## Abstract

An all-optical regeneration scheme for DQPSK signals is proposed and analyzed. In the regenerator, an incoming DQPSK signal is demodulated to two parallel OOK signals by one-symbol delay interferometers. After the amplitude noise is removed by 2R (reamplifying and reshaping) regenerators and the power levels are suitably amplified, the OOK signals modulate the phase of clocked probe pulses in the subsequent all-optical phase modulators by which the noise-reduced (D)QPSK signal is generated. The alteration of phase data encoded on the pulses in the regeneration process can be undone by suitable encoding or decoding. Numerical simulation for short-pulse RZ-DQPSK signals at 160 Gbit/s (80 Gsymbol/s) shows that reduction in both phase and amplitude noise can be obtained by the regeneration scheme where fiber-based 2R amplitude regenerators and phase modulators using self- and cross-phase modulation, respectively, are employed.

©2010 Optical Society of America

## 1. Introduction

All-optical signal regenerators are expected to overcome distance limitations in high-speed signal transmission imposed by accumulation of signal distortion and noise. The use of all-optical regenerators, which does not rely on optical/electrical/optical conversion nor on electrical signal processing, leads to cost- and energy-efficient long-distance ultrahigh-speed optical signal transmission. A number of experimental demonstrations and theoretical studies have been reported for more than a decade, most of which have discussed the regeneration of on-off keying (OOK) signals [1]. Considering that signals in advanced modulation formats such as differential phase shift keying (DPSK), differential quadrature phase shift keying (DQPSK), and other multi-level formats are becoming used in long-distance transmission, all-optical regenerators capable of processing such signals will be highly desired.

Recently several schemes of DPSK signal regeneration and regenerative wavelength conversion have been proposed and demonstrated. Reduction of phase and amplitude noise is obtained by the use of phase-sensitive amplifiers [2,3] or by converting the phase information to/from the amplitude information and performing the regeneration operation on the amplitude [4–6]. Averaging of phase fluctuations over neighboring bits can also lead to phase-noise reduction [7–9]. Phase preserving amplitude-only regeneration, which is effective in reducing the nonlinear phase noise (the Gordon-Mollenauer effect [10]), has also been demonstrated [11,12].

Besides the regeneration of binary phase-shift keying (PSK) signals, that of M-ary PSK signals with M≥4 is interesting and beneficial because the transmission distance of such multi-level signals is severely limited by noise and distortion owing to the small minimum distance between signal points in the constellation [13]. Several papers have discussed all-optical regeneration of (D)QPSK signals. In [14], a scheme using two parallel phase-sensitive amplifiers has been proposed. Under the perfect optical-phase locking between the incoming QPSK signal and a locally generated pump light, it is numerically shown that both of amplitude and phase noise on the incoming signal can be reduced. In [15], we have proposed a DQPSK signal regeneration scheme where the DQPSK signal is converted to two parallel OOK signals by a pair of delay interferometers and the noise on the OOK signals is suppressed by all-optical 2R (reamplifying and reshaping) amplitude regenerators. The regenerated OOK signals are used as control pulses in subsequent all-optical phase modulators for probe clock pulses to generate phase-remodulated output pulses. This scheme is an extension of the all-optical DPSK signal regenerator whose performance was experimentally demonstrated in [6]. This type of regenerator maps the phase difference between consecutive pulses in the incoming signal to the absolute phase of the outgoing regenerated pulse so that the phase data encoded on the signal is altered by the regeneration. The conversion of the phase data can be undone by suitable encoding or decoding in the electrical domain [4,5].

In this paper, we describe the principle of operation and present a numerical analysis of the performance of the DQPSK regenerator proposed in [15]. In Section 2 we explain the operation of the regenerator and derive the strength of noise suppression required to the 2R amplitude regenerators for sufficient suppression of phase noise of the output signal. In Section 3 we discuss the encoder or decoder for undoing the phase data conversion by the regenerator. In Section 4 numerical simulation is performed of the regenerator for short-pulse return-to-zero (RZ)-DQPSK signals at 160 Gbit/s (80 Gsymbol/s) speed, where cascaded fiber-based all-optical 2R amplitude regenerators are used for amplitude noise suppression. Discussion and conclusion are given in Section 5.

## 2. Operation principle

Figure 1
shows the block diagram of the DQPSK signal regenerator proposed in this paper. Incoming
DQPSK signals are demodulated to OOK signals by the use of two parallel one-symbol delay
interferometers (DIs). The optical phase difference in two arms in the DIs are set at
θ_{DI}=π/4 and – π /4 for the upper and
lower DIs, respectively, which is the same way as in typical DQPSK receivers [16]. In the regenerator shown in Fig. 1, signals emerging from one of the output ports of each DI are used,
power level of which takes high or low value depending on the optical phase difference between
the consecutive input symbols. The subsequent 2R amplitude regenerators remove amplitude
fluctuations of the high-level signals and suppress the low-level signals to zero. The
amplitude-stabilized OOK pulses are then amplified to prescribed levels and fed to all-optical
phase modulators in which the phase of clock (probe) pulses is modulated by π or
π/2 in proportion to the power of the OOK pulses. In this way the four-level phase
difference between adjacent symbols of input DQPSK signals can be regenerated and mapped to the
absolute phase of the output pulses.

The proposed regenerator uses only one output port of each DI for DQPSK to OOK conversion. It may seem better that both output ports of each DI producing complementary OOK signals are used [4,5]. The two OOK signals can drive a Mach-Zehnder Interferometer (MZI) all-optical phase modulator operating in push-pull mode. When pure phase modulation is applied to the probe signal in each arm of the MZI, however, the amplitude noise of the OOK signal is transferred to the phase noise of the MZI output signal [17]. To suppress the phase noise we need to put either two 2R amplitude regenerators at both of the output ports of each DI or a phase-preserving amplitude limiter prior to the DI [18]. In the former case, we need four 2R regenerators in total in the DQPSK regenerator. To avoid this complexity, we use the scheme shown in Fig. 1 where only one output port of each DI is used and the regenerated OOK pulses drive two all-optical phase modulators connected in series. The situation is different when semiconductor optical amplifiers (SOAs) are used for the modulation of the probe signal in the push-pull mode MZI [4,5]. If cross-gain modulation instead of cross-phase modulation (XPM) dominates in the SOA, which corresponds to the case that the linewidth enhancement factor of the SOA is small, phase fluctuation of the output signal can be made small without the use of 2R amplitude regenerators [19]. Use of the saturation behavior of the SOA further reduces amplitude fluctuation of the output signal. A drawback of the SOA-based regenerators is the limitation in operation speed caused by finite carrier lifetime.

The crucial component in the regenerator shown in Fig. 1 is the 2R amplitude regenerators after the DIs. For the phase noise of the output signal to be smaller than that of the input signal, the 2R regeneration strength must be sufficiently large. The noise reduction factor required to the 2R regenerator can be estimated in the following way.

We first assume that the incoming pulses have a complex amplitude of the form ${E}_{n}^{in}$=(A_{s}+ΔA_{n})
exp[i(ϕ_{n}+Δϕ_{n})] where A_{s} and
ϕ_{n} (ϕ_{n}- ϕ_{n-1} = 0,
π/2, π, or 3π/2) are an amplitude and phase of the pulse,
respectively, and ΔA_{n} and Δϕ_{n} are
amplitude and phase fluctuations of the pulse. The complex amplitude of the pulses at the output
ports of the DIs is given by

_{DI}=π/4 and –π/4 for the upper and lower DIs shown in Fig. 1. A factor 2 in the denominator in Eq. (1) accounts for the loss by the 3dB coupler that delivers the incoming signal to the two DIs. The power of the output signal from the DIs is expressed as

In the first-order approximation under the conditions $|\Delta {A}_{n-1,\text{\hspace{0.17em}}n}|<<{A}_{s}$ and $|\Delta {\varphi}_{n-1,\text{\hspace{0.17em}}n}|<<1,$ Eq. (2) can be linearized with respect to the small fluctuations as

The first term in Eq. (3) gives the average
value while the second and third terms represent noise. Here we denote the output powers ${\left|{E}_{DI}^{}\right|}^{2}$ at the upper (θ_{DI}=π /4) and lower
(θ_{DI}=-π/4) DIs as P_{DI,1} and P_{DI,2},
respectively. The average of P_{DI,1} and P_{DI,2} takes either of two power
levels depending on the differential phase ${\varphi}_{n}-{\varphi}_{n-1}$ as (${\overline{P}}_{DI,1}$,${\overline{P}}_{DI,2}$)=(P_{L}, P_{L}), (P_{H}, P_{L}),
(P_{H}, P_{H}), and (P_{L}, P_{H}) for ${\varphi}_{n}-{\varphi}_{n-1}$=0, π/2, π, and 3π/2, respectively, where ${P}_{H}={A}_{s}^{2}(2+\sqrt{2})/8$ and ${P}_{L}={A}_{s}^{2}(2-\sqrt{2})/8$. The 2R amplitude regenerators after the DIs reduce power fluctuations about
P_{H} and suppress P_{L} together with noise to zero. The output pulse power
from the upper (lower) 2R regenerator is amplified to a level giving π
(π/2) phase modulation to the probe pulse in the subsequent all-optical phase
modulators. The absolute phase of the probe pulse leaving the regenerator
ϕ_{out} becomes, therefore, ϕ_{out}=0, π,
3π/2, and π/2 for the input differential phase ${\varphi}_{n}-{\varphi}_{n-1}$=0, π/2, π, and 3π/2, respectively.

Now we discuss the phase noise of the output pulse from the DQPSK regenerator in the case of
input differential phase ${\varphi}_{n}-{\varphi}_{n-1}$=π with ϕ_{out}=3π/2 in more detail.
As will be shown later, for the input pulses corrupted by amplified spontaneous emission (ASE),
the output phase noise becomes largest at this input differential phase. In this case,
P_{DI,1} and P_{DI,2} are

We then assume that the 2R regenerator suppresses the power fluctuations relative to the average power by a factor of r (<1). The output signal powers from the 2R regenerators can be respectively expressed as

We assume that the output pulses from the 2R regenerators, after being amplified, modulate the
phase of the probe pulse by an amount proportional to the pulse power in the subsequent phase
modulators. P_{2R,1} and P_{2R,2} are therefore assumed to be amplified by
proportionality factors k_{1} and k_{2}, respectively, so that
k_{1}P_{H}=π and k_{2}P_{H}=π/2 are
satisfied. The phase fluctuation of the output probe pulse is then given by

Equation (6) indicates that both of the
amplitude and phase fluctuations of the input pulses, ΔA_{n} and
Δϕ_{n}, cause phase fluctuation in the output pulse. This differs
from the case of DPSK regenerators, where the phase fluctuation in the input pulses is not
translated to the phase of the output pulses in the first-order approximation because the DI for
DPSK to OOK signal demodulation is operated at extremes in the interferometer response as
described by output power versus the phase difference [6,17]. Assuming that there is no correlation
between fluctuations of adjacent incoming pulses

_{n}-ϕ

_{n-1}=π can be similarly calculated with the results:

In the analytical calculation, we assume that the 2R amplitude regenerators suppress the
low-power-level (P_{L}) pulses fed from the DIs to zero including their power
fluctuation. Fiber-based 2R regenerators discussed in Section 4 perform such extinction ratio
enhancement.

When the input signal is degraded by circular Gaussian noise such as ASE, which is the
two-dimensional noise having equal variances in amplitude and phase directions, ${\sigma}_{Ain}^{2}/{A}_{s}^{2}={\sigma}_{\varphi in}^{2}$ is satisfied. The phase noise enhancement/reduction factor by the regenerator
then becomes ${\sigma}_{\varphi out}^{2}/{\sigma}_{\varphi in}^{2}$=0, $4{\pi}^{2}{r}^{2}(2-\sqrt{2})$, ${\pi}^{2}{r}^{2}(6-\sqrt{2})$, and ${\pi}^{2}{r}^{2}(2-\sqrt{2})$ for the input differential phase
ϕ_{n}-ϕ_{n-1} = 0, π/2, π, and
3π/2, respectively. Because the largest ${\sigma}_{\varphi out}^{2}/{\sigma}_{\varphi in}^{2}$ is$\underset{\xaf}{{\pi}^{2}{r}^{2}(6-\sqrt{2})}$ at ϕ_{n}-ϕ_{n-1}=π, the
factor r must be smaller than $1/(\pi \sqrt{6-\sqrt{2}})$ for the phase noise of all the output symbols to be smaller than the input
phase noise. The 2R amplitude regenerators after the DIs are therefore required to have the
noise reduction factor 1/r larger than 10log_{10} (π$\sqrt{6-\sqrt{2}}$) = 8.3dB. This estimation assumes an ideal condition that the probe pulses
before the phase modulation have no phase noise. Larger strength of 2R amplitude regenerators
will be required when the original phase noise in the probe pulses is taken into account.

It is noted that Eqs. (7)-(9) hold even when the amplitude and phase fluctuations have non-zero correlations, that is, $\u3008\Delta {A}_{n}\Delta {\varphi}_{n}\u3009\ne 0.$ Eqs. (7)-(9) can therefore be used for estimating the regenerator performance against different types of noise other than ASE such as nonlinear phase noise. The above analysis, however, only discusses the relation between the variances of phase fluctuations of the input and output signals based on the first-order approximation that is satisfied when the fluctuations are small. We have to carry out more fully statistical analyses taking the actual characteristics of the 2R amplitude regenerators into account for the accurate evaluation of noise reduction and of error generation probability by the regenerator.

## 3. Precoding and decoding for undoing data alteration in the regeneration process

As it is noted in Introduction, the DQPSK signal regenerator studied in this paper maps the phase difference between neighboring pulses in the incoming pulse train to the absolute phase of the outgoing pulse with the fluctuation in the phase difference suppressed. This means that the phase data encoded on the signal is converted by the regeneration. Precording or decoding that undoes the conversion is needed when the regenerator is used in practical systems.

First we discuss the precoder or decoder that is needed in DQPSK transmission systems in the
absence of the regenerator. Figure 2(a)
shows a typical DQPSK system where a precoder is located before the modulator.
a_{n} and b_{n} (=0 or 1) are original data and q_{n} and p_{n}
(=0 or 1) are the output data from the precoder, where real and imaginary parts of the complex
amplitude of the light emitted from the source are modulated in proportion to 2q_{n}-1
and 2p_{n}-1, respectively. The receiver consists of two parallel one-symbol DIs and
balanced detectors. The optical phase difference given to signals traveling in the upper and
lower arms in the DIs are set at θ_{DI}=π/4 or
–π/4. The output signal from the balanced detectors are given by
cos(ϕ_{n}-ϕ_{n-1}+θ_{DI}), where
ϕ_{n}-ϕ_{n-1} is the phase difference between consecutive
pulses fed to the receiver. Received data c_{n} and d_{n} take values 1 or 0
according to the output signal from the balanced detector that is positive or negative,
respectively.

Temporal transition of the data q_{n} and p_{n} driving the modulator and the
symbol phase difference ϕ_{n}-ϕ_{n-1} are related by the
left two columns in Table 1(a)
. The symbol phase difference leads to the output data c_{n} and
d_{n} given in the right column in Table 1(a).
For the output data c_{n} and d_{n} to be identical to the original data
a_{n} and b_{n}, respectively, the precoder (precoder 1 shown in Fig. 2(a)) should produce q_{n} and p_{n}
obeying the transition depending on a_{n} and b_{n} as given in Table 1(b). The logic transition rules of the precoder are
then given by [13, 16]

It is noted that a decoder can be used after the detectors as shown in Fig. 2(b). The decoder should have the same logic operation as the precoder.

Now we consider that the DQPSK signal regenerator is inserted in the system as shown in Fig. 3
. The phase difference between consecutive symbols incoming to the regenerator
ϕ_{n}-ϕ_{n-1} and the absolute phase of the output symbol
Θ_{n} are related by the left two columns in Table 2(a)

, as discussed in the previous section. The absolute phase Θ_{n}
is represented by two logical variables e_{n} and f_{n} (=0 or 1) as shown in
the right column in Table 2(a), where

By using the precoder 2, (e_{n}, f_{n}) becomes equal to (q_{n},
p_{n}) so that the same relation between the transition of (q_{n},
p_{n}) and the output data (c_{n}, d_{n}) as shown in Table 1 (a) is satisfied. Note that
other assignments between Θ_{n} and (e_{n}, f_{n}) are
possible with different phase bias in the right-hand side of Eq. (11) such as $\mathrm{exp}\left[i\left({\Theta}_{n}-\pi /4\right)\right]$ instead of $\mathrm{exp}\left[i\left({\Theta}_{n}-3\pi /4\right)\right]$. Then the required transition rules of the precoder 2, (12a) and (12b), are
changed.

When the number of regenerators inserted in the system is more than one, the same number of precoders (precoder 2) should be inserted before the modulator. As in the case of the DQPSK system without regenerators, decoders having the same logical operation as the precoders can be used after the detectors in the receiver instead of using the precoders in the transmitter.

## 4. Numerical simulation

Numerical simulation of the regenerator is performed for short-pulse RZ-DQPSK signals (duty ratio=0.2) at a speed of 160 Gb/s (80 Gsymbol/s). A 2R amplitude regenerator based on power-dependent spectral broadening in a highly nonlinear fiber (HNLF) and off-centered filtering is used for amplitude noise suppression because it can be operated at speeds faster than 100 GSymbol/s without any pattern dependency and with superior performance of noise reduction of both mark and space levels [20]. The all-optical phase modulator after the 2R amplitude regenerators also uses fiber nonlinearity. XPM between the control and probe pulses with different wavelengths that walk through with each other in a nonlinear fiber induces chirp-free phase shifts in the probe pulses [21]. The 2R regenerators are cascaded, by which strong amplitude noise suppression and desirable wavelength shift of the regenerated OOK signals are obtained. The phase-remodulated probe pulses have the same wavelength as the incoming signal to the regenerator. The number of regenerator stages used in the simulation is three instead of two in the case of the DPSK regenerator [6] because stronger noise suppression is needed in the DQPSK signal regenerator. The required noise reduction factors are estimated to be 8.3dB, as was derived in Section 2, and 6.5dB [6] for DQPSK and DPSK regenerations, respectively.

Figure 4(a) shows the three-cascaded fiber-based 2R amplitude regenerators. HLNFs with nonlinear and loss coefficients of γ=12/W/km and α=0.5dB/km, respectively, and dispersion of −0.35ps/nm/km are used. Effects of dispersion slope are ignored in this simulation. The length of each HNLF is 750m. The optical bandpass filters (OBPFs) for spectrum slicing after the HNLFs have a Gaussian shape with a bandwidth of 180 GHz. Noise bandwidth of the ASE and noise figure of the erbium-doped fiber amplifier (EDFA) before each HNLF are 2nm and 6dB, respectively. Figure 4(b) shows wavelength allocation of the input and output signals at each stage of the 2R regeneration. Wavelength shifts at 1st, 2nd, and 3rd stages are +4, −2, and +4nm, respectively, so that the total wavelength shift by the three-stage regeneration is 6nm.

Figure 5
shows examples of eye patterns of input and output OOK pulse trains from the cascaded
regenerator. The input signal is a 256-bit OOK signal emerging from one of the DIs when a DQPSK
signal with an optical signal to noise ratio (OSNR) of 25dB/0.1nm noise bandwidth is launched to
the DI. The initial DQPSK pulses have a duration of 2.5ps. Averaged powers of the signals
entering the nonlinear fibers are optimally chosen and are 255, 85, and 175mW at the 1st, 2nd,
and 3rd stages, respectively. The ratio of the average to the standard deviation of the pulse
energy within the bit duration for mark bits is 6.5, 9.6, 20 and 82 for the input and output
signals from the 1st, 2nd, and 3rd stages, respectively. The overall noise reduction is
10log_{10}(82/6.5)=11dB, which is larger than the required value (8.3dB) estimated in
the previous section. The zero-level noise is also well suppressed. There are several notes
about the eye patters shown in Fig. 5. First, the eye
patters are those if the pulses were detected with electrical bandwidth of 60GHz. The bandwidth
restriction gives eye diagrams displaying energy fluctuation of the pulses more clearly. The
actual width of the pulses fed to the phase modulators is almost the same as the input pulses
~2.5ps. Second, the space-level noise reduction is not uniform in the multiple stages. The
space-level noise is strongly suppressed in the 1st and 3rd stages, but not in the 2nd stage. In
the wavelength arrangement shown in Fig. 4, the 2nd stage
regeneration having a smaller wavelength offset principally plays a role of mark-level noise
reduction, but not of space-level noise reduction.

In the nonlinear fiber for all-optical phase modulation, a complete collision between the control and probe pulses induces an approximate phase shift of

to the probe pulse through the effect of XPM, where U, D, and Δλ are the control pulse energy, dispersion of the fiber, and wavelength difference between the control and probe pulses, respectively. Equation (13) is derived by integrating the equationover the length of the fiber, where E_{c}, E

_{s}, δ, and τ are the complex amplitudes of the control and probe pulses, difference in inverse group velocities of control and probe pulses, and the time coordinate moving at the group velocity of the probe pulse. For the complete collision, the integration interval can be taken as ($-\infty ,\text{\hspace{0.05em}}\infty $). In the numerical simulation, HNLFs with γ=12/W/km, α=0.5dB/km, and length of 1.5km are used. In order to suppress excessive broadening of the control pulses while to acquire complete walk through between the control and probe pulses, the fiber dispersion is set at an anomalous value of 1.4ps/nm/km. The wavelength separation between the control and probe signals is Δλ =6nm so that the estimated walk off time in the HNLF is 12.5ps, which is equal to the symbol period. Equation (13) predicts that the required control pulse energy is 1.1pJ for phase shift of π, corresponding to an average power of 44mW for 80Gbit/s OOK pulse trains. In numerical simulation, where fiber loss is considered, the optimum control pulse power fed to the first HNLF is 50mW, somewhat larger than the predicted value. The control pulse power to the second HNLF for phase modulation of π/2 is 25mW. The peak power of each control pulse fed to these HNLFs is higher than that of the fundamental soliton of 2.5ps width in the HNLFs. The control pulses therefore behave as higher-order solitons exhibiting initial pulse narrowing in the fiber. Although the pulse shape varies in the fiber, the induced phase shift given to the probe pulses does not have large chirp because of the walk-through nature in the interaction between the control and probe pulses. ASE from the EDFAs for the control-pulse amplification is introduced in the simulation although its amount is small because the output power from the last stage of the 2R amplitude regenerators is ~20mW, which requires power amplification by a factor at most 2 or 3. Timing of the control and probe pulses are adjusted so that the pulse trains are precisely time interleaved at the entrance of each HNLF. An ideal clock extraction and noise-free clock pulse generation is assumed with the probe pulse width of 2.5ps.

Figure 6 shows constellation diagrams of input and output signals of the DQPSK regenerator, where the input signal is corrupted by ASE noise. The input signal has OSNR of (a) 26dB/0.1nm, (c) 24dB/0.1nm, and (e) 22dB/0.1nm. The number of symbols used in the simulation is 1024. When the noise is small, the regenerator works well and scattering of signal points in the constellation due to noise is clearly diminished as shown in Fig. 6(b). When the input noise becomes larger to 22dB/0.1nm, for example, the demodulated OOK signals exiting the DIs have larger amplitude noise and unsuppressed noise remains at the output of the 2R amplitude regenerators. This leads to large phase noise in the output signal as shown in Fig. 6(f). At the input OSNR of 24dB/0.1nm, standard deviations of the phase fluctuations of the output pulses are 4.5, 3.4, 2.4, and 1.6 degrees for the pulses that acquire phase modulation of 3π/2, π, π/2, and 0, respectively, at the phase modulator. They are all smaller than the phase fluctuation of the input pulses, 6.3 degrees. Figure 7(a) shows the waveform of the DQPSK signal at the entrance of the regenerator, which is a part of the 1024 pulse train used in the simulation. Figure 7(b) and (c) are the waveforms at the output of one of the DIs and at the output of one of the cascaded 2R amplitude regenerators, respectively, showing good noise suppression and enhancement of extinction by the amplitude regenerator. Figure 7(d) shows the output probe signal after phase remodulation and removal of control pulses. It is shown in this figure that not only the phase noise but also the amplitude noise of the output signal is strongly suppressed if the quality of the clock pulses is high. The suppressed amplitude noise greatly reduces generation of nonlinear phase noise in subsequent transmission after the regenerator [10].

Figure 8
shows the standard deviation of phase fluctuations of the input and output pulses versus
OSNR of the input DQPSK signal. A theoretical estimation of the input phase fluctuation given by ${\sigma}_{\varphi \text{\hspace{0.17em}}in}^{2}={r}_{duty}\text{\hspace{0.17em}}B/(2\cdot OSNR\cdot 12.5GHz)$ is plotted by a dash-dotted curve, where r_{duty} and B are the duty
ratio of the pulse (=0.2) and the noise bandwidth (250GHz), respectively. This expression is
derived from an expression of phase variance of a constant-envelope signal fluctuated by ASE ${\sigma}_{\varphi \text{\hspace{0.17em}}}^{2}=N\text{\hspace{0.17em}}B/(2{P}_{s})$ [22] together with relations $N={P}_{N,0.1nm}/12.5GHz$, ${P}_{s}={P}_{ave}/{r}_{duty}$, and OSNR=${P}_{ave}/{P}_{N,0.1nm}$, where N, P_{s}, P_{N,0.1nm}, and P_{ave} are the
noise power density, signal (peak) power, noise power within 0.1nm bandwidth, and averaged
signal power, respectively. The dotted curve is the standard deviation of the input phase
fluctuation numerically obtained by the simulation using 1024 pulses. The agreement between the
two curves is reasonable considering the approximate nature of the theoretical estimation.

Solid curves in Fig. 8 are the standard deviations of phase fluctuations of the output pulses. The curves are plotted separately for the pulses that undergo different phase modulations (3π/2, π, π/2, and 0) in the regenerator. In the simulation, data patterns and noise realization are varied randomly as the OSNR is varied. Figure 8 shows that the output phase noise is made smaller than the input phase noise for input OSNR larger than ~23dB/0.1nm. For input OSNR smaller than the value, erroneous discrimination between low an high power levels takes place in the 2R regenerators so that the phase fluctuation in the output signal can be significantly larger than that of the input signal.

In high-speed long-distance transmission, PSK signals are impaired not by pure ASE but rather
by the nonlinear phase noise (NLPN). We then examine the effectiveness of the DQPSK regenerator
to such a signal degraded by the NLPN. Here we simulate the NLPN, caused by the self-phase
modulation (SPM), simply by applying power dependent phase rotation to the signal, that is, the
degraded signal is generated by multiplying the ASE-added signal q(t) by a complex factor $\mathrm{exp}\text{\hspace{0.05em}}(i{k}_{NLP}{\left|q(t)\right|}^{2})$, where the nonlinear phase rotation coefficient k_{NLP} determines the
strength of the NLPN. We again use a train of short pulses with duration of 2.5ps modulated in
DQPSK format as the signal. The bit rate is 160Gb/s (80Gsymbol/s). In this simulation, the
average input signal power and OSNR are fixed at 1mW and 26dB/0.1nm, respectively. Figure 9(a)
is a constellation of the NLPN-impaired input DQPSK signal with
k_{NLP}=0.1rad/mW. The regenerator successfully removes the noise as found in the output
signal constellation shown in Fig. 9(b). For larger
k_{NLP}, however, the regeneration fails as shown in Figs. 9(c) and (d). Figure 10
shows standard deviations σ_{ϕout} and
σ_{ϕin} of the input and output phase noise versus
k_{NLP}. Data patterns and noise realization are randomly varied as k_{NLP} is
varied similarly to the simulation in Fig. 8. The erratic
fluctuation of σ_{ϕout} for large k_{NLP} is due to the
erroneous discrimination of the two power levels by the 2R amplitude regenerators. For the
NLPN-degraded signals, the lower-power-level pulses exiting the DIs have larger power
fluctuations as compared to the case of ASE-degraded signals. The optimum average signal input
power to the first stage of the 2R amplitude regenerator is a little smaller than that in the
case of regeneration of the ASE-degraded signals (210 versus 255mW). This indicates that precise
optimization of the regenerator performance depends on the noise statistics of the input
signal.

## 5. Discussion and conclusion

A scheme of an all-optical DQPSK signal regeneration is proposed and analyzed. In the regenerator, the phase noise of the incoming signal is converted to amplitude noise by one-symbol delay interferometers and is suppressed by 2R amplitude regenerators. The amplitude-stabilized OOK pulses are used as control pulses in modulating phase of clocked probe pulses. The alteration of the phase data encoded on the signal in the process of regeneration can be undone by inserting an encoder or a decoder in the transmitter or in the receiver. The encoder/decoder has to be adapted to the number of regenerators located in the signal path.

The regenerator operation is numerically demonstrated where cascaded fiber-based 2R amplitude regenerators using SPM and also fiber-based all-optical phase modulators using XPM are employed. It was shown that short-pulse RZ DQPSK signal at 160 Gbit/s (80 Gsymbol/s) can be successfully regenerated unless the amount of noise is so large that it leads to decision errors by the 2R amplitude regenerators. Although the number of stages of the 2R amplitude regenerators is three in the numerical simulation, it is expected to be reduced to two or even one if further optimization of the regenerator is attempted [23,24]. Use of bidirectional configuration will also reduce the number of required fiber spools [25]. Other regeneration schemes such as cascaded four-wave mixing in fibers may also be used [26]

Considering its use in real systems, the regenerator has a number of problems to be solved. Because of the long lengths of the fibers, it is difficult to achieve correct time alignment of control and probe pulses interacting in the HNLFs for the all-optical phase modulation. Possible solutions to this issue will be to use the same HNLFs bidirectionally for the two 2R amplitude regeneration paths, to extract the clock after the 2R amplitude regeneration, and to perform the XPM-based phase modulation in a single HNLF by interacting the probe and the two control pulse streams with suitable wavelength allocation. In either case, much shorter fibers having high nonlinearity are strongly desired. Bismuth and chalcogenide glass fibers are candidates for such high nonlinearity fibers [24,27]. When we use XPM for the all-optical phase modulation, a major issue is its polarization sensitivity. This will be avoided if we use a circular birefringence twisted fibers with which XPM operation independent of control signal polarization can be obtained [28]. For successful operation of the regenerator, we need a high-speed, low timing-jitter, and low phase-noise clock pulse source [29].

Because the regenerator is a single-channel device, a number of regenerators will be used in parallel in WDM transmission environments so that integration of the regenerators together with devices for multiplexing and demultiplexing is highly desired. Realization of the regenerator scheme in semiconductor optical amplifier (SOA)-based [4,5] or silicon nanowaveguide-based [30] systems is a subject of further study, which will include attempts to enhance operation speeds in these systems.

## Acknowledgements

This work was supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (B)20360171.

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