Abstract

The application of the Hamilton-Jacobi equation to isotropic optical materials leads to the well-known eikonal equation which provides the surfaces normal to the ray trajectories. The symmetry between the coordinates x=(x 1,x 2,x 3) and the momenta p=(p 1,p 2,p 3) in the Hamiltonian formulation of Geometrical Optics establishes a dual Hamilton-Jacobi equation for “wavefronts” in the momentum space. This equation is also an eikonal equation when the refractive index distribution has spherical symmetry. In this case, another spherical symmetric refractive index distribution may exist such that the ray trajectories in the coordinates and momentum space are exchanged (examples of this case are given: Maxwell fish-eye, Eaton lens and Luneburg lens). The relationship between the wavefronts in the coordinate and momentum space is also analyzed. Curved orthogonal coordinates are considered as well.

© 2006 Optical Society of America

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References

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  3. J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach, (Cambridge University Press, 1998).
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    [CrossRef]
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  7. M. A. Alonso and G. W. Forbes, "Generalization of Hamilton's formalism for geometrical optics," J. Opt. Soc. Am. A 12, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744.
    [CrossRef]
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    [CrossRef]
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  13. H. A. Buchdahl, "Luneburg lens: unitary invariance and point characteristic," J. Opt. Soc. Am. 73, 490- (1983).
    [CrossRef]
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    [CrossRef]
  16. D. J. Struik, Lectures on Classical Differential Geometry, (Dover, New York, 1988).
  17. D. E. Blair, Inversion Theory and Conformal Mapping, (American Mathematical Society, 2000).

1991

G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

1978

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

Forbes, G. W.

G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

Am. J. Phys.

G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991).
[CrossRef]

H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys.,  46, 840-843 (1978).
[CrossRef]

Other

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

H. A. Buchdahl, "Luneburg lens: unitary invariance and point characteristic," J. Opt. Soc. Am. 73, 490- (1983).
[CrossRef]

E. W. Weisstein "Gradient." From MathWorld. http://mathworld.wolfram.com/Gradient.html.

H. A. Buchdahl, "Rays in gradient-index media: separable systems," J. Opt. Soc. Am. 63, 46- (1973).
[CrossRef]

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

D. E. Blair, Inversion Theory and Conformal Mapping, (American Mathematical Society, 2000).

Wikipedia contributors, "Canonical transformation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June 17, 2006).

O. N. Stravoudis, The Optics of Rays, Wavefronts and Caustics, (Academic, New York, 1972).

M. A. Alonso and G. W. Forbes, "Generalization of Hamilton's formalism for geometrical optics," J. Opt. Soc. Am. A 12, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744.
[CrossRef]

Wikipedia contributors, "Legendre transformation," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Legendre_transformation&oldid=66180751 (accessed May 7, 2006).

M. Born, and E. Wolf, Principles of Optics, 5th ed, (Pergamon, Oxford, 1975).

S. Cornbleet, Microwave and Geometrical Optics, (Academic, 1994).

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005).

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer-Verlag, New York, 1989).

J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach, (Cambridge University Press, 1998).

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Figures (4)

Fig. 1.
Fig. 1.

Phase space trajectories of rays in the Maxwell fish-eye lens projected on the coordinate plane x 1,x 2 (left) and projected on the momentum plane p 1,p 2 (right). These trajectories coincide with the ones of the Eaton lens if the coordinates and momenta are exchanged ( x p ). The same color of the dots indicates corresponding points of the trajectories. The bold arrow tip is the vector p and the hollow arrow tip is the vector x .

Fig. 2.
Fig. 2.

Retro reflected rays in a Eaton lens with index n=1 outside the sphere x 2=1.

Fig. 3.
Fig. 3.

Phase-space trajectories of rays with x 3=p 3=0 in the Luneburg lens; projected on the x 1, x 2 plane (left) and projected on the momentum plane p 1, p 2 (right). The bold arrow tip is vector p and the hollow arrow tip is vector x .

Fig. 4.
Fig. 4.

Trajectories of rays in the Luneburg lens in the plane x 3=0 (n=1 for x 2 >1).

Equations (25)

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d x i dt = H p i d p i dt = H x i i = 1 , 2 , 3
H ( x 1 , x 2 , x 3 , p 1 , p 2 , p 3 ) = ( n 2 ( x 1 , x 2 , x 3 ) p 1 2 p 2 2 p 3 2 ) 2
p i = V x i P i = V X i
H ( x 1 , x 2 , x 3 , V x 1 , V x 2 , V x 3 ) = 0
x i = W p i P i = W X i
H ( W p 1 , W p 2 , W p 3 , p 1 , p 2 , p 3 ) = 0
V ( x ) W ( p ) = x · p
x i = T p i X i = T P i
H ( T p 1 , T p 2 , T p 3 , p 1 , p 2 , p 3 ) = 0
V ( x , X ) T ( p , P ) = x · p X · P
( V x 1 ) 2 + ( V x 2 ) 2 + ( V x 3 ) 2 = 1 x 1 2 ( W p 1 ) 2 = 1 p 1 2 p 2 2 p 3 2
± W = 1 2 ( 1 p 2 2 p 3 2 ) arcsin ( p 1 1 p 2 2 p 3 2 ) + 1 2 p 1 1 p 1 2 p 2 2 p 3 2 + f ( p 2 , p 3 )
d x i d t = p i dp i d t = x i i = 1 , 2 , 3
d 2 x i d t 2 + x i = 0 d 2 p i d t 2 + p 1 = 0 i = 1 , 2 , 3
( x 1 x 2 x 3 p 1 p 2 p 3 ) = ( cos t sin t sin t cos t ) ( x 10 x 20 x 30 p 10 p 20 p 30 )
x i 2 + p i 2 = x i 0 2 + p i 0 2 i = 1 , 2 , 3
( x i x j 0 + p j p i 0 ) 2 + ( x i p j 0 p j x i 0 ) 2 = ( x i 0 x j 0 + p j 0 p i 0 ) 2
( x i p j 0 x j p i 0 ) 2 + ( x i x j 0 x j x i 0 ) 2 = ( x i 0 p j 0 + x j 0 p i 0 ) 2
( p i p j 0 p j p i 0 ) 2 + ( p i x j 0 + p j x i 0 ) 2 = ( x i 0 p j 0 x j 0 p i 0 ) 2
H ( ξ 1 , ξ 2 , ξ 3 , ζ 1 , ζ 2 , ζ 3 ) = n 2 ( ξ 1 , ξ 2 , ξ 3 ) ζ 1 2 h 1 2 ( ξ 1 , ξ 2 , ξ 3 ) ζ 2 2 h 2 2 ( ξ 1 , ξ 2 , ξ 3 ) ζ 3 2 h 3 2 ( ξ 1 , ξ 2 , ξ 3 )
1 h 1 2 ( ξ 1 , ξ 2 , ξ 3 ) ( V ξ 1 ) 2 + 1 h 2 2 ( ξ 1 , ξ 2 , ξ 3 ) ( V ξ 2 ) 2 + 1 h 3 2 ( ξ 1 , ξ 2 , ξ 3 ) ( V ξ 3 ) 2 = n 2 ( ξ 1 , ξ 2 , ξ 3 )
i = 1 3 ζ i 2 h i 2 ( W ζ 1 , W ζ 2 , W ζ 3 ) = n 2 ( W ζ 1 , W ζ 2 , W ζ 3 )
1 h 1 2 ( ζ 1 , ζ 2 , ζ 3 ) ( W ζ 1 ) 2 + 1 h 2 2 ( ζ 1 , ζ 2 , ζ 3 ) ( W ζ 2 ) 2 + 1 h 3 2 ( ζ 1 , ζ 2 , ζ 3 ) ( W ζ 3 ) 2 = f 2 ( ζ 1 , ζ 2 , ζ 3 )
( V ξ 1 ) 2 + ( V ξ 2 ) 2 + ( V ξ 3 ) 2 = η 2 ( ξ 1 , ξ 2 , ξ 3 )
ζ 1 2 + ζ 2 2 + ζ 3 2 = η 2 ( W ζ 1 , W ζ 2 , W ζ 3 )

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