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Classical and quantum electromagnetic momentum in anisotropic optical waveguides

Denis Kopylov, Manfred Hammer

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.

Classical and quantum electromagnetic momentum in anisotropic optical waveguides

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.

Paper Structure

This paper contains 6 sections, 3 theorems, 69 equations, 3 figures.

Table of Contents

  1. Mathematical identities
  2. Maxwell equations in the frequency domain for straight dielectric waveguides
  3. Derivation of orthogonality conditions
  4. Energy orthogonality condition
  5. Momentum orthogonality condition
  6. Connection between energy and momentum orthogonality conditions

Key Result

Lemma 1

For a transverse-evanescent field $\mathbf{X} (\mathbf{r} )$ with $\lim_{|\mathbf{r} _\perp|\to\infty} \mathbf{X} (\mathbf{r} )=0$, the following equality holds: where $\mathbf{r} _\perp \equiv (x,y)$, $d\mathbf{r} _\perp \equiv dxdy$ and $X_z(\mathbf{r} )\equiv \mathbf{X} (\mathbf{r} ) \cdot \mathbf{v} _z$.

Figures (3)

  • Figure 1: (a) Optical channel WG with translational invariance along its axis $z$. (b) Schematic illustration of an optical pulse in a WG, propagating through the plane at $z_0$. It transfers the energy $F(z_0, t_0, t_1)=F(z_0)$ and momentum $G(z_0, t_0, t_1)=G(z_0)$. In the quantum case, the minimal transferred excitation corresponds to a single-photon pulse.
  • Figure 2: Results for an X-cut TFLN strip WG. (a) WG schematic; crystal and channel coordinates are $\{X, Y, Z\}$ and $\{x, y, z\}$, respectively. (b) WG cross-section; parameters: $t = 0.6\,\upmu\hbox{\rm m}$, $c = 1\,\upmu\hbox{\rm m}$, $w = 0.3445\,\upmu\hbox{\rm m}$, sidewall angle $\phi = 30^\circ$, orientation of the channel axis $z$ at angle $\theta = 30^\circ$ versus the crystal $Y$-axis, dispersion for SiO$_2$ ($n_{\hbox{\scriptsize{\rm s}}}$ and $n_{\hbox{\scriptsize{\rm c}}}$) and TFLN media (tensor permittivity $\tensor{\varepsilon}_{\hbox{\scriptsize{\rm f}}}$), with air cover ($n_{\hbox{\scriptsize{\rm a}}}$) are as in Refs. EFH_2022EFH_2023Herzinger_1998. (c) Effective indices $\eta_m(\omega) = k_m(\omega)c/\omega$ of guided modes as a function of vacuum wavelength $\lambda = 2\pi c/\omega$.
  • Figure 3: Electric profile components of hybrid (a) first and (b) second order modes at $\lambda = 1.55\,\upmu\hbox{\rm m}$, colormap: black-negative, blue-zero, white-positive.

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof