The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing fields are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric fields, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcar\'e sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action flux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.
The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing fields are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric fields, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcar'e sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action flux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.
Abstract This tutorial provides a joint theoretical and experimental overview of heterodyne wave mixing spectroscopy, focusing mainly on four-wave mixing (FWM). This powerful and versatile time-resolved nonlinear optical spectroscopy technique enables the investigation of individual localized single photon emitters, as well as microscopy of extended samples, e.g. two-dimensional transition metal dichalcogenides. Starting with the fundamental theory of optically driven two-level systems, we motivate the utility of wave mixing spectroscopy via a discussion on homogeneous and inhomogeneous linewidths which can be independently measured using FWM. We then provide a detailed overview of the heterodyne wave mixing setup operated by one of the authors (JK) at Institut Néel in Grenoble, supported by theoretical modeling of the signal detection process. Throughout the paper we elaborate on important benefits of heterodyne wave mixing spectroscopy, e.g. background-free detection, measurement of the full signal field including amplitude and phase, and investigation of coupling mechanisms in few-level systems. Within the context of the latter point we discuss the significance of two-dimensional FWM spectra. This tutorial is dedicated to students, young researchers, as well as experts in the field of nonlinear spectroscopy in general and FWM in particular. It explains the fundamental concepts and building blocks required to operate a heterodyne wave mixing experiment both from the experimental and theoretical side. This joint approach is helpful for theoreticians who want to accurately and quantitatively model wave mixing signals, as well as for experimentalists who aim to interpret their recorded data.
Optical phase conjugation is an important nonlinear process, with many applications in the areas of optical communications and optical processing. This is as a result of the wave-front correction properties that phase conjugation offers and this has generated much interest in the area. A common method of producing phase conjugated wavefronts is the four-wave mixing interaction and many publications on the steady state solution to this problem have appeared over the last decade. More recently, however, interest has been focussed upon the temporal behaviour of four-wave mixing systems, with instabilities and chaos being both demonstrated by Gauthier et al [1] and predicted by Królikowski et al [2]. To date, the analysis for the temporal behaviour of anisotropic four wave mixing, in the transmission grating regime, has involved the direct numerical integration of the equations describing four wave mixing. This analysis however, does not utilise the symmetries of the four wave mixing process, and thus results in a more complex problem. Through exploitation of these symmetries the complexity of the problem has been reduced from one containing four complex variables, to one containing three real variables. This has been accomplished through the use of the Special Unitary Group 2, a group providing a two dimensional matrix, whose elements are functions of three real variables. The multiplication of this matrix, together with one containing the boundary conditions for the problem, thus enables the four complex beam amplitudes to be reexpressed in terms of three real quantities. Using this technique, the temporal nature of anisotropic four wave mixing has been studied, and shown under certain conditions, to exhibit chaotic behaviour when an electric field is present. The effect of any material absorption on the chaotic nature is also demonstrated.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.
A local-field theory describing optical phase conjugation in condensed media in the special case of degenerate four wave mixing is established. The aim of the theory is to form the framework for microscopic studies of optical phase conjugation (i) of evanescent fields in the context of near-field optics, (ii) in mesoscopic films and quantum wells, (iii) in small particles, and (iv) in lossy media where the field penetration depth is comparable to or (substantially) less than the vacuum wavelength of the driving field. The aforementioned goal makes it necessary to abandon both the slowly varying envelope- and the electric dipole approximations usually adopted in phase conjugation studies where spatially slowly decaying or modulated fields are mixed. By keeping in the interaction Hamiltonian the term of second order in the vector potential and in the current-density operator the term of first order in the vector potential new microscopic field-matter interaction processes of particular importance in the present context are included. The physics of the various nonlinear microscopic processes is analysed, and systematised by presenting in diagrammatic form the nonlocal electrodynamics hidden in the nonlinear constitutive relation. A new nonlocal conductivity tensor, enabling one to describe the degenerate four wave mixing process among the prevailing local fields, is presented and the eigensymmetries of its various parts are analysed. Starting from the general local-field theory a degenerate four wave mixing response tensor of relevance for media exhibiting two-dimensional translational invariance is established and discussed. In the last part of the paper an integral equation allowing one to obtain the phase conjugated local field inside and outside the nonlinear medium is established, discussed and formally solved, and it is pointed out that this equation for systems with two-dimensional translational invariance often can be analysed analytically and numerically using methods previously developed in theoretical studies of the linear local-field electrodynamics of mesoscopic films.
The application of the adiabatic geometric phase (AGP) to nonlinear frequency conversion may help to develop new types of all-optical devices, which leads to all-optical modulation of the phase front of one wave by the intensity of other waves. In this paper, we develop the canonical Hamilton equation and a corresponding geometric representation for two schemes of four-wave mixing (FWM) processes (ω1 + ω2 = ω3 + ω4 and ω1 + ω2 + ω3 = ω4), which can precisely describe and calculate the AGP controlled by the quasi-phase matching technique. The AGPs of the idler (ω1) and signal (ω4) waves for these two schemes of FWM are studied systematically when the two pump waves (ω2 and ω3) are in either the undepleted or in the depleted pump cases, respectively. The analysis reveals that the proposed methods for calculating the AGP are universal in both cases. We expect that the analysis of AGP in FWM processes can be applied to all-optically shaping or encoding of ultrafast light pulse.
A large-signal theory of noncollinear degenerate four-wave mixing (DFWM) is presented, taking into account the pump depletion and the full nonlinear polarization density appropriate for the noncollinear geometry. Phases of all of the interacting fields, as well as the amplitudes, are obtained as functions of the spatial coordinate. Owing to the completeness of the set of analytic solutions, considerable physical insights into the physics of DFWM are obtained. The DFWM exhibits a variety of rich behaviors, including hysteresis and bistability, multistability, and multiple isolated branches. The authors show that all of these behaviors can be explained in a unified way, that is, in terms of mode hopping characteristics.>