luneburg lens: a new solution,” Opt. Eng. 32, 2118–2121 (1993).
[CrossRef]

1992 (1)

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

1983 (1)

1958 (1)

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

1954 (2)

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
[CrossRef]

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

1886 (1)

L. Matthiesen, “Über die Beziehungen welche zwischen dem Brechungsindex des Kernzentrums der Krystallinse und den Dimensionen des Auges bestehen,” Pfluegers Arch. 27, 510–528 (1886).
[CrossRef]

1854 (1)

J. C. Maxwell, “On the general laws of optical instruments,” Q. J. Pure Appl. Math. 2, 233–247 (1854).

Campbell, M. C. W.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

Doric, S.

S. Doric, “Generalized non-full-aperture Luneburg lens: a new solution,” Opt. Eng. 32, 2118–2121 (1993).
[CrossRef]

S. Doric, E. Munro, “General solution of the non-full-aperture Luneburg lens problem,” J. Opt. Soc. Am. 73, 1083–1086 (1983).
[CrossRef]

Fernald, R. D.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

Fletcher, A.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
[CrossRef]

Gutman, A. S.

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

Jagger, W. S.

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

Kröger, R. H. H.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

Land, M. F.

M. F. Land, “Optics and vision in inverterbrates,” in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, New York, 1981), Vol. 7/6B, pp. 471–594.
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 208–213.

Matthiesen, L.

L. Matthiesen, “Über die Beziehungen welche zwischen dem Brechungsindex des Kernzentrums der Krystallinse und den Dimensionen des Auges bestehen,” Pfluegers Arch. 27, 510–528 (1886).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, “On the general laws of optical instruments,” Q. J. Pure Appl. Math. 2, 233–247 (1854).

Mikaelian, A. L.

A. L. Mikaelian, “Self-focusing media with variable index of refraction,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Chap. V, pp. 281–345.

Morgan, S. P.

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

Munro, E.

Murphy, T.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
[CrossRef]

Wagner, H.-J.

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

Welford, W. T.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 39–47, 231–242.

Winston, R.

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 39–47, 231–242.

Young, A.

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
[CrossRef]

J. Appl. Phys. (2)

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

A. S. Gutman, “Modified Luneberg lens,” J. Appl. Phys. 25, 855–859 (1954).
[CrossRef]

J. Comp. Physiol. A (1)

R. H. H. Kröger, M. C. W. Campbell, R. D. Fernald, H.-J. Wagner, “Multifocal lenses compensate for chromatic defocus in vertebrate eyes,” J. Comp. Physiol. A 184, 361–369 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

S. Doric, “Generalized non-full-aperture Luneburg lens: a new solution,” Opt. Eng. 32, 2118–2121 (1993).
[CrossRef]

Pfluegers Arch. (1)

L. Matthiesen, “Über die Beziehungen welche zwischen dem Brechungsindex des Kernzentrums der Krystallinse und den Dimensionen des Auges bestehen,” Pfluegers Arch. 27, 510–528 (1886).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. Fletcher, T. Murphy, A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
[CrossRef]

Q. J. Pure Appl. Math. (1)

J. C. Maxwell, “On the general laws of optical instruments,” Q. J. Pure Appl. Math. 2, 233–247 (1854).

Vision Res. (1)

W. S. Jagger, “The optics of the spherical fish lens,” Vision Res. 32, 1271–1284 (1992).
[CrossRef] [PubMed]

Other (4)

M. F. Land, “Optics and vision in inverterbrates,” in Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, New York, 1981), Vol. 7/6B, pp. 471–594.
[CrossRef]

W. T. Welford, R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989), pp. 39–47, 231–242.

A. L. Mikaelian, “Self-focusing media with variable index of refraction,” in Progress in Optics XVII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Chap. V, pp. 281–345.

R. K. Luneburg, Mathematical Theory of Optics (Brown U. Press, Providence, R.I., 1944), pp. 208–213.

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Figures (6)

Fig. 1
Fig. 1

Illustration of the Luneburg lens as a radiation concentrator from an extended far-field source of angular extent 2θ a onto a spherical-cap absorber (of arc length 2θ a in cross section). The three wave fronts traced are at the two extreme angles plus a nominal incidence angle of zero.

Fig. 2
Fig. 2

Sample ray trajectory and illustration of variables: (a) r 1 > 1, (b) r 1 < 1.

Fig. 3
Fig. 3

(a) Refractive index relative to that of the environment, n/n env, against radial position r for the modified Luneburg lens with r 1 = 2.55; (b) (n/n env)2 against r 2 for the same case.

Fig. 4
Fig. 4

Refractive index at the core of the modified Luneburg lens, n(0), as a function of r 1.

Fig. 5
Fig. 5

Ray trajectories in modified Luneburg lenses: (a) r 1 = 0.4, (b) r 1 = 2.55.

Fig. 6
Fig. 6

Spherical gradient-index lenses as radiation concentrators, illustrated for an incident extended far-field source of angular extent 2θ a (in analogy to Fig. 1 for the Luneburg lens, r 1 = 1). The three wave fronts traced are at the two extreme angles plus a nominal incidence angle of zero. (a) r 1 = 2.55 (r 1 greater than unity), also corresponding to the fish-eye lens, (b) r 1 = 0.4 (r 1 less than unity).

Equations (34)

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nr=2nenv1+r2
n2r=2-r2.
n2r=nenv22-r2.
C2D=C3D.
rnrsinϕ=h
r*1drrn2r2-h2+r*r1drrn2r2-h2=π-sin-1hh  r11,
r*1drrn2r2-h2=2 cos-1h+sin-1h/r12h  r11,
1n*dnnn2r2-h2+1/r1n*dnnn2r2-h2=sin-1hh  r11,
1n*dnnn2r2-h2=sin-1h/r12h  r11,
tanϕ=r dθdr.
θb-θa=h rarbdrrn2r2-h2.
n2r=1+r12-r2r12,
nr=exp1πρ1sin-1h/r1dhh2-ρ2,  ρ=nrr.
 ndl=const.,
dl=dr1+r dθdr21/2=rndrn2r2-h2.
r*1n2rdrn2r2-h2+r*r1n2rdrn2r2-h2=1-h2+Fr1  r11,
r*1n2rdrn2r2-h2=Gr1-1-h22  r11,
Gr1=r12-h22.
Fr1=π1+r124r1.
Cmax=nabs2sin2θa,
Aaperture=π0θadsin2θcosθ0θadsin2θ=πcos2θa/2,
Aabsorber=4π sin2θa/2
Aaperture=2 0θadsinθcosθ0θad[sinθ=2θasinθa
Aabsorber=2θa.
Ωaperture=π sin2θa.
E´=4π2 sin2θa/2.
Ωabsorber=π sin2π/2=π,
n2Ωabsorber=n2π sin2π/2=n2π.
Aabsorber=4πr12 sin2θa/2.
E´=4π2 sin2θa/2=Aabsorbern2Ωabsorber=4π2n2r12 sin2θa/2,
θaπ2-sin-11r1,
θout=sin-11/r1.
Cmax=sin2θoutsin2θa=1r12 sin2θa
Cmax=nabs2sin2θa=1r12 sin2θa

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