Second harmonic generation tensor components

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). 

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. 

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) ... 

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 same way as in the two previous sections, the first hyperpolarizabilities associated to three different nonlinear effects were calculated the static response and two dynamic effects, the dc-Pockels response characterized by P(-(b o),0) and the second harmonic generation response, p(-2 to, ). Again, in the dynamic case ffl=1.16 eV. The results are listed in Tables XIV, XV, and XVI for the ID clusters, and in Tables XVII, XVIII, and XIX for the 2D and 3D clusters. Similarly to the dipole moment case, only the major tensor component, Paaa> is considered. 

Optical second-harmonic generation experiments give a more detailed description of the anchoring at a microscopic scale [68,69 see also Chapter 5]. The molecule/surface interaction determines the orientational distribution in a thin surface layer extending up to 1 nm. The bulk uniaxial order develops on top of this layer via a transition layer of thickness which is well described by the usual mean-field theory, possibly including non-uniaxial components of the tensor order parameter. 

The HRS technique [25-27] involves the detection of the incoherently scattered second harmonic generated by the molecule in solution under irradiation with a laser of wavelength 2, leading to the mean value of the x tensor product. By analysis of the polarization dependence of the second harmonic signal, which can be evaluated selecting the polarization of the incident and scattered radiation, it is possible to obtain information about the single components of the quadratic hyperpolarizability tensor jS. Unlike EFISH, HRS can also be used for ionic molecular species and for nondipolar molecules such as ocmpolar molecules. In this chapter, the quadratic hyperpolarizability measured with an incident wavelength 2 by the EFISH and HRS techniques will be indicated as /l i(EFISH) and / (HRS), respectively. 

The zero-field and field-induced optical second harmonic generation (SHG) was investigated for the nematic and smectic A phases of various liquid crystals. The components of the cubic non-linear susceptibility tensor were measured for substances with different molecular structure. The phase-matched SHG was observed for all the compounds investigated. The directions of the phase synchronisms as well as the corresponding non-linear susceptibilities were determined for the ee-o and oe-o interactions. The zero-field phase-matched SHG was observed for the oe-o interaction. It was accounted for by a multipolar mechanism. 

The summation runs over repeated indices, /r, is the i-th component of the induced electric dipole moment and , are components of the applied electro-magnetic field. The coefficients aij, Pijic and Yijki are components of the linear polarizability, the first hyperpolarizability, and the second hyperpolarizability tensor, respectively. The first term on the right hand side of eq. (12) describes the linear response of the incident electric field, whereas the other terms describe the nonhnear response. The ft tensor is responsible for second order nonlinear optical effects such as second harmonic generation (SHG, frequency AotAAin, frequency mixing, optical rectification and the electro-optic effect. The ft tensor vanishes in a centrosymmetric envirorunent, so that most second-order nonlinear optical materials that have been studied so far consists of non-centrosyrmnetric, one-dimensional charge-transfer molecules. At the macroscopic level, observation of the nonlinear optical susceptibility requires that the molecular non-symmetry is preserved over the physical dimensions of the bulk stmcture. 

From the measurement of the intensity of the generated second harmonic for a dipolar molecule and with knowledge of the value of the dipole moment, an experimental value for / can be obtained. The relation of this value to the hyperpolarizability tensor components, in the completely general case... 

Most of the studies on push-pull phthalocyanines have been carried out by means of Electric Field Induced Second Harmonic (EFISH) Generation experiments in solution. Since only one experimental condition is favorable for EFISH, namely, parallel polarizations for all optical and static fields, these experiments lead to only one observable the vector component along the charge transfer axis assuming that the dipole moment vector and the vector part of the third-rank tensor along tlie molecular z-axis are collinear). 

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