In this article a computer model of the polariscope is regarded as a 3D scene. In this case, the interference pictures are the result of rendering. The light rays pass through several well-specified polariscope blocks. When developing a suitable renderer, algorithms are selected and estimated for calculating the behavior of the beams based on their physical correctness, speed, etc. A plane parallel plate of an anisotropic crystal is the main block of the scene that affects the resulting image. This article discusses the calculation of the interaction of light with this plate only. Two approaches to calculate orthoscopic interference pictures of optically anisotropic transparent crystals are considered. One is described in many well-known books and bases on definite simplifications. The other is a direct physically based modeling of a light ray path through a plane parallel plate made of a uniaxial crystal taking into account all losses of intensity while passing boundaries between media. The purpose of this paper is to estimate a difference between values obtained via different approaches Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Mineralogists watch orthoscopic and conoscopic interference pictures (patterns,
figures) while look at anisotropic transparent minerals through a polarizing microscope. A
polariscope is a simplest optical device that allows to see such pictures. Mineralogists
[
A virtual polariscope (VP) is our development of a computer model of a polariscope
[
Specialists make decisions on minerals taking into account colors they see. There
are standard color charts that support this process. Such charts are called Michel-Levy
Color Charts (or nomograms) [
In modern conditions, many charts are digitized and available on the Internet, see
[
The polariscope consists of a small number of blocks, each of which performs a
well-specified task of processing polarized light rays. Every microscope includes a lens.
We have already had to develop a computer model of a lens made of an isotropic
transparent material that physically correctly processes rays of linear polarized light, because
the developers did not pay attention to this issue or paid very little attention, see [
Comparison of Two Approaches to Calc. Interference Pictures 3
The idea of this article arose when reading the work [
The module for calculating the physically correct passage of light in the studied
mineral sample is one of the main ones in forming the final image. In this article, based
on a direct physically correct simulation of the process of passing a light beam in a
sample, the effect of reflections on the result is estimated based on the algorithm from
[
Paper Structure The second section of the article describes the polariscope and sets forth the preparation of interference orthoscopic pictures based on the qualitative method. The third section is devoted to the calculation of pictures by direct modeling. The calculation results are also presented here, and the influence of the error due to ignoring reflections is estimated. The calculations were performed for two transparent nonabsorbing optically uniaxial crystals of calcite and quartz. Finally, conclusions are made and further plans considered.
Consider a monochrome light source with a wavelength . For a real source, analogous
calculations should be performed for all wavelengths of the spectrum. The qualitative
algorithm is described as in the books [
Polariscope In Fig. 2 left, the following components are marked: 1. Source of parallel unpolarized light (inside the case). 2. Polarizer P . The polarizer has a preferred direction dP . 3. Plane-parallel plate P P from an optically uniaxial crystal with the following characteristics: h is a thickness; dO is the direction vector of the optical axis; refraction indices for each wavelength : no( ) is for an ordinary ray, ne( ) is for an extraordinary one. 4. Analyzer A (second polarizer). It has a privileged direction dA. 5. Screen (camera, eye) for fixing the image. It should be noted that converting the spectrum of light that reaches the screen to color is not the task of this article.
The case of crossed polarizer and analyzer is considered, i.e. the vectors dP and
dA are perpendicular. The optical axis of the studied crystal dO is directed along the
bisector of the angle between the vectors dP and dA. The light rays fall perpendicular
to the sample. With these parameters, ordinary and extraordinary rays move in the same
direction. Similarly the book [
– Rays(1 ! 2): arrows indicating parallel beams of unpolarized monochrome light before the Polarizer. – Rays(2 ! 3): arrows indicating parallel rays of linear polarized light from the Polarizer to the plate. Let Ri denote the ray incident on the plate. Oscillations of the electric vector E are carried out in the direction dP , and its amplitude is equal to E. – Rays(3 ! 3): the rays inside the plate. At the entrance to the plate, two families of linearly polarized rays are formed, polarized in mutually perpendicular planes and corresponding to two refractive indices: the ordinary ray Rot and the extraordinary ray Ret with equal amplitudes – Rays(3 ! 4). Inside the plate, the Rot and Ret rays pass through the optical paths noh and neh, respectively. At the entrance to the plate, both types of rays had the same phase, and then the phase difference appears at the output and is equal p2 2
E: =
Amplitudes and polarizations of the rays Roti and Reti are preserved.
– The Analyzer converts the oscillation directions of the rays Roti and Reti to the
direction dA and outputs the rays Rotia and Retia, respectively, see Fig. 3. Intensity
losses in the Analyzer are not taken into account.
– Rays(4 ! 5) are obtained as a result of interference of Rotia and Retia rays, since
they go in the same direction and their polarization planes coincide and we apply
classical formula [
Ires = I (Rotia) + I (Retia) + 2pI (Rotia)I (Retia) cos :
The resulting intensity is calculated using the formula [
Ires = E2 sin2( ): 2 Let E = 1, then Ires = sin2( 2 ). In such a way, we simplify farther calculations. 3
We proceed to direct modeling of the passage of rays of linearly polarized light. Also
consider a monochrome light source with a wavelength of . Fig. 3 shows the rays
that occur when light falls on the plate. For numerical experiments, let us select two
minerals: calcite with huge birefringence equal to 0.187635 and quarts with a weak
difference between indices of refraction of ordinary and extraordinary rays, not more
than 0.009657. Indices of refraction are computed using Sellmeier approximation [
Calculations are as follows. (3) (4)
Front Plane Incident ray Ri is isotropic linear polarized. An amplitude of its
electrical component is equal to 1. Intensity I (Ri) = 1 also. The only reflected ray Rir is
isotropic linear polarized too. An ordinary ray Rot and an extraordinary ray Ret are
transmitted into a plate.
Inside Plate Both rays coincide. They are linear polarized in perpendicular planes.
Back Plane Rot falls on a back plane and generates up to three linear polarized rays:
reflected ordinary ray Rotor, reflected extraordinary ray Roter, and transmitted isotropic
linear polarized ray Roti that preserves the polarization of the ray Rot. Analogously
Ret generates up to three linear polarized rays: reflected ordinary ray Retor, reflected
extraordinary ray Reter, and transmitted isotropic linear polarized ray Reti that
preserves the polarization of the ray Ret. The rays Roti and Reti coincide but polarized in
different planes. Note that intensities I(Roter) and I(Retor) are equal to zero because
of selected directions dP , dA, and dO. Table 1 presents the results of calculating the
intensities by both methods: column 1 contains the name of the quantity; columns 2
and 3 contain values for calcite and quartz calculated by the direct modeling method;
columns 4 and 5 are values calculated according to the qualitative approach taken from
the books [
Thus at a wavelength of 380 nm, for calcite, total losses are 0.003717, and for
quartz, 0.094762. Recall that after the Polarizer, the intensity was equal to 1. In Fig. 4
are graphs showing the dependence of the intensities on the wavelength in the visible
range of 380–780 nm. Obviously, the calculated graphs repeat those obtained using
a qualitative assessment. It is also seen that, due to energy losses during reflections,
the calculated curves are located lower. Comparing the values I(Rotia) and I(Retia) in
Table 1, we can conclude that the discrepancies are greater, the greater the birefringence
no ne.
In general, this work is part of a project to develop a computer model of a polariscope:
the functions of a modeling block for the interaction of polarized light with transparent
optically anisotropic crystals were investigated. At the same time, a comparison was
made with an approximate qualitative calculation method widely used in the literature,
and estimates of the errors that were obtained depending on the birefringence of the
mineral were obtained. A polariscope is considered a 3D scene, and the visualization
process is a scene rendering. It should be noted that the anisotropic object of the scene
is encapsulated: linearly polarized rays fall on it, and it gives out linearly polarized
rays. In general, rendering of scenes is carried out not by an ordinary isotropic ray, but
by a linearly polarized ray. Such a renderer is able to calculate images of interference
patterns [
Note that in this paper we do not fulfill a tone reproduction, i.e. converting the light representation from the spectrum to RGB, since this is a separate task that we plan to investigate in the next stage of the development.
The practical aspect of the computer model of the polariscope in a modern environment that has arisen in connection with the coronavirus pandemic can be noted. The mode of distance work and study often becomes the main one. In such conditions, various computer models of devices can prepare people for work with them.