Abstract
In the present work, the optical chiral properties of a resonant hybrid photonic crystal (RHPC) are studied taking into account the spin-orbit effect of the 2D light-hole exciton. In these systems, the space inversion breakdown of the elementary cell preserves unidirectional exciton–polariton propagation and optical activity occurs, due to the pure dissipative and off-diagonal elements of the material polarization tensor. The non-Bloch solutions of Maxwell equations in such kind of systems are pointed out and their effect on the light propagation and optical rotation is briefly discussed. Finally, the dispersion relation properties of the light-hole exciton polariton are compared with the optical response in a rather large N-cluster of elementary cells \((N=32)\). The physical parameter values adopted in the calculation are close to \(YVO_4/GaAs\) systems.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There are no external data associated with the manuscript.]
References
D. Schiumarini, A. D’Andrea, N. Tomassini, J. Opt. 18, 035101 (2016)
A. D’Andrea, N. Tomassini, Phys. Rev. A. 94, 013840 (2016)
D. Schiumarini, A. D’Andrea, Eur. Phys. J. 92, 92 (2019)
A. Figotin, I. Vitebskiy, Phys. Rev. B 67, 165210 (2003)
A. Figotin, I. Vitebskiy, Slow wave phenomena in Photonic Crystals, arXiv:0910.1220v4 [physics.optics] and A. Figotin, I. Vitebskiy, Laser and Photonics Reviews, Wiley Online Library 5, 201 (2011)
Z.S. Yang, N.H. Kwong, R. Binder, Arthur L. Smirl, J. Opt. Soc. Am. B 22, 2144 (2005)
D. Schiumarini, A. D’Andrea, Eur. Phys. J. B 93, 209 (2020)
F. Jonsson, C. Flytzanis, Phys. Rev. Lett. 97, 193903 (2006)
L.V. Kotova, A.V. Platonov, V.N. Kats, V.P. Kochereshko, S.V. Sorokin, S.V. Ivanov, L.E. Golub, Phys. Rev. B. 94, 165309 (2016)
K. Cho, Reconstruction of Macroscopic Maxwell Equations, STMP, volume 237, (Springer-Verlag GmbH Germany, part of Springer Nature, 2018)
L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of continuous media, 2nd edn. (Butterworth-Heinemann, Oxford, 1984)
G. Bastard, E.E. Mendez, L.L. Chang, L. Esaki, Phys. Rev. B 26, 1974 (1982)
N. Tomassini, A. D’Andrea, L. Pilozzi, D. Schiumarini, Phys. Rev. B. 75, 085317 (2007)
N. Tomassini, L. Pilozzi, D. Schiumarini, A. D’Andrea, Theory of optical response in semiconductor heterostructures. Recent Res. Devel. Appl. Phys. 6, 569 (2003)
M.V. Durnev, M.M. Glazov, E.L. Ivchenko, Phys. Rev. B 89, 075430 (2014)
In order to cure the former asymmetry between Schr\(\ddot{{\rm o}}\)dinger and Maxwell solutions, we will have to come back to Schr\(\ddot{{\rm o}}\)dinger equation for a second cycle of calculation (this computation is in progress and results will be discussed in a paper in preperation)
L. Pilozzi, A. D’Andrea, R. Del Sole, Phys. Rev. B 54, 10763 (1996)
L. Pilozzi, A. D’Andrea, K. Cho, Phys. Rev. B 69, 205311 (2004)
E.L. Ivchenko, A.N. Poddubny, Phys. Solid State 55, 905–923 (2013)
Acknowledgements
The authors are indebted to Norberto Tomassini for the critical reading of the manuscript, and to the ISM-CNR project Complex Optics in Mesoscopic Materials (COMMa).
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AD conceived the study. DS developed the numerical codes and the graphical parts. AD and DS contributed to analyzing the results, to write and to revise the paper.
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A Appendix
A Appendix
The Cartesian electric field components of light-hole exciton–polariton in a quantum well are TM \(\left( E_{x},E_{z}\right) \) and TE \(\left( E_{y}\right) \) [3, 7] while for heavy-hole exciton (\(P_z (Z,\omega ) \rightarrow 0.0\)) are TM \(\left( E_{x}\right) \) and TE \(\left( E_{y}\right) \) [3]; in this case, the resultant field is expressed by the following relations:
where \(q=\omega /c\) and the Green’s functions are: \(g_w^ \pm (Z,Z') = \theta \left( {Z - Z'} \right) \,e^{ik_z (Z - Z')} \pm \theta \left( {Z' - Z} \right) \,e^{ - ik_z (Z - Z')} \), while the quantities \(W_{\alpha ,\beta }\) for \(\alpha ,\beta =x,y,z\) are the normalization constant of the Cartesian components of the Green’s functions [3]. The self-consistent solutions are obtained by substituting the polarization components, given in Eq. (6), in the Maxwell Eqs. (A1)–(A3), and finally, by solving the transfer matrix by using the normal Maxwell’s boundary conditions, because additional boundary conditions (ABCs) are not necessary.
It is well known, from the model calculation on 2D Bragg quantum wells [18, 19], that, using the no-escape boundary conditions, the Wannier exciton model of 3D bulk semiconductors becomes a 2D Wannier exciton on the (x, y)-plane, while a Frenkel exciton model is well suited to describe its propagation along z-axis [13, 14]. Moreover, since the periodicity is:, \(d=L_w+2L_B>>{a_B}^{2D}=a_B /4\), where \(a_B\) is 3D exciton Bohr radius and \(L_B\) the barriers thickness, also the dipole–dipole interaction between next near neighbourings becomes negligible small for rather large barrier thicknesses (\(L_B>>{a_B}^{2D}\)); therefore we can adopt an impurity model for 2D Wannier exciton and consider only the band effect due to the exciton–polariton periodicity. Notice that, from the point of view of Maxwell equation in the Coulomb gauge, the former approximation means to neglect a very small part of the longitudinal component of the total electric field [10] or otherwise to embody its small effect in non-radiative homogeneous broadening \(\varGamma _\mathrm{NR}(\omega )\). In a two band model, Schr\(\ddot{\mathrm{o}}\)dinger equation of Wannier exciton, perfectly confined in a 2D quantum well, is similar to a non-adiabatic hydrogenic atom, namely:
where \(\overrightarrow{r} = \overrightarrow{r}_e - \overrightarrow{r}_h \) and \(\overrightarrow{R} = \frac{{m_e \,\overrightarrow{r}_e + \overrightarrow{m}_h \,\overrightarrow{r}_h }}{M}\) are relative and center-of-mass coordinates, respectively, while M and \(\mu \) are the total and relative mass of Wannier exciton.
Notice that, by imposing the so called no-escape boundary conditions at barrier-well interfaces:
the Schr\(\ddot{\mathrm{o}}\)dinger solution could take into account e-h boundary states only. Finally, the solution of Schr\(\ddot{\mathrm{o}}\)dinger equation is obtained by a variational energy minimization as a function of Bohr radius parameter value (\(a_{B}\), [12]).
The dispersion curves and the optical response of the uniaxial layers can be computed from the material equations:
where
is the left (L) and right (R) dielectric tensor of each anisotropic layer with respect to the isotropic slab, namely:
and
where \(\alpha \) and \(\beta \) are the angles of rotation of \(\hat{C}\)-axis in-plane (x, y) and in the plane of incidence (x, z) respectively. The optical response of both (L/R) uniaxial layers is obtained by solving the Maxwell equations as an eigenvalue problem, and to use the Maxwell’s boundary conditions at the interfaces, for computing the optical transfer matrix of the elementary cell [3].
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Schiumarini, D., D’Andrea, A. Optical activity in resonant hybrid photonic crystals and clusters. Eur. Phys. J. B 94, 150 (2021). https://doi.org/10.1140/epjb/s10051-021-00146-3
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DOI: https://doi.org/10.1140/epjb/s10051-021-00146-3