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4
i l St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.2 Научно-технические ведомости СПбГПУ. Физико-математические науки. 15 (3.2) 2022
Conference materials UDC 535.8
DOI: https://doi.org/10.18721/JPM.153.222
Research on the character of laser radiation propagation in a differential Anderson cuvette
A. A. Goldberg 1 e, I. D. Kochetkov \ V. V. Davydov 1 2, 3 1 Peter the Great Saint-Petersburg Polytechnic University, Saint Petersburg, Russia;
2 The Bonch-Bruevich Saint Petersburg State University of Telecommunications, Saint Petersburg, Russia;
3 All-Russian Research Institute of Phytopathology, Moscow Region, Russia H [email protected]
Abstract. The necessity of studying the nature of the propagation of laser radiation in the Anderson differential cuvette is substantiated in order to determine the optimal design parameters of a small-sized differential type refractometer. The construction of the Anderson differential cuvette is considered. A new method for studying the nature of the propagation of laser radiation in the differential Anderson cuvette is proposed. The trajectory of movement of the maximum of the laser radiation directive pattern in the cuvette, as well as beyond it (up to the sensor of the photodiode ruler on which the registration takes place) is plotted. An equation is obtained to study the changes in the nature of the propagation of laser radiation from various parameters of the differential cuvette, the reference liquid medium and the investigated liquid medium. A polynomial of the 12th degree is formed to obtain an analytical solution of the equation from the refractive index of the investigated medium.
Keywords: laser radiation, refraction, liquid, refractive index, Anderson's cuvette, refraction, displacement, polynomial
Citation: Goldberg A. A., Kochetkov I. D., Davydov V. V., Research on the character of laser radiation propagation in a differential Anderson cuvette, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. 15 (3.2) (2022) 118-123. DOI: https://doi. org/10.18721/JPM.153.222
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)
Материалы конференции УДК 535.8
DOI: https://doi.org/10.18721/JPM.153.222
Исследование характера распространения лазерного излучения в дифференциальной кювете Андерсона
А. А. Гольдберг 1Н, И. Д. Кочетков 1, В. В. Давыдов 1 2 3 1 Санкт-Петербургский Политехнический университет Петра Великого, Санкт-Петербург, Россия;
2 Санкт-Петербургский государственный университет телекоммуникаций
им. профессора М.А. Бонч-Бруевича, Санкт-Петербург, Россия;
3 Всероссийский научно-исследовательский институт фитопатологии, Московская область, Россия
Аннотация. Обоснована необходимость исследования характера распространения лазерного излучения в дифференциальной кювете Андерсона для определения оптимальных параметров конструкции малогабаритного рефрактометра дифференциального типа. Рассмотрена конструкция дифференциальной кюветы Андерсона. Предложена новая методика исследования характера распространения лазерного излучения в дифференциальной кювете Андерсона. Построена траектория движения максимума диаграммы направленности лазерного излучения в кювете, а также за её пределами (до сенсора фотодиодной линейки, на котором происходит регистрация). Получено уравнение для исследования изменения характера распространения лазерного излучения от различных параметров дифференциальной кюветы, эталонной и исследуемой жидкой
© Goldberg A. A., Kochetkov I. D., Davydov V. V., 2022. Published by Peter the Great St.Petersburg Polytechnic University. 118
среды. Сформирован полином 12-й степени для получения аналитического решения уравнения относительно показателя преломления исследуемой среды.
Ключевые слова: лазерное излучение, рефракция, жидкость, показатель преломления, кювета Андерсона, преломление, смещение, полином
Ссылка при цитировании: Гольдберг А. А., Кочетков И. Д., Давыдов В. В. Исследование характера распространения лазерного излучения в дифференциальной кювете Андерсона // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 15. № 3.2. С. 118-123. DOI: https://doi.org/10.18721/ JPM.153.222
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)
Introduction
Reliable express control is becoming an important element in scientific research, industrial production and environmental monitoring [1-7]. The use of express control is necessary to obtain reliable information about the state of the environment at the sampling site in order to make an adequate decision [2, 8-11]. In addition, it is necessary to obtain further confirmation of the detected deviation in the sample on high-resolution devices [10-12]. In this case, the measurements carried out in express mode should not make irreversible changes to the sample of the medium, which will change its composition and physical structure [2, 11-15]. Taking into account these conditions, the number of methods and devices for express control is limited [1, 2, 6, 10-12, 16-18]. One of the devices that allows express control of the state of liquid medium with high precision at the sampling site is a refractometer.
High requirements for conducting experiments, the manufacture of complex mediums, for example, medical suspensions or medicaments for injection into veins, requires measurements of the refractive index n of a liquid medium with high precision over a large range of values [17-25]. A differential refractometer based on an Anderson cuvette, for measurements in which a reference liquid is used, is one of the possible solutions of these express control problems. The problem of measuring n with an error of 0.0001 over a large range of values is related to the lack of relations between different parameters in the design of the refractometer and the Anderson cuvette. This significantly limits the use of this type of refractometer.
Anderson cuvette design and profile of laser radiation propagation
In the constructions of modern differential refractometers, three types of Anderson cuvettes
(in the form of a square and two types in the form of a rectangle), which are made of quartz glass, are mainly used for measurements. On figure 1 is shown the propagation of laser radiation in the Anderson cuvette and after exiting it to the sensor of the photodiode ruler (the sensor is located at a distance e from the wall of the cuvette). The use of a photodiode ruler for registering laser radiation in refractometers is currently the most optimal solution [18-25].
In the construction of the differential refractometer designed by us, in contrast to those that were used earlier, it is proposed to use the measured value L (by shifting the maximum of laser radiation on the
© Гольдберг А. А., Кочетков И. Д., Давыдов В. В., 2022. Издатель: Санкт-Петербургский политехнический университет Петра Великого.
Fig. 1. Anderson cuvette and laser radiation propagation. Fragment A denotes the area for which the equation is derived to determine n
m
photodiode ruler from the entry point y on the lateral surface of the Anderson cuvette of laser radiation) to determine the n value. The measurement error of 5n in this case will be determined
/ m m
by ensuring that the maximum of laser radiation is registered on one photosensitive sensor. To determine the minimum value 5nm, it is necessary to establish the dependence between the change in the value L on the photodiode ruler on the parameters of the Anderson cuvette, the distance e, and the values n and n .
' m s
The equation for research of laser radiation direction change in the time it dissemination in optical part of refractometer and ratio control
To derive the equation and then verify the mathematical relations at the control points, the description of the nature of the change in the displacement of laser radiation on the photodiode ruler was divided into 4 parts: Lx, L2, L and L4 (fragment A, Fig. 1).
For laser radiation in the considered fragment A (Fig. 1), the following relations were written down:
sin aj nq sin a2 nm sin a3 n? sin a4 na :-= — ; B :-= — ; C :-=—; d :-= —
sin ßj ns sin ß2 nq sin ß3 nm sin ß4 nq
a =a3 +P2, L = etgP4, L2 = ^tgP3,4 = Ktga 3, K = (y - b + L4)tga1,tga1 =—,
b
L4 = \AB\ • sin(a1 -P1) =—-— sin(a1 -P1) = d (sin a1 - cos a1tgP1). cos P1
These relations allow us to express the values of L1, L2, L3 and L4 in terms of the parameters of the Anderson cuvette, the distance e, and also the values of the refractive indexes of the reference liquid ns and the investigated liquid medium nm.
L4 = d sin aj
J -
n cos a
4
2 2-2 nq - n, sin a.
tgaj - tgß2
L3
— = ga3 =■
n, sin a tgaj- s
V
m s
>tga3 = tg (aj ß2 )=T t
J + tga,tgß2
ijjr.
n2 - n]sin2 a, sin a,Jn2 - n]sin2 a, - n sin a, cos a.
m s
J + tga,
4
2 2-2 n - n sin a.
cosa^/n2, - n2sin2 a, + n, sin2 a.
m s
m s
sin a3 = sin (a, - ß2 ) = sin a, cos ß2 - sin ß2 cos a, =
sin a
L n . 2 ns . sin a, / r
= sin auIJ —-sin a,--sin a, cos a, =-1IJ,
V n n n v
K m m m
sin a, (^n2m - n^ sin2 a, - ns cos a, )
nm - n] sin2 a, - ns cos a
tgß3=h-=
^n2q - sin2 a, (n2m + n2s cos2 a, - n, sin2 a, - 2ns cos a, ^Jn^- n, sin2 a,)
1 (^nl - ns2sin2 a, - nscos a, )
L, = etgß4; L = tgß4 = e
sin a,
n
(
, -
sin a
V na
1U'
nl - n, sin2 a, - ns cos a
h )
h (fi*
sin aAJn2 - n2sin2 a, - n„ cos a
n2a - sin2 a, (n2m - ns2 sin2 a, + n2 cos2 a, - 2ns cos a^n2m - n, sin2 a, )
ns sin a,
This allowed us to obtain the following equation for L:
( (
L = L, + L2 + L3 + L4 = sin aj
ns cos a,
J — s
2 2 • 2 n - n sin a.
(in
+ (\ln2 - n sin2 a, - ns cos a
■ )•
- sin2 a, (n2 - n2s sin2 a, + n2 cos2 a, - 2ns cosajyJn2m - n2 sin^aj)
+
(1)
+
- sin2 aj (n
/n^ - sin2 a, ( n^ + n2 cos2 a, - n2 sin2 a -
K
, - 2ns cos a, -yj,
AA
2 2-2 n - n sin a
m s
+
' )
+
cos a, ,jnm - n2 sin2 a, + ns sin2 a,
J J
A control check was performed for the obtained equation (1). The refractometers use laser radiation with X = 632.8 nm. For X = 632.8 nm, the value of n = 1.537826, n = n = 1.000273,
? q ? a air 3
d = 0.5 mm, dx = 1 mm, e = 20 mm, y = 4 mm. The test results are presented in table 1.
Table 1
Calculation results of beam offset distance L
n and n values (n = n ) s m K s m/ Cuvette (size 50 mm x 50 mm)
n = 1.000273 air L = 0.171129
n t = 1.327412 water L = 0.082450
nth , = 1.361513 ethanol L = 0.071131
n t , = 1.437762 petrol L = 0.043715
Notations: L — displacement of the laser radiation after leaving the cuvette, ns — refractive index of the reference liquid, nm — refractive index of the investigated liquid.
To derive the analytical equation nm (d, L, n , e, K,, d,, n , a,), the following notation is introduced: ( a
sin a
= d
, -
n cos a
4
2 2-2 n - n sin a,
q s 1
= t< L - T< = L_L± = A
sin a
sin a
x/nmm^nsrsîn^- ns cos a,=y,; /2=^m a,,
A = /
d
K
,Jn2a - /22 ^n2 - /22 cos a, (/ + «s cos a, ) + n. sin" a
In this case equation (1) is reduced to the following form.
Asin a.
K
/2 A2 - /22 >q2 - /22 /2tga, + ns'
(2)
After various conversions equation (2) can be represented as a polynomial P(f2). The resulting equation (3) is a polynomial of the 12th degree fromf2, in which n is located. If we solve it for f2, we will get an analytical expression for conducting research explicitly and calculating critical points (maximums and boundary conditions). This will be the subject of our further work.
Conclusion
It is worth noting that the obtained equation (1) allows us to evaluate the possibility of measuring the n value with an error of 0.0001 when changing the distance L on the photodiode ruler corresponding to the distance between photosensitive sensors (the ruler design contains 1024 sensors). Using equation (1), it is possible to determine experimentally the parameters of the cuvette, the distance e, as well as the value n , so that when nm changes by 0.0001, the maximum of laser radiation moves by one photosensitive sensor. It takes a lot of time and resources. There will also be difficulties when switching from one nm measurement range to another (for example, from the range from 1.34 to 1.35 to the range from 1.52 to 1.53).
In the case of using an analytical solution for nm, these difficulties can be identified and measurement techniques can be developed to eliminate them.
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THE AUTHORS
GOLDBERG Artemiy A.
[email protected] ORCID: 0000-0002-1573-4619
KOCHETKOV Igor D.
[email protected] ORCID: 0000-0002-5046-6234
DAVYDOV Vadim V.
[email protected] ORCID: 0000-0001- 9530- 4805
Received 04.08.2022. Approved after reviewing 12.08.2022. Accepted 13.08.2022.
© Peter the Great St. Petersburg Polytechnic University, 2022
