=Paper=
{{Paper
|id=Vol-2744/short44
|storemode=property
|title=Comparison of Two Approaches to Calculate Orthoscopic Interference Pictures (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short44.pdf
|volume=Vol-2744
|authors=Victor Debelov,Roman Shelepaev
}}
==Comparison of Two Approaches to Calculate Orthoscopic Interference Pictures (short paper)==
https://ceur-ws.org/Vol-2744/short44.pdf
Comparison of Two Approaches to Calculate
Orthoscopic Interference Pictures?
Victor Debelov1[0000−0002−7577−4700] and
Roman Shelepaev2[0000−0003−4132−360X]
1
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6, Ac.
Lavrentieva ave., Novosibirsk, 630090, Russia
debelov@oapmg.sscc.ru
https://icmmg.nsc.ru
2
Sobolev Institute of Geology and Mineralogy SB RAS, 3, Ac. Koptyuga ave. Novosibirsk,
630090, Russia
rshel@igm.nsc.ru
Abstract. 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 develop-
ing 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 af-
fects the resulting image. This article discusses the calculation of the interaction
of light with this plate only. Two approaches to calculate orthoscopic interfer-
ence 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
Keywords: Polarized Light, Anisotropic Crystal, Orthoscopic Pictures.
1 Introduction
Mineralogists watch orthoscopic and conoscopic interference pictures (patterns, fig-
ures) while look at anisotropic transparent minerals through a polarizing microscope. A
polariscope is a simplest optical device that allows to see such pictures. Mineralogists
[1] note that the polariscope may be one of the most underestimated tools in gemol-
ogy. Most gemologists use it to quickly determine if the stone at hand is isotropic or
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
?
The study was supported by state assignment project IGM SB RAS and state contract with
ICMMG SB RAS (0315-2019-0001). Publication was supported by RFBR grant 19-01-
00402.
2 V. Debelov and R. Shelepaev
anisotropic or, at best, to determine the optic character of gemstones. With some small
additions, one can determine both optic character and the optic sign of a gemstone. It
is also the preferred tool – next to the microscope – for separating synthetic amethyst
from its natural counterparts (although with recent synthetics that may prove difficult).
A virtual polariscope (VP) is our development of a computer model of a polariscope
[2]. In order to compute orthoscopic and conoscopic pictures the VP is represented as a
3D scene. A process of calculation of a picture is considered as rendering of this scene.
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) [3], prepared for each light source used in polarizing
microscopes, and help to determine a mineral type, specimen thickness, birefringent
power, etc. Note that cameras and printing equipment for printing charts also must
comply with certain requirements for correct color reproduction.
In modern conditions, many charts are digitized and available on the Internet, see
[3]. Their use requires calibrated monitors. There are also programs that allow you to
calculate and build a chart directly on the computer for the selected mineral (ortho-
scopic wedge pattern) and the selected standard light source, for example, the program
SICC [4]. The development of such programs is relevant, since new light sources are
emerging, which are characterized by specific spectra. Obviously, the resulting interfer-
ence colors will not match the Michel-Levy nomograms corresponding to other sources.
Fig. 1 shows two examples of color charts calculated using the SICC program for dif-
ferent standard lighting sources CIE D65 and CIE D33.
Fig. 1. Different views of color chart for different light sources.
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 trans-
parent 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 [5]. A
mineral sample is also a VP unit that processes light rays. We implemented its separate
computer model to examine it alone. We count that the existing applications of com-
puter analogs of microscopes were quite suitable for energy calculations only, and the
corresponding programs provided acceptable accuracy of the results. For a polariscope,
it is necessary to consider polarized light, and for calculating interference patterns, the
exact phase of the incoming light wave is important.
Comparison of Two Approaches to Calc. Interference Pictures 3
The idea of this article arose when reading the work [6], in which the author used a
common method for calculating interference colors. This approach (let us call it qual-
itative) is well described in [7] and a number of other books aimed at mineralogists,
for example, [8]. The same approach is used in the mentioned computer programs. This
calculation does not take into account the energy loss caused by light reflections from
the boundaries between media.
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
[9].
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 cal-
culation results are also presented here, and the influence of the error due to ignor-
ing reflections is estimated. The calculations were performed for two transparent non-
absorbing optically uniaxial crystals of calcite and quartz. Finally, conclusions are made
and further plans considered.
2 Interference Orthoscopic Pictures
Consider a monochrome light source with a wavelength λ. For a real source, analogous
calculations should be performed for all wavelengths of the spectrum. The qualitative
algorithm is described as in the books [7, 8]. Let us look at the process of getting a
picture, based on Fig. 2.
2.1 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 char-
acteristics: h is a thickness; dO is the direction vector of the optical axis; refraction
indices for each wavelength λ: no (λ) is for an ordinary ray, ne (λ) is for an extraor-
dinary 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 [7] only transparent non-absorbing crystals are considered.
4 V. Debelov and R. Shelepaev
Fig. 2. Left: a photo of polariscope. Right: a model of polariscope.
2.2 Qualitative Approach. Transformations of Light Rays
Recall that a ray of monochrome light with a wavelength λ is considered, therefore,
we will assume that the corresponding quantities depend on the parameter λ, which
is omitted. Note that no reflected rays are considered. The following stages of light
transformations are marked in the Fig. 2 right.
– 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 √
2
E. (1)
2
– Rays(3 → 4). Inside the plate, the Rot and Ret rays pass through the optical paths
no h and ne h, 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
2π
δ= h(no − ne ). (2)
λ
Comparison of Two Approaches to Calc. Interference Pictures 5
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 [7]
p
Ires = I(Rotia ) + I(Retia ) + 2 I(Rotia )I(Retia ) cos δ. (3)
The resulting intensity is calculated using the formula [7]
δ
Ires = E 2 sin2 ( ). (4)
2
Let E = 1, then Ires = sin2 ( 2δ ). In such a way, we simplify farther calculations.
3 Direct Modeling of Light Propagation
We proceed to direct modeling of the passage of rays of linearly polarized light. Also
consider a monochrome light source with a wavelength of λ. 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 [10].
Calculations are as follows.
Fig. 3. Interaction of light ray with a plate and Analyzer. Red arrows denote the fact of reflection
only, the directions are wrong.
Front Plane Incident ray Ri is isotropic linear polarized. An amplitude of its electri-
cal 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.
6 V. Debelov and R. Shelepaev
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: re-
flected 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 pre-
serves 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 [7, 8]. The sign ‘-‘ means that the corresponding value is not calculated by
corresponding method. Iout is the total intensity of both rays after exiting the plate be-
fore entering the Analyzer. Ires is the intensity taking into account interference after
exiting the Analyzer.
Table 1. Calculated values for λ=380 nm, plate thickness is 0,01 mm.
Intensity Calcite Quartz Books calcite Books quartz
1 2 3 4 5
I(Ri ) 1 1 1 1
I(Rir ) 0.052688 0.048560 - -
I(Rot ) 0.467292 0.476043 - -
I(Ret ) 0.480020 0.475397 - -
I(Rotor ) 0.030568 0.022809 - -
I(Reter ) 0.019181 0.023393 - -
I(Roti ) 0.436724 0.453235 - -
I(Reti ) 0.460839 0.452004 - -
Iout 0.897563 0.905239 - -
I(Rotia ) 0.218362 0.226617 - -
I(Retia ) 0.230419 0.226002 - -
no − ne 0.187635 0.009657 0.187635 0.009657
δ 31.024889 -1.596758 31.024889 -1.596758
Ires 0.034026 0.464369 0.037743 0.512980
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 .
Comparison of Two Approaches to Calc. Interference Pictures 7
Fig. 4. Plots of Ires (λ) for calcite (red), quartz (blue) calculated via qualitative method (solid) or
direct modeling (dashed). Plate thickness is 0,01 mm.
4 Conclusions
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 [11], and its development serves as a guiding thread for our researches.
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.
8 V. Debelov and R. Shelepaev
The practical aspect of the computer model of the polariscope in a modern envi-
ronment 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.
References
1. Polariscope, http://gemologyproject.com/wiki/index.php?title=Polariscope. Last accessed 13
Jul 2020
2. Vasilyeva, L.F., Debelov, V.A., Shelepaev, R.A.: On development of polariscope.
In: International Conference SCVRT-2018 Proceedings, pp. 325–332 (2018)
https://elibrary.ru/download/elibrary 37151167 22447183.pdf. Last accessed 10 Jul 2020 (In
Russian)
3. The Michel-Levy Color Chart relates Birefringence, sample thickness, and interference color,
http://microscopy.berkeley.edu/courses/tlm/plm/m-l chart.html. Last accessed 14 Jul 2020
4. Sakai, H.: Execution logs by RNIA software tools,
http://www.mns.kyutech.ac.jp/ sakai/RNIA. Last accessed 10 Jul 2020
5. Debelov, V.A., Kushner, K.G., Vasilyeva, L.F.: Lens for a computer model
of a polarizing microscope. Mathematica Montisnigri XLI, p. 151 (2018).
http://www.montis.pmf.ac.me/vol41/12.pdf. Last accessed 10 July 2020
´
6. Sorensen, B.E.: A revised Michel-Le vy interference colour chart based on first-
principles calculations. Eur. J. Mineral. 25(1), 5–10 (2013). https://doi.org/(10.1127/0935-
1221/2013/0025-2252)
7. Born, M., Wolf, E.: Principles of Optics: Electromagnetic theory of propagation, interference
and diffraction of light. Cambridge Univ. Press, (1980)
8. Bloss, F.D.: Introduction to the methods of optical crystallography. Holt, Rinehart and Win-
ston, NY (1961)
9. Debelov, V.A., Kozlov, D.S.: A local model of light interaction with transparent crys-
talline media. IEEE Transactions on Visualization and Computer Graphics 19(8), 1274-–1287
(2013). https://doi.org/(10.1109/TVCG.2012.304)
10. Wilkie, A., Tobler, R.F., Purgathofer, W.: Raytracing of dispersion effects in trans-
parent materials. In: 8th International Conference WSCG’2000 Proceedings (2000).
http://wscg.zcu.cz/wscg2000/Papers 2000/W7.pdf.gz. Last accessed 10 Jul 2020
11. Debelov, V.A., Vasilyeva, L.F.: Visualization of interference pictures of 3D scenes includ-
ing optically isotropic transparent objects. Scientific visualization 12(3), 119–136 (2020).
https://doi.org/(10.26583/sv.12.3.11)