Laser and Non Linear Optics by Imran Aziz

Introduction to Nonlinear OpticsIntroduction to Nonlinear Optics

MOHAMMAD IMRAN AZIZMOHAMMAD IMRAN AZIZAssistant ProfessorAssistant Professor

PHYSICS DEPARTMENTPHYSICS DEPARTMENTSHIBLI NATIONAL COLLEGE, AZAMGARH SHIBLI NATIONAL COLLEGE, AZAMGARH

(India).(India).

[email protected][email protected]

How to make a laser in How to make a laser in three easy steps …three easy steps …

• • Pick a medium that has the potential for optical gain – i.e., Pick a medium that has the potential for optical gain – i.e., anan

amplifying medium.amplifying medium. • • Select a means of putting energy into that medium – i.e., Select a means of putting energy into that medium – i.e.,

anan excitation system.excitation system. • • Construct an optical feedback system for stimulating Construct an optical feedback system for stimulating

furtherfurther emission, i.e., an emission, i.e., an optical resonator.optical resonator.


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

h

20/ aeEat

220 / mea

esu102 [email protected][email protected]

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

13

2Application: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

13

2


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

Ecn

I 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([email protected][email protected]

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 [email protected][email protected]

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


Origin of Nonlinearities in Origin of Nonlinearities in OpticsOptics

The fast response of media to an The fast response of media to an electromagnetic wave in visible and electromagnetic wave in visible and near IR is caused by a displacement near IR is caused by a displacement of electrons, both free ones in metals of electrons, both free ones in metals and bound ones in dielectrics.and bound ones in dielectrics.



Origin of Nonlinearities in Origin of Nonlinearities in OpticsOptics

The fast response of media to an electromagnetic wave in visible and near IR is caused by a displacement of electrons, both free ones in metals and bound ones in dielectrics.


1. Free electronsThe motion of electron in the field of a light wave:

)(exp)( 0 rktiEtE

)(exp)( 0 rktiHtH

(1

is described by an equation:

HV

cE

m

e

dt

rd 1

2

2

(2)

Because EV

, the vector product HV

is proportional to

.2E

The solution of (2) can be found in a form:

...)3()2()1( EEEEEEr (3)

where

)1( is linear,

..., )3()2( are nonlinear polarisabilities.The induced electrical dipole moment is

equal tored

(4)


2. Bound electronsFor the case of bound electron the equation has the following

form:)(2 2 tE

m

eFrrr NL

(5)

where the term

NLF

takes into account real anharmonisity of the

oscillator: ... rrrbrraFNL

Considerin NLF

as a small term the solution of (5) can be

presented as:

(3)...)3()2()1( EEEEEEr


3. Macroscopic characteristicsTo describe the media response for the electromagnetic field one must calculate a polarization vector , which is a dipole moment of a unit volume.

P

rNedNP

Where N is the concentration of electrons.If a nonlinear dependence of on takes place the

vectors and can be presented in the form:d

E

d

P

...)( )3()2()1( EEEEEEddEdd NLL

(7

)...)( )3()2()1( EEEEEEPPEPP NLL

(8)

where are tensors of 2 rank, are tensors of 3 rank

and so on. are nonlinear susceptibilities

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

)2()( nn

(6)


4. Local field factorIn a microscopic model of nonlinearity (we presented two such models) it is important to describe correctly microscopic and macroscopic values. For crystals of cubic symmetry:

)(n)(n

3

2)()(

2)1()1( n

N (9)

where the term in brackets is so-called Lorentz factor (local field factor). For nonlinear susceptibility in particular for quadratic nonlinearity:

3

2)(

3

2)(

3

2)()()(

22

12

212

21)2(

21)2(

n

nnN

(10)


5. How high is the nonlinearityIf the response of the media is caused by electrons in nonresonant case for the following ratio is valid: )(n

An

n

E

1)(

)1(

(11)

where is an interatomic field. For hydrogen AE .109 cmVEA One can see from this that appreciable nonlinear effects can be observed at relatively high light intensities, which are the features of pulse lasers. The nonlinear optics experiments became real after innovation of Q-switched laser with pulse duration of 10-8 s and intensities of 1010-1011 W/cm2. Now femtosecond lasers became available, which generate pulses with duration of 6-30·10-15 s at the intensity up to 1017-1020 W/cm2. In this case the electric field in the light wave exceeds the value of EA. It opens completely new branch of optics: physics of superstrong fields.

Besides the above electronic nature of nonlinear response a strong nonlinearity can be caused by an anharmonisity of atomic oscillation in molecules, orientation of polar molecules in an electric field, heating of medium. The slower is a mechanism responsible for nonlinearity the stronger is the nonlinearity.Let us present the values of characteristic time constants and the values of for different mechanism of nonlinear polarization.

)(n

Mechanism nonresonantelectronic

resonantelectronic

orientation inliquid crystals

Time constant, s 10-14 10-7-10-8 1-10-1

(2), esu 10-9 10-6-10-8

(3), esu 10-14-10-15 10-10 10-1-10-2



III. Optical III. Optical HarmonicHarmonic

GenerationGeneration

The high intensity light wave induces the nonlinear polarization in a medium. The wave of polarization is a source for new electromagnetic waves.


1. Second-harmonic generationFirst of all we should notice that the tensor , for centrosymmetric media is equal to zero.

)2(

The same is valid for all even order .2,)( mnn

EEPNL

)2()2(

The operation of symmetry transforms the terms from (12) in the following way:

PP

EE

)2()2(

(12)

(13)

Then , that can not take place under nonzero .)2(

)2()2()2()2( ))(( NLNL PEEEEP


For a simplicity we assume that the medium is isotropic. Then the polarization:

...... 3)3(2)2()1()3()2()1( EEEPPPP (14)The incident waves propagating in z-direction can be presented as: )cos( 11101 zktEE

)cos( 22202 zktEE (15)

]})()cos[(

])()cos[(

)](2cos1[5.0)](2cos1[5.0{

)]cos()cos(2

)(cos)(cos[

)]cos()cos([

21212010

21212010

2222011

210

)2(

22112010

2222

201122

10)2(

222201110

)2()2(

zkktEE

zkktEE

zktEzktE

zktzktEE

zktEzktE

zktEzktEPNL

(16)


A spectrum of polarization waves contains new frequencies:

E , E1 2

ωP

ω

ω1 ω20

0ω -2 ω1 2ω1 2ω2ω +2 ω1

.0,,,2,2 121221


2. Third-harmonic generationIf the medium possesses cubic nonlinearity, under the action of two monochromatic waves and the polarization would contain the components with frequencies:

.2,2,3,3 122121

1 2 )3(P


IV. Wave Nonlinear IV. Wave Nonlinear OpticsOptics

As the optical harmonic generation takes place both induced waves of polarization and free running electromagnetic waves of harmonics are propagating in the medium. If the dimensions of the medium are much larger than pumping wavelength the phase matching determines the efficiency of the energy transfer from the pumping wave to harmonics. Let us consider the phase matching conditions for the case of second harmonic generation.


1. Maxwell equationsThe propagation of the light in the medium is described by Maxwell equations:

4div

0div

14rot

1rot

D

B

t

D

cj

cH

t

B

cE

NLPPE

PED

HB

44

4)1(

(17)

For optical range

where

(18)

0,0,1 j (19)

Combining first and second equations from (17) one may obtain so-called wave equation:

2

2

2

2

2

41

t

D

ct

E

cE

(20)


Inserting (18) into (20) we are getting:

2

2

2

)1(2

2

2

2

441

t

P

ct

P

ct

E

cE

NL

(21)

The nonlinear polarization term in the right hand side of (21) plays a role of a source of electromagnetic waves

2. Phase mismatchFor quadratic media and relatively low nonlinearity the plane wave solution of (21) for the intensity of the second harmonic looks like:

)0( )2(

2

2

22

22

2)2(

2)(

)(sin][

cnnc

znn

n

II

(22)


)(2

2 22

nnc

kkk Phase mismatch

Δkz/2

1

-2π 2π-π 0 π

I2/I2

max


For the case of the exact phase matching the energy of the pumping wave can be completely transferred into second harmonic

0

I

I2

I

z2LL


3. Phase matchingHow the condition or can be realized? In an isotropic medium with normal dispersion > and

02 2 kkk 02 nn2n n

k never equals to zero

But in birefringent uniaxial crystal there are two beams ordinary and extraordinary. For so-called negative crystal no>ne. If pumping wave is ordinary one and second harmonic is extraordinary one the material dispersion ( > ) can be compensate for the difference in refractive indices for o and e beams:

2n n

eo nn 2 ne

2ω

no

2ω

ne

ω

no

ω

Directions ofphase matching


For the process of third-harmonic generation the condition of phase matching looks the same: 0k

As it was mentioned already and values for the fast nonresonant electronic polarization do not much differ for many materials and the only way to enhance the efficiency of nonlinear energy transformation is to phase match the interacting waves.

)2( )3(


V. Other Nonlinear V. Other Nonlinear EffectsEffects1. Modulation of a refractive index

Cubic nonlinearity causes not only wave generation with new frequency but also appearance of a wave of nonlinear polarization with the frequency of pumping wave:

1111 )()(),,;()( 1

2

11111)3(

1 EEPNL (23)

As a result of such selfaction a nonlinear refractive index n2I appears at the frequency :1

Innn 20 ),,;( 1111)3(

2 n (24)For the fast nonresonant nonlinearity n2 is relatively small:n2~10-13 cm2/kW.For slower mechanisms of the nonlinearity n2

can be much larger in particular for liquid crystals: n2~0.1 cm2/kW.


2. SelffocusingIf the intensity of a laser beam is high enough instead of diffraction an opposite effect of selffocusing takes place. Phase velocity depends on the intensity through nonlinear refractive index: Vph=c/n0+n2I (25)If n2 > 0 the phase velocity at the axis of the beam is lower and nonlinear medium is working as a lens.

Z

Phase front


VI. Nonlinear VI. Nonlinear OpticalOptical

DiagnosticsDiagnosticsNonlinear susceptibilities and are tensors and they inherit the symmetry properties of the crystalline medium. It means that nonlinear optical effects are structure sensitive. It can be employed to study different structure transformations. A lot of such experiments were done. I will mention just one related with laser induced melting of semiconductors.

)2( )3(

R

t

Ge, Si

laser pulse

RL

RS

R(t)

meltingpoint


Probing Nd:YAG

laser beam ( )ωNonlin

ear

reflection (2

)ω

Powerful laser beamwhich melts the surface

(Ruby, Nd:YAG , Eximer)

Semiconductor

R

t

A B3 5

laser pulse

RL

RL

RS

meltingpoint

L

L

NL

RS

NL

linearreflection

nonlinearreflection

Idea of experiment

Metal in liquid stateR

t

A B2 6

laser pulse

RS

meltingpoint

L

RS

NL

linearreflection

nonlinearreflection

RL

NL

Semiconductor in liquid state

1. Nonlinear optical diagnostics of phase transitions


VIII. ConclusionsVIII. Conclusions1.Nonlinear optics is an attractive and fast

developing part of modern optics.2.Nonlinear effects are structure sensitive in

their nature. It can be used for time-resolved monitoring of structural transformation (up to femtosecond time resolution).

3.Artificial photonic media on the base of porous semiconductors open new exciting possibilities for the control of nonlinear optical processes.

Documents

Laser and Non Linear Optics by Imran Aziz