nlo1

Introduction to Nonlinear OpticsIntroduction to Nonlinear Optics

H. R. KhalesifardH. R. KhalesifardInstitute for Advanced Studies in Institute for Advanced Studies in

Basic SciencesBasic SciencesEmail: [email protected]: [email protected]

ContentsContents

1.1. IntroductionIntroduction

2.2. The essence of nonlinear opticsThe essence of nonlinear optics

3.3. Second order nonlinear phenomenaSecond order nonlinear phenomena

4.4. Third order nonlinear phenomenaThird order nonlinear phenomena

5.5. Nonlinear optical materialsNonlinear optical materials

6.6. Applications of nonlinear opticsApplications of nonlinear optics

IntroductionIntroduction

Question:Question:

Is it possible to change Is it possible to change the color of a the color of a monochromatic light?monochromatic light?

Answer:Answer:

Not without a laser lightNot without a laser light

output

NL

O s

am

ple

input

Stimulated emission, The Stimulated emission, The MASER and The LASERMASER and The LASER

(1916) The concept of stimulated emission (1916) The concept of stimulated emission Albert EinsteinAlbert Einstein

(1928) Observation of negative absorption or (1928) Observation of negative absorption or stimulated emission near to resonant stimulated emission near to resonant wavelengths, wavelengths, Rudolf Walther LadenburgRudolf Walther Ladenburg

(1930) There is no need for a physical system to (1930) There is no need for a physical system to always be in thermal equilibrium, always be in thermal equilibrium, Artur L. Artur L. SchawlowSchawlow

h

E1

E2

AbsorptionE1

E2h

Spontaneous Emission

E1

E2hh h

Stimulated Emission

LASER(MASER)

Light (Microwave) Amplification by

Stimulated Emission of Radiation

The MaserThe Maser

Two groups were working on Maser in 50s

Alexander M. Prokhorov and Nikolai G. Bassov (Lebedev institute of Moscow)

Charles H. Townes, James P. Gordon and Herbert J. Zeiger (Colombia University)

Left to right: Prokhorov, Townes and Basov at the Lebede institute (1964 Nobel prize in Physics for (1964 Nobel prize in Physics for developing the “Maser-Laser principle”) developing the “Maser-Laser principle”)

Townes (left) and Gordon (right) and the ammonia maser they had built at Colombia University

The LASERThe LASER

(1951) (1951) V. A. FabrikantV. A. Fabrikant ““A method for the application of A method for the application of electromagnetic radiation (ultraviolet, visible, infrared, and electromagnetic radiation (ultraviolet, visible, infrared, and radio waves)radio waves)” patented in Soviet Union.” patented in Soviet Union.

(1958) (1958) Townes Townes andand Arthur L. Schawlow Arthur L. Schawlow, “, “Infrared and Infrared and Optical Masers,Optical Masers,” Physical Review” Physical Review

(1958) (1958) Gordon GouldGordon Gould definition of “ definition of “LaserLaser” as “” as “Light Light Amplification by Stimulated Emission of RadiationAmplification by Stimulated Emission of Radiation””

(1960) (1960) Schawlow Schawlow andand Townes Townes U. S. Patent No. 2,929,922 U. S. Patent No. 2,929,922

(1960) (1960) Theodore MaimanTheodore Maiman Invention of the first Invention of the first Ruby LaserRuby Laser (1960) (1960) Ali JavanAli Javan The first The first He-Ne LaserHe-Ne Laser

Maiman Maiman and the and the first ruby first ruby laserlaser

Ali Javan and Ali Javan and the first He-the first He-Ne LaserNe Laser

Properties of Laser BeamProperties of Laser Beam

A laser beam A laser beam Is intenseIs intense Is CoherentIs Coherent Has a very low divergenceHas a very low divergence Can be compressed in time up to few Can be compressed in time up to few

femto second femto second

Applications of Laser Applications of Laser

(1960s) (1960s) “A solution looking for a problem”“A solution looking for a problem”

(Present time) (Present time) Medicine, Research, Medicine, Research, Supermarkets, Entertainment, Industry, Military, Supermarkets, Entertainment, Industry, Military, Communication, Art, Information technology, …Communication, Art, Information technology, …

Start of Nonlinear OpticsStart of Nonlinear Optics

Nonlinear optics Nonlinear optics started by the started by the discovery of Second discovery of Second Harmonic Harmonic generation shortly generation shortly after demonstration after demonstration of the first laserof the first laser..

((Peter FrankenPeter Franken et al et al 19611961))

2. The Essence of Nonlinear 2. The Essence of Nonlinear Optics Optics

When the intensity When the intensity of the incident of the incident light to a material light to a material system increases system increases the response of the response of medium is no medium is no longer linearlonger linear

Input intensity

Output

Response of an optical Response of an optical MediumMedium

The response of The response of an optical an optical medium to the medium to the incident electro incident electro magnetic field is magnetic field is the induced the induced dipole moments dipole moments inside the inside the mediummedium

h

hh

h

Nonlinear SusceptibilityNonlinear Susceptibility

The general form of polarization The general form of polarization

lkj)(

ijklkj)(

ijkj)(

ijii EEEχEEχEχPP 3210 lkj)(

ijklkj)(

ijkj)(

ijii EEEχEEχEχPP 3210

Dipole moment per unit volume or polarization

jijii EPP 0 jijii EPP 0

Nonlinear PolarizationNonlinear Polarization

Permanent Permanent PolarizationPolarization

First order First order polarization:polarization:

Second order Second order PolarizationPolarization

Third Order Third Order PolarizationPolarization

jiji EP )1(1 jiji EP )1(1

kjijki EEP )2(2 kjijki EEP )2(2

lkjijkli EEEP )3(3 lkjijkli EEEP )3(3

How does optical nonlinearity How does optical nonlinearity appear appear

The strength of the The strength of the electric field of the electric field of the light wave should be light wave should be in the range of atomic in the range of atomic fieldsfields

N

a0

e

h20/ aeEat

220 /mea

esu102 7atE

Nonlinear Optical Nonlinear Optical InteractionsInteractions

The E-field of a laser beamThe E-field of a laser beam

22ndnd order nonlinear polarization order nonlinear polarization

C.C.)(~ tiEetE

)C.C.(2)(~ 22)2(*)2()2( tieEEEtP

2)2(

22ndnd Order Nonlinearities Order Nonlinearities The incident optical fieldThe incident optical field

Nonlinear polarization contains the following Nonlinear polarization contains the following termsterms

..)(~

21

21 CCeEeEtE titi ..)(~

21

21 CCeEeEtE titi

(OR) )(2)0(

(DFG) 2)(

(SFG) 2)(

(SHG) )2(

(SHG) )2(

*22

*11

)2(

*21

)2(21

21)2(

21

22

)2(2

21

)2(1

EEEEP

EEP

EEP

EP

EP

(OR) )(2)0(

(DFG) 2)(

(SFG) 2)(

(SHG) )2(

(SHG) )2(

*22

*11

)2(

*21

)2(21

21)2(

21

22

)2(2

21

)2(1

EEEEP

EEP

EEP

EP

EP

1

2)2(

1

2213

Sum Frequency GenerationSum Frequency Generation

132

Application:Tunable radiation in the UV Spectral region.

Application:Tunable radiation in the UV Spectral region.

Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification.

Application:The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification.

2

1

1

2)2(

2

1213

Difference Frequency Difference Frequency GenerationGeneration

132

Phase Matching Phase Matching

)2(

2

•Since the optical (NLO) media are dispersive, The fundamental and the harmonic signals have different propagation speeds inside the media.

•The harmonic signals generated at different points interfere destructively with each other.

•Since the optical (NLO) media are dispersive, The fundamental and the harmonic signals have different propagation speeds inside the media.

•The harmonic signals generated at different points interfere destructively with each other.

SHG ExperimentsSHG Experiments

We can use a We can use a resonator to resonator to increase the increase the efficiency of SHG.efficiency of SHG.

Third Order NonlinearitiesThird Order Nonlinearities

When the general form of the incident electric When the general form of the incident electric field is in the following form,field is in the following form,

The third order polarization will have 22 The third order polarization will have 22 components which their frequency dependent components which their frequency dependent are are

tititi eEeEeEtE 321321)(

~ tititi eEeEeEtE 321

321)(~

3,2,1,,),2(),2(

)(),(,3,

kjijiji

kjikjiii

3,2,1,,),2(),2(

)(),(,3,

kjijiji

kjikjiii

The Intensity Dependent The Intensity Dependent Refractive Index Refractive Index

The incident optical fieldThe incident optical field

Third order nonlinear polarizationThird order nonlinear polarization

C.C.)()(~ tieEtE C.C.)()(~ tieEtE

)(|)(|)(3)( 2)3()3( EEP )(|)(|)(3)( 2)3()3( EEP

)(|)(|)(3)()( 2)3()1(TOT EEEP )(|)(|)(3)()( 2)3()1(TOT EEEP

The total polarization can be written as

One can define an effective susceptibility

)3(2)1(eff |)(|4 E

)3(2)1(eff |)(|4 E

The refractive index can be defined as usual

eff2 41 n eff

2 41 n

By definition

Innn 20 Innn 20

where

20 |)(|2

E

cnI

20 |)(|2

E

cnI

)3(20

2

2

12 cn

n )3(

20

2

2

12 cn

n

MechanismMechanism nn2 2 (cm(cm22/W)/W) (esu)(esu) Response time Response time (sec)(sec)

Electronic Electronic PolarizationPolarization 1010-16-16 1010-14-14 1010-15-15

Molecular Molecular OrientationOrientation 1010-14-14 1010-12-12 1010-12-12

ElectrostrictionElectrostriction 1010-14-14 1010-12-12 1010-9-9

Saturated Atomic Saturated Atomic AbsorptionAbsorption 1010-10-10 1010-8-8 1010-8-8

Thermal effectsThermal effects 1010-6-6 1010-4-4 1010-3-3

Photorefractive Photorefractive EffectEffect largelarge largelarge Intensity Intensity

dependentdependent

)3(1111

Typical values of nonlinear refractive index

MaterialMaterial 1111 1111 Response Response timetime

AirAir 1.2×101.2×10-17-17

COCO22 1.9×101.9×10-12-12 2 Ps2 Ps

GaAs (bulk room GaAs (bulk room temperature)temperature) 6.5×106.5×10-4-4 20 ns20 ns

CdSCdSxxSeSe1-x1-x doped doped glassglass

1010-8-8 30 ps30 ps

GaAs/GaAlAs GaAs/GaAlAs (MQW)(MQW) 0.040.04 20 ns20 ns

Optical glassOptical glass (1-100)×10(1-100)×10-14-14 Very fastVery fast

Third order nonlinear susceptibility of some material

Processes due to intensity Processes due to intensity dependent refractive index dependent refractive index

1.1. Self focusing and self Self focusing and self defocusingdefocusing

2.2. Wave mixingWave mixing

3.3. Degenerate four wave mixing Degenerate four wave mixing and optical phase and optical phase conjugation conjugation

Self focusing and self Self focusing and self defocusingdefocusing

The laser beam has Gaussian The laser beam has Gaussian intensity profile. It can induce a intensity profile. It can induce a Gaussian refractive index profile Gaussian refractive index profile inside the NLO sample.inside the NLO sample.

)3(

Wave mixingWave mixing

/2)Sin(2 0

n /2)Sin(2 0

n

Optical Phase ConjugationOptical Phase Conjugation

Phase conjugation mirrorPhase conjugation mirror

M

M

PCM

PCMs

Aberration correction by Aberration correction by PCMPCM

PCMAberrating medium

PCMs Aberrating

medium

What is the phase What is the phase conjugationconjugation

C.C.),(~ ti

ss eEtrE C.C.),(~ ti

ss eEtrE rikss

seAE .sε̂

rikss

seAE .sε̂

The signal wave

The phase conjugated wave

C.C.),(~ * ti

sc erEtrE C.C.),(~ * ti

sc erEtrE

Degenerate Four Wave Degenerate Four Wave MixingMixing

)3(

A1 A2

A3

A4

•All of the three incoming beams A1, A2 and A3 should be originated from a coherent source.•The fourth beam A4, will have the same Phase, Polarization, and Path as A3.

•It is possible that the intensity of A4 be more than that of A3

•All of the three incoming beams A1, A2 and A3 should be originated from a coherent source.•The fourth beam A4, will have the same Phase, Polarization, and Path as A3.

•It is possible that the intensity of A4 be more than that of A3

Mathematical BasisMathematical Basis

..)().(~ ).( CCerAtrE trki

iii ..)().(

~ ).( CCerAtrE trkiii

i

The four interacting waves

The nonlinear polarization

)).((*321

)3(*321

)3(NL 32166 trkkkieAAAEEEP )).((*

321)3(*

321)3(NL 32166 trkkkieAAAEEEP

The same form as the phase conjugate of A3

The same form as the phase conjugate of A3

Holographic interpretation of Holographic interpretation of DFWMDFWM

)3(

A1 A2

A3

A4

Bragg diffraction from

induced dynamic gratings

Bragg diffraction from

induced dynamic gratings