a taste of extreme nonlinear optics
DESCRIPTION
A Taste of Extreme Nonlinear Optics. George Stegeman, KFUPM Chair Professor., College of Engineering King Fahd Un. of Petroleum and Minerals, Saudi Arabia Professor Emeritus College of Optics and Photonics/CREOL University of Central Florida, USA. - PowerPoint PPT PresentationTRANSCRIPT
A Taste of Extreme Nonlinear Optics
There are two areas of ultrafast nonlinear optics under intense investigation because the phenomena observed cannot be explained using “classical nonlinear optics”. This has led to the birth of a field called “extreme nonlinear optics. The key processes are:1.the electromagnetic field-induced ionization of electrons, 2.their subsequent motion under the influence of the field and 3.their recombination with ions.
George Stegeman,KFUPM Chair Professor., College of Engineering
King Fahd Un. of Petroleum and Minerals, Saudi Arabia
Professor EmeritusCollege of Optics and Photonics/CREOL
University of Central Florida, USA
Two excellent reviews: T. Brabec and F. Kraus, “Intense Few –Cycle Laser Pulses: Frontiers of Nonlinear Optics”,Rev. Mod. Phys., 72, 545 (2000) F. Kraus and M. Ivanov, “Attosecond Physics”, Rev. Mod. Phys.81, 163 (2009)
“First” High Harmonic Generation Paper
1. Only odd harmonics.2. Plateau regions.3. Requires terra-watt petta-watt intensities
R. L.Carman, C. K. Rhodes and R. F. Benjamin, “Observation of harmonics in the visible andultraviolet created in CO2-laser-produced plasmas”, Phys. Rev. A, 24, 2649-2663 (1981).
240nm
CO2 laser (=10.6m , 10,600nm)
Carbon wire
Electron plasma
Intensity at focus 2.3x1015W/cm2
Authors correctly identified the sourceof the effect as the interaction of the radiation with free electrons.
“Sub-100nm” High Harmonic Generation
1. Only odd harmonics.2. Plateau regions.3. Requires terra-watt petta-watt intensities
Ar, Xe, Kr, Ne
More experiments followed with other noblegases like Xe, Kr etc., many in capillaries
Z. Chang, A. Rundquist, H. Wang, M. M. Murnane andH. C. Kapteyn “Generation of Coherent Soft X Raysat 2.7 nm Using High Harmonics (Ti:SAF laser)”Phys. Rev. Lett., 79, 2967 (1997).
Nd:YAG =1.06m
M. Ferray, A. L'Huillier, X. F, Li, L. A. Lompré, G, Mainfray and C. Manus, Let. to Editor.“Multiple-harmonic conversion of 1064 nm radiation in rare gases”, J. Phys. B: Atomic & Molecular Optics, 21, L31-L35, (1988).
Ar gas jet
Deals with bound electrons in atoms or molecules.The induced polarization which radiates is written as a perturbation expansion in terms ofincreasing powers of electric field
...{ 7)7(6)6(5)5(4)4(3)3(2)2()1(0 EEEEEEEPtotal
Nonlinear Optics
“Classical” Nonlinear Optics
Since the noble gases (He, Ar, Xe etc.) gases are centro-symmetric atoms, when excitedby a single beam:
...{ 7)7(5)5(3)3()1(0 EEEEPtotal
The range of validity of this expansion is that ,i.e. the series converges
....6)7(4)5(2)3()1( etcEEE
The usual criterion used is that the optical field should be a small fraction of the atomicfield binding the electrons to atoms.
The hydrogen atom’s energy levels etc. can be calculated so they are used as an approximatescaling factor.
Hydrogen Atom Fields
n=1 (ground state)
Atomic field Eat binding electron to proton (ionization field)
in hydrogen atom:
)watts/cm-peta (10
/10 ofintensity anfor EE
Volts/cm107.5Volts/m107.5 E
0.053nm 4
;4
E
2
216opticalatomic
911atomic
2
20
B2B0
atomic
cmW
emr
r
e
e
Optical FieldEoptical=0 atomic potential well
Eoptical>Eatomic
n=1 (ground state)Response (displacements) of free electronsto applied optical fields is much greater thanfor bound electrons, typically a small fractionof rB!
Regimes of Nonlinear Optics
Perturbative Regime Strong Field Regime
Extreme Nonlinear Optics
Bound Electrons Free Electrons
1012 W/cm2 1014 W/cm2 1016 W/cm2 1018 W/cm2
: 11
E=0 atomic potential well
E>0 atomic potential well
Optical Field
Tunneling: 1
Optical Field
1 :ssionDirect Emi
Optical Field
Electron Ionization Mechanisms
field goscillatin an in electron ofenergy
kinetic average time2
2
: ParameterKeldysh
20
2
cm
IeK
K
I
e
p
Multi-Photon Absorption: 1
Electron-Radiation Interaction
..2
1)( (t))(
..2
1)(:field Optical
2max
max0
0
e
L
tie
tiL
m
eEx
ccextxeEtxm
cceEtE
)(tx
nmr 053.0B
nmxInmxI 4.12 W/cm10 24.1 W/cm10 max215
max213
Electron is “free” since it spends most of its trajectory far from the ion (and its Coulomb fieldwhich decreases as 1/x2)
2
222
42
1 :energy"twitter " cycle 1over energy kinetic averaged Time
eL
em
EexmK
+
“a” (=1350) and “c” (=930) correspond to the same final energy. “b” (=1070) is a cut-off trajectory with the highest kinetic energy (3.17Ip).
“d” (=900) starts at the peak of the electric field where most electrons are produced but returns to the core with zero kinetic energy. “e” (=450) never returns to its parent ion.
Electrons emitted ation’s location (0,0).
Motion of Free Electrons in Electromagnetic Field
(electric field phase at the instant of ionization)
0 2
This recombination process leads to an ultrafast nonlinearity, time scale of an optical period.
The electrons which do not recombine with ions inside one cycle recombine on longertime scales.
1.During optical pulse, the electron excursion is still large and recombination occurs on a time scale which depends on the free electron density. Femtoseconds?
2.After the optical pulse, carriers diffuse and time scales of a 100ps – ns have been measured.
“Three Step” Process
M. Lewenstein, Ph. Balcou, M. Yu.Ivanov, A. L’Huillier and P. B. Corkum, “Theory of High-harmonic Generationby Low Frequency Laser Fields”, Phys.Rev. A, 49, 2117-2132, (1994).
1. Ionized electrons accelerated by strong optical fields.
2. Electrons gain energy from EM field, K – “jitter” energy”
3. Energy released when the trajectory brings electron back to ion and electron recombines with ion.
4. Maximum harmonic generated “mmax ”
5. This process repeats every half cycle of
the laser field, i.e. Tperiod/2.. Fourier
transform of capture event (and emission of radiation) m=1, 3, 5 etc.
6. Plateaus predicted.
ipIKm 17.30max
Overall good agreementwith experiment!
Electron-Ion Recombination
tieru g)(gg
rate ionization - )(
)(ψ
0
) ,(0c
00
tw
etw
i
ttKitii
cψ inpresent frequency a dt
d
)ψReal(ψ cg
2cg |ψψ| e
),( tpNL ),/( tpNL
0max
0 ... 3 1,
cyclelaser incident per
twiceoccurs ),(
Km
mm
tpNL
High Harmonic Generation From Few Cycle Pulses
Optimization Techniques
Goals
(1)
(2) (3)
Adaptive Control
C. Winterfeldt, Ch. Spielmann and G. Gerber, “Optimal control of high-harmonic generation”,Rev. Mod. Phys., 80, (1980)
Control of: 1. Pulse shape 2. Pulse Duration 3. Frequency chirp 4. Interaction geometry
Optimization of Specific Orders
Suppression of Specific Orders
Phase-Matching of Harmonics to Fundamental
20
2
0
2
0
0
2
21)1)](()([
:harmonic th'For
ak
mu
P
P
krN
m
mnmn
P
Pmk
kmkk
m
pq
atmeatm
atm
m
Dispersion of non-ionized gas Plasma dispersion
Waveguide dispersiona – inner capillary radiusupq – p’th root of (q-1)th
Bessel function
radius)electron (classical 4 2
0
2
cm
er
ee
Ion contribution neglected since freeelectron contribution is much largerand number densities the same
Impossible to wave-vector match over largerange of harmonics!
z
I(q)
k2 > k1mkmkk 0
Quasi-Phase Matching (QPM)
In the flat regions, either zero nonlinearityor phase-mismatch large enough so that no net increase occurs.
PPLN: Sign of the nonlinearity is periodicallyreversed.
QPM: Grating or periodically reversed nonlinearity
Grating Induced By Counter-Propagating Beam
4541373329 4541373329
3 counter pulses1 counter pulse
Harmonic Order Harmonic Order
Enh
ance
men
t Fac
tor
1
10
102
Inte
nsit
y (a
rb. u
nits
)
1
10
102
103
3 counter pulses1 counter pulse
Ar
Periodically Modulated Capillary Radius
Modulation depth 5-10%
150 m diameter
1.0 mm
Straight fiber
Helium
2 4 6 8 10Wavelength (nm)
Sign
al (
arb.
Uni
ts) Modulated
Hollow fiber
Nonlinear Birefringence in Gases at Ultra-High Intensities
z
y
x
Pump BeamProbe Beam
40
Air molecules
N2
N2
N2
N2+
N2N2
N2
N2N2
N2
O2O2
O2
O2 --
There have been some recent experiments in which the nonlinear birefringence of air and its major constituents have been measured at very high intensities with 90fs pulses . The interpretation is in dispute. The laser wavelengths was 800nm, non-resonant regime.
x
y
E
pE
450
Ipump>>Iprobe
O2 and N2 are linear molecules are randomly oriented and at atmospheric pressure and
temperature. The gas can be considered isotropic over a wavelength.Although strong ionization occured at their laser intensities due to multi-photon absorption, theyneglect the contribution to the ionized electrons since they are “isotropic”.
1 ,24
,843
,632
,42,21
,,
...
air ofeffect Kerr the todue as dinterprete and measured )( -)(
mm
pmmpxpxpxpxNLbir
NLpy
NLpx
NLbir
InAInAInAInAInAn
InInn
“Higher Order Kerr” (Optics Express, 17, 13429, 2009)
Values for pump-probe Kerr coefficients obtained by fitting to experimental data.
Experimental Results
Table from calculated 1 ,2,
mm
pmxNL
px Inn
Non
line
ar I
ndex
Cha
nge
Intensity (Terrawatt/cm2) I is a CW intensity.
Is the Interpretation Physical?
Linear refractive index of airis 300x10-6
This represents a 10% changein magnitude and a change in sign!
Usual perturbation expansionof NLO cannot work! But, the Keldysh parameter is about 0.1so that multi-photon absorptiondominates the ionization process.
Perturbative Regime Strong Field Regime
Extreme Nonlinear Optics
Bound Electrons Free Electrons
1012 W/cm2 1014 W/cm2 1016 W/cm2 1018 W/cm2
: 11
Higher Order Kerr Experiments
Electron-Ion Recombination: Effect on Laser Field
nmxcmWxNexeNP eeNL
x 6.4 /5x10 :At max214
00 Electron motion drivenby optical field
Appearance of ionized electrons just
outside (x0) the atom/molecules
Experiments in strong field regime have shown a blue shift in 0 with intensity.
0)(
v)(.}.{2
1
.}.{2
1),(
,2
phase,2])],([
])([
00
0,20
eeff
vaceefftINiznik
L
tizINnnikLx
Nn
IkNncceE
cceEztE
evac
eeffvac
)( -)()( ,,, InInIn NLprobey
NLprobex
NLprobebir
1. Ion recombination process is ultrafast
2. During optical pulse, recombination occurs on a relatively fast time scale. Femtoseconds?
3. After the optical pulse, carriers diffuse and time scales of a 100ps – ns have been measured.
Free Electron-Based Alternate Explanations
Third harmonic generation has been measured from an electron plasma, and the equivalent quantified by Suntsov et. al., Phys. Rev. A, 81, 033817(2010).
),,;3()3( xxxx
Contributions from electron plasmas was dismissed by the authors of the paper becauseit is isotropic. However a collective electron plasma should behave like a classical thirdorder nonlinear material and exhibit birefringence, intense optical fields, for example as
just discussed previously., via an effective third order nonlinearity n2eff . At the pulse
peak intensity of 50x1012W/cm2, plasma0.10 with .Can this electron plasma nonlinearity explain the nonlinear birefringence experiment?
Pump creates plasma string.Filament produces third harmonicdue to air + plasma nonlinearity.
n)(saturatio moleculeir electron/a 1
/2x10 ,TW/cm 20)(50at
/10)2.18.1(|~),,;3(|
:al. Suntsovet.by ion Extrapolat
3192
2223)3(,
cm
Vm
e
xxxxpl
022 / eeplasma mNe
),,;3(]9[][)(
)(),,;3(4
1
]9[][
)(E
4
)()3(P
0000)3(2
032
04
40
03
0000)3(
020
320
03
4
4
0
xxxxe
exxxx
xxxxxxxxxx
e
eNLx
INe
mk
k
m
INe
Plasma Nonlinearity
..)(Q)(Q)(Q8
1)3( 03
0)1(
0)1(
0)1(
0)3( cce
m
kq ti
xxxe
xxxxx
Assume 1. collective electron effects such as plasma oscillation etc. exist, i.e. acts like a classical isotropic medium, i.e. 2. nonlinear response of electrons is just that of a single electron (in an intense field)
X electron density (Ne) . 3. ultrafast response relative to pulse width of femtosecond lasers, i.e. 90 fs
pxNLbir nn ,20 3
2)(
kxxxx <0 – nonlinear response
20
00
)1(0
)1(0
)1(00
)1( )()(Q..)(Q
2
1)( )()(:field pump 0
e
xx
tixxx
ex
m
eEcceqE
m
eq
..)3(Q2
1)3( :field pump todue response THG 03
0)3(
0)3( cceq ti
xx
Need to estimate response at 0 from measured third order nonlinearity
..)(2
1)( :beam) (pump toduent displaceme Electron 0
0)1(
0)1( cceQqE ti
xxx
..)(E|)(E|}][
{8
6)(
:)( beam Probeon beam Pumpofeffect Nonlinear
00
20
320
0)3(
0)3(
,
ccem
e
m
kq
q
tixpx
ee
xxxxxp
px
.|)(E|)(E),,;(4
6
)(E][
)(E)(E
4
6)()( P
2000
)3(0
420
*0
4
4
0)3(
0
xpxpppxxxx
pxpxxxxxxe
xpeNLxp
k
m
NeQeN
),,;3(]9[][ :t ExperimenTHG From 0000)3(2
032
4
40
xxxxe
exxxx
Ne
mk
9|),,;3(
),,;(|
0000)3(
00)3(
xxxx
ppxxxx
Nonlinear Birefringence Experiment
xe
xxe
xm
eQtrE
m
eq E
][)( ),()( : BeamPump
20
0)1(
0)1(
Wcmnn
n
pxpx
px
/10)4.00.3();( |);(|
)TW/cm 2050(at /W cm 10)6-(9);(
2190air,20plas,2
22-160plas,2
]);();([3
2
,neglected) n(saturatio/W cm x10]10)2050(
)[6-(9=);(-
0air,20plas,2
216-8120plas,2
InInn
In
pxpxNLbir
px
THG Prediction of Electron Plasma n2
)!/cm10 glass,silica fused nlarger tha 2X( /109
|),,;3(|4
54),,;(
4
6);(
322glass
216
0000)3(
000
)3(
00plas,2
NWcmx
ccn xxxxppxxxxpx
n)(saturatio moleculeir electron/a 1/2x10 ,TW/cm 20)(50at
/10)2.18.1(|~),,;3(| :al. Suntsovet.by ionExtrapolat3192
22230000
)3(
cmN
Vm
e
xxxx
Assume: kxxxx is negative!! (other evidence suggests this.)
Experiment VS CW Theory
Theory is essentially “plane wave” but does take into account nonlinear beam narrowing. It doesnot include: 1. plasma saturation and the 2. evolution of the plasma density over the temporal pulse3. ultrafast nonlinearity which was verified experimentally
Because of these factors, we definitely over-estimate the mature plasma contribution. However, it appears that the shape of the curves is due to electron density, i.e. the photo-ionization process.
Conclusions
1. There has been a paradigm shift in our understanding of the interaction of matter with intense optical fields when the optical field becomes comparable to the fields inside atoms and molecules. The ionized electrons can dominate the nonlinearity!2. Coupled with shorter and shorter pulse lasers, dramatic new science and technology will emerge.3. Characterization (not discussed) and creation of ultrashort pulses depend s on nonlinear optics!