four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber

4954 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009

Four-Wave Mixing Between Coherent Signal andIncoherent Pump Light in Nonlinear Fiber

Yan Yan and Changxi Yang

Abstract—We report an experimental and theoretical investiga-tion of the four-wave mixing (FWM) process between a coherentsignal light and an incoherent pump light in a highly nonlinearfiber. In the experiment, two main phenomena are observed: 1) thepeak power of the spectrum of the coherent signal light decreasesalong the fiber rather than be amplified; 2) the spectrum of the co-herent signal is broadened. A theory of the FWM process betweenincoherent pump and coherent signal is proposed by consideringthe incoherence of the pump which has amplitude and phase fluctu-ations. The theoretical results agree with the experimental results.

Index Terms—Coherent light signal, four-wave mixing (FWM),incoherent light pump.

I. INTRODUCTION

F OUR-WAVE mixing (FWM) of coherent light in a highlynonlinear fiber has been studied comprehensively and its

application has been well developed, such as the fiber-opticalparametric amplifiers and wavelength converter in wavelength-division multiplexing (WDM) systems [1]–[5]. FWM involvingincoherent beams (incoherent FWM) attracts people’s interestsince incoherent FWM can provide polarization-independentgain [6]. Moreover, incoherent pump has high stimulated Bril-louin scattering (SBS) threshold in optical fiber, so high powerpump could be used to enhance the FWM conversion efficiency.However, in the case of coherent pump the complicated am-plitude or phase modulation is usually taken to suppress SBS.Using incoherent pump can simplify the FWM setup comparedwith the case using the coherent pump light [7], [8]. Jiang andChung experimentally observed that the wavelength conversionefficiency of incoherent FWM is higher than that of coherentFWM. They developed a theory of nondegenerate FWM withincoherent beams (both signal light and pump light are inco-herent light)[9], [10]. Gao et al. reported the wavelength conver-sion experiment with coherent pump light and incoherent signallight [11]. They measured the conversion efficiency and ana-lyzed the results using the theory in [9]. Tian et al. reported po-larization-independent wavelength conversion using a coherent

Manuscript received January 21, 2009; revised May 21, 2009. This workwas supported by the National Natural Science Foundation of China (Grant60878007). First published July 14, 2009; current version published September10, 2009.

The authors are with State Key Laboratory of Precision Measure-ment Technology and Instruments, Department of Precision Instrumentsand Mechanology, Tsinghua University, Beijing 100084, China (e-mail:[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/JLT.2009.2027213

Fig. 1. Experimental setup to investigate FWM process between inco-herent pump and coherent signal. CW Laser: Continuous wave laser (SantecMLS-8100). ASE: Amplified spontaneous emission (ASE) source (Optical-Wave BLS-C). EDFA: Erbium-doped fiber amplifier (HA300). PC: Polarizationcontroller. BPF: Band-pass filter (Santec OTF-930). HNL-DSF: Highly non-linear dispersion-shifted fiber. OSA: Optical spectrum analyzer (Agilent86142B). AWG: Array waveguide grating.

pump and incoherent signal [6]. In this paper we present in-coherent FWM experiments between incoherent pump and co-herent signal light. We observed that the peak power of the co-herent signal spectrum decreases along the fiber rather than beamplified. Meanwhile, the spectrum of the coherent signal isbroadened. We propose a model of FWM process involved co-herent signal and incoherent pump by taking into account thepump’s incoherent nature. The experimental results are well ex-plained by this model.

This paper is organized as follows. In Section II, the inco-herent FWM experiments with coherent signal and incoherentpump is presented. In Section III, we propose a model ofFWM process between incoherent pump and coherent signal,in which the stochastic characteristics of incoherent pump istaken into account. In Section IV, the comparison between theexperimental and theoretical results verifies our FWM modelinvolved incoherent pump and coherent signal. The conclusionis given in Section V.

II. EXPERIMENT

The experimental setup of FWM between incoherent pumpand coherent signal is shown in Fig. 1. A tunable contin-uous-wave (CW) laser is used as a coherent signal light source.The un-polarized incoherent pump is provided by a spec-trum-sliced amplified spontaneous emission (ASE) source, andits 3 dB bandwidth is about 0.4 nm. The pump is amplifiedby an erbium-doped fiber amplifier (EDFA). After passingthrough an optical tunable filter, the incoherent pump is cou-pled with the coherent signal into a 1-km-long highly nonlineardispersion-shifted fiber (HNL-DSF) through a 90:10 coupler.The zero-dispersion-wavelength (ZDW) of the nonlinear fiberis 1542.5 nm. At the end of the HNL-DSF, we use an opticalspectrum analyzer (OSA) to observe the spectrum of signal,

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YAN AND YANG: FOUR-WAVE MIXING BETWEEN COHERENT SIGNAL AND INCOHERENT PUMP LIGHT IN NONLINEAR FIBER 4955

Fig. 2. Spectrum of signal light when pump on (dash, green line) and pumpoff (solid, blue line). Comparison between them shows the peak power loss andbroadening of coherent light spectrum caused by incoherent pump.

Fig. 3. Signal spectrum bandwidth broadening and loss of peak power underdifferent levels of pump power. Signal wavelength �� is 1555.5 nm.

pump and ideal after FWM process. A polarization controller isplaced after the CW laser to adjust the signal polarization state.

Figs. 2 and 3 show the experimental results obtained from theOSA. The incoherent pump wavelength , which isset at the vicinity of ZDW in order to ensure high efficiencyof FWM process. Signal spectrum is on the right and its wave-length is . As Fig. 2 shows, after pushing up pump power, thegenerated idler light with frequency appears. Weobserve that the signal spectrum is broadened and its peak powerdecreases. Our observations are contrary to the case of the FWMprocess with coherent pump and coherent signal, in which thesignal peak power is amplified without spectrum broadening [1].We believe that these phenomena observed in our experimentsare caused by the incoherent nature of the pump.

Fig. 3 shows the spectrum broadening and the loss of peakpower of the signal at different levels of pump power. Theinput signal power is dBm (0.5 mW). With pump powerincreasing, the peak power of the signal spectrum attenuatesmore and the spectral broadening increases. In Fig. 3(b) thelargest loss of the signal peak power is dBm.

In the above experiment, two new phenomena are observedwhen the pump is incoherent. Contrary to the FWM process onlyinvolved coherent light, the peak power of the coherent signaldecreases and its spectrum is broadened.

III. THEORY OF FWM BETWEEN COHERENT SIGNAL

AND INCOHERENT PUMP

Our analysis shows that these phenomena are caused by fea-tures of the incoherent pump from two points of view. One isbroad bandwidth of pump light in spectral domain. The otherone is amplitude and phase fluctuation of the pump in timedomain.

A. Loss and Broadening of Signal Spectrum Induced by BroadBandwidth of the Pump

In the FWM process as shown in Fig. 4(a), two frequencycomponents of the pump with frequency and interactwith the signal with frequency to generate the idler with .The energy conversion requires that

(1)

Since the incoherent pump has numerous frequency compo-nents, and represent any two frequencies in the spec-trum of the incoherent light [9], and, therefore, the newly gen-erated via FWM process varies within a certain frequencyrange. That is why the shape of the idler spectrum is similar tothe shape of pump spectrum.

Fig. 4(b) shows another FWM process that also involves fourfrequencies. In this case, however, the idler frequency is

(2)

In (2), the idler frequencies locate around signal frequency ,since and are close to each other. This kind of idler alsohas similar shape as pump and appears as the broadening of thesignal spectrum. This is the first reason why the signal spectrumbroadens.

To distinguish these two sets of FWM processes and idlers,we define the FWM and idler in Fig. 4(a) as and .Define those in Fig. 4(b) as and .

transfers the energy from the incoherent pump lightto the signal and the and, therefore, amplifies the signalpower. , however, transfers the energy from the coherentlight to and, thus, consumes the signal power. The widegreen arrows in Fig. 4 show the energy transfer directions.

The powers of and are considered as the amountof increase and decrease of signal, respectively. If poweris larger than that of , the peak power of signal wouldattenuate along the fiber. Later, we show mathematically thatdue to different phase-matching conditions [4], these two idlerwaves do have different powers.

We assume that incoherent light is composed of numerousindependent frequency components. The electric field of inco-herent pump is described as [9]

(3)


Fig. 4. Illustration for two FWM processes: (a) Two frequency components ofthe pump interact with the signal to generate the idler � , which is defined as�� . (b) Two frequency components of the pump interact with the signal togenerate the idler � , which is defined as �� . Wide green arrows show thetransferring energy direction.

where is the complex envelope of one component ofpump light with angular frequency . For an incoherent light, it

is described as circular complex Gaussian random variable [9],[13].

The wave equation of idler wave is described as (4), shownat the bottom of the page, where and are theenvelopes of the idler and the signal wave, respectively, is thenonlinear parameter of nonlinear fiber, is the phase mis-match term, and is the fiber loss coefficient.

The solution of is given in [3]. Then the averageintensity of the idler wave with angular frequency can beobtained as (5), shown at the bottom of the page, where is thelength of the fiber. Assuming each component of the incoherentlight is independent of the others and using the properties ofcircular complex Gaussian random variable, (5) becomes

(6)

where is the efficiency of FWM process [4]

(7)

The total power of the idler power is

(8)

We use (5)–(9) to calculate the powers of and .Except for the mismatch term and , other parametersare same. Expressions of two different are [11]

(9)

(10)

(4)

(5)


Fig. 5. Theoretically calculated powers of �� and �� with the differentsignal wavelengths.

where is the zero-dispersion frequency of the highly non-linear fiber.

Fig. 5 shows the theoretical results of andunder the following conditions: dB km, ZDW of thefiber nm, ps km nm , and

W km . The incoherent pump center wavelengthis 1542.7 nm and its 3 dBm width is 0.4 nm. The signal wave-length varies from 1530 nm to 1600 nm. The incoherent pumppower is 47.2 mW, and the input signal power is 0.5 mW. Incalculation, is considerably bigger than when thesignal frequency is away from the center frequency of the in-coherent pump. Therefore, is smaller than ;thus, power is smaller than that of . This is the firstreason for the loss of signal peak power.

Although the loss of peak power and spectrum broadening ofsignal in FWM process has not been observed when the pumpis coherent [1], we can still find the counterpart of inthe FWM process with coherent pump and signal. As pump iscoherent and continuous, its spectrum density is treated as adelta function . So and in the (2) are equal, i.e.,

. Then represents pump cross phase modulation(XPM) to the signal [12]. So XPM with coherent pump is thedegenerate when the pump spectrum can be describedas a delta function.

B. Loss and Broadening of Signal Spectrum Induced by SignalIncoherent Increment

In this part, we demonstrate that not all energy of poweradds to the exact frequency of the signal because of pump in-coherent property. Instead, it adds to a frequency range around

signal frequency. Its width is nearly the same as the pump spec-trum width. This is another reason for the loss and broadeningof signal power.

The equation representing the increment of the signal ampli-tude along the fiber is similar to (4) [see equation (11), shown atthe bottom of the page]. Signal increment is relatedto the amplitude of the incoherent pump componentand the component amplitude , both of whichare complex Gaussian variables. Therefore, is alsoa complex Gaussian variable and carries the features of inco-herent light as well. Incoherent light has amplitude and phasefluctuation in time domain and is of a broadband spectrum cor-respondingly [13]. For this reason, in spectrum domain,power does not totally add to the exact frequency of the coherentsignal . Instead it adds a frequency range centered at andonly part energy of adds to the signal frequency .

We assume the loss of the signal power at its center frequencyas

(12)

where is a coefficient representing the percentageof power that adds to the center frequency of the signal.In simulation, we simulate the incoherent light in time domain,and then obtain its power spectrum density distribution. By con-sidering the width of the pump spectrum and the resolution ofOSA, we assume according to the simulation resultas shown in Fig. 6. Fig. 6(a) shows the numerical simulation re-sult of incoherent light power spectrum density distribution. Theincoherent pump’s energy spreads at a range around its centerfrequency and only ten percent of its energy locates at its centerfrequency. For comparison we also simulate the spectrum of acoherent light beam without any amplitude and phase fluctua-tion. As shown in Fig. 6(b), about seventy percent of the co-herent light energy is concentrated at its center frequency.

IV. THEORETICAL AND EXPERIMENTAL RESULT

We calculate the power of and compared it with the ex-perimental result. The peak power of the can be measureddirectly with an OSA. In our calculation, the parameters of theHNL-DSF are as follows: loss coefficient dB km,ZDW of the HNL-DSF nm, the third-order dis-persion ps km nm , nonlinear coefficientof the fiber W km . The measurement was per-formed by fixing the center wavelength of incoherent pump lightat 1542.7 nm and its 3 dBm width was 0.4 nm. The wavelengthof signal was adjusted from 1530 nm to 1600 nm. The inco-herent pump power was 47.2 mW, and the input signal power

(11)


Fig. 6. Comparison between incoherent light’s power density and coherentlight’s power spectral density. Sum of spectral power of all frequencies is 1.

Fig. 7. Experimental and theoretical results of �� power as a function ofsignal wavelength.

was 0.5 mW. Fig. 7 shows the experimental and theoretical re-sults of the power.

With and (12) we calculated the loss of the signalpeak power. Fig. 8 shows experimental and theoretical results.

We give further explanation for the experimental and theoret-ical results: Fig. 5 shows when the signal wavelength is closeto that of the pump, the efficiencies of and are

Fig. 8. Loss of the signal peak power at its center wavelength.

both high and nearly the same. So the loss in Fig. 8 is mainlycaused by the incoherent increment ; When the signalfrequency is further away from the pump wavelength, the dif-ference between efficiencies of and is notableand it becomes the main cause for the signal peak power loss.Comparing Figs. 6 and 8 we find the signal wavelength at 1566nm with largest peak power loss is determined by the largestpower difference between and , while incoherentincrement contributes less to the loss because theefficiency of decreases rapidly due to the phase-mis-match. When the signal wavelength deviates further from thepump wavelength, the efficiency of FWM between the pumpand the signal is small and so is the loss. Only considering bothmechanisms A and B discussed in Section II can we explain thecurve’s trend in the entire signal wavelength region as shown inFig. 8.

V. SUMMARY

In this paper, the FWM process between coherent lightsignal and incoherent light pump has been investigated. Thefacts are that the peak of the spectrum of signal attenuates andthat the spectrum of signal broadens. These phenomena areexplained by considering the incoherent pump with stochasticfluctuations of amplitude and phase in time domain as well asthe broad spectrum of pump in spectral domain. First, FWMprocess between the incoherent pump and the coherent signalgenerates idler whose frequency is around signal frequency,and this process consumes the signal power. Second, since theincrement of signal is incoherent, the energy transferred frompump to signal would not totally add to the exact frequency ofinput signal. The theoretical results agree with the experimentresults.

ACKNOWLEDGMENT

The authors would like to thank Dr. X. Xiao in the Depart-ment of Precision Instruments and Mechanology, Tsinghua Uni-versity, for his kind revision of this paper, and the reviewers fortheir helpful comments.

REFERENCES

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[2] J. Toulouse, “Optical nonlinearities in fibers: Review, recent examples,and systems applications,” J. Lightw. Technol., vol. 23, p. 3625, 2005.


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[12] G. P. Agrawal, Nonlinear Fiber Optics. New York: Academic, 2007.[13] J. W. Goodman, Statistical Optics. New York: Wiley, 1985.

Yan Yan received the B.E. degree from TsinghuaUniversity, Beijing, China, in 2008. He is currentlyworking toward the M.S. degree, supervised by Prof.C. Yang in the Department of Precision Instrumentand Mechanology, Tsinghua University, China.

His current research interests are nonlinear fiberoptics and waveguide optics.

Changxi Yang received the B.S. degree in physicsfrom Nankai University, Tianjin, China, in 1986,and the Ph.D. degree in physics from the Instituteof Physics, Chinese Academy of Sciences, Beijing,China, in 1992.

In 1992, he joined the Crystal Growth Group,Institute of Physics, Chinese Academy of Sciences.From 1994 to 2001, he visited and did researchat several universities and academic institutes, in-cluding the University of California, Santa Barbara;the National Laboratory of Metrology, Tsukuba,

Japan; the University of Exeter, U.K.; Bell Labs, Murray Hill, NJ; and theUniversity of Arkansas, Fayetteville. He is currently a Professor in the De-partment of Precision Instruments, Tsinghua University, Beijing. His currentresearch interests are nonlinear fiber optics, femtosecond fiber lasers, andsurface-enhanced vibrational spectroscopy.

four-wave mixing between coherent signal and incoherent pump light in nonlinear fiber

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