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Phase-diversity method using phase-shifting interference algorithms for digital coherent receivers
Optics Communications 356 (2015) 269–277
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Phase-diversity method using phase-shifting interference algorithms for digital coherent receivers Thang M. Hoang n, Mohamed M. Osman, Mathieu Chagnon, Meng Qiu, David Patel, Mohammed Sowailem, Xian Xu, David Plant Department of Electrical and Computer Engineering, McGill University, Montreal, Canada
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2015 Received in revised form 26 July 2015 Accepted 30 July 2015
We describe phase-diversity optical coherent receivers (CRx) using the phase-shifting-interference (PSI) framework. The goals of the analysis are several-fold. First, we show that coherent detection can be realized with optical hybrids that have an arbitrary number of branches and phase shifts using a closedform solution of the inphase-quadrature mapping. Second, we show that CRx with 2 4 90° hybrids using balanced detection (BD) and CRx with 2 3 120° hybrid using single-ended detection (SED) perform optimally compared to alternative configurations. A proof-of-concept WDM colorless 10 132-Gb/s PDM-QPSK transmission experiment is conducted. We demonstrate that an example of arbitrary phase diversity CRx with a 2 3 90° hybrid SED operates with a 0.3 dB signal-to-noise-ratio (SNR) penalty relative to conventional CRx at 6400 km and 4480 km for bit-error-rates below the threshold of 2 10 2 and the threshold of 3.8 10 3 respectively. The sensitivity degradation of the 2 3 90° hybrid SED with respect to the 2 4 90° hybrid BD in the shot noise limited regime at a distance of 4480 km is 3 dB, which matches well with the predicted penalty from the PSI analytical model. To the best of our knowledge, this paper is the first attempt to model a CRx using the PSI model. & 2015 Elsevier B.V. All rights reserved.
Keywords: Coherent receivers Phase-diversity Optical hybrid Phase shifting interference
1. Introduction Coherent fiber optic transmission systems are being developed to meet burgeoning capacity demands [1]. The ability to linearly map the in-phase (I) and quadrature (Q) components of the received optical signal to the electrical domain by mixing the received optical field with that of a local oscillator (LO) is the key advantage of a coherent receiver (CRx) [2]. This direct mapping enables digital signal processing (DSP), which then allows the use of spectrally efficient modulation formats as well as the mitigation of transmission impairments [2–9]. Furthermore, the recent interest in transparent networks has lead to the wide adoption of colorless reception using 2 4 90° hybrid balanced detection (BD) CRx to reject undesired directdetection components in wavelength-division multiplexing (WDM) channels [10–12]. Since coherent detection is expected to be soon deployed in metro and shorter-distance network, reducing complexity front-end is highly desirable. To reduce complexity, optical front ends with single-ended detection (SED), such that they are still applicable for colorless reception, have been recently reported [13–17]. These proposals n
Corresponding author. E-mail address: [email protected] (T.M. Hoang).
http://dx.doi.org/10.1016/j.optcom.2015.07.086 0030-4018/& 2015 Elsevier B.V. All rights reserved.
consist of using only a 120° hybrid. Those proposal offers several benefits. First, using SED may lower the total cost because of simpler transimpedance amplifier (TIA) and less radio-frequency connections between optical and electrical devices [17]. Second, compared to conventional 2 4 90° hybrid, 2 3 hybrid have broader optical bandwidth and larger fabrication tolerance [16,18]. Thus, there is a need for in-depth studies to explore the possibility in phase-diversity CRx. In this paper, we develop and experimentally validate a model of phase-diversity coherent detection using an optical hybrid with an arbitrary number of outputs and optical phase shifts (referred to as an arbitrary hybrid). This is achieved by applying phaseshifting interference (PSI) algorithms which originated in the field of optical measurement [19–21]. In Section 2, the theoretical description of traditional and alternate configurations of a CRx frond-end is reviewed. Section 3 describes the PSI signal processing and presents a general model and condition for an arbitrary phase-diversity CRx. In Section 4, a closed form expression for the signal-to-noise ratio (SNR) for a phase-diversity CRx under the assumption of an ideal receiver is presented. In the context of this paper, an “ideal receiver” means that there is no power imbalance, optical phase imbalance, and skew mismatch between the branches of the receiver. The analytical solution indicates that a 2 4 90° hybrid BD and a 2 3 120° hybrid SED are the best phase-diversity receivers.
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An arbitrary phase-diversity receiver deviating from the optimum demodulation suffers from a performance penalty which can be predicted using the PSI based analytic solution derived in this paper. The analytical model also indicates that the minimum number of optical phase-diverse branches per polarization to realize coherent receiver is three for colorless reception. Finally, in Section 5 a proof-of-concept experiment is carried out to validate our approach in full colorless reception of 10 132Gb/s PDM-QPSK WDM channels. The experiment uses a phasediversity CRx 2 3 90° hybrid SED, which was realized by using 3 out of 4 output branches per polarization of a 2 4 90° hybrid. Using a LO power of 10 dBm, we evaluate the bit-error-rate (BER) and the SNR of the center channel at various transmission distances for two different phase-diversity schemes at a received signal power of 10 dBm per channel. The penalty, in terms of transmission distance and SNR, at the hard-decision forward-error-correction (HD-FEC) threshold of 3.8 × 10−3 of the simplified CRx is 500 km and 0.3 dB SNR relative to conventional CRx using balanced detection. Next, we restrict our measurements to fixed distances of 6400 km and 4480 km corresponding respectively to BER at the soft-decision forward-error-correction threshold (SDFEC) threshold of 2 10 2 and the HD-FEC threshold of 3.8 10 3 and sweep the received signal power with a fixed LO power of 5 dBm. This set of measurement is used to evaluate the differences in sensitivity performance between phase-diversity methods employing PSI algorithms when used for colorless reception. The experimental outcomes show that the simplified front-end class can achieve colorless reception with less than 0.3 dB of SNR penalty with respect to a conventional 2 4 90° hybrid BD CRx in typical sensitivity operating regimes. At low received signal powers, where shot noise plays a critical role, the sensitivity penalty of the simplified CRx with respect to the BD CRx is 3 dB at 4480 km transmission distance. These results are shown to be in good agreement with the proposed PSI-model. We conclude that the PSI model is effective for predicting performance for generalized phase-diversity CRx front ends.
2. Brief description of a coherent receiver (non-PSI) This section provides a brief review of the traditional homodyne polarization-diversity and phase-diversity CRx. The section is included in order to elucidate the PSI framework outlined in Section 3 below. For brevity, only one polarization is considered in the following development, but it is feasible to achieve also polarization diversity by splitting the received signal into two orthogonal polarization components, using phase diversity method to detect the field of each orthogonal polarization and utilizing DSP to combine them at electrical domain. In fact, we present both phasediversity and polarization diversity (DP-QPSK 132 Gb/s per channel) transmissions in the experiment. Also, it is noted that there are possibilities of combining polarization and phase diversity using PSI concept. We show one of such options in Appendix C. In the following model, the photocurrent from the photo-detector (PD) of an arbitrary branch i of the optical hybrid can be generally described as
() ()
⁎ Ii (t ) = RS |ES (t )|2 + RLO |ELO (t )|2 + 2 RS RLO R {ES t ELO t e jδ i }
= IS (t ) + ILO (t ) + 2 IS ( t ) ILO ( t ) cos (φ ( t ) + δi )
(1)
where ES and ELO are respectively the signal and LO fields at the input of the optical hybrid; φ is the phase difference between those fields; δi is the optical phase shift of either the signal or the LO introduced in that branch by the optical hybrid; RS and RLO are the total responsivities of the PD with respect to the signal and LO; t is the time; and R and n denote the real part and conjugate
operation of complex numbers. The total responsitivities are obtained using the responsitivity of the PD RPD and the power coupling coefficient S 2 of the corresponding branch such that RS = RPD SS 2 and RLO = RPD SLO 2. This description can be extended to a WDM system with M transmission channels by calculating the total signal field as M ES = ∑ j = 1 ESj, where ESj is the optical field of channel j. In this study, we assign the term “in-band” to any field that is within the bandwidth of the channel of interest ESI, whereas fields outside this band are designated as “out-of-band” (OOB). For notational convenience, we drop the time variable t from this point on. It should be noted that Eq. (1) assumes that the contributions from thermal noise and shot noise are negligible. Accordingly, the first and second terms (IS + ILO ), which arise from the self-beating of the signal and LO and includes all of the direct detection of the incoming fields in baseband, need to be eliminated in order to accurately capture the linearly mapped portion IS (t ) ILO (t ) cos (φ (t ) + δ i ). This is achieved by using phase diversity which is the concept of differing optical hybrid architectures to ensure diverse phase differences between the signal and the LO fields [22–25]. The solution to remove the undesirable self-beating term is to choose a 90° hybrid with BD. The in-phase (I) and quadrature (Q) components can then be retrieved with a pair of identical PDs which returns the signal mixed with the opposite phase of the LO as depicted by
In − phase = I1 − I3 = 4 IS ILO cos φ, Quadrature = I4 − I2 = 4 IS ILO sin φ
(2)
where I1 to I4 are the four outputs of the 90° hybrid with phase differences in the linear mixing term of: 0, 1/2π, π, and 3/2π. Our objective in this study is to determine methods of reducing complexity and component count for a CRx. One method to achieve this is to use an additional PD to directly detect IS in supplement to a 90° hybrid SED [26]. The unwanted IS is then subtracted from the outputs of the SED with DSP. Although this scheme is also applicable for colorless reception, it has the following two primary limitations. First, it is challenging to estimate IS due to optical path delays and/or polarization mismatches between the SED and the direct-detection front-ends. Second, two additional ADCs are required. In contrast, the works in [13–17] investigate another alternative using a 120° hybrid. Let us define I1, I2, and I3 as the three output photocurrents of the hybrid with phase differences (δi of Eq. (1)) in the linear mixing term of 0, 2/ 3π, and 4/3π, respectively. Then, by applying Eq. (1), the photocurrents of the in-phase (II) and quadrature (IQ) components for this case with the 120° hybrid can be obtained as
II,120 = I1 − I2/2 − I3/2,
IQ ,120 =
3 (I3 − I2 ) 2
(3)
This scheme allows for colorless reception with simpler optical designs compared to the former scheme. There is also a complexity increase in the electrical domain because the scheme requires analog circuits for scaling by an irrational factor and/or two additional ADCs for DSP. It is noted that the original phase diversity concept in 1980s aims to ease phase-locking condition for ASK, PSK, FSK format and twofold, threefold, fourfold diversity schemes was compared extensively. While 120° hybrid (using 3 3 coupler) and 90° hybrid (using 2 4 coupler) are widely used in the time of 1980s, their sole purpose is to detect one dimensional modulation format at that time [22,23]. On the other hand, the present-day phase-diversity idea's target is to retrieve full optical field for advanced modulation format (such as QAM) of renewed interest in CRx [24,13,15]. In the following section, we pursue our objective of studying
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
the use of PSI algorithms to describe a phase-diversity CRx.
3. Phase shifting interference (PSI) framework The problem of I–Q (or field) extraction is analogous to the concept of phase extraction in the field of optical interferometry measurements [19–21]. More precisely, the measurement to receive signals that are phase diverse using a CRx is a subset of PSI. For instance, I/Q extraction methods for a conventional 90° hybrid BD and the recently proposed 120° hybrid SED in optical communications shown in Eqs. (2) and (3) are replicas of an N-least square PSI (four and three steps), where the optical phase shifts of N-branches are equally spaced over 2π [19]. Table 1 shows the comparison between two schemes of phase-diversity CRx and N-least square PSI. A rationale behind this phenomenon is that the two categories share the same physical principle: to provide diverse phase either in time (CRx) or in space (PSI interferogram). PSI is a system of approaches to extract the object phase using phase-shifting interferometry. A phase-diversity coherent receiver relies on the optical hybrid, which in essence is an optical phase-shifting interferometer. The concept of phase diversity has not been studied extensively for field detection except for balanced detection and the 120° hybrid proposed recently [1,5,13–17]. We map the concept of PSI to a CRx in order to develop phase-diversity coherent detection for arbitrary hybrid configurations to positively influence CRx design. The description of an arbitrary optical hybrid is made with a set of equations such that Si (ES + ELO e jδi ), where i = 1, … , N and N is the total number of optical branches before the PDs, δi and Si (for simplicity, Si = SLOi = SSi ) are respectively the optical phase shift and coupling coefficient at the ith branch. We rewrite the photocurrent in Eq. (1) as
(
Ii = Ri ES
2
+ ELO
2
)
+ 2 ES ELO cos ( φ + δi ) ,
i = 1, ‥ , N
(4)
Si2 RPD
where Ri = is the total responsitivity with the responsitivity of every photodetector RPD in units of A/W at the ith branch with
271
Table 2 Examples of phase-diversity CRx that can be explained by PSI with their characteristic optical phase shift diagrams; N: number of optical branches per 1 hybrid/ polarization; Δδ is the optical phase shift difference between consecutive branches; δi is the optical phase shift at corresponding branch; bi and ai are weighting factors. Class 4B is the conventional 90° hybrid BD and class 3A is the 120° hybrid SED. Other classes are listed as examples of CRx with arbitrary optical hybrid in this article. N
Δδ
3
4
Class
b1, …, bN a1, …, aN
2π /3
3A
3 /2 [0, − 1, 1] 1/2 [2, − 1, − 1]
π/2
3B
3/4 [1, − 2, 1] 3/4 [1, 0, − 1]
π/2
4A
1/2 [1, − 3, 1, 1] 1/2 [1, 1, − 3, 1] [0, − 1, 0, 1] [1, 0, − 1, 0] 1/2 [1, − 3, 1, 1] 1/2 [3, − 1, − 1, − 1]
δi diagrams
4B 4C
respect to the incoming field. The objective of using the PSI algorithm is to recover the field through I–Q component mapping using signal processing either in the analog or the digital domain. A type of PSI which can be employed is a linear combination of phase-diverse branches. For instance, we can extract the I–Q component directly from the superposition of the output PD currents as described by the following equation: N
II =
∑ Ii ai, i=1
N
IQ =
∑ Ii bi
for N ≥ 3
(5)
i=1
where ai and bi are weighting factors. Table 2 lists some examples of optical hybrids, where the phase shift difference Δδ between two consecutive branches is the same. Thus, for practical implementation, one can regard PSI as a “spatial FIR” filter to retrieve quadrature components as depicted by Eq. (5). The weighting factors ai and bi should satisfy the following conditions and are
Table 1 Comparison between phase-diversity CRxs and PSI algorithms. In PSI algorithms column, Io, Ir and φ are the intensity of object, the intensity of reference plane wave, and the searched phase, respectively. Phase diversity CRxs
PSI algorithms
Type
Balanced detection 90° hybrid
4-least square PSI
Current/intensity
I1 = IS + ILO + 2 IS ILO cos φ
I1 = Io + Ir + 2 Ir Io cos φ
I3 = IS + ILO − 2 IS ILO cos φ
I3 = Io + Ir − 2 Io Ir cos φ
I2 = IS + ILO − 2 IS ILO sin φ
I2 = Io + Ir − 2 Io Ir sin φ
I4 = IS + ILO + 2 IS ILO sin φ
I4 = Io + Ir + 2 Io Ir sin φ
Configuration
In = I1 − I3 Quad = I4 − I2 4 PDs
I4 − I2 ) I1 − I3 4 interferograms
Type
Single-ended detection 120° hybrid
3-least square PSI
Current/intensity
I1 = IS + ILO + 2 IS ILO cos φ
I1 = Io + Ir + 2 Ir Io cos φ
Extraction
φ = tan−1(
2π ) 3 4π cos (φ + ) 3
I2 = IS + ILO + 2 IS ILO cos (φ + I3 = IS + ILO + 2 IS ILO Extraction
In = I1 − I2/2 − I3/2
Configuration
Quad = 3 PDs
3 /2 (I3 − I2 )
2π ) 3 4π I2 = Io + Ir + 2 Io Ir cos (φ + ) 3 ( − ) I I 3 3 2 φ = tan−1( ) 2I1 − I2 − I3 3 interferograms I3 = Io + Ir + 2 Io Ir cos (φ +
272
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277 ⁎ II = RPD Re {Esig − I ELO },
discussed in more detail in Eq. (A.3):
⎤ ⎡ ⎤ ⎤ ⎡ ⎡ N N ∑i = 1 ai ⎥ ⎢ ∑i = 1 bi ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0⎥ ⎥ ⎢ ⎢ N N ⎢∑ ai sin δi ⎥ = ⎢ ∑ bi cos δi ⎥ = ⎢ ⎥ i=1 ⎥ ⎢ N⎥ ⎥ ⎢ ⎢ i=1 ⎥ ⎢ 2⎥ ⎥ ⎢ N ⎢ N a cos b sin − δ δ ∑ ∑ i i⎦ i⎦ ⎣ ⎦ ⎣ i=1 ⎣ i=1 i
(8)
Therefore, the in-band complex signal can be written as ISI = II + jIQ , which leads to a calculated power of < ISI 2〉, where <〉 denotes the ensemble average operator:
(6)
Physically, as given in detail in Appendix A, the first constraint in Eq. (6) enforces the removal of the signal–signal and the LO–LO interfering terms IS + ILO . Namely, in the case of a 90° hybrid with BD, the first constraint has the identical interpretation as the common-mode rejection ratio (CMRR) metric [10]. The second expression in Eq. (6) ensures the orthogonality of the detected quadrature components. Finally, the last condition takes into account the amplitude mapping of the field scaled by the value of the PD responsitivity RPD ES ∥ ELO . We note that the analytical solution in the presence of receiver imperfections (namely, power imbalance, phase imbalance and skew mismatch) will not be addressed in this paper.
4. Analytical SNR expression for IQ retrieval using an arbitrary hybrid in a colorless WDM scenario In the colorless WDM framework, we can re-model Eq. (4) for the ith branch by considering only the beating of the signal of interest with the LO:
⎞ ⎛ M Ii = Ri ⎜⎜ ∑ |ESj |2 + |ELO |2 + 2|ESI ||ELO | cos (φ + δi ) ⎟⎟ ⎠ ⎝ j=1 = IS + ILO + 2 ISI ILO cos (φ + δi )
⁎ IQ = RPD Im {Esig − I ELO }
< ISI
2
2 > = RPD Esig − I
2 2 σnoise = 2σASE − LO + − total
PSI algorithms satisfying the requirements of Eq. (6) and therefore corresponding to CRx configurations are able to effectively cancel the self-induced power interfering terms and accurately detect quadrature components. This technique is most effective with the assumption of ideal hardware such that there is no power or phase imbalance in the hybrid and no skew mismatch between the channels. Assuming we are interested in recovering the optical field in the absence of noise (including amplified stimulated emission (ASE) noise), then the I/Q currents of the inband signal field Esig − I are
2
2 = RPD Psig − I PLO
(9)
N a2 i =1 i
2
N
)
2 + ∑ i = 1 bi σshot − total
N
(∑
N a2 i =1 i
2
N
+ ∑i = 1 bi N
) 2eB R
e PD (P S
+ PLO )
(10)
Then from Eqs. (9) and (10), the closed form expression for a signal-to-noise ratio (SNR) per symbol can be obtained as 2
< ISI >
SNR =
2 σnoise − total
2 R PD Psig − I PLO
= 2 2R PD NASE PLO Be +
(
N ∑ i = 1 a i2
N
2
+ ∑i = 1 bi N
) 2eB R
e PD (P S
+ PLO )
R PD Psig − I PLO
=
(∑ +
N a2 i =1 i
N
2
+ ∑i = 1 bi
) 2eB (P e
N
S
+ PLO )
(11)
Using the definition of optical signal-to-noise ratio (OSNR) [3,4]:
Psig − I 2NASE Bref
(12)
where Bref is the reference bandwidth and the factor of 2 accounts for both polarization of ASE. The SNR in Eq. (11) due to detection with an arbitrary hybrid can be related to the OSNR by
1
SNR = Be + OSNR⁎Bref
(
⎧ ⎫ N + ∑i = 1 bi2 ⁎2eBe ⎪ P ⎪ ⎨ S + 1⎬ N⁎RPD ⁎Psig − I ⎪ PLO ⎪ ⎩ ⎭
N ∑i = 1 ai2
)
(13)
Eq. (13) is the closed-form performance prediction of a phasediversity CRx using the PSI framework for an arbitrary hybrid. The divergence in SNR between phase-diversity CRx is rooted on different weighting factors ai and bi for different hybrids. For example, using a 90° hybrid with BD results in the ratio (∑iN= 1 ai2 + ∑iN= 1 bi2 ) N
4.1. Perfect receiver without power imbalance, optical phase balance, and skew mismatch
(∑
2 = 2R PD NASE PLO Be +
OSNR = Because of the limited-bandwidth of the PDs and the ADCs, IS in the first term, which comprises the OOB signal–signal beating of all WDM channels at baseband is not the same as the in-bandinduced ISI in the last term of Eq. (7). Intuitively speaking, at any time t, as pointed out by Eq. (7), the functionality of a phase-diversity CRx is to detect three unknowns (IS, ISI, and φ) in the framework of colorless reception. In such cases, a phase-diversity CRx should have at least 3 PDs corresponding to 3 different optical phase shifts δi. Some PSI classes, which satisfy the conditions of Eq. (6), are listed in Table 2. The fifth column in the table represents the proper weighting factor to reconstruct I/Q in Eq. (5). Motivated by the above observation, we investigate various PSI algorithms for I/Q retrieval, each with a different optical hybrid design in order to find the optimal hybrids for a CRx. The following Section 4.1 presents an analytical model and Section 4.2 describes the electronic components needed to realize a phase-diversity CRx.
ELO
The most significant noise sources in CRx are the ASE-LO beat noise, shot noise, and the relative intensity noise (RIN) from the LO. As detailed in Appendix B, we note that for ideal receiver, the self-beating noise of the OOB channels plus the LO, and the RIN from both the LO and the signal vanish. In addition, we neglect the contribution from thermal noise. The expression for total noise, Eq. (B.7) of Appendix B, is
2R PD NASE PLO Be
(7)
2
becoming equal to 1. Then, the aforementioned SNR
expression applied to 90° hybrid with BD simplifies to the model commonly found in the literature [4,5,12]. Given a fixed OSNR, the ASE-LO beat noise normally dominates over the shot noise, which in turn makes phase-diversity receivers independent of the LO-to-signal-power-ratio (LOSPR) and the PSI algorithms. However, when the LOSPR is sufficiently large, the shot noise is responsible for degradations. As a consequence, at low received signal powers, different PSI weighting methods result in different performance degradations. We acknowledge from previous sections and from Appendix B that the class 3A, the class with the 120° hybrid SED, and class 4B, the class with the 90° hybrid BD, have the same shot noise contribution. The other PSI classes listed in Table 2, e.g., class 3B with the 90° hybrid SED, have
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
a 1.7 dB penalty in sensitivity relative to the optimum phase-demodulation (120° hybrid SED and 90° hybrid BD) in the shot noise limited regime. 4.2. PSI electronics hardware realization This section presents several examples of how PSI can be implemented with electronic components. Fig. 1 shows the architectures of phase-diversity coherent receiver using the PSI approach for class 3A (120° hybrid SED) and class 3B (90° hybrid SED). In the architecture shown in Fig. 1(a), (b), three ADCs are used to individually convert the three detected signals to digital signals. The linear operations given in Eq. (5) and in Table 2 are performed digitally to obtain the in-phase and quadrature signal components. In the architecture shown in Fig. 1(c), (d), linear operations to obtain the I/Q components from the three photocurrents are performed with analog scaling and subtraction circuits followed by two ADCs and an additional scaling/adding/ subtracting in digital domain (analogues to a balanced detection CRx). Even though this approach offers a simpler front-end (same number of ADCs relative to balanced detection and simple RF circuits), it limits the possibility of PSI realization in order to compensate for hardware impairments. In addition the approach does not provide the access to the direct detection term IS which could be beneficial for cross-phase modulation compensation [27].
5. Tb/s experiment and discussion The PSI theory, presented in the previous sections, for WDM transmission in colorless reception using QPSK modulation format was experimentally validated with the setup illustrated in Fig. 2. Ten channels spaced 50 GHz apart using external cavity lasers (ECLs) and centered at 193.4 THz were multiplexed using a 1:12 coupler. A dual-polarization QPSK signal was then bulk modulated onto all the channels using a Ciena WaveLogic 3 card and a randomly generated sequence of 218 bits. The PDM-QPSK signal at 33 Gbaud was root-raised cosine (RRC) pulse shaped having a roll-off factor of 0.1. The data was decorrelated by 4 ns relative to the neighboring channels. The multiplexed signals were amplified with a booster EDFA and then attenuated with a variable optical attenuator (VOA) to set the launch power at 4 dBm per channel. This launch power was kept constant throughout the experiment. Next, the resulting signal, which has an aggregated data rate of
273
1.32 Tb/s, was launched into a recirculating loop containing four spans of 80 km of SMF-28e þ fiber, each span being followed by an EDFA with a noise figure of 5 dB. Additionally, a gain flattening filter was inserted after the second in-line EDFA. No noise loading was performed after the recirculating loop. The signal out of the recirculating loop was then amplified to a constant power of 15 dBm and subsequently filtered using a wavelength and bandwidth tunable filter. The bandwidth of the filter was set to 5 nm to select all 10 channels for colorless realization, which were then attenuated using a VOA to achieve the desired total received power (PS, tot ) at the receiver. The receiver comprised a dual-pol 90° hybrid with its eight outputs connected to eight single-ended PIN þTIA PDs with a conversion gain of 1.3 V/mW. Another ECL was used as the LO and was set to the center channel frequency (193.4 THz). The output of this ECL was fed into the LO port of the hybrid. The signals were then captured by two real-time oscilloscopes with a 33 GHz bandwidth at 80 GSa/s and were processed offline. Using the above test bench, we experimented with two different reception scenarios: (1) a WDM configuration in which the total received power PS, tot and LO power were fixed to 0 dBm (or 10 dBm/channel) and 10 dBm, respectively; (2) another WDM configuration in which the transmission distances were fixed and the LO power was set to 5 dBm. The first scenario investigates the impact of transmission on a PSI-based CRx, while the second scenario evaluates the dynamic range of the PSI-based CRx. For the offline DSP, the sampling skew between each branch was first corrected and the power imbalance was compensated by weighting each of the 8 outputs by a factor proportional to its selfbeating term. Afterwards, IQ detection using PSI as described in Section 3 was performed. Further offline signal processing consisted of IQ imbalance correction using Gram–Schmidt orthogonalization procedure (GSOP), re-sampling to 2 samples per symbol, chromatic dispersion (CD) compensation, matched-filtering in frequency domain, frequency offset removal, polarization demultiplexing, and carrier phase estimation performed using decision directed least mean squares (LMS) algorithm with step sizes of 0.001 and 0.005 respectively [9]. After performing hard decision decoding on a symbol-by-symbol basis, the achieved electrical SNR was evaluated and the BER was calculated. It is noteworthy that in order to mimic ideal hardware, our experimental system is unique in two key aspects. First, a free-space-optics-based hybrid (Kylia) is used to exploit the smallest amount of phase imbalance. Second, power imbalance plus skew is mitigated in DSP. Together with the aforementioned DSP, GSOP is utilized to remove any
a
b
c
d
Fig. 1. Examples of CRx implementations with ideal 2 3 optical hybrids per polarization (2 3 120° and 2 3 90°). PSI processing to retrieve quadrature component is done with digital signal processing with the use of 3 ADCs in (a) and (b); and with analog circuits and 2 ADCs in (c) and (d).
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T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
Fig. 2. Experiment setup for WDM 1.32 Tb/s transmission (IL: Interleaver, ECL: External Cavity Laser, VOA: Variable Optical Attenuator, PM: Polarization Maintaining).
transmission at the HD-FEC threshold and a penalty of 0.3 dB and 0.2 dB in SNR at distances of 4480 km and 6400 km, respectively. At shorter distances, the degradation of the class 3B becomes larger. This can be understood from the explanations in Section 4.1, which showed that the shot noise contribution plays a more significant role in the short distance regime (or high OSNR) and/or colorless reception with a large number of WDM channels at high received powers. We then assess the impact of received signal power with
remaining phase imbalance. We first compared the transmission performance with two select classes of PSI: class 3B with a 90° hybrid and the conventional class 4B with a 90° hybrid. For both cases, the LO power was fixed to 10 dBm and the received signal power per channel was 10 dBm. For class 3B, we use only 6 out of the 8 outputs of the CRx for signal processing. The results are given in Fig. 3(a), (b). We observe that the class 3B achieves a similar performance compared to the class 4B BD, but with a penalty of 500 km in
a
b
c
d
Fig. 3. Experiment results of WDM 10 channels: 10 dBm received signal power per channel and LO power of 10 dBm (a) BER versus transmission distance, (b) SNR versus transmission distance; dynamic range with LO power of 5 dBm and transmission distances of 4480, 6400 km (c) BER versus received signal power per channel, (d) SNR versus received signal power per channel.
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
respect to the two PSI-based CRxs. In our experiment, the class 3B is not realized by a real 2 3 90° hybrid but by using three out of the four branches of a 2 4 90° hybrid. Here, the PSI weightings for I and Q are [1, 2, 1] and [1, 0, 1] instead of 3/4[1, 2, 1] and 3/4[1, 0, 1], as illustrated in Table 2. Therefore, in the shot-noise limited domain, the class 3B is expected to have a 3 dB penalty instead of a 1.7 dB penalty in terms of the power needed to achieve a similar performance. In Fig. 3(c), (d), BER versus Psig and SNR versus Psig are plotted at transmission distances of 4480 and 6400 km for a LO power of 5 dBm. We observe that both receivers are limited by shot-noise as indicated by the similar trend at signal power levels less than 14 dBm. Fig. 3(c) shows a 3 dB difference in power at the HD-FEC threshold after 4480 km. This was reflected in the PSI model which predicts the divergence in performance between PSI classes. We note from the curves of Fig. 3 (d) that the typical penalty of a class 3B CRx in the ASE-limited region is less than 0.3 dB at transmission distances of 4480 km and 6400 km. Although in our experiment the class 3B suffers a nonnegligible penalty compared to conventional BD in the shot-noise limited regime (expressed by either short distance or low received signal power), it should be noted that such a penalty is smaller in real implementations of class 3B for two reasons. First, a real 2 3 90° hybrid as would be used in class 3B (not utilizing 3 ports out of 2 4 90° hybrid as in our experiment), only suffers from 1.7 dB in power sensitivity compared to class 4B (described in Section 4.1). Second, 2 3 hybrids (90° or 120°) typically have lower fabrication deviations, resulting in the class 3B having less physical impairments (including power imbalance) than class 4B, which is built from 2 4 90° hybrids [18]. To equalize for power imbalance, a 2 3 90° hybrid uses a smaller scaling operation after each PD relative to a 2 4 90° hybrid. The scaling operation can yield larger shot noise (or additive receiver noise) contribution. Furthermore, among the sub-classes of class 3 PSI CRxs, the class 3A is the best. Therefore, the experimental results indicate that a simpler front end, at least with 2 3 120° hybrid, can achieve equivalent performance compared to conventional balanced detection. This indication matches well the recent experimental demonstrations [13].
6. Conclusion In summary, the themes produced by this study provide a comprehensive CRx model using the concept of PSI from the optical measurement domain. A framework for a colorless phasediversity CRx is found for an arbitrary hybrid with an analytical solution to predict performance with an ideal receiver. We conclude that 2 3 120° hybrid SED and 2 4 90° hybrid BD CRx are the best phase-diversity coherent receiver configurations in terms of rejection of receiver's noise. The performance penalty from other phase-diversity CRx flavors can be gauged from the optimum value using PSI model. We identify shot noise as the primary element responsible for the degradation disparity between the phase-diversity methods. Next, a proof-of-concept experiment is performed in which colorless 10 132-Gb/s PDM-QPSK WDM channels are successfully transmitted. Results show that an example of arbitrary phase-diversity CRx using a 2 3 90° hybrid can achieve colorless reception by sacrificing less than 0.3 dB in SNR in comparison to a conventional CRx BD at FEC thresholds of 3.8 10 3 and 2 10 2, respectively. We further conclude that shot noise has a considerable effect on performance in the regime of low received signal powers or short transmission distances. To the best of our knowledge, this is the first time such model been reported and experimentally demonstrated. The flexibility in CRx design is the attractiveness behind the presented PSI approach.
275
Acknowledgments The authors would like to thank Charles Laperle and Doug Charlton from Ciena for their guidance with a WaveLogic3 bulk modulator transmitter.
Appendix A. Weighting factor conditions In this appendix, we derive the general conditions for PSI weighting factors in order to retrieve the optical field from an arbitrary hybrid using the ideal hardware assumption. We start by using the relation in Eq. (5) to express the two quadrature components to arrive at N
N
∑ Ii ai = RPD |ES ||ELO | cos φ, ∑ Ii bi = RPD |ES ||ELO | sin φ i=1
i=1
(A.1)
Next, if we use Eq. (4) to represent photocurrents of an arbitrary optical branch, we obtain
(
Ii = Ri ES
2
+ ELO
2
)
+ 2 ES ELO cos ( φ + δi ) ,
i = 1, ‥ , N
= c0i + c1i cos (φ + δi ) = c0i + c1i cos φ cos δi − c1i sin φ sin δi
(A.2)
where c0i = Ri ( ES 2 + ELO 2); c1i = 2Ri ES ELO . For the sake of simplicity, we assume each PD has the same total responsitivity R1 = ⋯ = Ri = R (S1⋯ = Si = S ) or c01 = ⋯ = c0i = c0; c11 = … = c1i = c1. Next, we substitute Eq. (A.2) back into to Eq. (A.1) and sum the photo-currents:
⎤ ⎤⎡ ⎤ ⎡ ⎡ N N N ∑i = 1 cos δi ∑i = 1 sin δi ⎥ ⎢ c ⎥ ⎢ ∑i = 1 Ii ⎥ ⎢ N 0 ⎥ ⎢ c1 cos φ ⎥ ⎥ ⎢ N ⎢ N N N = ∑ ∑ ∑ ∑ I a a a cos a sin δ δ ( ) − ( ) i i i i⎥⎢ ⎥ ⎢ i=1 i i⎥ ⎢ i=1 i i=1 i=1 ⎥ ⎢ c1 sin φ ⎥ ⎥ ⎢ N ⎢ N N N ∑ ∑ ∑ ∑ I b b b cos b sin δ δ ( ) − ( ) i i⎦⎣ ⎦ ⎣ i=1 i i ⎦ ⎣ i=1 i i=1 i i=1 i In the 3 3 matrix of the above equation, the first column represents a superposition of direction detection terms while the second and third columns depict in-phase and quadrature components, respectively. Note that a similar matrix can be found in the literature on PSI [19–21] and a minimum number of phaseshifting branches is 3 in order to solve above linear system. A 90° hybrid BD and 120° hybrid SED are the least square solutions of the above matrix which turn the 3 3 matrix to a diagonal matrix. In the optical measurement domain, the N-least square PSI is commonly considered as the best algorithm in terms of noise rejection for N phase-shifting branches [19–21]. Without any loss of generality, in order to detect I/Q accurately, the weighting factors a and b must follow these conditions:
⎤ ⎤ ⎡ ⎤ ⎡ ⎡ N N ∑i = 1 ai ⎥ ⎢ ∑i = 1 bi ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ N N ⎥ ⎢∑ ai sin δi ⎥ = ⎢ ∑ bi cos δi ⎥ = ⎢ 1 i 1 i = = ⎥ ⎢ 1 ⎥ ⎥ ⎢ ⎢ N N ⎥ ⎢ 2S 2 ⎥ ⎥ ⎢ ⎢ ⎦ ⎣ ∑i = 1 ai cos δi ⎦ ⎣ ∑i = 1 − bi sin δi ⎦ ⎣ where
C = RPD ES ELO =
c1 2S2
(A.3)
. In Eq. (A.3), the first condition
indicates the ability to reject the signal–signal beating noise (or “Common Mode”). The second condition points out the capability to preserve orthogonality between quadrature components. The last condition describes the mapping process of the quadrature component's magnitude and is the least important condition. The first two constraints are important because they affect directly the quality of quadrature extraction. If we consider an ideal hybrid with equal power splitting and no excess loss, we can replace S2 by 1/N, which is the inverted
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number of optical branches. We remark that we distinguish between the fully detected optical field, included OOB WDM chanM nels (ES = ∑ j = 1 Esj ) in Eq. (A.1) and the desired in-band optical field (ESI) in colorless reception. In this case, an additional electrical filter is needed to filter out the desired channel in the electrical domain which is accomplished by the ADCs and/or PDs.
Appendix B. Noise contribution In this appendix, we evaluate the degradation due to two principal noise sources in general CRx: optical Gaussian ASE beat noise and shot noise. Thermal noise or transimpedance amplifier (TIA) noise is supposed to be negligible in the derivation. Relative intensity noise (RIN) is eliminated assuming ideal hardware. Shot noise is always present in any optical receiver. Considering e the elementary charge and Be the equivalent electrical bandwidth, the shot noise variance at arbitrary branch ith in Eq. (4), (7) is given by
⎛ ⎜ 2 σshot − i = 2eBe Ri ⎜ ⎝
2
M
∑ (Esig− I + nASE ) i=1
⎞ + |ELO |2 ⎟ ⎟ ⎠
= 2eBe Ri (PS + PLO )
(B.1)
For an ideal receiver, the signal and LO are equally split between N ports of an arbitrary hybrid. In other words, R1 = R2 = Ri = ⋯ = R . Owning to the statistical independence of the interactions between shot noise in each PD, one can regard the total shot noise as 2 2 σshot − total = σshot ⁎N = 2NeBe Ri (PS + PLO )
= 2eBe RPD (PS + PLO ) (⁎)
(B.2)
2 σshot
2 2 = σshot − 1 = ⋯ = σshot − i , (⁎) is the assumption of no RPD 1 2 S = N (or Ri = N ). Shot noise is identified as a non-
where excess loss stationary noise source. Optical ASE-LO beat noise, as we shall see shortly, is the beat term between the LO and optical noise field ASE nASE induced by optical amplifier during transmission.Starting with conventional 90° hybrid balanced detection, we get the photocurrents for I/Q with the perfect hardware realization assumption after low-pass filtering in electrical domain:
IQ = RPD Im { ( Esig − I + nASE ) (ELO
+ ishot 4 − ishot2
2 2 σnoise − total = 2σASE − LO +
(∑
N a2 i=1 i
N
)
2 + ∑i = 1 bi2 σshot − total
(B.7)
N
From Eqs. (B.6)–(B.7), a class 3A 120° hybrid has the same noise power as a conventional 90° hybrid balanced detection while class 3B, 4A, 4C 90° hybrid SEDs suffer from 1.7 dB increase in shot noise power. The least square PSI, where the angle separation Δδ = 2π /N , make the ratio
(∑iN= 1 ai2 + ∑iN= 1 bi2 ) N
equal to 1.
Appendix C. Example of combination of polarization diversity and phase diversity using PSI In this appendix, we show the possibility in combining polarization diversity and phase diversity using PSI. In particular, we demonstrate a pair of 2 3 90° hybrid and 2 2 90° hybrid, which is illustrated in Fig. C1, is capable of detecting polarization multiplexed signal. In the above figure, the incoming signal ES has two components with respect to polarization beam splitter (PBS): ESX and ESY. It is split by a 50:50 coupler before going to 2 3 90° hybrid of the top branch and 2 2 90° hybrid of the bottom branch. In addition, the LO continuous wave ELO is equally split into ELOX and ELOY as reference tone of the two hybrids. Since ESX and ELOX cannot beat with ESY and ELOY, we can model the output currents (neglecting the scale factor RPD) as follows:
I1 = ESX
2
+ ESY
2
+ ELOX
2
⁎ + 2R {ESX ELOX }I2 = ESX
I3 = ESX
2
+ ESY
2
+ ELOX
2
⁎ − 2R {ESX ELOX }
I4 = ESX
2
+ ESY
2
+ ELOY
2
⁎ + 2I {ESY ELOY }
I5 = ESX
2
+ ESY
2
+ ELOY
2
⁎ + 2R {ESY ELOY }
2
+ ESY
2
+ ELOX
2
(C.1)
The I/Q components of X and Y polarizations can be found using PSI and the fact that ELOX 2 = ELOY 2:
IXI = 3/4 ( I1 − I3 ) IXQ = 3/4 ( I1 − 2I2 + I3 )
II = RPD Re { ( Esig − I + nASE ) (ELO )⁎} + ishot1 − ishot 3 )⁎}
can derive a noise variance analytical expression for different PSI configurations using a similar procedure. For arbitrary class PSI
(B.3)
IYI = ⎡⎣ I5 − ( I1 + I3 )/2⎤⎦/2 IYQ = ⎡⎣ I4 − ( I1 + I3 )/2⎤⎦/2
Consequently, we can derive
(C.2)
inoise − I = iASE − LO − I + ishot1 − ishot 3, inoise − Q = iASE − LO − Q + ishot 4 − ishot2
(B.4)
From Eq. (B.4), the main noise contributors are ASE-LO beat noise and shot noise. The noise variance of the beat noise term between the LO and ASE field is defined as 2
2 2 2 2 2 σASE − LO − I = σASE − LO − Q = σASE − LO = RPD NASE ELO Be = RPD NASE PLO Be (B.5)
where NASE is the power spectral density of ASE noise. Without any impairment from the receiver, the self-beating common mode noise and the RIN from the LO or signal are perfectly cancelled. Hence, one can represent the total noise variance as 2 2 2 2 σnoise − I = σnoise − Q = σASE − LO + 2σshot 2 σshot − total 2 2 2 σnoise − total = 2σASE − LO + σshot − total (B.6) 2 mentioned in Eq. (B.2) with N ¼4. Furthermore, we
2 = σASE − LO + 2 where σshot − total
Fig. C1. An example of combination of polarization diversity and phase diversity using 2 3 90° hybrid and 2 2 90° hybrid.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase-diversity method using phase-shifting interference algorithms for digital coherent receivers Thang M. Hoang n, Mohamed M. Osman, Mathieu Chagnon, Meng Qiu, David Patel, Mohammed Sowailem, Xian Xu, David Plant Department of Electrical and Computer Engineering, McGill University, Montreal, Canada
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2015 Received in revised form 26 July 2015 Accepted 30 July 2015
We describe phase-diversity optical coherent receivers (CRx) using the phase-shifting-interference (PSI) framework. The goals of the analysis are several-fold. First, we show that coherent detection can be realized with optical hybrids that have an arbitrary number of branches and phase shifts using a closedform solution of the inphase-quadrature mapping. Second, we show that CRx with 2 4 90° hybrids using balanced detection (BD) and CRx with 2 3 120° hybrid using single-ended detection (SED) perform optimally compared to alternative configurations. A proof-of-concept WDM colorless 10 132-Gb/s PDM-QPSK transmission experiment is conducted. We demonstrate that an example of arbitrary phase diversity CRx with a 2 3 90° hybrid SED operates with a 0.3 dB signal-to-noise-ratio (SNR) penalty relative to conventional CRx at 6400 km and 4480 km for bit-error-rates below the threshold of 2 10 2 and the threshold of 3.8 10 3 respectively. The sensitivity degradation of the 2 3 90° hybrid SED with respect to the 2 4 90° hybrid BD in the shot noise limited regime at a distance of 4480 km is 3 dB, which matches well with the predicted penalty from the PSI analytical model. To the best of our knowledge, this paper is the first attempt to model a CRx using the PSI model. & 2015 Elsevier B.V. All rights reserved.
Keywords: Coherent receivers Phase-diversity Optical hybrid Phase shifting interference
1. Introduction Coherent fiber optic transmission systems are being developed to meet burgeoning capacity demands [1]. The ability to linearly map the in-phase (I) and quadrature (Q) components of the received optical signal to the electrical domain by mixing the received optical field with that of a local oscillator (LO) is the key advantage of a coherent receiver (CRx) [2]. This direct mapping enables digital signal processing (DSP), which then allows the use of spectrally efficient modulation formats as well as the mitigation of transmission impairments [2–9]. Furthermore, the recent interest in transparent networks has lead to the wide adoption of colorless reception using 2 4 90° hybrid balanced detection (BD) CRx to reject undesired directdetection components in wavelength-division multiplexing (WDM) channels [10–12]. Since coherent detection is expected to be soon deployed in metro and shorter-distance network, reducing complexity front-end is highly desirable. To reduce complexity, optical front ends with single-ended detection (SED), such that they are still applicable for colorless reception, have been recently reported [13–17]. These proposals n
Corresponding author. E-mail address: [email protected] (T.M. Hoang).
http://dx.doi.org/10.1016/j.optcom.2015.07.086 0030-4018/& 2015 Elsevier B.V. All rights reserved.
consist of using only a 120° hybrid. Those proposal offers several benefits. First, using SED may lower the total cost because of simpler transimpedance amplifier (TIA) and less radio-frequency connections between optical and electrical devices [17]. Second, compared to conventional 2 4 90° hybrid, 2 3 hybrid have broader optical bandwidth and larger fabrication tolerance [16,18]. Thus, there is a need for in-depth studies to explore the possibility in phase-diversity CRx. In this paper, we develop and experimentally validate a model of phase-diversity coherent detection using an optical hybrid with an arbitrary number of outputs and optical phase shifts (referred to as an arbitrary hybrid). This is achieved by applying phaseshifting interference (PSI) algorithms which originated in the field of optical measurement [19–21]. In Section 2, the theoretical description of traditional and alternate configurations of a CRx frond-end is reviewed. Section 3 describes the PSI signal processing and presents a general model and condition for an arbitrary phase-diversity CRx. In Section 4, a closed form expression for the signal-to-noise ratio (SNR) for a phase-diversity CRx under the assumption of an ideal receiver is presented. In the context of this paper, an “ideal receiver” means that there is no power imbalance, optical phase imbalance, and skew mismatch between the branches of the receiver. The analytical solution indicates that a 2 4 90° hybrid BD and a 2 3 120° hybrid SED are the best phase-diversity receivers.
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An arbitrary phase-diversity receiver deviating from the optimum demodulation suffers from a performance penalty which can be predicted using the PSI based analytic solution derived in this paper. The analytical model also indicates that the minimum number of optical phase-diverse branches per polarization to realize coherent receiver is three for colorless reception. Finally, in Section 5 a proof-of-concept experiment is carried out to validate our approach in full colorless reception of 10 132Gb/s PDM-QPSK WDM channels. The experiment uses a phasediversity CRx 2 3 90° hybrid SED, which was realized by using 3 out of 4 output branches per polarization of a 2 4 90° hybrid. Using a LO power of 10 dBm, we evaluate the bit-error-rate (BER) and the SNR of the center channel at various transmission distances for two different phase-diversity schemes at a received signal power of 10 dBm per channel. The penalty, in terms of transmission distance and SNR, at the hard-decision forward-error-correction (HD-FEC) threshold of 3.8 × 10−3 of the simplified CRx is 500 km and 0.3 dB SNR relative to conventional CRx using balanced detection. Next, we restrict our measurements to fixed distances of 6400 km and 4480 km corresponding respectively to BER at the soft-decision forward-error-correction threshold (SDFEC) threshold of 2 10 2 and the HD-FEC threshold of 3.8 10 3 and sweep the received signal power with a fixed LO power of 5 dBm. This set of measurement is used to evaluate the differences in sensitivity performance between phase-diversity methods employing PSI algorithms when used for colorless reception. The experimental outcomes show that the simplified front-end class can achieve colorless reception with less than 0.3 dB of SNR penalty with respect to a conventional 2 4 90° hybrid BD CRx in typical sensitivity operating regimes. At low received signal powers, where shot noise plays a critical role, the sensitivity penalty of the simplified CRx with respect to the BD CRx is 3 dB at 4480 km transmission distance. These results are shown to be in good agreement with the proposed PSI-model. We conclude that the PSI model is effective for predicting performance for generalized phase-diversity CRx front ends.
2. Brief description of a coherent receiver (non-PSI) This section provides a brief review of the traditional homodyne polarization-diversity and phase-diversity CRx. The section is included in order to elucidate the PSI framework outlined in Section 3 below. For brevity, only one polarization is considered in the following development, but it is feasible to achieve also polarization diversity by splitting the received signal into two orthogonal polarization components, using phase diversity method to detect the field of each orthogonal polarization and utilizing DSP to combine them at electrical domain. In fact, we present both phasediversity and polarization diversity (DP-QPSK 132 Gb/s per channel) transmissions in the experiment. Also, it is noted that there are possibilities of combining polarization and phase diversity using PSI concept. We show one of such options in Appendix C. In the following model, the photocurrent from the photo-detector (PD) of an arbitrary branch i of the optical hybrid can be generally described as
() ()
⁎ Ii (t ) = RS |ES (t )|2 + RLO |ELO (t )|2 + 2 RS RLO R {ES t ELO t e jδ i }
= IS (t ) + ILO (t ) + 2 IS ( t ) ILO ( t ) cos (φ ( t ) + δi )
(1)
where ES and ELO are respectively the signal and LO fields at the input of the optical hybrid; φ is the phase difference between those fields; δi is the optical phase shift of either the signal or the LO introduced in that branch by the optical hybrid; RS and RLO are the total responsivities of the PD with respect to the signal and LO; t is the time; and R and n denote the real part and conjugate
operation of complex numbers. The total responsitivities are obtained using the responsitivity of the PD RPD and the power coupling coefficient S 2 of the corresponding branch such that RS = RPD SS 2 and RLO = RPD SLO 2. This description can be extended to a WDM system with M transmission channels by calculating the total signal field as M ES = ∑ j = 1 ESj, where ESj is the optical field of channel j. In this study, we assign the term “in-band” to any field that is within the bandwidth of the channel of interest ESI, whereas fields outside this band are designated as “out-of-band” (OOB). For notational convenience, we drop the time variable t from this point on. It should be noted that Eq. (1) assumes that the contributions from thermal noise and shot noise are negligible. Accordingly, the first and second terms (IS + ILO ), which arise from the self-beating of the signal and LO and includes all of the direct detection of the incoming fields in baseband, need to be eliminated in order to accurately capture the linearly mapped portion IS (t ) ILO (t ) cos (φ (t ) + δ i ). This is achieved by using phase diversity which is the concept of differing optical hybrid architectures to ensure diverse phase differences between the signal and the LO fields [22–25]. The solution to remove the undesirable self-beating term is to choose a 90° hybrid with BD. The in-phase (I) and quadrature (Q) components can then be retrieved with a pair of identical PDs which returns the signal mixed with the opposite phase of the LO as depicted by
In − phase = I1 − I3 = 4 IS ILO cos φ, Quadrature = I4 − I2 = 4 IS ILO sin φ
(2)
where I1 to I4 are the four outputs of the 90° hybrid with phase differences in the linear mixing term of: 0, 1/2π, π, and 3/2π. Our objective in this study is to determine methods of reducing complexity and component count for a CRx. One method to achieve this is to use an additional PD to directly detect IS in supplement to a 90° hybrid SED [26]. The unwanted IS is then subtracted from the outputs of the SED with DSP. Although this scheme is also applicable for colorless reception, it has the following two primary limitations. First, it is challenging to estimate IS due to optical path delays and/or polarization mismatches between the SED and the direct-detection front-ends. Second, two additional ADCs are required. In contrast, the works in [13–17] investigate another alternative using a 120° hybrid. Let us define I1, I2, and I3 as the three output photocurrents of the hybrid with phase differences (δi of Eq. (1)) in the linear mixing term of 0, 2/ 3π, and 4/3π, respectively. Then, by applying Eq. (1), the photocurrents of the in-phase (II) and quadrature (IQ) components for this case with the 120° hybrid can be obtained as
II,120 = I1 − I2/2 − I3/2,
IQ ,120 =
3 (I3 − I2 ) 2
(3)
This scheme allows for colorless reception with simpler optical designs compared to the former scheme. There is also a complexity increase in the electrical domain because the scheme requires analog circuits for scaling by an irrational factor and/or two additional ADCs for DSP. It is noted that the original phase diversity concept in 1980s aims to ease phase-locking condition for ASK, PSK, FSK format and twofold, threefold, fourfold diversity schemes was compared extensively. While 120° hybrid (using 3 3 coupler) and 90° hybrid (using 2 4 coupler) are widely used in the time of 1980s, their sole purpose is to detect one dimensional modulation format at that time [22,23]. On the other hand, the present-day phase-diversity idea's target is to retrieve full optical field for advanced modulation format (such as QAM) of renewed interest in CRx [24,13,15]. In the following section, we pursue our objective of studying
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
the use of PSI algorithms to describe a phase-diversity CRx.
3. Phase shifting interference (PSI) framework The problem of I–Q (or field) extraction is analogous to the concept of phase extraction in the field of optical interferometry measurements [19–21]. More precisely, the measurement to receive signals that are phase diverse using a CRx is a subset of PSI. For instance, I/Q extraction methods for a conventional 90° hybrid BD and the recently proposed 120° hybrid SED in optical communications shown in Eqs. (2) and (3) are replicas of an N-least square PSI (four and three steps), where the optical phase shifts of N-branches are equally spaced over 2π [19]. Table 1 shows the comparison between two schemes of phase-diversity CRx and N-least square PSI. A rationale behind this phenomenon is that the two categories share the same physical principle: to provide diverse phase either in time (CRx) or in space (PSI interferogram). PSI is a system of approaches to extract the object phase using phase-shifting interferometry. A phase-diversity coherent receiver relies on the optical hybrid, which in essence is an optical phase-shifting interferometer. The concept of phase diversity has not been studied extensively for field detection except for balanced detection and the 120° hybrid proposed recently [1,5,13–17]. We map the concept of PSI to a CRx in order to develop phase-diversity coherent detection for arbitrary hybrid configurations to positively influence CRx design. The description of an arbitrary optical hybrid is made with a set of equations such that Si (ES + ELO e jδi ), where i = 1, … , N and N is the total number of optical branches before the PDs, δi and Si (for simplicity, Si = SLOi = SSi ) are respectively the optical phase shift and coupling coefficient at the ith branch. We rewrite the photocurrent in Eq. (1) as
(
Ii = Ri ES
2
+ ELO
2
)
+ 2 ES ELO cos ( φ + δi ) ,
i = 1, ‥ , N
(4)
Si2 RPD
where Ri = is the total responsitivity with the responsitivity of every photodetector RPD in units of A/W at the ith branch with
271
Table 2 Examples of phase-diversity CRx that can be explained by PSI with their characteristic optical phase shift diagrams; N: number of optical branches per 1 hybrid/ polarization; Δδ is the optical phase shift difference between consecutive branches; δi is the optical phase shift at corresponding branch; bi and ai are weighting factors. Class 4B is the conventional 90° hybrid BD and class 3A is the 120° hybrid SED. Other classes are listed as examples of CRx with arbitrary optical hybrid in this article. N
Δδ
3
4
Class
b1, …, bN a1, …, aN
2π /3
3A
3 /2 [0, − 1, 1] 1/2 [2, − 1, − 1]
π/2
3B
3/4 [1, − 2, 1] 3/4 [1, 0, − 1]
π/2
4A
1/2 [1, − 3, 1, 1] 1/2 [1, 1, − 3, 1] [0, − 1, 0, 1] [1, 0, − 1, 0] 1/2 [1, − 3, 1, 1] 1/2 [3, − 1, − 1, − 1]
δi diagrams
4B 4C
respect to the incoming field. The objective of using the PSI algorithm is to recover the field through I–Q component mapping using signal processing either in the analog or the digital domain. A type of PSI which can be employed is a linear combination of phase-diverse branches. For instance, we can extract the I–Q component directly from the superposition of the output PD currents as described by the following equation: N
II =
∑ Ii ai, i=1
N
IQ =
∑ Ii bi
for N ≥ 3
(5)
i=1
where ai and bi are weighting factors. Table 2 lists some examples of optical hybrids, where the phase shift difference Δδ between two consecutive branches is the same. Thus, for practical implementation, one can regard PSI as a “spatial FIR” filter to retrieve quadrature components as depicted by Eq. (5). The weighting factors ai and bi should satisfy the following conditions and are
Table 1 Comparison between phase-diversity CRxs and PSI algorithms. In PSI algorithms column, Io, Ir and φ are the intensity of object, the intensity of reference plane wave, and the searched phase, respectively. Phase diversity CRxs
PSI algorithms
Type
Balanced detection 90° hybrid
4-least square PSI
Current/intensity
I1 = IS + ILO + 2 IS ILO cos φ
I1 = Io + Ir + 2 Ir Io cos φ
I3 = IS + ILO − 2 IS ILO cos φ
I3 = Io + Ir − 2 Io Ir cos φ
I2 = IS + ILO − 2 IS ILO sin φ
I2 = Io + Ir − 2 Io Ir sin φ
I4 = IS + ILO + 2 IS ILO sin φ
I4 = Io + Ir + 2 Io Ir sin φ
Configuration
In = I1 − I3 Quad = I4 − I2 4 PDs
I4 − I2 ) I1 − I3 4 interferograms
Type
Single-ended detection 120° hybrid
3-least square PSI
Current/intensity
I1 = IS + ILO + 2 IS ILO cos φ
I1 = Io + Ir + 2 Ir Io cos φ
Extraction
φ = tan−1(
2π ) 3 4π cos (φ + ) 3
I2 = IS + ILO + 2 IS ILO cos (φ + I3 = IS + ILO + 2 IS ILO Extraction
In = I1 − I2/2 − I3/2
Configuration
Quad = 3 PDs
3 /2 (I3 − I2 )
2π ) 3 4π I2 = Io + Ir + 2 Io Ir cos (φ + ) 3 ( − ) I I 3 3 2 φ = tan−1( ) 2I1 − I2 − I3 3 interferograms I3 = Io + Ir + 2 Io Ir cos (φ +
272
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277 ⁎ II = RPD Re {Esig − I ELO },
discussed in more detail in Eq. (A.3):
⎤ ⎡ ⎤ ⎤ ⎡ ⎡ N N ∑i = 1 ai ⎥ ⎢ ∑i = 1 bi ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0⎥ ⎥ ⎢ ⎢ N N ⎢∑ ai sin δi ⎥ = ⎢ ∑ bi cos δi ⎥ = ⎢ ⎥ i=1 ⎥ ⎢ N⎥ ⎥ ⎢ ⎢ i=1 ⎥ ⎢ 2⎥ ⎥ ⎢ N ⎢ N a cos b sin − δ δ ∑ ∑ i i⎦ i⎦ ⎣ ⎦ ⎣ i=1 ⎣ i=1 i
(8)
Therefore, the in-band complex signal can be written as ISI = II + jIQ , which leads to a calculated power of < ISI 2〉, where <〉 denotes the ensemble average operator:
(6)
Physically, as given in detail in Appendix A, the first constraint in Eq. (6) enforces the removal of the signal–signal and the LO–LO interfering terms IS + ILO . Namely, in the case of a 90° hybrid with BD, the first constraint has the identical interpretation as the common-mode rejection ratio (CMRR) metric [10]. The second expression in Eq. (6) ensures the orthogonality of the detected quadrature components. Finally, the last condition takes into account the amplitude mapping of the field scaled by the value of the PD responsitivity RPD ES ∥ ELO . We note that the analytical solution in the presence of receiver imperfections (namely, power imbalance, phase imbalance and skew mismatch) will not be addressed in this paper.
4. Analytical SNR expression for IQ retrieval using an arbitrary hybrid in a colorless WDM scenario In the colorless WDM framework, we can re-model Eq. (4) for the ith branch by considering only the beating of the signal of interest with the LO:
⎞ ⎛ M Ii = Ri ⎜⎜ ∑ |ESj |2 + |ELO |2 + 2|ESI ||ELO | cos (φ + δi ) ⎟⎟ ⎠ ⎝ j=1 = IS + ILO + 2 ISI ILO cos (φ + δi )
⁎ IQ = RPD Im {Esig − I ELO }
< ISI
2
2 > = RPD Esig − I
2 2 σnoise = 2σASE − LO + − total
PSI algorithms satisfying the requirements of Eq. (6) and therefore corresponding to CRx configurations are able to effectively cancel the self-induced power interfering terms and accurately detect quadrature components. This technique is most effective with the assumption of ideal hardware such that there is no power or phase imbalance in the hybrid and no skew mismatch between the channels. Assuming we are interested in recovering the optical field in the absence of noise (including amplified stimulated emission (ASE) noise), then the I/Q currents of the inband signal field Esig − I are
2
2 = RPD Psig − I PLO
(9)
N a2 i =1 i
2
N
)
2 + ∑ i = 1 bi σshot − total
N
(∑
N a2 i =1 i
2
N
+ ∑i = 1 bi N
) 2eB R
e PD (P S
+ PLO )
(10)
Then from Eqs. (9) and (10), the closed form expression for a signal-to-noise ratio (SNR) per symbol can be obtained as 2
< ISI >
SNR =
2 σnoise − total
2 R PD Psig − I PLO
= 2 2R PD NASE PLO Be +
(
N ∑ i = 1 a i2
N
2
+ ∑i = 1 bi N
) 2eB R
e PD (P S
+ PLO )
R PD Psig − I PLO
=
(∑ +
N a2 i =1 i
N
2
+ ∑i = 1 bi
) 2eB (P e
N
S
+ PLO )
(11)
Using the definition of optical signal-to-noise ratio (OSNR) [3,4]:
Psig − I 2NASE Bref
(12)
where Bref is the reference bandwidth and the factor of 2 accounts for both polarization of ASE. The SNR in Eq. (11) due to detection with an arbitrary hybrid can be related to the OSNR by
1
SNR = Be + OSNR⁎Bref
(
⎧ ⎫ N + ∑i = 1 bi2 ⁎2eBe ⎪ P ⎪ ⎨ S + 1⎬ N⁎RPD ⁎Psig − I ⎪ PLO ⎪ ⎩ ⎭
N ∑i = 1 ai2
)
(13)
Eq. (13) is the closed-form performance prediction of a phasediversity CRx using the PSI framework for an arbitrary hybrid. The divergence in SNR between phase-diversity CRx is rooted on different weighting factors ai and bi for different hybrids. For example, using a 90° hybrid with BD results in the ratio (∑iN= 1 ai2 + ∑iN= 1 bi2 ) N
4.1. Perfect receiver without power imbalance, optical phase balance, and skew mismatch
(∑
2 = 2R PD NASE PLO Be +
OSNR = Because of the limited-bandwidth of the PDs and the ADCs, IS in the first term, which comprises the OOB signal–signal beating of all WDM channels at baseband is not the same as the in-bandinduced ISI in the last term of Eq. (7). Intuitively speaking, at any time t, as pointed out by Eq. (7), the functionality of a phase-diversity CRx is to detect three unknowns (IS, ISI, and φ) in the framework of colorless reception. In such cases, a phase-diversity CRx should have at least 3 PDs corresponding to 3 different optical phase shifts δi. Some PSI classes, which satisfy the conditions of Eq. (6), are listed in Table 2. The fifth column in the table represents the proper weighting factor to reconstruct I/Q in Eq. (5). Motivated by the above observation, we investigate various PSI algorithms for I/Q retrieval, each with a different optical hybrid design in order to find the optimal hybrids for a CRx. The following Section 4.1 presents an analytical model and Section 4.2 describes the electronic components needed to realize a phase-diversity CRx.
ELO
The most significant noise sources in CRx are the ASE-LO beat noise, shot noise, and the relative intensity noise (RIN) from the LO. As detailed in Appendix B, we note that for ideal receiver, the self-beating noise of the OOB channels plus the LO, and the RIN from both the LO and the signal vanish. In addition, we neglect the contribution from thermal noise. The expression for total noise, Eq. (B.7) of Appendix B, is
2R PD NASE PLO Be
(7)
2
becoming equal to 1. Then, the aforementioned SNR
expression applied to 90° hybrid with BD simplifies to the model commonly found in the literature [4,5,12]. Given a fixed OSNR, the ASE-LO beat noise normally dominates over the shot noise, which in turn makes phase-diversity receivers independent of the LO-to-signal-power-ratio (LOSPR) and the PSI algorithms. However, when the LOSPR is sufficiently large, the shot noise is responsible for degradations. As a consequence, at low received signal powers, different PSI weighting methods result in different performance degradations. We acknowledge from previous sections and from Appendix B that the class 3A, the class with the 120° hybrid SED, and class 4B, the class with the 90° hybrid BD, have the same shot noise contribution. The other PSI classes listed in Table 2, e.g., class 3B with the 90° hybrid SED, have
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
a 1.7 dB penalty in sensitivity relative to the optimum phase-demodulation (120° hybrid SED and 90° hybrid BD) in the shot noise limited regime. 4.2. PSI electronics hardware realization This section presents several examples of how PSI can be implemented with electronic components. Fig. 1 shows the architectures of phase-diversity coherent receiver using the PSI approach for class 3A (120° hybrid SED) and class 3B (90° hybrid SED). In the architecture shown in Fig. 1(a), (b), three ADCs are used to individually convert the three detected signals to digital signals. The linear operations given in Eq. (5) and in Table 2 are performed digitally to obtain the in-phase and quadrature signal components. In the architecture shown in Fig. 1(c), (d), linear operations to obtain the I/Q components from the three photocurrents are performed with analog scaling and subtraction circuits followed by two ADCs and an additional scaling/adding/ subtracting in digital domain (analogues to a balanced detection CRx). Even though this approach offers a simpler front-end (same number of ADCs relative to balanced detection and simple RF circuits), it limits the possibility of PSI realization in order to compensate for hardware impairments. In addition the approach does not provide the access to the direct detection term IS which could be beneficial for cross-phase modulation compensation [27].
5. Tb/s experiment and discussion The PSI theory, presented in the previous sections, for WDM transmission in colorless reception using QPSK modulation format was experimentally validated with the setup illustrated in Fig. 2. Ten channels spaced 50 GHz apart using external cavity lasers (ECLs) and centered at 193.4 THz were multiplexed using a 1:12 coupler. A dual-polarization QPSK signal was then bulk modulated onto all the channels using a Ciena WaveLogic 3 card and a randomly generated sequence of 218 bits. The PDM-QPSK signal at 33 Gbaud was root-raised cosine (RRC) pulse shaped having a roll-off factor of 0.1. The data was decorrelated by 4 ns relative to the neighboring channels. The multiplexed signals were amplified with a booster EDFA and then attenuated with a variable optical attenuator (VOA) to set the launch power at 4 dBm per channel. This launch power was kept constant throughout the experiment. Next, the resulting signal, which has an aggregated data rate of
273
1.32 Tb/s, was launched into a recirculating loop containing four spans of 80 km of SMF-28e þ fiber, each span being followed by an EDFA with a noise figure of 5 dB. Additionally, a gain flattening filter was inserted after the second in-line EDFA. No noise loading was performed after the recirculating loop. The signal out of the recirculating loop was then amplified to a constant power of 15 dBm and subsequently filtered using a wavelength and bandwidth tunable filter. The bandwidth of the filter was set to 5 nm to select all 10 channels for colorless realization, which were then attenuated using a VOA to achieve the desired total received power (PS, tot ) at the receiver. The receiver comprised a dual-pol 90° hybrid with its eight outputs connected to eight single-ended PIN þTIA PDs with a conversion gain of 1.3 V/mW. Another ECL was used as the LO and was set to the center channel frequency (193.4 THz). The output of this ECL was fed into the LO port of the hybrid. The signals were then captured by two real-time oscilloscopes with a 33 GHz bandwidth at 80 GSa/s and were processed offline. Using the above test bench, we experimented with two different reception scenarios: (1) a WDM configuration in which the total received power PS, tot and LO power were fixed to 0 dBm (or 10 dBm/channel) and 10 dBm, respectively; (2) another WDM configuration in which the transmission distances were fixed and the LO power was set to 5 dBm. The first scenario investigates the impact of transmission on a PSI-based CRx, while the second scenario evaluates the dynamic range of the PSI-based CRx. For the offline DSP, the sampling skew between each branch was first corrected and the power imbalance was compensated by weighting each of the 8 outputs by a factor proportional to its selfbeating term. Afterwards, IQ detection using PSI as described in Section 3 was performed. Further offline signal processing consisted of IQ imbalance correction using Gram–Schmidt orthogonalization procedure (GSOP), re-sampling to 2 samples per symbol, chromatic dispersion (CD) compensation, matched-filtering in frequency domain, frequency offset removal, polarization demultiplexing, and carrier phase estimation performed using decision directed least mean squares (LMS) algorithm with step sizes of 0.001 and 0.005 respectively [9]. After performing hard decision decoding on a symbol-by-symbol basis, the achieved electrical SNR was evaluated and the BER was calculated. It is noteworthy that in order to mimic ideal hardware, our experimental system is unique in two key aspects. First, a free-space-optics-based hybrid (Kylia) is used to exploit the smallest amount of phase imbalance. Second, power imbalance plus skew is mitigated in DSP. Together with the aforementioned DSP, GSOP is utilized to remove any
a
b
c
d
Fig. 1. Examples of CRx implementations with ideal 2 3 optical hybrids per polarization (2 3 120° and 2 3 90°). PSI processing to retrieve quadrature component is done with digital signal processing with the use of 3 ADCs in (a) and (b); and with analog circuits and 2 ADCs in (c) and (d).
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T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
Fig. 2. Experiment setup for WDM 1.32 Tb/s transmission (IL: Interleaver, ECL: External Cavity Laser, VOA: Variable Optical Attenuator, PM: Polarization Maintaining).
transmission at the HD-FEC threshold and a penalty of 0.3 dB and 0.2 dB in SNR at distances of 4480 km and 6400 km, respectively. At shorter distances, the degradation of the class 3B becomes larger. This can be understood from the explanations in Section 4.1, which showed that the shot noise contribution plays a more significant role in the short distance regime (or high OSNR) and/or colorless reception with a large number of WDM channels at high received powers. We then assess the impact of received signal power with
remaining phase imbalance. We first compared the transmission performance with two select classes of PSI: class 3B with a 90° hybrid and the conventional class 4B with a 90° hybrid. For both cases, the LO power was fixed to 10 dBm and the received signal power per channel was 10 dBm. For class 3B, we use only 6 out of the 8 outputs of the CRx for signal processing. The results are given in Fig. 3(a), (b). We observe that the class 3B achieves a similar performance compared to the class 4B BD, but with a penalty of 500 km in
a
b
c
d
Fig. 3. Experiment results of WDM 10 channels: 10 dBm received signal power per channel and LO power of 10 dBm (a) BER versus transmission distance, (b) SNR versus transmission distance; dynamic range with LO power of 5 dBm and transmission distances of 4480, 6400 km (c) BER versus received signal power per channel, (d) SNR versus received signal power per channel.
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
respect to the two PSI-based CRxs. In our experiment, the class 3B is not realized by a real 2 3 90° hybrid but by using three out of the four branches of a 2 4 90° hybrid. Here, the PSI weightings for I and Q are [1, 2, 1] and [1, 0, 1] instead of 3/4[1, 2, 1] and 3/4[1, 0, 1], as illustrated in Table 2. Therefore, in the shot-noise limited domain, the class 3B is expected to have a 3 dB penalty instead of a 1.7 dB penalty in terms of the power needed to achieve a similar performance. In Fig. 3(c), (d), BER versus Psig and SNR versus Psig are plotted at transmission distances of 4480 and 6400 km for a LO power of 5 dBm. We observe that both receivers are limited by shot-noise as indicated by the similar trend at signal power levels less than 14 dBm. Fig. 3(c) shows a 3 dB difference in power at the HD-FEC threshold after 4480 km. This was reflected in the PSI model which predicts the divergence in performance between PSI classes. We note from the curves of Fig. 3 (d) that the typical penalty of a class 3B CRx in the ASE-limited region is less than 0.3 dB at transmission distances of 4480 km and 6400 km. Although in our experiment the class 3B suffers a nonnegligible penalty compared to conventional BD in the shot-noise limited regime (expressed by either short distance or low received signal power), it should be noted that such a penalty is smaller in real implementations of class 3B for two reasons. First, a real 2 3 90° hybrid as would be used in class 3B (not utilizing 3 ports out of 2 4 90° hybrid as in our experiment), only suffers from 1.7 dB in power sensitivity compared to class 4B (described in Section 4.1). Second, 2 3 hybrids (90° or 120°) typically have lower fabrication deviations, resulting in the class 3B having less physical impairments (including power imbalance) than class 4B, which is built from 2 4 90° hybrids [18]. To equalize for power imbalance, a 2 3 90° hybrid uses a smaller scaling operation after each PD relative to a 2 4 90° hybrid. The scaling operation can yield larger shot noise (or additive receiver noise) contribution. Furthermore, among the sub-classes of class 3 PSI CRxs, the class 3A is the best. Therefore, the experimental results indicate that a simpler front end, at least with 2 3 120° hybrid, can achieve equivalent performance compared to conventional balanced detection. This indication matches well the recent experimental demonstrations [13].
6. Conclusion In summary, the themes produced by this study provide a comprehensive CRx model using the concept of PSI from the optical measurement domain. A framework for a colorless phasediversity CRx is found for an arbitrary hybrid with an analytical solution to predict performance with an ideal receiver. We conclude that 2 3 120° hybrid SED and 2 4 90° hybrid BD CRx are the best phase-diversity coherent receiver configurations in terms of rejection of receiver's noise. The performance penalty from other phase-diversity CRx flavors can be gauged from the optimum value using PSI model. We identify shot noise as the primary element responsible for the degradation disparity between the phase-diversity methods. Next, a proof-of-concept experiment is performed in which colorless 10 132-Gb/s PDM-QPSK WDM channels are successfully transmitted. Results show that an example of arbitrary phase-diversity CRx using a 2 3 90° hybrid can achieve colorless reception by sacrificing less than 0.3 dB in SNR in comparison to a conventional CRx BD at FEC thresholds of 3.8 10 3 and 2 10 2, respectively. We further conclude that shot noise has a considerable effect on performance in the regime of low received signal powers or short transmission distances. To the best of our knowledge, this is the first time such model been reported and experimentally demonstrated. The flexibility in CRx design is the attractiveness behind the presented PSI approach.
275
Acknowledgments The authors would like to thank Charles Laperle and Doug Charlton from Ciena for their guidance with a WaveLogic3 bulk modulator transmitter.
Appendix A. Weighting factor conditions In this appendix, we derive the general conditions for PSI weighting factors in order to retrieve the optical field from an arbitrary hybrid using the ideal hardware assumption. We start by using the relation in Eq. (5) to express the two quadrature components to arrive at N
N
∑ Ii ai = RPD |ES ||ELO | cos φ, ∑ Ii bi = RPD |ES ||ELO | sin φ i=1
i=1
(A.1)
Next, if we use Eq. (4) to represent photocurrents of an arbitrary optical branch, we obtain
(
Ii = Ri ES
2
+ ELO
2
)
+ 2 ES ELO cos ( φ + δi ) ,
i = 1, ‥ , N
= c0i + c1i cos (φ + δi ) = c0i + c1i cos φ cos δi − c1i sin φ sin δi
(A.2)
where c0i = Ri ( ES 2 + ELO 2); c1i = 2Ri ES ELO . For the sake of simplicity, we assume each PD has the same total responsitivity R1 = ⋯ = Ri = R (S1⋯ = Si = S ) or c01 = ⋯ = c0i = c0; c11 = … = c1i = c1. Next, we substitute Eq. (A.2) back into to Eq. (A.1) and sum the photo-currents:
⎤ ⎤⎡ ⎤ ⎡ ⎡ N N N ∑i = 1 cos δi ∑i = 1 sin δi ⎥ ⎢ c ⎥ ⎢ ∑i = 1 Ii ⎥ ⎢ N 0 ⎥ ⎢ c1 cos φ ⎥ ⎥ ⎢ N ⎢ N N N = ∑ ∑ ∑ ∑ I a a a cos a sin δ δ ( ) − ( ) i i i i⎥⎢ ⎥ ⎢ i=1 i i⎥ ⎢ i=1 i i=1 i=1 ⎥ ⎢ c1 sin φ ⎥ ⎥ ⎢ N ⎢ N N N ∑ ∑ ∑ ∑ I b b b cos b sin δ δ ( ) − ( ) i i⎦⎣ ⎦ ⎣ i=1 i i ⎦ ⎣ i=1 i i=1 i i=1 i In the 3 3 matrix of the above equation, the first column represents a superposition of direction detection terms while the second and third columns depict in-phase and quadrature components, respectively. Note that a similar matrix can be found in the literature on PSI [19–21] and a minimum number of phaseshifting branches is 3 in order to solve above linear system. A 90° hybrid BD and 120° hybrid SED are the least square solutions of the above matrix which turn the 3 3 matrix to a diagonal matrix. In the optical measurement domain, the N-least square PSI is commonly considered as the best algorithm in terms of noise rejection for N phase-shifting branches [19–21]. Without any loss of generality, in order to detect I/Q accurately, the weighting factors a and b must follow these conditions:
⎤ ⎤ ⎡ ⎤ ⎡ ⎡ N N ∑i = 1 ai ⎥ ⎢ ∑i = 1 bi ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎢ N N ⎥ ⎢∑ ai sin δi ⎥ = ⎢ ∑ bi cos δi ⎥ = ⎢ 1 i 1 i = = ⎥ ⎢ 1 ⎥ ⎥ ⎢ ⎢ N N ⎥ ⎢ 2S 2 ⎥ ⎥ ⎢ ⎢ ⎦ ⎣ ∑i = 1 ai cos δi ⎦ ⎣ ∑i = 1 − bi sin δi ⎦ ⎣ where
C = RPD ES ELO =
c1 2S2
(A.3)
. In Eq. (A.3), the first condition
indicates the ability to reject the signal–signal beating noise (or “Common Mode”). The second condition points out the capability to preserve orthogonality between quadrature components. The last condition describes the mapping process of the quadrature component's magnitude and is the least important condition. The first two constraints are important because they affect directly the quality of quadrature extraction. If we consider an ideal hybrid with equal power splitting and no excess loss, we can replace S2 by 1/N, which is the inverted
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T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
number of optical branches. We remark that we distinguish between the fully detected optical field, included OOB WDM chanM nels (ES = ∑ j = 1 Esj ) in Eq. (A.1) and the desired in-band optical field (ESI) in colorless reception. In this case, an additional electrical filter is needed to filter out the desired channel in the electrical domain which is accomplished by the ADCs and/or PDs.
Appendix B. Noise contribution In this appendix, we evaluate the degradation due to two principal noise sources in general CRx: optical Gaussian ASE beat noise and shot noise. Thermal noise or transimpedance amplifier (TIA) noise is supposed to be negligible in the derivation. Relative intensity noise (RIN) is eliminated assuming ideal hardware. Shot noise is always present in any optical receiver. Considering e the elementary charge and Be the equivalent electrical bandwidth, the shot noise variance at arbitrary branch ith in Eq. (4), (7) is given by
⎛ ⎜ 2 σshot − i = 2eBe Ri ⎜ ⎝
2
M
∑ (Esig− I + nASE ) i=1
⎞ + |ELO |2 ⎟ ⎟ ⎠
= 2eBe Ri (PS + PLO )
(B.1)
For an ideal receiver, the signal and LO are equally split between N ports of an arbitrary hybrid. In other words, R1 = R2 = Ri = ⋯ = R . Owning to the statistical independence of the interactions between shot noise in each PD, one can regard the total shot noise as 2 2 σshot − total = σshot ⁎N = 2NeBe Ri (PS + PLO )
= 2eBe RPD (PS + PLO ) (⁎)
(B.2)
2 σshot
2 2 = σshot − 1 = ⋯ = σshot − i , (⁎) is the assumption of no RPD 1 2 S = N (or Ri = N ). Shot noise is identified as a non-
where excess loss stationary noise source. Optical ASE-LO beat noise, as we shall see shortly, is the beat term between the LO and optical noise field ASE nASE induced by optical amplifier during transmission.Starting with conventional 90° hybrid balanced detection, we get the photocurrents for I/Q with the perfect hardware realization assumption after low-pass filtering in electrical domain:
IQ = RPD Im { ( Esig − I + nASE ) (ELO
+ ishot 4 − ishot2
2 2 σnoise − total = 2σASE − LO +
(∑
N a2 i=1 i
N
)
2 + ∑i = 1 bi2 σshot − total
(B.7)
N
From Eqs. (B.6)–(B.7), a class 3A 120° hybrid has the same noise power as a conventional 90° hybrid balanced detection while class 3B, 4A, 4C 90° hybrid SEDs suffer from 1.7 dB increase in shot noise power. The least square PSI, where the angle separation Δδ = 2π /N , make the ratio
(∑iN= 1 ai2 + ∑iN= 1 bi2 ) N
equal to 1.
Appendix C. Example of combination of polarization diversity and phase diversity using PSI In this appendix, we show the possibility in combining polarization diversity and phase diversity using PSI. In particular, we demonstrate a pair of 2 3 90° hybrid and 2 2 90° hybrid, which is illustrated in Fig. C1, is capable of detecting polarization multiplexed signal. In the above figure, the incoming signal ES has two components with respect to polarization beam splitter (PBS): ESX and ESY. It is split by a 50:50 coupler before going to 2 3 90° hybrid of the top branch and 2 2 90° hybrid of the bottom branch. In addition, the LO continuous wave ELO is equally split into ELOX and ELOY as reference tone of the two hybrids. Since ESX and ELOX cannot beat with ESY and ELOY, we can model the output currents (neglecting the scale factor RPD) as follows:
I1 = ESX
2
+ ESY
2
+ ELOX
2
⁎ + 2R {ESX ELOX }I2 = ESX
I3 = ESX
2
+ ESY
2
+ ELOX
2
⁎ − 2R {ESX ELOX }
I4 = ESX
2
+ ESY
2
+ ELOY
2
⁎ + 2I {ESY ELOY }
I5 = ESX
2
+ ESY
2
+ ELOY
2
⁎ + 2R {ESY ELOY }
2
+ ESY
2
+ ELOX
2
(C.1)
The I/Q components of X and Y polarizations can be found using PSI and the fact that ELOX 2 = ELOY 2:
IXI = 3/4 ( I1 − I3 ) IXQ = 3/4 ( I1 − 2I2 + I3 )
II = RPD Re { ( Esig − I + nASE ) (ELO )⁎} + ishot1 − ishot 3 )⁎}
can derive a noise variance analytical expression for different PSI configurations using a similar procedure. For arbitrary class PSI
(B.3)
IYI = ⎡⎣ I5 − ( I1 + I3 )/2⎤⎦/2 IYQ = ⎡⎣ I4 − ( I1 + I3 )/2⎤⎦/2
Consequently, we can derive
(C.2)
inoise − I = iASE − LO − I + ishot1 − ishot 3, inoise − Q = iASE − LO − Q + ishot 4 − ishot2
(B.4)
From Eq. (B.4), the main noise contributors are ASE-LO beat noise and shot noise. The noise variance of the beat noise term between the LO and ASE field is defined as 2
2 2 2 2 2 σASE − LO − I = σASE − LO − Q = σASE − LO = RPD NASE ELO Be = RPD NASE PLO Be (B.5)
where NASE is the power spectral density of ASE noise. Without any impairment from the receiver, the self-beating common mode noise and the RIN from the LO or signal are perfectly cancelled. Hence, one can represent the total noise variance as 2 2 2 2 σnoise − I = σnoise − Q = σASE − LO + 2σshot 2 σshot − total 2 2 2 σnoise − total = 2σASE − LO + σshot − total (B.6) 2 mentioned in Eq. (B.2) with N ¼4. Furthermore, we
2 = σASE − LO + 2 where σshot − total
Fig. C1. An example of combination of polarization diversity and phase diversity using 2 3 90° hybrid and 2 2 90° hybrid.
T.M. Hoang et al. / Optics Communications 356 (2015) 269–277
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