Abstract

In the framework of geometrical optics we consider a two-dimensional transparent inhomogeneous isotropic medium (dispersive or not). We show that (i) for any family belonging to a certain class of planar monoparametric families of monochromatic light rays given in the form f(x,y)=c of any definite color and satisfying a differential condition, all the refractive index profiles n=n(x,y) allowing for the creation of the given family can be found analytically (inverse problem) and that (ii) for any member of a class of two-dimensional refractive index profiles n=n(x,y) satisfying a differential condition, all the compatible families of light rays can be found analytically (direct problem). We present appropriate examples.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Arnaud, "Application of the mechanical theory of light to fiber optics," J. Opt. Soc. Am. 65, 174-181 (1975).
    [CrossRef]
  2. M. A. Hindy, "Refractive-index profile in fiber optics," Microwave Opt. Technol. Lett. 29, 252-256 (2001).
    [CrossRef]
  3. M. S. Dineen, "Extended summaries of several articles from the March Optics and Photonics News special issue on fiber optics," J. Opt. Netw. 1, 143-148 (2002).
  4. P. J. Sands, "Inhomogeneous lenses. VI. Derivative of paraxial coefficients," J. Opt. Soc. Am. 63, 1210-1216 (1973).
    [CrossRef]
  5. K. Maeda and J. Hamasaki, "A method of determining the refractive-index profile of a lenslike medium," J. Opt. Soc. Am. 67, 1672 (1977).
    [CrossRef]
  6. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Springer-Verlag, 1990), Vol. 6.
    [CrossRef]
  7. J. A. Grzesik, "Focusing properties of a three-parameter class of oblate, Luneburg-like inhomogeneous lenses," J. Electromagn. Waves Appl. 19, 1005-1019 (2005).
    [CrossRef]
  8. B. Wang, P. J. Bos, and C. D. Hoke, "Light propagation in variable-refractive-index materials with liquid-crystal-infiltrated microcavities," J. Opt. Soc. Am. A 20, 2123-2130 (2003).
    [CrossRef]
  9. M.-H. Wei and W.-C. Chen, "Theoretical analysis on the refractive-index distribution and bandwidth of gradient-index polymer optical fibers from a centrifugal field," Appl. Opt. 42, 2174-2180 (2003).
    [CrossRef] [PubMed]
  10. E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
    [CrossRef]
  11. K. S. Kunz, "Propagation of microwaves between a parallel pair of doubly curved conducting surfaces," J. Appl. Phys. 25, 642-653 (1954).
    [CrossRef]
  12. J. C. Minãno, "Refractive-index distribution in two-dimensional geometry for a given one-parameter manifold of rays," J. Opt. Soc. Am. A 2, 1821-1825 (1985).
    [CrossRef]
  13. G. Beliakov and D. Y. C. Chan, "Analysis of inhomogeneous optical systems by the use of ray tracing. I. Planar systems," Appl. Opt. 36, 5303-5309 (1997).
    [CrossRef] [PubMed]
  14. R. F. Rinehart, "A solution of the problem of rapid scanning for radar antennae," J. Appl. Phys. 19, 860-862 (1948).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. revised (Cambridge U. Press, 2002).
  16. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  17. S. Nemoto and T. Makimoto, "Refractive-index distribution for a prescribed ray path," J. Opt. Soc. Am. 69, 450-459 (1979).
    [CrossRef]
  18. F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005).
    [CrossRef]
  19. A. Fletcher, T. Murphy, and A. Young, "Solutions of two optical problems," Proc. R. Soc. London, Ser. A 223, 216-225 (1954).
    [CrossRef]
  20. G. Toraldo di Francia, "A family of perfect configuration lenses of revolution," Opt. Acta 1, 157-163 (1955).
    [CrossRef]
  21. S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
    [CrossRef]
  22. J. Sochacki, "Exact analytical solution of the generalized Luneburg lens problem," J. Opt. Soc. Am. 73, 789-795 (1983).
    [CrossRef]
  23. J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W.D.Niven, ed. (Cambridge U. Press, 1890), Vol. I, Solutions of Problems no. 2, pp. 76-79.
  24. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).
  25. J. Hietarinta, "Direct methods for the search of the second invariant," Phys. Rep. 147, 87-154 (1987).
    [CrossRef]
  26. G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

2005 (3)

J. A. Grzesik, "Focusing properties of a three-parameter class of oblate, Luneburg-like inhomogeneous lenses," J. Electromagn. Waves Appl. 19, 1005-1019 (2005).
[CrossRef]

F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005).
[CrossRef]

E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
[CrossRef]

2003 (2)

2002 (1)

2001 (1)

M. A. Hindy, "Refractive-index profile in fiber optics," Microwave Opt. Technol. Lett. 29, 252-256 (2001).
[CrossRef]

2000 (1)

G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

1999 (1)

S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
[CrossRef]

1997 (1)

1987 (1)

J. Hietarinta, "Direct methods for the search of the second invariant," Phys. Rep. 147, 87-154 (1987).
[CrossRef]

1985 (1)

1983 (1)

1979 (1)

1977 (1)

1975 (1)

1973 (1)

1955 (1)

G. Toraldo di Francia, "A family of perfect configuration lenses of revolution," Opt. Acta 1, 157-163 (1955).
[CrossRef]

1954 (2)

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

K. S. Kunz, "Propagation of microwaves between a parallel pair of doubly curved conducting surfaces," J. Appl. Phys. 25, 642-653 (1954).
[CrossRef]

1948 (1)

R. F. Rinehart, "A solution of the problem of rapid scanning for radar antennae," J. Appl. Phys. 19, 860-862 (1948).
[CrossRef]

Acosta, E.

Arnaud, J. A.

Beliakov, G.

Borghero, F.

F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. revised (Cambridge U. Press, 2002).

Bos, P. J.

Bozis, G.

F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005).
[CrossRef]

G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
[CrossRef]

Chan, D. Y. C.

Chen, W.-C.

Contopoulos, G.

G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

Dineen, M. S.

Elmabsout, B.

S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
[CrossRef]

Fletcher, A.

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

Garner, L.

Grigoriadou, S.

S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
[CrossRef]

Grzesik, J. A.

J. A. Grzesik, "Focusing properties of a three-parameter class of oblate, Luneburg-like inhomogeneous lenses," J. Electromagn. Waves Appl. 19, 1005-1019 (2005).
[CrossRef]

Hamasaki, J.

Hietarinta, J.

J. Hietarinta, "Direct methods for the search of the second invariant," Phys. Rep. 147, 87-154 (1987).
[CrossRef]

Hindy, M. A.

M. A. Hindy, "Refractive-index profile in fiber optics," Microwave Opt. Technol. Lett. 29, 252-256 (2001).
[CrossRef]

Hoke, C. D.

Kravtsov, Yu. A.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Springer-Verlag, 1990), Vol. 6.
[CrossRef]

Kunz, K. S.

K. S. Kunz, "Propagation of microwaves between a parallel pair of doubly curved conducting surfaces," J. Appl. Phys. 25, 642-653 (1954).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

Maeda, K.

Makimoto, T.

Maxwell, J. C.

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W.D.Niven, ed. (Cambridge U. Press, 1890), Vol. I, Solutions of Problems no. 2, pp. 76-79.

Minãno, J. C.

Murphy, T.

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

Nemoto, S.

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Springer-Verlag, 1990), Vol. 6.
[CrossRef]

Rinehart, R. F.

R. F. Rinehart, "A solution of the problem of rapid scanning for radar antennae," J. Appl. Phys. 19, 860-862 (1948).
[CrossRef]

Sands, P. J.

Smith, G.

Sochacki, J.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

Toraldo di Francia, G.

G. Toraldo di Francia, "A family of perfect configuration lenses of revolution," Opt. Acta 1, 157-163 (1955).
[CrossRef]

Vazquez, D.

Wang, B.

Wei, M.-H.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. revised (Cambridge U. Press, 2002).

Young, A.

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

Acta Geogr. Sin. (1)

G. Contopoulos and G. Bozis, "Complex force fields and complex orbits," Acta Geogr. Sin. 8, 1-14 (2000).

Appl. Opt. (2)

Celest. Mech. Dyn. Astron. (1)

S. Grigoriadou, G. Bozis, and B. Elmabsout, "Solvable cases of Szebehely's equation," Celest. Mech. Dyn. Astron. 74, 211-221 (1999).
[CrossRef]

J. Appl. Phys. (2)

K. S. Kunz, "Propagation of microwaves between a parallel pair of doubly curved conducting surfaces," J. Appl. Phys. 25, 642-653 (1954).
[CrossRef]

R. F. Rinehart, "A solution of the problem of rapid scanning for radar antennae," J. Appl. Phys. 19, 860-862 (1948).
[CrossRef]

J. Electromagn. Waves Appl. (1)

J. A. Grzesik, "Focusing properties of a three-parameter class of oblate, Luneburg-like inhomogeneous lenses," J. Electromagn. Waves Appl. 19, 1005-1019 (2005).
[CrossRef]

J. Opt. Netw. (1)

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

F. Borghero and G. Bozis, "A two-dimensional inverse problem of geometrical optics," J. Phys. A 38, 175-184 (2005).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

M. A. Hindy, "Refractive-index profile in fiber optics," Microwave Opt. Technol. Lett. 29, 252-256 (2001).
[CrossRef]

Opt. Acta (1)

G. Toraldo di Francia, "A family of perfect configuration lenses of revolution," Opt. Acta 1, 157-163 (1955).
[CrossRef]

Phys. Rep. (1)

J. Hietarinta, "Direct methods for the search of the second invariant," Phys. Rep. 147, 87-154 (1987).
[CrossRef]

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

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

Other (5)

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Springer-Verlag, 1990), Vol. 6.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. revised (Cambridge U. Press, 2002).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W.D.Niven, ed. (Cambridge U. Press, 1890), Vol. I, Solutions of Problems no. 2, pp. 76-79.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (47)

Equations on this page are rendered with MathJax. Learn more.

f ( x , y ) = c .
n x + γ n y = γ γ x γ y 1 + γ 2 n ,
γ ( x , y ) = f y f x .
ω ( x , y , γ ) = 0
γ x x + γ y y = 2 γ ( γ 2 x + γ 2 y ) 1 + γ 2 ,
d y d x = γ ( x , y ) ,
d y d x = 1 γ ( x , y )
μ ( x , y ) = ( 1 + γ 2 ) 1 2 exp ( Θ ( x , y ) ) .
Θ x = γ y 1 + γ 2 , Θ y = γ x 1 + γ 2 .
μ y γ + μ γ y + μ x = 0 .
μ γ d x μ d y = 0 ,
μ y = μ γ x + γ μ x ,
μ d x + μ γ d y = 0 ,
d x 1 = d y γ = ( 1 + γ 2 ) d n ( γ γ x γ y ) n .
γ y 1 + γ 2 d x + γ x 1 + γ 2 d y = d n n .
n = k 1 exp Θ ( x , y ) .
B ( x , y ) = k 2
n ( x , y ) = exp Θ ( x , y ) G ( B ( x , y ) ) .
f ( x , y ) = x y log sin ( π 4 x y ) = c ,
γ = tan ( x + y ) ,
Θ = y x .
μ = cos ( x + y ) exp ( y x ) ,
sin ( x + y ) exp ( y x ) d x cos ( x + y ) exp ( y x ) d y = 0 .
B ( x , y ) = exp ( y x ) ( sin ( x + y ) + cos ( x + y ) ) = k 2 .
n ( x , y ) = exp ( y x ) G [ exp ( y x ) ( sin ( x + y ) + cos ( x + y ) ) ] .
γ γ x γ y = ( n x + γ n y ) n ( 1 + γ 2 ) .
d x γ = d y 1 = d γ ( n x n + γ n y n ) ( 1 + γ 2 ) .
n y n d x n x n d y = d γ 1 + γ 2 .
n x x + n y y = n x 2 + n y 2 n .
γ = tan ( Φ + c 1 ) ,
Φ x = n y n , Φ y = n x n .
d y d x = 1 γ ( x , y ) ,
Ψ ( x , y ) = c 2
Ψ x = μ , Ψ y = γ μ .
γ = tan ( Φ + H ( Ψ ) ) ,
n = x 2 + y 2 ,
Φ x = 2 y x 2 + y 2 , Φ y = 2 x x 2 + y 2
Φ = 2 arctan y x + Φ 0 ,
γ = tan ( c 1 2 arctan y x ) .
( x 2 + y 2 ) cos ( c 1 2 arctan y x ) d x + ( x 2 + y 2 ) sin ( c 1 2 arctan y x ) d y = 0 ,
Ψ ( x , y ) = ( 1 3 x 3 x y 2 ) cos c 1 ( 1 3 y 3 x 2 y ) sin c 1 = c 2 .
Ψ = 1 3 ( x 2 + y 2 ) ( x + y γ ) ( 1 + γ 2 ) 1 2 .
γ = 2 x y Ψ 0 ( x 2 y 2 ) y 2 x 2 2 x y Ψ 0 ,
2 arctan γ ( x , y ) = 0 ,
2 log n ( x , y ) = 0 ,
y + 2 x γ 1 + γ 2 = Φ ( x 1 + γ 2 ) , Φ = arbitrary .
γ = 1 y ( x + x 2 + x y y 2 ) .

Metrics