Nonlinear optics electric field vectors

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

Light-matter interactions can be described via an induced polarization, i.e., the induced dipole moment per unit volume. Ultrafast laser pulses, which are used in laser scanning microscopes, have high enough intensity to induce a nonlinear polarization in various materials. For intense optical electric field E, the polarization vector P can be expanded in the power series (Boyd 1992)... 

The properties of a dielectric medium through which an electromagnetic (optical) wave propagates are completely described by the relation between the polarization density vector P(r, t) and the electric-field vector E r, t). It was suggested that P(r, t) could be regarded as the output of a system whose input was E(r, t). The mathematical relation between the vector functions P(r, t) and E(r, t) defines the system and is governed by the characteristics of the medium. The medium is said to be nonlinear if this relation is nonlinear. 

Optical nonlinearities can be explained by considering the interaction of strong electric fields with matter. If the fields have optical frequencies, the phenomena resulting from the nonlinear interactions are called nonlinear optical phenomena. Most texts on nonlinear optics (e.g., Refs. 22-25) begin the discussion of this area from considerations of macroscopic relations between the vector quantities P (the polarization vector), D (the displacement vector), and E (the electric field vector). Chemists, however, consider the molecular origin of physical phenomena, so the description of NLO phenomena that follows starts from consideration of the behavior of a single molecule in a strong electric field. 

Since p and E are vector quantities, a, P, 7, etc., are tensors. For example, the electric field vector in the first term will have three components in the molecular coordinate system. Each electric field component can contribute to polarization along each of the three directions in the molecular coordinate system. This triple contribution of electric field components leads to a total of nine elements to the second rank polarizability tensor. Similarly, there are 27 components to the P tensor and 81 components to 7. Molecular symmetry generally reduces these tensors to only a few independent elements. Unless the molecular coordinate system lacks an inversion center, the form of the odd-rank tensors such as P will lead to zero induced polarization in this representation of optical nonlinearities. For molecules such as benzene and polymers such as poly[bis(p-toluenesulfonate)diacetylene]... 

Since second-order nonlinear optical materials are anisotropic, their optical properties are described by tensors as discussed previously in Sect. 2.1.2. For a nonlinear optical process, the -th order nonlinear polarization is due to n interacting electric held vectors and is described by an (n -I-1) rank tensor composed of 3"+ tensor elements. In nonlinear optics, several fields with different frequencies l can be present simultaneously so that the electric field and the polarization are represented by... 

The theory of nonlinear optical processes in crystals is based on the phenomenological Maxwell equations, supplemented by nonlinear material equations. The latter connect the electric induction vector D(r,t) with the electric field vector E(r, t). In general, the relations are both nonlocal and nonlinear. The property of nonlocality leads to the so-called spatial dispersion of the dielectric tensor. The presence of nonlinearity leads to the interaction between normal electromagnetic waves in crystals, i.e. makes conditions for the appearance of nonlinear optical effects. 

Fig. 5.99. Coordinate system for the description of nonlinear optics in a uniaxial birefringent crystal. An incident wave with wavevector kx and k = kx, ky, 0) electric field vector E = Ex, Ey,0 generates in a KDP crystal the polarization P = 0, 0, P,(2co)]...

There have also been reports on the preparation of polar materials by a photo-electro-poling technique that combines the optically induced quadrupolar depletion of chromophores in the direction of the light electric vector with an additional field-induced orientation of dipolar chro-mophores. The latter allows the preparation of cold electrets, which are interesting for nonlinear optical applications, such as optical harmonic generation, wave mixing, etc. ... 

In order to introduce some notation, we first recall a few of the well-known properties of the interaction of light pulses with molecules in the linear approximation. Frequently, the signals in nonlinear optical experiments are expressed in terms of the polarization induced in the medium by the incident pulses. The complex linear polarization P t) vector for a distribution of identical two-level systems is obtained from an elementary calculation of the density matrix using the Liouville equation of a system perturbed by an electric field and proceeding as follows ... 

Nonlinear optical effects can be described within the framework of macroscopic electrodynamics (see, e.g. Bloembergen (49)), by applying the nonlinear relation between the induction vector T> and the strength E of the macroscopic electric field. When the value of E in the light wave is small compared to the intra-atomic electric fields, this nonlinear relation can be written in the form of the expansion... 

The importance of the hyperpolarizability and susceptibility values relates to the fact that, provided these values are sufficiently large, a material exposed to a high-intensity laser beam exhibits nonlinear optical (NLO) properties. Remarkably, the optical properties of the material are altered by the light itself, although neither physical nor chemical alterations remain after the light is switched off. The quahty of nonlinear optical effects is cmciaUy determined by symmetry parameters. With respect to the electric field dependence of the vector P given by Eq. (3-4), second- and third-order NLO processes may be discriminated, depending on whether or determines the process. The discrimination between second- and third-order effects stems from the fact that second-order NLO processes are forbidden in centrosymmetric materials, a restriction that does not hold for third-order NLO processes. In the case of centrosymmetric materials, x is equal to zero, and the nonhnear dependence of the vector P is solely determined by Consequently, third-order NLO processes can occur with all materials, whereas second-order optical nonlinearity requires non-centrosymmetric materials. 

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 ... 

There is a separation of charges in a nonlinear optical (NLO) medium under the application of an electric field. For weak electric fields, the applied electric vector, E, induces a polarization, P, in the material which is linearly dependent on the electric field, linear = E, where is the susceptibility tensor. The net polarization in the medium under the influence of several fields is a linear superposition of the effects of the same fields acting independently. At high intensities the optical fields can interact with each other through the higher order terms in the polarization vector. P can now be expanded as j (l) E + j (2) E E + j (3) E E E +. .. The nonlinear susceptibilities, etc., of... 

It is instructive to first discuss the parameter defining optical nonlinearity. In an electrically polarizable material with negligible magnetic susceptibility, the electric and magnetic field vectors satisfy the following equations (in SI units)... 

If laser light of sufficient intensity is propagating through an optically nonlinear material, and the time dependence of the electric vector component of the electromagnetic field is given by E = E0 sin(cot), then from Eq. (8.35), the polarization is given by... 

The vector of a permanent dipole moment Pe and polarizability tensor a y describe the linear (in field) electric and optical properties. The nonlinear properties are described by tensors of higher ranks (this depends on the number of fields included). For instance, the efficiency of mixing two optical waves of frequencies coi and CO2 is determined by polarization Py (co3) = E coi) Ey(co2) where E/(coi) and Ey((X)2)j are amplitudes of two interacting fields. Here is a third rank tensor of the electric hyperpolarizability. 

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