## Some experimental and theoretical studies in crystal optics.

##### Abstract

The contents of this thesis are divided into two separate parts, both of which are concerned with
optical phenomena for light propagation through source-free nonmagnetic media. Considered in
the first part are various theoretical investigations pertaining to nonmagnetic crystals. While the
second part focuses specifically on the experimental investigation of the quadratic electro-optic
effect in some crystals of the KDP family.
The opening theoretical chapters of this thesis are based aln10st entirely on four publications
which are the result of work completed in this PhD study. These papers~ in the order that they
appear in the following chapters, are: Systematic Eigenvalue Approach to Crystal Optics: An
Analytic Alternative to the Geometric Ellipsoid model (Gunning and Raab 1998a); Algebraic
Determination of the Principal Refractive Indices and Axes in the Electrooptic Effect (Gunning
and Raab 1998b); Physical Implications of the Use of Primitive and Traceless Electric
Quadrupole moments (Gunning and Raab 1997a); and Electric-Field-Induced Optical Activity
in Nonmagnetic Crystals (Gunning and Raab 1997b). In later chapters two separate experimental
projects on the quadratic electro-optic effect in KDP-type crystals are described. Some of the
results of these investigations have yet to be published, while others, for both the interferometric
and polarimetric experiments, appear in the papers: The Quadratic Electrooptic Coefficient gxxxxgyyxx
of DKDP Crystals (Gunning et al. 1998a); The Quadratic Electrooptic Effect and Estimation
of Antipolarization in ADP (Gunning et al. 1998b); and Studies of the Quadratic Electrooptic
Effect in KDP-type Crystals (Gunning et al. 1999).
The first chapter introduces a multipole eigenvalue theory of light propagation through a
nonmagnetic anisotropic dielectric. This theory, in the appropriate multipole order, explains a
range of optical phenomena, including circular birefringence and the linear birefringences of
Jones and Lorentz, not otherwise fully explained by the widely-used Fresnel ellipsoid or related
indicatrix ellipsoid. By means of this model the birefringences, optic axes, and eigenpolarizations
of light propagating through a transparent medium are fully accounted for. The theory presented
here concentrates on the electric-dipole description, which is the formal basis of the ellipsoid
models, and the description of the birefringences, optic axes, and eigenpolarizations is algebraic,
rather than geometric, and in terms of measurable crystal property tensors.
Chapter 2 follows the multipole approach of Chapter 1 with a systematic algebraic means to
derive expressions for the principal refractive indices and dielectric axes of a source-free
nonmagnetic crystal in a uniform electric field. This is seen as a preferred alternative to the more
common iterative numerical approaches to the problem. Results obtained in this chapter, to the
chosen order in the field, are again expressed algebraically in terms of crystal property tensors.
A number of illustrations of the approach are given for the linear electro-optic effect in crystals
of the symmetry point groups 43m, 3m, 42m, and 1. Also, to demonstrate the inadequacies of the
numeric method, the quadratic effect in 42m is investigated. Throughout these examples,
comparisons are drawn with published numerical results.
Chapters 3 and 4 start to allow for higher-order multipole moments, in contrast to the opening
two chapters which are limited to the electric-dipole description. Specifically, electromagnetic
effects to the order of electric quadrupole and magnetic dipole are considered. Since these effects,
such as chiral phenomena in fluids and crystals and gyrotropic birefringence in antiferromagnetic
crystals, require for their full description the inclusion of electric quadrupoles, the
nature of the electric-quadrupole moment is considered in Chapter 3. In many theories this
moment is defined to be traceless, but this chapter shows that when this definition is used in the
derivation of a wave equation to describe light propagation through an optically active uniaxial
medium, this equation as well as properties derived from it, in particular the refractive index of
the eigenpolarizations, are in general dependent on the origin used to define the Inoment. This
is physically unacceptable for the theoretical expression of an observed effect. It is shown that
if the primitive definition of the quadrupole monent is used in this theory instead, this difficulty
does not arise.
Also to the order of electric quadrupole and magnetic dipole, Chapter 4 presents a macroscopic
theory of electric-field-induced optical activity in source-free nonmagnetic crystals. In this
approach a wave equation, to this multipole order and including distortions linear in an applied
electric field, is derived which allows for the description of the linear electric-field-induced
circular birefringence. It also identifies other coexisting birefringences, both natural and fieldinduced
which are described in this multipole order. All these birefringences are expressed in
terms of measurable crystal property tensors. In this chapter, nonmagnetic crystal classes are
identified which exhibit this field-induced optical activity for particular light-propagation and
applied-field configurations for light waves with purely transverse fields. These crystals,
symmetry considerations, and other coexisting natural and field-induced birefringence are
presented in Table 4.1.
The first chapter of Part 2 of this thesis introduces the quadratic electro-optic effect in KDP-type
crystals and provides motivation for the present experimental inquiry of the effect. Tabulated are
previous experimental results for components of the quadratic electro-optic coefficients of
crystals of KDP, ADP, and DKDP investigated in this work. Also introduced is the
electrostrictive effect in these crystals, since it plays a part in the experimental determination of
quadratic electro-optic coefficients of these crystals, and values for coefficients of this effect are
listed. A multipole theory, based on that in Chapters 1 and 2 to the order of electric dipole and
allowing for perturbations quadratic in a low-frequency applied electric field, is presented which
describes fully the observed quadratic electro-optic effect in these crystals for the lightpropagation
and applied-field geometries of interest. The analytic results derived for the observed
induced refractive index changes for the polarization eigenvectors for a particular propagation
path through the medium are expressed in terms of the quadratic electro-optic coefficients gxxxx,gyyxx, and gxxxx,
One of the two methods proposed in this work for the measurement of coefficients of the
quadratic electro-optic effect in specimens of KDP-type crystals is the interferometric technique.
Chapter 6 serves to introduce this approach and to provide the theoretical basis for it, within the
context of determining small induced phase changes. After a number of experimental factors are
highlighted the method proposed for this work is explained. Specifically, this approach utilizes
a Michelson arrangement, invoking active stabilization against low-frequency noise influences,
and the coefficients are measured by phase-compensation means. This apparatus and its various
components is then fully described in Chapter 7, including the method for its operation. The
results of this interferometric research appear in Chapter 8 for the various light-propagation and
applied-field geometries used, and values are presented for the individual coefficients gxxxx, gyyx.x,
and gxxx.x in KDP and ADP, and also the electrostrictive coefficients Yxxyy and Yxxxxin the same
crystals.
An explanation of the second approach to measure quadratic electro-optic coefficients in this
work, the dynamic polarimetric technique, is given in Chapter 9. Two proposed means to
investigate the KDP-type crystals by this approach are described and explained theoretically, and
the layout of the apparatus and its various components is highlighted. Results of this polarimetric
investigtion for two light-propagation directions are given in Chapter 10 for investigations on
KDP, ADP, and DKDP single crystals. These give values for the difference in quadratic electrooptic
coefficients |gxxxx - gyyxx| and |-n03gxxxx|
3gxxx.x - n/g==xJ
Finally, in concluding this thesis a summary of the quadratic electro-optic and electrostrictive
results determined by the interferometric and polarimetric techniques is given and comparison
of these results is drawn with those published previously. Considered briefly are some further
conclusions that may be derived from these values for the nature of the electro-optic effect in
KDP, ADP, and DKDP.