Chapter 6 Passive Modelocking – MIT OpenCourseWare

 

 

Chapter 6

Passive Modelocking

As we have seen in chapter 5 the pulse width in an actively modelocked laser
is inverse proportional to the fourth root of the curvature in the loss modu-
lation. In active modelocking one is limited to the speed of electronic signal
generators. Therefore, this curvature can never be very strong. However, if
the pulse can modulate the absorption on its own, the curvature of the ab-
sorption modulationcan become large, or in other words the net gain window
generated by the pulse can be as short as the pulse itself. In this case, the
net gain window shortens with the pulse. Therefore, passively modelocked
lasers can generate much shorter pulses than actively modelocked lasers.

However, a suitable saturable absorber is required for passive modelock-
ing. Depending on the ratio between saturable absorber recovery time and fi-
nal pulse width, one may distinguish between the regimes of operation shown
in Figure 6.1, which depicts the final steady state pulse formation process.
In a solid state laser with intracavity pulse energies much lower than the sat-
uration energy of the gain medium, gain saturation can be neglected. Then
a fast saturable absorber must be present that opens and closes the net gain
window generated by the pulse immediately before and after the pulse. This
modelocking principle is called fast saturable absorber modelocking, see Fig-
ure 6.1 a).

In semiconductor and dye lasers usually the intracavity pulse energy ex-
ceeds the saturation energy of the gain medium and so the the gain medium
undergoes saturation. A short net gain window can still be created, almost
independent of the recovery time of the gain, if a similar but unpumped
medium is introduced into the cavity acting as an absorber with a somewhat
lower saturation energy then the gain medium. For example, this can be

225

226

CHAPTER 6. PASSIVE MODELOCKING

Image removed due to copyright restrictions.

Please see:

Kartner, F. X., and U. Keller. “Stabilization of soliton-like pulses with a slow saturable absorber.”
Optics Letters 20 (1990): 16-19.

Figure 6.1: Pulse-shaping and stabilization mechanisms owing to gain and
loss dynamics in passively mode-locked lasers: (a) using only a fast saturable
absorber; (b) using a combination of gain and loss saturation; (c) using a
saturable absorber with a finite relaxation time and soliton formation.

arranged for by stronger focusing in the absorber medium than in the gain
medium. Then the absorber bleaches first and opens a net gain window,
that is closed by the pulse itself by bleaching the gain somewhat later, see
Figure 6.1 b). This principle of modelocking is called slow-saturable absorber
modelocking.

When modelocking of picosecond and femtosecond lasers with semicon-
ductor saturable absorbers has been developed it became obvious that even
with rather slow absorbers, showing recovery times of a few picoseconds, one
was able to generate sub-picosecond pulses resulting in a significant net gain
window after the pulse, see Figure 6.1 c). From our investigation of active
modelocking in the presence of soliton formation, we can expect that such a
situation may still be stable up to a certain limit in the presence of strong
soliton formation. This is the case and this modelocking regime is called
soliton modelocking, since solitary pulse formation due to SPM and GDD
shapes the pulse to a stable sech-shape despite the open net gain window
following the pulse.

6.1. SLOW SATURABLE ABSORBER MODE LOCKING

227

6.1 Slow Saturable Absorber Mode Locking

Due to the small cross section for stimulated emission in solid state lasers,
typical intracavity pulse energies are much smaller than the saturation energy
of the gain. Therefore, we neglected the effect of gain saturation due to one
pulse sofar, the gain only saturates with the average power. However, there
are gain media which have large gain cross sections like semiconductors and
dyes, see Table 4.1, and typical intracavity pulse energies may become large
enough to saturate the gain considerably in a single pass. In fact, it is this
effect, which made the mode-locked dye laser so sucessful. The model for the
slow saturable absorber mode locking has to take into account the change
of gain in the passage of one pulse [1, 2]. In the following, we consider a
modelocked laser, that experiences in one round-trip a saturable gain and a
slow saturable absorber. In the dye laser, both media are dyes with different
saturation intensities or with different focusing into the dye jets so that gain
and loss may show different saturation energies. The relaxation equation of
the gain, in the limit of a pulse short compared with its relaxation time, can
be approximated by

The coefficient EL is the saturation energy of the gain. Integration of the
equation shows, that the gain saturates with the pulse energy E(t)

dg
dt

=

g |

−

2
A(t)
|
EL

E(t) =

t

TR/2

Z

−

2
A(t)
dt
|
|

(6.1)

(6.2)

when passing the gain

g(t) = gi exp [

E(t)/EL]

(6.3)

where gi is the initial small signal gain just before the arrival of the pulse. A
similar equation holds for the loss of the saturable absorber whose response
(loss) is represented by q(t)

q(t) = q0 exp [

E(t)/EA]

(6.4)

−

−

228

CHAPTER 6. PASSIVE MODELOCKING

where EA is the saturation energy of the saturable absorber. If the back-
ground loss is denoted by l, the master equation of mode-locking becomes

1
TR

∂
∂T

A = [gi (exp (

E(t)/EL)) A

−

lA

−

−

(6.5)

q0 exp (

−

E(t)/EA)] A + 1
Ω2
f

∂2
∂t2 A

Here, we have replaced the filtering action of the gain Dg = 1
as
Ω2
f
produced by a separate fixed filter. An analytic solution to this integro-
differential equation can be obtained with one approximation: the exponen-
tials are expanded to second order. This is legitimate if the population deple-
tions of the gain and saturable absorber media are not excessive. Consider
one of these expansions:

1
2

∂2
∂t2

¸

q0 exp (

E(t)/EA)

−

≈

q0

1
∙

−

(E(t)/EA) +

(E(t)/EA)2

.

(6.6)

¸

We only consider the saturable gain and loss and the finite gain bandwidth.
Than the master equation is given by

TR

∂A(T, t)
∂T

−

−

∙

=

g(t)

q(t)

l + Df

A(T, t).

(6.7)

The filter dispersion, Df = 1/Ω2
f , effectively models the finite bandwidth
of the laser, that might not be only due to the finite gain bandwidth, but
includes all bandwidth limiting effects in a parabolic approximation. Sup-
pose the pulse is a symmetric function of time. Then the first power of the
integral gives an antisymmetric function of time, its square is symmetric.
An antisymmetric function acting on the pulse A(t) causes a displacement.
Hence, the steady state solution does not yield zero for the change per pass,
∂A
the derivative 1
∂T must be equated to a time shift ∆t of the pulse. When
TR
this is done one can confirm easily that A(t) = Ao sech(t/τ ) is a solution of
(6.6) with constraints on its coefficients. Thus we, are looking for a “steady
state” solution A(t, T ) = Ao sech( t
τ + α T
).Note, that α is the fraction of
TR
the pulsewidth, the pulse is shifted in each round-trip due to the shaping by
loss and gain. The constraints on its coefficients can be easily found using

6.1. SLOW SATURABLE ABSORBER MODE LOCKING

229

t
τ

t
τ

T
TR

)

−

T
TR

t
τ

the following relations for the sech-pulse

t

E(t) =

dt
|

2 =
A(t)
|

W
2

µ

1 + tanh(

+ α

t
τ

E(t)2 =

2 + 2tanh(

+ α

sech2(

+ α

Z

TR/2
−
W
2

2

µ

¶

µ

T
TR

)
¶

t
τ

T
TR

)
¶

(6.8)

(6.9)

A(t, T ) =

∂
∂T
∂2
∂t2 A(t, T ) =

TR

1
Ω2
f

−

1
Ω2
f τ 2

1
µ

−

α tanh(

+ α

)A(t, T )

(6.10)

2sech2(

+ α

A(t, T ),

(6.11)

T
TR

)
¶

substituing them into the master equation (6.5) and collecting the coefficients
in front of the different temporal functions. The constant term gives the
necessary small signal gain

gi

1
”

−

W
2EL

+

2

W
2EL ¶

#

µ

= l + q0

1
”

−

W
2EA

+

2

W
2EA ¶

µ

# −

1
f τ 2 .
Ω2

(6.12)

The constant in front of the odd tanh
round-trip

−

function delivers the timing shift per

(6.13)

(6.14)

α =

= gi

∆t
τ

W
2EL −

”

2

W
2EL ¶

µ

q0

# −

”

W
2EA −

2

W
2EA ¶

.

#

µ

And finally the constant in front of the sech2-function determines the pulsewidth

1
τ 2 =

f W 2
Ω2
8

q0
E2
A −

gi
E2

µ
These equations have important implications. Consider first the equation for
the inverse pulsewidth, (6.14). In order to get a real solution, the right hand
side has to be positive. This implies that q0/E2
L. The saturable
absorber must saturate more easily, and, therefore more strongly, than the
gain medium in order to open a net window of gain (Figure 6.2).

A > gi/E2

L ¶

This was accomplished in a dye laser system by stronger focusing into
the saturable absorber-dye jet (Reducing the saturation energy for the sat-
urable absorber) than into the gain-dye jet (which was inverted, i.e. optically

230

CHAPTER 6. PASSIVE MODELOCKING

Gain

Saturable
Absorber

TR

Loss

Net gain

Gain

0

Time

TR

Dynamics of a laser mode-locked with a slow saturable absorber.

Figure 6.2: Dynamics of a laser mode-locked with a slow saturable absorber.

Figure by MIT OCW.

6.1. SLOW SATURABLE ABSORBER MODE LOCKING

231

pumped). Equation (6.12) makes a statement about the net gain before pas-
sage of the pulse. The net gain before passage of the pulse is

gi

q0

−

−

l =

1
f τ 2 + gi
Ω2

−

W
2EL −

”

2

W
2EL ¶

#

µ

2

(6.15)

q0

−

”

W
2EA −

W
2EA ¶

µ

.

#

Using condition (6.14) this can be expressed as

∙

∙

+

gi

−

−

−

q0

q0

(6.16)

l = gi

1
f τ 2 .
Ω2

W
2EA ¸

W
2EL ¸
This gain is negative since the effect of the saturable absorber is larger than
that of the gain. Since the pulse has the same exponential tail after passage
as before, one concludes that the net gain after passage of the pulse is the
same as before passage and thus also negative. The pulse is stable against
noise build-up both in its front and its back. This principle works if the
ratio between the saturation energies for the saturable absorber and gain
χP = EA/EP is very small. Then the shortest pulsewidth achievable with a
given system is

τ =

4
√q0Ωf

EA
W

>

2
√q0Ωf

.

(6.17)

The greater sign comes from the fact that our theory is based on the ex-
pansion of the exponentials, which is only true for W
< 1. If the filter 2EA dispersion 1/Ω2 f that determines the bandwidth of the system is again re- placed by an average gain dispersion g/Ω2 g and assuming g = q0. Note that the modelocking principle of the dye laser is a very faszinating one due to the fact that actually non of the elements in the system is fast. It is the in- terplay between two media that opens a short window in time on the scale of femtoseconds. The media themselves just have to be fast enough to recover completely between one round trip, i.e. on a nanosecond timescale. Over the last fifteen years, the dye laser has been largely replaced by solid state lasers, which offer even more bandwidth than dyes and are on top of that much easier to handle because they do not show degradation over time. With it came the need for a different mode locking principle, since the saturation energy of these broadband solid-state laser media are much higher 232 CHAPTER 6. PASSIVE MODELOCKING than the typical intracavity pulse energies. The absorber has to open and close the net gain window. 6.2 Fast Saturable Absorber Mode Locking The dynamics of a laser modelocked with a fast saturable absorber is again covered by the master equation (5.21) [3]. Now, the losses q react instantly on the intensity or power P (t) = 2 of the field A(t) | | q(A) = q0 1 + | , 2 A | PA (6.18) where PA is the saturation power of the absorber. There is no analytic solution of the master equation (5.21) with the absorber response (6.18). Therefore, we make expansions on the absorber response to get analytic insight. If the absorber is not saturated, we can expand the response (6.18) for small intensities q(A) = q0 2, γ A | | − (6.19) with the saturable absorber modulation coefficient γ = q0/PA. The constant nonsaturated loss q0 can be absorbed in the losses l0 = l + q0. The resulting master equation is, see also Fig. 6.3 TR ∂A(T, t) ∂T l0 + Df = g ∙ − ∂2 ∂t2 + γ 2 + j D2 A | | ∂2 ∂t2 − j δ 2 A | | ¸ A(T, t). (6.20)