Hamilton-Jacobi equation in momentum space

doi:10.1364/OE.14.009083

Optics Express

Editor: Michael Duncan
Vol. 14, Iss. 20 — Oct. 2, 2006
pp: 9083–9092

Hamilton-Jacobi equation in momentum space

Juan C. Miñano, Pablo Benítez, and Asunción Santamaría »View Author Affiliations

Optics Express, Vol. 14, Issue 20, pp. 9083-9092 (2006)
http://dx.doi.org/10.1364/OE.14.009083

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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 =(x₁,x₂,x₃) and the momenta p =(p₁,p₂,p₃) 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.

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(080.2720) Geometric optics : Mathematical methods (general)
(080.2740) Geometric optics : Geometric optical design

ToC Category:
Geometric Optics

History
Original Manuscript: June 30, 2006
Revised Manuscript: August 9, 2006
Manuscript Accepted: September 4, 2006
Published: October 2, 2006

Citation
Juan C. Miñano, Pablo Benítez, and Asunción Santamaría, "Hamilton-Jacobi equation in momentum space," Opt. Express 14, 9083-9092 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9083

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References

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).
G. W. Forbes, "On variational problems in parametric form," Am. J. Phys. 59, 1130-1140 (1991). [CrossRef]
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).
H. A. Buchdahl, "Kepler problem and Maxwell fish-eye," Am. J. Phys., 46, 840-843 (1978). [CrossRef]
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).

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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