Polaritons dispersion

Fig. 4. Polariton dispersion curves for a cubic crystal with three infrared active phonons 17>...

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. 

Therefore the dispersion of the LO plasmon-phonon states is formally equivalent to the dispersion of the TO photon-phonon states, with 4irne2/m replacing k2 c2. When the plasmon-phonon frequency to is plotted against fn instead of k, dispersion curves for the LO modes are obtained which are similar to the polariton dispersion curves, the TO phonons showing no dispersion with /n. 

Figure 1.3. Sketch of the polariton dispersion for a given direction K (notice the scale change to cover the entire Brillouin zone). The broken straight lines indicate the dispersion of the electromagnetic waves in the crystal far from the excitonic b transition. In the stopping band (hatched), only excitonic states with large wave vectors may be created, and the crystal reflection is "quasi-metallic .

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.

Figure 1 Phonon-polariton dispersion in LiTa03. The solid lines describe the dispersion of the upper and lower branches of the polariton, the dashed line describes the dispersion of light at frequencies below the phonon resonance, and the dotted line describes the dispersion of light at frequencies above the phonon resonance.

The experimental and theoretical investigation of higher order vibrational motion and relaxation in solids should now advance quite rapidly as, in fact, there are no lack of suitable problems and the methods to tackle them. We note in this context, and in anticipation of the section on vibrational energy propagation, that the existance of polariton bound states has been inferred from the production of additional gaps in the phonon-polariton dispersion curves of ionic crystals having a molecular subgroup. ... 

Figure 8. Polariton group velocity F, normalized to speed of light in vacuum C ( ) for v, polariton in NH Cl at 78 K, measured directly using a picosecond technique, as a function of polariton frequency in cm . Full line is calculated using known spectroscopic line positions and polariton dispersion relation [Eq. (132) (From Burstein and Martini. )]...

A generalization of biphonon theory beyond the Van Kranendonk model was made later (14)—(17). Subsequently, the effect of biphonons on polariton dispersion in the spectral region of two-particle states was investigated in a number of papers (18)—(22), and the contribution of biphonons to the nonlinear polarizability of a crystal was discussed in (23)-(25). Problems of the theory of local and quasilocal biphonons in disordered media were discussed in a number of papers (14), (26) -(28). The influence of anharmonicity in crystals on the spectra of inelastically scattered neutrons was considered by Krauzman et al. (29), Prevot et al. (30), and in Ref. (31). 

FlG. 6.6. Polariton dispersion in the Fermi resonance region, neglecting anharmonic-ity 2 (0) is the energy of the longitudinal vibration at k = 0 min and max are the minimum and maximum energy values in the band of two-particle states. 

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)). 

Inside the band of two-particle states a channel opens up for polariton decay into two phonons. This process leads to a broadening of the polariton line (see, e.g. (59), (60)), and also a change in the polariton dispersion law. The two most important effects in the latter case are (a) interference of scattering by a polariton and two-particle states (this can lead to drops in intensity of the Fano antiresonance type see (22)), and (b) the presence of singularities in the density of two-particle states (those, in particular, that correspond to quasibiphonons). 

FlG. 6.14. The polariton dispersion curve of ammonium chloride in the vicinity of the 2 4 two-phonon quasicontinuum (shaded region) obtained by near forward Raman scattering (filled circles). The dashed curve is the calculated dispersion curve in the absence of polariton Fermi resonance. Reprinted with permission from Mitin et al. (71). Copyright (1975), American Physical Society. 

When the frequency of the surface biphonon lies within the band of the surface polariton, Fermi resonance occurs and the dispersion curve of the po-lariton is subject to a number of essential changes (gaps appear, etc. (86)). Consequently, experimental research of surface polariton dispersion under these conditions could yield, like similar investigations of bulk polaritons, a great deal of interesting information, not only about the surface biphonons themselves, but about the density of states of surface phonons and the magnitude of their anharmonicity constants as well. 

Thus the cavity polariton dispersion has a simple interpretation. Let us calculate the electric fields from eqn (10.7) for the modes (10.16). Neglecting small terms of the order of q2 /n2, the fields Ei and Et are related in these modes by Ei = —Et cot (p. Then the y-component of the fields Eu,l is equal to zero. In other words, with accuracy up to small terms (of the order of q2 /k2) the total in-plane electric field in the polaritonic modes is parallel to the dipole moment Pi for any direction of the wavevector q, and the value of the Rabi splitting energy thus does not depend on the wavevector direction. 

The predicted in-plane isotropization of the polariton dispersion in an anisotro -pic crystalline organic microcavity can be observed in the spectra of reflection, transmission and photoluminescence. 

Now let us consider the wavevector broadening of the upper polariton states for large q. At large wavevectors the upper polariton dispersion curve tends to that of the cavity photon, and 5q 7o(A2e63/2/cj2h c3) Rabi splitting, for large q the upper cavity polariton branch contains the coherent states only. 

Just such a situation takes place for microcavity dispersion at the bottom of the lower and upper polariton branches in a microcavity with a = h/2M where M is the effective mass of the cavity polariton. Of course, specific features of the low-energy wavepackets stem from the fact that the polariton dispersion near the... 

If the polarizabilities a are known, eqn (12.78) determines the surface polariton dispersion as a function of the isotopic composition of the mixture. 

Fig. 6.6 Polariton dispersion in the Fermi resonance region, neglecting anharmonicity 191...

Fig. 6.14 The polariton dispersion curve of ammonium chloride in the vicinity of the 2rq two-phonon quasicontinuum 208...

The evaluation of the coupling constants g, the most tedious part of the polariton calculation, is described in more detail in Ref.30. Here we only show the resulting polariton dispersion curves for PTS again (Fig.3). 

Kojima, S., Tsumura, M., Takeda, M.W. and Nishizawa, S. (2003) Far-infrared phonon-polariton dispersion probed by terahertz time-domain spectroscopy. Phys. Rev. B, 67, 035102(1)-035102(5). 

Figure 3.8 A simplified sketch of the exciton-polariton dispersion.

Figure3.43 (a) Schematic representation ofthesefdrffiactionlMO-pump FWM experimental setup, (b) Two-pulse spectrally resolved FWM spectrum from bulk ZnO obtained by using ft polarization at a delay time of 0.01 ps. The dotted lines show the polariton dispersion A denotes the lower A polariton branch and B and C represent the mixed branches due to the upper A/lower B and upper B/lower C polaritons, respectively. (After Ref [211].)...

Figure8.17 A simple sketch ofthe cavity polariton dispersion. 6 is measured from the surface normal.

Figure 8.18 (a) Angle-resolved PL spectra at RT in the range of 0 -40° for a X-thick ZnO hybrid MC. The dotted line is the exciton mode. The solid lines are guides to the eye. (b) Experimental cavity polariton dispersion curve. The dashed lines represent the cavity and exciton modes. (Courtesy of R. Shimada [140].)... 

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