Photon polarization vector

Figure 1 Angles 6 and

In this equation, the spherical angles 6 and

defined relative to the photon momentum k, photoelectron momentum p, and photon polarization vector e, as indicated in Figure 1, fi i is a dipole photoelectron angular distribution parameter, yni and Sni are nondipole photoelectron angular distribution parameters. 

Fig. 6.1 S left) and R (right) enantiomers of DCPH. The directions of transition moments (jilg and (tHG of an R enantiomer are shown as weU as those of photon polarization vectors e defined asfii Q-e = The magnitudes of p, LG and (iHG are 2.02euo and 1.63eoo, respectively...

In Fig. 6.5a, the initial direction of K-electron rotation depends on the photon polarization vector, that is, clockwise (counterclockwise) direction for e+ (e ) excitation, which has been described in Sect. 6.3. However, the amplitudes of mt) temporally vary for both cases, due to the decrease of the overlap between the nuclear WPs moving on the relevant two adiabatic PESs as depicted later in Fig. 6.6. This is one of the characteristic behaviors that are absent in a frozen-nuclei model. As for nuclear motions, DCP vibrates during n-electron rotation as seen in Fig. 6.5b, but the behavior of Q(f) differs only slightly between e+ and e excitations. 

Here, Ho is the molecular Hamiltonian in the BO approximation, and V is the nonadiabatic coupling operator. U(t) = —/x e(r)cos(a)/t) is the pulse excitation operator. Here, e t) is the amplitude of the laser pulse with photon polarization vector e, and coi is laser central frequency. In Eq. 6.18, i] denotes the parameter depending on photon polarization direction of the linearly polarized laser pulse = 1 for the polarization vector e+, while r] = -l for e. ... 

This relation is applicable for excitation with photons of identical frequency (Fig. 3.1a). Thus, the excitation frequency of the virtual state is about one-half that of the TP excited state S . The term e is the complex polarization vector. This term is needed to describe the orientation and polarization affecting TP excitation [23, 238]. 

When the photons are linearly polarized, the Rabi frequencies = ( d p)/27i coincide with the above defined Rabi frequencies. We assume real transition dipole moments and amplitudes of the electromagnetic fields and use the following expressions for the field amplitudes, = fit/cen. = Jlle/ceQ. Here = e , > = e> e and e are the polarization vectors of strong pump and ASE fields. 

To simplify notation we will specify the photon state by the wavevector k, and snppress, unless otherwise needed, the polarization vector ff. 

Here V denotes the quantization volume, and e 1 is the unit polarization vector for the radiation mode characterized by wavevector k, polarization A and circular frequency co = c k where it appears, an overbar denotes complex conjugation. The polarization vector is considered a complex quantity specifically to admit the possibility of circular or elliptical polarizations. Associated with each mode (k, A) are a Hermitian conjugate pair of photon annihilation and creation operators, and k / , respectively, which operate eigenstates of //raci with m(k, A) photons (m being the mode occupation number) as follows... 

The arguments associated with each unit vector are now dropped for brevity. The polarization unit vectors e, refer to each photon involved in the interaction process. The polarization vectors are represented as above for each photon that is annihilated, but created photons carry the overbar to represent complex... 

It is most important to note that in many cases of harmonic emission, a more completely index-symmetric form of the polarizability tensor is implicated. Consider once again the prototypical example of optical nonlinearity afforded by harmonic generation. When any harmonic is generated from a plane-polarized beam, in an isotropic medium, it produces photons with the same polarization vector as the incident light. In such a case the radiation tensor pyk becomes fully index-symmetric, and arguments similar to those given above show that only the fully index-symmetric part of the hyperpolarizability tensor, 3p(—2m co, co), can be involved. This does not mean that the tensor itself is inherently fully index-symmetric, but it does mean that experiments of the kind described cannot determine the extent of any index antisymmetry. 

Once the summation over virtual photon wave-vectors and polarizations in Eq. (5.10) is performed, the result can be cast in terms of a retarded resonance electric dipole-electric dipole interaction tensor V, (a>, R) (Power and Thirunamachandran 1983 Andrews and Sherborne 1987), using the identity... 

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