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Optical eigenmode collapse

Michaël Mazilu

Published Apr 25, 2017• 0 citations
OpenAlex

Abstract

The optical eigenmode representation of light fields defines a natural orthogonal basis of solutions of Maxwell's equations taking into account the geometry and interactions involved in a problem. Formally, the optical eigenmodes are similar to the wave functions in quantum mechanics. Here, I put forward that, in a linear interaction at the single photon level, the electromagnetic field collapses into an optical eigenmode of the interaction in question. This is to satisfy the principle that no linear optical system can distinguish between pure states and their superposition for single photons. To exemplify this statement, we will consider the case of angular momentum and linear momentum transfer in optical scattering.

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The optical eigenmode representation of light fields defines a natural orthogonal basis of solutions of Maxwell's equations taking into account the geometry and interactions involved in a problem. Formally, the optical eigenmodes are similar to the wave functions in quantum mechanics. Here, I put forward that, in a linear interaction at the single photon level, the electromagnetic field collapses into an optical eigenmode of the interaction in question. This is to satisfy the principle that no linear optical system can distinguish between pure states and their superposition for single photons. To exemplify this statement, we will consider the case of angular momentum and linear momentum transfer in optical scattering.
When light scatters from an object, it can impart some physical quantity such as momentum or angular momentum. This can act as a measurement on the photon, which collapses on to an eigenstate of the measurement operator. However the corresponding operator is not the same as that describing the total linear or angular momentum in free space. Optical eigenmodes provide a powerful method to describe this interaction by expanding the field as a linear combination of some basis modes and examining the eigenvalues and eigenvectors of the quadratic measure in question. We extend this to the quantum case by writing the quantum operator corresponding to a given measurement such as energy, momentum or angular momentum as a superposition of creation and annihilation operators for each eigenmode. Upon measurement we find that the possible states of a single photon are simply the classical eigenmodes of the measurement. As an application, we examine the force and torque on a general, possibly anisotropic, material. By looking at eigenvalues of the measurement operator we show that the amount of a given quantity transferred in an interaction with matter is not in general the expected amount which a photon carries in free space, even at the single photon level. In particular the difference in linear or angular momentum from before and after is in general not equal to ~k or ~ which are the eigenvalues of these quantities in free space.
In linear optics, the concept of a mode or eigenmode is well established. Often these modes correspond to a set of fields that are mutually orthogonal with intensity profiles that are invariant as they propagate through a given optical system. More generally, using an eigenmode decomposition, one can define a set of orthogonal modes with respect to an optical measure given that is linear in the intensity of the fields or Hermitian in the fields themselves. However, if the intensity of the light is sufficiently large, the dipole response of an optical medium includes nonlinear terms that cause the eigenmode decomposition to break down. In this work, we introduce the eigenmode decomposition in the presence of these nonlinear source terms by introducing small perturbation fields whose interaction is mediated by some high-intensity background field. Unlike the eigenmodes of linear optics, these novel modes correspond to a set of orthogonal fields that are, in general, distributed across multiple wavelengths. Here, we study the definition and interaction of these eigenmodes for classical electromagnetic fields and multiphoton fields. In the context of classical fields, with our eigenmodes established, we highlight the influence of the high-intensity background field on the symmetry of the eigenmodes. At the multiphoton level, we show that the description of multiphoton fields is simplified by using the propagation eigenmodes while remaining equivalent to the standard approach.
One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev. A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of quantum mechanics, this wave function gives the probability to count a photon at any position in space.
Abstract In reporting and explaining the interactions of light with atoms and molecules, a photon-based description is not only appropriate but clearly essential when electronic or other kinds of quantum transition ensue. However, textbook treatments frequently go no further than the Planck relation, in representing the quantum nature of the light itself—often resorting to classical principles when discussing mechanism. Complete consistency and rigour can be achieved by treating both the matter and the radiation in a fully quantised form, which requires the electromagnetic fields to be cast in a quantum field representation. It proves possible to develop a rigorous approach to this theory that is simple to convey and apply, and which lends itself to a significantly enhanced level of appreciation of mechanism. This paper lays a concise foundation and exemplifies the application in three specific cases: absorption, emission and scattering. It is also shown how this formulation affords a basis for applications in higher-order, multiphoton and nonlinear optical processes.