Bin Li

April. 7th, 2003

SHG and Dispersion compensation

Monochromator

Mach Zehnder Interferometer

2PPE UHVChamber

Beam - splitter

Piezosystem

Largefix time delay

Smallfix time delay

Time Information and Alignment of Optics

Monochromator

N=1N=2 N=0

Grating

d

mm

m

tgLy

md

sin

d

yLmy m

m

22

Lwhen Lym

we can get the first order of diffraction Pattern at

d

Ly )(1

1

2

Select a single wavelength out of thefemto-second laser’s wave package.

E t ( )

t

E t( )

t

Our pulsed laser repetition rate is 83MHz, repetition time period is around 0.012 us, which is much faster than the response time of photo-diode detector. So we can treat the two different components as continuous wave.

Output is coherent interference signal of these two split beams:

2

2

2)]()([1

)(

T

T

dttEtET

I

The Delayed time can be generated by using scanning signal of piezo-system, one arm fixed, the other arm moves to a distance:

cL

I t( )

t

fsc

T 33.1

I t( )

t

So we can get the precise time delay between two beams from this signal;meanwhile, we also can see if the optics alignment in MZI is good or not.

Good

Bad

Dispersion Compensation

)()(

n

cv

)()( nc

L

v

Lt

After traveling finite distanceIn air or optics, the different components of femto–secondpulse will arrive at differenttime!

T

In order to make different wavelets in a same phase, we have to generatenegative dispersion!

Multiple Coating Reflection Mirror, thedeeper layers have smaller index.

One widely used method:

)(nL

A simple calculation: Two Layers Case

n0

n1

n2

0

1

Air

Where n1>n2

So when

)(sin1

211 n

nc

There should beTIR, but when themedium is thin, we have penetrationdepth:

Evanescent wave

1sin)(2

1

122

2

12

n

nn

1sin)(coscos

2

122

2

11

122

n

nnLn

Optical Length path:

Negative Dispersion !!

In Our Experiment, we combine the discrete negative dispersion (Chirp Mirror) with continuous positive dispersion (Wedge). But how do we know when the minimaldispersion occurs?

We do need another diagnostic signal to indicate the dispersion!!

Intensity Spectrum

When pump pulse and probe pulse are orthogonal polarization, we have Intensity cross-correlation:

No phase information!

Interference Signal

dttEtEI 221 )]()([)(

dttItIA probepumpc )()()(

)]()()][()([ 2

~

1

~

2

~

1

~

tEtEtEtE

The case when they have same polarization:

dtettdtettdtttdtttI ll ii )()(4

1)()(

4

1)()(

4

1)()(

4

1)( 2

~*

1

~*

2

~

1

~*

2

~

2

~*

1

~

1

~

So we have:

)()()0()0( 12

~

12

~

2211

AAAA

)0(11A )0(22A are average intensity of Beam 1 and Beam 2, they are constant. ,

)]([)( 12

~

12

~

AAAnd

Let’s consider the Fourier transformation:

)]()([)()(*

2

~

1

~)(

12

~

12

~

ttdteddeAA lii

)()(

)()(

2

~

1

~

*

2

~

1

~

EE

ll

When the two output beams are identical, we just get the spectral intensity of light:

)()(12

~

IA

For Gaussian pulse 2

0

)(

)( T

t

etI

2

02)()( TleI

Fourier Transformation is

Since the first order Interference signal has high background (peak tobackground ratio is 2:1), people are not using it as an indicator of phaseor dispersion. Instead, we use SSHG.

x

p-Polarization

s-Polarization

Selection Filter

Sample

Electron Photoemission

e

The filter will eliminate the fundamental, the second order interferometric correlation signal will be detected:

dttEtEG 22212 })]()({[)(

By using titi ll etettEtEtE

)(2

1)(

2

1)()()(

*~~~~

We get:})(Re{2})(Re{4)()( 2

~~

2 ll ii eCeBAG

})()({)(

})]()()[()({)(

)}()(4)()({)(

)]()([222

21

~

)]()([22

2121

~

22

21

42

41

21

21

tti

tti

ettdtC

ettttdtB

ttttdtA

where

Considering Identical fields case:

When time delay 0 , we will get the sum of all constructive interference terms:

dttG )(2)( 42

dttG )(16)0( 42

When time delay is large, the cross product terms vanish, so we have a background value:

The peak to background ratio is 8:1, and it is sensitive to the pulse phasemodulation, so people use it as diagnostic signal for quantitatively measur-ement of linear chirp!!

(Jean-Claude Diels, Wolfgang Rudolph, Ultrashort Laser Pulse Phenomena,Academic Press, 1996)

More discussion on dispersion

)()(2

)( n

cnk

Taylor Expansion at center frequency l :

orderhighertermcubicd

kd

d

dkkk lll ll

,)(|2

1)(|)()( 2

2

2

])([~

)( LktieAE (L is the optical length in Air or Optics.)

)(')( lll LkLkLk

Just consider up to 1st order , in frequency region:

]})(')[(exp{)(])([~

LkLc

nkiAeAE lll

Lkc

nLi

In time Region:

dEtiEFtE )()exp())(()(~~

1~

)])(

'([]))(

((exp[)(

]exp[

])(exp[]')(exp[])(exp[]exp[)(

~

~

Lc

nktL

c

nktiAtE

c

nLi

dc

nLiLkititiLikAtE

ll

ll

l

llllll

After rearrangement, we obtain:

Up to 1st order expansion of wave number k is just a time delay factor:

Lk

kc

nLL

k

c

nLLk

l

ll

l

ll )'()()'(

c

nk l

)(

c

L

d

dnl )|

)((

By using

We get

GVD (Group Velocity Dispersion) for Gaussian Pulse

Bring in the GVD term and neglect the Time Delay Factor:

])(|2

1exp[)()(' 2

2

2~

Ld

kdiAE ll

For femto second Laser source, under equal mode approximation, we get

)2

sin(

)2

sin()(

td

c

td

cN

tA

In real case, the laser amplification profile will make each mode has different amplitude, in order to make calculation simpler, it is good to use Gaussian Approximation:

2

0

)(2

1

)( T

t

etA

])(2

1exp[)()( 2

02)( TdtetAA l

ti l

By the way, If we use repetition rate: MHzd

cF 100~

2

and the pulse width: fsNc

dT 10~

20

We can compare these two normalized functions, they are pretty close.

(The number of Mode Locking is in the order of million !!)

f t( ) e

t2

2

g t( ) 106 sin 0.5 t

sin 106

0.50 t

6 4 2 0 2 4 60

0.2

0.4

0.6

0.8

11

0

f t( )2

g t( )2

66 t

Gaussian Shape is a fairlygood Approximation.

)]1()(2

1exp[]}[)(

2

1exp{)('

20

"2

02"2

02

~

T

LkiTLkiTE l

lll

])2

1()(2

1exp[)(' 2

20

"2

02

~

T

LkiTE l

l

So we will get:

When GVD is a small value )( 20

" TLkl , we have:

)1()(2

1

"2

0

2

0

~1

~ 2

0)]1()(2

1exp[)]('[)('

iaT

t

l ekT

Li

T

tEFtE

Conclusion: GVD term is the linear chirp of Gaussian Pulse.

Here we can see linear chirp:ld

kd

T

La

|2

2

20

})(Re{2})(Re{4)()( 2~~

2 ll ii eCeBAG

Previously, we have second order interferometric signal:

If we consider a linearly chirped Gaussian pulse: )1()( 2

0)(aj

T

t

et

)}2cos(]))(1(exp[

)cos(])(2

cos[])(4

3exp[4])(exp[21{)(

2

0

2

2

0

2

0

22

02

l

l

Ta

T

a

T

a

TG

There will be:

fsc

T l

ll 33.1

2

fsT 100 Let and pulse widthOptical cycle

By using linear chirp term: a1=0.1, a2=0.5, a3=2, a4=4, a5=16, we willget the following second harmonic interferometric correlation signal!

4 3 2 1 0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

252nd Order Interference Signal

time (in unit of pulse width)

Inte

nsity

24

4.479 104

G2 t a1( )

G2 t a2( ) 4

G2 t a3( ) 8

G2 t a4( ) 12

G2 t a5( ) 16

44 t

So from this signal, we can minimize the GVD, meanwhile we can estimate the pulse width of our femto second Laser.

Time Resolved Interferometric 2PPE Correlation

Ultrafast interferometric pump-probe techniques can be applied to metals or semiconductors, decay rates of hot-electron population and quantum phase and other underlying dynamics can be extracted by careful analysis the 2PPESignal.

Ef

Ev

l

l

l2m ECBM

Ef

Evac

EVBM

l

l

l

l

s

012T

122T

022T

11T

21T2

1

0

T1 population relaxation time, T2 coherence decay time.

MetalSemi-conductor

Quantum Perturbation Theory (First Order Approximation)

Before Apply pump-probe laser pulse, electrons are in non-interacting discreteEnergy Levels:

kkkkk EH 0

After introducing laser pulses, Hamiltonian becomes: )('0 tHHH Let

)()(

tHt

ti

kklkk tctiktat )()exp()()(

Put it in EquationdingeroSchr..

We have

tikkk

tikk

tiklkk

lll etHHtci

etceiktc ))(')((])())(([ 0

Multiplied by , then do integration, using the orthonormal condition. *n

So

k

ktkni

nknlnn ceHi

cnic l )(')(

rdtHH knnk3* )(''

rtEtEetH ))()(()(' where and

)]exp()exp([2

1)(

*~~

tititE ll

)()](exp[)()(~

ttitt

Using real electric field:

Assuming small dispersion,chirp term disappears !

)cos()()( tttE lNow

For Bosonic system, we have )(

aar

For Fermionic system, we have )(

CCr

Absorbing the constant factor into amplitude (here we can use Gaussian Approximation), considering just one dimensional dipole transition, the pump-probe perturbation term becomes:

))]}((cos[)()cos()({)('

CCtttttH ll

Hot electronic energy states is in a Fermionic system, and we consider the case The laser energy is just good for the transitions: between an initial energy state, which is below the Fermi energy and an intermediate energy state, which is an excited state above Fermi level, or between an intermediate energy state and aFinal state, which is above the Vacuum level and can be observed by Energy Analyzer.

Ef

Ev

l

l

m

E1

E2

Eint

So for electrical dipole moment operator , only two adjacent terms do not vanish at certain constrains!

11,11, '')()(

n

tinnn

tinnnln

n ceHi

ceHi

cnit

tcll

)]](cos[)()cos()([' 111, ttEttEH l

nnl

nnnn

)]](cos[)()cos()([' 111, ttEttEH l

nnl

nnnn

Here, I just do a simple calculation for dipole transition term; in fact, the more accurate results rely on the knowledge of the interacting quantum states of the system and the polarization of the electric field.

For example, transition from ),,( rnlm ),,(''' rmlnto

mnlllmln

knnk

ttEttErrde

rdtHH

,,*

'''3

3*

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

)(''

The transition term should be:

111000 )]](cos[)()cos()([

)(cettEttE

ici

t

tc till

l

011

222111

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

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

cettEttEi

cettEttEi

cit

tc

till

tilll

l

l

122222 )]](cos[)()cos()([)2(

)(cettEttE

ici

t

tc tilll

l

So, for a three-Level-Atom-System, with pump-probe radiationperturbation, we have:

By using initial condition: ,10 c ,021 cc and integration recurrently,

Theoretically, We can solve the derivative equation and get C0(t), C1(t), C2(t)

2

0

)(2

1

)( T

t

AetE

Letl 11

l 222 Estimate (Detuning of resonance)

Now we discuss the density operators, which are measurable quantities, and we can compare them with the 2nd order auto-correlation signal, then extract usefuldynamics out of them. Each component has:

tnminmnmmn

lecctata )(**)()(

tnminml

tnminmn

mmn ll eccnmiet

ccc

t

c

t

t

)(*)(*

* )()()(

Now we can calculate the 1st order derivative equations of , but we have to consider one more thing. Since level 1 and level 2 are above the Fermi levels, are unoccupied states, so their population densities will relax to zero quickly, meanwhile we have to consider the coherent decay between different polarizations induced by one photon pulse excitation or two-photon excitation . l

l2

Where we define 11T

21T as population relaxation time of level 1 and level 2;

and 122T

012T

022T

as 1st order decoherent time, as 2nd order decoherent time.

Then, we will get 9 first order derivative equations for population density (when m=n, diagonal terms), or coherence dynamics (when m!=n, off-diagonal elements).

)()( * tctc nm

Since only the energy of Level 2 is above the vacuum level, so the time resolved2PPE correlation signal is the dynamics of . Not just , but the average value of it. Here laser pulse is 10 femto-second, Energy Analyzer acquisition time is about 163.84 us, pulse repetition time is about 0.012 us, so our detected 2PPE signal is including about 13,500 different pump-probe coherent interactingprocesses with electrons. The signal amplitude is mainly dependent on the delay time between pump and probe pulses ------ , the relaxation time ----- T1 , and coherent time T2.

22 22

*22

*2

2222

1

22 )(1

t

ccc

t

c

Tt

From

),()](cos[)cos([)cos()()cos()( 2

)()(

222

2

0

2

0

tfteteAttEttE lT

t

lT

t

l

Set

By using 12222 ),( cetf

ici

t

c ti l

*12

*22

*2 ),()( cetfi

cit

c ti l

),()](cos[)cos([)cos()()cos()( 1

)()(

111

2

0

2

0

tfteteAttEttE lT

t

lT

t

l

)(]cos[),(21)( *

2122221

22 ccIttfTt

tml

So, we get

And the measured electron photon-emission signal can be denoted as:

AT

A

dttT

PPE0

22 ),(1

)(2

It only depends on the delay time between two pulses.

How can we solve the derivative function of ? 22

Firstly, we have to consider

*21

*2

1*212

11

112

2

*21 )(]

2

1

2

11[

)(

t

ccc

t

ccc

TTTt

cc

By using: 0122111 ),(),( cetf

icetf

ici

t

c titi ll

and*12

*22

*2 ),()( cetfi

cit

c ti l

So finally, we will get:

*20122112

*212

11

1122

21

*21

),()(),(

]2

1

2

11)([

)(

ccetfi

etfi

ccTTT

it

cc

titi ll

We see solving derivative equation of *21cc is not the end of story, it depends

on other variables, such as *20,2211, cc

So we can expect these nine elements of density matrix are dependent on each others, they only way to get the absolute solution for is to solve all these 6 dependent derivative equations (some of them are complex conjugates). We can plug in reasonable parameters and solve those equations to see how well the theoretic calculation match with real time experimental results !!!

22

After calculation, we obtained

*02111222

*122

11

112

212

*12

),()(),(

]2

1

2

11)([

)(

ccetfi

etfi

ccTTT

it

cc

titi ll

*02200111

*011

101

201

*01

),()(),(

]2

11)([

)(

ccetfi

etfi

ccTT

it

cc

titi ll

]),(

),([]2

11)([

)(

*012

*121

*022

102

202

*02

cctf

cctfei

ccTT

it

cc ti l

])()[,(

])()[,(1

**12

*122

**01

*011111

1

11

cceccetfi

cceccetfi

Tt

titi

titi

ll

ll

])()[,(1 **

12*122222

1

22 cceccetfi

Tttiti ll

t

t

t

t

t

t

)()()( 221100

Another Process: Fitting Procedure for the calculation of the relaxation time and decoherent time:

[W. Nessler, S. Ogawa, H. Nagano, H. Petek, etc, J. of Elec. Spec and Phenomena, 88-91 (1998) 495]

Simulation of TR-2PPE process by using Perturbation Theory is entangled Quite a few unknown quantities together, it is not easy to extract information from it, so there is a consideration from another point of view.

)}2cos(])(exp[

)cos(])(4

3exp[4])(exp[21{)(

2

0

2

0

2

02

l

l

T

TTG

From previous discussion, we know the 2nd order interferometric signal ofthe Gaussian pulse with negligible dispersion is:

It includes 0w (phase averaged component), 1w (1st order component),and 2w (2nd order component).

For Gaussian pulse

2

0

)(

)( T

t

et

, The 0w,1w, and 2w components have time

,2

0

)(4

3

T

t

e

,2

0

)(T

t

e 2

0

)(T

t

e

respectively.

Now if we think the first laser pulse excites the electron to intermediate level, then the 2nd pulse just works as a probe to get the dynamics of the intermediateEnergy level. So the 0w, 1w and 2w components should be convolution betweenPulsed signal and the coherent decay or in coherent decay.

So, same as SSHG, the pump-probe electron emission will have similar signal, but more. The same is the two pulse or two induced polarization interference, the additional part is the response of electron (coherent interaction, population relaxation).

dteecIt

T

t

fit2

022

))(2ln(4202 )(

, where Is the FWHM of Gaussian

Pulse. If you still want to use the notation of 2

0

)(T

t

e

, we have the identical

convolution:

dteecI T

tT

t

fit2

002

22 )(

)2ln4(

1

202 )(

dteecIt

T

t

fit2

012

))(2ln(3101 )(

0w (Phase Average Component) consists both coherent parts and incoherent, and background term, so:

)1()(2

012

21

1))(2ln(3

2

))(2ln(4

10

dteecdteeccIt

T

ttT

t

pafitpa

Fitting these three theoretical calculated curves with Fourier Transformation (2w, 1w, 0w) of experimental 2PPE correlation signal respectively, we can get the population decay time , decoherent time of first order , and 2nd order of intermediate level of our system.

11T

012T

022T

The following are Fourier Transformation terms:

c

c

c

c

dxxxIdxxxIc

I 4

4

224

4

2 ])2sin()([])2cos()([4

)(

c

c

c

c

dxxxIdxxxIc

I 2

2

222

2

2 ])sin()([])cos()([2

)(

c

c

pa dxxIc

I 2

2

)()(

1.0

0.8

0.6

0.4

0.2

0.0

inte

nsi

ty [n

orm

aliz

ed

]

-100 -80 -60 -40 -20 0 20 40 60 80 100Time delay [fs]

TR-2PPE Signal

Phase Average

1w Component

2w Component

Apply this fitting procedure to

TR-2PPE Signal

Our Experiment Data on TiO2 (110) surface

Clean Surface

200

150

100

50

0

2PP

E I

nten

sity

(C

PS

)

7.06.86.66.46.26.05.85.65.45.25.0

Hot Electron Final Energy (eV)

TiO2 Clean Surface at 110K

1:(5.8 eV)

2:(5.71 eV)

3:(5.9 eV)

5:(6.01 eV)

7:(6.12 eV)

Example

7-Channel Data Acquisition& Time-resolved Measurement

1.0

0.8

0.6

0.4

0.2

0.0

inte

nsity

[no

rmal

ized

]

150100500-50

Time delay [fs]

raw dataphase average1 envelope2 envelope

-6

-4

-2

0

2

4

red

uce

d re

sidu

als

150100500-50Time delay [fs]

0.40

0.35

0.30

0.25

0.20

inte

nsi

ty [n

orm

aliz

ed

]

w0 of 1025002X_offs. = 0.52fstau = 10.0 fscoh. = 5.9 fsinc. = 18.5 fssum = 1.316diff = 0.305Y_scal.= 0.5403

-10

-5

0

5 red

uce

d re

sidu

als

150100500-50Time delay [fs]

0.5

0.4

0.3

0.2

0.1

0.0

Inte

nsi

ty [n

orm

aliz

ed

]

w1 of 1025002X_offs. = 0.64 fstau = 10.0 fscoh. = 5.9 fsY_offs. = 5e-03Y_scal. = 0.5107

4

2

0

-2

red

uce

d re

sidu

als

150100500-50Time delay [fs]

0.12

0.10

0.08

0.06

0.04

0.02

0.00

inte

nsi

ty [n

orm

aliz

ed

]

w2 of 1025002X_offs. = 0.61 fstau = 10.0 fscoh. = 5.8 fsY_offs. = 2e-03Y_scal. = 0.1206

T1(1) : 19.5 fs

T2(01): 5.2 fs

T2(02): 1.8 fs

The Relaxation Time is pretty close to a constant (around 20 femto-seconds! )

Hot Electron Relaxation Time --- T1

0

5

10

15

20

25

2.4 2.5 2.6 2.7 2.8 2.9 3

Intermediate State Energy Level (eV)

Tim

e (

fs)