A simple and comprehensive electromagnetic theory uncovering the complete picture of light transport in birefringent crystals

Birefringence production of light by natural birefringent crystal has long been studied and well understood. Here, we develop a simple and comprehensive rigorous electromagnetic theory that allows one to build up the complete picture about the optics of crystals in a friendly way. This theory not only yields the well-known refraction angle and index of ellipse for birefringence crystal, but also discloses many relevant physical and optical quantities that are rarely studied and less understood. We obtain the reflection and transmission coefficient for amplitude and intensity of light at the crystal surface under a given incident angle and show the electromagnetic field distribution within the crystal. We derive the wavefront and energy flux refraction angle of light and the corresponding phase and ray refractive index. We find big difference between them, where the phase refractive index satisfies the classical index of ellipse and Snell’s law, while the ray refractive index does not. Moreover, we disclose the explicit expressions for the zero-reflection Brewster angle and the critical angle for total internal reflection. For better concept demonstration, we take a weak birefringent crystal of lithium niobate and a strong birefringent crystal tellurium as examples and perform simple theoretical calculations. In addition, we perform experimental measurement upon z-cut lithium niobate plate and find excellent agreement between theory and experiment in regard to the Brewster angle. Our theoretical and experimental results can help to construct a clear and complete picture about light transport characteristics in birefringent crystals, and may greatly facilitate people to find rigorous solution to many light-matter interaction processes happening within birefringent crystals, e.g., nonlinear optical interactions, with electromagnetic theory.

Surface and Subsurface Quality Assessment of Polished Lu2O3 Single Crystal Using Quasi-Brewster Angle Technique

The sesquioxide Lu2O3 single crystal has attracted tremendous attention as potential host material for high-power solid-state lasers. As polishing is the terminal process of conventional ultra-precision machining, the quality of polished crystal directly impacts the crucial performance indicators of optics. The high melting point of Lu2O3 single crystal makes crystal preparation difficult. Therefore, investigations on the surface/subsurface quality inspection of polished Lu2O3 single crystal are scarce. In this paper, we utilize the quasi-Brewster angle technique (qBAT) based on ellipsometry to inspect the quality of polished Lu2O3 single crystal, achieving fast, non-destructive, and high-sensitive surface/subsurface damage assessment. A systematic crystal processing scheme is designed and polished Lu2O3 crystal samples are obtained. To verify the results of qBAT, the surface and subsurface quality are tested using optical profilometer and transmission electron microscope, respectively. The consistency of the test results demonstrates the feasibility, high sensitivity, and accuracy of the qBAT. To our knowledge, this is the first time that the qBAT is applied to investigate the polished surface/subsurface quality of Lu2O3 single crystal. In conclusion, this method provides a powerful approach to the high-precision characterization of the surface/subsurface quality of Lu2O3 single crystal, and has significant potential for material property study and process optimization during ultra-precision machining.

Investigation of the parameters of the reflected wave near the Brewster angle

In free space, the relative permittivity is determined by the Brewster formula without taking into account dielectric and magnetic losses. In experimental studies, discrepancies in the angular position of the minimum of the reflected wave from dielectric materials are observed in comparison with calculations, which are known as deviations from Fresnel’s laws. By solving the task of inclined falling wave on an plate made of a dielectric material with complex of the dielectric and magnetic permittivity, the parameters of the reflected wave were calculated, according to which the angles corresponding to the minimum reflection were determined, depending on the dielectric losses of the material. From the condition that the reflected wave is equal to zero, a formula for determining the Brewster angle for a material with dielectric and magnetic losses was analytically obtained, the results of calculations for which coincided with the calculations for the reflected wave in the context of geometric optics. It is determined that in the general case, the conditions for determining the position of the minimum of the complex amplitude and the phase jump by 180° of electromagnetic waves do not coincide and can be found only when solving the task an falling wave on a plate with complex electrodynamic parameters of the material in the context of geometric optics.

Optical properties of two-dimensional materials with tilted anisotropic Dirac cones: theoretical modelling with application to doped 8-Pmmn borophene

The optical reflection, transmission and absorption properties of borophene, a newly discovered two-dimensional material with tilted anisotropic Dirac cones, are explored within a simple electronic band structure model of 8-Pmmn borophene, proposed by Zabolotskiy and Lozovik (2016 Phys. Rev. B 94 165403). It is assumed that the borophene layer is deposited on a dielectric substrate, such as Al2O3, and that the borophene's electron density is controlled by an external gate voltage. The reflectance, transmittance and absorbance of the borophene layer, the conduction band of which is filled with electrons up to the Fermi level, are calculated against the frequency of the incident radiation, as well as on the angle of its incidence on the layer. Considered are the two principal cases of the incident radiation polarization either parallel to or normal to the plane of incidence. We reveal that the optical characteristics of 8-Pmmn borophene are distinctly different for the above two cases at all angles of radiation incidence, excepting the grazing incidence, for which the borophene layer is found to behave like a mirror regardless of the wave polarization. The results obtained indicate the possibility of visualizing the borophene layer deposited on a dielectric substrate by observing the minimum reflectivity of this layer at a certain angle incidence (called the quasi-Brewster angle) of the p-polarized radiation, which may differ by a value of about ten degrees from the Brewster angle of the substrate.

Determination of the Brewster angle of the wave reflected from a plate with dielectric losses

In free space, the permittivity of materials is usually determined by the value of the Brewster angle using the angular dependences of the amplitude and phase of the wave reflected from the material plate. An expression corresponding to materials without dielectric and magnetic losses is used as a calculation model. Experimental studies of the parameters of the wave reflected from dielectric materials show the discrepancies with theoretical calculations known as deviations from the Fresnel laws. We present the results of determining the Brewster angle of the wave reflected from a plate made of a material with dielectric losses. The angular dependences of the amplitude and phase of the reflected wave were calculated using the numerical solution of the problem of falling at an arbitrary angle of a plane linearly polarized wave with an electric field vector lying in the plane of incidence on a plate of a dielectric material with complex values of the dielectric and magnetic permittivity. They were used to determine the angles corresponding to the minimum reflection coefficient depending on the dielectric losses of the plate material. The differences between the numerical calculations and the data obtained using the Brewster angle formula were noted, which increased with increasing dielectric losses of the material. From the condition that the modulus of the reflected wave amplitude is equal to zero, a different formula for calculating the Brewster angle for a material with losses is analytically obtained. The results of calculations using this formula coincided with the calculations for the reflected wave when solving the classical problem of the inclined incidence of a plane wave on a plate of a dielectric material in the framework of geometric optics. The results obtained can be used to determine the Brewster angle for a wave reflected from a plate with magnetic and dielectric losses.

Determination of the Optical Properties of Thylakoid Membranes from Whole Leaf Reflectance Measurements Using a Capacitor and Unbound Electron Model

This paper demonstrates that a capacitor equivalent along with unbound electrons can be used to model thylakoid membranes in grana stacks. From whole leaf reflectance measurements at normal incidences at 660nm wavelength and taken from the literature, refractive indices are obtained from the Fresnel equation for transverse electric (TE) and transverse magnetic (TM) polarization. The TE and TM polarizations for external reflectance depict the Brewster angle at which the magnitude of the reflected electric vector is zero; the internal reflections show that there is a narrow angle window of about 10 degrees before the internally refracted light goes into critical angle. The clustering and separation of reflection measurements with angle of incidence is explained using Fresnel equation; the cross-over angle is located beyond the Brewster angle for internal reflection. The predicted relaxation times from a capacitor and unbound electron model gave favorable comparisons against commonly reported fluorescence times in the 0.1 to 1 ns range (our results gave 0.5-0.8 ns). The di-electric constant for the membrane is estimated to be 5. The stacking number (number of grana layers) is consistent with the light penetration depth (skin depth). The magnetic permeability was shown to be close to that of vacuum and therefore the thylakoid lacks any magnetic properties as would be expected for such a transparent media. An in-vivo estimate based on thermal equilibrium of molecules for the permanent dipole moment of the chlorophyll molecule gave 2,025D (Debye).