The resonant nonlinearity

we express the operator of the electron-hole interband polarization Pw in terms of the electron and hole creation and annihilation operators in the envelope function approximation, following the standard procedure (18), (39)  

Here xe(z)- Xh(z) are electron and hole wavefunctions in the given IQW subbands (resonant with the FE), Ck and hk are annihilation operators for an electron and hole with the in-plane wavevector k in the subbands under consideration, S is the in-plane normalization area and dvc is the matrix element (28). We do not take into account the spin degeneracy, considering thus the polarization produced by electrons and holes with a given spin (thus, the final expression for the susceptibility should be multiplied by two). An analogous expression for the OQW polarization is 

Given the Hamiltonian, we can write the equations of motion for the Heisenberg operators. The polarization is obtained by averaging the expressions (13.51), (13.52) over the equilibirium density matrix. The result is expressed in terms of the polarization functions 

Average values of the four-operator terms are factorized in the Hartree-Fock approximation and are expressed in terms of the polarization functions and the populations defined by 

Here the averages with different wavevectors correspond to the intraband polarization, which is far off resonance and may be neglected. Since the electric field excites only states with the given total in-plane wavevector Qy, from now on we set k = Q. As a result, we obtain the equations for the polarization functions  

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The proper phase shift introduced to the signal wave allows direct measurements of the imaginary part of the resonant nonlinear optical susceptibility, i.e ... 

This phase shift is akin to the phase shift experienced by a damped harmonic oscillator driven in the vicinity of the oscillator s eigenfreqnency Hence, the presence of a vibrational resonance not only changes the amplitnde of the signal field, bnt also its phase. To incorporate this effect, the resonant nonlinear susceptibility is no longer real as it contains imaginary contributions ... 

A formal expression for the resonant nonlinear susceptibility can be obtained by describing the light-matter interactions in a density matrix formalism (Boyd 2003 Mukamel 1995), which is beyond the scope of this chapter. A third-order perturbative expansion of the system s density matrix yields the following form for the nonlinear susceptibility ... 

In the above discussion, we have only considered the effects due to the CTE-CTE repulsion, which contribute to the resonant nonlinear absorption (as well as to other resonant nonlinearities) by the CTE themselves. Here, however, we want to mention a more general mechanism by which the nonlinear optical properties of media containing CTEs in the excited state can be enhanced. This influence is due to the strong static electric field arising in the vicinity of an excited CTE, If, for example, the CTE (or CT complex) static electric dipole moment is 20 Debye, at a distance of 0.5 nm it creates a field Ecte of order 107 V/cm. Such strong electric fields have to be taken into account in the calculation of the nonlinear susceptibilities, because they change the hyperpolarizabilities a, / , 7, etc. of all molecules close to the CTE. For instance, in the presence of these CTE induced static fields, the microscopic molecular hyperpolarizabilities are modified as follows... 

One subject that attracted much attention is the nonlinear optical properties of these semiconductor nanoclusters [17], The primary objective is to find materials with exceptional nonlinear optical response for possible applications such as optical switching and frequency conversion elements. When semiconductors such as GaAs are confined in two dimensions as ultrathin films (commonly referred to as multiple quantum well structures), their optical nonlinearities are enhanced and novel prototype devices can be built [18], The enhancement is attributed mostly to the presence of a sharp exciton absorption band at room temperature due to the quantum confinement effect. Naturally, this raises the expectation on three-dimensionally confined semiconductor nanoclusters. The nonlinearity of interest here is the resonant nonlinearity, which means that light is absorbed by the sample and the magnitude of the nonlinearity is determined by the excited state... 

Different parameters are required to characterize the resonant and the nonresonant optical nonlinearity. This has often been a source of confusion in the literature, even today. For nonresonant processes, the magnitude of the nonlinearity is measured by either x(3) or n2. However, for resonant processes, x(3)> a2> or ni alone cannot measure the magnitude of the nonlinearity. For example, a different x(3) value can result from the same material when lasers with different pulse widths are used for the measurement. A complete characterization of the nonlinearity requires a set of parameters, including % 3), the ground state absorption coefficient, the laser pulse width, and the excited state relaxation time. In a simple two- or three-level system, once all these factors are properly taken into account, the best parameter for measuring the resonant nonlinearity is simply the ground state absorption cross section of the material. In the following section I focus on the resonant nonlinearity only as this is closely related to the photophysical properties. The discussion of nonresonant nonlinearity of semiconductor nanoclusters can be found elsewhere [17, 84-86],... 

For bulk semiconductors at room temperature, the mechanism for the resonant nonlinearity can be described by the band-filling model [82,87]. This is shown schematically in Figure 16b for a direct gap semiconductor such as CdS. Absorption of photons across the band gap, g, generates electrons and holes which fill up the conduction and valence band, respectively, due to the Pauli exclusion principle. If one takes a snap shot of the absorption spectrum before the electrons and holes can relax, one finds that the effective band gap, , increases (Figure 166), since transitions to the filled states are forbidden. The bleaching efficiency per photon absorbed can be derived as... 

With the basic mechanism understood, the resonant nonlinearity of semiconductor clusters can now be quantitatively analyzed. Since one trapped electron-hole pair can bleach the exciton absorption of the whole cluster, the bleaching efficiency per absorbed photon of a nanocluster is the same as that of a molecule, as described by Eq. (20). For a given rp and r, the resonant third-order optical nonlinearity of a nanocluster is simply determined by the (a - ax) term. 

One major difference between a semiconductor nanocluster and molecule is the size dependence of a. As discussed in Section II.B and illustrated in Figure 5, there is a rapid rise in the absorption cross section of the first excited state as the cluster size is reduced below the exciton size. The resonant third-order nonlinearity of a nanocluster is therefore predicted to increase with decreasing cluster size. In reality it is limited by the cluster size dispersion and the presence of surface defects [15]. To maximize the resonant nonlinearity, one needs (1) a sharp exciton absorption band (which means smaller and monodisperse clusters), and (2) semiconductor clusters with larger absorption coefficients, such as GaAs or PbS. 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

For measurements carried out closer to resonances, thermal contributions to the optical nonlinearity, given by the following formula, can compete with electronic contributions (54,55) ... 

The discussion in this chapter is limited to cyanine-like NIR conjugated molecules, and further, is limited to discussing their two-photon absorption spectra with little emphasis on their excited state absorption properties. In principle, if the quantum mechanical states are known, the ultrafast nonlinear refraction may also be determined, but that is outside the scope of this chapter. The extent to which the results discussed here can be transferred to describe the nonlinear optical properties of other classes of molecules is debatable, but there are certain results that are clear. Designing molecules with large transition dipole moments that take advantage of intermediate state resonance and double resonance enhancements are definitely important approaches to obtain large two-photon absorption cross sections. 

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