Wavevector incident

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... 

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. 

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 Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

Fig. 25. He energy loss sp tra from a Kr mono-layer taken along the f azimuth. With decreasing incident angle 9 phonons with larger wavevectors are sampled.

The zone axis coordinate system can be used for specifying the diffraction geometry the incident beam direction and crystal orientation. In this coordinate, an incident beam of wavevector K is specified by its tangential component on x-y plan = k x + k y, and its diffracted beam at Kt+gt, for small angle scatterings. For each point inside the CBED disk of g, the intensity is given by... 

Figure 1.3 The addition of waves scattered by an angle 2 from an atom at the origin and one at a vector r from the origin. The wavevectors k q and k jj are in the directions of the incident and diffracted beams, respectively, and k o = k h =l/...

The polarisation factor, C=D g. D j,. This is unity for polarisation, in which the electric field vector is perpendicular to the dispersion plane and cos(2 g) for polarisation in which the electric field vector lies in the dispersion plane. We may apply equation (4.20), which since h is real implies that Im(K o)=Im(K ), and, in order to express the solution simply, introduce the very important idea of deviation parameters, g and. These express the deviation of the incident and diffracted wavevectors from the kinematic assumption where k g = k ij =l/, and are defined as... 

Figure 4.18 The effect of spherical incident waves on the excitation of Bloch waves, (a) Reciprocal space the divergent incident beam has wavevectors ranging from P j O to P 2 O. (b) Real space energy is distributed throughout the Borrmann fan ABC. The beams generated outside the crystal are indicated...

If the specimen is moved away from the focal position, then this will cause a phase shift that depends on 6. If the wavenumber in the coupling fluid is k = 2n/Xo, then the z component of the wavevector is kz = k cos 6. Defocusing the specimen by an amount z causes a phase delay of 2zkz, or 2zk cos 0 (the factor of two arises because both the incident wave and the reflected wave suffer a change in path length). Expressing this phase delay as the complex exponential of a phase angle, the response of the microscope with a defocus z is... 

If the z direction is taken as the normal to the surface, and the x direction is taken as also lying in the plane of incidence, then in calculating the Christoffel matrix T for arbitrary crystallographic orientation the stiffness matrix must be transformed to these new coordinates. In the notation of (11.2), and for an incident wavevector (k x, k y, k z) in the fluid, the requirements of Snell s law become... 

In order to calculate V(x, z) for a spherical lens in the presence of a crack a summation must be made over ky. For each value of ky a double summation is first made over kx and k x. A wave is considered to be incident with components of wavevector k x and ky. It is then transmitted by the crack with components of wavevector kx and ky, and reflected with components —kx and ky transmitted and reflected waves may be summed in the same integration. If the axis of the lens is displaced a distance x from the crack, the resulting phase change is kxx for the incident wave and —kxx for the scattered wave. Then, by extension of eqn (12.2),... 

Fig. 2.54 Neutron reflectivity profile for a symmetric PS-dPMMA diblock (Mw 30 kg moP1) as a function of incident wavevector (Russell 1990). The inset shows the scattering length density (b/V, the neutron scattering length per unit volume) profile normal to the film surface that was used to calculate the reflectivity profile shown as the solid line, This is typical of a block copolymer film containing a multilayer stack, with lamellae parallel to the surface.

On one level it is a quantum effect, and can be described in terms of photon—phonon scattering. The incident NIR beam is a source of photons, and the energy from the piezotransducer provides a source of lattice phonons that propagate through the crystal. As in all collision processes, the twin principles of conservation of momentum and conservation of energy apply. The momentum of a quantum particle is linked to its wavevector by hk. The energy is linked to its frequency by hjj. 

Consider first the simpler case of acousto-optic interaction in an isotropic medium (i.e. a standard acousto-optic modulator). If the wavevector of the incident light is ko, that of the scattered light k+ or k and that of the phonon K, we have for conservation of momentum... 

The incident plane contains the incident and the reflected beam wavevectors. The parallel ( ) and perpendicular (T) polarization components of the electric field are defined relative to the plane of incidence. 

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