The continuum limit of k-space cavity angular momentum is controlled by an infinite range difference operator

TL;DR

This paper investigates how the angular momentum of wavepackets in k-space transitions from a non-local discrete sum to a localized continuum integral, revealing the role of an infinite range difference operator in this process.

Contribution

It introduces an infinite range difference operator that controls the continuum limit of k-space cavity angular momentum, bridging discrete and continuous analyses.

Findings

Discrete k-space analysis shows non-local angular momentum distribution.

Continuum limit involves an infinite range difference operator.

Conditions identified for the discrete sum to approximate the localized continuum distribution.

Abstract

A wavepacket (electromagnetic or otherwise) within an isotropic and homogeneous space can be quantized on a regular lattice of discrete k-vectors. Each k-vector is associated with a temporal frequency omega; together, k and omega represent a propagating plane-wave. While the total energy and total linear momentum of the packet can be readily apportioned among its individual plane-wave constituents, the same cannot be said about the packet's total angular momentum. One can show, in the case of a reasonably smooth (i.e., continuous and differentiable) wave packet, that the overall angular momentum is expressible as an integral over the k-space continuum involving only the Fourier transform of the field and its k-space gradients. In this sense, the angular momentum is a property not of individual plane-waves, but of plane-wave pairs that are adjacent neighbors in the space inhabited by the k-vectors, and can be said to be localized in the k-space. Strange as it might seem, this hallmark property of angular momentum does not automatically emerge from an analysis of a discretized k-space. In fact, the discrete analysis shows the angular momentum to be distributed among k-vectors that pair not only with nearby k-vectors but also with those that are far away. The goal of the present paper is to resolve the discrepancy between the discrete calculations and those performed on the continuum, by establishing the conditions under which the highly non-local sum over plane-wave pairs in the discrete k-space would approach the localized distribution of the angular momentum across the continuum of the k-space.