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Visnyk N'l'UU KP1 Seriia Radiolekhnika tiadioaparatobuduummia, "2020, Iss. 83, pp. 69—74
УДК 621.384.3
Computer Modelling Technologies of Optical System of Polarizing Thermal Imager
Kolobrodov V. H.
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
E-mail: Ikerrnо(Фикт. neI.
The energetic resolution is a main parameter of any thermal imager depending on the t.ransmit.tance of its optical system. The optical system of a polarizing thermal imager (PTI) consists of a polarizer, a phase shifter and a lens arranged in series on the optical axis. This paper proposes a method for calculation of the energic transmittance of the PTI's optical system for partially polarized radiation as a function of angular orientation of the polarizer and the phase ret.arder. The physical-mathematical model of transformation of partially polarized radiation within the optical system which depends on parameters of optical elements and their orientation in space is investigated. This model allowed us to determine the transmittance of the system "polarizer - phase shifter" system depending on the angle a between the optical axes of the polarizer and the phase ret.arder. The analysis of this method has shown that., for the natural radiation, the normalized transmittance of the optical system does not depend on the angular orientation of the phase ret.arder and is equal to 0.5. For the partially polarized radiation, the transmittance depends on the angle a: the maximum t.ransmit.tance value will be achieved in the case when the optical axis of the phase ret.arder lies in the transmittance plane of the polarizer (a = 0°). For an arbitrary degree of polarization, the transmittance decreases with increasing angle a. At an angle (a = 45°), the transmittance is equal to 0.5 and does not depend on the degree of polarization of the examined radiation. To calculate the characteristics of the partially polarized radiation using Stokes parameters, the intensity is to be measured at. the output, of the optical system for angles a equal tо 0°, 90°, 45°, and 135°. For such angles, the normalized transmittance for the degree of polarization of 0.5 is equal to 0.75, 0.25, 0.5, and 0.5, respectively. This feature of the PTFs optical system must, be taken into account, when calculating the temperature resolution and the maximum range of the thermal imager.
Key words: polarizing thermal imager: polarizer: phase ret.arder: energetic t.ransmit.tance: Stokes vector DOI: 10.20535/RADAP.2020.83.69-74
Introduction
Polarization is one of specific characteristics of the electromagnetic radiation field, besides its intensity, wavelength, and coherence fl 3]. Polarimetry measures the vector character of the radiation reflected or scattered from the object's surface and provides important information about the surface's orientation. shape and quality. The polarization properties of radiation detected from observation objects may differ properties of radiation obtained from backgrounds and are not correlated with the intensity and spectrum. As a rule, radiation from the target is partially polarized, while radiation from the background is natural [4. 5]. Tims. Polarimetrie images are very useful for increasing the signal from the target and suppressing background noise.
The main characteristics of polarized radiation are the intensity, the degree of polarization, the azimuth and ellipticity of polarization [6 8]. To measure these characteristics in the infrared (IR) region of
the spectrum, polarizing thermal imagers (PTI) are applied [9 11]. The main characteristic of any thermal imager is energetic resolution, which depends on the transmittance of its optical system. There are many monographs and articles in whichthe calculation and measurement of the transmittance of thermal imager lenses are explored [12 14]. At the same time, there is almost no scientific and technical information on calculation of the transmittance of the PTI's optical system consisting of the polarizer, phase retarder and lens, which are sequentially located on the optical axis.
1 Problem formulation
The purpose of this article is to develop a method for calculating the energetic transmittance of the optical system of a polarizing thermal imager for partially-polarized radiation depending on the angular orientation of the polarizer and the phase retarder.
2 Physical basics of the polarization of the thermal imager
Wo consider the optical system of PTI, which consists of the IR polarizer, quarter-wave rotardor and IR lens of the thermal imager, sequentially located on the optical axis (Fig. 1). Let a parallel beam of natural or partially polarized radiation with amplitudes En or Epp respectively is directed at the input of the optical system. At the output of the polarizer, the linearly polarized radiation is formed with the vector Eip, which is oriented at an angle a relative to the optical axis oo of the rotardor (Fig. 2). After passing through a quarter-wave rotardor. the optical axis of which is parallel to the surface of the retarder and makes an angle a with the vector Eip (plane of polarization), clS ct result of birefringence, the ordinary and extraordinary rays with amplitudes B^d Ee are formed in the retarder. They propagate in the same direction and have the following phase difference at the output from the retarder
2-k
Ay = k ■ Ad = — (na — ne)d,
A
(1)
First, let's consider the case when the optical system receives natural radiation with intensity I0 = I0o + Ioe, where I0o = |£0o|2 and I0e = |£0e|2 is the intensity of ordinary and extraordinary rays, respectively, and Iqo = Ioe = 0,5/o.
Let's determine the amplitudes of ordinary Eo1 and extraordinary Ee1 rays at the output of the retarder, if the plane of polarization of the ray Eip incident on the retarder forms an angle a with the optical axis of the crystal. From Fig. 2, we have
Eo1 = Eip sin a; Ee1 = Eip cos a.
The amplitudes (instantaneous values of field strengths) of ordinary and extraordinary rays change over time according to the law
Ee = Ee1 cos wt; Ea = Eo1 cos(ut — Ay). (2)
The system of equations (2) is represented as one equation, which does not depend on time t:
Eç
E,
= cos wi;
el
where d is a thickness of the retarder, Ad is the optical path difference between the ordinary and extraordinary-rays, na and ne are refractive indices for ordinary and extraordinary rays, respectively.
E0
=cos(wt — Ay) = cos wt cos Ay + sin wt sin Ay =
E01
=¥7cosA^ +V1 — (J*^} sinAy;
*
sin
Ea Ee / I Ee ,
— — — cosAy =W1 — ^^ sinAy.
Let us bring the right and left sides of the last relationship to the square
(a)
E„ A
S
s
Em "
(b)
Fig.
1. Scheme for research of the polarization of radiation (a) and its vector model (b)
Fig. 2. Vector model of obtaining olliptically polarized
light
(§L\2 —2 |L cosAy +( 2 cos2 Ay:
EP,
- sin2 Ay — sin2 Ay ( ——
. Ee1
or
( IT )2 —2 IT IT cos Ay+ T ^ )2 =sin2 Ay. (3)
The equation (3) is the equation of the ellipse, which is arbitrarily oriented relative to the optical axis oo of the retarder (Fig. 3). Therefore, the resulting field amplitude at the output of the retarder will form the elliptically polarized light, where Er = Ea + Ee. The semi-axes Eo1 and Ee1 of the ellipse, as well as its orientation depend on the angle a and the phase difference Ay ( ). The obtained equation ( ) is called a polarization ellipsoid, in which the ollipticity of the ellipse is a degree of polarization, the direction of the major axis of the ellipse is a direction of polarization 0P of the ellipse, and the circle inscribed in the ellipse is a natural component of radiation. The linearly polarized component in the direction of the polarization angle 0P
is the largest, while in the perpendicular direction it is zero.
By analysing the equation (3) for the case when the phase difference A < between ordinary Ea and extraordinary Ee rays is equal to n/2, we can determine the thickness of the retarder that provides this case. From formula (1):
л n X
A* = TK-ne)d =2; = 4(na -ne) ■
Eel 4o
(4)
El + El Eh + Eh
1.
4. a = 135°. Then Eol = Elp/V2 and Eei = and the equation (5) will be presented as El +El = Efp/2. This means that, in this case, the loft-circular polarized radiation with intensity I135° = 0.5/o is formed at the output of the retarder.
Consider the case when partially polarized radiation is directed into the input of the optical system
o
degree of polarization P, the direction of polarization 9p. By using the model of partially polarized radiation, according to which such radiation is a superposition of linearly polarized Ip and unpolarized radiation (natural radiation) In, one can obtain the resulting intensity [2,11,15]
Ir,
o
In + I®.
(6)
Fig. 3. Vector model of olliptically polarized light
A
the ordinary and extraordinary rays is oqnal to A d = (na — ne)d = -^A .
Therefore, the uniaxial crystal retarder that provides the condition (4), is called a retarder thickness of A/4 or a quarter wave retarder.
For snch a retarder, equation (3) lias the form
After passing the polarizer (Fig. 1), the intensity-will depend on the orientation angle 0P of the partially
the observation (measurement) angle < = a. Then, according to Malns's law
Ipp(<fi) = In(<p)+ Ip(<) = 22^ + c°s2(<p — dp), (7)
where 2In is a natural component; Ip cos2(< — 0p) is a linearly polarized component.
The degree of polarization is determined by the classical formula
P
(5)
l\\ - h h +h
(8)
The expression (5) represents the equation of the ellipse having the semi-axes
a = E0i = Eip sin a; b = Eei = Eip cos a .
Here, we can consider four cases for the oqnati-
a
plano of polarization of the radiation incident on the wave retarder and the optical axis of the retarder (2):
1. a = 0°. Then Ea = Eo1 = 0 and Eei = Elp, and the equation ( ) will be presented as Ee = Eel. This means that, in this case, the linearly polarized radiation with intensity /o° = I0e = 0.510 is formed at the output of the retarder.
where I\\ mid I± are intensity components of two mutually perpendicular polarization components, which are obtained from formula (7):
J,, (<)=i( ep)=in( ep) + ip( ep) = 1 in + ip, (9)
Xf)=i[eP+2) = in(ep+2) + ip (op+2) = 1 in.
(10)
From equations (C) - (10) we can obtain the degree of polarization of the partially polarized radiation
P
Ir,
^n + I p
Ip
I
pp
2. a = 90°. Then Eol = Eip and Ee = Eel =0, and the equation ( ) will be presented as Ea = Eo1. This means that, in this case, the linearly polarized radiation with intensity Iqo° = loo = 0.5/o is formed at the output of the retarder.
3. a = 45°. Then E0i = Elp/V2 wd Eei = Elp/V2, and the equation ( ) will be presented as E2a + El = Efp/2. This means that, in this case, the right-circular polarized radiation with intensity /45° = 0.5/0 is formed at the output of the retarder.
where
Ip=Plpp=Plo; In=/0 - Ip=/0(1 -p)■ (11)
Below we analyze four cases for the equation
a
plane of polarization of the radiation incident on the wave retarder and the optical axis of the retarder, when
ep=0.
The characteristics of the polarization image are determined using Stokes parameters, which are measured for angles a equal to 0°, 90°, 45°, mid 135°. For these angles, we have:
1. a = 0°. Then Iq° = 2In + IP = 1/q(1-P)+P/q = = 1/Q(1+P ).
2. a = 90°. Then I9q° = 2In = l-To(l-P).
3. a = 45°. Then I45° = 2^ + IP cos2 45° = 2/„ + + 1 = 1 [ /Q(1-P ) + P Iq] = 1Iq.
4. a = 135°. Then I135° = \l0.
An energetic transmittance of the optical system of PTI is defined as
I*
Iq '
(12)
o
optical system (polarizer); Ia is the radiation intensity at the output of the optical system (lens); tp , Thp , t0 are transmittances of the polarizer, phase retarder and lens, respectively.
By using the formulas obtained above, we can determine the transmittance of the system "polarizer -phase retarder" depending on the angle a between the optical axes of the polarizer and the phase retarder.
For natural radiation, we have:
Tp—hp TpThp(a) 2 Tp^~hp,
la. 7~pT hpTQ
s T
q
pThpl
~ pThpl
q
7,In + IP cos a
2Iq(1 -P)+PIq cos2 a
1(1-P)+P cos2 a
1. For natural radiation, the normalized transmittance of the optical system Tos,n of PT does not depend on the angle a and is equal to 0.5.
2. For partially polarized radiation, the transmittance of the optical system decreases as the angle a increases.
3. For partially polarized radiation, at an angle a = 45°, the transmittance Tos,n with an arbitrary-degree of polarization is equal to 0.5. This occurs due to the fact the circularly polarized radiation is formed when the angle a = 45°.
4. For partially polarized radiation, at observation angles a < 45°, the transmittance ros,n is greater than for the natural radiation, while at angles a > 45° -vice versa.
5. For small values of the polarization degree P <
0.1
the range 0.45 < Tos,n < 0.55.
(13)
where tp mid Thp are transmittances due to Fresnel losses on the input and output surfaces of the optical elements and absorption in the optical medium of the polarizer and the phase retarder, respectively.
It is clear from the formula (13) that the transmittance of the optical system for natural radiation does not depend on the angle a.
For partially polarized radiation with different angles a:
1- tos = TpThp(a = 0°)= TpThp 1 (1+P);
2- tos = Thp(a = 90°) = TpThp1 (1— P);
3. tos = Thp (a = 45° ) = TpThp;
4. tos = Thp (a =135°)= TpThp.
Let's determine the transmittance of the optical system for an arbitrary angle a. From the expressions (7), (11), and (12), we have
1 0,8 0,6 0,4 0,2 0
t_(os,h)
v2
з4 L)
4 l
a,
град
(14)
The charts of the normalized transmittance of the PTI's optical system Tos,n = tos/TpThpT0 in dependence on the angle a are shown in Fig. .
The analysis of the function (14) and its charts shows that:
0 10 20 30 40 50 60 70 80 90
Fig. 4. Dependence of the energetic transmittance of the optical system of a polarizing thermal imager Tos,n for the partially polarized radiation on the angle a between the plane of polarization of the radiation incident on the wave retarder and the optical axis of the
P P
P P
The increase the transmittance of the optical system of PTI, and hence the improvement the energetic resolution of PTI can be achieved by:
1. Reduction of Fresnel losses on reflection by using optical elements with low refractive index or applying antireflection coatings on surfaces.
2. Use a polarizer, phase retarder and IR lens with high transmittance.
Conclusions
The proposed physical-and-mathematical model of the optical system of a polarizing thermal imager enabled to develop a method of calculating the energetic transmittance of an optical system for the partially polarized radiation depending on the angular orientation of the polarizer and the phase retarder. The analysis of this method has shown that:
1. For natural radiation, the normalized transmittance Tos,n of the optical system does not depend on
и
q
и
the angular orientation of the phase retarder and is equal to 0.5.
2. For partially polarized radiation, the transmi-ttance Tos,n depends on the angle a:
2.1. The maximum value of the transmittance will occur in the case when the optical axis of the phase retarder lies in the plane of transmission of the polarizer (a = 0°). For an arbitrary degree of polarization,
a
2.2. At an angle a = 45° the transmittance is equal to 0.5 and does not depend on the degree of polarization of the examined radiation.
2.3. To calculate the characteristics of the partially polarized radiation using Stokes parameters, the intensity at the output of the optical system for angles a equal tо 0°, 90°, 45° and 135° is measured. For these angles, the normalized transmittance ros,n with the degree of polarization of 0.5 is equal to 0.75,0.25,0.5, and 0.5, respectively. This feature of the optical system of PTI must be taken into account when calculating the temperature resolution of the thermal imager.
References
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[3] Schott .1. R. (2009) Fundamentals of Polarimetrie Remote Sensing. SP1E Press. 244 p.
[4] Vollmer M.. Möllmann K.-P. (2018) Infrared Thermal Imaging: Fundamentals, Research and Applications. 2nd Edition. Wiley-VCH. 788 p.
[5] Vollmer M. (2001) Identilication and Suppression of Thermal Imaging. InfraMation Proceedings, University of Applied Sciences. Brandenbueg. ITC 104 Л.
[6] Prokhorov Л. M. (1984) Fizicheskii entsiklopedicheskii slouar' [Physical encyclopedic dictionary]. Soviet encyclopedia. Moscow. 944 p.
[71 Yang В.. Wu Т.. Chen W.. Li Y.. Knjazihhin .1.. AsundiA.. Van L. (2017) Polarization Remote Sensing Physical Mechanism. Key Methods and Application. The Intermation Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, China. Vol XL11-2/W7. pp. 956-960. D01:10.5194/isprs-archives-XLll-2-W7-955-2017.
[8] Zhao Y.. Yi C.. Kong S. G.. Pan Q.. Cheng Y. (2016) Multiband Polarization Imaging and Applications. SpringerVerlag Berlin Heidelberg. 194 p. DOltlO.1007/978-3-662-49373-1.
[9] Gurion K. P.. Yulla A. .1.. Videen G. W. (2014) Enhanced facial recognition for thermal imagery using Polarimetrie imaging. OSA Publishing. Optics Letters, Vol. 39. Iss. 13. pp. 3857 3859. DOltlO.1364/UL.39.003857.
[10] Mann A. (2009) Infrared Optics and Zoom Lenses. 2nd Edition. SP1E Press. 164 p.
[11] Zhang Y.. Shi Z. G.. Qiu T. W. (2017) Infrared small target detection method based on decomposition of polarization information. .Journal of Electronic Imaging, Vol. 26(3). DOltlO. 1117/1..I El.26.3.033004.
[12] Kaplan H. (2007) Practical Applications of Infrared Thermal Sensing and Imaging Equipment. 3rd Edition. SP1E Press. 236 p.
[13] Macleod H. A. (2018) Thin-Film Optical Filters. 5th Edition. Taylor&Francis Group. 664 p.
[14] Chrzanowski K. (2010) Testing thermal imagers. Practical guidebook. Military University of Technology. 00-908. Warsaw. Poland. 164 p.
[15] Kolobrodov V. H.. Mykytenko V. 1.. Tymchyk H. S. (2020) Poliaryzatsiina model teplokontrastnykh olyiektiv sposterezhennia [Polarization model of heat-contrast objects of observation]. Termoelektryka, Iss. 1. pp. 36-52.
Комп'ютерш технолог!!" моделювання оптично!" системы поляризащйного теплов!зора
Колобродив В. Г.
У статта досл!джуеться запропоповапий метод роз-рахупку епергетичпого коефщ!епта пропускания оптично! системи поляризагцйпого теплммзора (ПТ) для частково поляризовапого випромшюваппя в залежпоста в!д кутово! ор1ептацГ! поляризатора i фазово! пластинки.
Основною характеристикою будь-якого теплов!зора е епергетичпе роздглеппя, яке залежить в!д коефщ!епта пропускаппя його оптично! системи. Оптичпа система ПТ складаеться 1з посл!довпо розташовапих па оптичш oci поляризатора, фазово! пластинки i об'ектива. Дослужена ф!зико-математична модель перетвореппя частково поляризовапого випромшюваппя в такш оптнчшй систем! в залежпоста в!д параметр!в оптичпих елемептав та !х ор1ептацп у простор!. Така модель дозволила визпа-чити коефщ!епт пропускаппя системи «поляризатор фазова пластина» в залежпоста в!д кута м!ж оптпчпимн осями поляризатора i фазово! пластшш.
Досл1джеппя цього методу показали, що пормовапий коефщ!епт пропускаш1я оптичпо! системи для приро-дпього випромшюваппя по залежить в!д кутово! opien-тацп фазово! пластшш i дор1вшое 0,5. Для частково поляризовапого випромшюваппя коефкцепт пропускаппя заложить в!д кута о:, максимально значения коефкцента пропускаш1я буде у випадку, коли оптичпа в!сь фазово! пластшш лежить в площиш пропускашш поляризатора (а = 0°). Для дов!льного ступепя полярпзаци ¡3 зб!ль-шеппям кута о. коефщ!ент пропускания зменшуеться. При кута а = 45° коефщ!ент пропускания дор!внюе 0,5 i не залежить в!д ступепя полярпзацп досл1джувапого випромшюваппя.
Для розрахупку характеристик частково поляризовапого випромшюваппя з використаппям параметр!в Стокса вим!рюеться штепсившсть па виход! оптичпо! системи для кутав а р!вних 0°, 90°, 45° i 135°. Для таких кутав пормовапий коефщ!епт пропускаппя для ступепя полярпзацп 0,5 дор1вшое 0,75,0,25,0,5 i 0,5 в1дпов1дпо. Таку особлшмсть оптичпо! системи ПТ пеобх1дпо вра-ховувати при розрахупках температурного роздглеппя i максимально! далыюста дп теплов!зора.
Ключоег слова: поляризацшпий теплов1зор: поляризатор: фазова пластина: епергетичпий коефкцепт пропускаппя: вектор Стокса
74
Колобродов В. Г.
Компьютерные технологии моделирования оптической системы поляризационного тепловизора
Колобродов В. Г.
В статье исследуется предложенный метод расчета энергетического коэффициента пропускания оптической системы поляризационного тепловизора (ПТ) для частично поляризованного излучения в зависимости от угловой ориентации поляризатора и фазовой пластинки.
Основной характеристикой любого тепловизора является энергетическое разделение, которое зависит от коэффициента пропускания его оптической системы. Оптическая система ПТ состоит из последовательно расположенных на оптические оси поляризатора, фазовой пластинки и объектива. Исследована физико-математическая модель преобразования частично поляризованного излучения в такой оптической системе в зависимости от оптических элементов и их ориентации в пространстве. Такая модель позволила определить коэффициент пропускания системы «поляризатор - фазовая пластина» в зависимости от угла о. между оптическими осями поляризатора и фазовой пластины.
Исследования этого метода показали, что нормированный коэффициент пропускания оптической системы
для естественного излучения не зависит от угловой ориентации фазовой пластины и равен 0,5. Для частично поляризованного излучения коэффициент пропускания зависит от угла о:, максимальное значение коэффициента пропускания будет в случае, когда оптическая ось фазовой пластины лежит в плоскости пропускания поляризатора (а = 0°). Для произвольной степени поляризации с увеличением угла о. коэффициент пропускания уменьшается. При угле о. = 45° коэффициент пропускания равен 0,5 и не зависит от степени поляризации исследуемого излучения.
Для расчета характеристик частично поляризованного излучения с использованием параметров Стокса измеряется интенсивность на выходе оптической системы для углов а равных 0°, 90°, 45° и 135°. Для таких углов нормированный коэффициент пропускания для степени поляризации 0,5 равен 0,75,0,25,0,5 и 0,5 соответственно. Такую особенность оптической системы ПТ необходимо учитывать при расчетах температурного разделения и максимальной дальности действия тепловизора.
Ключевые слова: поляризационный тепловизор; поляризатор; фазовая пластина; энергетический коэффициент пропускания; вектор Стокса
