VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001

Quantum-Noise-Induced Order in Lasers Placed in Chaotic Oscillationby Frequency-Shifted Feedback

Jing-Yuan Ko, Kenju Otsuka, and Tamaki KubotaDepartment of Applied Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan

(Received 17 October 2000; revised manuscript received 16 January 2001)

A kind of chaotic oscillations featuring random switching between sustained relaxation oscillations(RO) and spiking oscillations (SO) has been demonstrated in lasers with frequency-shifted feedback.The presence of stochastic frequency locking between two periodicities of RO and SO motions andselective quantum-noise-induced ordering of chaotic spiking oscillations is demonstrated theoreticallyand experimentally.

DOI: 10.1103/PhysRevLett.86.4025 PACS numbers: 42.65.Sf, 05.45.Jn, 42.50.Lc, 42.55.–f

The effect of noise on nonlinear systems is an intrigu-ing general subject from viewpoints of nonlinear dynam-ics and applications. In various devices, the increase ofthe noise amplitude leads to a degradation of the outputsignal. In nonlinear systems, however, this is not alwaystrue and a finite amount of noise can induce a dynamicalstate which is more ordered. Examples of such a noise-induced order include, for instance, the synchronizationwith a weak periodic input signal in bistable systems (sto-chastic resonance) [1,2] and the minimalization of pulseinterval fluctuations in autonomous excitable oscillators(coherence resonance) [3,4]. These studies concerningnoise-induced ordering to date, however, have been re-stricted to the effect of “externally applied” artificial noise.In real nonlinear systems, intrinsic quantum noise alwaysexists and degrades their performances. Then, the ques-tion arises: Is there any macroscopic nonlinear system inwhich quantum noise can result in ordering? From thisviewpoint, lasers are expected to provide a promising sys-tem for investigating the effect of internal intrinsic quan-tum noise (spontaneous emission) on nonlinear dynamics.

Among many models of laser instabilities, instabilitiesin lasers with delayed feedback initiated by Lang andKobayashi [5] have been attracting much attention overthe past 20 years. Three universal routes to chaotic relax-ation oscillations, low-frequency fluctuations, and coher-ence collapse have been demonstrated in semiconductorlasers [6]. Recently, pump-noise-induced coherence reso-nance has been demonstrated experimentally [4]. Thenoise-induced chaotic burst generation was reported in mi-crochip lasers with delayed feedback, featuring randomswitchings between the stable and chaotic spiking oscil-lation states [7]. It has been numerically demonstratedin the model of lasers with incoherent delayed feedbackthat there coexist two types of chaotic motion, namely,sustained relaxation oscillation (RO) which is born fromthe lasing stationary solution and spiking oscillation (SO)which builds up from the nonlasing stationary solution andtheir intensity approaches zero during spikes, that dependson the spontaneous emission rate, and they exhibit gener-alized bistability [8].

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In this paper, we propose the model of lasers withfrequency-shifted delayed feedback and demonstrate,for the first time, quantum-noise(spontaneous emission)-induced ordering in the random switching regime betweenchaotic RO and SO states. Ordering is shown to arisefrom stochastic locking of the two periodicities of RO andSO motions both theoretically and experimentally.

The model equations of lasers with frequency-shiftedfeedback are given by [9–11]

dN�t��dt � �w 2 1 2 N�t� 2 �1 1 2N�t��E�t�2��K ,

(1)

dE�t��dt � N�t�E�t� 1 mE�t 2 tD� cosC�t�

1

q2´�N�t� 1 1� j�t� , (2)

df�t��dt � m�E�t 2 tD��E�t�� sinC�t� , (3)

C�t� � DVt 2 f�t� 1 f�t 2 tD�

2 �Vo 1 DV�2�tD , (4)

where E is the normalized field amplitude, N is thenormalized excess gain (population inversion density),w � P�Pth is the relative pump power normalized bythe threshold, f is the phase of lasing field, C is thephase difference between the lasing field and the feedbackfield, DV � Vi 2 Vo is the normalized frequency shift(Vo,i � vo,itp is the normalized frequency of the lasingfield and feedback field), m is the feedback coefficient,K � t�tp is the lifetime ratio, t and tD are the timeand delay time normalized by tp . Here the last term inEq. (2) expresses the quantum (spontaneous emission)noise where ´ is the spontaneous emission rate and j�t� isthe Gaussian white noise with zero mean and d correlatedin time �j�t�j�t0� � d�t 2 t0�. Under the condition ofDV � jVo 2 Vij ¿ m, the self-mixing modulation atthe detuning frequency DV appears in the short delaylimit [12]. In this case, the equations become equivalentto those of loss-modulated lasers [12,13]. When thefeedback coefficient m is increased, the phase differenceC becomes effective and complex dynamics are expectedto occur.

© 2001 The American Physical Society 4025

VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001

0 20 40 60

Time ( x 103)

0

2

4

E2

0.0

0.2

0.4

E2

0.00 0.01 0.02m

0

5

10

15

Pea

kV

alue

s

(a) (b)

(c)

FIG. 1. (a) Bifurcation diagram for n � 2 �DV � 2VR � 0.020 86�, w � 1.1125, K � 2 3 103, ´ � 0, and delay time tD � 1.(b) Periodic sustained relaxation oscillation (RO) at m � 0.0012. (c) Self-induced switching behavior at m � 0.0089.

From numerical simulations of Eqs. (1)–(4), we foundthat periodic sustained RO are resonantly excited atrelaxation oscillation frequency, VR � vRtp , when thefrequency shift DV is tuned to harmonics of relaxationoscillation frequency, i.e., DV � nVR�n � 2, 3, 4, . . .�in small m regime. As the feedback coefficient wasincreased, random switchings between RO and SO stateswere numerically obtained. Dwell times within the ROstate were found to decrease when the feedback coeffi-cient was increased. A bifurcation diagram and typicalwaveforms of periodic RO and switching behavior areshown in Fig. 1 in the absence of noise, i.e., ´ ! 0.Such a self-induced switching between chaotic RO andSO states differs from noise-induced random switchingbetween RO and SO states in microchip lasers [7] andpredicted generalized bistability between RO and SO, inwhich self-induced random switching does not take place[8]. This implies that the present switching behaviorbetween RO and SO can be self-induced deterministicallyin this model.

Next, let us examine the effect of quantum noise on thischaotic behavior. Figures 2(a)–2(c) show the evolution ofchaotic oscillations over time indicating random switch-ings between the two types of motion, i.e., RO and SO, atn � 2 for three different spontaneous emission rates, ´, ata fixed feedback coefficient, m. In the presence of weaknoise, i.e., small ´, random switching between RO and SOstates occurs as shown in Fig. 2(a), in which both RO andSO regimes indicate chaotic evolutions. As ´ is increased,the chaotic spiking oscillation (SO) is found to be selec-tively tamed accordingly and the periodicity appears in theSO regime, as can be seen in Fig. 2(b). However, this peri-odicity tends to be suppressed at larger values of ´ again asshown in Fig. 2(c). The spontaneous emission rate is givenby ´ � cst�pw2

onL (c: velocity of light; s: stimu-lated emission cross section; wo: lasing beam spot sizeaveraged over the cavity length L; n: refractive index).

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Therefore, it can be changed by using lasers with differentlengths (i.e., L). This point will be discussed later.

In order to quantitatively characterize such quantum-noise-induced ordering in chaotic lasers, we employ the in-dicators used in [4]: the standard deviations Ra and Rb ofscaled variables a � A��A and b � T��T , and the jointentropy H�a, b� � 2

PP�a, b� log2 P�a, b�, where A is

the amplitude of the intensity peaks, T is the time interval

FIG. 2. Numerical waveforms in the switching regime�m � 0.0125� for different spontaneous emission rates,´ � 10210 (a), 1028 (b), and 1027 (c). Other parameters arethe same as Fig. 1. Experimental waveforms in the switchingregime at n � 2 for LNP lasers with the thickness of 3 mm(d), 1 mm (e), and 0.3 mm (f). The relaxation oscillationfrequencies fR without feedback were 250 KHz for (d), (e) and2.5 MHz for (f ).

VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001

between peaks, and P�a, b� is the joint probability of a

and b. Ra thus indicates the degree of disorder in am-plitude and Rb indicates the degree of disorder in phase.H�a, b� indicates the degree of disorder in both time andamplitude of the peaks. Figures 3(a)–3(c) show these in-dicators as a function of the strength of spontaneous emis-sion. Here we used 5 000 000 data points (step size � 0.1)for statistical analysis. It is apparent that the minimumvalues for all quantities correspond to the same amountof quantum noise. Contrary to the previous results ofcoherence resonances observed in a semiconductor lasersubjected to optical feedback [4], noise-induced orderingoccurs in both amplitude and phase in this case.

Let us verify the theoretical results of quantum-noise-induced ordering experimentally. We carried out ex-periments using laser-diode (LD)-pumped microchipLiNdP4O12 (LNP) lasers with different cavity lengths toidentify the effect of quantum noise (spontaneous emis-sion) on chaotic dynamics quantitatively in the switchingregime because the spontaneous emission rate is inverseproportional to mode volume. We used 3-mm, 1-mm,and 0.3-mm-thick LNP lasers for this purpose, wherethe common experimental apparatus (LD, frequencyshifter, reflector) was employed. The input surfaces werecoated to be antireflective at the LD pump wavelength(808 nm) and highly reflective at the lasing wavelength.The output surfaces were coated to be 99% reflective atthe lasing wavelength. The experiment was carried outin single transverse TEM00 and single longitudinal mode

ε

0.4

0.6

0.8

Rα

0.3

0.4

0.5

Rβ

6.0

7.0

8.0

H(α

,β)

σ

(a)

(b)

(c)

(d)

(e)

(f)

10-11 10-1010-9 10-8 10-7

10-110-210-310-410-5

FIG. 3. Standard deviations of scaled variables, amplitude andphase, and joint entropy of pulsation peaks as a function ofspontaneous emission rate ´ (a), (b), (c) and pump noise s (d),(e), (f ) in the switching regime. Parameters are the same asFig. 2. For each run, 5 3 106 data points were used to analyzethe statistical properties.

oscillation regimes for all cases. The distance betweenthe laser and the reflector was 60 cm, where the delaytime of 4 ns is much smaller than the spiking pulse widthof 100 ns of the laser. (Therefore, the present systemcan be considered to be operated in a short delay limit.)When the round-trip frequency shift fs provided by twoacousto-optic modulators (one provides up-shift and theother provides down-shift) placed between the laser andthe reflector was tuned to harmonics of the relaxationoscillation frequency fR (i.e., fs � nfR , n � 2, 3, 4), theresonant excitation of a periodic sustained RO at the fre-quency fR was observed by weak feedback similar to thetheoretical result for all lasers. As the feedback ratio wasincreased by the variable attenuator, the periodic RO wasdestabilized, being interrupted at times by a chaotic SO,and random switching between the two types of motionappeared. Dwell times within the RO state were foundto be shortened by increasing the feedback ratio similarto the numerical result. Experimental results for n � 2for three lasers are shown in Figs. 2(d)–2(f). The ordereddynamics in spike amplitude and frequency similar toFig. 2(b) appears for the 1-mm-thick laser, while the otherlasers exhibit more wild chaotic variations of amplitudeand frequency in SO regimes. This result corresponds wellto theoretical prediction shown in Figs. 2(a)–2(c). Herespontaneous emission rates calculated from measured spotsizes wo [14], emission cross section, and fluorescencelifetime are 8.3 3 10210, 1.7 3 1028, and 6.5 3 1027

for 3-mm, 1-mm, and 0.3-mm lasers, respectively.Physically, this phenomenon of quantum-noise-

induced ordering can be interpreted as follows: spikingoscillations which build up from the nonlasing solutionof model equations without feedback, which is givenby E2

nl � 2´w��w 2 1�2 ø 1, from Eqs. (1)–(4) as-suming ´ ø 1 and m � 0 [8]. Therefore, spikingoscillations are strongly affected by the spontaneousemission noise. It was demonstrated analytically and ex-perimentally that in deeply modulated lasers, the spikingfrequency fSO depends on the spontaneous emission rate[15], while the relaxation oscillation frequency fR ��1�2p�

p�w 2 1��ttp is determined from the lasing sta-

tionary solution given by E2l w 2 1, from Eqs. (1)–(4)

assuming ´ ø 1 [8], and it is thus not affected by noise.This result may be applicable to the present system. Letus examine dynamics which is occurring in the switchingregions indicated by + for ordered states in Figs. 2(b)and 2(e) more precisely by joint time-frequency analy-sis of time series [16]. Results are shown in Fig. 4,where power spectra around A (RO) and B (SO) nearthe switching point are also shown. It should be notedthat the frequency ratio between fSO and fR is lockedto the rational number of 3�4 for both cases. This isevident from 3�4 locking indicated by ). This stronglysuggests that quantum-noise-induced stochastic frequencylocking between two periodicities of RO and SO resultsin ordering in the present chaotic dynamic system. To

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VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001

FIG. 4. Joint time-frequency analysis of time series in theswitching region, SO ! RO ! SO, indicated by + in Figs. 2(b)and 2(e). (a) Numerical result. (b) Experimental result. Ashort-time Fourier transform (power spectra) around A (RO) andB (SO) within Hanning window length [16] of 8.192 3 103 (a)and 102.4 msec (b) are also shown on the right. ) indicates3�4 locking.

confirm this idea, we omit the spontaneous emissionnoise in Eq. (2) and then introduce the additive Gaussianwhite noise to the relative pump power term in Eq. (1),i.e., replacing w by �w 1 sz �t�� where s is the noisestrength and z �t� is the Gaussian white noise whichsatisfies �z �t� � 0 and �z �t�z �t0� � d�t 2 t0�. Resultsare shown in Figs. 3(d)–3(f). As expected, ordering doesnot occur for such an externally applied noise in thepresent system because the pump noise does not affect thespiking oscillation frequency.

The same results have been obtained for n � 3 and4, although harmonic resonance and switching behaviorsoccur for a larger feedback coefficient, m, as n is increased.

In summary, a novel nonlinear system of lasers with fre-quency-shifted feedback has been proposed and its genericfeatures have been explored. Excitation of sustained RO byharmonic resonances and the nonstationary chaotic state,featuring random switching between RO and SO have beenpredicted and verified experimentally. Statistical analy-

4028

sis of the chaotic behavior has been carried out focus-ing on the effect of quantum noise theoretically. Thequantum-noise-induced ordering in chaotic SO regimes,featuring stochastic frequency locking between the two pe-riodicities of RO and SO, has been demonstrated theoreti-cally and experimentally.

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