1. Introduction

Optical fibers are widely used for transmission of information. Ideally, information carried by optical pulses would be transmitted undamaged. In reality, however, various impairments emerge naturally in the transmission media, which can lead to unrecoverable information losses. In high-bit rate optical fiber systems, the noise induced by optical amplifiers and polarization mode dispersion (PMD) caused by the birefringent disorder [1]–[7], are the two major sources of the transmission failure [8]–[11]. It is essential to quantify the achievable transmission rates in the contemporary communication systems.

The bit-error-rate (BER), the probability of detecting an error, serves to benchmark the reliability of the information transmission systems, as for a fixed fiber disorder, BER characterizes the transmission performance. Due to the thermodynamic fluctuations of pressure and temperature, however, the birefringent disorder undergoes slow, but significant changes in time [12, 13]. This, in turn, results in the fluctuations of BER and thus, we need to study the probability distribution of BER for precise evaluation of the system performance.

Considerable theoretical work has been devoted to the statistics of BER [14]–[18]. The main results consist in the emergence of an extended tail in the probability distribution function (PDF) of BER, the evidence of extreme information outage. We note that this finding was purely based on analytical investigation, not accompanied by numerical simulations. In addition, a fixed threshold was employed for determining whether the output signal is corrupted. Specifically, the decision level was set as a half of the input signal intensity. At the receiver, if the output signal intensity is below the decision level, the transmission system is considered to be impaired. In this case, if the output signal contains more intensity than the input pulse, it is not taken as a distorted signal. As a result, the BER for detecting the state “0” (empty time slot) as “1”(occupied slot) does not depend on the birefringent disorder. In practice, however, the technique for measuring BER is somewhat different. Generally, the decision threshold is dynamically optimized in order to attain the lowest error probability (see e.g., [19]–[23] and the references therein). In this case, the mathematical formulation for BER needs to be modified from the previous consideration.

In this paper, we investigate the BER based on the dynamically adjusted decision level and present the analytical form of the PDF of BER for both weak and strong birefringent disorder. We also conduct numerical simulations to achieve the PDF of BER for a given initial pulse profile, birefringence realizations, and the amplifier noise characteristics. For collecting the statistics of BER numerically, we first propagate an optical pulse through fiber for a given profile of birefringence and the additive noise. After retrieving the shape of the pulse at the receiver, we calculate the BER based on the dynamically adjusted threshold. Repeating this procedure for independent realizations of birefringent disorder, we obtain the PDF of BER. The results of numerical simulations agree qualitatively with the predictions in [14]–[18], and confirm our quantitative theoretical prediction, namely, the emergence of a long algebraic regime in the PDF for the weak birefringent disorder. An extended algebraic tail in the PDF of BER demonstrates that even weak fluctuation of birefringent disorder can result in strong deviation of BER from its typical value. Due to the low average BER of contemporary schemes, information loss is unlikely for the typical realization of disorder. The extended tail, however, shows that the reliability of communication systems is subject to the error rates much larger than the most probable value. Therefore, for complete evaluation of the system performance, the analysis of information loss must focus on extremely rare events such as the tail of the PDF instead of just considering the average value of the BER.

We conduct numerical simulations in a wide range of birefringent disorder, which will show the evolution of PDF with the disorder strength. As the disorder gets stronger, the length of algebraic tail becomes shorter, leading to a non-algebraic shape of the PDF. The characteristics of the PDF also depend on the size of electrical filter temporal width. In the regime of weak birefringent disorder, the PDF contains a longer tail if the electrical filter temporal width is smaller. We will collect the statistics of BER for various sizes of the filter and discuss its relevance in the shape of PDF.

The material in this paper is organized as follows. In Section 2, we briefly describe the theoretical set-up. In Section 3, the mathematical formulation for the BER is introduced based on the dynamically adjusted threshold. In Section 4 and 5, we perform analytical study on the PDF of BER for weak and strong fluctuation of fiber disorder, respectively. In Section 6, we present the results of direct numerical simulations and compare them to our theoretical prediction. Finally, in Section 7, we summarize our main results.

2. General relations

At relatively low optical power, the description for the envelope of electrical field subject to the PMD and amplifier noise can be found in [24]–[26]. Passing to the reference system rotating together with principle polarization axes, we obtain the following equation for the envelope of electromagnetic field:

where z, t,ξ are the position along the fiber, retarded time, amplifier noise, respectively. The envelope Ψ is a two-component complex field and each component represents two states of the light polarization. The birefringent disorder is represented by a random matrix m̂. Although the birefringence changes rapidly along the propagation length, it is practically frozen on all propagation-related time scales, i.e., t-independent.

The solution of Eq.(1) can be represented as

where Ψ₀(t) is the input pulse and V̂(z)≡Texp[${\int }_0^z$ dz ^′ aâ(z ^′)] is the ordered exponential, defined as the solution to the equation, ∂_zV̂=â(z)V̂, with the initial condition V̂(0)=1̂, where $\widehat{·}$ represents a matrix and 1̂ is the unit matrix. Due to the linearity of the system, the effect of chromatic dispersion d(z) can be incorporated by the initial data, and thus, we consider d(z)=0. Note that we do not consider the random effects of chromatic dispersion in this paper to focus on the influence of birefringent disorder and amplifier noise.

The matrix m̂ is a random Hermitian 2×2 traceless matrix. We can also express it as m̂=∑h_i (z)$\widehat{\sigma }$ _i , where $\widehat{\sigma }$ _i are Pauli matrices and h_i (z) are real-valued functions of z. We assume that the field h=〈h_i 〉 is zero on average and short-correlated in z characterized by

The amplifier noise ξ also follows zero mean Gaussian statistics and can be treated as δ-correlated in both time and space. Using Eqs. (3),(4), one can find that ϕ is a zero mean Gaussian field characterized by

Here, D_m ,D_ξ represent the intensity of birefringent disorder and additive (amplifier) noise, respectively.

3. BER as a functional of disorder

The total BER is an average of the two contributions: BER for “ones” detected as “zeroes”, B _1→0 and “zeroes” detected as “ones”, B _0→1, which are defined as

Here, P _0,1(I) are the PDFs of the output signal intensity corresponding to “zero” input bit (0-subscript) and to “one” input bit (1-subscript). I_d denotes the decision threshold.

The output intensity is defined as

where G(t) is a convolution of the electrical filter response function with the time window function. K is a linear operator that represents an optical filter. Here, we consider the limit of a very broad optical filter and thus, K can be treated as the identity operator (see e.g., [18] for the non-trivial optical filters). We also consider the case of rectangular G(t), i.e.,

In addition, we focus on the regime of weak amplifier noise and birefringent disorder intensities I ₀ T≫D_ξ Z and b ² T≫D_mZ, where b is the input pulse width (FWHM), T is the electrical filter temporal width, which can be considered as the width of the information detection slot due to Eq. (10), Z is the total propagation length, and I ₀ denotes the measured intensity of the unperturbed signal of “ones” (I ₀=${\int }_{-\infty }^{\infty }$ dtG(t)|KΨ₀|² where Ψ₀ is the input signal).

Following the mathematical approach explained in detail [14]–[18] (see e.g., Appendix II of [18]), we can use a saddle-point expression for the PDFs of “ones” and “zeroes”. Then we obtain

and with an exponentially small error we can derive

In order to obtain the BER based on the dynamically adjusted threshold, we first assume that the same number of “zeroes” and “ones” are sent on average. Now we need to optimize the decision threshold by minimizing the probability for having a penalty, which is achieved when P ₀(I)=P ₁(I). This, together with Eq. (12) results in I_d determination,

Thus, for the dynamically adjusted threshold, we obtain

We note that the output signal at the receiver is related to the initial signal as

Therefore, the BER defined by Eq.(14) is a functional of birefringent disorder parameterized by h(z).

4. Weak disorder

In the regime of weak birefringent disorder, b ² T≫D_mZ, we can expand the ordered exponential in Eq. (15) in a series of h(z):

It follows that we can write the output pulse intensity as

If Ψ₀(t) is real and t →-t symmetric, the r.h.s. of Eq. (18) is reduced to

Using this approximation together with Eq. (14), we obtain with an exponentially small error

where Γ(h)=µ H ², µ=-2Ψ₀(T/2)∂_t Ψ₀(T/2).

Therefore, from Eq. (20), we find the PDF of lnB, P(lnB),

5. Strong disorder

In the limit of strong disorder, we can also make a simple estimation following the last paragraph of [14] or the last appendix of [18], where a fixed threshold was employed. For the dynamically adjusted threshold which we consider here, the analysis and the results remain similar.

We first denote J≡J{h}=I{h,ξ=0}/I ₀ by the normalized intensity of the signal, without the amplifier noise. We now consider the birefringent disorder whose typical intensity remains small, but its fluctuation is strong leading to a seriously impeded signal. We also assume that the amplifier noise intensity remains small. This implies that max{D_ξ Z,D_mZ}≪J≪1. Then, we can replace the BER expression (20) by

where C~1. Since the statistics of ${\int }_0^Z$ dz h(z) is zero mean Gaussian with the variance D_mZ, the PDF of J follows

where F(J) is a non-negative real-valued function of J that does not depend on D_mZ, and for J~1, it is O(1). Using Eqs. (22), (23), we obtain the following universal asymptotic for the PDF of BER which is valid on the interval max{D_ξ Z,D_mZ}≪J≪1:

where C ₁=F(0) and C ₂~|F ^′(0)| are constants of the order one. The major factor in Eq. (24) is [-C ₁/(D_m Z)] supplemented by relatively weak dependence of B.

6. Numerical simulation

In order to demonstrate the dependence of birefringent disorder on the PDF of BER, we fix the additive noise intensity D_ξ =(0.03)² and consider various values of birefringent disorder intensity D_m =(0.2)² …(0.5)². We also perform the simulations for different sizes of the information slot T. The total propagation length is fixed Z=5 in the dimensionless unit and initially, we send a Gaussian type of pulse, exp(-t ²). Taking into account the dynamically adjusted threshold, the mathematical formulation for BER shows that we need to calculate φ which is the solution of Eq. (1) without the amplifier noise ξ (z, t). Hence, we numerically solve the homogeneous part of the Eq.(1) and calculate the BER following the relation (14). Repeating this procedure for 100,000 independent realizations of the birefringence disorder m̂ (z), we achieve the PDF of BER.

Figures 1(a), 1(b), 1(c), and 1(d) represent the PDFs of BER for the time slot size T=3.3,2.9,2.5,2.2, respectively. Here, we note that the FWHM of the input pulse, b~1.67 and thus, our selection of T varies from T~1.3b to 2b. The PDFs are plotted in the log-log scale, i.e., the y-axis represents lnP(lnB) and the x-axis is for lnB. The figures show that in the regime of relatively weak disorder, the PDF contains a long algebraic tail. As D_m gets stronger, however, the algebraic tail of the PDF becomes shorter and the PDF eventually attains a non-algebraic shape. The characteristics of the PDF also depend on the size of time slot. For relatively weak disorder, the PDF achieves a longer algebraic tail as the size of time slot decreases. This observation is consistent with our estimation for the slope of algebraic tail, α. Recall α from Eq.(21). We notice that as T is reduced, |Ψ₀(T/2)| and |∂_t Ψ₀(T/2)| increase, and thus, |α| decreases. This results in more observable algebraic tail. On the other hand, for stronger disorder, the deviation of PDF from the algebraic shape is more significant in the smaller time slot.

The quantitative comparison between the theoretical prediction and the numerical result requires to calculate the exponent α from Eq. (21), and estimate the slope of the algebraic tail from the numerical simulations. From Eq. (21), we notice that α contains 1/µ, µ=-2Ψ₀(T/2)∂_t Ψ₀(T/2). In our case, the detection window size T is large enough to contain most of the pulse energy. This implies that |Ψ₀(T/2)|, |∂_t Ψ₀(T/2)| are very close to 0, which imposes limitation on calculating α directly. Thus, we compare the ratio of slopes. For example, we take two different values of birefringent disorder, $D_m^1$ ,$D_m^2$ with a fixed noise intensity D_ξ , and fixed T. From Eq. (21), we can estimate the corresponding slopes of the algebraic tails, ${\alpha }_m^1$ ,${\alpha }_m^2$ , and find the ratio of the slopes ${\alpha }_m^1$ /${\alpha }_m^2$ =$D_m^2$ /$D_m^1$ which is our theoretical prediction. Then we calculate the ratio of slopes from the numerical simulations and compare it to this theoretical prediction.

In Fig. 2, we illustrate the relative disorder intensity $D_m^1$ /$D_m^2$ vs. the relative slope of the algebraic tails. More specifically, we calculate $D_m^1$ /$D_m^2$ from {$D_m^1$ =(0.2)²,$D_m^2$ =(0.25)²}, {$D_m^1$ =(0.25)²,$D_m^2$ =(0.3)²}, {$D_m^1$ =(0.3)²,$D_m^2$ =(0.35)²}, up to {$D_m^1$ =(0.45)²,$D_m^2$ =(0.5)²}. The relatively small ratio D ¹ _m /$D_m^2$ corresponds to the regime of weak disorder intensity. As it was mentioned before, the algebraic regime in the PDF is more extended if the size of detection window is smaller for weak disorder intensity. Simultaneously, for relatively strong intensity, the simulations demonstrate that the PDF contains a shorter length of algebraic regime as the smaller size of time slot is used. Thus, it is difficult to determine the linear regime and estimate the slope from the numerical simulation in the case of a relatively wide time window with the weak disorder and a narrow window with the strong disorder. This leads to the big discrepancy between the theory and the simulation in Fig. 2 for certain values of D ¹ _m /$D_m^2$ and T, where the algebraic regime cannot be clearly identified. For relatively weak disorder, the emergence of an algebraic tail is more evident as the size of T decreases. In this case, the numerical result is in better agreement with the theoretical prediction.

For the strong disorder, the quantitative comparison between the theory and the simulation is a difficult task due to many unknown parameters in the analytical prediction (24). Nevertheless, the computational results clearly demonstrate that the extended algebraic tail diminishes as the disorder becomes stronger and the PDF eventually attains a non-algebraic shape.

 

Fig. 1. The logarithm of the probability density vs. the logarithm of BER. From the blue to the black curve (leftmost to rightmost), each curve is the PDF of BER corresponding to D_m =0.2,0.25,0.3,0.35,0.4,0.45,0.5, respectively. Figures (a)-(d) represent the PDFs for detection window size, T=3.3…2.2, respectively. As the guide to the eye, the thick solid lines indicate the algebraic behavior of the PDF of BER.

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7. Conclusion

We investigated numerically and analytically the probability of outage which evaluates the reliability of optical fiber systems. Taking into account the two major impairments in information transmission systems (amplifier noise and birefringent disorder), we showed the emergence of an extended algebraic tail in the PDF of BER in the regime of weak disorder. This demonstrates that even a small fluctuation of fiber disorder can cause a substantial deviation of BER from its mean value.

In this paper, we employed a different measuring technique for BER from the previous considerations in [14]–[18]. We estimated the BER based on the dynamically adjusted threshold which aims to find the minimal probability for a given optical fiber system producing a penalty. Our numerical simulations confirm the analytical results in [14]–[18] qualitatively (only qualitatively due to the discrepancy between the measuring techniques for BER), and our theoretical prediction quantitatively. The results are also consistent with other numerical [27] and experimental observations [28]. In the regime of relatively weak disorder, the PDF of BER contains a long range of algebraic tail. As the disorder intensity exceeds certain limit and becomes stronger, the algebraic regime in the PDF gets shorter, leading to a non-algebraic shape of the PDF. The characteristics of the PDF also vary with the size of information detection slot. For weak disorder, the PDF has a more extended tail if the detection slot size is smaller. For relatively strong disorder, the PDF follows more significant non-algebraic shape if the smaller size of detection slot is used.

 

Fig. 2. The ratio of algebraic tail slopes (${\alpha }_m^1$ /${\alpha }_m^2$ ) vs. the ratio of corresponding disorder intensities ($D_m^1$ /$D_m^2$ ). The -+- (blue) line is for the theoretical prediction. The rest of the curves are obtained from numerical simulations for different sizes of detection window.

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We conducted direct Monte Carlo simulations to obtain the outage probabilities which correspond to extremely rare events. This was possible because our numerical simulations were incorporated by an analytical formulation of the BER. In the absence of analytical tools, we need to use the technique of importance sampling which allows the rare events to be efficiently simulated by concentrating the Monte Carlo simulation in the regions of our interest [29]. We remark that this technique can be combined with our simulation method used here in order to achieve even longer algebraic tail and suppress the big fluctuations in some of the PDFs.

The average BER of the current fiber communication systems is low, i.e., the information loss is unlikely for the typical birefringent disorder. Nevertheless, we showed, by generalizing the arguments and approach of [14]–[18] based on the dynamically adjusted decision threshold, that even the extremely rare events are relevant in the transmission performance. Therefore, for the complete evaluation of the system performance, it is essential to study the PDF of BER, specifically focusing on its tail.