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Advanced Photonics Research

Generalized Heterodyne Interferometry in Kerr Materials

Arnaud Rogemont, Rajath Sawant, Aurélien Coillet, Benoît Cluzel

Published Aug 28, 2025• 2 citations
Publisher

Abstract

Interferometry has long been used to measure the phase of light signals. Combined with a heterodyne detection scheme, it allows to simultaneously record amplitude and phase variations. In this work, these well‐known techniques are exploited to evaluate the nonlinear phase induced by the optical Kerr effect during the propagation of a laser pulse in a nonlinear medium. It is shown that the nonlinear index can easily be retrieved when the accumulated phase remains small, but counter‐intuitive results can be observed at higher powers. In particular, the combination of pulsed excitation and Kerr nonlinearity results in nonlinear variations of the amplitude of the interference pattern, whose shape depends on the pulse profile and the dispersion of the propagation medium.

DOI
10.1002/adpr.202500020

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Interferometry has long been used to measure the phase of light signals. Combined with a heterodyne detection scheme, it allows to simultaneously record amplitude and phase variations. In this work, these well‐known techniques are exploited to evaluate the nonlinear phase induced by the optical Kerr effect during the propagation of a laser pulse in a nonlinear medium. It is shown that the nonlinear index can easily be retrieved when the accumulated phase remains small, but counter‐intuitive results can be observed at higher powers. In particular, the combination of pulsed excitation and Kerr nonlinearity results in nonlinear variations of the amplitude of the interference pattern, whose shape depends on the pulse profile and the dispersion of the propagation medium.
Interferometry has long been used to measure the phase of light signals. Combined with a heterodyne detection scheme, it allows to simultaneously and unambiguously record amplitude and phase variations. In this work, we exploit these well-known techniques to evaluate the nonlinear phase induced by the optical Kerr effect during the propagation of a laser pulse in a nonlinear medium. We show that the nonlinear index can easily be retrieved when the accumulated phase remains small, but counter-intuitive results can be observed at higher powers.
Interferometry has long been used to measure the phase of light signals. Combined with a heterodyne detection scheme, it allows to simultaneously and unambiguously record amplitude and phase variations. In this work, we exploit these well-known techniques to evaluate the nonlinear phase induced by the optical Kerr effect during the propagation of a laser pulse in a nonlinear medium. We show that the nonlinear index can easily be retrieved when the accumulated phase remains small, but counter-intuitive results can be observed at higher powers.
In this complement we discuss several examples of optical phenomena in media where the refractive index depends nonlinearly on the intensity , known as optical Kerr media. This nonlinear effect exists in all materials, even isotropic ones, like glass or fused silica, but it is particularly marked in certain physical systems to be exemplified in Section 7B.1. After investigating the propagation of light through such media in Section 7B.2, we shall discuss three applications of the optical Kerr effect (which can be studied in any order). We begin by describing a bistable optical system , when this nonlinear medium is inserted in a Fabry–Perot cavity (Section 7B.3). We then study phase conjugate mirrors and examine their potential applications in adaptive optics (Section 7B.4). Finally, we discuss certain effects occurring during the propagation of an isolated wave, bounded either transversely or temporally, in a Kerr medium, and describe self-focusing effects (Section 7B.5) and self-phase-modulation effects (Section 7B.6). In particular, we shall show that nonlinear effects and dispersion effects can compensate to produce stable structures known as solitons , which maintain their shape during propagation. Examples of third-order nonlinearities Nonlinear response of two-level atoms We begin by studying a simple case of a nonlinear interaction, namely a two-level quantum system under the effects of a plane wave.
Accurate modeling of optical nonlinearities is crucial to describe macroscopic laser propagation in a medium, including sum frequency generation, spectral broadening due to self-phase modulation, various ionization processes and soliton formation. For incident laser light the response of the medium is given by the induced polarization of the microscopic system. The polarization is usually expanded in a Taylor series for the electric field amplitude, which is truncated after the first non-linear term being of third order for isotropic media. A third-order nonlinearity leads to the well-known optical Kerr effect, where the refractive index of the medium becomes intensity dependent via n = n0+ n2I. This leads to an inherent problem when modeling laser propagation in two or more spatial dimensions, linked to the formal divergence (n2> 0) of the refractive index for increasing intensity.
Conventional devices for measuring the amplitude of a light wave absorb the light, thus destroying it. Recently, quantum nondemolition (QND) methods based on the optical Kerr effect have been shown capable of measuring optical amplitude fluctuations with near quantum-limited sensitivity.1 In these experiments, amplitude fluctuations of a signal beam are transferred to the phase fluctuation quadrature of a probe beam which co-propagates with the signal beam, by the intensity-dependent refractive index, without absorption of the signal beam. A pure Kerr nonlinearity can cause only phase modulation of the light wave, and the amplitude of the signal wave is preserved. Such methods should more properly be called back-action evading measurement schemes since the quantity being measured is not perturbed by the back-action that quantum mechanics requires for any measurement. The signal-to-noise ratio achieved in these experiments was limited by the small magnitude of the Kerr nonlinearity relative to the excess phase noise imposed on the probe beam by thermal light scattering processes in the glass fibers that were employed.
Intense light beams propagating in a lossless, dispersionless, single-mode optical fiber are subject to the Kerr effect, i.e., to the intensity-dependent refractive index of the fiber’s fused-silica core. Classically, Kerr-effect-induced self-phase modulation (SPM) can be used for spectral broadening of a picosecond pulse for grating-pair pulse compression down to femtosecond duration. Quantum mechanically, Kerr-effect-induced four-wave mixing (FWM) has been used to produce squeezed-state light. We present a quantum propagation theory for a lossless, dispersionless fiber with the Kerr nonlinearity. The theory includes classical SPM and quantum FWM within their regions of validity. It introduces a material time constant for the Kerr interaction, limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any input field, to develop a coarse-grained time multitemporal mode field analysis. Explicit expressions are obtained for the first and the second output-field moments when the fiber’s input field is in an arbitrary Gaussian state. These results are used to obtain homodyne-detection noise spectra, which are employed, in turn, to seek experimentally accessible manifestations of the Kerr time constant.