Lorentz model

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

Let us now check the validity of the simple Lorentz model in order to explain the spectra of real solids. Figure 4.2 shows the dependence of the reflectivity on photon energy for a typical semiconductor. Si (Figure 4.2(a)), and for a typical insulator, KCl (Figure 4.2(b)). The Lorentz oscillator cannot quantitatively explain both spectra. In fact, we have supposed a single resonance frequency >o, but in the most general case a... 

We will now analyze the general optical behavior of a metal using the simple Lorentz model developed in the previous section. Assuming that the restoring force on the valence electrons is equal to zero, these electrons become free and we can consider that Drude model, which was proposed by R Drude in 1900. We will see how this model successfully explains a number of important optical properties, such as the fact that metals are excellent reflectors in the visible while they become transparent in the ultraviolet. 

The classical picture that describes these anisotropic effects based on the Lorentz model is illustrated in Fig. 9.8, which is a generalization of the spring model of Fig. 9.1 note that the spring stiffness depends on direction. 

These equations are identical with the high-frequency limit (9.13) of the Lorentz model this indicates that at high frequencies all nonconductors behave like metals. The interband transitions that give rise to structure in optical properties at lower frequencies become mere perturbations on the free-electron type of behavior of the electrons under the action of an electromagnetic field of sufficiently high frequency. 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

We shall call the frequency at which t = —2em and t" — 0 the Frohlich frequency coF the corresponding normal mode—the mode of uniform polarization—is sometimes called the Frohlich mode. In his excellent book on dielectrics, Frohlich (1949) obtained an expression for the frequency of polarization oscillation due to lattice vibrations in small dielectric crystals. His expression, based on a one-oscillator Lorentz model, is similar to (12.20). The frequency that Frohlich derived occurs where t = —2tm. Although he did not explicitly point out this condition, the frequency at which (12.6) is satisfied has generally become known as the Frohlich frequency. The oscillation mode associated with it, which is in fact the lowest-order surface mode, has likewise become known as the Frohlich mode. Whether or not Frohlich s name should be attached to these quantities could be debated we shall not do so, however. It is sufficient for us to have convenient labels without worrying about completely justifying them. 

We start our discussion of laser-controlled electron dynamics in an intuitive classical picture. Reminiscent of the Lorentz model [90, 91], which describes the electron dynamics with respect to the nuclei of a molecule as simple harmonic oscillations, we consider the electron system bound to the nuclei as a classical harmonic oscillator of resonance frequency co. Because the energies ha>r of electronic resonances in molecules are typically of the order 1-10 eV, the natural timescale of the electron dynamics is a few femtoseconds to several hundred attoseconds. The oscillator is driven by a linearly polarized shaped femtosecond... 

To describe the optical properties of the crystals over a wide temperature range and to derive the basic phenomenological parameters of the electronic transition, we fitted the R(co) and a(co) spectra using the Drude-Lorentz model. The Drude part describes the intraband transition of free carriers,... 

The relation between p and E is linear when E is small, but becomes nonlinear as E acquires values comparable with interatomic electric fields (typically, 105 to 108 V/m). This may be explained in terms of the simple Lorentz model in which the dipole moment is p = —ex, where x is the displacement of a mass with charge —e to which an electric force — eE is applied. If the restraining elastic force is proportional to the displacement (i.e., if Hooke s law is satisfied), the equilibrium displacement x is proportional to E P is then proportional to E, and the medium is linear. However, if the restraining force is a nonlinear function of the displacement, the equilibrium displacement x and the polarization density P are nonlinear functions of E and, consequently, the medium is nonlinear. 

From the point of view of theory, the formulae of Table 2.6 are equally applicable to both gas and condensed phase samples, as they include the local field factors, which account for local modifications to the Maxwell fields due to bulk interactions within the Onsager-Lorentz model. 

Theory of the dielectric function. The discussion of absorption properties of astrophysically relevant solids is frequently based on the classical Lorentz model for dielectric materials. This assumes that the electrons and ions forming the solid matter are located at fixed equilibrium positions in the solid, determined by internal forces. An applied electromagnetic field shifts the charged particles, labeled by... 

An increase in fractional free volume will reduce the number of polarisable groups per unit volume, and thereby reduce the relative permittivity of the polymer. Quantitatively, the effect may be estimated by means of the Clausius-Mossotti/Lorenz-Lorentz model for dielectric mixing (Bottcher, 1978) ... 

Just as the linear response of the medium displays a resonance in the vicinity of an electronic absorption, the non-linear properties also show resonant behaviour. These can be understood by extending the Lorentz model to the case of an anharmonic oscillator where the restoring force contains terms proportional to the square, cube, etc. of the displacement. Hence Equation (3.63) becomes ... 

Development of the ES FR was motivated by the seminal work by Evans, Cohen and Morriss. This paper focussed on constant energy dynamics and, based on results for the 2 dimensional Lorentz model, proposed that the measure of a trajectory was related to the exponential of the sum of the positive finite-time Lyapunov exponents of that trajectory. In this way they obtained a relationship for... 

Figure 37. Variation of the dielectric constant s/s2 of a metal-insulator colloid solution as a function of the metallic phase (silver) p (a) calculation based on the Lorentz model (b) calculation based on the fractal model.

Formally, one can think of the Raman transition probability being proportional to the elements of the polarizability tensor of a bound electron as the exciting frequency approaches the resonance frequency, these elements are enhanced in a Lorentz model of the bound electron. A common example of this mechanism is furnished by the ring-breathing (in-plane expansion) modes of porphyrins. Another mechanism, called vibronic enhancement, involves vibrations which couple two electronic excited states. In both mechanisms, the enhancement factors are nearly proportional to the intensities in the absorption spectrum of the adsorbate. 

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