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Self Pulsing, Chaos and Antiphase Dynamics in an Er3+
Doped Fiber Laser
Eric Lacot, Frédéric Stoeckel, Marc Chenevier
To cite this version:
Eric Lacot, Frédéric Stoeckel, Marc Chenevier. Self Pulsing, Chaos and Antiphase Dynamics in an Er3+ Doped Fiber Laser. Journal de Physique III, EDP Sciences, 1995, 5 (3), pp.269-279.
�10.1051/jp3:1995124�. �jpa-00249309�
Classification Physics Abstracts
42.81 42.55 42.65
Self Pulsing, Chaos and Antiphase Dynanlics in an Er~+
Doped Fiber Laser
Eric Lacot, Fr6d6ric Stoeckel et Marc Chenevier
Laboratoire de Spectrom6trie Physique (UA 08), associ£ au CNRS, Universit£ Joseph Fourier de Grenoble, B-P 87, 38402 Saint-Martin-d'H~res Cedex, France
(Received 14 April 1994, accepted 15 December1994)
Abstract. The temporal dynamics of an Er~+ doped fiber laser shows various interesting
modes of behaviours. In the transient regime, beating and antiphase phenomena between two orthogonal state of polarisation can be observed, while the total intensity exhibits regular relax- ation oscillations. In the stationary regime a Hopf bifurcation appears leading the laser from a cw working mode to different self pulsed working modes. Under a sinusoidal modulated pump- ing the temporal behaviour of the fiber laser exhibits a period doubling cascade leading to chaos.
A study of the influence of the laser parameters allow us to determine (in a reduce parameter spaces) a bifurcation diagram showing the boundary between the dilTerent working modes.
1. Introduction
In addition to their interest in telecommunication as laser source or optical amplifier [I], fiber
laser are also interesting systems for non linear dynamic study. Indeed, in laser and more
particularly in fiber laser, a wide variety of non linear effects such as generalised bistability, crises in the chaotic attractor... can be observed [2-5]. They also can be described by a lim- ited number of degrees of freedom and have short characteristic time in the microsecond-
miUisecond range permitting an easy way of studying temporal dynamics [6-8]. Although the presence of instabilities in lasers have been observed as early as the first experiment on the
ruby laser [9], the physical studies of this instabilities continue to be the subject of on going research [10, 11]. In this paper the dynamics of an Er~+ doped fiber laser have been analysed
both in the transient and the stationary regime and a wide variety of dynamical behaviours going from cw working mode to deterministic chaos via self pulsed working mode beating and antiphase phenomena between two orthogonal state of polarisation have been observed.
A study of the influence of the extrinsic parameters (pumping, cavity, ...) and of the intrinsic parameters (doping, fiber length, ...) of the fiber laser allow us to determine in reduce laser parameter spaces bifurcation diagrams showing the boundary between the different working
Q Les Editions de Physique 1995
mode. In each experimental condition, the laser parameters, I-e- the photon lifetime, the
population inversion lifetime and the pump parameter are determined by the study of the transient relaxation oscillation of the laser during the build up time [12, 13].
2. Experimental Set-Up
Our experimental set-up is the same as in [7]. The active medium is an erbium doped silica
fiber with a length between I and 10 meters, doped with 40 to 2 000ppm of Er~+, with a
numerical aperture of 0.16 and with a core diameter of 6.4 prn. The pump laser is a Kr+ laser
and the beam passes through an acousto-optic modulator (ACM), with a bandpass of
10 MHz, used to study the transient behaviour. The erbium doped fiber is pumped longitudi- nally through a x 10 microscope objective with a numerical aperture of 0.2 and through the
input dichroic mirror which is transparent at the pump wavelength (647 nm) and has a reflec- tion coefficient R~ > 999b at the laser operating wavelength (1.538 prn). The output mirror is also buttcoupled to the fiber and has reflection coefficient R~ between 0A and 0.9. After a
high pass filter blocking the residual krypton pump beam, a beam splitter is used to record the laser intensity either directly for the total intensity or through a polarizer adjustable in
rotation for the detection of one of the two orthogonal directions of the polarisation of the emitted light. In all cases, the laser intensities are recorded by germanium detectors with a response time of I ps. In typical operating condition, the threshold is of the order of 100 mW at the fiber input and the maximum power available is about 500 mW. The near infrared out- put power obtained at the fiber output is of the order of a few mW.
3. 3kansient Dynamic
When the Kr+ pump is applied to the Er~+ doped fiber as a step function, the total intensity (~~ of the fiber laser shows after a starting time t~, a damped relaxation oscillations with a
frequency m~ (Fig. lc). From the experimental study of the variation of the starting time t~
and the square of the relaxation frequency m( versus the pump power we can determine,
using a simple model of a monomode class-B laser [14, 15], the following laser parameter [13] : the lifetime of the photon in the cavity T the proportionality constant C~ which links the absorption probability of a pump photon m~ to the pump power P according to m~ =
C~ P, and the maximum value m~~ = N~~~ B T of the pump parameter, where N~~~ and B are
respectively the total population and the Einstein coefficient. Let's recall that the pump pa-
rameter m, which described also the laser gain, is defined as the ratio of the inversion popu-
lation produce by the pump power to the threshold inversion population gain and for a three level laser is given by [14, 15] :
(m i-I)
""NtotBT ~ (l)
where i is the lifetime of the upper level. For a fiber of 1.40 m long, doped with 120 ppm of E~+ and with a output coupling mirror of reflectivity R~
= 0.9, we have determined the
following laser parameter :
C~ = 3 s~ mVC T = 4.76 x 10~~
s m~~ = 1.48 (2)
a)
-2K/0JR
__ 2x/to~
~
2f
f
~
~
-
il~
~
~i
-' Q
~th 11
? 2
ime (ms)
Fig. I. Experimental transient behaviours of the intensities ( and I~ of the two orthogonal states of polarisation and of the total intensity (~~ in response of a stepped pumping at t = 0 for P
= 250 mW. The
steady state if a continuous working mode.
This means that for a pump power of P=400mW, the pump rate is equal to
m~=1280s~~ and therefore, the lifetime T~ of the population inversion defined by I / T~ = m~ + I / i is equal to T~
= 0.7 ms, taking i = lo ms.
With the above parameter (Eq. (2)) we can easily determined the threshold pump power
P~~ defined by m=1:
tX~~ +
~~ ~( tX~~ -1 ~~~
We determine a value of P~~ = 172 mW in good agreement with our experimental results.
In our experimental condition, we can notice that the value m~~ is small compare to values of a close to 4 or 5 which can be reached in Nd~+ doped fiber laser [6]. This small value, which is a disadvantage for a large dynamical study of the Er~+ doped fiber laser, is due to a
low value of the photon lifetime T due to a bad coupling between the fiber and the cavity
mirrors or a bad fiber quality. Indeed, with L=1.40m, Rim I, R~=0.9 and T= 4.76 x
10~~ s, we determined that the extemal losses due to the reflectivity of the output coupling
mirror are equal to q~ = 3.72 x10~~ m~~ while the intemal losses (scattering absorption
in the active medium...) are equal to q~ = 0.66 m~~. This bad cavity configuration is also
confirmed experimentally by the fact that photon lifetime T is approximately unchanged
when the reflectivity of the output mirror is changed from R~ = 0.9 to R~ = 0.4.
In an optical fiber doped with a few hundred of ppm, quenching effects are negligible and in order to increase the laser gain m, we have increase the doping concentration (I.e. N~~~).
Table I gives the laser parameters determined for two fibers of length L
= 3 m doped respec- tively with 25 ppm and 120 ppm with the same cavity mirrors.
Table I. Experimentally determined laser parameters for two fibers of length L = 3 m doped respecti- vely with 25 ppm and 120 ppm. N~~~ and B are respectively the total population and the Einstein coeffi-
cient, C~ = m~ IF where m~ is the probability absorption of a pump photon and P the input pump power T is the photon lifetime and a~ = N~~~B T is the maximum value of the pump parameter a.
NtoiB Cp T
(<') (f~miv~~> (s) "~"
120ppm 3.l1zI08 1,12 9.59k10-9 2,98
25 ppm 5,94xl07 3,8 2,31x10-8 1.38
The difference between the two values of Tare explain by the difference of the fiber qual- ity or by an unintentional different coupling between the cavity mirrors and the fiber. The
comparison of the N~~~B parameter allows us to find the concentration ratio (120 ppm / 25 ppm) of the two fibers and therefore to prove that despite the strongly multi- mode operation of the fiber laser, a simple model of monode laser is sufficient to describe in
first approximation the fiber laser. The comparison of the C~ parameters shows that for the
same input pump power P the average pump rate m~ is smaller in the more doped fiber.
Indeed, in fiber laser the pumping is done longitudinally and therefore for a higher doping, the pump power is more rapidly absorbed in the beginning of the fiber. So for the same fiber length and the same input pump power, the average absorption probability of a pump photon is smaller in a more doped fiber. Consequently the parameter m~ and N~~~ are coupled. Due to
this fact, taking the parameters from Table I we determine easily that for a pump power of 180 mW, the laser gain m is lower in the more doped fiber (while its the opposite for a pump
power of P
= 400 mW).
In conclusion, in an Er~+ doped fiber laser we must compare the experimental results not for a same value of the input pump power (P) but for a same value of the average pump rate
(m~). Indeed, for the same values of m~ the gain is always higher in the more doped fiber.
Therefore, m~ is a good control parameter for the dynamical study of the Er~+ doped fiber.
In order to increase the fiber laser gain m, we can also increase the photon lifetime T, either by improving the fiber mirror coupling by using an index matching liquid or by deter-
mining the optimal fiber length. Indeed, due to the fact that the Er~+ doped fiber laser is a three level laser and the pumping is done longitudinally, it exists, for a given doping and a
given input pump power, an optimal fiber length. This length is obtained when the absorption
on the laser transition has disappeared at the fiber output (I.e, when the population inversion is realised all along the fiber).
Figure 2 illustrates for a 80 ppm doped fiber and for an incident pump power of 450 mW the starting time t~~ and the relaxation frequency m~ versus the fiber length. We have deter-
mined the laser parameters for different fiber lengths, in the vicinity of L
= 4.5 m (Tab. Hi.
As seen previously, the decrease of the C~ parameter with the fiber length is due to the lon- gitudinally pumping. The comparison of the m~~ (or 7~ parameters shows that in our experi- mental condition, the optimal length is obtained for L
= 4.5 m for which the losses are mini- mized I-e- the photon lifetime T is maximum.
350 4
j O°
° ~oo ,,( ~--
"i .-~
l.2 .'.- 1-.
.,q O
'O~°~ Q j° O)
~ ~o ' ~
(1.0 I.."..-joo.-I- )-... ~ 250 ~~~; -..(."
~' n~
~~ QOO'
O ' a jO~
OOj O °*°°
.1°,~
~o0 ~
<" ""j~~'~'
'
j_
j
150
3 4 5 6 7 8 3 4 5 6 7 8
L (m) L (m)
Fig. 2. Experimental evolution of the starting time (t~) and of the relaxation frequency (m~) versus the fiber length (L) for a 80 ppm E~+ doped fiber and for an input pump power of P
= 450 mW.
Table II. Experimentally determined laser parameters for different length of a 80 ppm doped fiber laser N,~, and B are respectively the total population and the Einstein coeyficient, C~ = m~ IF where m~ is the probability absolption of a pump photon and P the input pump power T is the photon I@etime
and a~~~ = N,~, B T is the maximum value of the pump parameter a.
Ntot B~ cp T
~
(s-I) (s"I mwl) (s) ~°~
L
= 3 m 1.84J08 7.84 6.33xl0.9 1.16
L
= 3,5 m 1,84d08 7.281 6,54J0.9 1.20
L=4m 1,84J08 7,12 6,61x10-9 1.22
L = 4,5 m 1,84xl08 6,87 7.06d0-9 1.30
L = 5 m 1.84x108 5.92 6.65kl0'9 1.22
During the transient regime, as for the Nd~+ doped fiber laser [6], the response of the total intensity of an Er~+ doped fiber laser is the sum of the response of the intensity I~ and I~ of
two orthogonal states of polarizations of the emitted light (Fig, I). This Figure shows that each states of pol3Tisation is a superposition of two damped oscillations with two different
oscillations frequencies (beating). The fast one (m~) correspond to the relaxation frequency previously used to determine the laser parameter while the lower frequency m~ is associated
with a new relaxation mode which appears in the fiber due to a coupling between the two
orthogonal states of polarisation. The comparison of the Figures la and 16 shows that the high frequencies oscillations (relaxation frequency) are in phase while the low frequencies
are in opposite phase (antiphase phenomena) [16-18]. As this low oscillations have almost the same amplitude for the two polarisations, they destructively interfere and as a result, the
total intensity exhibits damped oscillation with only the high frequency (Fig. lc). Such be-
haviour (beating and antiphase phenomena) can be described in the frame of a multimode
laser theory including spatial hole buming [17-19] or by using a phenomenological model of two class-B laser coupled by their intensity and population inversion (cross saturation cou- pling) [6, 7, 13].
4. Stationary dynamic
Under continuous pumping, by increasing the pump power, the laser reachs a second thresh-
old where a Hopf bifurcation occurs leading the laser from a continuous working mode to a
self pulsed working mode. Figure 3 shows clearly the transitions between the cw and the self pulsed working mode. For a pump power between the threshold pump power P~h and a given
pump power P~(P~~ <P<P~) the Er~+ doped fiber laser exhibits in the stationary re-
gime, a cw working mode and the total intensity increases linearly with the pump power. For
P > P~ the fiber laser exhibits a self pulsed working mode and the shape of the pulses
changes from a near sinusoidal to a more narrower and deeper function when the pump in-
creases. As for the c-w- working mode, the transient behaviour preceding the self pulsed
working mode is also the sum of the transient behaviour of two orthogonal polarised intensi- ties, which presents beating and antiphase phenomena. However, Figure 4 shows that in this
case only the lowest frequency is damped and as a result the total intensity exhibits both in the transient and in the asymptotic regime, oscillations, with only the high oscillation fre-
quency.
By a study of the influence of the laser parameters (cavity and amplifier medium) on the dynamics of the Er~+ doped fiber laser we have found that the different working domains of the fiber laser (cw or self pulsing) can be represented in a two dimension parameter space (N~~~ B'T, m~ i) in which the boundary between the c-w- and the self pulsed working mode
(Hopf bifurcation) appears like a hyperbola branch similar to that for the laser threshold
(Fig. 5). These steady-state bifurcations have been obtained by changing the photon lifetime T (I.e. the intemal loss) for two lengths (L=3 m and L=1,4m) of a fiber doped with
120 ppm. In each experimental conditions, the photon lifetime has been changed by slightly moving the distance between the fiber and the output mirror, and the threshold pump power P~h, the Hopf pump power PM and the laser parameters are determined.
Experimentally, the variation of the starting time t~~ and the relaxation frequency m~ ver- sus the pump power (which allows us to determine the laser parameters) show any disconti-
nuity in the vicinity of the Hopf pump power. Consequently, and although in the monomode class-B laser model [14, 15] any self pulsed working mode exists, the previously determined laser parameters which are in good agreement with the experimental results below the Hopf pump power remain in first approximation valid above this pump parameter.
400
300 1-. ~.j~~....~~
I i
§ 200 ~~_~j_~~____
c~ p
100 ..j j
0
50 100 150 200 250
3
H
~
2
~
E p
~'.
_s l
o
Time (ms)
Fig. 3. Evolution of the dynamical behaviour of the total intensity l~~~ of the E~+ doped fiber laser when the pump power P linearly increase with time. P~ and P~ are respectively the threshold and the
Hopf pump powers.
Using the Figure 5 and the parameters of Table I, we can understand why in our experi-
mental conditions, such a fiber exhibits only a c-w- working mode. Indeed, we determined
that for a value for m~~ =1.38 the threshold is reached for m~1= 6.5 while the Hopf
bifurcation occurs for a value of m~ i which seems to be higher than 15 and then corresponds
to a pump power we cannot reach experimentally.
Figure 6 shows the coupled effect of a longitudinal pumping and a three level system.
Indeed, when the fiber length increases, the laser evolves from a cw working mode to a self
pulsed working mode with a maximum frequency and pulses amplitude for L
= 4.5 m. For an
higher length of the fiber, the laser tends again toward a cw working mode (not observed experimentally) by a decrease of the pulses repetition rate and of the modulation depth. The appearance and the progressive disappearance of the self pulsed working mode are explained by the progressive evolution of the laser gain parameter (m ) with the fiber length (see
Table II).
In lasers, chaos can appear either spontaneously either when one control parameter is modulated or when they are submitted to an injected signal [20]. We have applied a sinusoi-
dal modulation of the pump power near the relaxation frequencies (m~ and m~) and we have
then observed in the dynamic response non linear phenomena and more complex dynanfical
behaviours. Figure 7 shows a typical bifurcation diagram obtained by using periodic sam-
pling, of the laser intensity, synchronised at the pump frequency modulation. In such tech-
nique the sampling unit delivers a single value when the signal is T-periodic, n values when
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