The steady-state response of a nonlinear medium to several simultaneously applied monochromatic electromagnetic fields is described by the density matrix method. A Fourier series expansion in terms of ascending powers of the amplitudes of the applied fields is especially useful to describe the parametric response in spectral regions, where the absorption is small. As the resonances of the material system are approached, the general formalism exhibits the mixture and interference between parametric processes and single- and multiple-photon absorption and emission processes. previously discussed examples of the two-level and three-level system are generalized. The reaction of the general nonlinear medium on the electromagnetic fields is incorporated.
The steady-state response of a nonlinear medium to several simultaneously applied monochromatic electromagnetic fields is described by the density matrix method. A Fourier series expansion in terms of ascending powers of the amplitudes of the applied fields is especially useful to describe the parametric response in spectral regions, where the absorption is small. As the resonances of the material system are approached, the general formalism exhibits the mixture and interference between parametric processes and single- and multiple-photon absorption and emission processes. previously discussed examples of the two-level and three-level system are generalized. The reaction of the general nonlinear medium on the electromagnetic fields is incorporated.
A diagrammatic approach to the theory of nonlinear susceptiblities is developed. Including damping, the steady state nonlinear response functions to several simultaneously applied monochromatic electromagnetic fields are obtained in terms of linear response functions in the weak coupling system. Saturation phenomena in microwave resonance and in parametric processes can be understood as lower order responses in the system with higher spin temperature. The condition for the Raman emission is discussed in some details. Results are compared with previous theories, in particular with that of Bloembergen and Shen. The theory is extended to the case where the spacially inhomogeneous perturbation is imposed.
We have incorporated spectral diffusion in a system of three quantum states immersed in a thermal bath for the study of optical responses in four-wave mixing spectroscopy, using the density matrix formalism and the development of quantum relaxation theory. The primary objective of this work is to establish a generalized theoretical framework that evaluates how internal relaxation within all three quantum states—including a non-resonant third state—modifies the macroscopic nonlinear polarization of the medium. By introducing specific scaling parameters, we aim to isolate and quantify the influence of spectral diffusion in the third state, a feature typically omitted in existing literature. The results found allow giving explanations about the parametric amplification, as well as the possibility of generation of sensors in regions close to the degeneracy of optical frequencies. In our study we have been able to recognize previously published results for the optical susceptibility responses of absorption and scattering. Our model, unlike those already presented in this topic, assumes all states of the quantum system with internal relaxations. We found that the inclusion of spectral diffusion in the third state significantly impacts the system’s kinetic behavior, specifically eliminating “hole burning” effects in dispersion and parametric amplification in nonlinear absorption. These findings suggest that interband and intraband kinetic constants must be considered simultaneously to accurately model the ultra-fast relaxation times in complex molecular systems like organic dyes. The development of the problem is subject to rotating wave considerations in the selection of the Fourier component terms associated with both the coherences induced in the system and those coming from the canonical populations of the states. In addition, it involves the development of perturbation theory in the amplitudes of the incident electromagnetic fields associated with this nonlinear FWM type technique. Our treatments have been performed for frequency differences in the absorption range of organic dyes, without any restriction in the cases of near-zero tunings. Our proposal for the third state with spectral diffusion effects completely eliminates both the “hole burning” created in the dispersion and the parametric amplification generated in nonlinear absorption, when compared with results with the state outside resonance treated without spectral diffusion effects within the band. This highlights the importance of interband kinetic constants in comparison with those within the band.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
General expressions for the steady-state population distribution and complex susceptibilities of a three-level system interacting with two strong near-resonant electromagnetic fields are obtained using a density-matrix formalism. A complete treatment of the steady-state population distribution established in the three-level system, under the action of two strong fields with arbitrary resonance detunings is presented. The criterion for maximal populations of the exciting levels, as well as conditions in which the population of the exciting levels exceeds the population of the ground state are found. The modification of the absorption and emission lines with increasing intensity of the second (probe) field is considered. The emphasis in the present treatment, as distinct from Panock and Temkin (see IEEE J. Quantum Electron., vol.QE-13, p.476 (1977)) is on the consideration of the arbitrary distribution of the equilibrium populations of the system which exhibits a number of interesting features which have not been previously discussed.
In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.