Susceptibilities linear

Up to this point, we have calculated the linear response of the medium, a polarization oscillating at the frequency m of the applied field. This polarization produces its own radiation field that interferes with the applied optical field. Two familiar effects result a change in tlie speed of the light wave and its attenuation as it propagates. These properties may be related directly to the linear susceptibility The index of... 

Following the derivation of the linear susceptibility, we may now readily deduce the second-order... 

Here the oscillatory terms like e2,ta> and e 2lt<0 have been ignored, which is called the rotating wave approximation (RWA). Linear susceptibility yl et) is defined by... 

That is, the rate of energy absorption Q is linearly related to the imaginary part of linear susceptibility x( >). 

We consider a model for the pump-probe stimulated emission measurement in which a pumping laser pulse excites molecules in a ground vibronic manifold g to an excited vibronic manifold 11 and a probing pulse applied to the system after the excitation. The probing laser induces stimulated emission in which transitions from the manifold 11 to the ground-state manifold m take place. We assume that there is no overlap between the two optical processes and that they are separated by a time interval x. On the basis of the perturbative density operator method, we can derive an expression for the time-resolved profiles, which are associated with the imaginary part of the transient linear susceptibility, that is,... 

The constant a is the molecular polarizability, and the macroscopic constant X(1) is known as the linear susceptibility. The molecular polarizability is related... 

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

The equilibrium linear susceptibility is, in the absence of an external bias field, given by... 

Sample shape refers to the shape of the whole ensemble of nanoparticles, not to the shape of the individual particles. The linear susceptibility exhibits a sample shape dependence, while the zero-field specific heat and the dipolar fields do not. 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

For the hnear susceptibility, the zero-held specihc heat as well as the dipolar helds, the anisotropy dependence cancels out in the case of randomly distributed anisotropy (at least for sufficiently symmetric lathees). In other cases the anisotropy is a very important parameter as shown for the linear susceptibility in Figure 3.3 for an inhnite (macroscopic) spherical sample. The susceptibility is divided by Xiso = order to single out effects of anisotropy and dipolar... 

The specific heat for uncoupled spins does not depend on the orientations of the anisotropy axes however, the corrections due to the dipolar coupling do, as can be seen in Figure 3.4. As for the linear susceptibility, the effect of dipolar interaction is stronger in the case of parallel anistropy than for random anistropy. 

The general expression for the equilibrium linear susceptibility is given by Eq. (3.22) with the following expressions for the coefficients... 

With a non-linear susceptibility, the polarization is composed of non-linear terms such as eox E, eoX E as well as the linear term cqx E. It is clear that these elements in the polarization maybe oscillating at 2co and 3co respectively, giving rise to harmonics of the original frequency co. Because the higher order terms in the susceptibility are small compared with the first term, non-linear optical effects were not observed until after the invention of the laser in 1960. 

An applied electric field can also change a material s linear susceptibility, and thus its refractive index. This effect is known as the linear electro-optic (LEO) or Pocket s effect, and it can be used to modulate light by changing the voltage applied to a second-order NLO material. The applied voltage anisotropically distorts the electron... 

Non-linear second-order optical properties such as second harmonic generation (SHG) and the linear electrooptic effect are due to the non-linear susceptibility in the relation between the polarization and the applied electric field. SHG involves the... 

A colloid chemical approach to CdS/HgS/CdS spherical quantum wells was described [79]. Size-dependent third-order non-linear susceptibilities of CdS clusters were investigated [80]. Reviews appeared on size-quantized nanocrystalline semiconductor films [81] and on the quantum size effects and electronic properties of semiconductor microcrystallites [82]. 

The applied voltage in effect changes the linear susceptibility and thus the refractive index of the material. This effect, known as the linear electrooptic (LEO) or Pockels effect, modulates light as a function of applied voltage. At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different... 

In Equation 1, x is the linear susceptibility which is generally adequate to describe the optical response in the case of a weak optical field. The terms x and X are the second and third-order nonlinear optical susceptibilities which describe the nonlinear response of the medium. At optical frequencies (4)... 

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