Download E - Uni Regensburg/Physik

Spektroskopie und Photonik mit
Femtosekunden-Lasern
1. Femtosecond (fs) laser pulses:
general properties, generation,
light matter interaction (linear), dispersion
2. Nonlinear optics:
nonlinear light matter interaction,
new frequencies, THz spectroscopy with a single shot,
white light from lasers
3. Ultrafast spectroscopy:
ultrafast Ohm-meters, Ampere-meters,
magnetometers etc.
4. Ultrafast nano-optics:
plasmons, photonic crystals,
manipulation of light on-the-fly
Tobias Kampfrath
Fritz Haber Institute Berlin
1
Lecture #1
1. Femtosecond (fs) laser pulses
- general properties
- applications
2. Fs pulse generation
- laser
- mode locking
3. Pulse propagation in vacuum
4. Excursus: light matter coupling
- Lorentz model
5. Pulse propagation in matter
- dispersion
- dispersion compensation
6. Pulse shaping
2
What is a femtosecond laser pulse?
• electromagnetic pulse
with duration of
~1 to 1000fs
• here: 100-fs pulse,
centered at 800 nm
(400THz, 2.5fs per cycle)
• peak field can be ∼ 10 V/Å
• pulses come periodically
with frequency frep
Typically ~1 W average power for Ti:sapphire
⇒ pulse energy = 10 nJ at frep = 100 MHz,
pulse energy = 1 mJ at frep = 1 kHz etc.
(⇒ laser is on for 1s/d,
1s/300y)
3
What is a femtosecond laser pulse?
|A(t)|2
|A(t)|
carrier
oscillation
E (t ) = Re [ A(t ) exp(iωct ) ]
complexvalued
envelope
Pulses can have a quite complex temporal structure, e.g. chirp
4
How short is a femtosecond?
1 fs = 10-15 s
1 s:
light travels from earth to moon
100 fs:
light propagates only 30μm
(thickness of a human hair)
5
How short is a femtosecond?
8 fs:
one vibration of H-H bond of H2
H
H
10 fs:
∼ time between subsequent collisions
of an electron in a metal
500 fs:
bacteriorhodopsin turns from
cis into trans conformation
bild
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Femtosecond pulses - what for?
Femtosecond laser pulses have unique properties:
1. short duration
2. high peak intensities
3. stable repetition rate
7
Ultrafast time-domain spectroscopy
Pulse (“pump”) triggers ultrafast processes, e.g.
• molecular or lattice vibration
• spin precession
• electron-hole pairs
Pulse (“probe”) monitors ultrafast processes, e.g.
• pump-induced dynamics
• thermally induced random dynamics (noise spectroscopy)
delay
sample
detect
transmitted
probe
probe pulse
pump pulse
8
Ultrafast photonics
Nonlinear optics: light interacts with light
Ultrafast optical switches
• for controlling light with light
Generation of new wavelengths
• longest wavelenghts (terahertz pulses, λ~1mm):
probe low-energy resonances
• shortest wavelenghts (soft x ray, λ~1nm):
for x-ray diffraction and core-level spectroscopy
Generation of electron pulses
• for ultrafast electron diffraction
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Scanning microscopy
Two-photon microscopy
• two photons absorbed,
one emitted
• increases resolution by √2
Stimulated-emission depletion
(STED) microscopy
• exploits nonlinear saturation
of stimulated emission
• breaks diffraction limit
(∼ λ/2) by factor of ≈ 20
1μm
www.zeiss.de
10
Scanning microscopy
Two-photon microscopy
• two photons absorbed,
one emitted
• increases resolution by √2
Stimulated-emission depletion
(STED) microscopy
• exploits nonlinear saturation
of stimulated emission
• breaks diffraction limit
(∼ λ/2) by factor of ≈ 20
1μm
www.zeiss.de
11
Micromachining
reduction of pulse duration
τ = 20 ps
τ = 1 ps
τ = 150 fs
Absorbed laser pulse heats electrons at first
⇒ shorter laser pulses
- give higher (electron) temperatures for same energy
- minimize the heat-affected zones
- yields higher reproducibility
- give higher process efficiency
12
Frequency comb
pulse train (“comb”) in time domain
⇒ comb in frequency domain
Bartels et al., Science (2008)
application: measurement of optical frequencies with high accuracy
13
How to generate fs pulses?
Laser: self-sustained oscillator for light
light in
gain
light out
losses due to
outcoupling,
scattering,
absorption
light back
Lasing condition:
gain ≥ loss
We need:
1. light-amplifying medium
2. feedback
14
Gain by laser-pumped Ti:sapphire
Ti3+ level scheme:
levels split by coupling
to phonons
pump (514...532nm)
|2〉
|3〉 (long-lived)
broad emission
(650...1080nm)
|4〉 (short-lived)
|1〉
15
Gain by laser-pumped Ti:sapphire
Ti3+ level scheme:
levels split by coupling
to phonons
pump (514...532nm)
|2〉
|3〉 (long-lived)
broad emission
(650...1080nm)
|4〉 (short-lived)
|1〉
Light amplification by stimulated
emission of radiation (LASER)
Consider: Light oscillation by
stimulated emission of radiation
16
Fabry-Perot resonator for feedback
mirror
mirror
only wavelengths
λj = 2L/j
survive
⇒ frequency comb of modes
light
wave
mode comb
ω
Idea:
superposition of modes
should give a
wavepacket or pulse
17
Example for mode superposition
more modes...
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Example for mode superposition
even more modes...
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Example for mode superposition
energy-time uncertainty: Δt Δf ∼ 1
Ti:sapphire laser oscillator: m ∼ 106
20
The laser
mirror
mirror
light
wave
gain
medium
gain curve
mode comb
loss
21
The laser
mirror
mirror
light
wave
gain
medium
gain curve
mode comb
loss
Switch gain on:
only one mode
depletes gain medium
No pulse!
• reason: homogeneous gain profile
• How can we excite and phase-couple the other modes?
22
Active resonator modulation
mirror
mirror
modulator generates
side bands at modulation
frequency ωmod = c/2L
light
wave
gain
modulator
medium
gain curve
mode comb
side bands are
amplified,
new sidebands etc.
loss
23
Active resonator modulation
mirror
mirror
modulator generates
side bands at modulation
frequency ωmod = c/2L
light
wave
gain
modulator
medium
gain curve
mode comb
loss
side bands are
amplified,
new sidebands etc.
Mode locking
⇒ pulse
24
Active resonator modulation
mirror
mirror
modulator generates
side bands at modulation
frequency ωmod = c/2L
light
wave
gain
modulator
medium
gain curve
mode comb
loss
side bands are
amplified,
new sidebands etc.
Mode locking
⇒ pulse
25
Passive (self-) mode locking
assume we have pulse: oscillates in resonator with frequency c/2L
⇒ can be used for self-modulation of resonator
Several approaches:
1. Saturable absorber
becomes transparent
at high intensities
transmittance
intensity
2. Transient Kerr effect
leads to self-focusing
at high intensities
26
Fs laser in the lab
Ti:sapphire laser oscillator
• 10 fs pulse duration
• 10 nJ pulse energy
• 80 MHz repetition rate
∼ 106 coupled modes
27
Fs laser in the lab
intensity autocorrelation
intensity spectrum
28
Frequency comb
pulse train (“comb”) in time domain
⇒ comb in frequency domain
Bartels et al., Science (2008)
application: measurement of optical frequencies with high accuracy
29
History for short-time measurements
30
Pulse propagation
• Eq of motion for light in vacuum: Maxwell eqs
⇒
⎛ 2 1 ∂2 ⎞
⎜∇ − 2 2 ⎟ E = 0
c ∂t ⎠
⎝
• plane wave E = E(z,t)e ⇒
general solution:
(∂
2
z
− c −2 ∂ t2 ) E = 0
E = u−(z+ct) + u+(z−ct)
pulse propagates with ±c, no dispersion
Reality:
1. beam is not plane wave ⇒ e.g. Gouy phase shift in focus
2. Beam propagates in media
We only consider 2.
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Light-matter interaction
current j
piece of matter
light wave (E,B)
equations
of motion
coupled by
Lorentz force
∝ E + (v/c) × B
Light: Maxwell eqs,
j → (E,B)
j
(E,B)
Matter: Newton eqs,
j ← (E,B)
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Assumptions:
• particle (electron, ion), charge q,
in potential minimum at x=0
• harmonic approximation:
(⇒ linear theory)
∇V = ω02 x
• electric-dipole approximation:
E(x,t)=E(0,t), B=0
• friction force −2γ∂tx
⇒ Eq of motion
energy V
Model: Lorentz oscillator
x
0
L̂x = qE ( t ) with Lˆ = ∂ t2 + 2γ∂ t + ω02
−1
Fourier
⇒ x ( ω ) = L ( ω ) qE ( ω )
transformation
−1
L
(ω ) =
1
ω02 − ω 2 − 2iγω
Lorentz function
34
Electric susceptibility
dipole moment
Nq 2
P (ω ) =
= Nqx =
E (ω )
volume
L (ω )
matter response
P (ω ) = χ (ω ) E (ω )
light field
electric susceptibility
(response function)
• χ quantifies how easily matter can be polarized
(vacuum: χ = 0, glass: 0.1 @800nm)
• ω Im χ(ω) quantifies locally absorbed light power
• time domain: χ(t) is polarization response to δ(t)
• j(t) = ∂P(t)/∂t ⇒ j(ω) = −iωχ(ω)E(ω) ⇒ σ(ω) = −iωχ(ω)
35
Example: ω0 = 25, γ = 1
1
0.5
0.9
0.4
0.8
0.3
0.7
0.2
0.6
0.1
χ
0.5
0
0.4
-0.1
0.3
-0.2
0.2
-0.3
0.1
-0.4
0
18
20
22
24
26
28
30
32
3
1
Re χ
18
20
22
24
26
28
30
32
22
24
26
28
30
32
0.9
0.8
2.5
0.7
2
arg χ
0.6
0.5
1.5
Im χ
0.4
1
0.3
0.2
0.5
0.1
0
18
20
22
24
26
28
frequency
30
32
0
18
20
frequency
36
Example: Frozen CO
37
Maxwell eqs in matter
Maxwell eqs ⇒
go to ω space:
⇒
⎛ 2 1 ∂2 ⎞
4π ∂ 2
⎜∇ − 2 2 ⎟ E = 2 2 P
c ∂t ⎠
c ∂t
⎝
P (ω ) = χ (ω ) E (ω )
⎡⎣∇ 2 + k 2 ( x , ω ) ⎤⎦ E ( x , ω ) = 0
n ( x, ω ) ω
k ( x, ω ) =
c
dispersion relation
medium homogeneous,
only one dimension z,
wave from z = −∞:
Helmholtz eq, describes
linear optics in isotropic space
n= ε
refractive index
ε = 1 + 4πχ
dielectric function
E ( z , ω ) = E ( 0, ω ) exp ⎡⎣ik (ω ) z ⎤⎦
38
Pulse propagation in homogenous medium
E ( z , t ) ∝ ∫ dω E ( 0, ω ) exp ⎡⎣ik (ω ) z − iωt ⎤⎦
n (ω ) ω
k (ω ) =
c
superposition of
plane harmonic waves
dispersion relation determines
pulse evolution
⎡ ( ω − ωc ) 2 ⎤
• assume Gaussian input: E ( 0, ω ) ∝ exp ⎢ −
⎥
2
Δω
⎢⎣
⎥⎦
• Taylor expansion:
k (ω ) ≈ kc + kc′ ⋅ (ω − ωc ) + 12 kc′′ ⋅ (ω − ωc )
2
39
Pulse propagation in homogenous medium
2
⎡
kc′ z − t ) ⎤
(
E ( z , t ) ∝ exp [ikc z − iωct ] exp ⎢ −
⎥
−2
⎢⎣ 4Δω − 2ikc′′z ⎥⎦
carrier wave
Gaussian envelope:
• group velocity 1/ kc′
• disperses if kc′′ ≠ 0
• chirp
2π
2Δω −2 − ikc′′z
conserves energy
medium
kc′′
pulse
in
pulse
out
kc′′′
40
Chromatic dispersion
kc′′z
⎡ 1
1 ⎤
−
output pulse duration ≈ ⎢
⎥ z ≈ kc′′z ⋅ (ω2 − ω1 )
⎣ v(ω2 ) v(ω1 ) ⎦
• Example: 10-fs-pulse as input
⇒ output pulse duration = 50THz 2000fs2 = 100fs
• application: chirped-pulse amplification
41
Dispersion compensation
Goal: longer path for red than blue
angular
dispersion
prism or
grating pairs
“chirped“
mirrors
local Bragg
reflections
42
More flexible: LCD pulse shaper
(C) T. Brixner, U Würzburg
Fourier plane with
amplitude and phase mask
Can create nearly arbitrary waveforms
43