Issue #7/2022

Video Range System for Outer Space Survey During the Night Optoelectronic Instruments & Devices

**I. V. Znamensky, E. O. Zotiev, I. I. Oleinikov, K. G. Popov**Video Range System for Outer Space Survey During the Night Optoelectronic Instruments & Devices

DOI: 10.22184/1993-7296.FRos.2022.16.7.512.521

The calculation result of the threshold illumination of a ground-based optoelectronic system (OES) in the visible range of 0.45–0.85 microns is provided. We have developed the method and performed the light calculation of the OES entrance pupil from the signal of a spherical space object (SO) illuminated by the Sun during the night. The dependence of the signal-to-noise ratio on the distance from the Earth to the SO is shown.

The calculation result of the threshold illumination of a ground-based optoelectronic system (OES) in the visible range of 0.45–0.85 microns is provided. We have developed the method and performed the light calculation of the OES entrance pupil from the signal of a spherical space object (SO) illuminated by the Sun during the night. The dependence of the signal-to-noise ratio on the distance from the Earth to the SO is shown.

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

Video Range System for Outer Space Survey During The Night

I. V. Znamensky, E. O. Zotiev, I. I. Oleinikov, K. G. Popov

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

The calculation result of the threshold illumination of a ground-based optoelectronic system (OES) in the visible range of 0.45–0.85 microns is provided. We have developed the method and performed the light calculation of the OES entrance pupil from the signal of a spherical space object (SO) illuminated by the Sun during the night. The dependence of the signal-to-noise ratio on the distance from the Earth to the SO is shown.

Keywords: optoelectronic system, visible range, zenith sky brightness, matrix photodetector, threshold illumination, accumulation time, signal-to-noise ratio, atmosphere, photon, overall brightness coefficient.

Received on: 01.07.2022

Accepted on: 04.08.2022

Introduction

The search for space debris, as well as tracking of space objects (SO) is performed by the celestial sphere survey using the ground operating complexes containing the antenna pedestals rotating at an angular velocity of up to 15 degrees per second (° / s) and having a wide field of vision. In this case, the most important criteria are the scanning speed and threshold sensitivity.

A comparative analysis of modern photodetector cameras is given in [1, 2]. To speed up the celestial sphere survey, the photodetectors shall have the largest possible format and high resolution. Moreover, it is necessary to consider that the space debris has small dimensions and weak luster, but relatively high velocity. Therefore, high sensitivity at the high frame rate is required. For the purpose of object photometry, a high dynamic range of the video channel is also necessary [1].

The development of back-illuminated sensors, such as GSense4040(BSI) [3] or GSense6060(BSI) [4] made by GPixelInc (PRC), allow to solve the problem of space monitoring. In addition, the sCMOS sensor structure with a high dynamic range has a large potential well capacitance, despite the small pixel output size. In addition, the sCMOS(BSI) sensors have high quantum efficiency (more than 90%).

The read-out noise value becomes the dominant parameter that determines the detection limit and establishes the signal-to-noise ratio at very low illumination levels [5, 6]. The sCMOS array (Scientific Complementary Metal-Oxide Semiconductor) technology has become widely popular in various fields of science and technology due to the combination of the following specifications: extremely low noise level, high frame rate, wide dynamic range, high quantum efficiency, high resolution and large field of vision.

The sensor of GSense6060(BSI) sCMOS array with 6144 × 6144 format has a large field of vision and high resolution simultaneously with the low read-out noise and a satisfactory frame rate. The read-out noise value is negligible, even compared to the CCD arrays with the highest performance. The low read-out noise 3e of the sCMOS sensor is complemented by a high dynamic range of 90.6 dB at 11 frames per second and a high quantum efficiency of 95%. In this article, the parameters of GSense6060(BSI) [4] will be used for the energy calculation of the optical channel for nighttime SO surveys.

Calculation of the energy characteristics of the OES night channel

The zenith sky brightness is located in the range of 15–19.25m[ang.s]–2 (approximately 1.1 · 10–1 – 2.2 · 10–3 cd m‑2 [4]. As a general approximation, it is possible to accept 2.7 · 10–3 cd m–2 (19.00m [ang.s]–2) as a representative of a truly black sky, although there may be any parts of the sky darker than this in an uncut section.

The darkest sky on Earth has a zenith brightness of approximately 22 stellar arcs per second (1.71 · 10–4 cd m–2). The sky background on a clear moonless night being a combination (in descending order) of natural airglow, zodiacal light, and scattered starlight, is changed depending on solar activity [7].

The given stellar magnitude outside the atmosphere is determined as [7]

μv = 12,58 – 2,5 lg(BV),

where BV is the light brightness of the sky background, cd · m–2.

The expression for μv was obtained at small spatial angles ΔΩ, av., then the illumination is EV = BV ΔΩ.

The relations between the energy and light characteristics is determined as

,

where S(λ) is the spectral sensitivity of the receiver in the λ1‑λ2 band, Δλ=λ2‑λ1, Lλ is the night sky radiance in the range of 0.45–0.85 µm, W · cm‑2 · av‑1.

The photon count rate nb, s‑1, on the array element, due to the background radiation of the sky in the spectral range Δλ, has the following view [8]

nb = Lλ τatm(λ) (a / Fl)2 Тopt Sl / Eq,

where: τatm(λ) is the average atmospheric transmissivity in the spectral range Δλ; a is the pixel size of the array element (square side), cm; Fl is the back focal length of the receiving lens, cm; Тopt = Tl · TIF is the transmittance factor of the receiving optics, Tl and TIF are the transmittance factors of the lens and the interference filter, respectively; Sl = π(Dl / 2)2 is the receiving lens area with the diameter Dl, cm; Eq = hс / λ0 · 10–6 is the quantum energy, J, h = 6,6256 · 10–34 is the Planck’s constant, с = 3 · 108 m / s is a light velocity; λ0 is the average wavelength, µm, λ0 = (λ1 + λ2) / 2.

The maximum survey time τac is limited by the capacity of the pixel potential well Сe [8] and occurs at ns = 0. The minimum survey time τac is limited by the array parameter and occurs at the maximum photon count rate nsmax. The inequality for the survey time is determined as

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

where Kz is the safety factor, Kz= 0,9–0,95;

Nre is the number of read-out noise electrons; η is the quantum efficiency of the array;

ns is the count rate of signal photons on the array element,

nd is the count rate of dark electrons, nd = id / e id is the dark current of the array, e is the elementary charge, e = 1,6 · 10–19 C.

Using the dependence τac = f(ns), we have plotted the graph shown in Fig. 1. The graph represents the dependence of the accumulation time on the count rate of signal photons on a logarithmic scale, calculated for the initial data indicated above in the spectral band of 0.45–0.85 μm.

The signal-to-noise ratio by power at the array output is determined with due regard to the geometric noise as follows [8]:

Qp = (ηnsτac)2 / (σΣ)2, (2)

where (σΣ)2 is the total noise dispersion, σΣ = [(σnt)2 + [(σng)2]0,5;

(σnt)2 is the temporal noise dispersion,

(σnt)2 = [ητac(nb + ns) + ndτac + (Nre)2];

(σng)2 is the geometric noise dispersion.

The temporal noise shall include the following: Johnson thermal noise, shot noise, and read-out noise. The array-based photodetectors are characterized by geometric noise occurred due to the non-uniformity of parameters of the receiver individual elements and the signal readout circuits from these elements. To reduce geometric noise to an acceptable level, the special signal processing is performed in the form of non-uniformity compensation or correction that can be carried out prior to convertion of the analog signals to the digital ones.

As shown in [6], when receiving weak signals, it is necessary to use an accumulation time of no more than 200 ms to reduce geometric noise, and it is proposed to combine several frames to increase the signal-to-noise ratio.

In the case of adjustment, it is recommended to reduce the geometric noise to the level of temporal noise or less [9]. We take σng = σnt, then Kg = 2 and (σΣ)2 = Kg(σnt)2.

The signal photon count rate is based on the following expression (2):

ns = QpKg / (2ητac)(1 + A), (3)

where A = {1 + 4[ητacnb + ndτac + (Nre)2] / (KgQp)}1 / 2;

Based on the ratio (2), we have plotted a graph of the power-based signal-to-noise ratio versus the signal photon count rate ns (Fig. 2). The calculation was made for the initial data used in Fig. 1.

It can be seen from Fig. 2 that at ns = 106 and more, the signal-to-noise ratio is almost unchanged due to the decrease in the survey time to the value determined by the array parameter equal to 2 μs.

When transforming expression (2) with due regard to (1), it is possible to write

.

Having transformed the numerator with due regard to CeKz >> Nre, nt / (ηnsmax) → 0, then

Qpmax ≈ CeKz / [Kg(nb / nsmax + 1)2].

Since for the night channel nb / nsmax << 1, it is finally possible to obtain the following

Qpmax ≈ CeKz / Kg. (4)

Therefore, the maximum power-based signal-to-noise ratio is limited by the potential well capacitance and geometric noise.

The pixel size shall be paired with the main lobe of the point spread function. However, even in this case, there is a loss of part of the received signal energy. This loss is considered by the coefficient χ. For the selected specifications of the lens and array χ = 0.901.

While using the additive rule for the random variable dispersions [10], we can determine the root-mean-square deviation of the number of noise photoelectrons arriving at the array pixel:

σΣ = {Kg[ητac(nb + ns) + ndτac + (Nre)2]}1 / 2,

where the factor Kg = 2 takes into account geometric noise.

The noise power Pn [W], reduced to an array pixel, at which Qp = 1, is determined as:

Рn = σΣЕq / ητac.

The expression for the threshold illumination value Eth [W / cm2] of the lens is as follows:

Eth = Рn / Toptχπ(Dl / 2)2.

If we assume that the minimum current-based signal-to-noise ratio at which the signal is detected is Qi = 7, then we can determine the minimum illumination value at the entrance pupil:

Emin = Psmin / [Toptχπ(Dl / 2)2],

where Psmin is the minimum optical power at the entrance pupil, Psmin = Еqnsmin, nsmin is the signal photon count rate, determined on the basis of expression (3) at Qp = 49.

We will represent the illumination at the entrance pupil in the form of brightness (stellar magnitude) [7]

mmin = –2,5lg(Eνmin) – 13,99,

where Eνmin is the minimum light illumination at the entrance pupil [lx].

To estimate the complex sensitivity, it seems convenient to use the stellar magnitude m. This allows the OES to be calibrated by the stars, since their magnitude is known with the high accuracy.

Figure 3 shows the dependence of the current-based signal-to-noise ratio on the SO brightness. As can be seen from Fig.3, the signal-to-noise ratio does not change from m = 10 to a negative value due to the dynamic range limitation strained by the potential well capacitance and geometric noise in accordance with the expression (4).

Illumination of the device entrance pupil by the SO radiation illuminated by the Sun

We will consider the signal at the entrance pupil of the device from the SO, illuminated by the Sun, in the visible range during the night. When performing calculations, we use a blackbody at a temperature of 6000 K as the Sun. The SO is located at a distance of 400 km from the Earth, it has a radius of rkо = 0,5 m and a reflection coefficient ρ = 0.5.

The Sun brightness BS(T, Δλ), W · m‑2 · avg‑1, has the following form [8]:

BS(T, Δλ) = R(T, Δλ) · 104 / π.

Then we will determine the SO brightness КО Bko(T, Δλ) [W · m‑2 · avg‑1] when conducting surveys from the Earth

Bko(T, Δλ) = BS(T, Δλ) ρKdb(rS / RS-ko)2,

where: rS = 6,9599 · 108 m is the radius of the Sun; ρ is the reflection coefficient of the SO, Kdb is the overall brightness coefficient for the sphere, RS-ko is the distance from the Sun to the SO, m. Kdb depends on the angle γ between the directions of the Sun-SO and the SO-OES, and is determined as follows [11]:

Kdb = (2 / 3π)[sinγ + (π - γ)cosγ];

Kdb = 0,5 при γ = 80°.

The entrance pupil illumination Еinp [W / cm2] is determined by the following expression

Еinp = 10–4 πBko(T, Δλ) τatm(λ) (rko / DE-ko)2, (5)

where: rko is the SO radius, DE-ko is the Earth-SO distance [m].

The radiation flux incident on the array can be different depending on the ratio between the image sizes of the radiation source (SO) and the pixel. The image area of the radiation source Sim is [12]:

Sim = S0(Fl / DE-ko)2.

If in the plane of the photodetector array receiving area the source image area Sim is greater than the pixel area Sp = a2, i. e. Sim > Sp, then the radiation flux incident on a pixel is limited by its size. This case corresponds to the ambient background. The background power per pixel is equal to the following

Pb = Lλ τatm(λ) Sl Topt(a / Fl)2.

If the source image area Sim is less than the pixel area Sp, i. e. Sim < Sp, then the radiation flux incident on the pixel is limited by the size of the entrance pupil. This case corresponds to the point signal receipt. The signal power per pixel is equal to the following:

Ps = Jτatm(λ) ToptχSl / (DE-ko)2, where J = S0Bko(T, Δλ).

The current-based signal-to-noise ratio Qi is determined based on the following expression (2)

Qi = EinpSlτacAl / [Kg{τac(η(nb + ns) + nd) + (Nre)2}]0,5, (6)

where Al = Toptχη / Eq, τmin ≤ τac ≤ τmax, ns = EinpSobAl / η, Einp is the signal illumination at the entrance pupil of the lens.

By using expressions (5) and (6), as well as the expression for the SO brightness, we will calculate the signal-to-noise ratio of the flux for various ranges (Fig. 4) in the case of the SO sun outage in the spectral range of 0.45–0.85 μm.

Based on the formulas provided, an energy calculation of the OES, operating during the night, was performed for two ranges of the SO and at r = 0.5. At a distance of 70,000 km, the current-based signal-to-noise ratio is 11.4. The calculation results are shown in the table.

Conclusion

In order to detect space debris during the space monitoring process, a special method was developed for the energy calculation of an optoelectronic system in the visible spectral range during night. Its application has made it possible to determine the optimal parameters of instrumental equipment for the celestial sphere surveys. The formulas are obtained for a cylindrical object illuminated by the Sun.

The maximum possible power-based signal-to-noise ratio, limited by the potential well capacity and geometric noise, is found.

The functional dependence of the current-based signal-to-noise ratio on the stellar magnitude at the entrance pupil is obtained. Its analysis demonstrates that at m ≤ 10 the signal-to-noise ratio is unchanged, since it has reached its maximum value, and at m > 10 the signal-to-noise ratio depends on the SO magnitude.

A graph of the dependence of the signal-to-noise ratio on the SO distance is plotted for two sphere radii of 0.1 and 0.5 m. At a range of 70,000 km and a sphere radius of 0.5, the current-based signal-to-noise ratio is 11.4.

The stellar magnitude at the entrance pupil is determined at a sphere radius of 0.5: for a SO distance of 400 km, it is equal to 4.64, and for a SO distance of 40,000 km, it is equal to 14.64. Thus, when the range is changed by a factor of 100, the stellar magnitude is changed by 10.

AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Tech. Sc., Leading Researcher, Joint-Stock

Company «Scientific and Production Corporation «Precision Instrumentation Systems» (JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0002–0612–1255

Zotyev Evgenii Olegovich, Head of the Scientific and Technical Complex, JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0002–2923–7779

Oleinikov Igor Igorevich, Doctor of Technical Sciences, Head of Department, Deputy General Designer, JSC «NPK «SPP»; Moscow, Russia.

Popov Konstantin Gennadyevich, Cand. of Tech.Sc., Assistant Head of Department, JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0001–8183–4231

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, E. O. Zotiev, I. I. Oleinikov, K. G. Popov

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

The calculation result of the threshold illumination of a ground-based optoelectronic system (OES) in the visible range of 0.45–0.85 microns is provided. We have developed the method and performed the light calculation of the OES entrance pupil from the signal of a spherical space object (SO) illuminated by the Sun during the night. The dependence of the signal-to-noise ratio on the distance from the Earth to the SO is shown.

Keywords: optoelectronic system, visible range, zenith sky brightness, matrix photodetector, threshold illumination, accumulation time, signal-to-noise ratio, atmosphere, photon, overall brightness coefficient.

Received on: 01.07.2022

Accepted on: 04.08.2022

Introduction

The search for space debris, as well as tracking of space objects (SO) is performed by the celestial sphere survey using the ground operating complexes containing the antenna pedestals rotating at an angular velocity of up to 15 degrees per second (° / s) and having a wide field of vision. In this case, the most important criteria are the scanning speed and threshold sensitivity.

A comparative analysis of modern photodetector cameras is given in [1, 2]. To speed up the celestial sphere survey, the photodetectors shall have the largest possible format and high resolution. Moreover, it is necessary to consider that the space debris has small dimensions and weak luster, but relatively high velocity. Therefore, high sensitivity at the high frame rate is required. For the purpose of object photometry, a high dynamic range of the video channel is also necessary [1].

The development of back-illuminated sensors, such as GSense4040(BSI) [3] or GSense6060(BSI) [4] made by GPixelInc (PRC), allow to solve the problem of space monitoring. In addition, the sCMOS sensor structure with a high dynamic range has a large potential well capacitance, despite the small pixel output size. In addition, the sCMOS(BSI) sensors have high quantum efficiency (more than 90%).

The read-out noise value becomes the dominant parameter that determines the detection limit and establishes the signal-to-noise ratio at very low illumination levels [5, 6]. The sCMOS array (Scientific Complementary Metal-Oxide Semiconductor) technology has become widely popular in various fields of science and technology due to the combination of the following specifications: extremely low noise level, high frame rate, wide dynamic range, high quantum efficiency, high resolution and large field of vision.

The sensor of GSense6060(BSI) sCMOS array with 6144 × 6144 format has a large field of vision and high resolution simultaneously with the low read-out noise and a satisfactory frame rate. The read-out noise value is negligible, even compared to the CCD arrays with the highest performance. The low read-out noise 3e of the sCMOS sensor is complemented by a high dynamic range of 90.6 dB at 11 frames per second and a high quantum efficiency of 95%. In this article, the parameters of GSense6060(BSI) [4] will be used for the energy calculation of the optical channel for nighttime SO surveys.

Calculation of the energy characteristics of the OES night channel

The zenith sky brightness is located in the range of 15–19.25m[ang.s]–2 (approximately 1.1 · 10–1 – 2.2 · 10–3 cd m‑2 [4]. As a general approximation, it is possible to accept 2.7 · 10–3 cd m–2 (19.00m [ang.s]–2) as a representative of a truly black sky, although there may be any parts of the sky darker than this in an uncut section.

The darkest sky on Earth has a zenith brightness of approximately 22 stellar arcs per second (1.71 · 10–4 cd m–2). The sky background on a clear moonless night being a combination (in descending order) of natural airglow, zodiacal light, and scattered starlight, is changed depending on solar activity [7].

The given stellar magnitude outside the atmosphere is determined as [7]

μv = 12,58 – 2,5 lg(BV),

where BV is the light brightness of the sky background, cd · m–2.

The expression for μv was obtained at small spatial angles ΔΩ, av., then the illumination is EV = BV ΔΩ.

The relations between the energy and light characteristics is determined as

,

where S(λ) is the spectral sensitivity of the receiver in the λ1‑λ2 band, Δλ=λ2‑λ1, Lλ is the night sky radiance in the range of 0.45–0.85 µm, W · cm‑2 · av‑1.

The photon count rate nb, s‑1, on the array element, due to the background radiation of the sky in the spectral range Δλ, has the following view [8]

nb = Lλ τatm(λ) (a / Fl)2 Тopt Sl / Eq,

where: τatm(λ) is the average atmospheric transmissivity in the spectral range Δλ; a is the pixel size of the array element (square side), cm; Fl is the back focal length of the receiving lens, cm; Тopt = Tl · TIF is the transmittance factor of the receiving optics, Tl and TIF are the transmittance factors of the lens and the interference filter, respectively; Sl = π(Dl / 2)2 is the receiving lens area with the diameter Dl, cm; Eq = hс / λ0 · 10–6 is the quantum energy, J, h = 6,6256 · 10–34 is the Planck’s constant, с = 3 · 108 m / s is a light velocity; λ0 is the average wavelength, µm, λ0 = (λ1 + λ2) / 2.

The maximum survey time τac is limited by the capacity of the pixel potential well Сe [8] and occurs at ns = 0. The minimum survey time τac is limited by the array parameter and occurs at the maximum photon count rate nsmax. The inequality for the survey time is determined as

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

where Kz is the safety factor, Kz= 0,9–0,95;

Nre is the number of read-out noise electrons; η is the quantum efficiency of the array;

ns is the count rate of signal photons on the array element,

nd is the count rate of dark electrons, nd = id / e id is the dark current of the array, e is the elementary charge, e = 1,6 · 10–19 C.

Using the dependence τac = f(ns), we have plotted the graph shown in Fig. 1. The graph represents the dependence of the accumulation time on the count rate of signal photons on a logarithmic scale, calculated for the initial data indicated above in the spectral band of 0.45–0.85 μm.

The signal-to-noise ratio by power at the array output is determined with due regard to the geometric noise as follows [8]:

Qp = (ηnsτac)2 / (σΣ)2, (2)

where (σΣ)2 is the total noise dispersion, σΣ = [(σnt)2 + [(σng)2]0,5;

(σnt)2 is the temporal noise dispersion,

(σnt)2 = [ητac(nb + ns) + ndτac + (Nre)2];

(σng)2 is the geometric noise dispersion.

The temporal noise shall include the following: Johnson thermal noise, shot noise, and read-out noise. The array-based photodetectors are characterized by geometric noise occurred due to the non-uniformity of parameters of the receiver individual elements and the signal readout circuits from these elements. To reduce geometric noise to an acceptable level, the special signal processing is performed in the form of non-uniformity compensation or correction that can be carried out prior to convertion of the analog signals to the digital ones.

As shown in [6], when receiving weak signals, it is necessary to use an accumulation time of no more than 200 ms to reduce geometric noise, and it is proposed to combine several frames to increase the signal-to-noise ratio.

In the case of adjustment, it is recommended to reduce the geometric noise to the level of temporal noise or less [9]. We take σng = σnt, then Kg = 2 and (σΣ)2 = Kg(σnt)2.

The signal photon count rate is based on the following expression (2):

ns = QpKg / (2ητac)(1 + A), (3)

where A = {1 + 4[ητacnb + ndτac + (Nre)2] / (KgQp)}1 / 2;

Based on the ratio (2), we have plotted a graph of the power-based signal-to-noise ratio versus the signal photon count rate ns (Fig. 2). The calculation was made for the initial data used in Fig. 1.

It can be seen from Fig. 2 that at ns = 106 and more, the signal-to-noise ratio is almost unchanged due to the decrease in the survey time to the value determined by the array parameter equal to 2 μs.

When transforming expression (2) with due regard to (1), it is possible to write

.

Having transformed the numerator with due regard to CeKz >> Nre, nt / (ηnsmax) → 0, then

Qpmax ≈ CeKz / [Kg(nb / nsmax + 1)2].

Since for the night channel nb / nsmax << 1, it is finally possible to obtain the following

Qpmax ≈ CeKz / Kg. (4)

Therefore, the maximum power-based signal-to-noise ratio is limited by the potential well capacitance and geometric noise.

The pixel size shall be paired with the main lobe of the point spread function. However, even in this case, there is a loss of part of the received signal energy. This loss is considered by the coefficient χ. For the selected specifications of the lens and array χ = 0.901.

While using the additive rule for the random variable dispersions [10], we can determine the root-mean-square deviation of the number of noise photoelectrons arriving at the array pixel:

σΣ = {Kg[ητac(nb + ns) + ndτac + (Nre)2]}1 / 2,

where the factor Kg = 2 takes into account geometric noise.

The noise power Pn [W], reduced to an array pixel, at which Qp = 1, is determined as:

Рn = σΣЕq / ητac.

The expression for the threshold illumination value Eth [W / cm2] of the lens is as follows:

Eth = Рn / Toptχπ(Dl / 2)2.

If we assume that the minimum current-based signal-to-noise ratio at which the signal is detected is Qi = 7, then we can determine the minimum illumination value at the entrance pupil:

Emin = Psmin / [Toptχπ(Dl / 2)2],

where Psmin is the minimum optical power at the entrance pupil, Psmin = Еqnsmin, nsmin is the signal photon count rate, determined on the basis of expression (3) at Qp = 49.

We will represent the illumination at the entrance pupil in the form of brightness (stellar magnitude) [7]

mmin = –2,5lg(Eνmin) – 13,99,

where Eνmin is the minimum light illumination at the entrance pupil [lx].

To estimate the complex sensitivity, it seems convenient to use the stellar magnitude m. This allows the OES to be calibrated by the stars, since their magnitude is known with the high accuracy.

Figure 3 shows the dependence of the current-based signal-to-noise ratio on the SO brightness. As can be seen from Fig.3, the signal-to-noise ratio does not change from m = 10 to a negative value due to the dynamic range limitation strained by the potential well capacitance and geometric noise in accordance with the expression (4).

Illumination of the device entrance pupil by the SO radiation illuminated by the Sun

We will consider the signal at the entrance pupil of the device from the SO, illuminated by the Sun, in the visible range during the night. When performing calculations, we use a blackbody at a temperature of 6000 K as the Sun. The SO is located at a distance of 400 km from the Earth, it has a radius of rkо = 0,5 m and a reflection coefficient ρ = 0.5.

The Sun brightness BS(T, Δλ), W · m‑2 · avg‑1, has the following form [8]:

BS(T, Δλ) = R(T, Δλ) · 104 / π.

Then we will determine the SO brightness КО Bko(T, Δλ) [W · m‑2 · avg‑1] when conducting surveys from the Earth

Bko(T, Δλ) = BS(T, Δλ) ρKdb(rS / RS-ko)2,

where: rS = 6,9599 · 108 m is the radius of the Sun; ρ is the reflection coefficient of the SO, Kdb is the overall brightness coefficient for the sphere, RS-ko is the distance from the Sun to the SO, m. Kdb depends on the angle γ between the directions of the Sun-SO and the SO-OES, and is determined as follows [11]:

Kdb = (2 / 3π)[sinγ + (π - γ)cosγ];

Kdb = 0,5 при γ = 80°.

The entrance pupil illumination Еinp [W / cm2] is determined by the following expression

Еinp = 10–4 πBko(T, Δλ) τatm(λ) (rko / DE-ko)2, (5)

where: rko is the SO radius, DE-ko is the Earth-SO distance [m].

The radiation flux incident on the array can be different depending on the ratio between the image sizes of the radiation source (SO) and the pixel. The image area of the radiation source Sim is [12]:

Sim = S0(Fl / DE-ko)2.

If in the plane of the photodetector array receiving area the source image area Sim is greater than the pixel area Sp = a2, i. e. Sim > Sp, then the radiation flux incident on a pixel is limited by its size. This case corresponds to the ambient background. The background power per pixel is equal to the following

Pb = Lλ τatm(λ) Sl Topt(a / Fl)2.

If the source image area Sim is less than the pixel area Sp, i. e. Sim < Sp, then the radiation flux incident on the pixel is limited by the size of the entrance pupil. This case corresponds to the point signal receipt. The signal power per pixel is equal to the following:

Ps = Jτatm(λ) ToptχSl / (DE-ko)2, where J = S0Bko(T, Δλ).

The current-based signal-to-noise ratio Qi is determined based on the following expression (2)

Qi = EinpSlτacAl / [Kg{τac(η(nb + ns) + nd) + (Nre)2}]0,5, (6)

where Al = Toptχη / Eq, τmin ≤ τac ≤ τmax, ns = EinpSobAl / η, Einp is the signal illumination at the entrance pupil of the lens.

By using expressions (5) and (6), as well as the expression for the SO brightness, we will calculate the signal-to-noise ratio of the flux for various ranges (Fig. 4) in the case of the SO sun outage in the spectral range of 0.45–0.85 μm.

Based on the formulas provided, an energy calculation of the OES, operating during the night, was performed for two ranges of the SO and at r = 0.5. At a distance of 70,000 km, the current-based signal-to-noise ratio is 11.4. The calculation results are shown in the table.

Conclusion

In order to detect space debris during the space monitoring process, a special method was developed for the energy calculation of an optoelectronic system in the visible spectral range during night. Its application has made it possible to determine the optimal parameters of instrumental equipment for the celestial sphere surveys. The formulas are obtained for a cylindrical object illuminated by the Sun.

The maximum possible power-based signal-to-noise ratio, limited by the potential well capacity and geometric noise, is found.

The functional dependence of the current-based signal-to-noise ratio on the stellar magnitude at the entrance pupil is obtained. Its analysis demonstrates that at m ≤ 10 the signal-to-noise ratio is unchanged, since it has reached its maximum value, and at m > 10 the signal-to-noise ratio depends on the SO magnitude.

A graph of the dependence of the signal-to-noise ratio on the SO distance is plotted for two sphere radii of 0.1 and 0.5 m. At a range of 70,000 km and a sphere radius of 0.5, the current-based signal-to-noise ratio is 11.4.

The stellar magnitude at the entrance pupil is determined at a sphere radius of 0.5: for a SO distance of 400 km, it is equal to 4.64, and for a SO distance of 40,000 km, it is equal to 14.64. Thus, when the range is changed by a factor of 100, the stellar magnitude is changed by 10.

AUTHORS

Znamenskiy Igor Vsevolodovich, Cand. of Tech. Sc., Leading Researcher, Joint-Stock

Company «Scientific and Production Corporation «Precision Instrumentation Systems» (JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0002–0612–1255

Zotyev Evgenii Olegovich, Head of the Scientific and Technical Complex, JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0002–2923–7779

Oleinikov Igor Igorevich, Doctor of Technical Sciences, Head of Department, Deputy General Designer, JSC «NPK «SPP»; Moscow, Russia.

Popov Konstantin Gennadyevich, Cand. of Tech.Sc., Assistant Head of Department, JSC «NPK «SPP»; Moscow, Russia.

ORCID‑0000–0001–8183–4231

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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