Second-harmonic generation tensors

Figure I. Comparative quantities for selected tensor components of second harmonic generation (left) and the linear electro-optic effect (right) (measured at 1.06 pm wavelength). (The strain free quantity, f, was measured at 0.633 pm wavelength except in the case of GaAs which was measured at - 0.9 pm). (Reproduced with permission from Ref. 8. Copyright 1982, Laser Focus.)...

Nonvanishing Components of Second-Order Susceptibility Tensor for Second-Harmonic Generation in Electric-Dipole Approximation for Achiral and Chiral Isotropic (i.e. isotropic in the plane of the film) Films0... 

By measuring the temperature dependence of second-harmonic generation (SHG) of the neodymium laser wavelength at 1.06 pm in sodium nitride for five different radiation geometries which correspond to the five nonvanishing components of the nonlinear susceptibility tensor, Vogt etal. 3 2) could determine these tensor components and the coherence length 1 = 4 2 re the... 

Electric Field Induced Second-Harmonic Generation. An essential aspect of the development of materials for second-order nonlinear optics is the determination of the p tensor components. The technique that has been developed to accomplish this is called electric field induced second harmonic generation (EFISH) (13,14). 

The dielectric tensor describes the linear response of a material to an electric field. In many experiments, and particularly in optical rheometry, anisotropy in is the object of measurement. This anisotropy is manifested as birefringence and dichroism, two quantities that will be discussed in detail in Chapter 2. The nonlinear terms are responsible for such effects as second harmonic generation, electro-optic activity, and frequency tripling. These phenomena occur when certain criteria are met in the material properties, and at high values of field strength. 

For second harmonic generation (SHG), the tensor is y(2)(—2co co, co) (useful for frequency doubling and parametric down-conversion) while for the linear electrooptic or Pockels71 effect the tensor is y(2)(— co co, 0) (useful for Q-switching of lasers, for phase or amplitude modulators, and for beam deflectors) for optical rectification the tensor is y 2>(0 00, —co) for frequency mixing the tensor is y(2)(— co3 oolr co2) (useful for frequency up-converters, optical parametric oscillators, and spectroscopy). 

Non-linear optical techniques, such as second harmonic generation (SHG), have recently been used as surface probes. Bulk materials with inversion symmetry do not generate second harmonic signals, while surfaces and interfaces cannot have inversion symmetry, so the total SHG signal will come from the surface region for many systems. The components of the non-linear polarizability tensor have been used to determine the orientation of chemisorbed molecules. 

The basis of NLO-effects arising from susceptibilities of second order, is the interaction of three electric fields with a material. The practical implementation of optical devices requires strong, coherent and monochromatic radiation and hence, laser technology. Not all of the interacting fields need to be optical fields, however. In devices that make use of the Pockels effect, an externally applied electric field is used to alter reversibly the refractive index of a material. In a second harmonic generation (SHG) process two photons of circular frequency w can be transformed into one photon of frequency Iw. SHG is the NLO effect used most for the evaluation of /3-tensor elements in solution. 

The odd order susceptibilities are nonzero in all materials. However, owing to the fact that x is a third rank tensor, the second order susceptibility is nonzero only in noncentrosym-metric materials, that is, materials possessing no center of symmetry. The focus of this paper is on second order processes, and the relationships between the bulk susceptibility, second harmonic generation, and the linear electro-optic effect. For second harmonic generation, Xijl is symmetric in ij, leading to the relationship between the second harmonic coefficient dijk and the bulk second order susceptibility x 2)[i2l... 

Since xl" is third-rank tensor, it possesses non-zero components only in the case of molecules which have no centre of symmetry the precetUng process of second harmonic generation will not interest us any further here. It will be remembered, howevor, that a static dectric Rdd acts in a manner to lower the symmetry of bodies and can thus endow an isotropic body, naturally possessing a centre of symmetry, with the symmetry of an axially symmetric body without a centre of symmetry. In this way, a naturally isotropic body without centre of symmetry, during the time it is acted on by a static electric field E, has the atnlity to cause a doubling of the oscillation frequency of an incident laser beam, since now we have in equation (29) ... 

On the assumption of total symmetry of the tensor of third-order nonlinear polarizability c(— co coi, cog, cog), its non-zero and independent elements are the same as those of Table 12. Direct theoretical calculations of c = c(0 0,0,0) have been performed for the atoms of inert gases and some simple molecules. Values of the tensor elements = c(— cu cu, 0,0) have been determined for numerous molecules from static Kerr effect studies and values of c = c(— cd ot>,coi — col) from measurements of optical birefringence induced by laser li t. Measurements of second-harmonic generation by gases in the presence of a static electric field yield the tensor elements c " = c( — 2co co, to, 0), which can also be obtained from second-harmonic scattering in centro-symmetric liquids. The elements of the tensor c = c(— 3co co, co, co)... 

For certain macroscopic nonlinear parameters the tensor notation can be simplified due to the intrinsic symmetry of the experiment, e.g., second-harmonic generation and the linear electro-optic effect. Let us first consider SHG. The second-order contribution to the polarization is given by Eq. (9). 

Under the influence of an optical pump, the molecular angular distribution described by Equation 12.4 can be considerably modified. In turn, this results in modification of the X ijkl tensor components. Further, we discuss the influence of a polarized pump beam on third-order nonlinear phenomena such as third harmonic generation (THG) [(described by (-3a),ft>,w,a>) coefficient], electric field induced second harmonic generation (EFISH) [x / kl -2(0, (o, o), 0)] and degenerate four-wave mixing (DFWM) X kl ... 

The most traditional experimental determination of p is the electric field-induced second harmonic (EEISH) method, which requires the molecules to be aligned in solution by an electric field, by means of their static dipole moment (po). The EEISH signal is therefore proportional to po and to p <>c (projection of p on po), which is assumed to be equal to p in most cases. The bulk NLO properties are frequently evaluated as the efficiency of a powdered sample in second-harmonic generation (SHG), or as the d components of the x tensor. 

The second harmonic generation is a coherent technique giving the fast, electronic in origin, second-order NLO susceptibility (-2co co,co) at a given, measurement frequency co. Here, we limit the discussion to poled films, with °o mm symmetry, which exhibit two nonzero x tensor components the diagonal A zzz( 2co co,co) and the off diagonal xzz( 2co co,co), where Z is the poling (preferential orientation) direction. Usually, thin films are deposited on one side of substrate only (thin film deposited on both sides is discussed in Swalen and Kaj-... 

In the case of second-harmonic generation, the second-order nonlinear susceptibility tensor elements are symmetric in their last two Cartesian indices and are unchanged by the permutation of their second and third frequency arguments because they are identical. Thus, Eq. (28) can be rewritten in the simplified form... 

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... 

Section 4.1.1 reviews second harmonic generation (SHG) for para-nitroaniline (PNA), Section 4.1.2 the polarizability and second hyperpolarizability of nitrogen and benzene, Section 4.1.3 the second hyperpolarizability of Cgo, Section 4.2 the excited state polarizability of pyrimidine and r-tetrazine. Section 4.3 three-photon absorption, and finally, in Section 4.5 the electronic g-tensor and the hyperfine coupling tensor are reviewed as examples of open shell DFT response properties. 

To illustrate a case to be revisited in detail later, we explicitly derive the rate for coherent second-harmonic generation in a system containing M molecules. Using the general expression Eq. (52) for the radiation tensor together with the Golden Rule, and retaining a sum over the emitted harmonic in the matrix elements, we first obtain an expression of the form... 

---