Piezooptic and Electrooptic Constants

405 Pages·1996·20.008 MB·English

by  D. F. Nelson (auth.), D. F. Nelson (eds.)

Ref. p. 3311 5.1 Introduction 1
5 Piezooptic and electrooptic constants of crystals
5.1 Introduction
D.F. NELSON
5.1.1 Definitions
The piezooptic and electrooptic eficts are the alterations of the optical propagation constants
of a medium caused by mechanical or electrical stress respectively. The piezooptic effect, sometimes
called the elastooptic or photoelastic effect, exists in all media. The linear electrooptic (Pockels) effect
exists in piezoelectric crystals; the quadratic electrooptic (Kerr) effect exists in all media. The reader is
referred to reviews ofthepiezooptic effect [57nl, 58vl,61rl, 64cl,66ml, 68fl, 7Owl,71dl, 72b2,72cl,
73~1, 76~1, 76~1, 79n1, 79t1, 80d1, 8011, 80n1, 84~1, 88k1, 89m1, 92x1] and the electrooptic effect
[57nl, 61rl,64cl, 66kl,66ml, 72wl,74kl, 79n1, 8011,8Onl, 84~11.
5.1.2 List of symbols
Note: The summation convention for repeatedi ndices is used.
A, p, v, Q,c s = 1 .. . 6; k, i, j, 1,m , n = 1 .. . 3.
Unit propagation vector (wave normal) of an acoustic wave, Eqs. (48) and (50)
ai
Piezoelectric stresst ensor that couples to P, a&j = {Helij/Eo, Eqs. (14) and (16)
akij
Piezoelectric strain tensor that couples to P, b, = SkldlijlEO) Eqs. (13) and (16)
btij
Stress-optic coefficient, Eqs. (25), (39), and (40)
C
Elastic stiffness tensor, Eqs. (9) and (12)
Cijkl
Normalized electric displacement vector, Eq. (1)
di
Electric displacement vector, Eq. (11)
Di
Piezoelectric strain tensor that couples to E, d&j = ehs&ij, Eqs. (13) and (16) *)
d,,
Electrooptic susceptibility tensor, Eqs. (5 1) and (55) *)
&m
Unit electric field vector, Eqs. (17) .. . (19)
;i Electric field vector
Piezoelectric stresst ensor that couples to E, Eqs. (9) and (10)
eijk
Linear electrooptic tensor that couples to D, Eqs. (2), (6 f), and (13). .. ( 16)
Linear electrooptic coefficients, Eq. (8 c)
9 Ik
Piezoelectric strain tensor that couples to D, gkij = pkl d&,, Eqs. (13) and (16)
gkij
Piezoelectric stresst ensor that couples to D, h kij= Pu eiij/s,, Eqs. (14) and (16)
htij
k Extinction coefficient
Quadratic electrooptic tensor coupling to polarization, Eqs. (58c), (58d), and (60b)
Mijkl
Quadratic electrooptic coefficients, Eq. (62b)
Mu
*) Note that the same letter is used as the conventional symbol for different properties.
Land&-Bdmstein
New Series 111130A
2 5.1 Introduction [Ref. p. 33 1
Electrostrictive tensor coupling to electric field, Eqs. (59a) and (60a)
N Number of concordant measurementso f a quantity
n Refractive index
Unperturbed refractive index
Complex refractive index
Change in refractive index induced by a dielectric impermeability change,E qs. (17) .. . (24)
Rotooptic tensor, Eqs. (2), (5), (6 c), and (57 a)
*ijkl
Rotooptic coefficients, Eq. (8e)
%
Linear polarization, Eq. (10)
pi
P Hydrostatic (or isostatic) pressure
Elastooptic tensor, Eqs. (2), (6 a), (12a), (14), and (56)
Pijkl
Elastooptic coefficients, Eq. (8d)
Ph
Nonlinear polarization, Eq. (5 1)
%
Piezooptic tensor, Eqs. (2), (6 b), (12b), and (13)
%jkl
Piezooptic coefficients, Eq. (8 f)
%
Electrostrictive tensor coupling to polarization, Eqs. (59 b) and (60b)
Qijk,
Linear electrooptic tensor that couples to E, Eqs. (2), (6d), and (13) .. . (16)
rijk
Linear electrooptic coefficients, Eq. (8 a)
rRk
Antisymmetric (rotation) part of the displacement gradient, Eq. (3)
Rkl
Quadratic electrooptic tensor coupling to electric field, Eqs. (SSa), (58b), (6Oa), and
Rijkl
(61)
R Quadratic electrooptic coefficients, Eq. (62a)
AP
s Standardd eviation
Unit propagation vector (wave normal) of a light wave, Eqs. (17) .. . (24)
4
$kl Elastic COtllp!iallCe kINOr, Cijkl Sk,,,,=” (q, Sj,+ &, 6j,)/2, Eq. (12 b)
Strain tensor equal to the symmetric part of the displacement gradient, Eq. (4)
sk,
In the tables: the following values are for constant strain
(9
Stresst ensor, Eq. (9)
In the tables: the following values are for constant stress
(‘;I)
Curie temperature,f erroelectric (or antiferroelectric)
T,
T Transition temperature( for special definition seer espective figures)
1.2.3~ Ti, TL
Displacement gradient 3 &4kl&, = R k,+ Sk1
uk,l
V Coefficient of variation
Optical absorption coefficient
Dielectric impermeability tensor equal to the inverse of the relative dielectric permittivity
ii j
tensor, Eq. (1)
Change in impermeability tensor from an elastic or electric perturbation, Eq. (2)
ABij
Dielectric permittivity tensor, EG= Eo(~i+j Xij)
Eij
Permittivity of free space
co
Complex dielectric constant
z
Imaginary part of dielectric constant
Ei
Real part of dielectric constant
E’
Relative dielectric permittivity tensor = .sijlso
Kij
Wavelength
1
Inverse of relative electric susceptibility tensor, Eq. (15)
5ij
Complex piezooptic coefficient tensor, Eq. (67)
nijkl
Linear electrooptic tensor that couples to P, Eqs. (2), (6e), and (13) . . . (16)
&jk
eu, Linear electrooptic coefficients, Eq. (8 b)
Electrical conductivity
0
Relative electric susceptibility tensor, Eq. (10)
xij
Elastooptic susceptibility tensor, Eqs. (5 1) and (56)
Xij@I)
Rotooptic susceptibility tensor, Eqs. (5 1) and (57)
Photon energy
LandobB6mrtcin
New Series 111130A
Ref. p. 3311 5.1 Introduction 3
5.1.3 Index ellipsoid
Piezooptic and electrooptic constants are traditionally defined as the coefficients in perturbations of the
index ellipsoid [1894Pl, 06~11.T he index ellipsoid is given by [61rl, 75bl]
Pijdidj = 1) (1)
where j? is the optical frequency dielectric impermeability tensor and d is the electric displacement
vector of the light normalized to make the right side of Eq. (1) unity. The summation convention for re-
peated indices is used. The index ellipsoid has the meaning that the lengths of the major and minor axes
of an ellipse formed by the intersection of a plane with the ellipsoid are the inverse squareso f the refiac-
tive indices for the two light waves propagating in a direction normal to the plane.
The coordinate system used in Eq. (1) is a rectangular Cartesian system (the “crystallographic
coordinate system”) fixed in the laboratory but oriented by convention relative to the symmetry
directions of the crystal in its unperturbed state. The coordinate system in which the dielectric
impermeability tensor p is diagonal (& = &r = plZ = 0) is called the principal coordinate system. In
the principal coordinate system the diagonal elements of /3 are equal to the inverse squares of the
principal refractive indices. In the cubic, hexagonal, tetragonal, trigonal, and orthorhombic crystal
systemsc rystal symmetry requires the principal coordinate system of the impermeability tensor to coin-
cide with the crystallographic coordinate system. In monoclinic crystals symmetry requires pZ3= plZ = 0
in the crystallographic coordinate system; in triclinic crystals symmetry allows &+&r+ /In =t=0 in that
system.
5.1.4 Linear perturbations of index ellipsoid
The piezooptic and electrooptic effects produce perturbations of the dielectric impermeability tensor
which require the replacemento f /? + p + Ap in Eq. (1) and which may be visualized as a deformation of
the index ellipsoid. If only linear perturbations are considered the increment in the dielectric impermea-
bility may be expressedi n any of the forms
Apij =P~~klskl+ OijklR~l+r ake,, (24
= q$l Ttcl+O ijklRt+i r&K 3 CW
=P$I&’ oijM&+ &Ptc, cw
= q$&+ q,R,+ &A CW
=P&& + oijti&+f$R, (24
= q~~,T,+ oij,Rkl+f~Dk. cm
The first term in each form of Eq. (2) representst he traditional piezooptic effect which may be regarded
as causedb y either the strain S, or the stress TM.T he secondt erm representsa predicted and confirmed
piezooptic effect causedb y the rotation Rkl. An alternative terminology is to call it the rotooptic efict.
The rotation Rkl is the antisymmetric combination of displacement gradients *),
(3)
in contrast to the strain which is the symmetric combination of displacement gradients,
sld = yk,l) = 3 (%,I + %,k)* (4)
*) Note that R,, is not what is conventionally called the infinitesimal rotation tensor which is 6, + u k ,].
Landolt-BBmstein
New Series 111130A
4 5.1 Introduction [Ref. p. 33 1
The third term in Eq. (2) is the linear electrooptic effect which may be regarded as causedb y a low
frequency or static electric field E, linear polarization P, or electric displacementD .
The quantitypij,, is called the tensol; qijkli s called thepiezooptic tensol; rijk, ~jjk, and~j~ are
elustooptic
called linear electrooptic tensors and oijk,m ay be called the rotooptic tensor. From the origin of the index
ellipsoid it can be seent hat the i and j indices of each of these tensors correspond to the directions of the
electric displacement vectors of the output and input light waves respectively. The superscripts on any
coefficient in Eq. (2) indicate which independent variables must be held constant if that coefficient is to
represent the entire perturbation. However, the superscript R (rotation held fixed) has been omitted for
simplicity. Also, all superscriptsh ave been dropped from o since it is the samet ensor in each form of Eq.
(2). The latter is true since there is no coupling in the linear stress-strain-electricf ield equation to the rota-
tion. All these material tensors are functions of the light frequency, the perturbing field
frequency, and the temperature. The slight difference that should arise between piezooptic constants
measuredu nder isothermal (quasistatic) conditions and under adiabatic (very high frequency) conditions
has been ignored in the above formulation becauset he expected difference in values is less than typical
present-daye xperimental uncertainty.
51.5 Rotooptic effect
The need for including rotation in characterizing the piezooptic effect was only realized relatively
recently [70N2, 7lN4]. That study showed that the rotation part of a deformation reorients the linear
optical anisotropy (the dielectric permittivity, or its inverse the dielectric impermeability) and so produces
a piezooptic contribution independent of the contribution arising from strain. Anaxial (cubic) crystals
which lack birefringence have vanishing rotooptic coefficients. In strongly birefringent crystals the
rotooptic coefficients are comparable in magnitude to the elastooptic coefficients. The relation of
the rotooptic tensor oijk,t o the dielectric impermeability tensor pij (expressed in the crystallographic
coordinate system) is
Oijkl = 4[k Bi]j + 6j[k Bl]i 9
where S,, is the Kronecker delta and A,$,, = i (A$, - A,&). The predictions of this equation have been
verified in Wile [7ON3, 76G1, 77N1, 80Gl], calcite [72N3], sodium nitrite [72H2], sodium nitrate
[72Kl], lithium acetate [73Vl], gadolinium molybdate [77Sl], zinc oxide [76Sl], barium titanate
[9lG2], potassium dihydrogen phosphate [83A3], potassium dideuterium phosphate [83A3], rubidium
dihydrogen phosphate [83A3], sodium bismuth molybdate [86Al], and potassium niobate [9lG2,
93221.
Since the rotooptic tensor o can be calculated from the refractive indices (and the orientation of prin-
cipal axes in the caseo f monoclinic or triclinic crystals), it may be calculated accurately and so measure-
ments of it may be used to check experimental accuracy.N ote also that Eq. (5) gives the algebraic signs
for the componentso f o. These signs may be used to determine the signs of the componentso f p if appro-
priate crystal orientations are studied [72N3].
51.6 Interchange symmetry
Each of the material tensors of Eq. (2) possessesc ertain interchange symmetry that follows from the
mechanicso f the interaction. For static elastic and electric perturbations thesei nterchange symmetriesa re
Pijkl = P(ij)( kl)3 (W
q(ij)( II) 3 (6b)
qijkl =
(64
Oijkl = O(a) PI] 9
Landolt-B6rnstein
New Series 111130A
4 5.1 Introduction [Ref. p. 33 1
The third term in Eq. (2) is the linear electrooptic effect which may be regarded as causedb y a low
frequency or static electric field E, linear polarization P, or electric displacementD .
The quantitypij,, is called the tensol; qijkli s called thepiezooptic tensol; rijk, ~jjk, and~j~ are
elustooptic
called linear electrooptic tensors and oijk,m ay be called the rotooptic tensor. From the origin of the index
ellipsoid it can be seent hat the i and j indices of each of these tensors correspond to the directions of the
electric displacement vectors of the output and input light waves respectively. The superscripts on any
coefficient in Eq. (2) indicate which independent variables must be held constant if that coefficient is to
represent the entire perturbation. However, the superscript R (rotation held fixed) has been omitted for
simplicity. Also, all superscriptsh ave been dropped from o since it is the samet ensor in each form of Eq.
(2). The latter is true since there is no coupling in the linear stress-strain-electricf ield equation to the rota-
tion. All these material tensors are functions of the light frequency, the perturbing field
frequency, and the temperature. The slight difference that should arise between piezooptic constants
measuredu nder isothermal (quasistatic) conditions and under adiabatic (very high frequency) conditions
has been ignored in the above formulation becauset he expected difference in values is less than typical
present-daye xperimental uncertainty.
51.5 Rotooptic effect
The need for including rotation in characterizing the piezooptic effect was only realized relatively
recently [70N2, 7lN4]. That study showed that the rotation part of a deformation reorients the linear
optical anisotropy (the dielectric permittivity, or its inverse the dielectric impermeability) and so produces
a piezooptic contribution independent of the contribution arising from strain. Anaxial (cubic) crystals
which lack birefringence have vanishing rotooptic coefficients. In strongly birefringent crystals the
rotooptic coefficients are comparable in magnitude to the elastooptic coefficients. The relation of
the rotooptic tensor oijk,t o the dielectric impermeability tensor pij (expressed in the crystallographic
coordinate system) is
Oijkl = 4[k Bi]j + 6j[k Bl]i 9
where S,, is the Kronecker delta and A,$,, = i (A$, - A,&). The predictions of this equation have been
verified in Wile [7ON3, 76G1, 77N1, 80Gl], calcite [72N3], sodium nitrite [72H2], sodium nitrate
[72Kl], lithium acetate [73Vl], gadolinium molybdate [77Sl], zinc oxide [76Sl], barium titanate
[9lG2], potassium dihydrogen phosphate [83A3], potassium dideuterium phosphate [83A3], rubidium
dihydrogen phosphate [83A3], sodium bismuth molybdate [86Al], and potassium niobate [9lG2,
93221.
Since the rotooptic tensor o can be calculated from the refractive indices (and the orientation of prin-
cipal axes in the caseo f monoclinic or triclinic crystals), it may be calculated accurately and so measure-
ments of it may be used to check experimental accuracy.N ote also that Eq. (5) gives the algebraic signs
for the componentso f o. These signs may be used to determine the signs of the componentso f p if appro-
priate crystal orientations are studied [72N3].
51.6 Interchange symmetry
Each of the material tensors of Eq. (2) possessesc ertain interchange symmetry that follows from the
mechanicso f the interaction. For static elastic and electric perturbations thesei nterchange symmetriesa re
Pijkl = P(ij)( kl)3 (W
q(ij)( II) 3 (6b)
qijkl =
(64
Oijkl = O(a) PI] 9
Landolt-B6rnstein
New Series 111130A
Ref. p. 3311 5.1 Introduction 5
@ijk = @(ij)k 9 (64
v&k =.&j)k 9 (60
where subscripts in ( ) may be interchanged and subscripts in [] may be interchanged after multiplication
of the particular perturbation by - 1 [see Eqs. (S)].
For dynamic perturbations the exact interchange symmetry between i and j in each of these
tensors is lost (a “dispersive asymmetry”) since the input and output light waves then have different
frequencies. However, because these frequencies are typically very close together, the dispersion in
the values of these tensors between the frequencies is negligible and the interchange symmetry
between i and j is an excellent approximation in transparent regions of materials. The paper [77Nl]
shows that if a dispersive asymmetry were observable, it would have to be accompaniedb y a change in
the relative strength of anti-Stokes and Stokesp rocesses( sum and difference frequency generation) in a
dynamic elastooptic interaction.
The adequacy of andP(ij in characterizing the piezooptic effect was predicted [71N4] from a
o(ij)[kl]
combination of the mechanics of the interaction, the estimated size of the resultant contributions, and
crystal symmetry. The characterization of the piezooptic effect by the most general form of a fourth rank
tensor possessingc rystal symmetry but having no interchange symmetry of any kind has also been dis-
cussed [74A4]. Data have been published [76Gl] supporting this lower symmetry. A dispersive asym-
metry (p2332+ p3& was reported in that work on rutile. Further work [77Nl, SOGl], however, did not
support that finding.
5.1.7 Matrix notation
For tabular presentation of the allowed forms of the material tensors a matrix or contracted notation
is used. It is made possible by the interchange symmetries, Eqs. (6). In order that the notation
applies to antisymmetric as well as symmetric pairs of tensor indices the association of a six-dimen-
sional index (denoted by a Greek letter) to a pair of three-dimensional indices (denoted by Latin letters)
is made via
A 1 2 3 4 5 6
ij 1, 1 2,2 3,3 2,3 3, 1 1,2. (7)
The matrix notation for the material tensors is now defined by
r, = (84
rijk = rjik )
@b)
&k = @ijk = @jik 3
fhk =Ajk =.f$) 03~)
PQ Puk (84
=Pijkl =Pjikl = =Pjilk 3
(84
05 = oijkl = ojikl = - Oijlk = - Ojilk )
(8f)
’ (k, 1) qhp = qijti = qjikl = qijk = qjilk 3
where
n (k, 1) = 1 if k, I= 1, 1; 2,2; 3,3
1
if k,l=2,3; 3, 1; 1,2. (8g)
=?
The factor IZ( k, 1) in the equation for q is needed to permit matrix multiplication of qh,,w ith the usual
matrix form of the stresst ensor.
Land&-Biirnstein
New Series 111/30A
Ref. p. 3311 5.1 Introduction 5
@ijk = @(ij)k 9 (64
v&k =.&j)k 9 (60
where subscripts in ( ) may be interchanged and subscripts in [] may be interchanged after multiplication
of the particular perturbation by - 1 [see Eqs. (S)].
For dynamic perturbations the exact interchange symmetry between i and j in each of these
tensors is lost (a “dispersive asymmetry”) since the input and output light waves then have different
frequencies. However, because these frequencies are typically very close together, the dispersion in
the values of these tensors between the frequencies is negligible and the interchange symmetry
between i and j is an excellent approximation in transparent regions of materials. The paper [77Nl]
shows that if a dispersive asymmetry were observable, it would have to be accompaniedb y a change in
the relative strength of anti-Stokes and Stokesp rocesses( sum and difference frequency generation) in a
dynamic elastooptic interaction.
The adequacy of andP(ij in characterizing the piezooptic effect was predicted [71N4] from a
o(ij)[kl]
combination of the mechanics of the interaction, the estimated size of the resultant contributions, and
crystal symmetry. The characterization of the piezooptic effect by the most general form of a fourth rank
tensor possessingc rystal symmetry but having no interchange symmetry of any kind has also been dis-
cussed [74A4]. Data have been published [76Gl] supporting this lower symmetry. A dispersive asym-
metry (p2332+ p3& was reported in that work on rutile. Further work [77Nl, SOGl], however, did not
support that finding.
5.1.7 Matrix notation
For tabular presentation of the allowed forms of the material tensors a matrix or contracted notation
is used. It is made possible by the interchange symmetries, Eqs. (6). In order that the notation
applies to antisymmetric as well as symmetric pairs of tensor indices the association of a six-dimen-
sional index (denoted by a Greek letter) to a pair of three-dimensional indices (denoted by Latin letters)
is made via
A 1 2 3 4 5 6
ij 1, 1 2,2 3,3 2,3 3, 1 1,2. (7)
The matrix notation for the material tensors is now defined by
r, = (84
rijk = rjik )
@b)
&k = @ijk = @jik 3
fhk =Ajk =.f$) 03~)
PQ Puk (84
=Pijkl =Pjikl = =Pjilk 3
(84
05 = oijkl = ojikl = - Oijlk = - Ojilk )
(8f)
’ (k, 1) qhp = qijti = qjikl = qijk = qjilk 3
where
n (k, 1) = 1 if k, I= 1, 1; 2,2; 3,3
1
if k,l=2,3; 3, 1; 1,2. (8g)
=?
The factor IZ( k, 1) in the equation for q is needed to permit matrix multiplication of qh,,w ith the usual
matrix form of the stresst ensor.
Land&-Biirnstein
New Series 111/30A
6 5.1 Introduction [Ref. p. 331
51.8 Crystal symmetry
The form of each of the material tensors appearing in Eq. (6) is also restricted by crystal symmetry. The
electrooptic tensors lCUJkQ, ij)k, and&,, have the same interchange symmetry as the piezoelectric stress
tensor e,.ij,( note, however,t he different order of indices). Further, the contracted (matrix) notation for rA,,
Table A. Allowed tensor components of the elastooptic and piezooptic tensors psi and qsl in various
crystal classes.
The following notation is used in this table:
A nonzero component;j oined componentsa re equal.
l
o A component equal to the negative of the one to which it is joined.
o A component which is equal to the solid dot componentj oined to it forpiP, or which is twice the solid
dot componentj oined to it for qb,, .
Q A component which is equal to minus the solid dot componentj oined to it for px,,, or which is minus
twice the solid dot componentj oined to it for qlp .
x A component equal to$(p,, -p,& or (q,, -q,&.
.. .
.. 0
Triclinic .. Mo.n oc linic Orthorhombic
both dosses .. all .c loses oil dosses
. . . . . .. . . . l ......
. . . . . I I. . . . . l ......
......
. . . 0 . . . .
......
. . . . . . . l
......
. . . . l . . l
......
. . . 0 . . l .
Tetragonol Trigonal
dosses 5,TaL/m classes Smm,Z lm. 422. Ummm classes 3.3 classes 3m, 32,j rn
. . .
XI::! X
- . . . . F-. . .
. . . . . . . . .
. .
. . . X . . .
. . \. .
co . . l . . l .
1 I
Hexogonol Cubic
classes 6,6,6/m classes 6m2,6mm., 622., G/m. mm classes 23, m3 classes 53m.4 32. m3m
X I . . .
- . . . .
. . . . .
. . . \. .
. . . . . x
1
Isotropic
Landolt-B6rnstein
NW Series 1W30A
Ref. p. 3311 5.1 Introduction 7
Table B. Allowed tensor components of the rotooptic tensor oap in various crystal classes (n, , n2, n3
are the principal refractive indices).
Cubic system - all elements vanish
Hexagonal, tetragonal, trigonal Orthorhombic
. . .
. . .
. . .
044 . .
* 055 *
. . 066
04= -055 = (n,* - n;Z)/2 04 = (n;Z - n;2)/2
055= (n p - n;2)/2
066= (q2 - ny2)/2
Monoclinic Triclinic
. . . .
015 * . . . *14 *15 *16
. . . . . . . . . *24 *25 *26
. . . . 035 . . . . *34 *35 *36
. . . *44 * *46 . . . *44 *45 *46
. . . . 055 . . . . *54 *55 OS6
_. . . *64 - *66 _* . * *64 *65 *66
6ljl/2
04.4 = (P33 - P22Y2 *hp = [Aj 4k - Pkj h + Pil skj - Pik
P&2
055 = @II -
066 = (P22 - Pm
035 = - 015 = 2064 =-2o46 = PI3
&k, andfhk for the various crystal classesi s the samea s that of eti in Sect. 3.1 of III/29 b. The electro-
optic tensors are zero for nonpiezoelectric crystal classes.T he form allowed by crystal symmetry for the
elastooptic tensor phP and the piezooptic tensor qhp is given in Table A for each of the crystal classes.
Table A follows Bhagavantam [42bl] who corrected a number of errors in Pockels’ original work [06pl].
Table B gives the form of the rotooptic tensor oh&a llowed by Eq. (5) for each crystal system.A ll crystal
classesw ithin a given crystal system have the samea llowed form for the rotooptic tensor.
5.1.9 Constitutive relations
The material tensorsi n the expansionso f Eq. (2) are related through the constitutive relations for the stress
tensor T, the polarization P, and the electric displacementD . This linear terms of these relations are
Tu = c&Sk, - ebjE k ,
(9)
Pi = &oX : Ej f e$&) (10)
Oi=EoEi+Pi,
(11)
where cE is the elastic stiffness tensor at zero electric field, e is the piezoelectric stress tensor
coupling to the electric field E, and xs is the linear electric susceptibility at zero strain. These equations
apply to pyroelectrics (crystals including ferroelectrics that possessa spontaneousp olarization) in their
Landolt-BSmstein
New Series 111130A
Ref. p. 3311 5.1 Introduction 7
Table B. Allowed tensor components of the rotooptic tensor oap in various crystal classes (n, , n2, n3
are the principal refractive indices).
Cubic system - all elements vanish
Hexagonal, tetragonal, trigonal Orthorhombic
. . .
. . .
. . .
044 . .
* 055 *
. . 066
04= -055 = (n,* - n;Z)/2 04 = (n;Z - n;2)/2
055= (n p - n;2)/2
066= (q2 - ny2)/2
Monoclinic Triclinic
. . . .
015 * . . . *14 *15 *16
. . . . . . . . . *24 *25 *26
. . . . 035 . . . . *34 *35 *36
. . . *44 * *46 . . . *44 *45 *46
. . . . 055 . . . . *54 *55 OS6
_. . . *64 - *66 _* . * *64 *65 *66
6ljl/2
04.4 = (P33 - P22Y2 *hp = [Aj 4k - Pkj h + Pil skj - Pik
P&2
055 = @II -
066 = (P22 - Pm
035 = - 015 = 2064 =-2o46 = PI3
&k, andfhk for the various crystal classesi s the samea s that of eti in Sect. 3.1 of III/29 b. The electro-
optic tensors are zero for nonpiezoelectric crystal classes.T he form allowed by crystal symmetry for the
elastooptic tensor phP and the piezooptic tensor qhp is given in Table A for each of the crystal classes.
Table A follows Bhagavantam [42bl] who corrected a number of errors in Pockels’ original work [06pl].
Table B gives the form of the rotooptic tensor oh&a llowed by Eq. (5) for each crystal system.A ll crystal
classesw ithin a given crystal system have the samea llowed form for the rotooptic tensor.
5.1.9 Constitutive relations
The material tensorsi n the expansionso f Eq. (2) are related through the constitutive relations for the stress
tensor T, the polarization P, and the electric displacementD . This linear terms of these relations are
Tu = c&Sk, - ebjE k ,
(9)
Pi = &oX : Ej f e$&) (10)
Oi=EoEi+Pi,
(11)
where cE is the elastic stiffness tensor at zero electric field, e is the piezoelectric stress tensor
coupling to the electric field E, and xs is the linear electric susceptibility at zero strain. These equations
apply to pyroelectrics (crystals including ferroelectrics that possessa spontaneousp olarization) in their
Landolt-BSmstein
New Series 111130A