Abstract
Gaussian beams include a number of field or wavefunctions which have a clear quantum mechanical analogue: coherent states, correlated coherent states and discrete modes for quantum oscillators. These are used to model optical fibers and to describe the output of laser devices. When these beams leave their source and travel freely through space, they loose their coherence and aberrate. The source of this aberration is purely geometrical, and is termed spherical aberration. We describe this process in the framework of the Fermat-Hamilton formulation of optics, studying the behaviour of the center of the beam, its width, and the way in which the initial uncorrelation of position and momentum is lost.
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References
D. Marcuse, Light Transmission Optics, (Van Nostrand, New York, 1972).
K.B. Wolf, The Heisenberg-Weyl ring in quantum mechanics. In Group Theory and its Applications, Vol. III, E.M. Loebl, Ed. (Academic Press, New York, 1975).
M. Nazarathy and J. Shamir, Fourier optics described by operator algebra, J. Opt. Soc. America 70 150–158 (1980); ib. First-order optics—a canonical operator representation: lossless systems, J. Opt. Soc. America 72, 356–364 (1982).
H. Bacry and M. Cadilhac, Metaplectic group and Fourier optics, Phys. Rev. A23, 2533–2536 (1981).
J.R. Klauder, Continuous representation theory. II. Generalized relation between quantum and classical dynamics, J. Math. Phys. 5, 177–1187 (1964).
R. Glauber, Coherent states of the quantum oscillator, Phys. Rev. Lett. 10, 84 (1963).
V.V. Dodonov, E.V. Kurmyshev, and V.I. Man'ko, Generalized uncertainty relation in correlated coherent states, Phys. Lett. 79A, 150–152 (1980).
J.A. Arnaud, Beam and Fiber Optics, (Academic Press, New York, 1976).
M. Leontovich and V. Fock, Solution of the problem of electromagnetic waves along the earth suface by the method of parabolic equations, Sov. J. Phys. 10, 13 (1946).
V.I. Man'ko, Possible applications of the integrals of the motion and coherent state methods for dynamical systems. In Quantum Electrodynamics with Outer Fields (in Russian), (Tomsk State University and Tomsk Pedagogical Institute, Tomsk, USSR, 1977); pp. 101–120.
K.B. Wolf, Canonical transforms. I. Complex linear transforms, J. Math. Phys. 15, 1295–1301 (1974).
K.B. Wolf, Integral Transforms in Science and Engineering, (Plenum, New York, 1979).
M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).
H. Goldstein, Classical Mechanics, (Addison-Wesley, Reading, Mass. 1950).
M.H. Stone, Linear transformations in Hilbert space. I. Geometrical aspects, Proc. Nat. Acad. Sci. U.S. 15, 198–200 (1929); II. Analytical aspects; ibid. 423–425; III. Operator methods and group theory; ibid. 16, 172–175 (1930); J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104, 570–578 (1931).
J.H. Van Vleck, The correspondence principle in the statistical interpretation of quantum mechanics, Proc. Nat. Acad. Sci. USA, 14, 178–188 (1982).
V.V. Dodonov, V.I. Man'ko, and V.V. Semionov, The density matrix of the canonical transformed Hamiltonian in the Fock basis, P.N. Lebedev Institute of Physics preprint N54, Moscow, 1983.
V.I. Man'ko and K.B. Wolf, The influence of aberrations in the optics of gaussian beam propagation. Reporte de Investigación del Depto. de Matemáticas, Universidad Autónoma Metropolitans, preprint 4, # 3 (1985).
G. Eichmann, Quasi-geometric optics of media with inhomogeneous index of refraction, J. Opt. Soc. America 61, 161–168 (1971).
M. GarcÃa-Bullé, W. Lassner, and K.B. Wolf, The metaplectic group within the Heisenberg-Weyl ring. Reporte de Investigación del Depto. de Matemáticas, Universidad Autónoma Metropolitans, preprint 4, # 1 (1985); to appear in J. Math. Phys.
K.B. Wolf, A euclidean algebra of hamiltonian observables in Lie optics, Kinam 6, 141–156 (1985).
J.W. Goodman, Introduction to Fourier Optics, (Mc Graw-Hill, New York, 1968), Section 2–3.
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Man'ko, V.I., Wolf, K.B. (1986). The influence of spherical aberration on gaussian beam propagation. In: Sánchez Mondragón, J., Wolf, K.B. (eds) Lie Methods in Optics. Lecture Notes in Physics, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16471-5_8
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DOI: https://doi.org/10.1007/3-540-16471-5_8
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