Download Dispersion and Ultrashort Pulses

Dispersion and Ultrashort Pulses
Angular dispersion and group-velocity dispersion
Phase and group velocities
Group-delay dispersion
Negative groupdelay dispersion
Pulse compression
Chirped mirrors
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/d
out (blue) > out (red )
Dispersion also disperses a pulse in time:
vgr(blue) < vgr(red)
Both of these effects play major roles in ultrafast optics.
“Chirp”
d2n/d2
Calculating the group velocity
vg d /dk
Now, is the same in or out of the medium, but k = k0 n, where k0 is
the k-vector in vacuum, and n is what depends on the medium.
So it's easier to think of as the independent variable:
v g [dk / d ]
1
Using k = n() / c0, calculate: dk /d = ( n + dn/d ) / c0
vg = c0 / ( n + dn/d) = (c0 /n) / (1 + /n dn/d )
Finally:
dn v g = v phase / 1 +
n
d
So the group velocity equals the phase velocity when dn/d = 0,
such as in vacuum. Otherwise, since n usually increases with , dn/d
> 0, and:
vg < vphase.
Calculating 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 0.
Use the chain rule :
dn dn d 0
=
d d 0 d
Now, 0 = 2 c0 / , so :
Recalling that :
we have :
or :
d 0 2 c0
2 c0
02
=
=
=
2
2
d
(2 c0 / 0 )
2 c0
c dn v g = 0 / 1 +
n n d 2
c0 2 c0 dn 0 vg = / 1+
n0 d 0 2 c0 n c0 0 dn v g = / 1 n
n
d
0 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 positive, that is, near a resonance. So vg < vphase for these frequencies!
Spectral Phase and Optical Devices
Recall that the effect of a linear passive
optical device (i.e., lenses, prisms, etc.) on
a pulse is to multiply the frequency-domain
field by a transfer function:
E˜ in( )
E˜ out( ) = H ( ) E˜ in ( )
where H() is the transfer function
of the device/medium:
H()
E˜ out( )
Optical device
or medium
exp[ ( ) L / 2] for a medium
H ( ) = BH ( ) exp[i H ( )]
Since we also write E() = S() exp[-i()], the spectral phase of the
output light will be:
out ( ) = H ( ) + in ( )
Note that we CANNOT add the temporal phases!
out (t) H (t) + in (t)
We simply add
spectral phases.
The Group-Velocity Dispersion (GVD)
The phase due to a medium is:
() = n() k L = k() L
To account for dispersion, expand the phase (k-vector) in a Taylor series:
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.”
The effect of group velocity dispersion
GVD means that the group velocity will be different for different
wavelengths in the pulse.
vgr(blue) < vgr(red)
Because ultrashort pulses have such large bandwidths, GVD is a
bigger issue than for cw light.
Calculation of the GVD (in terms of wavelength)
d 0 02
=
d 2 c0
Recall that:
d 0 d
02 d
d
=
=
d d d 0 2 c0 d 0
dn v g = c / n 0
d
0 and
Okay, the GVD is:
d
d
1
02 d 1 dn 02 d dn n
=
n
=
0
0
2
c
d
c
d
v
2
c
d
d
2
0
0 0 0
0 0
g 02
=
2 c02
Simplifying:
dn
d 2 n dn 0
2
d
d
d
0
0
0
GVD k ( 0 ) =
d n
2 c02 d 02
3
0
2
Units:
ps2/km or
(s/m)/Hz or
s/Hz/m
Dispersion parameters for various materials
GVD in optical fibers
Note that
fiber folks
define GVD
as the
negative of
ours.
Sophisticated cladding structures, i.e., index profiles have been
designed and optimized to produce a waveguide dispersion that
modifies the bulk material dispersion
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:
t =
k ( 0 ) L
L
=
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 + H 1 [ 0 ] + H 2 [ 0 ]2 / 2! + ...
phase
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 + H2 = 0
i.e.,
d 2
d 2
+
0
d 2 H
d 2
= 0
0
Propagation of the pulse manipulates it.
Dispersive pulse
broadening
is unavoidable.
If 2 is the pulse 2nd-order spectral phase on entering a medium, and
k”L is the 2nd-order spectral phase of the medium, then the resulting
pulse 2nd-order phase will be the sum: 2 + k”L.
A linearly chirped input pulse has 2nd-order phase: 2,in
Emerging from a medium, its 2nd-order phase will be:
2,out
03 d 2 n
/2
/2
L
= 2
+ GDD = 2
+
2
2
2
2
+
+
2 c0 d 0
/2
= 2
+2
(This result
pulls out the
in the
Taylor
Series.)
This result, with
the spectrum,
can be inverse
Fouriertransformed to
yield the pulse.
A positively chirped pulse will broaden further;
a negatively chirped pulse will shorten.
Too bad material GDD is always positive in the visible and near-IR…
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 an optical element introduces angular dispersion.
Optical
element
Input
beam
Optic
axis
Here, there is negative
GDD because the blue
precedes the red.
Well need to compute the projection onto the optic axis (the
propagation direction of the center frequency of the pulse).
Negative GDD
Taking the projection of k ( )
()
() << 1
onto the optic axis, a given
frequency sees a phase
delay of ():
z
( ) = k ( ) roptic axis
= k ( ) z cos[ ( )]
= ( / c) z cos[ ( )]
Optic
axis
Were considering only the GDD due to
dispersion and not that of the prism itself.
So n = 1 (that of the air after the prism).
d / d = ( z / c) cos( ) ( / c) z sin( ) d / d 2
d
z
d z
d
z
z
d 2
d = sin( )
sin( )
cos( ) sin( ) 2
2
d
c
d c
d
c
c
d
d 2
But << 1, so the sine terms can be neglected, and cos() ~ 1.
Angular dispersion yields negative GDD.
d
d 2
2
0
z
0
c
d d 0 2
The GDD due to angular dispersion is always negative!
Also, note that it doesnt matter where the angular dispersion
came from (whether a prism or a grating).
And the negative GDD due to prism dispersion is usually much
greater than that from the material of the prism.
A prism pair has negative GDD.
How can we use dispersion to introduce negative chirp conveniently?
Lsep
d 2
d 0
2
Always
negative!
dn
4Lsep
2 2c d
3
0
2
Assume Brewster
angle incidence
and exit angles.
30 d 2 n
+ Lprism
2
2
2
c
d
0
0
This term assumes
that the beam grazes
the tip of each prism
Vary Lsep or Lprism to tune the GDD!
Always
positive (in
visible and
near-IR)
This term allows the beam
to pass through an additional
length, Lprism, of prism material.
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
traverse more glass.
It’s routine to stretch and then compress ultrashort pulses by factors
of >1000.
What does the pulse look like inside a
pulse compressor?
If we send an unchirped pulse
into a pulse compressor, it
emerges with negative chirp.
Note the unintuitive color variation of the pulse after the first prism.
To see the effect on a positively chirped pulse, read right to left!
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.
Input beam
The output path is
independent of prism
position.
Output beam
The required separation between prisms
in a pulse compressor can be large.
The GDD
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)
Its best to use highly dispersive glass, like SF10, or gratings.
But compressors can still be > 1 m long.
Four-prism pulse compressor
Also, alignment is critical, and many knobs must be tuned.
Wavelength
tuning
Wavelength
tuning
Prism
Prism
Prism
Wavelength
tuning
Prism
Fine GDD
tuning
Wavelength
tuning
Coarse GDD tuning
(change distance between prisms)
All prisms and their incidence angles must be identical.
! $
"# # % ! "!! " #! " " "% " ! " #! ! "
#! #! ! #! ! "!$ ! !! ! " $ !
!! ! ! Two-prism pulse
compressor
Coarse GDD tuning
Wavelength
tuning
Roof
mirror
Periscope
Prism
Wavelength tuning
Prism
Fine GDD
tuning
This design cuts the size and alignment issues in half.
Single-prism pulse compressor
Corner cube
Periscope
Prism
GDD tuning
Roof
mirror
Wavelength
tuning
Beam magnification is always one in
a single-prism pulse compressor!
M1
din
dout
M2
M d out / din
1
1
M1 =
= M3 =
M2
M4
M tot = M 1 M 2 M 3 M 4 = 1
The total dispersion in a single-prism
pulse compressor is always zero!
The dispersion
depends on the
direction through
the prism.
D1
D2
D1 = M 2 D2 = D3 = M 4 D4
Dtot
D3
D1
D2
=
+
+
+ D4 = 0
M 2 M 3M 4
M 3M 4
M4
So the spatial chirp and pulse-front tilt are also identically zero!
Diffraction-grating pulse compressor
The grating pulse compressor also has negative GDD.
Grating #2
d 2
d 2
0
Lsep
03
2 c 2 d 2 cos 2 ( )
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
2nd- and 3rd-order phase terms for prism
and grating pulse compressors
Grating compressors offer more compression than prism compressors.
''
'''
Piece of glass
Note that the relative signs of the 2nd and 3rd-order terms are opposite
for prism compressors and grating compressors.
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:
input2 + prism2 + grating2 = 0
input3 + prism3 + grating3 = 0
This design was used by Fork and Shank at Bell Labs in the mid 1980s
to achieve a 6-fs pulse, a record that stood for over a decade.
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).
The (transmission)
grism equation is:
a [sin(m) – n sin(i)] = m
n Chirped mirror coatings also yield
dispersion compensation.
Longest wavelengths
penetrate furthest.
Such mirrors
avoid spatiotemporal
effects, but
they have
limited GDD.
Chirped mirror coatings
Longest
wavelengths
penetrate
furthest.
Doesnt work
for < 600 nm