chapter 8 electro-optic ceramics 학번 : 2003127004 성명 : 장 성 수

Chapter 8Electro-Optic Ceramics

학번 : 2003127004 성명 : 장 성 수

8.1 Background Optics Electromagnetic Wave Theory of Light

- James Clerk Maxwell, Andre Ampere, Karl Gauss, Michael Faraday

Maxwell’s Equations1. How an electromagnetic wave originates from an

accelerating charge and propagates in free space2. An electromagnetic wave in the visible part of the spectrum

may be emitted when an electron changes its position relative to the rest of an atom, involving a change in dipole moment

3. Light can be emitted from a single charge moving at high speed under the influence of a magnetic field

The radiation is emitted naturally from regions of the Universe An electromagnetic wave in free space

1. An electric field E and a magnetic induction field B which vibrate in mutually perpendicular directions in a plane normal to the wave propagation direction

2. The E vector of a single sinusoidal plane-polarized wave propagating in the z direction (Fig. 1)

8.1 Background Optics (cont.)

Radiation from a single atom persists in phase and polarization for a time of the order of s

- A random mixture of polarizations and phase as well as a wide range of wavelengths

The light can be obtained with a specified polarization, a coherence persisting for times in the neighborhood of s and a line-width of less that 10Hz at frequency of Hz

Fig. 8.1 E vector of a wave polarized in the y-z plane and propagating in the z direction towards the observer

810

510

1410

8.1.1 Polarized Light

The various forms of polarization- Could be understood by considering a plane polarized wave

traveling in the z direction- One polarized in the x-z plane and the other in the y-z plane

(Fig.2)

Fig. 8.2 The two components of a right elliptically polarized wave


A phase difference can be established between the two components

- By passage through a medium with an anisotropic refractive index such that the velocity of the y-z wave is greater than that of the x-z wave

- The y-z wave to lead the x-z wave by a distance - , by a phase angle , will remain constant

when the light emerges into an isotropic medium Electric fields

- X wave : (8.1) - Y wave : (8.2)

where , are amplitude k : propagation number

is the angular frequency

z /2 z

)sin(0 tkzEE xx )sin(0 tkzEE yy

xE0 yE0

)/2( k


For simplicity, we put z=0 in equations (8.1) and (8.2) (8.3)

(8.4)

Substituting from equation (8.3) into (8.4)

(8.5)

; Ellipse with semiminor and semimajor axes Eoy and Eox

(8.6)

xx EEt 0/)sin(

)sin()sin( 00 tEtEE xxx )sin()sin( 00 tEtEE yyy

sin)cos(cos){sin(0 ttEE yy

2

00

2

0

2

0sincos))((2)()(

y

y

x

x

y

y

x

x

E

E

E

E

E

E

E

E

220

00 cos2)2tan(

oyx

yx

EE

EE


1. When

(8.7)

2. When

,

(8.8)

1)()( 2

0

2

0

y

y

x

x

E

E

E

E2/

0

02)()(00

2

0

2

0

y

y

x

x

y

y

x

x

E

E

E

E

E

E

E

E0)( 2

00

y

y

x

x

E

E

E

E

xx

yy EE

EE

0

0


Fig. 8.3 How elliptically polarized light depends on the phase difference between plane-polarized components

Fig. 8.3 : The paths described by the tip of E for the various values of

The sense of the polarization is taken to be the sense of rotation of the E vector for coming light (Fig. 8.3)

- (a) & (e) : plane-polarized light- (b) : right elliptically polarized light- (f) : left elliptically polarized light

8.1.2 Double Refraction The optical and electro-optical properties of dielectrics

- Determined by their refractive indices or, their permittivities In an isotropic dielectric (such as glass)

- The induced electrical polarization is always parallel to the applied electrical field and the susceptibility is an scalar

- The three components of the polarization

(8.9)

(8.10)

(8.11)

)( 1312110 zyxx ExExExP )( 2322210 zyxy ExExExP )( 3332310 zyxz ExExExP

zyxx EEED 131211 zyxy EEED 232221 zyxz EEED 333231

jiji ED

8.1.2 Double Refraction

In an anisotropic dielectric the phase velocity of an electromagnetic wave

- Depends on its polarization and its direction of propagation - The solution to Maxwell’s electromagnetic wave equations for a plane wav

e show that it is the vectors D and E The classical example of an anisotropic crystal is calcite (CaCO3)

- First recorded observation of ‘double refraction’- The particular arrangement of atoms in calcite, light generally propagates a

t a speed depending on the orientation of its plane of polarization relative to the crystal structure

- For one particular direction, the optic axis, the speed of propagation is dependent of the orientation of the plane of polarization

Crystal may have two optic axes- Orthorhombic, monoclinic and triclinic crystals are biaxial- Hexagonal, tetragonal and trigonal crystals are uniaxial- Cubic crystals are isotropic

8.1.2 Double Refraction

The situation can be analyzed more closely by considering a source of monochromatic light located at S in Fig. 8.4

- One with a spherical surface and the other with an ellipsoidal surface- A wavefront is the locus of points of equal phase, i.e. the radii (e.g. SO an

d SE), which are proportional to the ray speeds and inversely proportional to the refractive indices

- Fig. 8.4 is a principal section of the wavefront surface- The ordinary or o rays : the electrical displacement component of the wave

vibrates at right angles to the principal section travel at a constant speed irrespective of direction

- The extraordinary or e rays : the electric displacement component lies in the principal section travel at a speed which depends on direction

- The refractive index ne for an e ray propagating along SX is one of the two principal refractive indices of a uniaxial crystal; the other, no refers to the o rays

8.1.2 Double Refraction The velocity of the o ray is less t

han that of the e ray, except for propagation along the optic axis in Fig. 8.4

- ne-no < 0, a situation defined as negative birefringence

- TiO2 : -0.18 ~ +0.29- Ferroelectrics : -0.01 ~ -0.1

The optical theory of crystals in terms of the relative impermeability tensor , which is a second-order

(8.12)

Fig. 8.4 A principal section of the wavefront surfaces for a uniaxial crystal; the dots on SO represent the E (and D) vectors which are normal to the plane of the paper

ijij rB )( 1

1jiij xxB

8.1.2 Double Refraction The representation quadric for the relative impermeability tensor (8.13)

B1 etc. are the principal relative impermeabilities

(8.14)

n1 is the refractive index for light whose dielectric displacement is parallel to x1

(8.15)

Equation (8.15) is known as the optical indicatrix. This is an ellipsoidal surface, as shown in Fig.8.5, and n1, n2 and n3 are the principal refractive indices of the crystal

1233

222

211 xBxBxB

etcn

B r2

11

11

1)(

12

3

23

22

22

21

21

n

x

n

x

n

x

8.1.2 Double Refraction In Fig. 8.5, if OP is an arbitrary directi

on, the semiminor and semimajor axes OR and OE of the shaded elliptical section normal to OP are the refractive indices of the two waves propagated with fronts normal to OP

For a uniaxial crystal the indicatrix is symmetrical about the principal symmetry axis of the crystal-the optic axis

(8.16)

Fig. 8.5 The optical indicatrix

12

0

23

20

22

20

21

n

x

n

x

n

x

8.1.3 The Electro-Optic Effect Fig. 8.6 : The actual response of

a dielectric to an applied field- The intense fields associated wit

h high power laser lightlead to the non-linear optics technology

- The electro optic effect has its origins in the non-linearity

- The permittivity measured for small increments in field depends on the biasing field E0 from which it follows that the refractive index depends on E0

(8.17) no : the value measured under zero bi

asing fielda,b : constants ...2

000 bEaEnn

Fig. 8.6 Non-linearity in the D versus E relationship

8.1.3 The Electro-Optic Effect Kerr Effect :

- Experiments on glass and detected electric field-induced optical anisotropy- A quadratic dependence of n on Eo is known ‘Kerr-Effect’

Pockels Effect : a linear electro-optic effect in quartz The small changes in refractive index caused by the application of an elec

tric field- Can be described by small changes in the shape, size and orientation of the o

ptical indicatrix- Can be specified by changes in the coefficients of the indicatrix

(8.18)(8.19)

: Pockels electro-optic coefficients, : Kerr coefficients

- (8.20) (8.21)

lkijklkijkij EERErB

0

k

ijkijk

rf

ijB

lkijklkijkij PPgPfB

))(( 00

lk

ijklijkl

Rg

ijkfijkrijklR ijklg

8.1.3 (a) The Pockels Electro-Optic Effect

Single crystal BaTiO3 (T < Tc) - At temperature below Tc, BaTiO3 belongs to the tetragonal crystal class (sym

metry group 4mm)- Optically uniaxial and the optic axis is the x3

- When an electric field is applied in an arbitrary direction, the representation quadric for the relative impermeability

(8.22)where

(8.23)in which k takes the value 1,…,3

- Considering of crystal symmetry,- The electric field E is directed along the x3 axis, so E1=E2=0

(8.24)

1)( jiijij xxBB

233

21

222

21

211

21

)1

()1

()1

( xErn

xErn

xErn

kkkkkk

kijkij ErB

1222 621513432 kkkkkk ErxxErxxErxx

51421323 , rrrr

1)1

()1

()1

( 2333

20

2213

20

2113

20

xErn

xErn

xErn


- In comparison with equation (8.22)

(8.22)

since

(8.25)

- The induced birefringence is , where

(8.26)

where (8.27)

eeo nnnn ,0

Ern

Ern e

332

132

0)

1(,)

1(

ErnnErnn ee 333

133

002

1,

2

1

nEr

n

nrnnnnnn

eeoee )(

2

1)( 13

3

30

333

0

Ern c3

2

1

133

30

33 rn

nrr

ec


Policrystalline ceramic - The form of the electro-optic tensor for 6mm symmetry is identical with th

at for the 4mm symmetry- The induced birefringence for a field directed along the x3 axis

, where(8.22)

Table 8.1 Properties of some electro-optic materials

Ern c3

2

1 1333 rrrc

8.1.3 (b) The Kerr Quadratic Electro-Optic Effect

Single crystal BaTiO3 (T > Tc) - At temperature above 130℃, BaTiO3 is cubic (symmetry group m3m)- An electric field is applied in an arbitrary direction the representation quad

ric is perturbed

(8.29)where

- Because the material is isotropic, the electric field can be directed along the x3 axis without any loss in generality, and E1=E2=0

- Symmetry requires R11=R22=R33, R12=R13=R23=R31=R32, R44=R55=R66, and the remaining components to be zero

(8.30)

1 ijij BB

1)1

()1

()1

( 211

22

32

122

22

212

22

1 ERn

xERn

xERn

x

22

123

112

1nERnnnnn

lkijklij EERB

211

333

2

1ERnnnnn

8.1.3 (b) The Kerr Quadratic Electro-Optic Effect

- The induced birefringence

(8.31)

(8.32)

Polycrystalline ceramic - Polycrystalline ceramic (6mm) the form of the electro-optic tensor is

the same as that for m3m symmetry except that

(8.33)

(8.34)

21211

3

220

23

)(2)1(

)1211(

2Pgg

nPRRn

r

2)1211

3130 (

2

1ERRnnnnnn e

21211

3

)(2

ERRn

n

21211

3

)(2

Pggn

n

8.1.4 Non-Linear Optics Second harmonic generation

- The non-linearity in the response of a dielectric to an applied field

(8.35)- The linear susceptibility is much greater than the coefficients of

the higher-order terms etc.- The higher-order terms are significant only when strong fields in the

range are applied- Suppose that laser light of sufficient intensity is incident on a non-

linear optical material and that the time dependence of the electric field is given by

(8.36)

: the response expected from a linear dielectric

)sin(0 tEE

...)( 33

2210 EEEP

12

1128 1010 Vm

...})(sin)(sin)sin({0 3303

220201 tEtEtEP

)}3sin()sin(3{4

1)}2cos(1{

2

1)sin( 3

0302

020010 ttEtEtE

)sin(010 tE

8.1.4 Non-Linear Optics

: a constant polarization which would produce a voltage across the material, i.e. rectification

: a variation in polarization at twice the frequency of the incident wave

- The process of frequency doubling The wavelength of the polarization wave is given

(8.37)where c : the velocity of light in a vacuum : the frequency of the incident light, n1 : the refractive index at that frequency

(8.38)

where n2 : the refractive index at the second harmonic frequency

2020

2

1E

)2cos(2

1 2020 tE

12 n

cp

p

12 n

cp

8.1.4 Non-Linear Optics Frequency mixing

- ‘mix’ frequencies when light beams differing in frequency are made to follow the same path through a non-linear medium

- If two waves with electric field components and

follow the same path through a dielectric

(8.39)

- The first two terms in braces describe SHG and the last term describes waves of frequency and

- The output of a wave of frequency is known as ‘up-conversion’ and has been used in infrared imaging

- Fig. 8.7 : infrared laser light reflected from an object can be mixed with a suitably chosen infrared reference beam to yield an up-converted frequency in the visible spectrum

)sin( 11 tE

)( 2210 EEP

)}sin()sin(2)(sin)(sin{ 2121222

2122

120 ttEEtEtE

21

)sin( 22 tE

21

8.1.4 Non-Linear Optics

Fig. 8.7 Up-conversion of infrared to the visible frequency range

8.1.5 Transparent Ceramics For a ceramic

- As an electro-optic material, it must be transparent- Ceramic dielectrics are mostly white and opaque, due to the scattering of

incident light- Scattering occurs because of discontinuities in refractive index which will

usually occur at phase boundaries and, if the major phase itself is optically anisotropic at grain boundaries

- For transparency, a ceramic should consist of a single phase fully dense material which is cubic or amorphous

- Rayleigh’s expression

(8.41)- For the intensity of light scattered through an angle by a dispersion

of particles of radius r and refractive index np in a matrix of refractive index nm, where np-nm is small

- : the incident intensity, : measured at a distance x from the particles, wavelength of the scattered light

2422

2

)()(cos1

m

mp

o n

nnrr

xI

I

I

0I I

chapter 8 electro-optic ceramics 학번 : 2003127004 성명 : 장 성 수

Documents