Nonlinear optics (NLO)

(1)

Linear Optics and Nonlinear Optics

Linear Optics

z The optical properties, such as the refractive index and the absorption coefficient are independent of light intensity.

z The principle of superposition holds.

z The frequency of light cannot be altered through the medium.

z Light cannot interact with light;

Æ two beams of light in the same region of a linear optical medium can have no effect on each other.

Æ Thus light cannot control light.

Nonlinear optics (NLO)

z The refractive index, and consequently the speed of light in an optical medium, does change with the light intensity.

z The principle of superposition is violated.

z Light can alter its frequency as it passes through a nonlinear optical material (e.g., from red to blue!).

z Light can interact with light via the medium Æ Thus light can control light.

26. Nonlinear optics and light modulation

(2)

0

2 1

2 3

1 2 3 0 1 0 2 0 3

2 3

L

L Polarization :

Susceptibility :

P E

E E

P P P P E E E

ε χ

χ χ χ

ε χ ε χ

χ

χ ε

=

= + + +

= + + + = + + +

Nonlinear optics

(3)

Second-order Nonlinear optics Second-order Nonlinear optics

Second-harmonic generation (SHG) and rectification

(0)

), 2 ( )

(

2 2 2

P P P ω ±ω = ω

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

→

∝

→

= E(

ω

) P₂ E²(

ω

)

E optical

{ } ^{ ^} ^{ ^}

{

E E

}

n E _electric _DC

P P

E E

P E

E P

2 , 2

2 2

) 0 ( )

( ) 0 ( )

( (0),

) ( ) ( )

(2 , ) ( ) 0 ( )

( , ) 0 ( (0)

∝ Δ

→

∝

→

∝

→

∝

→

ω ω

ω

{

^but, ⁽⁰⁾ ⁽ ⁾

}

)

( )

0

( , E

ω

E E

ω

E

E = electrical DC + optical >>

Three-wave mixing

2

2 0 2

P = ε χ E

optical optical E

E

E = (

ω

₁) + (

ω

₂)

{ } { }

{ }

{

⁽ ⁾ ⁽ ⁾

}

) (

, ) ( ) ( )

(

, ) ( )

(2 ,

) ( )

(2

2 1

2

2 1

2

2 2 2

2 1

2

2 2

ω ω

E E

P

E E

P

E P

∝

−

∝ +

∝

→

∝

→

Æ SHG

Æ Frequency up-converter

Æ Parametric amplifier, parametric oscillator Æ Index modulation by DC E-field

ÆFrequency doubling Æ Rectification

(4)

Third-order Nonlinear optics Third-order Nonlinear optics

Third-harmonic generation (THG)

{

⁽ ⁾ ⁽ ⁾

}

^, ⁽³ ⁾

^{

⁽ ⁾

^}

)

( ² ₃ ³

3 ω E ω E ω P ω E ω

P ∝ ∝

Optical Kerr effect

→

∝

→

= E(

ω

) P₃ E³(

ω

)

E optical

3

3 0 3

P = ε χ E

Æ Self-phase modulation

Æ Frequency tripling

) ( )

( ) ( )

( ²

3 ω E ω E ω I ω E ω n I ω

P ∝ ∝ → Δ ∝ Æ Index modulation by optical Intensity

) (

)

( ₀ ₀

0 n I k nL

n

n = + Δ →ϕ =ϕ + Δϕ = Δ

{ } { }

₀

0 n I(x) n I(x) n

n

n = + Δ → Δ > Æ Self-focusing, Self-guiding (Spatial solitons)

{ } { }

₀

0 n I(x) n I(x) n

n

n = +Δ → Δ < Æ Self-defocusing

Electro-optic (EO) Kerr effect

{

^but, ⁽⁰⁾ ⁽ ⁾

}

)

( )

0

( , E

ω

E E

ω

E

E = electrical DC + optical >>

2

DC , 2

DC

3( ) E(0) _electric, E( ) n E(0) _electric

P ∝ → Δ ∝

→

ω ω

Æ Index modulation by DC E²

(5)

Third-order Nonlinear optics Third-order Nonlinear optics

Four-wave mixing

3

3 0 3

P = ε χ E

optical optical

optical E E

E

E = (

ω

₁) + (

ω

₂) + (

ω

₃)

(

₁, ₂, ₃

)

³ 6³ 216 terms

3

3 ∝ → ± ± ± → =

→ P E ω ω ω

Æ Frequency up-converter

Æ Degenerate four-wave mixing

) ( ) ( ) ( )

(

: P₃ ω₁ ω₂ ω₃ ω₄ E ω₁ E ω₂ E ω₃ example

One + + ≡ ∝

→

) ( ) ( ) ( )

- (

: ^P₃ ω₁ ω₂ ω₃ ω₄ Ê ω₁ Ê ω₂ Ê^* ω₃ example

Another + ≡ ∝

→

4 3

2

1 ω ω ω

ω = = =

→ If

ω ω

ω₁ = ₂ = ₃ → ₄ =3

→ If Æ THG

4 3

2

1 ω ω ω

ω + = +

→

directions opposite

in traveling

are them among

waves two

s plane wave Assume

→

) ( ) ( ) ( )

( ₄ ^*

3 ω ω E ω E ω E ω

P = ∝

→ ^Æ Optical phase conjugation

(6)

2 2 0 2

P = ε χ E

1 2

2 2

0 1 0 2

2 2

0 2 0 1 0 2

cos

cos cos

1 1

cos cos 2

2 2

o

o o

o o o

E E t

P P P

E t E t

E E t E t

ω

ε χ ω ε χ ω

ε χ ε χ ω ε χ ω

=

= +

= + +

Constant (DC) term Æ Optical rectification

Second harmonic term Æ 2ω

: Only for non-centro-symmetry crystals

26-2. Second harmonic generation (SHG) 26-2. Second harmonic generation (SHG)

( )

2 1

cos 1 cos 2

θ 2 θ

⎧ = + ⎫

⎨ ⎬

⎩ ⎭

2 2

2 0 2 0 0 2 0 2 2

1 1

( ) cos 2 (0) (2 )

2 2

P t = ^⎧ ⎨ ε χ E ^{⎫ ⎧} ⎬ ⎨ + ε χ E ^⎫ ⎬ ω t = P + P ω

⎩ ⎭ ⎩ ⎭

[GaAs. CdTe, InAs, KDP, ADP, LiNbO₃, LiTaO₃, …]

(7)

SHG does not occur in isotropic, centrosymmetry crystals

(8)

Second harmonic generation Second harmonic generation

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

P

₂

E

P

₂

(t)

E(t)

2 2

2 0 2 0 0 2 0 2 2

1 1

( ) cos 2 (0) (2 )

2 2

P t = ^⎧⎨ ε χ E ^{⎫ ⎧}⎬ ⎨+ ε χ E ^⎫⎬ ωt = P + P ω

⎩ ⎭ ⎩ ⎭

(9)

Second harmonic generation Second harmonic generation

o

cos E = E ω t

( )

2

2 0 2

1 cos 2

2 1 2

o

P E t

E

ε χ ω

ε χ

= +

1 2

P = P + P

1 0 1 _o

cos

P = ε χ E ω t

(10)

Second harmonic generation Second harmonic generation

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

2 ( / 2)

ω → ω λ → λ

(11)

Second harmonic generation

(12)

Phase matching (index matching) in SHG Phase matching (index matching) in SHG

( )

2

2 2 2

2 0

k k k

n n

c c

n n

c

ω ω

ω

Δ = −

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ − ⎜ ⎟ ⎝ ⎠

⎛ ⎞

= − ⎜ ⎟ =

⎝ ⎠

Output intensity after second harmonic generation

2

sin ,

2

2 I ∝ c ⎛ ⎜ ⎝ L k Δ ⎞ ⎟ ⎠ Δ = k k

_ω

− k

_ω

Phase matching : Δk=0

O-ray surface (n_ω)

E-ray surface (n_2ω)

Optic axis

Direction of Matching

(Δk=0)

(13)

Frequency mixing by three-wave mixing Frequency mixing by three-wave mixing

frequency up-converter

parametric amplifier

parametric oscillator

( ω ω

1

+

2

⇒ ω

3

)

( ω ω

3

−

1

⇒ ω

2

→ ω ω

3

−

2

⇒ ω

1

)

( ω

3

⇒ ω ω

1

+

2

→ ω ω (

3ω

−

² →2

⇒

idler, or parameter,

ω

1

)

중개자

)

(14)

Parametric interaction Parametric interaction

{ } { }

( ) ( ) ( ) ( )

1 2

1 1 2 2

1 1 1 2 2 2

2

2 0 2

2 2

( ) ( )

cos cos

1 1

exp( ) exp( ) exp( ) exp( )

2 2

1+ 1 1 , 2+ 2 2 , 1 3 , 1 3

o o

E E E

E t E t

E i t i t E i t i t

P E

ω ω

ω ω ω ω

ε χ

ω ω ω ω ω ω ω ω ω ω ω ω

= +

= + − + + −

− = +

⇒ = =

=

Æ Frequency conservation

Æ Momentum (phase) matching

(15)

Nonlinearity of the refractive index Nonlinearity of the refractive index

2 2

1 1

2

n

o

E

n rE R

= + +

Second-order nonlinearity (P₂)

Æ Linear electro-optic coefficient (r) Æ Pockels effect (E: DC field)

third-order nonlinearity (P₃)

Æ Quadratic electro-optic coefficient(R) Æ Kerr effect

0

2

1 2 3

L

Polarization : Susceptibility :

P E

E E

ε χ

χ χ χ χ

=

= + + +

) 1

( ⁺ χ

n =

(16)

Nonlinearity of the refractive index Nonlinearity of the refractive index

2 2 2

1 1

o

r RE

n E

n = + +

(17)

Pockels effect (Linear electro-optic effect) Pockels effect (Linear electro-optic effect)

: half-wave plate to make 2

3

o HW

o

V rn

λ π

= Φ =

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

3

3 3

2 2 2

2 2

2 2 2

o

o o

L n

L r n E

rn EL rn V ϕ

π λ

π π

λ λ

Φ = Δ

⎛ ⎞

= ⎜ ⎟ Δ

⎝ ⎠

⎛ ⎞ ⎛ ⎞

≅ ⎜ ⎝ ⎟ ⎜ ⎠ ⎝ ⎟ ⎠

⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ = ⎜ ⎟

⎝ ⎠ ⎝ ⎠

SA FA

V

L

Pockels cell

(18)

Pokels electro-optic modulator

(19)

Pockels effect (Q-switch in laser cavity)

(20)

Third-order nonlinear effect Third-order nonlinear effect

In media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric field is reversed.

The dominant nonlinearity is then of third order,

3 3 0 3

P = ε χ E

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

P

₃

E

(21)

26-5. Electro-optic Kerr effect 26-5. Electro-optic Kerr effect

Kerr cell

(22)

All-optical Kerr effect All-optical Kerr effect

3 2

0 2

2 n ⎛ R n ⎞ E n I

Δ = ⎜ ⎟ ≡

⎝ ⎠ n I ( ) = n ⁰ + n I ²

Self-phase modulation

The phase shift incurred by an optical beam of power P and cross-sectional area A, traveling a distance L in the medium,

Self-focusing (Optical Kerr lens)

(23)

All-optical Kerr effect : Spatial solitons All-optical Kerr effect : Spatial solitons

0 2

( )

n I = n + n I In linear medium, wave is spreading.

In optical Kerr medium, wave can be guided.

Self-guided beam = spatial soliton

(24)

26-8. Optical phase conjugation

(25)

Phase Conjugation and Time Reversal

( ) ^t [ ( ) ^e

ⁱ⁽ ^t ^kz⁾

]

E

₁

r , = Re ψ r

^ω ⁻

( ) ^t [ ( ) ^e

ⁱ⁽ ^t ^kz⁾

]

E

₂

r , = Re ψ

^∗

r

^ω ⁺

( ) ( )

^{( )}

2

, Re

ⁱ ^t ^kz

E r t = ^⎡ ⎣ ψ r e

^{⎡ − − ⎤}^⎣^ω ^⎦

^⎤ ⎦ Phase conjugation

Time reversal

Incident

(26)

Four-wave mixing (third-order nonlinearity) Four-wave mixing (third-order nonlinearity)

Superposition of three waves of angular frequencies ω

₁

, ω

₂

, and ω

₃

3

3 0 3

P = ε χ E

(as sum of 6³ = 216 terms)

{ }

4 1 2 3

3 4 1 2

r r r r

If

k k k k

ω ω ω ω

= + −

→ + = +

+ = +

(27)

Four wave mixing by phase conjugation Four wave mixing by phase conjugation

Nonlinear Medium

(FWM) A

₃

signal

A

₁

Pump beam

A

₂

Pump beam

A

₄

PC output

*

4 1 2 3

A ∝ A A A

It looks like a mirror,

but it’s quite strange.

(28)

Phase conjugate mirror (PCM)

(29)

Image restoration by phase conjugation Image restoration by phase conjugation

Optical reciprocity.

(30)

26-6. Faraday effect (Linear magneto-optic effect) 26-6. Faraday effect (Linear magneto-optic effect)

Rotation of the polarization plane:

: V; Verdet constant

2 1.0083

e

VBd

e dn m c d Bd

dn Bd d

β

λ λ λ λ

=

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠

Optical isolator

Faraday rotator

d

Some other effects for optical modulation

(31)

AO (acousto-optic) effect : interaction of optical & acoustic waves

Brillouin scattering : collision between photons and acoustic phonons

26-7. Acousto-optic effect

Photoelasticity : change in n of crystal due to mechanical stress

K K’

K

_s

'

: momentum conservation k = ± k k

s

'

: energy

conservation

ω = ± ω ω

s

(32)

λ

_s

Φ

_D

θ θ

K K’

AO effect : Bragg condition AO effect : Bragg condition

K K’

K

_s

'

: momentum conservation k = ± k k

s

'

: energy conservation ω = ± ω ω

s

Æ Doppler effect for light

(33)

AO effect : Raman-Nath regime

(34)

(참고) Stimulated Inelastic Scattering (참고) Stimulated Inelastic Scattering

Stimulated Raman Scattering (SRS)

: interaction between photon and optical-phonon (induced by incident photon) Photon with hν energy incident on molecule having vibration frequency ν_m Æ molecules absorb energy from photon

Æ optical phonon is induced Æ photon is scattered

Æ It occurs in forward direction Stimulated Brillouin Scattering (SBS)

: interaction between photon and acoustic-phonon Æ photons scatter from acoustic wave Æ It occurs only in backward direction