INDEX wave vector

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

The x-dependent separation constant is the y-component of the wave-vector, and acts as an effective index profile, cf. figure 5,... 

In regard to Equation (5.4), we have to note that without the above mentioned assumptions the nonlinearities will contain weighting factors that are proportional to the corresponding wave-vector mismatch and inversely proportional to refractive index, thus suggesting that the THG signal is sensitive to the refractive index interface(s) as well. In order to differentiate between contrast mechanisms in THG imaging of soft tissue materials it would be important to know the relationships of the corresponding linear and nonlinear optical parameters. A nonlinear optical... 

We first consider the interaction of two laser beams with the same frequency inside a PR medium. When two coherent laser beams intersect inside a PR medium (Fig. 1), a periodic variation of the intensity is formed due to interference, which will induce a periodic charge distribution and lead to the formation of volume index grating through a PR mechanism. The grating wave vector is given by... 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

Pi(t) are the electrical potentials of the leads, the index k is the wave vector, but can be considered as representing an other conserved quantum number, a is the spin index, but can be considered as a generalized channel number, describing e.g. different bands or subbands in semiconductors. Alternatively, the tight-binding model can be used also for the leads, then (186) should be considered as a result of the Fourier transformation. The leads are assumed to be noninteracting and equilibrium. 

As usual, the index k includes the wave vector k and the polarizations e [ and e2 perpendicular to k, and al (at) is the creation (annihilation) operator for photons of energy h(ok = he k 1. We must write in this formalism the operators, vector potential, and electric field which are involved in our calculations ... 

Figure 2.20. Right part The polariton dispersion at a few tens of reciprocal centimeters below the bottom of the excitonic band, vs the wave vector, or the refractive index n = ck/w (notice the logarithmic scale). The arrows indicate transitions with creation of one acoustical phonon, with linear dispersion in k (with a sound velocity of 2000 m/s). For the transitions T, Tt, T3 the final momentum is negligible compared to the initial momentum, and the unidimensional picture suffices. For the transitions between T3 and the point A, the direction of the final wave vectors should be taken into account. Left part The density of states m( ) (2.141) of the polaritons in the same energy region. This diagram explains why the transitions T, will be much slower than the transitions around T3 and the point A. The very rapid increase of m( ) at a few reciprocal centimeters below E0 shows the effect of the thermal barrier.

Here we look for a solution of the form, HIH,(r, t) = e(n)(r, t)exp(ik r — kost), where k s is different from ks, which is given by a combination of the incoming wave vectors, due to the frequency dispersion of the refractive index of the optical sample. 

The diagrammatic representations of Eq. (2-26) are hardly altered from 4-7, except that the momentum conservation condition is incorporated into the rule that demands each line to have a wave vector index and the sum of the wave vectors of outgoing lines be equal to the sum of the wave vectors of incoming lines [52], The momentum of each of the diagrams 4-7 is still conserved (at a possibly nonzero value) because each contains the (or x 2-1) amplitude only once. 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

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