Angular dispersion and group-velocity dispersion
Phase and group velocities
Group-delay dispersion
Negative group-
delay dispersion
Pulse compression
Chirped mirrors
Dispersion and Ultrashort Pulses
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.
Group delay
dispersion or
Chirp
d2n/dλ2
Angular dispersion
dn/dλ
Both of these effects play major roles in ultrafast optics.
Dispersion also disperses a pulse in time:
Dispersion disperses a pulse in space (angle):
vgr(blue) < vgr(red)
θout(blue) > θout(red)
When two functions of different
frequency interfere, the result is beats.
0 0 21exp( ) (Re{ exp( }) )tot E i Et t i tωω= +X taking E0 to be real.
2
1 ave
aveω
ωω ω
ωω
= + ∆
= − ∆
Adding waves of two different frequencies yields the product of a
rapidly varying cosine (ωave) and a slowly varying cosine (∆ω).
0 0
0
0
( ) Re{ exp ( ) exp ( )}
Re{ exp( )[exp( ) exp( )]}
Re{2 exp( )cos( )}
ave ave
ave
a
t
v
ot
e
t E i t t E i t t
E i t i t i t
E i t t
ω ω
ω ω
ω ω
ω
ω ω
⇒ = + ∆ + − ∆
= ∆ + − ∆
= ∆
X
0( ) 2 cos( )cos( )avetot t E t tω ω⇒ = ∆X
1 12 2 2 2
ave
ωω
ωωωω
+ −= ∆ =andLet:
When two functions of different
frequency interfere, the result is beats.
Individual
waves
Sum
Envelope
Intensity
time
2π/∆ω
In phase Out of phaseIn phaseOut of phase In phase
When two waves of different frequency
interfere, they also produce beats.
0 2 2
2
0 1 1
1 2
2 21 1
0
1
0
( , ) Re{ }
2 2
2 2
( , ) Re{ exp ( ) ex
exp ( )
p
ex ( )
(
ptot
t
ave
ave
avot e ave ave
E i k z t
k k
z t
k
z t
E
k
k
i k z
E i z kz t t E i z
t
kz
k k
k
ω
ω
ω
ωω
ω
ωω
ω
ω
= +
+ −= ∆ =
+ −= ∆ =
= + ∆ − −∆ + −∆
−−X
X
Let and
Similiarly, and
So:
[ ]0
0
0
)}
Re{ exp ( ) exp[ ( )] }
Re{2 cos( )}
exp ( )
exp ( )
cos( ) 2 cos( )
ave
ave ave
ave ave
ave ave
i k z t
i k z
t t
E i kz t i kz t
E kz t
E kz t
t
k z t
ω ω
ωω
ω
ω
ω
ω
ω
− + ∆
= ∆ −∆ + − ∆ −∆
= ∆ −∆
= − ∆ −∆
−
−
taking E0 to be real.
Traveling-Wave Beats
Indiv-
idual
waves
Sum
Envel-
ope
Inten-sity
In phase Out of phaseIn phaseOut of phase In phase
z
time
In phase Out of phase
Seeing Beats It’s usually very difficult to see optical
beats because they occur on a time
scale that’s too fast to detect. This is
why we say that beams of different
colors don’t interfere, and we only
see the average intensity.
However, a sum of many
frequencies will yield a train
of well-separated pulses:
Indiv-
idual
waves
Sum
Intensity
2π/∆ωmax Pulse separation: 2π/∆ωmin
0 cos( , ) 2 Puls () e )(tot ave avek zz t E z tt k ωω= ∆ −∆−X
Group Velocity
vg ≡≡≡≡ dωωωω /dk
Light-wave beats (continued):
X tot(z,t) = 2E0 cos(kavez–ωavet) Pulse(∆kz–∆ωt)
This is a rapidly oscillating wave: [cos(kavez–ωavet)]
with a slowly varying amplitude: [2E0Pulse(∆kz–∆ωt)]
The phase velocity comes from the rapidly varying part: v = ωave / kave
What about the other velocity—the velocity of the pulse amplitude?
Define the group velocity: vg ≡ ∆ω /∆k
Taking the continuous limit,
we define the group velocity as:
carrier wave
amplitude
Group velocity is not equal to phase velocity
if the medium is dispersive (i.e., n varies).
0 1 0 2
1 1 2 2
vg
c k c k
k n k n k
ω −∆≡ =
∆ −
Evaluate the group velocity for the case of just two frequencies:
0 01 21 2
1 2
1 2 0
, v v
, v v /
g
g
c ck kn n n
n k k n
n n c n
φ
φ
−= = = = = =
−
≠ ≠ =
If phase velocity
If
where k1 and k2 are the k-vector magnitudes in vacuum.
Phase and Group Velocities
UnrealisticUnrealistic
In vacuum
Common
case for
most
materials
Unrealistic
Possible
vg ≡ dω /dk
Now, ω 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 themedium. So it's easier to think of ω as the independent variable:
Using k = ω n(ω) / c0, calculate: dk /dω = (n + ω dn/dω) / c0
vg = c0 / (n + ω dn/dω) = (c0 /n) / (1 + ω /n dn/dω )
Finally:
So the group velocity equals the phase velocity when dn/dω = 0, such
as in vacuum. But n usually increases with ω, so dn/dω > 0, and:
vg < vφ
Calculating the group velocity
[ ] 1v /g dk dω
−≡
v v / 1g
dn
n dφ
ωω
= +
The group velocity is less than the phase
velocity in non-absorbing regions.
vg = (c0 /n) / (1+ ω dn/dω) = vφ / (1+ ω dn/dω)
Except in regions of anomalous dispersion (near a resonance and which
are absorbing), dn/dω is positive. So vg < vφ for most frequencies!
0
0
2
0 0 0 00 0 2 2
0 0 0
0
00
2 22 /
(2 / ) 2
v / 1
2v / 1
g
g
ddn dn
d d d
d c cc
d c c
c dn
n n d
cc
n
λω λ ω
λ π π λλ π ω
ω ω π λ π
ωω
π
=
− − −= = = =
= +
= +
Use the chain rule :
Now, , so :
Recalling that :
we have: 2
0
0 0 02
dn
n d c
λλ λ π
− or :
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 λ0.
0 0
0
v / 1g
c dn
n n d
λλ
= −
Recall that the effect of a linear passive
optical device (i.e., windows, filters, etc.) on
a pulse is to multiply the frequency-domain
field by a transfer function:
˜ E out(ω) = H (ω) ˜ E in (ω)
where H(ω) is the transfer function of the device/medium:
( ) ( ) exp[ ( )]H HH B iω ω ϕ ω= −
Since we also write E(ω) = √S(ω) exp[-iϕ(ω)], the spectral phase of the output light will be:
ϕout (ω ) = ϕH (ω) +ϕ in (ω ) We simply add
spectral phases.
Spectral Phase and Optical Devices
Note that we CANNOT add the temporal phases!
φout (t) ≠ φH (t) + φin (t)
H(ω)˜ E in(ω) ˜ E out(ω)
Optical device
or medium
exp[ ( ) / 2]Lα ω−for a material with
absorption
coefficient α(ω)
~
The Group-Velocity Dispersion (GVD)
The phase due to a medium is: ϕΗ(ω) = n(ω) k0 L = k(ω) L
To account for dispersion, expand the phase (k-vector) in a Taylor series:
[ ] [ ]210 0 0 0 02
( ) ( ) ( ) ( ) ...k L k L k L k Lω ω ω ω ω ω ω ω′ ′′= + − + − +
is the group velocity dispersion.
00
0
( )v ( )
kφ
ωω
ω= 0
0
1( )
v ( )g
k ωω
′ =
1( )
vg
dk
dω
ω
′′ =
1( )
vg
dk
dω
ω
′′ =
The first few terms are all related to important quantities.
The third one in particular: the variation in group velocity with frequency
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)
Recall that:
2
0 0
02
d
d c
λ λω π
−=
2
0 0
0 0 02
dd d d
d d d c d
λ λω ω λ π λ
−= =
0 0
0
v /g
dnc n
dλ
λ
= −
2
00
0 0 0 0
1 1
v 2g
d d dnn
d c d c d
λλ
ω π λ λ
−= −
2
002
0 0 02
d dnn
c d d
λλ
π λ λ −
= −
2 2
002 2
0 0 0 02
dn d n dn
c d d d
λλ
π λ λ λ −
= − −
and
3 2
00 2 2
0 0
( )2
d nGVD k
c d
λω
π λ′′≡ =
Okay, the GVD is:
Simplifying:Units:
ps2/km or
(s/m)/Hz or
s/Hz/m
GVD in optical fibers
Sophisticated cladding structures, i.e., index profiles, have been
designed and optimized to produce a waveguide dispersion that
modifies the bulk material dispersion
Note that
fiber folks
define GVD
as
proportional
to the
negative of
the definition
we’ve been
using.
psec2 / cm
psec / km - nm
"Dispersion
parameter"
λ=
gV
L
d
d
L
1D
"kc22λ
π−=
GVD yields group delay dispersion (GDD).
The delay is just the medium length L divided by the velocity.
The phase delay:
00
0
( )v ( )
kφ
ωω
ω=
0
0
1( )
v ( )g
k ωω
′ =
1( )
vg
dk
dω
ω
′′ =
The group delay:
The group delay dispersion (GDD):
so:0
0
0 0
( )( )
v ( )
k LLtφ
φ
ωω
ω ω= =
0 0
0
( ) ( )v ( )
gg
Lt k Lω ω
ω′= =
1( )
vg
dGDD L k L
dω
ω
′′= =
so:
so:
Units: fs2 or fs/Hz
GDD = GVD L
Dispersion parameters for various materials
Manipulating the phase of light
Recall that we expand the spectral phase of the pulse in a Taylor Series:
2
0 1 0 2 0( ) [ ] [ ] / 2! ...ϕ ω ϕ ϕ ω ω ϕ ω ω= + − + − +
So, to manipulate light, we must add or subtract spectral-phase terms.
2
0 1 0 2 0( ) [ ] [ ] / 2! ...H H H Hϕ ω ϕ ϕ ω ω ϕ ω ω= + − + − +
and we do the same for the spectral phase of the optical medium, H:
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:
ϕ2 + ϕH2 = 0d2ϕdω 2
ω 0
+d2ϕ H
dω 2
ω 0
= 0i.e.,
group delay group delay dispersion (GDD)phase
Propagation of the pulse manipulates it.
Dispersive pulse
broadening
is unavoidable.
If ϕ2 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: ϕ2 + k”L.
A linearly chirped input pulse has 2nd-order phase:
Emerging from a medium, its 2nd-order phase will be:
3 2
02, 2 2 2 2 2 2
0 0
/ 2 / 2
2out
d nGDD L
c d
λβ βϕ
α β α β π λ= + = +
+ +
2, 2 2
/ 2in
βϕ
α β=
+
(This result
pulls out the
½ in the
Taylor
Series.)
Since GDD is always positive (for transparent materials in
the visible and near-IR), a positively chirped pulse will
broaden further; a negatively chirped pulse will shorten.
This result, with
the spectrum,
can be inverse
Fourier-
transformed to
yield the pulse.
Posing the problem
We have a short pulse
(initially, with zero chirp).
α(ω), n(ω)
It traverses
through a block
of something
transparent (with
known n,α).
What does it
look like when
it emerges?
?
To analyze:
start with Ein(t) (known)
convert to Ein(ω)propagate forward
in the frequency
domain: Eout(ω)
back to time domain: Eout(t)
(ignore absorption)
This is still a quadratic in ω
( )( )2 2
0
40, 0
Gt
E z E eω ω
ω−
−= = ⋅Spectral amplitude of the form:
( ) ( )zikz eEzE ω−= ⋅=ω 0,
After propagation:
( ) ( )2 2
0
0 exp4
GtE ik zω ω
ω −
= ⋅ − −
Taylor expansion of k(ω) at ω = ω0:
( ) ( ) ( ) ( )2 2
20
0 0 0 0
"( , ) exp '
4 2
Gt ik zE z E ik z ik z
ω ωω ω ω ω ω ω
−= ⋅ − − − ⋅ − − ⋅ −
Time-domain field at z is found via inverse Fourier transform:
( ) ( )∫∞
∞−
ω ωωπ
= dezEztE ti,2
1,
Propagation of Gaussian pulses
generally positive
( )( )
pp
0'
1
ω
ω=ω
=ωdk
d
kVg
( )( )
p
ppϕ ω
ω=ω
kV
where: ( ) 2"2+= ziktzt
GG
( )( )( ) ( )
( )
2
0
0 0 0
/ V1( , ) exp / V
g
GG
t zE t z E i t z
t zt zφ
ωω ω
π
− = ⋅ − −
pulse width increases with propagation
phase velocity
group velocity -
speed of pulse envelope
Equivalent relations: ( )( )
( )( )
ωωω
ωω
ωϕ
ddn
n
c
n
cg
+== V V
Group velocity dispersion
Propagation of Gaussian pulses
( )( )
2
2
1 ′′ = =
g
d k dk
d d Vω
ω ω ω
pulse width increases with z chirp parameter β
Gaussian part of the exponential:
( )( )2222
2 "2 1
1exp
GG t
kzi
zt=ξ
ξ−
ξ+
τ−
t - z/Vg
SF6 glass
n = 1.78
n' = 2×10−5 psec
n" = 8×10−9 psec2z = 0 µm
∆ω∆τ = 0.441
z = 100 µm
∆ω∆τ = 0.474
z = 200 µm
∆ω∆τ = 0.561
z = 300 µm
∆ω∆τ = 0.682
z = 400 µm
∆ω∆τ = 0.821
z = 500 µm
∆ω∆τ = 0.972
Group velocity dispersion (GVD)
2
"2
Gt
k=ξ units of (length)-1 pulse width doubles after
propagation through a
length √3/ξ
• group velocity dispersion k" distorts pulses
• typical materials have k" > 0, which induces an up chirp
• shorter pulses distort much more readily (larger bandwidth)
z = 0 z = 100 µm
Group velocity dispersion (GVD)
Is it reasonable to neglect absorption?
Dispersion vs. absorption
Fractional change in pulse energy:
( )1
(0)rel
U zU z
Uα∆ = − ≈ ⋅
Fractional change in pulse duration:
( )2
2
"Re21
)0(
)(
⋅≈−=∆
G
relt
kzt
ztt
Example:
Typical fiber optics have: α ~ 0.11 / km ( = 1 dB/km)
and: k" ~ 21 psec2 / km at λ = 1 mm
Thus, in 10 m of fiber, a 100 fsec pulse experiences:
- 0.2% absorption loss
- pulse width broadening by a factor of ~900!
In the non-resonant regime: Dispersion is Everything
If we include absorption, then k has an imaginary part: Im(k) = α/2
0.5 0.7 0.9 1.1
1.75
1.76
1.77
n(λ)
-1e-4
-5e-5
0
dn/dλµm
−1
1e-7
3e-7
5e-7
0.5 0.7 0.9 1.1
d2n/dλ2
µm
−2
sapphire, L ~ 1 cm
At λ = 800 nm:chirp parameter α = 2ξL
= 3.2×10-7
(per round trip)
It is small, but not zero!
Dispersion in a laser cavity
0.5 0.6 0.7 0.8 0.9 1 1.1
1e-7
2e-7
3e-7
λ (µm)
cm−1
2
"2
Gt
k=ξ
Material: Al2O3
Pulse width: tp = 100 fsec
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.
Negative GDD
Device