WHY ???? Ultrashort laser pulses. (Very) High field physics Highest peak power, requires highest...
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Transcript of WHY ???? Ultrashort laser pulses. (Very) High field physics Highest peak power, requires highest...
WHY????
Ultrashort laser pulses
(Very) High field physics
Highest peak power, requires highest concentration of energy
E L I
Create … shorter pulses (attosecond)Create x-rays (point source)Imaging
High fields high nonlinearities high accuracy
F=ma0~ 31 Å
1015 W/cm2, 800 nm
20
Electrons ejected by tunnel ionization can be re-captured by the next half optical cycle of opposite sign. The interaction of the returning electron with the atom/molecule leads to high harmonic generation and generation of single attosecond pulses.
-1
0
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To do this you need to control a single cycle
Resolve very fast events
- “Testing” Quantum mechanics
Probing chemical reactions
Pump probe experiments
All applicatons require propagation/manipulation of pulses
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MANIPULATION OF THIS PULSE
Chirped pulse
LEADS TO THIS ONE:
Propagation through a medium with time dependent index of refraction
Pulse compression: propagation through wavelength dependent index
Why do we need the Fourier transforms?
Construct the Fourier transform of
“Linear” propagation in frequency domain
“Non-Linear” propagation in time domain
Actually, we may need the Fourier transforms (review)
0
Properties of Fourier transforms
Shift
Derivative
Linear superposition
Specific functions: Square pulse Gaussian Single sided exponential
Real E(E*(-
Linear phase
Product Convolution
Derivative
Description of an optical pulse
Real electric field:
Fourier transform:
Positive and negative frequencies: redundant information Eliminate
Relation with the real physical measurable field:
Instantaneous frequency
0 z
t t
What is important about the Carrier to Envelope Phase?
Slowly Varying Envelope Approximation
Meaning in Fourier space??????
Forward – Backward Propagation
Maxwell Equation
s = t – n/c Zr = t + n/c z
20 0
02
n nE P
z c t z c t t
2 2 2 20
02 2 2 2
nE P
z c t t
No scattering
i s i rE e e FΕ Ε
No coupling between EF & EB
22 2
024
cE P
r s n s r
22 2i s i s
0 F2
ce 2 i P e
r s r 4 n s r
F FΕ Ε
No linear assumption
Slowly varying envelope
22
02
22
0
2
1
2
F
F
ci P
r n
Pn
FΕ
22
2F FP P
t
Study of linear propagation
(Maxwell second order)
Solution of 2nd order equation
22
02
( ) ( , ) 0E zz
0( )P E 0( ) (1 ( ))
( )( , ) ( , ) ik zE z E 0 e
( ) ( )2 20k
Propagation through medium
No change in frequency spectrum
To make F.T easier shift in frequencyExpand k value around central freq l
l
( )( , ) ( , ) lik zz 0 e ε ε
( )1( , ) ( ,0) ( )
2lik z i tt z e e d
z
Z=0
1( , ) ( , ) ( )
2i tE t z E z e d
Dispersion includedk real
10
gz v t
ε ε
Study of linear propagation
Expansion orders in k(Material property
l
l
2| 22
1( , ) ( ,0) (1 | ( ) ) ( )
2l
dkiik z i td d k
t z e e e i z dd
ε ε
II)( )( , ) ( , ) lik zz 0 e ε ε
ll| ( )| ( )( , )
22
2 l
1 d kdk i zi z ik z2d d0 e e e
ε
l
l
| ( )( , ) ( | ( ) ) l
dk 2i z 2 ik zd2
1 d k0 e 1 i z e
2 d
ε
22
2
( ) 1( ) ( )
2ixtt
x x e d xt
ε ε
2 2
2 2
10
2g
i d k
z v t d t
ε ε ε
Second
Study of linear propagation
Propagation in the time domain
PHASE MODULATION
n(t)or
k(t)
E(t) = (t)eit-kz
(t,0) eik(t)d (t,0)
DISPERSION
n()or
k()() ()e-ikz
Propagation in the frequency domain
Retarded frame and taking the inverse FT:
PHASE MODULATION
DISPERSION
Application to a Gaussian pulse
Evolution of a single pulse in an ``ideal'' cavity
Dispersion
Kerr effect
Kerr-induced chirp
20 0
02
n nE P
z c t z c t t
2 2 2 20
02 2 2 2
nE P
z c t t
22
2F FP P
t
Study of propagation from second to first order
From Second order to first order (the tedious way)
( ) ( )kz kz
2 2 2 20 i t i t
02 2 2 2
ne P e
z c t t
2 2 22
2 2 2 2 2
22
0 0 02
1 2ik 2ik
c z c t c t z
P i P Pt t
01 i cP
z c t 2
(Polarization envelope)
Pulse duration, Spectral width
Two-D representation of the field: Wigner function
-2 -1 0 1 2
-2
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0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cy
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cyGaussian Chirped Gaussian
Wigner Distribution
Wigner function: What is the point?
Uncertainty relation:
Equality only holds for a Gaussian pulse (beam) shape free of anyphase modulation, which implies that the Wigner distribution for aGaussian shape occupies the smallest area in the time/frequencyplane.
Only holds for the pulse widths defined as the mean square deviation