p18 electro optics

7/27/2019 P18 Electro Optics

1/26

18. Electro-optics

18. Electro-optics


2/26

(Introduction)

Linear Optics and Nonlinear OpticsLinear 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;

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

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 superposit ion 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

Thus light cannot see other lights,but, light can control other lights via the nonlinear medium.


3/26

0

21

2 31 2 3 0 1 0 2 0 3

2 3

"

"

"

Polarization :Susceptibili ty :

P E

E E

P P P P E E E

== + + +

= + + + = + + +

+===+=+== 1)1(

0

000

c

vnEEED

(Introduction) Nonlinear effects in Optics

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


4/26

(Introduction)

Second-order Nonlinear effects

Second-harmonic generation (SHG) and rectif ication

(0)

),2()(

2

22

P

PP =

Electro-optic (EO) effect (Pockells effect)

= )()( 22 EPEE optical

{ } { } { }{ }

DCelectricEnEEPP

EEPEEPEP

EP

,22

22

2

2

2

2

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

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

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

EEEEEopticalDCelectrical

>>+=

Three-wave mixing

22 0 2P E

opticaloptical EEE )()( 21 +=

{ } { }

{ }{ })()()(,)()()(

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

21212

21212

2

2

221

2

12

2

2

EEPEEP

EPEP

EP

+

SHG

Frequency up-converter

Parametric amplif ier, parametric oscillator

Index modulation by DC E-field

Frequency doubling

Rectification


5/26

Third-harmonic generation (THG)

{ })()3(,)()()( 332

3 EPEEP

Optical Kerr effect

= )()( 33 EPEE optical

33 0 3P E

Self-phase modulation

Frequency tr ipling

)()()()()()(2

3

InEIEEP Index modulation by optical Intensity

)()( 000 nLkInnn =+=+=

{ } { } 00 )()( nxInxInnn >+= Self-focusing, Self-guiding (Spatial soli tons)

{ } { } 00 )()( nxInxInnn >+=

2

DC,

2

DC,3 )0()()0()( electricelectric EnEEP Index modulation by DC E2

(Introduction)

Third-order Nonlinear effects


6/26

Four-wave mixing

33 0 3P E

opticalopticalopticalEEEE )()()( 321 ++=

( ) terms2166,, 333213

3 = EP

Frequency up-converter

Degenerate four-wave mixing

)()()()(: 32143213 EEEPexampleOne ++

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

2143213 EEEPexampleAnother +

4321 ===

If

34321 === If THG

4321 +=+

waves among them are

traveling in opposite directions

If we assume two

plane waves

)()()()( *43 EEEP = Optical phase conjugation

(Introduction)

Third-order Nonlinear effects


7/26

18.1 Principles of Electro-optic effects18.1 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 :

The refractive index changes in proportion to the appl ied electric field.

Quadratic electro-opt ic effect or Kerr effect :

The refractive index changes in proportion to the square of the appliedelectric field.


8/26

Pockels effect and Kerr effect

0

21 2 3 "

Polarization :

Susceptibil ity :

P E

E E

== + + + )1( +=n

02

( ) REE rE + +310

3 210 022

( )n E Rnrnn E E

Pokels Effect Kerr Effect


9/26

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

31( )

2

n E n dn n n Er+ =

2

3123

1( )( )

( ) 1

2 ( )

n

d E dn

r dn rn dEd

E

E E

rE

dn

=

= = =

= +

Pockels coefficient (linear electro-optic coefficient)


10/26

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

R


11/26

Electro-optic modulators and switchesElectro-optic modulators and switches

Phase modulators ( Pockels cell) 31

( )2

n E n dn n n Er+ =


12/26

Phase modulators ( Pockels cell)


13/26

Dynamic wave retarders

SA

(n1)

FA

(n2)

V

LPockels cell


14/26

Intensity modulators : Use of an interferometer


15/26

Intensity modulators : Use of crossed polarizers


16/26

Scanners : electro-optic prisms

Position switch


17/26

Directional couplers


18/26

Spatial light modulators (SLM)


19/26

Q-switching lasers


20/26

18.2 Electro-optics of anisotropic media18.2 Electro-optics of anisotropic media

11 22 332 2 2

1 2 3

1 1 1; ;

n n n = = =

ij ji

where =


21/26

Pockels and Kerr coefficients

( 32 = 9 elements )

( 33 = 27 elements )

( 34 = 81 elements )

Impermeabil ity at E = 0

: Linear E-O (Pockels) coefficients

: Quardratic E-O (Kerr) coefficients


22/26

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 as a single indexI= 1, 2,..., 6.

(k, l), k, l= 1,2,3 as a single indexK= 1, 2,..., 6.


23/26

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)


24/26

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

3( ) (0)ij ij ijE r E = +

2 2 2

11 1 22 2 33 3( ) ( ) ( ) 1E x E x E x + + =

113 13 123 63 133 53

223 23 13 213 63 233 43

333 33 13 313 53 323 43

; 0; 0

; 0; 0

; 0; 0

r r r r r r

r r r r r r r

r r r r r r r

= = = = == = = = = =

= = = = = =

E

3( ) 0ijOnly r E for i j =

( )

( )

( )

2 2

11 13 1 13 12

2 2

22 13 2 13 22

2 2

33 13 3 33 32

1(0)

1(0)

1(0)

o

o

e

r E x r E xn

r E x r E xn

r E x r E xn

+ = +

+ = +

+ = +


25/26

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.


26/26

HomeworkHomework

Derive their f inal pr incipal refractive indices, in DETAIL step-by-step.

p18 electro optics

Documents