Three-wave interactions

F ure 2.21 Different types of three-wave interaction in a nonlinear media (at the photon level) (a) sum-frequency generation and (b) the optical parametric generation. 

The following three-wave interactions in crystals with a square nonlinearity (x 5 0) are possible ... 

Non-linear optical interactions occur in materials with high optical intensities and have been used to produce coherent light over a wide range of frequencies from the far infra-red to the ultra-violet. The three wave mixing process is of particular interest as it can be used for optical parametric amplification and optical second harmonic generation (SHG) and occurs in non-centrosymmetric materials. 

IT Complexes may be described in terms of Mulliken s charge-transfer theory (23-25). This theory attributes the complex formation of stable aromatic molecules to an electron donor (D)-electron acceptor (A) interaction which is approximately represented by the combination of three wave functions ... 

A quantitative treatment of tt complex formation is, however, more complicated, since it is generally recognized that all three wave functions are necessary for an accurate description of the bond. For instance, it has been pointed out by Orgel (27) that n complex stability cannot solely be the result of n electron donation into empty metal d orbitals, since d and ions (Cu+, Ag+, Ni , Rh+, Pt , Pd++) form some of the strongest complexes with poor bases such as ethylene, tt Complex stability would thus appear to involve the significant back-donation of metal d electrons into vacant antibonding orbitals of the olefin. Because of the additional complication of back-donation plus the uncertainty of metal surface orbitals, it is only possible to give a qualitative treatment of this interaction at the present time. 

DC electric fields. DC generation is known as optical rectification. The actual phenomena that will be observed depend on the experimental conditions and whether or not phase matching has been achieved. Three-wave mixing processes in which two beams interact to generate a third beam require the mixing medium to have a non-zero In an isotropic medium, reversal of the... 

Of these, the pure electron-electron Coulomb interaction (4.14a) appears to be the obvious choice and, indeed, has been widely used [12,14,16]. The electron-electron contact interaction (4.14b), which only acts if both electrons are at the position of the ion (in effect, a three-body contact interaction), has also been frequently employed [15], Both interactions have been compared in various regards in [17,18,40]. More recently, the Coulomb interaction (4.14c), which is only effective if the second (bound) electron is located at the position of the ion, and the electron-electron contact interaction (4.14d), which is not restricted to the position of the ion, have also been studied [27]. The interactions (4.14b) and (4.14c) are effective three-body interactions, which attempt to take into account that the effective electron-electron interaction will depend on the positions of the electrons relative to the ion. An alternative interpretation, which formally leads to the same results, is to consider a two-body interaction Vi2 in (4.17) and a wave function (rlip ) in (4.18) that is extremely strongly localized at the position of the ion for details, see [27]. 

In Fig. 4.8 the effect of the initial-state wave functions is explored, for the case where the crucial electron-electron interaction is the two-body Coulomb interaction (4.14a) and for the case where this interaction is the two-body contact interaction (4.14d), which is not restricted to the position of the ion. In both cases, the form factor includes the function (4.23), which favors momenta such that pi + p2 is large. This is clearly visible for the contact interaction (4.14d) and less so for the Coulomb interaction (4.14a) whose form factor also includes the factor (4.19), which favors pi = 0 (or p2 = 0)- We conclude that (i) the effect of the specific bound state of the second electron is marginal and (ii) that a pure two-body interaction, be it of Coulomb type as in (4.14a) or contact type as in (4.14d), yields a rather poor description of the data. A three-body effective interaction, which only acts if the second electron is positioned at the ion, provides superior results, notably the three-body contact interaction (4.14b), cf. the left-hand panel (d). This points to the significance of the interaction of the electrons with the ion, which so far has not been incorporated into the S-matrix theory beyond the very approximate description via effective three-body interactions such as (4.14b) or (4.14c). 

Many of the different susceptibilities in (18)-(21) correspond to important experiments in linear and non-linear optics. The argument in parentheses again describes the kind of interacting waves. TWo waves interact in a first-order process as described above in (9), three waves in a second-order process, and four in a third-order process. x ° describes a possible zeroth-order (permanent) polarization of the medium t- (0 0) is the first-order static susceptibility which is related to the relative permittivity (dielectric constant) at zero frequency, e,.(0), by (22). 

Section 111 deals with nonreactive shock waves. The thread here is composed of three simple equations that describe the conservation of mass, momentum, and energy across the shock front. In this section we learn how to deal quantitatively with shock waves interacting with material interfaces and other shock waves. 

It is well know that when a wave interacts with and is scattered by a point object, the outcome of this interaction is a new wave, which spreads in all directions. If no energy loss occurs, the resultant wave has the same frequency as the incident (primary) wave and this process is known as elastic scattering. In three dimensions, the elastically scattered wave is spherical, with its origin in the point coinciding with the object as shown schematically in Figure 2.20. 

Thus, with each polymer segment, we can associate an orientation and a wave vector q. In the same way, with each interaction line, we can associate an orientation and a wave vector. Each wave vector can be considered to be transferred in the direction of the arrow which defines the corresponding orientation. We note that a two-body interaction transfers one wave vector q, whereas a three-body interaction transfers two wave vectors q and q". We also remark that the orientation of each line is arbitrary, but if the vector q is associated with an orientation, the vector - q must be associated with the opposite orientation. On a polymer, it is generally convenient to choose the direction in which s increases to define the orientations. Finally, we note that a factor (27i)"d corresponds to the introduction of each wave vector. 

However, in general, it is more convenient to work in reciprocal space by calculating the Fourier transforms of the restricted partition functions. In this case, the contribution of a diagram depends on a certain number of internal vectors. Applying Fixman s method is equivalent to summing the contributions over these wave vectors. The number Q of independent wave vectors is called the number of loops. In particular, for a diagram with p two-body interactions and p three-body interactions, we have... 

Fig. 14.19. Generation of a two-body interaction from two three-body interactions-in a divergent diagram (see the discussion in Section 6.4). The figure shows seven diagrams belonging to three different types. The contributions associated with these types are the same if each external leg bears a zero wave vector and if we neglect the constraints concerning the lengths of the chains. The origins of these chains are in A and B.

---