Complex polarization unit vector

For elliptically polarized light, the complex polarization unit vectors are often more convenient. They are defined by combining the real Cartesian polarization unit vectors, e, with the complex absolute phase part, exp[i,], in Eq. (2.3) (see [12]). 

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... 

The state of polarization is determined by the pair of complex numbers e and e2 the quantities ei 2 and e2 2 represent probability densities of a definite (linear or circular) polarization of the photon as determined by the unit vectors Xi and x2- Since ej and e2 are related by the normalization condition... 

Within a phase factor and amplitude, c 1 = e(2> is the vectorial part of the complex description of right and left circularly polarized radiation. The basis unit vectors c 1 c 1, and c 3 form the 0(3) cyclic permutation relations ... 

T We can ensure the validity of Eq. (1.23) by writing Ak as a product of a unit vector ik, called the polarization, and a complex scalar amplitude Ak... 

Here c.c. refers to the complex conjugate and ej (t) and Uj are the temporal envelope function and unit vector of the polarization of the jth electric field. The frequencies of the five fields are assumed to be identical, i.e., ojj = co0 for j = 1, 2, 3, 4, and 5. This can be experimentally achieved by using time-delayed pulses generated from a common laser oscillator. Although the femtosecond pulses generated in the laboratory have a finite width, for the sake of simplicity the laser pulses are assumed to be impulsive in this section, i.e. ... 

Here, (> and /> are initial and final states of the supermolecular complex, 11 is the induced incremental polarizability (a second-rank tensor), fio and fij are unit vectors in the direction of the electric polarization of incident and scattered waves, which are often specified in the form of subscripts V and H, for vertical and horizontal, (aif = (Ej- — Ei)lh is the energy difference of initial and final state in units of angular frequency, Pi T) is the population of the initial state (a function of temperature), <5 (to) is Dirac s delta function, and the summation is over all initial and final states of the collisional complex. 

Eq and Bq are constant complex amplitudes, and e- and 2 shall be constant unit vectors. As a consequence the electric field vector always points in the direction of Cy and such an electromagnetic wave is said to be linearly polarized. By superpositions of two such linearly polarized waves with different phases and amplitudes so-called elliptically polarized waves may be constructed. However, we do not need to further discuss this possibility here. 

Expressions for polarized spectral intensities are of the form 8 0(0/)-8 for one-photon processes and of the form 8 8 (4)O(0/) -8 8 for two-photon processes (Table 2). The nature and possible complex conjugation of the unit vectors 8 and 8 and the nature of the second-rank tensor 0(0/) and the... 

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... 

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