비선형 광학

(1)

Nonlinear Optics Lab

. .

HanyangHanyang Univ.Univ.

비선형 광학

(Nonlinear Optics)

(2)

. .

비선형 광학

...

) ( )

( )

( t =

⁽¹⁾

E t +

⁽²⁾

E

²

t +

⁽³⁾

E

³

t +

P χ χ χ

Polarization(분극) of medium :

linear 2^nd order 3^rd order

Wave equation :

2 2 2

2 2 2 2

t P t

E c

E n

∂

= ∂

∂

− ∂

∇

new wave source term

- Mostly, the 1^stlinear term is dominant, and the higher order terms are negligible : linear medium - In some medium, the 2^ndor 3^rdorder term is comparable to the 1^st term : nonlinear medium

- Invention of Laser Î high intensity : easy to observe the nonlinear effects

(3)

. .

2^nd order nonlinear effect

. . )

( t E

₁

e

¹

E

₂

e

²

C C E =

⁻ⁱ^ω ^t

+

⁻ⁱ^ω ^t

+

) ( )

( ⁽²⁾ ²

) 2

( t E t

P =

χ

[ ² ²

1 2^* ⁽ ⁾

^. ^. ]

) (

2 1 2 2

2 2 2

1 ) 2

( ₁ ₂ ₁ ₂ ₁ ₂

C C e

E E e

E e

E

ⁱ ^t

+

ⁱ ^t

+

ⁱ ^t

+

ⁱ ^t

+

= χ

⁻ ^ω ⁻ ^ω ⁻ ^ω ⁺^ω ⁻ ^ω ⁻^ω

[

2 2^*

]

* 1 1 ) 2

2

(

E E + E E

+ χ

SH(second harmonics) SF(sum frequency) DF(difference frequency)

OR(optical rectification)

- SHG(second harmonic generation) - SFG(sum frequency generation)

- DFG(difference frequency generation) - Optical rectification

- Parametric oscillation - Electro-optic effect

ω ω

ω + → 2

3 2

1

ω ω

ω + →

3 2

1

ω ω

ω − →

1

0

1

− ω →

ω

1

0 ω

ω

³

+ ω →

¹

ω

²

ω → +

(4)

. .

2차 비선형 효과의 조건 1 : noncentrosymmetric medium

1. Centro symmetric media

- Potential energy for electric dipole :

4 ...

) 2

( =

₀² ²

+ m Bx

⁴

+ m x

x

V ω

- Equation of motion :

) ( )

(i i

Nex P = −

x

m t

eE Bx

x x

x _&& + 2 γ _& + ω

₀²

+

³

= − ( ) /

Lorentz picture of an atom

- Perturbation expansion method :

. . )

( t E

₁

e

¹

E

₂

e

²

E

₃

e

³

C C E =

⁻ⁱ^ω ^t

+

⁻ⁱ^ω ^t

+

⁻ⁱ^ω ^t

+

)

...

3 ( 3 )

2 ( 2 )

1

(

+ + +

= x x x

x λ λ λ

) ( )

( t E t E

Î

→ λ

(5)

. .

에 대한 계수는 0

Î

x _&&

⁽¹⁾

+ 2 γ x _&

⁽¹⁾

+ ω

₀²

x

⁽¹⁾

= − eE ( t ) / m λ

n

0 2

⁽²⁾ ₀² ⁽²⁾

) 2

(

+ x + x =

x _&& γ _& ω

0 ]

[

2

⁽³⁾ ₀² ⁽³⁾ ⁽¹⁾ ³

) 3

(

+ x + x + B x =

x _&& γ _& ω

Î

x

⁽²⁾

= 0

: Centrosymmetric medium 에서 2차 비선형 효과는 발생하지 않음

2. Noncentrosymmetric media

- Potential energy for electric dipole :

3 ...

) 2

( =

₀² ²

+ m Dx

³

+ m x

x

V ω

- Equation of motion :

m t

eE Bx

x x

x _&& + 2 γ _& + ω

₀²

+

³

= − ( ) /

(6)

. .

m t

eE x

x

x _&&

⁽¹⁾

+ 2 γ _&

⁽¹⁾

+ ω

₀² ⁽¹⁾

= − ( ) /

0 ]

[

2

⁽²⁾ ₀² ⁽²⁾ ⁽¹⁾ ²

) 2

(

+ x + x + D x =

x _&& γ _& ω

0 2

⁽³⁾ ₀² ⁽³⁾ ³⁽¹⁾

) 3

(

+ x + x + Bx =

x _&& γ _& ω

Î

ω ω ω

_j

ω

_j

γ

j j

j

i E

m e L

E m

x e

) 2

(

₀² ²

) 1 (

−

− −

=

−

=

) ( ) 2 (

) / ) (

2 (

₂

2 2 )

2 (

j j

j

L L

E m e x D

ω ω = − ω

) ( ) ( ) (

) / ( ) 2

(

2 1

2 1 2 2

1 ) 2 (

ω ω

ω ω ω

ω L L L

E E m e x D

− +

= +

) (

) ( ) (

) / ( ) 2

(

2 1

* 2 1 2 2

1 ) 2 (

ω ω

ω ω ω

ω − = − − −

L L

L

E E m

e x D

) (

) ( ) 0 (

) / ( 2 )

( ) ( ) 0 (

) / ( ) 2

0 (

2 2

* 2 2 2

1 1

* 1 1 2 )

2 (

ω ω

ω

ω − −

− −

= L L L

E E m e D L

L L

E

m

e

x D

(7)

. .

2^nd order susceptibility :

) ( ) 2 (

) /

) ( , , 2

(

₂

2 3

) 2 (

i i

i

L L

D m

e N

ω ω ω

ω ω

χ = −

) ( ) ( ) (

) /

) ( ,

, (

2 1

2 3

2 1 2 1

) 2 (

ω ω

ω ω ω

χ L L L

D m

e N

− +

= +

) (

) ( ) 0 (

) /

) ( ,

, 0 (

1 1

2 3

1 1

) 2 (

ω ω ω

ω

χ − = − −

L L

L

D m

e N

: SHG

: SFG

) (

) ( ) (

) /

) ( ,

, (

2 1

2 3

2 1 2 1

) 2 (

ω ω

ω ω ω

χ − = − − −

L L

L

D m

e

N

: DFG

: OR

(8)

. .

ex) 2

^nd

harmonic generation

_{optic axis}

2 2 2

2 2

t P t

E E

^NL

∂ + ∂

∂

= ∂

∇ με μ

Wave equation :

) ) (

( )

(ω₁ ω₂ ω₃

E E

E

E = + +

Total electric field :

.]

. )

( 2 [

1

( )

)

(

E z e C C

E

^ω^j

=

_j ⁱ ^ω^j^t⁻^k^j^z

+

where,

z k

e

i

z E

d dz i

dE

⁽² ⁾ ₍ ₎ ₂ ₍ ₎

)]

(

[

^Δ

−

=

^ω

ω

ε

ω μ

^where,

^Δ ^k ⁼ ^k

₃⁽²^ω⁾

⁻ ² ^k

₁⁽^ω⁾

Î

ω ω

2ω

2차 비선형 효과의 조건 2 : Phase matching condition

(9)

. .

2 2

2 4 ) ( 2

2 2

0 0

) 2 (

) 2 / (

] sin 2 [

1 kl l kl

n E I d

Δ

⎟⎟ Δ

⎠

⎜⎜ ⎞

⎝

= ⎛

^ω

ω

ε μ ε

μ

2^nd harmonic intensity :

Phase matching condition :

) ( )

2

( ^ω

2 k

^ω

k =

^Î

n

⁽²^ω⁾

= n

⁽^ω⁾

θ_m optic axis

) (ω

k

결정과 광의 편광이 특정 각도 조건을 만족해야 함.

(10)

. .

2차 비선형 효과의 응용

- New wave generation : SHG, SFG, DFG

- Twin photons generation : parametric down conversion - Modulator, switch : electro-optic effect

(11)

. .

Parametric down conversion

(12)

. .

Electrooptic modulator

input

polarizer electrooptic crystal quarter wave plate polarizer V

(13)

. .

3^rd order nonlinear effect

- THG(third harmonic generation) - Phase conjugate wave generation - Self focusing

- Saturated absorption - Optical bistability

ω ω

ω

ω + + → 3

(14)

. .

Phase conjugate wave generation by four wave mixing

pump signal

pump conjugate

2

0

1

+ k = k

4

0

3

Phase matching condition :

= + k k

Nonlinear crystal

.

~ .

C C e

E

_s

=

_s ⁻ⁱ^ω^t

+

Signal wave :

.

~

*

.

C C e

E

_c

=

_s ⁻ⁱ^ω^t

+

where,

E

_s^*

= e ˆ

^*

A

_s^*

e

⁻^ik^s^r

Conjugate wave :

where, ^ik ^r

s s

e

s

A e E = ˆ

wave front reversal beam direction reversal

(15)

. .

응용 : Aberration correction

wave front

ordinary mirror PCM(phase conjugated mirror)

(16)

. .

(17)

. .

응용 : optical resonator

2회 왕복시 마다 자동적으로 위상이 일치

Î 공진기 길이에 무관하게 항상 공진조건 만족

1

2 3

9 87

6 5 4

Phase change :

φ

R

φ φ

₂

=

₁

+

l R

l

φ φ φ

φ φ

φ

₃

=

₂

+ =

₁

+ +

α φ

φ φ

α φ

φ

₄

= −

₃

+ = − (

₁

+

_R

+

_l

) +

…

φ

l

φ α

φ

₈

= −

₇

+ =

₁

−

1 8

9

φ φ φ

φ = +

_l

=

(18)

. .

Self focusing

I n n

n =

₀

+

₂

2 > 0 n

2 < 0 n

Self focusing

Self defocusing

(19)

. .

Optical bistability

I

s

I / 1

0

= + α α

I

1 ²

I

₃

I

2

2 0 2

1

1 1 2 / ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + +

=

I

s

I TI C

I

2

3

TI

I =

I

3

I

1

응용 : optical memory

비선형 광학

. .