动力学混沌【混沌动力学3】

Optics Communications 285(2012)143–148

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Optics Communications

j o u r na l h o me p a g e :ww w. e l s e v i e r. c o m /l oc a te /op t c o m

Chaotic dynamics of erbium-doped fiber laser with nonlinear optical loop mirror

Lingzhen Yang ⁎, Li Zhang, Rong Yang, Li Yang, Baohua Yue, Ping Yang

Institute of Optoelectronic Engineering, Department of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan, 030024, China

a r t i c l e i n f o a b s t r a c t

We experimentally and numerically demonstrate the chaotic dynamics of the erbium-doped fiber laser with a nonlinear optical loop mirror. When the polarization controllers are fixed at an appropriate orientation, we observe that the fiber laser exhibits a period-doubling route to chaos with increasing the pump power in the experiment and simulation. The numerical simulation shows a good agreement with the experimental results. The results show experimentally and numerically that the chaotic dynamics of the erbium-doped fiber laser is related to the polarization state and the pump power of light in the cavity.

2011Elsevier B.V. All rights reserved.

Article history:

Received 13April 2011

Received in revised form 20July 2011Accepted 10September 2011

Available online 28September 2011Keywords:Chaos

Fiber laser

Nonlinear optical loop mirror (NOLM)

1. Introduction

Due to its natural con fidentiality, anti-interference and non-predictability, chaos has been extensively applied in the fields of se-cure communication [1], chaotic sensor [2]and imaging encryption [3]. It is well known that the erbium-doped fiber lasers are relatively simple and compact. Chaotic properties of many kinds of fiber lasers have been proposed and demonstrated [4–6]. The basic methods of the chaotic generation in erbium-doped fiber lasers include the pump modulation, the loss modulation, the nonlinear Kerr effect and the nonlinear optical loop mirror (NOLM)[7–13]. The NOLM has been used as fast saturable absorber to passively mode-locked laser oscillators [14]or sagnac interferometer [15]. A.L. Steele numer-ically analyzed the optical instabilities in a nonlinear optical fiber loop mirror with feedback, and the results showed that the NOLM with feedback can lead to the period-doubling route to chaos [16]. C.A. Merchant theoretically used the NOLM with optical feedback generat-ing chaos [17]. In 2005, Lai W.J. analyzed the dynamics of a NOLM and a nonlinear amplifying loop mirror (NOLM–NALM) fiber ring laser, and they foresaw the possibility of the chaotic operation of the laser [18]. However, the works did not extend further into the chaotic dynamics of NOLM –NALM fiber ring laser.

In this paper, we study the chaotic dynamics of the erbium-doped fiber ring laser with a NOLM without the optical injection by experi-mental validation and numerical simulation. When the polarization controllers are adjusted to an appropriate position, we can obtain the period-doubling route to chaos with increasing the pump power. The experimental and numerical results show that the chaotic

dynamics of fiber laser with NOLM is related to the polarization state and the pump power of the light in the cavity. 2. Experiment 2.1. Experimental setup

The experimental setup of the erbium-doped fiber ring laser (EDFRL)with a NOLM is shown in Fig. 1. The fiber ring cavity consists of a wavelength division multiplexer (WDM),an erbium-doped fiber (EDF)with a length of 9m, and a polarization independent isolator (ISO).A 980-nm laser diode (LD)with the maximum pump power of 250mW is used to pump EDF through a 980/1550-nmWDM. The ISO ensures the unidirectional propagation of light. The NOLM con-structed with single mode fiber (SMF)is connected together with the ring cavity through a 2×2fiber coupler (70:30,C 1). Two polariza-tion controlers (PCs)and a 30-m SMF are inserted inside the NOLM. The PCs are set to change the polarization states of the light waves, and determine the interference in the NOLM. The output of the fiber laser is taken from the 10%output of fiber coupler (C2). The output light in the laser cavity is monitored by a 500MHz bandwidth and 5Gbps sampling ratio digital oscilloscope together with a 2GHz photo detector (PD).The chaotic spectrum is measured with a RF spectrum analyzer. 2.2. Experimental results

The cavity round-trip time of the fiber laser is 200.8ns corre-sponding to the loop length of the fiber laser with 40.16m in our ex-periment. We adjust the polarization controllers and the pump power to observe the output dynamics of the fiber laser. In order to achieve

⁎Corresponding author. Tel.:+[1**********]1.

E-mail address:of fice-science@tyut.edu.cn(L.

Yang).

0030-4018/$–see front matter 2011Elsevier B.V. All rights reserved. doi:

10.1016/j.optcom.2011.09.029

144L. Yang et al. /Optics Communications 285(2012)143–148

Fig. 1. Experimental setup of the EDFRL with a

NOLM.

the chaotic output, the PCs are carefully adjusted. When we appropri-ately select the orientations of the two polarization controllers to an appropriate combination, the period-one, period-two, and quasi-periodicity states are observed separately for the pump power to be 42.8mW, 45.1mW, and 46.9mW, which are shown in Figs. 2–4respectively. The time series, the power spectra, the autocorrelation traces, and the phase portraits of different oscillation states in Figs. 2–4show the route of period-doubling to quasi-periodicity states.

The output of the fiber laser begins to enter a chaotic state, when we further increase the pump power to 62.5mW. The chaos still exists when the pump power is increased to 240.0mW. The first and second columns of Fig. 5illustrate the time series, the power spectra, the autocorrelation traces, and the phase portraits for pump powers of 77.0mW and 240.0mW respectively.

From the experiment, we find that the chaotic state does exist in a very small region. In addition, if we adjust polarization controllers to other positions, we observe the fiber laser could demonstrate various dynamic characteristics such as multi-pulse, mode-locking.

Fig. 2. Pump power of 42.8mW for period one. Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase

portrait. Fig. 3. Pump power of 45.1mW for period two. (a)Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase

portrait.

Fig. 6shows the optical spectrum of the fiber laser. The center wavelength is 1603nm. The chaotic optical spectrum exhibits a strongest sideband in Fig. 6, indicating the existence of the strong self-phase modulation on the chaotic trains. The output power of the fiber laser is shown in Fig. 7with a threshold power of 33mW. 3. Numerical simulation 3.1. Numerical model

The numerical analysis of the chaotic dynamics of EDFRL with a NOLM is shown in Fig. 8based on the model of EDFRL proposed by Abarbanel [12]and Lexis [19].

The propagation equation of the electrical field E (z , t ) =ε(z , t ) e i (k 0z -ω0t ) in the EDFRL can be described by ∂εx ; y ðz ; τÞ

¼gn ðτÞεx ; y þL x ; y εx ; y þN x ; y εx ; y :ð1Þ

The retarded time is given by τ=t -z /v g , where v g is the group velocity of the waves, g is the gain parameter, n (τ) population inver-sion, and ω0is the light wave angular

frequency.

Fig. 4. Pump power of 46.9mW for quasi-periodicity state. (a)Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase portrait.

L. Yang et al. /Optics Communications 285(2012)143–148145

Fig. 5. Chaotic states of the fiber laser with pump powers of 77.0mW and 240.0mW. (a)Time series, (b)power spectra, (c)autocorrelation traces, and (d)phase

portraits.

The linear operator L x , y includes the linear birefringence, group-velocity dispersion (GVD),and gain dispersion

2ik 0n x Àn y Δi gn ðτÞω2T 22

i ω−β2ω−L x ; y ¼ÆÀÆ; ð2Þ001þωT 2where Δ=n 0(n x -n y ) is the differential birefringence, ωis the signal

angular frequency, and T 2is the decay time of the fast fluctuations of polarization.

The nonlinear operators are

22 2

N x εx ¼i γ εx ðz ; τÞ þ εy ðz ; τÞ εx ðz ; τÞ;

22 2

N y εy ¼i γ εy ðz ; τÞ þ εx ðz ; τÞ εy ðz ; τÞ;

3where γis the nonlinear coef ficient of the fi

ber.

ð3a Þ

ð3b Þ

Fig. 6. Chaotic spectrum of the fiber laser with pump power of 240.0

mW. Fig. 7. Output power of the fiber laser versus pump power.

146L. Yang et al. /Optics Communications 285(2012)143–148

Fig. 8. Numerical model of the EDFRL with a

NOLM.

In Fig. 8, the NOLM is added to the EDFRL; this adds a second time delay to the EDFRL. When the light arrives at the entrance to the NOLM, the input electric field is split into two counter propagating fields which return in coincidence to recombine at the coupler. The optical path lengths experienced by the two propagating fields are exactly the same, since they follow the same route but in opposite directions. The transmission equation of the NOLM can be described by

p ðt þτD Þ¼TP ðt Þ;

ð4Þ

where T is the transitivity of the NOLM, T ¼2k ð1−2k Þf 1þcos ½ð1−2k Þγl NOLM P ðt Þ g ;

ð5ÞτD is the round trip time of the NOLM, and k is the coupling ratio. After the light from the NOLM propagates over the round trip time τD , it will experience a Jones matrix J PC with the polarization controller, then reenter the main ring.

The dynamics of the EDFRL with NOLM can be described by the evolution equations of electrical field ε(t ) and integrated population l A inversion w ðτÞ¼

l −A 1

∫n ðz ; τÞdz . The overall propagation map includ-

ing all the passive and active parts of the ring is εðz ¼l A þl F ; τþτR Þ¼RJ PC TP ðt Þ

ð6Þ

dw ðτÞτR n 2 2gl A w ðτ o ¼Q −n ðτÞþ1þj εðτÞj e Þ

À1; ð7Þ

1

where R =diag (R x , R y ) is the polarization-dependent attenuation, τR is ring round-trip time, l A is the length of EDF, and l F is the length of SMF in the whole cavity. Transmission operator is P (t )=ε(z =l A , t ), Q represents pump strength, and T 1is the lifetime of the excited states.

3.2. Numerical results

We use the following parameters for the simulation unidirec-tional:l A =9m, l F =31.16m, n x −n y =1.8×10−6, n −10=1.46,β2=−20ps 2/km,k =0.7,γ=3×10−3W −1km , T 1=10ms, T 2=1ps, and τR =200.8ns. As the orientations of the two polarization control-lers are selected to be an appropriate combination, we get the period-doubling route to chaos in numerical simulation. The time series, the power spectra, the autocorrelation traces, and phase portraits of different oscillation states in Figs. 9–11show the periodic states for the pump power of 41.7mW, 46.2mW, and 47.0mW separately, where the period-one, period-two, and quasi-periodicity states are obtained respectively. Compared with the Figs. 2–4, the time series, the power spectra, the autocorrelation traces, and the phase portraits have a good agreement. If we further increase the pump power to 68.0mW, the fiber laser will present chaotic state as shown in Fig. 12. The first and second columns of Fig. 5illustrate the time series,

Fig. 9. Pump power of 41.7mW for period one. (a)Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase

portrait.

the power spectra, the autocorrelation traces, and the phase portraits for the pump power to be 68.0mW and 240.0mW respectively. 4. Discussion

In terms of the procedure of the route of period-doubling, quasi-periodic states to chaotic states in the experiment and simulation, we can find random intensity fluctuations in time series, bandwidth enhancement of power spectra, and randomly distributed dots in the phase portrait. The autocorrelation trances for the pump power of 240mW are similar to δfunction with some side lobes. The time interval between the two neighboring side lobes is 200.8ns which is the round-trip time of the cavity corresponding to the length of the EDFRL with NOLM. The cavity round-trip time is estimated as L /v , where L is the cavity length and v =c /n is the speed of light in the fiber, c is the speed of light in vacuum, and n is the index of refraction of erbium-doped fiber. We believe that the high side lobes are in-duced by the repetitive interference of light in the cavity, which suffer a nonlinear phase shift. When the pump power is 240.0mW, the bandwidth of Fig. 12is wider than that of Fig. 5, except for the trailing of power spectra in Fig. 5, which are caused by the limitation of band-width of the oscilloscope.

In terms of the experimental and numerical results, we can get that the chaotic states are not only related to the pump power

but

Fig. 10. Pump power of 46.2mW for period two. (a)Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase portrait.

L. Yang et al. /Optics Communications 285(2012)143–148147

also to the polarization state. If we fix the pump power to a certain value which must be higher than the threshold power, the polariza-tion controllers will determine the output states. For different polari-zation states, we find that the EDFRL with NOLM presents various dynamic characteristics, such as multi-pulse, mode-locking. The cha-otic states can be observed in the proper position of the polarization controllers. Consequently, the chaotic generation is related to the po-larization state and pump power. It is an intrinsic property of such a nonlinear cavity whose output exhibits the period-doubling route to chaos under larger nonlinear phase shift of light in the cavity [20].

5. Conclusion

We have demonstrated the chaotic dynamics behavior of the EDFRL with a NOLM in numerical simulation and experiment. By changing the pump power and the polarization states in the cavity, the chaotic dynamics has been observed. When the polarization con-trollers are fixed in a certain state, we can achieve the period-doubling route to chaos without external optical injection as the pump power increasing. The experimental and numerical

results

Fig. 11. Pump power of 47.0mW for quasi-periodicity state. (a)Time series, (b)power spectrum, (c)autocorrelation trace, and (d)phase

portrait.

Fig. 12. Chaotic states of the fiber laser with pump powers of 68.0mW and 240.0mW. (a)Time series, (b)power spectra, (c)autocorrelation traces, and (d)phase portraits.

148L. Yang et al. /Optics Communications 285(2012)143–148

con firm that the chaotic behaviors of the EDFRL with NOLM are relat-ed to the polarization state and the pump power of the light in the cavity.

Acknowledgments

This work is supported in part by the Natural Science Foundation for Young Scientists of Shanxi Province, China and the National Natural Science Foundation of China under Grant Nos. 2008021008and 60908014. References

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