Optimization of the switch-on and switch-off transition in a commercial laser

Optimization of the switch-on and switch-off transitionin a commercial laser

X. Hachair, S. Barland, J. R. Tredicce, and Gian Luca Lippi

The response of a Class B laser to a rapid change in one of its parameters is known to be accompaniedby delay and ringing. It has been theoretically and numerically shown that the transition can be modifiedby using adequate functional shapes for the control parameter (e.g., the laser pump) in order to steer thelaser from one point of operation to another. Here we experimentally show the implementation of theseideas in a commercial device: a semiconductor laser. We establish a procedure for optimizing a controlledswitch-on and switch-off and obtain a clean, fast, and reliable square pulse, either in a single shot or ina repetitive sequence. The generality of this procedure promises a wide field of application for a varietyof laser systems. © 2005 Optical Society of America

OCIS codes: 140.2020, 140.3460, 140.5560, 140.0140.

1. Introduction

Over the years, lasers have evolved from a scientificcuriosity and source of radiation for laboratory spec-troscopic investigations to flexible tools that are usedin the most varied settings and applications: bar-codereading, machining, range-finding and measure-ment, health, telecommunications, etc. Such a diver-sity of goals also requires diversification of thedevices’ features and specialization in their design.

Most lasers used for technological applications be-long to so-called Class B,1 which includes all solid-state, semiconductor, and CO2 lasers. All thesedevices have in common the long lifetime of the ex-cited state (relative to both the medium polarizationlifetime and to the photon lifetime in the cavity).Hence they are particularly suitable for efficient en-ergy conversion. Indeed, thanks to slow intrinsic re-laxation, the amount of energy that can be convertedinto an electromagnetic field in these devices is largerthan for lasers belonging to other classes (e.g., ClassA,1 which groups among others Ar� and He–Ne la-sers). The high energy conversion efficiency is, how-

ever, accompanied by a higher degree of complexity inlaser behavior.2–9

To use such devices for applications, their intrinsi-cally complex dynamics need to be mastered. In thiscontext the successful control of unstable periodicoscillations has been demonstrated,10–13 while apromising line of investigation makes use of chaoticoscillations to encrypt messages.14–18 The encryp-tion�decoding procedure is rendered possible by thesynchronization of complex oscillators—such as semi-conductor lasers—in a chaotic state.19–23

As the preceding examples demonstrate, recent ef-forts toward improving the performance of modernoptical devices are based on some form of control, beit directed at constructing particular temporal pulsesequences (e.g., for the optimization of chemical re-actions or quantum control24–30) or at obtaining par-ticular beam shapes in lasers31–39 or in other passivenonlinear optical systems.40–46 The techniques testedso far are many and varied, and the degree of interestin this field is attested to by a recent special issue ofthe Journal of the Optical Society of America B.47

In the past few years we have concentrated on aform of control aimed at modifying the transitionbetween states of operation (e.g., the output powerlevel) of a Class B laser. Our motivation stems fromconsidering that the advantages of these devices areaccompanied by shortcomings that cannot be avoid-ed: an intrinsically delayed response of the laser to allchanges in control parameter values (e.g., the pump),accompanied by damped oscillations toward the newstate; in some devices (e.g., Nd:YAG lasers48) thespikes are particularly large, and the ringing can last

The authors are with Institut Non Linéaire de Nice, Unité Mixtede Recherche 6618, Centre National de la Recherche Scientifique(Paris), Université de Nice-Sophia Antipolis, 1361 Route des Lu-cioles, F-06560 Valbonne, France (e-mail for G. L. Lippi, [email protected]).

Received 25 June 2004; accepted 1 December 2004.0003-6935/05/224761-14$15.00/0© 2005 Optical Society of America

1 August 2005 � Vol. 44, No. 22 � APPLIED OPTICS 4761

for quite a long time. It is obvious that many fieldsof application could benefit from removal of suchoscillations and from shortened delays; opticaltelecommunications—to name one—is indeedplagued by a sluggish laser response that gives rise toso-called memory effects49–51 (also known as inter-symbol interference) that cause speed limitations intransmission.52–54 We call steering a technique thatguides the laser through a suitably engineered pathin phase space, continuously connecting the targetregime of operation to the preceding one.55

In previous publications we have shown that mod-ifying the pump change from a step function (e.g.,throwing a switch) to an appropriately chosen, morecomplex one55 affects delay time and the amplitude ofthe oscillations of a CO2 laser.56–58 Applying the samesteering concepts to semiconductor lasers, we couldpredict substantial improvements in the device’s re-sponse to direct modulation of the current injected inthe junction, thus allowing for a strong potential gainin optical telecommunication performance.59–63 Fi-nally, a scheme that we proposed for quasi-smoothfunction steering64 has been successfully imple-mented on a Nd:YVO4 laser,65 thus demonstratingthe potentials of this technique. Similar ideas havebeen applied by other authors to reduce timing jitterin the pulses’ output of a laser with a saturable ab-sorber.66

The purpose of the present research is to confronta recurrent problem in applying the steering con-cepts: a lack of reliable data for modeling the actualdevice. Indeed, although the steering principle isquite straightforward,55 its applicability may be lim-ited by the amount of information available on thedevice in question. This problem has been discussedin Ref. 65 and is all the more important if one intendsto render steering a practically implementable tech-nique. Here we experimentally demonstrate the suc-cessful application of steering to an off-the-shelf,packaged semiconductor laser integrated with elec-tronics for which little information is available.Rather than trying to determine the parameters forthe whole system, a lengthy and complex procedure,we choose a more straightforward approach and testan empirical procedure that lets us reach our goal inmany fewer steps. This empirical choice offers theadditional advantage of rendering steering applicablealso to all those devices where a satisfactory modelcannot be established, because either a commerciallyengineered device is too complex for a managablemathematical description or standard techniques,67

valid for some devices, are not amenable to general-ization. In this way we broaden the class of systemsthat can benefit from the advantages of the tech-nique.

This approach requires, however, performing somepreliminary tests to verify whether the device is eli-gible for steering, i.e., whether its general featuressatisfy the conditions in which steering can be suc-cessfully applied.55 We illustrate these tests on thedevice that we have chosen as a sample.

For completeness in Section 2 we briefly review the

basics of the steering principle. The experimentalsetup is presented in Section 3, and the eligibilitytests are reported in Section 4. In Section 5 we dis-cuss implementation of the procedure and determi-nation of the optimum steering values applied to ourspecific device. The quality of the resulting signaloutput by the laser is analyzed on the basis of errorsexpressly introduced in the steering function to testthe robustness of the technique in our device (Section6). In Section 7 we discuss the switch-off of the laser,while we present in Section 8 application of the opti-mal switch-on–switch-off sequence to the laser. Wediscuss the robustness of steering relative to variousparameters (laser temperature and aging) in Section9. Conclusions in Section 10 complete the paper.

2. Principle of Transient Steering

The typical response of a Class B laser to a modifi-cation in the values of its operating parameters ischaracterized by a delay followed by an overshoot anddamped oscillations toward the new state.68 The sim-plest idea consists in applying piecewise constant lev-els to the pump supplied to the laser by insertingsome intermediate steps between the initial and thefinal pump value. To switch on the laser withoutoscillations and in the shortest possible time, we firstapply a large pump that speeds up the switch-onfollowed by a low value (temporarily below threshold)tailored to use up just the right amount of accumu-lated population whose excess would otherwise causeovershoot and damped oscillations.64

For an illustration of the previous qualitative de-scription, we turn to a basic rate equation model fora single longitudinal and transverse-mode Class Blaser1,64,69:

I � (D � 1)I, (1)

D � ��[D(1 � I) � P], (2)

where I and D represent the field intensity and thepopulation inversion, respectively, whereby I and Dare their time derivatives; � denotes the ratio be-tween the population decay rate and the cavity losses(for the field intensity), P is the pump rate, and timeis normalized to the cavity losses.

The direct integration of these equations provides agood illustration of principle. Figure 1(a) shows theusual switch-on (dashed curve) obtained from Eqs. (1)and (2), where the pump is suddenly brought frombelow to above threshold at t � 0 (the parameters aregiven in the Fig. 1 caption). The delayed responsewith ringing appears clearly in Fig. 1. One sees, as-sociated with the intensity oscillation, the populationinversion [Fig. 1(b), dashed curve] that evolves be-yond its stationary value [Dss � 1 in the normaliza-tion of Eqs. (1) and (2)] and converges toward it in anoscillatory fashion. The oscillations in population andfield intensity are, as is well known, in quadrature,the former (representing the source term) anticipat-ing the latter.

4762 APPLIED OPTICS � Vol. 44, No. 22 � 1 August 2005

The application of the composite switch shape forthe pump (Fig. 6 below), with appropriately chosenpump values and durations for the steps, provides afaster and oscillation-free switch-on of the field inten-sity [Fig. 1(a), solid curve]. One way of understandingthis result rests on noting that in these conditions theaccompanying evolution of the population inversionshows an overshoot [Fig. 1(b), solid curve] during thetime leading to t1, which is necessary to start theamplification process, accompanied by disappearanceof the excess population (the quantity D � 1) exactlyat time t2 when the laser intensity reaches its finalvalue. In this way the equilibrium situation is at-tained simultaneously for both physical variables (Iand D), and the transition is completed. (A more for-mal discussion is given in Ref. 64.)

As discussed elsewhere,55 the possibility of attain-ing our goal in this fashion is intimately related to thephase-space topology of the system, which is bestillustrated in a representation where time is factoredout (Fig. 2). Indeed this graphic form convincinglyshows the existence of an unstable fixed point (a sad-dle located at D � 1.2, I � 0 in Fig. 2) toward whichthe trajectory initially points and from which it thendeviates in its evolution from the initial state �D� 0.9, I � 0� to the final one �D � 1, I � 0.2�. Thefeatures illustrated by Fig. 2 (saddle point and stablefocus55) are necessary for the correct operation of oursteering procedure. Hence we experimentally recon-struct the phase portrait of our system as a test of itscompatibility with our steering scheme.

3. Experimental Setup

We apply our steering technique to a Thorlabs modu-latable semiconductor laser emitting at � � 670 nm,

which lases on a small number of longitudinal modesover the full current injection interval. (At steadystate, lasing normally occurs on one or two modes formost pump current values.) The laser head includeselectronics whose detailed schematics are not avail-able. However, the amount of current, our controlparameter, injected into the laser junction is con-trolled by an externally applied voltage. The band-pass of the electronics built into the device is specifiedby the manufacturer at 500 kHz for transistor–transistor logic modulation �0 ↔ 5 V�. Since in ourmeasurements we use only part of the input voltagerange (lower state, Voff � 0.941 V, upper state, Von� 2.549 V, with the laser threshold at Vth � 1.78 V)the effective bandpass can be experimentally seen toincrease to a few megahertz.

The experimental setup is as follows (Fig. 3). ALeCroy 9100 arbitrary function generator, controlledby a PC with LabVIEW, provides the steering func-tion for the current. The resulting laser light is sentto a detector (Thorlabs DET210) with a rise time of1 ns, whose output goes to a 500 MHz HP 54616Bdigital oscilloscope with a general purpose interfacebus (GPIB). The cycling over the parameter values forthe steering function and the acquisition and storageof the corresponding data is automatized with the

Fig. 1. Numerical integration of the direct current step (dashedcurve) and of the optimally steered transition (solid curve): (a) laserintensity, (b) population inversion. Parameters: � � 2.5 � 10�2,Poff � 0.9, Pon � 1.2; for the steered transition (see Fig. 6 below fordefinitions) P1 � 1.8, P2 � 0.9, t1 � 1.46 � 10�6 s, t2 � 3.61� 10�6 s.

Fig. 2. Numerically reconstructed phase space for the switch-on.Parameters are as in Fig. 1.

Fig. 3. Experimental setup: DC, stabilized laser power supply(MVP driver by Thorlabs); Laser System, Thorlabs modulatablesemiconductor laser; bias, electronic adder (built into the lasersystem); D, Si p-i-n detector (Thorlabs DET 210); L, focusing lens;Oscill., HP 54616B with the GPIB; PFG, programmable func-tion generator (LeCroy 9100); PC, personal computer runningLabVIEW.


help of a dedicated program. All interfacing and con-trols are run through LabVIEW.

The presence of the electronic voltage control of theinjected current, integrated and sealed in the pack-age, together with the paucity of information pro-vided by the manufacturer, renders the modeling ofthe system and the measurement of its parametervalues a challenging and lengthy process. Similarconditions are often encountered even when the laseris not packaged with electronics, since most manu-facturers and suppliers do not provide the user withmany of the technical details necessary for accuratemodeling. The empirical, model-free approach thatwe have chosen bypasses this problem by quickly andeasily determining the optimal operating conditions.

Before proceeding with this step, however, we needto test whether the global device (laser with electron-ics) fulfills the same generic topological properties asthose shown in Fig. 2—a necessary condition for theapplicability of our steering technique.70

4. Phase-Space Reconstruction

The response of our device to a current step is shownin Fig. 4. We see that the optical field is emitted witha time delay and converges toward the steady emis-sion state with damped oscillations. In spite of thesimilarity between this temporal response of the out-put power and that of a Class B laser,55 this is notsufficient to prove that steering is successful in thissystem. A counterexample is a second-order ac elec-trical circuit with underdamped features71: It alsoreaches steady state with damped oscillations, but itstrajectory does not evolve in a phase space with thesame topological properties as those of a Class Blaser.55 In other words we need to test whether theglobal device (laser plus electronics) still possesses asaddle point. We achieve this goal by reconstructinga phase space for our system in a two-dimensionalprojection. One variable is directly available, the la-ser power measured by the detector, whereas the

second has to be reconstructed from the time series.Specifically we need to recognize whether the oscil-lations and the delay that we see in Fig. 4 are simplythe compounded consequence of the finite bandwidthof the electronics (delay) and the presence of a reso-nance (oscillations) or whether the system can bedescribed by a phase space similar to that of Fig. 2.

For the second variable it is useful to introduce thefollowing quantity by analogy with the generic rateequations model for Class B lasers [Eqs. (1) and (2)]:

s � 1 � c(I�I), (3)

where I is our measured laser power, I is its timederivative (easily obtained from the digitized data),and c is a positive constant that we adjust; s becomesthe equivalent of D in Eqs. (1) and (2). Note that thischoice is made only for convenience and does notrestrict the generality of the procedure. Indeed alter-native representations may be used just as well, suchas plotting I versus I or phase-space plots resultingfrom embedding techniques.72 The reason for thepresent choice is that its graphic form shows morestraightforwardly the topological phase-space fea-tures that we are looking for,73 but the same infor-mation is contained in any of the equivalentrepresentations as well. Hence the analogy betweenD in the generic Class B laser model used in theintroduction of the auxiliary variables, Eq. (3), doesnot introduce restrictions in our approach.

The need for a constant c in Eq. (3) comes from thefact that the power values measured in the experi-ment are not normalized quantities, contrary to thevariables in Eqs. (1) and (2); therefore we need anadjustable constant for closer comparison.

Figure 5 shows the experimentally reconstructedtrajectory for c � 0.6, plotted in the �s, I� space, whilethe inset shows the equivalent trajectory in the Iversus D representation, numerically obtained fromEqs. (1) and (2). The similarity between the curves isquite striking: The experimentally reconstructed tra-jectory possesses not only the spiraling motionaround the final stable point of operation (typical alsoof damped oscillators) but also the portion of trajec-tory that shoots away (nearly) horizontally, which ischaracteristic of all systems modeled by bilinearlycoupled variables [as in Eqs. (1) and (2); more detailsare given elsewhere64]. This portion of trajectory isthe signature of the presence of a saddle point in thephase space, which acts as an attractor in one direc-tion and as a repellor in the other.64,76 This picturesuggests that the device possesses the same globalproperties as a Class B laser and that the steeringtechnique with two prepulses inserted between theinitial and final pump value55 should successfullymodify the laser transient behavior.

The experimentally reconstructed phase-space tra-jectory (Fig. 5) does not end directly in the stable,final point: the spiraling stops and the trajectorydrops down in power. The physical causes for suchbehavior can be discussed at length77 but are not

Fig. 4. Laser turn-on subject to a stepwise current switch (asschematically shown in the inset). Laser threshold: Vth � 1.78 V;Voff � 0.941; Von � 2.549 V. These values hold throughout thepaper. All symbols concerning the laser pumping levels (and du-rations of steering steps, when applicable) are defined in Fig. 6below. The switch-on step is applied at t � 30 ns.


relevant to this research, since we prove that steeringis not sensitive to this detail. The apparent crossingof the phase-space trajectory in Fig. 5 tells us that theactual space is higher dimensionally and that we aredisplaying simply its two-dimensional projection. Themain phase-space features appear satisfied, and wecan go ahead in applying the steering procedure.

5. Laser Steering

The implementation of the so-called patterned cur-rent front [or direct modulation through patternedcurrent fronts60 (DMPCF)], shown in Fig. 6, requiresdetermination of the appropriate values for the levelsand their duration. Hence we need to establish asimple and effective procedure for determining theappropriate parameter values: V1, V2, t1, t2, as de-fined in Fig. 6.

We mention in passing that fitting the data directlyto Eqs. (1) and (2) does not provide reliable resultsbecause of the great sensitivity of a fit to the relax-ation time, 1��, which cannot be determined with

sufficient accuracy from the data. More complex mod-els could obviously be considered, but, as mentionedabove, this requires a substantial time investment fordetermination of the experimental parameters, atime investment that is rendered obsolete by our em-pirical procedure.

With the understanding that results from thepresent procedure need to be tested for quality, reli-ability, and robustness (in particular with respect toboth noise and aging, see Section 9), we determinethe best parameter values by using a grid scan andchoosing the �V1, V2, t1, t2� set that yields the bestpossible switch-on. Two procedures, one simpler butmore cumbersome, and a second, leaner one, whichrequires some educated guesses, are outlined in Sub-sections 5.A and 5.B. The specific application of thesecond one to our laser is in Subsection 5.C.

Note that we are searching for steering parametervalues that will not need adjustment from time totime but that will be determined once and for all foreach device. As we stress in Section 9 we have exper-imentally tested this to be true for our laser over along period of time and in different conditions of op-eration. Hence the values in Subsection 5.C will holdnominally forever for our laser.

A. General Procedure

To easily determine the best parameters, we programthe function generator with Labview to run in a loop,sending pulses to the laser while gradually varyingthe parameter values one by one. Since the time res-olution of our arbitrary function generator is quitecoarse (the minimum distance between points is10 ns in increments of 10 ns), we fix the duration of t1and t2 and run the loops on the amplitudes V1 and V2,which have a much finer adjustment (�0.4% of therange).78

An estimate of the total number of trials, of thetime, and of the disk storage space necessary to per-form them can be easily obtained in the followingway. We call �1 and �2 the number of tests for theduration of each plateau, t1 and t2, respectively, ac-cording to the definition

�j �tj, max � tj.min � �t

�t (j � 1, 2), (4)

where tj, max and tj.min are the longest and the shortesttime for which the level Vj ought to be applied and �tis the time step used for the increment. We furtherdefine �1 and �2, the number of steps in the controlvoltage, as

�j �Vj, max � Vj, min � �V

�V (j � 1, 2). (5)

The maximum number of trials, assuming an equallylikely (i.e., unweighted) combination of all the previ-ous possibilities, is given by �max � �1 � �2 � �1� �2.

Each of these numbers is determined by various

Fig. 5. Reconstructed phase space for direct switch-on (Fig. 4).Variables are as in Eq. (3). The inset shows an equivalent numer-ical picture obtained from Eqs. (1) and (2); compared with Fig. 2, afaster relaxation �� 0.05� has been used, and the larger amountof stimulated emission (typical of a semiconductor laser) has beensimulated by considering a larger initial value (by nearly 4 ordersof magnitude) for laser intensity I; c � 0.6 is for the numericaltrajectory.

Fig. 6. Shaped current function for optimal switch-on. The volt-age values for the added plateaus and their respective durationsare identified by labels 1 and 2, respectively (see text for discus-sion).


constraints. In our case the number of steps in thecontrol voltage must satisfy the following condi-tions55: V1, min � Von � V1 � Vsat � V1, max and V2, min

� 0 � V2 � Voff � V2, max, where Voff and Von are definedin Fig. 6 and Vsat is the maximum voltage allowed bysaturation. (In a different device Vj, min and Vj, max

would be determined by equivalent constraints.) Fur-thermore the voltage step is determined by the capa-bilities of the function generator and amounts to�V � �Vsat � 0��Nbits, which for 8 bits and Vsat

� 5 V provides �V � 19.6 mV. Substituting the val-ues of Von and Voff, we obtain, following Eq. (5), �1

� 124 and �2 � 48, for our system.79

The number of time steps in our case is stronglylimited by the generator’s resolution ��t � 10 ns�.The longest duration tj, max for which each level can beapplied equals at most the time the laser intensitytakes to cross, for the first time, its asymptotic emis-sion value.55,64 (In reality this choice overestimatesthe actual time interval but can be used to obtain amaximum number of steps.) An equally pessimisticestimate for the minimum duration for each level istj, min � 0. From Fig. 4 we estimate this interval to be�120 ns, which can equal at most �t1 � t2�max. Givenour �t we can consider that �j is at most 13 (for eachj � 1, 2).80

With these numbers our worst-case estimate gives�max � 8.7 � 105. The number of points to be acquiredby the oscilloscope in each run can be easily fixed at1000 without loss of resolution. (Indeed one couldeven accept a smaller number.) Thus we are led to anestimate of the maximum data volume to approxi-mately 870 Mbytes (storing in binary the laser out-put intensity with an 8-bit resolution; the time can bereconstructed from the sequence of points based onthe sampling time). Estimating the global time toperform a loop81 at �60 ms, we obtain a total mea-surement time for a sequence lasting �52,200 s.

Although the measurement time is already verylong, at this level the main obstacle appears to be ananalysis of the data volume. A first level of decision onwhich set of parameter values provides the best laserresponse can be realized automatically by settingsome thresholds on the intensity around (i.e., aboveand below) the asymptotic value Ion: If values areregistered outside this interval the data are dis-carded and the corresponding parameter values aremarked as unsuitable.82 This prefiltering can be ex-tremely useful in removing most of the runs from theset of interesting parameter values; the actual num-ber of sets available depends on the tightness of thewindow, hence, among other things, on the noise levelpresent in the system. The final choice among a smallnumber of trials instead is best done by the experi-menter, since a fully automatic search for the bestconditions requires complex decisions easily made byeye but difficult to encode. (This will be clearer oncethe errors are discussed in Section 6.) In particular,noisy signals can be analyzed quite efficiently by look-ing at the data, whereas automatic discriminationbecomes quite laborious.

Note that the procedure outlined above, yielding aworst-case estimate of the number of trials, is neces-sary only in those cases in which no educated guessabout the system’s response is possible. However,this procedure will need to be performed only once fornominally identical devices, since a reduced param-eter search will suffice to compensate for the residualfluctuations that exist from one laser to the next.

B. Speedier Procedure

The procedure described in Subsection 5.A is a worst-case scenario that can be blindly applied, but it re-quires a certain effort in the measurement timeconsumed, data taking, and analysis of the results.(The storage volume is not a serious issue nowadays.)It is, however, possible to strongly reduce the numberof loops either (a) by performing a preliminarycoarser scan of the parameter range and�or (b) bymaking use of some considerations about the sys-tem’s behavior and our steering operation.

A first reduction in the size of the parameter spaceto be explored in detail is obtained by simply perform-ing a coarse preliminary scan of those variables thatcan be changed in the experiment with a finer step(here the Vj terms); this choice effectively narrows thenumber of possible combinations. In our experimenta coarse scan with �Vc � 10 � �V reduces by a factorof 100 the number of trials, thus also the correspond-ing measurement time and the volume of the data tobe stored and analyzed.

A second reduction can be obtained through phys-ical considerations based on the principle of our steer-ing technique. Since the laser is switched on, it mustreceive a net energy increase from the source. Thefirst step (labeled 1) provides additional energy, whilethe second (labeled 2) subtracts the excess (relative tothe asymptotic operation value, see Section 2). Henceit is reasonable to expect that t2 t1, all the whilesatisfying the condition t1 � t2 t�,83 where t� is themaximum total time for which we can apply the pat-terned current front successfully. (If t� is too long, thelaser output power overshoots and damped ringingappears again.) Although straightforward in a laserwhose characteristics are well reproduced by a mod-el,55,60,64 identification of the appropriate value of t� isnot immediate in a complex device. However, it iscertain that t� cannot be longer than the time t�max thatthe laser takes to reach the value of the output powerequal to the asymptotic one. (We see a posteriori thatt� for our device is approximately equal to the timethat the laser field takes to grow out of the sponta-neous emission noise.)

The considerations above suggest that a priori,without knowing the value of t�, one should searcharound �t��2� t1 with the constraint t2 � t� � t1. Giventhe uncertainty on the actual value of t�, one searchesa few values of t2 such that t1 � t2 t�max. The amountof reduction in the interval of parameter space to besearched depends on the specific details of the systemunder consideration.84 In our experiment, since ourarbitrary function generator has a poor time resolu-


tion �t�max � 10 � �t�, we use three values of t2 for eachchosen value of t1.

The global reduction obtained from all consider-ations above can be expected to amount to at least afactor of 1000, relative to the estimate in Subsection5.A, thus reducing for the preliminary scan the datavolume to less than 1 Mbit and the measurementtime to less than 1 h. An analysis of the data providesa sufficiently narrow interval of parameters to beexplored again, and in all likelihood another scan ofthe same size, with much finer scales, focused on thepromising values of the tj terms and the Vj termssuffices to converge to the optimal set.

C. Application of the Procedure

Given that we are choosing Von � 2.549 V and Voff� 0.941 V (where Vth � 1.78 V), the ranges to explorewith �Vc � 10�V � 196 mV are 2.646 V V1 5 V (providing Nc, 1 � 13) and 0.09 V V2 0.882 V (providing Nc, 2 � 5). Their combination re-quires 65 runs. For the time values we can read fromFig. 4 that t�max � �150–30� ns since the laser reachesthe asymptotic output power at t � 150 ns while theswitch-on is applied at t � 30 ns. Keeping in mind theconditions discussed in Subsection 5.B, we can com-pile the combinations of times shown in Table 1.From it we see that the number of combinations is�1 � �2 � 13, yielding a total number of � � 845 runsfor the preliminary parameter space search.

The results of the preliminary run restrict thesearch to the intervals 5.606 V V1 4.802 V and0.098 V V2 4.296 V with 60 ns t� 70 ns, thusleaving only four possibilities for t1 and t2 (Table 1).The number of runs is now with the finest �V� 19.6 mV, � � 400. Repeating a run of measure-ments, we easily isolate the best parameter values:V1, opt � 4.706 V, V2, opt � 0.235 V, t1, opt � 40 ns, andt2, opt � 20 ns. Figure 7 shows the best switch-on (solidcurve) and compares it with the standard squarefront current switch (dashed curve). We immediatelynotice that the oscillations are entirely removed andthe rise time has been shortened, so that the laserreaches its asymptotic power directly and in a shortertime.

The fact that we easily obtain a faster, oscillation-free switch-on proves that sole knowledge of the gen-eral parameter space topology suffices for us to applyour steering technique55,60 and that the procedure we

are following is very efficient. The generality of theseideas is proved by the fact that an empirical proce-dure can be successfully applied to a commercial de-vice whose detailed features are not known andwhose detailed modeling would require more thantwo variables. Hence it is reasonable to expect thatnumerous devices may be successfully steered in thisway.

One important point concerns the uniqueness ofthe combination of parameter values that providesuccessful steering. The procedure that we employdoes not provide the answer to this question, since itis always possible that some combinations of param-eter values may have been overlooked or that it wasnot available in the scan discretization. However, aformal optimization theory provides the answer63: Ithas been shown that the solution in the form that wehave chosen, i.e., the DMPCF pulse shape,59 providesa unique and optimal response to the steering prob-lem. Hence the empirically found set of parametervalues is unique (within the discretization error).

In practice, given the unavoidable impedance mis-match due to the packaged device (with input imped-ance of �24 k�), in our experiment we could notobtain the time optimal steering, as defined in formalmathematical terms.63 Nevertheless, as shown else-where,55,61,64 different suboptimal shapes of the steer-ing function may exist that allow for a margin offlexibility, all the while the goal is attained. Such arethe functional shapes that are useful in practicalcases.

6. Steering Errors

No real device is devoid of fluctuations, drifts, orother sources of errors that affect either the short-term behavior (fluctuations) or its long-term oper-ation (drifts). It is therefore important toexperimentally address the question of the tech-nique’s sensitivity to errors in the steering functionor to noise in the initial conditions. Indeed its ro-

Table 1. Tested Durationsa

t1 �ns� t2 �ns� t� �ns�

70 50, 40, 30 120, 110, 10060 40, 30, 20 100, 90, 8050 40, 30, 20 90, 80, 7040 40, 30, 20 80, 70, 6030 30 60

aFixing the value of t1 and keeping in mind the constraints (t2

t1, using three values of t2 for every t1, and choosing 60 ns t� 120 ns), we obtain these values. The total number of combina-tions is 13.

Fig. 7. Laser switch-on. Solid curve, optimal steering; dashedcurve, stepwise switch. The inset shows the shape of the steeringfunction experimentally applied to the laser. The switch is appliedat t � 30 ns. The two steering levels are V1, opt � 4.706 V andV2, opt � 0.235 V with durations t1, opt � 40 ns and t2, opt � 20 ns,respectively.


bustness to such perturbations is crucial in deter-mining its usefulness. To test this robustness, weintroduce variations in voltage levels V1 and V2 (thetime step �t being too coarse to do the same with thestep durations).

A. Error Analysis

Figure 8 shows the response of the system to varia-tions in the amplitude of the first plateau V1. Thesolid bold curve shows the nearly optimal switch-onwhere we have expressely introduced a deviationin V2 from its optimal value. Comparison with Fig.7 shows how this deviation, |V2 � V2, opt|� 0.135 mV, introduces some damped ripples in theoutput power. They are the consequence of an exces-sive reduction in the energy stored in the laser duringthe patterned current front, which does not allow thedevice to reach the correct operating condition in adirect way: At the end of the steering period the la-ser’s internal state is below its asymptotic one. Thusthe switch from V2 to Von (Fig. 6) acts on a smallerscale in the same way that the steplike switch-ondoes and causes a (small) overshoot and ringing.

Inspection of all the curves (solid thin curves) ofFig. 8 shows that the deviations from the optimumswitch can occupy a band as large as one third of theasymptotic output power value. This is true, however,when we observe the superposition of all trajectoriessimultaneously. Indeed, in spite of the large excur-sions in V1 that we introduce, even for the V1 valuesthat deviate most from V1, opt, we find a global im-provement relative to the unsteered switch-on (dot-ted curve).

Some of the temporal traces realized by the steer-ing function have unusual properties; i.e., they arethe exclusive result of the incorrect juxtaposition ofdifferent trajectory pieces.55 We are referring to thosetraces where the laser does not switch on as fast asthe (nearly) optimal trace (solid bold curve) and that

show a certain amount of delayed oscillation. In thesecases the amount of energy supplied to the laser dur-ing the first step is insufficient �V1 � V1, opt�. Hence,when the steering period comes to an end, the laserhas not reached its asymptotic power value andjumps up on a smaller scale but in the same way asit would when switching on with a stepwise switch,until it relaxes to the correct output power (with theusual oscillations).85

The sensitivity of the system to errors in V2 is muchsmaller (Fig. 9), and the rise time is practically inde-pendent of the parameter values of the second pre-pulse. This is easily understandable, since the firstprepulse reduces the transient time by bringing thelaser close to threshold in a shorter time,86 while themain function of the second prepulse is to subtractexactly the amount of excess population inversionpresent in the laser.87 The excess, or defect, in popu-lation at the moment the parameter is switched to Vonis indeed the source of all oscillations. Note that forthe sake of investigating the influence of steeringerrors on V1 and V2, we have introduced large varia-tions in the optimal parameter values, well beyondthe level of internal fluctuations of a typical device.Indeed the intrinsic fluctuations (noise in the laserand in all the apparatus) do not appear to have adetectable influence on the quality of the transientsteering for our system.

B. A posteriori Analysis of the Optimization Procedure

The robustness analysis also provides some a poste-riori checks on the procedure followed to identify theoptimal parameter values (Section 5). Figure 8 showsthat an error in V2 approximately equal to 0.13 V issufficiently large to give a recognizable oscillationeven when V1 is (nearly) optimal (thick solid curve).The other thin curves show the influence of largedeviations in V1 and give an ensemble demonstrationof the picture that one obtains from a preliminaryscan run over the parameter interval correspondingto that of Fig. 8. It is immediately obvious from Fig.8 that it is quite easy to spot the interval of values of

Fig. 8. Response of the system to variations in amplitude: boldcurve, nearly optimal transition (V2 � 0.1 V, all other parametervalues are the optimal ones specified in the caption of Fig. 7). Allother curves are obtained with V2 � 0.1 V and 3 V � V1 � 5 Vvaried in 0.2 V steps. Here and in Fig. 9 the dashed curve corre-sponds to the switch obtained with a stepwise function (as in Fig.4).

Fig. 9. Sensitivity of the system: bold curve, optimal transition.All other curves are obtained with V1 � 4.7 V and 0.2 V � V2

� 2.6 V varied in 0.2 V steps.


V1 that ought to be searched with the fine �V value.(Figure 8 is obtained with a voltage step equal to the�Vc given in Subsection 5.C.) We also see how anautomatic recognition (directly on the oscilloscope’spatterned trigger or incorporated in the data-acquisition program) could be effective in quite anumber of runs. The same remarks hold for largeexcursions in V2 values, which in Fig. 9 extend evenbeyond the range considered in Subsection 5.C. In-terestingly, we remark that even when the parame-ter values for V1 and V2 are far from their optimalvalues, the response of our laser is always improvedcompared with that obtained with the application ofa step signal.

The lowered sensitivity to the value of V2 impliesthat wider intervals may need to be checked after thepreliminary run. (Examination of one trace at a timestill allows for a clear identification of the best inter-val in our experiment, however.) In general terms,although this lowered sensitivity to one parameteramounts to a disadvantage when we are searching itsoptimal value, at the same time it also means that theprecision required in identifying it is correspondinglylower. Hence one expects that for devices whose re-sponse varies widely with the values of V1 and V2(such is expected to be the case of solid-state lasers;see Ref. 65 as an example) the identification of theoptimal parameter ranges will be simple: An auto-matic recognition successfully discards a large num-ber of possibilities without the need for data storageand for an examination by the experimenter. On theother hand, nominally identical devices of this kindmay turn out to be different enough to need a widerexploration of their parameter space for optimization.Less sensitive systems may need a bit more work forthe first optimization, but, being more insensitive (atleast in some parameters, e.g., V2), they are less likelyto suffer from smaller fluctuations in their optimalvalues.

7. Optimized Switch-off

So far, we have discussed the transient when thelaser is switched on. However, transients occur everytime the system passes from one steady state to an-other, regardless of the relative height of the initialand the final values. Having already analyzed ingreat detail the procedure for the on-transition, wenow limit ourselves to a rapid application of the sameideas to the laser switch-off. The general steeringconcepts hold.

The problem has already been discussed in detail,60

and it was shown that the principle that holds for aswitch-on (or switch-up) also applies to the switch-off(or switch-down) of a laser. It is a well-known factthat the settling down on the lower state is muchslower than the corresponding process at switch-on.Indeed, when the lower state is below threshold,there is a long delay due to the exponential tails of therelaxation of the population inversion that imposes along waiting time before the correct state is recov-ered.60

The question arises as to whether one can just as

well steer the switch-off of our device. Following aprevious analysis,60 we apply the symmetric form(Fig. 10) of the steering function already used for theswitch-on. We recall60 that the function of the steplabeled 3 is to obtain a faster switch-off of the opticalfield. This, however, is achieved at the expense of anexcessive decrease in population. To recover the de-sired condition, we apply the step labeled 4 just forthe duration t4 necessary to bring the population backto level Doff in a shorter time [corresponding to thesteady state of Poff in Eq. (2)].60

Note that since the detector measures the opticalfield intensity the recovery of the population duringthe latter step cannot be directly visualized. A way ofnoting its presence is to apply a steered switch-onfollowing the switch-off with a variable time delaybetween the events. If parameter values V3, V4, t3,and t4 for the switch-off are not correct (or in partic-ular if t3 � t4 � 0), there is a correlation betweensuccessive transitions for time delays below a deter-mined value. The usefulness of steering the laser offis to sharpen the fall of the output field and to (dras-tically) reduce such a minimum delay.60 This is ex-actly what one expects from the principle, and itprovides the criteria for the determination of V4 andt4 in a simple laser. In optimizing the switch-on, inSection 5, we take care to apply a front with a suffi-ciently long wait to allow full relaxation to the off-state.60

Determination of the best values of the parameters�Vj, tj, j � 3, 4� is obtained following the procedureoutlined above. Here we discuss the results. The di-rect steplike switch-off of the laser is shown in Fig. 11(solid curve). When level 3 is added, with optimalparameter values (Fig. 11, dashed curve) the laserintensity shuts down much faster. Thus we obtain afront with a sharper edge and complete switch-off atan earlier time.

As mentioned above, the main purpose of level 4 isto speed up the recovery of the system toward thebelow-threshold operating point rather than improvethe shape of the laser’s power drop. On this point ourobservations (Fig. 11, dotted curve) agree with expec-tations (see above). An effective way of determiningthe best value for level 4 is the following: After opti-mization of the switch-on and the best choice of level

Fig. 10. Shaped current function for optimal switch-off. The volt-age values for the added plateaus and their respective durationsare identified by labels 3 and 4, respectively (see text for discus-sion).


3, the electronics is adjusted to send repeated pulsesat a low rate. If the chosen levels 1–3 are correct, oneobtains a clean, relaxation-free switch-on and a sharpswitch-off. When the repetition rate is gradually in-creased, the minimum waiting time is recognized asthat below which the laser switch-on shape is modi-fied. Changing the value of the 4 level, one can iden-tify the condition that minimizes this time.

In our device we improve the switch-off by sharp-ening the fall of the laser power and by removingsome rebounds (better seen in Fig. 12, dotted curve,but recognizable also in Fig. 11, solid curve, at timevalues until and even exceeding 100 ns). The pres-ence of level 4 instead does not substantially shortenthe dead time, i.e., the minimum wait before theswitch-on necessary to remove all correlation be-tween successive switches (here the minimum time is�4 �s). This observation does not agree with what isexpected from a laser and is possibly due to the elec-tronic circuitry integrated in our device.

8. Optimized On–Off Switching

The combination of steered switch-on and switch-offallows us to generate clean, very short (on the timescale of our device), square-looking pulses (or pulsesequences). The improvement introduced by thesteered DMPCFs is remarkable and avoids both ring-ing in the upper state and rebounds in the lowerstate, as shown in Fig. 12.

This result suggests that the modified fronts thatwe introduce in our steering may allow the user totailor pulses in a wide variety of practical devices tosuit particular needs. Each situation must be indi-vidually examined to solve the specific technical dif-ficulties that accompany the realization of such frontsin semiconductor lasers, solid-state lasers, or complex

laser systems. However, the potential benefits areclearly visible, and the simplicity of our approach,which does not require the addition of feedback loopsor of additional optical devices, makes the generalidea rather attractive.

9. Robustness

The final point that remains to be considered is therobustness of our technique. Indeed one has to checkthe sensitivity of the device’s response to externalperturbations: changes in all those parameters thatare not controlled by the experimenter and that donot enter directly into determination of the optimalsteering.

Various parameters play a role in the response of asemiconductor laser to changes in the injected cur-rent. A crucial one is the temperature. Note that ourlaser is not temperature stabilized and is encased bythe manufacturer in a steel cylindrical housing,which needs to be kept electrically isolated from theground: We hold the laser on a kinematic mount witha plastic ring, which provides simultaneously electri-cal and thermal insulation, relative to the large ther-mal mass represented by the mechanical mount andthe optical table. Hence the thermal losses are exclu-sively of a radiative and a convective nature. Giventhe fact that the laser parameters vary somewhatwith temperature, it is important to estimate thetemperature range over which the oscillations can besuccessfully removed.

Our experimental results show that the parametervalues �V1, opt � V4, opt, t1, opt � t4, opt� provide resultsidentical with those of Fig. 12 (solid curve) for tem-perature values of the outside surface of the case (incontact with air) measured with a thermocouple be-tween 20 and 34 °C. Although we cannot deduce thetemperature of the junction (certainly much higherthan that of the housing), we know that T � 34 ºCmeasured in this case corresponds to the asymptotictemperature after hours of operation, with the lasersteered by the function generator with the parameter

Fig. 11. Experimental switch-off: solid curve, stepwise; dashedcurve, with level 3; dotted curve, with both levels 3 and 4. Sche-matics of the electrical signal is shown in the corresponding insets.The addition of level 3 noticeably speeds up the switching andremoves the residual modulation visible in the bottom trace. Level4 does not contribute visibly to speeding up the laser intensityswitch-off, as expected from the theory.60 Parameter values: V3

� 0.039 V, V4 � 1.568 V, t3 � 40 ns, and t4 � 20 ns.

Fig. 12. Improvement introduced by the steered DMPCF: solidcurve, steered off–on–off sequence with optimal electrical steeringas obtained from Figs. 7 and Fig. 11: dotted curve, same sequencebut with simple square switching.


values already specified and at a 5% duty cycle. (Theroom is stabilized at 20 °C.) The 14 °C temperaturedifference that we measure in this case correspondsto the whole range of accessible temperatures withthe given duty cycle, from the moment the laser isturned on to full thermalization. Given that the steer-ing function operates correctly in the whole range, weconclude that the temperature does not interfere withthe correct steering of our device.

A second factor, extremely important, is the agingof the system, which can modify the optimal steeringvalues through changes in the laser characteristics.We have repeated our measurements over a timeinterval of �3 years, using the optimal parametersdetermined at the beginning of the experiment. In themeantime the laser has been used by students fordifferent purposes, including periodic fourth-year-level (one-semester-long) laboratory experiments,where the device is often handled rather roughly.During all this time the optimal steering values haveremained unchanged. Hence we conclude that laser-aging issues are not of particular concern in steeringour device.

After approximately 2 years of operation an abruptchange in the optimal values has been reported. Webelieve it to be due to a malfunction in our program-mable function generator that occasionally sendslarge overvoltage spikes ��10 V� over extremelyshort time intervals. It is possible that the protectioncircuit built into the electronics may have been un-able to entirely suppress one spike, which may havepartially damaged the laser, thus changing the opti-mal values. Since then the new optimal steering val-ues have remained constant with minimal change inlaser response (Fig. 13). Although the change in theoptimum values of the steering parameters is sizable,the fact that the same values both hold for a long time

and, at times, are subjected to rough handling andthat the new values appear to be again very reliablesuggests that laser-aging issues are not of particularconcern in steering our device.

Finally, note that the quality of the pulse shape(Fig. 12) remains the same whether we generate asingle pulse or a whole sequence with variable fre-quency, provided the maximum frequency imposedby the independence of pulses (Section 7) is not ex-ceeded. Hence the steering parameters chosen pro-vide high-quality fronts and pulses with a largedegree of versatility.

10. Conclusions

We have shown that the transient between differentoutput power levels in a laser can be effectively mod-ified even when the characteristics of the device arenot known. We have outlined a technique that allowsthe user (1) to recognize the potential for device steer-ing with simple functions and (2) to determine theoptimal steering levels to be applied to the laserthrough the power source. We have shown furtherthat the switch-off can be improved in the same wayas the switch-on and that clean square pulses (orpulse sequences) can be generated with our steeringtechnique. The limitations imposed by our program-mable function generator have proved the most con-straining factor (a) in obtaining the shortest pulsesand (b) in achieving optimal response. Indeed, as for-mally proved,63 the optimal result is achieved by set-ting the voltage levels during the transient to theirextremum values (maximum and minimum) and ad-justing the time accordingly. The poor time resolutionof the generator has obliged us to make a suboptimalchoice, where we have set the steering plateau dura-tions as close as possible to the best ones and havedetermined the voltage levels to optimize the laserresponse. Although this choice is not ideal, the suc-cessful steering shows that the technique can be im-plemented even with the limitations imposed byexternal constraints.

A discussion of the costs of a general procedure interms of experimental time, data-storage volume,and analysis has been offered together with a discus-sion and demonstration of how the effort can be re-duced with the help of appropriate strategies. Wehave further shown that errors on the steering levelsmust be quite large before they affect the quality ofthe transient response and that the unavoidable fluc-tuations present in our system do not affect the steer-ing in any visible way. Finally we have shown thatour device is rather insensitive to environmental(thermal) conditions and to aging.

The successful application of general steering to alaser system without needing a detailed modelingeffort demonstrates the strong potential for usingthis technique on various existing lasers. In the caseof telecommunication lasers this advantage trans-lates into faster data bit rates and higher precision intargeting the 1 and 0 logical values, thus offeringpotentially better bit error rates.

Fig. 13. Laser optical response for optimal steering before (dottedcurve) and after (solid curve) the abrupt change in steering values(probably) due to an electrical spike that modified the laser’s char-acteristics. The response to steering is practically unchanged, butthe new optimal values are V1, opt� � 4.237 V, V2, opt� � 0.686 V,t1, opt� � 40 ns, and t2, opt� � 30 ns.


We are grateful to N. Dokhane, H. Maurer, andJ.-L. Meunier for discussions.

References and Notes1. J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni,

“Instabilities in lasers with an injected signal,” J. Opt. Soc. Am.B 2, 173–183 (1985).

2. L. A. Lugiato, L. M. Narducci, and N. B. Abraham, eds., In-stabilities in Active Optical Media, J. Opt. Soc. Am. B 2, 1–264(1985).

3. D. K. Bandy, A. N. Oraevksy, and J. R. Tredicce, eds., Nonlin-ear Dynamics of Lasers, J. Opt. Soc. Am. B 5, 876–1215 (1988).

4. N. B. Abrahan and W. J. Firth, eds., Transverse Effects inNonlinear Optics, J. Opt. Soc. Am. B 7, 948–1157 (1990).

5. N. B. Abrahan and W. J. Firth, eds., Transverse Effects inNonlinear-Optical Systems, J. Opt. Soc. Am. B 7, 1264–1373(1990).

6. L. A. Lugiato, ed., Nonlinear Optical Structures, Patterns,Chaos, Chaos Solitons Fractals 4, 1251–1844 (1994).

7. M. Brambilla, A. Gatti, and L. A. Lugiato, “Optical patternformation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998).

8. R. Neubecker and T. Tschudi, eds., Pattern Formation in Non-linear Optical Systems, Chaos Solitons Fractals 10, 615–925(1999).

9. F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, “Pattern for-mation and competition in nonlinear optics,” Phys. Rep. 318,1–83 (1999).

10. S. Bielawski, D. Derozier, and P. Glorieux, “Experimentalcharacterization of unstable periodic orbits by controllingchaos,” Phys. Rev. A 47, R2492–R2495 (1993).

11. V. N. Chizhevsky and P. Glorieux, “Targeting unstable peri-odic orbits,” Phys. Rev. E 51, R2701–R2704 (1995).

12. V. N. Chizevsky, E. V. Grigorieva, and S. A. Kashchenko,“Optimal timing for targeting periodic orbits in a loss-drivenCO2 laser,” Opt. Commun. 133, 189–195 (1997).

13. L. A. Kotomtseva, A. V. Naumenko, A. M. Samson, and S. I.Turovets, “Targeting unstable orbits and steady states inClass B lasers by using simple off�on manipulations,” Opt.Commun. 136, 335–348 (1997).

14. G. D. VanWiggeren and R. Roy, “Optical communication withchaotic waveforms,” Phys. Rev. Lett. 81, 3547–3550 (1998).

15. G. D. VanWiggeren and R. Roy, “Communicating with chaoticlasers,” Science 279, 1198–1200 (1998).

16. J.-P. Goedgebuer, L. Larger, and H. Laporte, “Optical crypto-system based on synchronization of hyperchaos generated by adelayed feedback tunable laser diode,” Phys. Rev. Lett. 80,2249–2252 (1998).

17. L. Larger, J.-P. Goedgebuer, and F. Delorme, “Optical encryp-tion system using hyperchaos generated by an optoelectronicwavelength oscillator,” Phys. Rev. E 57, 6618–6624 (1998).

18. G. D. VanWiggeren and R. Roy, “Chaotic communication usingtime-delayed optical systems,” Int. J. Bifurcation Chaos Appl.Sci. Eng. 9, 2129–2156 (1999).

19. L. M. Pecora and T. L. Carroll, “Synchronization in chaoticsystems,” Phys. Rev. Lett. 64, 821–824 (1990).

20. R. Roy and K. S. Thornburg, Jr., “Experimental synchroniza-tion of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).

21. T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Shimizu,“Observation of synchronization in laser chaos,” Phys. Rev.Lett. 72, 3502–3505 (1994).

22. K. S. Thornburg, Jr., M. Möller, R. Roy, T. W. Carr, R.-D. Li,and T. Erneux, “Chaos and coherence in coupled lasers,” Phys.Rev. E 55, 3865–3869 (1997).

23. M. Möller, B. Forsmann, and W. Lange, “Instabilities in cou-pled Nd:YVO4 microchip lasers,” J. Opt. B: Quantum Semi-classical Opt. 10, 839–848 (1998).

24. D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond

temporal probes using spectrally tailored noisy quasi-cw laserlight,” Phys. Rev. A 55, 761–771 (1997).

25. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V.Swyfried, M. Strehle, and G. Gerber, “Control of chemical re-actions by feedback-optimized phase-shaped femtosecond la-ser pulses,” Science 282, 919–922 (1998).

26. R. J. Levis, G. M. Menkir, and H. Rabitz, “Selective bonddissociation and rearrangement with optimally tailored,strong-field laser pulses,” Science 292, 709–713 (2001).

27. D. Meshlach and Y. Silberberg, “Coherent quantum control oftwo-photon transitions by a femtosecond laser pulse,” Nature396, 239–242 (1998).

28. T. C. Weinacht, J. Ahn, and P. H. Bucksbaum, “Controlling theshape of a quantum wavefunction,” Nature 397, 233–235(1999).

29. T. Brixner, N. H. Damrauer, P. Niklaus, and G. Gerber, “Pho-toselective adaptive femtosecond quantum control in the liquidphase,” Nature 414, 57–60 (2001).

30. J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M.Motzkus, “Quantum control of energy flow in light harvesting,”Nature 417, 533–535 (2002).

31. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J.Firth, “Spatial soliton pixels in semiconductor devices,” Phys.Rev. Lett. 79, 2042–2045 (1997).

32. D. Hochheiser, J. V. Moloney, and J. Lega, “Controlling opticalturbulence,” Phys. Rev. A 55, R4011–R4014 (1997).

33. M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar,“Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).

34. M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T.Tchudi, “Origin and control of dynamics of hexagonal patternsin a photorefractive feedback system,” Chaos Solitons Fractals10, 701–707 (1999).

35. V. Raab, A. Heuer, J. Schultheiss, N. Hodbson, J. Kurths, andR. Menzel, “Transverse effects in phase conjugate laser mir-rors based on stimulated Brillouin scattering,” Chaos SolitonsFractals 10, 831–838 (1999).

36. C. Simmendinger, M. Münkler, and O. Hess, “Controlling com-plex temporal and spatiotemporal dynamics in semiconductorlasers,” Chaos Solitons Fractals 10, 851–864 (1999).

37. G.-L. Oppo, R. Martin, A. J. Scroggie, G. K. Harkness, A. Lord,and W. J. Firth, “Control of spatiotemporal complexity in non-linear optics,” Chaos Solitons Fractals 10, 865–874 (1999).

38. P.-Y. Wang and P. Xie, “Eliminating spatiotemporal chaos andspiral waves by weak spatial perturbations,” Phys. Rev. E 61,5120–5123 (2000).

39. G. J. de Valcàrcel and K. Staliunas, “Excitation of phase pat-terns and spatial solitons via two-frequency forcing of a 1:1resonance,” Phys. Rev. E 67, 026604 (2003).

40. R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, “Stabi-lization, selection, and tracking of unstable patterns by Fou-rier space techniques,” Phys. Rev. Lett. 77, 4007–4010 (1996).

41. G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, andW. J. Firth, “Elimination of spatiotemporal disorder by Fourierspace techniques,” Phys. Rev. A 58, 2577–2586 (1998).

42. A. V. Mamaev and M. Saffman, “Selection of unstable patternsand control of optical turbulence by Fourier plane filtering,”Phys. Rev. Lett. 80, 3499–3502 (1998).

43. S. Juul Jensen, M. Schwab, and C. Denz, “Manipulation, sta-bilization, and control of pattern formation using Fourier spacefiltering,” Phys. Rev. Lett. 81, 1614–1617 (1998).

44. T. Ackemann, B. Giese, B. Schäpers, and W. Lange, “Investi-gations of pattern forming mechanisms by Fourier filtering:properties of hexagons and the transition to stripes in ananisotropic system,” J. Opt. B: Quantum Semiclassical Opt. 1,70–76 (1999).

45. G. K. Harkness, G.-L. Oppo, E. Benkler, M. Kreuzer, R. Neu-becker, and T. Tschudi, “Fourier space control in an LCLV


feedback system,” J. Opt. B: Quantum Semiclassical Opt. 1,177–182 (1999).

46. E. Benkler, M. Kreuzer, R. Neubecker, and T. Tschudi, “Ex-perimental control of unstable patterns and elimination ofspatiotemporal disorder in nonlinear optics,” Phys. Rev. Lett.84, 879–882 (2000).

47. Y. Fainman, K. Kitayama, and T. W. Mossberg, eds., Innova-tive Physical Approaches to the Temporal or Spectral Control ofOptical Signals, J. Opt. Soc. Am. B 19, 2740–2823 (2002).

48. A. E. Siegman, Lasers (University Science, 1986).49. M. Danielsen, “A theoretical analysis for gigabit�second pulse

code modulation of a semiconductor laser,” IEEE J. QuantumElectron. QE-12, 657–660 (1976).

50. P. Torphammar, R. Tell, H. Eklund, and A. R. Johnston, “Min-imizing pattern effects in semiconductor lasers at high ratepulse modulation,” IEEE J. Quantum Electron. QE-15, 1271–1276 (1979).

51. P. Colet, C. Mirasso, and M. San Miguel, “Memory diagram ofsingle-mode semiconductor lasers,” IEEE J. Quantum Elec-tron. 29, 1624–1630 (1993).

52. R. Olshansky and D. Fye, “Reduction of dynamic linewidth insingle-frequency semiconductor lasers,” Electron. Lett. 20,928–929 (1984).

53. K. Petermann, “Analysis of reduced chirping of semiconductorlasers for improved single-mode-fiber transmission capacity,”Electron. Lett. 21, 1143–1145 (1985).

54. L. Bickers and L. D. Westbrook, “Reduction of transient laserchirp in 1.5 �m DFB lasers by shaping the modulation pulse,”IEE Proc. J 133, 155–162 (1986).

55. G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta,H. Grassi, and L. M. Hoffer, “Phase space techniques for steer-ing laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2,375–381 (2000).

56. P. A. Porta, L. M. Hoffer, H. Grassi, and G. L. Lippi, “Controlof turn-on in Class B lasers,” Int. J. Bifurcation Chaos Appl.Sci. Eng. 8, 1811–1819 (1998).

57. G. L. Lippi, P. A. Porta, L. M. Hoffer, and H. Grassi, “Controlof transients in ‘lethargic’ systems,” Phys. Rev. E 59, R32–R35(1999).

58. P. A. Porta, L. M. Hoffer, H. Grassi, and G. L. Lippi, “Analysisof nonfeedback technique for transient steering in Class Blasers,” Phys. Rev. A 61, 033801 (1–10) (2000).

59. N. Dokhane and G. L. Lippi, “Chirp reduction in semiconduc-tor lasers through injection current patterning,” Appl. Phys.Lett. 78, 3938–3940 (2001).

60. N. Dokhane and G. L. Lippi, “Improved direct modulationtechnique for faster switching of diode lasers,” IEE Proc.-JOptoelectron. 149, 7–16 (2002).

61. G. L. Lippi, N. Dokhane, X. Hachair, S. Barland, and J. R.Tredicce, “High speed direct modulation of semiconductor la-sers,” in Semiconductor Lasers and Optical Amplifiers forLightwave Communication Systems, R. P. Mirin and C. S.Menoni, eds., Proc. SPIE 4871, 103–114 (2002).

62. N. Dokhane and G. L. Lippi, “Faster modulation of single-mode semiconductor lasers through patterned current switch-ing: numerical investigation,” IEE Proc.-J Optoelectron. 151,61–68 (2004).

63. H-J. R. Kim, G. L. Lippi, and H. Maurer, “Minimizing thetransition time in lasers by optimal control methods. Single-mode semiconductor laser with homogeneous transverse pro-file,” Physica D 191, 238–260 (2004).

64. G. L. Lippi, S. Barland, and F. Monsieur, “Invariant integraland the transition to steady states in separable dynamicalsystems,” Phys. Rev. Lett. 85, 62–65 (2000).

65. B. Ségard, S. Matton, and P. Glorieux, “Targeting steadystates in a laser,” Phys. Rev. A 66, 053819 (1–5) (2002).

66. J. B. Khurgin, F. Jin, G. Solyar, C.-C. Wang, and S. Trivedi,“Cost-effective low timing jitter passively Q-switched diode-

pumped solid-state laser with composite pumping pulses,”Appl. Opt. 41, 1095–1097 (2002).

67. M. Bruensteiner and G. C. Papen, “Extraction of VCSEL rate-equation parameters for low-bias system simulation,” IEEE J.Sel. Top. Quantum Electron. 5, 487–494 (1999).

68. For simplicity, in this discussion we assume that the laser isswitched on from below threshold. One can easily introduce ageneralization to consider the transition between states wherethe laser is active with different output power levels. (This hasbeen discussed for telecommunication semiconductor lasers.60)

69. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).70. The step that we describe in detail in Section 4 is necessary for

laser systems, i.e., devices that package together both the laserand some controls (e.g., electronic, optical, or others). If onedesires to optimize a Class B laser with no additional elements,the general considerations of Refs. 55, 60, and 64 apply di-rectly. The procedure that we propose in this paper can still beapplied and is likely to provide faster results, but the testsdescribed in detail in Section 4 become unnecessary.

71. Although some degree of steering is possible in an oscillator,one can show that the time necessary to attain the new equi-librium state cannot be shortened, contrary to what happens inClass B lasers, and in the more complex device that we analyzehere.

72. Embedding techniques amount to generating N-dimensionalspaces starting from a one-dimensional data sequence by con-structing vectors with elements shifted by an arbitrary num-ber of elements. For a discussion see Refs. 74 and 75.

73. Compare Fig. 14 in Ref. 60 with Fig. 1.55 From the latter thepresence of the saddle point is much more readily recognized.

74. F. Takens, “Detecting strange attractors in turbulence,” inDynamical Systems and Turbulence, Vol. 898 of Springer Lec-ture Notes in Mathematics, D. A. Rand and L.-S. Young, eds.(Springer-Verlag, 1981), pp. 366–381.

75. J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos andstrange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).

76. H. G. Solari, M. A. Natiello, and G. B. Mindlin, NonlinearDynamics (Institute of Physics, 1996).

77. N. Dokhane, “Amélioration de la modulation logique directedes diodes laser par la technique de l’espace des phases,” Ph.D.dissertation (in French) (Université de Nice-Sophia Antipolis,Valbonne, France, 2000).

78. On the basis of general principles63 one prefers the oppositeapproach, fixing V1 and V2 and varying t1 and t2. If the instru-mentation allows it, this is certainly preferred. However, wesee that even the opposite one, imposed by our generator,produces successful results.

79. In general, it is not meaningful to attempt a number of levelsmuch larger than 100, no matter what system is taken intoconsideration. Maximum estimates of the number of trials cantherefore be based on this worst-case assumption if other in-formation is missing.

80. If the generator’s time resolution is small compared with thetime interval to be explored, we come back to the consider-ations already made79 and consider that 100 trials are morethan enough for a scan of the parameter space. In such a case,if no other restrictions are applied, the amount of data to beanalyzed may rapidly become too large to be practicable. A wayof reducing the portion of parameter space to analyze is pre-sented in Subsection 5.B.

81. The time estimate for one loop can be obtained on the basis ofthe following considerations. For 1 kbyte the typical accesstime for a hard disk is nowadays �15 �s, while a standardGIPB interface is currently capable of transferring the sameamount of data in �150 �s. Choosing a large safety factor inthe duty cycle, to make sure that the measurement is notaffected by external factors influencing the laser (e.g., heating,memory effects), we can set the signal frequency to �10 kHz;


thus the waiting time to synchronize the cycles is at most100 �s. In addition we have to take into account the timeLabview takes to update the parameters and send them to thegenerator, and to activate it, and for the oscilloscope to triggerand store the data in memory. Given the speed of computerclocks, even if the program is not written in an efficient way,the bottleneck of the operation is the arbitrary waveform gen-erator’s reaction time in responding and sending out the sig-nal. In our measurements this time was particularly long (3 s)because of the very old technology of the apparatus. Withsufficient error margin we can generically assume moderngenerators to be �50 times faster; therefore we arrive at anestimated cycle time of �0.06 s.

82. Although we have not used this option in our measurements,most modern oscilloscopes offer a window discrimination orsmart trigger on the data acquired. This option can be used forprefiltering the data. Alternatively this filtering can be done onthe computer, once the data are transferred from a lower-classoscilloscope or from an older model, before the information isstored on the hard disk.

83. This statement is valid with the assumption that the range ofvalues over which the steering parameter (voltage in our sys-tem) can be varied above and below threshold covers a com-parable range. In our device the laser response abovethreshold grows considerably more for 1.8 V V 3.5 V than

for 3.5 V V 5 V. Thus we can consider that the range ofbelow-threshold voltages �Vbt�1.8 V is of the same order asthe main contribution in the above-threshold interval �Vat

� 1.7 V. This response can be tested a priori in each system,and a weighted version of our statement can be used as aneducated guess for determining a reasonable interval of ratiosbetween t1 and t2 to be tested.

84. In all cases in which the time resolution of the arbitrary func-tion generator is sufficiently fine, one would effectively invertthe roles of the scans on the time values, t1 and t2, and voltagevalues, V1 and V2. In this latter case the number of voltagevalues to be chosen could also be rather small, since one wouldimmediately start by considering in a preliminary run onlythose that are sufficiently close to Vmax for V1 and to Vmin for V2.

85. This situation is represented graphically by the trajectory la-beled A in Fig. 7 in Ref. 62 when the target point in the phasespace is approached from below.

86. This corresponds to the part of the composite trajectory thatapproaches the saddle point in phase space,55 starting from theinitial operating point (with the laser switched off). It is in theneighborhood of this point that the laser field grows rapidly outof the intrinsic noise (not shown in the figures in Ref. 55).

87. In the phase-space picture55,60 this phase corresponds to aim-ing at the fixed point, thereby removing the oscillations.


Documents

Optimization of the switch-on and switch-off transition in a commercial laser