Transcript The Generation of Ultrashort Laser Pulses The importance of bandwidth More than just a light bulb Laser modes and mode-locking Making shorter and shorter.

The Generation of Ultrashort Laser Pulses
The importance of bandwidth
More than just a light bulb
Laser modes and mode-locking
Making shorter and shorter pulses
Pulse-pumping
Q-switching and distributed-feedback lasers
Passive mode-locking and the saturable absorber
Kerr-lensing and Ti:Sapphire
Active mode-locking
Other mode-locking techniques
Limiting factors
Commercial lasers
But first: the progress has been amazing!
10ps
Nd:glass
Nd:YAG
SHORTEST PULSE DURATION
Dye
S-P Dye
Nd:YLF
Diode
1ps
CW Dye
Color
Center
100fs
Cr:LiS(C)AF
Er:fiber
CP M Dye
Nd:fiber
Cr:YAG
Cr:forsterite
10fs
w/Compression
Ti:sapphire
1965
1970
1975
1980
1985
YEAR
1990
1995
2000
2005
The shortest
pulse vs. year
(for different
media)
Continuous vs. ultrashort pulses of light
A constant and a delta-function are a Fourier-Transform pair.
Irradiance vs. time
Spectrum
time
frequency
time
frequency
Continuous beam:
Ultrashort pulse:
Long vs. short pulses of light
The uncertainty principle says that the product of the temporal and
spectral pulse widths is greater than ~1.
Irradiance vs. time
Spectrum
time
frequency
time
frequency
Long pulse
Short pulse
For many years, dyes have been the broadband
media that have generated ultrashort laser
pulses.
Ultrafast solid-state laser media have
recently replaced dyes in most labs.
Laser power
Solid-state laser media have broad bandwidths and are convenient.
Light bulbs,
lasers, and
ultrashort pulses
But a light bulb is also broadband.
What exactly is required to make an ultrashort pulse?
Answer:
A Mode-locked Laser
Okay, what’s a laser, what are modes, and what does it mean to
lock them?
Before
After
Absorption
Unexcited
molecule
Spontaneous
Emission
Stimulated
Emission
Excited
molecule
Stimulated emission leads to a chain
reaction and laser emission.
If a medium has many excited molecules, one photon can become
many.
Excited medium
This is the essence of the laser. The factor by which an input beam is
amplified by a medium is called the gain and is represented by G.
The laser
A laser is a medium that stores energy, surrounded by two mirrors.
A partially reflecting output mirror lets some light out.
I0
R = 100%
I3
I1
Laser medium
with gain, G
I2
R < 100%
A laser will lase if the beam increases in intensity during a round trip:
that is, if I3  I 0
Usually, additional losses in intensity occur, such as absorption, scattering, and reflections. In general, the laser will lase if, in a round trip:
Gain > Loss
This called achieving Threshold.
Calculating the gain:
Einstein A and B coefficients
2
1
In 1916, Einstein considered the various transition rates between
molecular states (say, 1 and 2) involving light of irradiance, I:
Absorption rate = B N1 I
Spontaneous emission rate = A N2
Stimulated emission rate = B N2 I
Laser gain
Neglecting spontaneous emission:
dI
dI
 c
 BN 2 I - BN1I
dt
dz
 B  N 2 - N1  I
The solution is:
[Stimulated emission minus absorption]
Proportionality constant is the
absorption/gain cross-section, 
I ( z )  I (0) exp   N 2  N1  z
There can be exponential gain or loss in irradiance. Normally, N2 < N1,
and there is loss (absorption). But if N2 > N1, there’s gain, and we
define the gain, G:
G  exp   N 2  N1  z
If N2 > N1:
g   N2  N1 
If N2 < N1 :
   N1  N2 
How to achieve laser threshold
In order to achieve threshold, G > 1, that is, stimulated emission
must exceed absorption:
Inversion
B N2 I > B N1 I
N2 > N1
This condition is called Inversion.
It does not occur naturally.
Energy
Or, equivalently,
“Negative
temperature”
Molecules
In order to achieve inversion, we must hit the laser medium very
hard in some way and choose our medium correctly.
Why inversion is impossible
in a two-level system
2
N2
1
N1
Write rate equations for the densities of the two states.
Absorption
Stimulated emission
dN 2
 BI ( N1  N 2 )  AN 2
dt
Spontaneous
emission
If the total number of
dN1
 BI ( N 2  N1 )  AN 2
dt
d N

 2 BI N  2 AN 2
dt
molecules is N:
N  N1  N2
N  N1  N2
2 N2  ( N1  N2 )  ( N1  N 2 )
d N

 2 BI N  AN  AN
dt
 N  N
Why inversion is impossible
in a two-level system (cont’d)
2
N2
1
N1
d N
 2 BI N  AN  AN
dt
In steady-state:
0  2BI N  AN  AN
 ( A  2BI )N  AN
 N  AN /( A  2BI )
 N  N /(1  2BI / A)
N
 N 
1  I / I sat
where:
I sat  A / 2B
Isat is the saturation intensity.
N is always positive, no matter how high I is!
It’s impossible to achieve an inversion in a two-level system!
Why inversion is possible
in a three-level system
Assume we pump to a state 3 that
rapidly decays to level 2.
Spontaneous
emission
dN 2
 BIN1  AN 2
dt
Absorption
dN1
  BIN1  AN 2
dt
d N
 2 BIN1  2 AN 2
dt
3
2
Fast decay
Pump
Transition
Laser
Transition
1
The total number
of molecules is N:
N  N1  N2
N  N1  N2
2N2  N  N
2N1  N  N
d N
  BIN  BI N  AN  AN

dt
Level 3
decays
fast and
so is zero.
Why inversion is possible
in a three-level system
(cont’d)
3
2
Fast decay
Pump
Transition
Laser
Transition
d N
1
  BIN  BI N  AN  AN
dt
In steady-state: 0   BIN  BI N  AN  AN
 ( A  BI )N  ( A  BI ) N
 N  N ( A  BI ) /( A  BI )
1  I / I sat
 N  N
1  I / I sat
where:
I sat  A / B
Isat is the saturation intensity.
Now if I > Isat, N is negative!
Why inversion is easy
in a four-level system
Now assume the lower laser level 1
rapidly decays to the ground level 0.
As before:
3
Fast decay
2
Pump
Transition
dN 2
 BIN 0  AN 2
dt
dN 2
 BI ( N  N 2 )  AN 2
dt
Because
N1  0,
N   N2
d N

 BIN  BI N  AN
dt
At steady state:
0  BIN  BI N  AN
1
0
Laser
Transition
Fast decay
The total number
of molecules is N :
N  N0  N 2
N0  N  N 2
Why inversion is easy
in a four-level system
(cont’d)
3
Fast decay
2
Pump
Transition
0  BIN  BI N  AN
1
0
 ( A  BI )N   BIN
Laser
Transition
Fast decay
 N  BIN /( A  BI )
 N  ( BIN / A) /(1  BI / A)

I / I sat
N   N
1  I / I sat
where:
I sat  A / B
Isat is the saturation intensity.
Now, N is negative—always!
3
What about the
saturation intensity?
Fast decay
2
Pump
Transition
I sat  A / B
A is the excited-state relaxation rate: 1/t
1
0
Laser
Transition
Fast decay
B is the absorption cross-section, , divided by
the energy per photon, ħw:  / ħw
Both  and t
depend on the
molecule, the
frequency, and
the various
states involved.
ħw ~10-19 J for visible/near IR light
I sat
w

t
t ~10-12 to 10-8 s for molecules
 ~10-20 to 10-16 cm2 for molecules (on
resonance)
105 to 1013 W/cm2
The saturation intensity plays a key role in laser theory.
Two-, three-, and four-level systems
It took laser physicists a while to realize that four-level systems are
best.
Two-level
system
Four-level
system
Three-level
system
Molecules
accumulate
in this level.
Fast decay
Fast decay
Pump
Transition
Pump
Transition
Laser
Transition
Pump
Transition
At best, you get
equal populations.
No lasing.
Laser
Transition
Level
empties
fast!
If you hit it hard,
you get lasing.
Laser
Transition
Fast decay
Lasing is easy!
A dye’s energy levels
Dyes are big molecules, and they have complex energy level structure.
S1: 1st excited
electronic state
Energy
S2: 2nd excited
electronic state
Pump Transition
S0: Ground
electronic state
Lowest vibrational and
rotational level of this
electronic “manifold”
Excited vibrational and
rotational level
Laser Transition
Dyes can lase into any (or
all!) of the vibrational/
rotational levels of the S0
state, and so can lase very
broadband.
Lasers modes: The Shah function
The Shah function, III(t), is an infinitely long train of equally
spaced delta-functions.
t

III(t ) 
  (t  m)
m 
The symbol III is pronounced shah after the Cyrillic character III, which is
said to have been modeled on the Hebrew letter
(shin) which, in turn,
may derive from the Egyptian
a hieroglyph depicting papyrus plants
along the Nile.
The Fourier transform of the Shah function



III(t)
 t  m) exp(iwt )dt


 m 



t
   t  m) exp(iwt )dt
m  


 exp(iwm)
m 
If w = 2np, where n is an integer, every term is
exp(2mnp i) = 1, and the sum diverges;
otherwise, cancellation occurs. So:
F {III(t )}  III(wp 
F {III(t)}
w
2p
The Shah function
and a pulse train
Convolution
An infinite train of identical pulses
(from a laser!) can be written:
E (t )  III(t / T )  f (t )
where f(t) is the shape of each pulse and T is the time between
pulses.



   (t / T  m) f (t  t) dt
m 
Set t’ /T = m or t’ = mT



m 
f (t  mT )
The Fourier transform of an infinite train of pulses
An infinite train of identical pulses can be written:
E(t) = III(t/T) * f(t)
where f(t) represents a single pulse and T is the time between pulses.
The Convolution Theorem states that the Fourier Transform of a
convolution is the product of the Fourier Transforms. So:
E (w ) 
III(wT / 2p ) F (w 
A train of pulses results from a single pulse bouncing back and forth
inside a laser cavity of round-trip time T. The spacing between
frequencies—called laser modes—is then w = p/T or n = 1/T.
Mode-locked vs. non-mode-locked light
Mode-locked pulse train:
E (w ) 

 F (w)  (w  2p m / T )
m 
A train of
short pulses

 F (w )
  (w  2p m / T )  F (w  III(wT / 2p )
m 
Non-mode-locked pulse train:
E (w ) 
Random phase for each mode

 F (w) exp(i
m 
m
)  (w  2p m / T )

 F (w )  exp(im )  (w  2p m / T )
m 
A mess…
Generating short pulses = mode-locking
Locking the phases of the laser modes yields an ultrashort pulse.
Locked modes
Intensities
Numerical simulation of mode-locking
Ultrafast lasers often have thousands of modes.
A generic ultrashort-pulse laser
A generic ultrafast laser has a broadband gain medium, a pulseshortening device, and two or more mirrors:
Mode-locker
Many pulse-shortening devices have been proposed and used.
Pulsed Pumping
Pumping a laser medium with a short-pulse flash lamp yields a
short pulse. Flash lamp pulses as short as ~1 µs exist.
Unfortunately, this yields a pulse as long as the excited-state
lifetime of the laser medium, which can be considerably longer
than the pump pulse.
Since solid-state laser media have lifetimes in the microsecond
range, it yields pulses microseconds to milliseconds long.
Long and potentially
complex pulse
Q-switching
Output intensity
Abruptly allowing the
laser to lase.
100%
Cavity Gain
Preventing the laser
from lasing until the
flash lamp is finished
flashing, and
Cavity Loss
Q-switching involves:
0%
Time
The pulse length is limited by how fast we can switch and the
round-trip time of the laser and yields pulses 10 - 100 ns long.
Q-Switching
How do we Q-switch a laser?
Q-switching involves preventing lasing until we’re ready.
A Pockels’ cell switches (in a few nanoseconds) from a quarterwave plate to nothing.
Before switching
0° Polarizer
Mirror
Pockels’ cell as
wave plate w/
axes at ±45°
Light becomes circular on the first
pass and then horizontal on the next
and is then rejected by the polarizer.
After switching
0° Polarizer
Mirror
Pockels’ cell as
an isotropic
medium
Light is unaffected by the
Pockels’ cell and hence is
passed by the polarizer.
   N0  N1 
Passive mode-locking:
the saturable absorber
For a twolevel system
N


1  I / I sat
Like a sponge, an absorbing medium can
only absorb so much. High-intensity spikes
burn through; low-intensity light is absorbed.
 (I) 
0
1  I Isat
 0  N
The effect of a saturable absorber
Intensity
First, imagine raster-scanning the pulse vs. time like this:
Short time (fs)
k=1
k=2
k=3
k=7
Notice that the weak pulses are suppressed,
and the strong pulse shortens and is amplified.
After many round trips, even a slightly saturable absorber can yield
a very short pulse.
Passive
mode-locking:
the
saturable
absorber
High-intensity spikes
(i.e., short pulses) see
less loss and hence
can lase while lowintensity backgrounds
(i.e., long pulses) won’t.
Passive mode-locking with a slow
saturable absorber
What if the absorber responds slowly (more slowly than the pulse)?
Then only the leading edge will experience pulse shortening.
This is the most common situation, unless the pulse is many ps long.
Gain saturation shortens the pulse
trailing edge.
The intense spike uses up the laser gain-medium energy,
reducing the gain available for the trailing edge of the pulse
(and for later pulses).
Saturable
gain and
loss
Lasers lase when
the gain exceeds
the loss.
The combination of
saturable absorption
and saturable gain
yields short pulses
even when the
absorber is slower
than the pulse.
The Passively Mode-locked Dye Laser
Pump
beam
Saturable
absorber
Gain medium
Passively mode-locked dye lasers yield pulses as short as a few
hundred fs.
They’re limited by our ability to saturate the absorber.
Some common dyes and their
corresponding saturable absorbers
Colliding pulses have a higher peak intensity.
Intensity
Two pulses colliding
Single pulse
Longitudinal position, z
And higher intensity in the saturable absorber is what CPM lasers require.
The colliding-pulse modelocked (CPM) laser
A Sagnac interferometer is ideal for
creating colliding pulses.
Saturable
absorber
Gain medium
Beamsplitter
CPM dye lasers produce even shorter pulses: ~30 fs.
A lens and a lens
A lens is a lens
because the phase
delay seen by a
beam varies with x:
x
L(x)
In both cases, a
quadratic variation
of the phase with x
yields a lens.
f(x) = n k L(x)
Now what if L is
constant, but n
varies with x:
f(x) = n(x) k L
x
n(x)
Kerr-lens mode-locking
A medium’s refractive index
depends on the intensity.
n(I) = n0 + n2I
If the pulse is more intense in
the center, it induces a lens.
Placing an aperture at the
focus favors a short pulse.
Losses are too high for a lowintensity cw mode to lase, but not
for high-intensity fs pulse.
Kerr-lensing is the mode-locking mechanism of the Ti:Sapphire laser.
Kerr-lensing is a type of saturable absorber.
If a pulse experiences additional focusing due to high intensity and
the nonlinear refractive index, and we align the laser for this extra
focusing, then a high-intensity beam will have better overlap with
the gain medium.
High-intensity pulse
Ti:Sapph
Low-intensity pulse
This is a type of saturable absorption.
Mirror
Additional focusing
optics can arrange
for perfect overlap of
the high-intensity
beam back in the
Ti:Sapphire crystal.
But not the lowintensity beam!
Modeling Kerr-lens mode-locking
Titanium Sapphire (Ti:Sapphire)
Ti:Sapphire is
currently the
workhorse laser
of the ultrafast
community,
emitting pulses as
short as a few fs
and average
power in excess
of a Watt.
Al2O3 lattice
oxygen
aluminum
Titanium Sapphire
Absorption and emission
spectra of Ti:Sapphire
It can be
pumped with a
(continuous)
Argon laser
(~450-515 nm)
or a doubledNd laser (~532
nm).
(nm)
Upper level
lifetime:
3.2 msec
Ti:Sapphire lases from
~700 nm to ~1000 nm.
Mechanisms that limit pulse shortening
The universe conspires to lengthen pulses.
Gain narrowing:
G(w) = exp(-aw2), then after N passes, the spectrum will narrow
by GN(w) = exp(-Naw2), which is narrower by N1/2
Group-velocity dispersion:
GVD spreads the pulse in time. And everything has GVD…
All fs lasers incorporate dispersion-compensating components.
We’ll spend several lectures discussing GVD!!
Etalon effects:
This yields multiple pulses, spreading the energy over time,
weakening the pulses.
The Ti:Sapphire laser including
dispersion compensation
Adding two prisms compensates for dispersion in the Ti:Sapphire
crystal and mirrors.
Ti:Sapphire
gain medium
cw pump beam
Slit for
tuning
Prism dispersion
compensator
This is currently the workhorse laser of the ultrafast optics community.
Commercial fs lasers
Ti:Sapphire
Coherent:
Mira (<35 fs pulse length, 1 W ave power),
Chameleon (Hands-free, ~100 fs pulse length),
Spectra-Physics:
Tsunami (<35 fs pulse length, 1 W ave power)
Mai Tai (Hands-free, ~100 fs pulse length)
Very-short-pulse commercial fs lasers
Ti:Sapphire
KM Labs
< 20 fs and < $20K
Femtolasers
As short as 8 fs!
Commercial fs lasers (cont’d)
Ytterbium Tungstate
(Yb:KGW)
Ytterbium doped laser
materials can be directly
diode-pumped, eliminating
the need for an intermediate
(green) pump laser used in
Ti:Sapphire lasers.
They also offer other
attractive properties, such as
a very high thermal efficiency
and high average power.
Amplitude Systemes
Model
t-Pulse 20
t-Pulse 100
t-Pulse 200
Pulse energy (nJ)
20
100
200
Average power (W)
1
1
2
Repetition rate (MHz)
50
10
10
Active mode-locking
Any amplitude modulator can preferentially induce losses for
times other than that of the intended pulse peak. This produces
short pulses.
It can be used to start a Ti:Sapphire laser mode-locking.
Gain switching
Modulating the gain rapidly is essentially the same as active
mode-locking.
This method is a common one for mode-locking semiconductor
lasers.
Synchronous pumping
Pumping the gain medium with a train of
already short pulses yields a train of
even shorter pulses.
Short pulses (ps)
Pump
beam
The laser roundtrip time must
precisely match
that of the train of
pump pulses!
Saturable
absorber
Gain medium
Trains of 60 ps pulses from a Nd:YAG laser can yield <1 ps
pulses from a sync-pumped dye laser.
Hybrid mode-locking
Hybrid mode-locking is any type of mode-locking incorporating
two or more techniques simultaneously.
Sync-pumping and passive mode-locking
Active and passive mode-locking
However, using two lousy methods together doesn’t really work
all that much better than one good method.
Diode lasers use hybrid mode-locking
Autocorrelation
Autocorrelation
Haneda, et al, UP 2004
Spectrum
Spectrum
Additive-pulse mode-locking
Nonlinear effects in an external cavity can yield a phasedistorted pulse, which can be combined in phase with the
pulse in the main cavity, yielding cancellation in the wings,
and hence pulse-shortening.
Early fiber lasers used this mechanism.
The soliton laser
Nonlinear-optical effects can compensate for dispersion, yielding a
soliton, which can be very short and remain very short, despite
dispersion and nonlinear-optical effects.
Commercial fs fiber lasers
Erbium
Menlo Systems
150 fs; 150 mW
IMRA America
Frequency-doubled
Ultrafast Q-switching using distributed
feedback
When two beams cross at an angle, their intensity is sinusoidal.
Intensity fringes
When energy is deposited sinusoidally in space, the actual gain (g) goes
quadratically with the energy deposited, yielding a type of very fast Qswitching. Using several stages, fs pulses have been created this way.
Traveling-wave excitation
Pump lasers for ultrafast lasers
Previously, only the Argon Ion laser was available, but much more stable
intracavity-frequency-doubled solid-state lasers are now available.