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

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xxxxxxxxxx

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mk

k

m

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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)(

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00

20

320

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xpxpppxxxx

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k

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xxxxe

exxxx

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mk

9|),,;3(

),,;(|

0000)3(

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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

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pxpxNLbir

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THG Prediction of Electron Plasma n2

)!/cm10 glass,silica fused nlarger tha 2X( /109

|),,;3(|4

54),,;(

4

6);(

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0000)3(

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)3(

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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!