p18 electro optics
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18. Electro-optics
18. Electro-optics
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(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.
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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
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(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
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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
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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
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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.
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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
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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)
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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 switchesElectro-optic modulators and switches
Phase modulators ( Pockels cell) 31
( )2
n E n dn n n Er+ =
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Phase modulators ( Pockels cell)
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Dynamic wave retarders
SA
(n1)
FA
(n2)
V
LPockels 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 media18.2 Electro-optics of anisotropic media
11 22 332 2 2
1 2 3
1 1 1; ;
n n n = = =
ij ji
where =
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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
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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 as a single indexI= 1, 2,..., 6.
(k, l), k, l= 1,2,3 as a single indexK= 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 = 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
+ = +
+ = +
+ = +
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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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HomeworkHomework
Derive their f inal pr incipal refractive indices, in DETAIL step-by-step.
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