Geometric phases in astigmatic optical modes of arbitrary order

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 wav...

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Bibliographic Details
Published in:Journal of mathematical physics 2010-08, Vol.51 (8), p.1
Main Authors: Habraken, Steven J. M., Nienhuis, Gerard
Format: Article
Language:English
Subjects:
AHARONOV-BOHM EFFECT
ANGULAR MOMENTUM
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
DEGREES OF FREEDOM
DIFFERENTIAL EQUATIONS
ELECTROMAGNETIC RADIATION
EQUATIONS
Exact sciences and technology
FUNCTIONS
GEOMETRY
HARMONIC OSCILLATORS
Mathematical methods in physics
MATHEMATICAL OPERATORS
MATHEMATICAL SOLUTIONS
MATHEMATICS
MECHANICS
OPTICAL MODES
OPTICS
ORBITAL ANGULAR MOMENTUM
OSCILLATION MODES
PARTIAL DIFFERENTIAL EQUATIONS
PHASE SHIFT
Physics
QUANTUM MECHANICS
QUANTUM OPERATORS
Quantum physics
RADIATIONS
Schrodinger equation
SCHROEDINGER EQUATION
Sciences and techniques of general use
SPACE
Studies
TRANSFORMATIONS
VISIBLE RADIATION
WAVE EQUATIONS
WAVE FUNCTIONS
WAVE PACKETS
Wavelet transforms
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Summary: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 Schrödinger 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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3456078