propagation of short pulses jörgen larsson, fysiska instutionen lunds tekniska högskola

Propagation of short pulses

Jörgen Larsson,

Fysiska Instutionen

Lunds Tekniska Högskola

Distributed feedback laser

Femtosecond X-Ray and Electron Source at the SLAC LinacFemtosecond X-Ray and Electron Source at the SLAC Linac

• 80 fs• 8-10 keV• 1x107 ph./pulse (2% bw.)• 10 Hz

Bending compress. Bending compress. <100 <100 fsecfsec

~1 Å~1 Å

Add 12-meter Add 12-meter chicanechicane compressor in linac at 1/3-point compressor in linac at 1/3-point

(9 GeV)(9 GeV)

Add 12-meter Add 12-meter chicanechicane compressor in linac at 1/3-point compressor in linac at 1/3-point

(9 GeV)(9 GeV)

Damping RingDamping Ring

9 ps9 ps 0.4 ps0.4 ps

50 ps50 ps

1 GeV1 GeV 20-50 GeV20-50 GeV

FFTBFFTB

undulator

femtosecond laser(synchronized)

Short-pulse accelerator sourcesSPPS

www-ssrl.slac.stanford.edu/jhhome.html

Femtosecond fiber laser

Splitters/Couplers

Femtosecond fiber laser

Wavelength Division Multiplexers

Bragg grating

Bandpass filter for fiber optics

Manufacturing Bragg gratings

Representation of short pulsesGaussian pulses

tjt eeEt 02

*)( 0E

CarrierEnvelopeAmplitudeFrequency

2220

020

2)(

2trr eE

ct

cI(t)

E

Representing ”chirp”)(

0

20

2

*)( attjt eeEt E

btdt

btt

dtt 2

)()( 0

20

Time-bandwidth product- Transform limited pulses

441.0)2ln(2

2

2ln2

)2ln(2

a

avt FWHMFWHM

If the pulse is chirped it is wider in the temporal domain441.0FWHMFWHM vt

Gaussian pulse

Propagation of a range of frequencies

4

)(

0

20

02

)*))(

eEeeEt tjtF(F(E

xikeeExE )(4

)( 20

),(

cnk /)()(

After the pulse has propagated the distance x

...)()(

24

1)(

)(

6

1)(

)(

2

1)(

)()()( 4

04

43

03

32

02

2

00

0000

d

kd

d

kd

d

kd

d

dkkk

')(

0

kd

dk

'')(

0

2

2

kd

kd

4

)(2)(

2

'')(')(

20

00),(

eeeeExE

xk

ixikxik

200 ))(

2

''

4

1()(')(

),(

x

kixikxik

eExE


4

)(

0

20

02

)*))(

eEeeEt tjtF(F(E

200 ))(

2

''

4

1()(')(

),(

x

kixikxik

eExE

)),(2

00 ))(2

''

4

1()(')(

xk

ixikxikeExtE F(

)),(2

0)''

24

1('

xk

ixikikxti eEextE

F(

)),(2

0)''

24

1(')(

xk

ixikkxti eEextE

F(


)),(2

0)''

24

1(')(

xk

ixikkxti eEextE

F(

)''24

1(4

)'( 2

)''24

1(2

')''

24

1()''

24

1('

2

2

xkixk

xki

xikxk

ixk

ixik

)),(

2

2

0)''

24

1(2

')''

24

1(

)''24

1(4

)'()(

xk

i

xikxk

i

xkixk

kxti

eEextE

F(


))''21

()(

1xik

x

)),(

2

2

0)''

24

1(2

')''

24

1(

)''24

1(4

)'()(

xk

i

xikxk

i

xkixk

kxti

eEextE

F(

22

0 )(4

1')(2)')(()(),(

xxkxxkxkxti eEextE

F(

2

20

)')(2()(4

1(

)')(()(),(

xikx

xxkxkxti eEextE

F(

220 )(')(2)')(()(

2

2),( txxkxxkxkxti eeExtE

2

0 )')(()(

2

2),( xktxkxtieExtE


20 ))(()(

),( gv

xtx

v

xti

eExtE

'

1)(,

)()(

0

00

0 kdk

dv

kv g

20 )')(()(

2

2),( xktxkxtieExtE

Group velocity dispersion


)()( 0

n

cv

ddkdk

dvg

1)( 0

d

cnd

d

dk ))((

d

dnn

d

dk )()(

))(

)(1()( 0

d

dn

nvvg

00)(

1''

2

2

gvd

d

d

kdk

GVD wavelength form

• Whiteboard

Group velocity dispersion compensation

• All matererials show a positive GVD in the visible range – hence various set-ups have been deviced for GVD compensation

GVD compensation using prisms

P1P2

Short wavelengths are deflected more

Longer wavelengths travel a longer optical path

Prism pair with negativ group velocity dispersion

laser in

Chirped mirrors

Dispersion compensation using Bragg gratings

Time-varying refractive indesSelf-phase modulation

xdt

tdn

cdt

xtnctd

dt

tdt

)())(()()( 0

0

00

))((

0

00),(

xtnctieExtE

)()( 20 tInntn

FIG 2.18

Time-varying refractive indesSelf-phase modulation

Passive modelockingKerr lens

High intensitysmall losses

Low intensitieslarge losses

n=n1+n2I

The beams spatial profile creates the "Kerr lens"

I

x

Titanium sapphirecrystal

Laser beam

Aperture

Matrix representaiton of beam propagation

Beam propagation in a linear cavity

´10

11

201

10

11

201

´ 1

1

21

n

n

n

n

r

rL

R

L

Rr

r

Stability analysis of Kerr-lens mode-locked laser

• Propagation matrix´´ 1

1

n

n

n

n

r

r

DC

BA

r

r

0

0

'´ r

r

DC

BA

r

rn

n

n

12

1

DAGeneral stability

criterion

2-mirror cavity

22 1

R

Lg

11 1

R

Lg

10 21 ggStability criterion

11122 21

R

L

R

LDA

Beampropagation in a folded cavity(out of plane – approx for in-plane)

P1P2

C MCM2

CM1OC

P1,P2 prismsCM1, CM2 curved mirror, krökt spegel(these are transparent for the pump radiation)M mirror, spegelC crystal, kristallOC output coupler utkopplingsspegelL lens for the pump laser

Lpump from Nd-laser

´10

111

201

10

1211

201

10

21

10

01

1

1

12

n

n

CMCMr

rOCCM

R

CMCM

R

CMM

10

211

201

10

2111

201

10

11

10

01

´21

MCM

R

CMCM

R

CMOC

r

r

CMCMn

n

Beampropagation including Kerr-lens

P1P2

C MCM2

CM1OC

P1,P2 prismsCM1, CM2 curved mirror, krökt spegel(these are transparent for the pump radiation)M mirror, spegelC crystal, kristallOC output coupler utkopplingsspegelL lens for the pump laser

Lpump from Nd-laser

´10

111

201

10

111

101

10

211

201

10

21

10

01

1

1

12

n

n

CMKLCMr

rOCCM

R

CMKL

f

KLCM

R

CMM

10

211

201

10

211

101

10

111

201

10

11

10

01

´21

MCM

R

CMKL

f

KLCM

R

CMOC

r

r

CMKLCMn

n

propagation of short pulses jörgen larsson, fysiska instutionen lunds tekniska högskola

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