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Photonics

Generation of Coherent and Spatially Squeezed States of an Electromagnetic Beam in a Planar Inhomogeneous Dielectric Waveguide

Anna V. Bogatskaya, Andrey E. Schegolev, N. V. Klenov, A. M. Popov

Published Jul 30, 2019• 3 citations
Publisher

Abstract

We use slow-varying amplitude approximation (SVA) for the wave equation to study both analytically and numerically propagation of an electromagnetic beam in the waveguide structure with parabolic susceptibility spatial dependence. Such a structure is similar to the harmonic oscillator in quantum mechanics. We analyze this structure as a single mode guide and introduce the notion of number of “photons” in the mode. In particular, we pay special attention to the possibility of effective build-up of the coherent and spatially squeezed vacuum states of the mode that can be of interest for a number of practical applications. The way to provide these types of mode excitation is suggested. Several applications for controlling the mode composition of an electromagnetic wave in the parabolic index-gradient waveguide for various frequency ranges are considered.

DOI
10.3390/photonics6030084

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We use slow-varying amplitude approximation (SVA) for the wave equation to study both analytically and numerically propagation of an electromagnetic beam in the waveguide structure with parabolic susceptibility spatial dependence. Such a structure is similar to the harmonic oscillator in quantum mechanics. We analyze this structure as a single mode guide and introduce the notion of number of “photons” in the mode. In particular, we pay special attention to the possibility of effective build-up of the coherent and spatially squeezed vacuum states of the mode that can be of interest for a number of practical applications. The way to provide these types of mode excitation is suggested. Several applications for controlling the mode composition of an electromagnetic wave in the parabolic index-gradient waveguide for various frequency ranges are considered.
Abstract Here we use approaches and concepts from quantum mechanics in order to analyze the propagation of classical electromagnetic waves in the elements of integrated optical circuits. These waveguide circuits are considered as potential wells between complex-shapes barriers formed by opaque media. This allowed us to construct in the framework of the slow-varying amplitude approximation for the wave equation an analogy between coherent oscillations in a quantum system and the redistribution in space of the field strength of a classical wave. We also demonstrated the methods to control the mode composition of a classical light in a spatially inhomogeneous waveguide structure. The proposed description was based on the analogy with Rabi-type oscillations in quantum mechanics.
For pt.I see ibid., vol.10, p.136-43 (1989). In the preceding paper the authors used the Lagrangian formalism to show the close relationship between the classical mechanics of the harmonic oscillator and the ray optics of the parabolic index optical waveguide. The authors have now studied the corresponding quantum mechanical and wave optical problems and, in particular, have studied the time evolution of a coherent state and have shown its close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide. The corresponding transitions to classical mechanics and ray optics have been shown. The authors have also presented an analysis of the recently developed squeezed states and have shown its correspondence to the waveguide problem. They have demonstrated that the relationship between classical and quantum mechanics is the same as between ray optics and wave optics. All such analyses should be of considerable pedagogical value.
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.
The authors discuss the harmonic oscillator problem in the domains of classical mechanics and the parabolic index optical waveguide problem in the domains of ray optics. While discussing the classical mechanics and the ray optics of the two problems the authors have used the Lagrangian formalism which allows them to see the close relationship between the two problems. In part II of the paper the authors study the corresponding quantum mechanical and wave optical problems. In particular whilst looking at the time evolution of a coherent state the authors demonstrate that it has a close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide.
We derive c-number Bloch-Maxwell (B-M) type equations from a quantum theory for 2-level atoms occupying a macroscopic slab-like region V of width L stimulated by a coherent-state electromagnetic field of radial frequency (omega) plus a broad-band correlated squeezed vacuum field with carrier frequency (omega) p. The squeezed vacuum field is in principle of arbitrary 3-dimensional geometry including geometries created by an optical parametric oscillator. The c-number spatially dependent B-M equations are solved self-consistently and 'exactly' in terms of a nonlinear dispersive refractive index m((omega) ) while the slab V acts, also self-consistently, as a natural Fabry-Perot cavity of finite spacing L. Very preliminary results for optical bistability in the squeezed vacuum and more exact results for anomalous dispersion and absorption in the squeezed vacuum in linearized approximation are reported. Finally we compute the effect of the dielectric on the free-field modes of the squeezed vacuum.
A new approach to investigating a broad class of dynamic states for a quantum oscillator is suggested. It is based on an invariant transformation of the equation to a new time determined by the quantum dispersion of the corresponding state. The squeezed states of a quantum system generated by the ground-state wave function are constructed. In coordinate representation, these states are described by a self-similar wave function localized near a classical trajectory. The statistics of the squeezed state of light is analyzed in the single-mode approximation. The parametric excitation of squeezed states for a quantum harmonic oscillator is considered.