Chapter 8 Electro-Optic Ceramics 학번 : 성명 : 장 성 수

8.1 Background Optics  Electromagnetic Wave Theory of Light -James Clerk Maxwell, Andre Ampere, Karl Gauss, Michael Faraday  Maxwell’s Equations 1. How an electromagnetic wave originates from an accelerating charge and propagates in free space 2. 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  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  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

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

8.1.1 Polarized Light  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

8.1.1 Polarized Light  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)

8.1.1 Polarized Light 1. When (8.7) 2.When, (8.8)

8.1.1 Polarized Light 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)

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 wave show that it is the vectors D and E  The classical example of an anisotropic crystal is calcite (CaCO 3 ) - First recorded observation of ‘double refraction’ - The particular arrangement of atoms in calcite, light generally propagates at 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 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 and 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 n e for an e ray propagating along SX is one of the two principal refractive indices of a uniaxial crystal; the other, n o refers to the o rays

8.1.2 Double Refraction  The velocity of the o ray is less than that of the e ray, except for propagation along the optic axis in Fig n e -n o < 0, a situation defined as negative birefringence - TiO2 : ~ Ferroelectrics : ~ -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

8.1.2 Double Refraction  The representation quadric for the relative impermeability tensor (8.13) B1 etc. are the principal relative impermeabilities (8.14) n 1 is the refractive index for light whose dielectric displacement is parallel to x 1 (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

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

8.1.3 The Electro-Optic Effect  Fig. 8.6 : The actual response of a dielectric to an applied field - The intense fields associated with 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 E 0 from which it follows that the refractive index depends on E 0 (8.17) no : the value measured under zero biasing field a,b : constants 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 electric field - Can be described by small changes in the shape, size and orientation of the optical 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)

8.1.3 (a) The Pockels Electro-Optic Effect  Single crystal BaTiO 3 (T < Tc) - At temperature below Tc, BaTiO3 belongs to the tetragonal crystal class (symmetry group 4mm) - Optically uniaxial and the optic axis is the x 3 - 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 x 3 axis, so E 1 =E 2 =0 (8.24)

8.1.3 (a) The Pockels Electro-Optic Effect -In comparison with equation (8.22) (8.22) since (8.25) - The induced birefringence is, where (8.26) where (8.27)

8.1.3 (a) The Pockels Electro-Optic Effect  Policrystalline ceramic - The form of the electro-optic tensor for 6mm symmetry is identical with that 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

8.1.3 (b) The Kerr Quadratic Electro-Optic Effect  Single crystal BaTiO 3 (T > Tc) - At temperature above 130 ℃, BaTiO 3 is cubic (symmetry group m3m) - An electric field is applied in an arbitrary direction the representation quadric 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 R 11 =R 22 =R 33, R 12 =R 13 =R 23 =R 31 =R 32, R 44 =R 55 =R 66, and the remaining components to be zero (8.30)

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)

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

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, n 1 : the refractive index at that frequency (8.38) where n 2 : the refractive index at the second harmonic frequency

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

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 n p in a matrix of refractive index n m, where n p -n m is small - : the incident intensity, : measured at a distance x from the particles, wavelength of the scattered light