1. Introduction

At high data rates (typically ≥10Gbits/s), polarization-mode dispersion (PMD) and polarization-dependent loss (PDL) can severely impair the performance of optical systems. While many bit-error-ratio (BER) evaluations have focused on various optical links with PMD or PDL alone, performance evaluations for systems with both PMD and PDL have been attracting growing theoretical attention in recent years. Interaction between PMD and PDL is a complicated problem [1], which needs to be studied to prevent or compensate the impairment. Considering that the accumulated PDL caused by various optical components (e.g., optical amplifiers, filters and isolators) can result in partially polarized ASE noise before the receiver, Refs. [2, 3, 4] further investigated the PDL induced polarized noise effects on the signal-to-noise ratio (SNR) and the Q factor in ON-OFF keying (OOK) systems with both PMD and PDL. Recently, Ref. [5] proposed an analytical approach to evaluate BER in a simplified system consisting of both PMD and PDL, ignoring the influences of filter and pulse shaping. In this work, we take into account these influences and extend the BER calculation from the OOK system to the binary differential phase-shift-keying (2-DPSK) system. Our results show that the optical SNR (OSNR) of 2-DPSK can have ~5dB, rather than 3dB, improvement from that of OOK. A similar conclusion was obtained through experiments [6]. We indicate that the additional 2dB improvement is mainly attributed to the partial cancellation between the noise-signal beatings produced by two photo detectors connecting the outputs of the 1-bit delay interferometer in Fig. 1. Moreover, we show that the PDL-induced partially polarized noise can significantly improve system performance and reduce BER variation caused by the random couplings between signal polarization, PDL and PMD vectors.

Although many communication systems with forward-error correction (FEC) can tolerate raw BERs up to 10^-3, typical BERs for optical systems are in the range of 10^-9–10^-15 or below, where the usual Gaussian fitting Q-factor approximation does not work well. In this case, the BER can be obtained by calculating the characteristic function, or equivalently the moment-generating function (MGF) [7, 8]. By using the Karhunen-Loève series expansion (KLSE) method to describe the filtered amplified spontaneous emission (ASE) noise, Ref. [9] further proposed a computationally efficient approach to accurately evaluate BER via MGF for an optically preamplified system with chromatic dispersion, arbitrary signal pulse shape and pre-and post-detection filtering. Ref. [10] extended Ref. [9]’s performance evaluation by taking into account the coupling between two orthogonal ASE noise components caused by the PDL in an OOK system with negligible PMD. Here, we extend Ref. [10]’s calculation to the system with both PMD and PDL by combining methods used in Refs. [5] and [9]. We reach this by expressing the filtered current in Ref. [9]’s one dimensional (1D) OOK model with Dirac bra-ket notations and their transformations and generalizing this form to various models. [Here “1D” means the signal and the noise are treated as scalar variables, while “2D” means both are considered as vector variables.] Based on this, accurate MGFs for systems with either unpolarized or partially polarized noise are derived. With negligible PMD, our MGF for the case of partially polarized noise reduces to the result of Ref. [10]. Furthermore, the effects of the partially polarized noise on the system performance are studied by comparing BERs in different polarization systems.

 

Fig. 1. Low-pass equivalent system with lumped PMD, PDL1, PDL2 and DPSK balanced receiver.

Download Full Size | PDF

2. System modeling and BER calculation

Figure 1 shows the low-pass equivalent model used for our study. The optical signal s⃗_in(t) is launched into the fiber system with lumped chromatic dispersion (CD) and lumped PMD-PDL1 (first-order) [5] in the linear regime. [Assuming the input power of the 2-DPSK system is low enough, the impact of the ASE-induced nonlinear phase noise (the Gordon-Mollenauer phase noise) has been neglected.] Then it is amplified by a flat gain amplifier G. The normalized ASE noise added at “4” in Fig. 1 is considered as additive white Gaussian noise n⃗_in(t) with two-sided power spectral density $N_0=n_{\mathrm{sp}}\frac{G-1}{G}hv$, where n_sp ≥1 is the spontaneous-emission or population-inversion parameter and hn is the photon energy. We assume that G≫1 so that N0≈n_sphnv [9, 11]. At the output of the second PDL component (PDL2), the ASE noise is partially polarized and the signal is further distorted. They are optically filtered prior to the DPSK balanced receiver. Finally the detected current at “7” in Fig. 1 is electrically filtered by the postdetection filter and sampled at the time t_k. The balanced direct-detection with responstivity R=1 yields the current at “7” in Fig. 1 which is given by [11, 12]

where T_b is the bit-time interval, c.c. or […]* stands for complex conjugation and s⃗o(t) [n⃗o(t)] is the signal (noise) at the output of the optical filter (“6” in Fig. 1).

The effect of the partially polarized ASE noise on a 2-DPSK system can be studied by comparing the BERs of two special cases of Fig. 1, i.e., the system with unpolarized ASE noise (assuming PDL2 is negligible) and the system with partially polarized ASE noise (assuming PDL1 is negligible). In both cases, the non-negligible PDL values are assumed to be the same. According to Ref. [9], the amplitude of the input signal s⃗_in(t) in Fig. 1 can be assumed to be a periodic repetition of signal $d(t)={\displaystyle {\sum }_{i=0}^{N-1}{a}_{i}p(t-i{T}_b)}$ with period NT_b, i.e., $\left|{\overrightarrow{s}}_{in}(t)\right|=s_{in}(t)={\displaystyle {\sum }_{n=-\infty }^{\infty }d(t-nN{T}_b)}$ . Here p(t) determines the elementary input pulse shape and a_i determines the logic value of the ith bit. In this paper, the input pulse shape is assumed to be $p\left(t\right)=\sqrt{2E_b⁄T_b}\cos \left[\frac{\pi }{2}{\cos }^2\left(\frac{\pi t}{T_b}\right)\right],$, where E_b is the optical energy per transmitted bit [11]. To include the intersymbol interference (ISI) effect, {a_i} for the OOK model is assumed to be a 2⁵-bit de Bruijn sequence [9, 11], i.e., 0000 0111 0111 1100 1011 0101 0011 0001. Repetition of this sequence yields all possible configurations of a 5-bit string from 00000 to 11111. For the balanced DPSK receiver shown in Fig. 1, a_i (∈{e^j0,ejπ}) is determined by requiring the received codes at sampling instants t_k(t_k=t₀+kT_b,k=0, …,N-1), normalized as “0” or “1” with no signal distortion, form a de Bruijn sequence (see inset of Fig. 2). Therefore, the input s⃗_in(t) can be expanded in Fourier series,

For simplicity, we assume all Fourier components of input signal are polarized in the same direction represented by a constant unit vector |p⃗_si=[x,y]^T in 2D Jones space.

On the other hand, the ASE noise n⃗_in(t) added at “4” in Fig. 1 can be decomposed along two orthonormal Jones vectors (e.g., |e⃗x〉 and |e⃗_y〉) and can be expressed in a Fourier series using a Karhunen-Loève expansion [9], e.g.,

In (3), we assume the overall duration of the impulse response of the optical (bandwidth B_o) and electrical (bandwidth B_r) filters is $T_0=\mu \left(\frac{1}{B_o}+\frac{1}{B_r}\right)$ and the noise contribution to the photoelectric current i(t_k) in (1) is caused by the noise input within the time interval (t_k-T_o, t_k) [9]. Dimensionless fitting parameter m must be determined iteratively [10]. As usual, the expansion coefficients (N_in)_i,m (i=x,y) are treated as complex independent and identically distributed (i.i.d.) random variables (r.v.) with Gaussian pdfs of zero mean and variance σ²=N₀/(2T_o) [9, 10].

In this work, the photoelectric currents are expressed within Dirac notation representation. Originally, Dirac notations were used to present quantum states. Considering that mathematically such notations are just vectors in multi-dimensional complex spaces, one can also use them to represent electric fields (states) in optical communication systems, such as the polarization states discussed in Refs. [3, 5, 13, 14]. Here, we further use Dirac notations to represent the polarization and magnitude parts of the signal and noise in an optical system. Results thus obtained show clearly the influence of each optical element. Therefore, to consider BERs of different models discussed in this paper, we only need to concern those related to the different parts in these models. In Appendix A, the basic concepts and features of the Dirac bra-ket notation are introduced by considering the electrically filtered current in the 1D OOK system of Ref. [9]. For a 1D 2-DPSK system, the expression of the filtered current is the natural extension of the 1D OOK result. Dirac notations used to describe the polarization properties of PMD and PDL were introduced in Ref. [3, 5, 13, 14].

2.1. BER in a 2-DPSK system with PMD, PDL and unpolarized noise

In this subsection, we consider the BER of one special case of Fig. 1, i.e., the effect of PDL2 on the ASE noise has been neglected. As a result, the ASE noise at “5” in Fig. 1 is unpolarized. Based on the filtered current y(t_k)=y_ss+y_nn+y_ns given by (35), (40) and (42) in Appendix B, its MGF can be obtained by averaging over the “canonical” noise at “4” in Fig. 1. By using the formula

and noticing that random variables (Z_i)m (i=α0,α⊥;m=-M, …,M) in (40) have zero mean and variance σ² [10], the MGF of the filtered current y(t_k) can be written as

where ${\text{β}}_m=2{\sigma }^2{\lambda }_m^D$ [see formulas between (25) and (26)], y_ss is given by (35) and |b̃ D_m |2 by (44). In this work the noise average is denoted as 〈.〉 and the Hermitian inner product as 〈.|.〉. Mean and variance of the detected current are given, respectively, by

As shown in Ref. [9], for a given MGF, the BER can be obtained using

where y_th is the detection threshold, + and C+ correspond for y_ss < y_th, while - and C_ for y_ss > y_th [9]. Substituting (5) into (7), we obtain ${\mathrm{BER}}_{y_{\mathrm{th}}}\left(t_k\right)=\frac{\pm 1}{2\pi j}{\int }_{C_{\pm }}ds\frac{e^{-\left(y_{\mathrm{th}}-y_{\mathit{ss}}\right)s}}{s}\times \prod _{m=-M}^M{e}^{\frac{s^{2}2{\sigma }^2{\mid {\tilde{b}}_m^D\mid }^2}{1-s{\beta }_m}}⁄{\left(1-s{\beta }_m\right)}^2$ for the binary DPSK with unpolarized noise. Averaging BERs over all bits in the de Bruijn sequence, we have (t_k=t₀+kT_b)

For a 2D OOK system, its MGF can be treated as a special case of (5). Actually, by setting the DPSK-induced factors Dⁱ (i=ss,nn,ns) introduced in (27) to be unity, we can get the reduced y_ss from (35), |b̃_m|² from (44), λ_m from (23) and then the MGF from (5) for the 2D OOK system.

2.2. MGF of a 2-DPSK system with PMD, PDL and partially polarized noise

In this subsection, we consider the second special case of Fig. 1, i.e., the PDL1 has been neglected. Due to PDL2, the noise at “5” of Fig. 1 is partially polarized.

The effect of the partially polarized noise on a negligible PMD system was investigated in Ref. [10], where the PDL (i.e., PDL2 in Fig. 1) component was represented by a matrix

When K is Hermitian, it can be diagonalized with its two real eigenvalues $k_0=\sqrt{1+\mathrm{DOP}}$ and $k_{\perp }=\sqrt{1-\mathrm{DOP}}$ [10].

To connect our MGF calculation with Ref. [10]’s result, we indicate that the Jones matrix of PDL2 in Fig. 1 relates with K of Ref. [10] by

with PDL2-induced degree of polarization $\mathrm{DOP}=\frac{1-e^{-2{\alpha }_k}}{1+e^{-2{\alpha }_k}}$, which yields $\mathrm{DOP}{\mid }_{{\alpha }_k\to 0}=0$ (unpolarized noise) and $\mathrm{DOP}{\mid }_{{\alpha }_k\to \infty }=1$ (completely polarized noise). Since k₀≥1, matrix K contains both PDL and polarization-dependent gain (PDG), whereas T_PDL2 has pure PDL effect.

With the help of (4), the MGF of a system with partially polarized noise can be obtained from (35), (45) and (46). Therefore we have (β_m=2σ²/λ^D_m)

To take into account the PDG effect, T_PDL2 (10) needs to be replaced by matrix K (9). In this case the signal-signal beating y_ss in (35) still hold, except ${\Theta }_{ll\text{'}}=e^{\alpha }\left[c_{l{\alpha }_0}^{*}{c}_{l\text{'}{\alpha }_0}+{\left(\frac{k_{\perp }}{k_0}\right)}^2{c}_{l{\alpha }_{\perp }}^{*}{c}_{l\text{'}{\alpha }_{\perp }}\right]$ in (36) should be replaced by (Θ_k)_ll′=k² ₀Θ_ll′. Also (45) and (46) now become $y_{\mathrm{nn}}=\sum _{m=-M}^M\left[k_0^2{\mid {Z_k}_0\mid }^2+k_{\perp }^2{\mid {Z_k}_{\perp }\mid }^2\right]{\lambda }_m^D$ and $y_{\mathrm{ns}}=k_0〈{Z_k}_0\mid {\tilde{b}}_{k_0}^D〉+k_{\perp }〈{Z_k}_{\perp }\mid {\tilde{b}}_{k_{\perp }}^D〉+c.c..$. They modify the MGF (11) as

When a system is OOK modulated and its PMD is negligible, formula (12) should return to the MGF given by (45) of Ref. [10]. This is true because, according to (37) and (43), |(b̃^D_i)m|² in MGF (12) now reduces to (i=k₀:+;i=k⊥:-)

[cf. |b_m|² near (24)] and the signal-signal beating term (34) now yields

with R_ss given by (22). Substituting (13) and (14) into (12) and replacing λ^D_m with λ_m for the OOK system, we obtain (β_m=2σ²λ_m)

which is same as (45) of Ref. [10], provided that p⃗_s·k⃗₀ equals to -cos(2θ) of Ref. [10], where q is the polar angle between signal polarization and the assumed x direction in the 2D Jones space. In fact, if the assumed y axis in Ref. [10] is set in the “direction” of |k⃗₀〉, then we have p⃗_s·k⃗₀=-cos(2θ). (See formulas between (10) and (11) in Ref. [5] and notice that the Stokes vector k⃗₀ is the mapping of its Jones vector |k⃗₀〉.)

3. Results and discussion

In this section, we first validate our analytical and numerical evaluations by considering some special cases of DPSK and OOK systems, as their BERs obtained fromMonte Carlo and numerical simulations had been discussed in Ref. [11]. Then we analyze the polarized noise effects on the system performance by evaluating BERs in different polarization systems. In the following specific applications, we neglect the chromatic dispersion (CD) effect. When necessary, this effect can be easily included by using the relevant formulas given in appendices.

3.1. Special cases of OOK and 2-DPSK systems

In the following numerical calculation, the optical filter in Fig. 1 is assumed to be a Fabry-Pérot type with its low-pass transfer function H_o(f)=1/(1+_j2f/B_o). The transfer function of the electrical filter in Fig. 1 is a fifth-order Bessel type, i.e., H_r(f)=945/(jF⁵+15F⁴-j105F3-420F²+ j945F+945) with F=2.43f/B_r [11]. In this subsection, we further assume the ASE noise is unpolarized. For the DPSK model given by Fig. 1, its BER is obtained by using (5)–(8). For the OOK model, because the balanced detection in Fig. 1 is replaced by the OOK direct detection in Fig. 5 (a) (Appendix A), its MGF is treated as a special case of (5), i.e., it is obtained from (5) with all the DPSK induced phase factors in (27) being reduced to one. The detection threshold y_th and initial detection time t₀=t_k-kT_b (k=0, …,N-1) are adjusted to get the minimal BER (8). For a system using RZ-2-DPSK format, its optimum threshold is fixed at y_th≈0. This is because, for the cases discussed in this work, the filtered signal-signal beating is always symmetrically “balanced” with respect to y_th≈0 [see the dashed curve in inset of Fig.2 (a)]. Therefore it is reasonable to minimize the BER at y_th≈0. For an OOK system, its optimum threshold is always changed because the currents of marks and spaces are directly affected by any system variation, such as E_b/N₀ (i.e., OSNR [9]), τ/T_b [the normalized differential group delay (DGD)], etc.

 

Fig. 2. (a) BER versus E_b/N₀ (OSNR) with α=τ=0 and (b) PMD-induced power penalty as a function of normalized DGD τ/T_b with α=0 for the OOK ({B_o,B_r}={1.8/T_b,0.65/T_b}) and the 2-DPSK ({B_o,B_r}={2.2/T_b,0.65/T_b}) systems. B_o (B_r) is the 3dB bandwidth of the Fabry-Pérot optical filter (fifth-order Bessel electrical filter), respectively [11]. BERs are evaluated using (5)–(8). Inset of (a): Time dependent filtered current caused by signal-signal beating in the OOK (solid) and the binary DPSK (dashed) systems. Crosses in (a): Monte Carlo simulation results of Ref. [11]. Stars (DPSK) and squares (OOK) in (b): numerical results of Ref. [11]. Also in (b) the power splitting ratio of PMD γ=0.5 and the required BER is 10^-9.

Download Full Size | PDF

As shown in Fig. 2 (a), our calculated BER versus E_b/N₀ (OSNR) curve for the DPSK (solid) agrees well with the Monte Carlo results (crosses) given by [11]. The dashed (OOK) curve is ~5dB, rather than the conventional prediction of 3dB [see the inset of Fig. 2 (a)], away from the DPSK (solid) one. Similar experimental results were given by Ref. [6]). (cf. the 2-DPSK curve at B_o=2.2/T_b and the OOK curve at B_o=1.8/T_b in Fig. 3(e) of Ref. [6]). This is mainly because, compared with OOK direct detection, the DPSK balanced detection not only strengthens the signal-signal beating (3dB) but also weakens the noise related beating (~1.5dB). The later can be seen by calculating the current variance (6), which is caused by the noise-noise beating (the first term) and noise-signal beating (the second term). The first term ${\displaystyle {\sum }_{m=-M}^{M}2{\beta }_m^2}$ is a constant. For the DPSK (OOK) system of Fig. 2 (a), it is 0.0005 (0.0008), respectively. The second term of (6), which is ~0.011 (~0.020) for DPSK (for OOK marks), is more important than the first term, as it is related with the transformed signal. Consequently the equivalent OSNR y_ss/Δy for DPSK (OOK) is $\sqrt{30.5}\approx 5.52(\sqrt{15.3}\approx 3.91)$, respectively.

 

Fig. 3. BER versus DOP for the 2-DPSK system with E_b/N₀=12dB and (a) τ/T_b=0 and (b) τ/T_b=0.3. The BERⁱ_PDL1 (i=pa,or) is obtained using (5)–(8) for the system with unpolarized noise, while the BERⁱ_PDL2 (i=pa,or) is calculated using (7), (8) and (11) for the case of partially polarized noise. pa (or) means the input signal polarization |p⃗_s〉 is parallel (orthogonal) to the minimum attenuation direction |α⃗₀〉 of PDL1 (or |k⃗0 of PDL2). The PDL-induced degree of polariztion is given by $\mathrm{DOP}=\sqrt{\left(1-e^{-2x}\right)⁄\left(1+e^{-2x}\right)}$, where x=a (x=a_k) is the PDL value of PDL1 (PDL2), respectively. Insets: the pdf as a function of BER for systems with unpolarized noise (dashed) and partially polarized noise (solid) at DOP=0.28 or α=α_k≈2.5dB. To show clearly the two pdf curves in the inset of (a), the dashed pdf curve (unpolarized noise) is shifted up by 0.3.

Download Full Size | PDF

Figure 2(b) shows our calculated PMD-induced power penalties at 10^-9 BER. They are almost the same as those given in Ref. [11]. In the inset of Fig. 2(a), the filtered signal-signal beating (dashed) in the 2-DPSK system is obtained using (35). For the OOK direct detection, its current (solid) is obtained as a special case of (35) where the DPSK-induced phase fatcors need to be reduced to unity.

3.2. Effect of polarized noise on the optical performance

In this subsection, the effect of the polarized noise on BER is studied by comparing the BER in the system with unpolarized ASE noise (Fig. 1 with negligible PDL2) and the BER with partially polarized noise (Fig. 1 with negligible PDL1).

Figure 3(a) shows BER versus DOP for these two systems using RZ-2-DPSK format with negligible PMD (τ=0,θ_ατ=0). For the first system where the noise is unpolarized, BER^pa_PDL1 (assuming |p⃗s〉‖|α⃗0〉) and BER^or_PDL1 (assuming |p⃗s〉⊥|α⃗₀〉), obtained from (5), (7) and (8) with PDL1-induced $\mathrm{DOP}=\sqrt{\left(1-e^{-2\alpha }\right)⁄\left(1+e^{-2\alpha }\right)}$, are plotted as thin and thick dashed curves. As shown, the BER^pa_PDL1 (thin dashed) keeps unchanged, because the signal is polarized in the zero attenuation direction |α⃗₀〉 and the two orthogonal components of the ASE noise are not affected by PDL1. On the other hand, when the signal is polarized in the maximum attenuation direction (|p⃗s〉 ⊥ |α⃗₀〉), the signal will be severely distorted when the PDL1-induced DOP becomes large. Therefore we get the thick dashed curve in Fig. 3(a). For the second system where the noise is partially polarized, we get BER^pa_PDL2 (thin solid) and BER^or_PDL2 (thick solid) versus PDL2-induced DOP curves, obtained from (7), (8) and (11). As shown in Fig. 3(a), BER^pa_PDL2 (BER^or_PDL2) is lower than BER^pa_PDL1 (BER^or_PDL1), respectively. This is because, no matter how the signal is polarized (|p⃗_s〉‖|k⃗₀〉 or |p⃗_s〉 ⊥ |k⃗₀〉), the overall noise in the second system (no PDL1) is always smaller than that in the first system (no PDL2). Moreover BER^pa_PDL2 (thin solid) shows that although in the zero attenuation direction (|k⃗₀〉) both noise and signal are not changed, the noise attenuation in the orthogonal direction (|k⊥0〉) can also improve the optical performance. Besides, at DOP=0.28 (α=α_k≈2.5dB), the ratio BER^or_PDL2/BER^pa_PDL2≈75 is much smaller than the ratio BER^or_PDL1/BER^pa_PDL1≈600.

The inset of Fig. 3 (a) shows the statistical features of the BER fluctuation due to the coupling between |p⃗_s〉 and |α⃗₀〉 (dashed) and the coupling between |p⃗_s〉 and |k⃗₀〉 (solid), obtained by setting DOP=0.28 and randomly changing parameters ${\theta }_i(\mathrm{weighting}\mathrm{factor}\frac{\sin {\theta }_i}{2},i=s\tau ,\alpha \tau )$) and (φ_ατ-φ_sτ) (weighting factor $\frac{1}{2\pi }$) with ~2×10⁵ random realiztions. (To show clearly the two pdf curves in this inset, the dashed pdf curve for the unpolarized noise case is shifted up by 0.3.) Each pdf can be approximated as a rectangular pulse. This means, BERs with random directional couplings determined by θ_i(i=sτ,ατ) and (φ_ατ-φ_sτ) are homogeneously distributed between thick and thin curves. As shown in this inset, the BER variation range and its average value for the case of unpolarized noise (dashed) are about one order of magnitude larger than those for the partially polarized noise case (solid). Therefore the PDL effect on the ASE noise can improve the system performance and reduce the BER variance caused by the directional coupling between the signal polarization and the PDL vector.

In Fig. 3(a), the dotted horizontal line near 10^-7, calculated by using the 1D balanced DPSK model discussed in Appendix A, is used to verify BER^pa_PDL2. Actually BER^pa_PDL2 approaches closely to this horizontal line when DOP>0.6. We also consider the PDG effect by replacing (11) with (12) in the BER calculation. The two curves thus obtained (BER^or_K and BER^pa_K, not shown) coincide almost exactly with the two solid curves (BER^or_PDL2 and BER^pa_PDL2) in Fig. 3(a), respectively. Mathematically one can prove that, for a 2-DPSK format with y_th≈0, by scaling transformation s→sk²₀ [cf. the discussion between (11) and (12)], the BER obtained using (11) and (7) is the same as the BER using (12) and (7). Physically, K is equivalent to PDL2 followed by an ideal amplifier with G=k₀>1, which does not cause further signal and noise distortion and therefore no additional effect on the BER.

Similar BER calculations for 2-DPSK systems with τ/T_b=0.3 are depicted in Fig. 3(b). As shown, the PMD increases the BER and further reduces the BER fluctuation caused by the PMD-PDL-induced directional coupling (BER^or_PDL1/BER^pa_PDL1≈66 and BER^or_PDL2/BER^pa_PDL2≈12 with DOP=0.28). In the inset of Fig. 3(b), we show the pdf versus BER with DOP=0.28 for the cases of unpolarized and partially polarized noise. Due to the PMD-PDL interaction, each pdf can be approximated as a “smoothed” rectangular pulse. Note that, in this inset, the BER range of each pdf curve is different from the corresponding gap between BER^or_i and BER^pa_i (i=PDL1,PDL2) in Fig. 3(b). As an example, for a system with partially polarized noise, the BER variation range of the solid curve in the inset of Fig. 3(b) is determined by BERs obtained by setting θ_ατ=0 with θ_sτ=π (~6×10^-5) and θ_sτ=0 (~8×10^-7), whereas the thick (thin) solid curve in Fig. 3(b), obtained using θ_ατ=θ_sτ=π/2 with (φ_ατ-φ_sτ)=π [(φ_ατ-φ_sτ)=0], yields BER^or_PDL2≈3×10^-5 (BER^pa_PDL2≈1.8×10^-6) at DOP=0.28.

For an OOK system, because the BER depends sensitively on the detection threshold, the noise polarization effects are rather complicated. However if the BER is calculated using optimal threshold, one can get the BER versus DOP curves in Fig. 4, which are similar to those shown in Fig. 3(a).

 

Fig. 4. BER versus DOP for the OOK format with E_b/N₀=18dB and τ/T_b=0. Calculations of BER^j_i (i=pa,or, j=PDL1,PDL2) are explained in the caption of Fig. 3. Also, for the OOK system, the DPSK induced factors Dⁱ (i=ss,nn,ns) detailed in (27) should be reduced to unity.

Download Full Size | PDF

4. Conclusion

The Dirac bra-ket notations are used to express the electrically filtered currents in optical systems using DPSK balanced detection and OOK direct detection, with non-negligible first-order PMD and PDL and with arbitrary filtering and pulse shaping. This makes it convenient to evaluate the MGFs and BERs for different optical systems. Our theoretical and numerical calculations for the special cases disussed in subsections 2.2 and 3.1 agree well with the relevant results given by Refs. [6, 10, 11]. We also show that, for a given BER, 2-DPSK requires ~5dB lower input OSNR (at “4” in Fig. 1) than OOK, mainly because the DPSK balanced detection not only strengthens the singal-singal beating (3dB) but also weakens the noise related beating (~1.5dB). By comparing BERs of different polarization systems, the effect of polarized noise on the optical system is studied. In the 2-DPSK system, the PDL-induced noise attenuation can significantly improve the system performance and reduce the BER variance caused by the PMD-PDL-induced directional coupling. For the OOK system, a similar conclusion can be obtained, provided the detection threshold is kept optimal.

Appendix A: Expressions of the filtered current: from 1D OOK to 1D DPSK

In this part, we first use Dirac notations and the related transformation matrices to briefly express the filtered photoelectric current in the 1D OOK system of Ref. [9] [cf. Fig. 5 (a)]. Then we show that such form of expression can be easily generalized to the case of 1D balanced DPSK shown in Fig. 5 (b).

By inserting a polarizer before or after the optical filter of Fig. 5(a) or (b), the signal and noise can be aligned in the same direction. In this case, the 2D vector forms of signal (2) and noise (3) can be simplified as

(16)$$s_{\mathrm{in}}\left(t\right)=\sum _{l=-\infty }^{\infty }{\left[s_{\mathrm{in}}\left(t\right)\right]}_l=\sum _l{\left(S_{\mathrm{in}}\right)}_l{e}^{-j\frac{2\pi lt}{N{T}_b}},n_{\mathrm{in}}\left(t\right)=\sum _{m=-\infty }^{\infty }{\left[n_{\mathrm{in}}\left(t\right)\right]}_m=\sum _m{\left(N_{\mathrm{in}}\right)}_m{e}^{-j\frac{2\pi m\left(t-t_k+T_0\right)}{T_0}},$$

where ${\left[s_{\mathrm{in}}\left(t\right)\right]}_l\equiv {\left(S_{\mathrm{in}}\right)}_l{e}^{-j\frac{2\pi lt}{N{T}_b}}$ and ${\left[n_{\mathrm{in}}\left(t\right)\right]}_m\equiv {\left(N_{\mathrm{in}}\right)}_m{e}^{-j\frac{2\pi m\left(t-t_k+T_0\right)}{T_0}}$. In (16), due to the optical filter response h_o(t) [or H_o(f)], only those components with frequencies within the filter bandwidth B_o need to be considered. Because of this the Dirac bra-ket notations for input signal and noise at time t_k can be introduced as

(17)$$\mid s_{\mathrm{in}}\left(t_k\right)〉={[{\left(S_{\mathrm{in}}\right)}_{-L}{e}^{\frac{j2\pi L{t}_k}{N{T}_b}},\dots ,{\left(S_{\mathrm{in}}\right)}_L{e}^{\frac{-j2\pi L{t}_k}{N{T}_b}}]}^T,〈s_{\mathrm{in}}\left(t_k\right)\mid =[{\left(S_{\mathrm{in}}\right)}_{-L}^{*}{e}^{\frac{-j2\pi L{t}_k}{N{T}_b}},\dots ,{\left(S_{\mathrm{in}}\right)}_L^{*}{e}^{\frac{j2\pi L{t}_k}{N{T}_b}}]$$
$$\mid n_{\mathrm{in}}\left(t_k\right)〉=\mid N_{\mathrm{in}}〉={[{\left(N_{\mathrm{in}}\right)}_{-M},\dots ,{\left(N_{\mathrm{in}}\right)}_M]}^T,〈n_{\mathrm{in}}\left(t_k\right)\mid =[{\left(N_{\mathrm{in}}\right)}_{-M}^{*},\dots ,{\left(N_{\mathrm{in}}\right)}_M^{*}]$$

with

(18)$$L=\eta B_{o}N{T}_b,M=\eta B_o{T}_0,T_0=\mu \left(\frac{1}{B_o}+\frac{1}{B_r}\right).$$

Dimensionless fitting parameters h and m must be determined iteratively [10].

 

Fig. 5. Low-pass equivalent (a) OOK system and (b) 2-DPSK system in the absence of PMD and PDL, assuming both signal and noise are aligned in the same direction.

Download Full Size | PDF

At the output of the optical filter, the signal and noise can be written as

(19)$$\mid s^o\left(t_k\right)〉=O_{\mathrm{ss}}{\Phi }_{\mathrm{CD}}\mid s_{\mathrm{in}}\left(t_k\right)〉,〈s^o\left(t_k\right)\mid =〈s_{\mathrm{in}}\left(t_k\right)\mid {\Phi }_{\mathrm{CD}}^{†}{O}_{\mathrm{ss}}^{†}$$
$$\mid n^o\left(t_k\right)〉=O_{\mathrm{nn}}\mid N_{\mathrm{in}}〉,〈n^o\left(t_k\right)\mid =〈N_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}$$

with ${\left(O_{\mathrm{nn}}\right)}_{\mathrm{mm}\text{'}}={\delta }_{m,m\text{'}}{H}_o\left(\frac{m}{T_o}\right)$, ${\left(O_{\mathrm{ss}}\right)}_{\mathrm{ll}\text{'}}={\delta }_{l,l\text{'}}{H}_o\left(\frac{1}{N{T}_b}\right)$ and ${\left({\Phi }_{\mathrm{CD}}\right)}_{\mathrm{ll}\text{'}}={\delta }_{l,l\text{'}}{H}_{\mathrm{CD}}\left(\frac{1}{N{T}_b}\right)$ [9, 11]. Here we assume the chromatic dispersion (CD) comes from the fiber. Its transfer function is $H_{\mathrm{CD}}\left(f\right)=e^{-j2{\pi }^2{\beta }_2{f}^{2}L}$ with β₂=-λ² D(λ)/(2πc). For a nondispersion shifted fiber at λ=1550nm, the dispersion parameter D≈17ps/(km·nm). As usual, the Hermitian conjugate of a matrix A is defined as (A ^†)_ij=A*_ji. The photoelectric current prior to the electrical filter is i_1D(t)=〈s^o(t)+n^o(t)|s^o(t)+n^o(t)〉 (R=1). Due to the response function of the electrical filter, h_r(t) [or H_r(f)], the filtered current at sampling time t_k consists of three parts, i.e., the signal-signal beating, noise-noise beating and noise-signal beating

(20)$$y\left(t_k\right)=y_{\mathrm{ss}}+y_{\mathrm{nn}}+y_{\mathrm{ns}},$$

where

(21)$$y_{\mathrm{ss}}=〈s^o\left(t_k\right)\mid R_{\mathrm{ss}}\mid s^o\left(t_k\right)〉,$$
$$y_{\mathrm{nn}}=〈n^o\left(t_k\right)\mid R_{\mathrm{nn}}\mid n^o\left(t_k\right)〉=〈N_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}{R}_{\mathrm{nn}}{O}_{\mathrm{nn}}\mid N_{\mathrm{in}}〉=〈Z\mid \Lambda \mid Z〉,$$
$$y_{\mathrm{ns}}=〈n^o\left(t_k\right)\mid R_{\mathrm{ns}}\mid s^o\left(t_k\right)〉+c.c.=〈N_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}{O}_{\mathrm{ss}}{\Phi }_{\mathrm{CD}}\mid s_{\mathrm{in}}\left(t_k\right)〉+c.c.=〈Z\mid b\left(t_k\right)〉+c.c.,$$

(22)$${\left(R_{\mathrm{ss}}\right)}_{ll\text{'}}\equiv H_r\left(\frac{l\text{'}-l}{N{T}_b}\right),{\left(R_{\mathrm{nn}}\right)}_{mm\text{'}}\equiv H_r\left(\frac{m\text{'}-m}{T_0}\right),{\left(R_{\mathrm{ns}}\right)}_{\mathrm{ml}}\equiv H_r\left(\frac{l}{N{T}_b}-\frac{m}{T_0}\right),$$

with m=-M, …,M, l=-L, …,L. In the y_nn term given by (21), because R_nn is a Hermitian matrix satisfying R^†_nn=R_nn, O†_nnR_nnO_nn is also Hermitian and it can be diagonalized by a orthogonal and unitary transformation U

(23)$$\Lambda \equiv U^{†}{O}_{\mathrm{nn}}^{†}{R}_{\mathrm{nn}}{O}_{\mathrm{nn}}U,$$

with Λmm′=δ_m,m′l_m and U being composed of eigenvectors |λ_m〉 of λ_m. Thus we have

(24)$$\mid Z〉=U^{†}\mid N_{\mathrm{in}}〉,$$
$$\mid b\left(t_k\right)〉=U^{†}{O}_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}{O}_{\mathrm{ss}}{\Phi }_{\mathrm{CD}}\mid s_{\mathrm{in}}\left(t_k\right)〉=U^{†}{O}_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}\mid s^o\left(t_k\right)〉\equiv B\mid s^o\left(t_k\right)〉,$$

and ${\mid b_m\mid }^2=\sum _{l,l\text{'}=-L}^L{s}_l^{o*}\left(t_k\right)B_{\mathrm{ml}}^{*}{B}_{\mathrm{ml}\text{'}}{s}_{l\text{'}}^o\left(t_k\right)\left(m=-M,\dots ,M\right)$. The transformed signal |b(t_k)〉 is the equivalent field to beat with the “canonical” noise |Z〉, although this field itself is noiseless. In this context, |b_m|2 is the square magnitude of the m-th component of the received noiseless signal. It only contributes to noise-signal beating. Notice that (20) holds not only for the 1D model of Fig. 5(a), but also for the other models discussed in this work. Also notice that, unlike the matrix operators Φ_CD, O_ss and O_nn in (19), which are used to represent the real effects of the corresponding optical elements, the receiver related operators R_i (i=ss,nn,ns) in (21) are only used to express the results of filtered photoelectric currents. For example, at the input of the electric filter, the current due to signal-signal beating is $i_{\mathrm{ss}}=\sum _{l,l\text{'}=-L}^L{S}_l^{o*}{S}_{l\text{'}}^o{e}^{-j2\pi t\left(l^{\prime }-l\right)⁄N{T}_b}$. Therefore the filtered current associated with this beating becomes $y_{ss}={\displaystyle {\sum }_{l,l\text{'}}{S}_l^{o*}{S}_{l\text{'}}^o{e}^{-j2\pi t(l\text{'}-l)/N{T}_b}}$ R_ss[(l′-l)/(NT_b)], which is mathematically equivalent to the first term of (21).

For the 1D system using DPSK balanced receiver shown in Fig. 5(b), the photoelectric current prior to the electrical filter is given by i^D _{1 D}(t)=[〈s^o(t+T_b)+n^o(t +T_b)|s^o(t)+n^o(t)i+ c.c.]/2 [11, 12]. Thus each filtered component in (21) needs to be modified as

(25)$$y_{\mathrm{ss}}\left(t_k\right)=[〈s^o\left(t_k+T_b\right)\mid R_{\mathrm{ss}}\mid s^o\left(t_k\right)〉+c.c.]⁄2=〈s^o\left(t_k\right)\mid R_{\mathrm{ss}}^D\mid s^o\left(t_k\right)〉$$
$$y_{\mathrm{nn}}\left(t_k\right)=[〈n^o\left(t_k+T_B\right)\mid R_{\mathrm{nn}}\mid n^o\left(t_k\right)〉+c.c.]⁄2=〈N^o\mid R_{\mathrm{nn}}^D\mid N^o〉=〈Z\mid {\Lambda }^D\mid Z〉$$
$$y_{\mathrm{ns}}\left(t_k\right)=[〈n^o\left(t_k+T_b\right)\mid R_{\mathrm{ns}}\mid s^o\left(t_k\right)〉+〈n^o\left(t_k\right)\mid R_{\mathrm{ns}}\mid s^o\left(t_k+T_b\right)〉+c.c.]⁄2$$
$$=[〈N_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}^D\mid s^o\left(t_k\right)〉+c.c.]=[〈Z\mid b^D\left(t_k\right)〉+c.c.]$$

where Λ^D≡U^†O^†_nnR^D_nnO_nnU with (Λ^D)_mm′=δ_m,m′λ^D_m and

(26)$$\mid b^D\left(t_k\right)〉=U^{†}{O}_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}^D\mid s^o\left(t_k\right)〉\equiv B^D\mid s^o\left(t_k\right)〉,$$

(27)$${\left(R_{\mathrm{ss}}^D\right)}_{ll\text{'}}={\left(R_{\mathrm{ss}}\right)}_{ll\text{'}}{D}_{ll\text{'}}^{ss},{\left(R_{\mathrm{nn}}^D\right)}_{mm\text{'}}={\left(R_{\mathrm{nn}}\right)}_{mm\text{'}}{D}_{mm\text{'}}^{nn},{\left(R_{\mathrm{ns}}^D\right)}_{ml}={\left(R_{\mathrm{ns}}\right)}_{ml}{D}_{ml}^{ns}$$
$$D_{\mathrm{ll}\text{'}}^{\mathrm{ss}}=\frac{e^{j\frac{2\pi l}{N}}+e^{-j\frac{2\pi l\text{'}}{N}}}{2},D_{mm\text{'}}^{nn}=\frac{e^{j\frac{2\pi m{T}_b}{T_0}}+e^{-j\frac{2\pi m\text{'}{T}_b}{T_0}}}{2},D_{\mathrm{ml}}^{\mathrm{ns}}=\frac{e^{j\frac{2\pi m{T}_b}{T_0}}+e^{-j\frac{2\pi l}{N}}}{2}.$$

Eq. (26) yields ${\mid b_m^D\mid }^2=\sum _{l,l\text{'}=-L}^L{s}_l^{o*}\left(t_k\right){\left(B^D\right)}_{\mathrm{ml}}^{*}{\left(B^D\right)}_{\mathrm{ml}\text{'}}{s}_{l\text{'}}^o\left(t_k\right)\left(m=-M,\dots ,M\right)$. Its physical meaning is similar to |b_m|² discussed above.

Therefore, from 1D OOK to 1D 2-DPSK, the receiver related matrices in (22) need to be multiplied by the DPSK induced phase factors Dⁱ (i=ss,nn,ns) given by (27).

Appendix B: Filtered current in DPSK system with unpolarized ASE noise

Key points of our study on PMD and PDL effects are the eigenvectors of their Jones matrices and the related transformations between them. In the frequency domain, the Jones matrix T_PMD of the lumped first-order PMD is given by [1, 3, 4, 5, 13]

(28)$$T_{\mathrm{PMD}}\left({\omega }_l\right)=\exp (j{\omega }_l\overrightarrow{\tau }·\overrightarrow{\sigma }⁄2),$$

where ω_l=2πl/(NT_b), σ⃗ is the standard physics notation for Pauli spin matrices satisfying σ⃗×σ⃗=2jσ⃗ [5]. τ⃗ is a 3D real Stokes vector. In this work, its magnitude τ (i.e., DGD) and direction τ⃗₀=τ⃗/τ are assumed to be frequency independent. Matrix (28) has two orthonormal eigenvectors (|τ⃗₀〉 and |τ⃗⊥〉) in 2D Jones space, indicating the directions of slow (|τ⃗₀〉) and fast (|τ⃗⊥〉) principal state of polarizations (PSPs) with matrix elements [5]

(29)$$〈{\overrightarrow{\tau }}_0\mid T_{\mathrm{PMD}}\left({\omega }_l\right)\mid {\overrightarrow{\tau }}_0〉=e^{j{\omega }_l\frac{\tau }{2}},〈{\overrightarrow{\tau }}_{\perp }\mid T_{\mathrm{PMD}}\left({\omega }_l\right)\mid {\overrightarrow{\tau }}_{\perp }〉=e^{-j{\omega }_l\frac{\tau }{2}},〈{\overrightarrow{\tau }}_0\mid T_{\mathrm{PMD}}\left({\omega }_l\right)\mid {\overrightarrow{\tau }}_{\perp }〉=0.$$

Here |τ⃗₀〉 is the Jones vector corresponding to Stokes vector τ⃗₀ [5]. (For the input signal presented in the form of (2), we introduce the PMD Jones matrix (28) so that the PMD vector τ⃗ points in the direction of the slow PSP, which is the usual convention. Note that, although the PMD Jones matrix in Ref. [5] was introduced in a different way, formula (11) in Ref. [5] still can be used here, provided the PMD vector τ⃗ in this formula now is replaced with -τ⃗). Similarly, for a given transform matrix T_PDL1=exp(-α/2)exp(α⃗·σ⃗/2) connecting the input and output of PDL1, its two orthonormal eigenvectors (|α⃗0i and |α⃗⊥〉) in 2D Jones space represent the minimum (zero) and maximum attenuation states which yield [5]

(30)$$〈{\overrightarrow{\alpha }}_0\mid T_{\mathrm{PDL}1}\mid {\overrightarrow{\alpha }}_0〉=1,〈{\overrightarrow{\alpha }}_{\perp }\mid T_{\mathrm{PDL}1}\mid {\overrightarrow{\alpha }}_{\perp }〉=e^{-\alpha },〈{\overrightarrow{\alpha }}_0\mid T_{\mathrm{PDL}1}\mid {\overrightarrow{\alpha }}_{\perp }〉=0.$$

Like (17) in Appendix A, input (2) can be represented by a Dirac bra ${[{\left(S_{\mathrm{in}}\right)}_{-L}{e}^{j\frac{2\pi L{t}_k}{N{T}_b}},\dots ,{\left(S_{\mathrm{in}}\right)}_L{e}^{-j\frac{2\pi L{t}_k}{N{T}_b}}]}^T|{\overrightarrow{p}}_s〉$, i.e., ${\mid \overrightarrow{s}}_{\mathrm{in}}\left(t_k\right)〉=\mid s_{\mathrm{in}}\left(t_k\right)〉\otimes \mid {\overrightarrow{p}}_s〉$, where ⊗ stands for tensor product. |p⃗s〉 is the unit Jones vector used to represent the input signal polarization. Because of this, each element of |s⃗_in(t_k)〉 is a 2D vector. Assuming the optical filter has no effect on the signal polarization, we express the signal at “6” in Fig. 1 as

(31)$$\mid {\overrightarrow{s}}^o\left(t_k\right)〉=\mid s^o\left(t_k\right)〉\otimes \mid {\overrightarrow{p}}_s〉=O_{\mathrm{ss}}P(\overrightarrow{\tau },\overrightarrow{\alpha }){\Phi }_{\mathrm{CD}}\mid s_{\mathrm{in}}\left(t_k\right)〉\otimes \mid {\overrightarrow{p}}_s〉,$$

where P(τ⃗,α⃗)ll′=δ_l,l′TPDLTPMD. (The possible PDL effect of the optical filter on signal and noise can be included into PDL2, which is the case discussed in subsection 2.2 and Appendix C.)With the help of (29), (30) and the completeness relations |k⃗₀〉〈k⃗₀|+|k⃗⊥〉〈k⃗⊥|=1(k⃗=τ⃗,α⃗) in the 2D Jones space, we have

(32)$$T_{\mathrm{PMD}}=\left(\mid {\overrightarrow{\tau }}_0〉〈{\overrightarrow{\tau }}_0\mid +\mid {\overrightarrow{\tau }}_{\perp }〉〈{\overrightarrow{\tau }}_{\perp }\mid \right)T_{\mathrm{PMD}}\left(\mid {\overrightarrow{\tau }}_0〉〈{\overrightarrow{\tau }}_0\mid +\mid {\overrightarrow{\tau }}_{\perp }〉〈{\overrightarrow{\tau }}_{\perp }\mid \right)=e^{j\frac{{\omega }_l\tau }{2}}\mid {\overrightarrow{\tau }}_0〉〈{\overrightarrow{\tau }}_0\mid +e^{-j\frac{{\omega }_l\tau }{2}}\mid {\overrightarrow{\tau }}_{\perp }〉〈{\overrightarrow{\tau }}_{\perp }\mid$$
$$T_{\mathrm{PDL}1}=\left(\mid {\overrightarrow{\alpha }}_0〉〈{\overrightarrow{\alpha }}_0\mid +\mid {\overrightarrow{\alpha }}_{\perp }〉〈{\overrightarrow{\alpha }}_{\perp }\mid \right)T_{\mathrm{PDL}1}\left(\mid {\overrightarrow{\alpha }}_0〉〈{\overrightarrow{\alpha }}_0\mid +\mid {\overrightarrow{\alpha }}_{\perp }〉{\overrightarrow{\alpha }}_{\perp }\mid \right)=\mid {\overrightarrow{\alpha }}_0〉〈{\overrightarrow{\alpha }}_0\mid +e^{-\alpha }\mid {\overrightarrow{\alpha }}_{\perp }〉〈{\overrightarrow{\alpha }}_{\perp }\mid ,$$

which means the polarization matrix element P(τ⃗,α⃗)ll′ projects the input polarization |p⃗_s〉 into two orthogonal directions, i.e., $P{(\overrightarrow{\tau },\overrightarrow{\alpha })}_{\mathrm{ll}\text{'}}{\mid \overrightarrow{p}}_s〉={\left({P_{\alpha }}_0\right)}_{\mathrm{ll}\text{'}}\mid {\overrightarrow{p}}_s〉+{\left({P_{\alpha }}_{\perp }\right)}_{\mathrm{ll}\text{'}}\mid {\overrightarrow{p}}_s〉$ with

(33)$${\left({P_{\alpha }}_0\right)}_{ll\text{'}}{\mid \overrightarrow{p}}_s〉={\delta }_{l,l\text{'}}\mid {\overrightarrow{\alpha }}_0〉\left[〈{\overrightarrow{\alpha }}_0\mid {\overrightarrow{\tau }}_0〉〈{\overrightarrow{\tau }}_0\mid {\overrightarrow{p}}_s〉e^{\frac{j{\omega }_l\tau }{2}}+〈{\overrightarrow{\alpha }}_0\mid {\overrightarrow{\tau }}_{\perp }〉〈{\overrightarrow{\tau }}_{\perp }\mid {\overrightarrow{p}}_s〉e^{\frac{-j{\omega }_l\tau }{2}}\right]\equiv {\delta }_{l,l\text{'}}{c}_{l{\alpha }_0}\mid {\overrightarrow{\alpha }}_0〉$$
$${\left({P_{\alpha }}_{\perp }\right)}_{ll\text{'}}\mid {\overrightarrow{p}}_s〉={\delta }_{l,l\text{'}}{e}^{-\alpha }\mid {\overrightarrow{\alpha }}_{\perp }〉\left[〈{\overrightarrow{\alpha }}_{\perp }\mid {\overrightarrow{\tau }}_0〉〈{\overrightarrow{\tau }}_0\mid {\overrightarrow{p}}_s〉e^{\frac{j{\omega }_l\tau }{2}}+〈{\overrightarrow{\alpha }}_{\perp }\mid {\overrightarrow{\tau }}_{\perp }〉〈{\overrightarrow{\tau }}_{\perp }\mid {\overrightarrow{p}}_s〉e^{\frac{-j{\omega }_l\tau }{2}}\right]\equiv {\delta }_{l,l\text{'}}{e}^{-\alpha }{c}_{l{\alpha }_{\perp }}\mid {\overrightarrow{\alpha }}_{\perp }〉.$$

Therefore one can express the signal (31) as

(34)$$\mid {\overrightarrow{s}}^o\left(t_k\right)〉=O_{\mathrm{ss}}\left[\left({P_{\alpha }}_0+{P_{\alpha }}_{\perp }\right){\mid \overrightarrow{P}}_s〉\right]{\Phi }_{\mathrm{CD}}\mid s_{\mathrm{in}}\left(t_k\right)〉=\left[\left({P_{\alpha }}_0+{P_{\alpha }}_{\perp }\right)\mid {\overrightarrow{P}}_s〉\right]\mid s^0\left(t_k\right)〉,$$

with |s^o(t_k)〉 given by (19). Because each of the three (2L+1)-dimensional matrices in (34) is diagonal, it can be commutated with any of other two matrices in (34). The electrically filtered signal-signal beating in (20) can be obtained easily by replacing |s^o(t_k)i in the first part of (25) by |s⃗^o(t_k)〉 and using (34), which yields

(35)$$y_{\mathrm{ss}}=〈{\overrightarrow{s}}^o\left(t_k\right)\mid R_{\mathrm{ss}}^D\mid {\overrightarrow{s}}^o\left(t_k\right)〉=〈s^o\left(t_k\right)\mid {\tilde{R}}_{\mathrm{ss}}^D\mid s^o\left(t_k\right)〉,$$

where the receiver related matrix is defined as ${\left({\tilde{R}}_{\mathrm{ss}}^D\right)}_{\mathrm{ll}\text{'}}={\left(R_{\mathrm{ss}}^D\right)}_{\mathrm{ll}\text{'}}{e}^{-\alpha }{\Theta }_{\mathrm{ll}\text{'}}(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s)$ with

(36)$${\Theta }_{ll\text{'}}=e^{\alpha }\left[c_{l{\alpha }_0}^{*}{c}_{l\text{'}{\alpha }_0}+e^{-2\alpha }{c}_{l{\alpha }_{\perp }}^{*}{c}_{l\text{'}{\alpha }_{\perp }}\right]\equiv {\Theta }_{\mathrm{ll}\text{'}}^{{\alpha }_0}+{\Theta }_{\mathrm{ll}\text{'}}^{{\alpha }_{\perp }},$$

(37)$${\Theta }_{\mathrm{ll}\text{'}}^i=e^{\pm \alpha }\left[e^{j\frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}}\tilde{A}({\overrightarrow{\tau }}_0,\pm {\overrightarrow{\alpha }}_0)+e^{-j\frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}}\tilde{A}\left(-{\overrightarrow{\tau }}_0,\pm {\overrightarrow{\alpha }}_0\right)\pm e^{j\frac{\left({\omega }_l-{\omega }_{l\text{'}}\right)\tau }{2}}\frac{\tilde{C}+j\tilde{D}}{4}\pm e^{-j\frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}}\frac{\tilde{C}-j\tilde{D}}{4}\right]$$
$$=\frac{e^{\pm \alpha }}{2}[\cos \frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}[1\pm \left({\overrightarrow{\tau }}_0·{\overrightarrow{p}}_s\right)\left({\overrightarrow{\tau }}_0·{\overrightarrow{\alpha }}_0\right)]+j\sin \frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}({\overrightarrow{\tau }}_0·{\overrightarrow{p}}_s\pm {\overrightarrow{\tau }}_0·{\overrightarrow{\alpha }}_0)$$
$$\pm \left(\cos \frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\tilde{C}-\sin \frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\tilde{D}\right)]\left(+:i={\alpha }_0;-:i={\alpha }_{\perp }\right)$$

and R^D_s s being given by (27). In (37), Ã(τ⃗₀,α⃗₀)=(1+τ⃗₀·α⃗₀)(1+τ⃗₀,p⃗s)/4, C̃=p⃗s·[τ⃗₀×(α⃗₀×τ⃗₀)] and D̃=τ⃗₀·(p⃗_s×α⃗₀). Phase factor $e^{j\frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}}$ in (37) is caused by the beating between the PMD induced slow mode of ω_l and slow mode of ω_l′, while $e^{-j\frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}}$ is due to the beating between fast modes of ω_l and ω_l′. Other phase factors $e^{j\frac{\left({\omega }_l-{\omega }_{l\text{'}}\right)\tau }{2}}$ and $e^{-j\frac{\left({\omega }_l-{\omega }_{l\text{'}}\right)\tau }{2}}$ are related with the beatings between fast and slow modes due to the PMD-PDL interaction [5]. Note that in (29)–(37), as mentioned in Ref. [5], a vector in Dirac notation |p⃗_s〉 or |k⃗_i〉 (k=τ,α; i=0,⊥) is used to represent a 2D Jones vector, while a vector in standard vector notation p⃗_s or k⃗_i stands for a 3D Stokes vector. Relationships connecting these two different kinds of vectors are 〉u⃗|σ⃗·A⃗|u⃗〉=u⃗〉·A⃗ and |u⃗〉〈u⃗|=[1+σ⃗·u⃗]/2 [3, 5, 13, 14], which are used to get (35)–(37). Here |u⃗〉 is any unitary Jones vector and A⃗ is any Stokes vector. When the Poincaré sphere’s polar axis is set in the direction of u⃗₀, we obtain $\cos \frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\tilde{C}-\sin \frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\tilde{D}=\sin {\theta }_{\alpha \tau }\sin {\theta }_{s\tau }\cos \left({\phi }_{\alpha \tau }-{\phi }_{s\tau }+\frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\right)$ and

(38)$${\Theta }_{ll\text{'}}(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s)=sh\alpha \sin {\theta }_{\alpha \tau }\sin {\theta }_{s\tau }\cos \left({\phi }_{\alpha \tau }-{\phi }_{s\tau }+\frac{\left({\omega }_l+{\omega }_{l\text{'}}\right)\tau }{2}\right)$$
$$+\cos \frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}\left(\mathrm{ch}\alpha +\mathrm{sh}\alpha \cos {\theta }_{\alpha \tau }\cos {\theta }_{s\tau }\right)+j\sin \frac{\left({\omega }_{l\text{'}}-{\omega }_l\right)\tau }{2}\left(\mathrm{ch}\alpha \cos {\theta }_{s\tau }+\mathrm{sh}\alpha \cos {\theta }_{\alpha \tau }\right)$$

where θ_ατ(θ_sτ) and φ_ατ (φ_sτ) are polar and azimuthal angles between α⃗ (p⃗_s) and τ⃗ in 3D Stokes space. For a system with given τ and α, the PMD-PDL-induced factor $e^{-\alpha }{\Theta }_{ll\text{'}}^i(\overrightarrow{\tau },\overrightarrow{\alpha };\overrightarrow{p}s)$ (i=α₀,α⊥) is determined by the relative directional relations between the input signal polarization p⃗_s, the PMD vector τ⃗₀ and the PDL vector α⃗₀. In this context, we called it the PMD-PDL-induced directional coupling factor. For a system with α=τ=0 (and θ_ατ=0), (38) yields e-αΘll′ (τ⃗,α⃗p⃗_s)=1.

Because of the random property of the noise polarization, the input ASE noise at “4” in Fig. 1 cannot be factorized as its magnitude part and its direction part. However, we can decompose it into two orthogonal components, e.g., $\mid {\overrightarrow{N}}_{\mathrm{in}}〉=\mid N_{\mathrm{in}}^{{\alpha }_0}〉\otimes \mid {\overrightarrow{\alpha }}_0〉+\mid N_{\mathrm{in}}^{{\alpha }_{\perp }}〉\otimes \mid {\overrightarrow{\alpha }}_{\perp }〉$. Similarly, the corresponding “canonical” noise at “4” in Fig. 1 can be written as

(39)$$\mid \overrightarrow{Z}〉=U^{†}\mid {\overrightarrow{N}}_{\mathrm{in}}〉=\mid Z_{{\alpha }_0}〉\otimes \mid {\overrightarrow{\alpha }}_0〉+\mid {Z_{\alpha }}_{\perp }〉\otimes \mid {\overrightarrow{\alpha }}_{\perp }〉.$$

As introduced in Appendix A, U ^† is the Hermitian matrix of U, which is used to diagonalize O†_nnR^D_{n n}O_nn. Thus the filtered current caused by noise-noise beating yields

(40)$$y_{\mathrm{nn}}=〈{\overrightarrow{N}}_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}{R}_{\mathrm{nn}}^D{O}_{\mathrm{nn}}\mid {\overrightarrow{N}}_{\mathrm{in}}〉=〈\overrightarrow{Z}\mid {\Lambda }^D\mid \overrightarrow{Z}〉=\sum _{m=-M}^M\left({\mid {\left({Z_{\alpha }}_0\right)}_m\mid }^2+{\mid {\left({Z_{\alpha }}_{\perp }\right)}_m\mid }^2\right){\lambda }_m^D,$$

where Λ^D and λ^D_m are given between (25) and (26).

For the same reason, the transformed field |b^D〉=B^D|so(t_k)i in (25) now becomes

(41)$$\mid {\overrightarrow{b}}^D\left(t_k\right)〉=B^D[P(\overrightarrow{\tau },\overrightarrow{\alpha })\mid {\overrightarrow{p}}_s〉]\mid s^o\left(t_k\right)〉=B^D[{P_{\alpha }}_0\mid {\overrightarrow{p}}_s〉]\mid s^o\left(t_k\right)〉+B^D[{P_{\alpha }}_{\perp }\mid {\overrightarrow{p}}_s〉]\mid s^o\left(t_k\right)〉$$
$$\equiv \mid {\tilde{b}}_{{\alpha }_0}^D〉\otimes \mid {\overrightarrow{\alpha }}_0〉+\mid {\tilde{b}}_{{\alpha }_{\perp }}^D〉\otimes \mid {\overrightarrow{\alpha }}_{\perp }\mid$$

with B^D given by (26) and P_i|p⃗_s′ (i=α₀,α⊥) by (33). So the electrically filtered noise-signal beating can be written as $y_{\mathrm{ns}}=〈{\overrightarrow{N}}_{\mathrm{in}}\mid O_{\mathrm{nn}}^{†}{R}_{\mathrm{ns}}^D\left[\left({P_{\alpha }}_0+{P_{\alpha }}_0\right)\mid {\overrightarrow{p}}_s〉\right]\mid s^o\left(t_k\right)〉+c.c.$ or

(42)$$y_{\mathrm{ns}}=〈\overrightarrow{Z}\mid {\overrightarrow{b}}^D〉+c.c.=〈{Z_{\alpha }}_0\mid {\tilde{b}}_{{\alpha }_0}^D〉+〈{Z_{\alpha }}_{\perp }\mid {\tilde{b}}_{{\alpha }_{\perp }}^D〉+c.c.$$

with ${\left({\tilde{b}}_i^D\right)}_m=\sum _{l=-L}^L{\left(B^D\right)}_{\mathrm{ml}}{c}_{\mathrm{li}}{s}_l^o\left(t_k\right)\left(i={\alpha }_0,{\alpha }_{\perp }\right)$ and cl_i given by (33). Thus we obtain

(43)$${\mid {\left({\tilde{b}}_i^D\right)}_m\mid }^2=\sum _{l,l\text{'}}{s}_l^{o*}{\left(B^D\right)}_{\mathrm{ml}}^{*}{\left(B^D\right)}_{\mathrm{ml}\text{'}}{s}_{l\text{'}}^o{e}^{-\alpha }{\Theta }_{\mathrm{ll}\text{'}}^i(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s)$$

with Θill′ (τ⃗,α⃗p⃗_s) given by (36) and

(44)$${\mid {\tilde{b}}_m^D\mid }^2={\mid {\left({\tilde{b}}_{{\alpha }_0}^D\right)}_m\mid }^2+{\mid {\left({\tilde{b}}_{{\alpha }_{\perp }}^D\right)}_m\mid }^2=\sum _{l,l\text{'}}{s}_l^{o*}{\left(B^D\right)}_{\mathrm{ml}}^{*}{\left(B^D\right)}_{\mathrm{ml}\text{'}}{s}_{l\text{'}}^o{e}^{-\alpha }{\Theta }_{\mathrm{ll}\text{'}}(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s).$$

In formulas (35)–(36) and (43)–(44), the directional factors $e^{-\alpha }{\Theta }_{\mathrm{ll}\text{'}}^i(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s)\left(i={\alpha }_0,{\alpha }_{\perp }\right)$ not only affect the signal-signal beating as well as the noise-signal beating but also determine how the noise-signal beating is “projected” into the two orthogonal directions (|α⃗₀〉, |α⃗⊥〉). For example, when t=0, we can take τ⃗₀‖α⃗₀ (i.e., θατ=0) and get ${\Theta }_{\mathrm{ll}\text{'}}^i(\overrightarrow{\tau },\overrightarrow{\alpha };{\overrightarrow{p}}_s)=\frac{e^{\pm \alpha }}{2}\left(1\pm \cos {\theta }_{s\tau }\right)$, which yields ${\mid {\left({\tilde{b}}_{{\alpha }_0}^D\right)}_m\mid }^2={\mid b_m^D\mid }^2\frac{1+\cos {\theta }_{s\tau }}{2}$ and ${\mid {\left({\tilde{b}}_{{\alpha }_{\perp }}^D\right)}_m\mid }^2={\mid b_m^D\mid }^2\frac{e^{-2\alpha }\left(1-\cos {\theta }_{s\tau }\right)}{2}$ (cf. |bDm|2 given in Appendix A). Obviously, the “projected” noise-signal beating in each direction is determined by signal polarization p⃗_s and the PDL vector α⃗0.

Notice that in Appendix A |s^o(t_k)〉 is the optical field at the output of the optical filter. But in Appendix B it is not the magnitude of |s⃗_o(t_k)〉. In fact it is the magnitude of the optically filtered field without the influence of PMD and PDL. The effect of PMD and PDL has been included in the directional factors Θill′ (i=α₀,α⊥) given by (37), since the matrices P(τ⃗,α⃗) and O_ss in (34) and (41) can commutate with each other.

Appendix C: Filtered current in DPSK system with partially polarized noise

In this case, as the signal is transformed by the matrices similar to the PMD and PDL1 in Appendix B, the signal related terms y_ss in (35) and |b̃^D〉 in (42) are not affected, except α should be replaced by a_k. For the filtered noise-noise beating, we introduce (P_PDL2)_mm′≡dmm′_TPDL2. According to (32), it can be written as ${\left(P_{\mathrm{PDL}2}\right)}_{\mathrm{mm}\text{'}}={\delta }_{\mathrm{mm}\text{'}}[{\mid \overrightarrow{k}}_0〉〈{\overrightarrow{k}}_0\mid +e^{-{\alpha }_k}\mid {\overrightarrow{k}}_{\perp }〉〈{\overrightarrow{k}}_{\perp }\mid ],$, where |k⃗₀〉 and |k⃗⊥〉 are two orthonormal eigenvectors of T_PDL2. Also for this special case, we denote the “canonical” noise at “4” in Fig. 1 by |Z⃗_k〉=U†|N⃗_in〉. Similar to (39), it can be written as $\mid {\overrightarrow{Z}}_k〉=\mid Z_{k_0}〉\otimes \mid {\overrightarrow{k}}_0〉+\mid {Z_k}_{\perp }〉\otimes \mid {\overrightarrow{k}}_{\perp }〉$. Therefore, the filtered noise-noise beating y_nn have the form

(45)$$y_{\mathrm{nn}}=〈{\overrightarrow{N}}_{\mathrm{in}}\mid P_{\mathrm{PDL}2}^{†}{O}_{\mathrm{nn}}^{†}{R}_{\mathrm{nn}}^D{O}_{\mathrm{nn}}{P}_{\mathrm{PDL}2}\mid {\overrightarrow{N}}_{\mathrm{in}}〉$$
$$=〈{\overrightarrow{Z}}_k\mid P_{\mathrm{PDL}2}^{†}{\Lambda }^D{P}_{\mathrm{PDL}2}\mid {\overrightarrow{Z}}_k〉=\sum _{m=-M}^M\left[{\mid {Z_k}_0\mid }^2+{\left(\frac{k_{\perp }}{k_0}\right)}^0{\mid {Z_k}_{\perp }\mid }^2\right]{\lambda }_m^D.$$

In (45), as the value of α_k is assumed to be frequency independent, P_PDL2 is a constant diagonal matrix, which means U†P_PDL2U=_PPDL2. Due to the effect of the partially polarized noise, the noise-signal beating in (42) now becomes

(46)$$y_{\mathrm{ns}}=〈{\overrightarrow{Z}}_k\mid P_{\mathrm{PDL}2}^{†}\mid {\overrightarrow{b}}^D〉+c.c.=〈{Z_k}_0\mid {\tilde{b}}_{k_0}^D〉+\frac{k_{\perp }}{k_0}〈{Z_k}_{\perp }\mid {\tilde{b}}_{k_{\perp }}^D〉+c.c.,$$

where ${\mid {\left({\tilde{b}}_{k_0}^D\right)}_m\mid }^2$ and ${\mid {\left({\tilde{b}}_{k_{\perp }}^D\right)}_m\mid }^2$ are same as ${\mid {\left({\tilde{b}}_{{\alpha }_0}^D\right)}_m\mid }^2$ and ${\mid {\left({\tilde{b}}_{{\alpha }_{\perp }}^D\right)}_m\mid }^2$ in (43), except α now becomes α_k.

Acknowledgment

The authors acknowledge the financial support from Canadian funding agencies: NSERC and the Centers of excellence program: AAPN. They thank an anonymous reviewer for insightful comments and suggestions. They also thank Dr. John Cameron for reading the manuscript.

References and links

1. B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortion in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Select. Topics Quantum Electron. 6, 317–329 (2000). [CrossRef]  

2. I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, “A receiver model for optical fiber communication systems with arbitrarily polarized noise,” J. Lightwave Technol. 23, 1478–1490 (2005). [CrossRef]  

3. A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. 22, 1856–1871 (2004). [CrossRef]  

4. M. Shtaif and O. Rosenberg, “Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems,” J. Lightwave Technol. 23, 923–930 (2005). [CrossRef]  

5. L. Chen, Z. Zhang, and X. Bao, “Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach,” Opt. Express 15, 2106–2119 (2007). [CrossRef]   [PubMed]  

6. P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photon. Technol. Lett. 15, 840–842 (2003). [CrossRef]  

7. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990). [CrossRef]  

8. P. A. Humblet and M. Azizoglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991). [CrossRef]  

9. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering,” J. Lightwave Technol. 18, 1493–1503 (2000). [CrossRef]  

10. J. L. Rebola and A. V. T. Cartaxo, “Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach,” IEE Proc. Optoelectron. 152, 251–262 (2005). [CrossRef]  

11. J. Wang and J. M. Kahn, “Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection,” J. Lightwave Technol. 22, 362–371 (2004). [CrossRef]  

12. L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, “WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD,” Opt. Fiber Technol. 12, 276–281 (2006). [CrossRef]  

13. P. Lu, L. Chen, and X. Bao, “Polarization mode dispersion and polarization dependent loss for a pulse in single-mode fiber,” J. Lightwave Technol. 19, 856–859 (2001). [CrossRef]  

14. H. Kogelnik, L. E. Nelson, and J. P. Gordon, “Emulation and inversion of polarization-mode dispersion,” J. Lightwave Technol. 21, 482–495 (2003). [CrossRef]