Energy conservation in self-phase modulation

TL;DR

This paper investigates the apparent energy conservation paradox in self-phase modulation (SPM) by analyzing transient energy storage in the medium, reconciling it with the spectral broadening phenomena observed in ultrashort laser pulses.

Contribution

It introduces a framework to understand energy conservation in SPM through transient energy storage, resolving a longstanding paradox with a new theoretical approach.

Findings

Transient energy storage explains spectral broadening in SPM.

Dephasing leads to an effective imaginary susceptibility component.

Temporal intensity variation in FWM offsets frequency shifts.

Abstract

Spectral broadening of ultrashort laser pulses is simultaneously described by either self-phase modulation (SPM) or four-wave mixing (FWM). The latter implies the instantaneous conservation of both the photon number and energy, while the former describes a time-dependent frequency shift, implying a violation of the energy conservation if the number of photons is to be conserved in each time slice. We resolve this paradox by considering the transient energy storage in the propagation medium, that can be calculated in the SPM formalism via the dephasing between the incident pulse and the medium polarization leading to an effective imaginary part in the third-order susceptibility. In parallel, considering the temporal variation of the incident intensity in FWM offsets the instantaneous frequency.