Nonlinear bulk polarization

The nonlinear bulk polarization density is given by an expression analogous to Eq. (4) ... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

The tenns beyond a S in Eq. (6.94) are not linear in W, so they are referred to as the nonlinear polarization and give rise to NLO effects. You shonld now see why nonlinear polarization becomes more important at higher electric field strengths, since it scales with higher powers of the field. For most materials, ag > (P/2) > (K/6)r so the first few observations of NLO effects were made prior to the invention of lasers and their associated large electric fields. The bulk polarization, P, which you will recall is the dipole moment per unit volnme, can then be expressed as... 

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. 

The first two terms are electric quadrupole in character while the last term is magnetic dipolar. Under excitation by a single plane wave, the first term vanishes. In a homogeneous medium the second term vanishes by Gauss Law. The third term describes the induced polarization which is along the propagation direction. It can only radiate at the discontinuity of the surface. The full expression for the second-order nonlinear polarization in an isotropic medium is then written as the sum of the surface and bulk polarizations [78] ... 

Phosphates showing a bulk polarization (i.e. ferroelectric phases) may be used for nonlinear optical processes see Nonlinear Optical Materials) such as second harmonic generation and electro-optic switching. KTP (Section 5.2.2) and related phases (NH4T10P04 and KTi0As04) are very efficient nonlinear materials. The ferroic phosphates described above also show nonlinear properties. KDP materials are inferior to KTP types but they find use in electro-optics as they are very transparent over a wide frequency range. 

To fulfill the need for understanding what structures will allow enhancement of optical nonlinearity, we have coupled ab-initio theoretical calculations of optical nonlinearity with synthesis of sequentially built and systematically derivatized model compounds, and the measurement of their optical nonlinearities. Now I would like to discuss very briefly our efforts to compare microscopic optical nonlinearities. An expression, similar to the expansion of the bulk polarization as a function of the applied field, can be written for the induced dipole moment. Naturally, the nonlinear term Y, for example, is the third derivative of the induced dipole moment with respect to the applied field. Also, using the Stark energy analysis, one can write the nonlinear terms 3 (and Y) as a sum over all excited states terms involving transition-dipoles and permanent dipoles, similar to what one does for polarizability. Consequently, the two theoretical approaches are (i) the derivative method and (ii) the sum-over-s1j tes method. We have used the derivative method at the ab-initio level. We correlate the predictions of these calculations with measurements on systematically derivatized and sequentially built model compounds. Some conclusions of our theoretical computations are as follows ... 

The variation in absorption due to the electric field modulation (Equation 19.16) is a nonlinear optical effect. We now consider the origin of nonlinear behavior in materials. In a classical description [89-91], the electric field interacts with the charges (q) in an atom through the force (qF). which displaces the centre of the electron density away from the nucleus. This results in charge separation and thus in a field-induced dipole pi. For an assembly of atoms, the average summation over all atoms ultimately gives rise to the bulk polarization P vector of the material. P opposes the externally applied field and is given by ... 

Second-harmonic generation for nonlinear optics, ferroelectricity, and piezoelectricity are all properties that are dependent on the pre.sence, magnitude, and orientation of bulk polarity in crystals and films. Therefore, the issue of how to design a polar solid from basic principles remains a challenge that has immense potential relevance to materials science. Obviously, a polar solid is guaranteed if a pure enantiomer is used as a component of a compound. However, the presence of polarity does not in any way imply that optimal packing will occur and, further-... 

The nonlinear optical properties of polydiacetylenes are subject of an increasing interest in last years (1-4). This is due to the fact that polydiacetylenes are a one-dimensionnal system of higly polarizable conjugated tt elections The polarization depends strongly on electron delocalization length Moreover, the electronic origin of hyperpolarizability implies short response times On the other hand, the wave dispersed measurements of molecular hyperpolarizabilities yield information about forbidden electronic transitions In fact the bulk polarization can be developped in external electric field power series ... 

The conjugated quasi-one dimensional polymers are characterized by a strong delocalization of electrons. This highly polarizable electronic cloud responds nonlinearly to the exiting external field. The resultant bulk polarization can be developed into the external field power series and its component along.x is given by... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

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... 

Takmg advantage of the synunetry changes induced by the presence of a surface. Many nonlinear teclmiques rely on the fact that the surface breaks the centrosynuuetrical nature of the bulk. The use of polarized light can also discriminate among dipole moments in different orientations. 

Unlike in bulk nonlinear spectroscopy experiments, the signal in nonlinear microscopy is generated within a volume that is on the order of an optical wavelength. The axial extent of this volume is often referred to as the interaction length, which denotes the length within which the incident fields interact to produce a nonlinear polarization in the material. Such microscopic interaction lengths yield signal interference profiles that can differ markedly from those observed in macroscopic spectroscopy. 

In this paper it has been attempted to provide an introductory overview of some of the various nonlinear optical characterization techniques that chemists are likely to encounter in studies of bulk materials and molecular structure-property relationships. It has also been attempted to provide a relatively more detailed coverage on one topic to provide some insight into the connection between the macroscopic quantities measured and the nonlinear polarization of molecules. It is hoped that chemists will find this tutorial useful in their efforts to conduct fruitful research on nonlinear optical materials. 

In this equation, d2u) represents the angle of the radiated SH light with respect to the surface normal, 7(co) is the pump intensity, and e(2co) is the polarization at the SH frequency. The vectors e(co) and e(2co) are related to the unit polarization vectors e(co) and e(2co) in medium 2 by Fresnel coefficients. The effective surface nonlinear susceptibility incorporates the surface nonlinear susceptibility x( and the bulk magnetic dipole contributions to the nonlinearity. The result simplifies since, for isotropic media, there are only three nonzero independent elements of xf These are x%, X% = X%.> and XsfL where 1 =... 

The model ascribes the effective polarization (Pseff(2a>)) of Eq. (3-5) to a linear combination of the bulk nonlinear polarization (Pj(2a>)) of Eq. (3.3) and a surface nonlinear polarization PNS(2o>). For a cubic medium excited by a single beam of frequency co, the bulk nonlinear polarization induced by (cu) becomes... 

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