Transcript Slide 1

Bin Li
April. 7th, 2003
Mach Zehnder Interferometer
Piezo
system
2PPE UHV
Chamber
Large
fix time delay
Beam - splitter
Small
fix time delay
SHG and Dispersion
compensation
Monochromator
Time Information and Alignment of Optics
d  sin  m  m  
Monochromator
N=2
N=1
y m  L  tg m
N=0
2
 ym  m 
1
L
  L2  y m 2
d
when y m  L
Grating
d
Select a single wavelength out of the
femto-second laser’s wave package.
we can get the first
order of diffraction
Pattern at
L
y1 ( )   
d
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:
T
1
I ( ) 
T
2
2
[
E
(
t


)

E
(
t
)]
dt

T
2
The Delayed time  can be generated by using scanning signal of piezo
-system, one arm fixed, the other arm moves to a distance: L  c  
I( t )
T 

c
t
 1.33 fs
Good
I( t )
t
Bad
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.
Dispersion Compensation
v ( ) 
c
n ( )
After traveling finite distance
In air or optics, the different
components of femto–second
pulse will arrive at different
time!
t ( ) 

L L
  n ( )
v c
L  n( )
T
In order to make different wavelets in a same phase, we have to generate
negative dispersion!
One widely used method:
Multiple Coating Reflection Mirror, the
deeper layers have smaller index.
A simple calculation: Two Layers Case
Where n1>n2

Evanescent wave
1
0
n2
So when
n1
n2
1   c  sin ( )
n1
1
n0
Air
 
Optical Length path:
n2 L  n2
2

cos 1
There should be
TIR, but when the
medium is thin, we
have penetration
depth:
1
2


 cos 1
n1 2
( ) sin 2 1  1
n2
n2 (

Negative Dispersion !!
n1 2
) sin 2  1  1
n2
In Our Experiment, we combine the discrete negative dispersion (Chirp Mirror) with
continuous positive dispersion (Wedge). But how do we know when the minimal
dispersion 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:
Ac ( ) 

I
pump
(t   )  I probe (t )dt

No phase information!
Interference Signal
The case when they have same polarization:
I ( ) 

2
[
E
(
t


)

E
(
t
)]
dt
1
2


~ 
~ 
~ 
~ 
 [ E 1 (t   )  E 2 (t )][ E 1 (t   )  E 2 (t )]
So we have:
~*
~*
~*
~
1 ~
1 ~
1 ~
1 ~*
il
I ( )    1 (t )   1 (t )dt    2 (t )   2 (t )dt    1 (t   )   2 (t )  e dt    1 (t   )   2 (t )  e il dt
4
4
4
4
~ 
~ 
 A11 (0)  A22 (0)  A12 ( )  A12 ( )
A11 (0) , A22 (0) are average intensity of Beam 1 and Beam 2, they are constant.
~ 
~ 
And A12 ( )  [ A12 ( )]
Let’s consider the Fourier transformation:
~
 ~
A12 ( )   A12 ( )  e
i 

d   d  e

~*
~

  1 (  l )   2 (  l )
~
~ 
 E 1 ( ) E 2 ( )
i ( l )

~
~*
[  dt   1 (t   )   2 (t )]

When the two output beams are identical, we just get the spectral intensity
of light: ~ 
A12 ( )  I ( )
For Gaussian pulse
I (t )  e
(
Fourier Transformation is
t 2
)
T0
I ()  e
( l ) 2 T02
Since the first order Interference signal has high background (peak to
background ratio is 2:1), people are not using it as an indicator of phase
or dispersion. Instead, we use SSHG.
Sample
p-Polarization
e
Electron
Photoemission
x
s-Polarization
Selection Filter
The filter will eliminate the fundamental, the second order interferometric
correlation signal will be detected:
G2 ( ) 

2 2
{[
E
(
t


)

E
(
t
)]
} dt
2
 1

~
~
1~
1 ~*
il t
By using E (t )  E (t )  E (t )   (t )  e
  (t )  e il t
2
2
We get:
~
G2 ( )  A( )  4 Re{ B( )  e
where
A( ) 
il 
~
}  2 Re{C ( )  e 2il  }

4
4
2
2
dt
{

(
t


)


(
t
)

4

(
t


)

2
1
2 (t )}
 1

~
B( ) 

i[1 ( t  )  2 ( t )]
2
2
dt
{

(
t


)

(
t
)[

(
t


)


(
t
)]

e
}
2
1
2
 1

~
C ( ) 

2 i[1 ( t  )  2 ( t )]
2
2
dt
{

(
t


)

(
t
)

e
}
1
2


Considering Identical fields case:
When time delay   0 , we will get the sum of all constructive interference terms:
G2 (0)  16  4 (t )dt
When time delay is large, the cross product terms vanish, so we have a
background value:
G2 ()  2   4 (t )dt
The peak to background ratio is 8:1, and it is sensitive to the pulse phase
modulation, 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
k ( ) 
2

n ( ) 

c
n( )
Taylor Expansion at center frequency  l :
dk
1 d 2k
2
k ( )  k (l ) 
|l (  l ) 
|
(



)
 cubic  term, higher  order
l
2 l
d
2 d
~
E ( )  A  e i[t k ( ) L ]
(L is the optical length in Air or Optics.)
Just consider up to 1st order k ( ) L  k l L  k l ' L(  l ) , in frequency region:
~
E ( )  A  e
i [ 
In time Region:
L n
 k ( ) L ]
c
~
 A  exp{i[( k l  n 
~


c
) L  k l ' (   l ) L]}
~
E (t )  F ( E ( ))   exp( i  t )  E ( )d
1

After rearrangement, we obtain:

~
E (t )  A  exp[ ik l L  i l t ]   exp[ i (   l )t ]  exp[ i (   l )k l ' L]  exp[ i(   l )

 exp[ i l
nL
]
c
~
E (t )  A  exp[ i l (t  (
kl
l

n( )
n( )
) L]   [t  (k l '
L)]
c
c
Up to 1st order expansion of wave number k is just a time delay factor:
k
k
nL
nL
)  ( l L  )  (k l ' l ) L
c
l
c
l
  n( )
kl 
c
  (k l ' L 
By using
We get
  ( 
dn( )
L
|l ) 
d
c
nL
]d
c
GVD (Group Velocity Dispersion) for Gaussian Pulse
Bring in the GVD term and neglect the Time Delay Factor:
1 d 2k
2
E ' ( )  A( )  exp[ i
|
(



)
L]
l
2 l
2 d
~
For femto second Laser source, under equal mode approximation, we
get
c
sin( N
t)
2
d
A(t ) 
c
sin(
t)
2d
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:
1 t 2
A(t )  e
A( ) 

 ( )
2 T0
1
i ( l )t
2
2
A
(
t
)

e
dt

exp[

(



)

T
]
l
0

2
c
~ 100 MHz
2d
2d
T0 
~ 10 fs
Nc
F
By the way, If we use repetition rate:
and the pulse width:
(The number of Mode Locking is in the order of million !!)
We can compare these two normalized functions, they are pretty close.
1
1
 t
f ( t )  e
0.8
f ( t)
g( t )
6
g ( t)  10
2 0.6


2
sin 0.5  t
6
sin 10
2
2
 0.50  t
0.4
Gaussian Shape is a fairly
good Approximation.
0.2
0
0
6
6
4
2
0
t
2
4
6
6

So we will get:
k l" L
1
1
2
2
"
2 2
E ' ( )  exp{ (  l ) [T0  i  k l L]}  exp[  (  l ) T0 (1  i  2 )]
2
2
T0
~
When GVD is a small value (kl" L  T02 ), we have:
kl" L 2
1
2 2
E ' ( )  exp[  (  l ) T0 (1  i  2 ) ]
2
2T0
~
1 t
L
E ' (t )  F 1 [ E ' ( )]  exp[  ( ) 2 (1  i 2 k l" )]  e
2 T0
T0
~
~
1 t
 ( ) 2 (1 ai )
2 T0
Conclusion: GVD term is the linear chirp of Gaussian Pulse.
Here we can see linear chirp:
L d 2k
a 2
|
2 l
T0 d
Previously, we have second order interferometric signal:
~
G2 ( )  A( )  4 Re{ B( )  e
il 
~
}  2 Re{C ( )  e 2il  }
If we consider a linearly chirped Gaussian pulse:
 (t )  e
(
t 2
) (1 aj )
T0
There will be:
 2
a2  3  2
a 
G2 ( )  {1  2 exp[ ( ) ]  4 exp[ 
( ) ] cos[ ( ) 2 ] cos( l )
T0
4 T0
2 T0
 2
2
 exp[ (1  a )( ) ] cos( 2 l )}
T0
2 l
T

  1.33 fs and pulse width T0  10 fs
Let Optical cycle l
l c
By using linear chirp term: a1=0.1, a2=0.5, a3=2, a4=4, a5=16, we will
get the following second harmonic interferometric correlation signal!
2nd Order Interference Signal
24 25
24
So from this signal, we can
minimize the GVD, meanwhile
we can estimate the pulse width
of our femto second Laser.
23
22
21
20
19
18
17
G2( t  a1)
16
G2( t  a2)  4
15
Intensity
14
G2( t  a3)  8
13
G2( t  a4)  12 12
11
G2( t  a5)  16
10
9
8
7
6
5
4
3
2
4.47910
4
1
0
4
4
3
2
1
0
t
time (in unit of pulse width)
1
2
3
4
4
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 2PPE
Signal.
Evac
Ev
T12
2
 l
2l 
m
 l
 l
T212
 l
s
ECBM
1
T11 T202
01
2
T
Ef
Ef
 l
 l
0
EVBM
Metal
Semi-conductor
T1 population relaxation time,
T2 coherence decay time.
Quantum Perturbation Theory (First Order Approximation)
Before Apply pump-probe laser pulse, electrons are in non-interacting discrete
Energy Levels:
H 0k  Ekk  kk
H  H 0  H ' (t )
 (t )   ak (t )k   exp( ikl t )ck (t )k
After introducing laser pulses, Hamiltonian becomes:
Let
Schr o dinger Equation i  (t )  H (t )
..
Put it in
t
We have
 [c (t )(ik )e
k
k
Multiplied by
l
 n*,
ikl t

 ck (t ) e
ikl t
i
]   ck (t )( H 0  H ' (t )) k e ikl t

then do integration, using the orthonormal condition.

So
where
i
cn  i( n  nl )cn   H ' nk e i ( n k )l t ck
 k
H ' nk    H ' (t ) k d r
*
n
Using real electric field:
3

and
Now
~*
1 ~
E (t )  [ exp( il t )   exp( il t )]
2
 (t )   (t ) exp[ i (t )]   (t )
E (t )   (t ) cos(l t )

For Bosonic system, we have
For Fermionic system, we have


r  (a  a )


H ' (t )  e  ( E (t ) E (t   ))  r
~
Assuming small dispersion,
chirp term disappears !


 
r  (C  C )
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 a
Final state, which is above the Vacuum level and can be observed by Energy
Analyzer.
E2
Ev
 l
m
Eint
Ef
 l
E1
Absorbing the constant factor into amplitude (here we can use Gaussian
Approximation), considering just one dimensional dipole transition, the
pump-probe perturbation term becomes:


H ' (t )  { (t ) cos(l t )   (t   ) cos[l (t   )]}(C  C )
So for electrical dipole moment operator , only two adjacent terms do not vanish
at certain constrains!
cn (t )
i
i
il t
 i(n  nl )cn  H ' n,n1 e cn1  H ' n,n1 e il t cn1
t


H 'n,n1  [ Enn1 (t ) cos(l t )  Enn1 (t   ) cos[l (t   )]]
H 'n,n1  [ Enn1 (t ) cos(l t )  Enn1 (t   ) cos[l (t   )]]
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
nlm (r , ,  )
to
n'l 'm' (r , ,  )
The transition term should be:
H 'nk   n* H ' (t )k d 3 r

 e  d r 
3
*
n 'l 'm '



[ r  ( E (t ) cos l t  E (t   ) cos l (t   ))]l , n , m
So, for a three-Level-Atom-System, with pump-probe radiation
perturbation, we have:
c0 (t )
i
 i0 c0  [ E1 (t ) cos(l t )  E1 (t   ) cos[l (t   )]]  e il t c1
t

c1 (t )
i
 i (1  l )c1  [ E2 (t ) cos(l t )  E2 (t   ) cos[l (t   )]]  e il t c2
t

i
 [ E1 (t ) cos(l t )  E1 (t   ) cos[l (t   )]]  eil t c0

c2 (t )
i
 i (2  2l )c2  [ E2 (t ) cos(l t )  E2 (t   ) cos[l (t   )]]  eil t c1
t

Let E (t )  Ae
1 t
 ( )2
2 T0
By using initial condition:
1  1  l
Estimate
 2  2  2l
c0  1, c1  c2  0,
(Detuning of resonance)
and integration recurrently,
Theoretically, We can solve the derivative equation and get C0(t), C1(t), C2(t)
Now we discuss the density operators, which are measurable quantities, and we
can compare them with the 2nd order auto-correlation signal, then extract useful
dynamics out of them. Each component has:
mn  a(t )m a(t )*n  cmcn*ei ( mn) t
l
 (t )
c
c
 mn
 ( m c n*  c m n )  e i ( m  n )l t  i (m  n) l  c m  c n*  e i ( m  n )l t
t
t
t
*
*
Now we can calculate the 1st order derivative equations of cm (t )cn (t ) ,
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 twol
photon excitation 2 .
l
Where we define T11 T12 as population relaxation time of level 1 and level 2;
and T201 T212 as 1st order decoherent time, T202 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, offdiagonal elements).
Since only the energy of Level 2 is above the vacuum level, so the time resolved
2PPE correlation signal is the dynamics of  22 . Not just  22 , 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 interacting
processes 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.
From
Set
 22
c2 *
c2 *
1
  2  22 
 c2  c2 (
)
t
t
t
T1
E1 (t ) cos(l t )  E1 (t   ) cos(t   )  A1[e
(
E2 (t ) cos(l t )  E2 (t   ) cos(t   )  A2 [e
By using
t 2
)
T0
(
t 2
)
T0
cos(l t )  e
(
t  2
)
T0
(
t  2
)
T0
cos(l t )  e
c2
i
 i 2 c2  f 2 (t ,  )  eil t c1
t

c2 *
i
*
( )  i 2c2  f 2 (t ,  )  e il t c1*
t

cos[l (t   )]  f1 (t ,  )
cos[l (t   )]  f 2 (t ,  )
So, we get
 22 (t )
1
2
  2  22  f 2 (t ,  ) cos[l t ]  I m (c1c2* )
t
T1

And the measured electron photon-emission signal can be denoted as:
TA
1
2 PPE( ) 
 22 (t ,  )dt

TA 0
It only depends on the delay time between two pulses.
How can we solve the derivative function of
 22 ?
Firstly, we have to consider
(c1c2* )
1
1
1
c
c
 [ 12  1  2 ]c1c2*  1 c2*  c1 ( 2 )*
t
T2 2T1 2T1
t
t
By using:
c1
i
i
 i l t
 i1c1  f 2 (t ,  )e c2  f1 (t ,  )eil t c0
t


and
(
c2 *
i
)  i 2 c2*  f 2 (t ,  )  e il t c1*
t

So finally, we will get:
 (c1c2* )
1
1
1
*
 [i ( 1   2 )  12 

]
c
c
1 2
t
T2
2T11 2T12

i
i
f 2 (t ,  )e il t ( 11   22 )  f1 (t ,  )e il t c0 c2*


*
c
c
We see solving derivative equation of 1 2 is not the end of story, it depends
on other variables, such as
11, 22,c0c2*
So we can expect these nine elements of density matrix are dependent on
each others, they only way to get the absolute solution for  22 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 !!!
After calculation, we obtained
11
1
i
  1 11 
f1 (t ,  )[ e il t c1c0*  e il t (c1c0* )* ]
t
T1


i
f 2 (t ,  )[ e il t c2 c1*  e il t (c2 c1* )* ]

 22
1
i
  2  22 
f 2 (t ,  )[ e il t c2 c1*  e il t (c2 c1* )* ]
t
T1

 00 (t )
11 (t )
 22 (t )
t

t

t
 (c1c0* )
1
1
*
 [i ( 1   0 )  01 
]
c
c
1
0
t
T2
2T11

i
i
f1 (t ,  )e il t ( 11   00 ) 
f 2 (t ,  )e il t c2 c0*


 (c2 c1* )
1
1
1
 [i (  2  1 )  12 

]c2 c1*
1
2
t
T2
2T1
2T1

i
i
f 2 (t ,  )e il t (  22  11 ) 
f1 (t ,  )e il t c2 c0*


 (c2 c0* )
1
1
i il t
*
*
 [i ( 2  0 )  02 
]
c
c

e
[
f
(
t
,

)
c
c
2
0
1
2
1
t
T2
2T12

 f 2 (t ,  )c1c0* ]
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.
From previous discussion, we know the 2nd order interferometric signal of
the Gaussian pulse with negligible dispersion is:
G2 ( )  {1  2 exp[ (
 exp[ (

T0

T0
) 2 ]  4 exp[ 
3  2
( ) ] cos(l )
4 T0
) 2 ] cos( 2l )}
It includes 0w (phase averaged component), 1w (1st order component),
and 2w (2nd order component).
For Gaussian pulse
(
e
t 2
)
T0
,e
3 t
 ( )2
4 T0
 (t )  e
(
, e
t 2
)
T0
(
t 2
)
T0
, The 0w,1w, and 2w components have time
respectively.
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).
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 intermediate
Energy level. So the 0w, 1w and 2w components should be convolution between
Pulsed signal and the coherent decay or in coherent decay.
I
fit
2
( )  c
2
0

e

 t
T20 2
e
t
 4 ln(2 )( ) 2

dt, where

Pulse. If you still want to use the notation of
convolution:
1
2
I 2fit
 ( )  c0

e


( 4 ln 2 )
2
(
e
t 2
)
T0 ,
 t
T20 2
 Is the FWHM of Gaussian
(
e
we have the identical
t 2
)
T0
dt

e
I1fit ( )  c01

 t
T20 1
e
t
3 ln(2 )( ) 2

dt

0w (Phase Average Component) consists both coherent parts and incoherent,
and background term, so:
   t
T11
I pafit ( )  c0pa (1  c1  e

e
t
 4 ln(2 )( ) 2

   t
T201
dt  c2  e
e
t
3 ln(2 )( ) 2

dt )

Fitting these three theoretical calculated curves with Fourier Transformation
(2w, 1w, 0w) of experimental 2PPE correlation signal respectively, we can
1
get the population decay time T1 , decoherent time of first order T 01 , and
2
02
nd
2 order T2 of intermediate level of our system.
The following are Fourier Transformation terms:
I 2 ( ) 
4c


[


4c

4c

I ( x) cos( 2x)dx]2  [ 


4c

4c
I ( x) sin( 2x)dx]2
I 2 ( ) 
2c

I pa ( ) 
c
[



 


2c

2c
I ( x) cos(x)dx]2  [ 



2c


2c

I ( x) sin( x)dx]2
2c
I ( x)dx
2c
1.0
TR-2PPE Signal
intensity [normalized]
0.8
Apply this fitting
procedure to
TR-2PPE Signal
0.6
1w Component
Phase Average
0.4
0.2
0.0
-100
2w Component
-80
-60
-40
-20
0
Time delay [fs]
20
40
60
80
100
Our Experiment Data on TiO2 (110) surface
Example
Clean Surface
2PPE Intensity (CPS)
200
TiO2 Clean Surface at 110K
7-Channel Data Acquisition
& Time-resolved Measurement
150
1:(5.8 eV)
100
2:(5.71 eV)
3:(5.9 eV)
5:(6.01 eV)
50
7:(6.12 eV)
0
5.0
5.2
5.4
5.6
5.8
6.0
6.2
Hot Electron Final Energy (eV)
6.4
6.6
6.8
7.0
intensity [normalized]
1.0
0.35
0.30
4
2
0
-2
0.25
reduced residuals
w0 of 1025002
X_offs. = 0.52fs
tau
= 10.0 fs
coh.
= 5.9 fs
inc.
= 18.5 fs
sum
= 1.316
diff
= 0.305
Y_scal.= 0.5403
0.40
raw data
phase average
1 envelope
2 envelope
T1(1) : 19.5 fs
-4
0.20
50
Time delay [fs]
100
0.4
0.3
w1 of150
1025002
X_offs. = 0.64 fs
tau
= 10.0 fs
coh.
= 5.9 fs
Y_offs. = 5e-03
Y_scal. = 0.5107
5
0
0.2
T2(01): 5.2 fs
-5
0.1
0.0
0.12-50
intensity [normalized]
0.6
0.4
-10
0
50
Time delay [fs]
100
0.10
0.08
w2 of150
1025002
X_offs. = 0.61 fs
tau
= 10.0 fs
coh.
= 5.8 fs
Y_offs. = 2e-03
Y_scal. = 0.1206
4
2
0.06
0
0.04
0.02
0.00
-50
reduced residuals
intensity [normalized]
0.8
-6
0
reduced residuals
Intensity [normalized]
0.5
-50
T2(02): 1.8 fs
-2
0
50
Time delay [fs]
100
150
0.2
0.0
-50
0
50
Time delay [fs]
100
150
Hot Electron Relaxation Time --- T1
25
Time (fs)
20
15
10
5
0
2.4
2.5
2.6
2.7
2.8
2.9
3
Intermediate State Energy Level (eV)
The Relaxation Time is pretty close to a constant (around 20 femto-seconds! )