Noise Suppression Mechanisms in Regenerators Based on XGC in an SOA With Subsequent Optical...

IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 18, NO. 2, MARCH/APRIL 2012 935

Noise Suppression Mechanisms in RegeneratorsBased on XGC in an SOA With Subsequent

Optical FilteringWei Hong, Minghao Li, Xinliang Zhang, Junqiang Sun, and Dexiu Huang

Abstract—A theoretical model is presented to analyze the noisesuppression characteristics in cross-gain-compression (XGC)-based regenerators, where an optical bandpass filter (BPF) is cas-caded after a semiconductor optical amplifier (SOA). Analyticalexpressions of variance are obtained for in-phase and quadraturenoises after SOA and after filter. The result shows that the physi-cal mechanism behind XGC-based noise suppression is that gainchange in the SOA is out of phase with the input power fluctu-ation, which leads to a negative cross-correlation term in the ex-pression of in-phase noise variance at SOA output. On the otherhand, nonlinear phase noise introduced by refractive index changein the SOA will contributes to quadrature noise, which is propor-tional to the square of the linewidth enhancement factor ∼α2

N . Itis proposed that a slightly blue-shifted optical BPF can convertphase/quadrature noise-related frequency chirp into amplitudefluctuation that can cancel a part of the amplitude fluctuation afterSOA, thus lead to further reduction of in-phase noise. This mech-anism can be used to alleviate the negative effect of SOA-inducednonlinear phase noise and enhance the overall noise suppressionperformance of XGC-based regenerators, which is experimentallyand numerically demonstrated at 10 Gbit/s.

Index Terms—All-optical regeneration, cross-gain compression(XGC), nonlinear phase noise, optical filter, semiconductor opticalamplifier (SOA).

I. INTRODUCTION

CROSS-GAIN compression (XGC) utilizes the cross-modulation effect of two synchronous inverted-in-logic

signals in a saturated semiconductor optical amplifier (SOA)to suppress amplitude noise in both signals [1]. This mech-anism has the advantage of stability, simplicity, and intrinsicimmunity to pattern effects. Noise suppression of two inverted-in-logic signals with different wavelengths [1], [2], differentpropagation directions, [3] or orthogonal polarizations [2] hasbeen reported. However, theoretical investigation has not beenseen in literatures till now.

Manuscript received November 14, 2010; revised March 14, 2011; acceptedApril 1, 2011. Date of publication June 20, 2011; date of current version March2, 2012. This work was supported by National Natural Science Foundationof China under Grant 60707005 and the Hi-Tech Research and DevelopmentProgram of China under Grant 2007AA03Z414.

The authors are with the Wuhan National Laboratory for Optoelectronics andthe School of Optoelectronics Science and Engineering, Huazhong Universityof Science and Technology, Wuhan 430074, China (e-mail: [email protected] (corresponding author Wei Hong); [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2011.2143697

It is known that one of the typical applications of optical fil-ters in all-optical signal processing is to cascade after nonlinearelement, such as SOA or highly nonlinear fiber, in order to selectthe desired part of the cross/self-phase modulation-broadenedspectrum [4]. Detuned optical filtering can successfully compen-sate the pattern effects induced by slow recovery of SOA carrierdensity [5], leading to ultrafast wavelength conversion [6], orextract the chirp dynamics to clamp the amplitude of return-to-zero (RZ) signals [7]. In principle, with appropriate opticalfiltering after XGC interaction in an SOA, it is also possible tosuppress amplitude noise further.

In this paper, noise suppression mechanisms in XGC-basedregenerators with an SOA and a cascaded optical bandpass fil-ter (BPF) are investigated theoretically and experimentally. Asmall-signal theoretical model of SOA [4], [8], [9] is used tocharacterize noise suppression in the SOA; while discrete convo-lution [10] and a spectrum slice technique are used to character-ize noise suppression due to subsequent filtering. The analyticalresults are then used to evaluate noise suppression performanceof XGC-based regenerators. It is found that improved noisesuppression can be obtained with a slightly blue-shifted opticalBPF. We also experimentally demonstrate filtering-enhancednoise suppression of polarization-shift-keyed (PolSK) signalsusing XGC-based regenerators. After that, time-domain simula-tion based on a wideband dynamic model of SOA is performed,which agrees with the theoretical model and experimental re-sults quite well. The results provide physical insight into themechanism of XGC effect and the enhanced noise suppressioneffect by using appropriate filtering after XGC interaction in theSOA.

II. NOISE SUPPRESSION FORMULIZATION

For the convenience of description, the schematic diagramof XGC-based regeneration enhanced by subsequent optical fil-tering is given in Fig. 1. The two logically inverted signals atdifferent wavelengths or orthogonal polarization states are syn-chronized and injected into an SOA simultaneously. That is tosay, a mark (or a space) and a space (or a mark) propagate atthe same time in the SOA. The total power of the two signalsdrives SOA into deep saturation and leads to suppression of am-plitude fluctuation in both signals at the output of SOA. Afterthat, a tunable BPF (TBPF) is cascaded after the SOA, whichcan be detuned with respect to the central frequency of the sig-nal. In this section, the noise suppression mechanism will be

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936 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 18, NO. 2, MARCH/APRIL 2012

Fig. 1. Schematic diagram of noise suppression in XGC-based regeneratorswith SOA and subsequent optical filtering.

characterized by formulizing noise suppression in the SOA andthe filter, respectively.

A. Input Signal and Noise

At the input of the SOA, the two logically inverted signalscontaining additive noise can be expressed as

Ein,1/2(t) = ⇀e 1/2 [A1/2(t) + εin,1/2(t)]

× exp[j(ω1/2t + ϕ1/2(t))]. (1)

For typical XGC applications, the central frequencies of the twosignal ω1 and ω2 will be different or polarization unit vectors⇀e 1 and ⇀

e 2 will be in orthogonal directions. εin,1/2(t) representsthe complex slowly varying envelope of additive noise fieldover signals 1 and 2. In lightwave systems, this additive noisemainly contains amplified spontaneous emission (ASE) noise,and εin,1/2(t) is usually modeled by bandwidth limited zero-mean complex Gaussian random process with power spectraldensity symmetric with respect to ω1/2 . A1/2(t) and ϕ1/2(t)are the amplitude and the phase of the noise-free signal forsignals 1 and 2, respectively. For simplicity, we assume thatthe amplitude and phase of the noise-free signal are constantamong all marks or spaces. The constant phase can be neglectedas it plays negligible roles in both XGC interaction and opti-cal filtering. In the following analytical formulizations, we willconcentrate on XGC interaction between two nonreturn-to-zero(NRZ) formatted signals and consider those bits that signal 1contains mark while signal 2 contains space at the same time.In these bits, the optical fields of signals 1 and 2 at the input ofthe SOA can be expressed as

Ein,1/2(t) = ⇀e 1/2 [Am/s + εin,1/2(t)] exp

(jω1/2t

)(2)

where m and s denoting mark and space, respectively. For thosebits that signal 1 contains space while signal 2 contains mark,the whole deduction can still apply as long as the subscript mand s changes alternatively.

In deducing signal and noise power at the input of SOA, wefirst neglect the components related to Δω for XGC interactionbetween different wavelengths, as Δω is always much largerthan the carrier response bandwidth. These components dimin-ish for XGC interaction between orthogonal polarizations. Wealso neglect the noise–noise beat term as signal–noise beat termis much larger than noise–noise beat term for typical ASE lim-ited systems [8], [9]. If we assume that the optical power in aspace is much smaller than that in a mark, signal–noise beat ina space can be further neglected. Thus, the total optical power

at the input of the SOA can be written as

Pin(t) ≈ A2m + A2

s + δPin(t) = Pin + δPin(t)

δPin(t) = 2Am Re {εin,1(t)} (3)

where Re{·} represents the real part of a complex quantity. In(3), Pin is the averaged total power which represents a staticterm and δPin(t) is the time-varying power fluctuation whichrepresents a dynamic term inducing carrier density modulationaround the steady-state value in the SOA.

B. Frequency Response of SOA

To investigate the noise suppression characteristics in theSOA, a small-signal analytical model of SOA is adopted in thispaper. Although intraband carrier process, the internal loss ofactive waveguide, and the ASE noise are all neglected in thissimplified model, it is yet capable of capturing the main featuresof gain saturation dynamics [4] and has been successfully usedto study the noise characteristics in an SOA [8], [9]. In thissimplified model, gain dynamics of SOA is described by thefollowing equation [4]:

dh

dt=

g0 − h

τc−

∑

i=1,2

(eh − 1)Pi

P isatτc

(4)

where h(t) = lnG(t) is the logarithm of gain. Pi(i = 1, 2)represents the input power of signals 1 and 2, which equalsA2

m + δPin(t) and A2s , respectively, in our case. τc is the carrier

lifetime of SOA. g0 = aΓ(Iτc/qV ) − Ntr)L is the unsaturatedgain, where a is the differential gain, Γ is the confinement factor,I is the injection current, q is the electron charge, V is the volumeof the active region, L is the length of active region, and Ntr isthe transparent carrier density. P i

sat = hωiA/Γaτc (i = 1, 2) isthe saturation power for signals 1 and 2, where h is the Planckconstant and A is the cross-sectional area of the active region.

In the small-signal model of SOA, h(t) can be expressedas h(t) = h + δh(t) in correspondence with the input powerPin(t) = Pin + δPin(t), where h can be obtained from steady-state solution of (4). By subtracting the steady-state part from(4), the relationship between the small-signal gain modulationδh(t) and input power fluctuation δPin(t) can be obtained, fromwhich it is found that the SOA can be treated as a linear differ-ential system with input excitation of δPin(t), output responseof δh(t), and frequency response of [8], [9]

H(ω) = − (eh − 1)τe

P 1satτc

· 11 + jωτe

(5)

where τe is the effective carrier lifetime defined by τ−1e =

τ−1c [1 +

∑i=1,2 ehPi/P

isat ].

It is worth noting that, although the small-signal model isobtained by linearization of (4), it can still handle nonlineargain and phase dynamics in the SOA. Because the linearizationis only valid near one operation point of SOA with δPin(t) �Pin(t), the relationship of h(t) and Pin (t) in the entire operationrange is still nonlinear.

HONG et al.: NOISE SUPPRESSION MECHANISMS IN REGENERATORS BASED ON XGC IN AN SOA WITH SUBSEQUENT OPTICAL FILTERING 937

C. Noise Suppression in the SOA

The output field of SOA can be represented by

Eout,1/2(t) = ⇀e 1/2 [Am/s + εin,1/2(t)] exp

(h + δh(t)

2

)

× exp[j(ω1/2t − φ − δφ(t)] (6)

where h and φ are the constant gain and phase change due toPin , while δh(t) and δφ(t) are the gain and phase fluctuationdue to δPin(t). Using the linewidth enhancement factor αN ,phase fluctuation can be expressed by gain fluctuation as δφ =−(1/2)αN δh [9]. Noting that |δh| � 1 and |δφ| � 1, (6) canbe approximated to a form with in-phase and quadrature noisecomponents:

Eout,1/2(t) = ⇀e 1/2

{Am/se

h/2 + εout,1/2(t)}

exp(jω1/2t

)

εout,1/2(t) =12Am/se

h/2 · δh(t) + eh/2 · Re{εin,1/2(t)

}

︸︷︷︸in-phase

+ j

⎡

⎢⎢⎣

12αN Am/se

h/2 · δh(t) + eh/2 · Im{εin,1/2(t)

}

︸︷︷︸quadrature

⎤

⎥⎥⎦ (7)

where εout,1/2(t) is the complex slowly varying envelope ofoutput noise field after SOA over signals 1 and 2, respectively.The constant phase change of φ is omitted just for simplicity.We can derive the autocorrelation of in-phase and quadraturecomponents of output noise, respectively, which in turn can bederived by calculating the autocorrelation of each individualterm and cross correlation between each two terms. The vari-ance of in-phase and quadrature output noises can be obtainedby evaluating the corresponding autocorrelation at τ = 0 asfollows:

σ2in-phase,1 =

14A2

m ehσ2δh +

12ehσ2

ε i n , 1+

12ehRδP i n ,δh(0)

(8a)

σ2quadrature,1 =

14α2

N A2m ehσ2

δh +12ehσ2

ε i n , 1(8b)

σ2in-phase,2 =

14A2

s ehσ2

δh +12ehσ2

ε i n , 2(8c)

σ2quadrature,2 =

14α2

N A2s e

hσ2δh +

12ehσ2

ε i n , 2(8d)

where σ2ε in , 1 / 2

is the variance of εin,1/2(t), σ2δh is the variance

of δh(t), and RδP in ,δh(0) represents the cross correlation ofδPin(t) and δh(t) at τ = 0. The expression for σ2

δh , σ2ε in , 1 / 2

,

and RδP in ,δh(0) in (8) can be further obtained by using thetheory of spectral analysis of linear systems with random inputand Wiener–Khinchin relation [11] and expressed in a form ofinverse Fourier transform of the corresponding power spectraldensity at τ = 0 as follows:

σ2ε in , 1 / 2

=12π

∫ +∞

−∞Sε i n , 1 / 2 (ω)dω (9a)

Fig. 2. Contribution of different noise terms in (8a) and (8b) to the total noisein a mark at SOA output.

σ2δh =

A2m

π

∫ −∞

+∞Sε in , 1 (ω) |H(ω)|2 dω (9b)

RδP in ,δh(0) =A2

m

π

∫ +∞

−∞Sε in , 1 (ω)Re {H(ω)}dω. (9c)

Detailed deduction of (8) and (9) can be found in Appendix A.Equations (8a)–(8d) reveal that the in-phase noise power in

an output mark contributes from a induced gain variation term,a amplified in-phase noise term, and a cross-correlation term,while the cross-correlation term is absent for the quadraturenoise in an output mark, as well as both in-phase and quadraturenoises in an output space. The reason can be attributed to theassumption that only the in-phase noise in a mark contributes togain fluctuation (through signal–noise beat). The induced phasevariation term of (1/4)α2

N A2m/se

hσ2δh in (8b) and (8d) repre-

sents the so-called nonlinear phase noise, which is proportionalto ∼ α2

N .By substituting the expression of H(ω) [see (5)] into (9c),

it can be easily found that the cross-correlation term in (8a) isnegative. The physical basis lies in the fact that the gain fluctua-tion in the SOA is out of phase with the input power fluctuation.Fig. 2 plots the contribution from different terms in (8a) and (8b)to the total noise in a mark at SOA output. Without special notifi-cations, the parameters used in the analytical model are listed asfollows. Differential gain a = 2 × 10−20 m−1 , cross-sectionalarea A = 0.24μm2 , SOA length L = 500μm, mode confine-ment factor Γ = 0.35, linewidth enhancement factor αN = 4,carrier lifetime τs = 200 ps, group refractive index ng = 3.5,transparent carrier density Ntr = 5.0 × 1023 m−3 , and injec-tion current I = 250mA. To give an example but not affectingthe physical understanding, we consider the case of XGC be-tween orthogonal polarizations in a polarization-independentSOA, with signals 1 and 2 both centered at 1550-nm with opti-cal signal-to-noise ratio (OSNR) of 15 dB defined within 0.1 nm.The power spectral density of input noise Sε in , 1 / 2 (ω) is assumedto be uniform within bandwidth of 2πBn and Bn = 100GHz

Sε in , 1 / 2 (ω) ={

S0,1/2 |ω − ω0 | ≤ πBn

0 otherwise. (10)

Fig. 2 shows that major contribution to the in-phase noisein a mark at SOA output is the amplified noise term and the


cross-correlation term, while the induced gain variation termis about one order of magnitude smaller. The negative cross-correlation term eventually leads to reduction of the in-phasenoise in a mark after propagation in the SOA. It is worth notingthat the induced phase variation term makes a significant contri-bution to the output quadrature noise, which is α2

N times largerthan the induced gain variation term. So, one inherent disadvan-tage of XGC-based noise suppression is that it suppresses theamplitude noise while adds nonlinear phase noise at the sametime, as long as αN is nonzero.

It is known that quadrature noise contributes to noise–noisebeat in a direct detection receiver, while in-phase noise con-tributes to both signal–noise beat and noise–noise beat. Al-though signal–noise beat is dominant over noise–noise beat innormal cases, noise–noise beat can be comparable to signal–noise beat if the quadrature noise is sufficiently large. In XGC-based regenerators, large amount of nonlinear phase noise willbe generated, which will directly add to quadrature noise andsignificantly increase noise–noise beat in the receiver. As a re-sult, signal quality improvement due to reduction of in-phasenoise might be partly canceled by the increase in nonlinearphase noise. As a result, the overall noise suppression perfor-mance will be abated comparing to the ideal case where nononlinear phase noise is generated in the SOA.

The previous discussion can only be applied to XGC-based re-generation of ON–OFF-keyed (OOK), optical frequency-shift-keyed, and PolSK signals, where the signals inherently containintensity-modulated components and can be directly detected.The situation will be different with XGC-based differentialphase-shift-keyed (DPSK) regenerators. In DPSK receivers, thereceived signal is usually demodulated to intensity-modulatedsignal using a 1-bit delayed interferometer before being detectedin a photodiode. During this process, quadrature noise will bepartly converted to in-phase noise, which will contribute to bothsignal–noise beat and noise–noise beat. Consequently, nonlinearphase noise will have more significant influence on XGC-basedDPSK regenerators.

It is reported that signal intensity, which is correlated withnonlinear phase noise, can be used to compensate nonlinearphase noise generated due to the Gordon–Mollenauer effect intransmission fiber [12], [13]. Alternatively, optical phase con-jugation can also be utilized [14]. In principle, similar methodscan also be used in XGC-based regenerators to suppress nonlin-ear phase noise generated in the SOA. However, the complexityof the regenerator will also be increased at the same time. Tocounteract the possible performance degradation due to nonlin-ear phase noise, the ultimate method is using an SOA with zeroαN , e.g., a quantum dot (QD) SOA [15]. However, if SOA withnonzero αN is used, further reducing in-phase noise helps torestore the overall noise suppression performance, which is themajor concern of this paper.

In a general case, noise in a space will also contributes to totalgain variation but at a much lower level than the noise in a mark.As a result, a much smaller negative cross-correlation term alsoexists for the output in-phase noise in a space. This might be thereason why XGC-mechanism is not so effective in suppressingnoise in a space, which has been experimentally demonstrated in

Fig. 3. Noise compression factor as a function of average input power for(a) different noise bandwidths and (b) carrier lifetimes of the SOA.

[16]. So, we will concentrate on the in-phase noise suppressionin a mark in the following quantitative investigation.

To give a quantitative evaluation of noise suppression perfor-mance in XGC-based regenerators, we define a noise compres-sion factor for the in-phase noise as follows:

Nc =(1

2 σ2ε in , 1 / 2

)/Pin

σ2in-phase,1/2/Pout

=12 ehσ2

ε in , 1 / 2

σ2in-phase,1/2

(11)

where (1/2)σ2ε in , 1 / 2

is the power of in-phase noise at the input of

the SOA. Pout is the average power at SOA output. Equation (11)shows that in-phase noise compression can be represented bythe power ratio of the amplified in-phase noise and the total out-put in-phase noise. Fig. 3(a) plots the noise compression factoras a function of average input power for different noise band-widths, while the overall input noise power is kept unchanged.It is found that noise with a narrower spectral bandwidth canbe more effectively suppressed, which is in accordance with thecommon understanding of high-pass filter behavior of a satu-rated SOA [17]. Fig. 3(b) plots the results for different carrierlifetimes of the SOA. It seems that the SOA with shorter carrierlifetime is preferred for XGC-based noise suppression. How-ever, with shorter carrier lifetime, carrier density in the SOAcan be recovered faster. As a result, larger carrier density fluctu-ations will be introduced with the same input power fluctuations,which in turn results in larger nonlinear phase noise [8]. Conse-quently, tradeoff should be made between the two aspects.


D. Noise Suppression by an Optical BPF

In deducing noise variance after optical filtering, we assumethat the filter is a Gaussian-type optical BPF with impulse re-sponse of

hf (t) =B0√2π

exp[− 1

2(B0t)2

]exp(jωf t) (12)

where ωf is the central frequency of the BPF, B0 =πB3 dB/

√ln 2, and B3 dB is the 3-dB bandwidth of the BPF.

It is worth noting that an optical BPF is usually cascade afterSOA with nominal function of selecting specific wavelengthor limiting the noise band in XGC-based regenerators [1]–[3];however, additional effect of the filter has not been mentioned.Bandwidth of the BPF used in these cases is always large enoughto ensure signal integrality. As a result, to focus on the behaviorof the noise, we can assume that the signal field will only be at-tenuated by exp{−(1/2)[(ωf − ω1/2)/B0 ]2} due to the centralfrequency mismatch of the input signal with respect to the filter.That is to say, we will not include the effect of filter on signalitself in the analytical investigations. The filtered noise field canbe obtained by the convolution of the input noise field (outputnoise field of the SOA) and the impulse response of the filter

εfiltered,1/2(t)ejω1 / 2 t

=∫ +∞

−∞

⎡

⎢⎢⎣

12Am/se

h/2 · δh(τ) + eh/2 · Re{εin,1/2(τ)

}


+ j12αN Am/se

h/2 · δh(τ) + jeh/2 · Im{εin,1/2(τ)

}


⎤

⎥⎥⎦

× ejω1 / 2 τ hf (t − τ)dτ (13)

where εfiltered,1/2(t) is the complex slowly varying envelopeof noise field after filtering. In (13), the noise is consideredto be centered at the center wavelength of the signal. How-ever, as the spectral distribution of noise might be equal toor even broader than the bandwidth of the filter, it is nec-essary to include spectral distribution of the noise into our

analysis. We can further assume that the noise contributes from2N + 1 spectrum slices. In this case, δh(τ) and εin,1/2(τ) in

(13) can be substituted by∑N

n=−N δh(n)(τ) exp(jωnτ) and∑N

n=−N ε(n)in,1/2(τ) exp(jωnτ), respectively, where ωn = nΔω

(n = 0, ±1, ±2, . . . , ±N ) and Δω is the width of each spec-trum slice. Thus, (13) can be written as (14) shown at the bottomof this page.

Using discrete convolution as in [10], the integral in (14)can be analytically obtained for a Gaussian filter with im-pulse response specified in (12), which is shown by (15) atthe bottom of the next page. In (15) we use the approximationej (ω1 / 2 +nΔω )t ≈ ejω1 / 2 t as nΔω � ω1/2 (noise bandwidth ismuch smaller than the center frequency of the signal) anddenote k1(ωn ) = exp{−(1/2)[(ωf − ω1/2 − ωn )/B0 ]2} andk2(ωn ) = (ωf − ω1/2 − ωn )/B2

0 . Equation (15) shows that thequadrature noise before optical filter (after SOA) will be con-verted to in-phase noise after optical filter.

The variance of in-phase and quadrature noise after filteringcan also be obtained by evaluating the corresponding autocor-relation at τ = 0, respectively. In deducing these expressions, itis noted that:

1) in ASE-limited systems, additive noise contributing fromdifferent spectral components can be considered statisti-cally independent and unrelated. As SOA and filter areboth linear systems, different spectral components of allthe involved random noise can be considered statisticallyindependent and unrelated;

2) the in-phase and quadrature components of Gaussian noisecan be considered statistically independent and unrelated;

3) according to (3), δh(n)(t) and its derivative δh(n)′(t) isunrelated to Im{ε(n)

in,1/2(t)};

4) as δh(n)(t) is real, its power spectral density is an evenfunction with respect to ω. Using spectral analysis oflinear systems with random input, the cross correlationof δh(n)(t) and its derivative δh(n)′(t) equals zero atτ = 0 [11];

5) as Re/Im{ε(n)in,1/2(t)} is real, it is the same as 4) that the

cross correlation of Re/Im{ε(n)in,1/2(t)} and its derivative

Re/Im{ε(n)in,1/2

′(t)} equals zero at τ = 0;

6) as SOA is a linear system, the cross correlation of δh(n)(t)and Re/Im{ε(n)

in,1/2′(t)} equals zero at τ = 0.

εfiltered,1/2(t)ejω1 / 2 t =∫ +∞

−∞eh/2

⎡

⎢⎢⎢⎢⎣

N∑

n=−N

[12Am/sδh

(n)(τ) + Re{

ε(n)in,1/2(τ)

}]


+ jeh/2N∑

n=−N

[12αN Am/sδh

(n)(τ) + jeh/2 · Im{

ε(n)in,1/2(τ)

}]


⎤

⎥⎥⎥⎥⎦

× exp[j(ω1/2 + ωn )τ ]hf (t − τ)dτ. (14)


Consequently, variance of the in-phase and quadrature noisein a mark after filtering can be deduced respectively as

σ2filtered−inphase,1 =

14A2

m ehN∑

n=−N

R(n)δh (0)k2

1 (ωn )

+ ehN∑

n=−N

R(n)εr , 1

(0)k21 (ωn )

+ Am ehN∑

n=−N

R(n,n)εr , 1 ,δh(0)k2

1 (ωn )

+14A2

m ehα2N

N∑

n=−N

R(n)δh ′ (0)

× k21 (ωn )k2

2 (ωn )

+ ehN∑

n=−N

R(n)ε ′

i , 1(0)k2

1 (ωn )k22 (ωn )

+ Am αN ehN∑

n=−N

R(n,n)εr , 1 ,δh ′(0)

× k21 (ωn )k2(ωn ) (16)

σ2filtered-quad,1 =

14A2

m ehα2N

N∑

n=−N

R(n)δh (0)k2

1 (ωn )

+ ehN∑

n=−N

R(n)εi , 1

(0)k21 (ωn )

+14A2

m ehN∑

n=−N

R(n)δh ′ (0)k2

1 (ωn )k22 (ωn )

+ ehN∑

n=−N

R(n)ε ′

r , 1(0)k2

1 (ωn )k22 (ωn )

+ Am ehN∑

n=−N

R(n,n)ε ′

r , 1 ,δh ′(0)k21 (ωn )k2

2 (ωn )

(17)

where R(n)X (τ) denotes the autocorrelation of X(n)(t) and

R(n,n)X Y (τ) denotes the cross correlation between X(n)(t) and

Y (n)(t). For convenience, we use subscripts εr,1 and εi,1 todenote Re {εin,1(t)} and Im {εin,1(t)} , respectively. For a co-propagation space, the cross-correlation terms in (16) and (17)will not exist, which results from the assumption that gain fluc-tuation is induced by power fluctuations in a mark.

According to the Wiener–Khinchin relation [11], the autocor-relations and cross correlations in (16) and (17) can be givenby the inverse Fourier transform of the corresponding powerspectral density in each spectrum slice. That is,

R(n)X (τ) =

[12π

∫ +∞

−∞SX (ω)rect

(ω − ωn

Δω

)ejωτ dω

](18)

R(n,n)X Y (τ) =

[12π

∫ +∞

−∞SX,Y (ω)rect

(ω − ωn

Δω

)ejωτ dω

]

(19)

where rect (ω/Δω) ={

1 −Δω/2 < ω < Δω/20 otherwise is a unit

rectangular function. SX (ω) is the power spectral density ofX(t), and SX Y (ω) is the cross power spectral density of X(t) andY(t). Equations (18) and (19) can be substituted in (16) and (17)accordingly. If the number of spectrum slice is infinite, thenwe get (1/Δω)rect[(ω − ωn )/Δω] → δ(ω − ωn ) and Δω →dωn . Thus, the summation with respect to n in (16) and (17)evolves into integration with respect to dωn . Then, we get

σ2filtered-inphase,1 =

14A2

m eh σ2δh +

12eh σ2

ε in , 1+

12eh RδP in ,δh(0)

+14A2

m ehα2N σ2

δh ′ +12eh σ2

ε ′in , 1

+12αN Am eh RδP in ,δh ′(0) (20a)

σ2filtered-quad,1 =

14A2

m ehα2N σ2

δh +12eh σ2

ε in , 1

+14A2

m eh σ2δh ′ +

12eh σ2

ε ′i n , 1

+12eh RδP ′

in ,δh ′(0) (20b)

εfiltered,1/2(t)

= eh/2N∑

n=−N

{12Am/sδh

(n)(t)+Re{ε

(n)in,1/2(t)

}}k1(ωn ) + eh/2

N∑

n=−N

{12αN Am/sδh

(n) ′(t) + Im{

ε(n)in,1/2

′(t)}}

k2(ωn )k1(ωn )


+jeh/2N∑

n=−N

{12αN Am/sδh

(n)(t)+Im{ε

(n)in,1/2(t)

}}k1(ωn)−jeh/2

N∑

n=−N

{12Am/sδh

(n) ′(t)+Re{ε

(n)in,1/2

′(t)}}

k2(ωn )k1(ωn )


(15)


where

σ2δh =

A2m

π

∫ +∞

−∞Sε i n , 1 (ω) |H(ω)|2 k2

1 (ω)dω (21a)

σ2ε in , 1 / 2

=12π

∫ +∞

−∞Sε i n , 1 / 2 (ω)k2

1 (ω)dω (21b)

RδP in ,δh(0) =A2

m

π

∫ +∞

−∞Sε i n , 1 (ω)k2

1 (ω)Re {H(ω)}dω (21c)

σ2δh ′ =

A2m

π

∫ +∞

−∞k2

2 (ω)Sε i n , 1 (ω) |H(ω)|2 k21 (ω)ω2dω

(21d)

σ2ε ′

in , 1 / 2=

12π

∫ +∞

−∞k2

2 (ω)Sε i n , 1 / 2 (ω)k21 (ω)ω2dω (21e)

RδP in ,δh ′(0) =τeA

2m

π

∫ +∞

−∞k2(ω)Sε i n , 1 (ω)k2

1 (ω)

× Re {H(ω)}ω2dω (21f)

RδP ′in ,δh ′(0) =

A2m

π

∫ +∞

−∞k2

2 (ω)Sε i n , 1 (ω)k21 (ω)

× Re {H(ω)}ω2dω. (21g)

Detailed deduction of (20) and (21) can be found in Appendix B.Contributions of different terms in (20a) to the filtered in-

phase noise in a mark can be found in Fig. 4(a) and (b), where the3-dB bandwidth of the Gaussian filter is assumed to be 100 GHz.The first three terms at the right-hand side of (20a) are similarto the variance of the in-phase noise before optical filtering [see(8a)], while the coefficient of k2

1 (ω) in the expression of eachvariance or autocorrelation [see (21a)–(21c)] accounts for thespectral transmittance of the optical BPF. Fig. 4(a) shows thatthese three terms have almost no dependence on filter detuningas long as the detuning is not large enough to degrade filtertransmittance significantly. The last three terms at the right-handside of (20a) represent the contribution from additional effect ofthe filter, where terms 4 and 5 represent in-phase noise directlyconverted from nonlinear phase noise and amplified quadraturenoise, respectively; term 6 represents the cross correlation ofthe in-phase noise converted from nonlinear phase noise andthe input power fluctuation, as shown in Fig. 4(b). It is worthnoting that term 6 in (20a) can make a major contribution to thetotal in-phase noise, but whose magnitude and sign depends onthe detuning of the central frequency of the filter with respectto the central frequency of the signal (through k2(ω)). It showsthat term 6 in (20a) can be negative with central frequency ofthe filter blue shifted from the central frequency of the signal(ωf > ω1/2), while it will be positive with a red-shift detuning(ωf < ω1/2). The absolute value of this term is increased withlarger detuning of the filter.

The results indicate that enhanced in-phase noise suppres-sion is possible with a slightly blue-shifted optical filter, whichoriginates from the negative cross correlation of the in-phasenoise converted from nonlinear phase noise and the power fluc-tuation at SOA input. While a red-shifted optical BPF will make

Fig. 4. Contribution from individual noise term in (20a) to the total in-phasenoise in a mark at filter output. (a) First three noise terms. (b) Last three noiseterms.

this cross-correlation term positive, leading to elevation of in-phase noise and degrade the regeneration performance. Similarnoise suppression/elevation mechanism is not provided for acopropagating space due to the lack of similar negative cross-correlation term in our case.

We show contributions of different terms in (20b) to the fil-tered quadrature noise in a mark in Fig. 5(a) and (b). The majordifference between filtering effect on in-phase and quadraturenoises is that no quadrature noise term is significantly detuningdependent, and there is no difference between blue-shift andred-shift detuning. The results show that filtering-induced ad-ditional noise suppression is only effective with in-phase noisebut not with quadrature noise. As a result, we will concen-trate on in-phase noise suppression in the following quantitativeinvestigations.

We calculate and plot the noise compression factor Nc asa function of the average input power without filter and withdifferent detuning of the filter, as shown in Fig. 6. The 3-dBbandwidth of the Gaussian filter is 100 GHz. The results show astraightforward relationship between the contribution of term 6in (20a) and the additional in-phase noise suppression/elevationwith an optical filter after XGC interaction in the SOA. Withoutdetuning, the negative contribution from the blue side is can-celed by that from the red side. Thus, term 6 in (20a) is nearlyzero. The effect of the filter is limiting the spectral distribution


Fig. 5. Contribution from individual noise term in (20b) to the total quadraturenoise in a mark at filter output. (a) First two noise terms. (b) Last three noiseterms.

Fig. 6. Noise compression factor Nc as a function of the average input powerwithout filter and with different detuning of the filter.

of the noise. When the filter is slightly detuned from the centralfrequency of the signal, the additional effect of filtering willemerge. Enhancement of the in-phase noise suppression per-formance can be obtained with blue-shift detuning of the BPF,just as expected. We note that the enhancement is larger withlarger blue-shift detuning of the filter, while it diminishes withred-shift detuning.

Fig. 7 shows the noise compression factor Nc as a functionof different filter detuning, as well as the transmittance of thefilter. Although larger blue-shift detuning is preferred for better

Fig. 7. Noise compression factor Nc (left Y-axis) and filter transmittance(right Y-axis) as a function of filter detuning for different linewidth enhancementfactors of the SOA.

noise suppression performance, more power loss of the signalshould be afforded. In fact, distortion of the signal will also beinduced by filter detuning if the bandwidth of the signal is notsmall enough. Signal distortion due to filter detuning will bediscussed in Sections III and IV. Although larger enhancementof the noise suppression performance can be obtained with largerαN of SOA, as shown in Fig. 7, it is still hardly to say larger αN

is preferred because SOA-induced nonlinear phase noise givenby term 1 in (20b) (proportional to α2

N ) might increase muchfaster.

Besides, although the analytical formalism is made with NRZsignals, we do not think there are any essential differences be-tween NRZ and RZ case in an XGC-based regenerator using anSOA and a BPF. In fact, both NRZ and RZ signals have been usedin the experimental demonstration of XGC effect in literatures,as in [1], [2], and [3], respectively. In analytical formalism withRZ signals, it should be noted that an SOA operates around aquasi-steady state in RZ case, where input signal power and car-rier density of SOA changes periodically with time. As Am/s in(3) is time dependent in RZ case, all of the autocorrelations withregard to δh(t) and cross correlations with regard to δh(t) andδPin(t) will be time dependent, and the expression can only begiven at specific time instance. We can refer to [8, Appendix A]for the analytical deduction in RZ case. On the other hand, it isworth noting that the spectrum of the RZ signal will be asym-metric and red shifted after propagation in a saturated SOA [4].It might be more complicated in filtering-induced quadrature toin-phase noise conversion in RZ case.

E. Noise Suppression Mechanisms by Optical Filtering

We have pointed out in the previous section that filtering-induced in-phase noise suppression originates from the negativecross correlation of the in-phase noise converted from nonlin-ear phase noise and the power fluctuation at SOA input. Theconversion mechanism is described in detail as follows. AfterXGC interaction in the SOA, on one hand, the in-phase noiseof the original signal is suppressed as a result of the gain sat-uration in the SOA. On the other hand, nonlinear phase noisewill be introduced due to carrier density fluctuation induced by


Fig. 8. Schematic diagram of transmission spectrum of an optical BPF andconversion of frequency chirp at its positive/negative slope.

power fluctuation of the input signal, which directly adds to thetotal quadrature noise at the output of the SOA. It is known thatthe phase fluctuation δφ is related to the frequency chirp δωby δω = −dδφ/dt [4], which can be translated into amplitudefluctuation at the slope of an optical filter. Fig. 8 plots the trans-mission spectrum of an optical filter in the frequency domainand conversion of frequency chirp at its positive/negative slope.It is shown that frequency chirp will be translated into amplitudefluctuation in phase with the chirp at the positive slope of the op-tical filter, while it will be translated into amplitude fluctuationout of phase with the chirp at the negative slope. Consideringthat the frequency chirp generated in an SOA is out of phasewith the original input power fluctuation [4], the positive slopecan function as a mechanism to translate the induced frequencychirp into amplitude fluctuation that can cancel a part of the am-plitude fluctuation after SOA, while the effect of the negativeslope of an optical filter is quite on the contrary. To make theeffect of the positive slope prevails that of the negative slope,the filter should be blue shifted from the central frequency ofthe signal.

To conclude, in XGC-based regenerators, quadrature noisewill be increased due to SOA-induced nonlinear phase noise, aslong as the αN of the SOA is nonzero. We found that a slightlyblue-shifted optical BPF cascaded after SOA can suppress thein-phase noise further, thus lead to alleviation of the negativeeffect of SOA-induced nonlinear phase noise and enhance theoverall regeneration performance of XGC-based regenerators.At least, the optical BPF cascaded after SOA should not be redshifted with the central frequency of the signal, in order not todegrade the regeneration performance.

III. EXPERIMENTAL DEMONSTRATION

A. Experimental Setup

To investigate the influence of noise suppression characteris-tics of SOA and optical filter on the performance of XGC-basedregenerators, and to give a preliminary demonstration of thetheory given earlier, XGC-based regeneration of PolSK signalis experimentally investigated. The whole experimental setupis shown in Fig. 9. The continuous wave (CW) light source

Fig. 9. Experimental setup for XGC-based regeneration of PolSK signal.BPG: bit pattern generator, VOA: variable optical attenuator, PMF: polarization-maintaining fiber, PC: polarization controller, ISO: isolator, OC: optical coupler,PBS: polarization beam splitter, OSA: optical spectrum analyzer, CSA: opticaloscilloscope Tektronix CSA8000.

is a tunable distributed-feedback laser diode with center wave-length set at 1549.3-nm. It is known that PolSK signal can beobtained by using a phase modulator as in [2]. However, dueto the lack of a phase modulator, we use the scheme describedin [18] to obtain 10-Gbit/s PolSK signal in the experiment. TheCW light is first phase modulated by a LiNbO3 Mach–Zehndermodulator to generate 10-Gbit/s NRZ-PSK signal, which is thendemodulated by a 1-bit delayed interferometer consisting of apolarization-maintaining fiber (PMF) and a polarization beamsplitter (PBS) to generate PolSK signal. The differential groupdelay of the PMF is 100.77-ps, which approximately equals thebit period of the incoming signal. In the demodulation stage,the input signal is directed at 45◦ with respect to the fast andslow axes of the PMF using a polarization controller (PC1).At PMF output, two 1-bit-delayed and orthogonally polarizedsignals will be obtained. They are directed by PC2 to arrive at45◦ with respect to the two axes of PBS1 and interfere along thetwo axes, respectively, to give two orthogonal and complementOOK data streams, which propagate back along the two arms ofPBS1 and combines into 10-Gbit/s NRZ-PolSK signal. PolSKsignals generated using this scheme have intensity dips that in-herit from the intensity dips in the original phase-modulatedsignal at transitions between “0” and “π,” which can be seenin inset (a) of Fig. 9. The 50/50 optical coupler (OC1) here isused as a circulator to separate the output from the input. Avariable optical attenuator (VOA1) and a subsequent Erbium-doped fiber amplifier (EDFA1) act as an ASE noise loader. Inthe 2R regeneration unit, the noisy PolSK signal is first am-plified by EDFA2, and then regenerated by the combinationof SOA (CIP SOA-NL-OEC-1550) and a tunable optical filter(Santac OTF-30M) with 3-dB bandwidth of 0.32-nm. The SOAutilized in this experiment is biased at 210 mA, where it pro-vides ∼22-dB gain at 1550-nm with ∼25-ps recovery time.XGC between orthogonally polarized components of the PolSKsignal eventually lead to noise suppression of both components.According to the theory given in the previous section, the sub-sequent optical filter not only removes out-of-band noise butalso can suppress the in-phase noise further, which results in


Fig. 10. (a)–(e) Experimental results of XGC-based regeneration without andwith an optical TBPF. The input power of SOA is about +1-dBm and SNR ismeasured at received optical power of −9.5-dBm.

enhancement of overall regeneration performance. The regen-erated PolSK signal is discriminated by PC3 and PBS2, andanalyzed by an optical spectrum analyzer (Anritsu MS9710 C)and an optical oscilloscope (Tektronix CSA8000) with VOA3adjusting the received optical power.

B. Results and Discussion

In the experiment, the generated PolSK signal contains twologically inverted and orthogonally polarized components: oneis in NRZ format; the other is in RZ format, but with dutycycle close to 100%, which can be seen in the left and righteye diagrams in Fig. 10(a), respectively. To distinguish betweenthe two components of the PolSK signal in our case, we callthem NRZ component and RZ component, respectively. How-ever, it is worth noting that this experiment can only verify XGCand filtering effect on NRZ signals. To verify those on RZ sig-nals, one should demodulate RZ-DPSK signals instead. We usesignal-to-noise-ratio (SNR) to quantitatively evaluate the noisesuppression performance of XGC-based regenerator in the ex-periment. The reason is as follows. We know that in-phase noisecontributes to both signal–noise beat and noise–noise beat, whilequadrature noise contributes to noise–noise beat in the receiver.If the signal power in a mark is much larger than that in a spaceand noise power is much lower than that of the signal, which isthe normal case, then the dominant noise term in the receivedsignal would be signal–noise beat in a mark, which is propor-tional to in-phase noise, as shown by (3). That is to say, SNRof the received signal gives an approximate measure of the in-phase noise. Although signal distortion due to detuned filteringwill also influence SNR, the influence is symmetric with blue-and red-shift detuning of the filter. Any asymmetry of measuredSNR with blue- and red-shift detuning would be a manifestationof filtering-induced additional noise suppression effect.

The measured SNR of NRZ and RZ components of the origi-nal PolSK signal is only 6.2 and 10.4, respectively, as shown inFig. 10(a). After propagation in the SOA synchronously, SNRof both components are improved due to noise suppression in asaturated SOA, as shown in Fig. 10(b). When the subsequent0.32-nm filter is applied with central wavelength aligned withthe central wavelength of the signal, we find another SNR im-

provement. We also find that it is very different between the casesof red-shift and blue-shift detuning of the filter. With 0.1-nmblue-shift detuning, improved noise suppression performancecompared to the zero-detuning case can be observed, whilewith 0.1-nm red shift, regeneration performance degradation isshown instead. These results can be found in Fig. 10(c)–(e).

It is found that, for larger blue-shift detuning of the filter,enhanced regeneration performance is not observed in our ex-periment, which can be attributed to degradation of the signalwith filter detuning close to or larger than the signal bandwidth.As a whole, the experimental results agree with our theoreticalanalysis quite well.

Note that polarization perturbation in the PMF may bringwaveform instability of the received signal and result in errorin the measured results. This problem can be solved by usingdynamically optimized PCs with endless control before and afterthe PMF.

To a certain extent, the intensity dips of the original PolSKsignal can also be regarded as a special kind of intensity noisethat only present at bit transitions. Due to gain dynamics ofthe SOA, overshoots and fluctuations appear at the leading andfalling edges of the regenerated signal, as shown in Fig. 10(b).Thus, elimination of the overshoots and fluctuations by using analigned or blue-shift-detuned filter, as shown in Fig. 10(c) and(d), can be regarded as an enlarged embodiment of the mecha-nism described in Section II-E. However, it is worth noting that,as SNR is evaluated near the center of each bit slot, we do notthink the major contribution of the SNR improvement comesfrom elimination of the distortions at pulse leading and fallingedges.

IV. NUMERICAL DEMONSTRATION

We also take PolSK signal as an example to numericallydemonstrate the enhanced noise suppression effect with a cas-caded optical BPF after XGC interaction in the SOA. A wide-band dynamic model of SOA similar to that described in [19]is adopted, except that two orthogonal polarization states areincluded and gain dynamics introduced by intraband relaxationis simply described by a gain compression factor as in [20].The simulated SOA is a polarization-independent device withsmall-signal gain of about 27-dB at 1540-nm when biased at200-mA. The original 10-Gbit/s PolSK signal is also gener-ated by demodulating a degraded NRZ-DPSK signal as in theexperiment described in Section III. However, there are no inten-sity dips in the original DPSK signal and the converted PolSKsignal. Some amount of band-limited white Gaussian noise isadded to the clean DPSK signal in the field domain to produceintensity fluctuation that is typical for an optical signal with afinite OSNR. The bandwidth of the noise is 50-GHz. In the sim-ulation, the average signal input power to the SOA is +5-dBm,which is high enough to drive SOA into deep saturation. Thecascaded optical filter is Gaussian type with 3-dB bandwidthof 0.32-nm. We investigate RZ component of the PolSK sig-nal before SOA, after SOA, and after filter, respectively, withfilter detuned from −0.2 to +0.2 nm, and calculate SNR andnoise standard deviation for marks and spaces, respectively. The


Fig. 11. (a) Calculated SNR. Noise standard deviation for (b) marks (c) andspaces, respectively, with different detuning of the cascaded optical filter.

results are shown in Fig. 11(a)–(c), with typical eye diagramsgiven as insets in Fig. 11(a).

Fig. 11(a) clearly shows that signal quality is improved afterpropagation in a saturated SOA, and another improvement canbe achieved only when the cascaded filter is aligned with thecenter wavelength of the signal or slightly blue shifted. WithαN of 6, the improvement is larger with blue-shift detuningof 0.1-nm than that without detuning of the filter. The simula-tion result is in agreement with both the experimental results inSection III and the theoretical results in Section II. The cal-culated noise standard deviation for marks and copropagatingspaces is given in Fig. 11(b) and (c), respectively. For the co-propagating spaces, noise standard deviation is symmetricallyreduced with blue- and red-shift detuning, and the result doesnot change with different αN , which is apparently nonrele-vant to the asymmetric SNR enhancement with different filterdetuning shown in Fig. 11(a). Considering that the noise band-width is 50-GHz in the simulation, the symmetric reduction ofnoise standard deviation can be attributed to reduction of theeffective noise passband. While for marks, the noise standarddeviation is asymmetric with blue- and red-shift detuning andlarger asymmetry is observed with larger αN . The result revealsthat the additional effect of filtering is only effective with noisein marks and the asymmetric SNR enhancement with differentwavelength detuning of the filter can be only attributed to theasymmetric noise suppression effect obtained in marks, just as

Fig. 12. (a) Simulated waveform change before and after filter. (b) Frequencychirp induced by phase fluctuation.

predicted by theory. Moreover, enhanced asymmetry of SNRenhancement with larger αN of the SOA is also in agreementwith the theoretical results in Fig. 7.

It is worth noting that, in Fig. 11(a), the calculated SNRdecreases when wavelength detuning changes from −0.1 to−0.2 nm, which seems not agree with the analytical resultsshown in Fig. 7. The reason can be attributed to exclusion ofthe filtering effect on signal itself in the analytical investigation.In practices, Detune of the filter with respect to the center fre-quency of the signal will inevitably result in degradation of thesignal to some extent. In the blue-shift case, although filtering-induced exclusion of out-of-band noise and suppression of in-band in-phase noise will elevate SNR, distortion of the signalwill degrade SNR on the other hand. In the simulation, wave-length detuning of ±0.2-nm (25-GHz) is actually larger than thedouble-sideband bandwidth of the signal (20-GHz). In this case,large degradation of the signal itself will be expected, whicheventually results in an overall reduction of SNR. Besides, inred-shift case, filtering will not only degraded signal quality, butalso elevate in-phase noise. Both will result in SNR reduction.

The simulated waveform change before and after filter andthe frequency chirp induced by phase fluctuation, as shown inFig. 12(a) and (b), picture the enhanced in-phase noise suppres-sion mechanism and demonstrate our theoretical analysis. Onesegment of temporal waveform is zoomed out to take a closelook at the additional noise suppression effect by optical filter-ing. It is shown that the frequency chirp shown in Fig. 12(b) isout of phase with the power fluctuation before filter (after SOA),as shown by solid line in Fig. 12(a). If the frequency chirp shownin Fig. 12(b) encounters the positive slope of the filter, whichcorresponds to blue-shifted detuning (see Fig. 7 for reference),it will be translated into amplitude fluctuation that is in phasewith the chirp but out of phase with the power fluctuation andresults in cancellation of the power fluctuation to some extend,as shown by dash and dotted line in Fig. 12(a). However, if thefrequency chirp shown in Fig. 12(b) encounters the negativeslope of the filter, which corresponds to the red-shift case (seeFig. 7 for reference), it will be translated into power fluctuation


that is out of phase with the chirp but in phase with the powerfluctuation, and results in enhancement of the power fluctuationas shown by dash-dotted line in Fig. 12(a).

V. CONCLUSION

In this paper, we present a theoretical model to analyze thenoise suppression characteristics in XGC-based regenerators,where a small-signal model of SOA is adopted to characterizethe noise suppression in the SOA, while discrete convolutionand a spectrum slice technique are used to characterize the noisesuppression due to subsequent filtering. Analytical expressionsare obtained for in-phase and quadrature noises after SOA andafter filter.

It is found that the mechanism behind XGC-based regenera-tion lies in the gain change out of phase with the input powerfluctuation, which leads to a negative cross-correlation term inthe expression of in-phase noise variance at SOA output. Onthe other hand, it is worth noting that nonlinear phase noisewill also be introduced by input power fluctuation during XGCinteractions, which is proportional to α2

N . In a direct detec-tion receivers, large nonlinear phase noise will directly elevatenoise–noise beat in a photodiode. In this case, the regenerationperformance due to in-phase noise suppression might be partlycanceled. The addition of nonlinear phase noise can be avoidedby using an SOA with zero α2

N , e.g., a QD SOA. Or we can en-hance in-phase noise suppression, so as to combat the increaseof noise–noise beat due to nonlinear phase noise by decreasingsignal–noise beat. It is proposed that a slightly blue-shifted op-tical BPF can convert phase-noise-related frequency chirp intoamplitude fluctuation that can cancel a part of the amplitudefluctuation after XGC interaction in the SOA, and lead to fur-ther reduction of in-phase noise, while a red-shifted filter willincrease the in-phase noise and degrade the regeneration perfor-mance. The reason why XGC effect is not effective with space isfound to be the small contribution to signal–noise beat in a space.We also present experimental demonstration and time-domainnumerical simulation of filtering-enhanced noise suppression ofPolSK signals using XGC-based regenerators. Both experimen-tal and numerical results agree with the analytical results quitewell. The results provide physical insight into the mechanism ofXGC effect and the enhanced noise suppression effect by usingappropriate filtering after XGC interaction in the SOA.

When XGC is used to regenerate phase-modulated signals,nonlinear phase noise generated in the SOA might directly leadto degradation the phase information. Besides, filtering effect ofa delayed interferometer usually used to demodulate the phase-modulated signal at the receiver end plays a more important rolein noise conversion from quadrature to in phase than an ordinaryoptical BPF. Consequently, nonlinear phase noise will have moresignificant influence on XGC-based DPSK regenerators, whichwill be discussed in another paper separately.

ACKNOWLEDGMENT

The authors would like to thank Prof. X. Li with theDepartment of Electrical and Computer Engineering, McMasterUniversity for useful discussion and valuable advices.

APPENDIX A

Here, we will give detailed deduction of each term in (8a),while the other terms in (8b)–(8d) can be deduced similarly andwill not be presented any more. For convenience, we use Cij

(i = 1, 2, . . . , 4 corresponds to (8a)–(8d); j = 1, 2, 3 is theterm number) to denote each terms in (8) accordingly. We willdeduce the expression of each term in (8a) from calculating theautocorrelation of in-phase component in (7) at τ = 0

σ2inphase,1 = C11 + C12 + C13

=14A2

m ehRδh(0) + ehRεr , 1 (0)

+ Am ehRεr , 1 ,δh(0) (A1)

where we use subscript εr,1 to denote Re{εin,1(t)

}. For Gaus-

sian noise, Rεr , 1 (0) = (1/2)Rε in , 1 (0). Due to the Wiener–Khinchin relation [11], each auto/cross correlation in (A1) canbe expressed in the form of inverse Fourier transform of thecorresponding power spectral density. That is to say

Rδh(0) = σ2δh =

12π

∫ +∞

−∞Sδh(ω)dω (A2)

Rεr , 1 (0) = σ2εr , 1

=12σ2

ε in , 1=

12

[12π

∫ +∞

−∞Sε in , 1 (ω)dω

]

(A3)

Rεr , 1 δh(0) =1

2AmRδP in δh(0)

=1

2Am

[12π

∫ +∞

−∞SδP in δh(ω)dω

]. (A4)

As SOA is a linear system with input exitation of δPin(t),output response of δh(t), and frequency response of H(ω), thepower spectral density of δh(t) can be expressed by

Sδh(ω) = |H(ω)|2 SδP in (ω) (A5)

according to the theory of linear system with random input. Onthe other hand, according to (3), the autocorrelation function ofδPin(t) is RδP in (τ) = 2A2

m Rε in , 1 (τ). As power spectral densityis the Fourier transform of autocorrelation, the relationship ofcorresponding power spectral density is

SδP in (ω) = 2A2m Sε in , 1 (ω). (A6)

Then, we get

Sδh(ω) = 2A2m |H(ω)|2 Sε in , 1 (ω). (A7)

Using (A2) and (A7), we get

C11 =14A2

m ehσ2δh

=14A2

m eh

[A2

m

π

∫ +∞

−∞|H(ω)|2 Sε in , 1 (ω)dω

]. (A8)


Using (A3), we get

C12 =12ehσ2

ε i n , 1

=12eh

[12π

∫ +∞

−∞Sε i n , 1 (ω)dω

]. (A9)

According to the theory of linear system with random input,the cross power spectral density of δPin(t) and δh(t) can beexpressed as

SδP in ,δh(ω) = H(ω)SδP i n (ω). (A10)

Using (A4), (A6), and (A10), we get

C13 =12ehRδP i n ,δh(0)

=12eh

[A2

m

π

∫ +∞

−∞H(ω)Sε i n , 1 (ω)dω

]. (A11)

If we assume that Sε in , 1 (ω) is an even function with respect toω, the contribution of the imaginary part of H(ω) (which is anodd function with respect to ω) to the integral is zero. Equation(A11) can be written as

C13 =12ehRδP in ,δh(0)

=12eh

[A2

m

π

∫ +∞

−∞Re {H(ω)}Sε i n , 1 (ω)dω

]. (A12)

APPENDIX B

Similarly, we use Fij (i = 1, 2; j = 1, 2, . . ., 6) to denote eachterms in (20a) and (20b) accordingly. First, we give detaileddeduction of each term in (20a). Still, we assume that Sε in , 1 (ω)is an even function with respect to ω.

By substituting (18) into (16), F11 can be expressed as

F11 =14A2

m ehN∑

n=−N

×[

12π

∫ +∞

−∞Sδh(ω)rect

(ω − ωn

Δω

)dω

]k2

1 (ωn ).

(B1)

Exchanging the order of summation and integration and let-ting N → ∞, (B1) can be rewritten as

F11 =14A2

m eh

{12π

∫ +∞

−∞dωSδh(ω)

limN →∞

[N∑

n=−N

1Δωn

(ω − ωn

Δω

)k2

1 (ωn )Δωn

]}

. (B2)

In this case, Δω → dωn and (1/Δω)rect[(ω − ωn )/Δω] →δ(ω − ωn ). The summation with respect to n changes into

integration with respect to dωn

F11 =14A2

m eh

[12π

∫ +∞

−∞dωSδh(ω)

×[∫ +∞

−∞δ(ω − ωn )k2

1 (ωn )dωn

]]

=14A2

m eh

[12π

∫ +∞

−∞Sδh(ω)k2

1 (ω)dω

]. (B3)

Using (A7), F11 can be expressed as

F11 =14A2

m eh σ2δh

=14A2

m eh

[A2

m

π

∫ +∞

−∞|H(ω)|2 Sε in , 1 (ω)k2

1 (ω)dω

].

(B4)

Following the same procedure, the expression of F12 and F13can be obtained similarly

F12 =12eh σ2

ε in , 1

=12eh

[12π

∫ +∞

−∞Sε in , 1 (ω)k2

1 (ω)dω

](B5)

F13 =12eh RδP in ,δh(0)

=12eh

[A2

m

π

∫ +∞

−∞H(ω)Sε in , 1 (ω)k2

1 (ω)dω

]. (B6)

Comparing the expression of F11– F13 and the expressionof C11–C13 , it is found that the transmittance of the filter isinvolved in the spectral integration in F11– F13 .

By substituting (18), F14 can be expressed as

F14 =14A2

m ehα2N

·N∑

n=−N

[12π

∫ +∞

−∞Sδh ′(ω)rect

(ω − ωn

Δω

)dω

]

× k21 (ωn )k2

2 (ωn ). (B7)

If N → ∞, (B7) changes into

F14 =14A2

m ehα2N

[12π

∫ +∞

−∞Sδh ′(ω)k2

1 (ω)k22 (ω)dω

]. (B8)

To obtain the power spectral density of δh′(t), in (B8) let usconsider a linear system of time-domain first-order diffrentia-tion, with input excitation of δh(t), output response of δh′(t),and frequency response of jω. The power spectral density ofδh′(t) can be expressed by

Sδh ′(ω) = ω2Sδh(ω). (B9)


Using (B8), (B9), and (A7), we get

F14 =14A2

m ehα2N σ2

δh ′

=14A2

m ehα2N

[A2

m

π

∫ +∞

−∞k2

2 (ω) |H(ω)|2 Sε i n , 1 (ω)

× k21 (ω)ω2dω

]

. (B10)

Similarly, we get

F15 =12eh σ2

ε ′in , 1

=12eh

[12π

∫ +∞

−∞k2

2 (ω)Sε i n , 1 / 2 (ω)k21 (ω)ω2dω

].

(B11)

By substituting (19), F16 can be expressed as

F16 =12αN eh

N∑

n=−N

R(n,n)δP i n ,δh ′(0)k2

1 (ωn )k2(ωn )

=12αN eh

N∑

n=−N

[12π

∫ +∞

−∞SδP i n ,δh ′(ω)

×rect(

ω − ωn

Δω

)dω

]k2

1 (ωn )k2(ωn ). (B12)


F16 =12αN eh

[12π

∫ +∞

−∞SδP i n ,δh ′(ω)k2

1 (ω)k2(ω)dω

].

(B13)

To obtain the cross power spectral density of δPin(t) andδh′(t) in (B13), we consider a linear system of cascading SOAand time-domain first-order differentiation in series, the fre-quency response of which can be given by H1(ω) = jωH(ω).Then, we get

SδP in ,δh ′(ω) = H1(ω)SδP i n (ω). (B14)

Using (A6), (B13), and (B14), we get

F16 =12eh RδP in ,δh ′(0)

=12eh

[A2

m

π

∫ +∞

−∞H1(ω)Sε i n , 1 (ω)k2

1 (ω)k2(ω)dω

].

(B15)

Using (5)

H1(ω) = − (eh − 1)P 1

satτc· 11 + (ωτe)

2

[(ωτe)

2 + jωτe

]. (B16)

k2(ω) in the integration of (B15) is an odd function whenωf = ω1 ; thus, the contribution of the real part of H1(ω) tothe integration is zero in this case. However, as the effectivecarrier lifetime τe of a common SOA is ∼100 ps, ωτe in (B16)is ∼104 . That is to say, if ωf �= ω1 , the real part of H1(ω) is 104

larger than the imaginary part. So, we can omit the imaginarypart of H1(ω) and rewrite it as

H1(ω) = τeRe {H(ω)}ω2 . (B17)

Substituting (B17) into (B15), we eventually get

F16 =12eh RδP in ,δh ′(0)

=12eh

[A2

m τe

π

∫ +∞

−∞k2(ω)Sε in , 1 (ω)

×Re {H(ω)} k21 (ω)ω2dω

]. (B18)

The deduction of F21–F24 is almost the same as that of F11 ,F12 ,F14 , and F15 ; so, we only give the deduction of F25 here. Sub-stituting (19), F25 can be expressed as

F25 =12eh

N∑

n=−N

R(n,n)δP ′

in ,δh ′(0)k21 (ωn )k2

2 (ωn )

=12eh

N∑

n=−N

[12π

∫ +∞

−∞SδP ′

in ,δh ′(ω)rect(

ω − ωn

Δω

)dω

]

× k21 (ωn )k2

2 (ωn ). (B19)


F25 =12eh

[12π

∫ +∞

−∞SδP ′

in ,δh ′(ω)k21 (ω)k2

2 (ω)dω

]. (B20)

As SOA can be regarded as a linear system, if the input excitationis the derivative of δPin(t), that is, δP ′

in(t), the output responsewould be the derivative of δh(t), that is, δh′(t). So, the followingrelationship holds:

SδP ′in ,δh ′(ω) = H(ω)SδP ′

in(ω). (B21)

Similar to (B9)

SδP ′in

(ω) = ω2SδP in (ω). (B22)

Using (B20)–(B22) and (A6), we get

F25 =12eh RδP ′

in ,δh ′(0)

=12eh

[A2

m

π

∫ +∞

−∞k2

2 (ω)Sε in , 1 (ω)

× Re {H(ω)} k21 (ω)ω2dω

]. (B23)

REFERENCES

[1] G. Contestabile, R. Proietti, N. Calabretta, and E. Ciaramella, “Reshapingcapability of cross-gain compression in semiconductor amplifiers,” IEEEPhoton. Technol. Lett., vol. 17, no. 12, pp. 2523–2525, Dec. 2005.

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[4] G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectralbroadening of optical pulse in semiconductor laser amplifiers,” IEEE J.Quantum Electron., vol. 25, no. 11, pp. 2297–2306, Nov. 1989.


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[6] Y. Liu, E. Tangdiongga, Z. Li, H. de Waardt, A. M. J. Koonen, G. D. Khoe,X. Shu, I. Bennion, and H. J. S. Dorren, “Error-free 320-Gb/s all-opticalwavelength conversion using a single semiconductor optical amplifier,”J. Lightw. Technol., vol. 25, no. 1, pp. 103–108, Jan. 2007.

[7] J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-opticalregenerator using wavelength shift induced by cross-phase modulation inhighly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett.,vol. 17, no. 2, pp. 423–425, Feb. 2005.

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[10] J. Dong, S. Fu, X. Zhang, P. Shum, L. Zhang, and D. Huang, “Analyticalsolution for SOA-based all-optical wavelength conversion using transientcross-phase modulation,” IEEE Photon. Technol., Lett., vol. 18, no. 24,pp. 2554–2556, Dec. 2006.

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[15] J. Wang, Y. Jiao, R. Bonk, W. Freude, and J. Leuthold, “Regenerativeproperties of bulk and quantum dot SOA based all-optical Mach–Zehnderinterferometer DPSK wavelength converters,” in Proc. Int. Conf. Photon.Switching, Oct. 16–18, 2006, pp. 1–3.

[16] G. Contestabile, R. Proietti, and N. Calabretta and E. Ciaramella, “Cross-gain compression in semiconductor optical amplifiers,” J. Lightw. Tech-nol., vol. 25, no. 3, pp. 915–921, Mar. 2007.

[17] K. Sato and H. Toba, “Reduction of mode partition noise by using semi-conductor optical amplifiers,” IEEE J. Sel. Topics Quantum Electron.,vol. 7, no. 2, pp. 328–333, Mar./Apr. 2001.

[18] G. Contestabile, A. D’Errico, and E. Ciaramella, “Cross-gain compressionin semiconductor optical amplifiers,” IEEE Photon. Technol. Lett., vol. 16,no. 9, pp. 2138–2140, Sep. 2004.

[19] W. Hong, M. Li, X. Zhang, J. Sun, and D. Huang, “Dynamic analysis ofall-optical wavelength conversion of differential phase-shift keyed signalsbased on semiconductor optical amplifier Mach–Zehnder interferometer,”J. Lightw. Technol., vol. 27, no. 24, pp. 5580–5589, Dec. 2009.

[20] M. L. Nielsen, J. Mørk, R. Suzuki, J. Sakaguchi, and Y. Ueno, “Exper-imental and theoretical investigation of the impact of ultra-fast carrierdynamics on high-speed SOA-based all-optical switches,” Opt. Exp.,vol. 14, pp. 331–347, 2006.

Wei Hong received the M.S. degree in science from the Department of Physicsand the Ph.D. degree in physical electronics from the Department of optoelec-tronic engineering, Huazhong University of Science and Technology (HUST),Wuhan, China, in 1996 and 2003, respectively.

She is currently an Associate Professor at the Wuhan National Laboratoryfor Optoelectronics and the School of Optoelectronics Science and Engineering,HUST. Her research interests include semiconductor optoelectronic devices andall-optical signal processing.

Minghao Li received the M.S. degree from the Department of Electronicsand Information Engineering, Huazhong University of Science and Technology(HUST), Wuhan, China, in 2005, where he is currently working toward thePh.D. degree in optoelectronic information engineering at the School of Op-toelectronic Science and Engineering and the Wuhan National Laboratory forOptoelectronics.

His research interests include high-speed all-optical signal processing, 2Rregeneration, and advanced modulation format.

Xinliang Zhang received the Ph.D. degree in physical electronics from theHuazhong University of Science and Technology (HUST), Wuhan, China, in2001.

He is currently a Professor at the Wuhan National Laboratory for Opto-electronics and the School of Optoelectronic Science and Engineering, HUST.He is the author or coauthor of more than 70 journal and conference papers.His current research interests include all-optical signal processing and relatedcomponents.

Junqiang Sun received the Ph.D. degree in electronic physics and optoelectron-ics from the Huazhong University of Science and Technology (HUST), Wuhan,China, in 1994.

From September 2000 to September 2001, he was a Research Associate inthe Department of Electrical and Electronic Engineering, Hong Kong Univer-sity of Science and Technology. From June 2005 to December 2005, he wasa Research Fellow at the School of Information Technology and Engineering,University of Ottawa, Canada. He is currently a Professor at the Wuhan NationalLaboratory for Optoelectronics and the School of Optoelectronic Science andEngineering, HUST. He is the author and coauthor of more than 100 papersin refereed journals and conference proceedings. His current research interestsinclude all-optical signal processing, all-optical wavelength conversion, fiberlasers and amplifiers, photonic generation of microwave signals, and opticalnetwork technologies.

Dexiu Huang was born in Hunan Province, China, in October 1937. He receivedthe B.E. degree from the Department of Radio Engineering, Huazhong Instituteof Technology (now Huazhong University of Science and Technology), Wuhan,China, in 1963.

Since 1963, he has been with the Huazhong University of Science and Tech-nology as an Assistant Professor, a Lecturer, an Associate Professor, and aProfessor in 1963, 1978, 1986, and 1990, respectively. Prior to 1972, he wasengaged in research on semiconductor devices and passive devices in radio en-gineering. From 1972 to 1981, he performed research on solid-state lasers andapplications. From 1981 to 1983, he was a Visiting Scientist with the OregonGraduate Center, focusing on semiconductor optoelectronic devices. Since then,he has been in the field of optical communication performing research on semi-conductor optoelectronics devices and some passive devices. He is currently aProfessor and the Dean of the College of Information Science and Engineeringand the Associate Director of the Wuhan National Laboratory for Optoelectron-ics, Huazhong University of Science and Technology. He is the author of fivebooks, namely, Semiconductor Optoelectronics (Sichuan, China: Univ. Electron.Sci. Technol. China Press, 1994), Semiconductor Lasers and Their Applications(Tianjin, China: Natl. Defense Ind. Press, 2001), Introduction of InformationScience (Beijing, China: Chinese Electrical Power Press, 2001), Fiber Optics(Tianjin, China: Natl. Defense Industry Press, 1995), and Fiber Technology andApplications (Sichuan, China: Univ. Electron. Sci. Technol. China Press, 1995).He is the author and coauthor of more than 200 papers.

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