Issue #6/2021

Comparative Analysis of Threshold Sensitivity of IR-Systems in Different Spectral Range

**I. V. Znamensky, E. O. Zot’ev, S. Yu. Yudin**Comparative Analysis of Threshold Sensitivity of IR-Systems in Different Spectral Range

DOI: 10.22184/1993-7296.FRos.2021.15.6.484.500

The result of calculating the threshold illumination of an optoelectronic system (OES) in various infrared sub-ranges in the range of 0.9–5.3 µm is presented. A technique has been developed and the illumination of the entrance pupil of the OES from the signal of the monitoring object (OM) of a cylindrical shape illuminated by the Sun and due to the intrinsic radiation of the OM surface is calculated. A program for energy calculation has been developed and its interface has been presented.

The result of calculating the threshold illumination of an optoelectronic system (OES) in various infrared sub-ranges in the range of 0.9–5.3 µm is presented. A technique has been developed and the illumination of the entrance pupil of the OES from the signal of the monitoring object (OM) of a cylindrical shape illuminated by the Sun and due to the intrinsic radiation of the OM surface is calculated. A program for energy calculation has been developed and its interface has been presented.

Теги: accumulation time atmosphere ir range matrix photodetector optoelectronic system overall brightness coefficient photon signal-to-noise ratio threshold illumination коэффициент габаритной яркости матричные фпу обзор космического пространства характеристики оптико-электронных систем

Comparative Analysis of Threshold Sensitivity of Ir Systems in Different Spectral Range

I. V. Znamensky, E. O. Zot’ev, S. Yu. Yudin

Research and Production Corporation “Рrecision Systems and Instruments” (JSC NPK SPP)

Moscow, Russia

The result of calculating the threshold illumination of an optoelectronic system (OES) in various infrared sub-ranges in the range of 0.9–5.3 µm is presented. A technique has been developed and the illumination of the entrance pupil of the OES from the signal of the monitoring object (OM) of a cylindrical shape illuminated by the Sun and due to the intrinsic radiation of the OM surface is calculated. A program for energy calculation has been developed and its interface has been presented.

Keywords: Optoelectronic system, IR range, matrix photodetector, threshold illumination, accumulation time, signal-to-noise ratio, atmosphere, photon, overall brightness coefficient.

Received on: 18.03.2021

Accepted on: 17.08.2021

INTRODUCTION

At present, systems operating in the infrared range are widely used in optoelectronic systems (OES) for ground-based space surveillance and tracking space objects. Operation in the IR range is associated with several reasons:

Decreasing the step and increasing the format is a common trend for almost all world developers and manufacturers of IR matrices. Reducing the step and increasing the format leads to a significant increase in the range of object recognition [1].

Reviews of the state and prospects for the development of matrix photodetectors have been considered by many specialists, for example [2]. To solve specific problems, programs for calculating an OES with a matrix photodetector in the IR range have been developed. Thus, in [3], the calculation of the threshold sensitivity of the OES in various spectral infrared ranges is considered and the illumination of the entrance pupil from a space object (SO) of a spherical shape is determined.

This article presents an algorithm and a calculation program for determining the threshold illumination of an OES in the spectral range of 0.9–5.3 μm with the refinement of the sub-ranges. The calculation of the signal from the SO of a cylindrical shape (rocket) under the illumination of the sun is carried out, and the results are presented using the dimension brightness coefficient (DBC), the graph of which is presented. A comparative analysis of the threshold sensitivity of IR systems in different spectral ranges and different temperatures of the rocket skin has been carried out. The developed method for calculating IR systems uses an approach based on the estimation of the photon count rate. The optimal observation time is found for a given background level and matrix parameters.

SELECTING THE SPECTRAL RANGE

When receiving the intrinsic radiation of the rocket skin, the receiving spectrum has the form of an absolute black body (ABB) spectrum at the rocket temperature.

For the case of the absence of a rocket torch when receiving radiation from a rocket illuminated by the Sun, the receiving spectrum has a spectrum of an absolutely black body at a temperature of Т = 6000 K [3]. Radiation from the Sun is reflected from the rocket skin and is received by the receiving lens of the system.

To optimize the S / N ratio, taking into account the transparency of the atmosphere, the following sub-ranges were selected: 0.9–1.3 µm; 1.4–1.8 µm; 1.8–2.6 µm; 0.9–2.6 µm; 3.2–4.2 µm; 4.4–5.2 µm; 3.0–5.2 µm. These IR sub-bands coincide with the transparency windows of the atmosphere, which should provide the highest signal-to-noise ratio.

Fig. 1 shows the dependence of the spectral transmittance of the atmosphere on the radiation wavelength in the range 0.9–5.3 μm at an elevation angle of 15°. The graph is calculated and built using a program developed by the authors.

When choosing a spectral range, it is necessary to take into account both the transparency windows of the atmosphere and the characteristics of commercially available matrices containing a built-in interference filter (IF). The Falcon matrix by SCD (Israel) can operate in the range of 1.0–5.45 μm, has a built-in factory IF in the range of 3.6–4.9 μm. The cooled IF is not replaceable, since it is installed in a sealed case. Selex ES manufactures a similar Eagle matrix. In this case, the required IF sub-range can be selected.

It is also important to assess the level of the external background in order to maximize the SNR. In Fig. 2 shows the dependence of the spectral brightness of the sky background at an elevation angle of 15° in the range of 0.9–5.3 μm. In the calculations of the mid-IR range, we use the parameters of the Falcon matrix, for the near-IR range, we use the same parameters.

The initial data for calculating the threshold sensitivity in different spectral sub-ranges are displayed on the input panel for the initial data (see Appendix): lens diameter Dl = 250 mm; focal length Fl = 720 mm; transmission coefficient of receiving optics Kopt = 0,7; maximum accumulation time 20 ms; elevation angle β = 15°; meteorological visibility range Sm = 20 km; relative air humidity V = 80%; air temperature T = 22 °C. The results of calculating the IR system in different spectral ranges are presented in Table 1.

The minimum illumination created by the background per pixel corresponds to the infrared sub-ranges of 1.8–2.6 and 3.2–4.2 μm. Therefore, in these ranges, the signal accumulation time will be maximum and equal to 20 ms.This accumulation time provides a frame rate of 50 Hz.

When calculating IR systems, three restrictions were taken:

The maximum and minimum accumulation times limit the dynamic range of the input signals.

When choosing a spectral range, the level of the external background is important, at which the signal-to-noise ratio is maximized. The threshold illumination at the entrance pupil is minimal in the sub-ranges of 1.8–2.6 and 3.2–4.2 μm. The situation with background radiation is somewhat worse in the 1.4–1.8 μm sub-range. The results obtained do not take into account the spectral transmission of the atmosphere.

The minimum illumination by the entrance pupil signal, recalculated outside the atmosphere (see Table 1) at Qp = 100 in the spectral range of 3.2–4.2 μm, is 0.937 ∙ 10–15 W / cm2. In the ranges 1.4–1.8 μm it is 1.685 ∙ 10–15 W / cm2 and in the range 1.8–2.6 μm it is 1.856 ∙ 10–15 W / cm2. Here Qp is the signal-to-noise power ratio.

The lowest sensitivity has a spectral range of 4.4–5.2 µm, which is 66.8 times worse than the range 3.2–4.2 µm. The final conclusion about the best spectral range can be made only by knowing the spectral distribution of the signal from the target.

The initial data for calculating the signal from the rocket are as follows: rocket radius Rrо = 0.5 m; rocket length Lrо = 2 m; the distance from the Earth’s surface to the rocket is calculated for 3 values: DE-rо = 360, 160, 50 km at three temperatures of 300, 500 and 700 K, respectively; the reflection coefficient of the rocket is Кr = 0.5; the angle between the direction to the Sun and the normal to the plane orthogonal to the longitudinal axis of the SO ξ1 = 30°; the angle between the direction to the receiver and the perpendicular to the plane perpendicular to the longitudinal axis of the SO ξ2 = 30°; the angle between the direction to the Sun from the SO and the direction to the receiver γ = 90°. The rest of the parameters are the same as those used to calculate the threshold sensitivity.

SIGNAL FROM A SPACE OBJECT

IN THE FORM OF A ROCKET

The spectral composition of the rocket radiation consists of three components:

intrinsic thermal radiation of the rocket body,

thermal radiation of the rocket torch,

solar radiation reflected from the rocket.

1. Intrinsic thermal radiation of the rocket body

The intrinsic thermal radiation of the rocket, which has a temperature T, is calculated for the blackbody in a given spectral range by the formula [4]:

, Вт ∙ см–2,

where С1 = 3.7415 ∙ 104 [W∙cm– 2 ∙ µm4];

С2 = 1.43879 ∙ 104 µm ∙ K;

T is the blackbody temperature;

K, λ is the wavelength, µm;

λ1, λ2 are the boundaries of the spectral range [μm];

ε is the emissivity, we take ε = 0.5.

The dependence of the height of the rocket Hro above the Earth’s surface on the distance Dro to it at a constant elevation angle β is found in the form [5]:

where RE is the Earth’s radius of 6371 km.

At an elevation angle β = 15° and a range of Dro = 360 km, the height of the rocket above the Earth’s surface is equal to Hro = 102.5; at a range of 120 km – Hro = 43.27 km and at a range of 50 km – Hro = 13.12 km.

For a rocket located at an altitude of Hro = 102.5 km from the Earth, its temperature is 300 K. When the rocket enters the denser layers of the atmosphere (<70 km), its temperature rises. At Hro = 43.27 km, we take the temperature Т = 500 K, and at Hro = 13.12 km, we take the temperature Т = 700 K. These data will be used in further calculations.

At heights of 50–55 km and below, a glow (plasma formation) appears, and the signal increases sharply.

Table 2 shows the calculation of the energy luminosity of an absolute blackbody with an emissivity of 0.5 for three temperatures: 300, 500, and 700 K. The calculation of the illumination of the entrance pupil of the objective and the S / N ratio by current at the corresponding distances of 360, 120, and 50 km are also presented in Table 2.

We choose a flight trajectory such that when the range to the rocket decreases, the flight altitude decreases and the temperature of the rocket skin rises.

Analysis of data from Table 2 showed the following. At Т = 300 K in the near infrared range 0.9–1.3; 1.4–1.8; 1.8–2.6; 0.9–2.6 S / N ratio <<1. In the mid-IR range 3.2–4.2 and 3.0–5.2, S / N ratio <7 and, therefore, the accumulation of frames will be required, and in the sub-range 4.4–5.2, S / N<1.

At Т = 500 K, in the near-IR sub-range of 0.9–1.3, the S / N ratio is <1, and in the 1.4–1.8 sub-range, the S / N ratio is <7 and, therefore, the accumulation of frames will be required. In the sub-ranges 1.8–2.6 and 0.9–2.6, the S / N ratio is >>1. In the mid-IR range, the S / N ratio in all sub-ranges is >>1.

At Т = 700 K in all sub-bands, the S / N ratio >>1.

In accordance with Wien’s displacement law [3], the spectral brightness of blackbody radiation per unit wavelength reaches a peak at the wavelength λmax = 22898 / T. Therefore, the maximum brightness of radiation at the selected temperatures occurs at the corresponding wavelengths:

Consequently, from the point of view of optimizing the reception of a signal from the OC with a temperature of 300 and 500 K, the used sub-bands are not optimal. At high velocities of more than 5 MAX and a decrease in flight altitude, the temperature of the rocket skin reaches 2000–2500 K [4].

Table 3 shows the calculation of the energy luminosity of a blackbody with an emissivity of 0.5 for a rocket skin temperature of 2000 and 2500 K. It follows from Wien’s displacement law that the maximum radiation brightness occurs: at Т = 2000 K at λ = 1.449 μm, which corresponds to the near infrared range; at Т = 2500 K at λ = 1.1592 μm, which also corresponds to the near-IR range. The magnitude of the luminosity of the OC, presented in Table 3, will be sufficient for reliable signal reception. Analyzing the data in Table 3, we find that the maximum luminosity lies in the sub-range of 0.9–2.6 μm with the largest spectral band of 1.7 μm.

2. Rocket torch radiation

The combustion process produces carbon dioxide and water vapor. Therefore, information on their spectra is of interest in relation to the assessment of the radiation from rocket torches and aircraft turbojet engines.

Fig. 3 shows the spectrum of the flame of a Bunsen burner [4]. A strong emission band is observed at a wavelength of 4.4 µm, and a weaker and broader band at 2.7 µm. The 2.7 µm band is formed by the superposition of several bands of water vapor and carbon dioxide, and the 4.4 µm band is a carbon dioxide band. In a flame, the emission band is wider than the corresponding absorption band, and the emission band is shifted towards longer wavelengths.

The spectra of different types of flame are very similar. All are characterized by a strong emission band in the range of 4–5 μm and a weaker band at λ < 3 μm. It is important to note that the spectral brightness of the emission band cannot exceed the spectral brightness of a blackbody at the same wavelength and temperature.

The spectrum of the rocket torch at an altitude of 40 km from the Earth’s surface is similar to the emission spectrum of the Bunsen burner (Fig. 4). The spectrum contains 3 maxima (in normalized form): 1) at a wavelength of 2.6 μm – an amplitude of 0.59; 2) at a wavelength of 4.25 μm – an amplitude of 0.74; 3) at a wavelength of 4.6 µm – an amplitude of 0.93. The radiation spectrum of the torch is located in the range of 1.0–6.1 microns.

Thus, due to variations in temperature and pressure inside the flame, the radiation spectrum of various rockets is somewhat modified, but insignificantly. In this case, the spectral regions of the atmosphere 4.2–4.4 μm and 2.6–3.2 μm are regions of strong absorption of radiation due to water vapor and carbon dioxide. Unfortunately, because of the lack of absolute values of spectrum intensity rocket torch, to evaluate the level of the signal at the entrance pupil of the receiving lens is difficult.

Basic Mathematical Relations for Determining the Threshold Illumination at the Entrance Pupil of the Device

An essential feature of the proposed technical calculation is finding the permissible accumulation time, and then we find the corresponding signal photon counting rate for a given S / N ratio.

The photon count rate nb [s–1] at the matrix element associated with atmospheric radiation in the spectral range Δλ has the form [6]

nb = LλΔλ (a / Fl)2 Тopt Sl / Eq,

where Lλ is the spectral brightness of the daytime sky [W ∙ cm–2 ∙ µm–1 ∙ sr–1];

Δλ = λ2 – λ1 is the band of the spectral range [µm];

λ1, λ2 are the boundaries of the spectral range; a is pixel size of the matrix element (side of the square) [cm];

Fl is the focal length of the receiving lens [cm];

Topt = Tl ∙ TIF is the transmittance of the receiving optics,

Tl and TIF are the transmittance of the lens and the interference filter, respectively;

Sl = π(Dl / 2)2 is the area of the receiving lens with a diameter Dl [cm];

Eq = h ∙ c / λ0 ∙ 10–6 is quantum energy [J];

h = 6,6256 ∙ 10–34 [J ∙ s] is Planck’s constant,

с = 3 ∙ 108 m / s is speed of light;

λ0 is average wavelength [μm], λ0 = (λ1 + λ2) / 2.

In [6], the graphs of the dependence of Lλ for discrete elevation angles are presented: 0; 7.2; 30 deg. The graphs were digitized, and spline-approximation was performed [7] for any angle value. At angles greater than 30°, the background level of the atmosphere decreases. Therefore, in calculations at elevation angles greater than 30°, the background level is taken to be equal to its value at 30°.

If the IR lens is not specially cooled, then it is a background source, and it can be considered as an absolute black body (ABB) with temperature T. In this case, we will assume that the field diaphragm is installed in a cooled matrix receiver.

The law of blackbody emission for the photon emission density F(λ) [s–1 ∙ cm–2 ∙ µm–1 ∙ sr–1] has the form [4]

F(λ) = (С3 / λ4) / [exp(С2 / λT) – 1].

Here С3 = 1.88365 ∙ 1023 [s–1 ∙ cm–2 ∙ µm3];

С2 = 1.43879 ∙ 104 [μm ∙ K];

T is the blackbody temperature [K].

The count rate of photons from the objective nl [s–1] in a given spectral range [8]

,

where Spix = a2 is pixel area [cm2]; Kra is the emissivity of the objective, Kra = 1 – Tl.

The maximum observation time τac, measured in seconds, is limited by the pixel accumulation capacity Сe [3]:

τac ≤ (СeKz – Nre) / [η(nb + nl + ns) + nd], (1)

where Kz is the safety factor, Kz = 0,7÷0,8; Nre is the number of readout noise electrons; η is the quantum efficiency of the matrix; ns is the counting rate of signal photons at the matrix element, nd is the counting rate of dark electrons, nd = id / e, id is the dark current of the matrix, e is the electron charge, e = 1.6 10 · 10–19 Kl.

Using the obtained function τac = f(ns), a graph of the dependence of the accumulation time on the counting rate of signal photons is plotted on a logarithmic scale (Fig. 5). The calculation was performed for the initial data indicated above in the spectral band 3.2–4.2 μm.

The signal-to-noise ratio in terms of the power at the output of the matrix is determined in the form [3]

Qp = (η · ns · τac)2 / [η · F · τac(nb + nl + ns) + ndτac + (Nre)2]. (2)

The signal photon counting rate is found from the expression (2):

ns = (FQp / 2ητac)(1 + A), (3)

where A = {1 + (4 / FQp)[ητac(nb + nl) + ndτac+ (Nre)2]}1 / 2,

F = 1–2, is the noise gain.

Based on relation (3), a graph of the dependence of the S / N ratio in terms of power on the count rate of signal photons ns is constructed (Fig. 6). The initial data for the calculation are taken from Fig. 5. It can be seen (Fig. 6) that at ns = 1010 and more, the S / N ratio practically does not change due to a decrease in the observation time in accordance with expression (1).

Typically, the receiving lens satisfies the Fraunhofer diffraction principle in the focal plane. In this case, the pixel size must be matched with the main petal. But even in this case, there is a loss of part of the received signal energy. This loss is taken into account by the χ coefficient. For the selected characteristics of the lens and matrix, we take χ = 0.837.

Using the rule of addition of the variances of random variables [7], we find the root-mean-square deviation of the number of noise electrons read from the matrix pixel:

σΣ = [η · F · τac(nb + nl + ns) + ndτac + (Nre)2]1 / 2.

The reduced noise power Pn [W] to the matrix pixel, at which Qp = 1, is defined as

Рn = σΣЕq / ητac.

The expression for the threshold illumination Eth [W / cm2] of the objective is:

Eth = Рn / Toptχπ(Dl / 2)2.

If we assume that the minimum current S / N ratio at which the signal is detected is Qi = 7 [3], then we can determine the value of the minimum illumination Emin = Psmin / [Toptχπ(Dl / 2)2], Psmin = Еqnsmin; nsmin – signal photon counting rate, is determined from expression (3) at Qp = 49.

3. Illumination of the Entrance Pupil by the Radiation of the Sun Reflected From the Rocket

Let us consider the signal at the entrance pupil of the device from a cylindrical rocket illuminated by the Sun in different spectral ranges. When calculating as the Sun, we use the ABB at a temperature of 6000 K. The rocket is located at a distance of 360 km from the Earth, has a radius r = 0.5 m, a length H = 2 m and a reflection coefficient ρ = 0.3.

We find the brightness of the Sun BS(T, Δλ) [W · m–2 · sr–1]

BS(Т, Δl) = R(Т, Δl) · 104 / π.

Then we determine the brightness of the rocket in the direction of the OES Bro(Т, Δl) [W · m–2 · sr–1]

Bro(Δl) = BS(Т, Δl) ρKdb(RS / LS-ro)2,

where: RS = 6.9599 · 108 [m] is the radius of the Sun; ρ is the reflection coefficient of the rocket, Kdb is the dimensional brightness coefficient (DBC) [9] at ρ = 1, equal to the ratio of the area of the rocket illuminated by the Sun visible from the OES to the total area of the rocket illuminated by the Sun, LS-ro is the distance from the Sun to the rocket [m], Kdb depends on the angles γ, ξ1, ξ2 and is defined as Kdb = Sro / S0 (S0 = 2rH, S0 is the projection of the area of the rocket illuminated by the Sun [m2], Sro is the effective area of the rocket [m2]) [8].

Sro = S0 cos(ξ1) cos(ξ2)[(π − δ + 0.5sin(2δ)cosδ + sin3δ] / π,

where cosδ = cosγ / [cosξ1 cosξ2] + tgξ1 tg ξ2 for γ ≥ (ξ1 + ξ2) and γ < 180° – |ξ1– ξ2|;

γ is the angle between the directions: the Sun-rocket and the OES rocket;

ξ1, ξ2 are the angles between the plane perpendicular to the longitudinal axis of the rocket and the direction to the Sun and OES, respectively;

δ is the angle in the plane perpendicular to the longitudinal axis of the rocket between the projections of the Sun – rocket and rocket – OES directions onto it.

The illumination of the entrance pupil of the objective Еen.p, W / cm2, is determined by the expression:

Еen.p = 10–10Broτatm(λ)S0 / (DE-ro)2,

where: DE-ro is the distance between the Earth and the rocket, km,

τatm(λ) is the average transmittance of the atmosphere in the spectral range Δλ.

The S / N ratio for current Qi is

Qi = EinpSobτacAl / [F{τac(A0 + ηns) + (Nre)2}]0,5, (4)

Where Al = Koptχη / Eq,

A0 = η(nb + nl) + nd,

τmin ≤ τac ≤ τmax;

ns = EinpSobAl / η,

Einp is the signal illumination at the entrance pupil of the lens.

Using dependence (4), we calculate the signal-to-noise ratio for the current at various values of the illumination of the entrance pupil of the lens (Fig. 8) in the spectral range of 3.2–4.2 µm.

Up to the illumination value at the entrance pupil of the lens~10–15 W / m2, the S / N ratio increases linearly, because the accumulation time remains practically unchanged. With an illumination of 5 ∙ (10–14–10–12) W / m2, the S / N ratio increases linearly. At higher values than 5 ∙ 10–12 W / m2, although the illumination increases, the S / N ratio will not increase, i. e. the accumulation time reaches a minimum value and the pixel charge capacity is saturated.

INTERFACE OF THE PROGRAM

The interface of the developed program for calculating the IR system (RIKS‑1) is shown in Fig. 9. The input data for the program are the parameters of the photodetector matrix, the receiving lens, the controlled object and the atmosphere. With the help of the program, initial data are entered, and the results of the calculation can be displayed on the monitor screen and on paper. It is possible to display the graphs of the atmospheric transmission, spectral brightness of the sky background, and DBC.

By the command “Graphatm.” the monitor displays the dependence of the spectral transmittance of the atmosphere on the radiation wavelength. The spectral transmittance of the atmosphere was calculated in the range 0.9–5.3 µm using experimental tables [6] using spline approximation [7].

By the command “Chart of Backgraund”, the monitor displays the dependence of the spectral brightness of the sky background at the selected elevation angle in the specified spectral range of radiation wavelengths.

When using the “Graph TAU” command, the monitor will display the dependence of the accumulation time on the counting rate of signal photons (Fig. 5), and on the “Graph Qp” command, the monitor will display the dependence of the S / N ratio by power on the count rate of signal photons (Fig. 6).

By the command “Graph_DBC”, the monitor displays the dependence of the rocket’s DBC on the angle γ at the given angles ξ1, ξ2.

When using the command “Result1” the screen will display: spectral brightness of the background; the illumination created by the background per pixel; background power per pixel; threshold signal power (at S / N ratio = 1) when one pixel is illuminated and when the signal is divided by 4 pixels; signal power at a S / N ratio by power equal to 100 and 10000; accumulation time at S / N ratio = 1.

When using the command “Result2” the screen will display: threshold number of ph-el; average number of ph-el background; counting rate of background + thermal ph-el; the count rate of signal photons per pixel at Qi = 10; signal power at the lens input at Qi = 10; signal power per pixel at Qi = 10; threshold illumination at the entrance pupil, minimum signal power, minimum illumination at the entrance pupil at Qi = 10; average atmospheric transmittance in a given spectral range; illumination at the entrance pupil, due to sunlight reflected from the SO; signal-to-noise ratio for current for a signal from a SO illuminated by the Sun; luminosity from own radiation of SO; illumination at the entrance pupil from the intrinsic radiation of the SO; Qi for the signal from the SO, due to its own radiation. Result 1 and Result 2 can be printed.

At the command “Source data”, the screen will display all the initial data, as well as the range to the upper edge of the atmosphere and the altitude of the space object to the Earth.

CONCLUSIONS

The developed method for calculating the illumination of the OES entrance pupil from the signal of a cylindrical space object, illuminated by the Sun and having its own radiation of sheathing, made it possible to establish the following.

The threshold illumination of the device at the entrance pupil is minimal in the sub-range of 1.8–2.6 μm, and the highest in the sub-range of 4.4–5.2 μm.

The intrinsic radiation of the rocket skin changes with the density of the atmosphere and, as a consequence, its temperature. At a temperature of 300 K (D = 360 km), the rocket radiation is small and, therefore, the S / N ratio is small. At T = 500 K (D = 160 km), SO detection is possible in all sub-ranges except 0.9–1.3 μm and 1.4–1.8 μm. At T = 700 K (D = 50 km), it is possible to detect SO in all sub-bands.

The maximum illumination of the entrance pupil, reflected by the solar radiation from the SO sheathing, is located in the sub-range of 0.9–2.6 μm and, therefore, the S / N ratio is maximum, and the minimum S / N ratio is located in the sub-range of 4.4–5.2 μm. This result is influenced by the spectral distribution of the signal and the transmission of the atmosphere.

When the rocket is flying in dense layers of the atmosphere at low altitude, the temperature of the skin (also of the torch) is Т = 2000 or 2500 K. At these temperatures, the luminosity of the SO is quite high, this allows it to be detected in all sub-ranges.

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ABOUT AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Tech. Sc., igorznamenskii@yandex.ru; Leading Researcher, Research and Production Corporation “Рrecision Systems and Instruments” (JSC NPK SPP); Moscow, Russia.

ORCID‑0000-0002-0612-1255

Zotev Evgeny Olegovich, Head of the Scientific and Technical Complex, JSC NPK SPP; 06-2@npk-spp.ru; Moscow, Russia.

Yudin Sergey Yurievich, morenos@npk-spp.ru, Head of Department, JSC NPK SPP; Moscow, Russia.

Contribution by the Members of the Team of Authors

The article was prepared on the basis of many years of work by all members of the team of authors.

ConflIct of Interest

The authors claim that they have no conflict of interest. All authors took part in writing the article and supplemented the manuscript in part of their work.

I. V. Znamensky, E. O. Zot’ev, S. Yu. Yudin

Research and Production Corporation “Рrecision Systems and Instruments” (JSC NPK SPP)

Moscow, Russia

The result of calculating the threshold illumination of an optoelectronic system (OES) in various infrared sub-ranges in the range of 0.9–5.3 µm is presented. A technique has been developed and the illumination of the entrance pupil of the OES from the signal of the monitoring object (OM) of a cylindrical shape illuminated by the Sun and due to the intrinsic radiation of the OM surface is calculated. A program for energy calculation has been developed and its interface has been presented.

Keywords: Optoelectronic system, IR range, matrix photodetector, threshold illumination, accumulation time, signal-to-noise ratio, atmosphere, photon, overall brightness coefficient.

Received on: 18.03.2021

Accepted on: 17.08.2021

INTRODUCTION

At present, systems operating in the infrared range are widely used in optoelectronic systems (OES) for ground-based space surveillance and tracking space objects. Operation in the IR range is associated with several reasons:

- low background level of solar radiation;

- the presence of “windows” of transparency of the atmosphere;

- industrial development of matrices with a large number of elements while reducing their size, low intrinsic noise and high quantum efficiency.

Decreasing the step and increasing the format is a common trend for almost all world developers and manufacturers of IR matrices. Reducing the step and increasing the format leads to a significant increase in the range of object recognition [1].

Reviews of the state and prospects for the development of matrix photodetectors have been considered by many specialists, for example [2]. To solve specific problems, programs for calculating an OES with a matrix photodetector in the IR range have been developed. Thus, in [3], the calculation of the threshold sensitivity of the OES in various spectral infrared ranges is considered and the illumination of the entrance pupil from a space object (SO) of a spherical shape is determined.

This article presents an algorithm and a calculation program for determining the threshold illumination of an OES in the spectral range of 0.9–5.3 μm with the refinement of the sub-ranges. The calculation of the signal from the SO of a cylindrical shape (rocket) under the illumination of the sun is carried out, and the results are presented using the dimension brightness coefficient (DBC), the graph of which is presented. A comparative analysis of the threshold sensitivity of IR systems in different spectral ranges and different temperatures of the rocket skin has been carried out. The developed method for calculating IR systems uses an approach based on the estimation of the photon count rate. The optimal observation time is found for a given background level and matrix parameters.

SELECTING THE SPECTRAL RANGE

When receiving the intrinsic radiation of the rocket skin, the receiving spectrum has the form of an absolute black body (ABB) spectrum at the rocket temperature.

For the case of the absence of a rocket torch when receiving radiation from a rocket illuminated by the Sun, the receiving spectrum has a spectrum of an absolutely black body at a temperature of Т = 6000 K [3]. Radiation from the Sun is reflected from the rocket skin and is received by the receiving lens of the system.

To optimize the S / N ratio, taking into account the transparency of the atmosphere, the following sub-ranges were selected: 0.9–1.3 µm; 1.4–1.8 µm; 1.8–2.6 µm; 0.9–2.6 µm; 3.2–4.2 µm; 4.4–5.2 µm; 3.0–5.2 µm. These IR sub-bands coincide with the transparency windows of the atmosphere, which should provide the highest signal-to-noise ratio.

Fig. 1 shows the dependence of the spectral transmittance of the atmosphere on the radiation wavelength in the range 0.9–5.3 μm at an elevation angle of 15°. The graph is calculated and built using a program developed by the authors.

When choosing a spectral range, it is necessary to take into account both the transparency windows of the atmosphere and the characteristics of commercially available matrices containing a built-in interference filter (IF). The Falcon matrix by SCD (Israel) can operate in the range of 1.0–5.45 μm, has a built-in factory IF in the range of 3.6–4.9 μm. The cooled IF is not replaceable, since it is installed in a sealed case. Selex ES manufactures a similar Eagle matrix. In this case, the required IF sub-range can be selected.

It is also important to assess the level of the external background in order to maximize the SNR. In Fig. 2 shows the dependence of the spectral brightness of the sky background at an elevation angle of 15° in the range of 0.9–5.3 μm. In the calculations of the mid-IR range, we use the parameters of the Falcon matrix, for the near-IR range, we use the same parameters.

The initial data for calculating the threshold sensitivity in different spectral sub-ranges are displayed on the input panel for the initial data (see Appendix): lens diameter Dl = 250 mm; focal length Fl = 720 mm; transmission coefficient of receiving optics Kopt = 0,7; maximum accumulation time 20 ms; elevation angle β = 15°; meteorological visibility range Sm = 20 km; relative air humidity V = 80%; air temperature T = 22 °C. The results of calculating the IR system in different spectral ranges are presented in Table 1.

The minimum illumination created by the background per pixel corresponds to the infrared sub-ranges of 1.8–2.6 and 3.2–4.2 μm. Therefore, in these ranges, the signal accumulation time will be maximum and equal to 20 ms.This accumulation time provides a frame rate of 50 Hz.

When calculating IR systems, three restrictions were taken:

- the number of accumulation electrons, limited by the value of the accumulation capacity of the pixel, 1.1 ∙ 107 electrons;

- the maximum accumulation time, limited by the frame duration, 20 ms;

- the minimum accumulation time, limited by the matrix parameters, ~ 10 μs.

The maximum and minimum accumulation times limit the dynamic range of the input signals.

When choosing a spectral range, the level of the external background is important, at which the signal-to-noise ratio is maximized. The threshold illumination at the entrance pupil is minimal in the sub-ranges of 1.8–2.6 and 3.2–4.2 μm. The situation with background radiation is somewhat worse in the 1.4–1.8 μm sub-range. The results obtained do not take into account the spectral transmission of the atmosphere.

The minimum illumination by the entrance pupil signal, recalculated outside the atmosphere (see Table 1) at Qp = 100 in the spectral range of 3.2–4.2 μm, is 0.937 ∙ 10–15 W / cm2. In the ranges 1.4–1.8 μm it is 1.685 ∙ 10–15 W / cm2 and in the range 1.8–2.6 μm it is 1.856 ∙ 10–15 W / cm2. Here Qp is the signal-to-noise power ratio.

The lowest sensitivity has a spectral range of 4.4–5.2 µm, which is 66.8 times worse than the range 3.2–4.2 µm. The final conclusion about the best spectral range can be made only by knowing the spectral distribution of the signal from the target.

The initial data for calculating the signal from the rocket are as follows: rocket radius Rrо = 0.5 m; rocket length Lrо = 2 m; the distance from the Earth’s surface to the rocket is calculated for 3 values: DE-rо = 360, 160, 50 km at three temperatures of 300, 500 and 700 K, respectively; the reflection coefficient of the rocket is Кr = 0.5; the angle between the direction to the Sun and the normal to the plane orthogonal to the longitudinal axis of the SO ξ1 = 30°; the angle between the direction to the receiver and the perpendicular to the plane perpendicular to the longitudinal axis of the SO ξ2 = 30°; the angle between the direction to the Sun from the SO and the direction to the receiver γ = 90°. The rest of the parameters are the same as those used to calculate the threshold sensitivity.

SIGNAL FROM A SPACE OBJECT

IN THE FORM OF A ROCKET

The spectral composition of the rocket radiation consists of three components:

intrinsic thermal radiation of the rocket body,

thermal radiation of the rocket torch,

solar radiation reflected from the rocket.

1. Intrinsic thermal radiation of the rocket body

The intrinsic thermal radiation of the rocket, which has a temperature T, is calculated for the blackbody in a given spectral range by the formula [4]:

, Вт ∙ см–2,

where С1 = 3.7415 ∙ 104 [W∙cm– 2 ∙ µm4];

С2 = 1.43879 ∙ 104 µm ∙ K;

T is the blackbody temperature;

K, λ is the wavelength, µm;

λ1, λ2 are the boundaries of the spectral range [μm];

ε is the emissivity, we take ε = 0.5.

The dependence of the height of the rocket Hro above the Earth’s surface on the distance Dro to it at a constant elevation angle β is found in the form [5]:

where RE is the Earth’s radius of 6371 km.

At an elevation angle β = 15° and a range of Dro = 360 km, the height of the rocket above the Earth’s surface is equal to Hro = 102.5; at a range of 120 km – Hro = 43.27 km and at a range of 50 km – Hro = 13.12 km.

For a rocket located at an altitude of Hro = 102.5 km from the Earth, its temperature is 300 K. When the rocket enters the denser layers of the atmosphere (<70 km), its temperature rises. At Hro = 43.27 km, we take the temperature Т = 500 K, and at Hro = 13.12 km, we take the temperature Т = 700 K. These data will be used in further calculations.

At heights of 50–55 km and below, a glow (plasma formation) appears, and the signal increases sharply.

Table 2 shows the calculation of the energy luminosity of an absolute blackbody with an emissivity of 0.5 for three temperatures: 300, 500, and 700 K. The calculation of the illumination of the entrance pupil of the objective and the S / N ratio by current at the corresponding distances of 360, 120, and 50 km are also presented in Table 2.

We choose a flight trajectory such that when the range to the rocket decreases, the flight altitude decreases and the temperature of the rocket skin rises.

Analysis of data from Table 2 showed the following. At Т = 300 K in the near infrared range 0.9–1.3; 1.4–1.8; 1.8–2.6; 0.9–2.6 S / N ratio <<1. In the mid-IR range 3.2–4.2 and 3.0–5.2, S / N ratio <7 and, therefore, the accumulation of frames will be required, and in the sub-range 4.4–5.2, S / N<1.

At Т = 500 K, in the near-IR sub-range of 0.9–1.3, the S / N ratio is <1, and in the 1.4–1.8 sub-range, the S / N ratio is <7 and, therefore, the accumulation of frames will be required. In the sub-ranges 1.8–2.6 and 0.9–2.6, the S / N ratio is >>1. In the mid-IR range, the S / N ratio in all sub-ranges is >>1.

At Т = 700 K in all sub-bands, the S / N ratio >>1.

In accordance with Wien’s displacement law [3], the spectral brightness of blackbody radiation per unit wavelength reaches a peak at the wavelength λmax = 22898 / T. Therefore, the maximum brightness of radiation at the selected temperatures occurs at the corresponding wavelengths:

- at Т = 300 K at λ = 9.66 μm, which corresponds to the far IR range;

- at Т = 500 K at λ = 5.796 μm, which corresponds to the mid-IR range;

- at Т = 700 K at λ = 4.14 μm, which corresponds to the mid-IR range.

Consequently, from the point of view of optimizing the reception of a signal from the OC with a temperature of 300 and 500 K, the used sub-bands are not optimal. At high velocities of more than 5 MAX and a decrease in flight altitude, the temperature of the rocket skin reaches 2000–2500 K [4].

Table 3 shows the calculation of the energy luminosity of a blackbody with an emissivity of 0.5 for a rocket skin temperature of 2000 and 2500 K. It follows from Wien’s displacement law that the maximum radiation brightness occurs: at Т = 2000 K at λ = 1.449 μm, which corresponds to the near infrared range; at Т = 2500 K at λ = 1.1592 μm, which also corresponds to the near-IR range. The magnitude of the luminosity of the OC, presented in Table 3, will be sufficient for reliable signal reception. Analyzing the data in Table 3, we find that the maximum luminosity lies in the sub-range of 0.9–2.6 μm with the largest spectral band of 1.7 μm.

2. Rocket torch radiation

The combustion process produces carbon dioxide and water vapor. Therefore, information on their spectra is of interest in relation to the assessment of the radiation from rocket torches and aircraft turbojet engines.

Fig. 3 shows the spectrum of the flame of a Bunsen burner [4]. A strong emission band is observed at a wavelength of 4.4 µm, and a weaker and broader band at 2.7 µm. The 2.7 µm band is formed by the superposition of several bands of water vapor and carbon dioxide, and the 4.4 µm band is a carbon dioxide band. In a flame, the emission band is wider than the corresponding absorption band, and the emission band is shifted towards longer wavelengths.

The spectra of different types of flame are very similar. All are characterized by a strong emission band in the range of 4–5 μm and a weaker band at λ < 3 μm. It is important to note that the spectral brightness of the emission band cannot exceed the spectral brightness of a blackbody at the same wavelength and temperature.

The spectrum of the rocket torch at an altitude of 40 km from the Earth’s surface is similar to the emission spectrum of the Bunsen burner (Fig. 4). The spectrum contains 3 maxima (in normalized form): 1) at a wavelength of 2.6 μm – an amplitude of 0.59; 2) at a wavelength of 4.25 μm – an amplitude of 0.74; 3) at a wavelength of 4.6 µm – an amplitude of 0.93. The radiation spectrum of the torch is located in the range of 1.0–6.1 microns.

Thus, due to variations in temperature and pressure inside the flame, the radiation spectrum of various rockets is somewhat modified, but insignificantly. In this case, the spectral regions of the atmosphere 4.2–4.4 μm and 2.6–3.2 μm are regions of strong absorption of radiation due to water vapor and carbon dioxide. Unfortunately, because of the lack of absolute values of spectrum intensity rocket torch, to evaluate the level of the signal at the entrance pupil of the receiving lens is difficult.

Basic Mathematical Relations for Determining the Threshold Illumination at the Entrance Pupil of the Device

An essential feature of the proposed technical calculation is finding the permissible accumulation time, and then we find the corresponding signal photon counting rate for a given S / N ratio.

The photon count rate nb [s–1] at the matrix element associated with atmospheric radiation in the spectral range Δλ has the form [6]

nb = LλΔλ (a / Fl)2 Тopt Sl / Eq,

where Lλ is the spectral brightness of the daytime sky [W ∙ cm–2 ∙ µm–1 ∙ sr–1];

Δλ = λ2 – λ1 is the band of the spectral range [µm];

λ1, λ2 are the boundaries of the spectral range; a is pixel size of the matrix element (side of the square) [cm];

Fl is the focal length of the receiving lens [cm];

Topt = Tl ∙ TIF is the transmittance of the receiving optics,

Tl and TIF are the transmittance of the lens and the interference filter, respectively;

Sl = π(Dl / 2)2 is the area of the receiving lens with a diameter Dl [cm];

Eq = h ∙ c / λ0 ∙ 10–6 is quantum energy [J];

h = 6,6256 ∙ 10–34 [J ∙ s] is Planck’s constant,

с = 3 ∙ 108 m / s is speed of light;

λ0 is average wavelength [μm], λ0 = (λ1 + λ2) / 2.

In [6], the graphs of the dependence of Lλ for discrete elevation angles are presented: 0; 7.2; 30 deg. The graphs were digitized, and spline-approximation was performed [7] for any angle value. At angles greater than 30°, the background level of the atmosphere decreases. Therefore, in calculations at elevation angles greater than 30°, the background level is taken to be equal to its value at 30°.

If the IR lens is not specially cooled, then it is a background source, and it can be considered as an absolute black body (ABB) with temperature T. In this case, we will assume that the field diaphragm is installed in a cooled matrix receiver.

The law of blackbody emission for the photon emission density F(λ) [s–1 ∙ cm–2 ∙ µm–1 ∙ sr–1] has the form [4]

F(λ) = (С3 / λ4) / [exp(С2 / λT) – 1].

Here С3 = 1.88365 ∙ 1023 [s–1 ∙ cm–2 ∙ µm3];

С2 = 1.43879 ∙ 104 [μm ∙ K];

T is the blackbody temperature [K].

The count rate of photons from the objective nl [s–1] in a given spectral range [8]

,

where Spix = a2 is pixel area [cm2]; Kra is the emissivity of the objective, Kra = 1 – Tl.

The maximum observation time τac, measured in seconds, is limited by the pixel accumulation capacity Сe [3]:

τac ≤ (СeKz – Nre) / [η(nb + nl + ns) + nd], (1)

where Kz is the safety factor, Kz = 0,7÷0,8; Nre is the number of readout noise electrons; η is the quantum efficiency of the matrix; ns is the counting rate of signal photons at the matrix element, nd is the counting rate of dark electrons, nd = id / e, id is the dark current of the matrix, e is the electron charge, e = 1.6 10 · 10–19 Kl.

Using the obtained function τac = f(ns), a graph of the dependence of the accumulation time on the counting rate of signal photons is plotted on a logarithmic scale (Fig. 5). The calculation was performed for the initial data indicated above in the spectral band 3.2–4.2 μm.

The signal-to-noise ratio in terms of the power at the output of the matrix is determined in the form [3]

Qp = (η · ns · τac)2 / [η · F · τac(nb + nl + ns) + ndτac + (Nre)2]. (2)

The signal photon counting rate is found from the expression (2):

ns = (FQp / 2ητac)(1 + A), (3)

where A = {1 + (4 / FQp)[ητac(nb + nl) + ndτac+ (Nre)2]}1 / 2,

F = 1–2, is the noise gain.

Based on relation (3), a graph of the dependence of the S / N ratio in terms of power on the count rate of signal photons ns is constructed (Fig. 6). The initial data for the calculation are taken from Fig. 5. It can be seen (Fig. 6) that at ns = 1010 and more, the S / N ratio practically does not change due to a decrease in the observation time in accordance with expression (1).

Typically, the receiving lens satisfies the Fraunhofer diffraction principle in the focal plane. In this case, the pixel size must be matched with the main petal. But even in this case, there is a loss of part of the received signal energy. This loss is taken into account by the χ coefficient. For the selected characteristics of the lens and matrix, we take χ = 0.837.

Using the rule of addition of the variances of random variables [7], we find the root-mean-square deviation of the number of noise electrons read from the matrix pixel:

σΣ = [η · F · τac(nb + nl + ns) + ndτac + (Nre)2]1 / 2.

The reduced noise power Pn [W] to the matrix pixel, at which Qp = 1, is defined as

Рn = σΣЕq / ητac.

The expression for the threshold illumination Eth [W / cm2] of the objective is:

Eth = Рn / Toptχπ(Dl / 2)2.

If we assume that the minimum current S / N ratio at which the signal is detected is Qi = 7 [3], then we can determine the value of the minimum illumination Emin = Psmin / [Toptχπ(Dl / 2)2], Psmin = Еqnsmin; nsmin – signal photon counting rate, is determined from expression (3) at Qp = 49.

3. Illumination of the Entrance Pupil by the Radiation of the Sun Reflected From the Rocket

Let us consider the signal at the entrance pupil of the device from a cylindrical rocket illuminated by the Sun in different spectral ranges. When calculating as the Sun, we use the ABB at a temperature of 6000 K. The rocket is located at a distance of 360 km from the Earth, has a radius r = 0.5 m, a length H = 2 m and a reflection coefficient ρ = 0.3.

We find the brightness of the Sun BS(T, Δλ) [W · m–2 · sr–1]

BS(Т, Δl) = R(Т, Δl) · 104 / π.

Then we determine the brightness of the rocket in the direction of the OES Bro(Т, Δl) [W · m–2 · sr–1]

Bro(Δl) = BS(Т, Δl) ρKdb(RS / LS-ro)2,

where: RS = 6.9599 · 108 [m] is the radius of the Sun; ρ is the reflection coefficient of the rocket, Kdb is the dimensional brightness coefficient (DBC) [9] at ρ = 1, equal to the ratio of the area of the rocket illuminated by the Sun visible from the OES to the total area of the rocket illuminated by the Sun, LS-ro is the distance from the Sun to the rocket [m], Kdb depends on the angles γ, ξ1, ξ2 and is defined as Kdb = Sro / S0 (S0 = 2rH, S0 is the projection of the area of the rocket illuminated by the Sun [m2], Sro is the effective area of the rocket [m2]) [8].

Sro = S0 cos(ξ1) cos(ξ2)[(π − δ + 0.5sin(2δ)cosδ + sin3δ] / π,

where cosδ = cosγ / [cosξ1 cosξ2] + tgξ1 tg ξ2 for γ ≥ (ξ1 + ξ2) and γ < 180° – |ξ1– ξ2|;

γ is the angle between the directions: the Sun-rocket and the OES rocket;

ξ1, ξ2 are the angles between the plane perpendicular to the longitudinal axis of the rocket and the direction to the Sun and OES, respectively;

δ is the angle in the plane perpendicular to the longitudinal axis of the rocket between the projections of the Sun – rocket and rocket – OES directions onto it.

The illumination of the entrance pupil of the objective Еen.p, W / cm2, is determined by the expression:

Еen.p = 10–10Broτatm(λ)S0 / (DE-ro)2,

where: DE-ro is the distance between the Earth and the rocket, km,

τatm(λ) is the average transmittance of the atmosphere in the spectral range Δλ.

The S / N ratio for current Qi is

Qi = EinpSobτacAl / [F{τac(A0 + ηns) + (Nre)2}]0,5, (4)

Where Al = Koptχη / Eq,

A0 = η(nb + nl) + nd,

τmin ≤ τac ≤ τmax;

ns = EinpSobAl / η,

Einp is the signal illumination at the entrance pupil of the lens.

Using dependence (4), we calculate the signal-to-noise ratio for the current at various values of the illumination of the entrance pupil of the lens (Fig. 8) in the spectral range of 3.2–4.2 µm.

Up to the illumination value at the entrance pupil of the lens~10–15 W / m2, the S / N ratio increases linearly, because the accumulation time remains practically unchanged. With an illumination of 5 ∙ (10–14–10–12) W / m2, the S / N ratio increases linearly. At higher values than 5 ∙ 10–12 W / m2, although the illumination increases, the S / N ratio will not increase, i. e. the accumulation time reaches a minimum value and the pixel charge capacity is saturated.

INTERFACE OF THE PROGRAM

The interface of the developed program for calculating the IR system (RIKS‑1) is shown in Fig. 9. The input data for the program are the parameters of the photodetector matrix, the receiving lens, the controlled object and the atmosphere. With the help of the program, initial data are entered, and the results of the calculation can be displayed on the monitor screen and on paper. It is possible to display the graphs of the atmospheric transmission, spectral brightness of the sky background, and DBC.

By the command “Graphatm.” the monitor displays the dependence of the spectral transmittance of the atmosphere on the radiation wavelength. The spectral transmittance of the atmosphere was calculated in the range 0.9–5.3 µm using experimental tables [6] using spline approximation [7].

By the command “Chart of Backgraund”, the monitor displays the dependence of the spectral brightness of the sky background at the selected elevation angle in the specified spectral range of radiation wavelengths.

When using the “Graph TAU” command, the monitor will display the dependence of the accumulation time on the counting rate of signal photons (Fig. 5), and on the “Graph Qp” command, the monitor will display the dependence of the S / N ratio by power on the count rate of signal photons (Fig. 6).

By the command “Graph_DBC”, the monitor displays the dependence of the rocket’s DBC on the angle γ at the given angles ξ1, ξ2.

When using the command “Result1” the screen will display: spectral brightness of the background; the illumination created by the background per pixel; background power per pixel; threshold signal power (at S / N ratio = 1) when one pixel is illuminated and when the signal is divided by 4 pixels; signal power at a S / N ratio by power equal to 100 and 10000; accumulation time at S / N ratio = 1.

When using the command “Result2” the screen will display: threshold number of ph-el; average number of ph-el background; counting rate of background + thermal ph-el; the count rate of signal photons per pixel at Qi = 10; signal power at the lens input at Qi = 10; signal power per pixel at Qi = 10; threshold illumination at the entrance pupil, minimum signal power, minimum illumination at the entrance pupil at Qi = 10; average atmospheric transmittance in a given spectral range; illumination at the entrance pupil, due to sunlight reflected from the SO; signal-to-noise ratio for current for a signal from a SO illuminated by the Sun; luminosity from own radiation of SO; illumination at the entrance pupil from the intrinsic radiation of the SO; Qi for the signal from the SO, due to its own radiation. Result 1 and Result 2 can be printed.

At the command “Source data”, the screen will display all the initial data, as well as the range to the upper edge of the atmosphere and the altitude of the space object to the Earth.

CONCLUSIONS

The developed method for calculating the illumination of the OES entrance pupil from the signal of a cylindrical space object, illuminated by the Sun and having its own radiation of sheathing, made it possible to establish the following.

The threshold illumination of the device at the entrance pupil is minimal in the sub-range of 1.8–2.6 μm, and the highest in the sub-range of 4.4–5.2 μm.

The intrinsic radiation of the rocket skin changes with the density of the atmosphere and, as a consequence, its temperature. At a temperature of 300 K (D = 360 km), the rocket radiation is small and, therefore, the S / N ratio is small. At T = 500 K (D = 160 km), SO detection is possible in all sub-ranges except 0.9–1.3 μm and 1.4–1.8 μm. At T = 700 K (D = 50 km), it is possible to detect SO in all sub-bands.

The maximum illumination of the entrance pupil, reflected by the solar radiation from the SO sheathing, is located in the sub-range of 0.9–2.6 μm and, therefore, the S / N ratio is maximum, and the minimum S / N ratio is located in the sub-range of 4.4–5.2 μm. This result is influenced by the spectral distribution of the signal and the transmission of the atmosphere.

When the rocket is flying in dense layers of the atmosphere at low altitude, the temperature of the skin (also of the torch) is Т = 2000 or 2500 K. At these temperatures, the luminosity of the SO is quite high, this allows it to be detected in all sub-ranges.

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ABOUT AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Tech. Sc., igorznamenskii@yandex.ru; Leading Researcher, Research and Production Corporation “Рrecision Systems and Instruments” (JSC NPK SPP); Moscow, Russia.

ORCID‑0000-0002-0612-1255

Zotev Evgeny Olegovich, Head of the Scientific and Technical Complex, JSC NPK SPP; 06-2@npk-spp.ru; Moscow, Russia.

Yudin Sergey Yurievich, morenos@npk-spp.ru, Head of Department, JSC NPK SPP; Moscow, Russia.

Contribution by the Members of the Team of Authors

The article was prepared on the basis of many years of work by all members of the team of authors.

ConflIct of Interest

The authors claim that they have no conflict of interest. All authors took part in writing the article and supplemented the manuscript in part of their work.

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