3. NTHU
Defined of ultrafast
What is ultrafast ?
The range of ultrafast ?
‘’ ultrashort’’ refers to the femtosecond(fs) to picosecond(ps)
range.
3
Milli- Micro- Nano- Pico- Femto- Atto-
Time(s) 10e-3 10e-6 10e-9 10e-12 10e-15 10e-18
frequency 1kHz 1MHz 1GHz 1THz 1PHz 1EHz
4. NTHU
Goal of pulse measurement
4
* ( )1
( ) Re{ ( ) } { ( ) ( ) }, ( ) ( )
2
o o oj t j t j t j t
E t a t e a t e a t e a t a t e
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X: 0.5811
Y: 0.6262
( )t
( )a t
It is straightforward to get carrier frequency by spectrometer,
we focus on measuring the complex envelope function
5. NTHU
Difficult
The laser pulse duration cannot be easily measured by
optoelectronic methods, since the response time of
phtodetector and oscilloscopes are at best of the order of
200(fs)
5(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
6. NTHU
Definition
For a given power spectrum , the pulse is :
Transform-limited (TL), if
Chirped, if is nonlinear
6
2
( )A
( ) 0
( )
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Chirped TL
7. NTHU
Pulse measurement method
Because the pulses are so short that no existing electronics
are capable of resolving them, so the common approach is
to measure the ultrashort pulse by itself
Auto-correlation
Cross-correlation
7
*
12 1 2( ) ( )f a t a t dt
*
( ) ( )f a t a t dt
8. NTHU
Outline
Defined of ultrafast
Mathematic introduce
Correlation & Convolution
Pulse measurement methods
Field autocorrelation
Cross correlation
Intensity autocorrelation
FAQ
8
10. NTHU
Field autocorrelation trace formula
10
0
1( ) ( ) 1 Re{ ( ) }j
FA outI P G e
11 01 ( ) cos( ( ))GG R
when
1
*
( )
1 12
( ) ( )
( ) ( )
( )
Gja t a t
G G e C
a t
Is the normalized field autocorrelation function of ( )a t
11. NTHU
Example A TL pulse with two smallside lobes
11(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
12. NTHU
How to retrieve G1 from the field autocorrelation
Perform Fourier transform for trace:
Extract the component centered at :
Shift to the baseband:
Perform inverse Fourier transform:
12
{ ( )}FA FAI F I
, 0( ) ( )oFA FAI I
0,0 , 0( ) ( )FA FAI I
1
,01( ) { ( )}FAG F I
11 0( ) 1 ( ) cos( ( ))FA GI G
(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
13. NTHU
Limitation
FA function is nothing but power spectrum of the field
envelope a(t):
As a result
NO spectral phase information , then we cannot
distinguish transform-limited pulse with
long chirped pulse with and even
incoherent noise
13
2
1{ ( )} ( )F G A
( )
( )TLI t ( ) 0
( )chirpI t
2
2
( )
2
( )noiseI t
19. NTHU
Field-cross-correlation
The field cross-correlation function of and
19
0
1,( ) ( ) 2Re{ ( ) }j
Fx out s r xI P U U G e
1, 0 1,2 ( ) cos( ( ))tot x G xU G
*
1, ( ) ( ) ( )x s rG a t a t C
( )sa t ( )ra t
2
( )i iT
U a t dt
20. NTHU
Field cross-correlation
For very short reference pulse
20
r st t
0 1,2 ( ) cos( ( ))tot s G xU a
1, 0 1,( ) 2 ( ) cos( ( ))FX tot x G xI U G
1, ( ) ( ) ( ) ( )x s sG a t t a
21. NTHU
Field cross-correlation
Perform Fourier transform for the trace
Extract the component centered at
Shift to the baseband
21
0
1,( ) 2Re ( ) j
FX tot xI U G e
*
1, ( ) ( ) ( )x s rG a t a t
{ ( )} ( )FXFXF I I
* *
0 0 0 0( ) [ ( ) ( ) ( ) ( )]s r s rA A A A
0
0, ( )FX oI
0
*
,0 , 0( ) ( ) ( ) ( )FX FX s rI I A A
22. NTHU
Field cross-correlation
The exact complex spectrum of the signal pulse can be
derived by:
If the complex spectrum of the reference pulse is
known
Bandwidth of the reference pulse is broader than that of the
signal pulse
22
,0
*
( )
( )
Fx
s
r
I
A
A
( )rA
23. NTHU
Poor signal-to-background contrast
23
Cross-correlation Field-autocorrelation
0
1,( ) ( ) 2Re{ ( ) }j
Fx out s r xI P U U G e
(Assume: TL Gaussian, ),s r s rU U t t
-10 -8 -6 -4 -2 0 2 4 6 8 10
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
cross-correlation
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Field-autocorrelation
24. NTHU
Outline
Defined of ultrafast
Mathematic introduce
Correlation & Convolution
Pulse measurement methods
Field autocorrelation
Cross correlation
Intensity autocorrelation
FAQ
24
25. NTHU
Second harmonic generation (SHG)
25
NLO
material
0 02
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x 10
-9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x 10
-9
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
2 2
( )j t j t
e e
2
2 ( )a a t
27. NTHU
Fringe-resolved intensity autocorrelation
…Intensity autocorrelation
……Intensity-field correlation
……Squared-field autocorrelation
27
2 0 0( ) 1 2 ( ) 4 ( ) cos( ( )) ( ) cos(2 ( ))FRIA f gI G f g
2 2
( ) ( )
( )
( )
I t I t
G R
I t
*
2
[ ( ) ( )] ( ) ( )
( )
2 ( )
I t I t a t a t
f C
I t
* 2
2
[ ( ) ( )]
( )
( )
a t a t
g C
I t
28. NTHU
Comparison between TL &chirped pulses of the same I(t)
28(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
29. NTHU
How to retrieve G2 from the Intensity autocorrelation
Perform Fourier transform for trace:
Extract the component centered at :
Remove the Dirac-function component
Perform inverse Fourier transform:
29
{ ( )}FAIA FAIAI F I
,0 ( ) ( 0)FRIA FRIAI I
1
,02 ( ) { ( )}FRIAG F I
(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
32. NTHU
Deconvolution factor
I(t)
1.41 1.54 1
32
decR
2
2( / )pt t
e
2
sec ( / )ph t t ( / )pt t
If this factor is know, or assumed, the time duration (Intensity width)
of a pulse can be measured using an Intensity autocorrelation
The deconvolution factor, defined as:
/decR t
34. NTHU
Noncollinear
34
0(2)
2 ( , ) ( ) ( ) j
a t a t a t e
2
(3)
2 2( ) ( , ) ( ) ( ) ( )IAI a t dt I t I t G
(Shang-Da Yang, Ultrafast Optics, Lecture slide 05)
