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Annual Meeting Optical Society of America

Quantum mechanical Fresnel relations

Jagdish Rai

Published Jan 1, 1991• 0 citations
Publisher

Abstract

Several authors1,2 have studied the lossless beam splitter on a quantum mechanical basis. In this approach the action of the beam splitter is described by unitary transformations that act on the operators for the radiation fields present. In this paper we study quantum mechanically the general case of reflection and transmission from a boundary. We obtain the quantum analog of the classical Fresnel relations. We find that the general operators that describe reflection and transmission at a boundary are a series of unitary operators involving appropriate radiation fields. The generalization for the nonlinear media presents many interesting situations where the classical laws of reflection and refraction are modified. This provides an interesting description of phase conjugation.

DOI
10.1364/oam.1991.thf2

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Several authors1,2 have studied the lossless beam splitter on a quantum mechanical basis. In this approach the action of the beam splitter is described by unitary transformations that act on the operators for the radiation fields present. In this paper we study quantum mechanically the general case of reflection and transmission from a boundary. We obtain the quantum analog of the classical Fresnel relations. We find that the general operators that describe reflection and transmission at a boundary are a series of unitary operators involving appropriate radiation fields. The generalization for the nonlinear media presents many interesting situations where the classical laws of reflection and refraction are modified. This provides an interesting description of phase conjugation.
The lossless beam splitter is classically well understood. The purpose of this work is to enable one to describe the action of a lossless plane-parallel beam splitter on an incident light beam in purely quantum-mechanical terms. Such a description may be obtained in terms of a unitary transformation that connects the incoming radiation-field operator to the outgoing one. This transformation, which we shall derive and discuss here, provides an elegant means to address issues and concepts that are intrinsically quantum-mechanical.
Using quantum theory, we study the propagation of an optical field in an inhomogeneous dielectric, and apply this scheme to traveling optical fields in a waveguide. We introduce a field-atom interaction Hamiltonian and derive the refractive index using quantum optics. We show that the transmission and reflection of optical fields at an interface between different materials can be described with normalized Fresnel coefficients and that this representation is related to the beam splitter operator. We then study the propagation properties of the optical fields for two types of slab waveguides: step-index and graded-index. The waveguides are divided into multiple layers to represent the spatial dependence of the optical field. We can evaluate the number of photons in an arbitrary volume in the waveguide using this procedure. Using the present method, the quantum properties of weak optical fields in a waveguide are revealed, while coherent states with higher amplitudes reduces to representation of classical waveguide optics.
A quantum theory of the lossless beam splitter is given in terms of the quasi mode theory of macroscopic canonical quantization used to treat problems in cavity quantum electrodynamics and quantum optics. A Heisenberg picture approach to quantum scattering theory is applied, in which the input and output operators that are related via the scattering operator are linked to quantum optical measurements described via multi-time quantum correlation functions. In the application to the beam splitter the Heisenberg equations of motion for the input operators associated with the quasi mode annihilation operators are formally solved in a rotating picture to show that the unitary transform of the incident quasi mode annihilation operator (via the scattering operator) is just a linear combination of the incident and reflected quasi mode annihilation operators, in accordance with assumptions made in previous treatments of the beam splitter. The results depend on conservation of the transverse component of the wave vector, which follows from the form of the quasi mode-quasi mode coupling constants, and on conservation of the unperturbed energy, which follows from scattering theory. The applicability of quantum scattering theory to the beam splitter is justified in the usual situation where integrated one photon and two photon detection rates are finite for incident light field states of interest.
Abstract When a light wave is reflected at the interface between two semi-infinite media, the relative phase between the reflected and incident amplitudes is a fundamental optical effect. Another fundamental phase relation arises in a symmetric beam splitter between its reflection and transmission amplitudes. Both effects originate from the phase structure of wave scattering, although their underlying connection is not always clearly understood. As a pedagogical investigation, this paper systematically analyzes these phase relations using simple one-dimensional potential models in quantum mechanics and provides intuitive physical explanations. For a symmetric square barrier, the reflected and transmitted amplitudes necessarily differ in phase by ±π/2, with the reflection phase referenced at the interface on the incident side and the transmission phase referenced at the interface on the exit side. The result is further extended to general symmetric barriers, for which the detailed form of the barrier determines only whether the phase difference is +π/2 or -π/2. The role of this phase structure in the second-order interference of two identical particles is then examined. Through this sequence of analysis, half-wave loss at a reflecting interface, the phase relation in a symmetric beam splitter, and multiparticle quantum interference are incorporated into a unified wave-scattering framework. This treatment provides a useful pedagogical reference for related topics in quantum mechanics and optics courses.
This is a review of a very interesting (in the authors' view) phenomenon — the operation of a nonlinear light beam splitter. The beam splitter is a flat interface between two transparent dielectrics, at least one of which exhibits Kerr nonlinearity, i.e., its refractive index depends on the transmitted radiation intensity. Interestingly, quantum and classical theories make directly opposite predictions about the phase fluctuations of the output radiation of this device. In classical theory, the phases remain unchanged; in quantum theory, the phases fluctuate in accordance with the amplitude–phase uncertainty relation. The origin of this difference is established at the fundamental level. A further remarkable point about this quantum paradox is that not only is the source beam split in two but one can also create conditions where the two split parts are respectively dominated by amplitude noise and phase noise, thus allowing the selection of photon fluctuations. Results of original studies are summarized and further developed.