Doubling direct-detection data rate by polarization multiplexing of 16-QAM without active polarization control

Moshe Nazarathy; Amos Agmon

doi:10.1364/OE.21.031998

1. Introduction

Short-reach access and metro networks (e.g., point-to-point interconnects between data centers at ~100 km or long-reach PONs) are emerging cost-sensitive transmission applications wherein capacity requirements are high, yet coherent detection is prohibitive. Improvements in the capacity-complexity (cost) tradeoff would be welcome in this field; in particular cost-effective increase in the data rate over direct-detection (DD) links would be a laudable goal.

A family of self-coherent systems, mainly for OFDM DD [1,2], targets this application by coherent multiplexing of the I and Q field quadratures, based on remote pilot transmission and low-cost DD, without requiring a local oscillator (LO) laser in the receiver (Rx). The remote pilot is effectively used as a local oscillator, emulating coherent detection in the direct detection process. The remotely transmitted Pilot (P) and Data (D) signals (Fig. 1) beat in the DD photo-current, forming the desired DxP product, plus spurious DxD and PxP products which are mitigated by various methods. The relative spectral locations of the P and D components set a tradeoff between performance, complexity, spectral efficiency and whether just band-pass filtering or more advanced DSP is required to mitigate the DxD spurious signal (possibly also mitigation of transmitter-side quadratic non-linearities [3]).

Fig. 1 Received spectra for a self-heterodyne link with pilot and spectral gap. (a). Spectrum of the received optical field, comprising a Pilot (P) and modulated Data (D), partitioned into two spectral slots (the transmitted field spectrum is similar). The fields are taken here as scalar, for simplicity, but more generally the P and D terms may be resolved in X and Y polarization components.(b). Spectrum of the directly-detected photo-current, comprising three mixing terms. The desired term is PxD, falling in the [W,2W] spectral band. The PxP term is at DC, whereas the DxD spurious term is at baseband, [-W,W]. The spectral guardband ensures that the spurious PxP and DxD terms are spectrally disjoint from the desired PxD term, which may be electrically filtered out and demodulated to baseband.

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The challenge of enhancing self-coherent systems with polarization (POL) multiplexing (MUX) capability still eludes a satisfactory solution. The difficulty is that the remotely transmitted pilot POL randomly rotates in the fiber, and once this random POL signal reaches the Polarization Beam Splitter (PBS) in the Rx front-end (Fig. 2), the P signal at either of the PBS branches may (nearly) vanish. The received PxD terms then fades out, causing outages in either of the two X and Y POL components. To achieve POL-MUX DD despite the “pilot POL-fading” effect, various approaches have been explored [4–7], e.g., launching two orthogonally polarized pilots at different frequencies along with sophisticated large-rank MIMO Rx DSP.

Fig. 2 Self-heterodyne receiver options. (a) Conventional “reference” polarization-multiplexed direct-detection (laserless) receiver with transmitted optical pilot and spectral guard-band, suited to detect polarization diversity signals. The polarization is separated into its orthogonal components which are electrically demodulated to baseband and A/D converted. After some pre-DSP, the two polarization signals are 2x2 MIMO processed in order to demultiplex the two polarizations, followed by some post-DSP. This Rx functions intermittently in the POL-MUX mode, due to the pilot polarization fading impairment. However, this receiver may be adapted to be reliably used in the POL-DIV mode, by transmitting the same data on both polarizations and replacing the 2x2 MIMO by a 2x1 coherent “maximal ratio” combiner of the two polarizations. (b): Novel self-heterodyne POL-MUX receiver. Robustness to the polarization fading impairment is achieved by adding a third “X x Y” redundant measurement of the state of polarization. The 2x2 MIMO processing of the Rx in (a) is now replaced by 3x2 MIMO processing in order to recover the two independently transmitted polarization components. Abbreviations: OS: Optical Splitter, ODC: Optical Directional Coupler, EQZ: Equalizer

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Another POL-MUX approach by Infinera [8] features an optical pre-processing front-end comprising a relatively complex Photonic Integrated Circuit, 8 ADCs (if equalization is done digitally rather than in analog fashion) and 4x2 complex-valued MIMO processing.

We propose a far simpler Rx structure mitigating pilot POL-fading, endowing the DD self-coherent Rx with POL-MUX capability, with 3 ADCs and 3x2 MIMO processing. Our concept was briefly introduced in [9]. The novel principle is based on 2x3 optical front-end, redundantly projecting onto a non-orthogonal 3-dimensional basis, followed by DSP reconstruction. Here we elaborate on the theory of operation and present more extensive simulation results.

For ease of introduction, the concept is initially presented in sections 2-4 in continuous-time (all signals functions of t), under the assumption of a memory-less (frequency-flat) fiber channel free of Inter-Symbol Interference (ISI). Section 5 extends the treatment to more realistic optical channels which are not frequency-flat, e.g. in the presence of Polarization Mode Dispersion (PMD), Chromatic Dispersion(CD) and electrical analog filtering, which impart memory to the channel. POL-demultiplexing by frequency-domain 3x2 MIMO equalization is introduced for a (DFT-Spread) OFDM system and simulation results are presented, indicating that the POL fading impairment is effectively mitigated with smallSignal to Noise Ratio SNR) penalty and even with a small SNR gain, depending on the incoming POL state.

2. Review of conventional self-heterodyne direct-detection Rx – the POL fading problem

Let us consider the spectral allocation of Fig. 1 for a single channel in direct-detection link with pilot and spectral gap, referred to here as self-heterodyne (self-HET). The Transmitter (Tx) transmits a pilot at optical frequency $ν_{p}$ with POL components P_x, P_y of equal amplitude, along with independent or dependent data signals, D_x, D_y, falling in the optical passband $[ν_{p} + W, ν_{p} + 2 W]$ . The system requires a bandwidth of 2W (the spectral support of the overall transmitted signal is $[ν_{p}, ν_{p} + 2 W]$ ), just half of which is used to carry the modulated signal. A conventional DD POL-Diversity (POL-DIV, as opposed to POL-MUX, as explained in caption of Fig. 2) Rx for such transmitted signal features received signals with spectra shown in Fig. 1(a). This structure will be henceforth referred to as “reference POL-DIV self-HET Rx. It essentially comprises a PBS feeding two photo-diodes (PD) on the X and Y POL arms. The PDs are used to opto-electronically mix the data (D) and pilot (P) components.

The spectral gap $[- W, W]$ introduced in the electrical domain is used to absorb the spurious $D_{X / Y} \times D_{X / Y}$ and $P_{X / Y} \times P_{X / Y}$ mixing products generated in the quadratic photo-detection process. The useful PxD signal is generated in the electrical band [W,2W] and is electrically demodulated to baseband, by either analog or digital means. This is well-known but it is reviewed in the caption of Fig. 1 and generalized further below for two polarizations. If the modulation format for the D signal is OFDM, then the self-heterodyne scheme is referred to as direct-detection OFDM, however, the D signal may have any modulation format (e.g., Nyquist single-carrier), as long as it is confined to the same spectral band.

The analysis to follow points to a key impairment arising in this reference POL-MUX self-HET Rx; this limitation is subsequently mitigated by the proposed Rx of Fig. 2(b). The key impairment precluding POL-MUX is Pilot Polarization Fading: the pilot SOP may randomly transform into or close to either the X or Y linear POLs, depleting the other POL axis of LO, precluding $P \times D$ detection along the other POL axis. To analyze this effect, setting the motivation for our improved Rx, we introduce the following model.

The analytic-signal of the band-pass optical field for the respective X and Y POLs is expressed as,

d_{X | Y} (t) = {\underset{˜}{d}}_{X | Y} (t) e^{j 2 π (ν_{p} + 1.5 W) t}

centered onto the midpoint

ν_{p} + 1.5 W

of the spectral slot

[ν_{p} + W, ν_{p} + 2 W]

(here

|

designates “or” and complex-envelopes (CE) signals are denoted by undertildes). The remote LO (pilot tone, P) as received at the Rx input, is expressed as:

p_{X | Y} (t) = {\underset{˜}{p}}_{X | Y} e^{j 2 π ν_{p} t} .

The received photo-currents are obtained by absolute-squaring the optical fields. We are just interested in the mixing component in the SSB passband

[W, 2 W]

(with respect to a frequency origin at the optical carrier, or DC in the electrical domain), hence the DxD and PxP terms,

{| {\underset{˜}{d}}_{X | Y} (t) |}_{}^{2}, {| {\underset{˜}{p}}_{X | Y} (t) |}_{}^{2}

are blocked by electrical band-pass filters (BPF) in the IQ demod + BPF of the Analog Front-End (AFE), which lets through just the PxD cross-terms:

\begin{array}{l} u (t) = {| {\underset{˜}{d}}_{X} (t) + {\underset{˜}{p}}_{X} |}_{B P F}^{2} \propto Re {{\underset{˜}{d}}_{X} (t) {\underset{˜}{p}}_{X}^{*} e^{j 3 π W t}} \\ v (t) = {| {\underset{˜}{d}}_{Y} (t) + {\underset{˜}{p}}_{Y} |}_{B P F}^{2} \propto Re {{\underset{˜}{d}}_{Y} (t) {\underset{˜}{p}}_{Y}^{*} e^{j 3 π W t}} \end{array}

Next, IQ electrical detection (bandpass demodulation with cos and sin at RF frequency 1.5W in the AFE), amounts to extracting the CEs of the passband signals, yielding the following digital domain complex signals, corresponding to the two orthogonal received polarizations:

\underset{˜}{U} [t] = {\underset{˜}{p}}_{X}^{*} {\underset{˜}{d}}_{X} = p_{0} \cos θ_{p} {\underset{˜}{d}}_{X}; \underset{˜}{V} [t] = {\underset{˜}{p}}_{Y}^{*} {\underset{˜}{d}}_{Y} = p_{0} e^{- j ϕ_{p}} \sin θ_{p}

where the received SOP of the LO (pilot) was expressed as

[{\underset{˜}{p}}_{X}^{}, {\underset{˜}{p}}_{Y}^{}] = p_{0} [\cos θ_{p}, e^{j ϕ_{p}} \sin θ_{p}]

The pilot fading problem is now evident. Consistent detection of both POL signals in Eq. (4) is precluded, as either of the two components of the pilot LO (Eq. (5)) may randomly fade away, nulling out either of the two PxD products in Eq. (4). Thus, self-coherent reception is severely degraded as either

\underset{˜}{U} [t] or \underset{˜}{V} [t]

may occasionally (nearly) vanish. This fading occurs whenever

θ_{p} ~ U [0, 2 π)

approaches

θ_{p} = \frac{1}{2} π k; k = 0, 1, ...

, nulling out one of the two pilot components in Eq. (5). This is unlike in a coherent receiver, wherein the LO components along the two polarizations are fixed, whereas here the two LOs trade power with each other.

To mitigate the pilot POL fading impairment, one possible solution is to give up POL-multiplexing, i.e., not transmit independent data at all over the two complex envelopes ${\underset{˜}{d}}_{X}, {\underset{˜}{d}}_{Y}$ , losing a factor of two in spectral efficiency. POL-DIV operation may be adopted, launching the same complex data ${\underset{˜}{d}}_{0}$ , over both polarizations: ${\underset{˜}{d}}_{X} = {\underset{˜}{d}}_{Y} = {\underset{˜}{d}}_{0}$ . The two signals in Eq. (4) now become linearly dependent:

\underset{˜}{U} [t] = {\underset{˜}{p}}_{X}^{*} {\underset{˜}{d}}_{0} = p_{0} \cos θ_{p} {\underset{˜}{d}}_{0}; \underset{˜}{V} [t] = {\underset{˜}{p}}_{Y}^{*} {\underset{˜}{d}}_{0} = p_{0} e^{- j ϕ_{p}} \sin θ_{p} {\underset{˜}{d}}_{0}

As

θ_{p}

varies, since the two coefficients of

\underset{˜}{U} [t], \underset{˜}{V} [t]

are

90^{\circ}

out-of-phase, whenever one of them fades, the other one attains maximum absolute value, thus POL-DIV Rx processing is able to recover the complex data, irrespective of pilot POL fading. E.g., as proposed in [10], a 2x1 Dual-Input Single-Out (DISO) adaptive filter may be used in the Tx for coherent combining of the two signals of Eq. (6) with optimal combining coefficients,

{\tilde{c}}_{u} = \cos θ_{p}, {\tilde{c}}_{v} = e^{j ϕ_{p}} \sin θ_{p}

, yielding a recovered signal:

\begin{array}{l} \underset{˜}{r} [t] \equiv {\tilde{c}}_{u} \underset{˜}{U} [t] + {\tilde{c}}_{v} \underset{˜}{V} [t] = {\tilde{c}}_{u} p_{0} \cos θ_{p} {\underset{˜}{d}}_{0} + {\tilde{c}}_{v} p_{0} e^{- j ϕ_{p}} \sin θ_{p} {\underset{˜}{d}}_{0} \\ = \underset{˜}{r} [t] = p_{0} {\underset{˜}{d}}_{0} (\cos^{2} θ_{p} + \sin^{2} θ_{p}) = p_{0} {\underset{˜}{d}}_{0} \end{array}

The data

{\underset{˜}{d}}_{0}

is now reliably reconstructed, independent of the pilot and data received SOPs. Nevertheless, the penalty is that POL-MUX capability has been given up, in exchange for robustness in the wake of LO POL fading. Half the spectral efficiency is now lost, however, ~3 dB of SNR is gained due to the coherent combining processing gain.

We next propose a novel Rx structure, enabling to recover the second polarization, even when the two polarizations carry independent data, at the expense of modest investment in extra opto-electronic hardware and DSP.

3. Overview of the novel self-heterodyne direct-detection Rx supporting POL-MUX

The proposed self-coherent Rx with POL-MUX (Fig. 2(b)) is quite efficient in hardware. It comprises a PBS, two optical splitters, an optical directional coupler (ODC), 4 PDs, 3 Trans-Impedance Amplifiers (TIA) and 3 ADCs. Relative to the conventional POL-DIV Rx of Fig. 1(a), which supports just half the data-rate, we have just added a third optical path comprising an ODC, the balanced PD pair, a TIA and an ADC. The only change in DSP is to use a 3x2 POL-demux MIMO filter, rather than the conventional 2x2 MIMO. This entails using two extra complex-multipliers (CM) (increasing the CMs count from 4 to 6). The front-end optical elements are further amenable to photonic integration over a cost-effective passive photonic integrated circuit structure comprising just the PBS (e.g., a grating coupler) 2 OSs and ODC. However, even discrete-component realizations may be cost-effective, as the PBS, OS and ODC + balanced PD components are readily available with fiber pigtails in discrete form.

This relatively simple receiver structure enables doubling the spectral efficiency by concurrently transmitting independent data on the X and Y orthogonal polarizations.

We briefly overview here the principle of operation of the novel POL-MUX DD Rx (the explanation is further elaborated and quantified in the next section).

As already illustrated in Fig. 2(b) we have added a new path, featuring a $180^{\circ}$ hybrid, comprising a 3 dB coupler combining the signals from the X and Y POL arms, feeding a photo-diode (PD) balanced pair, a TIA and an extra ADC. This extra path generates a $X \times Y$ mixing product enabling to mitigate pilot POL-fading. The ${X, X \times Y, Y}$ ADC outputs yield baseband signals, $\underset{˜}{U} [t], \underset{˜}{W} [t], \underset{˜}{V} [t]$ after some pre-DSP. The 2x2 MIMO POL-demux DSP of a conventional coherent receiver is replaced here by 3x2 MIMO processing, as shown in Fig. 2(b), yielding two decoupled POL streams, ${\underset{˜}{d}}_{X} [t], {\underset{˜}{d}}_{Y} [t]$ optionally processed in a conventional Rx DSP back-end by any of the scalar self-coherent methods [1,2], applied to each of the two decoupled POLs. The new 3x2 MIMO equalizer has its coefficients adaptively adjusted, e.g. by means of a Least-Mean-Square (LMS) data-aided algorithm, further simulated below.

The key idea is that if the pilot fades out along one of the two polarization axes, it must then become strong along the other polarization axis, thus it is now going to be the new path, $X \times Y$ that will generate a reliable $P \times D$ signal. E.g. if the pilot fades along X (precluding detection of the X-data via $P_{X} \times D_{X}$ ) then the pilot must be strong along Y (i.e., $P_{Y}$ is strong), then $X \times Y$ now contains a strong $D_{X} \times P_{Y}$ term. Thus, rather than appearing on the X output, the reconstructed data, $D_{X}$ , now appears on new $X \times Y$ output. Similarly, if P fades along Y (precluding detection of the Y-data via $P_{Y} \times D_{Y}$ ) then $P_{X}$ must be strong, then the $X \times Y$ path contains a term $P_{X} \times D_{Y}$ , enabling to reliably detect $D_{Y}$ . These have been limiting cases wherein the pilot completely faded along one of the two POL axes. In these cases the 3x2 MIMO equalizer routes the $X \times Y$ output to the X output port whereas the Y output is routed to the Y output port. In contrast, when the pilot has nearly equal powers in its two polarization components, the three X, Y and $X \times Y$ measurements generated by the optical front-end are redundantly used to reliably recover received POL. The next section analyzes the process of reliable extraction of two POL components from the three inputs, irrespective of the data and pilot SOPs, establishing that in certain cases, it is possible to even improve the SNR relative to a 2x2 conventional system (which would only function well when the pilot has nearly equal powers in the two orthogonal received polarization).

4. Theory of operation

Our novel solution to mitigating POL-fading in direct detection with pilot + guardband is to add a $180^{\circ}$ hybrid yielding a third $" X \times Y "$ POL measurement electro-optically mixing signals from the two polarizations (Fig. 2(b)).

The 50/50 ODC outputs generate the sum and difference ${\underset{˜}{E}}_{X \times Y}^{\pm} (t) = {\underset{˜}{E}}_{X} \pm {\underset{˜}{E}}_{Y}$ of the two optical fields ${\underset{˜}{E}}_{X}^{} (t) = {\underset{˜}{d}}_{X} (t) e^{j 3 π W t} + {\underset{˜}{p}}_{X}^{}, {\underset{˜}{E}}_{Y} = {\underset{˜}{d}}_{Y} (t) e^{j 3 π W t} + {\underset{˜}{p}}_{Y}^{}$ fed into the ODC input ports. The new XxY branch output cross-term, $\underset{˜}{w} (t)$ , is given by the difference of the two PD photo-currents, reducing to the conjugate product of the two CE signals fed into the two $180^{\circ}$ hybrid ports:

\begin{array}{l} w (t) = \frac{1}{4} {{| {\underset{˜}{E}}_{X \times Y}^{+} (t) |}^{2} - {| {\underset{˜}{E}}_{X \times Y}^{-} (t) |}^{2}}_{B P F} = Re {{\underset{˜}{E}}_{X} {\underset{˜}{E}}_{Y}^{*}}_{B P F} \\ = Re {_{({\underset{˜}{d}}_{X} e^{j 3 π W t} + {\underset{˜}{p}}_{X}^{}) (^{{\underset{˜}{d}}_{Y} e^{j 3 π W t} + {\underset{˜}{p}}_{Y}^{}) *}} B P F} \\ = Re {_{({\underset{˜}{d}}_{X} e^{j 3 π W t} + {\underset{˜}{p}}_{X}^{}) (^{{\underset{˜}{d}}_{Y} e^{j 3 π W t} + {\underset{˜}{p}}_{Y}^{}) *}} B P F} \\ = Re {{\underset{˜}{d}}_{X} {\underset{˜}{p}}_{Y}^{*} e^{j 3 π W t}} + Re {{\underset{˜}{d}}_{Y} {\underset{˜}{p}}_{X}^{*} e^{j 3 π W t}} = Re {[{\underset{˜}{d}}_{X} {\underset{˜}{p}}_{Y}^{*} + {\underset{˜}{d}}_{Y} {\underset{˜}{p}}_{X}^{*}] e^{j 3 π W t}} \end{array}

Electrical IQ demod (including demodulation down to baseband) removes the

e^{j 3 π W t}

RF sub-carrier in the last expression, yielding the CE:

\underset{˜}{W} [t] = {\underset{˜}{p}}_{Y}^{*} {\underset{˜}{d}}_{X} [t] + {\underset{˜}{p}}_{X}^{*} {\underset{˜}{d}}_{Y} [t]

This third new front-end measurement now complements the two measurements of Eq. (4) to provide a complete set of three projections,

\underset{\underset{˜}{R}}{\underset{︸}{[\begin{matrix} \underset{˜}{U} \\ \underset{˜}{W} \\ \underset{˜}{V} \end{matrix}]}} = \underset{P}{\underset{︸}{[\begin{matrix} {\underset{˜}{p}}_{X}^{*} & 0 \\ {\underset{˜}{p}}_{Y}^{*} & {\underset{˜}{p}}_{X}^{*} \\ 0 & {\underset{˜}{p}}_{Y}^{*} \end{matrix}]}} \underset{d}{\underset{︸}{[\begin{matrix} {\underset{˜}{d}}_{X} \\ {\underset{˜}{d}}_{Y} \end{matrix}]}} = \underset{P}{\underset{︸}{[\begin{matrix} \cos θ_{p} & 0 \\ e^{- j ϕ_{p}} \sin θ_{p} & \cos θ_{p} \\ 0 & e^{- j ϕ_{p}} sin θ_{p} \end{matrix}]}} \underset{\underset{˜}{d}}{\underset{︸}{[\begin{matrix} {\underset{˜}{d}}_{X} \\ {\underset{˜}{d}}_{Y} \end{matrix}]}}

of the received data vector

\underset{˜}{d} \equiv {[{\underset{˜}{d}}_{X}^{}, {\underset{˜}{d}}_{Y}^{}]}^{T}

onto three (non-orthogonal) axes, expressed in matrix form as,

{\underset{˜}{R}}_{[3 \times 1]} = P_{[3 \times 2]} {\underset{˜}{d}}_{[2 \times 1]}

, where

{\underset{˜}{R}}_{[3 \times 1]}

is the measurements vector,

P_{[3 \times 2]}

is the analysis matrix and

{\underset{˜}{d}}_{[2 \times 1]}

is the received Jones vector. The 3x2 MIMO DSP task is to process the three projections

\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}

, in order to recover

\underset{˜}{d}

and subsequently,

{\underset{˜}{d}}^{T x}

, the Jones vector at the Tx. The task at hand is to “invert” the rectangular analysis matrix,

P_{[3 \times 2]}

, going from the three measurements back to the Jones vector,

{\underset{˜}{d}}_{[2 \times 1]}

. The analysis matrix is readily shown to have full column rank, i.e., its two columns are linearly independent for any pilot SOP. This implies that the 2-vector to 3-vector mapping,

P

, is injective (one-to-one), thus is invertible on its image (range). Any received 3-vector (including noise, throwing it outside the image of

P

) may be projected onto the image of

P

, then inverted. The matrix

P

admits a left-inverse, the Moore-Penrose Pseudo-Inverse (PI), given by

P_{[2 \times 3]}^{-} = {(P_{[2 \times 3]}^{†} P_{[3 \times 2]}^{})}^{- 1} P_{[2 \times 3]}^{†}

satisfying the left-inversion relation

P_{[2 \times 3]}^{-} P_{[3 \times 2]}^{} = 1_{[2 \times 2]}

.

The 3 rows of the analysis matrix, P are three vectors onto which the received Jones 2-vector, $\underset{˜}{d}$ , is projected, yielding the three “measurements”, $\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}$ . The three vectors are referred to in modern linear algebra as a frame [11], a redundant representation, as in 2-dimensional space, two orthogonal projections would suffice (whereas we have here three vectors, namely the three rows of $P$ . Frames are sets of over-complete (redundant) dependent vectors, generalizing the concept of linear algebra base (here the three rows of $P$ are not independent, but they span $ℂ^{2}$ ). Given a properly specified frame, a vector to be measured may always be reconstructed from the frame coefficients, namely the projections of the given vector onto the frame vectors. In the current finite-dimensional context, the $P \underset{˜}{d}$ matrix multiplication projects $\underset{˜}{d}$ -vector onto the three frame vectors, yielding the three frame coefficients $\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}$ . Here $\underset{˜}{W}$ is a new measurement, further to the two orthogonal components, $\underset{˜}{U}, \underset{˜}{V}$ , which are usually measured.

The full column rank of $P$ (having its two columns independent) is necessary in order to have its three rows form a frame, enabling a unique inverse transformation reconstructing the $\underset{˜}{d}$ vector from the projection measurements. The independence of the two columns of $P$ implies that there is a unique vector, $\underset{˜}{d}$ , yielding the measured frame coefficients $\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}$ , thus the frame projection may be always inverted from these three inner products of the vector $d$ with the three rows of $P$ .

Back to the matrix-oriented view, the PI is readily evaluated in terms of the Moore Penrose formula of Eq. (11). We first evaluate the Gram matrix and its inverse, which are invertible here, since $P_{[3 \times 2]}^{}$ has full column rank:

G_{[2 \times 2]}^{} \equiv P_{[2 \times 3]}^{†} P_{[3 \times 2]}^{} = p_{0} [\begin{matrix} 1 & e^{j ϕ_{p}} \cos θ_{p} \sin θ_{p} \\ e^{j ϕ_{p}} \cos θ_{p} \sin θ_{p} & 1 \end{matrix}]

G_{[2 \times 2]}^{- 1} \equiv {(P_{[2 \times 3]}^{†} P_{[3 \times 2]}^{})}^{- 1} = \frac{8 p_{0}^{- 1}}{7 + \cos [4 θ_{p}]} [\begin{matrix} 1 & - e^{j ϕ_{p}} \cos θ_{p} \sin θ_{p} \\ - e^{j ϕ_{p}} \cos θ_{p} \sin θ_{p} & 1 \end{matrix}]

Finally, the pseudo-inverse is given by:

\begin{array}{l} P_{[2 \times 3]}^{-} = {(P_{[2 \times 3]}^{†} P_{[3 \times 2]}^{})}^{- 1} P_{[2 \times 3]}^{†} \\ = \frac{8 p_{0}^{- 1}}{7 + \cos [4 θ_{p}]} [\begin{matrix} \cos θ_{p} & e^{j ϕ_{p}} \sin^{3} θ_{p} & - e^{j 2 ϕ_{p}} \cos θ_{p} \sin^{2} θ_{p} \\ - e^{- j ϕ_{p}} \cos^{2} θ_{p} \sin θ_{p} & \cos^{3} θ_{p} & e^{j ϕ_{p}} \sin θ_{p} \end{matrix}] \end{array}

The three columns

| p^{- (i)} 〉, i = 1, 2, 3

(designated using Dirac notation) of

P_{[2 \times 3]}^{-} \equiv [| p^{- (1)} 〉, | p^{- (2)} 〉, | p^{- (3)} 〉]

form a dual frame which may be used to reconstruct the received polarization vector,

\underset{˜}{d}

, from the frame coefficients,

\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}

:

{\underset{˜}{\hat{d}}}_{[2 \times 1]} = P_{[2 \times 3]}^{-} {\underset{˜}{R}}_{[3 \times 1]} = \underset{˜}{U} | p^{- (1)} 〉 + \underset{˜}{W} | p^{- (2)} 〉 + \underset{˜}{V} | p^{- (3)} 〉

Thus, the three dual frame vectors are linearly combined with the measured coefficients,

\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}

to reconstruct the original 2-vector projected onto the frame. Using Dirac bra-ket notation, the dual frame column vectors,

| p^{- (1)} 〉, | p^{- (2)} 〉, | p^{- (3)} 〉

, are bi-orthogonal to the original frame row vectors,

〈 p^{(1)} |, 〈 p^{(2)} |, 〈 p^{(3)} |

, which in turn are the rows of the analysis matrix,

P_{[3 \times 2]}^{} \equiv [〈 p^{(1)} |, 〈 p^{(2)} |, 〈 p^{(3)} |]

. The bi-orthogonality is formally expressed as

〈 p^{(i)} | p^{- (j)} 〉 = δ_{i j}

. Computationally, we just apply the

P_{[2 \times 3]}^{-}

pseudo-inverse matrix onto the received vector,

R_{[3 \times 1]} \equiv {[\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}]}^{T}

. Equivalently, the dual frame vectors,

| p^{- (i)} 〉, i = 1, 2, 3

, are weighted by the frame coefficients in order to reconstruct

{\underset{˜}{d}}_{[2 \times 1]}

(Eq. (15)). As

P_{[2 \times 3]}^{-}

is a left-inverse of

P_{[3 \times 2]}^{}

, namely

P_{[2 \times 3]}^{-} P_{[3 \times 2]}^{} = 1_{[2 \times 2]}

, we obtain ideally perfect reconstruction:

{\underset{˜}{\hat{d}}}_{[2 \times 1]} = P_{[2 \times 3]}^{-} R_{[3 \times 1]} = P_{[2 \times 3]}^{-} P_{[3 \times 2]}^{} {\underset{˜}{d}}_{[2 \times 1]} = 1_{[2 \times 2]} {\underset{˜}{d}}_{[2 \times 1]} = {\underset{˜}{d}}_{[2 \times 1]}

An intuitive sense why the third middle branch (the W-signal) resolves the remote-LO POL fading may be obtained by inspecting pathological POL-fading cases, such as ${\underset{˜}{p}}_{Y}^{} = 0$ ,( $θ_{p} = 0$ ) in Eqs. (5), (17). In this case the X-branch peaks out, $\underset{˜}{U} = p_{0}^{} {\underset{˜}{d}}_{X}$ , whereas the Y-branch completely fades out, $\underset{˜}{V} = 0$ , but the new third branch (Eq. (9)) comes to the rescue, recovering the lost Y-data:

\underset{˜}{W} = {{\underset{˜}{p}}_{Y}^{*} {\underset{˜}{d}}_{X} [t] + {\underset{˜}{p}}_{X}^{*} {\underset{˜}{d}}_{Y} [t] |}_{{\underset{˜}{p}}_{Y}^{} = 0} = p_{X}^{*} {\underset{˜}{d}}_{Y} = p_{0}^{} {\underset{˜}{d}}_{Y} .

In this case the PI reconstruction matrix now routes

\underset{˜}{U}

and

\underset{˜}{W}

to the respective

{\hat{\underset{˜}{d}}}_{X}^{}, {\hat{\underset{˜}{d}}}_{Y}^{}

outputs, as may be readily verified by setting

θ_{p} = 0

in Eq. (14), yielding:

{\underset{˜}{\hat{d}}}_{[2 \times 1]} = \underset{P_{[2 \times 3]}^{-}}{\underset{︸}{p_{0}^{- 1} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}]}} \underset{\underset{˜}{R}}{\underset{︸}{[\begin{matrix} \underset{˜}{U} \\ \underset{˜}{W} \\ 0 \end{matrix}]}} = p_{0}^{- 1} [\begin{matrix} \underset{˜}{U} \\ \underset{˜}{W} \end{matrix}] = p_{0}^{- 1} [\begin{matrix} p_{0}^{} {\underset{˜}{d}}_{X} \\ p_{0}^{} {\underset{˜}{d}}_{Y} \end{matrix}] = [\begin{matrix} {\underset{˜}{d}}_{X} \\ {\underset{˜}{d}}_{Y} \end{matrix}]

A similar derivation dually applies to the case that the pilot is received Y-polarized. In effect, by projecting onto a non-orthogonal base (the three rows of the P-matrix) we secure resilience to the fading of the remote LO (pilot) for any SOP.

A unique feature of our solution is that the measurement frame is not fixed but it varies with the SOP of the pilot. Assume, without loss of generality, that the pilot is received linearly polarized, $ϕ_{p} = 0$ . Then, the analysis matrix, P, (Eq. (10)), becomes real-valued and the three frame vectors (rows of the P matrix) reduce to,

〈 p^{(1)} | = [\cos θ_{p}, 0], 〈 p^{(2)} | = [\sin θ_{p}, \cos θ_{p}], 〈 p^{(3)} | = [0, \sin θ_{p}]

These three frame vectors are plotted in Fig. 3 for nine pilot tilt angles,

θ_{p} \in \frac{π}{8} ℤ

ranging from 0 to

π / 2

. The extreme frames on the diagonal

θ_{p} = 0, π / 2

are seen to degenerate from proper three-vectors frames to orthogonal bases, each containing just two vectors. Following the color code, the U (V)-measurement and its associated vector is red (blue), while the new W-measurement is green. At

θ_{p} = 0 (θ_{p} = π / 2)

the Y (X) pilot fades out thus the blue (red) vector is gone from the frame, but the missing vector (be it red or blue) is replaced by the green vector, as generated by the third measurement. The 3-vectors frame now degenerates to an orthogonal basis of two vectors, in these extreme cases. The system is invertible at any

θ_{p}

value (SOP of the pilot). It is just that in the extreme cases, the three vectors frame degenerates to a two-vector orthogonal base, whereas in all other cases, projections onto proper frames are generated.

Fig. 3 Frames graphical representations. Assuming that the pilot is received linearly polarized, we plot here in $ℝ^{2}$ the triplets of frame vectors, for various settings of the pilot polarization angle, $θ_{p} \in \frac{π}{8} ℤ$ , in the range $0 \leq θ_{p} \leq π / 2$ . The angle $θ_{p}$ parameterizing each frame is indicated by a black marker arrow pointing inward on the circle circumference. For the extreme angles $θ_{p} = 0, π / 2$ , corresponding to the respective frames in the upper-left and lower right corners (surrounded by rounded squares), the triplet of frame vectors degenerates to a pair of vectors. Actually, at these pilot angles, there occurs fading of the vertical (horizontal) POL component, $\underset{˜}{V} = 0 (\underset{˜}{U} = 0)$ , such that the corresponding blue (red) vector vanishes, but the missing vector is supplanted by the green vector corresponding to the $\underset{˜}{W}$ measurement, which is always perpendicular to the surviving vector, such that there are left two orthogonal vectors, the green one and the red (blue) one, forming in these extreme cases an orthogonal base (whereas in all other cases the processing is based on projecting the vector of the three measurements onto a proper frame of three analysis vectors). The “best” frame, is the one at $θ_{p} = π / 4$ , as depicted in the center of the array, providing the highest processing gain. An interesting geometric property is that the angle $θ_{p}$ , as indicated by the black marker and the angle made by the green vector, corresponding to the new measurement, $\underset{˜}{W}$ , are symmetrical around a ray at $π / 4$ .

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The pseudo-inversion procedure described above is able to reconstruct the received Jones vector, $\underset{˜}{d} \equiv {[\begin{matrix} {\underset{˜}{d}}_{X}^{} & {\underset{˜}{d}}_{Y}^{} \end{matrix}]}^{T}$ , with components which are linear mixtures of the two transmitted polarizations:

\underset{˜}{d} \equiv [\begin{matrix} {\underset{˜}{d}}_{X}^{} \\ {\underset{˜}{d}}_{Y}^{} \end{matrix}] = J {\underset{˜}{d}}_{}^{Tx} = [\begin{matrix} J_{X X} & J_{X Y} \\ J_{Y X} & J_{Y Y} \end{matrix}] [\begin{matrix} {\underset{˜}{d}}_{X}^{Tx} \\ {\underset{˜}{d}}_{Y}^{Tx} \end{matrix}] .

with

J

the fiber link Jones matrix. But we require the Jones vector back at the Tx, rather than the Jones vector at the Rx. Thus, once we acquire

\underset{˜}{\hat{d}}

by pseudo-inversion of the three measurements, we must linearly back propagate via the inverse Jones matrix, in order to estimate

{\underset{˜}{\hat{d}}}_{}^{Tx}

:

{\underset{˜}{\hat{d}}}_{}^{Tx} = J^{- 1} \underset{˜}{\hat{d}} = J_{[2 \times 2]}^{- 1} P_{[2 \times 3]}^{-} R_{[3 \times 1]} = C_{[2 \times 3]}^{} R_{[3 \times 1]}

where we introduced the cascade of the inverse Jones matrix and the pseudo-inverse,

C_{[2 \times 3]}^{} \equiv J_{[2 \times 2]}^{- 1} P_{[2 \times 3]}^{-},

which is a 2x3 MIMO equalizer matrix, as it acts on the three measurements

R_{[3 \times 1]} \equiv {[\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}]}^{T}

, in order to generate our estimate of the transmitted Jones vector

{\underset{˜}{\hat{d}}}_{}^{Tx} = C_{[2 \times 3]}^{} R_{[3 \times 1]} = C_{[2 \times 3]}^{} {[\underset{˜}{U}, \underset{˜}{W}, \underset{˜}{V}]}^{T} .

The two outputs of the 3x2 MIMO equalizers are estimates of the two independently transmitted X and Y signals. In the proposed system, there is no need to separately evaluate the pseudo-inverse and the Jones inverse and take their product. The 2x3 MIMO equalizer matrix is generated at once, adaptively estimated using a data-aided LMS algorithm, directly acquiring and tracking the six elements of the matrix $C_{[2 \times 3]}^{}$ by adaptively adjusting the 3x2 MIMO equalizer taps such that they effectively form a matrix reconstructing a vector ${\underset{˜}{\hat{d}}}_{}^{Tx}$ close (in the mean-square sense) to the transmitted training data-vector, ${\underset{˜}{d}}^{Tx} \equiv {[{\underset{˜}{d}}_{X}^{Tx}, {\underset{˜}{d}}_{Y}^{Tx}]}^{T}$ .

5. Numeric simulations

Extension to optical channels with memory: The analysis heretofore assumed for simplicity a frequency-flat (memoriless) optical channel. Over a frequency-colored OFDM link (e.g. for long-haul transmission, in the presence of CD and PMD effects, which induce channel memory) one should resort to a frequency-domain version of proposed scheme, whereby 3x2 MIMO processing is separately applied onto each OFDM sub-carrier. Each of the three X,Y and XxY paths are fed into to M-point FFTs. Corresponding index sub-carriers from the three FFT outputs are then input into 3x2 MIMO modules, yielding POL-demultiplexed X and Y signals for each sub-carrier. The 6 complex taps of each 3x2 MIMO module per sub-carrier are adjusted by a data-aided LMS algorithm. An OFDM cyclic prefix (CP) of proper length is assumed. This provides the complete specification of the OFDM POL-demux algorithm.

In the simulations below for the optical amplified POL-MUX self-HET OFDM link of Fig. 4, we assume OFDM with M = 128 subcarriers (128 samples OFDM symbol), and a CP of 2 samples. The simulation results are presented in Figs. 5 and 6 numerically demonstrating that the proposed Rx of Fig. 2(b), based on the novel 2x3 photonic FE and 3x2 MIMO DSP with LMS data-aided tracking, is capable of robustly carrying POL-MUX double rate transmission of two independent 16-QAM OFDM streams, while mitigating POL fading over arbitrarily rotated SOPs in the fiber link.

Fig. 4 Simulated short-reach, 117 km, optically amplified POL-MUX OFDM link. Link parameters are marked in the figure. Independent OFDM signals are transmitted onto the two POLs with equal powers of 2 dBm each, 5 dBm total power, which equals the total pilot power, which is also equi-partitioned between the two orthogononal pilot POL components, each at 2 dBm, as the pilot is launched at an angle of 45º relative to the Tx principal axes. The polarization transformation in the fiber (not explicitly shown in the figure) is modeled as a rotation of the SOP by an angle $α$ . Both optical amplifier noise and Rx thermal noise are accounted for. The assumed link parameters imply that the receiver operates in the beat-noise limited regime, i.e., thermal noise is negligible relative to the SOA induced signal x ASE noise. An optical filter is further assumed at Rx input, bringing the ASE x ASE noise term to a negligible level.

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Fig. 5 Received constellations for the Rx of Fig. 2(b) on the link of Fig. 4, for a received LO SOP aligned with the X-axis and the Tx data launched with equal powers in its X,Y POLs. (a,b,c): Input ports X,Y, XxY of the 3x2 MIMO equalizer, as generated by the three optical front-end measurements. The PxD signal is seen to totally fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs(detected at a $\pm 45^{\circ}$ relative to the P direction). Nevertheless, the third branch signal (c), (itself a diverse mixture of the transmitted X, Y POLs) enables recovery of the two transmitted X and Y 16-QAM constellations (e,f), by adaptive 3x2 MIMO. The three vectors frame degenerates in this case to an orthogonal base of two vectors, as shown in (d).

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Fig. 6 Received constellations for the Rx of Fig. 2(b) on the link of Fig. 4, for a received LO SOP as specified in (d) for the set of LHS sub-figures, namely received at $+ {22.5}^{\circ}$ with respect to the X-axis and in (k) for the set of RHS sub-figures ( $+ 45^{\circ}$ ). (a,b,c): Input ports X,Y, XxY of the 3x2 MIMO equalizer, as generated by the three optical front-end measurements. The PxD signal is seen to partially fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs. Nevertheless, the power of the third branch signal (c), is maintained, preventing noise enhancement while recovering the transmitted X and Y 16-QAM constellations (e,f), by adaptive 3x2 MIMO.

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Figure 5 presents the various IQ plane constellations for the worst case that the SOP rotation is the fiber is $α = - 45^{\circ}$ such that the pilot, launched at $45^{\circ}$ relative to the principal axes of the Tx, is received aligned with the X-axis. In this case, the $\underset{˜}{V}$ POL measurement along the Y-axis completely fades out, but the third POL measurement, $\underset{˜}{W}$ , enables recovering the faded POL as explained in the last section. The detected SNR at the output of the 3x2 MIMO stage, as measured in our simulation, is 17.9 dB, lower by just 0.4 dB than a reference SNR_ref = 18.3 dB. The reference system is defined as follows: A POL-DIV self-HET Tx transmitting the same signal on both POLs (such that the Tx signal POL is at $45^{\circ}$ deg with respect to the PBS principal axes, i.e., aligned with the transmitted pilot). The energy transmitted per bit is normalized to be the same in our system as in the reference system, therefore, since our POL-MUX system carries twice the data rate, then its power is twice as large as that of the reference system (which is set at half the power of the system under test).

The worst case then occurs when the received pilot SOP is aligned with one of the Rx PBS axes (Fig. 5), yielding SNR lower by just 0.4 dB than the reference SNR, a small penalty worth paying for doubling data throughput.

The most favorable case (Fig. 6(g)-6(m), corresponding to null SOP rotation angle in the fiber, $α = 0$ , such that the received SOP is identical to the transmitted SOP) occurs when the pilot SOP is received at $45^{\circ}$ with respect to the PBS principal axes, In this case SNR attains its optimal value over all pilot POL angles, at 19.21 dB, exceeding by 0.91 dB the reference case with same energy per bit. The source of the SNR improvement is that the redundant frame projection onto the frame of three vectors provides processing gain. An interim case is shown in Fig. 6(a)-6(f). In this case, the fiber SOP rotation angle is $α = - {22.5}^{\circ}$ , such that the received SOP makes an angle of $+ {22.5}^{\circ}$ with the X-axis and the Y-signal partially fades out. Nevertheless the redundant 3x2 processing recovers “clean” 16-QAM constellations for the X and Y components with SNR >18.3dB, just a 0.9 dB degradation relative to that for to the optimal SOP.

For comparison, we also present in Fig. 7 the received constellations for the naïve 2x2 MIMO Rx of Fig. 2(a) based on straightforward detection on the two PBS outputs (lacking the proposed XxY cross-POL signal). Surprisingly, for an ideal SOP of $+ 45^{\circ}$ whereby both X and Y pilot POLs are detected with equal pilot powers, the 3x2 Rx is seen to provide 1.0 dB SNR gain relative to the 2x2 Rx (cf. Fig. 6(l), 6(m) vs. Figure 7(g), 7(h)). We conjecture that this SNR gain stems from the third XxY cross signal, averaging out the signal x ASE noise terms. A thorough signal vs. noise propagation analysis is deferred to future work. Comparing Fig. 6(e), 6(f), vs. Figure 7(c), 7(d) for the case of pilot SOP at $+ {22.5}^{\circ}$ , corresponding to partial pilot fading, the 3x2 Rx yield a 2.8 dB advantage vs. the 2x2 Rx, consistent with the pilot fading visible in Fig. 6(b).

Fig. 7 Received constellations for a naïve 2x2 MIMO Rx (Fig. 2(a)) over the link of Fig. 4, conventionally processing just the 2 PBS output signals, lacking the proposed XxY cross-POL signal. Pilot SOPs with regard to PBS X axis are: $+ {22.5}^{\circ}$ in sub-figures (a-d) incurring moderate fading conditions as in Fig. 6. Cases (e-h) correspond to no fading, as the pilot SOP is received at $+ 45^{\circ}$ . These constellations are to be compared with the improved ones of Figs. 6,7 for the novel 3x2 Rx.(a,b): Input ports X,Y of the 2x2 MIMO equalizer, as generated by the two optical front-end measurements. The PxD signal is seen to partially fade out on the Y port (b), whereas the X signal (a) is a mixture of the transmitted X,Y data POLs. (d,e): adaptive 2x2 MIMO equalizer outputs for the X and Y ports. The measured SNR is 18.4/15.5 dB for X/Y respectively, lower than that attained with the novel 3x2 receiver at the same pilot SOP, as shown in Fig. 6(e), 6(f), (SNR there is 20.4/18.3 dB). Evidently, even moderate fading conditions incur significant SNR degradation for the naïve 2x2 Rx. (e-h): constellations for received SOP at $+ 45^{\circ}$ . Compared to the SNR for proposed 3x2 Rx in Fig. 6(l), 6(m), the SNR attained with the conventional 2x2 Rx is lower by 1.0 dB, even in this case which is most favorable for conventional 2x2 detection.

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6. Comparisons with alternative schemes

Adding the third XxY path in the Rx in Fig. 2(b) may be deemed a reasonable price to pay (relative to the naïve Rx of Fig. 2(a), which is affected by POL-fading) enabling spectral efficiency doubling by signaling over both polarizations with direct-detection, making this proposed Rx particularly useful in optical access applications which may not tolerate the higher cost of a fully coherent Rx. The enhanced spectral efficiency may be alternatively applied to reduce electrical bandwidth rather than transmitting higher data rates, hence cutting Rx costs for cost-sensitive short-reach links.

Traditionally, for direct-detection receivers, polarization multiplexing has been contemplated to be performed by active polarization controllers, which have attained very high performance, see [12,13], however such devices are currently not yet cost-effective, hence the polarization control approach has not been adopted in practical use.

Briefly reviewing other approaches for POL-MUX over direction detection, Refs [4,6,7]. use either 4x4 MIMO or a simplified version of block diagonal 4x4 MIMO or alternatively switch among various MIMO levels depending on the received states of polarization. Ref [5]. incurs some reduced spectral efficiency due to pilot carriers and requires some higher rank MIMO processing. These schemes are affected by higher DSP complexity than the proposed scheme.

A similar principle enabling to decode the data irrespective of the POL alignment of the Rx, features in [8], restricted to DQPSK direct-detection with POL-MUX (namely mixing X and Y POLs to generate new redundant terms as in our scheme), however based on a more complex optical front-end based on 8-way rather than 3-way redundancy. Another characteristic of our scheme (not enjoyed by the scheme of [8]) is that our photonic integrated circuit tolerances are relaxed - there is no need to accurately control the optical path lengths on the waveguides –slacks in couplings and phase-shifts are adaptively corrected in DSP.

Note that the spectral efficiency of our scheme may be further improved by resorting to various DSP methods [1,2] in order to separate out the DxD term while reducing or eliminating the wasteful spectral gap.

DSP complexity: The 3x2 MIMO equalizer requires 6 complex multipliers per sample, for POL-demultiplexing a frequency-flat optical link, which is also the case treated in [7]. If implemented with 4 real multipliers per complex multiplier, as was done in Table 1 of [7], then our 3x2 MIMO equalizer would require 24 real multipliers which is more favorable than the other entries in Table 1 of [7] (the minimal value there being 32 real multipliers).

Over a frequency-colored link (e.g. for long-haul transmission, in the presence of CD and PMD which induce channel memory) one should use the frequency-domain version of proposed scheme, whereby 3x2 MIMO processing is applied onto each sub-carrier. In this case the 6 complex multipliers per sample direct contribution of the MIMO processing remains effectively valid (as the number of 3x2 MIMO modules equals the number M of subcarriers in the OFDM symbol, however each of these 3x2 MIMO modules operates M times slower). However, the complexity count should now also comprise a new FFT to be added in the third XxY branch in addition to the two FFTs already present on the X and Y branches for conventional OFDM detection. In our system, as M = 128, the complexity per sample of the 128-FFT is 0.5M log(2,M) / M = 3.5 complex multipliers. However, this extra FFT complexity should not be tallied in the comparison vs [7], as the extra FFT is associated with frequency-colored channels, which are outside the scope of [7] (the complexity in [7] would further increase if that scheme were to be adapted to channels with memory, which has not been done there).

7. Conclusion

In this paper we enabled POL-MUX advanced modulation formats over direct-detection, without active POL control, at very low extra complexity. The direct-detection receiver features a simple optical front-end and just slightly more complex DSP than a reference self-het receiver with dual polarization paths (e.g. as used in polarization diversity). The proposed scheme is modulation format-transparent e.g., supports POL-MUX m-QAM for higher capacity at short-reach, either (DFT-Spread) OFDM or Nyquist-shaped single-carrier.

The system is DSP-compatible with prior-art system variants [1,2] which aim at reducing or eliminating the wasteful spectral gap. Our solution may be useful in addressing the need for low-cost high-capacity links in access networks and short-reach optical interconnects in the range of ~10-100 km. It would also be interesting to explore comparisons with coherent detection over longer metro links, and in particular relation of the self-heterodyne technique with simplified coherent heterodyne technique in [14].

Acknowledgment

We acknowledge the kind support of the Israel Science Foundation (ISF) and of the Piano + OTONES trans-national EU photonics program.

References and links

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2. D. Hsu, C. Wei, H. Chen, C. Song, I. Lu, and J. Chen, “74. 4% SSII Cancellation in an EAM-based OFDM-IMDD Transmission System” in OFC 2013, 2013, p. OM2C.7.

3. J. Leibrich, A. Ali, and W. Rosenkranz, “Decision Feedback Compensation of Transmitter / Receiver Nonlinearity for DD-OFDM” in European Conference of Optical Communication (ECOC) (2011), p. We.8.A.5. [CrossRef]

4. D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108 Gb/s OFDMA-PON With Polarization Multiplexing and Direct Detection,” J. Lightwave Technol. 28(4), 484–493 (2010). [CrossRef]

5. C. C. Wei, C.-T. Lin, C.-Y. Wang, and F.-M. Wu, “A Novel Polarization Division Multiplexed OFDM System with a Direct-detection BLAST-Aided Receiver” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013(2013), p. JTh2A.49. [CrossRef]

6. N. Cvijetic, N. Prasad, M. Cvijetic, and T. Wang, “Cvijetic Efficient and Robust MIMO DSP Equalization in POLMUX OFDM transmission with direct-detection” in ECOC 2011(2011).

7. N. Cvijetic, N. Prasad, D. Qian, and T. Wang, “Block-Diagonal MIMO Equalization for Polarization-Multiplexed OFDM Transmission With Direct Detection,” IEEE Photon. Technol. Lett. 23(12), 792–794 (2011). [CrossRef]

8. J. Rahn, G. Goldfarb, H.-S. Tsai, W. Chen, S. Chu, B. Little, J. Hryniewicz, F. Johnson, C. Wenlu, T. Butrie, J. Zhang, M. Ziari, J. Tang, A. Nilsson, S. Grubb, I. Lyubomirsky, J. Stewart, R. Nagarajan, F. Kish, and D. F. Welch, “Low-Power, Polarization Tracked 45.6 GB/s per Wavelength PM-DQPSK Receiver in a 10-Channel Integrated Module” in OFC 2010(2010), p. OThE2.

9. M. Nazarathy and A. Agmon, “Doubling Direct-detection Data Rate by Polarization Multiplexing of 16-QAM without a Polarization Controller” in ECOC 2013(2013), p. Mo.4.C.4.

10. A. Agmon, M. Nazarathy, D. M. Marom, S. Ben-Ezra, A. Tolmachev, R. Killey, P. Bayvel, L. Meder, M. Hübner, W. Meredith, G. Vickers, P. C. Schindler, R. Schmogrow, D. Hillerkuss, W. Freude, and J. Leuthold, “Bi-directional Ultra-dense Polarization-muxed/diverse OFDM/WDM PON with Laserless Colorless 1Gb/s ONUs Based on Si PICs and <417 MHz mixed-signal ICs” in OFC 2013(2013), p. OTh3A.6.

11. O. Christensen, Frames and Bases. Birkhauser, 2008.

12. B. Koch, R. Noé, V. Mirvoda, D. Sandel, V. Filsinger, and K. Puntsri, “40-krad / s Polarization Tracking in 200-Gb / s PDM-RZ-DQPSK Transmission Over 430 km,” IEEE Photon. Technol. Lett. 22(9), 613–615 (2010). [CrossRef]

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14. Z. Dong, X. Li, J. Yu, and J. Yu, “Generation and transmission of 8 × 112-Gb/s WDM PDM-16QAM on a 25-GHz grid with simplified heterodyne detection,” Opt. Express 21(2), 1773–1778 (2013). [CrossRef] [PubMed]

Doubling direct-detection data rate by polarization multiplexing of 16-QAM without active polarization control

Abstract

1. Introduction

2. Review of conventional self-heterodyne direct-detection Rx – the POL fading problem

3. Overview of the novel self-heterodyne direct-detection Rx supporting POL-MUX

4. Theory of operation

5. Numeric simulations

6. Comparisons with alternative schemes

7. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Equations (22)

Optics Express