Electric field second harmonic generation theory

One of the most important theoretical contributions of the 1970s was the work of Rudnick and Stern [26] which considered the microscopic sources of second harmonic production at metal surfaces and predicted sensitivity to surface effects. This work was a significant departure from previous theories which only considered quadrupole-type contributions from the rapid variation of the normal component of the electric field at the surface. Rudnick and Stern found that currents produced from the breaking of the inversion symmetry at the cubic metal surface were of equal magnitude and must be considered. Using a free electron model, they calculated the surface and bulk currents for second harmonic generation and introduced two phenomenological parameters, a and b , to describe the effects of the surface details on the perpendicular and parallel surface nonlinear currents. In related theoretical work, Bower [27] extended the early quantum mechanical calculation of Jha [23] to include interband transitions near their resonances as well as the effects of surface states. 

In 1996, Munn extended the microscopic theory of bulk second-harmonic generation from molecular crystals to encompass magnetic dipole and electric quadrupole effects [96] and included all contributions up to second order in the electric field or bilinear in the electric field and the electric field gradient or the magnetic field. This was accomplished by replacing the usual polarization of Refs. 72 and 84 by an effective polarization as well as by defining an effective quadrupole moment. Consequently, the self-consistently evaluated local electric field and electric field gradient were expressed in terms of various molecular response coefficients and lattice multipole tensor sums (up to octupole). In this... 

ELECTRIC FIELD INDUCED SECOND HARMONIC GENERATION 12.3.1 Theory... 

The Hartree-Fock method for periodic systems nowadays represents a routine approach coded in several ab initio computer packages. We may analyze the total energy, its dependence on molecular conformation, the density of states, the atomic charges, etc. Also calculations of first-order responses to the electric field (polymers are of interest for optoelectronics) have been successful in the past. However, non-linear problems (like the second harmonic generation, see Chapter 12) still represent a challenge. On the one hand, the experimental results exhibit wide dispersion, which partly comes from market pressure. On the other hand, the theory itself has not yet elaborated reliable techniques. 

Saha, S. K., and G. K. Wong. 1979. Phase-matched electric-field-induced second-harmonic generation in a nematic hquid crysfal. Opt Commun. 30 119 see also Ou-Yang, Z.-C., and Y.-Z. Xie. 1986. Theory of second-harmoitic generation in hquid crystals. Phys. Rev. A. 32 1189. 

The argument Rp implies structure relaxation in the field, and P" means the nuclear relaxation part of P, while the subscript oc oo invokes the so-called infinite optical frequency (lOF) approximation. In principle, this procedure allows one to obtain most of the major dynamic vibrational NR contributions in addition to the purely static ones of Eqs.4.5. 7. The linear term in the electric field expansion of Eq. (4) gives the dc-Pockels effect the quadratic term gives the optical Kerr Effect and the linear term in the expansion of beta yields dc-second harmonic generation (all in the lOF approximation). For laser frequencies in the optical region it has been demonstrated that the latter approximation is normally quite accurate [29-31]. In fact, this approximation is equivalent to neglecting terms of the order with respect to unity (coy is a vibrational frequency). In terms of Bishop and Kirt-man perturbation theory [32-34] all vibrational contributions through first-order in mechanical and/or electrical anharmonicity, and some of second-order, are included in the NR treatment [35]. 

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