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Abstract
This study analyzes the bit-error-rate characteristics of Stokes-vector modulation–demodulation systems employing optically preamplified receivers. We derive unified analytical formulae for the bit-error rate of binary, quad, and octal modulation formats, in which Stokes vectors are modulated in a binary manner per dimension. The receiver sensitivities estimated from the derived bit-error-rate formulae are examined by analyzing the probability-density function of noise in the Stokes-vector direct-detection receiver. In addition, the bit-error-rate formulae enable a comprehensive comparison of the sensitivity among various optical receivers, such as coherent, direct-detection, optically differential direct-detection, and Stokes-vector direct-detection receivers.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A Stokes-vector receiver can track random fluctuations in the state of polarization (SOP) of an optical signal by using low-complexity digital signal processing (DSP), even if it is based on direct detection (DD) [1,2]. Such a DD receiver followed by DSP offers an effective solution for high-capacity, cost-effective optical communication systems, because it allows us to employ the multilevel SOP-modulation format, the constellation of which is designed in the three-dimensional (3D) Stokes space [3–5]. In particular, an optically preamplified Stokes-vector receiver can compensate for its inherent branching loss and achieve sensitivity as high as the quantum-noise limit [6,7].
In this study, the 1D, 2D, and 3D Stokes-vector modulation formats are theoretically analyzed. Owing to our assumption that the modulation multiplicity is two per dimension, the 1D, 2D, and 3D modulation formats realize binary (1 bit/symbol), quad (2 bit/symbol), and octal (3 bit/symbol) modulations, respectively. When these formats are demodulated using optically preamplified Stokes-vector DD receivers, analytical expressions for the bit-error rate (BER) are derived; this, to the author’s best knowledge, is the first study to report on such a finding.
When the amplifier gain significantly exceeds unity, the receiver sensitivity can be estimated via the input number of photons per bit and input-referred amplifier noise in a manner independent of the amplifier gain, branching loss in the receiver, and receiver circuit noise. The quantum-noise-limited receiver sensitivities estimated from the derived BER formulae are examined by analyzing the probability-density function (PDF) of noise in the optically preamplified Stokes-vector DD receiver. Further, we discuss the effectiveness and limits of the Gaussian noise model for analyzing the BER characteristics. In addition, by using the derived BER formulae, the BER characteristics in the Stokes-vector modulation–demodulation system are comprehensively compared with those in a coherent system, an optically differential DD system, and an intensity modulation/DD (IMDD) system.
The remainder of this paper is organized as follows. Section 2 summarizes modulation and detection schemes for the Stokes vector, which are used in the BER analyses. Section 3 describes the derivation process of the analytical BER formulae for the 1D, 2D, and 3D Stokes-vector modulation formats. Section 4 presents the numerical analysis of the PDF of the receiver noise and elucidates the effectiveness and limits of the Gaussian noise model for analyzing the BER characteristics. In Sec. 5, we make a comprehensive comparison of the BERs among coherent, IMDD, optically differential DD, and Stokes-vector DD systems. Finally, Sec. 6 concludes the paper.
2. Modulation and detection schemes for the Stokes vector
2.1 Stokes vector
Let $E_x$ and $E_y$ be the $x$- and $y$-polarization components of the signal’s electric field, respectively. Then, Stokes vector $ \textbf {S}$ is defined as [8]
Conversely, by changing the basis of the electric-field vector on the $x$–$y$ plane, the Stokes vector can be transformed into
2.2 Modulation formats
Figure 1 illustrates the constellation maps analyzed in this study of the 1D binary, 2D quad, and 3D octal modulation formats in the Stokes space. In these formats, the modulation level per dimension is two. In this paper, we denote the average number of photons per symbol as $n_s$, and the average number of photons per bit as $n$. For the 1D, 2D, and 3D modulation formats, $n=n_s$, $n_s/2$, and $n_s/3$, respectively. The receiver sensitivity is given as the average number of photons per bit, $n$, at which a desired value of the BER is guaranteed, and is used to compare the performance of various multilevel modulation formats.
In the 1D modulation format (Fig. 1(a)), the $y$-polarization component, $E_y$, is modulated in the binary phase-shift-keying (BPSK) format such that $E_y= \pm \sqrt {n/2}$, whereas the $x$-polarization component, $E_x$, remains a continuous wave satisfying $E_x=\sqrt {n/2}$. Consequently, Eq. (1) demonstrates that $S_1=S_3=0$, and $S_2$ is modulated in a binary manner between $-n$ and $+n$.
In the 2D modulation scheme (Fig. 1(b)) [5], $E_y$ is modulated in the quadrature phase-shift-keying (QPSK) format such that $E_y = (\pm 1 \pm i)\sqrt {n/2}$, whereas $E_x$ is a constant equal to $\sqrt {n}$. Equation (1) demonstrates that the four constellation points constitute a square on the 2D $S_2$–$S_3$ plane, and each of the values of $S_2$ and $S_3$ is modulated in a binary manner such that $S_2, S_3=\pm \sqrt {2}n$.
The 1D and 2D modulation formats have the advantage that dispersion compensation is possible in the digital domain, because the unmodulated $E_x$ acts as a local oscillator light in the coherent receiver [5]. Therefore, the 1D and 2D modulation–demodulation schemes are sometimes called self-coherent schemes.
In the 3D modulation format (Fig. 1(c)) [9], $E_y$ is QPSK modulated, and $|E_x|^2$ and $|E_y|^2$ are intensity-modulated in the push–pull mode such that $|E_x|^2/|E_y|^2=(\sqrt {3}+1)/(\sqrt {3}-1)$ or $|E_x|^2/|E_y|^2=(\sqrt {3}-1)/(\sqrt {3}+1)$. Consequently, the eight constellation points constitute a cube, and each of the values of $S_1$, $S_2$, and $S_3$ is modulated in a binary manner such that $S_1, S_2, S_3 = \pm \sqrt {3}n$.
2.3 Optically preamplified Stokes-vector DD receiver
Figure 2 shows the configuration of the optically preamplified Stokes-vector DD receiver, which is obtained from Eq. (2) [6] and is used for the BER analyses described in Sec. 3. After passing through an optical preamplifier, which is SOP-independent, the incoming signal is equally split into three branches. In the first, the polarization beam splitter (PBS) splits the signal into its $x$- and $y$-polarization components. In the second, the PBS following a half-wave ($\lambda /2$) plate splits the signal into the $45^\circ$- and $135^\circ$-linearly-polarized components. In the third, the PBS following a quarter-wave ($\lambda /4$) plate splits the signal into the right- and left-circularly-polarized components. Balanced photodiode (BPD) outputs of $I_1$, $I_2$, and $I_3$ result in the following elements of the Stokes vector: $S_1$, $S_2$, and $S_3$, respectively. The BPD outputs are sent to the DSP circuit for polarization tracking and symbol discrimination [2,3]. In the following sections, we assume that random fluctuations of SOP are fully compensated in the digital domain.
3. BER formulae
First, we consider the model of the optical preamplifier for the Stokes-vector DD receiver. When the optical preamplifier with a gain $G$ is installed in front of the Stokes-vector receiver, the number of output photons is given as $Gn_s+(G-1)n_{sp}$ per polarization, where $n_{sp} (\geq 1)$ denotes the spontaneous emission factor of the optical preamplifier [10,11], as shown in Fig. 3(a). Then, dividing the number of output photons by the known gain $G$, we can estimate the the number of input photons as $n_s+(G-1)n_{sp}/G$. This estimated number of input photons is approximately given as $n_s+n_{sp}$ when $G \gg 1$, which means that the signal having $n_{s}$ photons, together with the input-referred amplifier noise having $n_{sp}$ photons, is considered to be incident on the noise-free optical preamplifier, as shown in Fig. 3(b). Since the signal-to-noise ratio at the optical preamplifier input, $n_s/n_{sp}$, in Fig. 3(b) is not affected by the amplifier gain and optical loss of the receiver, the BER characteristics are analyzed in terms of $n_s$ and $n_{sp}$ [6,7]. In the following BER analyses, however, we use $n$ instead of $n_s$, because the number of input photons per bit is a better measure to make a comparison among receiver sensitivities of various multilevel modulation formats.
Fig. 3. Model of the optical preamplifier for the Stokes-vector DD receiver. (a): The input-output relation of the optical preamplifier. (b): The concept of the input-referred amplifier noise. We consider that the signal having $n_s$ photons and the input-referred amplifier noise having $n_{sp}$ photons are incident on the noise-free optical preamplifier.
Next, we derive the BER formulae for the 1D, 2D, and 3D Stokes-vector modulation formats based on a method similar to that used for analyzing the differential BPSK/QPSK signals [12–14].
According to Eq. (2), each element of the Stokes vector is expressed as the intensity difference between two orthogonal SOPs. We express signals $s_1$ and $s_2$, the SOPs of which are orthogonal to each other, as
In order to analyze the Stokes parameter $S_1$, let $s_1$ and $s_2$ be $E_x$ and $E_y$, respectively. Meanwhile, for analyzing $S_2$, $s_1$ and $s_2$ should be $E_{45^{\circ }}$ and $E_{135^{\circ }}$, respectively, and for analyzing $S_3$, $s_1$ and $s_2$ should be $E_L$ and $E_R$, respectively.In Eqs. (3) and (4), $A_{1,2}$ represents the complex amplitudes of the signals, and real axes of the complex plane representing $s_{1,2}$ are taken as the directions of the vectors $A_{1,2}$. On the other hand, $x_{1i,2i}$ and $x_{1q,2q}$ are the in-phase and quadrature components of the input-referred optical amplifier noise, respectively, which can be regarded as Gaussian random variables. The correlation relations for $x_{1i,2i}$ and $x_{1q,2q}$ are represented as
Given $|A_2|^2 \le |A_1|^2$, errors occur when $R_1^2<R_2^2$. Consequently, the error probability is calculated as
In the 1D modulation format (Fig. 1(a)), the constellation point is assumed to be set at $[0, 1, 0]^T n$, where $E_x=E_y=\sqrt {n/2}$. Then, given $S_2=|E_{45^{\circ }}|^2-|E_{135^{\circ }}|^2$, we have
In such a case, Eq. (15) is reduced to the following simple form: For the 2D modulation (Fig. 1(b)), we consider the $S_2$ modulation and assume that the constellation point is set at $[0, \sqrt {2}, \sqrt {2}]^T n$, where $E_x$ and $E_y$ satisfy Consequently, we have The $S_3$ modulation yields the same BER result.In the 3D modulation format (Fig. 1(c)), we consider the $S_1$ modulation and focus on the constellation point at $[\sqrt {3}, \sqrt {3}, \sqrt {3}]^T n$, where $E_x$ and $E_y$ are expressed as
In this way, Eq. (15) is commonly used to calculate the BER characteristics of the three formats of the Stokes-vector modulation; the values of $a^2$ and $b^2$ are summarized in Tab. 1.
Table 1. Values of $a^2$ and $b^2$ in the BER formula given by Eq. (15).
Figure 4 shows the BERs as a function of the average number of photons per bit, $n$, when $n_{sp}=1$. The black, red, and blue solid curves represent the BERs calculated using Eq. (15) and the values in Tab. 1. The black, red, and blue colors indicate the 1D, 2D, 3D modulation formats, respectively. (The green curve shows the BER obtained when the PDFs of the modulation signals are approximated as Gaussian distributions, as discussed in Sec. 4.) Conversely, the circles represent computer-simulation results. We determined that the BER curves calculated using the derived BER formulae are in perfect agreement with those obtained through computer simulations.
Fig. 4. BER characteristics as a function of the average number of photons per bit. The black, red, and blue colors correspond to the 1D, 2D, and 3D modulation formats, respectively. The solid curves are obtained using the BER formulae, whereas the circles represent simulation results. The green curve shows the BER obtained when the PDFs of the modulation signals are approximated as Gaussian distributions.
In addition, note that the BER characteristics of the 2D and 3D modulation formats are very similar, and those of the 1D modulation format are much better, as discussed in Sec. 4 with respect to the PDF of the modulation signals.
4. PDF of noise
The difference in the BER characteristics between the 1D modulation format and 2D/3D modulation formats, as shown in Fig. 4, is derived from the difference in the PDF of the signals.
The blue curves in Figs. 5, 6, and 7 show the computer-simulation results of the PDFs for the 1D, 2D, and 3D modulation signals, respectively. Figure 5 shows the PDF of the 1D modulation signal along the $S_2$ axis when $n_s=20$ and $n_{sp}=1$. Figure 6 shows the PDF of the 2D modulation signal along the $S_2$ or $S_3$ axis when $n_s=40$ and $n_{sp}=1$. The PDF of the 3D modulation signal shown in Fig. 7 was obtained along the $S_1$, $S_2$, or $S_3$ axis, when $n_s=60$ and $n_{sp}=1$. In all the three aforementioned cases, the photon number per bit, $n = 20$.
Fig. 5. Probability-density function of the 1D modulation signal along the $S_2$ axis. The blue curve denotes the computer-simulation result, and the red curve shows the result obtained via the Gaussian approximation, when $n_s=20$ and $n_{sp}=1$.
Fig. 6. Probability-density function of the 2D modulation signal along the $S_2$ or $S_3$ axis. The blue curve denotes the computer-simulation result, and the red curve shows the result obtained through the Gaussian approximation, when $n_s=40$ and $n_{sp}=1$.
Fig. 7. Probability-density function of the 3D modulation signal along the $S_1$, $S_2$, or $S_3$ axis. The blue curve denotes the computer-simulation result, and the red curve shows the result obtained through the Gaussian approximation, when $n_s=60$ and $n_{sp}=1$.
By disregarding the spontaneous–spontaneous beat noise generated from direct detection of the optical amplifier noise, the PDF of the input-referred Stokes vector can be approximated as a 3D Gaussian distribution, the variance per dimension of which is $2n_{s}n_{sp}$ [6,7]. The red curves in Figs. 5, 6, and 7 denote the PDFs obtained under the Gaussian approximation, where $n_{sp}=1$. As observed, the Gaussian approximation is almost valid for the 2D and 3D modulation signals, whereas the PDF for the 1D modulation signal has a steeper slope than that obtained through the Gaussian approximation when $S_2$ approaches zero. The parameter $S_0$ for the 1D modulation signal approaches zero as $S_2$ becomes close to zero. In this region, it is not possible to ignore the spontaneous–spontaneous beat noise, and the Gaussian approximation fails. In contrast, for the 2D and 3D modulation signals, $S_0$ never approaches zero even if each Stokes parameter is reduced to zero; hence, the Gaussian approximation is valid.
The steeper slope of the PDF of the 1D modulation format contributes toward the much better BER characteristics of this format than those of the 2D and 3D modulation formats. The green curve in Fig. 4 shows the BER for the 1D, 2D, and 3D modulation formats obtained via the Gaussian approximation of the PDF. In this calculation, we employed the BER formula under the Gaussian approximation, expressed as [10]
where $\hbox {erfc} \left (* \right )$ denotes the complementary error function. We determined that the BER curves for the 2D and 3D modulation formats are close to their respective Gaussian approximations; however, that for the 1D modulation format significantly differs from its corresponding approximation.5. Comparison of BER characteristics
Figure 8 shows the comparison among the BER characteristics of the binary IMDD system (black curve), coherent BPSK/QPSK system (red curve), optically differential BPSK system (blue solid curve), optically differential QPSK system (blue broken curve), 1D Stokes-vector modulation system (green solid curve), 2D Stokes-vector modulation system (green broken curve), and 3D Stokes-vector modulation system (green dash-dotted curve). For the IMDD and optically differential DD systems, we considered only the single-polarization component of the preamplifier noise as an ideal case; however, the effect from the preamplifier noise in the orthogonal SOP is negligibly small [15].
Fig. 8. Comprehensive comparison of the BER characteristics. Black, IMDD; red, coherent BPSK/QPSK; blue solid curve, differential BPSK; blue broken curve, differential QPSK; green solid curve, 1D Stokes; green broken curve, 2D Stokes; and green dash-dotted curve, 3D Stokes.
All the BER curves in Fig. 8 were calculated using analytical formulae. The BER formulae for the systems other than the Stokes-vector modulation–demodulation system are summarized as follows. The BER of the coherent BPSK/QPSK system is calculated as [11]
The BER formula for the binary IMDD system is calculated as [16]Thus, the receiver sensitivities of the various coherent and noncoherent systems are evaluated comprehensively, and the following points should be noted.
- (2) The sensitivity difference between the optically differential BPSK and Stokes-vector 1D modulation formats is 3 dB. In addition, the sensitivity difference between the optically differential QPSK format and Stokes-vector 2D modulation format is 3 dB. This 3-dB sensitivity difference is due to the fact that only half the signal power is used for 1D/2D modulation of the Stokes vector. Although the Stokes-vector modulation–demodulation system has a lower receiver sensitivity than the optically differential DD system by 3 dB, its dispersion compensation capability is advantageous.
- (3) The IMDD and Stokes-vector 1D modulation–demodulation systems have almost the same receiver sensitivity. This is because Eq. (29) is approximated as $p_e \simeq (1/2)\exp (-n/2)$, which is the same as the BER formula of Eq. (18) for the Stokes-vector 1D modulation format, in the region where $n$ is significantly larger than unity. The receiver sensitivity of the Stokes-vector 1D modulation–demodulation system is lower than that of the coherent system by about 3 dB. However, it has the advantage that dispersion compensation can be achieved in the digital domain, even if it is based on DD.
6. Conclusion
In this study, we analyzed the BER characteristics of Stokes-vector modulation–demodulation systems, which employ optically preamplified receivers. This is the first study to derive the analytical BER formulae for the 1D binary, 2D quad, and 3D octal Stokes-vector modulation formats, thus enabling a comprehensive comparison among the receiver sensitivities of the various optical communication systems: the coherent, IMDD, optically differential DD, and Stokes-vector DD systems. The difference between the receiver sensitivities of the 1D and 2D/3D Stokes-vector modulation formats was also examined by analyzing the PDF of the noise in the receiver.
Disclosures
The author declares that there are no conflicts of interest related to this article.
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