Passive modelocking - the Keller Group

Ultrafast Laser Physics
Ursula Keller / Lukas Gallmann
ETH Zurich, Physics Department, Switzerland
www.ulp.ethz.ch
Chapter 8: Passive modelocking
Ultrafast Laser Physics
ETH Zurich

Pulse-shaping in passive modelocking
U. Keller, Ultrafast solid-state lasers, Landolt-Börnstein, Group VIII/1B1, edited by G. Herziger, H. Weber, R. Proprawe,
pp. 33-167, 2007, ISBN 978-3-540-26033-2

Slow saturable absorber and dynamic gain saturation
loss
gain
pulse
time
Conditions for stable modelocking:
1) at the beginning loss is larger than gain:
l 0
> g 0
2) absorber saturates faster than the gain:
3) absorber recovers faster than the gain:
E sat ,A
< E sat ,L
⇔ A A
σ A
τ A
< τ L
< A L
σ L

Colliding pulse modelocking (CPM)
Amplifier jet: Rhodamine 6G (R6G)
Saturable absorber jet: DODCI (3,3-Diethylocadicarbocyanin)
Rhodamine 6G (R6G) σ L
= 1.36 ⋅10 −16 cm 2 τ L
= 4 ns
DODCI σ A
= 0.52 ⋅10 −16 cm 2 τ A
= 2.2 ns
Photo-isomer σ A
= 1.08 ⋅10 −16 cm 2 τ A
≈ 1 ns

Colliding pulse modelocking (CPM)
CPM Rhodamine 6G-laser Typical Best [86Val]
FWHM pulse duration
τ p
100 fs 27 fs
Emission wavelength
λ L
620 nm 620 nm
Average power
P av
2 ⋅ 25 mW 2 ⋅ 20 mW
Resonator length
L
3.3 m 3.3 m
Repetition rate
f rep
100 MHz 100 MHz
Pulse peak power in resonator
P peak
100 kW 190 kW
Output coupling transmission
T out
Pump power
Amplifier jet thickness (at focus) and
pressure
Absorber jet thickness (at focus) and
pressure
Prisms for GDD-compensation
2.5 % 3.5 %
5 W at 514
nm
4 W at 514 nm
80 m
300 m nozzle, 40
psi
30 – 50 m
120 m nozzle, 15
psi
Quartz glass

Loss modulation for a slow and fast saturable absorber
dq( t)
dt
= − q( t)
− q 0
τ A
− q t
( ) P ( t)
E sat ,A
slow saturable absorber:
τ p
> τ A
neglect recovery within pulse duration follows “immediately” incoming power
dq( t)
dt
≈ − q t
( ) P ( t)
E sat ,A
0 = − q( t)
− q 0
τ A
− q t
( ) P ( t)
E sat ,A
T R
∫
P ( t) = E p
f ( t) , where f t
0
( )dt = 1
( )
( )
P sat ,A
= E sat ,A
, ⇒ P t = I t A
τ A
P sat ,A
I sat ,A
( ) = q 0
exp − E p
q t
⎡
⎢
⎣
t
⎤
f ( t′
)d t′
E sat ,A
∫ ⎥
0 ⎦
q( t) =
q 0
1+ I A
t ( ) I sat ,A

Slow saturable absorber and dynamic gain saturation
q( t) = σ A
N A
d A
e −E p( t) E sat ,A
P ( t) = E p
f ( t)
E p
( t) = A L
A t′
t
∫
−∞
( ) 2 d t′
loss
gain
pulse
time
g( t) = σ L
N L
d L
e −E p( t) E sat ,L
Assume slow saturable absorber and slow gain saturation
net gain window:
g T
( t) = g t
( ) − q( t) = g 0
e −E p t
( ) E sat ,L
− q A
e −E p( t) E sat ,A
− l R
resonator loss:
s-parameter:
l 0
= q A
+ l R
s ≡ E sat,L
E sat,A
q A
= σ A
N A
d A

Slow saturable absorber and dynamic gain saturation
g T
( t) = g( t) − q( t) = g 0
e −E p( t) E sat ,L
− q A
e −E p( t) E sat ,A
− l R
e x ≈ 1+ x + x2
2
g T
(E) ≈ (g 0
− q A
− l R
) + (q A
− g 0
s )⋅ E + 1 2
g 0
s 2 − q A
For stable pulse generation we need to have g T < 0 at both beginning and end of the pulse.
(see later that this does not always need to be the case - see soliton modelocking)
⎛
⎝
⎜
⎞
⎠
⎟ ⋅ E 2
loss
gain
g T ( 0) ≤ 0
g T
Shorter pulses for larger max
( E p ) ≤ 0
g T
pulse
time
g T max = g T
( E max ), with
∂g T
∂E E=Emax
= 0

Slow saturable absorber and dynamic gain saturation
g T
(E) ≈ (g 0
− q A
− l R
) + (q A
− g 0
s )⋅ E + 1 2
⎛
⎝
⎜
g 0
s 2 − q A
⎞
⎠
⎟ ⋅ E 2
For stable pulse generation we assume to have g T < 0 at both beginning and end of the pulse.
g T
max
loss
gain
pulse
time
g T ( 0) ≤ 0
g T
max
E
Shorter pulses for larger
( p ) ≤ 0
g T
g T max = g T
g T max = l 0
2
( E max ), with
( 1− 1 s) 2
1− 1 s 2
( )
∂g T
∂E E=Emax
= 0
optimize s-parameter: goal in CPM
s
3 < s = σ AA L
σ L
A A
< 12

Colliding pulse modelocking (CPM)
jet thickness 30-50 µm
jet thickness ≈80 µm
Amplifier jet: Rhodamine 6G (R6G)
Saturable absorber jet: DODCI (3,3-Diethylocadicarbocyanin)
3 < s = σ AA L
σ L
A A
< 12
Optimization of s-parameter:
• mode size: factor of 4
• colliding pulse in absorber (bi-directional pulse propagation in ring laser): factor of 2 - 3
need thin absorber < spatial extend of pulses (100 fs pulses)
d
s ≈ 12
A
< c n τ p
≈ 20 µm

Slow saturable absorber and dynamic gain saturation
Master equation:
∂A( T ,t)
T R
∂T
⎛
= ⎜ g t
⎝
( ) − q( t) + g 0
Ω g
2
d 2
dt 2 + t D
d
dt
⎞
⎟
⎠
A T ,t
( ) = 0
SAM (self-amplitude modulation) time shift of pulse
A out
( t) = exp ⎡⎣ −q( t)
⎤⎦ A in
t
ΔA 2
≈ −q( t) A( T ,t)
( )
q( t) = q A
exp − A t
⎛
L
σ A
A ( t′
) 2 ⎞
⎜
A A
hν
∫ d t′
⎟
⎝
−∞ ⎠
q A : unsaturated loss of the absorber
Solution: A ( t ) = A sech ⎛ t ⎞
Rhodamin 6G:
0 ⎝
⎜
τ ⎠
⎟ τ ≤ 4 π Δν g
Δν g
≈ 4 ⋅10 13 Hz
With SPM: factor of 2 shorter (Martinez numerically)
1
ΔA 3
= t D
H. A. Haus, “Theory of modelocking with a slow saturable absorber,”
IEEE J. Quantum Electron. 11, 736, 1975
d
dt A T ,t ( )
τ p
= 1.76 ⋅τ ≤ 56 fs

Passive mode locking with an ideally fast saturable absorber
loss
loss
gain
gain
pulse
pulse
time
time
Semiconductor and dye
lasers:
Dynamic gain saturation
G.H.C. New,
Opt. Com. 6, 188, 1974
Solid-state lasers (e.g. Ti:sapphire)
Kerr lens modelocking (KLM)
Ideally fast saturable absorber
Opt. Lett. 16, 42, 1991

Ultrashort pulse generation with modelocking
FWHM pulse width (sec)
10 ps
1 ps
100 fs
10 fs
1 fs
1960
A. J. De Maria, D. A. Stetser, H. Heynau
Appl. Phys. Lett. 8, 174, 1966
Nd:glass
first passively modelocked laser
Q-switched modelocked!
Science 286, 1507, 1999!
dye laser
27 fs with ≈10 mW
compressed
Ti:sapphire laser
≈5.5 fs with ≈200 mW
1960
1970
1970 1980 # 1990
1990
2000
2000
Year
Year#
Kerr lens modelocking (KLM):
200 ns/div#
discovered - initially not understood (“magic modelocking”)
D. E. Spence, P. N. Kean, W. Sibbett, Opt. Lett. 16, 42, 1991
50 ns/div#
KLM mechanism explained for the first time:
U. Keller et al., Opt. Lett. 16, 1022, 1991#
KLM!
c

World record pulse duration at ETH
Tages Anzeiger 7./8. Juni 1997

How does such a short pulse look like?
λ/c = 2.7 fs @800 nm#
“The shortest pulses (≈5 fs at 800 nm)
in the VIS-IR spectral region
are still generated with
KLM Ti:sapphire lasers”
1.5 µm #
speed of light#
“under laser lab conditions KLM is stable” 1 fs 0.3 µm
spatial extent of a 5-fs pulse

Kerr Lens Modelocking (KLM)
D. E. Spence, P. N. Kean, W. Sibbett, Optics Lett. 16, 42, 1991
Effective Saturable Absorber
Saturation fluence
Fast Self-Amp. Modulation
Loss
Gain
Loss
Pulse fluence on absorber
Pulse
Time

Kerr Lens Modelocking (KLM)
First Demonstration: D. E. Spence, P. N. Kean, W. Sibbett, Optics Lett. 16, 42, 1991
Explanation: U. Keller et al., Optics Lett. 16, 1022, 1991
•
Advantages of KLM
very fast thus shortest pulses
very broadband thus broader tunability
Disadvantages of KLM
not self-starting
critical cavity adjustments
(operated close to the stability limit)
saturable absorber coupled to cavity
design (limited application)

Kerr Lens Modelocking (KLM)
q( t) = q 0
− γ A
A( t) 2
g( t) = g
Ideally fast saturable absorber and cw gain saturation for solid-state lasers
(no dynamic pulse-to-pulse gain saturation)

Ideally fast saturable absorber modelocking
cw Argon-Ionen
Pumplaser
f=12.5 cm
R=10cm
Ti:Saphir, 4mm dick
R=10cm
Fused Quartz Prismen
60 cm
Master equation:
T = 3 - 10 %
Auskopplung
T R
∂A
∂T = g ⎛
⎜ 1+ 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2

Master equation without SPM and GDD
∂A
T R
∂T = g ⎛
⎜ 1+ 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A = 0
( ) = A sech t
0 ⎝
⎜
τ
A t
⎛
⎞
⎠
⎟
τ =
4D g
γ A
E p
Limit of very long pulses (cw limit):
g =
l 0
1 + 1
Ω g 2 τ 2 ⇒ g = l 0
, for τ → ∞
H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Structures for additive pulse modelocking,”
J. Opt. Soc. Am. B 8, 2068, 1991

Master equation without SPM and GDD
∂A
T R
∂T = g ⎛
⎜ 1+ 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A = 0
( ) = A sech t
0 ⎝
⎜
τ
A t
⎛
⎞
⎠
⎟
τ =
4g
Ω g 2 γ A
E p
= 4D g
γ A
E p
E p
= P t
Comparison with active modelocking:
τ 4 = 2g
M
1
ω 2 2
m
Ω g
2
∫ ( ) dt = ∫ A dt =
2 A0
Mω m 2 ⇔ γ A A 0
τ 2
2
∫
t
sech 2 ⎛ ⎞
⎝
⎜
τ ⎠
⎟dt = 2A 2 0
τ
H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Structures for additive pulse modelocking,”
J. Opt. Soc. Am. B 8, 2068, 1991

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2

Linearized operators: self-phase modulation (SPM)
n 2
> 0
I(t)
( )
I t
φ 2
( t) = −kn 2
I ( t) L K
= −kn 2
L K
A( t) 2 ≡ −δ A( t) 2
leading edge
SPM: red
Pulsfront
Gaussian Pulse
Zeitabhängige Intensität
trailing Pulsflanke edge
SPM: blue
( t)
t
ω ( 2 t)
ω 0
0
Verbreiterung des Spektrums
Spectral broadening ω 2
t
t
δ ≡ kn 2
L K
( ) = dφ ( t ) 2
dt
= −δ dI ( t)
dt
E ( L K
,t) = A 0,t
( )exp ⎡⎣ iω 0
t + iφ ( t)
⎤⎦ = A( 0,t)exp
⎡ iω 0t − ik n
ω
⎣
0
A( L K
,t) = e −iδ A 2 A( 0,t)e −ik n ω 0
( )L K δ A 2

Linearized operators: group delay dispersion (GDD)
A ( z,Δω ) = A ( 0,Δω )e −iΔknz = A 0,Δω
( )e −i ⎡⎣ k n ω 0 +Δω
( )−k n ω 0
( )
⎤ ⎦ z
k n
( ω ) − k n ( ω 0 ) ≅ + k n
′ Δω + 1 2 k′′
Δω 2 n
+ ...
A ( z, Δω ) ≅ A 0, Δω
Δ AGDD
⎛
( )exp −i 1 2 k′′
nΔω 2
⎝
⎜
⎞
⎠
⎟
k n ′′ Δω 2

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
Here normalize pulse envelope as follows:
P ( t) = A( t) 2
Therefore:
δ → δ ≡ δ A L
γ A
→ γ A
≡ γ A
A A

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
( ) = A sech t
0 ⎢ ⎝
⎜
τ
A t
⎡
⎣
g − l 0
+ 1 − β 2
τ 2
⎛
⎞ ⎤
⎠
⎟ ⎥
⎦
g
Ω g
2
2
1
τ = γ A
A 0 2
⎛ g
⎜ 2 − β 2
2
⎝ Ω g
1+iβ
− 2βD
τ 2 = 0
( ) − 3βD
β = 1 ⎧
⎪
2
3 D g
D ± 9 ⎛ D g ⎞
⎨
⎝
⎜
D ⎠
⎟
⎩⎪
2
⎞
⎟
⎠
+ 8 + 4 δ A 2 0τ 2
D
⎫
⎪
⎬
⎭⎪
no SPM and GDD: β = 0
A ( t ) = A sech ⎛ t ⎞
0 ⎝
⎜
τ ⎠
⎟
g − l 0
+ g 1
= 0
2
Ω g
τ 2
4g
τ = = 4D g
Ω 2 g
γ A
E p
γ A
E p
D
δ = D g
γ A
H. A. Haus, J. G. Fujimoto, E. P. Ippen, “Structures for additive pulse modelocking,”
J. Opt. Soc. Am. B 8, 2068, 1991

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
( ) = A sech t
0 ⎢ ⎝
⎜
τ
A t
⎡
⎣
⎛
⎞ ⎤
⎠
⎟ ⎥
⎦
1+iβ
β = − 3 2 χ ± ⎛ 3
⎝
⎜
2 χ ⎞
⎠
⎟
2
+ 2
H. A. Haus, J. G. Fujimoto, E. P. Ippen,
J. Opt. Soc. Am. B 8, 2068, 1991
τ n
≡ E p
2D g
τ
D n
≡
D
g Ω = D 3β
2
g
D g 2 − β = δ + γ D A n
2
δ D n
− γ A
≡ 1 χ

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
( ) = A sech t
0 ⎢ ⎝
⎜
τ
A t
⎡
⎣
⎛
⎞ ⎤
⎠
⎟ ⎥
⎦
1+iβ
τ n
= 2 − β 2 − 3βD n
γ A
= −2D n
+ D n
β 2 − 3β
δ
H. A. Haus, J. G. Fujimoto, E. P. Ippen,
J. Opt. Soc. Am. B 8, 2068, 1991
τ n
≡ E p
2D g
τ
D n
≡
D
g Ω = D 3β
2
g
D g 2 − β = δ + γ D A n
2
δ D n
− γ A
≡ 1 χ

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
( ) = A sech t
0 ⎢ ⎝
⎜
τ
A t
⎡
⎣
⎛
⎞ ⎤
⎠
⎟ ⎥
⎦
1+iβ
Stability of solution (a simple criteria):
l cw
> g ⇒ g − l cw
< 0
l cw
> 1
T R
T R
∫
0
l ( t)dt
l cw
≈ l 0
g − l 0
= − ( 1− β 2
)
g
Ω 2 g
τ + 2βD < 0
2 τ 2
S ≡ ( 1− β 2
) − 2βD n
> 0
τ n
≡ E p
2D g
τ
D n
≡
D
g Ω = D 3β
2
g
D g 2 − β = δ + γ D A n
2
δ D n
− γ A
≡ 1 χ

Master equation with SPM and GDD
T R
∂A
∂T = g ⎛
⎜ 1 + 1 2
⎝ Ω g
∂ 2
∂t 2
⎞
⎟
⎠
A − l A + γ A 2 0 A
A + iD ∂ 2
∂t A − iδ A 2 A − iψ A = 0
2
( ) = A sech t
0 ⎢ ⎝
⎜
τ
A t
⎡
⎣
⎛
⎞ ⎤
⎠
⎟ ⎥
⎦
1+iβ
S ≡ ( 1 − β 2
) − 2βD n
> 0
H. A. Haus, J. G. Fujimoto, E. P. Ippen,
J. Opt. Soc. Am. B 8, 2068, 1991
τ n
≡ E p
2D g
τ
D n
≡
D
g Ω = D 3β
2
g
D g 2 − β = δ + γ D A n
2
δ D n
− γ A
≡ 1 χ

Master equation with SPM and GDD
H. A. Haus, J. G. Fujimoto, E. P. Ippen,
J. Opt. Soc. Am. B 8, 2068, 1991

Self-starting modelocking?
Fast Absorber
without Dynamic Gain Saturation
Modelocking Driving Force
Active
Modelocking
Slow Absorber
with dynamic gain saturation
Dispersion
Limit
( 1 / Pulse width )
shorter pulse width
E. P. Ippen, “Principles of passive modelocking,” Appl. Phys. B 58, 159, 1994

Ultrashort pulse generation with modelocking
A. J. De Maria, D. A. Stetser, H. Heynau
Appl. Phys. Lett. 8, 174, 1966
Q-switching instabilities
continued to be a problem until 1992"
200 ns/div#
SESAM#
Nd:glass
first passively modelocked laser
Q-switched modelocked!
50 ns/div#
#
KLM!
First passively modelocked
(diode-pumped) solid-state laser
without Q-switching
U. Keller et al.
Opt. Lett. 17, 505, 1992
IEEE JSTQE 2, 435, 1996
Nature 424, 831, 2003 #
1960 1970 1980 1990 2000
Year#
Flashlamp-pumped
solid-state lasers !
Diode-pumped solid-state lasers
(first demonstration 1963)#

SESAM modelocking
U. Keller et al. Opt. Lett. 17, 505, 1992
IEEE JSTQE 2, 435, 1996#
Progress in Optics 46, 1-115, 2004"
Nature 424, 831, 2003#
SESAM solved Q-switching problem
for diode-pumped solid-state lasers
Gain!
Output
coupler!
SESAM
SEmiconductor Saturable Absorber Mirror
#
self-starting, stable, and reliable # modelocking of
diode-pumped ultrafast solid-state # lasers

KLM vs. Soliton modelocking
loss
loss
gain
gain
pulse
pulse
time
Kerr lens modelocking (KLM)
Fast saturable absorber
D. E. Spence, P. N. Kean, W. Sibbett
Opt. Lett. 16, 42, 1991
time
Soliton modelocking
“not so fast” saturable absorber
F. X. Kärtner, U. Keller,
Opt. Lett. 20, 16, 1995

Passive mode locking with slow saturable absorbers
loss
loss
gain
gain
pulse
pulse
time
Semiconductor lasers:
Dynamic gain saturation
G.H.C. New,
Opt. Com. 6, 188, 1974
time
Ion-doped solid-state lasers:
Constant gain saturation:
soliton modelocking
F. X. Kärtner, U. Keller,
Opt. Lett. 20, 16, 1995

Pulse-shaping
VECSEL
KLM
Solid-state
and SESAM
Gain window can be
up to 20 times longer
than the pulse
before mode locking
becomes unstable
fast/broadband sat. abs.#
critical cavity adjustment:
KLM better at cavity
stability limit #
typically not self-starting
“not so fast” sat. abs
absorber independent of
cavity design
self-starting
For stable pulse generation it was initially assumed that we need g T < 0 at both
beginning and end of the pulse. This is however not the case!
See SESAM modelocking with slow saturable absorber and soliton modelocking

Slow saturable absorber modelocking
R. Paschotta, U. Keller, Appl. Phys. B 73, 653, 2001
absorber delays pulse!
loss
Fully saturated absorber:#
leading edge of pulse#
negligible loss for#
has significant loss from #
trailing edge of pulse#
the saturable absorber#
time
Dominant stabilization process:!
Picosecond domain: absorber delays pulse#
The pulse is constantly moving backward and #
can swallow any noise growing behind itself#
Femtosecond domain: dispersion in soliton modelocking#

Soliton modelocking: GDD negative, n 2 > 0
Outputcoupler
Group
Delay Velocity
Dispersion
Self-
Phase-
Modulation
Gain
Slow
Absorber
A out ( T ,t) = e −q(t ) A in ( T ,t)
Master equation:
( ) = ⎡⎣ 1− q( t)
⎤⎦ A in ( T ,t)
( ) = −q( t) A( T ,t)
A out
T ,t
⇒ ΔA T ,t
T R
∂
∂T A T ,t
⎛ ⎝
⎜
( ) = iD ∂ 2
( ) 2
− iδ A T ,t
2
∂t
⎞
⎠
⎟ A T ,t
⎛
⎝
⎜
( ) + g − l + D g
∂ 2
∂t 2 − q t
( )
⎞
⎠
⎟ A T ,t
( ) = 0
( ) = A sech t
⎝
⎜ 0 ⎝
⎜
τ
A T ,t
⎛
⎛
⎞ ⎞
⎠
⎟
⎠
⎟ exp ⎡
⎢ iφ 0
⎣
T
T R
⎤
⎥ + continuum τ = 4 D
⎦
δ ⋅ F p,L

Soliton modelocking: GDD negative, n 2 > 0
F. X. Kärtner, U. Keller, Optics Lett. 20, 16, 1995
Invited Paper: F. X. Kärtner, I. D. Jung, U. Keller, IEEE JSTQE, 2, 540, 1996
Soliton #
Perturbation #
Theory:#
A(T,t) =
⎛
⎜
⎝
⎛
Asech t ⎜
⎝ τ
⎞ ⎞
⎟ ⎟
⎠ ⎠
⎢ T
⎟ exp⎢
i Φ 0 T
⎣ ⎢ R
{
soliton
⎡
⎤
⎥
⎥
⎦ ⎥
+
small perturbations
{
“continuum”
only GDD & SAM (no SPM)
Continuum
Loss
GDD
Continuum
spreading
GDD
Gain
Gain
Pulse
Pulse
Frequency
Time
Frequency domain
Time domain
Stabilization: Dispersion spreads continuum out where it sees more loss

A T ,t
Soliton modelocking: GDD negative, n 2 > 0
Solution: stable soliton pulses
⎛
( ) = A sech t ⎝
⎜ 0 ⎝
⎜
τ
⎛
⎞ ⎞
⎠
⎟
⎠
⎟ exp ⎡
⎢ iφ 0
⎣
T
T R
≈ 0
⎤
τ = 4 D
⎥ + continuum δ ⋅ F p,L
⎦
Continuum
Loss
GDD
Continuum
GDD
Gain
Gain
Pulse
Pulse
Frequency
Frequency domain
Time
Time domain
Stabilization: Dispersion spreads continuum out where it sees more loss

Experimental confirmation: example Ti:sapphire laser
T=3%
output
coupler
SF 10
prism
45 cm
5 µm etalon
SF 10
prism
ROC
10cm
Ti:sapphire
4mm, 0.15%
doping
40cm
ROC
10cm
Argon pump
d
R=10cm
Absorber
(A-FPSA) SESAM
SESAM: LT-GaAs
Impulse response measured with 300 fs pulses
clearly a slow saturable absorber
No KLM: cavity operated in the middle of cavity
stability regime (and use etalon for bandwidth limitation)
Reflection, %
98.4
98.2
98.0
97.8
97.6
97.4
97.2
0
A
= 10 ps
10 20 30
Time delay, ps
40
50
I. D. Jung, F. X. Kärtner, L. R. Brovelli, M. Kamp, U. Keller, "Experimental verification of
soliton modelocking using only a slow saturable absorber," Opt. Lett., vol. 20, pp. 1892-1894, 1995

Experimental confirmation: example Ti:sapphire laser
Solution: soliton pulse
A s ( T ,t) = F p,L
soliton phase per resonator roundtrip
φ 0
= D τ 2
= δ ⋅ F p,L
4τ
Stability
(soliton perturbation theory):
continuum loss l c is
larger than soliton loss l s
Normalized abs. recovery
lc/q0 , l s/q0
1.0
0.8
0.6
0.4
0.2
2τ sech
⎛
⎝
⎜
Kontinuumverluste
continuum loss
soliton loss
Solitonverluste
T
t ⎞
τ ⎠
⎟ e iφ 0
T R
τ FWHM
= 1.76 ⋅τ = 1.76 4 D
τ FWHM
stabiles
stable
regime
Regime
unstabiles
unstable
regime
Regime
1000
800
600
400
δ ⋅ F p,L
Pulse Width τ FWHM , fs
w = τ A
τ
q 0
φ 0
= τ A
q 0
D ∝
1 D
0.0
0
10 20 30
Normalized Absorber Recovery Time w
40
200
I. D. Jung, F. X. Kärtner, L. R. Brovelli, M. Kamp, U. Keller, "Experimental verification of
soliton modelocking using only a slow saturable absorber," Opt. Lett., vol. 20, pp. 1892-1894, 1995

Experimental confirmation: example Ti:sapphire laser
τ FWHM
1.0
0.8
= 1.76 ⋅τ = 1.76 4 D
0.6
δ ⋅ F p,L
0.4
lc/q0 , l s/q0
0.2
0.0
τ FWHM
stabiles unstabiles
Kontinuumverluste
stable unstable
continuum loss
Regime regime regime Regime
Solitonverluste
soliton loss
1000
800
600
400
200
0 10 20 30 40
Normalized Absorber Recovery Time w
Pulse Width τ FWHM , fs
Autocorrelation, a. u.
- D = 200 fs 2
- D = 300 fs 2
- D = 500 fs 2
- D = 800 fs 2
FWHM
= 312 fs
- D = 1000 fs 2
- D = 1200 fs 2
-2000 -1000 0 1000 2000
Time, fs
FWHM
=391 fs
FWHM
= 469 fs
I. D. Jung, F. X. Kärtner, L. R. Brovelli, M. Kamp, U. Keller, "Experimental verification of
soliton modelocking using only a slow saturable absorber," Opt. Lett., vol. 20, pp. 1892-1894, 1995
Intensity, a. u.
- D = 200 fs 2
- D = 300 fs 2
- D = 500 fs 2
- D = 800 fs 2
- D = 1000 fs 2
- D = 1200 fs 2 f = .68THz
802
804
806 808
Wavelength, nm
f = .97 THz
f = .85 THz
810
812

Experimental confirmation: example Ti:sapphire laser
τ FWHM
1.0
0.8
= 1.76 ⋅τ = 1.76 4 D
0.6
δ ⋅ F p,L
0.4
lc/q0 , l s/q0
0.2
0.0
τ FWHM
stabiles unstabiles
Kontinuumverluste
stable unstable
continuum loss
Regime regime regime Regime
Solitonverluste
soliton loss
1000
800
600
400
200
0 10 20 30 40
Normalized Absorber Recovery Time w
Pulse Width τ FWHM , fs
Autocorrelation, a. u.
- D = 200 fs 2
- D = 300 fs 2
- D = 500 fs 2
- D = 800 fs 2
FWHM
= 312 fs
- D = 1000 fs 2
- D = 1200 fs 2
-2000 -1000 0 1000 2000
Time, fs
FWHM
=391 fs
FWHM
= 469 fs
High dynamic range autocorrelation: 330 fs
Opt. Lett. 20, 1889, 1995
I. D. Jung, F. X. Kärtner, L. R. Brovelli, M. Kamp, U. Keller, "Experimental verification of
soliton modelocking using only a slow saturable absorber," Opt. Lett., vol. 20, pp. 1892-1894, 1995
Autocorrelation, a.u.
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
-3 -2 -1 0 1 2 3
Time, ps

Coupled cavity modelocking
Chapter 8.5: for historical reasons summarized here - not used much anymore
The soliton laser
L. F. Mollenauer, R. H. Stolen, "The soliton laser," Optics Lett., vol. 9, pp. 13-15, 1984.
Additive pulse modelocking (APM)
K. J. Blow and D. Wood, "Modelocked lasers with nonlinear external cavities,"
J. Opt. Soc. Am B, vol. 5, pp. 629-632, 1988
P. N. Kean, X. Zhu, D. W. Crust, R. S. Grant, N. Landford, W. Sibbett, "Enhanced modelocking
of color center lasers," Optics Lett., vol. 14, pp. 39-41, 1989 9
E. P. Ippen, H. A. Haus, L. Y. Liu, "Additive Pulse Modelocking," J. Opt. Soc. Am. B,
vol. 6, pp. 1736-1745, 1989
Resonant passive modelocking (RPM)
U. Keller, W. H. Knox, and H. Roskos, "Coupled-Cavity Resonant Passive Modelocked
Ti:Sapphire Laser," Optics Lett., vol. 15, pp. 1377-1379, 1990
H. A. Haus, U. Keller, and W. H. Knox, "Theory of Coupled-Cavity Modelocking with
a Resonant Nonlinearity," J. Opt. Soc. Am. B, vol. 8, pp. 1252-1258, 1991