Nonlinear Optics: Phenomena, Materials and Devices

- Honors senior undergraduate and graduate level course.- Approximately 24-26 lecture hours + 3 seminars.- Lectures 2-4:15 Saturday, Sunday, Tuesday and Wednesday- Designed to provide a working knowledge of Nonlinear Optics.- Requires an understanding of Maxwell’s equations and their solutions- Understanding of quantum mechanics would be useful, The quantum concepts will be introduced when needed.- Lecture notes and schedule available on website.- Textbook “Nonlinear Optics: Phenomena, Materials and Devices”, authors George and Robert Stegeman, will be published in early 2012 by J. Wiley and Sons.

Lecture 1 Introduction and Linear Susceptibility - optical polarization expansions, orders of magnitude of nonlinearities, linear susceptibility, local field effects

Lecture 2 Second Order Susceptibility – coupled wave theory – nonlinear polarization and interactionsLecture 3 Second Harmonic Fields - coupled wave equations – wave-vector matching – solutionsLecture 4 Practical Second Harmonic Generation - optimization – beams –QPMLecture 5 Tunable Frequencies – Optical parametric amplifiers and oscillators – applications – materialsLecture 6 Quantum Theory of Susceptibilities – 1st, 2nd and 3rd order susceptibilities

Lecture 7 Nonlinear Index and Absorption – simple two & three level models – frequency dispersion – examplesLecture 8 Third Order Nonlinearities Due to Electronic Transitions: Materials – molecules - glasses – semiconductorsLecture 9 Miscellaneous (Slower) Third Order Nonlinearities: Materials -vibrational, electrostrictive, liquid crystal, electrostrictive, cascading effectsLecture 10 Ramifications and Applications of Nonlinear Refraction - self-focusing and defocusing - instabilities - solitons - bistability - all-optical switchingLecture 11 Multi-Wave Mixing - degenerate and non-degenerate four wave mixing – three wave mixing - nonlinear spectroscopyLecture 12 Stimulated Scattering - stimulated Raman and Brillouin scatteringLecture 13 Extreme Nonlinear Optics - Ultra-Fast and Ultra-High IntensityLecture 14 “Overflow”- Terminology- Fields written as

- Superscript “roof” or “hat” (example ) emphasizes a complex quantity- The unit vector is written as and has components where i=x, y, z. - The “Einstein” notation is used for summations over repeated indices. For example, - Quantities with a “bar” on top, e.g. refers to individual properties of isolated molecules in a single molecule’s frame of reference.- SI (mks) units are used throughout

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1..)e(E

2

1),( ][ E

aie

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Examples of Nonlinear Optics

Examples of behavior associated with nonlinear optics

Soliton generation and modulation instability

Increasing input intensity

Harmonic generation Intensity dependent transmission

Nonlinear Interferometry

Classical Expansion of the Nonlinear Polarization

with ...]),(),(),(

),,;,,(ˆ

),(),( ),;,(ˆ

),();(ˆ[),(

(3)

)2(

)1(0

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tdtdrdrdt""rEt''rEt"tt't"rr'rr

tdrdtrEttrrtrP

kj

t t tijk

kjt t

ijk

tjiji

fieldsinput lity with susceptibiorder th "" - ˆ )(.. nnn

i

tdtdrdrdt""rEt''rEt"tt't"rr'rrtrP kjt t

ijki

),(),( ),;,(ˆ),( e.g. )2(

0)2(

The second order polarization is created at time t and position by two separateinteractions of the total EM field with the medium at time and position , and at time and position in a material in which (2)0. This form includes nonlocality in space and time.

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r

t' 'r

t""r

),()2( trPi

r

'r "r

),( t''rE j

),( t""rEk

Frequency Spectrum Due to Non-Instantaneous Response

e.g. 2 level system, excited state with lifetime ; excited with pulse t<<

After passage of pulseBefore incidence of pulse

Evolution of frequency spectrum with time

Polarization Expansion: Nonlinearity Localized in Space

),(E)(E .;.)e(E2

1),( .;.),(

2

1),( *ti

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mj

m

ωm

mii

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.]),(E),(E),(E),, ;(ˆ4

1

),(E),(E) ;(ˆ2

1

),(E);(ˆ[),(P

)((3)

)(,

)2(

)1(0

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qpm

tip

pkm

mjpmijk

pm

ti

mm

mjmij

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pm

m

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ezez

Summation over all of the input frequencies present

qpm

),(E),(E) ;(ˆ),(E),(E) ;(ˆ4

1

sfrequenciefor ),(E),(E) ;(ˆ4

1 e.g.

1)1(

2)2(

1,2)2(

2)2(

1)1(

2,1)2(

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,)2(

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kjijkkjijk

tip

pkm

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pm

pm

Note that all possible combinations of 1+2 are needed!

Magnitudes of Nonlinear Susceptibilities

- Depends on the number of electrons in an atom (or molecule), distribution of energy levels etc.- Consider hydrogen atom with one electron

will give a minimum valueanalytical solutions known for energy, levels etc.

The atomic Coulomb field binding the electron to the proton in its lowest orbit is

mVr

eE /10

412

2B0

atomic

Assume that the perturbation expansion for the nonlinear polarization is valid up to E 0.1Eatomic

.m/V10 101

P

P 13)2(

atomic)2()2(

)1(

(2)

(1)

EE

etc. /m10 10P

P 2225)3(2)3(

)1(

2)3(

)1(

(3)

(1)

VEE atomic

Simple Model: Electron on a Spring (1)

2 ways to calculate the nonlinear susceptibilities

Electron as an anharmonic oscillator –resonance frequencies are given by energydifference between ground and excited states simple but approximate

Quantum mechanics – electric dipole transitions between atomic (molecular )energy levels

exact but complicated

Electron on a Spring Model

For 3D, need 3 springs with the spring displacements . The electron motionis that of a simple harmonic oscillator along these directions. The directions are chosenparallel to the axes which diagonalize the molecular polarizability tensor and hence . This does not imply that the higher order susceptibilities are diagonalized along these axes!

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)1(ijij

.2

1)( )()()()( m

im

im

iim qqkqV

)(2)()()(

)()()(

)(

)()(

)(

)()( ][

2

1

:Mechanics Classical

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mi

miim

i

mim

im

imi

mim

iimi

mm

i qmqkq

qqq

q

qk

q

VF

Field Induced Electron Displacement

Simple Harmonic Oscillator Potential

em

iimg mk )(The spring constant is given in terms ofthe resonance frequencies by .

)(miik

The decay of the spring motion (lifetime of excited state) means that there is damping of theSHO and the driven SHO equation, with the electron charge, is given by

2 )(2)(1)(i

mimg

mimge

mie Eeqqmqm

Inertial force Spring restoring force Electromagnetic driving force

)(

)( ..

2

1 Assuming

)()()()(

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ie

aimi

timi

mi

Dm

eQcceQq a

212212)( 22)(22

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Linear Susceptibility

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1);(ˆ

)();(ˆ)(

1)()(

..)(2

1..

2

1)(

)(0

2)1(

)1(0)(

2

)(

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mi

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ai

tiai

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ccecceQeNtP aa

.4)[(

2)( );(ˆ

componentsmaginary and eal into gDecomposin

22222

122

0

2)1(

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m

eN

m mgamg

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m

eN

])[(2 );(ˆ:For

22

1

0

2)1(

Local Field Effects (1)

Maxwell’s equations in the material and the usual boundary conditions at the interface arevalid for spatial averages of the fields over volume elements small on the scale of a wavelength,but large on the scale of a molecule. The “averaged” quantities also include the refractive index,the Poynting vector, and the so-called Maxwell field which has been written here as . Itis the Maxwell field that satisfies the wave equation for a material with averaged refractiveindex n. At the site of a molecule the situation can be quite complex since the dipoles induced by theMaxwell electric fields on all the molecules create their own electric fields which must beadded to the “averaged” field to get the total (“local”) field acting on a molecule.It is very difficult to calculate the “local” field accurately because it depends on crystalsymmetry, intermolecular interactions, etc. Standard treatments like Lorenz-Lorenz are onlyapproximately valid even for isotropic and cubic crystal media.

),( trE

),(loc trE

Local Field Effects (2)

Dipole moments of the molecules induced by the Maxwell field produce a Maxwellpolarization in the material. Consider a spherical cavity around the molecule of interest tofind the local field acting on the molecule. Assuming that the effects of the induced dipolesinside the cavity average to zero, the polarization field outside the cavity induces chargeson the walls of the cavity which produce an additional electric field on the molecule in thecavity.

).,(3

1),(),( ),(

3

1),( )1(

0loc

0dipoles trPtrEtrEtrPtrE

. ),(),(] 3

11[

)],(3

1),([),(

)],(3

1),([),(

)1(

)1(

0

)1(

)1(

0

)1(

trEαNtrPαN

trPtrEαNtrP

trPtrEαtrp

),,(3

2),(

3

1

2

1 :Mossotti Clausius

)1( trENtrP

N

r

r

r

,3

2 );,(

3

2)(),( )1(

loc

rar ftrEtrE

++++

+

+

+

+

+

+

+

- -- ---- ----

Linear Susceptibility with Local Field Effects

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2)()( );(ˆ

22222

122

0

)1(2)1(

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