Issue #1/2022

Regarding the Possibility of Detecting Space Objects in the Spectral Range of 8–12 mm

**I. V. Znamensky, A. T. Tungushpaev**Regarding the Possibility of Detecting Space Objects in the Spectral Range of 8–12 mm

DOI: 10.22184/1993-7296.FRos.2022.16.1.44.58

The result of calculating the threshold illumination of an optical-electronic system (OES) in the far infrared range of 8–12 μm is presented. A technique has been developed and a calculation has been made of the illumination of the OES entrance pupil from the signal of a space object (SO) of a cylindrical shape, illuminated by the Sun, and due to the own radiation of the SO.

The result of calculating the threshold illumination of an optical-electronic system (OES) in the far infrared range of 8–12 μm is presented. A technique has been developed and a calculation has been made of the illumination of the OES entrance pupil from the signal of a space object (SO) of a cylindrical shape, illuminated by the Sun, and due to the own radiation of the SO.

Теги: accumulation time atmosphere ir range matrix photodetector optoelectronic system overall brightness coefficient. photon signal-to-noise ratio threshold illumination атмосфера время накопления ик диапазон коэффициент габаритной яркости матричный фотоприемник оптико-электронная система отношение сигнал-шум пороговая освещенность фотон

Regarding the Possibility of Detecting Space Objects in the Spectral Range of 8–12 Μm

I. V. Znamensky, A. T. Tungushpaev

Precision Instrumentation Systems Scientific and Production Corporation JSC, Moscow, Russia

The result of calculating the threshold illumination of an optical-electronic system (OES) in the far infrared range of 8–12 μm is presented. A technique has been developed and a calculation has been made of the illumination of the OES entrance pupil from the signal of a space object (SO) of a cylindrical shape, illuminated by the Sun, and due to the own radiation of the SO.

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

Received on: 20.12.2021

Accepted on: 28.01.2022

Introduction

Currently, IR radiation is widely used in optoelectronic systems (OES) of ground-based complexes for space surveillance and tracking of space objects (SO). This choice is due to several reasons, since the following is observed in the IR range [1]: low level of solar radiation background; existence of “windows” of transparency of the atmosphere; availability of industrially produced detectors based on matrices with a large number of elements, low intrinsic noise and high quantum efficiency; regulations for the operation of the OES at night and daytime.

Taking into account the radiation of the Sun and the temperature regime of natural radiation sources on the Earth in the wavelength range of 3 microns or less, reflected radiation dominates-this is the so-called backlight region of the spectrum. In the wavelength range of 7 μm and more, the intrinsic radiation of objects and backgrounds prevails. The wavelength range of 3–5 μm is, as it were, a transitional region. The use of the 8–12 µm range instead of the 3–5 µm range is dictated by the fact that in it the interference from the radiation of sky inhomogeneities is about 10 times less. This is especially important when capturing and tracking low-flying targets. Since in the IR range of 8–12 µm, OES are less affected by solar flare, the best solution for using this range is daytime operation [1].

In [2, 3], the state of and prospects for the development of matrix photodetectors (PD) for various IR subranges are considered. In [4], the calculation of the threshold sensitivity of the OES in the near and mid-IR subranges is considered, the illumination of the entrance pupil from a space object (SO) of a spherical shape is determined. In [5], the threshold illumination of the entrance pupil from a cylindrical SO in the near and mid-IR subranges and the signal-to-noise ratio (S / N) were determined. This article presents an algorithm and a program for determining the threshold illumination of the OES in the far infrared range of 8–12 µm (LWIR). Calculation of the signal from the SO of cylindrical shape (hereinafter – SO) is carried out. A comparative analysis of the SO’s own radiation and radiation when it is illuminated by the Sun is made.

Calculation of threshold illumination at the entrance pupil

When receiving the own radiation of the casing of the SO, the receiving spectrum has the form of the spectrum of an absolute 6 000 K [6, 10]. The blackbody at the temperature of the SO equal to 300 K [1]. When receiving radiation from SO illuminated by the Sun, the receiving spectrum has a blackbody spectrum at a temperature of T = 6 000 K [6, 10]. The solar radiation is reflected from the casing of the SO and enters the receiving lens of the OES.

The IR range of 8–12 μm lies in the atmospheric transparency window, which has prospects for using space monitoring in this range. Fig. 1 shows the dependence of the spectral transmittance of the atmosphere on the radiation wavelength in the range of 8–12 μm at an elevation angle β = 20°.

The dependence was plotted using a PC program [6, 8] developed on the basis of the model of the dependence of atmospheric transmission on altitude [7]. The resulting spectral resolution is quite sufficient for engineering calculations of the atmospheric transmission.

When choosing the spectral range for OES, it is necessary to take into account both the atmospheric transmittance and the characteristics of commercially available matrices containing a built-in cooled interference filter (IF). The cooled IF is non-replaceable, as it is installed in a hermetically sealed case. In calculations of the far IR range, we use the following matrix parameters.

The initial data for calculating the threshold sensitivity in a given spectral range are displayed on the panel for entering the initial data of the program: lens diameter Dl = 1 100mm; focal length Fl = 2 200 mm; the transmission coefficient of the receiving optics Kopt = 0.7; maximum accumulation time 2 ms; elevation angle β = 20°; relative air humidity V = 55%; air temperature T = 25 °C; matrix format 1280 × 1024; quantum efficiency 0.55.

Table 1 shows the results of the calculation of the IR system in two spectral subranges. In the 8–12 µm subrange, the power of the background radiation flux generated at a pixel is higher compared to the 8–10 µm range (because the spectral range is 2 times wider). Also, for both subranges, the background power is three orders of magnitude higher than the threshold power, so there is a background limitation mode.

When calculating IR systems, the following limitations arise:

The first two parameters indicated affect the accumulation time. But in the far IR range, the real accumulation time, determined by the background level, does not exceed 1 ms.

The maximum and minimum accumulation times limit the dynamic range of the input signals. The real maximum accumulation time ensures a frame rate of at least 200 Hz.

The threshold illumination at the entrance pupil is minimal in the 8–10 µm subrange and is 1.4152 ∙ 10–16 W / cm2. The results obtained in terms of the maximum S / N ratio do not take into account the spectral transmission of the atmosphere and the level of the input signal.

The minimum illumination by the signal of the entrance pupil, recalculated outside the atmosphere (see Table 1), at Qi = 7 in the spectral range of 8–10 µm is 3.8776 ∙ 10–15 W / cm2. Calculations showed that the minimum illumination of the entrance pupil, recalculated outside the atmosphere for the spectral range of 8–12 µm, should be 1.598 times greater than for the range of 8–10 µm.

It is important to note that the threshold number of photoelectrons (ph-el) generated in the detector is the same for both the 8–10 µm and 8–12 µm subranges. Therefore, for the same energies of the input signals from two different spectral subranges, the values of the S / N will be equal. This result was obtained as a result of the fact that the background power per pixel for the range of 8–12 µm is 1.905 times greater than in the range of 8–10 µm, but the accumulation time is 2.11 times shorter.

The final conclusion about the best spectral subrange can only be made by knowing the spectral distribution of the signal from the target and calculating the S / N ratio.

The initial data used to calculate the signal from the SO: the radius of the SO Rrо = 0.5 m; length SO Lro = 4 m; distance from the Earth’s surface to SO: DE-rо = 350, 150, 50 km at 3 temperatures of 300, 500 and 700 K, respectively; reflection coefficient SO ρro = 0.3; 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 normal 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 above to calculate the threshold sensitivity.

Own thermal radiation of the SO body

The intrinsic thermal radiation of the SO, having a temperature T, is calculated for a blackbody in a given spectral range according to the formula [5, 8]:

, [W / cm2],

where С1 = 3.7415 104 [W cm‑2 μm4], С2 = 1.43879 104 [μm K], T is the temperature of the blackbody [K], λ is the wavelength [μm], λ1, λ2 are the boundaries of the spectral range [μm], ε is the emissivity, accepted ε = 0.7.

The dependence of the height of the SO Hro above the Earth’s surface on the range of the Dro to the SO at a constant elevation angle β, adjusted for the height of the lifting of the PD, is in the form [7]:

where RE is the Earth’s radius of 6371 km, hk is the height of the PD rise [km].

At the elevation angle β = 20° and range Dro = 350 km, the height of the SO above the Earth’s surface is Hro = 128 km; at a range of 150 km Hro = 52.9 km and at a range of 50 km Hro = 17.3 km.

For a SO located at a height of Hro = 128 km from the Earth, the temperature is 300 K. When the SO enters the denser layers of the atmosphere (Hro < 70 km), its temperature rises. At Hro = 52.9 km, we take the temperature Т = 500 K, and at Hro = 17.3 km, we take the temperature Т = 700 K.

We use these data in further calculations.

At altitudes below 50 km, when the SO speed is more than 5 MAX, a glow (plasma formation) occurs, and the signal strength increases sharply due to an increase in the temperature of the SO shell [6]. For laminar flow and SO flight in the stratosphere (at altitudes above 11 km), the surface temperature due to aerodynamic heating is determined by the formula in Kelvin [6]:

Tlam = 216,7(1 + 0,164 М2) + 273,

where М is the Mach number. At М = 2.4 Tlam = 694.4 K; at М = 3.8 Tlam = 1002.9 K.

Table 2 shows the calculation of the energy brightness of a blackbody with an emissivity of 0.7 for three temperatures: 300, 500 and 700 K, i. e. B (T = 300), B (T = 500), B (T = 700). The calculation of the illumination of the entrance pupil of the lens and the current S / N ratio for the corresponding ranges of 350, 120 and 50 km are also presented in Table. 2. Here, the brightness for the Lambertian emitter has the form:

B (T) = R (T) / π. (1)

We choose such a flight trajectory that with a decrease in the range to the SO, the flight altitude decreases and an increase in the temperature of its skin is observed.

From the data in Table 2 shows that at Т = 300 К, the current S / N ratio in the IR range of 8–12 µm is 94.07, and in the 8–10 µm range, the S / N ratio is 76.65. Therefore, reliable detection and signal reception are ensured. At Т = 500 К and Т = 700 К, the S / N ratio increases even more. At D = 50 50 km, the S / N ratio is limited by the minimum accumulation time. The right columns of Table 2 (solar illumination) are calculated at the same temperature of 300 K, but for different distances.

In accordance with Wien’s displacement law [6], the spectral brightness of blackbody radiation per unit wavelength reaches a peak at the wavelength λmax:

λmax = 2 898 / T. (2)

From expression (2) it follows that the maximum radiation brightness occurs at Т = 300 K at a wavelength of 9.66 μm, which corresponds to the far infrared range (LWIR). Therefore, from the point of view of optimizing the reception of a signal from a SO with a temperature of 300 K, the subranges used are optimal.

Basic mathematical relations for calculating the energy characteristics of the OES

Illumination of the Entrance Pupil of the Device by SO Own Radiation

Of interest are the expressions for the individual radiation components, i. e. luminosities of sources entering the entrance pupil of the IR system. These components will be:

luminosity of radiation due to the self-radiation of the SO and the background reflected from it;

luminosity of radiation due to the luminosity of the natural radiation of the atmosphere.

The total energy luminosity of the SO and the atmospheric path E0 in the spectral range Δλ has the form [1]:

, (3)

where: E (Tro) is the intrinsic luminosity of the SO at the temperature Tro; εro is the emissivity of the SO, εro = 1 – ρro is the reflection coefficient of the SO; E (Tat) is the luminosity of the atmosphere at the temperature Tat; τatm is the atmospheric transmittance; Δλ = λ2 – λ1 is the band of the spectral range, µm; λ1, λ2 are the boundaries of the spectral range.

The atmospheric background brightness B (Tat) in the spectral range Δλ is determined from expression (3) taking into account (1) in the form:

. (4)

The brightness of the intrinsic radiation of the SO B (Tro) in the spectral range Δλ is determined from expression (3) taking into account (1) in the form

. (5)

An essential feature of the proposed energy calculation is the calculation of the allowable accumulation time. Therefore, we find the corresponding count rate of signal photons for a given S / N ratio.

The photon counting rate nb [c‑1] on the matrix element, associated with atmospheric radiation in the spectral range Δλ, has the form [4, 5]:

nb = B (Tat) (a / Fl)2 Тopt Sl / Eq,

where B (Tat) has the dimension of [W / (cm2 sr)]; a is the 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 transmittances of the objective 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 the quantum energy, [J]; h = 6.6256 ∙ 10–34 [J ∙ s], is the Planck’s constant; с = 3 ∙ 108 m / s is the speed of light; λ0 – average wavelength [µm], λ0 = (λ1 + λ2) / 2.

If there is no cooling system in the design of the IR lens, then the lens is a background source, and it can be considered as a blackbody with temperature T. In this case, we will assume that the field diaphragm is installed in a cooled matrix receiver.

The blackbody radiation law for the photon emission density F(λ), [s‑1 ∙ cm‑2 ∙ μm‑1 ∙ sr‑1], has the form [4, 5]:

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].

Photon counting rate from the lens nl [s–1], in a given spectral range [4, 5]:

,

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

The maximum observation time τac [s], is limited by the pixel accumulation capacity Сe and is calculated at ns = 0:

τac = (Сe Kz – Nre) / [η(nb + nl) + nd],

where Kz is safety factor, Kz = 0.8–0.9; Nre is the number of reading noise electrons; η is the quantum efficiency of the matrix; nd is the dark electron count rate, nd = id / e, id is the matrix dark current, e is the electron charge, e = 1.6 ∙ 10–19 Kl.

Accordingly, with a ratio of S / N ≥ 1, the observation time τac has the form [5]:

τac ≤ (Сe Kz – Nre) / [η(nb + nl + ns) + nd], (6)

where ns is the count rate of signal photons on the matrix element.

Using the dependence τac = f (ns), a graph (Fig. 2) was plotted for the dependence of the accumulation time on the signal photon count rate (on a logarithmic scale, the initial data are indicated above, the spectral range is 8–12 μm).

The S / N ratio in terms of power at the output of the matrix is determined taking into account the geometric noise in the form [4]:

Qp = (η ∙ ns ∙ τac)2 / (σΣ)2 (7)

where (σΣ)2 is the total noise variance,

σΣ = [(σnt)2 + [(σng)2]0.5,

(σnt)2 is the temporal noise dispersion,

(σnt)2 = [η ∙ F ∙ τac (nb + nl + ns) + nd τac + (Nre)2],

(σng)2 is the dispersion of geometrical noise.

Temporal noise includes Johnson thermal noise, shot noise, and reading noise. Matrix PDs are characterized by geometric noise arising from the inhomogeneity of the parameters of individual elements of the receiver and the circuits for reading signals from these elements. PDs used to detect a low-temperature emitter operate almost in the background-limited mode. To reduce geometric noise to an acceptable level, special signal processing is performed in the form of compensation or correction of inhomogeneity, which can be performed before converting analog signals to digital. When correcting, they seek to reduce the level of geometric noise to the level of temporal noise or less [1]. We accept σng = σnt.

The signal photon count rate is found from expression (7):

ns = (F Qp / η τac) (1 + A), (8)

where A = {1 + (2 / F Qp) [η τac (nb + nl) + nd τac + (Nre)2]}1 / 2;

F = 1–2 is the noise factor.

Based on relation (7), a plot of the power S / N ratio versus the signal photon count rate ns was plotted (Fig. 3). The calculation was made for the initial data that were used for the calculation in Fig. 2. From Fig. 3 it can be seen that when the signal photon count rate is less than ns = 1011, the power S / N ratio increases linearly, and at ns = 1012 and more, the S / N ratio practically does not change due to a decrease in the observation time in accordance with expression (6).

The position of the focal plane of a conventional receiving objective satisfies Fraunhofer diffraction conditions. In this case, the pixel size must be associated with the main lobe (diffraction maximum). But even in this case, there is a loss of part of the energy of the received signal. This loss is taken into account by the coefficient χ. For the selected characteristics of the lens and matrix, χ = 0.837.

Using the rule for adding dispersions of random variables [9], we find the root-mean-square deviation of the number of noise photoelectrons arriving at the matrix pixel:

σΣ = 1,41 [η ∙ F ∙ τac (nb + nl + ns) + nd τac + (Nre)2]1 / 2.

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

Рn = σΣ Еq / η τac.

The expression for the value of the threshold illumination Eth, [W / cm2], the lens has the form:

Eth = Рn / Topt χ π(Dl / 2)2.

If we assume that the minimum current S / N ratio at which the signal is detected is Qi = 7 [4], then we can determine the value of the minimum illumination:

Emin = Psmin / [Topt χ π(Dl / 2)2],

where Psmin is the minimum optical power at the entrance pupil, Psmin = Еqnsmin; nsmin is the signal photon count rate, determined from expression (7) at Qp = 49.

Accordingly, in the general case, the illumination of the entrance pupil due to the self-radiation of the SO has the form:

Ein = Ps / [Topt χ π(Dl / 2)2]. (9)

Illumination of the Entrance Pupil of the Device by the Radiation of SO Illuminated by the Sun

Let us consider the signal at the entrance pupil of the device from the SO illuminated by the Sun in different spectral ranges. When calculating as the Sun, we use a blackbody at a temperature of 6 000 K. The SO is located at a distance of 350 km from the Earth, has a radius Rrо = 0.5 m, a length Lro = 4 m and a reflection coefficient ρ = 0.3.

Find the brightness of the Sun BS(T, Δλ), [W m‑2 sr‑1] [10]:

BS(T, Δλ) = R(Т, Δλ) ∙ 104 / π.

Then we determine the brightness of the SO in the direction to the OES Bro(Т, Δλ) [W m‑2 sr‑1]:

Bro(Δλ) = BS(T, Δλ) ρ Kdb (RS / LS–ro)2, (10)

where: RS = 6.9599 108 [m] is the radius of the Sun; ρ is the reflection coefficient of the SO, Kdb is the coefficient of dimension brightness [5] at ρ = 0.3, equal to the ratio of the area of the SO visible from the OES, illuminated by the Sun, to the total area of the SO, LS-ro is the distance from the Sun to the SO [m], Kdb depends on the angles γ, ξ1, ξ2 and is defined as Kdb = Sro / S0, where S0 = 2Rrо Lro, S0 is the projection of the SO area illuminated by the Sun [m2], Sro is the effective area of the SO, [m2] [7]:

Sro = S0 cos(ξ1) cos(ξ2)[(π–δ + 0,5 sin(2δ) cosδ + sin3δ] / π,

where with γ ≥ (ξ1 + ξ2)

and γ < 180°–|ξ1 – ξ2|,

γ is the angle between the directions of the Sun-SO and SO-OES;

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

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

The illumination of the entrance pupil of the lens Еinp, [W / cm2], is determined by the expression:

Еinp = 10–10 Bro τatm (λ) S0 / (DE-ro)2, (11)

where: DE-ro is the Earth–SO distance [km], τatm(λ) is the average atmospheric transmittance in the spectral range Δλ.

The current S / N ratio Qi is determined from expression (7):

Qi = EinpSobτacAl / [2F{τac(A0 + ηns) + (Nre)2}]0,5, (12)

where Al = Koptχ η / Eq, A0 = η(nb + nl) + nd, τmin ≤ τac ≤ τmax, ns = Einp Sob Al / η, Einp is the signal illumination at the entrance pupil of the lens.

Using expressions (11), (12) and the brightness of the SO, determined by expression (10), the current S / N ratio was calculated for different distances (Fig. 4) with solar illumination of the SO in the spectral range of 8–12 μm. For the SO brightness determined by expression (5), the current S / N ratio was calculated for different distances (Fig. 4) for the SO own radiation in the spectral range of 8–12 µm. Graphs are constructed for elevation angle β = 20°.

The current S / N ratio with solar illumination of the hull is much less than the S / N ratio due to its own radiation from the SO hull. Therefore, the signal from the solar illumination can be neglected.

Conclusion

The calculation of the threshold illumination of an optoelectronic system (OES) in the far infrared range of 8–12 µm was performed in several stages.

The threshold illumination at the entrance pupil for the spectral subranges 8–12 µm and 8–10 µm was considered, and the subrange 8–10 µm with the lowest illumination equal to 1.4152 ∙ 10–16, W / cm2 was determined.

The main mathematical relations for the energy calculation of the OES in the far infrared range (LWIR) are presented.

Illumination was calculated at the entrance pupil of the OES from the SO, located at 3 heights, at casing temperatures of 300, 500 and 700 K, respectively. Based on the calculation results, it was found that the current S / N ratio for both spectral subranges is approximately equal to and much greater than 1.

The current S / N ratio due to the self-radiation of the hull casing is much greater than the S / N ratio due to solar illumination and provides a detection range of up to 1,200 km at an elevation angle of 20°.

The spectral transmittance of the atmosphere was calculated in the range of 8–12 µm using experimental tables [8] using spline approximation [9], which allows one to calculate the atmospheric transmission depending on the elevation angle.

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ABOUT AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Technical Sciences, Leading Researcher, Joint-Stock Company “Scientific and Production Corporation “Precision Instrumentation Systems” (JSC “NPK “SPP”; Moscow, Russia.

ORCID‑0000-0002-0612-1255

Tungushpaev Albert Tolevjanovich, Doctor of Technical Sciences, Senior Researcher, Lleading Researcher, Joint-Stock Company Scientific and Production Corporation

“Precision Instrumentation Systems” (JSC “NPK”SPP”; Moscow, Russia.

ORCID‑0000-0003-2068-473X

Contribution by the members

of the team of authors

The article was prepared on the basis 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, A. T. Tungushpaev

Precision Instrumentation Systems Scientific and Production Corporation JSC, Moscow, Russia

The result of calculating the threshold illumination of an optical-electronic system (OES) in the far infrared range of 8–12 μm is presented. A technique has been developed and a calculation has been made of the illumination of the OES entrance pupil from the signal of a space object (SO) of a cylindrical shape, illuminated by the Sun, and due to the own radiation of the SO.

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

Received on: 20.12.2021

Accepted on: 28.01.2022

Introduction

Currently, IR radiation is widely used in optoelectronic systems (OES) of ground-based complexes for space surveillance and tracking of space objects (SO). This choice is due to several reasons, since the following is observed in the IR range [1]: low level of solar radiation background; existence of “windows” of transparency of the atmosphere; availability of industrially produced detectors based on matrices with a large number of elements, low intrinsic noise and high quantum efficiency; regulations for the operation of the OES at night and daytime.

Taking into account the radiation of the Sun and the temperature regime of natural radiation sources on the Earth in the wavelength range of 3 microns or less, reflected radiation dominates-this is the so-called backlight region of the spectrum. In the wavelength range of 7 μm and more, the intrinsic radiation of objects and backgrounds prevails. The wavelength range of 3–5 μm is, as it were, a transitional region. The use of the 8–12 µm range instead of the 3–5 µm range is dictated by the fact that in it the interference from the radiation of sky inhomogeneities is about 10 times less. This is especially important when capturing and tracking low-flying targets. Since in the IR range of 8–12 µm, OES are less affected by solar flare, the best solution for using this range is daytime operation [1].

In [2, 3], the state of and prospects for the development of matrix photodetectors (PD) for various IR subranges are considered. In [4], the calculation of the threshold sensitivity of the OES in the near and mid-IR subranges is considered, the illumination of the entrance pupil from a space object (SO) of a spherical shape is determined. In [5], the threshold illumination of the entrance pupil from a cylindrical SO in the near and mid-IR subranges and the signal-to-noise ratio (S / N) were determined. This article presents an algorithm and a program for determining the threshold illumination of the OES in the far infrared range of 8–12 µm (LWIR). Calculation of the signal from the SO of cylindrical shape (hereinafter – SO) is carried out. A comparative analysis of the SO’s own radiation and radiation when it is illuminated by the Sun is made.

Calculation of threshold illumination at the entrance pupil

When receiving the own radiation of the casing of the SO, the receiving spectrum has the form of the spectrum of an absolute 6 000 K [6, 10]. The blackbody at the temperature of the SO equal to 300 K [1]. When receiving radiation from SO illuminated by the Sun, the receiving spectrum has a blackbody spectrum at a temperature of T = 6 000 K [6, 10]. The solar radiation is reflected from the casing of the SO and enters the receiving lens of the OES.

The IR range of 8–12 μm lies in the atmospheric transparency window, which has prospects for using space monitoring in this range. Fig. 1 shows the dependence of the spectral transmittance of the atmosphere on the radiation wavelength in the range of 8–12 μm at an elevation angle β = 20°.

The dependence was plotted using a PC program [6, 8] developed on the basis of the model of the dependence of atmospheric transmission on altitude [7]. The resulting spectral resolution is quite sufficient for engineering calculations of the atmospheric transmission.

When choosing the spectral range for OES, it is necessary to take into account both the atmospheric transmittance and the characteristics of commercially available matrices containing a built-in cooled interference filter (IF). The cooled IF is non-replaceable, as it is installed in a hermetically sealed case. In calculations of the far IR range, we use the following matrix parameters.

The initial data for calculating the threshold sensitivity in a given spectral range are displayed on the panel for entering the initial data of the program: lens diameter Dl = 1 100mm; focal length Fl = 2 200 mm; the transmission coefficient of the receiving optics Kopt = 0.7; maximum accumulation time 2 ms; elevation angle β = 20°; relative air humidity V = 55%; air temperature T = 25 °C; matrix format 1280 × 1024; quantum efficiency 0.55.

Table 1 shows the results of the calculation of the IR system in two spectral subranges. In the 8–12 µm subrange, the power of the background radiation flux generated at a pixel is higher compared to the 8–10 µm range (because the spectral range is 2 times wider). Also, for both subranges, the background power is three orders of magnitude higher than the threshold power, so there is a background limitation mode.

When calculating IR systems, the following limitations arise:

- number of accumulation electrons, limited by the value of the accumulation capacity of the pixel 1.3 ∙ 107 el;

- number of background electrons;

- maximum accumulation time, limited by the duration of the frame;

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

The first two parameters indicated affect the accumulation time. But in the far IR range, the real accumulation time, determined by the background level, does not exceed 1 ms.

The maximum and minimum accumulation times limit the dynamic range of the input signals. The real maximum accumulation time ensures a frame rate of at least 200 Hz.

The threshold illumination at the entrance pupil is minimal in the 8–10 µm subrange and is 1.4152 ∙ 10–16 W / cm2. The results obtained in terms of the maximum S / N ratio do not take into account the spectral transmission of the atmosphere and the level of the input signal.

The minimum illumination by the signal of the entrance pupil, recalculated outside the atmosphere (see Table 1), at Qi = 7 in the spectral range of 8–10 µm is 3.8776 ∙ 10–15 W / cm2. Calculations showed that the minimum illumination of the entrance pupil, recalculated outside the atmosphere for the spectral range of 8–12 µm, should be 1.598 times greater than for the range of 8–10 µm.

It is important to note that the threshold number of photoelectrons (ph-el) generated in the detector is the same for both the 8–10 µm and 8–12 µm subranges. Therefore, for the same energies of the input signals from two different spectral subranges, the values of the S / N will be equal. This result was obtained as a result of the fact that the background power per pixel for the range of 8–12 µm is 1.905 times greater than in the range of 8–10 µm, but the accumulation time is 2.11 times shorter.

The final conclusion about the best spectral subrange can only be made by knowing the spectral distribution of the signal from the target and calculating the S / N ratio.

The initial data used to calculate the signal from the SO: the radius of the SO Rrо = 0.5 m; length SO Lro = 4 m; distance from the Earth’s surface to SO: DE-rо = 350, 150, 50 km at 3 temperatures of 300, 500 and 700 K, respectively; reflection coefficient SO ρro = 0.3; 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 normal 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 above to calculate the threshold sensitivity.

Own thermal radiation of the SO body

The intrinsic thermal radiation of the SO, having a temperature T, is calculated for a blackbody in a given spectral range according to the formula [5, 8]:

, [W / cm2],

where С1 = 3.7415 104 [W cm‑2 μm4], С2 = 1.43879 104 [μm K], T is the temperature of the blackbody [K], λ is the wavelength [μm], λ1, λ2 are the boundaries of the spectral range [μm], ε is the emissivity, accepted ε = 0.7.

The dependence of the height of the SO Hro above the Earth’s surface on the range of the Dro to the SO at a constant elevation angle β, adjusted for the height of the lifting of the PD, is in the form [7]:

where RE is the Earth’s radius of 6371 km, hk is the height of the PD rise [km].

At the elevation angle β = 20° and range Dro = 350 km, the height of the SO above the Earth’s surface is Hro = 128 km; at a range of 150 km Hro = 52.9 km and at a range of 50 km Hro = 17.3 km.

For a SO located at a height of Hro = 128 km from the Earth, the temperature is 300 K. When the SO enters the denser layers of the atmosphere (Hro < 70 km), its temperature rises. At Hro = 52.9 km, we take the temperature Т = 500 K, and at Hro = 17.3 km, we take the temperature Т = 700 K.

We use these data in further calculations.

At altitudes below 50 km, when the SO speed is more than 5 MAX, a glow (plasma formation) occurs, and the signal strength increases sharply due to an increase in the temperature of the SO shell [6]. For laminar flow and SO flight in the stratosphere (at altitudes above 11 km), the surface temperature due to aerodynamic heating is determined by the formula in Kelvin [6]:

Tlam = 216,7(1 + 0,164 М2) + 273,

where М is the Mach number. At М = 2.4 Tlam = 694.4 K; at М = 3.8 Tlam = 1002.9 K.

Table 2 shows the calculation of the energy brightness of a blackbody with an emissivity of 0.7 for three temperatures: 300, 500 and 700 K, i. e. B (T = 300), B (T = 500), B (T = 700). The calculation of the illumination of the entrance pupil of the lens and the current S / N ratio for the corresponding ranges of 350, 120 and 50 km are also presented in Table. 2. Here, the brightness for the Lambertian emitter has the form:

B (T) = R (T) / π. (1)

We choose such a flight trajectory that with a decrease in the range to the SO, the flight altitude decreases and an increase in the temperature of its skin is observed.

From the data in Table 2 shows that at Т = 300 К, the current S / N ratio in the IR range of 8–12 µm is 94.07, and in the 8–10 µm range, the S / N ratio is 76.65. Therefore, reliable detection and signal reception are ensured. At Т = 500 К and Т = 700 К, the S / N ratio increases even more. At D = 50 50 km, the S / N ratio is limited by the minimum accumulation time. The right columns of Table 2 (solar illumination) are calculated at the same temperature of 300 K, but for different distances.

In accordance with Wien’s displacement law [6], the spectral brightness of blackbody radiation per unit wavelength reaches a peak at the wavelength λmax:

λmax = 2 898 / T. (2)

From expression (2) it follows that the maximum radiation brightness occurs at Т = 300 K at a wavelength of 9.66 μm, which corresponds to the far infrared range (LWIR). Therefore, from the point of view of optimizing the reception of a signal from a SO with a temperature of 300 K, the subranges used are optimal.

Basic mathematical relations for calculating the energy characteristics of the OES

Illumination of the Entrance Pupil of the Device by SO Own Radiation

Of interest are the expressions for the individual radiation components, i. e. luminosities of sources entering the entrance pupil of the IR system. These components will be:

luminosity of radiation due to the self-radiation of the SO and the background reflected from it;

luminosity of radiation due to the luminosity of the natural radiation of the atmosphere.

The total energy luminosity of the SO and the atmospheric path E0 in the spectral range Δλ has the form [1]:

, (3)

where: E (Tro) is the intrinsic luminosity of the SO at the temperature Tro; εro is the emissivity of the SO, εro = 1 – ρro is the reflection coefficient of the SO; E (Tat) is the luminosity of the atmosphere at the temperature Tat; τatm is the atmospheric transmittance; Δλ = λ2 – λ1 is the band of the spectral range, µm; λ1, λ2 are the boundaries of the spectral range.

The atmospheric background brightness B (Tat) in the spectral range Δλ is determined from expression (3) taking into account (1) in the form:

. (4)

The brightness of the intrinsic radiation of the SO B (Tro) in the spectral range Δλ is determined from expression (3) taking into account (1) in the form

. (5)

An essential feature of the proposed energy calculation is the calculation of the allowable accumulation time. Therefore, we find the corresponding count rate of signal photons for a given S / N ratio.

The photon counting rate nb [c‑1] on the matrix element, associated with atmospheric radiation in the spectral range Δλ, has the form [4, 5]:

nb = B (Tat) (a / Fl)2 Тopt Sl / Eq,

where B (Tat) has the dimension of [W / (cm2 sr)]; a is the 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 transmittances of the objective 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 the quantum energy, [J]; h = 6.6256 ∙ 10–34 [J ∙ s], is the Planck’s constant; с = 3 ∙ 108 m / s is the speed of light; λ0 – average wavelength [µm], λ0 = (λ1 + λ2) / 2.

If there is no cooling system in the design of the IR lens, then the lens is a background source, and it can be considered as a blackbody with temperature T. In this case, we will assume that the field diaphragm is installed in a cooled matrix receiver.

The blackbody radiation law for the photon emission density F(λ), [s‑1 ∙ cm‑2 ∙ μm‑1 ∙ sr‑1], has the form [4, 5]:

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].

Photon counting rate from the lens nl [s–1], in a given spectral range [4, 5]:

,

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

The maximum observation time τac [s], is limited by the pixel accumulation capacity Сe and is calculated at ns = 0:

τac = (Сe Kz – Nre) / [η(nb + nl) + nd],

where Kz is safety factor, Kz = 0.8–0.9; Nre is the number of reading noise electrons; η is the quantum efficiency of the matrix; nd is the dark electron count rate, nd = id / e, id is the matrix dark current, e is the electron charge, e = 1.6 ∙ 10–19 Kl.

Accordingly, with a ratio of S / N ≥ 1, the observation time τac has the form [5]:

τac ≤ (Сe Kz – Nre) / [η(nb + nl + ns) + nd], (6)

where ns is the count rate of signal photons on the matrix element.

Using the dependence τac = f (ns), a graph (Fig. 2) was plotted for the dependence of the accumulation time on the signal photon count rate (on a logarithmic scale, the initial data are indicated above, the spectral range is 8–12 μm).

The S / N ratio in terms of power at the output of the matrix is determined taking into account the geometric noise in the form [4]:

Qp = (η ∙ ns ∙ τac)2 / (σΣ)2 (7)

where (σΣ)2 is the total noise variance,

σΣ = [(σnt)2 + [(σng)2]0.5,

(σnt)2 is the temporal noise dispersion,

(σnt)2 = [η ∙ F ∙ τac (nb + nl + ns) + nd τac + (Nre)2],

(σng)2 is the dispersion of geometrical noise.

Temporal noise includes Johnson thermal noise, shot noise, and reading noise. Matrix PDs are characterized by geometric noise arising from the inhomogeneity of the parameters of individual elements of the receiver and the circuits for reading signals from these elements. PDs used to detect a low-temperature emitter operate almost in the background-limited mode. To reduce geometric noise to an acceptable level, special signal processing is performed in the form of compensation or correction of inhomogeneity, which can be performed before converting analog signals to digital. When correcting, they seek to reduce the level of geometric noise to the level of temporal noise or less [1]. We accept σng = σnt.

The signal photon count rate is found from expression (7):

ns = (F Qp / η τac) (1 + A), (8)

where A = {1 + (2 / F Qp) [η τac (nb + nl) + nd τac + (Nre)2]}1 / 2;

F = 1–2 is the noise factor.

Based on relation (7), a plot of the power S / N ratio versus the signal photon count rate ns was plotted (Fig. 3). The calculation was made for the initial data that were used for the calculation in Fig. 2. From Fig. 3 it can be seen that when the signal photon count rate is less than ns = 1011, the power S / N ratio increases linearly, and at ns = 1012 and more, the S / N ratio practically does not change due to a decrease in the observation time in accordance with expression (6).

The position of the focal plane of a conventional receiving objective satisfies Fraunhofer diffraction conditions. In this case, the pixel size must be associated with the main lobe (diffraction maximum). But even in this case, there is a loss of part of the energy of the received signal. This loss is taken into account by the coefficient χ. For the selected characteristics of the lens and matrix, χ = 0.837.

Using the rule for adding dispersions of random variables [9], we find the root-mean-square deviation of the number of noise photoelectrons arriving at the matrix pixel:

σΣ = 1,41 [η ∙ F ∙ τac (nb + nl + ns) + nd τac + (Nre)2]1 / 2.

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

Рn = σΣ Еq / η τac.

The expression for the value of the threshold illumination Eth, [W / cm2], the lens has the form:

Eth = Рn / Topt χ π(Dl / 2)2.

If we assume that the minimum current S / N ratio at which the signal is detected is Qi = 7 [4], then we can determine the value of the minimum illumination:

Emin = Psmin / [Topt χ π(Dl / 2)2],

where Psmin is the minimum optical power at the entrance pupil, Psmin = Еqnsmin; nsmin is the signal photon count rate, determined from expression (7) at Qp = 49.

Accordingly, in the general case, the illumination of the entrance pupil due to the self-radiation of the SO has the form:

Ein = Ps / [Topt χ π(Dl / 2)2]. (9)

Illumination of the Entrance Pupil of the Device by the Radiation of SO Illuminated by the Sun

Let us consider the signal at the entrance pupil of the device from the SO illuminated by the Sun in different spectral ranges. When calculating as the Sun, we use a blackbody at a temperature of 6 000 K. The SO is located at a distance of 350 km from the Earth, has a radius Rrо = 0.5 m, a length Lro = 4 m and a reflection coefficient ρ = 0.3.

Find the brightness of the Sun BS(T, Δλ), [W m‑2 sr‑1] [10]:

BS(T, Δλ) = R(Т, Δλ) ∙ 104 / π.

Then we determine the brightness of the SO in the direction to the OES Bro(Т, Δλ) [W m‑2 sr‑1]:

Bro(Δλ) = BS(T, Δλ) ρ Kdb (RS / LS–ro)2, (10)

where: RS = 6.9599 108 [m] is the radius of the Sun; ρ is the reflection coefficient of the SO, Kdb is the coefficient of dimension brightness [5] at ρ = 0.3, equal to the ratio of the area of the SO visible from the OES, illuminated by the Sun, to the total area of the SO, LS-ro is the distance from the Sun to the SO [m], Kdb depends on the angles γ, ξ1, ξ2 and is defined as Kdb = Sro / S0, where S0 = 2Rrо Lro, S0 is the projection of the SO area illuminated by the Sun [m2], Sro is the effective area of the SO, [m2] [7]:

Sro = S0 cos(ξ1) cos(ξ2)[(π–δ + 0,5 sin(2δ) cosδ + sin3δ] / π,

where with γ ≥ (ξ1 + ξ2)

and γ < 180°–|ξ1 – ξ2|,

γ is the angle between the directions of the Sun-SO and SO-OES;

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

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

The illumination of the entrance pupil of the lens Еinp, [W / cm2], is determined by the expression:

Еinp = 10–10 Bro τatm (λ) S0 / (DE-ro)2, (11)

where: DE-ro is the Earth–SO distance [km], τatm(λ) is the average atmospheric transmittance in the spectral range Δλ.

The current S / N ratio Qi is determined from expression (7):

Qi = EinpSobτacAl / [2F{τac(A0 + ηns) + (Nre)2}]0,5, (12)

where Al = Koptχ η / Eq, A0 = η(nb + nl) + nd, τmin ≤ τac ≤ τmax, ns = Einp Sob Al / η, Einp is the signal illumination at the entrance pupil of the lens.

Using expressions (11), (12) and the brightness of the SO, determined by expression (10), the current S / N ratio was calculated for different distances (Fig. 4) with solar illumination of the SO in the spectral range of 8–12 μm. For the SO brightness determined by expression (5), the current S / N ratio was calculated for different distances (Fig. 4) for the SO own radiation in the spectral range of 8–12 µm. Graphs are constructed for elevation angle β = 20°.

The current S / N ratio with solar illumination of the hull is much less than the S / N ratio due to its own radiation from the SO hull. Therefore, the signal from the solar illumination can be neglected.

Conclusion

The calculation of the threshold illumination of an optoelectronic system (OES) in the far infrared range of 8–12 µm was performed in several stages.

The threshold illumination at the entrance pupil for the spectral subranges 8–12 µm and 8–10 µm was considered, and the subrange 8–10 µm with the lowest illumination equal to 1.4152 ∙ 10–16, W / cm2 was determined.

The main mathematical relations for the energy calculation of the OES in the far infrared range (LWIR) are presented.

Illumination was calculated at the entrance pupil of the OES from the SO, located at 3 heights, at casing temperatures of 300, 500 and 700 K, respectively. Based on the calculation results, it was found that the current S / N ratio for both spectral subranges is approximately equal to and much greater than 1.

The current S / N ratio due to the self-radiation of the hull casing is much greater than the S / N ratio due to solar illumination and provides a detection range of up to 1,200 km at an elevation angle of 20°.

The spectral transmittance of the atmosphere was calculated in the range of 8–12 µm using experimental tables [8] using spline approximation [9], which allows one to calculate the atmospheric transmission depending on the elevation angle.

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ABOUT AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Technical Sciences, Leading Researcher, Joint-Stock Company “Scientific and Production Corporation “Precision Instrumentation Systems” (JSC “NPK “SPP”; Moscow, Russia.

ORCID‑0000-0002-0612-1255

Tungushpaev Albert Tolevjanovich, Doctor of Technical Sciences, Senior Researcher, Lleading Researcher, Joint-Stock Company Scientific and Production Corporation

“Precision Instrumentation Systems” (JSC “NPK”SPP”; Moscow, Russia.

ORCID‑0000-0003-2068-473X

Contribution by the members

of the team of authors

The article was prepared on the basis 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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