Self-starting of feedback control in lasers witha tendency to Q-switch

Nicolas Joly1, Serge Bielawski*

Laboratoire de Physique des Lasers, Atomes et Mol�eecules, UMR CNRS 8523, Centre d’ �EEtudes et de Recherches Lasers et Applications,

Universit�ee des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, France

Received 16 January 2003; received in revised form 8 March 2003; accepted 11 March 2003

Abstract

We show that control of Q-switch instabilities using derivative feedback [Phys. Rev. A 47 (1993) 3276] can fail in a

range of conditions. This problem appears in particular when the allowed perturbations on the pump power are limited

in size. From a numerical bifurcation analysis, it appears that this problem arises from the coexistence of a spurious

stable periodic solution together with the desired stabilized state. From this bifurcation analysis, we deduce a protocol

that solves the problem, and permits to bring the system to steady-state behavior. It consists in applying the feedback

before slowly increasing the pump power. This technique is expected to find applications in the suppression of Q-switch

in mode-locked lasers.

� 2003 Elsevier Science B.V. All rights reserved.

PACS: 42.60.Mi; 42.65.Sf

Keywords: Laser instabilities; Control; Mode-locking; Q-switch

Electronic feedback is a promising solution for

suppressing unwanted Q-switch instabilities that

appear when some nonlinearities, such as saturablelosses, are introduced in the cavity of a laser [1–4].

In the case of continuous-wave lasers [2], it has

been demonstrated that stabilization can be ob-

tained by applying on the pump power a correc-

tion proportional to the derivative of the laser

output power. Experimentally, Q-switch suppres-

sion has thus been demonstrated on a multimodeNd-doped fiber laser. More recently, it has been

demonstrated that this technique can also be used

without modification for stabilizing mode-locked

lasers [3,4]. Theoretically, the demonstration has

been performed (i) on the basis of a rate-equation

model [3] similar to the continuous-wave case [2]

and (ii) on the basis of Haus master equation [4].

Experimentally, the feasibility has been evidencedon a picosecond YVO4 laser [3]. As an important

characteristics of this feedback, the needed cor-

rections of the pump power for keeping stabiliza-

Optics Communications 220 (2003) 171–177

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +33-3-2033-6450; fax: +33-3-

2043-4084.

E-mail address: serge.bielawski@univ-lille1.fr (S. Bielawski).1 Present address: Optoelectronics Group, Department of

Physics, University of Bath, Claverton Down Bath BA2 7AY,

UK.

0030-4018/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0030-4018(03)01347-6

tion of the steady state are extremely small (the

correction tends to zero in the absence of noise).

The preceding works focused essentially on the

fact that the desired unstable state (continuous-

wave state [2] or regular mode-locking [3,4]) can be

stabilized by an adequate feedback. However, thefact that a state is made stable, does not necessarily

imply that it will be actually reached when the

feedback loop is switched on. Indeed, if the desired

stable state coexists with another unwanted state,

the state eventually reached will depend on the full

history of the system, and the stabilized state can

be still out of reach.

In this paper we show that this problem be-comes important when one imposes limitations on

the amplitude of the parameter perturbations. This

is motivated by the fact that the high pump powers

used do not cease to increase each year, and that it

is thus less and less easy to modulate the high di-

ode laser currents at relatively high speed (kHz to

MHz). Since the perturbations needed to maintain

control are small (arbitrary small in the absence ofnoise), one would be tempted to modulate only

part of the diodes of the array, or only an addi-

tional diode, or modulate the full array with an –

easier to realize – current source that is modulable

with a small amplitude.

On the basis of a rate-equation model, we will

show how a straightforward application of the

control on a Q-switching laser can fail in these sit-uations, and propose a modification of the control

strategy that solves the problem. We expect that all

the qualitative results also apply to mode-locked

lasers subjected to Q-switch instabilities.

Controlling Q-switch instabilities with deriva-

tive feedback [2–4] consists to measure the laser

power I and apply perturbations on the pump

power that are of the form

f ðtÞ ¼ bdI=dt: ð1Þ

To evaluate the consequences of amplitude limi-

tations of the pump power corrections, we con-

sider that these corrections are limited to values

between fmin < 0 and fmax > 0. Let us denote A0

our reference pump rate (pump power in units of

its value at threshold), and write

AðtÞ ¼ 0 if A0 þ f ðtÞ < 0 ð2Þ

else:

AðtÞ ¼ A0 þ f ðtÞ if fmin < f ðtÞ < fmax; ð3ÞAðtÞ ¼ A0 þ fmin if f ðtÞ < fmin; ð4ÞAðtÞ ¼ A0 þ fmax if f ðtÞ > fmax: ð5Þ

Two interesting cases are to be considered in par-

ticular: (i) fmin ¼ �A0 and A0 þ fmax slightly larger

than A0 and (ii) fmin and fmax small. The first case

corresponds to the possibility of full-range pa-rameter perturbations on the pump diode with

operating conditions near the maximum power of

the pump laser. The second case appears typically

when one tries to modulate part of the pump di-

odes. We choose to treat here essentially the sec-

ond case, although, the results presented also

apply to the former one.

First we consider a laser described by a rate-equation model characterized by an output power

I and a gain variable D [2]:

dI=dt ¼ fIðI ;DÞ; ð6ÞdD=dt ¼ fDðI ;D;AðtÞÞ; ð7Þwithout limiting ourselves to a particular model in a

first step. A typical class-B laser subjected to Q-

switch has a steady-state solution ðIst;DstÞ that is anunstable focus. From a elementary linear stabilityanalysis, one shows easily that this focus can always

be stabilized if ð�bÞ is sufficiently large [2]. Since

this result comes from a local analysis, it is inde-

pendent of the limitations of the feedback ampli-

tude fmin and fmax. As a first conclusion, the

limitations in the allowed pump variations [Eqs.

(2)–(5)] have no effect on the stability of the steady

state. A similar reasoning can be applied to mode-locked lasers: if one is able to stabilize the laser

steady state by using a sufficiently high value of�b,a change in allowed pump power variation range

does not affect the stability of the steady state. This

is also a direct consequence of the fact that the

perturbations required for stabilization tend to zero

when the system approaches steady state.

In order to illustrate how this stabilized stea-dy-state can be out of reach, it is worth con-

sidering the effect of starting from a situation

where the laser is already in a Q-switch regime,

and then applying the feedback gain by increas-

ing b. Let us consider in this case a particular

172 N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177

four-level model of class-B laser with nonlinear

losses:

dIdt

¼ fIðI ;DÞ ¼ ½D� 1� kðIÞ�I þ gD; ð8Þ

dDdt

¼ fDðI ;D;AðtÞÞ ¼ c½AðtÞ � D� ID�; ð9Þ

where the time is expressed in units of the cavity

lifetime sc, and c is the inverse of upper state life-

time ck in units of sc (c 1). g is the spontaneousemission factor. This parameter is usually very

small (we will take g ¼ 1 10�10), and does not

significantly affect the stability analysis far from

laser threshold. However, using a nonzero value

will be necessary at the end of the paper, when we

will consider the laser startup from the ‘‘off ’’ state

(I ¼ 0). kðIÞ ¼ a=ð1þ bIÞ describes the nonlinear

losses responsible for Q-switch, with a the low-power losses and b the saturability.

From a linear stability analysis, we can easily

find that the nontrivial steady state is stable when

the following condition is satisfied:

�b > �bth ¼1

cokoI

þ Ist þ 1

Ist; ð10Þ

if we neglect the spontaneous emission parameter

g. This approximation is justified because its in-

fluence is negligible far above threshold [5]. Note

that Ist is approximately equal to A� 1 when

a 1, and has to be calculated otherwise.We have integrated these equations for various

values of the parameters, with c small since we

concentrated here on class-B lasers (c ¼ 10�3 �10�5), and found systematic appearance of the

several features that are presented in this paper.

Only the precise values of the parameters differ,

but the qualitative results are unchanged.

The numerical examples presented in this paperfor illustration correspond to a typical Fabry–

Perot Yb3þ laser of optical length L ¼ 1:5 m, and

5% round-trip losses. Using the value 1=ck ¼ 2 ms

for the upper state lifetime, we deduce the model

adimensional parameter c ¼ 10�4. For the satura-

ble absorber, we have taken a ¼ 0:05, that corre-sponds to a modulation depth (difference between

losses for I ¼ 0 and I ! 1) of 0.25%. b is a freeexperimental parameter that is typically chosen by

changing the waist ws on the saturable absorber (b

scales as 1=w2s ). We have chosen a value of b ¼

0:05. We suppose that the waist in the laser me-

dium is such that the desired operating point is

associated with a pump power that is five times the

threshold value. These physical parameters are

given for illustration, however, note that all theresults presented below are in a large measure in-

dependent of the precise values of the parameters.

Here we present the results in situation where

the feedback is limited to fmin ¼ �0:25 and fmax ¼0:25 with A0 ¼ 5, as would be the case if the

feedback control is applied on one tens of the

pump diodes. If one starts from an uncontrolled

situation (b ¼ 0) where Q-switch is present, andthen applies progressively the feedback (by in-

creasing �b) beyond the stabilization threshold

�bth, the obtained result depends on the values of

c, a and b. Control is found to be efficient only for

very low values of a and b, i.e., very close to the Q-

switch instability. However, if we increase a and bto values as small as a ¼ 0:05 and b ¼ 0:05, we findthat progressive application of the control doesnot lead to continuous-wave behavior (Fig. 1) even

when the steady state is stabilized (for �b >�bth 16).

It is also important to note that the problem

remains similar in the case of an instantaneous

switch of the feedback loop (from b ¼ 0 to b >bth), or for the startup of the laser (namely an in-

stantaneous switch of A0 from 0 to the prescribedvalue A0 ¼ 5).

In order to obtain a more global understanding

of the mechanism and find a solution to the prob-

lem, it is useful to draw the bifurcation diagram of

the periodic and stationary solutions as a function

of the pump parameters b and A0. First, for refer-

ence (Fig. 2(a)), we have considered the nonpro-

blematic situation where no technical constraint isimposed on A: A0 þ fmin ¼ 0 and A0 þ fmax ¼ þ1.

We verify that the stationary is stable as expected

when (�b) exceeds a threshold value (�bth). In

addition, the periodic solution (Q-switch) disap-

pears through a Hopf bifurcation. In consequence,

starting from the uncontrolled situation (point A)

and increasing ð�bÞ leads to the stable stationary

state (point B). In contrast, with severe limitationsof the pump power (fmin ¼ �0:25 and fmax ¼ 0:25),we find that – although the stationary still becomes

N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177 173

stable as in Fig. 2(a) – the stable periodic solution

still exists, and an unstable solution appears (Fig.

2(b)). Hence, if we start from the uncontrolled sit-

uation and increase slowly ð�bÞ, we remain on the

Q-switch branch (path A ! B0). Note that this

problem is also met when pump limitations are notsevere. For instance, the qualitative features of the

diagram Fig. 2(b) are identical if we allow pump

power values ranging from 0 to 2 times the pre-

scribed value (i.e., fmin ¼ �A0 and fmax ¼ 5). In

summary, at this point the bifurcation diagrams

give us the following information: failure of the

application of the control is due to the coexistence

of a nondesired branch of periodic solution (origi-nating from the Q-switch branch) together with the

solution stabilized by feedback.

To obtain a solution to this problem, it is worth

examining the bifurcation diagrams versus the

prescribed pump power A0 (Fig. 3). Indeed, we

know that, in the system with feedback (Figs. 3(c)

and (d)), the steady-state solution can be made

stable from A ¼ 0 to any arbitrary value of A. Anatural strategy is hence to (i) ensure stabilization

Fig. 2. Bifurcation diagram versus feedback gain (�b) showingthe coexistence of the Q-switch solutions together with the

stabilized state. Solid lines indicate stable solutions; thin lines:

stable steady states; thick lines: periodic (Q-switch) solutions.

Dotted lines: unstable steady states; dashed lines: unstable pe-

riodic solutions. The maximal values of I are represented. (a)

bifurcation diagram versus feedback gain b without limitations

on the values of f ðtÞ, except AðtÞ > 0, i.e., fmin ¼ �A0 and

fmax ¼ þ1. (b) Same diagram with pump limitations (fmin ¼�0:25 and fmax ¼ 0:25, see text). In each diagram the arrows

represents the path followed by the system during the sweep of

b. Other parameters are: A0 ¼ 5, c ¼ 10�4, g ¼ 10�10, a ¼ 0:05,

b ¼ 0:05.

Fig. 1. Integration of Eqs. (8), (9) and (2)–(5) demonstrating

failure of Q-switch suppression, in spite of the stabilization of

the steady state. The feedback gain is slowly increased from

zero to high values (a). (b) and (c) represent the evolutions of

the laser power and pump rate, respectively. (d–f) Expanded

views. Evolution of laser power I before application of control,

when b ¼ 0 (d), and for a large value of the feedback gain b (e).

(f) Pump power evolution corresponding to (e). Note that the

steady state is never reached although it is stable when ð�bÞ >ð�bthÞ 16. Parameters are A0 ¼ 5, c ¼ 10�4, g ¼ 10�10, a ¼0:05, b ¼ 0:05.

Fig. 3. Bifurcation diagrams versus prescribed pump power A0

in the case of pump limitations (fmin ¼ �0:25, fmax ¼ 0:25),

same line type conventions as in Fig. 2. (a) Without control

ðb ¼ 0Þ; (b) associated expanded view of the region of usual

pump powers; (c) with control ð�b ¼ 30Þ; (d) associated ex-

panded view. Note that in (c) the steady state is stable

throughout the range of A0 represented. Same parameters as in

Fig. 2.

174 N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177

of the steady state from A ¼ 0 to the desired pump

power Aop0 , by choosing a sufficiently high value of

ð�bÞ, and then (ii) to continue the branch of stable

steady state, slowly, from a low value (under the

laser threshold, e.g., A0 ¼ 0) to Aop0 . Experimen-

tally, this solution consists of starting from thelaser in the ‘‘off’’ state with the feedback loop ap-

plied, and then increasing slowly the pump power

up to the desired pump power Aop0 . This protocol is

illustrated by the path A00 ! B00 in Fig. 3(d). The

slow increase of Aop0 should be applied at laser

startup, and also each time the laser would be

accidentally strongly perturbed and driven again

in the Q-switched regime.A test of such a protocol is presented in Fig. 4.

The value of b is first fixed to )30. Then the pump

power is increased slowly and linearly in time

dA0

dt¼ �; ð11Þ

starting from A0 ¼ 0 and A0 is kept constant when it

reaches a prescribed value (here Aop0 ¼ 5). This lin-

ear evolution is not important, and other types can

be used (e.g., exponential), the main point is that

this evolution must be slow enough to follow clo-

sely the stable steady-state branch of the bifurca-tion diagram displayed in Fig. 3(d) (A00 ! B00). � isthus taken small with respect to the gain relaxation

rate c (here � ¼ 10�5 ¼ 0:1c). Stabilization is suc-

cessfully obtained with this protocol (Fig. 4), in

contrast to the case where one attempts to stabilize

the laser after its pump is switched on (Fig. 1). Note

that a spike or a burst of spikes is observed after the

laser threshold is crossed, a phenomenon expect-able from the studies on slow passages through a

bifurcation [6]. We observe that the spike�s size canbe decreased at will by decreasing the sweep speed �or increasing the feedback gain b.

At this point it is important to emphasize that the

slow increase of A0 is essential. Indeed the same

procedure (first feedback application, and then

pumppower application)with instantaneous switchfrom A0 ¼ 0 to A0 ¼ Aop

0 cannot lead to control (if

the stabilized steady-state coexists with the Q-

switch solution). This arises from a strong topo-

logical constraint in phase space (see Appendix A).

We can conclude that the protocol of sweeping

the reference pump rate A0 must work in general

provided (i) it is sufficiently slow, and (ii) the

steady-state branch is stable for all values of A0

below the desired operating value Aop0 . This does

not represent important constraints in many cases.

First the sweep rate can be made as slow as needed

and thus found by successive trials. Then – pro-

vided the laser does not display a bistable behavior

– it is possible in lasers modeled by Eqs. (8) and (9)to choose b sufficiently high so that a stable stea-

dy-state solution exists for A0 2 ½0;Aop0 �. Note,

however, that, in the case where the nonlinearities

Fig. 4. Efficient protocol for reaching the stabilized steady-state

(integration of Eqs. (8), (9) and (2)–(5)). The feedback gain is

taken to a value sufficiently high to allow stabilization of the

steady state in the interval ½0;Aop0 � (b ¼ �30 here). Then the

desired pump rate A0 is increased slowly from an under

threshold value (A ¼ 0 here), to the desired value (A0 ¼ 5 in this

example). Note the difference in efficiency with the naive pro-

tocol illustrated in Fig. 1. Same parameters as in Fig. 2.

N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177 175

are strong enough to induce bistability, no general

results can be easily obtained and further studies

are necessary.

In conclusion, when attempting to suppress

Q-switch in lasers using feedback control, stabil-

ization of the unstable continuous-wave state (orregular mode-locking state) is not a sufficient con-

dition for ultimately reaching a stable behavior.

This problem is omnipresent when one limits the

allowed (fast) pump parameter perturbations. The

problem is that a stable periodic regime coexists

with the target stabilized state.We thus find that the

path in parameter space should be chosen carefully.

An efficient stabilization protocol consists of firstapplying the feedback, and then increasing the

pump power, sufficiently slowly, from under the

laser threshold to the desired operating value. Ex-

perimentally, it should be useful to automatically

reiterate this protocol after detecting an accidental

Q-switch behavior (that can be due, e.g., to an ac-

cidental perturbation of the laser cavity). We expect

that this protocol can be useful for stabilizing con-tinuous-wave lasers (e.g., Yb- and Er-doped lasers)

as well as mode-locked lasers. The potential success

of this protocol should allow to use a feedback loop

acting on part on the pump diodes or an additional

low-power diode. A similar problem is expected in

the case of feedback control on the intracavity los-

ses. From the fundamental point of view, an im-

portant open question concerns the possibility ofderiving analytically scaling laws for the maximum

allowed speed of the pump (parameter �), as a func-tionof themainparametersof the laser (inparticular

c, g, fmin, and fmax). This will probably require

to consider the passage through the threshold

bifurcation of the laser by perturbationmethods [6].

The Centre d��EEtudes et de Recherches Lasers et

Applications is supported by the Minist�eere charg�eede la Recherche, the R�eegion Nord-Pas de Calais

and the Fonds Europ�eeen de D�eeveloppement�EEconomique des R�eegions.

Appendix A. Failure of the method if A0 is switched

instantaneously

In the solution proposed, the slow increase of

the prescribed pump power is essential. It is not

possible at this point to find an analytical ap-

proximation of the limit speed (maximal value of

�) from the laser parameters. However, it is pos-

sible to show that an instantaneous switch of the

pump parameter A from zero to a prescribed value

Aop0 (i.e., � ! þ1), cannot allow to reach the sta-

bilized steady-state (if the stabilized state coexist

with the Q-switch solution).

This arises from a strong topological con-

straint in phase space. Indeed, when the self-

starting problem occurs, two periodic orbits

exists: a stable one, associated with Q-switch, and

an unstable one (see Fig. 2(b)). Moreover, each

branch of periodic orbit is connected via a Hopfbifurcation to the steady state. In consequence,

the two periodic orbits encircle the steady state

(Fig. 5).

Laser startup consists of taking an initial

condition at the point ðI ¼ 0;D ¼ 0Þ. Reaching

the steady target steady-state would require the

trajectory to cross the two periodic orbits. This is

impossible since determinism forbids crossing ofphase-space trajectories [7]. In other words, with

a sudden pump switch-on, starting from the ‘‘off’’

state ðI ¼ 0;D ¼ 0Þ the stable steady state is out

of reach. This argument also applies for any

initial condition outside the unstable periodic

orbit.

Fig. 5. Phase portrait showing the impossibility to reach the

stable steady state (point B00), starting from the laser in the ‘‘off’’

state (point A00). The periodic orbit labeled (QS) is associated

with the stable Q-switch regime. Note that the point B00 is en-

circled by an unstable periodic orbit, that is too small to be

distinguished from B00 on the figure. A0 ¼ 5, �b ¼ 30. Other

parameters are identical to the ones of Fig. 2.

176 N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177

References

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N. Joly, S. Bielawski / Optics Communications 220 (2003) 171–177 177