Transcript Hyperbolic Secant Squared Pulse Shape

Pulsed Lasers
A pulse forming mechanism is needed otherwise
lasers run “continuous wave” (CW)
Three types of pulsed operation
1. Gain switched (micro or millisecond pulses
typically) turn gain on and off (flash lamps,
modulate pump)
2. Q-switched (nanosecond pulses) modulate cavity
loss on times scales > round trip time
3. Modelocked (picosecond to femtosecond pulses)
modulate cavity loss periodically at roundtrip time
ADVANCES IN SHORT PULSE GENERATION
The “Mode” in “Mode-locking” is longitude
Models
Log(Iout/Iin)
Gain
Gain region
No absorption
Below energy
gap
Loss region
0
Absorption
Many longitudal modes can co-exists in the same cavity
Mode-Lock
The relative phases of the simultaneously lasing cavity
modes must be “locked” at proper values.
Roughly:
How to achieve Mode-lock
Laser with many modes but without mode-locking
• Relative phases of the modes are random
• Output looks completely random but repeats itself
in time equal to 2L/c
• Slow photodetectors see only the average power
Q: which of the above are still true under mode-lock condition?
How to achieve Mode-lock
Design a cavity so that the round-trip loss of the
light is BIG, unless the modes are “lock” to
generate short pulses.
Two Types of techniques:
1. Saturated absorbers due to limited number
of absorbing particles (e.g., dye molecules)
2. Nonlinear effect: Kerr lens
Optical Kerr Effect
A change in the refractive index of a material in response to an
electric field.
The change is proportional to the square of the electric field
All materials show Kerr effect
Optical Kerr Effect is the special case where the electric field is
from the light itself
Kerr Lens Mode-locking
• Kerr Lens & Aperture gives
increased transmission at
high Intensity
• Increased transmission at
high intensity
• Short, intense pulse
preferred in laser
• Kerr effect instantaneous
Dispersion in Optics
The dependence of the refractive index on wavelength has two
effects on a pulse, one in space and the other in time.
Dispersion disperses a pulse in space (angle):
“Angular dispersion”
dn/dl
Dispersion also disperses a pulse in time:
“Chirp”
d2n/dl2
Both of these effects play major roles in ultrafast optics.
In most of the region, n increases as frequency of the light increase
angular frequency.
Group Velocity:
The velocity with which the envelope of the wave propagate
through space.
v g   dk / d  
1
: angular frequency.
k: angular wave number
k = 2 / l
 is the same in or out of the medium;
but k = k0n, where k0 is k in vacuum, and n is
what depends on the medium.
A Simple Derivation of Group Velocity:
v =  / k
Group Velocity in terms of Optical dispersion
  dn 
v g  v phase / 1 

 n d 
Vphase = c0 / n
Conclusion: group velocity has nothing to
do with the true velocity of the light
The group velocity is less than the phase
velocity in non-absorbing regions.
vg = c0 / (n +  dn/d)
Except in regions of anomalous dispersion (which are absorbing), dn/d is
negative, that is, near a resonance. So vg > vphase for these frequencies!
Group Velocity vs. Wavelength
We more often think of the refractive index in terms of wavelength,
so let's write the group velocity in terms of the vacuum wavelength l0.
 c0   l0 dn 
v g    / 1 

n
n
d
l
  
0 
Consider the simplest case: dn/dl0 =  = constant
d (1 / vg )
dl0
dn / dl0  

0
c0
This means vg is independent of wavelength
The effect of group velocity dispersion
GVD means that the group velocity will be different for different
wavelengths in the pulse.
early
times
late
times
vgr(yellow) < vgr(red)
d
k ( ) 
d
1 
 
 v g 
is the “group velocity dispersion.”
Calculation of the GVD (in terms of wavelength)
d l0 l02

d 2 c0
Recall that:
d l0 d
l02 d
d


d d d l0 2 c0 d l0

dn 
v g  c /  n  l0

d
l
0 

and
Okay, the GVD is:
d
d
1
l02 d 
dn 
l02 d  1 
dn  

n

l

n

l
 
 

0
0


2
2

c
d
l
d
l
v
2

c
d
l
c
d
l
0
0 
0 
0
0  
0 
 g 
l02

2 c02
Simplifying:
 dn
d 2 n dn 
 l0



2
d
l
d
l
d
l
0
0
 0
l d n
GVD  k (0 ) 
2 c02 d l02
3
0
2
Units:
s2/m or
(s/m)/Hz or
s/Hz/m
GVD yields group delay dispersion (GDD).
We can define delays in terms of the velocities and the medium length L.
The phase delay:
0
k (0 ) 
v (0 )
so:
k (0 ) L
L
t 

v (0 )
0
The group delay:
1
k (0 ) 
v g (0 )
so:
L
t g (0 ) 
 k (0 ) L
v g (0 )
The group delay dispersion (GDD):
d
k ( ) 
d
1 
 
 v g 
so:
d
GDD 
d
GDD = GVD L
1
  L  k ( ) L
 v g 
Units: fs2 or fs/Hz
Manipulating the phase of light
Recall that we expand the spectral phase of the pulse in a Taylor Series:
 ( )  0  1 [  0 ]  2 [  0 ]2 / 2!  ...
and we do the same for the spectral phase of the optical medium, H:
H ( )  H 0  H1 [  0 ]  H 2 [  0 ]2 / 2!  ...
phase delay
group delay
group delay dispersion (GDD)
So, to manipulate light, we must add or subtract spectral-phase terms.
For example, to eliminate the linear chirp (second-order spectral phase),
we must design an optical device whose second-order spectral phase
cancels that of the pulse:
2  H 2  0
i.e.,
d 2
d 2

0
d 2 H
d 2
 0
0
So how can we generate negative GDD?
This is a big issue because pulses spread
further and further as they propagate through
materials.
We need a way of generating negative GDD to
compensate.
Angular dispersion yields negative GDD.
Suppose that some optical element introduces angular dispersion.
If frequency 0 propagates
a distance L to plane S’, 
then frequency  sees
a phase delay of ()
Optical
element

Input
beam
L
P0

d
0 L


d 2 
c
2
0
2
d 
d 
  0 
The GDD due to angular dispersion is always negative!
A prism pair has negative GDD
How can we use dispersion to introduce negative chirp conveniently?


Lsep
d 
l dn
  4Lse p
2
2 
d  0
2 c dl
2
Always
negative!
3
0
2

l 30 d 2 n
  Lprism
2
2

2

c
d
l
l0
l0
This term assumes
that the beam grazes
the tip of each prism
Vary the second term to tune the GDD!
Assume Brewster
angle incidence
and exit angles.
Always
positive (in
visible and
near-IR)
This term allows the beam
to pass through an additional
length, Lpriam, of prism material.
Adjusting the GDD maintains alignment.
Any prism in the compressor can be translated perpendicular to the
beam path to add glass and reduce the magnitude of negative GDD.
Remarkably, this does
not misalign the beam.
New beam path
through prism
New prism position
Original path
through prism
Original and new
path out of the prism
Original prism position
Pulse Compressor
This device has negative group-delay dispersion and hence can
compensate for propagation through materials (i.e., for positive chirp).
The longer wavelengths
have a longer path.
It’s routine to stretch and then compress ultrashort pulses by factors
of >1000
Ti:Sapphire laser
• Optically active atoms TiO3 <1% by Weight
• Host solid is sapphire (Al2O3)
• Ti interacts with solid so the E2, E1 broadened
significantly (inhomogeneous broadening).
The atoms in the solid vibrate and interact
with the Ti atoms.
• Gain bandwidth huge (100 THz)
• Can use as tunable laser
• Makes for ultra short pulses
Titanium Sapphire Ti3+: Al2O3
E 3/2 excited state
2T
2 ground
state
Spin forbidden: long emission lifetime 300 ms
Reasonable extinction coefficient e14000
Huge Stokes shift, Broad emission spectrum compared to dyes
Revolutionized laser industry in ability to make tunable short pulses
Tuning Range and Power of Ti:Sapphire
Longer wavelengths
Less damaging
900 is often good
Compromise between
Power and viability
100 femtosecond pulses
10 nm FWHM bandwidth
Excite essentially every dye
With this wavelength range
end
Output coupler
1. Pumped by Argon ion or doubled Nd laser
2. Birefringence filter used for color tuning
END
Introduction to Non-linear Optical effects
•NLO effects have been observed since 19thC
•Pockels effect
•Kerr effect
•High fields associated with laser became available in the 1960s
and gave rise to many new NLO effects
•second harmonic generation (SHG)
•third harmonic generation (THG)
•stimulated Raman scattering
•self-focussing
•In NLO we are concerned with the effects that the light itself
induces as it propagates through the medium
•In linear optics the light is deflected or delayed but its
frequency (wavelength) is unchanged
Pulse Compression Simulation
Using prism and grating pulse compressors vs. only a grating compressor
Resulting intensity vs. time
with only a grating compressor:
Note the cubic
spectral phase!
Resulting intensity vs. time
with a grating compressor
and a prism compressor:
Brito Cruz, et al., Opt. Lett., 13, 123 (1988).
Pulse compressors achieve amazing results, but,
if not aligned well, they can introduce spatiotemporal distortions, such as “spatial chirp.”
Propagation through a prism pair produces a beam with no angular
dispersion, but the color varies spatially across the beam.
Care must be taken to cancel out this effect with the 3rd and 4th prisms.
E(t )  exp[i(0   x)t ]
Prism pairs are inside nearly every
ultrafast laser, so we’re just asking
for spatial chirp.
Color
varies
across
beam
Spatial chirp is difficult to avoid.
Simply propagating through a tilted window causes spatial chirp!
Different colors have different
refractive indices and so have
different refraction angles.
n()
Color
varies
across
beam
Because ultrashort pulses are so broadband, this distortion is
very noticeable—and problematic!
Angular dispersion also causes pulse
fronts to tilt.
Phase fronts are perpendicular to the direction of propagation.
Because group velocity is usually less than phase velocity,
pulse fronts tilt when light traverses a prism.
E(t )  I (t   x)
Pulse-front tilt and angular dispersion are manifestations of the same
effect and their magnitudes are directly proportional to each other.
Angular dispersion causes pulse-front tilt
even when group velocity is not involved.
Diffraction gratings also yield a pulse-front tilt.
The path is simply
shorter for rays that
impinge on the near
side of the grating.
Of course, there’s
angular dispersion,
too.
Since gratings have about ten times the dispersion of prisms, they
yield about ten times the tilt.
Chirped mirror coatings offer an alternative to
prisms and gratings for dispersion compensation.
Longest wavelengths
penetrate furthest.
Doesn’t work
for < 600 nm
Such mirrors
avoid spatiotemporal
effects, but
they have
limited GDD.
The required separation between prisms
in a pulse compressor can be large.
The resulting negative GDD is proportional to the prism separation
and the square of the dispersion.
Different prism
materials
Compression of a 1-ps,
600-nm pulse with 10 nm
of bandwidth (to about 50
fs).
Kafka and Baer,
Opt. Lett., 12,
401 (1987)
It’s best to use highly dispersive glass, like SF10, or gratings.
Diffraction-grating pulse compressor
The grating pulse compressor also has negative second-order phase.
Grating #2

Lse p
d 2
l30
 
2
d  0
2c 2 d 2 cos2 (' )
where d = grating spacing
(same for both gratings)
Note that, as in the prism
pulse compressor, the
larger Lsep, the larger
the negative GDD.

Lsep
Grating #1
Compensating 2nd and 3rd-order spectral phase
Use both a prism and a grating compressor. Since they have 3rd-order
terms with opposite signs, they can be used to achieve almost arbitrary
amounts of both second- and third-order phase.
Prism compressor
Grating compressor
Given the 2nd- and 3rd-order phases of the input pulse, input2 and input3,
solve simultaneous equations:
input 2   prism 2  grati ng 2  0
input 3  prism 3  grati ng 3  0
This design was used by Fork and Shank at Bell Labs in the mid 1980’s
to achieve a 6-fs pulse, a record that stood for over a decade.
Spectral Phase and Optical Devices
We simply add spectral phases.
 out ( )   H ( )   in ( )
E˜ in( )
H()
Optical device
or medium
H() = 2 L / l = k() L
The phase due to a medium is:
E˜ out ( )
k() L = k0 n() L
To account for dispersion, expand
k() = k(0) + 1[01/22 [02
The Group-Velocity Dispersion (GVD)
k ( ) L  k ( 0 ) L  k ( 0 )    0  L 
0
k (0 ) 
v (0 )
1
k (0 ) 
v g (0 )
1
2
k ( 0 )    0  L  ...
2
d

k ( ) 
d
1
 
 v g 
The first few terms are all related to important quantities.
The third one is new: the variation in group velocity with frequency:
d
k ( ) 
d
1
 
 v g 
is the “group velocity dispersion.”