nonlinear dynamics of a modulated bidirectional solid-state ring laser

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F. Rawwagah and S. Singh Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. B 1785

Nonlinear dynamics of a modulated bidirectionalsolid-state ring laser

Fuad Rawwagah and Surendra Singh

Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701

Received November 28, 2005; accepted April 26, 2006; posted May 23, 2006 (Doc. ID 66216)

We investigate the dynamical behavior of a class-B, bidirectional, solid-state ring laser with a square-wavemodulated pump. Our treatment includes the coupling of oppositely directed traveling wave modes via back-scattering in addition to their coupling via the gain medium. We find that depending on the pump ratio and thedepth and frequency of modulation, the intensity waveforms of the two oppositely directed modes may exhibitperiodic, quasi-periodic, and chaotic behavior. We also find that although the periodic waveforms of mode in-tensities are antisynchronized, chaotic waveforms may be synchronized or unsynchronized. A detailed map ofdifferent operating regimes as functions of frequency and depth of modulation is presented. Curves are pre-sented to illustrate the behavior. © 2006 Optical Society of America

OCIS codes: 140.1540, 140.3580, 270.3100, 270.3430.

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. INTRODUCTIONnteraction of atoms and field in a laser may produce areat variety of periodic or aperiodic intensity waveformsn the output of a laser. In particular, lasers with modu-ated pump or loss are capable of generating nonlinear os-illations as well as chaotic waveforms, depending on theperating parameters of the laser. Chaos in single-modeolid-state lasers with modulated loss or gain was inves-igated theoretically by Khanin and coworkers,1–5 and pe-iod doubling and chaotic emission were observed experi-entally by Klische and coworkers6,7 in a standing-wavedP5O14 laser with a periodically modulated pump.Nonlinear dynamics of solid-state ring lasers operating

n a single mode in both directions have also been inves-igated, both experimentally and theoretically.8–14 Lari-ntsev and coworkers have investigated relaxation oscil-ations and dynamical chaos in bidirectional monolithicd:YAG ring lasers in a series of papers.8–10 Using a co-

inusoidal pump modulation, they investigated the ap-earance of dynamical chaos and attributed its appear-nce to parametric interactions between self-modulationnd relaxation oscillations.One major reason for the instability of a single-mode bi-

irectional ring laser with a homogeneously broadenedain medium is the coupling of oppositely traveling wavesue to backward scattering of one wave (backscattering)n the direction of the other off the optical elements insidehe cavity.11 This coupling leads to a spatial modulation ofain via a population-inversion grating. If this grating issmall modulation of the overall population inversion, it

an be approximated by a sinusoidal function. Zeghlachend Mandel used this approximation and adiabatic elimi-ation of the polarization to derive and solve the equa-ions of motion for a CO2 ring laser.12 They found that al-hough the bidirectional steady-state operation isnstable, the unidirectional operation can be stable or un-table, depending on the operating parameters of the gainedium and the resonator.In this work we investigate the nonlinear dynamics of a

0740-3224/06/091785-8/$15.00 © 2

ingle-mode bidirectional solid-state ring laser (SSRL)nder the influence of square-wave modulation of laserain. This is different from previous investigations thatonsidered sinusoidal modulation of gain. It is knownrom earlier studies15 that the shape of modulation canrofoundly affect nonlinear behavior. Indeed, withquare-wave modulation, we find a phase-space portraitn the parameter space spanned by the frequency andepth of modulation that is different from that with sinu-oidal modulation.

The paper is organized as follows. In Section 2 weresent the equations of motion for the slowly varyingeld amplitudes and population inversion for the SSRLased on semiclassical laser theory. We consider the solu-ion of these equations for a Nd:YAG ring laser with gainodulation in Section 3 and numerically explore the dy-amical behavior they predict over a wide range of modu-

ation frequencies and depths. Its behavior at differentevels of excitations is characterized in terms of Lyapunovxponents and spectral densities for the counterpropagat-ng waves. The results are summarized in Section 4.

. EQUATIONS OF MOTIONonsider an optically pumped SSRL that supports one

ongitudinal mode in each direction. These modes interactith a collection of two-level atoms. To describe the dy-amical evolution of this system, we use a semiclassicalpproach in which the field is treated classically usingaxwell’s equations, and the atoms of the active medium

re treated quantum mechanically using density matrix-quations of motion. Assuming that the mode frequencies1 and �2 are close to the atomic transition frequency0��1��0��2��, we can take both modes to haveearly the same propagation constant. We can then writehe electric field as the sum of two oppositely directedraveling waves:

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1786 J. Opt. Soc. Am. B/Vol. 23, No. 9 /September 2006 F. Rawwagah and S. Singh

E�z,t� = ��E1�t�ei�kz−�t� + E2�t�e−i�kz+�t��, �1�

here the unit vector � denotes the direction of field po-arization. Owing to the interaction of the field with at-ms, the complex amplitudes Ej�t� evolve in time but on aong scale compared with an optical period 2� /�. This

eans the fractional change in Ej in an optical period ismall, �dEj�t� /dt � � � �Ej�t��.

In class B lasers, such as a YAG or Ti:sapphire laser,he decay rate for atomic polarization is large comparedith the cavity and population decay rates. Hence, atomicolarization follows the field and population inversiondiabatically. Under these circumstances, the dynamicalariables for the laser are the two field amplitudes Ej andopulation-inversion density D, which is the difference inhe populations of the upper and lower atomic levels pernit volume. Now the electric fields of the two waves over-

ap inside the gain medium such that the atoms respondo the superposed field leading to a spatially varying in-ersion density. Consequently, we can expand inversionensity D�z , t� in a Fourier series of the form

D�z,t� = Do�t� + D+2�t�ei2kz + D−2�t�e−i2kz + D+4�t�ei4kz . . . .

�2�

ere Do ,D±2, . . . are the Fourier coefficients. These coeffi-ients are slowly varying functions of time and are ex-ected to get progressively smaller for higher spatial har-onics. We will, therefore, neglect Fourier coefficients

eyond the second harmonic. The equations of motion forhe field amplitudes and the Fourier components thenave the form

dE1

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* − D+�1 + s��E1�2 + �E2�2�� . . . . �3d�

ere, for simplicity, we have written D±�D±2. The equa-ion for D− is obtained by noting that D−=D+

*. In the equa-ions of motion, � �s−1� is the cavity-field amplitude decayate, and b�s−1� is the backscattering coefficient repre-enting mode coupling due to scattering of light from oneode in the direction of the other by optical elements in

he cavity. W is the rate at which population density in-reases, and s is the saturation parameter given by

s =4�2

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. �4�

ere � is the electric dipole matrix element for the atomicransition and �� and �� are the population and polariza-ion decay rates, respectively. Note that s has the dimen-

ion of the inverse square of the electric field.These equations can be written in a more compact form

y noting that the amplitude gain coefficient��2D0 /2�o�� at threshold is determined by theondition16

��2Dth

2�o � ��

= �.

hen by introducing dimensionless variables

fj = sEj, do = Do/Dth, d+ = D+/Dth, � = �t, a = ��/�,

�5�

e can write Eqs. (3) more compactly as

f1 = �do − 1�f1 + ib

�+ d+�f2, �6�

f2 = �do − 1�f2 + ib

�+ d−�f1, �7�

do = a�r − do�1 + �f1�2 + �f2�2� − d+f1*f2 − d−f1f2

*�, �8�

d+ = a�− dof1f2* − d+�1 + �f1�2 + �f2�2��. �9�

he pump ratio r is the laser excitation level relative tohreshold

r = W/Wth. �10�

he laser is said to be operating above the threshold if r1 and below threshold if r1. Equations (6)–(9) form

he basis of our investigations.

. RESULTS AND DISCUSSIONhe steady-state solutions of Eqs. (6)–(9) without theackscattering �b=0� are well known.17–19 In this case theaser supports unidirectional operation. A linear stabilitynalysis of a more general set of equations for a bidirec-ional solid-state ring laser equations in the general caseut without backscattering has also been carried out.12

With backscattering coupling of modes, which is alwaysresent in a real system, time-independent steady-stateolutions do not exist. However, when the pump ratio r isot too far above threshold �r−1 � �1, an approximateime-dependent solution, where the mode intensities ex-ibit out-of-phase sinusoidal oscillations (self modula-ion), can be written down.20–22 This solution with inten-ities given by

I1 � �f1�2 ��r − 1�

2�1 + cos bt�,

�11�

I2 � �f2�2 ��r − 1�

2�1 − cos bt�

orresponds to a limit-cycle behavior. Note that the fre-uency of self-modulation is numerically equal to theackscattering coefficient b /2�. We explored the systemynamics without pump modulation for several different

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alues of the backscattering coefficient b /�=100, 150,nd 170 kHz. In all cases, the mode intensities show out-f-phase (phase difference �) sinusoidal oscillations ingreement with Eq. (11). We refer to this regime as theelf-modulation regime. Typical intensity waveforms inhis case are shown in Fig. 1(a). The sum of the intensitiestays constant in time so that the system trajectory in the1−I2 plane is a straight line I1+I2=const., as shown inig. 1(b). The frequency of intensity oscillations, deter-ined by taking the fast fourier transform (FFT) of the

ntensity waveforms, is found to be equal to the back-cattering coefficient b /2� as expected. The backscatter-ng coupling between the modes is clearly responsible forhese antiphase oscillations, as setting b=0 eliminateshese oscillations and the laser reverts to unidirectionalperation.17–19

When the depth of pump modulation is small, the en-elope of intensity self-modulation oscillations is modu-ated at the frequency of pump modulation. The ampli-ude of intensity envelope modulation increases as theepth of pump modulation is increased, and new frequen-ies appear in the spectrum of intensity waveforms. How-ver, for deep gain modulation, analytic solution (even ap-roximate ones) are not known, even in special cases. Theynamical behavior that these equations predicted ineneral was, therefore, explored numerically.

Recall that the field amplitudes fj and the populationomponent d+ are complex quantities. The equations ofotion [Eqs. (6)–(9)] were transformed into eight autono-ous equations for real variables. These equations were

olved numerically in double precision using a FORTRAN

7 code based on the eighth order Runge–Kutta methodith a fixed step size. The step size and time of integra-

ig. 1. Time evolution of the counterpropagating mode intensi-ies in the self-modulation regime for pump ratio r=1.5 and back-cattering coefficient b /2�=100 kHz. The two signals are sinu-oidal and out of phase by �.

ion were chosen such that a further variation of these pa-ameters did not change the maximal Lyapunov exponentithin certain limits. Lyapunov exponents were computedsing the approach followed by Wolf et al. in Ref. 23. Theethod can be explained as follows. First, the system of

ight equations of real variables is linearized to obtain anight-dimensional linearized system of equations of theorm

d�

dt= J�, �12�

here J is the Jacobian �88� matrix and � is an eight-imensional vector. The linearized system is then solvedor eight different initial conditions. The initial conditionsre chosen such that the vectors �i and �j, correspondingo any two initial conditions i and j, are orthonormal. Af-er a number of iterations, the magnification of each vec-or is computed. The set of vectors � are then orthonor-alized using Graham–Schmidt orthonormalization

rocedure. This guarantees that only one vector has aomponent in the direction of most rapid expansion andeeps vectors oriented properly in the phase space. Next,he product of magnifications for each vector i is com-uted. This product over a large number of iterations ofhe equations of motion for a total time t leads to 2�it,here �i are Lyapunov exponents. The full spectrum ofxponents was also used to compute the information di-ension of the attractor. The information dimension of

he strange attractor is computed via24

DI = j +�i=1

j�i

��j+1�, �13�

here �=� /� and j is defined by the condition that

�i=1

j

�i � 0, �i=1

j+1

�i 0. �14�

n addition to the Lyapunov exponents and informationimension, we also calculated power spectra using a FFToutine for the mode intensities.

ig. 2. Regions of dynamical behavior in the parameter spacepanned by modulation depth hm and frequency m. Figures 2–12n this paper are for pump ratio r=1.5 and b /2�=115 kHz.

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In the calculations reported here, we used parametersorresponding to a Nd:YAG gain medium ��=4.17103 s−1� inside a ring cavity of perimeter L=70 cm (re-

ractive index n=1) and a round-trip loss factor of �=1.his leads to a cavity decay rate of 2�=�c /nL=4.3106 s−1.The pump ratio r was modulated by a square-wave sig-

al of amplitude hm

r = ro + hm sign�cos �2� m�/��, �15�

here the sign�x� function is +1 if its argument x is posi-ive and −1 if it is negative. The calculations were per-ormed for several pump ratios, but here we report the re-ults only for the pump ratio ro=1.5. These results areypical for pump ratios ro�3.0. In all calculations, theackscattering coefficient was taken to be b /2�115 kHz. The depth of modulation was varied over theange 0�hm�ro=1.5 in steps of 0.02, and the frequencyf modulation was varied from 2 kHz m50 kHz inteps of 2 kHz. This procedure was equivalent to dividinghe m−hm plane into cells of size �hm=0.02 and � m2 kHz. Dynamical response of the system is summarized

n the two maps shown in Figs. 2 and 3. In Fig. 2, weecord the dynamical response as periodic, quasi-periodic,r chaotic. In Fig. 3,we record the state of synchronizationf the two intensity waveforms. These maps were con-tructed by examining the Lyapunov exponents, powerpectrum, and phase space portrait in the I1−I2 plane forach cell. Such diagrams are useful when the system isxamined experimentally and serve as guides to selectinghe appropriate operating parameters for studying a par-icular dynamical regime.

From Fig. 2 we see that the system exhibits periodic,uasi-periodic, and chaotic behavior. Periodic behavior isharacterized by vanishing Lyapunov exponents and apectrum that consists of one or a few sharp lines. Quasi-eriodic behavior is characterized by one or more positiveyapunov exponents and a spectrum dominated by amall number of strong spectral lines. The chaotic behav-or is characterized by positive Lyapunov exponents and aroad continuous spectrum.We find that for low modulation depths �hm�0.04� the

ystem exhibits periodic oscillations independent of pump

ig. 3. Regions of dynamical behavior according to the synchro-ization of the intensity waveforms in the hm− m plane.

odulation frequency m. In this regime the intensityower spectrum consists of a strong self-modulation peakt b /2�.Quasi-periodic behavior appears, approximately, in theodulation depth range 0.04�hm�0.34 and low fre-

uency range 2� m8 kHz. In this regime, once theransition from quasi-periodic to chaotic behavior occursaround point C), the system never reverts to quasi-eriodic behavior as hm increases. In the 10� m14 kHz frequency range, the transition to chaos occurs

t smaller modulation depths, but the system switchesack to quasi-periodic behavior before making a finalransition to chaos. For higher frequencies � m�16 kHz�,he transition to chaos occurs at a higher depth of modu-ation. For example, for m=20 kHz, the system exhibitsuasi-periodic behavior for modulation depths as large as.74. For modulation frequencies greater than about8 kHz, the system shows quasi-periodic behavior regard-ess of the value of the modulation depth.

We also find that the low-frequency scenario to chaos isifferent from the high-frequency scenario when synchro-ization of intensity waveforms is considered. This can beppreciated by following the transition to chaos along twoaths ABCDE (low frequency) and FGHIJ (high fre-uency), shown in Figs. 2 and 3. Along each path, the fre-uency of modulation m remains fixed but the depth ofodulation hm increases.Along the low frequency path ABCDE � m=2 kHz� we

ncounter strictly antisynchronized self-modulation oscil-ations for small depths of pump modulation (Point A).he phase space trajectory in the I1−I2 plane is thetraight line I1+I2=const. (Fig. 1) As the depth of pumpodulation increases, the system enters the quasi-

eriodic regime (Point B), where the envelope of self-odulation oscillation itself develops a small modulation

t a pump modulation frequency. Accordingly, the system

ig. 4. (a) Pump modulation and intensity waveforms (b) theower spectrum, and (c) the trajectory in the I1−I2 plane in theuasi-periodic regime (Point B in Figs. 2 and 3). Notice thetrong presence of self-modulation and modulation frequencies inhe power spectrum.

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rajectories are confined to a narrow band near the line1+I2=const. (Fig. 4) As hm increases further, the enve-ope of self-modulation is strongly modulated. The bandn the I1−I2 plane expands toward the origin. The spec-rum is still dominated by the self-modulation frequency,ut the pump modulation frequency begins to grow intrength, and new frequencies start appearing in thepectrum, corresponding to more complex modulation ofhe envelope of self-modulation oscillations. The systemrajectory still has a sharp upper bound I +I =const.

ig. 5. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane at thenset of chaos (Point C in Figs. 2 and 3).

ig. 6. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane in thehaotic regime (Point D in Figs. 2 and 3). Note the disappearancef the self-modulation frequency at 115 kHz from the powerpectrum.

1 2

ith further increase in hm, the system makes a suddenransition to chaos (Point C). The intensity dynamics inhis case are unsynchronized and the system trajectory iso longer bound by I1+I2=const. This is the region of un-ynchronized chaos. (Fig. 5) Continuing further, past hm0.42, a new trend emerges. The system trajectory con-

erges toward the I1=I2 line. (Fig. 6, point D). This meanshe chaotic dynamics of the mode intensities tend to syn-hronize. This transition is accompanied by the disap-earance of the self-modulation frequency b /2� from thepectrum. Until this transition [Fig. 6(b)], the self-odulation frequency was a prominent feature of the

pectrum. As the self-modulation frequency disappears,he pump modulation frequency becomes stronger in thepectrum. For large modulation depths (Point E), the in-ensity dynamics tend to synchronize and the degree ofynchronicity increases. Examples of intensity waveformnd spectrum in this regime (Point S) are shown in Fig. 7.Along the high-frequency path FGHIJ � m=12 kHz� we

ncounter a different route to chaos. Beginning with an-isynchronized periodic and quasi-periodic (Point F) oscil-ations at low depths of modulation, we encounter unsyn-hronized quasi-periodic intensity waveforms as hmncreases. The system appears to switch between antisyn-hronized and synchronized bursts (Fig. 8, Point G). Onncreasing hm further, the system enters the unsynchro-ized chaotic regime (Fig. 9, Point H). This unsynchro-ized chaos evolves into synchronized quasi-periodic be-avior (Point I and Q). As we move toward point J,ynchronized quasi-periodic behavior evolves into syn-hronized chaotic behavior. Synchronization finallyreaks down at sufficiently high modulation depths, buthe memory of synchronization remains in that the sys-em trajectories still crowd around the line I1=I2.

Buried in the chaotic region are regions where theower spectrum is continuous without any dominant fre-uencies. This regime may be referred to as the regime of

ig. 7. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane for syn-hronized chaos (Point S in Figs. 2 and 3). Note the absence ofhe self-modulation frequency at 115 kHz from the powerpectrum.

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ell-developed chaos (Fig. 10, Point W).We also characterize system dynamics in terms of theaximal Lyapunov exponent. The maximal Lyapunovmax=�max/� normalized to the cavity decay rate �, as a

unction of modulation depth hm, is shown in Fig. 11 foreveral different modulation frequencies. We find that theyapunov exponent �max, in general, starts with smallalues close to zero for small modulation depths hm. Asm reaches a threshold value, the exponent increases rap-

dly, forming a peak for some pump modulation frequen-

ig. 8. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane foruasi-periodic signals (Point G in Figs. 2 and 3).

ig. 9. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane for un-ynchronized chaos (Point H in Figs. 2 and 3). Note the presencef dominant frequencies against a continuous background in theower spectrum.

ies. This threshold is different for different frequencies.n interesting feature may be observed at m=10 kHz.or this frequency, it can be seen from Fig. 2 that the sys-

em switches back and forth between quasi-periodic andhaotic regimes as hm is increased before making a finalransition to chaos. This behavior is reflected in �max ver-us hm for this frequency (Fig. 11). The peak around hm0.3 in the curve corresponds to the chaotic block embed-ed in the quasi-period region of the map around theoint �hm=0.3, m=10 kHz�. We find that the Lyapunovxponent �max for higher modulation frequencies � m50 kHz� approaches zero, a characteristic of periodic dy-

amics consistent with the state diagram in Fig. 2.We also calculated the dimension of the strange attrac-

or underlying the chaotic dynamics using the spectrumf Lyapunov exponents and Eq. (13). The information di-ension DI a function of modulation depth hm is shown inig. 12 for several different modulation frequencies. Alance at the curves of DI versus hm reveals that they fol-ow trends similar to those of �max versus hm. The valuesf DI for higher frequencies � m�50 kHz� are insensitiveo hm and are very close to 2. This is an indication thathe system at these high frequencies no longer has atrange attractor.

The dynamical portrait of the system with square-waveodulation emerging from Figs. 2 and 3 is different from

hat for a monolithic ring laser with sinusoidal modula-ion of pump reported by Lariontsev et al.9 This differenceay be traced to the fact that a square wave is a sum ofany Fourier frequency components. Therefore, when a

aser pump is modulated by a square wave, the system re-ponds, in effect, to a series of simultaneously appliedhased and definite amplitude sinusoidal modulations ofain. The difference in the mode-synchronization map be-ween sinusoidal and square-wave modulation also hintst interesting possibilities for chaos synchronization andontrol studies.

ig. 10. (a) Pump modulation and intensity waveforms, (b) theower spectrum, and (c) the trajectory in the I1−I2 plane for well-eveloped chaos (Point W in Figs. 2 and 3). Note the absence ofny dominant frequencies in the power spectrum.

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. CONCLUSIONSn this paper we have studied the solutions of the equa-ions of motion for a single-mode class B bidirectional ringaser with backscattering. We find that without pump

odulation, the laser exhibits antisynchronized periodicscillations (self-modulation) of mode intensities. In thebsence of backscattering, only unidirectional operation isossible. With square-wave pump modulation, the systemhows periodic, quasi-periodic, and chaotic regimes of op-ration. For an average pump ro=1.5 and m�38 kHz, theystem changes from periodic to quasi-periodic and even-ually to chaotic regime as the depth of modulation hm isncreased from 0 to the maximum value ro. In making thisransition, the system may switch between the quasi-eriodic and chaotic regimes, as indicated in the stabilityiagram.A detailed map of the dynamical response of the system

n the m−hm parameter space has been constructed byystematically characterizing the dynamical behavior inerms of power spectra and Lyapunov exponents. Theroximity of regions of synchronized and unsynchronizedhaos in the mode intensities offers exciting possibilitiesor chaos control by feedback mechanisms. This and otherheoretical predictions summarized in this map can beested in experiments on homogenously broadened class Basers such as a Nd:YAG ring laser.

CKNOWLEDGMENTSe gratefully acknowledge many helpful exchanges with. G. Lariontsev on the investigations reported here.

ig. 11. Variations of the largest Lyapunov exponent �max/�ith modulation depth hm for different modulation frequencies.

ig. 12. Variations of the information dimension DI with modu-ation depth hm for different modulation frequencies.

uad Rawwagah also acknowledges partial support fromhe International Center for Scientific Culture Worldaboratory in Switzerland.

*Current address, Department of Physics, Yarmoukniversity, Irbid 211-63, Jordan.

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