Generalized 4 × 4 matrix formalism for light propagation in anisotropic stratified media: study of surface phonon polaritons in polar dielectric heterostructures

Abstract

We present a generalized 4×4 matrix formalism for the description of light propagation in birefringent stratified media. In contrast to previous work, our algorithm is capable of treating arbitrarily anisotropic or isotropic, absorbing or non-absorbing materials and is free of discontinous solutions. We calculate the reflection and transmission coefficients and derive equations for the electric field distribution for any number of layers. The algorithm is easily comprehensible and can be straight forwardly implemented in a computer program. To demonstrate the capabilities of the approach, we calculate the reflectivities, electric field distributions, and dispersion curves for surface phonon polaritons excited in the Otto geometry for selected model systems, where we observe several distinct phenomena ranging from critical coupling to mode splitting, and surface phonon polaritons in hyperbolic media.

Layer-resolved absorption of light in arbitrarily anisotropic heterostructures

“…13 in Ref. 25 ). In non-birefringent media, the two modes are separated into p-polarized (q i1 and q i3 ) and s-polarized (q i2 and q i4 ) waves by analyzing the xcomponent of their electric fields.…”

“…The 4 × 4 transfer matrix formalism comprising the calculation and sorting of the eigenmodes and the treatment of singularities (Section II A), the calculation of reflection and transmission coefficients (Section II B) and of the electric fields (Section II C) is based on our previous work 25 , and therefore is here only briefly summarized arXiv:2002.03832v2 [physics.optics] 3 Apr 2020 in order to provide the necessary framework for the calculation of the layer-resolved absorption (Section III).…”

“…In isotropic layered media, a 2 × 2 transfer matrix fully describes any light-matter interaction, and with knowledge of the local electric and magnetic fields, the optical power flow can be described by the Pointing vector S 21,22 . However, what already proves intricate in isotropic multilayers, becomes even more cumbersome when the materials are uniaxial or even biaxial requiring a 4 × 4 transfer matrix formalism [23][24][25] , as it is the case for many state-of-theart nanophotonic materials like hexagonal boron nitride (hBN) 26 , molybdenum trioxide (MoO 3 ) 27 , or BP 28 . In consequence, to the best of our knowledge, previous approaches aiming at the analytical computation of light absorption in anisotropic multilayers are restricted to special cases [29][30][31][32] , whereas a fully generalized formalism applicable to any number of layers of media with arbi-trary permittivity has not been proposed so far.…”

Surface Polariton‐Like s‐Polarized Waveguide Modes in Switchable Dielectric Thin Films on Polar Crystals

“…Clearly, each spectrum features a distinct resonance dip that corresponds to a p‐polarized SPhP (blue and green curves) or an s‐polarized waveguide mode (red and orange curves). All data were fitted with a global fitting procedure (see the Experimental Section for details), and the resulting theoretical spectra calculated by means of a 4 × 4 transfer matrix formalism are also shown in Figure b, nicely reproducing the experimental data. (The low amplitude of the waveguide mode in c‐GST arises due to a prism coupling gap below the gap of critical coupling, see the Experimental Section for details).…”

“…Global Fitting Procedure : The spectra in Figure b, the dispersion curves in Figures e,f, , and , and the electric field distributions in Figure were all calculated with a 4 × 4 transfer matrix formalism that accounts for the anisotropy of the c‐cut hexagonal 4H‐SiC substrate. For the calculations, most of the material parameters and system variables were set to values from literature or in advance determined values (including the phonon frequencies cm −1 , cm −1 , cm −1 , cm −1 where o stands for ordinary and e for the extraordinary crystal direction, the damping γ SiC = 3.75 cm −1 , , the spectral width of the FEL Δω = 5.56 cm −1 , the angular width of the excitation beam Δθ = 0.24°, and all nine incident angles θ = 26°–34°).…”