Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Effect of external mismatch and internal mismatch caused by pointing error on performances of space chaos laser communication system

Mi Li, Cheng Zeng, Jizhao Lei, You Guo, Mengnan Li, Jianjie Yu, Yuechun Shi, and Xiangfei Chen

Open Access Open Access

Abstract

In order to evaluate the effect of pointing error on space chaos laser communication system, we conduct bit error rate (BER) analysis with external mismatch and internal mismatch caused by pointing error. Based on BER formulae, numerical simulations are conducted to investigate the effect of external and internal mismatches on performance of the system under different boresight and jitter. Our results indicate that jitter will affect BER more than boresight and internal mismatch will affect the performance of system more than external mismatch. These results are significant for optimizing space chaos laser communication system design.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The application of chaos laser communication has attracted wide attention due to its unique features like unpredictable, pseudorandom, high level of robustness and so on [1–4]. After the chaos synchronization has been implemented in the circuit in 1990 [5], the potential of chaos laser communication has been demonstrated by laboratory demonstrations [6–7]. Then, the field experiment based on chaos synchronization over 120 km of optical fiber link has been undertaken with bit error rate (BER) of 10−7 in 2005 [8]. Contrary to fiber link, space channel will introduce other channel noise and distortions which deteriorate chaos synchronization [9]. Space chaos laser communication system will be affected or even not work normally when signal distortions exceed to a certain level in space channel. In 2002, [10] reported the demonstration of self-synchronizing chaos communication over a 5 km free space laser link with the BER of 1.92×10−2, which shows the potential of chaos synchronization applied in space channel. Space communication systems are vulnerable to pointing error, which consists of boresight and jitter, due to building sway, building vibration and thermal expansion [11]. Pointing error has been widely studied in traditional space laser communication system. Reference [12] reported the performance of BER under exponentiated Weibull turbulence channels with nonzero boresight pointing error in free space communication system. Reference [13] reported a generalized pointing error model with different jitters for the horizontal and vertical misalignment as well as boresight error and computed ergodic capacity for free space links under Log-normal and Gamma-Gamma channels. And [14] reported the asymptotic outage performance of free space optical communication system under exponentiated Weibull channels with generalized pointing error which follows a Beckmann distribution.

However, as far as we know, most of the investigations of pointing error effect are based on traditional space laser communication. While the effect of pointing error on performance of space chaos laser communication has not been reported yet. In fact, pointing error will affect the synchronous demodulation of receiver from two aspects. On the one hand, pointing error has no influence on chaotic carrier generation loop at the emitter, but will be introduced to chaotic carrier generation loop by space channel at the receiver. This mismatch caused by pointing error will occur between two optoelectronic feedback loops and will affect optoelectronic oscillator gain match. On the other hand, it is well to know that the power source for optoelectronic feedback loop should match relatively at the emitter and receiver. The optical signal generated by laser diode at the emitter will be affected by pointing error when it propagates in space channel. Meanwhile, the optical signal generated at the receiver will not propagate in space channel. This mismatch will occur between the power generated by laser diode at the emitter and receiver. In conclusion, there are two kinds of mismatches caused by pointing error, and we name them as internal mismatch and external mismatch respectively. With the effect of internal mismatch and external mismatch, the degree of mismatch in synchronous carrier cancellation will increase to a certain extent. Then the BER of chaos communication system will increase correspondingly and the quality of communication system will decrease as well.

In this paper, external and internal mismatches caused by pointing error are analyzed in detail. And considering the effect of pointing error, BER formulae of space chaos laser communication is derived. To better explain the effect of pointing error on space chaos laser communication system, we simulate and discuss the BER of communication system from the aspects of internal and external mismatches caused by pointing error under different boresight and jitter. We find the BER order of magnitude with internal mismatch is larger than the BER order with external mismatch. And the increase of boresight and jitter can greatly deteriorate the performance of space chaos laser communication system. That means the effect of external mismatch and internal mismatch on chaos synchronization caused by pointing error should be considered carefully in optimizing space chaos laser communication system. In conclusion, the results of this paper are significant for optimizing space chaos laser communication system design.

2. Principle

2.1 Configuration of space chaos communication system with pointing error

The configuration of space chaos laser communication system is showed in Fig. 1. The chaotic carrier c1(t) is generated by optoelectronic feedback structure (Loop 1) [15] composed of Mach–Zehnder modulator (MZ1), fiber delay line, avalanche photodiode (APD3) and RF amplifier (RFA1) at the emitter. Message signal m(t) emitted from laser diode (LD3) will be loaded into chaotic system by a 2×2 coupler at the emitter. Then the signal light with chaotic carrier will be amplified by erbium-doped optical fiber amplifier (EDFA1) and propagate in space channel. The light will be amplified again by EDFA2 and split into two parts by a 1×2 coupler at the receiver. More specifically, one part will be detected by APD1 and the other one will be the feedback signal of optoelectronic feedback structure (Loop 2) which is the same structure and parameters with the emitter loop. The chaotic carrier c2(t) generated by optoelectronic feedback structure (Loop 2) will be detected by APD2. Considering that receiver and emitter having the same structures and parameters, chaotic carrier c2(t) will be basically identical with c1(t). Therefore, the message signal m(t) will be demodulated from chaotic carrier through power combiner.

 figure: Fig. 1.

Fig. 1. Space chaos laser communication system with internal and external mismatches caused by pointing error. Pointing error has no influence on chaotic carrier generation loop (optoelectronic feedback Loop 1) at the emitter, while pointing error will be introduced to chaotic carrier generation loop (optoelectronic feedback Loop 2). The difference in feedback structure will introduce mismatch to internal gain of chaotic structure which we name the mismatch as internal mismatch. The power generated by LD2 will not propagate in space channel and is a fixed value. And the power generated by LD1 will be affected by pointing error. This mismatch caused by pointing error will occur before APD1 and APD2 (circle 1 and 2) and we name the mismatch as external mismatch.

Download Full Size | PDF

However, space chaos laser communication is different from ground chaos laser communication. Specifically, as for space chaos laser communication, the signal will propagate in space channel with pointing error. Pointing error will bring random misalignment between beam center and center of detector. The detector at the receiver will receive less power because of the limited area and distance deviation of the beam center. The power at the receiver will be affected by pointing error.

As for traditional space laser communication, pointing error effect has been studied deeply [12–14]. Bit error rate (BER) is used to measure the confidentiality of communication systems. BER will be affected by pointing error in space chaos laser communication system. Chaos laser communication is a long-term process and the ensemble average BER is used in the long-term communication system [16]. We can use the ensemble average BER to depict the performance of space chaos laser communication system. Synchronous cancellation is the key of chaos laser communication system. As mentioned above, power at the receiver will attenuate due to pointing error effect. The affected power will further affect chaos synchronization and performance of communication system.

Specifically, the light at the emitter will propagate in space channel and be amplified at the receiver. Then the power affected by pointing error will be split into two parts which is showed in Fig. 1. Two kinds of mismatches will be introduced. One mismatch caused by pointing error will occur between two optoelectronic feedback loops (Loop 1 and Loop 2 in Fig. 1). And it will affect optoelectronic oscillator gain match. The other mismatch will occur before APD1 and APD2 (circle 1 and 2 in Fig. 1). And it will affect current amplitude of APD.

In order to explain this problem more clearly, based on [17], we firstly give the expression of signal at APD1 and APD2 without pointing error.

$${P_x} = \frac{1}{4}{G_{EDFA2}}{P_T}{\cos ^2}\left( {\frac{\pi }{{2{V_{\pi 1}}}}x(t )+ {\phi_1}} \right) + \frac{1}{4}{G_{EDFA2}}d{P_T}m(t )$$
$${P_y} = {P_2}{\cos ^2}\left( {\frac{\pi }{{2{V_{\pi 2}}}}y(t )+ {\phi_2}} \right)$$
where d is the masking efficiency, PT=GEDFA1αP1 is power amplified by EDFA1 and attenuated by atmosphere at the receiver, GEDFA1 is the gain factor of EDFA1 at the emitter, α=αlossDr2/2W2 is geometric loss, αloss is loss efficiency, Dr is receiving aperture, W = W0+θL/2 is beam radius at receiver, W0 is the beam radius at the emitter, θ is the divergence angle, L is the link length [18], P1 is the power generated by LD1, GEDFA2 = 2/(GEDFA1α) is the gain factor of EDFA2 at the receiver, P2 is the power generated by LD2 and is equal to P1/2, Vπ1 is dynamic half wave voltage at the emitter, ϕ1 is fixed phase shift at the emitter, Vπ2 and ϕ2 are correspond parameters of the receiver, x(t) is the feedback voltage of MZ1 at time t, y(t) is the feedback voltage of MZ2 at time t.

If the differences are relatively small between these two optoelectronic feedback structures, y(t) will be approximately equal to x(t). And ϕ2 will be equal to ϕ1+π/2 for decryption. Then two chaotic carriers can be removed and the signal is obtained.

However, the received power PT will be affected by pointing error. As for Gaussian beam, based on [19], the received power PT is the function of distance deviation of the beam center r and can be expressed as PT (r) at the receiving terminal.

$${P_T}(r )= {P_T}\exp \left( { - \frac{{2{r^2}}}{{{W^2}}}} \right)$$
where r is instantaneous radial displacement between the beam center and the detector center.

Pointing error is composed of boresight and jitter. Boresight introduces a fixed displacement of beam center. While jitter introduces a random shift near beam center. Both boresight and jitter will affect the power detected by receiving terminal. The probability density function (PDF) of radius deviation can be given as Rice distribution [11].

$$\rho (r )= \frac{r}{{\sigma _s^2}}\exp \left( { - \frac{{{r^2} + {A^2}}}{{2\sigma_s^2}}} \right){I_0}\left( {\frac{{rA}}{{\sigma_s^2}}} \right)$$
where σs=σjL is jitter error and A = AjL is boresight error, σj and Aj are scale parameters of jitter and boresight respectively, L is link length, I0 is the modified Bessel function of order zero.

More specifically, Fig. 2 shows that the actual beam direction R2 will deviate from ideal beam direction center R1 with the effect of boresight. And jitter will make beam direction center shift randomly nearby R2. Figure 2(a) shows probability distribution of the radial position under different Aj and fixed σj, while Fig. 2(b) shows probability distribution of the radial position under different σj and fixed Aj. PDF curve shifts right when Aj becomes larger. And the PDF curve will be less concentrated when σj increases. In conclusion, pointing error will introduce a larger probability of directional deviation and deteriorate the receiving power.

 figure: Fig. 2.

Fig. 2. Pointing error principle. (a) PDF of boresight Aj. (b) PDF of jitter σj.

Download Full Size | PDF

As mentioned above, the received power PT will be affected by pointing error. Then the power PT in Eq. (1) should be modified as PT(r). And Eq. (1) can be rewritten as

$${P_x} = \frac{1}{4}{G_{EDFA2}}{P_T}(r ){\cos ^2}\left( {\frac{\pi }{{2{V_{\pi 1}}}}x(t )+ {\phi_1}} \right) + \frac{1}{4}{G_{EDFA2}}d{P_T}(r )m(t )$$
where x(t) is time-dependent voltage applied to MZ1 and will not be affected by pointing error.

In addition, power PT(r) will affect the feedback voltage of Loop 2 in Fig. 1. So the time-dependent voltage y(t) applied to MZ2 will be affected by pointing error and can be rewritten as y(t,r). The laser diode LD2 serves as the energy source for optoelectronic feedback structure at the receiver. The power P2 generated by LD2 at the receiver will not propagate in space channel and will not be affected by pointing error. The power P2 detected by APD2 is a fixed value. And the expression of signal detected by APD2 can be modified as

$${P_y} = {P_2}{\cos ^2}\left( {\frac{\pi }{{2{V_{\pi 2}}}}y({t, r} )+ {\phi_2}} \right)$$
Based on Eq. (5) and Eq. (6), there exist two mismatches. One is the mismatch of 1/4GEDFA2PT(r) and P2 at external chaotic loop structure (circle 1 and circle 2 in Fig. 1) and we name this mismatch as external mismatch. And the other is the mismatch of x(t) and y(t,r) at internal chaotic loop structure (between Loop 1 and Loop 2) and we name this mismatch as internal mismatch. Then the power Px and Py will be detected by APD1 and APD2. Based on [20], they can be converted to current signal Sx and Sy which can be written as below.
$${S_x} = {K_1}(r )\left\{ {{{\cos }^2}\left( {\frac{\pi }{{2{V_{\pi 1}}}}x(t )+ {\phi_1}} \right) + dm(t )} \right\}$$
$${S_y} = {K_2}{\cos ^2}\left( {\frac{\pi }{{2{V_{\pi 2}}}}y({t, r} )+ {\phi_2}} \right)$$
where K1(r) is the current amplitude of APD1 and will be affected by pointing error, K2 is the current amplitude of APD2 and will not be affected by pointing error.

These two current signals detected by APD1 and APD2 will then pass through the power combiner. Then message signal K1(r)dm(t) will be demodulated.

2.2 BER formulae analysis

In the description above we have found two mismatches caused by pointing error. One is the mismatch of current amplitude K1(r) and K2. The mismatch will affect the amplitude of chaotic carrier so the two carriers cannot be cancelled completely. The external mismatch of K1(r) and K2 is outside the chaotic loop structure. The other is internal mismatch of x(t) and y(t,r). It intervenes the chaotic process in the loop structure. System performance will be affected due to external mismatch and internal mismatch caused by pointing error. Here the BER performance affected by the two mismatches will be analyzed in detail.

Based on [20], traditional BER can be written as

$$BER = \frac{1}{2}erfc\left( {\frac{u}{{2\sqrt 2 }}} \right) = \frac{1}{2}erfc\left( {\frac{{{K_1}d}}{{2\sqrt {2\left\langle {{n^2}} \right\rangle } }}} \right)$$
where u is the signal-to-noise ratio, d is the masking efficiency, <n2> is the time average of mismatch noise.

However, considering the external and internal mismatches caused by pointing error, Eq. (9) will be modified.

In order to demonstrate the effect of two mismatches on BER more clearly, external mismatch and internal mismatch are discussed separately. Firstly, internal mismatch is ignored and we only discuss external mismatch. Secondly, external mismatch is ignored and we only discuss internal mismatch. At last the two mismatches will be taken into account jointly.

We firstly analyzed the effect of external mismatch. The power P2 in formula of K2 will not be affected by pointing error. While the power PT(r) in formula of K1(r) will be affected by pointing error. Based on [17,20], K1(r) and K2 can be rewritten as K1E(r) and K2E under only external mismatch for convenience

$$\begin{aligned} {K_{1E}}(r) &= \frac{1}{4}{G_{EDFA2}}{P_T}(r)\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s},\\ {K_{2E}} &= {P_2}\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s} \end{aligned}$$
where G is gain factor of APD, e is elementary charge, η is quantum efficiency, hp is Planck constant, ν is laser light frequency, Ts is signal light pulse time.

Current detected by APD1 is affected by pointing error, so K1E(r) is changed with Aj and σj. K1E(r) should be equal to K2 without pointing error, however, Aj and σj introduce the external mismatch ΔKE(r) = K2E – K1E(r).

To show the effect of external mismatch, internal mismatch is not considered. So we ignore the effect of pointing error on chaotic loop. Based on [20], optoelectronic oscillator gain of the emitter and receiver β1 and β2 can be rewritten as β1E and β2E under only external mismatch for convenience

$$\begin{aligned} {\beta _{1E}} &= \frac{{\pi {A_1}{G_1}}}{{2{V_{\pi 1}}}}\frac{1}{2}{P_1},\\ {\beta _{2E}} &= \frac{{\pi {A_2}{G_2}}}{{2{V_{\pi 2}}}}\frac{1}{4}{P_T}{G_{EDFA2}} \end{aligned}$$
where A1 is the responsivity of the chaotic feedback at the emitter, G1 is amplification factor of RFA1 at the emitter. A2, G2 are corresponding parameters at the receiver.

In addition, there are still some device mismatches. And they will affect chaotic synchronization. Based on [20], synchronization error <ɛE2> under only external mismatch can be expressed as below.

$$\left\langle {\varepsilon_E^2} \right\rangle = \frac{1}{3}{\left( {\frac{{\Delta T}}{\tau }} \right)^2} + \left( {1 - \frac{\pi }{4}} \right){\left( {\frac{{\Delta \tau }}{\tau }} \right)^2} - 2\left( {1 - \frac{\pi }{4}} \right)\frac{{\Delta T}}{\tau }\frac{{\Delta \tau }}{\tau }$$
where ΔT is mismatch of time delay, τ is high cutoff response time, Δτ is mismatch of high cutoff response time.

Based on [20], chaotic mismatch noise < nE2(r)> can be modified as below.

$$\left\langle {n_E^2(r )} \right\rangle = \frac{1}{2}{K_{1E}}^2(r )\left[ {\left\langle {\varepsilon_E^2} \right\rangle + {{({\Delta \phi } )}^2} + \frac{1}{4}{{\left( {\frac{{\Delta {K_E}(r )}}{{{K_{1E}}(r )}}} \right)}^2}} \right]$$
where Δϕ is mismatch of offset phase.

Message signal will also be affected by pointing error. K1d can be modified as K1E(r)d under only external mismatch. And BER model under only external mismatch can be rewritten as BERchaosE.

$$BE{R_{chaosE}} = \frac{1}{2}erfc\left( {\frac{{{K_{1E}}(r )d}}{{2\sqrt {2\left\langle {n_E^2(r )} \right\rangle } }}} \right)$$
So external mismatch caused by pointing error will affect message signal and chaotic mismatch noise, but have no influence on synchronization error.

We have discussed the effect of external mismatch caused by pointing error on BER. Further, internal mismatch will be discussed. The power P1 in formula of β1 will not be affected by pointing error. While the power PT(r) in formula of β2(r) will be affected by pointing error. β1 and β2(r) can be rewritten as β1I and β2I(r) under only internal mismatch for convenience.

$$\begin{aligned} {\beta _{1I}} &= \frac{{\pi {A_1}{G_1}}}{{2{V_{\pi 1}}}}\frac{1}{2}{P_1},\\ {\beta _{2I}}(r) &= \frac{{\pi {A_2}{G_2}}}{{2{V_{\pi 2}}}}\frac{1}{4}{P_T}(r ){G_{EDFA2}} \end{aligned}$$
Based on Eq. (15), the difference between PT and PT(r) will introduce internal mismatch ΔβI(r)=β1I–β2I(r) which will be affected by boresight and jitter. Based on [20], both internal mismatch and device mismatches will affect chaotic synchronization error <ɛ2>. Synchronization error under only internal mismatch can be rewritten as <ɛI2(r)> for convenience.
$$\begin{aligned} \left\langle {\varepsilon_I^2(r)} \right\rangle &= \frac{1}{3}{\left( {\frac{{\Delta T}}{\tau }} \right)^2} + {\left( {\frac{{\Delta {\beta_I}(r)}}{{{\beta_{1I}}}}} \right)^2} + \left( {1 - \frac{\pi }{4}} \right){\left( {\frac{{\Delta \tau }}{\tau }} \right)^2} \\ &- 2\left( {1 - \frac{\pi }{4}} \right)\frac{{\Delta {\beta _I}(r)}}{{{\beta _{1I}}}}\frac{{\Delta \tau }}{\tau } - 2\left( {1 - \frac{\pi }{4}} \right)\frac{{\Delta T}}{\tau }\frac{{\Delta \tau }}{\tau } \end{aligned}$$
To better discuss the effect of internal mismatch, external mismatch is ignored. K1 and K2 can be written as K1I and K2I under only internal mismatch for convenience
$$\begin{aligned} {K_{1I}} &= \frac{1}{4}{G_{EDFA2}}{P_T}\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s},\\ {K_{2I}} &= {P_2}\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s} \end{aligned}$$
Based on [20], chaotic mismatch noise < nI2(r)> will be affected by internal mismatch and can be expressed as
$$\left\langle {n_I^2(r )} \right\rangle = \frac{1}{2}K_{1I}^2\left[ {\left\langle {\varepsilon_I^2(r )} \right\rangle + {{({\Delta \phi } )}^2}} \right]$$
External mismatch is ignored so pointing error will not affect message signal. Message signal K1(r)d under only internal mismatch can be modified as K1Id. BERchaosI(r) under only internal mismatch can be written as below
$$BE{R_{chaosI}}(r )= \frac{1}{2}erfc\left( {\frac{{{K_{1I}}d}}{{2\sqrt {2\left\langle {n_I^2(r )} \right\rangle } }}} \right)$$
In the above part we have discussed the effect of external mismatch and internal mismatch separately. In the real system, these two mismatches should be considered together. Both external and internal mismatches are affected by pointing error. So the current K1(r) of APD1 and current K2 of APD should be expressed as below.
$$\begin{aligned} {K_1}(r) &= \frac{1}{4}{G_{EDFA2}}{P_T}(r)\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s},\\ {K_2} &= {P_2}\frac{{Ge\eta }}{{{h_p}\upsilon }}{T_s} \end{aligned}$$
And chaotic loop gain β1 and β2(r) can be expressed as below
$$\begin{aligned} {\beta _1} &= \frac{{\pi {A_1}{G_1}}}{{2{V_{\pi 1}}}}\frac{1}{2}{P_1},\\ {\beta _2}(r) &= \frac{{\pi {A_2}{G_2}}}{{2{V_{\pi 2}}}}\frac{1}{4}{P_T}(r ){G_{EDFA2}} \end{aligned}$$
Based on Eq. (20) and Eq. (21), external mismatch and internal mismatch can be expressed as below
$$\begin{aligned} \Delta K(r ) &= {K_2} - {K_1}(r ),\\ \Delta \beta (r) &= {\beta _1} - {\beta _2}(r )\end{aligned}$$
Then synchronization error <ɛ2(r)> will be affected by pointing error and should be modified as below
$$\begin{aligned} \left\langle {{\varepsilon^2}(r)} \right\rangle &= \frac{1}{3}{\left( {\frac{{\Delta T}}{\tau }} \right)^2} + {\left( {\frac{{\Delta \beta (r)}}{{{\beta_1}}}} \right)^2} + \left( {1 - \frac{\pi }{4}} \right){\left( {\frac{{\Delta \tau }}{\tau }} \right)^2}\\ &- 2\left( {1 - \frac{\pi }{4}} \right)\frac{{\Delta \beta (r)}}{{{\beta _1}}}\frac{{\Delta \tau }}{\tau } - 2\left( {1 - \frac{\pi }{4}} \right)\frac{{\Delta T}}{\tau }\frac{{\Delta \tau }}{\tau } \end{aligned}$$
Further, both external and internal mismatches have influence on chaotic mismatch noise < n2(r)>, which is
$$\left\langle {{n^2}(r )} \right\rangle = \frac{1}{2}K_1^2(r )\left[ {\left\langle {{\varepsilon^2}(r )} \right\rangle + {{({\Delta \phi } )}^2} + \frac{1}{4}{{\left( {\frac{{\Delta K(r )}}{{{K_1}(r )}}} \right)}^2}} \right]$$
And message signal will be affected by external mismatch and should be equal to K1(r)d. BERchaos(r) affected by external and internal mismatches should be modified as below
$$BE{R_{chaos}}(r )= \frac{1}{2}erfc\left( {\frac{{u(r )}}{{2\sqrt 2 }}} \right) = \frac{1}{2}erfc\left( {\frac{{{K_1}(r )d}}{{2\sqrt {2\left\langle {{n^2}(r )} \right\rangle } }}} \right)$$
When deviation r between beam center and detector center affected by pointing error changes continuously, the BERchaos(r) given above varies continuously as well. So the average BER of the system affected by pointing error can be expressed as integral of BERchaos(r) and PDF of Aj and σj which is givgen by Eq. (4). The ensemble average BER should be modified as
$$\begin{aligned} BER &= \int_0^{ + \infty } {BE{R_{chaos}}(r )\rho (r )dr} \\ &= \int_0^{ + \infty } {\frac{1}{2}erfc\left( {\frac{{{K_1}(r )d}}{{2\sqrt {2\left\langle {{n^2}(r )} \right\rangle } }}} \right)\frac{r}{{\sigma _s^2}}\exp \left( { - \frac{{{r^2} + {A^2}}}{{2\sigma_s^2}}} \right){I_0}\left( {\frac{{rA}}{{\sigma_s^2}}} \right)dr} \end{aligned}$$
We have discussed the BER under only external mismatch, only internal mismatch and both external and internal mismatches. Further, simulation results and discussion will be given based on analysis above.

3. Simulation results and discussion

Based on the modified BER formulae in the second chapter, the effect of different boresight and jitter on external mismatch and internal mismatch will be analyzed in detail. Then the effect of external mismatch and internal mismatch on BER and system performance will be discussed. The numerical simulation results are based on these parameters: power of the emitter P1=4 mW, power of the receiver P2=2 mW, gain factor of EDFA1 GEDFA1=2000, loss efficiency αloss=1, receiving aperture Dr=0.6 m, masking efficiency d=1.2, divergence angle θ=40 µrad, link length L=38000 km, APD gain factor G=100, quantum efficiency η=0.75, data rate is 1 Gb/s, time delay mismatch ΔT=1 ps, high cutoff response time τ=25 ps, high cutoff response time mismatch rate Δτ/τ=0.01, offset phase mismatch Δ<ι>ϕ = 0.02 rad.

Pointing error effect can be divided into boresight and jitter, of which Aj and σj will be used to depict angular offset respectively. The effect of boresight will be discussed firstly and the effect of jitter will be discussed secondly. At last the joint effect of both will be discussed. Pointing error effect will bring external mismatch and internal mismatch. To figure out which mismatch will deteriorate the performance of space chaos laser system more, external mismatch and internal mismatch will be discussed separately.

3.1 External mismatch and internal mismatch affected by boresight

We firstly discuss the effect of boresight Aj on external mismatch under jitter σj=3 µrad. Specifically, boresight Aj is set to be 1 µrad, 3 µrad, and 5 µrad separately. Aj = 1 µrad corresponds to a relatively faint boresight while Aj = 5 µrad corresponds to a relatively strong boresight. Internal mismatch is ignored when external mismatch is discussed. Figure 3(a) shows probability distribution of current amplitude K. The blue line, green line and purple line represent corresponding PDF of K1(r) with Aj set to be 1 µrad, 3 µrad, and 5 µrad separately. When Aj becomes larger, current amplitude K1 is more likely to descrease and PDF curve will shift left. On the contrary, the current amplitude K2 detected by APD2 is a fixed value based on Eq. (10). The PDF of current amplitude K2 can be considered as a δ-function. Average K1 under different Aj is used to discuss the effect of Aj on current amplitude K1. The average K1 under different Aj is calculated by integral of current amplitude K1. In detail, the average K1i can be calculated by Eq. (27) as below.

$$Average \;{K_{1i}} = \int_0^{ + \infty } {{K_{1i}}(r )\rho (r )dr}, i = 1,2,3$$
It should be noted that average K1i is a fixed value under the fixed Aj. So the PDF of average K1i will also be a δ-function. Average current amplitudes K1i will be K11, K12 and K13 with Aj set to be 1 µrad, 3 µrad, and 5 µrad respectively. Average external mismatch rate can be calculated as ΔK/K=(K2-K1i)/K2 (i=1,2,…,n). The red line depicts the relationship between external mismatch rate and current amplitude K in Fig. 3(a).

 figure: Fig. 3.

Fig. 3. (a) External mismatch rate ΔK/K=(K2-K1)/K2 versus current amplitude K (left ordinate) under jitter σj=3µrad and different boresight Aj. PDF versus current amplitude K (right ordinate). (b) External mismatch rate versus BER under jitter σj=3µrad.

Download Full Size | PDF

When there is no pointing error effect, K1 should be equal to K2 and the two δ-function curves will be coincident. External mismatch rate will be equal to zero. The δ-function of average K1 will shift left under a greater Aj. At the same time, external mismatch rate ΔK will become larger as well. The average external mismatch rate can be directly found as the intersection of the external mismatch rate curve and average K1 δ-function with the property of the δ-function. Average external mismatch rates are 7.2%, 10.2% and 16.5% with Aj set to be 1 µrad, 3 µrad, and 5 µrad. The increase rates of average external rates are not evenly when Aj increases evenly. The increase rate when Aj changes from 1 µrad to 3 µrad is less than the rate when Aj changes from 3 µrad to 5 µrad.

To better discuss the effect of external mismatch on BER, the relationship between external mismatch rate and BER is depicted in Fig. 3(b). BER will deteriorate when external mismatch becomes larger. When Aj is set to be 1 µrad, 3 µrad, and 5 µrad, log(BER) will be −5.96, −5.19, −4.42 respectively. That is to say, BER will deteriorate 1.54 dB when external mismatch rates increase from 7.2% to 16.5%. So larger external mismatch is induced and space chaos laser communication system will be affected as Aj increases.

Then internal mismatch is discussed with external mismatch ignored. Boresight Aj is set to be 1 µrad, 3 µrad, and 5 µrad as well. The results are showed in Fig. 4. The loop gain β1 is a fixed value at the emitter based on Eq. (15). So the PDF of loop gain β1 can be considered as a δ-function. While the loop gain β2(r) at the receiver is the function of radial position r based on Eq. (15). The blue line, green line and purple line represent corresponding PDF of β2(r) with Aj set to be 1 µrad, 3 µrad, and 5 µrad respectively. When Aj becomes larger, β2 is more likely to decrease and PDF curve will shift left. Average β2 under different Aj is used to discuss the effect of Aj on loop gain β2(r). The calculation of average β2i is similar to average current amplitude K1i and is given by Eq. (28) as below.

$$Average \;{\beta _{2i}} = \int_0^{ + \infty } {{\beta _{2i}}(r )\rho (r )dr}, i = 1,2,3$$
Average loop gains β2 are β21, β22 and β23 respectively with Aj set to be 1 µrad, 3 µrad, and 5 µrad. And average internal mismatch rate can be calculated as Δβ/β=(β1-β2i)/β1 (i=1,2,…,n).

 figure: Fig. 4.

Fig. 4. (a) Internal mismatch rate Δβ/β=(β1-β2)/β1 versus loop gain β (left ordinate) under jitter σj=3µrad and different boresight Aj. PDF versus loop gain β (right ordinate). (b) Internal mismatch rate versus BER under jitter σj=3µrad.

Download Full Size | PDF

The red line depicts the relationship between internal mismatch rate and loop gains β in Fig. 4(a). When there is no pointing error effect, β2 should be equal to β1 and the two δ-function curves will be coincident. And internal mismatch rate is equal to zero. The δ-function of average β2 will shift left under a greater Aj. At the same time, internal mismatch rate will become larger as well. And the average internal mismatch rate can be directly found as the intersection of the red internal mismatch rate curve and average β2 δ-function. Average internal mismatch rates are 7.2%, 10.2% and 16.5% with Aj set to be 1 µrad, 3 µrad, and 5 µrad. The relationship between internal mismatch rate and BER is depicted in Fig. 4(b). When Aj is set to be 1 µrad, 3 µrad, and 5 µrad, log(BER) will be −3.54, −3, −2.45 respectively. That is to say, BER will deteriorate 1.09 dB when internal mismatch rates increase from 7.2% to 16.5%.

The influences of only external or internal mismatch are discussed above. However, both external mismatch and internal mismatch have influence on BER in real space chaos laser communication system. So the relationship between boresight Aj and BER under only external mismatch, only internal mismatch and both is showed in Fig. 5. BER will all deteriorate under different mismatch when Aj increases from 1 µrad to 7 µrad. More specifically, log(BER) will deteriorate from −5.96 to −3.73 with only external mismatch, from −3.54 to −1.99 with only internal mismatch and from −3.26 to −1.81 with both. To illustrate the effect of mismatches on BER more clearly, the mismatch rate and log(BER) are marked respectively on curves with Aj set to be 1 µrad, 3 µrad, and 5 µrad. Though the value of internal mismatch rate is equal to external mismatch rate, the two kinds of mismatches have different influence on space chaos communication system. The attenuation of BER with only internal mismatch, which is 1.09 dB, is slightly less than the value 1.54 dB with only external mismatch. However, the BER order of magnitude with only internal mismatch is roughly 2 dB larger than the BER order of magnitude with only external mismatch when Aj increases from 1 µrad to 5 µrad. When joint effect of both mismatches taking into account, log(BER) will deteriorate from −3.26 with 7.2% external and internal mismatches to −2.24 with 16.5% external and internal mismatches. The curve of BER with only internal mismatch is closer to the curve with two mismatches than the curve of BER with only external mismatch.

 figure: Fig. 5.

Fig. 5. Boresight Aj versus BER under different mismatch. The mismatch rate and log(BER) are marked respectively on curves with Aj set to be 1 µrad, 3 µrad, and 5 µrad.

Download Full Size | PDF

To explain which mismatch will affect the system more under different boresight, we further calculate average signal K1(r)d (only external mismatch) or signal K1d (only internal mismatch), average chaotic mismatch noise < n(r)2> and average signal-to-noise ratio (SNR). Boresight Aj will set to be 1 µrad, 3 µrad and 5 µrad. The data is shown in the table below.

Pointing error will affect the signal and noise at the receiver then affect SNR. BER will be further affected by SNR. As for signal, the increase of pointing error will deteriorate signal with only external mismatch based on Eq. (10) and Eq. (14). And the increase of pointing error will not affect the signal with only internal mismatch based on Eq. (17) and Eq. (19). Comparing these two mismatches effects, the signal with only external mismatch is slightly less than the signal with only internal mismatch under fixed jitter and different boresight. As for noise, the increase of pointing error will deteriorate signal with only external mismatch or internal mismatch based on Eq. (13) and Eq. (18). However, the noise with only internal mismatch is much bigger than the noise with only external mismatch in Table 1. Therefore, we can see from the Table 1 that the SNR with only external mismatch is 39.09 and the SNR with only internal mismatch is 29.54 when Aj=1 µrad. And the SNR with only external mismatch is 25.54 and the SNR with only internal mismatch is 16.54 when Aj=5 µrad.

Tables Icon

Table 1. Average signal (only external mismatch) / signal (only internal mismatch), average noise and average SNR data under fixed jitter σj and different boresight Aj.

View Table | View all tables in this article

It is obvious that SNR with only external mismatch is larger than that with only internal mismatch. That means BER with only internal mismatch is worse than that with only external mismatch. Therefore, internal mismatch will affect system performance more than external mismatch under fixed jitter and different boresight. In the system design, internal mismatch caused by pointing error should be taken into account.

3.2 External mismatch and internal mismatch influenced by jitter

The effect of boresight is discussed above, the influence of jitter will be discussed likewise as below. Jitter σj is set to be 1 µrad, 3 µrad, and 5 µrad with fixed boresight Aj = 3 µrad. σj = 1 µrad corresponds to a relatively faint jitter while σj = 5 µrad corresponds to a relatively strong jitter. External mismatch will be discussed firstly with internal mismatch ignored. Figure 6(a) shows the relationship between PDF and current amplitude K. When σj becomes larger, current amplitude K1 is more likely to decrease and PDF curve will be less concentrated. Average external mismatch rates are 4.9%, 10.2% and 20.7% respectively with σj set to be 1 µrad, 3 µrad, and 5 µrad. The increase rates of average external rates are not even when σj increases evenly. When σj increases from 1µrad to 3 µrad and 3 µrad to 5 µrad, the increase mismatch rates are 5.3% and 10.5% respectively. The increase rates caused by σj are larger than 3% and 6.3% caused by Aj.

 figure: Fig. 6.

Fig. 6. (a) External mismatch rate ΔK/K=(K2-K1)/K2 versus current amplitude K (left ordinate) under boresight Aj=3µrad and different jitter σj. PDF versus current amplitude K (right ordinate). (b) External mismatch rate versus BER under Aj=3µrad.

Download Full Size | PDF

To better discuss the effect of external mismatch on BER, the relationship between external mismatch rate and BER is depicted in Fig. 6(b). When σj is set to be 1 µrad, 3 µrad, and 5 µrad, log(BER) will be −14.36, −5.19, −3.38 respectively. When external mismatch rates increase from 4.9% to 20.7%, BER will deteriorate 10.98 dB. Meanwhile, BER will deteriorate 1.54 dB with only external mismatch when boresight increases from 1 µrad to 5 µrad based on the conclusion of 3.1 section. BER will deteriorate more when jitter increases than that (1.54 dB) when boresight increases. Larger external mismatch will be induced and space chaos laser communication system will be greatly affected when σj increases.

Then internal mismatch is discussed with external mismatch ignored. As showed in Fig. 7, average internal mismatch rates are 4.9%, 10.2% and 20.7%, which are the same as external mismatch rates in Fig. 6(a), with jitter σj set to be 1 µrad, 3 µrad, and 5 µrad. However, they have different influence on BER. The relationship between internal mismatch rate and BER is depicted in Fig. 7(b). When σj is set to be 1 µrad, 3 µrad, and 5 µrad, log(BER) will be −7.9, −3, −1.95 respectively. That is to say, BER will deteriorate 5.95 dB when internal mismatch rate increases from 4.9% to 20.7%. Meanwhile, BER will deteriorate 10.98 dB with only external mismatch when boresight increases from 1 µrad to 5 µrad in Fig. 6(b). It should be noted that the deterioration of BER with only internal mismatch (5.95 dB) is greatly less than that with only external mismatch (10.98 dB).

 figure: Fig. 7.

Fig. 7. (a) Internal mismatch rate Δβ/β=(β1-β2)/β1 versus loop gain β (left ordinate) under boresight Aj=3µrad and different jitter σj. PDF versus loop gain β (right ordinate). (b) Internal mismatch rate versus BER under Aj=3µrad.

Download Full Size | PDF

Actually, both external mismatch and internal mismatch should be considered in a real system. The relationship between jitter σj and BER with only external mismatch, only internal mismatch and both is showed in Fig. 8. BER will deteriorate 10.98 dB with 15.8% external mismatch rate and log(BER) will be influenced around −14.4 to −3.4 when σj is set to be 1 µrad to 5 µrad. Similarly, BER will deteriorate 5.95 dB with 15.8% internal mismatch rate and log(BER) will be influenced around −8 to −2 when σj is set to be 1 µrad to 5 µrad. The BER order of magnitude with only internal mismatch is 6 dB larger than BER order of magnitude with only external mismatch under σj = 1 µrad and 1.28 dB larger under σj = 5 µrad. The division between BERs with only internal mismatch, only external mismatch and both will be less when σj is set to be larger. BER order of magnitude will be larger with the joint effect of external and internal mismatches. And log(BER) will deteriorate from −7.16 with 4.9% external and internal mismatches to −1.8 with 16.5% external and internal mismatches when σj increases from 1 µrad to 5 µrad.

 figure: Fig. 8.

Fig. 8. Jitter σj versus BER under different mismatch with boresight Aj=3µrad. The mismatch rate and log(BER) are marked respectively on curves with σj set to be 1 µrad, 3 µrad, and 5 µrad.

Download Full Size | PDF

To explain which mismatch will affect the system more under different jitter, we further calculate average signal (only external mismatch) or signal (only internal mismatch), average chaotic mismatch noise and average SNR. The data is shown in the table below.

Similar to Table 1, Table 2 also shows that the SNR of system with only external mismatch is better than that with only internal mismatch under the same pointing error. It means internal mismatch will also affect system performance more than external mismatch under fixed boresight and different jitter. Based on the results of Table 1 and Table 2, it indicates that internal mismatch will always affect system performance more than external mismatch regardless of different boresight or jitter.

Tables Icon

Table 2. Average signal (only external mismatch) / signal (only internal mismatch), average noise and average SNR data under fixed boresight Aj and different jitter σj.

View Table | View all tables in this article
Tables Icon

Table 3. Average distance deviation of the beam center r under different jitter σj and fixed boresight Aj.

View Table | View all tables in this article
Tables Icon

Table 4. Average distance deviation of the beam center r under different boresight Aj and fixed jitter σj.

View Table | View all tables in this article

Boresight component of pointing error can be controlled and assumed to be zero in some cases at present. If we consider zero boresight error with A=0, the probability function of nonzero boresight pointing error of Eq. (4) will be simplified to the zero boresight pointing error as below.

$$\rho (r )= \frac{r}{{\sigma _s^2}}\exp \left( { - \frac{{{r^2}}}{{2\sigma_s^2}}} \right)$$
The Eq. (29) is also often used in the field of space laser communication when boresight component can be controlled better [21]. To show the differences between zero boresight and nonzero boresight, we analyze the relationship between jitter σj and BER under boresight Aj = 0 and Aj = 3 µrad. Both external mismatch and internal mismatch are considered in Fig. 9. BER will deteriorate 10.72 dB with 17.04% external mismatch and internal mismatch rates and log(BER) will be influenced around −12.7 to −2 when σj is set to be 1 µrad to 5 µrad with zero boresight. Similarly, BER will deteriorate 5.36 dB with 15.8% external mismatch and internal mismatch rates and log(BER) will be influenced around −7 to −1.8 when σj is set to be 1 µrad to 5 µrad with boresight Aj = 3 µrad. These results indicate that the performance of space chaos laser communication system will be improved to some extent due to the zero boresight. Therefore, controlling boresight component closer to zero in real system design will be helpful for optimizing the performance of space chaos laser communication system.

 figure: Fig. 9.

Fig. 9. Jitter σj versus BER under different boresight. The mismatch rate and log(BER) are marked respectively on curves with σj set to be 1 µrad, 3 µrad, and 5 µrad.

Download Full Size | PDF

In addition, it should be noted that the effort to make boresight closer to zero will improve the system performance more obviously in the case of less jitter shown in Fig. 9. However, when the jitter is large, such as 7 µrad or more than, controlling boresight from 3 µrad to zero will be helpless for system design. That also means the size of jitter should be fully considered when we improve the system performance by reducing boresight as much as possible.

The influences of boresight and jitter have been discussed separately above. Further, the joint effect of boresight and jitter will be discussed. Three-dimensional histogram is used to depict the effect of boresight Aj and jitter σj with both external and internal mismatches more clearly. BER versus Aj and σj is showed in Fig. 10. Jitter σj is set from 1 µrad to 10 µrad and boresight Aj is set from 0 µrad to 10 µrad. Shown in Fig. 10, the space chaos laser communication system will work best with zero Aj and less σj. BER will deteriorate greatly (> 5 dB) when σj increases from 1 µrad to 2 µrad with Aj = 1 µrad, and BER will deteriorate 2 dB when Aj increases from 1 µrad to 2 µrad with σj = 1 µrad.

 figure: Fig. 10.

Fig. 10. BER versus boresight Aj and jitter σj.

Download Full Size | PDF

Then log(BER) will be −1.085 with Aj = 1 µrad and σj = 10 µrad, while log(BER) will be −1.526 with Aj = 10 µrad and σj = 1 µrad. As for Aj = 10 µrad and σj = 10 µrad, log(BER) will be −0.948. When jitter or boresight is strong, the performance of space chaos communication system will be greatly affected.

In the real system, both internal mismatch and external mismatch should be considered. The relationship of signal and noise will affect BER finally. Pointing error will not affect signal or noise directly. In fact, pointing error will affect average distance deviation of the beam center r then affect signal and noise. Therefore, we calculate average distance deviation of the beam center r under different jitter and fixed boresight shown in Table 3.

Then we calculate average distance deviation of the beam center r under different boresight and fixed jitter shown in Table 4.

Average distance deviation of the beam center will increase 55.98 when jitter increases from 1 µrad to 3 µrad and increase 29.72 when boresight increases from 1 µrad to 3 µrad. Average distance deviation of the beam center will increase 82.55 when jitter increases from 3 µrad to 5 µrad and increase 51.60 when boresight increases from 3 µrad to 5 µrad. In conclusion, the increase of jitter will make beam center deviate more. The above results indicate that the increase of jitter σj will make BER deteriorate more than the increase of boresight Aj. As a result, the decrease of σj can greatly improve the performance of space chaos laser communication system.

As we all known, pointing error has been widely studied by researchers in traditional space laser communication system. Reference [11] reported that diversity order of the free space optical communication system over the composite lognormal fading channel will not be affected by boresight component. Reference [12] investigated the performance of free space optical links over exponentiated Weibull turbulence channel with nonzero boresight pointing error and reported that the diversity gain is different between the zero boresight and nonzero boresight cases. Reference [22] developed a closed-form expression to calculate channel gain for inter-satellite links and reported that the diversity gain is related to the ratio of equivalent beam radius to the standard jitter at the receiver. However, the previous works have not compared the specific impact of jitter and boresight on the system performance. Therefore, the result that jitter will affect BER more than boresight is useful for optimizing space chaos laser communication system design, and may also offer a reference for traditional space laser communication system.

In summary, the increase of boresight and jitter will increase external and internal mismatches then deteriorate the performance of communication system. Jitter will affect BER more than boresight. Internal mismatch will affect the performance of space chaos system more than external mismatch. The quality of space chaos laser communication system will decrease if the external and internal mismatches are not considered in system design.

4. Conclusion

In summary, the effect of pointing error on performance of space chaos laser communication is analyzed in detail. The effect of pointing error on performance of space chaos laser communication system is different from traditional space laser communication system. Pointing error will make receiving power attenuate then bring the mismatches of amplitude current and optoelectronic oscillator gain to chaotic synchronous demodulation process. We name the mismatch of amplitude current as external mismatch caused by pointing error and we name the mismatch of optoelectronic oscillator gain as internal mismatch caused by pointing error. We conduct bit error rate (BER) analysis with external mismatch and internal mismatch caused by pointing error of space chaos laser communication system. Considering that pointing error is composed of boresight and jitter, we analyze the effect of boresight and jitter on the BER of space chaos laser communication system respectively based on the derived BER formulae. The results of numerical simulation indicate that the performance of system will be affected more by internal mismatch than external mismatch under different boresight or jitter. The effect of internal mismatch caused by pointing error should be emphatically considered in system design. Further, the joint effect of boresight and jitter is discussed. BER will deteriorate greatly (> 5 dB) when jitter σj increases from 1 µrad to 2 µrad with boresight Aj = 1 µrad. While BER will deteriorate 2 dB when boresight Aj increases from 1 µrad to 2 µrad with jitter σj = 1 µrad. That indicates that the system will be affected more by jitter than boresight. So the decrease of jitter can greatly improve the performance of space chaos laser communication system. All in all, the increase of boresight and jitter will increase external and internal mismatches then deteriorate the performance of system. Jitter will affect BER more than boresight and internal mismatch will affect the performance of system more than external mismatch. The results of this paper are significant for optimizing space chaos laser communication system design.

Science and technology make our life better. We hope Wuhan will recover soon. Love you, Wuhan, Love you, China.

Funding

Joint Funds of Space Science and Technology (6141B060307); Six Talent Peaks Project in Jiangsu Province (KTHY-003); Suzhou Technology Innovative for Key Industries Program of China (SYG201729); National Natural Science Foundation of China (61205045); Fundamental Research Funds for the Central Universities (021314380152).

Disclosures

The authors declare no conflicts of interest.

References

1. G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef]  

2. X. F. Li, W. Pan, B. Luo, and D. Ma, “Chaos synchronization and communication of cascade-coupled semiconductor lasers,” J. Lightwave Technol. 24(12), 4936–4945 (2006). [CrossRef]  

3. W. L. Zhang, W. Pan, B. Luo, X. H. Zou, M. Y. Wang, and Z. Zhou, “Chaos synchronization communication using extremely unsymmetrical bidirectional injections,” Opt. Lett. 33(3), 237–239 (2008). [CrossRef]  

4. Y. Z. Liu, Y. Y. Xie, Y. C. Ye, J. P. Zhang, S. J. Wang, Y. Liu, G. F. Pan, and J. L. Zhang, “Exploiting optical chaos with time-delay signature suppression for long-distance secure communication,” IEEE Photonics J. 9(1), 1–12 (2017). [CrossRef]  

5. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990). [CrossRef]  

6. J. P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80(10), 2249–2252 (1998). [CrossRef]  

7. S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001). [CrossRef]  

8. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]  

9. C. Williams, “Chaotic communications over radio channels,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 48(12), 1394–1404 (2001). [CrossRef]  

10. N. F. Rulkov, M. A. Vorontsov, and L. Illing, “Chaotic free-space laser communication over a turbulent channel,” Phys. Rev. Lett. 89(27), 277905 (2002). [CrossRef]  

11. F. Yang, J. L. Chen, and T. A. Tsiftsis, “Free-space optical communication with nonzero boresight pointing errors,” IEEE Trans. Commun. 62(2), 713–725 (2014). [CrossRef]  

12. X. Yi and M. W. Yao, “Free-space communications over exponentiated Weibull turbulence channels with nonzero boresight pointing errors,” Opt. Express 23(3), 2904–2917 (2015). [CrossRef]  

13. H. Alquwaiee, H. C. Yang, and M. S. Alouini, “On the asymptotic capacity of dual-aperture FSO systems with generalized pointing error model,” IEEE Trans. Wireless Commun. 15(9), 6502–6512 (2016). [CrossRef]  

14. R. Boluda-Ruiz, A. García-Zambrana, C. Castillo-Vázquez, B. Castillo-Vázquez, and S. Hranilovic, “Outage performance of exponentiated weibull FSO links under generalized pointing errors,” J. Lightwave Technol. 35(9), 1605–1613 (2017). [CrossRef]  

15. Gastaud Gallagher, “NHR,” Multi-gigahertz encrypted communication using electro-optical chaos cryptography (Diss. Georgia Institute of Technology, 2007).

16. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009). [CrossRef]  

17. M. Li, Y. F. Hong, S. Wang, Y. J. Song, and X. Sun, “Radiation-induced mismatch effect on performances of space chaos laser communication systems,” Opt. Lett. 43(20), 5134–5137 (2018). [CrossRef]  

18. M. Li, Y. F. Hong, Y. J. Song, and X. P. Zhang, “Effect of controllable parameter synchronization on the ensemble average bit error rate of space-to-ground downlink chaos laser communication system,” Opt. Express 26(3), 2954–2964 (2018). [CrossRef]  

19. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, vol. 1 (SPIE Press, 2005).

20. Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Mismatch-induced bit error rate in optical chaos communications using semiconductor lasers with electrooptical feedback,” IEEE J. Quantum Electron. 41(2), 156–163 (2005). [CrossRef]  

21. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]  

22. T. Y. Song, Q. Wang, M. W. Wu, T. Ohtsuki, M. Gurusamy, and P. Y. Kam, “Impact of pointing errors on the error performance of intersatellite laser communications,” J. Lightwave Technol. 35(14), 3082–3091 (2017). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)
Fig. 1.
Fig. 1. Space chaos laser communication system with internal and external mismatches caused by pointing error. Pointing error has no influence on chaotic carrier generation loop (optoelectronic feedback Loop 1) at the emitter, while pointing error will be introduced to chaotic carrier generation loop (optoelectronic feedback Loop 2). The difference in feedback structure will introduce mismatch to internal gain of chaotic structure which we name the mismatch as internal mismatch. The power generated by LD2 will not propagate in space channel and is a fixed value. And the power generated by LD1 will be affected by pointing error. This mismatch caused by pointing error will occur before APD1 and APD2 (circle 1 and 2) and we name the mismatch as external mismatch.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Pointing error principle. (a) PDF of boresight Aj. (b) PDF of jitter σj.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) External mismatch rate ΔK/K=(K2-K1)/K2 versus current amplitude K (left ordinate) under jitter σj=3µrad and different boresight Aj. PDF versus current amplitude K (right ordinate). (b) External mismatch rate versus BER under jitter σj=3µrad.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. (a) Internal mismatch rate Δβ/β=(β1-β2)/β1 versus loop gain β (left ordinate) under jitter σj=3µrad and different boresight Aj. PDF versus loop gain β (right ordinate). (b) Internal mismatch rate versus BER under jitter σj=3µrad.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Boresight Aj versus BER under different mismatch. The mismatch rate and log(BER) are marked respectively on curves with Aj set to be 1 µrad, 3 µrad, and 5 µrad.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. (a) External mismatch rate ΔK/K=(K2-K1)/K2 versus current amplitude K (left ordinate) under boresight Aj=3µrad and different jitter σj. PDF versus current amplitude K (right ordinate). (b) External mismatch rate versus BER under Aj=3µrad.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. (a) Internal mismatch rate Δβ/β=(β1-β2)/β1 versus loop gain β (left ordinate) under boresight Aj=3µrad and different jitter σj. PDF versus loop gain β (right ordinate). (b) Internal mismatch rate versus BER under Aj=3µrad.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. Jitter σj versus BER under different mismatch with boresight Aj=3µrad. The mismatch rate and log(BER) are marked respectively on curves with σj set to be 1 µrad, 3 µrad, and 5 µrad.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. Jitter σj versus BER under different boresight. The mismatch rate and log(BER) are marked respectively on curves with σj set to be 1 µrad, 3 µrad, and 5 µrad.

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. BER versus boresight Aj and jitter σj.

View in Article | Download Full Size | PDF

Tables (4)

Tables Icon

Table 1. Average signal (only external mismatch) / signal (only internal mismatch), average noise and average SNR data under fixed jitter σj and different boresight Aj.

View Table | View all tables in this article
Tables Icon

Table 2. Average signal (only external mismatch) / signal (only internal mismatch), average noise and average SNR data under fixed boresight Aj and different jitter σj.

View Table | View all tables in this article
Tables Icon

Table 3. Average distance deviation of the beam center r under different jitter σj and fixed boresight Aj.

View Table | View all tables in this article
Tables Icon

Table 4. Average distance deviation of the beam center r under different boresight Aj and fixed jitter σj.

View Table | View all tables in this article

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

P x = 1 4 G E D F A 2 P T cos 2 ( π 2 V π 1 x ( t ) + ϕ 1 ) + 1 4 G E D F A 2 d P T m ( t )
P y = P 2 cos 2 ( π 2 V π 2 y ( t ) + ϕ 2 )
P T ( r ) = P T exp ⁡ ( − 2 r 2 W 2 )
ρ ( r ) = r σ s 2 exp ⁡ ( − r 2 + A 2 2 σ s 2 ) I 0 ( r A σ s 2 )
P x = 1 4 G E D F A 2 P T ( r ) cos 2 ( π 2 V π 1 x ( t ) + ϕ 1 ) + 1 4 G E D F A 2 d P T ( r ) m ( t )
P y = P 2 cos 2 ( π 2 V π 2 y ( t , r ) + ϕ 2 )
S x = K 1 ( r ) { cos 2 ( π 2 V π 1 x ( t ) + ϕ 1 ) + d m ( t ) }
S y = K 2 cos 2 ( π 2 V π 2 y ( t , r ) + ϕ 2 )
B E R = 1 2 e r f c ( u 2 2 ) = 1 2 e r f c ( K 1 d 2 2 ⟨ n 2 ⟩ )
K 1 E ( r ) = 1 4 G E D F A 2 P T ( r ) G e η h p υ T s , K 2 E = P 2 G e η h p υ T s
β 1 E = π A 1 G 1 2 V π 1 1 2 P 1 , β 2 E = π A 2 G 2 2 V π 2 1 4 P T G E D F A 2
⟨ ε E 2 ⟩ = 1 3 ( Δ T τ ) 2 + ( 1 − π 4 ) ( Δ τ τ ) 2 − 2 ( 1 − π 4 ) Δ T τ Δ τ τ
⟨ n E 2 ( r ) ⟩ = 1 2 K 1 E 2 ( r ) [ ⟨ ε E 2 ⟩ + ( Δ ϕ ) 2 + 1 4 ( Δ K E ( r ) K 1 E ( r ) ) 2 ]
B E R c h a o s E = 1 2 e r f c ( K 1 E ( r ) d 2 2 ⟨ n E 2 ( r ) ⟩ )
β 1 I = π A 1 G 1 2 V π 1 1 2 P 1 , β 2 I ( r ) = π A 2 G 2 2 V π 2 1 4 P T ( r ) G E D F A 2
⟨ ε I 2 ( r ) ⟩ = 1 3 ( Δ T τ ) 2 + ( Δ β I ( r ) β 1 I ) 2 + ( 1 − π 4 ) ( Δ τ τ ) 2 − 2 ( 1 − π 4 ) Δ β I ( r ) β 1 I Δ τ τ − 2 ( 1 − π 4 ) Δ T τ Δ τ τ
K 1 I = 1 4 G E D F A 2 P T G e η h p υ T s , K 2 I = P 2 G e η h p υ T s
⟨ n I 2 ( r ) ⟩ = 1 2 K 1 I 2 [ ⟨ ε I 2 ( r ) ⟩ + ( Δ ϕ ) 2 ]
B E R c h a o s I ( r ) = 1 2 e r f c ( K 1 I d 2 2 ⟨ n I 2 ( r ) ⟩ )
K 1 ( r ) = 1 4 G E D F A 2 P T ( r ) G e η h p υ T s , K 2 = P 2 G e η h p υ T s
β 1 = π A 1 G 1 2 V π 1 1 2 P 1 , β 2 ( r ) = π A 2 G 2 2 V π 2 1 4 P T ( r ) G E D F A 2
Δ K ( r ) = K 2 − K 1 ( r ) , Δ β ( r ) = β 1 − β 2 ( r )
⟨ ε 2 ( r ) ⟩ = 1 3 ( Δ T τ ) 2 + ( Δ β ( r ) β 1 ) 2 + ( 1 − π 4 ) ( Δ τ τ ) 2 − 2 ( 1 − π 4 ) Δ β ( r ) β 1 Δ τ τ − 2 ( 1 − π 4 ) Δ T τ Δ τ τ
⟨ n 2 ( r ) ⟩ = 1 2 K 1 2 ( r ) [ ⟨ ε 2 ( r ) ⟩ + ( Δ ϕ ) 2 + 1 4 ( Δ K ( r ) K 1 ( r ) ) 2 ]
B E R c h a o s ( r ) = 1 2 e r f c ( u ( r ) 2 2 ) = 1 2 e r f c ( K 1 ( r ) d 2 2 ⟨ n 2 ( r ) ⟩ )
B E R = ∫ 0 + ∞ B E R c h a o s ( r ) ρ ( r ) d r = ∫ 0 + ∞ 1 2 e r f c ( K 1 ( r ) d 2 2 ⟨ n 2 ( r ) ⟩ ) r σ s 2 exp ⁡ ( − r 2 + A 2 2 σ s 2 ) I 0 ( r A σ s 2 ) d r
A v e r a g e K 1 i = ∫ 0 + ∞ K 1 i ( r ) ρ ( r ) d r , i = 1 , 2 , 3
A v e r a g e β 2 i = ∫ 0 + ∞ β 2 i ( r ) ρ ( r ) d r , i = 1 , 2 , 3
ρ ( r ) = r σ s 2 exp ⁡ ( − r 2 2 σ s 2 )
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use