Invited Article: Polarization diversity and modulation for high-speed ...

Invited Article: Polarization diversity and modulation for high-speed optical communications: architectures and capacity William Shieh, Hamid Khodakarami, and Di Che Citation: APL Photonics 1, 040801 (2016); doi: 10.1063/1.4949568 View online: http://dx.doi.org/10.1063/1.4949568 View Table of Contents: http://scitation.aip.org/content/aip/journal/app/1/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High-speed all-optical terahertz polarization switching by a transient plasma phase modulator Appl. Phys. Lett. 96, 161103 (2010); 10.1063/1.3407514 High-speed Ge photodetector monolithically integrated with large cross-section silicon-on-insulator waveguide Appl. Phys. Lett. 95, 261105 (2009); 10.1063/1.3279129 Direct coupling of high-speed optical detector preamplifiers Rev. Sci. Instrum. 71, 3918 (2000); 10.1063/1.1290500 Voltage-tunable near-infrared photodetector: Versatile component for optical communication systems J. Vac. Sci. Technol. B 16, 2619 (1998); 10.1116/1.590244 Voltage tunable SiGe photodetector: A novel tool for crypted optical communications through wavelength mixing Appl. Phys. Lett. 70, 3194 (1997); 10.1063/1.119155

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APL PHOTONICS 1, 040801 (2016)

Invited Article: Polarization diversity and modulation for high-speed optical communications: architectures and capacity William Shieh, Hamid Khodakarami, and Di Che Department of Electrical and Electronic Engineering, The University of Melbourne, VIC, 3010, Australia

(Received 7 March 2016; accepted 3 May 2016; published online 19 July 2016) Polarization is one of the fundamental properties of optical waves. To cope with the exponential growth of the Internet traffic, optical communications has advanced by leaps and bounds within the last decade. For the first time, the polarization domain has been extensively explored for high-speed optical communications. In this paper, we discuss the general principle of polarization modulation in both Jones and Stokes spaces. We show that there is no linear optical device capable of transforming an arbitrary input polarization into one that is orthogonal to itself. This excludes the receiver self-polarization diversity architecture by splitting the signal into two branches, and then transferring one of the branches into orthogonal polarization. We next propose a novel Stokes vector (SV) detection architecture using four single-ended photodiodes (PD) that can recover a full set of SV. We then derive a closed-form expression for the information capacity of different SV detection architectures and compare the capacity of our proposed architectures with that of intensity-modulated directly-detected (IM/DD) method. We next study the 3-PD SV detection architecture where a subset of SV is detected, and devise a novel modulation algorithm that can achieve 2-dimensional modulation with the 3-PD detection. By using cost-effective SV receivers, polarization modulation and multiplexing offers a powerful solution for short-reach optical networks where the wavelength domain is quickly exhausted. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4949568]

Polarization is a fundamental property of electromagnetic wave propagating through optical media where both the electric and magnetic fields are predominantly perpendicular to the direction of travel, or the so-called “plane wave” is an accurate approximation. The optical signal can be modeled as a Jones vector and the channel can be modelled as a Jones matrix in a linear transmission system. Polarization has played an important role in many fields such as geology, chemistry, astronomy, photography, display, and wireless or optical communications. In wireless communications over multipath channels, there is a generally large discriminating effect against one of the polarizations, rending only one of the polarizations as the preferred carrier for the signal. In contrast, in optical fiber communications, the two polarizations experience a uniform loss which ideally lends themselves to being exploited for doubling the channel capacity. Despite the fact that polarization modulation or multiplexing for optical communication has been studied for more than 2 decades,1–3 the real-field application is rather recent when dual-polarization coherent detection modulation format becomes the de facto transport standard for high-speed long-haul transmission at 100 Gb/s and beyond.4,5 The reason for lack of progress in polarization multiplexed systems in the past is that they are either much more complicated or more expensive than single-polarization systems, or the overly sophisticated detection algorithms cannot be supported by the electronics then. The recent advances in electronic digital signal processing (DSP) is with no doubt the enabling technology responsible for the optical transport breakthrough in optical communication industry during the last decade. The polarization modulation and multiplexing in optical fiber aided by 2378-0967/2016/1(4)/040801/9

1, 040801-1

© Author(s) 2016.


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APL Photonics 1, 040801 (2016)

coherent detection has become quite mature and straightforward in the realm of communication. This is mainly because such a system can be modelled as a simple 2 × 2 Multiple Input Multiple Output (MIMO) channel with the channel matrix most often approximated as unitary. On the other front, the emergence of cloud computing and content distribution networks allows the majority of heavy-duty data and signal processing to take place at the dedicated large datacenters.6 As a result, a tremendous demand has emerged for the cost-effective high-speed interconnects for inter- or intra-datacenter applications. Those interfaces at high speeds of 100 Gb/s and beyond cannot directly adopt the conventional coherent detection which is cost-prohibitive. Neither can it use conventional direct modulation/direct detection (DM/DD) transport due to reach constraints at high transmission rates. Various DD modulation formats have been put forward to achieve 100 Gb/s.7–9 Among the modulation formats beyond conventional DM/DD, Stokes vector direct detection (SVDD)10–13 has recently gained much attraction thanks to its capability of achieving high data rate at reasonable reach. In this paper, we first lay out some fundamental properties of polarization in relation to polarization modulation. We then show that there is no existing device capable of transforming an arbitrary input polarization into the one orthogonal to itself. This excludes the receiver polarization diversity architecture by splitting the optical signal into two branches, and then transferring one of the branches to the orthogonal polarization. We next show an SV receiver architecture using 4 single-ended photo detectors (4-PD) that can recover a full set of SV without excess loss. We then derive a closed-form expression for the information capacity of different SV detection architectures and compare the capacity of our proposed architectures with that of intensity-modulated directly-detected (IM/DD) method. At last, we study a 3-PD SV receiver architecture, and find that it can recover the SV with the sign of the first element undetermined. We also devise a novel modulation algorithm that can achieve 2-dimensioanl transmission with 3-PD detection. Directly detected polarization modulation poses to play a major role in short-reach optical networks when the wavelength domain is being quickly used up. An optical signal propagating along an optical medium can be described in Jones space as Eout = UEin,

(1)

where Eout and Ein are respectively the optical signal at the input and output of the medium, and U is the Jones matrix describing the polarization coupling of optical signal traversing the medium. The optical signal E is represented as a 2-dimensional Jones vector given by  E = Ex

Ey

T

,

(2)

where subscript “T” stands for transpose. In coherent optical communication, the information is carried independently by the fields E x and E y each being a 2-dimensional complex value. As such, the coherent optical system can be considered as 4-dimensional modulation or multiplexing. The coherent receiver recovers the complex field of Eout, and the channel matrix U is estimated using either pilot-based14 or blind channel estimation,4 the input symbol can be estimated as Eˆ in = U−1Eout. As such, the coherent detection not only can gracefully rewind all the channel linear effects including both polarization and chromatic dispersion, but also can effectively perform polarization multiplexing, which ideally leads to two fold channel capacity. Nevertheless, for short-reach applications such as inter-datacenter interconnects or passive optical networks, coherent detection can be cost-prohibitive, and exploring the polarization with direct detection becomes a challenging problem. For direct detection, the problem is better treated in T Stokes space, where the signal is presented by a Stokes vector (SV) S = S1 S2 S3 defined as14  S0  |E x |2 + |E y |2     2 2   S1 =  |E x | − |E y |  .  S2 2 Re(E x · E y ∗)    ∗   S3 2 Im(E x · E y )

(3)


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 The total power S0 is not an independent component because S0 = S12 + S22 + S32. SV is looking at the relative phase between E x and E y , losing the absolute phase information of the field. In SV based communication, the information is carried by S1−3, and therefore polarization modulation on Stokes space can be considered as 3-dimensional (3-D) modulation. We note that some publications report detection of 4 independent channels using SV modulation.13 However, there is no contradictory as the 3-D modulation space here is referred to the case for which the information is carried by a single wavelength. On the other hand, if N wavelengths are used, the dimensionality of the Stokes space scales by the number of channels, i.e., 3 × N. In Ref. 13, 4 separate wavelengths are used, indicating the maximum dimension is 12, so there is no contradictory for their 4-D modulation. In Stokes space, the Stokes vectors at the output and input are related by a so-called 3 × 3 Muller matrix H as S R = HST .

(4)

The signal processing procedure in Stokes space is similar: the SV receiver will detect 3-D vector S R , Muller matrix H is estimated by pilot or blind channel estimation, and ST can be rewinded as H−1S R . To fully recover polarization modulation, polarization diversity detection is necessary where both polarizations are detected to avoid signal fading. In some direct detection schemes, such as self-coherent detection where the main carrier is transmitted at the transmitter,9,15,16 this may require splitting the incoming carrier into 2 branches in order to form a complete basis of polarization, as shown in Fig. 1. This is also known as the self-polarization diversity. In the self-polarization diversity, the signal is split into two branches, with one branch passing through a polarization rotator (PR) such as Faraday Rotator Mirror (FRM). The intention is to find a PR that enables E′ to be orthogonal to E so that these two branches form a complete polarization basis. We show below that such an approach to realize polarization diversity is not always valid in general because there is no linear device capable of rotating any input polarization to one orthogonal to itself. Namely, we are not able to find a rotation matrix F that satisfies the following constraint for arbitrary input polarization: E H FE = 0,

(5)

where the subscript “H” stands for Hermitian conjugate. F as a unitary matrix can be always diagonalized and represented as η 1 E = U H DU, D =   0

0  , η 2

(6)

where U is a diagonalization unitary matrix, D is a diagonal matrix with the diagonal elements η 1 and η 2 having magnitude of 1. Substituting Eq. (6) into Eq. (5), we have 2 E1H DE1 = 0 or η 1 · |E1x |2 + η 2 · E1y = 0,

(7)

FIG. 1. Self-polarization diversity detection by constructing two orthogonal polarizations using a polarization rotator such as Faraday rotator mirror (FRM). E and E′ are the optical fields of two diversity branches. The scaling factors are ignored in the figure.


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T where E1 = E1x E1y = UE. The only solution of Eq. (7) for any arbitrary E1x and E1y is that η 1 and η 2 are simultaneously zero. This is contradictory to the fact that the magnitude of η 1 and η 2 is equal to 1.  0It is1 instructive to study the particular case of using FRM where its rotation matrix F = −1 0 (Ref. 18) (the sign for F12 is changed to have the same coordinate at both input  0 1 T and output). Subsisting E = E x E y and F = −1 0 into Eq. (5) gives the condition of self∗ polarization diversity, which is Im(E x · E y ), or S3 = 0 on Poincare Sphere. This indicates that the only polarizations that can be transformed into orthogonal polarizations are those linear polarizations, and all the elliptical polarizations would come out to be not orthogonal to the input. In √ T particular the right/left circular polarization 1/ 2 1 ±i will come out from the FRM with no change in the polarization. In order to recover the polarization modulation, polarization diversity detection is necessary. As shown above, self-polarization diversity is not always valid in general. As such, polarizationdiversity detection is commonly done by using Stokes vector receivers,10–13 which will be further discussed in the following paragraphs. There exist various architectures for SV receivers.2,10 These architectures use either multiple expensive polarization beam splitters, or balanced receivers, or polarizers, or they may introduce excess loss, in the sense that some of the receiver power is thrown away. For instance, for the Creceiver in Ref. 2 that uses wave plates followed by polarizers, half of the signal power is removed by the polarizers. We will show a novel design that employs single-ended PDs and does not have excess loss. It is noted that single-ended PDs are preferred here because of its cost advantage compared to balanced-receivers and thus suitable for short-reach applications. Receiver A in Fig. 2(a) shows our proposed SV receiver comprising a polarization beam splitter, a 1 × 2 coupler, a 3 × 3 coupler, and 4 single-ended PDs. Assuming an ideal 3 × 3 coupler having 3 equal-power outputs with phase difference of 2π/3, it can be shown that the SV can be reconstructed with the photocurrent as

 S0    S1  S2    S3

 1  2  I1  −1   1−γ  I2  = M1   , M1 =  0  I  3   I4  0 

1

1

−1

−1

1 −√ γ  3 − γ

2 √ γ 0

1   −1   1  − √  , γ   3   γ 

(8)

where I1−4 is the photocurrent for PD1−4, γ is the power splitting ratio of the output port of the 1 × 2 coupler leading to the 3 × 3 coupler. It can be seen that there is no excess loss for receiver as all the optical power is detected. The power split ratio γ is a variable we can adjust. We will show later that optimal splitting ratio is 50% where the capacity of the SV modulation is maximized. As such, the 1 × 2 coupler is a 3-dB coupler achieving the optimal performance. The SV receiver using 3-balanced PDs,10 called Receiver B is also shown in Fig. 2(b) for comparison. In what follows, we quantify the information capacity of SV direct detection associated with certain receiver design such as Receivers A and B shown in Figs. 2(a) and 2(b). The transmission of a Stokes vector, S = [S1, S2, S3]T uses a real-value 3 × 3 MIMO channel for which the channel

FIG. 2. (a) Receiver A: SV detection with single-ended PDs and without excess insertion loss. (b) Receiver B: SV detection with balanced-PDs. PBS: Polarization beam splitter. B-PD: Balanced Photodiode.


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capacity is extensively explored in the literature for additive white Gaussian noise and different channel state information scenarios.18 A realistic model for this MIMO channel including amplified spontaneous noise (ASE), shot and thermal noise leads to a nonlinearly correlated noise with a generally non-Gaussian distribution. The distribution of the noise also depends on the optical power which makes the analysis extremely sophisticated. In this paper, we focus on the un-amplified system dominated by the thermal noise. Assuming an additive Gaussian noise at the receiver, the sampled received SV S R is given by S R = HST + n,

(9)

where H, ST , and n = [n1, n2, n3]T are 3 × 3 channel matrix, transmitted Stokes vector and additive noise vector, respectively. We denote the covariance matrix of the noise vector n as Kn . As the receiver has instantaneous knowledge of H, it reverses the channel effect to obtain ˆ S˜ R = H−1HST + H−1n = ST + H−1n = ST + n.

(10)

The information capacity for this real valued channel with correlated Gaussian noise is quantified as18 ) ( det(Knˆ + KS ) (bps/Hz), (11) C(H) = max log KS det(Knˆ ) where the maximization is over all realizations of covariance matrix KS for the distribution of the channel input ST . Here, Knˆ is the covariance matrix of the noise after de-rotation of the received Stokes vector by H−1, denoted by nˆ that is given by T Knˆ = H−1Kn H−1 . (12) In (11), we have already assumed the information baud rate can be twice as the electrical bandwidth. For a random polarization rotation, the ergodic capacity of the channel is given by CE = EH [C(H)] ,

(13)

where EH[·] denotes expectation over all channel realizations. In order to achieve the channel capacity, the full channel state information, including instantaneous knowledge of H, is required at both transmitter and receiver. We assume that this information is only available to the receiver using regular channel estimation via training sequences. However, this information is not available at the transmitter which makes it unable to tune the input covariance matrix to achieve the capacity. The noise covariance Kn is highly dependent on the detection scheme and how the vector S R is obtained using the photodetectors’ output currents. Considering the output currents of the photodetectors given by the vector I = [I1, I2, . . . , Ik ]T , the Stokes vector S R is obtained as S R = MI, where matrix M is the 3 × k matrix representing the mapping scheme of the photodetector output currents ˜ where n˜ is the vector of Gaussian to [S1, S2, S3]T . Thus, the noise vector n is given as n = Mn, noise from photodetectors, with covariance matrix Kn˜ = diag(σ n2 , σ n2 , . . . , σ n2 ) = σ n2 Ik , where σ n2 is the power of the accumulated electronic noise. Therefore, the covariance matrix of the noise vector corrupting the Stokes vector n is given by Kn = MKn˜ M H = σ n2 MM H . Substituting Kn into (12) T yields Knˆ = σ n2 H−1MM H H−1 . Then, the capacity of the Stokes vector direct detection reads as ( ) H −1 T 2 −1 + KS + *. det σ n H MM H ( ) /// (bps/Hz). C = max log .. (14) T KS −1 H −1 det σ n2 H MM H , When photocurrent to SV mapping results in a generally correlated additive noise to the received SV, in order to optimize the instantaneous capacity of the channel, we use the following approach. T The covariance matrix of the additive noise is decomposed as Knˆ = σ n2 H−1MM H H−1 = QΛQ H , where QQ H = I3. Note that for a 3 × 3 Stokes vector detection, HH H = ςI3, where ς is the channel gain factor, and the eigenvalues of Knˆ , the diagonal elements of Λ, are obtained by finding the eigenvalues of MM H , λ i ’s, multiplied by δ2n/ς, then det(Knˆ + KS ) = det(QΛQ H + KS ) = det (Q) det(Λ + Q H KS Q) det(Q H ) = det(Λ + Q H KS Q).


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As the channel information is not available at the transmitter, the optimum input (ST ) distribution that maximizes the ergodic capacity is Gaussian with uncorrelated modulation of elements, i.e.,  scaled identity matrix as KS = ρI3.18 Thus, det Λ + Q H KS Q = det (Λ + KS ) = ς13 (σ n2 λ i + ς ρ). i  Also, det (Knˆ ) = det (Λ) = ς13 σ n2 λ i . The achievable capacity is then obtained as i

C=

 i

log (1 + SNRi )(bps/Hz), SNRi =

ςρ , σ n2 λ i

(15)

where SNRi is the electrical signal-to-noise ratio of the ith component of SV. It is instructive to compare the result of SVDD systems with 3 parallel wavelength-divisionmultiplexed (WDM) IM/DD systems. For analytical simplicity,

we assume

the fiber link is unitary and ignore SVDD component loss, ρ = Si2 = 13 S02 = 31 P2 = 3 p2 , where P is the total received optical power, p = P/3 is the optical power per dimension on Stokes space, and ⟨ ⟩ stands for ensemble average. Thus, (15) is rewritten as

) ( ( )  

3ς p2 3 = log 1 + SNRd (bps/Hz), SNRd = ς p2 /σ n2 , C= log 1 + (16) 2 λi σn λ i i i where SNRd is the signal to noise ratio per dimension, and ς is equivalent to the photo-detector responsibility. Such defined SNRd would be equivalent to the SNR of the 3 parallel IM/DD WDM channels. For the reminder of the paper, SNR implies SNRd , or SNR per dimension for SVDD systems. The mapping matrix M for Receiver A is a truncation matrix of M1 in Eq. (8) by removing the top row. It can be easily shown that at the high SNR regime, the maximization of capacity C of Eq. (14) leads to a very simple form given by ∂ det MM H = 0, 0 < γ < 1. (17) ∂γ The solution for Eq. (17) reveals that the optimal power splitting ratio γ is 1/2 and 2/3 for SV Receiver A and B, respectively. Substituting the mapping matrix M at the optimal ratio into Eq. (15), the achievable capacity yields C A = 3 log(1 + SNRd /4), CB = 3 log(1 + SNRd /6),

(18)

where C A and CB are the achievable capacity for Receivers A and B, respectively. Fig. 3 shows the information capacity as a function of the power splitting ratio of the 1 × 2 coupler. It shows SVDD of Receiver A can achieve a spectral efficiency of 5.4, 14.1, and 23.9 bit/s/Hz at

FIG. 3. Information capacity of SVDD as a function of the power splitting ratio factor in the SV receiver. The solid line for Receiver A and the dashed lines for Receiver B in Fig. 2. SNR is the signal-to-noise ratio per dimension.


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FIG. 4. Information capacity of SVDD and IM/DD systems as a function of SNR per dimension. SVDD is assumed at the optimal splitting ratio.

the optimal splitting ratio at SNR of 10, 20, and 30 dB, respectively. The numerical simulation also confirms the optimal splitting ratio at high SNR of 30 dB is 1/2 and 2/3, for Receivers A and B, respectively. Fig. 4 shows the information capacity of the 2 receiver designs at the optimal splitting ratio. It can be seen that the Receiver A has a SNR advantage of 1.8 dB over Receiver B. Additionally, Receiver A uses a simple 3 × 3 coupler commonly having excess loss of ∼0.2 dB whereas B uses a sophisticated 90◦ hybrid commonly having loss of greater than 1 dB. We also add the information capacity of IM/DD systems in Fig. 4 assuming transmitted optical signal is exponentially distributed as studied in Ref. 19. It can be seen at the SNR of 20 dB, Receiver A of SVDD can achieve 14.1 bit/s/Hz information capacity compared to 6.1 bit/s/Hz for IM/DD systems, indicating a drastic capacity gain that can be achieved by using SVDD transmission. We know above that Receiver A using 4-PD can fully reconstruct the SV. The natural question is what the performance is if we use less number of PDs, and most interestingly, what the system performance is for 3-PD systems.20 It can be easily shown that part of the SV can be reconstructed from the 3 photo-currents I1−3 using the Receiver C that uses 3PDs which is shown in Fig. 5, where S0, S2, and S3 can be extracted. We can then recover S1 = ± S02 − S22 − S32, or SV can be also reconstructed except that the sign of S1 is unknown. Obviously we cannot transmit 3-D modulation as SV can no longer be completely reconstructed. Our study will be focused on 2-D modulation performance under the constraint of 3-PD SV receiver. At the transmitter, the SV is modulated T as 0 S2 S3 , or only S2 and S3 are encoded with data whereas S1 has a constant value of zero. Assuming we use pilot symbols to fully recover the channel matrix of H, the detected SV is ′ ′ ′T ±S1 S2 S3 . Therefore, the transmitted SV can be estimated as  Sˆ = Sˆ1

Sˆ2

Sˆ3

T

 = H−1 ± Sˆ1′

Sˆ2′

T Sˆ3′ .

(19)

FIG. 5. Receiver C: Stokes vector detection with 3 single-ended PDs. Mapping from photocurrents to SVs is shown to the right.


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FIG. 6. BER performance as a function of the channel matrix rotation angles for SV transmission using 3-PD receiver.

ˆ depending on the sign of Sˆ ′. Since the transmitted S1 = 0, we use Sˆ1 to There are two copies of S, 1 determine the correct sign of Sˆ1′. Once the sign is resolved, Sˆ2 and Sˆ3 will be computed accordingly using Eq. (19). Of course, in the presence of noise, there will be errors when using Sˆ1 to determine the correct sign of Sˆ1′, which results in sensitivity degradation.   − j φ/2

j φ/2

− cos(θ/2)e . Fig. 6 The optical channel is modelled by a Jones matrix of the form sin(θ/2)e cos(θ/2)e j φ/2 sin(θ/2)e − j φ/2 shows the system performance of 3-PD SV detection with varying θ and φ. It can be seen that there are spots of “fading” where the BER is very high. This corresponds to the situation where both positive and negative signs satisfy S1 = 0, and the two copies of the estimated transmitted Sˆ cannot be uniquely determined. We propose a digital polarization scrambling at the transmitter to resolve the occasional fading problem. At the transmitter, the transmitted signal is rotated by a sequence of random but known orthonormal matrices R for each symbol, or Ssc = RS, and at the receiver, the estimated Sˆ sc is recovˆ Those orthonormal matrices can be stored in the ered multiplexed by the inverse of R, Sˆ sc = R−1S. memory at the transmitter and receiver. This digital scrambling is equivalent to the situation where the signal transverses different channels with rapidly varying θ and φ at each symbol, effectively avoiding accidently dwelling at fading spot for a long period of time. Fig. 7 shows the BER performance after

FIG. 7. BER performance for 3-PD receiver using digital scrambling.


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digital polarization scrambling. The modulation scheme has a BER floor of ∼3 × 10−3, which can be corrected by using the state-of-the-art FEC. To the best of our knowledge, this is the first time to show successful polarization modulation when only subset of SV is detected using digital scrambling. We note that relatively large SNR is required for the 3-PD SV receiver, and further research is needed to improve its receive sensitivity. We have discussed the general principle of polarization modulation in both Jones and Stokes spaces. We show that there is no device that can transform an arbitrary input polarization into one orthogonal to itself. This excludes the receiver polarization diversity architecture by splitting the signal into two branches, and then transferring one of the branches into orthogonal polarization. We derive a closed-form expression for the information capacity of different SV direct detection architectures and compare the capacity of our proposed SV detection architecture to that of conventional IM/DD. We also study the 3-PD SV detection architecture where a subset of SV is detected, and devise a novel modulation algorithm that can achieve 2-D transmission with 3-PD detection. By using cost-effective SV receivers, the polarization modulation and multiplexing can potentially play a major role in short-reach networks when the wavelength domain is quickly exhausted. 1

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992). 2 S. Benedetto, R. Gaudino, and P. Poggiolini, “Direct detection of optical digital transmission based on polarization shift keying modulation,” IEEE J. Sel. Areas Commun. 13, 531–542 (1995). 3 S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992). 4 C. R. Fludger, T. Duthel, D. Van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, G.-D. Khoe, and H. De Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26, 64–72 (2008). 5 P. J. Winzer, G. Raybon, H. Song, A. Adamiecki, S. Corteselli, A. H. Gnauck, D. A. Fishman, C. R. Doerr, S. Chandrasekhar, and L. L. Buhl, “100-Gb/s DQPSK transmission: From laboratory experiments to field trials,” J. Lightwave Technol. 26, 3388–3402 (2008). 6 C. F. Lam, H. Liu, B. Koley, X. Zhao, V. Kamalov, and V. Gill, “Fiber optic communication technologies: What’s needed for datacenter network operations,” IEEE Commun. Mag. 48, 32–39 (2010). 7 W. Yan, L. Li, B. Liu, H. Chen, Z. Tao, T. Tanaka, T. Takahara, J. Rasmussen, and D. Tomislav, “80 km IM-DD transmission for 100 Gb/s per lane enabled by DMT and nonlinearity management,” in Optical Fiber Communication Conference, M2I. 4 (Optical Society of America, 2014). 8 J. Lee, N. Kaneda, T. Pfau, A. Konczykowska, F. Jorge, J.-Y. Dupuy, and Y.-K. Chen, “Serial 103.125-Gb/s transmission over 1 km SSMF for low-cost, short-reach optical interconnects,” in Optical Fiber Communication Conference, Th5A. 5 (Optical Society of America, 2014). 9 Q. Hu, D. Che, Y. Wang, and W. Shieh, “Advanced modulation formats for high-performance short-reach optical interconnects,” Opt. Express 23, 3245–3259 (2015). 10 D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for linear complex optical channels,” J. Lightwave Technol. 33, 678–684 (2015). 11 K. Kikuchi, “Electronic polarization-division demultiplexing based on digital signal processing in intensity-modulation direct-detection optical communication systems,” Opt. Express 22, 1971–1980 (2014). 12 M. Morsy-Osman, M. Chagnon, M. Poulin, S. Lessard, and D. V. Plant, “224 Gb/s 10-km transmission of PDM PAM-4 at 1.3 µm using a single intensity-modulated laser and a direct-detection MIMO DSP-based receiver,” J. Lightwave Technol. 33, 1417–1424 (2015). 13 J. Estarán, M. A. Usuga, E. P. da Silva, M. Piels, M. I. Olmedo, D. Zibar, and I. T. Monroy, “Quaternary polarizationmultiplexed subsystem for high-capacity IM/DD optical data links,” J. Lightwave Technol. 33, 1408–1416 (2015). 14 Q. Yang, Y. Tang, Y. Ma, and W. Shieh, “Experimental demonstration and numerical simulation of 107-Gb/s high spectral efficiency coherent optical OFDM,” J. Lightwave Technol. 27, 168–176 (2009). 15 M. Sjodin, P. Johannisson, M. Karlsson, Z. Tong, and P. A. Andrekson, “OSNR requirements for self-homodyne coherent systems,” IEEE Photonics Technol. Lett. 2, 91–93 (2010). 16 C. Xie, “PMD insensitive direct-detection optical OFDM systems using self-polarization diversity,” in Optical Fiber Communication Conference, OMM2 (Optical Society of America, 2008). 17 M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). 18 A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Sel. Areas Commun. 21, 684–702 (2003). 19 X. Li, R. Mardling, and J. Armstrong, “Channel capacity of IM/DD optical communication systems and of ACO-OFDM,” in IEEE International Conference on Communications, ICC’07 (IEEE, 2007), pp. 2128–2133. 20 M. Nazarathy and A. Agmon, “Doubling direct-detection data rate by polarization multiplexing of 16-QAM without active polarization control,” Opt. Express 21, 31998–32012 (2013).
