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arXiv:2512.16495

Classical and quantum electromagnetic momentum in anisotropic optical waveguides

Denis Kopylov, Manfred Hammer

quant-phphysics.opticsSubmitted Dec 18, 2025• Announced Dec 19, 2025
arXivHTMLPDF

Abstract

The guided modes supported by dielectric channel waveguides act as individual carriers of momentum. We show this by proving that the modes satisfy an orthogonality condition which relates to the momentum of the optical electromagnetic field, with a link to the more familiar power (energy) orthogonality. This result forms the basis for a rigorous, self-consistent procedure for the quantization of broadband guided electromagnetic fields in the typical channels used in integrated photonic circuits. Our work removes the existing theoretical gap between the classical solution of the Maxwell equations for guided fields and the intuitive understanding of photons in waveguides. The presented approach is valid for straight, lossless, and potentially anisotropic, dielectric waveguides of general shape, in the linear regime, and including material dispersion. Examples for the hybrid modes of a thin film lithium niobate strip waveguide are briefly discussed.

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The guided modes supported by dielectric channel waveguides act as individual carriers of momentum. We show this by proving that the modes satisfy an orthogonality condition which relates to the momentum of the optical electromagnetic field, with a link to the more familiar power (energy) orthogonality. This result forms the basis for a rigorous, self-consistent procedure for the quantization of broadband guided electromagnetic fields in the typical channels used in integrated photonic circuits. Our work removes the existing theoretical gap between the classical solution of the Maxwell equations for guided fields and the intuitive understanding of photons in waveguides. The presented approach is valid for straight, lossless, and potentially anisotropic, dielectric waveguides of general shape, in the linear regime, and including material dispersion. Examples for the hybrid modes of a thin film lithium niobate strip waveguide are briefly discussed.
These notes, intended to be self contained and tutorial, present a direct, macroscopic approach to quantizing light inside a linear-response dielectric material when both spectral dispersion and spatial nonuniformity are present, but the spectral region of interest is optically transparent so that explicit treatment of the underlying physics of the medium is not needed. The approach taken is based on the macroscopic Maxwell equations and a corresponding Hamiltonian, without the use of Lagrangians or any dynamical model for the medium, and uses a standard mode-based quantization method. The treatment covers: energy density and flux in a dispersive dielectric; a summary of the inverse permittivity formalism; a new derivation of the mode normalization condition; a direct proof of the nonorthogonality of the modes; examples of quantized field expressions for the general case and various special cases; the relationship between group velocity and energy flux; the band approximation and the continuum limit; and quantum optical treatment of waveguide modes.
The electromagnetic field is quantized in a material whose dielectric function varies with frequency and one spatial dimension. The dielectric function is assumed to have a known form and to be real over the range of frequencies important for a particular application, for example the propagation of an optical signal. General properties of the mode functions are derived and employed in the quantization procedure. Expressions are obtained for the energy and momentum density and current operators, and these are shown to satisfy the appropriate conservation and continuity relations. The general formalism is illustrated by application to the examples of a homogeneous dielectric and two different dielectrics with a sharp interface.
This work deals with a hot topic of using quantum approaches in advanced telecommunications. Here, the major trend involves exploiting a flow of individual photons for signal transmission, because the photon is impossible to split, measure, copy, or imperceptibly delete. Such manipulations destroy the photon, prohibiting it to reach the recipient. Before the field quantization can be implemented, a problem of the electromagnetic field propagation needs first to be solved within classical electrodynamics, with dynamic quantities then replaced by the corresponding operators. This work describes a procedure for the electromagnetic field quantization in a waveguide with superconducting walls. In an optical fiber with inhomogeneous refractive index, the modes are shown to be non-orthogonal, so that the field can only approximately be quantized using the basis of waveguide modes. Expressions for operators of the electric and magnetic fields are derived, which may prove helpful for calculating both dynamic and other characteristics of the electromagnetic field, including nonclassical states of the electromagnetic field.
A quantum mechanical analysis of the guided light in integrated photonics waveguides is presented. The analysis is made starting from one-dimensional (1D) guided vector modes by taking into account the modal orthonormalization property on a cross section of an optical waveguide, the vector structure of the guided optical modes and the reversal-time symmetry in order to quantize the 1D vector modes and to derive the quantum momentum operator and the Heisenberg equations. The results provide a quantum-consistent formulation of the linear and nonlinear quantum light propagations as a function of forward and backward creation and annihilation operators in integrated photonics. As an illustration, an application to an integrated nonlinear directional coupler is given, that is, both the nonlinear momentum and the Heisenberg equations of the nonlinear coupler are derived.
Quantization of the phenomenological Maxwell theory of the electromagnetic field in dispersive and absorbing dielectrics has been of increasing interest over the last years since it allows for a consistent formulation of quantum optics in the presence of dielectric bodies. We show, using a Dyson-type integral representation of the Green tensor, that the quantization scheme is consistent with QED for both isotropic and anisotropic media and therefore applies to arbitrary linear Kramers-Kronig media including amplifying media.
We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.