Wavevector

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

Electronic and optical excitations usually occur between the upper valence bands and lowest conduction band. In optical excitations, electrons are transferred from the valence band to the conduction band. This process leaves an empty state in the valence band. These empty states are called holes. Conservation of wavevectors must be obeyed in these transitions + k = k where is the wavevector of the photon, k is the... 

The quantity 1 + x is known as the dielectric constant, it is constant only in the sense of being independent of E, but is generally dependent on the frequency of E. Since x is generally complex so is the wavevector k. It is customary to write... 

Figure Al.6.16. Diagram showing the directionality of the signal in coherent spectroscopy. Associated with the carrier frequency of each interaction with the light is a wavevector, k. The output signal in coherent spectroscopies is detemiined from the direction of each of the input signals via momentum conservation (after [48a]).

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

On the other hand, when properties of the new field are being measured—such as its frequency, direction (wavevector), state of polarization and amplitude (or intensify)—one has a Class II spectroscopy. 

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... 

Consider all of the spectroscopies at third order s = 3). To be as general as possible, suppose the total incident field consists of the combination of three experimentally distinct fields (/ = 1, 2, 3). These can differ in any combination of their frequency, polarization and direction of incidence (wavevector). Thus the total field is written as... 

Wlien resonances, or near resonances, are present in the 4WM process, the ordering of the field actions in the perturbative treatment (equation (Bl.3.1)). can be highly significant. Though the tliree-coloiir generators (E.s e, E e.s, . . . ) have identical frequency and wavevector algebra, their associated susceptibility... 

Unlike the typical laser source, the zero-point blackbody field is spectrally white , providing all colours, CO2, that seek out all co - CO2 = coj resonances available in a given sample. Thus all possible Raman lines can be seen with a single incident source at tOp Such multiplex capability is now found in the Class II spectroscopies where broadband excitation is obtained either by using modeless lasers, or a femtosecond pulse, which on first principles must be spectrally broad [32]. Another distinction between a coherent laser source and the blackbody radiation is that the zero-point field is spatially isotropic. By perfonuing the simple wavevector algebra for SR, we find that the scattered radiation is isotropic as well. This concept of spatial incoherence will be used to explain a certain stimulated Raman scattering event in a subsequent section. 

This is not the case for stimulated anti-Stokes radiation. There are two sources of polarization for anti-Stokes radiation [17]. The first is analogous to that in figure B1.3.3(b) where the action of the blackbody (- 2) is replaced by the action of a previously produced anti-Stokes wave, with frequency 03. This radiation actually experiences an attenuation since the value of Im x o3 ) is positive (leading to a negative gam coefficient). This is known as the stimulated Raman loss (SRL) spectroscopy [76]. Flowever the second source of anti-Stokes polarization relies on the presence of Stokes radiation [F7]. This anti-Stokes radiation will emerge from the sample in a direction given by the wavevector algebra = 2k - kg. Since the Stokes radiation is... 

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.

Flere we model the pump beams associated with fields E(a> ) and (102) as plane waves with wavevectors Jti = and Jt, = feiwiv/fii (wil/r - The directions of tlie reflected and transmitted beams can... 

Figure B3.4.12. A schematic ID vibrational pre-dissociation potential curve (wide flill line) with a superimposed plot of the two bound fimctions and the resonance fimction. Note that the resonance wavefiinction is associated with a complex wavevector and is slowly increasing at very large values of R. In practice this increase is avoided by iismg absorbing potentials, complex scaling, or stabilization.

This relationship holds for wavevectors tliat probe tire appropriate size range, R [Pg.2684]

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