Analog nonlinear MIMO receiver for optical mode division multiplexing transmission Arnaldo Spalvieri,1 Pierpaolo Boffi,1 Simone Pecorino,1 Luca Barletta,2 Maurizio Magarini,1,∗ Alberto Gatto,1 Paolo Martelli,1 and Mario Martinelli1 1 Dipartimento
di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy 2 Institute for Advanced Study, Technische Universit¨ at M¨unchen, Lichtenbergstrasse 2a, D-85748 Garching, Germany ∗ [email protected]
Abstract: The complexity and the power consumption of digital signal processing are crucial issues in optical transmission systems based on mode division multiplexing and coherent multiple-input multiple-output (MIMO) processing at the receiver. In this paper the inherent characteristic of spatial separation between fiber modes is exploited, getting a MIMO system where joint demultiplexing and detection is based on spatially separated photodetectors. After photodetection, one has a MIMO system with nonlinear crosstalk between modes. The paper shows that the nonlinear crosstalk can be dealt with by a low-complexity and non-adaptive detection scheme, at least in the cases presented in the paper. © 2013 Optical Society of America OCIS codes: (040.1240) Arrays; (060.2330) Fiber optics communications; (060.4080) Modulation; (060.4230) Multiplexing; (080.4865) Optical vortices.
References and links 1. R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006). 2. A. Tarighat, R. Hsu, A. Shah, A. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45, 57–63 (2007). 3. T. Morioka, Y. Awaji, R. Ryf, P. J. Winzer, D. Richardson, and F. Poletti, “Enhancing optical communications with brand new fibers,” IEEE Commun. Mag. 50, s31–s42 (2012). 4. R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” IEEE Proc. 100, 1035–1055 (2012). 5. R. Dar, M. Feder, and M. Shtaif, “The underaddressed optical multiple-input, multiple-output channel: capacity and outage,” Opt. Lett. 37, 3150–3152 (2012). 6. R. Dar, M. Feder, and M. Shtaif, “The Jacobi MIMO channel,” IEEE Trans. Inf. Theory 59, 2426–2441 (2013). 7. V. Sleiffer, Y. Jung, B. Inan, H. Chen, R. van Uden, M. Kuschnerov, D. van den Borne, S. Jansen, V. Veljanovski, T. Koonen, D. Richardson, S. Alam, F. Poletti, J. Sahu, A. Dhar, B. Corbett, R. Winfield, A. Ellis, and H. De Waardt, “Mode-division-multiplexed 3×112-Gb/s DP-QPSK transmission over 80 km few-mode fiber with inline MM-EDFA and blind DSP,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper Tu.1.C.2. 8. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). 9. C. Koebele, M. Salsi, L. Milord, R. Ryf, C. Bolle, P. Sillard, S. Bigo, and G. Charlet, “40km transmission of five mode division multiplexed data streams at 100Gb/s with low MIMO-DSP complexity,” in European Conference
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25174
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Introduction
Mode Division Multiplexing (MDM) has attracted widespread attention as a means to increase the information throughput over a single fiber [1–4]. Considerations about the capacity of optical MDM can be found in [5, 6]. In recent experiments [7,8], MDM exploiting multiple-input multiple-output (MIMO) digital signal processing (DSP) in combination with coherent demultiplexing and detection has demonstrated its effectiveness in removing the intermodal crosstalk that originates during propagation. However, real-time coherent MIMO processing requires high complexity and power consumption at the receiver. A scheme performing optical demultiplexing before coherent detection, thus allowing reduced complexity compared to the fully coherent MIMO signal precessing, has been proposed in [9]. Even simpler schemes, that are fully noncoherent, have been considered in [10–12]. Such approaches rely upon the assumption of having groups of modes that are well separated in terms of propagation constants, so that the crosstalk between them, when used as independent channels, can be considered negligible at the receiver. The low crosstalk condition is achieved in [10] by considering a fiber with a core radius that is large enough to guarantee that the multiplexed mode groups remain well spatially separated after propagation over the fiber. Spatial detection is then carried out on each group independently by means of circular multi-segment Photode#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25175
tectors (PDs). Assuming linear superposition of the power distributions at the fiber output, a MIMO channel matrix is estimated, without implementing a real mode group division multiplexing transmission, in order to perform detection. On another hand, the requirement for well separated propagation constants is used as a prerequisite in [11] for experimental transmission of mode groups over a distance in the order of tens of meters. In this case, the short propagation distance introduces a relatively low crosstalk that can be easily mitigated by mode-selective spatial filtering. This spatial filtering is based on the use of a number of standard single-mode fibers equals to the number of the transmitted mode groups. As in [10], the experimental investigation done in [12] does not implement a real mode group division multiplexed transmission. In the considered experimental analysis the number of mode groups that can be used as separate spatial channels for on-off keying transmission over a 2 km long standard graded-index multimode fiber is firstly determined and the error performance for each mode group, when used as an individual channel, is then evaluated. A lower conversion efficiency is shown for higher-order modes with respect to the fundamental mode, due to imperfect mode conversion at the transmitter, that is responsible for performance degradation. This paper considers an MDM scheme that is candidate for application in those systems where cost, power consumption, and complexity, are relevant issues, such as local area networks and optical interconnects for data centers. In this context of short-range optical communication links (i.e., up to a few kilometers), the impact of chromatic dispersion and modal dispersion is negligible also for high bit rate [13]. The main motivation behind our work is the observation that the detection techniques proposed in [10–12] are designed by assuming a low level of intermodal crosstalk and, as such, they are not able to fully take into consideration its effects. To this aim, a first step toward the characterization of such a crosstalk has been done in [14,15]. In particular, a penalty in the order of 2 dB is shown for optical mode group demultiplexing without any MIMO processing compared to single channel transmission in a standard 50 μ mcore graded-index multimode fiber for lengths of 2 km and of 8 km. In the present paper, we propose a novel MDM scheme that is able to cope with the intermodal crosstalk. The paper focuses on two-mode multiplexing, and, starting from the mode coupling model of [5,6], shows that, after detection by two spatially separated concentric PDs [16], one gets a MIMO channel with nonlinear crosstalk, that, in the above mentioned applicative context, can be dealt with by MIMO processing of low complexity. 2.
System model
Let us consider MDM transmission in optical fiber based on the excitation of two modes, called mode 1 and mode 2, that are on-off keying modulated and characterized by the same energy. The MIMO alphabet A is therefore made by the four column vectors a = [a1 , a2 ]T , where the superscript T denotes vector or matrix transposition, and the bits a1 and a2 associated with modes 1 and 2, respectively, are assumed to be i.i.d. random variables taking their values with uniform probability in {0, 1}. In this paper it is assumed that the field distributions of fiber modes are given by the Laguerre-Gaussian functions [17]. This assumption, which holds for parabolic index profile, can be used also to approximate other fiber index profiles, still maintaining the here exploited geometric and symmetry properties of the field. To be concrete, in what follows the excitation of optical vortices (OVs) carrying orbital angular momentum (OAM), that have been proposed in [15], [18–21] in place of Linearly Polarized (LP) modes to achieve MDM in freespace, is considered. However, it is worth mentioning that the analysis presented here holds also for LP modes. Mode 1 is the fundamental mode OV01 , while mode 2 is OV11 . The two modes are characterized by the radial intensity profiles R21 (ρ ) and R22 (ρ ) that are the squares of the radial parts of the Laguerre-Gaussian field associated with modes OV01 and OV11 , respectively. Figure 1 shows the intensity patterns of the two modes and the two spatially separated
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25176
PD2 PD1
Fig. 1. From left to right: intensity pattern of OV01 and intensity pattern of OV11 before coupling. Rightmost picture: inner circle PD1 , outer circular annulus PD2 .
PDs matched to the two modes. Owing to mechanical perturbations (like twisting, bending, and vibrations) applied to the fiber, mode coupling during propagation can take place, leading to intermodal crosstalk. Moreover, mode coupling can be introduced at the transmitter due to imperfect mode conversion in the desired mode group. The experimental characterization of intermodal crosstalk made for LP [14] and OV [15] modes suggests that the strong random mode coupling regime [22,23] does not apply in short-range (i.e., up to lengths of a few kilometers) standard multimode fiber-optic links. In this case the crosstalk among mode groups is mainly induced by the imperfect generation of the multiplexed modes and is more relevant for higher-order modes. In the following we introduce the crosstalk matrix between the modes corresponding to OV01 and OV11 . More in detail, the fundamental mode OV01 is two-fold degenerate in the circularly polarized modes right with opposite handedness OVleft 01 and OV01 , that are associated with electrical fields E1 (ρ , θ ) and E2 (ρ , θ ), both with radial intensity profile R21 (ρ ). Moreover, OV11 represents the group +right −right −left of four modes OV+left , and OV11 , according to the different handedness 11 , OV11 , OV11 of circular polarization (left/right) and OAM (+/-), that are associated with the electrical fields E3 (ρ , θ ), E4 (ρ , θ ), E5 (ρ , θ ), and E6 (ρ , θ ), all with radial intensity profiles R2 (ρ ). Note that 2 the Laguerre-Gauss field distribution we are considering only focuses on the two-dimensional transverse components of the fields. The electrical field at the output of the fiber can be modelled as E(ρ , θ , a) = E(ρ , θ ) · C · a, (1) where · is the vector-matrix product, E(ρ , θ ) = [E1 (ρ , θ ), E2 (ρ , θ ), · · · , E6 (ρ , θ )], and C is a 6 × 2 matrix consisting of the two columns corresponding to the two transmitted modes of the 6 × 6 unitary crosstalk matrix [5, 6]. It is worth noting that a unitary crosstalk matrix is energy conserving. 3.
Mode separation by circular PDs
After mode coupling, direct detection by the two spatially separated PDs of Fig. 1 gives the two signals i = {1, 2}, (2) zi (a) = ||E(ρ , θ , a)||2 ρ d ρ d θ , Si
where Si is the surface of the i-th PD. By plugging (1) into (2) and by taking into account the beating between Ek and Eh given by Eq. (5) in the Appendix and following the calculations there reported, one obtains the nonlinear MIMO model z(a) = F · P · a + w(a),
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25177
Fig. 2. Realizations of intensity patterns after propagation with p1 = p2 = 0.8. From the upper to the lower row: a = [1, 0]T , a = [0, 1]T , and a = [1, 1]T .
where z(a) = [z1 (a), z2 (a)]T , w(a) is a N × 1 vector, and the product F · P is 2 R (ρ )ρ d ρ d θ R2 (ρ )ρ d ρ d θ p1 1− p2 S1 12 S1 22 · , 1− p1 p2 S2 R1 (ρ )ρ d ρ d θ S2 R2 (ρ )ρ d ρ d θ where coefficients 0 ≤ p1 ≤ 1 and 0 ≤ p2 ≤ 1 that characterize the crosstalk are reported in the Appendix. The term w(a) depends in a nonlinear manner on a: w(a) = a1 a2 wi, √ √ where i = [ 0.5, − 0.5]T and w is a random variable that is bounded as follows: √ w ≤ 2 2||F · i|| pmax (1 − pmin ), with pmax = max (p1 , p2 ) and pmin = min (p1 , p2 ). Figure 2 shows the intensity patterns of several realizations of the mode coupling given by (1). Note that in [10], where intermodal crosstalk is not considered, one has that P is the identity matrix and that w(a) is zero. 4.
Mode separation by optical demultiplexing
As an adversary to our proposed approach, we consider a scheme based on optical mode demultiplexing where the optical demultiplexer ideally separates the six modes comprised in the families OV01 and OV11 [13]. We assume direct detection after optical demultiplexing, therefore six PDs are used, one for each of six outputs of the demultiplexer. The PDs’ outputs can be collected in the vector: z(a) = P · a + w(a), where P is the 6 × 2 matrix containing the squared magnitude of elements ci, j of C and
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25178
PD2
z1
PD1
z2
y1 Transimpedance Transimpedance y2
a1 Electrical Network
Decision Logic a2
Fig. 3. Receiver block diagram.
⎧⎡ ∗ ⎤⎫ c1,1 c1,2 ⎪ ⎪ ⎪ ⎪⎢c∗ c ⎥⎪ ⎪ ⎨ ⎬ ⎢ 2,1 2,2 ⎥ w(a) = 2a1 a2 ℜ ⎢ . ⎥ , ⎪ ⎪⎣ .. ⎦⎪ ⎪ ⎪ ⎪ ⎩ ∗ ⎭ c6,1 c6,2 where ℜ{x} is the vector that collects the real parts of each element of vector x, and the superscript ∗ denotes complex conjugation. 5.
MIMO processing
Considering short reach without optical amplifiers, the noisy signal after photodetection is y = z(a) + n, where n is an additive white Gaussian noise (AWGN) vector with variance σn2 per PD. The optimal decision rule about the transmitted vector a is the maximum likelihood (ML) rule, which, when AWGN affects the received signal, is aˆ ML = arg min (y − z(a))T · (y − z(a)).
(3)
a∈A
The block diagram of the receiver based on spatially separated PDs is reported in Fig. 3. The block called electrical network produces the four differences (y − z(a)) by biasing y by the four biases z(a) and then squares the four results. The block called decision logic, that can be based on few logic elements operating at symbol frequency, compares the four squared distances and decides in favor of the vector a associated with the minimum squared distance. 6.
Simulation results
Hereafter the performance of the two schemes presented above is compared. Figure 4 reports the symbol error rate (SER) for p1 = p2 = p against the ratio R=
E{(∑Ni=1 zi )2 } , N σn2
where E{·} is the expectation operator, N is the number of PDs, and E{(∑Ni=1 zi )2 }/N is proportional to the mean square of the optical power that reaches each PD. We adopt E{(∑Ni=1 zi )2 }/N as a figure of merit because it depends on the input optical power only, while it is independent of the mode coupling generated during propagation. In the case of spatially separated PDs, we found that the optimal radius of the inner PD is fairly insensitive to p and R. Figure 4 shows that also the sensitivity of the SER to p is moderate. It is worth observing that, from the reported results, it appears that the ML rule for p = 1 is virtually optimal also for p = 0.9 and p = 0.8. The reason behind this can be seen from the decision boundary reported in Fig. 5. From Fig. 5 it is apparent that the error between [0, 1]T and [1, 0]T dominates the performance, and that the
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25179
0
10
−1
SER
10
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10
−3
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10
5
10
15 R[dB]
20
25
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15 R[dB]
20
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15 R[dB]
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Fig. 4. From top to bottom: SER versus R with p = 1, p = 0.9, and p = 0.8. Solid line: circular PDs with ML decision. Asterisks: circular PDs with suboptimal decision (ML rule for p = 1). Dashed line: 20 realizations of C and optical demultiplexing with ML detection.
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25180
1.5
z
2
1
a1=1 a2=1
a1=0 a2=1
0.5 a =1 1 a2=0
a1=0 a =0
0
2
0
0.5
1
1.5
z1 Fig. 5. Asterisks: z(a) with p = 1. Circles: 20 realizations of z(a) with p = 0.8. The solid line bounds the decision regions of the ML rule for p = 1. From the position of the circles in the plane one realizes that the optimal decision boundary for p = 0.8 is virtually the same as that for p = 1.
ML decision boundary for p = 1 remains virtually optimal for discriminating between [0, 1]T and [1, 0]T also when crosstalk is present. As a consequence, MIMO processing can be nonadaptive. With optical demultiplexing there are infinitely many matrices C that lead to the same value of p, the performance depending on the specific C, as the families of curves reported in Fig. 4 shows. In the Fig. for each considered value of p only 20 curves, out of the 104 obtained by randomly generating matrix C, have been reported for readability reasons. The lower and upper curves correspond, respectively, to the best and the worse performance we observed among the 104 random realizations of the matrix C, while the other 18 curves were obtained by randomly choosing among the remaining matrices C. From Fig. 4 we observe that also taking the worst case, optical demultiplexing outperforms spatially separated PDs, but the implementation of (3) is much more demanding in terms of signal processing. Specifically, besides the device for optical demultiplexing, ML detection after optical demultiplexing and photodetection needs a quite complex 6-dimensional MIMO processing that should adaptively match the specific matrix C. This adaptive matching to the matrix C explains the better performance achieved by the optical demultiplexing approach compared to our proposed non-adaptive scheme. 7.
Conclusions
A nonlinear MIMO system exploiting the inherent characteristic of optical modes in fiber in a direct detection scheme based on circular concentric PDs has been proposed and compared to MIMO based on optical demultiplexing with direct detection. The advantage of circular concentric PDs is that the receiver does not require high-speed sampling and DSP to detect the transmitted bits, being devised here a low cost, low consumption solution, not requiring any adaptive processing.
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25181
Appendix The intensity of electrical field in (1) is I(ρ , θ , a) = ||E(ρ , θ , a)||2 = E(ρ , θ ) · G(a) · EH(ρ , θ ) with
G1,1 (a) G(a) = C · a · a · C = G2,1 (a) c c C = 1,1 1,2 , c2,1 c2,2 H
H
G1,2 (a) G2,2 (a)
(4)
where the superscript H denotes transposition and complex conjugation. These matrices have been partitioned considering the two families of modes: for k, h ∈ {1, 2}, Gk,h (a) is an mk × mh matrix, while ck,h is an mk × 1 vector, with m1 = 2 and m2 = 4. Thanks to the circular symmetry of PDs, the beating between any two of the considered fields is such that 2 Ek (ρ , θ ) · Eh∗ (ρ , θ )ρ d ρ d θ = Si Rk (ρ , θ )ρ d ρ d θ k = h , (5) 0 k = h Si with i = {1, 2}. By using the above property and plugging (4) into (2), the output of i-th PD becomes zi (a) =
6
∑
6
∑ gk,h (a)
k=1 h=1
Si
Ek (ρ , θ ) · Eh∗ (ρ , θ )ρ d ρ d θ
= tr{G1,1 (a)} fi,1 + tr{G2,2 (a)} fi,2 ,
(6)
where tr{·} indicates the trace operator of a square matrix, G(a) = gk,h (a) and fi, j =
Si
R2j (ρ )ρ d ρ d θ .
The trace of G j, j (a) is tr G j, j (a) =
2
2
∑ ∑ ak ah cHj,k · c j,h .
k=1 h=1
By separating the linear and nonlinear part of this equation and plugging it into (6) one obtains H c1,1 · c1,2 , zi (a) = [ fi,1 fi,2 ] · P · a + 2a1 a2 [ fi,1 fi,2 ] · ℜ cH 2,1 · c2,2 and
||c1,1 ||2 P= ||c2,1 ||2
p1 ||c1,2 ||2 = 1 − p1 ||c2,2 ||2
1 − p2 . p2
In order to obtain the nonlinear MIMO model, the outputs of two PDs can be collected in the vector z (a) = F · P · a + w(a), z(a) = 1 z2 (a)
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25182
with F = [ fi, j ] and the nonlinear term w(a) = 2a1 a2 F · ℜ
cH 1,1 · c1,2 cH 2,1 · c2,2
.
(7)
According to the unitary property of C, H jφ cH 1,1 · c1,2 = −c2,1 · c2,2 = α e .
Substituting this relation in (7), the nonlinear term becomes √ w(a) = 2 2a1 a2 α cos(φ ) F · i √ √ where i = [ 0.5, − 0.5]T is the unit vector with slope −1. By using the assumption that the two modes have the same energy, the following relation holds F · i = γ i, with γ ≥ 0. Using Schwarz’s inequality, one obtains H α cos(φ ) ≤ min |cH 1,1 · c1,2 |, |c2,1 · c2,2 | p1 (1 − p2 ), p2 (1 − p1 ) , ≤ min leading to w(a) = a1 a2 w i, where pmax = max (p1 , p2 ) , pmin = min (p1 , p2 ), and √ √ w = 2 2α cos(φ ) γ ≤ 2 2γ pmax (1 − pmin ).
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25183
di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy 2 Institute for Advanced Study, Technische Universit¨ at M¨unchen, Lichtenbergstrasse 2a, D-85748 Garching, Germany ∗ [email protected]
Abstract: The complexity and the power consumption of digital signal processing are crucial issues in optical transmission systems based on mode division multiplexing and coherent multiple-input multiple-output (MIMO) processing at the receiver. In this paper the inherent characteristic of spatial separation between fiber modes is exploited, getting a MIMO system where joint demultiplexing and detection is based on spatially separated photodetectors. After photodetection, one has a MIMO system with nonlinear crosstalk between modes. The paper shows that the nonlinear crosstalk can be dealt with by a low-complexity and non-adaptive detection scheme, at least in the cases presented in the paper. © 2013 Optical Society of America OCIS codes: (040.1240) Arrays; (060.2330) Fiber optics communications; (060.4080) Modulation; (060.4230) Multiplexing; (080.4865) Optical vortices.
References and links 1. R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006). 2. A. Tarighat, R. Hsu, A. Shah, A. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links,” IEEE Commun. Mag. 45, 57–63 (2007). 3. T. Morioka, Y. Awaji, R. Ryf, P. J. Winzer, D. Richardson, and F. Poletti, “Enhancing optical communications with brand new fibers,” IEEE Commun. Mag. 50, s31–s42 (2012). 4. R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” IEEE Proc. 100, 1035–1055 (2012). 5. R. Dar, M. Feder, and M. Shtaif, “The underaddressed optical multiple-input, multiple-output channel: capacity and outage,” Opt. Lett. 37, 3150–3152 (2012). 6. R. Dar, M. Feder, and M. Shtaif, “The Jacobi MIMO channel,” IEEE Trans. Inf. Theory 59, 2426–2441 (2013). 7. V. Sleiffer, Y. Jung, B. Inan, H. Chen, R. van Uden, M. Kuschnerov, D. van den Borne, S. Jansen, V. Veljanovski, T. Koonen, D. Richardson, S. Alam, F. Poletti, J. Sahu, A. Dhar, B. Corbett, R. Winfield, A. Ellis, and H. De Waardt, “Mode-division-multiplexed 3×112-Gb/s DP-QPSK transmission over 80 km few-mode fiber with inline MM-EDFA and blind DSP,” in European Conference and Exhibition on Optical Communication, OSA Technical Digest (online) (Optical Society of America, 2012), paper Tu.1.C.2. 8. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6×6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012). 9. C. Koebele, M. Salsi, L. Milord, R. Ryf, C. Bolle, P. Sillard, S. Bigo, and G. Charlet, “40km transmission of five mode division multiplexed data streams at 100Gb/s with low MIMO-DSP complexity,” in European Conference
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25174
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Introduction
Mode Division Multiplexing (MDM) has attracted widespread attention as a means to increase the information throughput over a single fiber [1–4]. Considerations about the capacity of optical MDM can be found in [5, 6]. In recent experiments [7,8], MDM exploiting multiple-input multiple-output (MIMO) digital signal processing (DSP) in combination with coherent demultiplexing and detection has demonstrated its effectiveness in removing the intermodal crosstalk that originates during propagation. However, real-time coherent MIMO processing requires high complexity and power consumption at the receiver. A scheme performing optical demultiplexing before coherent detection, thus allowing reduced complexity compared to the fully coherent MIMO signal precessing, has been proposed in [9]. Even simpler schemes, that are fully noncoherent, have been considered in [10–12]. Such approaches rely upon the assumption of having groups of modes that are well separated in terms of propagation constants, so that the crosstalk between them, when used as independent channels, can be considered negligible at the receiver. The low crosstalk condition is achieved in [10] by considering a fiber with a core radius that is large enough to guarantee that the multiplexed mode groups remain well spatially separated after propagation over the fiber. Spatial detection is then carried out on each group independently by means of circular multi-segment Photode#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25175
tectors (PDs). Assuming linear superposition of the power distributions at the fiber output, a MIMO channel matrix is estimated, without implementing a real mode group division multiplexing transmission, in order to perform detection. On another hand, the requirement for well separated propagation constants is used as a prerequisite in [11] for experimental transmission of mode groups over a distance in the order of tens of meters. In this case, the short propagation distance introduces a relatively low crosstalk that can be easily mitigated by mode-selective spatial filtering. This spatial filtering is based on the use of a number of standard single-mode fibers equals to the number of the transmitted mode groups. As in [10], the experimental investigation done in [12] does not implement a real mode group division multiplexed transmission. In the considered experimental analysis the number of mode groups that can be used as separate spatial channels for on-off keying transmission over a 2 km long standard graded-index multimode fiber is firstly determined and the error performance for each mode group, when used as an individual channel, is then evaluated. A lower conversion efficiency is shown for higher-order modes with respect to the fundamental mode, due to imperfect mode conversion at the transmitter, that is responsible for performance degradation. This paper considers an MDM scheme that is candidate for application in those systems where cost, power consumption, and complexity, are relevant issues, such as local area networks and optical interconnects for data centers. In this context of short-range optical communication links (i.e., up to a few kilometers), the impact of chromatic dispersion and modal dispersion is negligible also for high bit rate [13]. The main motivation behind our work is the observation that the detection techniques proposed in [10–12] are designed by assuming a low level of intermodal crosstalk and, as such, they are not able to fully take into consideration its effects. To this aim, a first step toward the characterization of such a crosstalk has been done in [14,15]. In particular, a penalty in the order of 2 dB is shown for optical mode group demultiplexing without any MIMO processing compared to single channel transmission in a standard 50 μ mcore graded-index multimode fiber for lengths of 2 km and of 8 km. In the present paper, we propose a novel MDM scheme that is able to cope with the intermodal crosstalk. The paper focuses on two-mode multiplexing, and, starting from the mode coupling model of [5,6], shows that, after detection by two spatially separated concentric PDs [16], one gets a MIMO channel with nonlinear crosstalk, that, in the above mentioned applicative context, can be dealt with by MIMO processing of low complexity. 2.
System model
Let us consider MDM transmission in optical fiber based on the excitation of two modes, called mode 1 and mode 2, that are on-off keying modulated and characterized by the same energy. The MIMO alphabet A is therefore made by the four column vectors a = [a1 , a2 ]T , where the superscript T denotes vector or matrix transposition, and the bits a1 and a2 associated with modes 1 and 2, respectively, are assumed to be i.i.d. random variables taking their values with uniform probability in {0, 1}. In this paper it is assumed that the field distributions of fiber modes are given by the Laguerre-Gaussian functions [17]. This assumption, which holds for parabolic index profile, can be used also to approximate other fiber index profiles, still maintaining the here exploited geometric and symmetry properties of the field. To be concrete, in what follows the excitation of optical vortices (OVs) carrying orbital angular momentum (OAM), that have been proposed in [15], [18–21] in place of Linearly Polarized (LP) modes to achieve MDM in freespace, is considered. However, it is worth mentioning that the analysis presented here holds also for LP modes. Mode 1 is the fundamental mode OV01 , while mode 2 is OV11 . The two modes are characterized by the radial intensity profiles R21 (ρ ) and R22 (ρ ) that are the squares of the radial parts of the Laguerre-Gaussian field associated with modes OV01 and OV11 , respectively. Figure 1 shows the intensity patterns of the two modes and the two spatially separated
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25176
PD2 PD1
Fig. 1. From left to right: intensity pattern of OV01 and intensity pattern of OV11 before coupling. Rightmost picture: inner circle PD1 , outer circular annulus PD2 .
PDs matched to the two modes. Owing to mechanical perturbations (like twisting, bending, and vibrations) applied to the fiber, mode coupling during propagation can take place, leading to intermodal crosstalk. Moreover, mode coupling can be introduced at the transmitter due to imperfect mode conversion in the desired mode group. The experimental characterization of intermodal crosstalk made for LP [14] and OV [15] modes suggests that the strong random mode coupling regime [22,23] does not apply in short-range (i.e., up to lengths of a few kilometers) standard multimode fiber-optic links. In this case the crosstalk among mode groups is mainly induced by the imperfect generation of the multiplexed modes and is more relevant for higher-order modes. In the following we introduce the crosstalk matrix between the modes corresponding to OV01 and OV11 . More in detail, the fundamental mode OV01 is two-fold degenerate in the circularly polarized modes right with opposite handedness OVleft 01 and OV01 , that are associated with electrical fields E1 (ρ , θ ) and E2 (ρ , θ ), both with radial intensity profile R21 (ρ ). Moreover, OV11 represents the group +right −right −left of four modes OV+left , and OV11 , according to the different handedness 11 , OV11 , OV11 of circular polarization (left/right) and OAM (+/-), that are associated with the electrical fields E3 (ρ , θ ), E4 (ρ , θ ), E5 (ρ , θ ), and E6 (ρ , θ ), all with radial intensity profiles R2 (ρ ). Note that 2 the Laguerre-Gauss field distribution we are considering only focuses on the two-dimensional transverse components of the fields. The electrical field at the output of the fiber can be modelled as E(ρ , θ , a) = E(ρ , θ ) · C · a, (1) where · is the vector-matrix product, E(ρ , θ ) = [E1 (ρ , θ ), E2 (ρ , θ ), · · · , E6 (ρ , θ )], and C is a 6 × 2 matrix consisting of the two columns corresponding to the two transmitted modes of the 6 × 6 unitary crosstalk matrix [5, 6]. It is worth noting that a unitary crosstalk matrix is energy conserving. 3.
Mode separation by circular PDs
After mode coupling, direct detection by the two spatially separated PDs of Fig. 1 gives the two signals i = {1, 2}, (2) zi (a) = ||E(ρ , θ , a)||2 ρ d ρ d θ , Si
where Si is the surface of the i-th PD. By plugging (1) into (2) and by taking into account the beating between Ek and Eh given by Eq. (5) in the Appendix and following the calculations there reported, one obtains the nonlinear MIMO model z(a) = F · P · a + w(a),
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25177
Fig. 2. Realizations of intensity patterns after propagation with p1 = p2 = 0.8. From the upper to the lower row: a = [1, 0]T , a = [0, 1]T , and a = [1, 1]T .
where z(a) = [z1 (a), z2 (a)]T , w(a) is a N × 1 vector, and the product F · P is 2 R (ρ )ρ d ρ d θ R2 (ρ )ρ d ρ d θ p1 1− p2 S1 12 S1 22 · , 1− p1 p2 S2 R1 (ρ )ρ d ρ d θ S2 R2 (ρ )ρ d ρ d θ where coefficients 0 ≤ p1 ≤ 1 and 0 ≤ p2 ≤ 1 that characterize the crosstalk are reported in the Appendix. The term w(a) depends in a nonlinear manner on a: w(a) = a1 a2 wi, √ √ where i = [ 0.5, − 0.5]T and w is a random variable that is bounded as follows: √ w ≤ 2 2||F · i|| pmax (1 − pmin ), with pmax = max (p1 , p2 ) and pmin = min (p1 , p2 ). Figure 2 shows the intensity patterns of several realizations of the mode coupling given by (1). Note that in [10], where intermodal crosstalk is not considered, one has that P is the identity matrix and that w(a) is zero. 4.
Mode separation by optical demultiplexing
As an adversary to our proposed approach, we consider a scheme based on optical mode demultiplexing where the optical demultiplexer ideally separates the six modes comprised in the families OV01 and OV11 [13]. We assume direct detection after optical demultiplexing, therefore six PDs are used, one for each of six outputs of the demultiplexer. The PDs’ outputs can be collected in the vector: z(a) = P · a + w(a), where P is the 6 × 2 matrix containing the squared magnitude of elements ci, j of C and
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25178
PD2
z1
PD1
z2
y1 Transimpedance Transimpedance y2
a1 Electrical Network
Decision Logic a2
Fig. 3. Receiver block diagram.
⎧⎡ ∗ ⎤⎫ c1,1 c1,2 ⎪ ⎪ ⎪ ⎪⎢c∗ c ⎥⎪ ⎪ ⎨ ⎬ ⎢ 2,1 2,2 ⎥ w(a) = 2a1 a2 ℜ ⎢ . ⎥ , ⎪ ⎪⎣ .. ⎦⎪ ⎪ ⎪ ⎪ ⎩ ∗ ⎭ c6,1 c6,2 where ℜ{x} is the vector that collects the real parts of each element of vector x, and the superscript ∗ denotes complex conjugation. 5.
MIMO processing
Considering short reach without optical amplifiers, the noisy signal after photodetection is y = z(a) + n, where n is an additive white Gaussian noise (AWGN) vector with variance σn2 per PD. The optimal decision rule about the transmitted vector a is the maximum likelihood (ML) rule, which, when AWGN affects the received signal, is aˆ ML = arg min (y − z(a))T · (y − z(a)).
(3)
a∈A
The block diagram of the receiver based on spatially separated PDs is reported in Fig. 3. The block called electrical network produces the four differences (y − z(a)) by biasing y by the four biases z(a) and then squares the four results. The block called decision logic, that can be based on few logic elements operating at symbol frequency, compares the four squared distances and decides in favor of the vector a associated with the minimum squared distance. 6.
Simulation results
Hereafter the performance of the two schemes presented above is compared. Figure 4 reports the symbol error rate (SER) for p1 = p2 = p against the ratio R=
E{(∑Ni=1 zi )2 } , N σn2
where E{·} is the expectation operator, N is the number of PDs, and E{(∑Ni=1 zi )2 }/N is proportional to the mean square of the optical power that reaches each PD. We adopt E{(∑Ni=1 zi )2 }/N as a figure of merit because it depends on the input optical power only, while it is independent of the mode coupling generated during propagation. In the case of spatially separated PDs, we found that the optimal radius of the inner PD is fairly insensitive to p and R. Figure 4 shows that also the sensitivity of the SER to p is moderate. It is worth observing that, from the reported results, it appears that the ML rule for p = 1 is virtually optimal also for p = 0.9 and p = 0.8. The reason behind this can be seen from the decision boundary reported in Fig. 5. From Fig. 5 it is apparent that the error between [0, 1]T and [1, 0]T dominates the performance, and that the
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25179
0
10
−1
SER
10
−2
10
−3
10
−4
10
5
10
15 R[dB]
20
25
10
15 R[dB]
20
25
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15 R[dB]
20
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10
−3
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10
5
Fig. 4. From top to bottom: SER versus R with p = 1, p = 0.9, and p = 0.8. Solid line: circular PDs with ML decision. Asterisks: circular PDs with suboptimal decision (ML rule for p = 1). Dashed line: 20 realizations of C and optical demultiplexing with ML detection.
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25180
1.5
z
2
1
a1=1 a2=1
a1=0 a2=1
0.5 a =1 1 a2=0
a1=0 a =0
0
2
0
0.5
1
1.5
z1 Fig. 5. Asterisks: z(a) with p = 1. Circles: 20 realizations of z(a) with p = 0.8. The solid line bounds the decision regions of the ML rule for p = 1. From the position of the circles in the plane one realizes that the optimal decision boundary for p = 0.8 is virtually the same as that for p = 1.
ML decision boundary for p = 1 remains virtually optimal for discriminating between [0, 1]T and [1, 0]T also when crosstalk is present. As a consequence, MIMO processing can be nonadaptive. With optical demultiplexing there are infinitely many matrices C that lead to the same value of p, the performance depending on the specific C, as the families of curves reported in Fig. 4 shows. In the Fig. for each considered value of p only 20 curves, out of the 104 obtained by randomly generating matrix C, have been reported for readability reasons. The lower and upper curves correspond, respectively, to the best and the worse performance we observed among the 104 random realizations of the matrix C, while the other 18 curves were obtained by randomly choosing among the remaining matrices C. From Fig. 4 we observe that also taking the worst case, optical demultiplexing outperforms spatially separated PDs, but the implementation of (3) is much more demanding in terms of signal processing. Specifically, besides the device for optical demultiplexing, ML detection after optical demultiplexing and photodetection needs a quite complex 6-dimensional MIMO processing that should adaptively match the specific matrix C. This adaptive matching to the matrix C explains the better performance achieved by the optical demultiplexing approach compared to our proposed non-adaptive scheme. 7.
Conclusions
A nonlinear MIMO system exploiting the inherent characteristic of optical modes in fiber in a direct detection scheme based on circular concentric PDs has been proposed and compared to MIMO based on optical demultiplexing with direct detection. The advantage of circular concentric PDs is that the receiver does not require high-speed sampling and DSP to detect the transmitted bits, being devised here a low cost, low consumption solution, not requiring any adaptive processing.
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25181
Appendix The intensity of electrical field in (1) is I(ρ , θ , a) = ||E(ρ , θ , a)||2 = E(ρ , θ ) · G(a) · EH(ρ , θ ) with
G1,1 (a) G(a) = C · a · a · C = G2,1 (a) c c C = 1,1 1,2 , c2,1 c2,2 H
H
G1,2 (a) G2,2 (a)
(4)
where the superscript H denotes transposition and complex conjugation. These matrices have been partitioned considering the two families of modes: for k, h ∈ {1, 2}, Gk,h (a) is an mk × mh matrix, while ck,h is an mk × 1 vector, with m1 = 2 and m2 = 4. Thanks to the circular symmetry of PDs, the beating between any two of the considered fields is such that 2 Ek (ρ , θ ) · Eh∗ (ρ , θ )ρ d ρ d θ = Si Rk (ρ , θ )ρ d ρ d θ k = h , (5) 0 k = h Si with i = {1, 2}. By using the above property and plugging (4) into (2), the output of i-th PD becomes zi (a) =
6
∑
6
∑ gk,h (a)
k=1 h=1
Si
Ek (ρ , θ ) · Eh∗ (ρ , θ )ρ d ρ d θ
= tr{G1,1 (a)} fi,1 + tr{G2,2 (a)} fi,2 ,
(6)
where tr{·} indicates the trace operator of a square matrix, G(a) = gk,h (a) and fi, j =
Si
R2j (ρ )ρ d ρ d θ .
The trace of G j, j (a) is tr G j, j (a) =
2
2
∑ ∑ ak ah cHj,k · c j,h .
k=1 h=1
By separating the linear and nonlinear part of this equation and plugging it into (6) one obtains H c1,1 · c1,2 , zi (a) = [ fi,1 fi,2 ] · P · a + 2a1 a2 [ fi,1 fi,2 ] · ℜ cH 2,1 · c2,2 and
||c1,1 ||2 P= ||c2,1 ||2
p1 ||c1,2 ||2 = 1 − p1 ||c2,2 ||2
1 − p2 . p2
In order to obtain the nonlinear MIMO model, the outputs of two PDs can be collected in the vector z (a) = F · P · a + w(a), z(a) = 1 z2 (a)
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25182
with F = [ fi, j ] and the nonlinear term w(a) = 2a1 a2 F · ℜ
cH 1,1 · c1,2 cH 2,1 · c2,2
.
(7)
According to the unitary property of C, H jφ cH 1,1 · c1,2 = −c2,1 · c2,2 = α e .
Substituting this relation in (7), the nonlinear term becomes √ w(a) = 2 2a1 a2 α cos(φ ) F · i √ √ where i = [ 0.5, − 0.5]T is the unit vector with slope −1. By using the assumption that the two modes have the same energy, the following relation holds F · i = γ i, with γ ≥ 0. Using Schwarz’s inequality, one obtains H α cos(φ ) ≤ min |cH 1,1 · c1,2 |, |c2,1 · c2,2 | p1 (1 − p2 ), p2 (1 − p1 ) , ≤ min leading to w(a) = a1 a2 w i, where pmax = max (p1 , p2 ) , pmin = min (p1 , p2 ), and √ √ w = 2 2α cos(φ ) γ ≤ 2 2γ pmax (1 − pmin ).
#192628 - $15.00 USD Received 19 Jun 2013; revised 14 Aug 2013; accepted 4 Sep 2013; published 15 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025174 | OPTICS EXPRESS 25183
