Polarization centrosymmetric media

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

FIG. 7 Schematics of the SHG process at the surface of a sphere of a centrosymmetrical medium with a radius much smaller and of the order of the wavelength of light. The cancellation or the addition of the nonlinear polarization contribution is given explicitly and underlines the effect of the electromagnetic field and the surface orientation. 

The SHG signals arise from the second-order polarization induced in a non-centrosymmetric medium by the electric field E(a)) of the incident fundamental radiation given by the tensor equation... 

Figure 1.33. Plot of polarization response P to an incident electromagnetic wave of field strength E((o) at frequency o) in (a) a noncentrosymmetric medium and (b) a centrosymmetric medium. The Fourier components for the noncentrosymmetric response are also shown. Reproduced with permission from reference 43. Copyright 1984 VCH.)...

For a centrosymmetric medium incoming fields E (o) and -E[(o) must Induce polarizations P (2 ) and-P (2o)). respectively. This, however, is not consistent with equation (3.7.16) unless is zero, indicating that SHG is forbidden. At an interface the inversion symmetry is broken and SHG is no longer forbidden. Interfacial non-linear radiation is emitted by a sheet of coherently driven dipoles oscillating at the same frequency. Therefore, non-linear emission is coherent and has a well-defined direction, in contrast to linear optical processes like Raleigh and Raman scattering (see 1.7.3 and 1.7.8). 

Figure 3.1 shows a simplified picture of an interface. It consists of a multilayer geometry where the surface layer of thickness d lies between two centrosymmetric media (1 and 2) which have two different linear dielectric constants e, and e2, respectively. When a monochromatic plane wave at frequency co is incident from medium 1, it induces a nonlinear source polarization in the surface layer and in the bulk of medium 2. This source polarization then radiates, and harmonic waves at 2 to emanate from the boundary in both the reflected and transmitted directions. In this model, medium 1 is assumed to be linear. 

A special case where the surface contribution becomes dominant is realized for nonlinear optical effects of even orders (second order, in particular). If a medium possesses inversion symmetry, its optical properties must not be changed on the inversion operation r —s- —r. On the other hand, both E and P are the polar vectors, i.e., they change their sign on inversion. From Eq. (6.1) we conclude that both properties are compatible only if all even-order terms are identically equal to zero. This is not the case in the surface region of the crystal where the inversion symmetry is broken. This finding allows one to use second harmonic and sum-frequency generation as surface/interface specific optical tools at surfaces of centrosymmetric crystals. 

The main fundamental difference between these two processes is that second-harmonic generations involve a polarization that is even order in the electric field (i.e., P EE), whereas third-harmonic generations involve an odd-order polarization P—EEE). Consider a centrosymmetric nonlinear medium. If we invert the coordinate system, then Equation (11.82) becomes... 

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