26. Nonlinear optics and light modulationLinear Optics and Nonlinear Optics

Linear OpticspThe optical properties, such as the refractive index and the absorption coefficient are independent of light intensity.The principle of superposition holdsThe principle of superposition holds.The frequency of light cannot be altered through the medium.Light cannot interact with light;

two beams of light in the same region of a linear opticaltwo beams of light in the same region of a linear optical medium can have no effect on each other. Thus light cannot see light.

Nonlinear optics (NLO)

The refractive index, and consequently the speed of light in an optical medium does change with the light intensitymedium, does change with the light intensity.The principle of superposition is violated.Light can alter its frequency as it passes through a nonlinear optical material (e g from red to blue!)material (e.g., from red to blue!).Light can interact with light via the medium

Thus light can control light.

Nonlinear optics

02

Polarization :

Susceptibility :

P E

E E

ε χ=

+ + +21 2 3 LSusceptibility : E Eχ χ χ χ= + + +

χεεχεεχεεε +===→+=→+== 1 )1( 000 c

vnEEED

2 31 2 3 0 1 0 2 0 3L LP P P P E E Eε χ ε χχ ε= + + + = + + +

ε 0c

Second-order Nonlinear optics 22 0 2P Eε χ=

Second-harmonic generation (SHG) and rectification

)2()( PP ωωω =±→∝→= )()( 2 ωω EPEE(0)

),2()(

2

22

PPP ωωω =±

Electro-optic (EO) effect (Pockell’s effect)

→∝→= )()( 2 ωω EPEEoptical Frequency doubling

Rectification

EP 22 ∝→

{ })()0( but, )()0( ,

ωω EEEEEopticalDCelectrical

>>+=

{ } { } { }{ }

DCelectricEnEEPP

EEPEEPEP

,22

222

2

)0()()0()((0),)()()(2 ,)()0()(,)0((0)

∝Δ→∝→

∝∝∝→

ωωωωωωω

Index modulation by DC E-field

Three-wave mixing

opticalopticalEEE )()( 21 ωω +=

{ } { } ,)()(2,)()(2 22

2212

12

22

ωωωω EPEP

EP

∝∝→

∝→

SHG

{ }{ })()()(

,)()()(

21212

21212

ωωωωωωωω

EEPEEP

∝−∝+ Frequency up-converter

Parametric amplifier, parametric oscillator

Third-order Nonlinear optics 33 0 3P Eε χ=

Third-harmonic generation (THG)

{ } { })()3()()()( 32 ωωωωω EPEEP ∝∝→∝→= )()( 3 ωω EPEE { } { })()3(,)()()( 33 ωωωωω EPEEP ∝∝→∝→= )()( 3 ωω EPEEoptical

Frequency tripling

Electro-optic (EO) Kerr effectp ( )

{ })()0( but, )()0( ,

ωω EEEEEopticalDCelectrical

>>+=

22 2

DC ,

2

DC ,3 )0()()0()(electricelectric

EnEEP ∝Δ→∝→ ωω Index modulation by DC E2

Optical Kerr effect

)()()()()()( 23 ωωωωωω InEIEEP ∝Δ→∝∝ Index modulation by optical Intensity

Self-phase modulation)()( 000 nLkInnn Δ=Δ+=→Δ+= ϕϕϕ

{ } { } 00 )()( nxInxInnn >Δ→Δ+= Self-focusing, Self-guiding (Spatial solitons)

{ } { } 00 )()( nxInxInnn <Δ→Δ+= Self-defocusing

Third-order Nonlinear optics 33 0 3P Eε χ=

Four-wave mixing

opticalopticalopticalEEEE )()()( 321 ωωω ++=

( ) terms2166,, 33321

33 =→±±±→∝→ ωωωEP ( )3213

Frequency up-converter)()()()( : 32143213 ωωωωωωω EEEPexampleOne ∝≡++→

ωωωωω 3=→==→ If THG

)()()()-( : 3*

2143213 ωωωωωωω EEEPexampleAnother ∝≡+→

ωωωωω 34321 =→==→ If THG

Degenerate four-wave mixing4321 ωωωω ===→ If

4321 ωωωω +=+→

directions oppositein traveling are themamong waves two

splane waveAssume→

)()()()( *EEEP )()()()( 43 ωωωωω EEEP ∝=→ Optical phase conjugation

24-2. Second harmonic generation (SHG)

22 0 2P Eε χ= : Only for non-centro-symmetry crystals2 0 2χ

cosoE E tω=[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, …]

1 22 2

0 1 0 2cos coso o

P P P

E t E tε χ ω ε χ ω

= +

= + ( )2 1cos 1 cos 22

θ θ⎧ ⎫= +⎨ ⎬⎩ ⎭

2 20 1 0 2 0 2

1 1cos cos 22 2o o oE t E E tε χ ω ε χ ε χ ω= + +

2⎩ ⎭

1 1⎧ ⎫ ⎧ ⎫2 22 0 2 0 0 2 0 2 2

1 1( ) cos 2 (0) (2 )2 2

P t E E t P Pε χ ε χ ω ω⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

C (DC) S d h i tConstant (DC) termOptical rectification

Second harmonic term2ω

SHG does not occur in isotropic, centrosymmetry crystals

2P E 22 0 2P Eε χ=

2 , is or If isotropic centrosymmetricχ

2-.

both E and E give the same P polarizationthat means the molecules are not polarized by the sencond effectχ

→ +

→

Second harmonic generation

P22 2

2 0 2 0 0 2 0 2 21 1( ) cos 2 (0) (2 )2 2

P t E E t P Pε χ ε χ ω ω⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

P2(t)

E

E(t)

From Fundamentals of Photonics (Bahaa E. A. Saleh)

Second harmonic generation

cosoE E tω=

P P P+1 2P P P= +

1 0 1 cosoP E tε χ ω=

( )21 cos 2P E tε χ ω= ( )2 0 2

2

cos 221

oP E t

E

ε χ ω

ε χ

=

+ 0 22 oEε χ+

Second harmonic generation 2 / 2) (ω ω λ λ→ →

From Fundamentals of Photonics (Bahaa E. A. Saleh)

Second harmonic generation

Phase matching (index matching) in SHG

Output intensity after second harmonic generation

22sin , 2

2L kI c k k kω ωΔ⎛ ⎞∝ Δ = −⎜ ⎟

⎝ ⎠2⎜ ⎟⎝ ⎠

Optic axis

2k k kΔ = −

Phase matching : Δk=0 Direction of

Matching(Δk=0)2

2

2

2 2

k k k

n n

ω ω

ω ωω ω

Δ = −

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(Δk=0)

( )

2

2 2 0

c c

n n

ω ω

ω ωω

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞= − =⎜ ⎟⎝ ⎠

E-ray surface(n2ω)

( )2 cω ω ⎜ ⎟⎝ ⎠ O-ray surface

(nω)

Frequency mixing by three-wave mixingy g y g

frequency up-converter ( )1 2 3ω ω ω+ ⇒

parametric amplifier ( )3 1 2 3 2 1 ω ω ω ω ω ω→− ⇒ − ⇒pa a et c a p e ( )3 1 2 3 2 1

parametric oscillator ( )ω ω ω ω ω ω→⇒ + − ⇒( )2 idler, or parameter, ω → 중개자

parametric oscillator ( )3 1 2 3 2 1 ω ω ω ω ω ω→⇒ + − ⇒

Parametric interaction

1 2( ) ( )E E EE Eω ω= +

{ } { }1 1 2 2

1 1 1 2 2 2

cos cos1 1exp( ) exp( ) exp( ) exp( )2 2

o o

o o

E t E t

E i t i t E i t i t

ω ω

ω ω ω ω

= +

= + − + + −

22 0 2

2 2

P Eε χ=

( ) ( ) ( ) ( )2 22 21 1 1 2 2 2 1 3 1 3+ , + , , ω ω ω ω ω ω ω ω ω ω ω ω− = +⇒ = ==

Frequency conservation

Momentum (phase) matching

Nonlinearity of the refractive index

0Polarization : P Eε χ=2

1 2 3 LSusceptibility : E Eχ χ χ χ= + + +

)1( +

2 221 1 ErE R= + +

)1( χ+=n

2 2on

En

rE R+ +

Second-order nonlinearity (P2) third-order nonlinearity (P3)

Linear electro-optic coefficient (r)

Pockels effect (E: DC field)

Quadratic electro-optic coefficient(R)

Kerr effect( ) Kerr effect

Nonlinearity of the refractive index

21 1 r REE= + +2 2o

r REn

En

+ +

Pockels effect (Linear electro-optic effect)

Phase difference (SA & FA)by inducing DC E fieldL by inducing DC E-field

2 ϕΦ = ΔSA FA

L

22o

L nπλ

⎛ ⎞= Δ⎜ ⎟

⎝ ⎠

3222

o

orL n Eπ

λ

⎝ ⎠⎛ ⎞ ⎛ ⎞≅ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠V

3 3

2

2 2o

o orn EL rn V

λ

π πλ λ

⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Pockels cell

h lf l t t koλ

o oλ λ⎝ ⎠ ⎝ ⎠

: half-wave plate to make 32o

HWo

Vrn

π= Φ =

Pokels electro-optic modulator

Pockels effect (Q-switch in laser cavity)

Third-order nonlinear effectIn media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the p yelectric field is reversed. The dominant nonlinearity is then of third order,

33 0 3P Eε χ=

The third-order nonlinear material is called a Kerr medium.

P3

EE

Electro-optic Kerr effect : E=V/d

Kerr cell

All-optical Kerr effect : E=E(ω)

3 20 2

Rn n E n I⎛ ⎞Δ = ≡⎜ ⎟ 0 2( )n I n n I= +0 22n n E n IΔ ⎜ ⎟

⎝ ⎠ 0 2( )n I n n I+

S lf h d l tiSelf-phase modulationThe phase shift incurred by an optical beam of power P and cross-sectional area A, traveling a distance L in the medium,cross sectional area A, traveling a distance L in the medium,

Self-focusing (Optical Kerr lens)

All-optical Kerr effect : Spatial solitons

I li diIn linear medium,wave is spreading.

In optical Kerr medium,

( )n I n n I= +

wave can be guided.

0 2( )n I n n I= +

Self-guided beam = spatial soliton

24-6. Optical phase conjugation

Phase Conjugation and Time ReversalPhase Conjugation and Time Reversal

( ) ( ) ( )[ ]kztietE −= ωψ rr Re,1Incident

( ) ( ) ( )[ ]kztietE +∗= ωψ rr Re,2Phase conjugation ( ) ( )[ ]

( ) ( ) ( )Re i t kzE t e ωψ ⎡ − − ⎤⎣ ⎦⎡ ⎤r rTime re ersal ( ) ( ) ( )2 , ReE t eψ ⎣ ⎦⎡ ⎤= ⎣ ⎦r rTime reversal

Four-wave mixing (third-order nonlinearity)

Superposition of three waves of angular frequencies ω1, ω2, and ω3

33 0 3P Eε χ= (as sum of 63 = 216 terms)

If ω ω ω ω= +

{ }4 1 2 3

3 4 1 2

If ω ω ω ω

ω ω ω ω

= + −

→ + = +

{ }3 4 1 2

r r r r k k k k+ = +

Four wave mixing by phase conjugation

A1Pump beam

*4 1 2 3A A A A∝

NonlinearAMedium

(FWM)

A3signal

A4 (FWM)A4PC output

A2 Pump beam

It l k lik iIt looks like a mirror,but it’s quite strange.

Phase conjugate mirror (PCM)Conventional mirror

PCM

Image restoration by phase conjugation

Optical reciprocity.

24 4 Faraday effect (Linear magneto optic effect)

Some other effects for optical modulation

24-4. Faraday effect (Linear magneto-optic effect)

Rotation of the polarization plane: Faraday rotator : V; Verdet constantVBd

e dn Bd

β

λ

=

⎛ ⎞⎜ ⎟

Faraday rotator

2

1 0083

eBd

m c d

dn Bd

λ

λ

= ⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟

d

1.0083 Bdd

λλ

= ⎜ ⎟⎝ ⎠

Optical isolatorOptical isolator

24-5. Acousto-optic effect

AO (acousto-optic) effect : interaction of optical & acoustic wavesPhotoelasticity : change in n of crystal due to mechanical stress

( p ) pBrillouin scattering : collision between photons and acoustic phonons

K’

K

KKs

K

' sk k k= ±

: momentumconservation

': energy

sω ω ω= ±

: energy conservation

AO effect : Bragg condition

K’Ks

K

'k k k= ±

θθK K’ : momentum conservation

sk k k±

ΦD ' ti

sω ω ω= ±

λs: energy conservation

Doppler effect for light

AO effect : Raman-Nath regime

(참고) Stimulated Inelastic Scattering

Stimulated Brillouin Scattering (SBS)

: interaction between photon and acoustic-phonon

photons scatter from acoustic wave

It occurs only in backward direction

Stimulated Raman Scattering (SRS)

: interaction between photon and optical-phonon (induced by incident photon)

Photon with hν energy incident on molecule having vibration frequency νm

molecules absorb energy from photon

optical phonon is induced

photon is scattered

It i f d di tiIt occurs in forward direction