Characterization technique of crystalline materials for most efficient their application as active element of electro-optic cell in infocommunication systems
In this work we present the experimental technique suitable for the determination of all the linear electro-optic effect (LEOE) tensor components in crystal materials of any symmetry. We describe in details the sample geometries that are needed to determine a complete set of the LEOE tensor components as well as derive the corresponding equations. It gives a possible to perform the electro-optic characterization for all crystalline materials with aim of most efficient their application as active elements of electro-optic cells for infocommunication systems.
Photoelastic and Electro-Optic Properties of Crystals
1. Photoelasticity of Crystals. Introduction.- 1.1. Discovery of the Phenomenon of Photoelasticity.- 1.2. Mathematical Formulation and Neumann's Constants. Pockels' Contribution.- 1.3. A Brief Historical Survey.- 1.3.1. Amorphous Solids.- 1.3.2. Cubic Crystals.- 1.3.3. Uniaxial and Biaxial Crystals.- 2. Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography.- 2.1. Linear Transformations.- 2.1.1. Coordinate Transformations.- 2.1.2. Orthogonality Relations.- 2.1.3. The Determinant of the Matrix [?ij] of the Direction-Cosine Scheme.- 2.2. Matrix Algebra.- 2.2.1. Introduction.- 2.2.2. Matrix Algebra and Coordinate Transformations.- 2.2.3. Some Common Types of Matrices.- 2.2.4. Orthogonal Matrix.- 2.2.5. Matrix Operators and Transformation of Tensor Components.- 2.2.6. The Diagonalization of a Matrix.- 2.3. Vectors and Their Transformation Laws.- 2.3.1. Vector Components and Coordinate Transformations.- 2.3.2. Transformations of Coordinate Differences.- 2.3.3. Transformation Law of Vectors.- 2.4. Tensor Nature of Physical Properties of Crystals and the Laws of Transformation of Cartesian Tensors.- 2.4.1. Concept of a Tensor Property and Some Examples of Tensor Properties.- 2.4.2. Transformation Law of Cartesian Tensors.- 2.4.3. Physical Properties and Crystal Symmetry.- 2.5. Crystal Symmetry and Geometrical Crystallography. The 32 Point Groups.- 2.5.1. The 32 Crystallographic Point Groups: Their Symmetry Elements and Some Examples of Crystals.- 2.5.2. Some Symmetry Operations and Their Representation by Symbols.- 2.5.3. The 32 Crystallographic Point Groups in the Schonflies Notation. Geometric Derivation.- 2.6. Symmetry Operations and Their Transformation Matrices.- 2.7. Symmetry Elements of the 32 Point Groups.- 2.7.1. Symmetry Elements of the 32 Point Groups.- 2.7.2. Comments on the 32 Crystallographic Point Groups and Their Symmetry Elements as Listed in Tables 2.3 and 2.5a.- 2.8. Neumann's Principle and Effect of Crystal Symmetry on Physical Properties.- 3. Pockels' Phenomenological Theory of Photoelasticity of Crystals.- 3.1. Introduction.- 3.2. Phenomenological Theory, Stress-Optical and Strain-Optical Constants in Four- and Two-Suffix Notations qij and pij Matrices for the 32 Crystallographic Point Groups.- 3.2.1. The Assumptions Forming the Basis of Pockels' Theory.- 3.2.2. Mathematical Formulation of Photoelasticity in Terms of qijkl and pijkl.- 3.2.3. Mathematical Formulation of Photoelasticity in Terms of qij and pij.- 3.2.4. Crystal Symmetry and the Number of Photoelastic Constants.- 3.3. Derivation of the Nonvanishing and Independent Photoelastic Constants for the Various Crystal Classes by Different Methods.- 3.3.1. Classical Method.- 3.3.2. Tensor Method.- 3.3.3. Group Theoretical Method.- 4. Elasticity of Crystals.- 4.1. Introduction.- 4.2. Stress and Strain as Tensors.- 4.2.1. Stress as a Second-Rank Tensor.- 4.2.2. Strain as a Second-Rank Tensor.- 4.3. Hooke's Law.- 4.3.1. Generalized Form of Hooke's Law with Elastic Constants cij and sij and the Matrices of cij and sij for the 32 Point Groups.- 4.3.2. Generalized Form of Hooke's Law with Elastic Constants cijkl and sijkl.- 4.3.3. Interrelation between cijkl and cmn and between sijkl and smn.- 4.4. Experimental Methods of Determining cij and sij Christoffel's Equation and Its Use in Determining cij of Crystals.- 4.5. Ultrasonics.- 4.5.1. Introduction.- 4.5.2. Diffraction of Light by Liquids Excited Ultrasonically.- 4.5.3. Optical Methods of Determining the Ultrasonic Velocities and Elastic Constants of Transparent Solids Employing the Schaefer-Bergmann Pattern, the Hiedemann Pattern, and the Lucas-Biquard Effect.- 4.5.4. Mayer and Hiedemann's Experiments.- 4.5.5. Raman-Nath Theory of Diffraction of Light by Ultrasonic Waves.- 4.5.6. Doppler Effect and Coherence Phenomenon.- 4.6. Brillouin Effect and Crystal Elasticity.- 4.6.1. Introduction.- 4.6.2. Theory of Light Scattering in Birefringent Crystals.- 4.6.3. Concluding Remarks.- 5. Experimental Methods of Determining the Photoelastic Constants.- 5.1. Optical Behavior of a Solid under a Mechanical Stress, and Neumann's Constants.- 5.2. Derivation of Expressions for the Stress Birefringence in Terms of qij for Cubic and Noncubic Crystals.- 5.2.1. Stress Birefringence in Cubic Crystals.- 5.2.2. Stress Birefringence in Noncubic Crystals.- 5.2.3. Tensor Method of Deriving q?ijkl in Terms of qmnop.- 5.2.4. Expression for the Change of Thickness in Terms of sij for an Orthorhombic Crystal for a Specific Orientation.- 5.3. Experimental Determination of qij and pij by Optical Methods.- 5.3.1. Measurement of Stress Birefringence, and Relative Path Retardation.- 5.3.2. Measurement of Absolute Path Retardation by Interferometric Methods.- 5.3.3. Photoelastic Studies of Optically Active Crystals.- 5.4. Dispersion of qij by Spectroscopic Methods.- 5.4.1. Birefringent Compensator for Studying Very Small Changes in Double Refraction.- 5.4.2. Dispersion of the Individual Stress-Optical Coefficients q11 and q12 of Vitreous Silica.- 5.4.3. Interference-Spectroscope Method of Studying the Absolute Photoelastic Coefficients of Glasses and Their Variation with Wavelength.- 5.5. Elliptic Vibrations and Elliptically Polarized Light.- 5.5.1. Composition of Two Rectangular Vibrations Giving an Ellipse: Use of the Senarmont Compensator.- 5.5.2. Photometric Method for the Measurement of Photoelastic Birefringence.- 5.5.3. The Poincare Sphere and Its Application to the Study of the Photoelastic Behavior of Optically Active Crystals.- 5.6. Ultrasonic Methods of Studying the Elasto-Optic Behavior of Crystals.- 5.6.1. Introduction.- 5.6.2. Mueller's Theory.- 5.6.3. Experimental Determination of Pij/pkl by Three Different Methods Due to Mueller.- 5.6.4. Pettersen's Method of Determining Pij/pkl.- 5.6.5. Bragg Diffraction Method of Determining the Individual Values of pij.- 5.6.6. Borrelli and Miller's Method of Determining the pij of Glass.- 5.6.7. Technological Applications of the Acousto-Optic Effect.- 5.7. Brillouin Scattering and Photoelasticity of Crystals.- 6. Atomistic Theory of Photoelasticity of Cubic Crystals.- 6.1. Introduction.- 6.2. Mueller's Theory-A Brief Survey.- 6.3. Effect of Hydrostatic Pressure on the Index of Refraction n The Strain Polarizability Constant ?0.- 6.4. Anisotropy of Rj and ?itj.- 6.5. Thermo-Optic Behavior of Crystals and Photoelastic behavior.- 6.6. Pockels' Photoelastic Groups in Cubic Crystals and Mueller's Theory.- 6.7. Photoelastic Dispersion in Cubic Crystals ?0 as a Function of Crystalline Material, Wavelength of Light, and Temperature.- 6.8. Effect of Elastic Deformation on the Oscillator Strengths and Dispersion Frequencies of Optical Electrons.- 6.9. Temperature Dependence of Stress-Optical Dispersion.- 6.10. Reversal of the Sign of Stress Birefringence in Pure and Mixed Crystals.- 6.10.1. Pure Crystals.- 6.10.2. Mixed Crystals of KCl and KBr.- 6.11. Stress-Optical and Strain-Optical Isotropy in Cubic Crystals.- 6.12. Optic Axial Angle and Its Dispersion in Stressed Cubic Crystals of T and Th Classes.- 7. Piezoelectricity.- 7.1. Introduction.- 7.2. Direct and Converse Piezoelectric Effects.- 7.3. Mathematical Formulation, Piezoelectric Constants dijk in Tensor Notation, and dij in Two-Suffix Notation Relation between dijl and dij.- 7.4. Deduction of the Surviving dijk for Some Crystal Classes by Tensor Method, and the dij Matrices for the 21 Noncentrosymmetric Classes.- 7.5. Concluding Remarks.- 8. Electro-Optic Effects in Crystals: Pockels Linear Electro-Optic and Kerr Quadratic Electro-Optic Effects.- 8.1. Introduction.- 8.2. Demonstration of the Electro-Optic Effects, Linear and Quadratic.- 8.3. Historical Survey.- 8.3.1. Earlier Work.- 8.3.2. More Recent Work.- 8.4. Pockels' Phenomenological Theory of the Linear Electro-Optic Effect in Three- and Two-Suffix Notations, Rijk and rij.- 8.5. Derivation of the Relation between the Linear Electro-Optic Constants of a Crystal: Free and Clamped Constants.- 8.5.1. Discussion: Primary and Secondary Electro-Optic Effects, and Clamped and Unclamped Electro-Optic Coefficients.- 8.5.2. Methods of Obtaining the Primary and Secondary Linear Electro-Optic Effects.- 8.6. Kerr Quadratic Electro-Optic Effect: Pockels' Phenomenological Theory.- 8.7. Crystal Symmetry and the Number of Surviving Linear Electro-Optic Coefficients Rijk and rij and Their Deduction by Tensor Method: rij Matrices for the 21 Noncentrosymmetric Classes.- 8.7.1. Crystal Symmetry and the Surviving Linear Electro-Optic Constants.- 8.7.2. Tensor Method of Deducing the Nonvanishing Independent Rijk.- 8.8. Derivation of the Expressions for ? = f(rij) for Some Typical Crystal Classes and Orientations.- 8.8.1. Cubic System: Classes 23(T) and $$\bar 43m$$ (Td).- 8.8.2. Tetragonal System: Class $$\bar 42m$$ (D2d).- 8.8.3. Trigonal System: Class 32 (D3).- 8.9. Experimental Methods of Determining rij.- 8.9.1. General Description and Application to Some Typical Crystal Classes.- 8.9.2. Some Experimental Methods.- 8.9.3. Methods of Applying the Electric Field to the Crystal Prism.- 8.9.4. Experimental Determination of rij in Some Specific Cases of Crystals.- 8.10. Some Points of Interest on the Use of the Pockels Effect in Crystals, and Half-Wave Voltage V?/2.- 8.11. Some Technological Applications of Pockels Cells (Linear Electro-Optic Devices).- 8.11.1. Use of the Electro-Optic Effect in Technology.- 8.11.2. Some Applications of Electro-Optic Devices.- Author Index.
Birefringence of crystals and its temperature-variation
SummaryIt is suggested that the birefringence of crystalline bodies can be explained as arising from the existence of polarised electronic transitions, so that the probability of transition is different for different directions of the incident electric vector. As a result, in the dispersion formulæ for the three principal refractive indices of a biaxial crystal, the oscillator-strengths will be different, although the dispersion frequencies are the same. The application of the idea to the cases of calcite and aragonite (two strongly birefringent crystals) enables one to construct dispersion formulæ involving three ultra-violet frequencies at 1535, 1000 and 500 Å.U. The formulæ, which fit the dispersion data for both rays of calcite from 0·2 to 3 μ, show that a large part of the birefringence arises from the large anisotropy in the strength of the nearest ultra-violet frequency at 1535 Å.U. A similar result is also found for aragonite.The dispersion formulæ are also successful in explaining the thermooptic behaviour of both calcite and aragonite, when they are utilised in the author’s theory of thermo-optic behaviour, which has so far been applied only for isotropic solids. For anisotropic solids, an additional principle has to be used,viz., that, while the total oscillator strength does not alter with temperature, the individual strengths along the three principal directions can change, resulting in a mutual transfer. It is suggested that this transfer is related to the differences in thermal expansion along the three directions, which finds some support in the case of aragonite. A physical explanation of the transfer can be found, wholly or partly, in the tilting oscillations of the CO3 ion.The theory is successful in explaining the remarkable fact that, whiledn/dt for both indices of calcite is positive, that for all the indices of aragonite is negative, the difference being attributable to the much larger coefficient of thermal expansion of the latter.