why ???? ultrashort laser pulses. (very) high field physics highest peak power, requires highest...
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WHY????
Ultrashort laser pulses
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(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
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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.
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-1
0
1
To do this you need to control a single cycle
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Resolve very fast events
- “Testing” Quantum mechanics
Probing chemical reactions
Pump probe experiments
All applicatons require propagation/manipulation of pulses
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-1
0
1
MANIPULATION OF THIS PULSE
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Chirped pulse
LEADS TO THIS ONE:
Propagation through a medium with time dependent index of refraction
Pulse compression: propagation through wavelength dependent index
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Why do we need the Fourier transforms?
Construct the Fourier transform of
“Linear” propagation in frequency domain
“Non-Linear” propagation in time domain
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Actually, we may need the Fourier transforms (review)
0
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Properties of Fourier transforms
Shift
Derivative
Linear superposition
Specific functions: Square pulse Gaussian Single sided exponential
Real E(E*(-
Linear phase
Product Convolution
Derivative
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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
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0 z
t t
What is important about the Carrier to Envelope Phase?
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Slowly Varying Envelope Approximation
Meaning in Fourier space??????
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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)
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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
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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
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Propagation in the time domain
PHASE MODULATION
n(t)or
k(t)
E(t) = (t)eit-kz
(t,0) eik(t)d (t,0)
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DISPERSION
n()or
k()() ()e-ikz
Propagation in the frequency domain
Retarded frame and taking the inverse FT:
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PHASE MODULATION
DISPERSION
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Application to a Gaussian pulse
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Evolution of a single pulse in an ``ideal'' cavity
Dispersion
Kerr effect
Kerr-induced chirp
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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
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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)
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Pulse duration, Spectral width
Two-D representation of the field: Wigner function
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-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
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
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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