Cavity polariton dispersion

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. 

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].)... 

In Subsection 10.2.2, the Maxwell equations in an anisotropic microcavity are discussed and some important facts concerning Coulomb and mechanical excitons are summarized. In Subsection 10.2.3 the dispersion equation of cavity polaritons is derived. Its solutions for the cases of crystals with one and two molecules in the unit cell are discussed in Subsections 10.2.4 and 10.2.5, respectively. The main results are summarized in the conclusions. 

The determinant of this system of equations yields the dispersion law of the cavity polaritons. 

We introduce Q2 = q2 + k2 and write the dispersion equation of the cavity polaritons in the form ... 

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... 

Ti31 is the polariton splitting if only the semiconductor QW is present, 2 T23 if only the organic QW is present. Equation (13.104) gives the dispersion of the cavity polaritons. We assumed above that e e to neglect the radiative width of states. However, eqn (13.100) also contains information on the radiative widths. For example, taking for the factor r (eqn 13.98), the more exact expression r — e-i L(l + 2/ /q), we obtain for H(fc) = fV(k) — ifl"(fc) for the lowest cavity modes... 

FlG. 13.15. (a) Dispersion curve for cavity polariton and energy relaxation of Wan-nier-Mott excitons (b) coupling coefficient of the lower branch to cavity photon and to exciton. 

Fig. 13.15 Dispersion curve for cavity polariton and coupling coefficients 406...

Typical examples of the dispersion curves are shown in Fig. 10.2 for 99 = 7t/4, small positive detuning u = ivc — d>i = 35 meV, and different relations between Wy, W2 and Ac = 2 — uq. It is clear that when one of the coupling parameters IF is small, the spectrum consists of a doublet of polariton branches and of two branches which are close to the bare cavity photon and the bare exciton (Fig. 10.2a). When the Davydov splitting is small and W W2, then the pairs of the dispersion curves almost overlap (Fig. 10.2b). The electric fields in the overlapping curves are (with accuracy up to terms of the order of q2 IQ2) perpendicular to each other (see the discussion in Subsection 10.2.4). When all the parameters are of the same order, the spectrum consists of four well-pronounced polaritonic branches (Fig. 10.2c). 

In this section we examine the solutions of Maxwell s equations for a system with a broad and dispersionless electronic resonance. We show that these conditions result in the appearance of the end-points of the lower and upper polariton branches. These end-points restrict the intervals in which the polariton states have well-defined wavevectors. This consideration is applicable, in particular, to the disordered system of J-aggregates since each J-aggregate chain possesses rather narrow electronic transitions instead of broad dispersion (Fig. 10.3). The disorder present in the system does not influence the following arguments, since for small-cavity photon wavevectors, the system can be treated as effectively homogeneous. 

FlG. 10.5. The dispersion curves of the coherent polaritonic states (solid lines) and of uncoupled cavity photons and the molecular excitation (dashed lines). The crosses show the end-points of the part of the spectrum with well-defined wavevector. On the right, the broadened line of the molecular resonance is shown. The inset shows the excitonic weights (10.26) for upper ( cix ) 2) and lower ( cix ) 2) polaritonic branches. Reprinted with permission from Agranovich et al. (15). Copyright 2003, American Physical Society. 

Fig. 10.5 The dispersion curves of the coherent polaritonic states and of uncoupled cavity photon and the molecular excitation 287...

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