Generation of short pulses
Jörgen Larsson,
Fysiska Instutionen
Lunds Tekniska Högskola
Generation of short pulses
• Cavity modes
• Locked cavity modes
• Time-bandwidth product
• Active mode-locking
• Acousto-optic modulation
• Passive modelocking
• Hybrid modelocking techniques
• Kerr lens modelocking
• SESAM
• Synchrnously pumped dye lasers
• Distributed feedback lasers
• Fiber lasers
• Short-pulse accelerator sources
• Group velocity dispersion
• Group velocity dispersion compensation
• Prism compressor
• Chirped mirrors
Representation of short pulses
Gaussian pulses
E (t )  E0e
Amplitude
I(t) 
 0 r c
2
 at 2
*e
Envelope
j0t
Carrier
E (t ) 
2
 0 r c
2
Frequency
2  2 at 2
0
E e
Representing ”chirp”
E (t )  E0e
 at 2
*e
j (0t bt 2 )
  (0t  bt 2 )
 (t ) 

 0  2bt
dt
dt
Group velocity dispersion
Modes in a cavity
Mode spacing

Gain profile
(Gain) bandwidth

Intensity
50
(a)
Single Mode
40
30
20
10
0
50
(b)
Two Modes
Intensity
40
30
20
10
0
Intensity
50
(c)
8 Modes
Random Phases
40
30
20
10
0
8 modes
Phases=0 @ t=0
Intensity
50
40
30
20
10
0
(d)
Fresnel diagrams
t=t
t=0
1
E
m 2

2
2E
1
m
mE
(b)
(a)
t= 2
m
m

t=T= 

1
E
m
(c)
t
2
2E
(d)
m
mE
Time-bandwidth product
FOURIER TRANSFORM LIMITED
t
T=2L/c

time
1/T
Frequency
Time-bandwidth product- How short
pulses can we get?
E (t )  E0e
I(t) 
 0 r c
2
 at 2
*e
E (t ) 
2
j0t
 0 r c
2
2  2 at 2
0
E e
FWHM of the intensity in the temporal domain
e
 2 at1 2 2
 2at1 2
t FWHM
2
1

2
1
ln 2
2
 ln  2at1 2  ln 2  t1 2 
2
2a
ln 2
2
2a
Time-bandwidth product- How short
pulses can we get?
Next we determine the width in the spectral plane
F(E(t ))  F( E0e
at 2
I() ~ E ()  E e
2
*e
2
j0t
)  E e
( 0 ) 2

4a
( 0 )
4a
FWHM of the intensity in the spectral domain
1
e
2
2
(1 2 0 ) 2
4a
 2
FWHM  2 2a ln(2)
(1 2  0 ) 2
4a
1
 ln   (1 2  0 )  2a ln(2)
2
2a ln(2)
vFWHM 

Time-bandwidth product- How short
pulses can we get?
Now lets calculate the time-bandwidth product for a
gaussian (unchirped) pulse
t FWHM vFWHM 
2a ln(2)

ln 2 2 ln(2)
2

 0.441
2a

If the pulse is chirped it is wider in the temporal domain
tFWHM vFWHM  0.441
Time-bandwidth product- How short
pulses can we get?
Task for the interested student:
A Ti:Sapphire laser operating at 800 nm has a 120 nm
FWHM spectrum. What is the shortest pulse we can get
from this laser?
Classes of methods for modelocking
Active modelocking:
From an active component in the cavity
(typically an optic modulator driven by an
RF-frequency)
Passive Modelocking
From a passive component in the cavity
(Saturable absorber, kerr lens ......)
Active modelocking
Acousto-optic modulation
Active modelocking
Acousto-optic modulation
Active modelocking
Acousto-optic modulation
Generation of sidebands in an AOM
i (t kx)
• Optical wave
E  E0e
• Acoustic wave P  P0 sin(t  Kz )
• Optical wave in presence of acoustic wave
l
E  E0e
0
i (t  kx) i k '( t , z ) dx
 E0e
i (t  kx) i{
 E0e
i (t kx) ik
2n n sin( t  Kz )
}l )

n
i (t kx)iasin( t  Kz )
E  E0e
n sin( t  Kz )
l
n
a
2nl

Generation of sidebands in an AOM
(travelling wave)
E  E0ei (t kx)eiasin( t Kz )
If a<<1
E  E0ei (t kx) (1  ia sin(t  Kz))
Euler’s formulae
ia i ( t  Kz ) i ( t  Kz )
E  E0 e
(1  {e
e
})
2i
a i ({ }t kx Kz ) a i ({ }t  kx Kz )
i (t  kx)
E  E0 {e
 e
 e
)
2
2
i (t  kx)
Generation of sidebands in an AOM
(travelling wave-strong Rf- field)
E  E0ei (t kx)eiasin( t Kz )
i (t kx)
E  E0e
E  E0e
E  E0
i (t  kx)
m  

J
m  

J
(1  ia sin(t  Kz))
m
( a )e
m
( a )e
im( t  Kz )
(  m ) t i ( kx mKz )
Generation of sidebands in an
AOM (standing wave)
E  E0ei (t kx)eiasin( t ) cos( Kz )
If a<<1
E  E0ei (t kx) (1  ia sin(t ) cos(Kz))
Euler’s formulae
E  E0 e
i (t  kx)
ia i ( t ) i ( t )) 1 i ( Kz )
(1  {e
e
}  {e
 e i ( Kz )) })
2i
2
a
a
a
a
E  E0 ei (t  kx)  {ei ({  }t  kx)  Kz ) }  {ei ({ }t  kx)  Kz ) }  {ei ({  }t  kx)  Kz ) }  {ei ({ }t  kx) Kz ) }
4
4
4
4
Active modelocking
Fig 3.7
Active modelocking
Fig 3.8
Passive modelocking
Saturable absorber
Fig 3.12
Passive modelocking
Saturable absorber
Fig 3.13
Gain vs intensity
Fig 3.14
Passive modelocking
Passive modelocking-saturable absorber
Fig 3.17
Passive modelocking
Saturable absorber
Passive modelocking
Kerr lens
Titanium sapphire
crystal
Aperture
Low intensities
large losses
Laser beam
x
n=n1+n2I
I
High intensity
small losses
The beams spatial profile creates the "Kerr lens"
Passive modelocking - Saturable
semiconductor mirror (SESAM)
Synchronous pumping
Frequency filtering
Passive modelocking-saturable absorber
Fig 3.19
Hybrid modelocking
Fig 3.20
Hybrid modelocking
Fig 3.21
Titanium Sapphire energy level diagram
Passive modelocking-Kerr lens (early design)
Modern Titanium Sapphire laser
OC
CM1
CM2
P2
L
pump from Nd-laser
C
P1
P1,P2 prisms
CM1, CM2 curved mirror, krökt spegel
(these are transparent for the pump radiation)
M mirror, spegel
C crystal, kristall
OC output coupler utkopplingsspegel
L lens for the pump laser
M