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Y. Yan and C. Yang Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2059

Coherent light wave generated from incoherentpump light in nonlinear Kerr medium

Yan Yan* and Changxi Yang

State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments,Tsinghua University, Beijing 100084, China

*Corresponding author: yany04@mails.tsinghua.edu.cn

Received August 14, 2009; accepted September 4, 2009;posted September 14, 2009 (Doc. ID 115730); published October 13, 2009

We show theoretically and numerically that in an instantaneous-response Kerr medium, a coherent signallight can be generated spontaneously from an incoherent pump light in the degenerate four-wave mixing pro-cess. Our analysis shows that this phenomenon requires the same group velocities of pump wave and idlerwave as well as the convection (walk-off) between the signal wave and pump/idler waves. © 2009 Optical So-ciety of America

OCIS codes: 030.1640, 060.4370, 190.4380.

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. INTRODUCTIONn nonlinear optics, three-wave mixing and four-waveixing are parametric processes caused by quadratic and

ubic nonlinearities taking place in weakly nonlinear me-ia [1]. Of primary importance in practice are the para-etric processes in which one of the waves (the pumpave) is externally excited and has much larger ampli-

ude than its daughter waves. Two alternative theoreticalpproaches are usually considered to treat the process ac-ording to the relationship of the correlation time Tc ofhe interacting waves and the characteristic evolutionime t0 of the nonlinear interaction [2]. On one hand,hen Tc� t0, the interacting waves are assumed to be sta-

ionary, and their relative phases govern the coherentvolution of waves. On the other hand, when Tc� t0, theaves in the parametric process are considered incoher-nt, and their relative phases are not significant to theirnteractions since the effects can be averaged [3].

It was once commonly believed that the coherent soli-on structure only exists in the coherent regime of thearametric process, i.e., Tc� t0 [4–6]. Nevertheless, theuthors in [2,7] have demonstrated that in quadratic non-inear media, the coherent soliton structure of a signalave can be generated spontaneously and sustained fromn incoherent pump in the incoherent three-wave para-etric interaction �Tc� t0�, provided that the pump and

dler wave have the same group velocity, which is calledhe phase-locking mechanism, as well as different groupelocity between the signal and pump/idler wave, which isalled the convection between them [7–9]. Besides, thehase-locking mechanism and convection also result in anncoherent soliton structure in an instantaneous-responseonlinear Kerr optical fiber [10]. This phenomenonhanges the once generally accepted opinion that the in-oherent soliton only exists in a noninstantaneous nonlin-ar medium.

Recently, the four-wave mixing process with an inco-erent pump light and a coherent signal light in a Kerr

0740-3224/09/112059-5/$15.00 © 2

onlinear optical fiber has been investigated [11,12]. Theignal spectrum is broadened as wide as the pump spec-rum, indicating that the coherence of the signal light isowered by interacting with the incoherent pump. In thisaper, however, we demonstrate that in a Kerr nonlinearptical fiber the phase-locking mechanism and convectionetween the signal and idler/pump waves can lead topontaneous generation of a coherent signal wave from anncoherent pump wave in the four-wave mixing (FWM)rocess. The incoherent pump wave’s correlation time Tchat we consider is much smaller than the characteristicvolution time of nonlinear interaction t0.

. THEORYonsider a degenerate FWM process in a nonlinear opti-al fiber involving an incoherent pump wave and itsaughter waves, i.e., the signal wave and the idler wave.he three waves copropagate in the same direction. As-uming the spectral width of the three waves to be muchmaller than their respective carrier frequencies (��j�j, j=s , i ,p 2�p=�s+�i), one can apply the slowly vary-

ng envelope approximation for their amplitude and ob-ain the coupled amplitude equations [13],

DsAs = i��2�Ap�2 + �As�2 + 2�Ai�2�As + 2i�ApApAi* exp�i��z�,

DiAi = i��2�Ap�2 + 2�As�2 + �Ai�2�Ai + 2i�ApApAs* exp�i��z�,

DpAp = i���Ap�2 + 2�As�2 + 2�Ai�2�Ap + 2i�AiAsAp*

�exp�− i��z�, �1�

here Aj=Aj�z ,T� are the complex amplitude of waves inhich T is the time and z is the distance. Dj= ��z+�1j�ti�2j�tt+�j�. �1j are defined as 1/vj, where vj are theroup velocities of the three waves. �2j are the dispersionarameters of the three waves at � , representing the

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2060 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Y. Yan and C. Yang

roup-velocity dispersion (GVD). � is the nonlinear pa-ameter of the optical fiber. ��=�0s+�0i−2�0p is thehase mismatch term [13], where �0j are the propagationonstants of the three waves. �j are the loss parameters.

Assume the amplitudes of the signal and idler wavesre small enough to neglect their influence on the pump.or simplicity, also neglect the second-order dispersionarameter �2j and the higher-order dispersion of thehree waves. By defining T= t−�1pz to fix the referencerame on the pump wave, the following equations can bebtained from Eq. (1):

� �

�z+ ��1s

�

�T− 2i��Ap�z,T�2� + �s�As

= 2i�ApApAi* exp�i��z� �a�,

� �

�z+ ��1i

�

�T− 2i��Ap�z,T��2 + �i�Ai

= 2i�ApApAs* exp�i��z� �b�,

Ap�z,T� = Ap�0,T�exp�i��Ap�0,T��2z� �c�, �2�

here ��1s=�1s−�1p, ��1i=�1i−�1pTo gain an insight into the phase-locking mechanism,

rst we investigate a simpler case in which the incoherentump wave exhibits only pure random phase fluctuation,.e., Ap�0,T�=P1/2 exp�ifp�T��, where fp�T� is a randomariable and P is the pump’s power, �Ap�0,T��2. Integrat-ng Eq. (2b) along the characteristics of idler waves and to

ake use of Eq. (2c), one obtains

Ai�z,T� = 2i� exp�2i�z��0

z

exp�i��z�� � exp�− �i�z − z���

��Ap�0,T���2As*�z�,T��dz�,

T� = T + ��1i�z� − z�. �3�

ubstituting Eq. (3) into Eq. (2a),

�

�z+ ��1s

�

�T− 2i�P + �s�As�z,T� = 4�2�

0

z

M�z − z��

� �Ap*�0,T���2�Ap�0,T��2As�z�,T��dz�, �4�

here M�z−z��=exp�−�i�z−z���exp�i�k�z−z���.Noting that if ��1i=�1i−�1p=0, i.e., the idler wave and

ump wave have the same group velocity, one has T=T�.he term �Ap

*�0,T���2�Ap�0,T��2= �Ap�0,T��4 becomes P2,nd therefore the evolution of signal wave As is indepen-ent of the pump’s phasefluctuation. This result suggestshe possibility of generating a signal with low amplitudend phase fluctuation, i.e., a high degree of coherence.he same group velocity of pump and idler waves is calledhe phase-locking mechanism [7].

This conclusion can be checked by numerical simula-ion of Eq. (2). Figure 1 shows this in an FWM processith a phase-locking mechanism ���1i=0�. In the simula-

ion also assume that �� =0, the length of the fiber is

1s

=200 m, the nonlinear parameter �=5.8 W−1 km−1, andhe pump’s power is 1 W. In fact, when the single pumpavelength is near the zero-dispersion wavelength (ZDW)f a nonlinear optical fiber and the signal and idler wave-engths are close to the ZDW, the difference of the groupelocities of the interacting waves is usually neglected.s�z ,T�, fp�z ,T�, and fi�z ,T� are the phases of the threeaves. Figure 1(b) shows the output signal and idleraves, and Fig. 1(d) shows the phase of output signalave. They indicate that the input signal grows with aigh degree of coherence in the process. Figure 1(f) showshat the relationship of the phases of the pump and idleraves is fi�z ,T�=2fp�z ,T�+C, where C is a constant. Iteans that while the signal wave evolves coherently, the

dler wave absorbs the pump’s phase fluctuation, and itshase fluctuation is twice that of the pump. It indicateshat in the spectral domain, the width of the idler spec-rum should be twice wide as that of the pump spectrum.his phenomenon is well-known in the area of fiber opti-al parametric amplifiers [14–16]. Define the mutual co-erence function of pump and idler wave Q�z ,T�Ap�z ,T�Ap�z ,T�Ai

*�z ,T�. Figure 1(e) shows that Q�z ,T�as no phase fluctuation, which suggests that althoughhe pump and idler waves are incoherent, the phase-ocking mechanism makes them mutually coherent in thearametric process.We have investigated the case when the pump exhibits

nly pure phase fluctuation. Next we deal with a moreeneral situation. When the incoherent pump wave hasoth amplitude and phase fluctuation, a signal wave withhigh degree of coherence can be generated spontane-

usly in the FWM process under the conditions of thehase-locking mechanism as well as the convection be-ween the signal and the pump/idler wave.

Start with Eq. (2) and the phase-locking mechanism�1i=0. For simplicity, assume the phase mismatch term�=0 and neglect the loss of the three waves. Then Eq.

2) becomes

ig. 1. (Color online) Amplitude and phase of the waves withhase-locking mechanism ���1i=��1s=0�. (a) Amplitude of thenput signal wave; (b) amplitude of output signal and idleraves; (c) phase of pump fp�z ,T�; (d) phase of signal wave fs�z ,T�;

e) phase of idler wave fi�z ,T�; (f) 2fp�z ,T�− fi�z ,T�. Other param-ters used in simulation: length of fiber Z=200 m, nonlinear pa-ameter �=5.8 W−1 km−1 Pump power is 1 W.

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Y. Yan and C. Yang Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2061

�

�z+ ��1s

�

�T− 2i��Ap�2As = 2i�ApApAi

* �a�,

�

�z− 2i��Ap�2Ai = 2i�ApApAs

* �b�,

Ap�z,T� = Ap�0,T�exp�i��Ap�0,T��2z� �c�. �5�

he input incoherent pump field Ap�0,T� is described asn ergodic circular complex Gaussian random variable,ith zero mean �Ap�0,T��=0, and has an exponential au-

ocorrelation function �Ap�0,T�Ap*�0,T��=P0 exp�−�T� /Tc�

7,17]. Multiplying �� /�z+2i�P�T�� on both sides of Eq.5a) and substituting the conjugation of Eq. (5b) into it toliminate Ai, we obtain the equation that governs the evo-ution of the signal wave,

�

�z� �

�z+ ��1s

�

�T− 2i�P�T��As = 4�2�P�T��2As, �6�

here P�T�= �Ap�0,T��2 is the instantaneous power of thenput pump. We rewrite P�T�=P0�1+e�T�� and �P�T��2

P�2�1+�T��, where e�T� is the normalized fluctuation ofhe pump field P�T� with zero means �e�T��=0[7], and�T� is the normalized fluctuation of �P�T��2 with ��T��0.Equation (6) can be solved by means of spatial Fourier

xpansion

As�z,T� =1

2�

−�

+�

A�k,T�exp�ikz�dk. �7�

ubstituting Eq. (7) into Eq. (6) and noting that � /�zik, the expression of As�z ,T� can be obtained:

s�T,z� =1

2�

−�

+�

A�0,k�exp�f�k�z�dk,

f�k� = −i 4�2P�2�1 + m�T�� + k2 − 2�P0�1 + n�T��k�T

k��1sz

+ ik,

m�T� =�0

T

�t�dt n�T� =�0

T

��t�dt. �8�

This integral can be calculated by the steepest descentethod when z→� [7,18],

As�z,T� exp�f�k0�z�dk,

k0 = 2�P���T + n�T��/�T − ��1sz� 0 � T � ��1sz, �9�

here k0 is the saddle point of the f�k�. Finally we obtainhat the asymptotic solution of As is

As�z,T� exp�4P��

��1s

��T + n�T������1sz − T���� exp�i

2P0�

��1s�1 + m�T��T� . �10�

When the pump is coherent, i.e., m�T�=n�T�=0, P�P0, the asymptotic solution of As�T ,z� for Eq. (10) be-omes

As�z,T� exp�4P0�

��1s

�T���1sz − T��exp�i2P0�

��1sT� .

�11�

The first exponential term in Eq. (11) represents themplitude of the signal wave. It means that in the refer-nce frame of the pump waves, the peak of the signalave is at T=��1sz/2 due to the convection between the

ignal and pump/idler waves, and its amplitude is propor-ional to exp�2�Pz�. The second exponential term is thehase of the signal wave. When the pump is incoherent,�T� and m�T� represent the amplitude and phase fluc-uation of the signal wave, respectively. Fortunately, byirtue of the ergodic and Gaussian properties of randomeld Ap�0,T�, one has the inequalities �m�T���Tc /T , �n�T����Tc /T [7,19]. They indicate that m�T�0, n�T�→0 when T is large, and therefore the signal

ight may have a high degree of coherence. It can be in-erpreted physically as the result of the convection-nduced average process and the signal is no longer sen-itive to the fluctuation of the pump [7]. This kind ofveraging effect is also discussed in [20,21]Define the nonlinear characteristic length Lnl= t0vp

1/ ��P0� [13] and the effective correlation length Lceff

Tc��1/vp−1/vs�−1�=Tc / ���1s�. As pointed out in [7], spon-aneous generation of a coherent signal wave occurs when

ceff�Lnl. It requires a strong convection between the sig-al and pump waves, as well as a very small group veloc-

ty difference for the pump and idler waves. Besides, aignificant FWM process of the three waves requires ahase-match condition, i.e., ���0. These conditions areot easily achieved in the usual optical fiber at the sameime. Fortunately, we find that a photonic crystal fiberPCF) structure with two zero-dispersion wavelengthsroposed in [22] meets the requirements. In [22], the au-hors display the dispersion orders of the PCF from �2 to12 at a fixed wavelength. So, it is convenient to deter-ine the relative value of �1j and the phase mismatch ��

f any three waves with their frequency of 2�p=�s+�i.e use the nonlinear PCF1 proposed in [22] to perform

ur numerical simulation to verify the theoretical result.The frequency of the three waves and their dispersion

arameters are displayed in Table. 1. Figure 2 shows rela-ive value of �1 calculated with the data in [22].

Table 1. Frequencies, Dispersion Parameters, andthe Relative Value of �1j of the Three Waves

Idler Pump Signal

CentralFrequency(THz)

248.50 287.29 326.08

��1j=�1j−�1p (s/m) 4.29�10−15 0 4.33�10−13

�2�s2/m� 3.76�10−27 −3.49�10−28 5.30�10−27

�3�s3/m� −6.29�10−41 1.15�10−41 3.40�10−41

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2062 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 Y. Yan and C. Yang

The phase mismatch of the three waves ��=−1.94610−4 m−1. The nonlinear parameter of the fiber �23 W−1 km−1. We assume the fiber’s length is 1000 m.he input incoherent pump’s correlation time Tc is 400 fs,nd the corresponding spectral width is about 1.5 THz.he pump average power is 0.1 W. From the parametersbove, one has Lnl�400 m and Lc

eff�1 m. The ratio

ceff /Lnl�0.0025. Although in the theory we neglectecond-order and higher-order dispersion of the threeaves by considering the rapid fluctuation of pump fieldue to the short correlation time Tc, we also take into ac-ount the second-order dispersion and third-order disper-ion in the numerical simulation. The second-order dis-ersion lengths of the three waves are 50–200 m and thehird-order dispersion lengths are 1000–2000 m. Theourth-order dispersion lengths are above 40000 m. Sincehe fiber length we used in simulation is 1000 m, it is nec-ssary to consider �2j and �3j in the numerical simulation.

ig. 2. (Color online) Illustration of the relative �1 of the PCFersus frequency. The circles mark the �1 of the three waves.

ig. 3. (Color online) Numerical results of high-coherence signalenerated from incoherent pump. PCF length z=1000 m; pumpverage power 0.1 W; nonlinear parameter �=23 W−1 km−1. (a)mplitudes of the input signal wave and three output waves. The

nput signal amplitude is given in an arbitrary unit. (b) Numeri-al results of the normalized autocorrelation functions Rj�t� ob-ained from temporal profiles of the input signal wave and thehree output waves.

. NUMERICAL RESULTSigure 3(a) shows the amplitude profiles �Aj� obtained by

he numerical simulation of Eq. (2) with the above param-ters. The input incoherent pump wave’s temporal widths about 2 ns. The input signal wave is also incoherentight, which has same property as the incoherent pumpight. As expected from the theory, thanks to the phase-ocking mechanism and the convection between the signalnd pump/idler wave, the signal wave is with a high de-ree of coherence in the FWM process, while the idlerave absorbs the pump’s fluctuation. Figure 3(b) illus-

rates the temporal autocorrelation functions of the out-ut waves and the ratio of correlation time between theignal and the pump, Tc,s /Tc,p�200.

Figure 4 shows the comparison of the simulation re-ults involving and not involving the higher-order disper-ion of the three waves when the fiber length is 2000 m.he numerical result of the output signal shape withoutonsidering the higher dispersion parameter agrees wellith Eq. (11). Although the higher-order dispersion leads

o wave broadening and worse wave shapes than theheory predicts, the signal wave still has much higher co-erence than the pump and input signal waves.

. CONCLUSIONn conclusion, we have shown theoretically and numeri-ally that in a Kerr nonlinear optical fiber, a coherent sig-al wave can be spontaneously generated and sustainedrom an incoherent pump light in a degenerate four-waveixing process. This phenomenon requires the same

roup velocities of pump wave and idler wave, which isalled the phase-locking mechanism. This mechanismakes the incoherent pump wave and idler wave mutu-

lly coherent in the FWM process and serves as the basisf coherent evolution of the signal wave. Besides, the con-ection between the signal and pump/idler waves is alsoequired. The convection-induced average processmooths both the amplitude and phase of the signal wave,aking the signal wave less sensitive to the amplitudeuctuation of the pump.

ig. 4. (Color online) Numerical results of normalized ampli-ude and correlation function for output signal waves with andithout the second-order and third-order dispersion in simula-

ion. PCF length z=2000 m; pump average power 0.1 W; nonlin-ar parameter �=23 W−1 km−1. (a) With second-order and third-rder dispersion; (b) without second and third-order dispersion.

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Y. Yan and C. Yang Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2063

CKNOWLEDGMENTSe are very grateful to Antonio Picozzi in Université deourgogne for the fruitful discussion with him. We also

hank Dr. Xiaosheng Xiao in Tsinghua University for hisind revision of this paper and the reviewers for theirelpful comments. This work was supported by the Na-ional Natural Science Foundation of China (NSFC) grant0878007.

EFERENCES1. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008).2. A. Picozzi and M. Haelterman, “Parametric three-wave

soliton generated from incoherent light,” Phys. Rev. Lett.86, 2010–2013 (2001).

3. V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum,1970).

4. D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolutionof nonlinear three-wave interactions. I. Interaction in ahomogeneous medium,” Rev. Mod. Phys. 51, 275–310(1979).

5. S. Pitois, G. Millot, and S. Wabnitz, “Polarization domainwall solitons with counterpropagating laser beams,” Phys.Rev. Lett. 81, 1409–1412 (1998).

6. A. Picozzi and M. Haelterman, “Dispersion-induceddynamical transition in parametric solitary waves,” Phys.Rev. Lett. 84, 5760–5763 (2000).

7. A. Picozzi, C. Montes, and M. Haelterman, “Coherenceproperties of the parametric three-wave interaction drivenfrom an incoherent pump,” Phys. Rev. E 66, 56605–56605(2002).

8. C. Montes, A. Picozzi, and K. Gallo, “Ultra-coherent signaloutput from an incoherent cw-pumped singly resonantoptical parametric oscillator,” Opt. Commun. 237, 437–449(2004).

9. C. Montes, W. Grundkötter, H. Suche, and W. Sohler,“Coherent signal from incoherently cw-pumped singlyresonant Ti: LiNbO�3 integrated optical parametricoscillators,” J. Opt. Soc. Am. B 24, 2796–2806 (2007).

0. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot,

“Incoherent solitons in instantaneous response nonlinearmedia,” Phys. Rev. Lett. 92, 143906 (2004).

1. Y. Yan and Y. Changxi, “Four-wave mixing betweencoherent signal and incoherent pump light in nonlinearfiber,” J. Lightwave Technol. 27 (2009). doi: 10.1109/JLT.2009.2027213

2. K. Hammani, C. Finot, and G. Millot, “Emergence ofextreme events in fiber-based parametric processes drivenby a partially incoherent pump wave,” Opt. Lett. 34,1138–1140 (2009).

3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).4. F. S. Yang, M. E. Marhic, and L. G. Kazovsky, “cw fiber

optical parametric amplifier with net gain andwavelengthconversion efficiency,” Electron. Lett. 32,2336–2338 (1996).

5. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A.R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley,“Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photonics Technol.Lett. 15, 673–675 (2003).

6. K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Phase-conjugate pump dithering for high-quality idler generationin a fiber optical parametric amplifier,” IEEE PhotonicsTechnol. Lett. 15, 33–35 (2003).

7. J. W. Goodman, Statistical Optics (Wiley, 2000).8. H. Feshfach and P. M. Morse, Methods of Theoretical

Physics (McGraw-Hill, 1953).9. R. L. Stratonovich, Topics in the Theory of Random Noise

(Science Publishers, 1963).0. W. Astar, A. S. Lenihan, and G. M. Carter, “Polarization-

insensitive wavelength conversion by FWM in a highlynonlinear PCF of polarization-scrambled 10-Gb/s RZ-OOKand RZ-DPSK signals,” IEEE Photonics Technol. Lett. 19,1676–1678 (2007).

1. C. J. McKinstrie and C. Xie, “Polarization-independentamplification and frequency conversion in strongly-birefringent fibers,” Opt. Express 16, 16774–16797 (2008).

2. A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre,“Tailoring cw supercontinuum generation inmicrostructured fibers with two-zero dispersionwavelengths,” Opt. Express 15, 11553–11563 (2007).