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
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
Propagation of a range of frequencies
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(
Propagation of a range of frequencies
)),(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(
Propagation of a range of frequencies
))''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
Propagation of a range of frequencies
20 ))(()(
),( gv
xtx
v
xti
eExtE
'
1)(,
)()(
0
00
0 kdk
dv
kv g
20 )')(()(
2
2),( xktxkxtieExtE
Group velocity dispersion
Propagation of a range of frequencies
)()( 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
Propagation of a range of frequencies
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
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
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