Crystal Optics: Properties and Applications E-Book

Ashim Kumar Bain

• Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 962

Veröffentlichungsjahr: 2019

Beschreibung

Reviews the properties and applications of photo-elastic, acousto-optic, magneto-optic, electro-optic, and photorefractive materials

This book deals with the basic physical properties and applications of photo-elastic, acousto-optic, magneto-optic, electro-optic, and photorefractive materials. It also provides up-to-date information on the design and applications of various optoelectronic devices based on these materials.

The first chapter of Crystal Optics: Properties and Applications covers the basic concepts of crystal optics, such as index ellipsoid or optical indicatrix, crystal symmetry, wave surface, birefringence, and the polarization of light. Chapter 2 reviews the physical phenomena of crystal optics in isotropic and crystalline materials. It describes in detail research information on modern photoelastic materials and reviews the up-to-date photoelastic device applications. Chapter 3 develops the underlying theory of acousto-optics from first principles, formulating results suitable for subsequent calculations and design. The fourth chapter describes the basic principles of magneto-optic effects and mode of interaction with magnetic materials. The fifth chapter provides an understanding of the physical phenomenon of the linear and quadratic electro-optic effects in isotropic and crystalline materials. The last chapter collects many of the most important recent developments in photorefractive effects and materials, and pays special attention to recent scientific findings and advances on photorefractive materials and devices.

-Features up to date information on the design and applications of various optoelectronic devices
-Looks at the basic concepts of crystal optics, including the polarization of light, effects of reflection and transmission of polarization and light polarizing devices, and more
-Pays special attention to design procedures for the entire range of acousto-optic devices and various applications of these devices
-Provides research information on modern magneto-optic materials and reviews the up-to-date magneto-optic device applications?up to terahertz (THz) regime

Crystal Optics: Properties and Applications is an excellent book for the scientific community working in the field, including researchers, lecturers, and advanced students.

Leseprobe

Table of Contents

Cover

Preface

Overview

1 Crystal Optics

1.1 Introduction

1.2 Index Ellipsoid or Optical Indicatrix

1.3 Effect of Crystal Symmetry

1.4 Wave Surface

1.5 Birefringence

1.6 Polarization of Light

1.7 Changing the Polarization of Light

1.8 Effects of Reflection and Transmission on Polarization

1.9 Light Polarizing Devices

References

2 Photoelasticity

2.1 Introduction

2.2 Principle of Photoelasticity

2.3 History of Photoelasticity

2.4 Phenomenological Theory of Photoelasticity

2.5 Atomic Theory of Photoelasticity

2.6 Photoelastic Devices

2.7 Photoelastic Materials and Applications

References

3 Acousto‐Optics

3.1 Introduction

3.2 Short History of Acousto‐optics

3.3 Principle of Acousto‐optic Effect

3.4 Acousto‐optic Devices

3.5 Acousto‐optic Materials and Their Applications

References

4 Magneto‐optics

4.1 Introduction

4.2 Mode of Interaction

4.3 Magneto‐optic Materials Classification

4.4 Magneto‐optic Devices

References

5 Electro‐optics

5.1 Introduction

5.2 History of Electro‐optic Effects

5.3 Principles of Electro‐optic Effects

5.4 Phenomenological Theory of Electro‐optic Effect

5.5 Electro‐optic Devices

5.6 Electro‐optic Materials and Applications

5.7 Electro‐optic Plasmonic Materials and Applications

References

6 Photorefractive Effect

6.1 Introduction

6.2 Photorefractive Effect

6.3 Applications of Photorefractive Effect

6.4 Photorefractive Materials and Devices

References

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1 Notable events in the history of photoelasticity.

Table 2.2 POCs of LiNbO

3

:MgO and LiNbO

3

crystals in Brewster.

Table 2.3 The main POCs of CaWO

4

and LiNbO

3

crystals (in Brewster).

Table 2.4 The main EOCs of CaWO

4

and LiNbO

3

crystals (in Brewster).

Table 2.5 Stress‐optical coefficients

C

pq

(in Brewsters) of Li

2

Ge

7

0

15

at RT = 298...

Table 2.6 Piezo‐optic coefficients

C

pq

(in Brewsters) for some ferroelectric cryst...

Chapter 3

Table 3.1

Properties of different AOTF crystals.

Table 3.2 Infrared spectrometer for ExoMars main characteristics and resources.

Table 3.3 Characteristic of the SPICAM – IR spectrometer.

Chapter 4

Table 4.1 Properties of selected magneto‐optic materials.

Table 4.2 Measured transmittance for input/output port combinations at a wavelen...

Table 4.3 Characteristics of typical magneto‐optical recording materials.

Chapter 5

Table 5.1 Notable events in the history of electro‐optics.

Table 5.2 Summary of the VOAs performance.

Table 5.3 Key parameters of polarization controllers.

List of Illustrations

Chapter 1

Figure 1.1 The index ellipsoid. The coordinates (

x

,

y

,

z

) are the principal axe...

Figure 1.2 The indicatrix for a (positive) uniaxial crystal.

Figure 1.3 The two circular sections of the indicatrix and the two primary opti...

Figure 1.4 Ray surfaces of uniaxial crystals: (a) positive, (b) negative, (

OZ

) ...

Figure 1.5 Positive uniaxial medium wave surface. Inner fold (left), vertical s...

Figure 1.6 Negative uniaxial medium wave surface. Inner fold (left), vertical s...

Figure 1.7 Principal sections of the wave surface for a biaxial crystal.

Figure 1.8 Wave surface for a biaxial medium. Inner fold (left), vertical secti...

Figure 1.9 Side view of the double refraction of light by a calcite crystal.

Figure 1.10 Double refraction at the boundary of an anisotropic medium.

Figure 1.11 Separation of light waves by a birefringent crystal [3].

Figure 1.12 Schematic view of electromagnetic wave propagation along the

z

‐axis...

Figure 1.13 Propagation of

E

x

and

E

y

components along the

z

‐axis [4].

Figure 1.14 Orientation of

E

vector at various locations along the

z

‐axis [4].

Figure 1.15 The oscillation of

E

vector back and forth along the same 45° line ...

Figure 1.16 Propagation of the unequal

E

x

and

E

y

components along the

z

‐axis [4...

Figure 1.17 Orientation of

E

vector at various locations along the

z

‐axis (when...

Figure 1.18 Propagation of

E

x

and

E

y

components along the

z

‐axis when they are ...

Figure 1.19 Orientation of

E

vector in the

x–y

plane at a fixed time [4].

Figure 1.20 Circular orientation of

E

vector along the

z

‐axis [4].

Figure 1.21 Elliptical polarization of

E

vector in the

x–y

plane [4].

Figure 1.22 Linear polarizer transmits

E

x

component of the light oriented along...

Figure 1.23 Polarization of light (

y

‐component) by the absorbing polarizer (a),...

Figure 1.24 Transmission of

E

x

component of light through the PVA sheet Polaroi...

Figure 1.25 Transformation of linearly polarized light into circularly polarize...

Figure 1.26 Structure of CaCO

3

along the

a‐

,

b‐

, and

c‐

axis [...

Figure 1.27 Polarization of the incident, reflected, and transmitted (refracted...

Figure 1.28 The intensity of reflection coefficient for a light wave for air‐to...

Figure 1.29 The fraction of intensity reflection (

I

R

)

and transmission (

I

T

)

for...

Figure 1.30 The fraction of intensity reflection (

I

R

) and transmission (

I

T

) for...

Figure 1.31 The fraction of intensity reflection (

I

R

) and transmission (

I

T

) in ...

Figure 1.32 Distribution of

I

R

and

I

T

in terms of

E

s

and

E

p

of the incident lig...

Figure 1.33 Thin‐film plate polarizer [4].

Figure 1.34 Polarizing prism. (a) Glan–Thompson prism (b) Nicol prism [5].

Figure 1.35 Quarter‐wave plate conversion of linearly polarized light into circ...

Figure 1.36 Demonstration of optical isolation [6].

Figure 1.37 The variable attenuator configuration [7].

Figure 1.38 A half‐wave plate rotates polarization by 90° [8].

Figure 1.39 (a) Left to right polarization and (b) right to left polarization [...

Figure 1.40 Liquid crystal twisted nematic polarization rotator cell [10].

Chapter 2

Figure 2.1 A solid under a linear stress of stress‐optical measurements (

P

zz

is...

Figure 2.2 Optical indicatrix (index ellipsoid) of a positive crystal.

Figure 2.3 Single crystal photoelastic modulator (SCPEM) made of a 3 m crystal.

Figure 2.4 Course of amplitude (top) and phase (bottom) of current and deformat...

Figure 2.5 Setup of a fiber laser with SCPEM Q‐switch. Two photodetectors are u...

Figure 2.6 Voltage course on the crystal (1), current signal of the crystal (2)...

Figure 2.7 Harmonic vibrator of the combined three‐component photoelastic waveg...

Figure 2.8 Harmonic vibrator of the integrated three‐component photoelastic wav...

Figure 2.9 A schematic drawing of the three‐component photoelastic waveguide ac...

Figure 2.10 (a) The detection principle of photoelastic sensitive system in

y

‐a...

Figure 2.11 Photograph of the accelerometer on the mechanical vibrator.

Figure 2.12 (a) On‐axis sensitivity and (b) cross‐axis sensitivity versus vibra...

Figure 2.13 Orientation of the photoelastic rod relative to the direction of th...

Figure 2.14 Monolithic configuration of the Nd:YAG laser used as sensing elemen...

Figure 2.15 (a) Response of the sensor under the action of deadweight. (b) Zoom...

Figure 2.16 Sensitivity of photoelastic force sensor versus dimensions of the m...

Figure 2.17 Photoelastic modulator. Hinds Instruments Inc. [34].

Figure 2.18 Stress‐optical dispersion of Li

2

Ge

7

0

15

crystals (un‐irradiated) wit...

Figure 2.19 Stress‐optical dispersion of Li

2

Ge

7

0

15

crystals (un‐irradiated and ...

Chapter 3

Figure 3.1 Raman–Nath diffraction (

Q

 ≪ 1). The laser beam is incident roughly n...

Figure 3.2 Bragg diffraction (

Q

 ≫ 1)

.

At one particular incidence angle

θ

B

Figure 3.3 Schematic setup of a nonresonant acousto‐optic modulator. A transduc...

Figure 3.4 Schematics of the focused modulator setup (www.optoscience.com).

Figure 3.5 Plot of rise time versus spot time of three AO modulator materials (...

Figure 3.6 A typical modulation transfer function (www.optoscience.com).

Figure 3.7 Scheme of a system for analysis of radio‐signal phase with use of tw...

Figure 3.8 Details of construction of A–O modulator (a) and a CCD picture (b) t...

Figure 3.9 Acousto‐optic beam deflector (www.optoscience.com).

Figure 3.10 Spot profiles of AOBD (www.optoscience.com).

Figure 3.11 Plot of modulation transfer function (www.optoscience.com).

Figure 3.12 Acousto‐optic frequency shifter, (a) downshifting and (b) upshiftin...

Figure 3.13 A schematic of a typical LDV (http://www.pegasus‐optik.de/PDF/VAR‐S...

Figure 3.14 (a) Schematic of acousto‐optical

Q

‐switch, (b)

Q

‐switch output puls...

Figure 3.15

Q

‐switched fiber‐laser setup; the acousto‐optic modulator is define...

Figure 3.16 Laser setup, the dashed line defines the acousto‐optic modulator.

Figure 3.17 Setup for the characterization of the acousto‐optic superlattice mo...

Figure 3.18 Setup of the mode‐locked all‐fiber laser by using an acousto‐optic ...

Figure 3.19 Setup of the doubly‐active

Q

‐switching mode‐locking all‐fiber laser...

Figure 3.20

Q

‐switched distributed feedback‐fiber laser setup. Lengths are in m...

Figure 3.21 Noncollinear AOTF design. The zero‐order (undiffracted) and first‐o...

Figure 3.22 Different types of acousto‐optic tunable filters (AOTF): (a) collin...

Figure 3.23 Spectral tuning curves of noncollinear TeO

2

AOTF for (a) the visibl...

Figure 3.24 Spectral resolution of (a) collinear quartz AOTF for the visible an...

Figure 3.25 Schematic diagram of an infrared multispectral imaging instrument b...

Figure 3.26 Schematic diagram of the AOTF‐based fluorimeter: PD, photodiode; S,...

Figure 3.27 Near‐IR light source illuminates water droplets at 960 nm (a) and 8...

Figure 3.28 AOTF section of specific regions for excitation in confocal microsc...

Figure 3.29 Schematic diagram of the multiwavelength thermal lens spectrometer ...

Figure 3.30 Schematic diagram of the near infrared spectrophotometer based on t...

Figure 3.31 Relative intensity of the light transmitted through six sheets of p...

Figure 3.32 Relative intensity of the light transmitted through four sheets of ...

Figure 3.33 Acousto‐optical tunable filter based on an anisotropic crystal. (a)...

Figure 3.34 Schematic diagram of the spectropolarimeter based on AOTF. (a) Tung...

Figure 3.35 A block scheme diagram of the Smart‐Spectra system and ATFS.

Figure 3.36 Multi‐sine AOTF driver.

Figure 3.37 Block diagram of the sweeping frequency driver.

Figure 3.38 Diffraction efficiency of the sweeping frequency driver.

Figure 3.39 Study of the chlorophyll content of a sunflower leaf treated with h...

Figure 3.40 Schematic diagram of the AOTF‐based detector for HPLC: PMT (photomu...

Figure 3.41 Three‐dimensional graph plotting the chromatogram of mixture of pen...

Figure 3.42 Near‐IR‐FIA instrument: W, waste; C, carrier; Pump, peristaltic pum...

Figure 3.43 Absorption as a function of time and wavelength after the injection...

Figure 3.44 Absorption (0.75‐cm flow cell using ethanol as blank) as a function...

Figure 3.45 Schematic view of the ExoMars rover mast instruments: PanCam, navig...

Figure 3.46 Schematic representation of possible ISEM and PanCam joint observat...

Figure 3.47 The ISEM instrument, the photographs of the optical (a) and electro...

Figure 3.48 The three‐dimensional model open view of the ISEM optical box. The ...

Figure 3.49 ISEM optical scheme. The numbering is the same as in Figure 3.48: 2...

Figure 3.50 PanCam and ISEM calibration target and its location on the rover. L...

Figure 3.51 Optical scheme of the IR spectrometer. 1‐front‐end telescope object...

Figure 3.52 Complete IR channel of SPICAM at the vibration test bench. 1‐object...

Figure 3.53 The protopype's layout of AOTF.

Figure 3.54 The geometry of acoustic reflection on the facet.

Figure 3.55 Configuration of the acousto‐optic cell of the tunable filter.

Figure 3.56 Focusing of a laser beam to arbitrarily chosen focal points at 30 k...

Figure 3.57 Schematic diagram of complete acousto‐optic lens (AOL) 2‐photon mic...

Figure 3.58 The optical setup. Laser: Tsunami Ti:Sa laser. AOM: acousto‐optic m...

Figure 3.59 (a) Proton exchange waveguide and (b) Ti indiffused intersecting wa...

Figure 3.60 (a) Ti‐deep diffusion SAWG and (b) film‐loaded‐type SAWG.

Figure 3.61 (a) Conventional waveguide reflector and (b) 0‐gap directional coup...

Figure 3.62 Integrated AOTF.

Chapter 4

Figure 4.1 Circular polarized light.

Figure 4.2 Polarized light in the Faraday configuration.

Figure 4.3 Magnetic splitting of an absorption line causing Faraday effect [1]....

Figure 4.4 Polarized light in the Voigt configuration.

Figure 4.5 (a) Polar, (b) longitudinal, and (c) equatorial configuration of lig...

Figure 4.6 (a) Parallel alignment of magnetic moments of atoms in ferromagnetic...

Figure 4.7 Structure of yttrium iron garnet crystal.

Figure 4.8 Schematic of semiconductor core isolator.

Figure 4.9 Waveguide design with alternating segments of positive and negative ...

Figure 4.10 An electromagnetic wave propagating along the

z

‐direction in a Cart...

Figure 4.11 Solenoid coil schematic.

Figure 4.12 Typical Faraday rotator schematic.

Figure 4.13 Plan view layout.

Figure 4.14 Full polarization controller configuration.

Figure 4.15 Rendering of 2D MOSLM array composed of individual MEMS Faraday rot...

Figure 4.16 2D MOSLM array assembly schematic.

Figure 4.17 Spiral geometry combination.

Figure 4.18 Structure of the superconducting magneto‐optic modulator. (a) MSL g...

Figure 4.19 Simulation of the magnetic field distribution inside the MO modulat...

Figure 4.20 Mach–Zehnder design of a EuSe MO modulator.

Figure 4.21 Schematic diagram of the structure of the MF‐based modulator.

Figure 4.22 Operating principle of the magneto‐optical modulator based on the m...

Figure 4.23 Transmission spectra of the magneto‐optical modulator under differe...

Figure 4.24 (a) Dip wavelength and (b) transmission power versus the magnetic f...

Figure 4.25 Light intensity modulation by (a) square wave and (b) sine wave.

Figure 4.26 (a) Schematic diagram of FFPC. (b) Experimental transmission spectr...

Figure 4.27 Three kinds of circulators. (a) A windmill‐shaped circulator. (b) A...

Figure 4.28 (a) Schematic diagram of a T‐shaped circulator in a square‐lattice ...

Figure 4.29

E

z

‐field distributions of two degenerate modes.

Figure 4.30 Schematic diagram of the light circuit in the T‐shaped circulator. ...

Figure 4.31 Distributions of

E

z

–field in the T‐shaped

MPC

circulator operating ...

Figure 4.32 (a) Design of T‐shaped circulator before optimization process. (b) ...

Figure 4.33 Optimal design of the T‐shaped circulator.

Figure 4.34 Frequency splitting of dipole modes excited in an MO resonator.

Figure 4.35 Frequency responses of T‐circulator for excitation at ports 1, 2, a...

Figure 4.36 Distribution of

E

z

‐component of the electromagnetic field for T‐ ci...

Figure 4.37 (a) Mask layout of the six‐port circulator. (b) Schematic of the de...

Figure 4.38 The four different configurations of the six‐port circulator.

Figure 4.39 (a) Intrinsic spectrum of the two rings with no magnetic field. Tra...

Figure 4.40 Silicon‐ferrite photonic crystal for THz circulator. (a) Schematic ...

Figure 4.41 Design of free‐space optical isolator. The Faraday rotator is place...

Figure 4.42 Schematic of QPM Faraday rotation isolator.

Figure 4.43 Nonreciprocal‐phase‐shift isolator uses modified Mach–Zehnder inter...

Figure 4.44 Typical TM‐mode nonreciprocal‐loss waveguide isolators. (a) Gain gu...

Figure 4.45 Principle of nonreciprocal‐loss waveguide isolator.

Figure 4.46 (a) Schematic cross section of the waveguide isolator for 1.5 μm TM...

Figure 4.47 Transmission spectra of device for forward transmission (dashed lin...

Figure 4.48 Transmission intensity as a function of device length, measured for...

Figure 4.49 Isolation ratio as a function of a wavelength from 1.53 to 1.55 μm ...

Figure 4.50 Schematics of (a) MZI optical isolator and (b) MZI optical circulat...

Figure 4.51 Photographs of fabricated silicon waveguide optical isolators.

Figure 4.52 Measured transmittance of silicon waveguide isolator [187].

Figure 4.53 Measured transmittance spectra of a circulator with port pairs of (...

Figure 4.54 (a) Schematic structure of the proposed MMOPW. (b) Band structure o...

Figure 4.55 (a) Schematic structure of the proposed MMOPL and the model of meta...

Figure 4.56 (a) 3D view of the device including the directions of wave polariza...

Figure 4.57 Interaction between light and magnetic fields within a magneto‐opti...

Figure 4.58 Typical configuration of an optical current sensor with flint fiber...

Figure 4.59 Faraday rotator mirror.

Figure 4.60 Bulk optic current sensor with double reflection.

Figure 4.61 Bulk optic triangular shaped current sensor.

Figure 4.62 Experimental scheme of fiber‐optic magnetic sensor.

Figure 4.63 Scheme of the all‐fiber magnetometer.

Figure 4.64 Experimental arrangement used for measuring electric current on hig...

Figure 4.65 Schematic diagram of the fiber sensor.

Figure 4.66 Configuration of optical fiber magnetic sensor.

Figure 4.67 Structure diagram of magnetic filled HC PCF FP sensor.

Figure 4.68 Principle of thermomagnetic recording (Curie point writing): (a) be...

Figure 4.69 Example of (a) the temperature profile calculated for a TbFeCo amor...

Figure 4.70 (a) Thermomagnetic recording process. The field of the electromagne...

Figure 4.71 (a) Thermomagnetic recording by magnetic field modulation. (b) Pola...

Figure 4.72 Schematic diagram of an MO head.

Figure 4.73 Schematic configuration of the magnetic moments in ferrimagnetic RE...

Figure 4.74 Temperature dependence of the magnetization for a GdCoMo amorphous ...

Figure 4.75 Curie temperature as a function of Co content for RE–FeCo amorphous...

Figure 4.76 Temperature dependence of (a) the saturation magnetization and (b) ...

Figure 4.77 Perpendicular anisotropy energy per one RE atom substitution in Gd

1

...

Figure 4.78 MO Kerr rotation

θ

K

and ellipticity

η

K

of a Gd–Ni amorpho...

Figure 4.79 MO Kerr rotation

θ

K

and ellipticity

η

K

of Y–Fe and Y–Co a...

Figure 4.80 Contribution of RE to the Kerr spectra in various RE–Co amorphous f...

Figure 4.81 Spectra of MO Kerr effect for typical MO recording materials, where...

Figure 4.82 Interface wall appearing between two exchange‐coupled layers with o...

Figure 4.83 Magnetization states in exchange‐coupled RE–TM films with out‐of‐pl...

Figure 4.84 Kerr hysteresis loops to be observed in exchange‐coupled RE–TM film...

Figure 4.85 An example of measured magnetization curves for A‐type.

Figure 4.86 Effective anisotropy per unit area

K

eff

t

Co

as a function of Co lay...

Figure 4.87 Perpendicular magnetic anisotropy as a function of bilayer thicknes...

Figure 4.88 Images of marks recorded by LPM and MFM methods in TbFeCo amorphous...

Figure 4.89 Readout techniques for recorded marks smaller than the laser beam s...

Figure 4.90 Schematic illustration of CAD magnetic super resolution.

Figure 4.91 Schematic of the flying MO head using an SIL.

Figure 4.92 Schematic of hybrid recording with optical recording and GMR readou...

Chapter 5

Figure 5.1 A steady electric field is applied to a quartz prism parallel to the...

Figure 5.2 Dependence of the refractive index on the electric field: (a) Pockel...

Figure 5.3 Phase shift

ϕ

with respect to applied voltage V.

Figure 5.4 (a) Transverse and (b) longitudinal modulators.

Figure 5.5 An integrated optical phase modulator.

Figure 5.6 Phase retardation Γ with respect to applied voltage V.

Figure 5.7 An intensity modulator operating in a limited region near point B. I...

Figure 5.8 An integrated optical intensity modulator (or optical switch). A Mac...

Figure 5.9 (a) An optical intensity modulator using a Pockels cell placed betwe...

Figure 5.10 (a) An electro‐optic prism. The deflection angle

θ

is controll...

Figure 5.11 A position switch based on electro‐optic phase retardation and doub...

Figure 5.12 (a) Exchange of power between two parallel weakly coupled waveguide...

Figure 5.13 An integrated electro‐optic directional coupler.

Figure 5.14 Dependence of the coupling efficiency on the applied voltage V. Whe...

Figure 5.15 A spatial light modulator.

Figure 5.16 An electrically addressable array of longitudinal electro‐optic mod...

Figure 5.17 The electro‐optic spatial light modulator uses a photoconductive ma...

Figure 5.18 The Pockels readout optical modulator (PROM).

Figure 5.19 Schematic representation of the ridge waveguide modulator structure...

Figure 5.20 Schematic of BaTiO

3

ridge waveguide structure.

Figure 5.21 The geometry of the

c

‐axis optical modulator for optical switching ...

Figure 5.22 (a) Schematic cross section of the electro‐optic waveguide modulato...

Figure 5.23 (a) Schematic of the electro‐optic modulator based on BTO on SOI. L...

Figure 5.24 Schematic of a plasmonic modulator based on interference of SPPs la...

Figure 5.25 TE1000 electro‐optic tunable etalon.

Figure 5.26 Nanosecond speed PLZT optical switch [40].

Figure 5.27 Device schematic (a) top view and (b) perspective view.

Figure 5.28 Device configuration and test setup.

Figure 5.29 Second harmonic generation tuning curve.

Figure 5.30 (a) Images at the output facet for various applied voltages and (b)...

Figure 5.31 Measured and calculated beam diameter at the output facet.

Figure 5.32 Schematic of laser cavity using LiNbO

3

as

Q

‐switch.

Figure 5.33 Schematic of integrated optical modulator based on a Mach–Zehnder i...

Figure 5.34 Three axis ASTRIX fiber‐optic inertial measurement unit.

Figure 5.35 Artist's view of FSO (free space optics) communication demonstratio...

Figure 5.36 Space compatible 20 GHz intensity modulator.

Figure 5.37 (a) Intensity modulator, (b) phase modulator, (c) polarization modu...

Figure 5.38 Schematic of the integrated LiNbO

3

quadrature modulator.

Figure 5.39 (a) Schematic of the integrated LiNbO

3

coherent receiver and (b) Li...

Figure 5.40 Device design. (a) Schematics of a traditional ion‐diffused LN wave...

Figure 5.41 Fabricated optical devices and electrical contacts (a, b). (a) Fals...

Figure 5.42 DC electrical and optical characterization. (a) Measured transmissi...

Figure 5.43 Bandwidth and high‐speed data operation. (a) Simplified experimenta...

Figure 5.44 The configuration of the measurement system.

Figure 5.45 The structure of the sensor: (a) top view and (b) cross‐section vie...

Figure 5.46 The structure of the sensors: (a) two electrodes connected with ver...

Figure 5.47 Mono‐shielding electrode optimized as a grid type. (a) Schematic of...

Figure 5.48 (a) Schematic of CI‐based IOES. (b) Sensor after encapsulation.

Figure 5.49 The CPI‐based IOES with dipole antenna and electrode. (a) Schematic...

Figure 5.50 Schematic diagram of a symmetric Mach–Zehnder interferometer with m...

Figure 5.51 Schematic configuration of the LiNbO

3

‐based integrated optical sens...

Figure 5.52 Geometry of tapered antenna array.

Figure 5.53 Configuration of the integrated optical waveguide

E

‐field measureme...

Figure 5.54 Schematic of the sensor: (a) geometry of the tapered antenna and (b...

Figure 5.55 Electric field sensing scheme using UMZIs as optical retarders.

Figure 5.56 Unbalanced Mach–Zehnder electric field sensor.

Figure 5.57 Experimental electric field sensing scheme using matched UMZIs.

Figure 5.58 The processing flow of fabricating an on‐chip electro‐optic tunable...

Figure 5.59 (a) The top view of the EO tunable LN microresonator integrated wit...

Figure 5.60 (a) The resonant wavelength in the fabricated microresonator as a f...

Figure 5.61 (a) A schematic design of TFL and (b) a fiber‐version four‐stage TL...

Figure 5.62 (a) TLF responses when three stages were powered and (b) transmissi...

Figure 5.63 Measured outputs of a composite TFPF with different applied voltage...

Figure 5.64 Configuration of a Q‐switched DPSS laser.

Figure 5.65 Pulse width and repetition rate of a Q‐switch made from an opto‐cer...

Figure 5.66 Schematic of the VOA construction and an Eclipse™ VOA device.

Figure 5.67 Optical modulation of a VOA at 1 MHz.

Figure 5.68 Schematics of a polarization controller design and an Acrobat™ pola...

Figure 5.69 A screen print of the measurement of a PC.

Figure 5.70 Schematic of a sinusoidal filter and (b) Equinox™ VGTF.

Figure 5.71 Measured results of a 24 nm sine filter.

Figure 5.72 Equinox™ DGFF.

Figure 5.73 Fitting of a DGFF consists of five sine filters and a VOA.

Figure 5.74 Cross section of fabricated KTN buried waveguide, and near‐field pa...

Figure 5.75 Experimental setup for realizing EO phase modulators by retardation...

Figure 5.76 Optical output intensity against applied voltage.

Figure 5.77 An illustration of configuration of dynamic optical waveguide based...

Figure 5.78 The simulated electric field distribution is plotted in

x–y

c...

Figure 5.79 (a) A schematic sketch of transverse EO modulator; (b) picture of f...

Figure 5.80 The experimentally measured transmission quadratic EO coefficient a...

Figure 5.81 Structure and functions of KTN varifocal lens. (a) Structure of KTN...

Figure 5.82 Two‐dimensional convergence by two KTN lenses.

Figure 5.83 Typical configuration of an electro‐optic deflector. (a) EOD based ...

Figure 5.84 (a) Space‐charge‐controlled high‐speed KTN deflector and (b) struct...

Figure 5.85 (a) The experimental system used to measure the deflection angle as...

Figure 5.86 KTN optical scanner and its mode of operation.

Figure 5.87 Number of resolvable points for the optical scanner.

Figure 5.88 Multi‐pass scheme for KTN optical scanner.

Figure 5.89 On‐screen scanning trace giving an estimation of

N

.

Figure 5.90 Optical properties of highly doped transparent conducting oxides: (...

Figure 5.91 Strong dependence of AZO permittivity (real (a) and imaginary (b) p...

Figure 5.92 Dispersion of SPPs on silver/silica and ITO/silica interfaces. For ...

Figure 5.93 Schematic view of the horizontally arranged silicon‐based plasmonic...

Figure 5.94 Schematic view of a multilayer stack: TCO and insulator layers sand...

Figure 5.95 Change of carrier concentration (a) and TCO permittivity (b) accord...

Figure 5.96 Influence of ITO permittivity on the absorption coefficient and FoM...

Figure 5.97 (a) SPPAM: the modulator section consists of a stack of silver, ITO...

Figure 5.98 Design of nanowire‐based modulator with the size 25 × 25 nm

2

: SPP m...

Figure 5.99 Hybrid design of plasmonic/photonic absorption modulator.

Figure 5.100 (a) Schematic of the waveguide‐integrated silicon‐based nanophoton...

Figure 5.101 Increase of the absorption coefficient in a multilayer waveguide f...

Figure 5.102 (a) One of the simplest yet efficient structures includes only two...

Figure 5.103 (a) Modulator design [636] includes two additional metal layers: T...

Figure 5.104 (a) TCO plasmonic modulator along with integration scheme: the inp...

Figure 5.105 Suggested modulators designs are varied by compactness and ease of...

Figure 5.106 (a) TCO modulator based on a plasmonic slot waveguide. (b) High mo...

Chapter 6

Figure 6.1 Band diagram behavior.

Figure 6.2 Light application to media.

Figure 6.3 Energy‐level diagram of LiNbO

3

illustrating the processes of (a) pho...

Figure 6.4 Single‐level band scheme in the conventional model at low intensitie...

Figure 6.5 Simplified band scheme of possible charge transitions in a photorefr...

Figure 6.6 Mechanisms involved in the photorefractive effect, illustrated here ...

Figure 6.7 Hologram basics.

Figure 6.8 Hologram data recovery.

Figure 6.9 Two‐wave mixing is a form of dynamic holography.

Figure 6.10 Simplified electric field and refractive index distribution in a ph...

Figure 6.11 (a) Arrangement for recording light‐induced waveguide structures. (...

Figure 6.12 (a) CCD images of the output face of a 6.8 mm long Sodium polystyre...

Figure 6.13 Schematic diagram of the fabricating technique for array of 3D‐WGs ...

Figure 6.14 Experimental setup for fabricating an array of rectangular waveguid...

Figure 6.15 Experimental results for fabricating an array of rectangular wavegu...

Figure 6.16 Experimental setup for guiding tests of the waveguide array. T, tel...

Figure 6.17 Experimental results for guiding tests of the waveguide array. (a, ...

Figure 6.18 Schematic top view of the experimental setup. The recording beams a...

Figure 6.19 Grating configuration and angle convention used in this work. Top v...

Figure 6.20 Bragg wavelength as a function of the frequency applied to the acou...

Figure 6.21 Wavelength selectivity scan for three different grating vectors sep...

Figure 6.22 Measured diffraction signal as a function of the grating spacing at...

Figure 6.23 (a) Time development of the diffraction efficiency for tuning‐in an...

Figure 6.24 Schematic diagram of the experimental setup. BS represents a beam s...

Figure 6.25 Temporal evolutions of the transmitted power

I

t

and of the output p...

Figure 6.26 (a, b) Temporal evolution of the transmitted power

I

t

with an

ac

sq...

Figure 6.27 Temporal evolution of the output phase conjugate power

I

pc

with an

Figure 6.28 (a)

I

t

(V)/

I

t

(0) and (b)

I

pc

(V)/

I

pc

(0) as a function of the applied ...

Figure 6.29 Real‐time holographic interferometry with photorefractive BSO cryst...

Figure 6.30 Holographic interferograms of a mirror static: (a)

θ

 = 0, (b)

Figure 6.31 Holographic interferograms of a mirror slopping: (a)

θ

 = 0, (b...

Figure 6.32 Image of PZO and the connected mirror.

Figure 6.33 Schematic of the photorefractive interferometer based on the two‐wa...

Figure 6.34 Frequency response of the BSO photorefractive crystal at a laser in...

Figure 6.35 Comparison of the sensitivity of a photorefractive (a) and Michelso...

Figure 6.36 Multicolor holographic stereograms [149,150].

Figure 6.37 Schematic of the optical tomography testbed. The inset image compar...

Figure 6.38 (a) Image of the sample under test. (b) The normalized signals dete...

Figure 6.39 Illustration of the concept of coherence gated holographic imaging ...

Figure 6.40 Detected power at the end of the waveguide as a function of time fo...

Figure 6.41 Structure of the 2 × 2 wavelength switch.

Figure 6.42 The 1 × 2 wavelength switch and the experimental setup for demonstr...

Figure 6.43 Reflection spectrum measured at field‐on and field‐off states of th...

Figure 6.44 Fresnel zone plates; (a) concentric zone plate acts as a two‐dimens...

Figure 6.45 Michelson interferometer; experimental arrangement for fabricating ...

Figure 6.46 Experimental arrangement for studying the light focusing behavior a...

Figure 6.47 Polarizing microscope images of the patterned sample with (a) cross...

Figure 6.48 The observed laser beam images (a) without LC sample, (b) with LC s...

Figure 6.49 Microscopic images of the patterned sample at various applied volta...

Figure 6.50 Intensity profiles of the outgoing focused (in one direction,

x

‐dir...

Figure 6.51 Diffraction efficiency at various applied electric fields measured ...

Figure 6.52 Electro‐optical response time of the fabricated Fresnel lens monito...

Figure 6.53 Principles of operation of VBG filters – thin lines denote planes w...

Figure 6.54 Several VBG filters can be overlapped or multiplexed in the same vo...

Figure 6.55 Photograph of a diced wafer and a single VBG element. Manufacturing...

Figure 6.56 (a) Spectral response of a 50 GHz VBG filter for DWDM applications ...

Figure 6.57 Low‐cost package for a four‐channel WDM combiner. In this package o...

Figure 6.58 An integrated module based on VBG technology. This module contains ...

Figure 6.59 Spectral output of a four‐channel CWDM transmitter based on VBG ele...

Guide

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Crystal Optics

Properties and Applications

Ashim Kumar Bain

Copyright

Author

Prof. Ashim Kumar Bain

University of Birmingham

Electronic, Electrical and Systems Engineering

B15 2TT Edgbaston

United Kingdom

All books published by Wiley‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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Print ISBN: 978‐3‐527‐41385‐0

ePDF ISBN: 978‐3‐527‐82303‐1

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Preface

Since the last world war there has been a growing interest in all aspects of solid‐state studies; inevitably, this has stimulated the teaching of crystallography and the physical properties of crystalline solids as a part of courses covering a wide range of academic disciplines and technological interests. Crystal optics is one of the most widely used practical techniques employed in the study of the physical properties of crystalline materials. The effect of electric and magnetic fields, mechanical stress, and ultrasound waves on the optical properties of crystals are studied in electro‐optics, magneto‐optics, photoelasticity, acousto‐optics, and photorefractivity, which are based on the fundamental laws of crystal optics. The present book aims to provide the basic physical properties and applications of photoelastic, acousto‐optic, magneto‐optic, electro‐optic, and photorefractive materials.

The first chapter deals with the basic concepts of crystal optics, such as index ellipsoid or optical indicatrix, crystal symmetry, wave surface, birefringence, polarization of light, changing the polarization of light, effects of reflection and transmission of polarization, and light polarizing devices. The second chapter provides an understanding of the physical phenomenon of the photoelastic effects in isotropic and crystalline materials. It describes in detail research information on modern photoelastic materials and reviews the up‐to‐date photoelastic device applications. The third chapter develops the underlying theory of acousto‐optics from first principles, formulating results suitable for subsequent calculations and design. Special attention is given to designing procedures for the entire range of acousto‐optic devices, and various applications of these devices are also described. The fourth chapter describes the basic principles of magneto‐optic effects and mode of interaction with magnetic materials. It also describes in detail research information on modern magneto‐optic materials and reviews the up‐to‐date magneto‐optic device applications up to terahertz (THz) regime. The fifth chapter provides an understanding of the physical phenomenon of the linear and quadratic electro‐optic effects in isotropic and crystalline materials. It describes in detail modern electro‐optic materials and it also reviews the up‐to‐date electro‐optic device applications in both bulk and plasmonic waveguide technologies. The sixth chapter is a collection of many of the most important recent developments in photorefractive effects and materials. Special attention has been paid to describe the up‐to‐date review of recent scientific findings and advances in photorefractive materials and devices.

I sincerely hope that this book will be of real value to the students and researchers moving into the wide field of crystal optics.

Finally, I would like to thank the Wiley‐VCH publishing team for their outstanding support.

Birmingham, UK

January 2019

Ashim Kumar Bain

Overview

Crystal optics is the branch of optics that describes the behavior of electromagnetic waves in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating in. The phenomena characteristics of crystals that are studied in crystal optics include double refraction (birefringence), polarization of light, rotation of the plane of polarization, etc. The effect of electric, magnetic field, mechanical stress, and ultrasound waves on the optical properties of crystals are studied in electro‐optics, magneto‐optics, photoelasticity, acousto‐optics, and photorefractivity, which are based on the fundamental laws of crystal optics.

This book deals with the basic physical properties and applications of photoelastic, acousto‐optic, magneto‐optic, electro‐optic, and photorefractive materials. It also provides up‐to‐date information on design and applications of various optoelectronic devices based on these materials, such as photoelastic devices (modulator, Q‐switches, accelerometer, sensor), acousto‐optic devices (modulators, beam deflector, frequency shifter, Q‐switch, tunable filter), magneto‐optic devices (modulator, circulator, isolator, sensor, and magneto‐optical recording), electro‐optic devices (modulator, dynamic wave retarder, scanner, directional, coupler, deflector, tunable filter, Q‐switch, attenuator, polarization controller, sensor), and photorefractive devices (waveguides, Q‐switch, tunable filter, holographic interferometers, and holographic 3D stereograms). This book will be very useful for the scientific community including students, teachers, and researchers working in these fields. It will also find readership with non‐experts of the subject.

1Crystal Optics

1.1 Introduction

Crystal optics is the branch of optics that describes the behavior of electromagnetic waves in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on the direction in which the light is propagating. The characteristic phenomena of crystals that are studied in crystal optics include double refraction (birefringence), polarization of light, rotation of the plane of polarization, etc.

The phenomenon of double refraction was first observed in crystals of Iceland spar by the Danish scientist E. Bartholin in 1669. This date is considered the beginning of crystal optics. Problems of the absorption and emission of light by crystals are studied in crystal spectroscopy. The effect of electric and magnetic fields, mechanical stress, and ultrasound waves on the optical properties of crystals are studied in electro‐optics, magneto‐optics, photoelasticity, acousto‐optics, and photorefractivity, which are based on the fundamental laws of crystal optics.

Since the lattice constant (of the order of 10 Å) is much smaller than the wavelength of visible light (4000–7000 Å), a crystal may be regarded as a homogeneous but anisotropic medium. The optical anisotropy of crystals is caused by the anisotropy of the force field of particle interaction. The nature of the field is related to crystal symmetry. All crystals, except crystals of the cubic system, are optically anisotropic.

1.2 Index Ellipsoid or Optical Indicatrix

In isotropic materials, the electric field displacement vector D is parallel to the electric field vector E, related by D = ɛ0ɛrE = ɛ0E + P, where ɛ0 is the permittivity of free space, ɛr is the unitless relative dielectric constant, and P is the material polarization vector.

The optical anisotropy of transparent crystals is due to the anisotropy of the dielectric constant. In an anisotropic dielectric medium (a crystal, for example), the vectors D and E are no longer parallel; each component of the electric flux density D is a linear combination of the three components of the electric field E.

(1.1)

where i, j = 1, 2, 3 indicate the x, y, and z components, respectively. The dielectric properties of the medium are therefore characterized by a 3 × 3 array of nine coefficients {ɛij} forming a tensor of second rank known as the electric permittivity tensor and denoted by the symbol ɛ. Equation (1.1) is usually written in the symbolic form D = ɛE. The electric permittivity tensor is symmetrical, ɛij = ɛji, and is therefore characterized by only six independent numbers. For crystals of certain symmetries, some of these six coefficients vanish and some are related, so that even fewer coefficients are necessary.

Elements of the permittivity tensor depend on the choice of the coordinate system relative to the crystal structure. A coordinate system can always be found for which the off‐diagonal elements of ɛij vanish, so that

(1.2)

where ɛ1 = ɛ11, ɛ2 = ɛ22, and ɛ3 = ɛ33. These are the directions for which E and D are parallel. For example, if E points in the x‐direction, D must also point in the x‐direction. This coordinate system defines the principal axes and principal planes of the crystal. The permittivities ɛ1, ɛ2, and ɛ3 correspond to refractive indices.

(1.3)

are known as the principal refractive indices and ɛ0 is the permittivity of free space.

In crystals with certain symmetries two of the refractive indices are equal (nl = n2 ≠ n3) and the crystals are called uniaxial crystals. The indices are usually denoted n1 = n2 = no and n3 = ne. The uniaxial crystal exhibits two refractive indices, an “ordinary” index (no) for light polarized in the x‐ or y‐direction, and an “extraordinary” index (ne) for polarization in the z‐direction. The crystal is said to be positive uniaxial if ne > no and negative uniaxial if ne < no. The z‐axis of a uniaxial crystal is called the optic axis. In other crystals (those with cubic unit cells, for example) the three indices are equal and the medium is optically isotropic. Media for which the three principal indices are different (i.e. n1 ≠ n2 ≠ n3) are called biaxial. Light polarized at some angle to the axes will experience a different phase velocity for different polarization components and cannot be described by a single index of refraction. This is often depicted as an index ellipsoid.

The optical properties of crystals are described by the index ellipsoid or optical indicatrix. It is generated by the equation

(1.4)

where x1, x2, and x3 are the principal axes of the dielectric constant tensor and n1, n2, and n3 are the principal dielectric constants, respectively. Figure 1.1 shows the optical indicatrix of a biaxial crystal. It is a general ellipsoid with n1 ≠ n2 ≠ n3 representative of the optical properties of triclinic, monoclinic, and orthorhombic crystals.

Figure 1.1 The index ellipsoid. The coordinates (x, y, z) are the principal axes and (n1, n2, n3) are the principal refractive indices of the crystal.

1.3 Effect of Crystal Symmetry

In the case of cubic crystals, which are optically isotropic, ɛ is independent of direction and the optical indicatrix becomes a sphere with radius n. In crystals of intermediate systems (trigonal, tetragonal, and hexagonal), the indicatrix is necessarily an ellipsoid of revolution about the principal symmetry axis (Figure 1.2). The central section is perpendicular to the principal axis, and only this central section is a circle of radius n0. Hence, only for a wave normal along the principal axis is there no double refraction. The principal axis is called the optic axis and the crystals are said to be uniaxial. A uniaxial crystal is called optically positive (+) when ne > n0 and negative (−) when ne < n0.

Figure 1.2 The indicatrix for a (positive) uniaxial crystal.

For crystals of the lower systems (orthorhombic, monoclinic, and triclinic), the indicatrix is a triaxial ellipsoid. There are two circular sections (Figure 1.3) and hence two privileged wave normal directions for which there is no double refraction. These two directions are called the primary optic axes or simply the optic axes, and the crystals are said to be biaxial.

Figure 1.3 The two circular sections of the indicatrix and the two primary optic axes OP1, OP2 for a biaxial crystal.

1.4 Wave Surface

If a point source of light is situated within a crystal the wave front emitted at any instant forms a continuously expanding surface. The geometric locus of points at a distance v from a point O is called the ray surface or wave surface. Actually, the wave surface is a wave front (or pair of wave fronts) completely surrounding a point source of monochromatic light. This is also a double‐sheeted surface. In most crystalline substances, however, two wave surfaces are formed; one is called the ordinary wave surface and the other is called the extraordinary wave surface. In both positive and negative crystals, the ordinary wave surface is a sphere and the extraordinary wave surface is an ellipsoid of revolution.

1.4.1 Uniaxial Crystal

In uniaxial crystals, one surface is a sphere and the other is an ellipsoid of revolution touching one another along the optical axis – OZ as shown in Figure 1.4a,b. In positive (+) crystals (ne > n0) the ellipsoid is inscribed within the sphere (Figure 1.4a), whereas in negative (−) crystals (ne < no) the sphere is inscribed within the ellipsoid (Figure 1.4b).

Figure 1.4 Ray surfaces of uniaxial crystals: (a) positive, (b) negative, (OZ) optical axis of the crystal, (v0) and (ve) phase velocities of ordinary and extraordinary waves propagating in the crystals.

The dependence of the ray velocity of a plane wave propagating in a crystal on the direction of propagation and the nature of polarization of the wave leads to the splitting of light rays in crystals. In a uniaxial crystal, one of the refracted rays obeys the usual laws of refraction and is therefore called the ordinary ray, whereas the other ray does not (it does not lie in the plane of incidence) and is called the extraordinary ray. The three‐dimensional view of positive and negative uniaxial medium wave surfaces can be seen in Figures 1.5 and 1.6.

Figure 1.5 Positive uniaxial medium wave surface. Inner fold (left), vertical section (center), and outer fold (right).

Source: Latorre et al. 2012 [1]. Reproduced with permission of Springer Nature.

Figure 1.6 Negative uniaxial medium wave surface. Inner fold (left), vertical section (center), and outer fold (right).

Source: Latorre et al. 2012 [1]. Reproduced with permission of Springer Nature.

1.4.2 Biaxial Crystal

It is very difficult to imagine what shapes the biaxial wave surface will have. For wave normals that lie in any of the three principal planes of the indicatrix the situation is very similar to that described for the uniaxial crystal. Three cross‐sectional views of the wave surfaces for a biaxial crystal are given in Figure 1.7.

Figure 1.7 Principal sections of the wave surface for a biaxial crystal.

In biaxial media, the wave surface equations are fourth‐order polynomials with even powers only; that is, the surface is symmetrical with respect to the origin. Both surface folds intersect only at four symmetrical points, as can be seen in Figure 1.8. Note that this intersection does not yield a curve but only the four points.

Figure 1.8 Wave surface for a biaxial medium. Inner fold (left), vertical section (center), and outer fold (right).

Source: Latorre et al. 2012 [1]. Reproduced with permission of Springer Nature.

1.5 Birefringence

When a beam of nonpolarized light passes into a calcite or quartz crystal, the light is decomposed into two beams that refract at different angles. This phenomenon is called birefringence or double refraction. The ray for which Snell's law holds is called the ordinary or O‐ray, and the other is called the extraordinary or E‐ray (Figure 1.9).

Figure 1.9 Side view of the double refraction of light by a calcite crystal.

Birefringent materials are optically anisotropic (their properties depend on the direction a light beam takes across them) because their molecules do not respond to the incident light evenly in all directions. This arises from their molecular (bond strengths) and crystal (arrangement) structures. However, in all such materials there is at least one optic axis (some materials have two) along which propagating light can travel with no consequences to either (any) component of its electric vector. This axis serves as a kind of reference. Light traveling in any other direction through the crystal experiences two different refractive indices and is split into components that travel at different speeds and have perpendicular polarizations.

This effect of double refraction or birefringence is further demonstrated in Figure 1.10[2]. In Figure 1.10, subscript 0 indicates the incident wave, while 1 and 2 indicate the refracted waves. The refractive index for ordinary wave is denoted by no and is independent of the direction of propagation. The refractive index for extraordinary wave is denoted by ne(θ) and depends on the direction of propagation (θ) relative to the optic axis.

Figure 1.10 Double refraction at the boundary of an anisotropic medium.

The behavior of refractive index is usually described in terms of the refractive index surface, i.e. the index ellipsoid. In the case of the ordinary ray it is a sphere, while for the extraordinary ray it is an ellipsoid. That is, in terms of ellipsoid, this effect becomes a three‐dimensional body with cylindrical symmetry. The two indices of refraction are then identical (nx = ny), so that the plane intersecting perpendicular to the optical axis forms a circle. If z‐axis is considered as the axis of cylindrical symmetry (the optical axis of a uniaxial crystal), then for uniaxial crystal, the principal indices of refraction are

(1.5)

where ɛo is the dielectric constant in free space (∼8.85 × 10−12 F/m); ɛx, ɛy, and ɛz are the dielectric constants along x, y, and z‐axes. For uniaxial crystals ɛx = ɛy. It is also a known fact that refractive index and critical angle of materials are related by

(1.6)

where co and ce are the critical angles at the ordinary and extraordinary axes respectively. The critical angle of a material determines whether an internal ray will be reflected back into the material. As shown in Eq. (1.6), it is a function of the refractive index, and hence, the higher the refractive index the lower the critical angle.

In Figure 1.11a, the incident light rays giving rise to the ordinary and extraordinary rays enter the crystal in a direction that is oblique with respect to the optical axis and are responsible for the observed birefringent character. When an incident ray enters the crystal perpendicular to the optical axis, it is separated into ordinary and extraordinary rays, but instead of taking different pathways, the trajectories of these rays are coincident. Even though the ordinary and extraordinary rays emerge from the crystal at the same location, they exhibit different optical path lengths and are subsequently shifted in phase relative to one another (Figure 1.11b). In the case where incident light rays impact the crystal in a direction that is parallel to the optical axis (Figure 1.11c), they behave as ordinary light rays and are not separated into individual components by an anisotropic birefringent crystal. Calcite and other anisotropic crystals act as if they were isotropic materials (such as glass) under these circumstances. The optical path lengths of the light rays emerging from the crystal are identical, and there is no relative phase shift.

Figure 1.11 Separation of light waves by a birefringent crystal [3].

Source: Courtesy of Nikon.

1.6 Polarization of Light

Polarization generally just means “orientation.” It comes from the Greek word polos, for the axis of a spinning globe. Wave polarization occurs for vector fields. For light (electromagnetic waves) the vectors are the electric and magnetic fields, and the light's polarization direction is by convention along the direction of the electric field. Generally, we should expect fields to have three vector components, e.g. (x, y, z), but light waves only have two non‐vanishing components: the two that are perpendicular to the direction of the wave.

Electromagnetic waves are the solutions of Maxwell's equations in a vacuum:

1.7

In order to satisfy all four equations, the waves must have the E and B fields transverse to the propagation direction. Thus, if the wave is traveling along the positive z‐axis, the electric field can be parallel to the x‐axis and B‐field parallel to the y‐axis (Figure 1.12).

Figure 1.12 Schematic view of electromagnetic wave propagation along the z‐axis.

We shall call the two distinct waves Ex and Ey, where we denote these by vectors to remind that they point in (or oscillate along) a certain direction (the x‐ and y‐directions, respectively) as shown in Figure 1.13. The amplitude of the light wave describes how the wave propagates in position and time. Mathematically, we can write it as a “sine wave” where the angle of the sine function is a linear combination of both position and time terms:

(1.8)

Figure 1.13 Propagation of Ex and Ey components along the z‐axis [4].

Source: Courtesy of Semrock.

where A is called the “amplitude factor,” the variable λ is the wavelength, and the variable ν is the frequency. If a snapshot of the wave could be taken at a fixed time, λ would be the distance from one wave peak to the next. If one sits at a fixed point in space and counts the wave peaks as they pass by, ν gives the frequency of these counts, or 1/ν gives the time between peaks. The sign between the position and time terms determines the direction the wave travels: when the two terms have the opposite sign (i.e. the “−” sign is chosen), the wave travels in the positive z‐direction.