Wavevector Representations

The second way to achieve quadrahire is to introduce another field, E, (called a local oscillator) designed in frequency and wavevector to conjugate (go into quadrahire) in its complex representation with the new field of interest. Thus in the heterodyne case, the signal photons are derived fromcr. jy i. or Sj (lieterodyne) x x X... 

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 remainder of this contribution is organized as follows In the next section, the connection between the experimentally observed dynamic Stokes shift in the fluorescence spectrum and its representation in terms of intermolecular interactions will be given. The use of MD simulation to obtain the SD response will be described and a few results presented. In Section 3.4.3 continuum dielectric theories for the SD response, focusing on the recent developments and comparison with experiments, will be discussed. Section 3.4.4 will be devoted to MD simulation results for e(k, w) of polar liquids. In Section 3.4.5 the relevance of wavevector-dependent dielectric relaxation to SD will be further explored and the factors influencing the range of validity of continuum approaches to SD discussed. 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

The exciton-photon interaction is written in such a form that 2y yields the Rabi splitting energy in the perfect system. We chose to use the same number N of photon modes, and the wavevectors k are discrete with 2ir/Na increments. Our approach is to straightforwardly find the normalized polariton eigenstates Tj) (i is the state index) of the Hamiltonian (10.50) and then use them in the site-coordinate representation ... 

Since the irreducible representations of the group of translations are onedimensional and are determined by the given wavevector, among the set of quantum numbers characterizing a surface exciton there is always a quasicontinuous quantum number - the wavevector. This vector differs from the corresponding one in the case of bulk excitons it may be directed only along the crystal surface and assumes values only within a two-dimensional Brillouin zone. In... 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosymmetric media. Input waves at frequencies (o j and a 2, with corresponding wavevectors Aj(co j) and are approaching the interface from medium 1. Nonlinear...

Now specify a cartesian coordinate system with orthogonal unit vectors satisfying a 6y = in the laboratory reference frame. It is convenient to introduce a momentum representation for the fields through fourier transformation between x and k in physical applications the momentum vector k can often be identified with the wavevector of a photon. We choose the z-axis to be along the direction of k (i.e. k = k ej. A vector field U(k) may be decomposed into its longitudinal and transverse components... 

Thus these points in a small but well-defined region of k space include all possible irreducible representations of the translation group the vectors of the reciprocal lattice transform points in the Brillouin zone into equivalent points. The Brillouin zone therefore contains the whole symmetry of the lattice, each point corresponding to one irreducible representation, and no two points being related by a primitive translation. The smallest value of k ki, k2, kz) belonging to the rep is called the reduced wave-vector. The set oi reduced wavevectors is called the first Brillouin zone. 

Fig. 5.1. Schematic representation of the SHG experiment denoting the orientation of a rod-shape liquid crystal molecule at the substrate surface. k u)) and fc(2oi) denote the wavevector of the incoming fundamental and reflected second harmonic waves. On the right hand side a 5CB molecule and its orientation vector 63 are sketched. Directions x,y,z indicate the laboratory frame.

FIGURE 3 The energy versus momentum curve of an electron traveling in a periodic potential, (a) Extended zone representation, (b) reduced zone representation, and (c) the endpoints of the wavevector for a three-dimensional cubic box of dimension d. 

The set of one-electron functions transforming according to the n j-dimensional irrep d( ) is called the shell. For molecules, these shells are connected with irreps of the point-symmetry group. For a crystal, f3 = ( fe,7) - full irreducible representation of space group G, defined by the star of wavevector k and irrep 7 of the point group of this vector. Taking into consideration the spin states a a) we have 2n/ one-electron states in the shell. The functions ( )Q ([Pg.110]

Figure 2.23. Graphical representation of the five independent Fourier components of the dielectric tensor for a given wavevector q(hkl) from the reciprocal lattice.

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