nonlinear optics - ufox.cfel.de · nonlinear optics • interaction of light with materials with...
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Nonlinear optics IMPRS-UFAST core course
Giovanni Cirmi
10-14 December 2018
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Organizational notes
• CFEL/bldg. 99, SR O1.060
• 1- week course: 10-14 December 10:00-13:00
• break at 11:20-11:40
• I will show slides and refer to my notes
• I will propose few questions/exercises
2
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Suggested material
• My notes and material • Boyd, R. W., Nonlinear Optics • Shen, Y. R., The principles of nonlinear optics • Dmitriev, V. G., et al., Handbook of Nonlinear
Optical Crystals • Trebino, R., Frequency resolved optical
gating • Notes Franz X. Kärtner
https://ufox.cfel.de/sites/sites_cfelgroups/site_cfel-ufox/content/e16281/e16715/e16716/e16717/UFST_V1final_2015.pdf
• Material from past courses https://ufox.cfel.de/teaching/
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
4
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
5
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Nonlinear optics • interaction of light with
materials with nonlinear optical response
• photon energy (frequency, wavelength, color) may change
• nonlinear response of outer electrons in the atom potential
• Maxwell’s equations with nonlinear polarization vs. the electric field
P = ε0 χ 1 E
P = ε0[χ 1 E + χ 2 E2 + χ 3 E3 + ⋯ ]
linear optics:
nonlinear optics:
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Discovery of nonlinear optics • NLO started in 1961, with the discovery of SHG from a ruby laser
• not by chance, it started 1 year after the first laser was constructed
• but somebody at the journal did not get the point!
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The galloping horse
Muybridge's The Horse in Motion, 1878 8
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Femtoseconds and attoseconds
• 1 femtosecond (fs) = 10-15 seconds
• 1 attosecond (as) = 10-18 seconds
• age of the Universe: 13.8 billion years = 0.435x1018 seconds
• there are as many attoseconds in only 2.3 seconds, as many seconds in the whole history of the Universe
• fs and as pulses can be generated through nonlinear optics
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fs and as cameras
• How do electrons move?
• How does a laser work?
• How do chemical reactions occur?
• How do we see?
• How does photosynthesis occur?
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
11
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Maxwell’s equations
Ampere’s law
Faraday’s law
Gauss’s law
no magnetic monopoles
electric displacement
magnetic field
light speed in vacuum
12 see notes page 5-9
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Wave equation • calculate curl of Faraday’s law
• no free charges, no free currents, no magnetization
• neglect nonlinear polarization vs. linear
• plane waves
13
𝜕2E
𝜕z2 = 1
c2
𝜕2E
𝜕t2 + μ0𝜕2P
𝜕t2
in vacuum
in materials
solution in vacuum: E=E(z±ct). -> demonstration (notes page 9)
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Sinusoidal waves in vacuum
• in space: l: wavelength
k=2p/l : wavevector module
• in time: T: period
n=1/T : frequency
w=2pn : angular frequency
14
E = E0 cos (2p 𝑧
𝜆 - 2pnt+j) =
= E0 Re {exp [ i (2p 𝑧
𝜆 -2pnt+j) ] }
ln=c
E0: magnitude
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H. A. Haus, ”Fields and Waves in Optoelectronics”, Prentice Hall 1984
EM wave
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Linear media
16
𝜕2E
𝜕z2 = 1
c2 n2 𝜕2E
𝜕t2
E = E0 cos (2p n 𝑧
𝜆 - 2pnt+j) =
= E0 Re {exp [i (2p n 𝑧
𝜆 -2pnt+j) ] }
P = ε0 χ 1 E
n: refractive index (Sellmeier equations) wavelength: l/n frequency: n speed: c/n
linear polarization wave equation
plane wave solution
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EM spectrum
https://www.khanacademy.org/science/physics/light-waves/introduction-to-light-waves/a/light-and-the-electromagnetic-spectrum
Color Wavelength Frequency Photon
energy
Violet 380–450 nm 668–789 THz 2.75–3.26 eV
Blue 450–495 nm 606–668 THz 2.50–2.75 eV
Green 495–570 nm 526–606 THz 2.17–2.50 eV
Yellow 570–590 nm 508–526 THz 2.10–2.17 eV
Orange 590–620 nm 484–508 THz 2.00–2.10 eV
Red 620–750 nm 400–484 THz 1.65–2.00 eV
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Optical pulses
18
I=1
2ε0nc|𝐄|2
intensity:
t
carrier
phase dispersion: n=n(w) linear phase: transform limited pulse (short) nonlinear phase: chirped pulse (long)
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
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2nd order nonlinear optics noncentrosymmetric media
20
P = ε0[χ 1 E + 𝛘 𝟐 𝐄𝟐 + χ 3 E3 + ⋯ ] -> verify that |PNL|<<|PL| (notes page 21)
𝜕2E
𝜕z2 = 1
c2 n2 𝜕2E
𝜕t2 +χ(2)
c2
𝜕2𝐸2
𝜕t2
2nd order nonlinear wave equation
no analytic solution known guess: bichromatic field E=E1exp(iw1t)+E2exp(iw2t)
see notes page 19-22
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2nd order phenomena:
• second harmonic generation (SHG)
• optical parametric amplification (OPA)
• difference frequency generation (DFG)
• sum frequency generation (SFG)
• optical rectification (OR) 21
2nd order nonlinear optics generated new frequencies:
2w1, 2w2, w1+w2, w1-w2, 0 SHG
SFG
DFG
OR
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Second harmonic generation (SHG)
• start from bichromatic field (w2 = 2w1)
• substitute in nonlinear wave equation
• neglect frequencies different than w1 and w2
• use slowly varying envelope approximation (SVEA)
22
-> derivation: notes from page 24
𝑑E1
𝑑z= −i
ω1d
n1cE1
∗E2exp (−iΔkz)
𝑑E2
𝑑z= −i
ω1d
n2cE1
2exp (iΔkz)
coupled equations for SHG
1
2
Dk=k2-2k1 d =
1
2χ(2)
see notes page 24-27
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Second harmonic generation
• no pump depletion (E1=constant)
• integrate equation upon a distance z=L (crystal length)
• calculate intensity and efficiency
• maximum efficiency for Dk=0
23
2
η =I2
I1=
2ω12d2I1
n2n12c3ε0
L2sinc2ΔkL
2
see notes page 27-31
-> exercise: notes page 30. Required laser intensity for SHG
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Momentum and energy in SHG
• Dk=k2-2k1=0 is the phase matching condition
• momentum conservation between photons (not necessary but improves efficiency).
24
ħk2
ħk1 ħk1
SHG waves in phase matching
SHG waves out of phase matching
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• coherence length in SHG:
• photon energy is always conserved:
25
Momentum and energy in SHG -> exercise: notes page 30. Required laser intensity for SHG
ħw2
ħw1
ħw1
h
L
0 p 2p 3p Dk Dk Dk
Dk=0
Dk≠0 p
Dk Lc=
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Phase matching in SHG
• Dk=k2-2k1=(w2n2-2w1n1)/c=0, w2=2w1 => n2=n1 !!!
• not possible in standard materials with normal dispersion
• one method exploits birefringent materials
26
n
w w1 w2
n2
n1
n
w w1 w2
n2=n1
na nb
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Birefringent phase matching
• refractive indices depend on E-field direction (polarization)
• ellipsoid of the indices
• BBO (nE<nO):
27
nE
nE
nO
nO nO nO
x
y
z
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Birefringent phase matching
z: optical axis of BBO
k: propagation direction
k ⊥ EFF ⊥ ESH
28
nE, SH
nO,FF
nO,SH
x
y
z
EFF
ESH nE,SH(q)
q
k
q
nE,SH < nE,SH(q) < nO,SH
cos2(θ)
nO,SH2
+sin2(θ)
nE,SH2
=1
nE,SH2(θ)
FF: fundamental frequency SH: second harmonic
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Birefringent phase matching (BPM)
29
n
w w1 w2
n2=n1
q
-> exercise: notes page 39. Calculate phase matching angle for SHG in BBO
if nE,SH(q) = nO,FF then q is the phase matching angle
k
O. A. (optical axis z)
q
BBO crystals can be cut at
the desired angle q
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Quasi phase matching (QPM) • periodic modulation of χ(2) in the crystal
• efficiency grows after coherence length
30
p Dk
Lc=
η
L Lc 2Lc 3Lc
+ χ(2) - χ(2) + χ(2) - χ(2) + χ(2)
L L
phase matching condition
k2 − 2k1 −π
Λ= 0
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Optical parametric amplifiers (OPA)
• 3 photons: pump, signal, idler. Pump amplifies signal and generates idler (DFG)
• ħwp = ħws + ħwi
• phase matching: Dk=kp-ks-ki=0
31
ħwp
ħwi
ħws pump
seed
pump
signal
idler
input output
crystal
-> question: how many photons interact in SHG?
see notes page 44-48
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Coupled equations for OPAs
• follow same steps as for SHG
• SVEA approximation
32
𝑑Ei
𝑑z= −i
ωid
nicEs
∗Epexp (−iΔkz)
𝑑Es
𝑑z= −i
ωsd
nscEi
∗Epexp (−iΔkz)
1
2
𝑑Ep
𝑑z= −i
ωpd
npcEiEsexp (iΔkz) 3
G. Cerullo et al., Rev. Sci. Instr. 74,1 (2003)
see notes page 44-48
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Signal and idler in OPAs
• perfect phase matching (Δk=0)
• no pump depletion (Ep=constant)
33
Is L =1
4 Is0 exp (2ΓL)
Ii L =1
4 𝜔𝑖
𝜔𝑠 Is0 exp (2ΓL)
Γ2 =2ωiωsd2Ip
ninsnpε0c3
Ii L
𝜔𝑖=
Is L
𝜔𝑠
# photons idler = # photons signal G =
Is L
Is0=
1
4exp (2ΓL) gain:
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Collinear phase matching in OPAs
34
kp
ks ki
wpnp-wsns-wini=0
wp-ws-wi=0
momentum conservation (phase matching)
energy conservation
Dk=kp-ks-ki=0
see notes page 44-48
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Noncollinear phase matching in OPAs
35
• angle between pump and signal
• vectorial relationship Dk=kp-ks-ki=0
kp ks
ki
a
b
kp-kscos a-kicos b=0
Kssin a-kisin b=0
phase matching x-direction
energy conservation wp-ws-wi=0
phase matching y-direction
x
y
see notes page 48-52
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Ultrashort pulses with OPAs
• OPAs can produce few-cycle pulses in 2 cases
degenerate (collinear) OPA (DOPA): ws = wi = wp/2
lp = 400 nm, ls = li = 600-1000 nm in BBO
-> Question: what is the PM angle?
lp = 800 nm, ls = li = 1200-2000 nm in BBO
noncollinear OPA (NOPA)
lp = 400 nm, ls = 500-700 nm, a = 3.7o in BBO (visible NOPA)
• effects of crystal length:
increase gain
decrease bandwidth
36 see notes page 48-52
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Ultrashort pulses with OPAs
37
G. Cirmi et al., JOSA B 25, B62 (2008)
IR DOPA TL: 8.5 fs
NIR DOPA TL: 4.5 fs
VIS NOPA TL: 6 fs
see notes page 53-55
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3rd order nonlinear optics
• can occur in centrosymmetric media
– third harmonic generation (THG)
– intensity dependent refractive index
• self focusing (SF)
• self phase modulation (SPM)
38
3ħw
ħw
ħw
ħw
n=n0+n2I
white light generation (WLG)
see notes page 56-57
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Self focusing (SF) • higher intensity in the center => higher refractive
index => lens effect
• filament formation (SF vs. natural divergence ND)
39
z
z
SF SF ND ND
see notes page 58-59
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l (mm)
pump
SPM
0.8 0.5 2
Self phase modulation (SPM)
• n appears in the phase
• with n=n(I), new frequency components can be generated
40
E = E0 exp [i (2p n 𝑧
𝜆 -2pnt+j)]
see notes page 59-60
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White light generation (WLG)
SPM + SF: WLG
41
z
lens crystal sapphire/YAG
input beam
output beam
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Visible NOPA architecture
42
pump (input): 40 mJ, 400 nm, 70 fs signal: 5 mJ, 500-700 nm, 6 fs (after compressor)
BS: beam splitter DL: delay line SM: spherical mirror SAP: sapphire, 1-mm BBO 1: 1-mm, 29o BBO 2: 1-mm, 31o
Ti:sapphire laser 800 nm, 100 fs, 100 mJ, 1 kHz
BS
BBO 1 SHG
DL
99%
1% SAP WLG
BBO 2 OPA
lens
pump pump
SM
see notes page 61-63
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Pulse measurement
• how to measure the shortest possible events?
• with the same event -> autocorrelation
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DL, t
det SHG
tAC
tpulse
tAC~1.5 tpulse
see notes page 70-73
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
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Applications of fs pulses Pump-probe technique
• follow fs dynamics (physics, chemistry, biology, medicine)
• pump excites sample
• probe tests it
• pump-probe delay t is controlled
• probe transmission or reflection is measured vs. t
• dynamics can be followed
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sample t notes page 63-68
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Pump-probe setup
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Ti:sapphire laser 800 nm, 100 fs, 200 mJ, 1 kHz
BS
DL, t
OPA 1
OPA 2
MC
det
BS: beam splitter DL: delay line MC: mechanical chopper det: detection
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Pump-probe detection • Differential probe transmission (pump on - pump off)
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S =ΔT
T=
TP ON − TP OFF
TP OFF
detection scheme 1 (“old style”)
band pass filter
photo diode
oscilloscope, PC probe
t
S l1, probe
l2, probe
detection scheme 2 (“new style”)
spectrometer
PC probe
t
lprobe
S
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Pump-probe signals
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l S=
ΔT
T
PA
SE/PB
photoinduced absorption (PA)
S<0
stimulated emission (SE)
S>0
x
photo- bleaching (PB)
S>0
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Application of ultrafast fs spectroscopy: human vision
Image from: Eye clarity
Image from: http://www.chm.bris.ac.uk/motm/retinal/retinalh.htm
Cones: day vision Rods: night vision
Our retina contains photoreceptors called cones and rods
Image from: https://www.sciencenews.org/article/how-rewire-eye
light
zoom: Retina
Cones and rods are composed of disks containing rhodopsin When light arrives, rhodopsin changes structure (cis-trans) How fast is this process?
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The primary step of vision
Schoenlein R. W. et al., “The first step in vision: femtosecond isomerization of rhodopsin”, Science 254, 412 (1991) DOI: 10.1126/science.1925597
PA
PB
PA
pu
mp
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The primary step of vision
6 ps
PA
PB
PA
400 fs
PA
PB
PA
pu
mp
Schoenlein R. W. et al., Science 254, 412 (1991)
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t
Carrier envelope phase (CEP) • important for few-cycle pulses, for phenomena like HHG
• can be controlled in different ways
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CEP=0 CEP=p/2 CEP=p
see notes page 73-74
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
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High harmonic generation (HHG) • HHG can occur when focusing a fs pulse in gas (~1014 W/cm2)
• radiation goes to UV and X-rays (up to keV)
• needs vacuum
• attosecond pulses possible
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low orders plateau
cut-off
1 (FF)
3 5 7 9 11 13 15 107 105 103 101 99 … HH order
gas target
fs pulse (IR/VIS)
HHG as pulse (?) (XUV)
vacuum chamber
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High harmonic generation
• cut-off energy: ECO = Ip + 3.17 Up
• ponderomotive energy:
• scaling laws:
– cut-off ∝ wD-2
– efficiency ∝ wD5
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Up ∝ID
ωD 2
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3 step model
• semiclassical model of HHG
• E-field modifies atomic potential of electron
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1. ionization (tunnel) 2. acceleration 3. recombination
XUV emission
see notes page 80-83
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Attosecond pulses
• A broadband (octave) driver with ~1 cycle duration generates a continuum HHG spectrum, instead of peaks
• In time, this can give attosecond pulses
• attosecond science 57
plateau
1 (FF)
3 5 7 9 11 13 15 107 105 103 101 99 … HH order
time (as)
see notes page 83-86
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Contents
• Introduction and motivation
• Maxwell’s equations
• Nonlinear optics of 2nd and 3rd order
• Pump-probe technique
• High harmonic generation
• Pulse synthesis
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Pulse synthesis • new frontier of NLO
• synthesize more pulses together
59 Mücke O. D. et al., JSTQE 21, 8700712 (201´5) see notes page 86-89
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Pulse synthesis
• Long driver l -> high HHG cut-off energy
• Short driver l -> high HHG efficiency
• Efficient HHG at high energies?
• Isolated attosecond pulses
• Pump-probe fs-fs or as-fs or as-as
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THANKS!