Issue #1/2023

Estimated Bit Error Rate in the Atmospheric Optical Communication Channel Based on Scattered Radiation in the UV-wavelength Range in the Daytime and at Night

**M. V. Tarasenkov, S. A. Peshkov, E. S. Poznakharev**Estimated Bit Error Rate in the Atmospheric Optical Communication Channel Based on Scattered Radiation in the UV-wavelength Range in the Daytime and at Night

DOI: 10.22184/1993-7296.FRos.2023.17.1.46.56

A model of the atmospheric optical communication channel based on scattered radiation in the UV wavelength range is considered. The model is based on the Monte Carlo method algorithms for the local and modified double local estimate to calculate the impulse response of the optical communication channel. The bit error rate in the day and at night is estimated for the wavelength range from 200 to 400 nm and information coding using the digital pulse interval modulation (DPIM). The results demonstrate that the wavelength λ = 295 nm is better to arrange a long-range communication using the receiving system under study in the daytime, whereas the wavelength λ = 395 nm is better at night.

A model of the atmospheric optical communication channel based on scattered radiation in the UV wavelength range is considered. The model is based on the Monte Carlo method algorithms for the local and modified double local estimate to calculate the impulse response of the optical communication channel. The bit error rate in the day and at night is estimated for the wavelength range from 200 to 400 nm and information coding using the digital pulse interval modulation (DPIM). The results demonstrate that the wavelength λ = 295 nm is better to arrange a long-range communication using the receiving system under study in the daytime, whereas the wavelength λ = 395 nm is better at night.

Теги: atmosphere monte carlo method optical communication based on scattered radiation scattered laser radiation uv wavelength range атмосфера метод монте-карло оптическая связь на рассеянном излучении рассеянное лазерное излучение уф-диапазон длин волн

Estimated Bit Error

Rate in the

Atmospheric Optical Communication Channel Based on Scattered Radiation in the UV-wavelength Range in the Daytime and at Night

M. V. Tarasenkov, S. A. Peshkov, E. S. Poznakharev

Zuev Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences

A model of the atmospheric optical communication channel based on scattered radiation in the UV wavelength range is considered. The model is based on the Monte Carlo method algorithms for the local and modified double local estimate to calculate the impulse response of the optical communication channel. The bit error rate in the day and at night is estimated for the wavelength range from 200 to 400 nm and information coding using the digital pulse interval modulation (DPIM). The results demonstrate that the wavelength λ = 295 nm is better to arrange a long-range communication using the receiving system under study in the daytime, whereas the wavelength λ = 395 nm is better at night.

Keywords: atmosphere, scattered laser radiation, optical communication based on scattered radiation, UV wavelength range, Monte Carlo method

Received on: 03.08.2022

Accepted on: 07.11.2022

The atmospheric optical communication based on scattered radiation is one of the actively developing methods for data transmission over an open atmospheric channel. The first experimental and theoretical works devoted to arrangement of this type of communication were published in the 1960s in such papers as [1, 2]. In recent years, due to the occurrence of new laser sources, highly sensitive photomultiplier tubes (PMTs), and photodiodes, the studies of this communication type have been further developed [3]. The advantages of atmospheric optical communication based on scattered radiation are as follows: 1) its multiple-address broadcast, 2) possible reconfiguration of the communication channel to eliminate signal interruptions, 3) relatively low cost, 4) potentially high data transmission rate. The main disadvantage of this type of communication is the strong dependence of its quality on the atmospheric conditions. At present, the following areas of using atmospheric optical communication based on scattered radiation can be determined: 1) indoor communication at small distances [4, 5], 2) ground communication via an open atmospheric channel at the baseline distances from tens of meters to tens of kilometers [6–8], 3) communication with the unmanned aerial vehicles [9, 10]. In this article, the main attention will be paid to the ground atmospheric optical communication in the open atmosphere.

At present, quite a lot of theoretical and experimental works on the arrangement of ground optical communication based on scattered radiation have been published. Currently, a number of experimental works have been performed [11–15]. The vast majority of works use the sources in the wavelength range of λ = 250–270 nm. The UV LEDs [11, 12], Nd : YAG lasers [13], and MPL-F‑266 lasers [14] are used as such sources. To receive information, the PMTs MP1922 [11, 15] and PMTs Hamamatsu R7154 [12, 14] are most often used. Most papers consider the baseline distances up to 100 m (for example, [11–14]), however, there are also some papers with a large baseline distance (for example, up to 1 000 m in [15]). As a part of the experimental work, the researchers determine attenuation of the desired signal radiation, reduced to decibels (path loss), probability of erroneous symbol recording (bit error rate or BER), and duration of the received pulse (pulse width) depending on the communication conditions.

Within the framework of theoretical works [16–21], the data transmission channel quality is studied depending on the optical and geometric conditions and the maximum information transmission rate. In addition, the works [16–17] propose an approximation formula for the impulse response of an optical communication channel based on scattered radiation.

Another field of theoretical and experimental research is related to the influence of various information encoding methods on the quality of a communication channel [12, 22–30]. Within the framework of [12, 22, 23], the OOK (On-Off-Keying) data encoding is considered; it is an approach where "0" is set by the absence of a pulse, and "1" by its availability. The papers [12, 22, 23] consider the PPM (Pulse Position Modulation) data encoding, namely an approach according to which the encoded symbol depends on the pulse position in the time slot. In [12, 23–27], the DPIM (Digital Pulse Interval Modulation) coding is applied, an approach according to which information is encoded by a delay between the pulses. The papers [28] use coding at several wavelengths. In [29], information is encoded by controlling the sent radiation polarization. The papers [25, 30] show that the application of DPIM can significantly increase the information transfer rate compared to the PPM, however, the bit error rate will be somewhat higher. As a part of the experimental works previously performed at the Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences [8, 31], the DPIM information encoding method was also used, namely a delay between the pulses of 65 µs corresponded to the symbol "1", and of 67 µs – to the symbol "0". In the daytime, the sustainable optical communication was established at a wavelength of λ = 250 nm at a baseline distance of up to 1.3 km [31].

In general, the existing literature review shows that the previously published works almost do not consider the wavelengths in the range of λ>270 nm. We can do no more than mention the paper [16], where the wavelengths λ = 230–310 nm are considered. Therefore, below we consider how the bit error rate Pe depends on the optical and geometric conditions in the wavelength range from 200 to 400 nm.

Problem statement

and solution method

The problem was considered using the following statement (Fig. 1). Let there be a planar system consisting of an atmosphere and the ground surface. The atmosphere is a scattering and absorbing aerosol-gas medium. The atmosphere is divided into 32 homogeneous layers in each of which the molecular and aerosol scattering and absorption rates, as well as the phase function of aerosol scattering are given. The optical parameters of the atmosphere are determined by LOWTRAN‑7 mid-latitude summer models [32] of a cloudless sky and information from the papers [33–35] on molecular scattering and absorption. An impulse laser source S located on the ground surface is transmitting information through the atmosphere in the spectral intervals with the centers at wavelengths λ = 205, 215,…, 395 nm and a width Δλ = 10 nm. The information is encoded by the delay between pulses (DPIM). The source axis is located at the zenith angle θs from the vertical, the source divergence angle is νs, the energy of one pulse is Q0 = 0.3 mJ, and the duration is Δt0 = ns. It is assumed that the pulse shape is rectangular in time. At the baseline distance YN from the source, the receiving system D is located, the axis of which is oriented at the zenith angle θd, and the field of view angle is equal to νd. A coplanar channel circuit is considered (the axes of the source and the receiving system are located in the same plane). The receiving system is assumed to be ideal with an aperture area of Sd = 0.01 m2, containing a UFK‑4G‑4 PMT [36]. The PMT parameters are as follows: PMT gain ratio MPMT = 106, dark current IT = 10–15 A, frequency bandwidth Δf = 11 kHz. The spectral sensitivity values of the photocathode ΣK at the considered wavelengths are given in Table 1. The receiving system parameters correspond to the system used by us in [31].

The solar (or lunar) radiation is incident on the upper boundary of the atmosphere at a zenith angle θb and an azimuth angle ϕb. While knowing the optical and geometric conditions and the specifications of the transmit/receive equipment, it is required to determine the impulse response of the communication channel and the background radiance and, by using their values, to estimate the bit error rate Pe recording.

The desired signal in the optical communication circuit under consideration is entirely generated by scattered radiation. To simulate the impulse response, we used two programs of the Monte Carlo method (Figure 2): the singly scattered part of the impulse response h1(t) was simulated by a Monte Carlo program with local estimates at the points of "photon" collision with the medium (Figure 2a), and the multiply scattered part of the impulse response hmsc(t) was simulated by a Monte Carlo program with the modified double local estimates at the collision points (Figure 2b). A brief description of these algorithms is given below.

The algorithm with local estimates is as follows (Figure 2a). The photon wandering trajectories in the atmosphere begin in the source point in accordance with the optical parameters of the medium. The modeling algorithm for photon wandering can be considered as classical [37]. At each collision point of photons with the medium (marked as M1 and M2 in the figure), a local energy estimate is made in relation to the photon that will arrive at the receiver from the point Mi with a wandering time falling into the i-th time interval Ii, pnk. If the conditions and are met, is equal to:

, (1)

where i is the number of the time interval where the local estimate is made; qM is the photon energy arriving to the collision point M, rMR is the distance from the collision point M to the receiving system D, τMR is the optical path thickness from the collision point M to the receiving point D, σsa, M, σsm, M are the aerosol and molecular scattering coefficients at the collision point M, respectively, σt, M is the attenuation coefficient at the collision point M, ga(μM), gm(μM) are the phase functions of aerosol and molecular scattering at the collision point M, respectively, μM is the scattering angle cosine between the photon trajectory direction at the collision point before scattering and the direction to the receiver, с is the light velocity in the medium, ωd is the direction of the receiver’s optical axis, ωpnk is the direction from the receiver point to the collision point M.

Otherwise Ιipnk = 0.

Then the singly scattered part of the impulse response h1(t) is determined as follows:

(2)

where i = 1, ..., Ntime is the number of the time interval, Ntime is the number of time intervals, ti + 1, ti are the boundaries of the i-th time interval, P is the number of sets of trajectories, N is the number of trajectories in a set, Kpn is the number of collisions in the n-th trajectory of the p-th set.

An algorithm with modified double local estimates is made as follows (Figure 2b). The wandering of photons in the medium is simulated. For each collision point (marked as M1 in the figure), the receiver’s field of view is divided into the subregions corresponding to the time intervals for which the impulse response is sought. The algorithm for dividing the field of view into the subregions is described in [38]. The subregions are bounded by two ellipsoids of revolution which foci are the collision point M1 and the receiver system D, and the cone of the receiver’s field of view. A random direction ω is selected in the field of view of the receiving system. One ghost collision point is selected in each of the obtained subregions (marked in the figure as N1, N2, etc.). Such points are located on the ray determined by the direction ω. In each of the possible time intervals, there is a double local estimate of radiation arrived at the receiving system, having scattered at points M and N1 for the first time interval, etc. The estimate is given by the following formula:

, (3)

where μMi is the scattering angle cosine between the photon direction at the collision point M before scattering and direction to the ghost collision point Ni corresponding to the i-th time interval, μNi is the scattering angle cosine between the directions MNi and NiD, Pi, ei are the focal parameter and eccentricity of the ellipsoid of revolution, relevant to the time ti, b is the dot product of the vector ω and the unit vector aimed in the direction DM, is the optical thickness of the path from the collision point M to the ghost point Ni, is the optical thickness of the path from the ghost collision point Ni to the receiving system D.

Thus, the multiply scattered part of the impulse response hmsc(t) is determined by the formula (2) using the values of Iipnk obtained by the formula (3).

Having known the impulse response of the communication channel, it is possible to determine the average desired signal power:

, (4)

. (5)

The background radiation power is determined by the following formula:

, (6)

where is the value of the solar (lunar) constant, is the radiance value of the background radiation of the Sun (Moon) at the zenith angle of the Sun (Moon) θb and the relative azimuthal angle ϕb, measured from the direction of the receiving system axis.

The values of solar constants were taken from [39]. The values of the lunar constants for the full moon conditions were taken from [40].

The value of Ib was calculated by the Monte Carlo method of backward trajectories with the local estimates at the collision points, based on [37].

In [38, 41], to test the calculation programs for h1 and hmsc, a comparison was made with the results of [20, 42]. In [43], a test comparison was made with the results of calculations Ib from [44]. The comparison demonstrates that the difference between the results is within the limits of the statistical calculation error.

For the DPIM information coding method, the papers [24–27] propose a formula for calculating the bit error rate. If information encoding is performed similarly to our work [31], then the bit error rate is determined as follows:

, (7)

where is the probability of sent pulse recording, is the probability that the pulse absence is registered correctly.

At a high level of the desired signal, according to [25], the values P0/1 and P1/0 are determined by the following formulas:

, (8)

, (9)

, (10)

, (11)

, (12)

, (13)

, (14)

, (15)

where e is the elementary charge; I1 is the current generated by the background radiation, I2 is the current generated by the desired signal.

Results

To solve the problem set, a series of calculations was performed for the following optical and geometric conditions: λ = 200–400 nm with a step of 10 nm; meteorological range of visibility SM = 50 km that corresponds to the LOWTRAN‑7 model with the vertical aerosol optical depth (AOD) at a wavelength of λ = 550 nm АОТ550 = 0.16, YN = 0.05–50 km, θs = 85°, νs = 0.06 mrad, θd = 85°, νd = 1°, θb = 0°, calculations for the daytime and night. The selection of values θs and θd is justified by the fact that the received desired signal will be the largest for such conditions.

Using the developed programs of the Monte Carlo method and formulas (4)—(14), the bit error rate Pe recording was calculated depending on the wavelength λ and the baseline distance YN. The calculation results are given in Figure 3.

It can be seen from Figure 3a that the wavelength λ = 295 nm will be the best choice for arranging the long-distance communications in the daytime, since the distance at which Pe becomes more than 10% is the largest for such a wavelength. The reason for such result is that, on the one part, due to the significant absorption by ozone, there is little background radiation from the Sun at this wavelength. On the other part, the ozone concentration in the atmospheric ground layer is less than in the stratosphere, thus, the ground layer radiation at this wavelength is attenuated less than at other wavelengths in the range of λ = 200–300 nm.

It can be seen from Fig. 3 that communication during the night for most wavelengths is significantly better than in the daytime conditions. This is due to the fact that the lunar background is much less than the solar one. Figure 3b shows that the best wavelength for the long-distance communications at night would be λ = 395 nm, and the far UV band wavelengths (200–250 nm) would be the least suitable.

Conclusions

The performed calculations of the bit error rate in the daytime and night conditions have shown the following:

The wavelength λ = 295 nm is better than others for the arrangement of atmospheric long-distance optical communication based on scattered radiation during the day.

During the night, the best wavelength for arranging the long-distance communications will be λ = 395nm, and the far UV band wavelengths (200–250 nm) will be the least suitable.

The work was performed as a part of the public task of the Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences.

Authors

Tarasenkov M. V., Cand. phys.-math. sci. V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

ORCID: 0000-0002-8826

Field of scientific interests: analysis of regularities of imaging through the atmosphere, atmospheric correction of images in the visible and UV ranges, theoretical investigation of non-line-of-sight communication channels.

Poznakharev E. S., V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

Field of scientific interests: experimental and theoretical investigation of non-line-of-sight communication channels.

Peshkov S. A., V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

Field of scientific interests: theoretical investigation of non-line-of-sight communication channels.

Author contributions

Tarasenkov M. V.: idea, design, workflow management, discussions, suggestions and remarks, literary work preparation; Poznakharev E. S.: discussions, suggestions and remarks; Peshkov S. A.: calculations, processing of results, discussions.

Conflict of interest

The authors declare no conflicts of interest.

Rate in the

Atmospheric Optical Communication Channel Based on Scattered Radiation in the UV-wavelength Range in the Daytime and at Night

M. V. Tarasenkov, S. A. Peshkov, E. S. Poznakharev

Zuev Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences

A model of the atmospheric optical communication channel based on scattered radiation in the UV wavelength range is considered. The model is based on the Monte Carlo method algorithms for the local and modified double local estimate to calculate the impulse response of the optical communication channel. The bit error rate in the day and at night is estimated for the wavelength range from 200 to 400 nm and information coding using the digital pulse interval modulation (DPIM). The results demonstrate that the wavelength λ = 295 nm is better to arrange a long-range communication using the receiving system under study in the daytime, whereas the wavelength λ = 395 nm is better at night.

Keywords: atmosphere, scattered laser radiation, optical communication based on scattered radiation, UV wavelength range, Monte Carlo method

Received on: 03.08.2022

Accepted on: 07.11.2022

The atmospheric optical communication based on scattered radiation is one of the actively developing methods for data transmission over an open atmospheric channel. The first experimental and theoretical works devoted to arrangement of this type of communication were published in the 1960s in such papers as [1, 2]. In recent years, due to the occurrence of new laser sources, highly sensitive photomultiplier tubes (PMTs), and photodiodes, the studies of this communication type have been further developed [3]. The advantages of atmospheric optical communication based on scattered radiation are as follows: 1) its multiple-address broadcast, 2) possible reconfiguration of the communication channel to eliminate signal interruptions, 3) relatively low cost, 4) potentially high data transmission rate. The main disadvantage of this type of communication is the strong dependence of its quality on the atmospheric conditions. At present, the following areas of using atmospheric optical communication based on scattered radiation can be determined: 1) indoor communication at small distances [4, 5], 2) ground communication via an open atmospheric channel at the baseline distances from tens of meters to tens of kilometers [6–8], 3) communication with the unmanned aerial vehicles [9, 10]. In this article, the main attention will be paid to the ground atmospheric optical communication in the open atmosphere.

At present, quite a lot of theoretical and experimental works on the arrangement of ground optical communication based on scattered radiation have been published. Currently, a number of experimental works have been performed [11–15]. The vast majority of works use the sources in the wavelength range of λ = 250–270 nm. The UV LEDs [11, 12], Nd : YAG lasers [13], and MPL-F‑266 lasers [14] are used as such sources. To receive information, the PMTs MP1922 [11, 15] and PMTs Hamamatsu R7154 [12, 14] are most often used. Most papers consider the baseline distances up to 100 m (for example, [11–14]), however, there are also some papers with a large baseline distance (for example, up to 1 000 m in [15]). As a part of the experimental work, the researchers determine attenuation of the desired signal radiation, reduced to decibels (path loss), probability of erroneous symbol recording (bit error rate or BER), and duration of the received pulse (pulse width) depending on the communication conditions.

Within the framework of theoretical works [16–21], the data transmission channel quality is studied depending on the optical and geometric conditions and the maximum information transmission rate. In addition, the works [16–17] propose an approximation formula for the impulse response of an optical communication channel based on scattered radiation.

Another field of theoretical and experimental research is related to the influence of various information encoding methods on the quality of a communication channel [12, 22–30]. Within the framework of [12, 22, 23], the OOK (On-Off-Keying) data encoding is considered; it is an approach where "0" is set by the absence of a pulse, and "1" by its availability. The papers [12, 22, 23] consider the PPM (Pulse Position Modulation) data encoding, namely an approach according to which the encoded symbol depends on the pulse position in the time slot. In [12, 23–27], the DPIM (Digital Pulse Interval Modulation) coding is applied, an approach according to which information is encoded by a delay between the pulses. The papers [28] use coding at several wavelengths. In [29], information is encoded by controlling the sent radiation polarization. The papers [25, 30] show that the application of DPIM can significantly increase the information transfer rate compared to the PPM, however, the bit error rate will be somewhat higher. As a part of the experimental works previously performed at the Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences [8, 31], the DPIM information encoding method was also used, namely a delay between the pulses of 65 µs corresponded to the symbol "1", and of 67 µs – to the symbol "0". In the daytime, the sustainable optical communication was established at a wavelength of λ = 250 nm at a baseline distance of up to 1.3 km [31].

In general, the existing literature review shows that the previously published works almost do not consider the wavelengths in the range of λ>270 nm. We can do no more than mention the paper [16], where the wavelengths λ = 230–310 nm are considered. Therefore, below we consider how the bit error rate Pe depends on the optical and geometric conditions in the wavelength range from 200 to 400 nm.

Problem statement

and solution method

The problem was considered using the following statement (Fig. 1). Let there be a planar system consisting of an atmosphere and the ground surface. The atmosphere is a scattering and absorbing aerosol-gas medium. The atmosphere is divided into 32 homogeneous layers in each of which the molecular and aerosol scattering and absorption rates, as well as the phase function of aerosol scattering are given. The optical parameters of the atmosphere are determined by LOWTRAN‑7 mid-latitude summer models [32] of a cloudless sky and information from the papers [33–35] on molecular scattering and absorption. An impulse laser source S located on the ground surface is transmitting information through the atmosphere in the spectral intervals with the centers at wavelengths λ = 205, 215,…, 395 nm and a width Δλ = 10 nm. The information is encoded by the delay between pulses (DPIM). The source axis is located at the zenith angle θs from the vertical, the source divergence angle is νs, the energy of one pulse is Q0 = 0.3 mJ, and the duration is Δt0 = ns. It is assumed that the pulse shape is rectangular in time. At the baseline distance YN from the source, the receiving system D is located, the axis of which is oriented at the zenith angle θd, and the field of view angle is equal to νd. A coplanar channel circuit is considered (the axes of the source and the receiving system are located in the same plane). The receiving system is assumed to be ideal with an aperture area of Sd = 0.01 m2, containing a UFK‑4G‑4 PMT [36]. The PMT parameters are as follows: PMT gain ratio MPMT = 106, dark current IT = 10–15 A, frequency bandwidth Δf = 11 kHz. The spectral sensitivity values of the photocathode ΣK at the considered wavelengths are given in Table 1. The receiving system parameters correspond to the system used by us in [31].

The solar (or lunar) radiation is incident on the upper boundary of the atmosphere at a zenith angle θb and an azimuth angle ϕb. While knowing the optical and geometric conditions and the specifications of the transmit/receive equipment, it is required to determine the impulse response of the communication channel and the background radiance and, by using their values, to estimate the bit error rate Pe recording.

The desired signal in the optical communication circuit under consideration is entirely generated by scattered radiation. To simulate the impulse response, we used two programs of the Monte Carlo method (Figure 2): the singly scattered part of the impulse response h1(t) was simulated by a Monte Carlo program with local estimates at the points of "photon" collision with the medium (Figure 2a), and the multiply scattered part of the impulse response hmsc(t) was simulated by a Monte Carlo program with the modified double local estimates at the collision points (Figure 2b). A brief description of these algorithms is given below.

The algorithm with local estimates is as follows (Figure 2a). The photon wandering trajectories in the atmosphere begin in the source point in accordance with the optical parameters of the medium. The modeling algorithm for photon wandering can be considered as classical [37]. At each collision point of photons with the medium (marked as M1 and M2 in the figure), a local energy estimate is made in relation to the photon that will arrive at the receiver from the point Mi with a wandering time falling into the i-th time interval Ii, pnk. If the conditions and are met, is equal to:

, (1)

where i is the number of the time interval where the local estimate is made; qM is the photon energy arriving to the collision point M, rMR is the distance from the collision point M to the receiving system D, τMR is the optical path thickness from the collision point M to the receiving point D, σsa, M, σsm, M are the aerosol and molecular scattering coefficients at the collision point M, respectively, σt, M is the attenuation coefficient at the collision point M, ga(μM), gm(μM) are the phase functions of aerosol and molecular scattering at the collision point M, respectively, μM is the scattering angle cosine between the photon trajectory direction at the collision point before scattering and the direction to the receiver, с is the light velocity in the medium, ωd is the direction of the receiver’s optical axis, ωpnk is the direction from the receiver point to the collision point M.

Otherwise Ιipnk = 0.

Then the singly scattered part of the impulse response h1(t) is determined as follows:

(2)

where i = 1, ..., Ntime is the number of the time interval, Ntime is the number of time intervals, ti + 1, ti are the boundaries of the i-th time interval, P is the number of sets of trajectories, N is the number of trajectories in a set, Kpn is the number of collisions in the n-th trajectory of the p-th set.

An algorithm with modified double local estimates is made as follows (Figure 2b). The wandering of photons in the medium is simulated. For each collision point (marked as M1 in the figure), the receiver’s field of view is divided into the subregions corresponding to the time intervals for which the impulse response is sought. The algorithm for dividing the field of view into the subregions is described in [38]. The subregions are bounded by two ellipsoids of revolution which foci are the collision point M1 and the receiver system D, and the cone of the receiver’s field of view. A random direction ω is selected in the field of view of the receiving system. One ghost collision point is selected in each of the obtained subregions (marked in the figure as N1, N2, etc.). Such points are located on the ray determined by the direction ω. In each of the possible time intervals, there is a double local estimate of radiation arrived at the receiving system, having scattered at points M and N1 for the first time interval, etc. The estimate is given by the following formula:

, (3)

where μMi is the scattering angle cosine between the photon direction at the collision point M before scattering and direction to the ghost collision point Ni corresponding to the i-th time interval, μNi is the scattering angle cosine between the directions MNi and NiD, Pi, ei are the focal parameter and eccentricity of the ellipsoid of revolution, relevant to the time ti, b is the dot product of the vector ω and the unit vector aimed in the direction DM, is the optical thickness of the path from the collision point M to the ghost point Ni, is the optical thickness of the path from the ghost collision point Ni to the receiving system D.

Thus, the multiply scattered part of the impulse response hmsc(t) is determined by the formula (2) using the values of Iipnk obtained by the formula (3).

Having known the impulse response of the communication channel, it is possible to determine the average desired signal power:

, (4)

. (5)

The background radiation power is determined by the following formula:

, (6)

where is the value of the solar (lunar) constant, is the radiance value of the background radiation of the Sun (Moon) at the zenith angle of the Sun (Moon) θb and the relative azimuthal angle ϕb, measured from the direction of the receiving system axis.

The values of solar constants were taken from [39]. The values of the lunar constants for the full moon conditions were taken from [40].

The value of Ib was calculated by the Monte Carlo method of backward trajectories with the local estimates at the collision points, based on [37].

In [38, 41], to test the calculation programs for h1 and hmsc, a comparison was made with the results of [20, 42]. In [43], a test comparison was made with the results of calculations Ib from [44]. The comparison demonstrates that the difference between the results is within the limits of the statistical calculation error.

For the DPIM information coding method, the papers [24–27] propose a formula for calculating the bit error rate. If information encoding is performed similarly to our work [31], then the bit error rate is determined as follows:

, (7)

where is the probability of sent pulse recording, is the probability that the pulse absence is registered correctly.

At a high level of the desired signal, according to [25], the values P0/1 and P1/0 are determined by the following formulas:

, (8)

, (9)

, (10)

, (11)

, (12)

, (13)

, (14)

, (15)

where e is the elementary charge; I1 is the current generated by the background radiation, I2 is the current generated by the desired signal.

Results

To solve the problem set, a series of calculations was performed for the following optical and geometric conditions: λ = 200–400 nm with a step of 10 nm; meteorological range of visibility SM = 50 km that corresponds to the LOWTRAN‑7 model with the vertical aerosol optical depth (AOD) at a wavelength of λ = 550 nm АОТ550 = 0.16, YN = 0.05–50 km, θs = 85°, νs = 0.06 mrad, θd = 85°, νd = 1°, θb = 0°, calculations for the daytime and night. The selection of values θs and θd is justified by the fact that the received desired signal will be the largest for such conditions.

Using the developed programs of the Monte Carlo method and formulas (4)—(14), the bit error rate Pe recording was calculated depending on the wavelength λ and the baseline distance YN. The calculation results are given in Figure 3.

It can be seen from Figure 3a that the wavelength λ = 295 nm will be the best choice for arranging the long-distance communications in the daytime, since the distance at which Pe becomes more than 10% is the largest for such a wavelength. The reason for such result is that, on the one part, due to the significant absorption by ozone, there is little background radiation from the Sun at this wavelength. On the other part, the ozone concentration in the atmospheric ground layer is less than in the stratosphere, thus, the ground layer radiation at this wavelength is attenuated less than at other wavelengths in the range of λ = 200–300 nm.

It can be seen from Fig. 3 that communication during the night for most wavelengths is significantly better than in the daytime conditions. This is due to the fact that the lunar background is much less than the solar one. Figure 3b shows that the best wavelength for the long-distance communications at night would be λ = 395 nm, and the far UV band wavelengths (200–250 nm) would be the least suitable.

Conclusions

The performed calculations of the bit error rate in the daytime and night conditions have shown the following:

The wavelength λ = 295 nm is better than others for the arrangement of atmospheric long-distance optical communication based on scattered radiation during the day.

During the night, the best wavelength for arranging the long-distance communications will be λ = 395nm, and the far UV band wavelengths (200–250 nm) will be the least suitable.

The work was performed as a part of the public task of the Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences.

Authors

Tarasenkov M. V., Cand. phys.-math. sci. V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

ORCID: 0000-0002-8826

Field of scientific interests: analysis of regularities of imaging through the atmosphere, atmospheric correction of images in the visible and UV ranges, theoretical investigation of non-line-of-sight communication channels.

Poznakharev E. S., V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

Field of scientific interests: experimental and theoretical investigation of non-line-of-sight communication channels.

Peshkov S. A., V. E. Zuev Institute of Atmospheric Optics SB RAS, Tomsk, Russia.

Field of scientific interests: theoretical investigation of non-line-of-sight communication channels.

Author contributions

Tarasenkov M. V.: idea, design, workflow management, discussions, suggestions and remarks, literary work preparation; Poznakharev E. S.: discussions, suggestions and remarks; Peshkov S. A.: calculations, processing of results, discussions.

Conflict of interest

The authors declare no conflicts of interest.

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