2D polaritons

Thus, flK(co) is imaginary for K < (o/c and real for K > cu/c these two cases correspond, respectively, to a radiatively unstable 2D exciton and to a radiatively stable 2D polariton. Solution of equations (1.3) and (3.12) provides the complete description of the 2D polariton dynamics. We analyze below the 2D polariton at different orders of exciton-photon coupling. 

Figure 3.10. Scheme of the 2D polaritons and radiatively very unstable 2D excitons in the coupled system of an exciton K and an effective photon continuum (a) The two subsystems are not coupled, (b) The coupled system with a discrete state split off below the continuum, called the 2D polariton excitonic solutions exist only in a small segment of the Brillouin zone,... 

This split-off discrete state rejoins, for cK co0, the exciton energy ha>0 it behaves qualitatively in the same way as the lower branch of the 3D polariton.33 35 For this reason we call it the 2D polariton. It is the projection of the exciton K> on this 2D polariton (radiatively stable) that constitutes (1) the finite limit value of the curves AK t) for t- oo (Fig. 3.8), and (2) the weight of the discrete peak in the spectrum PK((o) (Fig. 3.9). The transition, in the 2D polariton branch, between the photon and the pure exciton characters occurs around the value K0 = co0/c in an area of width AK = r0/c (with ro = 15cm 1). Thus, the 2D polariton may be considered as a photon mode trapped in the 2D lattice, where it acquires its own dispersion.115,116,126 Therefore, the 2D polaritons cannot be excited by free photons, but they may be coupled to evanescent waves, by ATR for example.115,116... 

Note that disorder, which was neglected above, is also able to destroy coherent superradiant emission of ID and 2D polaritons. This can happen when the uncertainty of the exciton wavevector arising due to scattering of an exciton by disorder will be of the order of value Eexc/hc. A quantitative theory of this effect has been developed in the paper by Orrit et al. (23). 

A theory of 2D excitons and polaritons is presented for this type of surfaces, with continuity conditions matching 2D states their 3D counterparts in the bulk substrate, investigated in Sections I and II. This leads to a satisfactory description of the excitations (polaritons, excitons, phonons) and their theoretical interactions in a general type of real finite crystals A crystal of layered structure (easy cleavage) with strong dipolar transitions (triplet states do not build up long-lived polaritons). 

Figure 3.7. Dispersion of the 2D monolayer polariton real part (left) and imaginary part (right) of the excitonic energy renormalization RK(

Figure 3.8. The exciton decay in photon and polariton states The time evolution (in units of w0r) of a 2D exciton K created at r = 0 (Kid). This decay, illustrated for various wave vectors (in units of iu0/c), is purely exponential for K < to0/c, but exhibits very complex transient oscillatory behavior in the region K - oj0/c. For K > o>Jc the 2D exciton is radiatively stable.

The model of an isolated layer was refined by introducing substrate effects by coupling the surface 2D excitons to the bulk polaritons with coherent effects modulating the surface emission and incoherent k-dependent effects damping the surface reflectivity and emission, both effects being treated by a KK analysis of the bulk reflectivity. The excitation spectra of the surface emission allowed a detailed analysis of the intrasurface relaxation dominated by resonant Raman scattering, by vibron fission, and by nonlocal transfer of... 

The plane waves of a perfect 2D lattice diagonalize the electromagnetic interactions, giving rise to the excitonic dispersion through the Brillouin zone, and to the surface-exciton-polariton phenomenon around the zone center.148,126 The corresponding hamiltonian may be written as... 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

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. 

Note, however, that the dependence of the polariton energy on the wavevec-tor, which arose when only retardation is taken into account, is correct only if we can neglect the dependence of the energy of the Coulomb exciton Etl on k, arising from instantaneous Coulomb interaction. For example, if we apply this theory for 2D quantum well polaritons, the linear term in the dispersion of po-laritons will be cancelled because in this case the linear term as a function of the energy of the quantum well exciton on the wavevector has the same value with opposite sign. 

The appearance of an enhanced radiative width and renormalization of the exciton dispersion are the main effects arising in one- and two-dimensional structures under the influence of retardation (21). Qualitatively these effects are valid for Frenkel as well as for Wannier-Mott excitons. In contrast to 3D structure where in the exciton-photon interaction all three components of the momentum have to be conserved and as a result a picture with 3D polaritons arises, for structures of lower dimensionality only the in-plane momentum for 2D structures is conserved and only one component for the ID structure. An exciton in both cases is coupled to a continuum of photon states. There is no possibility of reversible strong... 

The numerical simulations performed in (32), (33) for one-dimensional microcavity containing organic material with diagonal disorder qualitatively confirm the conclusions made above on the structure of the polaritons spectrum in an organic microcavity. Future simulations for realistic 2D organic microcavities would be very topical. 

Finding polariton states in disordered planar microcavities microscopically is a difficult task which do not attempt here. As a first excursion into the study of disorder effects on polariton dynamics, here we will follow (32) to explore the dynamics in a simpler microscopic model of a ID microcavity. Such microcavities are interesting in themselves and can have experimental realizations from the results known in the theory of disordered systems (39) one can also anticipate that certain qualitative features may be common for ID and 2D systems (38). 

On physical grounds (see Section 10.3 and (15)) one should expect that low-energy polaritons in 2D organic microcavities would also be rendered strongly localized by disorder, as it would also follow from the general ideas of the theory of localization (39). Further work on microscopic models of two-dimensional polariton systems is required to quantify their localization regimes. 

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