Effect of controllable parameter synchronization on the ensemble average bit error rate of space-to-ground downlink chaos laser communication system

Mi Li; Yifeng Hong; Yuejiang Song; Xuping Zhang

doi:10.1364/OE.26.002954

1. Introduction

As space-to-ground laser communication has developed rapidly in these years, many countries are devoting to research the key techniques to manufacture the system with high performance [1–6]. So far, countries such as USA, Japan, Europe, and China have mastered the technique of a medium-low-data-rate system, and they are making great efforts on the more advanced techniques. For example, NASA’s Laser Communications Relay Demonstration (LCRD) has updated their plan for a high data rate optical communication service [7]. The European Data Relay System (EDRS) plans to realize a 1.8Gb/s communication link in the project of EDRS-C [8]. Also, Japan has realized a space-to-ground laser communication link with a small optical transponder whose mass is less than 6kg [9]. At the meantime, they tend to develop the laser communication system with higher information security [10–12]. In spite of the laser communication being securer than the microwave communication, the huge laser beam spot at the receiving terminal remains a risk of information divulgation. To overcome this weakness, we suppose that a chaos-based scheme can be ported to the space-to-ground laser communication to keep the information security. Compared with the traditional modulation schemes of laser communication, chaotic modulation owns higher confidentiality for its carrier wave being random and its demodulation requiring a high degree of parameters synchronization [13–19]. Practically, without the configuration parameters of the emitting terminal, the receiving terminal is unable to demodulate the transmitted chaotic optical signal. Therefore, it is an effective way to adopt this scheme in space-to-ground laser communication system to provide higher level of confidentiality.

Up to now, numerous researches have been carried out on the chaos laser communication. Reference [20] demonstrates a 1Gb/s commercial fiber-optic communication link. And it offers two schemes to produce chaotic dynamics in fiber-optic communication, one is based on the optoelectronic scheme, and the other is based on the all-optical scheme. The optoelectronic scheme has a more complex configuration which can provide with higher confidentiality. Besides, this kind of scheme has greater extensibility which is more possible to realize high-data-rate communication. With these two advantages, the optoelectronic scheme is selected to set up the subsystems of chaotic modulation and demodulation of the system in this paper. Based on the optoelectronic scheme showed in [20], reference [21] further gives a mathematical way to simulate the chaotic carrier versus the feedback strength of optoelectronic scheme, which is helpful for the adjustment of feedback strength to obtain the chaotic signal with enough complexity. Also, a 5km space chaos laser communication link is presented in [22], although the data rate is only 60Kb/s, it first proves the feasibility of space laser communication with chaos. Based on these formal results, we believe that chaotic dynamics can also be ported to an extra-long-distance space-to-ground chaos communication link.

However, the extra-long-distance space-to-ground chaos communication system needs a more precise configuration, so it is not easy to port the devices of the chaotic fiber-optic system directly to a space optical communication link. In fact, such an extra-long communication link will contribute to a big loss of optical power, which makes it more difficult for the chaotic signal to be demodulated. Besides, pointing error and special effects induced by the atmospheric turbulence should be further considered when analyzing this kind of communication link [23,24].

Currently, the inside of chaotic feedback has been researched by numerous works. But few works or reports have been done to analyze and predict the long-term communication performance of a space-to-ground chaos laser communication system. In this paper, a space-to-ground downlink transmission link is taken as an example to analyze this issue. Our work includes a new ensemble average mismatch-induced bit error rate (BER) calculation model under the effects of intensity scintillation and pointing error and analysis on how the controllable parameter synchronization will affecting the BER under different jitters and boresight errors.

2. Ensemble average BER calculation model

Using the optoelectronic-feedback-based subsystems of chaotic modulation and demodulation as the emitting terminal and the receiving terminal, chaotic dynamics can be ported to the space-to-ground laser communication (which is demonstrated in Fig. 1, the terminals are similar to the optoelectronic structure in [20]). In the emitting terminal, the loop of the fiber delay line, the avalanche photodiode (APD), the radio-frequency (RF) amplifier, and the Mach-Zehnder (MZ) modulator is used to generate the chaotic carrier wave. When the laser diode (LD) 2 inputs a continuous optical signal to the MZ modulator, the MZ modulator can generate the chaotic carrier wave. At the first bit time t₀, considering that there is no input message, the MZ modulator produces the chaotic carrier wave c(t₀). Then, after passing through the port 1 of the 2 × 2 coupler, c(t₀) will pass through the extra-long channel, and it can also pass the port 2 of the 1 × 2 coupler in the receiving terminal. Then, c(t₀) will serve as the initial value of the feedback for the MZ modulator in the receiving terminal to generate the chaotic carrier wave c(t₁) at the next bit time t₁, and this signal will be detected by APD 2 and become an electrical signal. At the meantime (the bit time t₁), in the emitting terminal, after c(t₀) passes through the port 2 of the 2 × 2 coupler, the MZ modulator in the emitting terminal will also generate the chaotic carrier wave c(t₁) for the structure of generating the chaotic carrier wave in the receiving terminal is the same as the one in the emitting terminal. At this time, LD 1 will input an impulse signal m(t₁) representing ‘0’ or ‘1’, and the combined signal c(t₁) + αm(t₁) will pass through the port 1 of the 2 × 2 coupler and be received at the receiver, where α is the masking efficiency. Then, after passing through the port 1 of the 1 × 2 coupler, the signal can be detected by APD 1, forming an electrical signal. Finally, with the parameters being properly tuned, the part of the chaotic carrier wave can be cancelled by the power combiner, and the original m(t₁) will be obtained.

Fig. 1 Optoelectronic-feedback-based space-to-ground chaos laser communication system. The blue spot stands for the ideal position of the receiving beam spot, whereas the green spot stands for the practical position of the receiving beam spot considering pointing error.

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However, some difference of the space-to-ground chaos communication system from the fiber chaos communication system should be considered carefully. Taking an example of a space-to-ground downlink transmission link shown in Fig. 1, the effects of intensity scintillation and pointing error can varies the value of the receiving optical power, thus affecting the communication quality. Also, certain parameter mismatch will occur because of the space radiation and the ambient temperature. For such a system, an ensemble average bit error rate (BER) calculation model can be used to analyze the communication performance.

As we know, the BER of a communication system is induced by different kinds of noise. In chaos communication system, mismatch noise is considered as the main part of the noise. It serves as a chaotic noise, which probability density function (PDF) can be modified by using the Gram–Charlier (GC) expansion. Thus, the mismatch-induced bit error rate (BER) of the system is given by [25]

B E R_{c h a o s} (Ρ_{i n}) = \frac{1}{2} e r f c (\frac{u}{2 \sqrt{2}}) + \sum_{k = 3}^{+ \infty} + a_{k} Ψ_{k} (u),

in which P_in is the receiving optical power which determines the signal-to-noise ratio (SNR) u, a_k is the GC coefficient, and Ψ_k is the kth correction of the Gaussian BER. With proper tuning of the feedback strength of chaotic system, the first two GC coefficients (higher orders of a_k can be neglected) can both converge to zero. u is denoted as

u = \frac{K α}{\sqrt{〈 n^{2} 〉}},

where K is the amplitude of the output electrical signal of the detector, α is the masking efficiency (which means the ratio of output power of laser diode 1 and laser diode 2), and〈n²〉is the root mean square amplitude of mismatch noise.

Due to the receiving optical power being small, APDs are used as the detectors in the space-to-ground chaos laser link instead of traditional photodiodes (PD) in the chaos fiber-optic link to amplify the signal. Thus, the expression of K in Eq. (2) can be derived as follow

K = \frac{1}{2} \cdot P_{i n} \cdot \frac{G η e}{h v},

in which 1/2 is the splitting ratio of the 1 × 2 coupler, G is the gain factor of the avalanche photodiodes (APD) in the receiving terminal, η is their quantum efficiency, e is the elementary charge, h is the Planck constant, and ν is the laser frequency. Here, it can be seen that K is a function of the receiving optical power P_in. That is to say, intensity scintillation and pointing error will have a further effect on its value in space-to-ground chaos laser communication scheme.

With the numerical expression of K above, the root mean square amplitude of the mismatch noise〈n²〉should be [25]

〈 n^{2} 〉 \approx \frac{1}{2} K^{2} [〈 ε^{2} 〉 + {(Δ ϕ)}^{2} + \frac{1}{4} {(\frac{Δ K}{K})}^{2}],

where Δϕ is the offset phase mismatch, ΔK is the detector mismatch, and〈ε²〉is the synchronization error that can be written as [25]

\begin{array}{l} 〈 ε^{2} 〉 = \frac{1}{3} {(\frac{Δ T}{τ})}^{2} + {(\frac{Δ β}{β})}^{2} + (1 - \frac{π}{4}) {(\frac{Δ τ}{τ})}^{2}, \\ - 2 (1 - \frac{π}{4}) \frac{Δ β}{β} \frac{Δ τ}{τ} - 2 (1 - \frac{π}{4}) \frac{Δ T}{τ} \frac{Δ τ}{τ} \end{array}

in which ΔT is the delay time mismatch, Δβ is the feedback strength mismatch, Δτ is the high cutoff response time mismatch. Furthermore, T, β, and τ are the corresponding parameters in the emitting terminal.

When chaotic dynamics is ported to a space-to-ground laser communication, atmospheric turbulence is an unavoidable issue to be considered for a long-term-use. For a downlink transmission link, atmospheric turbulence can induce the effect of intensity scintillation, which can cause the receiving optical power varying randomly. Also, the effect of pointing error is necessary to be taken into consideration for an extra-long-distance laser communication link, which can cause the beam center drifting randomly.

Normally, in a downlink transmission, intensity scintillation is the main effect induced by atmospheric turbulence. However, when the effect of pointing error is included, the deviation r of the beam center should be considered. Therefore, when pointing error is added, the traditional PDF of intensity scintillation can no longer be used, and the modified PDF of intensity scintillation (of a Gaussian beam) under pointing error can be modified as [26]

p_{i} (P_{i n}) = \frac{1}{\sqrt{2 π σ_{I}^{2} (r, L)}} \cdot \frac{1}{P} \cdot \exp (\frac{\ln {(\frac{P_{i n}}{〈 P_{i n} (0, L)} + \frac{2 r^{2}}{W^{2}} + \frac{σ_{I}^{2} (r, L)}{2})}^{2}}{2 σ_{I}^{2} (r, L)}) .

In Eq. (6), P_in is the receiving optical power, ζ is the zenith angle, 〈P_in(0,L)〉 = α_aP_TD2 r/2W² is the mean receiving optical power, α_a is the energy loss caused by atmosphere, P_T is the laser transmission power, D_r is the receiving diameter, W = W₀ + θL/2 is the beam radius at the receiver, W₀ is the transmission aperture, θ is the divergence angle, σ2 I(r,L) is the variance of intensity scintillation with a deviation of r and a link length of L = (H-h₀)sec(ζ), H is the altitude of emitting terminal, and h₀ is the altitude of receiving terminal.

Considering r and P_in are independent, the marginal PDF of intensity scintillation can be calculated by

p_{I} (P_{i n}) = \int_{0}^{+ \infty} p_{i} (P_{i n}) \cdot p_{p} (r) d r,

in which p_p(r) is the PDF of pointing error, and the deviation r conforms to the PDF of [27]

p_{p} (r) = \frac{r}{σ_{s}^{2}} \exp (\frac{r^{2} + s^{2}}{2 σ_{s}^{2}}) \cdot I_{0} (\frac{r s}{σ_{s}^{2}}) .

In Eq. (7), σ_s is the scale parameter that measures the degree of jitter, s is the boresight error, and I₀ is the modified Bessel function of the first kind with a zero order.

In light of the atmosphere condition varying much more slowly than the transmitted digital signals, the ensemble average BER of the space-to-ground chaos laser communication system under intensity scintillation and pointing error can be expressed as [28]

B E R = \int_{0}^{+ \infty} B E R_{c h a o s} (P_{i n}) \cdot p_{I} (P_{i n}) d P_{i n} .

The BER expression above can reflect the communication performance of the system for a long-term use. Numerical simulation can be conducted to predict the performance of the space-to-ground chaos laser communication system under such effects, which is meaningful to practical system configuration.

3. Numerical simulation and analyses

3.1 Simulation parameters setup

Since the optoelectronic-feedback-based chaotic dynamics can be ported to space-to-ground laser communication through proper designing, parameters of practical communication system can be used to conduct a numerical simulation. The parameters used are shown as follows: zenith angle ζ = 0°, altitudes of emitting terminal and receiving terminal are H = 38,000km, h₀ = 100m, laser wavelength λ = 1550nm, transmission aperture W₀ = 0.1m, divergence angle θ = 30μrad, transmission power P_T = 3w, data rate is 1Gb/s, mask efficiency α = 0.2, receiving diameter D_r = 0.3m, energy loss α_a = 1, quantum efficiency η = 0.75, and avalanche photodiode (APD) gain factor G = 20dB.

The default parameters used for calculating the amplitude of the mismatch noise are that feedback strength mismatch rate Δβ/β = 0.002, high cutoff response time mismatch rate Δτ/τ = 0.01, detector mismatch ΔK = 0.1μA, delay time mismatch ΔT = 1ps, and offset phase mismatch Δϕ = 0.01rad. During the process of our numerical simulation, the effects of three controllable mismatch parameters are analyzed on the ensemble average bit error rate (BER) of the system under intensity scintillation and pointing error, which are ΔK, ΔT, and Δϕ.

3.2 Results under different jitters

As one part of pointing error, whether the jitter is strong or faint can be depicted by its scale parameter σ_s. In this part, considering the space-to-ground link has a high requirement on the pointing error, the boresight error s is determined as s = 50m, and the BERs shown are calculated with σ_s from 20m to 100m, which is based on the interval introduced in [27].

According to Eq. (4), when the value of ΔK is far smaller than K, the term of ΔK/K will have almost no contribution on the amplitude of mismatch noise. This explains why the BER shown in Fig. 2 remains the same under different σ_s when ΔK is zero. Such phenomenon indicates that when the detector mismatch goes to zero, the effects of intensity scintillation and pointing error will almost exert no influence on the space-to-ground chaos laser communication system. Thus, we should reduce the value of ΔK in the receiving terminal to keep the anti-interference of the system against these effects. However, considering some unpredictable effects such as the background light and the device disturbance, it might be hard to keep the value of ΔK precisely remaining zero. Hence, intensity scintillation and pointing error should be taken into consideration when there is a nonzero detector mismatch. Through Eq. (4), it can be noticed that K conforms to the same probability density function (PDF) as the marginal PDF of intensity scintillation (which is modified by the effect of pointing error). According to Eqs. (3) and (7), a lower σ_s will lead to a higher probability to obtain a higher K (since a lower σ_s will lead to a higher probability to obtain a higher receiving optical power P_in), which can still make ΔK far smaller than K in the case that ΔK is relatively small. This explains why the BER surface with ΔK from 0μA to 0.1μA will become flatter with the decrease of σ_s, meaning there exists an acceptable scope of ΔK to obtain a stable communication performance for space-to-ground chaos laser communication system under a relatively faint jitter.

Fig. 2 BER versus detector mismatch ΔK under different scale parameters σ_s. Whether ΔK is positive of not contributes the same to the amplitude of mismatch noise. In fact, the curve of BER versus ΔK should be symmetrical about ΔK = 0. Therefore, we only plot the BER curves with positive ΔK.

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For an optoelectronic-feedback-based chaos communication system, an optimal BER can be obtained with a proper adjusting of delay time that depends on both the high cutoff response time mismatch and the total delay time. For the high cutoff response time mismatch always exists, the optimal delay time is usually nonzero, and its corresponding value to the parameter we used is displayed in Fig. 3. Moreover, it can be noticed that when σ_s is up to 100m, the BER is almost unchangeable in the interval of ΔT we select. And when σ_s reduces, the BER will undergo a sudden change with the increase of ΔT. To be specific, the BER of σ_s = 20m will suffer a sudden increase when ΔT grows to 0.8ps. Moreover, the value of ΔT of sudden change point will become larger with the increase of σ_s. This phenomenon indicates that the BER is more sensitive to the value of ΔT when there is a lower degree of jitter. Thus, when the jitter is faint, as the BER has a high requirement on the precise of ΔT, we should pay more attention to the control of delay time synchronization to keep a well-behaved communication performance.

Fig. 3 BER versus delay time mismatch ΔT under different scale parameters σ_s.

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According to the numerical results demonstrated in Fig. 4, with Δϕ varying from 0.02rad to 0.03rad, the BERs show no obvious change under σ_s from 20m to 100m. Through calculation, it can be found that, in Eq. (4), the part of the mismatch noise induced by Δϕ and〈ε〉will become far larger than the one induced by ΔK, thus making the BER almost be irrelevant to the effect of jitter. However, when Δϕ is from 0rad to 0.02rad, the part of the detector mismatch will engender a larger contribution to the amplitude of the mismatch noise, making the BER be more sensitive to the effect of jitter. And this explains the phenomenon that the BER will increase more significantly with the increase of σ_s under a smaller Δϕ.

Fig. 4 BER versus offset phase mismatch Δϕ under different scale parameters σ_s. The curve of BER versus Δϕ is symmetrical about Δϕ = 0 for the same reason as ΔK. Therefore, only the BER curves with positive Δϕ are plotted.

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3.3 Results under different boresight errors

Besides the effect of jitter, the boresight error is the other factor of pointing error. Especially in an extra-long-distance laser link, its value serves as a key factor to determine the communication performance. In this part, the scale parameter of jitter σ_s is determined (σ_s = 50m), and the BERs are calculated with s from 0m to 200m, which is also based on the interval introduced in [27].

It can be noticed that the features of the ensemble average BER in Fig. 5 are similar to the ones shown in the part above. In other words, the increase of the boresight error has a similar effect on the BER of the system with these parameters, and the reason lies in the PDF of pointing error and the marginal PDF of intensity scintillation.

Fig. 5 BERs versus detector mismatch ΔK, delay time mismatch ΔT, and offset phase mismatch Δϕ under different boresight errors s.

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Figures 6(a) and 6(b) display the PDFs of pointing error under different σ_s and different s. In fact, Fig. 6(a) shows the relationship between the PDF of pointing error and the degree of jitter, whereas Fig. 6(b) shows the relationship between the PDF of pointing error and the degree of boresight error. Both the increase of σ_s and s will move the PDFs towards the right, however, there still lies some difference, the peak value of the PDF in Fig. 6(a) will become lower with σ_s increasing, while the peak value of the PDF in Fig. 6(b) will tend to be stable with s increasing. Thus, it can be concluded that jitter and boresight error can contribute to the shape of the PDF of pointing error (Rayleigh distribution) differently.

Fig. 6 (a). PDF of pointing error under different scale parameters σ_s (s = 50m). Figure 6(b). PDF of pointing error under different boresight error s (σ_s = 50m). Figure 6(c). Marginal PDF of intensity scintillation considering pointing error under different scale parameters σ_s (s = 50m). Figure 6(d). Marginal PDF of intensity scintillation considering pointing error under different boresight error s (σ_s = 50m).

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However, according to Figs. 6(c) and 6(d), it can be noticed that the deviations of the marginal PDFs of pointing error do not show great difference. Both the increase of σ_s and s will make the marginal PDFs move towards the left since larger jitter and boresight error will contribute to a larger probability for the system to receive a lower optical power (it is more possible for the system to receive the optical power that deviates more from the center under a higher value of σ_s and s). Also, the similarity of the marginal PDF of Figs. 6(c) and 6(d) means the increase of σ_s and s within the scope we select will affect the PDF of intensity scintillation similarly. Hence, the concordance between the ensemble average BER shown in Fig. 5 and the figures in part B can be well explained, and we can obtain similar results of how the parameter synchronization can affect the ensemble average BER of space-to-ground chaos laser communication system.

The results shown in part B and part C demonstrate the characteristics of ensemble average BER of space-to-ground chaos laser communication system under intensity scintillation and pointing error, which have good reference value for a practical system configuration.

4. Conclusion

In this paper, an ensemble average bit error rate (BER) calculation model is established to measure the long-term communication performance of an extra-long space-to-ground chaos laser communication link. According to our results, the synchronization of some controllable parameters is required to be cared when the system is used under the effects of intensity scintillation and pointing error. Specifically, when the detector mismatch approaches zero, the system will have a good anti-interference against these effects. Besides, on the condition that the degree of jitter or the boresight error is relatively low, the BER of the system will have a high sensitivity to the synchronization of the delay time. Moreover, when the offset phase mismatch locates in a relatively lower interval, the BER of the system will become more sensitive to the effects of pointing error. With these results, as long as the parameter synchronization is controlled precisely, the space-to-ground chaos laser communication system can perform well in an extra-long transmission link under the effects of intensity scintillation and pointing error. Thus, we believe that chaotic modulation is an excellent scheme that is suitable to be applied as the next generation of encryption method in space-to-ground laser communication.

Funding

Jiangsu Provincial Natural Science Foundation of China (BK20151256); Suzhou Technology Innovative for Key Industries Program of China (SYG201729); National Natural Science Foundation of China (61205045, 61401279); National Training Program of Innovation and Entrepreneurship for Undergraduates (S201710284057).

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Effect of controllable parameter synchronization on the ensemble average bit error rate of space-to-ground downlink chaos laser communication system

Abstract

1. Introduction

2. Ensemble average BER calculation model

3. Numerical simulation and analyses

3.1 Simulation parameters setup

3.2 Results under different jitters

3.3 Results under different boresight errors

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (9)

Optics Express