Abstract

In the framework of geometrical optics, we consider the inverse problem consisting in obtaining refractive-index distributions n=n(u,v) of a two-dimensional transparent inhomogeneous isotropic medium from a known family f(u,v)=c of monochromatic light rays, lying on a given regular surface. Using some basic concepts of differential geometry, we establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive-index profiles compatible with this family. In particular, we study the refractive-index distribution producing, as light rays, a given family of geodesic lines on some remarkable surfaces. We give appropriate examples to explain the theory.

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    [CrossRef]
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    [CrossRef]
  5. F. Borghero and G. Bozis, “A two-dimensional inverse problem of geometrical optics,” J. Phys. A 38, 175–184 (2005).
    [CrossRef]
  6. F. Borghero and G. Bozis, “Two solvable problems of planar geometrical optics,” J. Opt. Soc. Am. A 23, 3133–3138(2006).
    [CrossRef]
  7. J. A. Arnaud, “Application of the mechanical theory of light to fiber optics,” J. Opt. Soc. Am. 65, 174–181 (1975).
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  12. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Vol. 6 of Springer Series on Wave PhenomenaSpringer-Verlag, 1990).
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  13. 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]
  14. 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).
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  16. 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).
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  17. 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).
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  18. K. S. Kunz, “Propagation of microwaves between a parallel pair of doubly curved conducting surface,” J. Appl. Phys. 25, 642–653 (1954).
    [CrossRef]
  19. K. S. Kunz, “Applications de la géométrie différentielle a l’optique de microondes,” Supplemento al Vol. IX, Serie IX del Nuovo Cimento N. 3, 322–332 (1952).
  20. G. Toraldo di Francia, “Realtá fisica di una varietá di Fermat pseudosferica,” Rend. Acc. Naz. Linc. 8, 489–494 (1950).
  21. G. Toraldo di Francia, “A family of perfect configuration lenses of revolution,” Opt. Acta 1, 157–163 (1955).
    [CrossRef]
  22. D. J. Struik, Lectures on Classical Differential Geometry(Dover, 1988).
  23. L. P. Eisenhart, An Introduction to Differential Geometry with Use of the Tensor Calculus (Princeton University, 1947).
  24. J. C. Maxwell, The Scientific Papers of James Clerk Maxwell I, W.D.Niven ed. (Cambridge University, 1890).
  25. A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. London Ser. A 223, 216–225 (1954).
    [CrossRef]
  26. J. Sochacki, “Exact analytical solution of the generalized Luneburg lens problem,” J. Opt. Soc. Am. 73, 789–795 (1983).
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  27. R. Mertens, “On the determination of the potential energy of a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 61, 252–253 (1981).
  28. T. Kotoulas, “Motion on a given surface: potentials producing geodesic lines as trajectories,” Rend. Sem. Fac. (Scienze Univ. Cagliari, Italy, 2006), Vol. 76, Fasc. 1-2, 1-15.
  29. G. Bozis and R. Mertens, “On Szebehely’s inverse problem for a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 65, 383–384 (1985).
    [CrossRef]
  30. T. Kotoulas, “Isoenergetic families of regular orbits on a given surface,” Differ. Geom. Dyn. Syst. 10, 206–214 (2008).

2008 (1)

T. Kotoulas, “Isoenergetic families of regular orbits on a given surface,” Differ. Geom. Dyn. Syst. 10, 206–214 (2008).

2006 (1)

2005 (3)

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]

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]

2003 (2)

2002 (1)

2001 (1)

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

1997 (1)

1985 (2)

J. C. Miñano, “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]

G. Bozis and R. Mertens, “On Szebehely’s inverse problem for a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 65, 383–384 (1985).
[CrossRef]

1983 (1)

1981 (1)

R. Mertens, “On the determination of the potential energy of a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 61, 252–253 (1981).

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 surface,” J. Appl. Phys. 25, 642–653 (1954).
[CrossRef]

1950 (1)

G. Toraldo di Francia, “Realtá fisica di una varietá di Fermat pseudosferica,” Rend. Acc. Naz. Linc. 8, 489–494 (1950).

Acosta, E.

Arnaud, J. A.

Beliakov, G.

Borghero, F.

F. Borghero and G. Bozis, “Two solvable problems of planar geometrical optics,” J. Opt. Soc. Am. A 23, 3133–3138(2006).
[CrossRef]

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 (Cambridge University, 2002).

Bos, P. J.

Bozis, G.

F. Borghero and G. Bozis, “Two solvable problems of planar geometrical optics,” J. Opt. Soc. Am. A 23, 3133–3138(2006).
[CrossRef]

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

G. Bozis and R. Mertens, “On Szebehely’s inverse problem for a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 65, 383–384 (1985).
[CrossRef]

Chan, D. Y. C.

Chen, W.-C.

Dineen, M. S.

Eisenhart, L. P.

L. P. Eisenhart, An Introduction to Differential Geometry with Use of the Tensor Calculus (Princeton University, 1947).

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.

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.

Hindy, M. A.

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

Hoke, C. D.

Kotoulas, T.

T. Kotoulas, “Isoenergetic families of regular orbits on a given surface,” Differ. Geom. Dyn. Syst. 10, 206–214 (2008).

T. Kotoulas, “Motion on a given surface: potentials producing geodesic lines as trajectories,” Rend. Sem. Fac. (Scienze Univ. Cagliari, Italy, 2006), Vol. 76, Fasc. 1-2, 1-15.

Kravtsov, Yu. A.

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

Kunz, K. S.

K. S. Kunz, “Propagation of microwaves between a parallel pair of doubly curved conducting surface,” J. Appl. Phys. 25, 642–653 (1954).
[CrossRef]

K. S. Kunz, “Applications de la géométrie différentielle a l’optique de microondes,” Supplemento al Vol. IX, Serie IX del Nuovo Cimento N. 3, 322–332 (1952).

Luneburg, R. K.

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

Maeda, K.

Makimoto, T.

Maxwell, J. C.

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell I, W.D.Niven ed. (Cambridge University, 1890).

Mertens, R.

G. Bozis and R. Mertens, “On Szebehely’s inverse problem for a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 65, 383–384 (1985).
[CrossRef]

R. Mertens, “On the determination of the potential energy of a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 61, 252–253 (1981).

Miñano, 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, Vol. 6 of Springer Series on Wave PhenomenaSpringer-Verlag, 1990).
[CrossRef]

Sands, P. J.

Smith, G.

Sochacki, J.

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry(Dover, 1988).

Toraldo di Francia, G.

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

G. Toraldo di Francia, “Realtá fisica di una varietá di Fermat pseudosferica,” Rend. Acc. Naz. Linc. 8, 489–494 (1950).

Vazquez, D.

Wang, B.

Wei, M.-H.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 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]

Appl. Opt. (2)

Differ. Geom. Dyn. Syst. (1)

T. Kotoulas, “Isoenergetic families of regular orbits on a given surface,” Differ. Geom. Dyn. Syst. 10, 206–214 (2008).

J. Appl. Phys. (1)

K. S. Kunz, “Propagation of microwaves between a parallel pair of doubly curved conducting surface,” J. Appl. Phys. 25, 642–653 (1954).
[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 (4)

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]

Microw. Opt. Technol. Lett. (1)

M. A. Hindy, “Refractive-index profile in fiber optics,” Microw. 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]

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]

Rend. Acc. Naz. Linc. (1)

G. Toraldo di Francia, “Realtá fisica di una varietá di Fermat pseudosferica,” Rend. Acc. Naz. Linc. 8, 489–494 (1950).

Z. Angew. Math. Mech. (2)

R. Mertens, “On the determination of the potential energy of a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 61, 252–253 (1981).

G. Bozis and R. Mertens, “On Szebehely’s inverse problem for a particle describing orbits on a given surface,” Z. Angew. Math. Mech. 65, 383–384 (1985).
[CrossRef]

Other (8)

T. Kotoulas, “Motion on a given surface: potentials producing geodesic lines as trajectories,” Rend. Sem. Fac. (Scienze Univ. Cagliari, Italy, 2006), Vol. 76, Fasc. 1-2, 1-15.

D. J. Struik, Lectures on Classical Differential Geometry(Dover, 1988).

L. P. Eisenhart, An Introduction to Differential Geometry with Use of the Tensor Calculus (Princeton University, 1947).

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell I, W.D.Niven ed. (Cambridge University, 1890).

K. S. Kunz, “Applications de la géométrie différentielle a l’optique de microondes,” Supplemento al Vol. IX, Serie IX del Nuovo Cimento N. 3, 322–332 (1952).

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

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2002).

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

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Tables (1)

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Table 1 Families of Geodesic Lines and Refractive Indices Where Q is an Arbitrary Function of Its Argument

Equations (52)

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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 ,
r = r ( u , v ) { x = x ( u , v ) , y = y ( u , v ) , z = z ( u , v ) } ,
f ( u , v ) = c ,
γ ( u , v ) = f v f u , ( f u 0 ) ,
δ n d s = 0 .
d s 2 = E d u 2 + 2 F d u d v + G d v 2 ,
E = r u 2 , F = r u · r v , G = r v 2 .
v = d v d u , T ( u , v , v ) = ( E + 2 F v + G ( v ) 2 ) 1 / 2 ,
δ n ( u , v ) T ( u , v , v ) d u = 0 .
v ( u , v ) = 1 γ ( u , v ) .
d d u [ ( n T ) v ] = ( n T ) v
T v = γ F G γ T .
T v = γ 2 E v 2 γ F v + G v 2 γ 2 T , d n d u n u 1 γ n v .
( n u 1 γ n v ) M + n ( M u 1 γ M v ) = n v T + n 2 γ 2 T ( γ 2 E v 2 γ F v + G v ) ,
M = γ F G γ T .
( G γ F ) n u + ( γ E F ) n v = W 2 Ψ n ,
Ψ = E γ 2 2 F γ + G , W = 2 g ( γ γ u γ v ) + W 3 γ 3 + W 2 γ 2 + W 1 γ + W 0 ,
g = E G F 2 > 0 , W 3 = 2 E F u E E v F E u , W 2 = 3 F E v 2 F F u + G E u 2 E G u , W 1 = 3 F G u + E G v 2 F F v 2 G E v , W 0 = 2 G F v G G u F G v .
n x + γ n y = Ω n ,
γ = f y f x , Γ = γ γ x γ y , Ω = Γ 1 + γ 2 ,
b 1 n u + b 2 n v = b 0 n ,
b 1 = G γ F , b 2 = γ E F , b 0 = W 2 ( E γ 2 2 γ F + G ) .
u n u + v n v = m n .
n u n = Δ 1 Δ 0 , n v n = Δ 2 Δ 0 ,
Δ 0 = v b 1 u b 2 , Δ 1 = v b 0 m b 2 , Δ 2 = m b 1 u b 0 .
( Δ 1 v Δ 2 u ) Δ 0 = Δ 1 Δ 0 v Δ 2 Δ 0 u .
γ = u v , E = 1 + v 2 , F = u v , G = 1 + u 2 .
n u n = u [ m u 2 + ( m 2 ) v 2 ] ( u 2 v 2 ) ( u 2 + v 2 ) , n v n = v [ ( m 2 ) u 2 + m v 2 ] ( u 2 v 2 ) ( u 2 + v 2 ) ,
n ( u , v ) = ( u 2 v 2 ) ( m 1 ) / 2 ( u 2 + v 2 ) 1 / 2 .
A A = b 0 b 1 γ u + b 2 γ v = B ( u , v ) .
γ v B u γ u B v = 0 ,
A ( γ ) = k 1 e B ( γ ) d γ , k 1 = const .
r ( u , v ) = { a ( u 2 + v 2 1 ) u 2 + v 2 + 1 , 2 b u u 2 + v 2 + 1 , 2 c v u 2 + v 2 + 1 } ,
B ( u , v ) = b 2 c 2 u ( u 2 + v 2 ) ( b 2 c 2 ) v ( c 2 u 2 + b 2 v 2 ) ,
B ( γ ) = b 2 c 2 ( 1 + γ 2 ) ( b 2 c 2 ) γ ( c 2 + b 2 γ 2 ) .
n ( u , v ) A ( γ ) = ( c 2 + b 2 γ 2 ) 1 / 2 γ b 2 b 2 c 2 .
Γ 11 2 f v 3 + ( Γ 11 1 2 Γ 12 2 ) f v 2 f u + ( Γ 22 2 2 Γ 12 1 ) f v f u 2 + Γ 22 1 f u 3 = f v 2 f u u 2 f u f v f u v + f u 2 f v v ,
Γ 11 1 = 1 2 g ( G E u 2 F F u + F E v ) , Γ 11 2 = 1 2 g ( E E v + 2 E F u F E u ) , Γ 12 1 = 1 2 g ( G E v F G u ) , Γ 12 2 = 1 2 g ( E G u F E v ) , Γ 22 1 = 1 2 g ( G G u + 2 G F v F G v ) , Γ 22 2 = 1 2 g ( E G v 2 F F v + F G u ) .
Γ 11 2 γ 3 + ( Γ 11 1 2 Γ 12 2 ) γ 2 + ( Γ 22 2 2 Γ 12 1 ) γ + Γ 22 1 + γ γ u γ v = 0 .
W 3 = 2 g Γ 11 2 , W 2 = 2 g ( Γ 11 1 2 Γ 12 2 ) , W 1 = 2 g ( Γ 22 2 2 Γ 12 1 ) , W 0 = 2 g Γ 22 1 .
( G γ F ) n u + ( γ E F ) n v = 0 .
k g = 1 g { u ( G γ F Ψ ) + v ( γ E F Ψ ) } ,
Ψ ( u , v ) = E γ 2 2 F γ + G ,
E = 1 + ( φ ) 2 , F = 0 , G = u 2 , φ = φ ( u ) ,
v = ± k 1 + ( φ ) 2 u u 2 k 2 d u , k = const. 0 .
f ( u , v ) = v + k 1 + ( φ ) 2 u u 2 k 2 d u = c ,
γ = u u 2 k 2 k 1 + ( φ ) 2 ,
k u n u + ( u 2 k 2 ) ( 1 + ( φ ) 2 ) n v = 0 ,
n ( u , v ) = Φ ( z ) , z = k v ( u 2 k 2 ) ( 1 + ( φ ) 2 ) u d u ,
γ γ u γ v = 1 n g ( E γ 2 2 γ F + G ) [ ( G γ F ) n u + ( γ E F ) n v ] 1 2 g ( W 3 γ 3 + W 2 γ 2 + W 1 γ + W 0 ) .

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