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Dispersion and how to control it

Group velocity versus phase velocity

Angular dispersion

Prism sequences

Grating pairs

Chirped mirrors

Intracavity and extra-cavity examples

2

pulse duration increases with z chirp parameter D� � � �

2

22 2 2

2 "exp 1 1 GG

ki ztt z

ª ºW« » � � [ [

� [« »¬ ¼

� �� �2

2

/ V( , ) exp

2 "g p

G

t zE t z

t ik z

ª º� Z« »v �« »�« »¬ ¼

Pulse propagation and broadening

After propagating a distance z, an (initially) unchirped Gaussian of initial duration tG becomes:

pulse duration tG(z)/tG

0

1

2

3

0 1 2 3propagation distance [z

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Chirped vs. transform-limitedA transform-limited pulse:• satisfies the ‘equal’ sign in the relation 'Q�'W t C• is as short as it could possibly be, given the spectral bandwidth• has an envelope function which is REAL (phase )�Z� = 0)• has an electric field that can be computed directly from S�Z�• exhibits zero chirp:

the same periodA chirped pulse:• satisfies the ‘greater than’ sign in the relation 'Q�'W t C• is longer than it needs to be, given the spectral bandwidth• has an envelope function which is COMPLEX (phase )�Z� z 0)• requires knowledge of more than just S�Z� in order to determine E(t)• exhibits non-zero chirp:

not the same period

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� � � �pp

0 '1

Z

Z Z

Zdkd

kVg

� � � �pp

pM ZZ

Zk

V

where: � � 2 "2� ziktztGG

� �� �� � � �

� � »»¼

º

««¬

ª¸̧¹

·¨̈©

§ ����

2

0

V/V/exp1),(

ztzt

ztizt

EztEG

pgpp

G

ZZZ

S M

pulse width increases with propagation

phase velocity

group velocity -speed of pulse envelope

Group velocity dispersion (GVD)(different for each material)

Propagation of Gaussian pulses

� � � �2

2

1§ ·cc ¨ ¸¨ ¸

© ¹g

d k dkd d V

ZZ Z Z

If (and only if) GVD = 0, then VI = Vg. For most transparent solids, VI > Vg

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GVD yields group delay dispersion (GDD)The delay is just the medium length L divided by the velocity.

The phase delay:

00

0

( )v ( )

kI

ZZZ

00

1( )v ( )g

k ZZ

c

The group delay:

so: 0

0 0

( )v ( )

k LLtII

ZZ Z

0 00

( ) ( )v ( )g

g

Lt k LZ ZZ

c so:

1( )vg

dkd

ZZª º

cc « »« »¬ ¼

The group delay dispersion (GDD):

1 ( )vg

dGDD L k Ld

ZZª º

cc « »« »¬ ¼

so:

Units: fs2 or fs/Hz

GDD = GVD×L

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0.5 0.7 0.9 1.1

1.75

1.76

1.77

n(O)

-1e-4

-5e-5

0

dn/dOPm�1

1e-7

3e-7

5e-7

0.5 0.7 0.9 1.1

d2n/dO2

Pm�2

sapphire, L ~ 1 cm

Dispersion in a laser cavity

0.5 0.6 0.7 0.8 0.9 1 1.1

1e-7

2e-7

3e-7

O (Pm)

cm��

2"2

Gtk [

Material: Al2O3Pulse width: tp = 100 fsec

At O = 800 nm:GDD � k L = 2×10-3 fs2

There is a small positive GDD added to the pulse on each round trip.

"

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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 for the positive GDD that comes from propagating through transparent materials.

Negative GDD Device

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index n(Z)

D

D'Ti

Ttsin sinT Ti tn

Beam width:cos'cos

t

i

D D TT

1 tan iphase

DdTc c

T

2' tan t

phasenDnT d

c cT

BUT: the pulse front travels at Vg, so the travel distance is less by:

� �g phased V V TI' � �

time to propagate the distance d1:

d1

must equal the time to propagate the distance d2:

d2

Group and phase velocities - refraction

phase fronts: perpendicular to propagationpulse fronts: not necessarily!

propagation direction

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Angular dispersion yields negative GDD.

Suppose that an optical element introduces angular dispersion.

We’ll need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse).

Inputbeam

Opticalelement D

Optic axis

Here, there is negative GDD because the blue precedes the red.

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Negative GDDT�Z�

Optic axis

( ) ( )

( ) cos[ ( )]( / ) cos[ ( )]

optic axisk r

k zc z

M Z Z

Z T ZZ T Z

�

G G

/ ( / ) cos( ) ( / ) sin( ) /d d z c c z d dM Z T Z T T Z �

Taking the projection of onto the optic axis, a givenfrequency Z sees a phase delay of M�Z��

( )k ZG

We’re considering only the GDD due to dispersion and not that of the prism itself. So n = 1 (that of the air after the prism).

22 2

2 2sin( ) sin( ) cos( ) sin( )d z d z d z d z dd c d c d c d c dM T T T TT T Z T Z TZ Z Z Z Z

§ · � � � �¨ ¸© ¹

T�Z������

But T << 1, so the sine terms can be neglected, and cos(T) ~ 1.

z

11

00

22

02

zd dd c d ZZ

ZM TZ Z

§ ·| � ¨ ¸¨ ¸

© ¹�

Angular dispersion yields negative GDD.

The GDD due to angular dispersion is always negative!

But recall:

In most dielectric materials, k" is positive! (and k" L = I�")

We can use angular dispersion to compensate for material dispersion:

k"total = k"material + k"angular

These two terms have opposite sign.

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In principle, we are free to specify:• the apex angle E

E

• the angle of incidence I0

I0

These are chosen using two conditions:• Brewster condition for minimum reflection loss (|| polarization)• minimum deviation condition (symmetric propagation)

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

Ref

lect

ance

incidence angle

||

n = 1.78

45 50 55 60 65 70 7545

46

47

48

49

50

devi

atio

n an

gle

incidence angle

n = 1.5E = 67.4º

Prisms

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A prism pair has negative GDD.

How can we use dispersion to introduce negative chirp conveniently?

Let Lprism be the path through each prism and Lsep be the prism separation.

This term accounts forthe angular dispersiononly.

d2MdZ 2

Z 0

| � 4LsepO0

3

2Sc2dndO O0

§�

©�¨�·�

¹�¸�

2

� LprismO0

3

2Sc2d2ndO2

O 0

This term accounts for the beampassing through a length, Lprism, of prism material.

Vary Lsep or Lprism to tune the GDD!

Always negative!

Always positive (in visible and near-IR)

Assume Brewsterangle incidenceand exit angles.

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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 doesnot misalign the beam.

Output beamInput beam

The beam’s output path is independent of prism displacement along the

axis of the apex bisector.

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It’s routine to stretch and then compress ultrashort pulses by factors of >1000.

Four-prism Pulse CompressorThis device, which also puts the pulse back together, has negativegroup-delay dispersion and hence can compensate for propagation through materials (i.e., for positive chirp).

Angular dispersion yields negative GDD.

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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 all the spatio-temporal distortions.

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Appl. Phys. Lett. 38, 671 (1981)

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The required separation between prisms in a pulse compressor can be large.

It’s best to use highly dispersive glass, like SF10.But compressors can still be > 1 m long.

Kafka and Baer, Opt. Lett., 12, 401 (1987)

Different prism materials

Compression of a 1-ps, 600-nm pulse with 10 nm of bandwidth (to about 50 fs).

The GDD the prism separation and the square of the dispersion.v

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Four-prism pulse compressorAlso, alignment is critical, and many knobs must be tuned.

All prisms and their incidence angles must be identical.

Fine GDD tuning

Prism

Wavelength tuning

Wavelength tuning

Prism

Coarse GDD tuning (change distance between prisms)

Wavelength tuning

Wavelength tuning

Prism

Prism

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Pulse compressors are notorious for their large size, alignment complexity, and spatio-temporal distortions.

Pulse-front tilt

Spatial chirp

Unless the compressor is aligned perfectly, the output pulse has significant: 1. 1D beam magnification 2. Angular dispersion3. Spatial chirp 4. Pulse-front tilt

Pulse-compressors have alignment issues.

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Why is it difficult to align a pulse compressor?

Minimum deviation

The prisms are usually aligned using the minimum deviationcondition.

The variation of the deviation angle is 2nd order in the prism angle.But what matters is the prism angular dispersion, which is 1st order!Using a 2nd-order effect to align a 1st-order effect is tricky.

Prism angleD

evia

tion

angl

e

Angu

lar d

ispe

rsio

n

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Two-prism pulse compressor

Prism

Wavelength tuning

Periscope

Wavelength tuning

Prism

Coarse GDD tuning

Roof mirror

Fine GDD tuning

This design cuts the size and alignment issues in half.

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Single-prism pulse compressor

Corner cube

Prism

Wavelength tuning

GDD tuningRoof mirror

Periscope

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Example:4 SF11 prismsLm=1 cm

0.75 0.8 0.85

-800

-400

0

400

800

Lp=14 cm

Lp=16 cm

Lp=18 cm

Wavelength (Pm)

)"

(fse

c2 )

)" depends on O! For very short pulses, third order dispersion is important.

Angular dispersion from a sequence of prisms

� �� �� � � �4

2 32 3 12 1 2 3

2�ª ºccc c c c cc cc ccc) � � � � �

¬ ¼p mL n n n n n n L n nc

O O O OS

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32

2" " 4 '2 m pL n L n

cO ª º| �¬ ¼S

)

SF6 glass at 800 nmn = 1.78n' = ���59 Pm��

n" = 0.22 Pm��

)" = 0 if 0633.0"n'n4

LL 2

p

m »»¼

º

««¬

ª

But then )"' | Lp × ���� fsec3/cm

Minimum value of Lp or Lm determines size of )"'

It is not possible to ideally compensate both 2nd and 3rd order dispersion with prisms alone.

Diffraction gratings!

When prisms are not sufficient

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E

E' > @)'cos(1 )'cos(

D�E�E�D�E

xCBAP

grating equation:

� � 2'sinsindc

ZS

� D�E�ED

A

B CZ

x = perpendicular grating separationd = grating constant (distance/groove)

Double pass:3/ 223 3

3/ 22 2 2 2 1 sin

3 sin

�

�ª º§ ·cc) � � � �« »¨ ¸

© ¹« »¬ ¼ª º§ ·ccc cc) � � � �)¨ ¸« »© ¹¬ ¼

x x rc d d c d

rcr d d

O O OES S

O O O ES

Diffraction-grating pulse compressor

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2nd- and 3rd-order phase terms for prism and grating pulse compressors

Piece of glass

M'' M'''

Grating compressors offer more compression than prism compressors.

Note that the relative signs of the 2nd and 3rd-order terms are oppositefor prism compressors and grating compressors.

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Compensating 2nd and 3rd-order spectral phaseUse both a prism and a grating compressor. Since they have 3rd-orderterms with opposite signs, they can be used to achieve almost arbitrary amounts of both second- and third-order phase.

This design was used by Fork and Shank at Bell Labs in the mid 1980’sto achieve a 6-fs pulse, a record that stood for over a decade.

M input 2 � M prism 2 � Mgrating 2 0

M input 3 � Mprism 3 � Mgrating 3 0

Given the 2nd- and 3rd-order phases of the input pulse, Minput2 and Minput3, solve simultaneous equations:

Grating compressorPrism compressor

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Pulse Compression Simulation

Resulting intensity vs. time with only a grating compressor:

Resulting intensity vs. time with a grating compressorand a prism compressor:

Note the cubic spectral phase!

Brito Cruz, et al., Opt. Lett., 13, 123 (1988).

Using prism and grating pulse compressors vs. only a grating compressor

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Chirped mirrors

A mirror whose reflection coefficient is engineered so that it has the form:

� � � � ir e I ZZ

so that � � 1 r Z and I�Z� is chosen to cancel out the phase of the incident pulse.

All 5 mirrors are chirped mirrors

8.5 fsec pulse from the laser

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Chirped mirror coatings

Longest wavelengths penetrate furthest.

Doesn’t work for < 600 nm

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Theory:Experiment:

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