Nonlinear optic

Although use of the density operator does not solve dissipative decay problems, it is still very useful in other areas such as nonlinear spectroscopy [33]. The starting point is very similar to Eq. (9.53), but now the field E(t) is not influenced by the quantum system. It is just an oscillating field that oscillates with the frequency of light 

Actually, it is shghtly more complicated as the molecule (or reacting system) has an orientation and a position in space, and the electric field has a polarization, but that is not essential to the fundamental idea. Formally, Eq. (9.53) can be turned into an integral equation  

The n term in the series represents an n order interaction with the light field. The polarization, here defined as Tr[/6/t], is the measurable quantity in an optical experiment. Each term in the series has a specific meaning. The first term, usually referred to as the susceptibility, gives rise to absorption. The second term, which averages to zero in an isotropic medium, gives rise to second-order processes the third term is the relevant term for virtually all nonlinear optical experiments, also called experiments for the name of the susceptibility matrix. Higher order terms are rarely considered in view of the complicated experimental configuration needed, and the equally involved mathematical analysis needed to separate it from lower order processes. Some experiments such as Stark fluorescence, reported for a few tautomeric compounds, can be classified as measurements. 

The total dipole moment p of a dielectric material contained in volume V is given by the volume integral 

At general number densities where the total field experienced by a molecule may be influenced by dipole moments induced on neighboring molecules, the susceptibility is given instead [1] by x = (iVa/eo)/(l — Noc/3sq). 

We now consider an electromagnetic wave with time dependence E = Egcoscoot incident upon a system of isotropically polarizable molecules. For simplicity, we assume the molecules undergo classical harmonic vibrational motion with frequency co in some totally symmetric mode Q. The normal coordinate then oscillates as Q = Qo cos(cot + 5), where S is the vibrational phase and Qo is the amplitude. If the molecular polarizability a is linear in Q (as a special case of Eq. 10.33), the vibrational motion will endow the molecule with the oscillating polarizability 

Ignoring the vibrational phases S (which will be random in an incoherently excited system of vibrating molecules), the polarization induced by the external field will be 

According to the classical theory of radiation [1], an oscillating dipole moment p will emit radiation with an electric field proportional to its second time derivative p. Equations 11.1 and 11.5 then imply that radiation will be scattered at the frequencies coq, coq — co, and coq + co, corresponding to Rayleigh, Stokes Raman, and anti-Stokes Raman scattering, respectively. The scattered electric fields are proportional to Eq, so that the Rayleigh and Raman intensities are linear in the incident laser intensity. Expressions similar to Eq. 11.5 are 

The electric field of a light beam induces a polarisation in a solid. For light beams of ordinary intensity the polarisation, P, is a linear function of the electric field, E  

The extra nonlinear terms are only high enough to be of importance in relatively few materials. In general, the magnitudes of the values of the dielectric susceptibilities decrease rapidly as the order increases, so that the second-order, x term is the most important nonlinear coefficient. Moreover, all even-order terms, including the second order, x term, are zero in centrosymmetric crystals. The second-order term, x has a nonzero value in noncentrosymmetric crystals, and it is these that are generally known as nonlinear optical materials. 

Second harmonic generation comes about in the following way. The sinusoidally varying electric field associated with a light beam can be written  

If two input waves are used, frequency mixing can occur. Suppose the crystal is irradiated with two beams simultaneously  

Substituting this into Equation (14.3) will yield a second-order polarisation, P  

When an electromagnetic wave interacts with atoms the electrons perform oscillations around their equilibrium position which results in the emission of radiation. If the intensity of the wave is sufficiently small the amplitude of these oscillations is small and the restoring force 

For higher intensities, however, as can be reached with lasers, the amplitude becomes so large that the restoring force is no longer linearly dependent on the displacement and higher order terms have to be included in (6.1), which has to be replaced by the sum 

We will now study this nonlinear behaviour and the applications of nonlinear optics in more detail. Several examples shall illustrate the subject [523-534]. 

Demtrbder, Laser Spectroscopy 1, DOI 10.1007/978-3-642-53859-9 6, Springer-Verlag Berlin Heidelberg 2014 

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SAMs are generating attention for numerous potential uses ranging from chromatography [SO] to substrates for liquid crystal alignment [SI]. Most attention has been focused on future application as nonlinear optical devices [49] however, their use to control electron transfer at electrochemical surfaces has already been realized [S2], In addition, they provide ideal model surfaces for studies of protein adsorption [S3]. 

Another approach is to use the LB film as a template to limit the size of growing colloids such as the Q-state semiconductors that have applications in nonlinear optical devices. Furlong and co-workers have successfully synthesized CdSe [186] and CdS [187] nanoparticles (<5 nm in radius) in Cd arachidate LB films. Finally, as a low-temperature ceramic process, LB films can be converted to oxide layers by UV and ozone treatment examples are polydimethylsiloxane films to make SiO [188] and Cd arachidate to make CdOjt [189]. 

Shen Y R 1984 The Principles of Nonlinear Optics (New York Wiley)... 

Shen Y R 1984 The Principles of Nonlinear Optics (New York Wiley) ch 21 for a clear discussion of the connection between the perturbative and nonperturbative treatment of photon echoes... 

An excellent and readable discnssion of all aspects of the interaction of light with matter, from blackbody radiation to lasers and nonlinear optics. 

A clear, comprehensive discussion of the many facets of nonlinear optics. The emphasis is on optical effects, such as hannonic generation. The treatment of nonlinear spectroscopy, although occupying only a fraction of the book, is clear and physically well-motivated. 

A valuable handbook describing the many uses of nonlinear optics for spectroscopy. The focus of the book is a unified treatment of and methods for modelling the signal. 

Shen Y R 1994 Nonlinear optical studies of surfaces Appi. Rhys. A 59 541... 

Shen Y R 1994 Surfaces probed by nonlinear optics Surf. Sc/. 299-300 551... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Butcher P N and Cotter D 1990 The Elements of Nonlinear Optics (Cambridge Cambridge University Press)... 

Mukamel S 1995 Principles of Nonlinear Optical Spectroscopy (New York Oxford University Press)... 

Kobayashi T 1994 Measurement of femtosecond dynamics of nonlinear optical responses Modern Noniinear Optics part 3, ed M Evans and S Kielich Adv. Chem. Rhys. 85 55-104... 

Laubereau A and Kaiser W 1978 Coherent picosecond interactions Coherent Nonlinear Optics ed M S Feld and V S Letokov (Berlin Springer) pp 271-92... 

Koroteev N I 1995 BioOARS—a novel nonlinear optical technique to study vibrational spectra of chiral biological molecules in solution Biospectroscopy 1 341-50... 

Akhmanov S A and Koroteev N I 1981 Methods of Nonlinear Optics in Light Scattering Spectroscopy (Moscow Nauka) (in Russian)... 

Because of the generality of the symmetry principle that underlies the nonlinear optical spectroscopy of surfaces and interfaces, the approach has found application to a remarkably wide range of material systems. These include not only the conventional case of solid surfaces in ultrahigh vacuum, but also gas/solid, liquid/solid, gas/liquid and liquid/liquid interfaces. The infonnation attainable from the measurements ranges from adsorbate coverage and orientation to interface vibrational and electronic spectroscopy to surface dynamics on the femtosecond time scale. 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

We now embark on a more fonnal description of nonlinear optical phenomena. A natural starting point for this discussion is the set of Maxwell equations, which are just as valid for nonlinear optics as for linear optics. 

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... 

Flaving now developed some of the basic notions for the macroscopic theory of nonlinear optics, we would like to discuss how the microscopic treatment of the nonlinear response of a material is handled. Wliile the classical nonlinear... 

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 linear and nonlinear optical responses for this problem are defined by e, 2, e and respectively, as indicated in figure Bl.5.5. In order to detemiine the nonlinear radiation, we need to introduce appropriate pump radiation fields E(m ) and (co2)- If these pump beams are well-collimated, they will give rise to well-collimated radiation emitted tlirough the surface nonlmear response. Because the nonlinear response is present only in a thin layer, phase matching [37] considerations are unimportant and nonlinear emission will be present in both transmitted and reflected directions. 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating die corresponding chiral surfaces [, and 70]. From the point of view of perfomiing surface and interface spectroscopy with nonlinear optics, we must first examhie the nonlinear response of tlie bulk liquid. Clearly, a chiral liquid lacks inversion synnnetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the pemiutation synnnetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

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]. 

Many of the fiindamental physical and chemical processes at surfaces and interfaces occur on extremely fast time scales. For example, atomic and molecular motions take place on time scales as short as 100 fs, while surface electronic states may have lifetimes as short as 10 fs. With the dramatic recent advances in laser tecluiology, however, such time scales have become increasingly accessible. Surface nonlinear optics provides an attractive approach to capture such events directly in the time domain. Some examples of application of the method include probing the dynamics of melting on the time scale of phonon vibrations [82], photoisomerization of molecules [88], molecular dynamics of adsorbates [89, 90], interfacial solvent dynamics [91], transient band-flattening in semiconductors [92] and laser-induced desorption [93]. A review article discussing such time-resolved studies in metals can be found in... 

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