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