Polaritons transverse

Surface plasmon-polaritons (SPP), also referred as to surface plasma waves, are special modes of electromagnetic field which can exist at the interface between a dielectric and a metal that behaves like a nearly-iree electron plasma. A surface plasmon is a transverse-magnetic mode (magnetic vector is perpendicular to the direction of propagation of the wave and parallel to the plane of interface) and is characterized by its propagation constant and field distribution. The propagation constant, P can be expressed as follows ... 

This is the background for the Lyddane-Sachs-Teller relation to be treated below. For transverse optical vibrations the origin of an is field is less obvious, but it is also present and its reaction on the eigenfrequency of the TO phonon later gives rise to the polaritons. 

One can show that for this solution B, P, and w are perpendicular to k and the z-direction. Phonons and polaritons for which condition (11.29) is fulfilled are called ordinary . Because of w 1 k there are by definition only transverse ordinary phonons. In Eq. (11.29) the right-hand side is... 

LiI03 is a uniaxial hexagonal crystal (factor group C6). Vibrations of species A, E, and E2 are allowed in the Raman effect, but only A and E, are infrared-active, therefore polariton dispersion is expected for the transverse phonons of these two species. The phonon and polariton spectra were investigated by Claus 26>27> and Otaguro et al. 28>29). Here we want to show two series of spectra recorded by Claus. 

Plasmon-phonon coupling represents mixing of two quasi-particles. The coupling of three quasi-particles has also been observed. The term plasmariton was used by Alfano 45) for a coupled state of a TO phonon and a dressed photon , namely, a photon surrounded by an electron cloud (a coupled state of a plasmon and a photon). The quasi-particle dressed photon is also called a transverse plasmon. Because the coupled state of a photon and a TO phonon has been termed polariton, a plasmariton can also be regarded as coupled state of a plasmon and a polariton. Earlier the term plasmariton was used in a more restricted sense, namely, when a partly transverse character of the plasmon is induced by an external magnetic field. 

We note that ionic crystals may have dielectric functions satisfying Eq. (4) for frequencies between their transverse and longitudinal optic phonon frequencies. SEW on such crystals are often called surface phonons or surface polaritons and the frequency range is the far IR. 

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

FIGURE 6 Evolution of the intensity of the reflectance structures with in-plane orientation of the electric field (a) [10-10] orientation, (b) [-12-10] orientation [8], Note that the increase of Ais is accompanied by a decrease of Bu and makes it easier to detect A2S. Arrows indicate the average positions of transverse excitonic polaritons The eigenenergies, which are different for the two polarisations due to the spin exchange interaction, can be obtained from a lineshape fitting of the reflectivity spectra. 

The dielectric is often assumed to be isotropic in order to simplify Eq. (8) by assuming transverse phonon-polaritons the extension to anisotropic media is straightforward (31). In the limit of very short pulse duration compared to the phonon-polariton oscillation period, the time-dependence of the excitation field can be treated as a delta function, and the phonon-polariton response is given by the impulse response function for the spatial excitation pattern used. If crossed excitation pulses are used, then it is simplest to describe the excitation and response in terms of the excitation wavevector or wavevector range. 

First of all we have to mention that the above described situation of resonance is not related to any quantum effects. Moreover, the role of the transverse electromagnetic field in crystal oscillations in the infrared part of the spectrum was discussed by means of the classical dynamics of crystal lattices a long time ago by Born and Ewald (2) (see also (3) and (4)), and later by a semiphenomenological approach in (5), (6). It is evident, however, that a quantum theory of polaritons in the region of electronic transitions can also be important particularly for the discussion of quantum effects. 

It follows from the above relation that the retarded interaction is important only in the vicinity of wavevectors k y/eoQ/c, i.e. in that part of the spectrum, where the frequencies of the Coulomb excitons are near to those of the transverse photons. When the retardation is ignored, the branches of the Coulomb excitons and the transverse photons intersect (Fig. 4.1a). This intersection is removed when the retardation is taken into account (Fig. 4.1b). In a similar way the dependence w(k) for polaritons can be found for crystals with different symmetries. 

Fig. 4.3. The dispersion of polariton in cubic crystals. Nongyrotropic crystals (a) The dependences of exciton and photon energy on wavevector, the retardation neglected (b) the same but with retardation taken into account. The symbols and L indicate longitudinal and transverse polarization of excitons (c) retardation neglected but dependence of the exciton energy on the wavevector taken into account here and in (d), (e), and (f) only the lower branch of the polaritons shown (d) the retardation and dependence of exciton energy on wavevector are taken into account. Gyrotropic crystals (e) Dispersion of excitons in the cubic gyrotropic crystals if retardation is neglected (f) the same when retardation is also taken into account Aq denotes the position of the bottom of the polariton energy.

In order to discuss more correctly the question of radiative width we keep in mind the fact that the operator (4.88) provokes not only a radiative damping of exciton states, but also changes their dispersion rule. To obtain this dispersion, we add to (4.88) operators of the free exciton and transverse photon fields, as was done in Sections 4.1 and 4.2, and diagonalize the total Hamiltonian so obtained. Recall for comparison that in the case of an ideal 3D crystal after such diagonalization of the total Hamiltonian the radiative width of new excitations (polaritons) disappeared. We show below that in ID and 2D the results are completely different. 

A is the wavelength of light with frequency oj = t o /h, C phonons of this kind strongly interact with the transverse photons. As a result, in the region of long wavelengths, new elementary excitations - phonon-polaritons (see Ch. 4) - are formed instead of C phonons and transverse photons. 

To take the interaction between phonons and photons into consideration, it is necessary to add to the Hamiltonian (6.32), the Hamiltonian Ho(a) of the free field of transverse photons and the Hamiltonian Hint for the interaction of the field of transverse photons with phonons. The linear transformation from the operators a and C to the polariton creation and annihilation operators, i.e. to the operators t(k) and p(k), diagonalizes the quadratic part of the total Hamiltonian. The two-particle states of the crystal, corresponding to the excitation of two B phonons, usually have a small oscillator strength and the retardation for such states can be neglected. In view of the afore-said, the quadratic part of the total Hamiltonian with respect to the Bose operators can be written in the form of the sum H0(B) + where... 

Polaritons in cubic crystals can be transverse or longitudinal and as we neglected the spatial dispersion, the polariton dispersion law, i.e. the dependence... 

FlG. 6.11. Polariton dispersion in the NH4CI crystal. Longitudinal-transverse splitting of the biphonon is observed in the frequency region v 1460 cm-1 (from (61)). 

The relations (7.27) and (7.33) express the dielectric tensor of the crystal in terms of polariton states. Below we apply it for the case of cubic crystals. In cubic crystals the normal waves p with small wavevectors are either transverse, or longitudinal. Therefore the product S m(k)S ]jin(k) in this medium always vanishes. In consequence (see also eqn 4.48)... 

The values a(w,k) and b(v, k) have resonances at frequencies corresponding to longitudinal and transverse polaritons. If one takes into account the dissipation, the imaginary parts of the polariton energies would appear in the denominators of these expressions. As longitudinal excitons do not interact with the transverse electric field, the resonances of a(v, k) coincide with the frequencies of longitudinal Coulomb excitons. 

The consideration of the strong dependence of dissipation of polaritons near exciton resonances will be performed below with the use of transverse dielectric tensor ej y(w, k). This tensor will be calculated assuming that the excitonic states, with complete account of the Coulomb interaction, are known. Since the derivation of the expression for k) is given in the monograph (3), see... 

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