Broadcast MIMO over multimode optical interconnects by modal beamforming

Amos Agmon; Moshe Nazarathy

doi:10.1364/OE.15.013123

1. Introduction

Multiplexing methods providing multiple parallel data channels over Multimode Fiber (MMF) or waveguides have long been subject to active research, based on wavelength, subcarrier, or angular/modal diversities. The advent of Multiple Input Multiple Output (MIMO) techniques in wireless transmission has inspired porting MIMO to guided multimode optical media [1–7]. It is noted that all MMF MIMO studies to date have addressed Point-To-Point (PTP) links with Informed Receivers (IR) estimating the channel for receiver-side MIMO processing, while the transmitter is uninformed of it. Moreover, in most works to date, the multiple input ports are driven by independent mutually incoherent optical sources, in effect operating under optical power superposition (see [7] for an exception, treating coherent optical detection MIMO, albeit of prohibitive complexity for short-range optical interconnects).

This paper extends previous MMF MIMO studies in three respects: (i): We explore, for the first time in optical MIMO, the consequences of an Informed Transmitter (IT) strategy, providing feedback paths from the (possibly distributed) Receivers (RX) to the Transmitter (TX), over which the RXs convey their estimates of the Channel State Information (CSI) to the TX (CSIT). (ii): We introduce a broadcast MMF MIMO system (with PTP links as a special case) over short-range (<100m) optical interconnects (iii): Our modulated optical input ports are mutually coherent but Direct-Detection (DD) is used at the output ports.

2. Optical zero-forcing beamforming system overview

We propose a novel optical Zero-Forcing BeamForming (ZFBF) technique, providing multiple decoupled virtual sub-channels to a large number of independent receivers at multiple optical taps or drops. Such broadcast capability, not available with the IR strategy, is potentially applicable to short-range photonic interconnect structures over multimode waveguides and MMF, in topologies such as buses and stars, i.e. short-reach “mini-PON” distribution networks, wherein each of the receive stations listens to its own targeted data tributary. Our method provides an optical equivalent to ZFBF techniques recently emerging in wireless communication for MIMO broadcast [8,9]. However, unique to the optical predicament, our approach resolves the complication of field interference cross-terms resulting from quadratic detection of mutually coherent modulated optical sources.

Our technique may also be viewed as an extension of the system in [10], which made use of a Spatial Light Modulator (SLM) to transmit 10 Gb/s over a 11 Km MMF. As in [10], we assume a feedback path is available from RX(s) to the TX, which is also equipped with a Spatial Light Modulator (SLM). However, [10] is essentially a Multiple Input Single Output (MISO) system making use of spatial diversity to mitigate ISI at a single output port, whereas our MIMO system feeds multiple ports, translating spatial diversity into multiplexing gain.

From an implementation point of view, an alternative to the SLM device required in the TX may be provided by leveraging recent silicon photonics progress to realize an equivalent MultiPort Modulator (MPM) as a parallel bank of modulators, each of the type described in [11], all fed by a single optical source (DFB or VCSEL) transmitting mutually coherent, independently modulated multiple optical signals coupled into the input facet of the MMF. The added ZFBF TX complexity enables the simplest possible uncoordinated OOK receivers.

Optical ZFBF concept: The SLM/MPM shapes the transmitted complex amplitude wavefront with a certain spatial resolution into field patterns referred to as PseudoModes (PM), each lighting up its own unique detector, while eliciting zero response in other photodetectors (Fig. 1). These beam-forming PM vectors are transmitted in superposition, each modulated by its own data tributary. As an application we propose a system transmitting 112 Gb/s over a single MMF link or short-reach PON (Fig. 2), realized as a massively parallel ZFBF MIMO system, mapping 32 data tributaries, each at 3.5 Gb/s, into 64 optical inputs, leading to 32 RXs, with no cross-talk between one another. In contrast, most MMF MIMO systems to date were 2×2; Exceptions are [4,5] using a large number of mutually incoherent optical sources.

Fig. 1. (Color online) ZFBF concept: Each PM A̰ ^(r) , r = 1,2,…,n_U is focusing light onto the r-th detector.

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The range limit of our method is set by intermodal dispersion, estimated to be limited to up to 100 meters at 3.5 Gb/s with modest or no equalization, for many types of Polymer Optical Fibers. As per [12] most data-center applications require <100m distances.

3. Quadratic MIMO channel model

We start by modeling the Quadratic MIMO (Q-MIMO) channel with Multi-Mode Detectors (MMD) for the case of mutually coherent input ports, and direct detection.

Let H denote the MMF random channel matrix (CM), describing the linear transformation of optical field from input to output, with D ₀, the number of (speckle / modal) Degrees of Freedom (DOF) at each detector. The complex modal fields vector incident at the n_R detectors inputs, is Ḛ^d =HḚ^s where the transmitted field vector Ḛ^smay be arbitrarily shaped in amplitude and phase by means of an SLM or MPM. It is noted that the elements of the input and output field vectors Ḛ^d,Ḛ^s are spatially-integrated and temporally-sampled quantities, representing the continuous-valued space-time quantities in discrete form.

In a MIMO system, multiple extended area detectors, called MultiMode Detectors (MMD) are coupled to the MMF end-facet, either directly attached to it or via intermediary short MMF leads. The origin of the MMD term is due to the fact that each such detector captures multiple field Degrees Of Freedom (DOF) over its aperture, be it speckle elements or orthogonal spatial transverse modes incident on the detector surface.

For simplicity we assume a uniform DOF number D ₀ per MMD.

The received field column vector Ḛ^d is partitioned into vector blocks Ḛ^d(r) , with the r-th vector containing the D ₀ field DOFs captured by the r-th detector:

Ḛ^d = [Ḛ^d(1)T,…,Ḛ^d(r)T,…,Ḛ^d(n_R)T]^T . Similarly, the CM H of size n_RD ₀×n_T is partitioned into n_R matrix blocks, H ^(r) , called partial CMs, such that Ḛ^d(r) = H ^(r) Ḛ^s, i.e. the r-th partial CM describes the linear transformation of field from the input vector to the field DOFs vector captured by the r-th MMD. The photocurrent generated by the r-th MMD is

I_{r}^{d} = \sum_{m = 1}^{D_{0}} {∣ {\underset{̃}{E}}_{m}^{d (r)} ∣}^{2} = {∥ {\underset{̃}{E}}^{d (r)} ∥}^{2} = {∥ H^{(r)} {\underset{̃}{E}}^{s} ∥}^{2} = 〈 {\underset{̃}{E}}^{s} ∣ H^{(r) †} H^{(r)} ∣ {\underset{̃}{E}}^{s} 〉 = 〈 {\underset{̃}{E}}^{s} ∣ G^{(r)} ∣ {\underset{̃}{E}}^{s} 〉

where for simplicity we assumed unity responsivity, equating total power with photocurrent, and in the first equality we used the fact that the total power received by the r-th MMD equals the sum of powers of its individual field DOFs, a consequence of the spatial orthogonality of either the speckle elements or the modes in the fiber lead of each MMD, as applicable. Each noiseless photocurrent is then expressed as a quadratic form, where G ^(r) = H ^(r)† H ^(r) is called the r-th Quadratic Channel Matrix(QCM). We further introduce MISO responsivities :

ρ_{r} {{\underset{̃}{E}}^{s}} \equiv \frac{I_{r}^{d} ({\underset{̃}{E}}^{s})}{{∥ {\underset{̃}{E}}^{s} ∥}^{2}} = \frac{〈 {\underset{̃}{E}}^{s} ∣ G^{(r)} ∣ {\underset{̃}{E}}^{s} 〉}{〈 {\underset{̃}{E}}^{s} | {\underset{̃}{E}}^{s} 〉}

The detected photocurrents I^d_r are corrupted by zero-mean additive gaussian noises N_r ~ N[0, σ² _I], independent of each other and of the transmitted signal: I_r = I^d_r + N_r.

Channel estimation procedure: The QCM is determined by a channel estimation procedure run once per coherence interval (CI), based on transmitting training sequences consisting of unit vectors, then two-element vectors with their components in-phase and in-quadrature:

Launching Ḛ^s = [δ_p], i.e. exciting just the p-th input port, p = 1,…,n_T, yields all the QCM diagonal elements: I^d_r(Ḛ^s) = G ^(r) _pp. Once the diagonal elements are estimated, the off-diagonal elements are derived by exciting all possible n_T (n_T - 1) / 2 distinct pairs of input ports in turn, launching: Ḛ^s = [δ_p + δ_t], yielding Re[G ^(r) _pt] = [I^d_r(Ḛ^s) - G ^(r) _pp - G ^(r) _tt] /2 followed by launching all possible pairs of signals in quadrature, Ḛ^s = [δ_p + jδ_t], yielding Im[G ^(r) _pt] = j[G ^(r) _pp+G ^(r) _tt-I_r^d(Ḛ^s)]/2. This completes the off-diagonal elements evaluation. Each training sequence is repeated N times per CI, and the responses are averaged, e.g. N=1000 repetitions provide ~30 dB noise reduction, incurring an overhead of the order of ~5% in multiGb/s scenarios, as the channel varies relatively slowly, CI≈ 25 msec. The channel estimation overhead is not excessive relative to the substantial increases in data-rate (up to ×32 i.e. thousands of %) attained with ZFBF MIMO relative to a SISO channel.

4. Optical Zero-Forcing-Beam-Forming

In this section we construct, a transmission alphabet of n_U PMs, A̰^(r),r = 1,…,n_U, to be loaded into the precoder, once per CI, satisfying the ZF Constraint (ZFC):

I_{r'}^{d} ({\underset{̃}{A}}^{(r)}) = 〈 {\underset{̃}{A}}^{(r)} ∣ {\underset{̃}{G}}^{(r')} ∣ {\underset{̃}{A}}^{(r)} 〉 = δ_{r' r}; r, r' \in {1,2, \dots n_{U}}, n_{U} \leq n_{R}

Physically, the r-th PM, A̰^(r), is an input vector lighting up the r-th detector while eliciting zero response in all other Active Detectors (AD) (forming a subset of the full detector set, selected once per each CI, the optimal selection of which is outside the scope). The currents of the non-active detectors are ignored. Here n_U is the “number of users” - the size of a subset of AD over which the ZFBF signaling is conducted in each CI, with the indexes r,r′ running over the ADs. The ADs are re-indexed from 1 to n_U, without loss of generality. PMs can be transmitted in superposition rather than one at a time. The total transmitted field is

{\underset{̃}{E}}^{S} = (\frac{E_{0}}{\sqrt{n_{U}}}) As = (\frac{E_{0}}{\sqrt{n_{U}}}) \sum_{r = 1}^{n_{U}} {\underset{̃}{A}}^{(r)} S_{r}

where A is a precoder matrix collecting the PMs in its columns, $\frac{E_{0}}{\sqrt{n_{U}}}$ is a scaling coefficient, and s = [s ₁,…,s_r,…s_{n_U}]^T is the vector of transmitted info symbols. The photocurrent in the r-th MMD due to the transmitted field (4) is seen to be solely dependent on the r-th transmitted info symbol, S_r indicating that the transmission has been essentially decoupled into n_U effective channels, referred to as pseudochannels:

I_{r}^{d} ({\underset{̃}{E}}^{s}) = (\frac{E_{0}^{2}}{n_{U}}) I_{r}^{d} ({\underset{̃}{A}}^{(r)}) {∣ S_{r} ∣}^{2} = (\frac{E_{0}^{2}}{n_{U}}) {∣ S_{r} ∣}^{2}

Taking the info symbols S_r to be i.i.d. with zero-mean and unity variance,⟨|S_r|²⟩=1, the average transmitted power is the sum of PM powers

P_{T} \equiv 〈 {∥ {\underset{̃}{E}}_{s} ∥}^{2} 〉 = \frac{E_{0}^{2} \sum_{r = 1}^{n_{U}} {∥ {\underset{̃}{A}}^{(r)} ∥}^{2}}{n_{U}} = E_{0}^{2} ρ_{eff}^{- 1}; ρ_{eff} \equiv n_{U} {(\sum_{r = 1}^{n_{U}} ρ_{r}^{- 1})}^{- 1} = {〈 ρ_{r} 〉}_{H}

where ρ_eff is the harmonic mean of the MISO responsivities over all pseudochannels, and ∥A̰^(r)∥ = ρ^-1 _r (from (3),(2)). Solving (6) for E ₀, under a fixed transmit power P_T , yields E ² ₀ = P_Tρ_eff. Let us now define the optical SNR at the r-th detector as γ_r =⟨I^d_r⟩/σ_I (its square is the electrical SNR). The optical SNR is then seen to be uniform over all MMDs

γ_{r} = \frac{(\frac{E_{0}^{2}}{n_{U}}) 〈 {∣ S_{r} ∣}^{2} 〉}{σ_{I}} = \frac{P_{T} ρ_{eff}}{(n_{u} σ_{I})} = γ_{U} ρ_{eff}

where γ_U =P_T/(n_Uσ_I) is the transmit SNR per user.

Optimizing the uniform SNR under an averaged transmit power constraint is equivalent to maximizing ρ_eff . This optimization decouples into n_U independent maximization problems over each of the ρ_r s.t. the ZFC (3).

Let H̅^(r) be the r-th interference submatrix obtained by collecting the partial CMs of all the ADs, except the r-th one: H̅^(r) =[H ^(1)T,…,H ^(r-1)T,H ^(r+1)T,…,H ^(n_U)T]^T . The ZFC (3) then implies that the r-th PM belongs to the right nullspace of this matrix:

{\bar{H}}^{(r)} {\underset{̃}{A}}^{(r)} = 0

An orthonormal basis of this nullspace is arrayed in the columns of a null interference basis matrix N̅^(r). The ZFC (8) is then equivalent to ∃a̰^(r): A̰^(r) = N̅^(r) a̅^(r). Substituting this expression into (2), the r-th MISO responsivity may be recast as:

ρ_{r} \equiv \frac{I_{r}^{d} ({\underset{̃}{A}}^{(r)})}{{∥ {\underset{̃}{A}}^{(r)} ∥}^{2}} = \frac{1}{{∥ {\underset{̃}{A}}^{(r)} ∥}^{2}} = \frac{〈 {\underset{̃}{a}}^{(r)} ∣ g^{(r)} ∣ {\underset{̃}{a}}^{(r)} 〉}{〈 {\underset{̃}{a}}^{(r)} | {\underset{̃}{a}}^{(r)} 〉}

where g ^(r) = N̅^(r)†G ^(r) N̅^(r) = (H ^(r) N̅^(r))^†(H ^(r) N̅^(r)), and a̰^(r) is unconstrained.

The last expression in (9) identified as a Rayleigh Quotient (RQ) [13] known to be optimized by selecting a̰^(r) as the dominant eigenvector of g ^(r) which coincides with the dominant right singular vector of H ^(r) N̅^(r), corresponding to the largest Singular Value (SV), denoted σ_r . The maximized RQ (9) then equals the dominant squared singular value: ρ^max _r = σ² _r , yielding ∥A̰^(r)∥² = σ^-2 _r, i.e. the optimal PM energies are given by the inverses of the squared singular values. This concludes the construction of the optimal PMs forming the columns of the precoder matrix, A, leading to maximum uniform SNR at all output ports, γ_r = ρ_eff γ_U , given by the “transmit SNR per user” γ_U times an effective power gain (6) equal, in the optimal case, to the harmonic mean of the squared SVs of the matrices H ^(r) N̅^(r).

Dimensionality considerations: The ZFC formulated in (8) for the r-th detector amounts to a set of (n_U -1)D ₀ homogeneous linear equations in n_T variables. Its solution space is the nullspace of H̅^(r), of dimension n_null ≡ n_T - (n_U - 1)D ₀, (as a random H̅^(r) matrix tends to have full-row rank almost surely). For non-trivial solutions we require n_T > (n_U -1)D ₀ , i.e. the number of transmit ports must exceed the number of zero-forced DOFs captured by all n_U - 1 zero-forced detectors. In our simulations we empirically found it necessary to maintain n_null / n_T ≥ 0.25 to obtain high pseudochannel responsivity (ρ^max _r) gains.

A rudimentary transmission strategy based on these concepts excites one detector at a time, by selecting one PM per symbol, mapped by log₂ n_U bits. A more advanced modulation format called here Orthogonal Vector Amplitude Modulation (OVAM), concurrently sends n_U bits per symbol, signaling over all decoupled pseudochannels, by launching a superposition of independently modulated PMs, each focused onto a different detector. Superposing multiple PMs, each driven by its independent data stream addressed to an individual RX, substantial multiplexing gain is extracted (e.g. 32× in our example), while keeping the RXs simple.

5. System performance

A short-reach (<100m) MMF PON operating at 112 Gb/s is illustrated in Fig. 2. For simplicity we assume On-Off Keying (OOK) transmission (extension to multi-level transmission is also possible). In the receivers terminating each output port, the photocurrent is one-bit quantized with a conventional OOK CDR (Clock-Data-Recovery) front-end.

We analyze the OVAM OOK BER performance vs. transmit SNR per bit (or equivalently, per user) parameterized by n_U, the number of ADs (or equivalently the number of bits per sym). As the received SNRs γ_r are uniform over r, the BER at each RX will be constant (uniform Quality of Service). With electrical AWGN, the BER is expressed as Gaussian-Q function

BER = Q [γ_{U} ρ_{eff}]

Fig. 2. Massively parallel transmission of 112 Gb/s over a single MMF Point-to-Multipoint ZFBF MIMO broadcast system. At the input, 32 × 3.5 Gb/s data streams are mapped into 64 optical input ports by means of the proposed ZFBF MIMO precoder. 32 active photodetectors (out of 64) are spread along the output facets of various branches. The r-th photodiode receives its own 3.5 Gb/s data stream from the r-th input, with no cross-talk from the other tributaries. The 32 uncoordinated, decoupled, conventional OOK receivers then detect an aggregate >100 Gb/s. A point-to-point >100 Gb/s link may be similarly constructed, with all 32 detectors co-located at the fiber output facet. All E-O modulators of the Multi-Port E-O Modulator or SLM are fed from a single laser.

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with Q-factor I^d _∆/(2σ_I) = ⟨;I_r^d⟩/σ_I = γ_r = γ_Uρ_eff where I^d _∆ is the “mark” current.

For the OVAM format, carrying 1 b/sym/active-user, the transmit SNR per user γ_U coincides with the transmit SNR per bit. In our simulations we drew 1000 channel matrix realizations and computed ρ_eff (6) as the harmonic mean of the n_U squared SVs σ² _r , in turn evaluated for each of the matrices H ^(r) N̅^(r). The BER per CM realizaton is then given by (10) The total performance is given by the average $\bar{B E R}$ , vs. γ_U (TX SNR per bit) over the CM ensemble.

We simulated two models for the statistics of the CM, H :

A more idealized Orthogonal channel̲, whereby the total received optical power is a fixed fraction η of the transmitted power, drawing matrices H = √ηU with U unitary isotropic. This channel lends itself to a simple analytic description: The SVs are all σ_r = η^1/2, yielding effective responsivity gains ρ_eff = η < 1, hence (10) yields $\bar{B E R}$ _ortho = Q[γ_Uη].
Remarkably, for this orthogonal channel the BER vs. SNR–per-bit is independent of the number of ADs (=number of users), as all decoupled pseudochannels attain constant loss through the CM. Hence, a linear growth (in n_U) of the aggregate bitrate at fixed BER may be provided by a linear growth in total TX power (i.e. constant power and bitrate per user), as opposed to M-ary PAM Single Input Single Output (SISO) systems, wherein a linear increase in bitrate requires approximately exponential increase in power (~3 dB optical per bit).
A more realistic CG-ZMSW channel, whereby H is drawn as a Circular Gaussian (CG) Zero-Mean Spatially White (ZMSW) matrix, with CG i.i.d. elements. This CG-ZMSW channel model is ubiquitous in wireless transmission. Its approximate applicability to multimode fiber transmission was established in [4] and [7].
The simulated BER performance vs. SNR for MIMO, MISO and SISO systems over the two types of channels is shown Fig. 2 for a 64 mode fiber. An insightful comparative performance of the tradeoff between the spectral efficiency gain and the required SNR at a fixed BER level is displayed in Fig. 2c, comparing ZFBF MIMO and MISO with conventional M-ary PAM SISO. Substantial multiplexing/diversity gain advantages are apparent for the ZFBF schemes.

6. Conclusions

ZFBF MIMO for MMF point-to-(multi)point configurations, provides substantial multiplexing gain (high spectral efficiencies) to simple uncoordinated receivers, at the expense of increased transmitter complexity and feedback from the receivers.

Fig. 3. (Color online) Performance of the ZFBF MIMO system described in Fig. 2. (a,b):BER vs. optical SNR per bit (10 log γ_U) for MIMO, MISO, SISO over MMF with η =10 dB loss, for the ZMSW and Ortho. Ch. (a): ZFBF MISO outperforms SISO by ~6 dB, both providing 1 b/sym (b): ZFBF MIMO: n_U = 2, 4, 8,16, 32 . The curves for n_U = 4, 8 almost coincide. Doubling the number of bits per symbol to 16, and then to 32 requires modest power increases in two steps of ~1 dB. (c): Spectral efficiency vs. SNR performance comparison of the novel ZFBF MIMO/MISO vs. multi-level SISO, for a fixed BER of 10^-4.

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Broadcast MIMO over multimode optical interconnects by modal beamforming

Abstract

1. Introduction

2. Optical zero-forcing beamforming system overview

3. Quadratic MIMO channel model

4. Optical Zero-Forcing-Beam-Forming

5. System performance

6. Conclusions

References and links

Cited By

Figures (3)

Equations (10)

Optics Express