## Abstract

Digital signal processing (DSP) combined with a phase and polarization diverse coherent receiver is a promising technology for future optical networks. Not only can the DSP be used to remove the need for dynamic polarization control, but also it may be utilized to compensate for nonlinear and linear transmission impairments. In this paper we present results of a 42.8Gbit/s nonlinear transmission experiment, using polarization multiplexed QPSK data at 10.7GBaud, with 4 bits per symbol. The digital coherent receiver allows 107,424 ps/nm of chromatic dispersion to be compensated digitally after transmission over 6400km of standard single mode fiber.

©2007 Optical Society of America

## 1. Introduction

Phase and polarization diverse coherent detection combined with digital signal processing (DSP) is a powerful combination for next generation optical receivers. Not only does it remove the need for dynamic polarization control [1], but also enables the detection of polarization multiplexed spectrally efficient formats [2]. Furthermore since the receiver is linear in field, dispersion from the fiber can be compensated using digital finite impulse response (FIR) filters [3], in principle allowing for transmission over standard fiber without optical dispersion compensation.

To date the longest distance reported for a 10Gbit/s system operating over standard fiber without optical dispersion compensation is 5120km using pre-distortion [4]. In this paper we demonstrate that a distance of 6400km can be achieved for a 42.8Gbit/s signal, by transmitting 10.7GBaud polarization multiplexed QPSK (PMQPSK) with 4 bits per symbol. This represents the longest reach achieved for transmission without optical dispersion compensation, not only for a 40Gbit/s system, but also for any 10GBaud system operating over standard fiber.

## 2. Coherent receiver architecture

#### 2.1 Overview

The coherent receiver illustrated in Fig. 1 is based on an intradyne system, using a local oscillator which is nominally at the same frequency as the incoming signal.

The I/Q coupler acts as a 90 degree hybrid, separating the incoming optical signal into inphase and quadrature components, corresponding to the real and imaginary parts of a complex signal the analysis of which is presented in the next subsection.

#### 2.2 Fiber I/Q coupler based hybrid

In our receiver we used a 3×3 coupler as the 90 degree hybrid. The coupling between three identical fibers, may be modeled by the following matrix equation [5]

Where **E** =(*E*
_{1}, *E*
_{2}, *E*
_{3})^{T} and *K* is a matrix in which all elements are equal to the coupling coefficient κ which determines the coupling between the three electric fields *E*
_{1}, *E*
_{2} and *E*
_{3}. The solution to this equation may be expressed in term of the matrix exponential such that **E**(*z*) = exp(*jKz*)**E**(*z* = 0) . If we choose a coupling distance L given by κ*L* = [2*nπ* + 2arctan(3)] / 3 , where *n* is an integer, the coupler splits power in the ratio 1:2:2, hence for input fields **E**(*z* = 0) = (*E _{sig}* ,0,

*E*

_{lo}exp - 3

*jπ*/ 4))

^{T}the output fields are given by

These resulting electric fields are then detected using a photodiode, with the corresponding photocurrents (*i*
_{1},*i*
_{2},*i*
_{3})^{T}(|*E*
_{1}(*L*)|^{2},|*E*
_{2}(*L*)|^{2},|*E*
_{3}(*L*)|^{2})^{T} given by

From Eq. (3) it can be seen that the coherently detected currents associated with *i*
_{1} and *i*
_{3} are in quadrature. Provided that |*E _{lo}*|

^{2}≫|

*E*|

_{sig}^{2}, if a DC block is employed, the directly detected terms may be neglected, allowing phase diversity using just two photodiodes.

## 3. Digital signal processing algorithms

#### 3.1 Overview

DSP is used to equalize the channel and recover the phase of the signal. This allows for chromatic dispersion and polarization dependent effects to be compensated, in addition to tracking the phase difference between the incoming signal and the local oscillator. To achieve this, the samples pass through four distinct stages. Firstly the data is resampled to give ~2 samples per symbol, after which the signals are equalized using FIR filters. Following this the phase of the signal is recovered, before finally a threshold is applied to recover the data bits.

#### 3.2 Details of the algorithms

Most of the distortion occurs due to chromatic dispersion. By using an FIR filter with a quadratic phase response of value equal and opposite to that imposed by the chromatic dispersion, this may be compensated. The FIR filter coefficients (512 taps) are calculated from a truncated impulse response of the chromatic dispersion with a Kaiser window function. Polarization tracking and compensation of the remaining distortions are achieved using four shorter filters **h**
_{xx}, **h**
_{xy}, **h**
_{yx} and **h**
_{yy}, having just 5 taps each (corresponding to the elements in the inverse Jones matrix). If **x** and **y** represent the incoming electrical signal for the two polarizations, then the output signals for the two polarization are *x*′ = *h*
_{xx} ∙**x** + **h**
_{xy} ∙**y** and *y*′ = *h*
_{yx} ∙**x** + **h**
_{yy} ∙**y**. These filters are initially adapted using the constant modulus algorithm[6], which exploits the fact that QPSK is “constant modulus” and so minimizes the errors *ε _{x}*=1-|

*x′*|

^{2}, and

*ε*=1-|

_{y}*y′*|

^{2}using the stochastic gradient such that

$${\mathbf{h}}_{\mathit{yx}}={\mathbf{h}}_{\mathit{yx}}+\mu {\epsilon}_{y}\mathit{y\prime}\bullet \stackrel{}{\overline{\mathbf{x}}}\phantom{\rule{6.em}{0ex}}{\mathbf{h}}_{\mathit{yy}}={\mathbf{h}}_{\mathit{yy}}+\mu {\epsilon}_{y}\mathit{y\prime}\bullet \stackrel{}{\overline{\mathbf{y}}}$$

where *μ* is the convergence parameter and **x̄** denotes the complex conjugate of **x**. Once the constant modulus algorithm has converged the offset frequency for the two polarizations are estimated using a 4th order nonlinearity to remove the QPSK data. The frequency estimated from the maximum of |*FFT*(*x*′^{4})| and |*FFT*(*y*′^{4})| respectively, with the maximum frequency offset that can be resolved being ±2.675 GHz, a quarter of 10.7GHz Having removed the offset frequency the receiver moves into a decision directed mode. The phase-corrected output symbols are given by *X* = exp(-*jϕ _{x}*)

*x*′ and

*Y*= exp(-

*jϕ*)

_{y}*y*′, where the phases are given by

where csgn(*X _{i}*) is the complex sign of the symbol

*X*defined by

_{i}The tap weights of the FIR taps adapted using a decision directed mode such that

$${\mathbf{h}}_{\mathit{yx}}={\mathbf{h}}_{\mathit{yx}}+\mu {\epsilon}_{y}\bullet \overline{\mathbf{x}}\phantom{\rule{6.em}{0ex}}{\mathbf{h}}_{\mathit{yy}}={\mathbf{h}}_{\mathit{yy}}+\mu {\epsilon}_{y}\bullet \overline{\mathbf{y}}$$

where the errors for the decision directed algorithm are given by ${\epsilon}_{x}=\mathrm{exp}\left(j{\varphi}_{x}\right)\left(\frac{\mathrm{csgn}\left(X\right)}{\sqrt{2}-X}\right)$ and $\phantom{\rule{.2em}{0ex}}{\phantom{\rule{.2em}{0ex}}\epsilon}_{y}=\mathrm{exp}\left(j{\varphi}_{y}\right)\left(\frac{\mathrm{csgn}\left(Y\right)}{\sqrt{2}-Y}\right)$

## 4. Experimental investigation into the nonlinear propagation of PMQPSK

The aim of the experiment was to determine, the maximum distance that can be achieved using 42.8Gbit/s polarization multiplexed QPSK, given that the fiber is nonlinear.

#### 4.1 Experimental setup

The experimental setup used (Fig. 2), consists of three distinct elements, the transmitter, the recirculating loop, and the coherent receiver. The DFB-based transmitter with wavelength of 1554.94nm and measured linewidth of 2MHz was operated at 10.7Gbaud, with a 2^{11}-1 PRBS, chosen to facilitate processing of the data in a realistic time frame. The data rate of 42.8Gbit/s was achieved using QPSK and polarization multiplexing. Within the recirculating loop the 80km fiber span generates 1342.8ps/nm of chromatic dispersion with a loss of 15.7dB. Two erbium doped fiber amplifiers, are used within the loop to overcome the loss of the fiber and loop components. At the receiver a DFB local oscillator with a linewidth of 100kHz was combined with the signal via two asymmetric 3 fiber couplers, with coupling ratios 1:2:2, to detect the in-phase and quadrature components of the two polarizations. On conversion into the electrical domain, the signal was digitized at 20GSa/s using a Tektronix TDS6154C digital storage oscilloscope with 8 bit resolution, with the waveforms then processed off-line using Matlab.

#### 4.2 Initial characterization

To determine the penalty due to transmission we performed back-to-back measurements which gave the required OSNR (ROSNR) at the FEC limit (BER=3×10^{-3}) as 10.2dB (in 0.1nm). Having determined the ROSNR we next determined how noise accumulated with transmission distance. Operating the loop with a launch power of -5dBm we determined that the OSNR after N recirculations was given by

With *P _{launch}* being the launch power in dBm, with 37.5 arising due to the experimentally determined parameter that -37.5dBm of optical noise (in 0.1nm) is added per recirculation.

#### 4.3 Characterizing the nonlinear transmission

To obtain the optimal fiber launch power for maximum distance, the BER was measured at distances (in km) of {1280, 2480, 3840, 5120} as the power level was increased from -15 to 5 dBm, in steps of 2dB. To determine the point of convergence, additional measurements were taken after {5720, 6400, 7040} km as the power was increased from -9 to -1 dBm again in 2 dB steps. The DSP algorithms used the same parameter set and converged in all cases. The BER was obtained by counting errors over a total of 73692 bits, from which a contour plot of BER=3×10^{-3} based on interpolation of 20logQ was obtained, which closely followed that obtained in preliminary tests.

From Fig. 3, we determine that the maximum reach of 6480km is obtained with a fiber launch power of -7dBm. At the optimum launch power after 6480km (81 recirculations), Eq. (8) gives the OSNR as 11.4dB, indicating that the OSNR penalty is 1.2 dB compared to the back-to-back performance. While these maximum distances are based on interpolation, the optimum point from Fig. 3 is close to a measurement point taken at 6400km with launch power of -7dBm. At this point, as illustrated in Fig. 4, the constellations are successfully recovered, with 176 errors counted over 73692 bits giving a BER of 2.4×10^{-3} with the 99% confidence interval being (2.1×10^{-3}, 2.7×10^{-3}) [8].

## 5. Conclusion

In this paper we have demonstrated 42.8Gbit/s per wavelength transmission with a record distance of 6400km over standard fiber with no optical dispersion compensation. A record total dispersion of 107,424ps/nm was compensated using digital signal processing, with an OSNR penalty of 1.2dB. This is the first time that 40Gbit/s per wavelength data has been transmitted over such a distance on standard fiber, without the use of optical dispersion compensation and exceeds the distance previously reported for 10Gbit/s systems.

## Acknowledgments

Financial support to SJ Savory and G Gavioli from the Leverhulme Trust is gratefully acknowledged, as is that received for the project from FP6 IST-IP NOBEL and EPSRC. The authors thank Dr R Griffin and Bookham for the loan of the DQPSK transmitter, AD Stewart and S Woods for their assistance and to Dr MG Taylor for valuable discussions.

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