Linear Optics and Nonlinear Optics

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18. Electro-optics

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(Introduction)

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 see other lights.

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 cannot see other lights,

but, light can control other lights via the nonlinear medium.

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0

2 1

2 3

1 2 3 0 1 0 2 0 3

2 3

"

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

=

= + + +

= + + + = + + +

ε χ χ ε

ε ε χ

ε ε

ε = + → = + → = = = +

= (1 ) 1

0 0

0

0 c

n v E

E E

D

(Introduction)

Nonlinear effects in Optics

Here we will discuss on electro-optic Pockels and Kerr effects

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(Introduction)

Second-order Nonlinear effects

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

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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

+Δ

ϕ

= 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

(Introduction)

Third-order Nonlinear effects

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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

ω

₁

ω

₂

ω

₃

ω

₄ E

ω

₁ E

ω

₂ E

ω

₃ example

One + + ≡ ∝

→

) ( ) ( ) ( )

- (

: P₃

ω

₁

ω

₂

ω

₃

ω

₄ E

ω

₁ E

ω

₂ E^*

ω

₃ example

Another + ≡ ∝

→

4 3

2

1

ω ω ω

ω

= = =

→ If

ω ω

ω

₁ = ₂ = ₃ → ₄ =3

→ If Æ THG

4 3

2

1

ω ω ω

ω

+ = +

→

waves among them are traveling in opposite directions

If we assume two plane waves

→

) ( ) ( ) ( )

( ₄ ^*

3

ω ω

E

ω

E

ω

E

ω

P = ∝

→ ^ÆOptical phase conjugation

(Introduction)

Third-order Nonlinear effects

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18.1 Principles of Electro-optic effects 18.1 Principles of Electro-optic effects

The electro-optic effect is the change in the refractive index

resulting from the application of a DC or low-frequency electric field.

Linear electro-optic effect or Pockels effect :

Æ The refractive index changes in proportion to the applied electric field.

Quadratic electro-optic effect or Kerr effect :

Æ The refractive index changes in proportion to the square of the applied electric field.

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Pockels effect and Kerr effect

0

2

1 2 3

"

Polarization : Susceptibility :

P E

E E

ε χ

χ χ χ χ

=

= + + + n = ( 1 + χ )

0

( ) E rE RE

2

η = η + +

1 3

0

3 2 1

0 2 2 0

( )

n E = n − rn E − Rn E

Pokels Effect Kerr Effect

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

1 3

( ) 2

n E = + n dn = − n r n E

2

1 3 3 2

( 1 )

( )

( ) 1

2 ( )

n

d E dn

r dn rn dE

d E

E E

rE

n d

η

η η

η

=

= − = ⇒ = −

= +

Pockels coefficient (linear electro-optic coefficient)

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Kerr effect (Quadratic electro-optic effect) Kerr effect (Quadratic electro-optic effect)

R

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Electro-optic modulators and switches Electro-optic modulators and switches

Phase modulators ( Pockels cell)

1

³

( ) 2

n E = +n dn = −n rn E

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Phase modulators ( Pockels cell)

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Dynamic wave retarders

SA (n1)

FA (n2)

V

L

Pockels cell

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Intensity modulators : Use of an interferometer

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Intensity modulators : Use of crossed polarizers

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Scanners : electro-optic prisms

Position switch

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Directional couplers

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Spatial light modulators (SLM)

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Q-switching lasers

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18.2 Electro-optics of anisotropic media 18.2 Electro-optics of anisotropic media

11 2 22 2 33 2

1 2 3

1 1 1

; ;

n n n

η η η

⇒ = = =

_ij _ji where η =η

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Pockels and Kerr coefficients

( 3² = 9 elements )

( 3³ = 27 elements )

( 3⁴ = 81 elements ) Impermeability at E = 0

: Linear E-O (Pockels) coefficients

: Quardratic E-O (Kerr) coefficients

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Symmetry in Pockels and Kerr coefficients

6 independent elements

(6 x 3) independent elements (6 x 6) independent elements

It is conventional to rename the pair of indices

(i, j), i, j = 1,2,3 Æ as a single index I = 1, 2,..., 6.

(k, l), k, l = 1,2,3 Æ as a single index K = 1, 2,..., 6.

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Pockels effect

The index ellipsoid is modified as a result of applying a steady electric field.

To determine the optical properties of an anisotropic material exhibiting the Pockels effect,

(that is, to find modified principal refractive indices⁾

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Example 18.2-1. Find the index change of uniaxial crystal by E = E

_z

( ) (0) 3

ij E ij r Eij

η =η +

2 2 2

11( )E x1 22( )E x2 33( )E x3 1

η +η +η =

113 13 123 63 133 53

223 23 13 213 63 233 43

333 33 13 313 53 323 43

; 0; 0

r r r r r r

r r r r r r r

= = = = =

= = = = = =

E

_ij3( ) 0

Only r E ≠ for i = j

( )

2 2

11 13 1 2 13 1

2 2

22 13 2 2 13 2

2 2

33 13 3 2 33 3

(0) 1

o

e

r E x r E x

n

r E x r E x

n

r E x r E x

n η

η η

⎛ ⎞

+ =⎜ + ⎟

⎝ ⎠

⎛ ⎞

+ =⎜ + ⎟

⎝ ⎠

⎛ ⎞

+ =⎜ + ⎟

⎝ ⎠

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Example 18.2-1.

E

When an electric field is applied along the optic axis of this uniaxial crystal, it remains uniaxial with the same principal axes,

but its refractive indices are modified.

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Homework Homework

Derive their final principal refractive indices, in DETAIL step-by-step.