Mid infrared polarization engineering via sub-wavelength biaxial hyperbolic van der Waals crystals

Mid-infrared (IR) spectral region is of immense importance for astronomy, medical diagnosis, security and imaging due to the existence of the vibrational modes of many important molecules in this spectral range. Therefore, there is a particular interest in miniaturization and integration of IR optical components. To this end, 2D van der Waals (vdW) crystals have shown great potential owing to their ease of integration with other optoelectronic platforms and room temperature operation. Recently, 2D vdW crystals of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoO}_{3}$$\end{document}MoO3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {V}_2 \hbox {O}_5$$\end{document}V2O5 have been shown to possess the unique phenomenon of natural in-plane biaxial hyperbolicity in the mid-infrared frequency regime at room temperature. Here, we report a unique application of this in-plane hyperbolicity for designing highly efficient, lithography free and extremely subwavelength mid-IR photonic devices for polarization engineering. In particular, we show the possibility of a significant reduction in the device footprint while maintaining an enormous extinction ratio from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoO}_{3}$$\end{document}MoO3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {V}_2$$\end{document}V2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_5$$\end{document}O5 based mid-IR polarizers. Furthermore, we investigate the application of sub-wavelength thin films of these vdW crystals towards engineering the polarization state of incident mid-IR light via precise control of polarization rotation, ellipticity and relative phase. We explain our results using natural in-plane hyperbolic anisotropy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoO}_{3}$$\end{document}MoO3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {V}_2$$\end{document}V2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_5$$\end{document}O5 via both analytical and full-wave electromagnetic simulations. This work provides a lithography free alternative for miniaturized mid-infrared photonic devices using the hyperbolic anisotropy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {MoO}_{3}$$\end{document}MoO3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {V}_2$$\end{document}V2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_5$$\end{document}O5.

This 4 X 4 matrix algebra describes the monochromatic plane wave propagation through the entire layer system. The incident and transmitted electromagnetic waves, for ith layer of material with di thickness, can be connected with partial transfer matrix Tip so that the ordered product of all Tip corresponds to the transfer matrix T. Furthermore, an incident matrix (La) represents the projection of in-plane component of the incident and reflected electromagnetic waves at the first interface. Likewise, Lf represents the amplitude of transmitted light from the last interface. Therefore, one can write the transfer matrix T as: The value of La, Tip, and Lf can be obtained from the solution of Maxwell's equations at the respective interfaces given as: From these coefficients, we evaluate the following:

S3: Analytical model for Fabry Perot modes in vdW thin films:
The high value of the real part of principle components of dielectric tensor of selected vdW crystals below the corresponding TO phonon frequencies results in the confinement of large wavelength electromagnetic waves in the sub-diffractional thickness of the film, forming the Fabry-Perot cavity. These Fabry-Perot modes are observed as a dip or a peak in the reflectance (transmittance/absorbance) spectrum. To evaluate these modes of absorption, we consider the fundamental condition of maxima in Transmittance/Absorbance in the Fabry Perot Here, m, d, n, and k0 (= ω/c) corresponds to the order of mode, the thickness of film, the refractive index in the particular crystal direction, and free space wave-vector, respectively.
Refractive index can be evaluated from the square root of the complex dielectric function predicted by Lorentz oscillator model 3 (Eq.1). Taking Г = 0 and placing it the Eq. S6, we will get: After squaring and rearranging the equation S7, it forms a quadratic equation given as: , the equation has one feasible physical solution represented by the following equation:

Mid-IR polarizers figure of merits in RB-2 spectral region:
We  Fig. 1(b)). Similarly, for α-V2O5 based thin film mid-IR polarizer, the transmission efficiency remains less than 55% over the entire RB-2 spectral region (shown in Fig. S4(b)) and hence cannot work as an efficient mid-IR polarizer in RB-2.

b) represent ER of the α-MoO3 thin film on KRS-5 substrate 6 in the transmission and reflection mode, respectively, as a function of frequency and thickness. Similarly, (c)&(d) represent ER of the α-V2O5 thin film on KRS-5 substrate in the transmission and reflection mode, respectively, as a function of frequency and thickness. It is observed that both the vdWs materials exhibit almost similar properties on the KRS-5 substrate and the silicon substrate.
(a) (b) (d) (c) S7: α-MoO3 and α-V2O5 based polarization rotator in transmission mode: Figure S7 (a)-(c) represent the phase difference between x-and y -components, angle of polarization state (ϕ)  and logarithmic ellipticity, respectively, of transmitted light through 1.1 μm thin film of α-MoO3. Similarly, (d)-(f) represent the phase difference between x-and y -components, angle of polarization state (ϕ) and logarithmic  ellipticity, respectively, of transmitted light through 1.0 μm thin film of α-V2O5. Here, scatter plots and