Subluminal and superluminal velocities of free-space photons

Abstract

We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals $c^2$. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.

Figures (3)

• Figure 1: Schematics of a localized electromagnetic wave packet propagating along the $z$-axis: (a) its plane-wave spectrum (momentum-space distribution) and (b) its real-space picture. The wavepacket is characterized by a subluminal group velocity $v_{g} < c$ and a superluminal phase velocity $v_{ph} > c$, such that $v_g v_{ph} = c^2$.
• Figure 2: (a) Propagation of a Gaussian-like wavepacket obtained from the exact solution of the scalar wave equation Vo2024JO [Eq. (3) therein], constructed as a superposition of Gaussian-like beams with a common Rayleigh range $z_R$ and frequencies distributed according to a Poisson-like spectrum $\propto \omega^s \exp(-s\omega/\omega_0)$ centered at $\omega_0 = ck_0$. The parameters are $k_0 z_R =10$ and $s=20$. The density plots show the intensity distributions $|\psi(x,y=0,z,t)|^2$ at different times, whereas the red dot marks propagation with the subluminal group velocity \ref{['group_velocity_beam']}. (b) Numerically calculated retardations of the wavepacket centroid \ref{['wavepacket_centroid']}, $Z_c -ct$, for the exact Vo2024JO and paraxial Caron1999JMO wavepacket solutions with similar parameters, compared with the theoretical prediction based on the group velocity \ref{['group_velocity_beam']}.
• Figure 3: Phase distribution in a $z$-propagating Gaussian beam \ref{['Gaussian_beam']} with $kw_0 = 5$ (shown in the $x>0$ half-plane) and in a plane wave with the same $k$ (shown in the $x<0$ half-plane). The larger spacing between phase fronts in the Gaussian beam corresponds to a superluminal phase velocity, Eq. \ref{['phase_velocity_beam']}.