Geometric phases in astigmatic optical modes of arbitrary order

Journal Article · Sun Aug 15 00:00:00 EDT 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3456078· OSTI ID:21476536

Habraken, Steven J. M.; Nienhuis, Gerard ^[1]

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1. Leiden Institute of Physics, P.O. Box 9504, 2300 RA Leiden (Netherlands)

The transverse spatial structure of a paraxial beam of light is fully characterized by a set of parameters that vary only slowly under free propagation. They specify bosonic ladder operators that connect modes of different orders, in analogy to the ladder operators connecting harmonic-oscillator wave functions. The parameter spaces underlying sets of higher-order modes are isomorphic to the parameter space of the ladder operators. We study the geometry of this space and the geometric phase that arises from it. This phase constitutes the ultimate generalization of the Gouy phase in paraxial wave optics. It reduces to the ordinary Gouy phase and the geometric phase of nonastigmatic optical modes with orbital angular momentum in limiting cases. We briefly discuss the well-known analogy between geometric phases and the Aharonov-Bohm effect, which provides some complementary insights into the geometric nature and origin of the generalized Gouy phase shift. Our method also applies to the quantum-mechanical description of wave packets. It allows for obtaining complete sets of normalized solutions of the Schroedinger equation. Cyclic transformations of such wave packets give rise to a phase shift, which has a geometric interpretation in terms of the other degrees of freedom involved.

• OSTI ID: 21476536
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 8; Other Information: DOI: 10.1063/1.3456078; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
AHARONOV-BOHM EFFECT
DEGREES OF FREEDOM
GEOMETRY
HARMONIC OSCILLATORS
MATHEMATICAL SOLUTIONS
OPTICAL MODES
OPTICS
ORBITAL ANGULAR MOMENTUM
PHASE SHIFT
QUANTUM MECHANICS
QUANTUM OPERATORS
SCHROEDINGER EQUATION
SPACE
TRANSFORMATIONS
VISIBLE RADIATION
WAVE FUNCTIONS
WAVE PACKETS
ANGULAR MOMENTUM
DIFFERENTIAL EQUATIONS
ELECTROMAGNETIC RADIATION
EQUATIONS
FUNCTIONS
MATHEMATICAL OPERATORS
MATHEMATICS
MECHANICS
OSCILLATION MODES
PARTIAL DIFFERENTIAL EQUATIONS
RADIATIONS
WAVE EQUATIONS