The implementation agreement of the Optical Internet Forum for a dual polarization (DP) I/Q downconverter defines strict requirements for the phase diversity network, resulting in a negligible penalty, but does not specify the extinction ratio (ER) of the polarization beam splitters (PBS) on which the polarization diversity network is based. We propose a novel metric, based on the Frobenius norm of the Jones receiver matrix, to accurately estimate the sensitivity penalty from receiver non-idealities, stablishing a precise interface for hardware specification. Results will be numerically verified for the reception of 112 Gbps DP-QPSK signals in a realistic receiver scenario with subsequent state-of-art DSP algorithms. The proposed metric highlights the benefits of the polarization diversity scheme based on two PBS, compared to the common alternative based on a PBS and a BS, as it achieves an improvement in the receiver sensitivity of at least 3 dB for the same ER. Furthermore, this paper shows than the sensitivity penalty is negligible for an ER higher than 16 dB and that it less than 2 dB for an ER of 8 dB. These results can have an important impact in monolithically integrated DP downconverters in which practical integration of PBS with high ER is still challenging.
© 2015 Optical Society of America
Polarization diversity in digital coherent receivers allows to compensate the detrimental effects of polarization mode dispersion (PMD) and dynamic signal’s state of polarization (SOP) variations in the optical channel. This makes it possible to double the transmission capacity by multiplexing two orthogonal polarizations. The Optical Internet Forum (OIF) has recently supported the digital coherent receiver with an implementation agreement for applications at 100 Gbps under Dual-Polarization (DP) QPSK transmission . The receiver relies on a polarization diversity network, based on polarization beam splitters (PBS), and two phase diversity downconverters, comprising 90° hybrids and balanced detection, for the demodulation of in-phase and quadrature (I/Q) components of each polarization.
Under an ideal receiver implementation, the digital signal processing (DSP) block is usually described as the inverse of the optical channel Jones matrix [2, 3], thus enabling perfect polarization demultiplexing. However, in a real DP I/Q receiver it is necessary to estimate the sensitivity degradation induced by non-ideal performance of its components. The penalty induced by imbalances in the phase diversity downconverters has been widely studied in the bibliography [4–6], and the sensitivity degradation has been quantified for practical realizations, particularly when orthogonalization digital algorithms, such as Gram-Schmidt orthogonalization procedure (GSOP), are carried out. Although the OIF defines strict requirements for the maximum amplitude and phase imbalance of the phase diversity downconverters (which result in a negligible penalty), no specifications for the extinction ratio (ER) of the PBS in the polarization diversity network exists. When the polarization diversity network is implemented by means of bulk optic components this is unimportant, because PBSs with ER much greater than 20 dB are available and the induced degradation is, in this case, negligible. However, when coming up to monolithically integrated receivers this becomes a crucial issue because high ER PBSs, with reasonable fabrication tolerances, are still lacking. Thus it seems necessary to develop a clear procedure to quantify the influence of PBS imbalances in DP coherent receiver performance. Up to the author’s knowledge, this procedure was still lacking in the literature.
In this paper, it will be shown that the Frobenius norm of the downconverter Jones matrix is an appropriate estimator of the penalty caused by a non-ideal polarization diversity network. This is an important result because such novel metric provides a straightforward evaluation of the influence of downconverter hardware impairments on the complete receiver performance. Furthermore, it will be shown that this metric allows to compute the downconverter induced power penalty avoiding the necessity of implementing all the DSP algorithms that are required in the complete receiver. This way, the Frobenius norm allows defining a clear boundary between the downconverter hardware and the subsequent receiver subsystems, thus stablishing a precise interface for hardware specification. Using this metric, the effects of finite ER of PBS on the receiver sensitivity will be compared for two well-known phase diversity architectures included in the OIF implementation agreement: the PBS + PBS scheme (PBS in the signal and local oscillator path) and the PBS + BS scheme (PBS in the signal path and a beam-splitter in the local oscillator path). Assessment of this metric will be carried out by comparison with results from numerical simulations of the complete receiver in a realistic 112 Gbps DP-QPSK transmission scenario including all the standard DSP algorithms. It will be shown that Frobenius norm clearly puts into evidence that the PBS + PBS scheme is far superior to PBS + BS, as it offers a 3 dB sensitivity improvement in an ideal situation. Although this fact is possibly known, it does not appear explicitly reported in the bibliography. Moreover, the PBS + BS scheme is commonly used [2, 7, 8] without any reference to its sensitivity degradation. Furthermore, the proposed metric can easily determine the ER specification target for a maximum allowable sensitivity penalty. In this paper it is shown that the sensitivity penalty for the PBS + PBS scheme is negligible for an ER higher than 16 dB and that it is only 0.6 dB for an ER of 12 dB. Even more, it will be shown that even using PBSs with a very low ER of 8 dB (a value that has been traditionally considered unacceptable for practical applications), this architecture does only suffer a 2 dB sensitivity penalty. Comparing these values with the 3 dB sensitivity penalty of the PBS + BS architecture (even in the ideal case of ER = inf), it is clearly highlighted the advantages of the PBS + PBS scheme. These results can also have an important impact in DP monolithically integrated downconverter design, where practical integration of PBS with high ER is still challenging. In fact, although electromagnetic simulation shows that monolithically integrated PBSs can theoretically exceed an ER of 20 dB, practical implementations commonly obtain an ER of 10-15 dB in the C-band [9–11] and rarely reach 20 dB .
The paper is organized as follows: in Section II the simplified matrix description of a DP I/Q downconverter is presented. The Frobenius norm and the condition number are also formulated as a convenient tool to measure the sensitivity penalty and the DSP demultiplexing algorithms convergence time, respectively. The performance comparison of polarization diversity schemes for digital coherent reception will be assessed in Section III. Finally, Section IV provides the main conclusions.
2. Theoretical background
2.1 Simplified matrix description of a dual polarization coherent optical communication system
In this subsection we provide a simplified matrix description of a DP coherent optical communication system, but focusing our attention on the receiver side (see Fig. 1).
The fiber link is modeled as a dispersion-free Jones matrix with negligible polarization dependent losses 
The functionality of the DP I/Q downconverter is specified by the OIF in . This block includes all the elements usually integrated within a single box in commercial integrated coherent receivers: two 90° hybrids (one for each polarization), one (or two) PBSs, eight photo-detectors and four transimpedance amplifiers. It must be noticed that the local oscillator (LO) is rarely integrated, so it is placed outside this block. On the other hand, the electric noise generated by the optoelectronic stage has been explicitly drawn in Fig. 1 for convenience.
The baseband electric signals at the output of the downconverter can be expressed as7, 13].
It can be shown that for an ideal device, and . Even more, for any receiver of practical interest (e.g. a receiver with OIF compliant 90° hybrids), it can be assumed that . Therefore, the DP I/Q downconverter can be characterized by a single complex matrix . Introducing (1) in (2) and imposing that , we obtain
In this simplified description of the system, the main function of the DSP unit is to compensate the impairments introduced by the fiber link and the downconverter. Regardless of the specific algorithms implemented in the DSP, it can be characterized by a matrix , so the received symbol is obtained as
From the previous expression it is clear that the DSP should implement the inverse of the product of the system matrix by the fiber transformation matrix, i.e. . In doing so, the received symbol will be a good approximation to the transmitted symbol
It is worth to note that, for an ideal downconverter, the DSP simply performs the inverse of the fiber transformation matrix (1) . This operation can be calculated with good accuracy, since is a unitary matrix (i.e. , where ‘T’ is used to denote the transpose) and therefore it is well-conditioned.
Nevertheless, since the assumption of ideal DP I/Q downconverters cannot be done in practice, in the following we will center our attention on the effects of the impairments of non-ideal downconverters (i.e. ) on the receiver performance.
2.2 Description of the DP I/Q downconverter
In this subsection we briefly describe the building blocks of a DP I/Q downconverter and how these blocks are interconnected. The functional diagram of this device, defined according to the OIF implementation agreement , is depicted in Fig. 2.
The system matrix directly relates the downconverter baseband electrical outputs with the modulated optical input signal as
From Fig. 2 it is clear that in the signal path, a PBS is always used to separate the input signal into two orthogonal polarizations, each of which is fed to the corresponding 90° hybrid.
On the contrary, in the LO path the OIF implementation agreement gives two options to split the LO power to the two 90° hybrids: i) a 3 dB power beam-splitter (BS3dB) or ii) a PBS. Although the first option is easier to implement and frequently found in bibliography [2, 7, 8], it significantly degrades the receiver performance, as it will be shown in the results section. From now on, we will refer to these two options as the PBS + BS receiver and the PBS + PBS receiver, respectively.
The PBS and the BS3dB are modeled using the Jones matrix formalism. Assuming for simplicity a perfectly symmetric device, the Jones matrices for each of the two output ports of the device can be written as
For a perfect PBS (a = 1 and b = 0), ER is infinite and IL = 0 dB, whereas for an ideal 3 dB beam splitter BS3dB (a = b = 1/√2), ER = 0 dB (no polarization selectivity at all) and IL = 0 dB.
It must be noticed that the implementation agreement does not give a target value for the ER of the PBS. We will explore this point in the next sections.
The 90° hybrids can be modeled from the scattering parameters, defined between its six ports, which can be obtained at a specific wavelength from an electromagnetic simulator. The hybrids are usually characterized via two figures of merit, directly related to their scattering parameters [5, 14], perfectly defined in the implementation agreement : i) the common-mode rejection ratio (CMRR), that measures from the output photocurrents the power imbalance between the hybrid outputs and ii) the phase error between I-Q axis.
The noise introduced by the optoelectronic stage (i.e. photodiodes and amplifiers) is not included in the definition of the system matrix of the DP I/Q downconverter (), but it will be considered in the simulation results of the next section.
It is worth to note that analytic closed expressions for the system matrix can be provided for both types of receivers in some simplified cases. For example, if we assume ideal 90° hybrids, lossless splitters and photodiode responsivities equal to 1, very simple expressions can be obtained:
On the contrary, when all possible non-idealities of the different building blocks of the receiver are modeled, it is probably easier to calculate the complete system matrix with any circuit-simulator software, since the analytical derivation of is rather involved. In this work we have used the same in-house simulator already employed in some of our previous works .
2.3 Analytic evaluation of the sensitivity penalty of a DP I/Q downconverter based on the Frobenius norm
The Frobenius norm (also known as Euclidean norm) of a matrix is defined as the square root of the sum of the absolute squares of its elements 
It is well known that the Frobenius norm is very useful to estimate the effect of noise over a linear system . Given a linear system of equations , the effect of the noise in the independent vector over the solution can be easily estimated as , where represents the noise root-mean square (RMS) value.
If we apply this concept to the receiver scheme described in the previous section, we obtain
Since is a unitary matrix, it can be shown that, so
Finally, using the ideal receiver as reference (), with Frobenius norm , the power penalty of a realistic (non-ideal) receiver can be easily calculated as
The above expression is one of the most important results of this work. It shows the additional signal power required at the receiver to compensate for the sensitivity degradation induced by the non-ideal performance of the downconverter. Its validity and accuracy will be checked through exhaustive numerical simulations in Section III.
2.4 Estimation of the convergence rate based on the condition number
The condition number in 2-norm of a square matrix is defined as 
It is well known that the condition number gives information about the accuracy of the matrix inversion operation. It is also known that the condition number is closely related to the convergence rates for conventional iterative optimization methods (e.g. the conjugate gradient algorithm) . This relation is also true for more complex algorithms, such as the stochastic gradient descent methods  to which belong the most popular algorithms to perform the polarization demultiplexing in the DSP: least-mean-square (LMS) or constant-modulus-algorithm (CMA) . We will numerically show in the next Section III the expected dependence between the convergence rate and the condition number of the downconverter system matrix. It is worth to note that exact convergence rate cannot be easily obtained from the condition number, since the convergence depends also on the algorithm, the optimization parameters, etc. Nevertheless, it is very useful for comparison between different receiver implementations sharing an identical DSP block.
3. Performance comparison of polarization diversity schemes for digital coherent reception
In this Section, the validity of the proposed metrics will be assessed by comparing them with simulation results in a realistic scenario, including polarization and phase diversity components and state-of-the-art DSP algorithms.
3.1 Simulation scenario
This paper focuses on DP-QPSK signal transmission at 112 Gbps (the symbol rate is 28 Gbaud/s). The arbitrary SOP of the signal at the receiver input is introduced with the simplified fiber optic model (1) from 10000 random channel realizations, being transmitted for each a sequence of 216 symbols. The amplified spontaneous emission (ASE) noise, modeled as an additive white Gaussian noise (AWGN), will degrade the incoming optical signal-to-noise ratio (OSNR) at the receiver.
To assure a realistic model of the coherent receiver, the polarization diversity network is based on non-ideal PBSs characterized by their ER with zero insertion losses. In the PBS + BS scheme the BS will be assumed ideal, whereas in the PBS + PBS scheme both PBS will have the same performance (equal ER).
As it refers to the 90° hybrid specification, an OIF compliant model has been selected with a CMRR of 20 dB and a phase error of 5° (specification limits for the operation in the C or L-band ).
Photodiodes are square-law detectors with additive shot noise. The thermal noise from the transimpedance amplifiers has been modeled with an input referred noise current density of 20 pA/√Hz [14, 18]. It has been considered that the power level of the LO, 10 dBm (unless otherwise specified), introduces a negligible relative intensity noise (RIN) contribution. The demodulated signals are synchronously sampled at the baud rate and digitized with 4 bit analog-to-digital converters (ADC). This ADC resolution offers a low quantization noise penalty, approximately 0.5 dB of OSNR penalty at a bit-error rate (BER) of 10−3 .
In the DSP block, only standard linear algorithms were used to determine the equalization matrix to correct system impairments. Amplitude and phase imbalances from the 90° hybrid based downconverters were corrected using the GSOP procedure [4, 14]. For the unitary channel matrix described by (1), the polarization demultiplexing was next performed with four 1-tap finite-impulse-response (FIR) filters arranged in a 2x2 butterfly configuration. Their coefficients were updated exploiting the properties of DP-QPSK signal from a blind CMA algorithm with a convergence parameter μ, unless otherwise specified, fixed to 10−3 [2, 3]. To prevent the well-known singularity problem of the CMA algorithm, where both outputs could converge to the same polarization, the matrix will be updated as soon as its determinant approaches zero with new weighting coefficients as proposed in . Once the equalizer has converged, a carrier phase recovery block from a Viterbi-Viterbi fourth power estimation will correct the phase offset of the demodulated signal prior to digital decision . The decoded sequence will be compared with the transmitted one to count the errors and calculate the BER. For each polarization diversity scheme, once the incoming OSNR is adjusted for a BER = 10−3, it will be also evaluated the convergence time of the polarization demultiplexing algorithm from the number of iterations/symbols (averaged over all channel realizations) required for the CMA error cost function to converge.
3.2 The Frobenius norm as a figure of merit of the power penalty due to non-ideal downconverter
Figures 3(a) and 3(b) depict the simulated BER versus signal power, in absence of optical ASE noise, as a function of certain ER values for the PBS + PBS and PBS + BS schemes, respectively. It has been considered a LO power of 20 dBm (shot-noise limited receiver) and 5 dBm (thermal-noise limited receiver) . These figures also show the BER performance for the ideal polarization diversity network implementation: i.e. for the PBS + PBS architecture using PBSs with infinite ER. It can be readily observed that the measured receiver sensitivity degradation is independent of the LO value and increases as the finite ER of non-ideal PBS is reduced. The sensitivity degradation induced by non-ideal 90° hybrids can be considered negligible since, as already published [4–6], the GSOP algorithm can compensate for practical imbalances. From Fig. 3(a), corresponding to the PBS + PBS architecture, a power penalty ΔPs of 0.25 dB, 0.6 dB and 1.6 dB is observed for ER values of 16 dB, 12 dB and 8 dB, respectively. For the PBS + BS architecture in Fig. 3(b), the power penalty is higher, increasing up to 3.5 dB, 4.1 dB and 5.8 dB for ER values of 16 dB, 12 dB and 8 dB, respectively.
Accordingly with the above, Fig. 4 shows with filled circles the sensitivity penalty versus the ER numerically obtained for each polarization diversity architecture in the coherent digital receiver under an AWGN channel model at a target BER = 10−3. The superimposed continuous line is evaluated from the Frobenius norm of the receiver matrix according to (12), thus confirming its validity as a figure of merit of the complete digital receiver.
For an ideal realization of each polarization diversity scheme, it is easily observed that the scheme PBS + BS will have an inherent sensitivity penalty of 3 dB respect to the PBS + PBS. This behavior can be expected from the 3 dB reduction of LO power per polarization component at the output of the BS that will be applied accordingly to the demodulated signal.
When facing a monolithic downconverter design, where only limited ER PBSs are available, important conclusions can be obtained from results in Fig. 4: using low quality PBSs (8 dB ER) the PBS + PBS architecture can still offer a 1.9 dB sensitivity improvement over the PBS + BS scheme using high quality PBSs (16 dB ER), which are usually not available in a monolithic platform.
3.3 Well-conditioning of the receiver matrix from the second-order condition number
Following the performance metrics commonly used in MIMO systems , with which the digital coherent receiver with polarization diversity is often compared [2, 3], the second-order condition number describes how well conditioned is the system matrix to perform its inversion at reception. Figure 5(a) shows the predicted condition number (in dB) versus the ER for each polarization diversity scheme. It can be observed that an ideal implementation (infinite ER) corresponds in both schemes to an ideal condition number equal to 0 dB, showing that it is not an appropriate metric for measuring the power penalty of the receiver.
On the contrary, the second-order condition number will be used here for its relationship with the convergence rate in stochastic gradient algorithms, such as the CMA algorithm used in this work. Thus, Fig. 5(b) depicts the average convergence time, normalized to that required in an ideal PBS + PBS realization with μ = 10−3, versus the condition number of each architecture. It can be seen that, irrespective of the polarization diversity scheme, receivers with a same condition number will need the same number of symbols to converge. This performance will be maintained with higher (lower) convergence parameter μ, although certainly the total convergence time will be reduced (increased).
It can be observed from both Fig. 5(a) and 5(b) that the unique PBS of the scheme PBS + BS will require twice ER (in dB) than the two PBS of the scheme PBS + PBS to perform the polarization demultiplexing at a same rate. Thus, the time required to converge in the schemes PBS + PBS with ER = 8 dB and PBS + BS with ER = 16 dB will be the same although, as it was previously discussed, the former can be easier monolithically integrated and has an improved sensitivity of 1.9 dB.
Nonidealities in the receiver front-end cause sensitivity degradation. The implementation agreement of the OIF for a DP I/Q downconverter defines strict requirements for the phase diversity network, resulting in a negligible penalty, but does not specify the ER of the PBSs on which the proposed polarization diversity networks are based. We have proposed a novel metric, based on the Frobenius norm of the receiver Jones matrix, to efficiently and accurately estimate the sensitivity penalty from receiver non-idealities. Thus, power penalty can be easily computed avoiding numerical simulation of the complete receiver (which should include subsequent DSP algorithms) and, consequently, a precise interface for hardware specification is stablished. Results have been verified in a realistic receiver scenario with state-of-the-art DSP algorithms under the reception of 112 Gbps DP-QPSK signals. The proposed metric highlights the benefits of the PBS + PBS compared to the PBS + BS scheme, as it achieves an improvement in the receiver sensitivity of at least 3 dB for a same ER. Furthermore, the estimation of the ER specification target for a maximum allowable sensitivity penalty is straightforward. Additionally, it has been verified numerically the usefulness of the second-order condition number of the system matrix as an estimator of the convergence rate of the digital demultiplexing algorithm. It has been found that the unique PBS of the PBS + BS receiver will require twice ER than the two PBSs of the PBS + PBS receiver to perform the polarization demultiplexing at a same rate. For instance, this paper shows than the sensitivity penalty for the PBS + PBS scheme is negligible for an ER higher than 16 dB and less than 2 dB for an ER of 8 dB. Thus this scheme improves in any case the receiver performance under an ideal implementation of the PBS + BS architecture with nearly the same digital polarization demultiplexing convergence time. These results can have an important impact in monolithically integrated DP downconverter design in which practical integration of PBS with high ER is still challenging.
This work was supported by the Spanish Ministry of Science under project TEC2013-46917-C2-1-R.
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