Second order nonlinear optics

We examine the optical properties of a nonlinear medium in which nonlinearities of order higher than the second are negligible, so that 

We consider an electric field E comprising one or two harmonic components and determine the spectral components of Pnl- In accordance with the first Born approximation, the radiation source S contains the same spectral components as Pnl, and so, therefore, does the emitted (scattered) field. 

Consider the response of this nonlinear medium to a harmonic electric field of angular frequency to (wavelength Ao = 2nco/u ) and complex amplitude E uj)  

The corresponding nonlinear polarization density Pnl is obtained by substituting (4.20) into Equation (4.19), 

The source S(t) = -pod2PNL(t)/dt2 corresponding to (4.22) has a component at frequency 2lo and complex amplitude S(2lo) = 4pb0u)2dE(uj)E u), which radiates an optical field at frequency 2co (wavelength Ao/2). Thus the scattered optical field has a component at the second harmonic of the incident optical field. Since the amplitude of the emitted second-harmonic light is proportional to S(2lo), its intensity is proportional to S 2uj) 2au)id I2, where I = E(uj) 2/2r/ is the intensity of the incident wave. The intensity of the second-harmonic wave is therefore proportional to d2, to 1/Ag, and to I2. Consequently, the efficiency of second-harmonic generation is proportional to / = P/A, where P is the incident power and A is the cross-sectional area. It is therefore essential that the incident wave have the 

---

The most significant symmetry property for the second-order nonlinear optics is inversion synnnetry. A material possessing inversion synnnetry (or centrosymmetry) is one that, for an appropriate origin, remains unchanged when all spatial coordinates are inverted via / —> - r. For such materials, the second-order nonlmear response vanishes. This fact is of sufficient importance that we shall explain its origm briefly. For a... 

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

The focus of the present chapter is the application of second-order nonlinear optics to probe surfaces and interfaces. In this section, we outline the phenomenological or macroscopic theory of SHG and SFG at the interface of centrosymmetric media. This situation corresponds, as discussed previously, to one in which the relevant nonlinear response is forbidden in the bulk media, but allowed at the interface. 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

The applications of this simple measure of surface adsorbate coverage have been quite widespread and diverse. It has been possible, for example, to measure adsorption isothemis in many systems. From these measurements, one may obtain important infomiation such as the adsorption free energy, A G° = -RTln(K ) [21]. One can also monitor tire kinetics of adsorption and desorption to obtain rates. In conjunction with temperature-dependent data, one may frirther infer activation energies and pre-exponential factors [73, 74]. Knowledge of such kinetic parameters is useful for teclmological applications, such as semiconductor growth and synthesis of chemical compounds [75]. Second-order nonlinear optics may also play a role in the investigation of physical kinetics, such as the rates and mechanisms of transport processes across interfaces [76]. 

Heinz T F 1991 Second-order nonlinear optical effects at surfaces and interfaces Noniinear Surfaoe... 

Reider G A and Heinz T F 1995 Second-order nonlinear optical effects at surfaces and interfaces recent advances Photonio Probes of Surfaoes ed P Halevi (Amsterdam Elsevier) pp 413-78... 

Fig. 1. Representative device configurations exploiting electrooptic second-order nonlinear optical materials are shown. Schematic representations are given for (a) a Mach-Zehnder interferometer, (b) a birefringent modulator, and (c) a directional coupler. In (b) the optical input to the birefringent modulator is polarized at 45 degrees and excites both transverse electric (TE) and transverse magnetic (TM) modes. The appHed voltage modulates the output polarization. Intensity modulation is achieved using polarizing components at the output.

Serious attempts to use LB films in commercial appHcations include the use of lead stearate as a diffraction grating for soft x-rays (64). Detailed discussion on appHcations of LB films are available (4,65). From the materials point of view, the abiHty to build noncentro symmetric films having a precise control on film thickness, suggests that one of the first appHcations of LB films may be in the area of second-order nonlinear optics. Whereas a waveguide based on LB films of fatty acid salts was reported in 1977, a waveguide based on polymeric LB films has not yet been commercialized. 

Acentricity greatly enhances the y-value (see 92 vs 91 and 90 or 101 vs 99 and 100, Fig. 8). Such a trend had been predicted for certain ranges of compounds by theory [137] however when the first hyperpolarizability, which determines second-order nonlinear optical properties, is maximized, y is predicted to be zero [138]. 

Bella SD (2001) Second-order nonlinear optical properties of transition metal complexes. Chem Soc Rev 30 355-366... 

A synthesis and physicochemical characterization, including molecular second-order nonlinear optical properties, of new push-pull-based chromophores 170 properly functionalized for polymerization and containing oxadiazole rings were reported <2002J(P2)1791>. 

Coordination and Organometallic Complexes as Second-order Nonlinear Optical Molecular Materials... 

In summary, we have briefly reviewed current research highlights from studies of second order nonlinear optical responses in organic and polymeric media. We have stressed how fundamental studies have led to microscopic understanding of important electronic states that comprise the origin of the large second order nonlinear responses in these... 

A specific set of experiments which must be mentioned, being directly associated with the main topic of this paper, is the work of Bergman, et. al. (22) dealing with the second-order nonlinear optical properties of polyvinylidene fluoride (PVF2). Nonvanishing the second-order nonlinear electric dipole susceptibility, is expected in PVF2 since it exhibits other properties requiring noncentrosymmetric microscopic structure. These properties appear... 

A novel second-order nonlinear optical medium which should offer considerable fabrication flexibility has been described. The physics of alignment of the highly nonlinearly polarizable moiety was discussed. However, observation of complex dynamical and thermal behavior indicates that an important role is played by the polymer liquid crystalline host. Additional properties of modified members of this family of lc polymers were consequently investigated. The explanations of guest alignment stabilization and thermal dependence of the alignability remain unresolved issues. 

The large molecular hyperpolarizability of the merocyanine chromophore (4,5) and the highly polar environment of the quasicrystals has prompted studies of the second order nonlinear optical properties of these materials (6). 

Second-Order Nonlinear Optical Signatures of Chirality... 

Chiral Materials for Second-Order Nonlinear Optical Applications... 

Second-Order Nonlinear Optics in Chiral Isotropic Bulk Media... 

In the following sections we will first in Section 2 briefly discuss the necessary background to understand optical activity effects in linear and nonlinear optics and to illustrate the similarities and differences between both types. In Section 3 we present a more thorough analysis of nonlinear optical effects in second-harmonic generation, both from a theoretical and an experimental point of view. Section 4 deals with experimental examples that illustrate the usefulness of nonlinear optical activity in the study of chiral thin films and surfaces. Finally, in Section 5 we give an overview of the role of chirality in the field of second-order nonlinear optics and show that chiral molecules can be useful for applications in this field. 

Figure 9.3 Schematic illustration of second-order nonlinear optical effects, (a) Second-harmonic generation. Two light fields at frequency go are incident on medium with nonvanishing / 2. Nonlinear interaction with medium creates new field at frequency 2 go. (b) Frequency mixing. One light field at frequency GO and one at frequency go2 is incident on nonlinear medium. Nonlinear interaction with medium creates new field at frequency goi + go2. (c) electro-optic effect. Static electric field E (0) applied over nonlinear medium changes phase of an incoming light field.

The proportionality constants a and (> are the linear polarizability and the second-order polarizability (or first hyperpolarizability), and x(1) and x<2) are the first- and second-order susceptibility. The quadratic terms (> and x<2) are related by x(2) = (V/(P) and are responsible for second-order nonlinear optical (NLO) effects such as frequency doubling (or second-harmonic generation), frequency mixing, and the electro-optic effect (or Pockels effect). These effects are schematically illustrated in Figure 9.3. In the remainder of this chapter, we will primarily focus on the process of second-harmonic generation (SHG). 

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 ... 

---