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On the photoelastic examination of vibrating bodies and the photoelastic effect in optically active media
J. Merh. I’hys. Solids, 1000, Vol. 8, pp.
ON THE
26 ta 38.
PHOTOELASTIC
Pergnmon
Press
EXAMISATION
BODIES AND ,THE PHOTOELASTIC ACTIVE By Hoy~l
HoHoway
A.
Prhltecl in Great Britain
Ltd., London.
OF VIBRATING
EFFECT
IN OPTICALLY
MEDIA*
F.
B.
College,
WOOD
of London?
University
bodies is discussed, with in~r~ret~tion of the ~~hotoel~tj~ stress patterns of ribcrting An expression is derived for the change in velocity of H particular referenre to quartz crystals. light wave propagateci along the optic axis due to stresses acting in the plane norrnsl to this axis. Such stresses should c:ruse an increase in the effective rotary power snd this is verified by experiment. This result offers an explmaticm of phenomena observed 1,s earlier workers.
Tiffs
IK
connexion
became
with
necessary
a photoelastic
to consider
examination
the manner
of
in which
vibrating the isoclinics
quartz
crystals
it
and isochromats
observable in a statically stressed isotropic medium are modified by the vibratory nature of the stress and the optical activity of the Z-cut specimens used for the experiments. The effects intensity polarizer
of vibration
of the light and analyzer
where I, is the incident
alone
may
easily
be taken
transmitted by a birefringent IS I = 1, sin2 ‘La sin2 (6/L) intensity,
into consideration, medium
CLthe acute angle between
between
The crossed (1)
the polarizer
axis and
a principal axis of the medium and 6 the relative phase retardation between the The transmitted intensity two plane polarized waves emerging from the specimen. is zero if Q = o or 4~ or if 6 = 2t~n. that is. if the principal axes of the medium are parallel to polarizer and analyser or the retardation is a whole number of periods. In a mechanirally stressed isotropic medium the principal axes of the artificial birefringence are parallel to the principal axes of stress, and the difference between the propagation velocities in any direction is proportional to the difference between the secondary principal stresses acting normal to the direction of propagation. The case usually considered is that of a lamina in a state of plane stress, uniform throughout the thickness, the light being propagated along the thickness. If the stress varies across the specimen two sets of dark fringes are observed. One set,, the isoelinie, joins points at which the principal stress directions are parallel
26
On the photoelastic t=xnminntionof vilmlting lx.dies
27
and analyser axes and its position is independent of the light wavelength. The other set, the isochromat, joins points at which the difference between the principal stresses causes a retardation of an integral number of periods and the position of the fringes for each non-zero value of n in equation (1) depends on the wavelength. The zero-order isochromat, for which there is no phase retardation and the principal stresses are equal, is common to all wavelengths. Considering, now, an isotropic lamina subjected to an oscillatory plane stress system caused by vibration of the lamina in a normal mode, then a fundamental characteristic of such vibrations is that the displacements at all points are in phase and hence the directions and ratio of the principal stresses remain fixed throughout the vibration cycle. The retardation will vary sinusoidally with the stress at the frequency of vibration and, writing 6 = A cos w t and finding the mean transmitted intensity over a whole period of the vibration, equation (1) becomes I, = I, sin2 2cr(1 - J, A)/2. (2) to the polarizer
This is zero when u = 0 or 4~ and also when A = 0 since .I, (0) = 1. Thus the isoclinic and the zero-order isochromat are unaffected by the vibration. The expression (1 - J, A) oscillates about the value one with decreasing amplitude as A increases. The values of A for the minima are given by A = 2krr where k is no longer integral but takes the values 0, 1.117, 2.121, 3.122, 4.123. Although these minima do not have zero intensity, apart from the zero-order isochromat, the position is unaffected by the intensity distribution due to the sin2 2 GLterm and they may be used in the same way as isochromats in the static case. Thus, when an isotropic body is vibrating in a normal mode of a type giving rise to a plane stress system, the usual photoelastic observations may be made except that the number of isochromats visible is limited by the decrease in visibility with order and that the retardation associated with any order is not an integral number of periods as in the static case. In these experiments on vibrating quartz crystals, however, owing to the smallness of the photoelastic effect and the thinness of the specimens used, the retardation could not be increased sufficiently to produce the first intensity minimum without danger of exceeding the breaking stress. Hence only the isoclinics and zero-order isochromat were available for study. 2.
THE PHOTOELASTICEFFECT IN QUARTZ
Under the action of a general stress system the equation of the index ellipsoid of a crystalline medium, referred to the orthogonal crystal axes, changes from a011x2 -t ao22y2 + aos3z2 = 1
to the general equation of an ellipsoid whose axes do not coincide with the coordinate axes : aI1 z2 + a22y2 +
a33
z2
-k
2%3
?.@ +
2a3,
zx
f
2%3
z?l
=
1
(4)
where aOii represent the squares of the principal velocities of light waves in the unstrained crystal relative to that *in vacua. In general, each of the coefficients aij is a linear function of all of the six stress components. POCKELS(1889) has shown that the strained index ellipsoid is obtained
2x
13. \\‘OOD
.A. F.
from the unstrained by suitable changes in the lengths of the axes and, provided that ati - noli and o,, are all small c~omparetl with noi, and (loii - uOJj, by small rotations rji about each of the coordinate anglt5 arr giver> by : tan 24, = 2aik ‘(uj, In
quartz
the coefiicients
by the following
-
asrs nkk) E
of the strained
equations
in tllrri.
l’he
L’olk, ‘(QO,, -
magnitudes
of theae (5)
fIOhJ.
ell~psoiti arc’ given in terms of the stresses
:
a11
=
v*o
+
"11
01
+
x12
02
+
n1303
+
n14
u'4
“22
=
v20
t-
n12
Ql
+
711
u2
+
r13
a3
-
=14
04
u33
=
1'2? +
7731(Jl
+
773102
+
733
'J3
u23
~~
x41(31
-
n41u2
+
"44(J4
7
(6)
u 31 =
7744 "6
(I12
7714 (J6 +
+
2n41
Us
(+rll -
7712) 06 I
where z’, aricf i’, are the ordinary coellicients
and extraordinary
and ut the stress components. uoll = uo2* = 2~*~, $3 is large but
Since may still be found from the formula out first and that it 1s calculated a state
of plane stress
given,
velocities,
the orientation provided
xi1 the stress
of the index
that the rotation
is considrrvd.
Since
rG6 _
ellipsoid
c$, is carried
from the first part of (,5). In the present
in the SY-plane
optic
discussion
nll -
n12 it is
ca\lly shown that in this case 4, := 8, the angle between the S and Y-axes and the It may further be shown that for suchh a stress system principal stress directions. the resultant effect of the rotations +1 and d2 is that the indes ellipsoid is turned through an angle (+2l + ~$2~)+about an axis in the S I’-plane which makes an angle -
%9 wit11 the S-axis.
cluartz
Irsing
this a11g1e IS found
I’o(.KEI.s’
to be about
optic axes of the now biaxial
values
for the stress
1 or Z per cent
optic
coefficients
of the angle
between
of the
quartz for a stress of 10~ dyne cm-* and will be ignored
III what follow\. On slll)stitllting the \-alues of the ellipsoid parameters from (6) and taking new S and I'-;iscs alolig the directions of the principal \trcsses, up and uu. (1) transforms to the sani(’ form as (3) :
-+- (5’2, $ The
differcIrce
between
n-31
the
\/(I22 -
(up i_ a,)) 22 = 1. principal +11
y
\-clocities (TN
-
(7)
in the
712) (uy -
Z-direction
Q/‘2vo.
is : (8)
Thus as regards the pure photoelastic effect a Z-cut lamina stressed in the XY-plane behaves as though isotropic to light propagated along the Z-axis because the section of the index ellipsoid by the wave front has axes parallel to the principal stress directions and the retardation is proportional to the difference between the principal stresses. Since quartz is piezoelectric there is a secondary photoclastir effect fact that the electric polarization produced by the stress causes further
due to the changes in
On the photoelastic examination of vibrating bodies the parameters magnitude
of the index
of the electric
ellipsoid
by virtue
polarization
‘29
of the electro-optic
will depend
on the electric
effect.
The
boundary
con-
ditions and will generally differ for static and oscillatory stresses. However, the effect is small in quartz, being a few per cent of the direct photoelastic effect and, more important, the symmetry of the direct effect is not altered by the secondary effect, for example the relation 7re8= ml1 - n12 is still valid. For these reasons it will be ignored in this discussion.
3.
EFFECT OF OPTICAL ACTIVITY
In a medium that is both birefringent and optically active two elliptically polarized waves of opposite hand may be propagated unchanged in any direction with slightly different velocities. The axes of the ellipses are parallel to those of the section
of the index
ellipsoid
although similar, lie oppositely, major axis of the other. In order to find the influence an incident conlponents
by the wave front and the ellipses themselves, so that the minor
axis of one is parallel
of such a medium
upon the polarization
to the state of
plane polarized wave, the wave is first resolved into plane polarized along the axes of the section of the index ellipsoid by the wave front
WKI then each of these is further resolved into two opposite elliptically polarized waves. lifter illtrod~lcing the appropriate retardations due to the passage through the medium, these are recombined to give the emergent wave. PKESTON (1901) IMS considered the general case of the intensity transmitted by an optically active birefringent
angles
crystal
between
polarizer
and analyser.
If the axes of these make
x and p respectively
with one axis of the section of the index ellipsoid bl then the ratio of the transmitted to the incident intensity is
the wave front, I I, =
c0S (a -
fi) cos S/2 -
zh
2
(1 + h2) sin (a 2cos2
(a
+
/T)
/?, sin S/2 sin2 S/2
where 6 is the relative retardation of the emergent elliptically polarized wa\-es and h is their axial ratio. Both terms of this expression are essentially positive and so the intensity zero when
can be zero only if they are simultaneously
(B and the second
when
a) = 77/2 + tan-’
i
A2
zero.
The first term is
tan S/2)
either (a + B) = m/2
(11)
or 6 = 21177.
(12)
Since (10) must always hold good for zero intensity axes must be misorientated by an angle
tan-’
-1
2h +
h2 * tan S/2
)
the polarizer
and analyser
A. Ir’. Ii. \\‘wI,
230
from the crossed position so that the efleedive rotation is p = tan-l
--%--
( 11_hZ’
tan 6,/2 . )
If h = 1, that is, if the waves are circularly pofarized, which is the ease when no birefringenee is present, then there results the well-known equation p = sjz.
(11)
Assun~ir~g that the polarizer and analyser have the correct relative settiug, then from (11) and (12) extinction can occur when {a) the bisector of the angle between the polarizer and analyser axes is parallel to that of the angle between the axes of the section of the index ellipsoid or (b) the retardation is a whole number of periods. Consideril~g, now, a Z-cut quartz lamina s~ti~ally stressed in the XY-plaae it is assumed that the natural optical activity is affected by the stress only to the same extent as the refractive index so that such a change is negligible compared with the birefringence introduced in the direction of the optic axis. In general the difference between the principal stresses, and their directions, will vary over the field of view so that it would be fortuitous if (lo) were satisfied sim~~Itar~eously with (I 1) or (12). If, however, the light is reflerted from the far side of the specimen so as to retrace its path exactly (lo) may be fulfilled automatictally. The rcllexion results in a complicated expression for the intensity since a left-handed elliptic* wave on incidence becomes a right-handed wave on reflexion ; but at points where the transmitted intensity would be zero if the analyser were correctly oriel~tated, that is, at just those points in which we are interested, the light emergent from the specimen must necessarily be plane polarized and will, on reflexion, retrace its path exactly and leave the specimen in the same state as that in which it entered. It may therefore be stopped by an analyser with its axis crossed relative to the polarizer. When observed under these conditions the specimen will exhibit two sets of dark fringes. One set joins points where the principal stress directions differ from those of the polarizer and analyser axes by half the effective rotation at the points concerned ; the other set joins points where the retardation is a whole member of periods. The sets correspond to isoclinies and isoehromats with these differences : (i)
The rotation must be known at any point on the isoclinic to find the principal stress directions. Since the rotation depends upon phase difference and therefore upon stress the interpretation is difficult. It will be shown however that for su~ciently thin crystals the rotation hardly varies from the natural value and hence the directions of the polarizer and analyser axes differ by a constant angle from the principal stress directions.
(ii)
Successive isochromats correspond to equal increments in the retardation but this does not now correspond to equal increments in stress owing to the inffuence of the optical activity, as will be seen later. In the work on vibrating crystals only the zero order isoehmmat is produced so this question does not arise.
On the photoelastie 4.
esamhurtion
of sibratin~
bodies
MAGNITVJXCOF CIUNGE IN EYSXCTIVE ROTATION
31
WITH STRESS
DHUDE (1905) has derived expressions for the propagation velocities vl, us and the axial ratio h of the two ellipticalIy polarized waves in a bi~fringent optically active medium. In what follows the coordinate axes are taken parallel to the principal axes of the index ellipsoid as in (3) but the principal velocities relative to that in wcuo are now denoted by u,_,a, and a, for mathematical convenience. DRUDE gives 2vz = a21 -f-
as8+ (a*,- a*$) cos g, cos g2 f
[W, -
&J* sin” g, sins g, + ~vJ~]$
h + I/h = [(azl - ~9~)~sin2 g, sin2 g, j- 4#]f/q
(W (I@
where g, and g, are the angles between the direction of p~pagation and each optic axis and 9 is a parameter proportiona to the rotary power in the absence of birefringence and varying with the direction of propagation. In the present case the direction of propagation is the Z-axis and since the optic axis must split symmetrically into two new optic axes on the application of stress g, and g, are simply equal to Ifr Q, the semi-angle between the optic axes. This has the value ~in-~ [(ozl - tz22)/(a21 - &,)]f and heuce sing, = + [(f82l - U~~)~(ff2~ - a$)]“,
(17)
sin g, =
WI
-
[b5
ff22)/(~2~ - a2,)]f,
-
eos g, c=.cos g, = + [(a22 - az,)/(a2,
- a2,)]*.
WI
Substitution of these values in (15) and (16) gives VI - u2 = (?/al) [I + (a, - %)*./(+Q]*,
(20)
h + r/h = 2 [l + fax - a2)~;(7j/f7,)~]**
1’21)
If d is the thickness of the lamina, T the period of the light wave, and c the velocity of light in WLWOthen the relative ~tardatio~~ of the emergent waves is S = 27Td(VI - v,f/Tc vi v2 N (2n d~/IW1)
[l -I- (aI -
4a/kd~I)2]*
(22)
from (20). By (14) the rotation in the unstressed specimen is equal to half the retardation : p. = 8,/Z
=
7+/Tca3,.
(23)
Hence
8 = 2& (1 + G)f
(24)
where
e = (a, -
(25)
a~)/(~~~~).
From (13) and (21) the effective rotation is P = tan-1
(
&%tanS/Z)
= tan-l ~~~~~~‘*I].
A. 1;. 1%.\voort
3” In
these expressions
6 is directly
stresses ; substituting
from
proportional
The resultant
retardation
the variation
of the retardation
thicknesses
of quartz,
to the dlffcrence
between the prlnrGpal
(8) into (25) :
is not proportional
0.5 mm
to t,he stress however.
6 and the effective
anti 3.06 mm.
The
first
Fig. (I) shows
rotation
p witft
E for two
thickness
is that
uf a %-cut
d : 3 06mm
n d=OfjOmm
disk used for experiments for the experiments
on vibrations
described
while the second is that of a Z-<*lit bar used
in the next section
to test this theory.
It will be
noticed that while pO and 6 are pro~~ortional to the thickness p is definitely not. For the smaller thickness p hardly varies from the natural value pO since (tan X p&,/K -+ pO as p. ---Iz0. Thus, for thin crystals, the rotation may he assumed and equal to the static value for all practical purposes. Owmg to the
constant
flatness of the curves provided notable.
when c is small the same will he true for thltker crystals Two other features of these curves are the stresses are small enough. First, for sufficiently high stresses the retardation tends asymptotically
to a straight line through the origin representing the retardation in the absence of optical activity. Second, whenever 6 = YI~ then p = 6 /2 as for the case of optical activity in the absence of hirefringence. Finally, perhaps the most i~~terest.ing feature of all in this theory is the fact that according to (24) the phase retardation between the two oppositely elliptically polarized waves can only incwase when the quartz becomes artificially birefrigent along the optic axis. Thus the increase in effective rotation given by (26) is independent of the sign of the stress and depends only upon the difference between the principal
stresses.
On the I>hotoelastic examirmtion of vibrating bodies 5.
The
TEST
ESPERIMESTAL
results of Section
P were tested
OF
ME:
33
THEORY
by investigating
the propagation
of light
through a Z-cut bar uniformly bent in the XY-plane. The bar had a rectangular section and its length was parallel to the X-axis. Fig. 3 shows how the stress was The polarizer axis was so applied and Fig. :3 shows the optical arrangement. orientated as to polarize the light in the plane of incidence on the half silvered mirror in order to avoid subsequent elliptic polarization. The thrust was provided by a calibrated
spring rather
than by weights
as the stresses had to act in the
horizontal plane owing to the geometry of the optical system. Before applying any stress the field of view was dark since the natural rotation of the quartz was cancelled
by the double
passage of the light through
PIG. 2. Apparatus
for applying
static bending stress to a quartz bar.
AndySCr
I-
Pamlkl
Half silvered mirror
monochromatic light
Specimen FIG.
the crystal.
silvered
on lower
surface
a. Optical apparatus for observation of photoelastic
stress patterns.
On applying a small stress the field brightened near the edges of the bar. At a slightly higher stress three distinct dark fringes could be seen parallel to the length of the bar, one along the axis and the other two symmetrically on either side. By rotating the bar and its bending jig about the Z-axis the latter fringes could be made to move outwards or, by a reverse rotation, coalesce with the centre fringe.
34
A. F. H. WOW
When the stress was increased still further the first-order isochromat appeared at the extreme edges of the bar. The thrust was increased slightly to a value of approximately 10’ dyne so that these lay clear of the edges and then a series of photographs was taken in which the applied stress remained constant but the bar had various orientations relative t.o the polarizer axis. It was noticed that the zero and first-order isochromats at the centre and near the edges respectively retained their position througtlout this series and also that the split ‘ isoclinic ’ was approximat.ely symmetrical throughout, although one side of the bar was in tompression and the other in tension. Fig. d (Plate) shows a photograph of a typical photoelastic stress pattern under these conditions and Fig. 5 an explanatory diagram.
FIG. .i. l’hotoel~tstic stressphltternin a statically stressed quartz bar.
In 11tc iihh(‘lt(*c of c@ic*al activity there would have been a solitary isoclinic fringe blackirlg ant the whole area over which the stress was substantially parallel to the axis, this being observed when the axis of the bar lay parallel to the polarizer or analyser axis. But from the theory of Sections 3 and 4 there is an effective rotation depending on the stress in the case of an optically active medium and darkness is only attained when there is a certain relation between polarizer orientation, principal stress directions and the magnitude of the stress, namely, that at any point on an isoclinic the stress results in an effective rotation equal to twice the angle between polarizer and analyser axes and the principal stress directions. The stress in a uniformly bent bar increases linearly from zero at the neutral axis t,o a maximum at the edges and so for a given applied stress and orientation of the bar relat.ive to the polarizer axis the isoclinic fringe consists of two lines parallel t,o the neutral
axis along the lines of appropriate
tensile and compressive
stress.
If the orientation of the bar is altered the fringes move outwards or inwards to the regions of appropriate stress at which the necessary conditions are still fulfilled. Alternatively if the orientation is fixed but the stress is varied the isoclinic fringes move so that they are always in regions of the same stress and therefore effective rotation. Although in this experiment owing
to poor
the absolute
mee~~anical construction
stress
was not known
of the bending
jig, a graph
the same accurately,
of rotation
against relative stress may be plotted from me~u~ments made on the series of photographs taken under conditions of constant applied thrust. For each effective rotation the distance between the two parts of the isoclinic was measured and taken to be proportional to the stress. Fig. 6 shows these experimental points. The curve is part of Fig. L for a thickness of 3.00 mm and has been drawn to pass through the experimental point for which the rotation is 180’ and the isoclinic coincides with the first-order isochromat. The values of l relate to the theoretical curve.
The agreement betweea the shapes of the two curves is very good.
As the thrust
was known to be about 10’ dyne for this experiment the stress in the region of the first order stress fringes was calculated from the geometry of the bar and the jig Co be about 8.41 x lOa dyne cm-z, and it was possible to estimate the value of x11 -
I%.
3712.
6.
I 2 0 Dependenceof rot&ion on phsse retardatiortshowing experinlenttrllydeternlined points.
From (24) E = (P/‘4&
-
1)f
(28)
and, since p0 = 78’ for the specimen used, c is 2.08 for the first-order isochromat. From (23) 71= p. asI Tc/m? (29) and substituting
in (27) gives =11 -
Taking a value of l/l-55 values quoted : nil -
xl2 =
2p,, a31 Tca/(u,
- q,) wd.
(30)
for the ordinary velocity and using the experimental
n12 = 10.3 x 10-l* dyne-l cm2 for X = 5461 A.
POCKELS’results give x11 -
TlZ =
13.95 x 10-ld dyne-’ cm2 for h = 5893 A
which is of the same order of magnitude.
6.
YHOTOELASTIC
EFFKT
IN VIBRATING
&TARTZ
CRYSTALS
The work described in the previous sections is part of an extensive study of the longitudinal vibrations of thin, circular, Z-cut quartz disks, the results of which will be published elsewhere. Some examples are given here, however, to show how the theoretical predictions of Sections 1-3 are borne out in practice. Z-cut disks
A. F, B.
36
WOOD
behave substa~ltially as tftough isotropic for normal modes of vibration
invofving
state of plane stress in the XY-plane.
in isotropic
media has been given by Of the multitude
LOVE
The theory of these vibrations
a
(1927).
of modes of tfns type tflerc are two series of symmetrical
modes
with simple displacements whose magnitude at any point depends only on the radial coordinate. In one series, type A, tfle displacements are everywflere along the radius v&or wfCle in tfle other, tyfje Z3, tfle dispfac~ements are everywhere normal to the radius vector. The principal st,resses at any point act &long the radius and its normal for type 14 and at 15” to these directions for type B. In each case the nodes of the longitudinal
displacements
consist of the centre of the disk together
witfl a number of concentric circles, the number increasing witfl mode order. Denoting successive modes by integers, including zero, the mth type ii mode has N. nodes ilk addition to the centre while t,he l/bthtype R has m + 1. The zero-order isochromat joins points at which thr principal stresses are precisely equal. In both types this occurs at the eentre and on a series of concentric circles which lie between the nodes, except that there is never one between tfle rcntre and the first nodal circle. In the case of the type R modrs the outermost circle of the isoc~hroniat is the periphery of the disk itself while for tyf’c 21 modes tfie outermost circie nlrnost coincides wit,h tfle periphery, cscept for tflc II) :- 0 mode. The qllartz
disks, all about 2.5
r111
diameter
and of’ various tfiic~kncss~s. were
cxcitcd into vif,ratiotl by a potential differericc of appropriate frecl”eli~y.~if)f)lied to six rfectrotfrs round the f)eripfrcry, thcsc being connected alternately in of>fjosite f)hasr so as to produce a field wit,11trigonal symmetry
in the XY-plane.
The orienta-
tion of tfle crystal axes rrl:rti\-e to the electrodes differed for types A and B modes since type -4 are dilatational modes and require a field along tfte S-asis while type 13 are shear modes and require a field along the Y-axis in order to excite suitable stresses. A suitable orientation was easily found by trial in every case. Tile crystals rested on a glass plate cwatecf with a rdteding layer this being composed of dielectric r~~~iltila~ers to avoid short circtnting tftc electrodes. In tfle figures sftowing the pflotoelastic~ strcsh patterns tfic f)oIarizer axis is always vertical. pflotoelastic
effects interference
Apart
from the
f’ringcs (*an be seen in some of the photographs,
these being formed by light rellr&ct
from the reflecting layer and the two crystal
surfaces. Fig. ‘7 (Plate)
shows the type ,4, 1~ = I, mode for a crystal wftirfi is 0.5 mm
tflick.
The isoclinic cross sflould have arms parallel to the polarizer and analyser axes but is misorien~ated by about 7”. The na.tural rotation of this specimen is x3”. Fig. 8 shows the same mode for a 2 mm tflick crystal. The natural rotation of this crystal is 51”, and the misorientation is increased to 26’. The thinner crystal was right-handed, and rotated tfle light vector anticlockwise when looking in t,he difettion of propagation while the thicker was left-handed so that the misorientations are both in the appropriate dire&on, that is, the isoclinics correspond to an orientation differing from the polarizer setting by half the natural rotation in the quart,z. Fig. 9 shows the type B, ~1 = 1, mode for a O-5 mm thick crystal. The isoclinic cross, whose arms should make an angle of 45’ with the polarizer axis, is misorientated by 7O as in the raw of Fig. 7. The radii of the zero-order isochromat rircles have been measured on photographs of the 0.5 mm thick specimen for type A and 23 modes, m = 0 to 8. In
On the photoelastic examination of vibrating bodies
37
every case the radii agreed with the theoretical values to well within the accuracy
of the measurements. In spite of this good agreement it is known from multiplebeam interference examination of the surface displacements normal to the plane of the disk that all of these modes are strongly coupled to flexural modes (TOIANSKY and WOOD 1958). However, these flexural displacements do not influence the longitudinal displacements and they do not distort the photoelastic patterns since the stresses involved are antisymmetric about the median plane of the disk and the optical changes due to the stresses in laminae at equal distances on opposite sides of this plane cancel out, to the first order, in a sufficiently thin disk. Fig. 10 shows the type B, m = 1, mode for a 3 mm thick crystal. For this thickness various second-order effects, both elastic and photoelastic, become significant and the stress pattern is correspondingly distorted. It is included, however, to demonstrate the increase of rotation for high stresses. In spite of the distortion it is seen that the isoclinic cross is misorientated by about 40” clockwise, the natural rotation being 78” and the crystal left-handed. Fig. 11 shows the crystal vibrating in the same mode at a much higher amplitude. Those parts of the isorlinic in between the zero-order isochromat have moved still further in the closewisc dircction indicating an increase in the effective rotation. It is. perhaps, surprising that the displaced isoclinic is so clearly defined, for it is actually moving to and fro at twice the frequency of vibration. The reason is that in the extreme position the isoclinic is momentarily at rest, the intensity of the rest of the field being then at a maximum, and the system acts stroboscopically. In the case of the 0.5 mm thick crystal there was no sign whatever of an increase in rotation with high stress ; a 1.0 mm thick crystal showed some signs of increased rotation at stresses just short of the breaking point. Thus, for thin quartz crystals of this type, the photoelastic stress patterns may be interpreted as though the quartz were isotropic and the stresses static except that the isoclinics correspond to a virtual polarizer setting differing by half the natural rotation in the quartz from the actual setting. 7.
COMMENTSON PHENOMENAOBSERVED
BY
EARLIER
WORKERS
(i) Experiments of MOENS and VERSCHAFFELT(192’7). A vibrating quartz bar was studied by placing it between polarizer and analyser, so that white light passed along the optic axis, the transmitted light being examined with a spectrometer. The dimensions of the bar were 8, 34 and 12 mm along the X, Y and Z-axes respectively and the bar was excited into longitudinal vibrations along the X-axis at a frequency of about 450 kc js by electrodes on the X-faces connected to a valve oscillator. Initially there was an extinction band in the spectrum of the transmitted light, due to the rotary dispersion of the quartz along the Z-axis, the position of which depended upon the precise relative setting of polarizer and analyser. Having moved this band to a suitable part of the spectrum the crystal was set into vibration and, as the amplitude was increased, the band was observed to move bodily towards the red end of the spectrum becoming less distinct as it did so. This occurred whatever the initial position of the band and indicated an increased rotation of some tens of degrees as shown by re-orientating the analyser to restore the band to its original position. The authors expressed surprise that the change in rotary power was in a fixed sense for an oscillatory stress. Moreover they stated that
A. F. Is. \VOOl)
38
the effect could be produced neither by a uniform compression perpendicular the Z-axis of a Z-cut specimen nor by m intense electric field. The change in rotation was clearly a manifestation of the effects discussed Sections
3 and 4 of this paper
rotation irrespective more precisely than
which always
result
of the sign of the st,rrss. this since the exprrimcntal
in an increase
to in
of the effective
It ih not possible to comment any conditions are not very well des-
cribed. The light admitted to the spectrometer for instance. was probably collected from a large area of the spec~imcn over which the magrutudc of the stress varied ~1s to their failure to product the effects by a statics tricc~hanical or considerably. electrical stress, this was probably due to their failure to appreciate thr enormous stresses present in a resonating piezoelectric crystal. They dimensions of the specimen IIMYI for the static experiments
do not mcntlon the but considering that
used for the dynamic cxpcriments, a force of about ~‘OOkg wt would be needed to produce rotations of the order of %-30” by uniform compression along thca S-axis, from the t3pcrimr~ntal results of Section 5. Using Po(.Kl~:I.s’ data for the electrooptic effect (1890) the same result, c~oultl b(a produc*ctl by a potential difference of several
h11ndrrd kilovoit,s
extremes
although.
suitable
specimen
(ii)
Experimerd
bc~twc~~rrtlic X-fac*c3.
as sun by
ita Section
simple
IGvitlcntly they did not go t,o these 5. the nr~ccssary stress is easily realized in 21
bcntfing.
of Pax ‘l’~~rr~~sc;
K.\o
(l!G:i).
It
was noticed
that
the high
order modes of a vibrating rectangular Z-cut quartz crystal often had the appearance of a regular arrav of red and green spots when the crystal was examinedin reflexion optical
by white plane polarized
TC’HENQ Kao
quartz
to principal
stress
from the directions the
isoclinic
colour,
that
Thus
colour
parameter
varictl
would be stopped
the general
appearance
of the high order
regular
array
differing
for each
by half the rotation axes.
wavelength,
Since
modes
and the c*omplemcntary
in the
white light
owing
point of t,fle crystal where there stress cluections and the rotation
one side and green on the other. many
tlirec,tions
of polarizchr and analyser
dispersion of quartz. ht any relation between the principal up.
Now this is the
used in thr Mark described in Sections 5 and 6 and so P.\s was observing, although hc did not realize it, the isoclinic patterns
corresponding used
light, usmg a crossed analyser.
arrangement
to the
was
rotary
was the correct for a particular
colour
would show
was t flat of diffuse isoclinics coloured red on Owing to the gridlike nat,ure of the isoclinics for
of a rcc*tarrgular
cLrysta1 the colours
appear
as a
of spots. ,~(‘KNoI~I.KI)(:~r~~~-,rs
The author encouragement Research
is grateful to Professor S. T~LANSKY in this work and to the Department
for a maintenance
for his keen interest and of Scientific and Industrial
grant. bFl
L)RUDE:, P.
1903
Love, A. E. H.
1927
MOENS,R. and
VERSCHAFFEL?‘. .J. E:.
1927
PAN TCNENG KAO
1!I35
POCKELS,F.
1xx9 1x90
PRESTON, T.
1901
TOLANSKY, S. and \Voon, A. F. R.
195X
(London : Longmans Green). (Cambridge : I-niversity Press). C.12. Acad. Sri. Pan’s, 184, 1615. C’.R. Acud. Sci. Paris, 200. 56X. ;Inn. Phys. Lpz. 28, 737. The Throry of Optics The .~~attmnutirul
S.
Theory of Elmlicity
Jahrb. J. Mitrer. Reil. 7, 221. The Theory oJLight (London : Macmillan). Phtpicfl 24, 308.
ON THE
26 ta 38.
PHOTOELASTIC
Pergnmon
Press
EXAMISATION
BODIES AND ,THE PHOTOELASTIC ACTIVE By Hoy~l
HoHoway
A.
Prhltecl in Great Britain
Ltd., London.
OF VIBRATING
EFFECT
IN OPTICALLY
MEDIA*
F.
B.
College,
WOOD
of London?
University
bodies is discussed, with in~r~ret~tion of the ~~hotoel~tj~ stress patterns of ribcrting An expression is derived for the change in velocity of H particular referenre to quartz crystals. light wave propagateci along the optic axis due to stresses acting in the plane norrnsl to this axis. Such stresses should c:ruse an increase in the effective rotary power snd this is verified by experiment. This result offers an explmaticm of phenomena observed 1,s earlier workers.
Tiffs
IK
connexion
became
with
necessary
a photoelastic
to consider
examination
the manner
of
in which
vibrating the isoclinics
quartz
crystals
it
and isochromats
observable in a statically stressed isotropic medium are modified by the vibratory nature of the stress and the optical activity of the Z-cut specimens used for the experiments. The effects intensity polarizer
of vibration
of the light and analyzer
where I, is the incident
alone
may
easily
be taken
transmitted by a birefringent IS I = 1, sin2 ‘La sin2 (6/L) intensity,
into consideration, medium
CLthe acute angle between
between
The crossed (1)
the polarizer
axis and
a principal axis of the medium and 6 the relative phase retardation between the The transmitted intensity two plane polarized waves emerging from the specimen. is zero if Q = o or 4~ or if 6 = 2t~n. that is. if the principal axes of the medium are parallel to polarizer and analyser or the retardation is a whole number of periods. In a mechanirally stressed isotropic medium the principal axes of the artificial birefringence are parallel to the principal axes of stress, and the difference between the propagation velocities in any direction is proportional to the difference between the secondary principal stresses acting normal to the direction of propagation. The case usually considered is that of a lamina in a state of plane stress, uniform throughout the thickness, the light being propagated along the thickness. If the stress varies across the specimen two sets of dark fringes are observed. One set,, the isoelinie, joins points at which the principal stress directions are parallel
26
On the photoelastic t=xnminntionof vilmlting lx.dies
27
and analyser axes and its position is independent of the light wavelength. The other set, the isochromat, joins points at which the difference between the principal stresses causes a retardation of an integral number of periods and the position of the fringes for each non-zero value of n in equation (1) depends on the wavelength. The zero-order isochromat, for which there is no phase retardation and the principal stresses are equal, is common to all wavelengths. Considering, now, an isotropic lamina subjected to an oscillatory plane stress system caused by vibration of the lamina in a normal mode, then a fundamental characteristic of such vibrations is that the displacements at all points are in phase and hence the directions and ratio of the principal stresses remain fixed throughout the vibration cycle. The retardation will vary sinusoidally with the stress at the frequency of vibration and, writing 6 = A cos w t and finding the mean transmitted intensity over a whole period of the vibration, equation (1) becomes I, = I, sin2 2cr(1 - J, A)/2. (2) to the polarizer
This is zero when u = 0 or 4~ and also when A = 0 since .I, (0) = 1. Thus the isoclinic and the zero-order isochromat are unaffected by the vibration. The expression (1 - J, A) oscillates about the value one with decreasing amplitude as A increases. The values of A for the minima are given by A = 2krr where k is no longer integral but takes the values 0, 1.117, 2.121, 3.122, 4.123. Although these minima do not have zero intensity, apart from the zero-order isochromat, the position is unaffected by the intensity distribution due to the sin2 2 GLterm and they may be used in the same way as isochromats in the static case. Thus, when an isotropic body is vibrating in a normal mode of a type giving rise to a plane stress system, the usual photoelastic observations may be made except that the number of isochromats visible is limited by the decrease in visibility with order and that the retardation associated with any order is not an integral number of periods as in the static case. In these experiments on vibrating quartz crystals, however, owing to the smallness of the photoelastic effect and the thinness of the specimens used, the retardation could not be increased sufficiently to produce the first intensity minimum without danger of exceeding the breaking stress. Hence only the isoclinics and zero-order isochromat were available for study. 2.
THE PHOTOELASTICEFFECT IN QUARTZ
Under the action of a general stress system the equation of the index ellipsoid of a crystalline medium, referred to the orthogonal crystal axes, changes from a011x2 -t ao22y2 + aos3z2 = 1
to the general equation of an ellipsoid whose axes do not coincide with the coordinate axes : aI1 z2 + a22y2 +
a33
z2
-k
2%3
?.@ +
2a3,
zx
f
2%3
z?l
=
1
(4)
where aOii represent the squares of the principal velocities of light waves in the unstrained crystal relative to that *in vacua. In general, each of the coefficients aij is a linear function of all of the six stress components. POCKELS(1889) has shown that the strained index ellipsoid is obtained
2x
13. \\‘OOD
.A. F.
from the unstrained by suitable changes in the lengths of the axes and, provided that ati - noli and o,, are all small c~omparetl with noi, and (loii - uOJj, by small rotations rji about each of the coordinate anglt5 arr giver> by : tan 24, = 2aik ‘(uj, In
quartz
the coefiicients
by the following
-
asrs nkk) E
of the strained
equations
in tllrri.
l’he
L’olk, ‘(QO,, -
magnitudes
of theae (5)
fIOhJ.
ell~psoiti arc’ given in terms of the stresses
:
a11
=
v*o
+
"11
01
+
x12
02
+
n1303
+
n14
u'4
“22
=
v20
t-
n12
Ql
+
711
u2
+
r13
a3
-
=14
04
u33
=
1'2? +
7731(Jl
+
773102
+
733
'J3
u23
~~
x41(31
-
n41u2
+
"44(J4
7
(6)
u 31 =
7744 "6
(I12
7714 (J6 +
+
2n41
Us
(+rll -
7712) 06 I
where z’, aricf i’, are the ordinary coellicients
and extraordinary
and ut the stress components. uoll = uo2* = 2~*~, $3 is large but
Since may still be found from the formula out first and that it 1s calculated a state
of plane stress
given,
velocities,
the orientation provided
xi1 the stress
of the index
that the rotation
is considrrvd.
Since
rG6 _
ellipsoid
c$, is carried
from the first part of (,5). In the present
in the SY-plane
optic
discussion
nll -
n12 it is
ca\lly shown that in this case 4, := 8, the angle between the S and Y-axes and the It may further be shown that for suchh a stress system principal stress directions. the resultant effect of the rotations +1 and d2 is that the indes ellipsoid is turned through an angle (+2l + ~$2~)+about an axis in the S I’-plane which makes an angle -
%9 wit11 the S-axis.
cluartz
Irsing
this a11g1e IS found
I’o(.KEI.s’
to be about
optic axes of the now biaxial
values
for the stress
1 or Z per cent
optic
coefficients
of the angle
between
of the
quartz for a stress of 10~ dyne cm-* and will be ignored
III what follow\. On slll)stitllting the \-alues of the ellipsoid parameters from (6) and taking new S and I'-;iscs alolig the directions of the principal \trcsses, up and uu. (1) transforms to the sani(’ form as (3) :
-+- (5’2, $ The
differcIrce
between
n-31
the
\/(I22 -
(up i_ a,)) 22 = 1. principal +11
y
\-clocities (TN
-
(7)
in the
712) (uy -
Z-direction
Q/‘2vo.
is : (8)
Thus as regards the pure photoelastic effect a Z-cut lamina stressed in the XY-plane behaves as though isotropic to light propagated along the Z-axis because the section of the index ellipsoid by the wave front has axes parallel to the principal stress directions and the retardation is proportional to the difference between the principal stresses. Since quartz is piezoelectric there is a secondary photoclastir effect fact that the electric polarization produced by the stress causes further
due to the changes in
On the photoelastic examination of vibrating bodies the parameters magnitude
of the index
of the electric
ellipsoid
by virtue
polarization
‘29
of the electro-optic
will depend
on the electric
effect.
The
boundary
con-
ditions and will generally differ for static and oscillatory stresses. However, the effect is small in quartz, being a few per cent of the direct photoelastic effect and, more important, the symmetry of the direct effect is not altered by the secondary effect, for example the relation 7re8= ml1 - n12 is still valid. For these reasons it will be ignored in this discussion.
3.
EFFECT OF OPTICAL ACTIVITY
In a medium that is both birefringent and optically active two elliptically polarized waves of opposite hand may be propagated unchanged in any direction with slightly different velocities. The axes of the ellipses are parallel to those of the section
of the index
ellipsoid
although similar, lie oppositely, major axis of the other. In order to find the influence an incident conlponents
by the wave front and the ellipses themselves, so that the minor
axis of one is parallel
of such a medium
upon the polarization
to the state of
plane polarized wave, the wave is first resolved into plane polarized along the axes of the section of the index ellipsoid by the wave front
WKI then each of these is further resolved into two opposite elliptically polarized waves. lifter illtrod~lcing the appropriate retardations due to the passage through the medium, these are recombined to give the emergent wave. PKESTON (1901) IMS considered the general case of the intensity transmitted by an optically active birefringent
angles
crystal
between
polarizer
and analyser.
If the axes of these make
x and p respectively
with one axis of the section of the index ellipsoid bl then the ratio of the transmitted to the incident intensity is
the wave front, I I, =
c0S (a -
fi) cos S/2 -
zh
2
(1 + h2) sin (a 2cos2
(a
+
/T)
/?, sin S/2 sin2 S/2
where 6 is the relative retardation of the emergent elliptically polarized wa\-es and h is their axial ratio. Both terms of this expression are essentially positive and so the intensity zero when
can be zero only if they are simultaneously
(B and the second
when
a) = 77/2 + tan-’
i
A2
zero.
The first term is
tan S/2)
either (a + B) = m/2
(11)
or 6 = 21177.
(12)
Since (10) must always hold good for zero intensity axes must be misorientated by an angle
tan-’
-1
2h +
h2 * tan S/2
)
the polarizer
and analyser
A. Ir’. Ii. \\‘wI,
230
from the crossed position so that the efleedive rotation is p = tan-l
--%--
( 11_hZ’
tan 6,/2 . )
If h = 1, that is, if the waves are circularly pofarized, which is the ease when no birefringenee is present, then there results the well-known equation p = sjz.
(11)
Assun~ir~g that the polarizer and analyser have the correct relative settiug, then from (11) and (12) extinction can occur when {a) the bisector of the angle between the polarizer and analyser axes is parallel to that of the angle between the axes of the section of the index ellipsoid or (b) the retardation is a whole number of periods. Consideril~g, now, a Z-cut quartz lamina s~ti~ally stressed in the XY-plaae it is assumed that the natural optical activity is affected by the stress only to the same extent as the refractive index so that such a change is negligible compared with the birefringence introduced in the direction of the optic axis. In general the difference between the principal stresses, and their directions, will vary over the field of view so that it would be fortuitous if (lo) were satisfied sim~~Itar~eously with (I 1) or (12). If, however, the light is reflerted from the far side of the specimen so as to retrace its path exactly (lo) may be fulfilled automatictally. The rcllexion results in a complicated expression for the intensity since a left-handed elliptic* wave on incidence becomes a right-handed wave on reflexion ; but at points where the transmitted intensity would be zero if the analyser were correctly oriel~tated, that is, at just those points in which we are interested, the light emergent from the specimen must necessarily be plane polarized and will, on reflexion, retrace its path exactly and leave the specimen in the same state as that in which it entered. It may therefore be stopped by an analyser with its axis crossed relative to the polarizer. When observed under these conditions the specimen will exhibit two sets of dark fringes. One set joins points where the principal stress directions differ from those of the polarizer and analyser axes by half the effective rotation at the points concerned ; the other set joins points where the retardation is a whole member of periods. The sets correspond to isoclinies and isoehromats with these differences : (i)
The rotation must be known at any point on the isoclinic to find the principal stress directions. Since the rotation depends upon phase difference and therefore upon stress the interpretation is difficult. It will be shown however that for su~ciently thin crystals the rotation hardly varies from the natural value and hence the directions of the polarizer and analyser axes differ by a constant angle from the principal stress directions.
(ii)
Successive isochromats correspond to equal increments in the retardation but this does not now correspond to equal increments in stress owing to the inffuence of the optical activity, as will be seen later. In the work on vibrating crystals only the zero order isoehmmat is produced so this question does not arise.
On the photoelastie 4.
esamhurtion
of sibratin~
bodies
MAGNITVJXCOF CIUNGE IN EYSXCTIVE ROTATION
31
WITH STRESS
DHUDE (1905) has derived expressions for the propagation velocities vl, us and the axial ratio h of the two ellipticalIy polarized waves in a bi~fringent optically active medium. In what follows the coordinate axes are taken parallel to the principal axes of the index ellipsoid as in (3) but the principal velocities relative to that in wcuo are now denoted by u,_,a, and a, for mathematical convenience. DRUDE gives 2vz = a21 -f-
as8+ (a*,- a*$) cos g, cos g2 f
[W, -
&J* sin” g, sins g, + ~vJ~]$
h + I/h = [(azl - ~9~)~sin2 g, sin2 g, j- 4#]f/q
(W (I@
where g, and g, are the angles between the direction of p~pagation and each optic axis and 9 is a parameter proportiona to the rotary power in the absence of birefringence and varying with the direction of propagation. In the present case the direction of propagation is the Z-axis and since the optic axis must split symmetrically into two new optic axes on the application of stress g, and g, are simply equal to Ifr Q, the semi-angle between the optic axes. This has the value ~in-~ [(ozl - tz22)/(a21 - &,)]f and heuce sing, = + [(f82l - U~~)~(ff2~ - a$)]“,
(17)
sin g, =
WI
-
[b5
ff22)/(~2~ - a2,)]f,
-
eos g, c=.cos g, = + [(a22 - az,)/(a2,
- a2,)]*.
WI
Substitution of these values in (15) and (16) gives VI - u2 = (?/al) [I + (a, - %)*./(+Q]*,
(20)
h + r/h = 2 [l + fax - a2)~;(7j/f7,)~]**
1’21)
If d is the thickness of the lamina, T the period of the light wave, and c the velocity of light in WLWOthen the relative ~tardatio~~ of the emergent waves is S = 27Td(VI - v,f/Tc vi v2 N (2n d~/IW1)
[l -I- (aI -
4a/kd~I)2]*
(22)
from (20). By (14) the rotation in the unstressed specimen is equal to half the retardation : p. = 8,/Z
=
7+/Tca3,.
(23)
Hence
8 = 2& (1 + G)f
(24)
where
e = (a, -
(25)
a~)/(~~~~).
From (13) and (21) the effective rotation is P = tan-1
(
&%tanS/Z)
= tan-l ~~~~~~‘*I].
A. 1;. 1%.\voort
3” In
these expressions
6 is directly
stresses ; substituting
from
proportional
The resultant
retardation
the variation
of the retardation
thicknesses
of quartz,
to the dlffcrence
between the prlnrGpal
(8) into (25) :
is not proportional
0.5 mm
to t,he stress however.
6 and the effective
anti 3.06 mm.
The
first
Fig. (I) shows
rotation
p witft
E for two
thickness
is that
uf a %-cut
d : 3 06mm
n d=OfjOmm
disk used for experiments for the experiments
on vibrations
described
while the second is that of a Z-<*lit bar used
in the next section
to test this theory.
It will be
noticed that while pO and 6 are pro~~ortional to the thickness p is definitely not. For the smaller thickness p hardly varies from the natural value pO since (tan X p&,/K -+ pO as p. ---Iz0. Thus, for thin crystals, the rotation may he assumed and equal to the static value for all practical purposes. Owmg to the
constant
flatness of the curves provided notable.
when c is small the same will he true for thltker crystals Two other features of these curves are the stresses are small enough. First, for sufficiently high stresses the retardation tends asymptotically
to a straight line through the origin representing the retardation in the absence of optical activity. Second, whenever 6 = YI~ then p = 6 /2 as for the case of optical activity in the absence of hirefringence. Finally, perhaps the most i~~terest.ing feature of all in this theory is the fact that according to (24) the phase retardation between the two oppositely elliptically polarized waves can only incwase when the quartz becomes artificially birefrigent along the optic axis. Thus the increase in effective rotation given by (26) is independent of the sign of the stress and depends only upon the difference between the principal
stresses.
On the I>hotoelastic examirmtion of vibrating bodies 5.
The
TEST
ESPERIMESTAL
results of Section
P were tested
OF
ME:
33
THEORY
by investigating
the propagation
of light
through a Z-cut bar uniformly bent in the XY-plane. The bar had a rectangular section and its length was parallel to the X-axis. Fig. 3 shows how the stress was The polarizer axis was so applied and Fig. :3 shows the optical arrangement. orientated as to polarize the light in the plane of incidence on the half silvered mirror in order to avoid subsequent elliptic polarization. The thrust was provided by a calibrated
spring rather
than by weights
as the stresses had to act in the
horizontal plane owing to the geometry of the optical system. Before applying any stress the field of view was dark since the natural rotation of the quartz was cancelled
by the double
passage of the light through
PIG. 2. Apparatus
for applying
static bending stress to a quartz bar.
AndySCr
I-
Pamlkl
Half silvered mirror
monochromatic light
Specimen FIG.
the crystal.
silvered
on lower
surface
a. Optical apparatus for observation of photoelastic
stress patterns.
On applying a small stress the field brightened near the edges of the bar. At a slightly higher stress three distinct dark fringes could be seen parallel to the length of the bar, one along the axis and the other two symmetrically on either side. By rotating the bar and its bending jig about the Z-axis the latter fringes could be made to move outwards or, by a reverse rotation, coalesce with the centre fringe.
34
A. F. H. WOW
When the stress was increased still further the first-order isochromat appeared at the extreme edges of the bar. The thrust was increased slightly to a value of approximately 10’ dyne so that these lay clear of the edges and then a series of photographs was taken in which the applied stress remained constant but the bar had various orientations relative t.o the polarizer axis. It was noticed that the zero and first-order isochromats at the centre and near the edges respectively retained their position througtlout this series and also that the split ‘ isoclinic ’ was approximat.ely symmetrical throughout, although one side of the bar was in tompression and the other in tension. Fig. d (Plate) shows a photograph of a typical photoelastic stress pattern under these conditions and Fig. 5 an explanatory diagram.
FIG. .i. l’hotoel~tstic stressphltternin a statically stressed quartz bar.
In 11tc iihh(‘lt(*c of c@ic*al activity there would have been a solitary isoclinic fringe blackirlg ant the whole area over which the stress was substantially parallel to the axis, this being observed when the axis of the bar lay parallel to the polarizer or analyser axis. But from the theory of Sections 3 and 4 there is an effective rotation depending on the stress in the case of an optically active medium and darkness is only attained when there is a certain relation between polarizer orientation, principal stress directions and the magnitude of the stress, namely, that at any point on an isoclinic the stress results in an effective rotation equal to twice the angle between polarizer and analyser axes and the principal stress directions. The stress in a uniformly bent bar increases linearly from zero at the neutral axis t,o a maximum at the edges and so for a given applied stress and orientation of the bar relat.ive to the polarizer axis the isoclinic fringe consists of two lines parallel t,o the neutral
axis along the lines of appropriate
tensile and compressive
stress.
If the orientation of the bar is altered the fringes move outwards or inwards to the regions of appropriate stress at which the necessary conditions are still fulfilled. Alternatively if the orientation is fixed but the stress is varied the isoclinic fringes move so that they are always in regions of the same stress and therefore effective rotation. Although in this experiment owing
to poor
the absolute
mee~~anical construction
stress
was not known
of the bending
jig, a graph
the same accurately,
of rotation
against relative stress may be plotted from me~u~ments made on the series of photographs taken under conditions of constant applied thrust. For each effective rotation the distance between the two parts of the isoclinic was measured and taken to be proportional to the stress. Fig. 6 shows these experimental points. The curve is part of Fig. L for a thickness of 3.00 mm and has been drawn to pass through the experimental point for which the rotation is 180’ and the isoclinic coincides with the first-order isochromat. The values of l relate to the theoretical curve.
The agreement betweea the shapes of the two curves is very good.
As the thrust
was known to be about 10’ dyne for this experiment the stress in the region of the first order stress fringes was calculated from the geometry of the bar and the jig Co be about 8.41 x lOa dyne cm-z, and it was possible to estimate the value of x11 -
I%.
3712.
6.
I 2 0 Dependenceof rot&ion on phsse retardatiortshowing experinlenttrllydeternlined points.
From (24) E = (P/‘4&
-
1)f
(28)
and, since p0 = 78’ for the specimen used, c is 2.08 for the first-order isochromat. From (23) 71= p. asI Tc/m? (29) and substituting
in (27) gives =11 -
Taking a value of l/l-55 values quoted : nil -
xl2 =
2p,, a31 Tca/(u,
- q,) wd.
(30)
for the ordinary velocity and using the experimental
n12 = 10.3 x 10-l* dyne-l cm2 for X = 5461 A.
POCKELS’results give x11 -
TlZ =
13.95 x 10-ld dyne-’ cm2 for h = 5893 A
which is of the same order of magnitude.
6.
YHOTOELASTIC
EFFKT
IN VIBRATING
&TARTZ
CRYSTALS
The work described in the previous sections is part of an extensive study of the longitudinal vibrations of thin, circular, Z-cut quartz disks, the results of which will be published elsewhere. Some examples are given here, however, to show how the theoretical predictions of Sections 1-3 are borne out in practice. Z-cut disks
A. F, B.
36
WOOD
behave substa~ltially as tftough isotropic for normal modes of vibration
invofving
state of plane stress in the XY-plane.
in isotropic
media has been given by Of the multitude
LOVE
The theory of these vibrations
a
(1927).
of modes of tfns type tflerc are two series of symmetrical
modes
with simple displacements whose magnitude at any point depends only on the radial coordinate. In one series, type A, tfle displacements are everywflere along the radius v&or wfCle in tfle other, tyfje Z3, tfle dispfac~ements are everywhere normal to the radius vector. The principal st,resses at any point act &long the radius and its normal for type 14 and at 15” to these directions for type B. In each case the nodes of the longitudinal
displacements
consist of the centre of the disk together
witfl a number of concentric circles, the number increasing witfl mode order. Denoting successive modes by integers, including zero, the mth type ii mode has N. nodes ilk addition to the centre while t,he l/bthtype R has m + 1. The zero-order isochromat joins points at which thr principal stresses are precisely equal. In both types this occurs at the eentre and on a series of concentric circles which lie between the nodes, except that there is never one between tfle rcntre and the first nodal circle. In the case of the type R modrs the outermost circle of the isoc~hroniat is the periphery of the disk itself while for tyf’c 21 modes tfie outermost circie nlrnost coincides wit,h tfle periphery, cscept for tflc II) :- 0 mode. The qllartz
disks, all about 2.5
r111
diameter
and of’ various tfiic~kncss~s. were
cxcitcd into vif,ratiotl by a potential differericc of appropriate frecl”eli~y.~if)f)lied to six rfectrotfrs round the f)eripfrcry, thcsc being connected alternately in of>fjosite f)hasr so as to produce a field wit,11trigonal symmetry
in the XY-plane.
The orienta-
tion of tfle crystal axes rrl:rti\-e to the electrodes differed for types A and B modes since type -4 are dilatational modes and require a field along tfte S-asis while type 13 are shear modes and require a field along the Y-axis in order to excite suitable stresses. A suitable orientation was easily found by trial in every case. Tile crystals rested on a glass plate cwatecf with a rdteding layer this being composed of dielectric r~~~iltila~ers to avoid short circtnting tftc electrodes. In tfle figures sftowing the pflotoelastic~ strcsh patterns tfic f)oIarizer axis is always vertical. pflotoelastic
effects interference
Apart
from the
f’ringcs (*an be seen in some of the photographs,
these being formed by light rellr&ct
from the reflecting layer and the two crystal
surfaces. Fig. ‘7 (Plate)
shows the type ,4, 1~ = I, mode for a crystal wftirfi is 0.5 mm
tflick.
The isoclinic cross sflould have arms parallel to the polarizer and analyser axes but is misorien~ated by about 7”. The na.tural rotation of this specimen is x3”. Fig. 8 shows the same mode for a 2 mm tflick crystal. The natural rotation of this crystal is 51”, and the misorientation is increased to 26’. The thinner crystal was right-handed, and rotated tfle light vector anticlockwise when looking in t,he difettion of propagation while the thicker was left-handed so that the misorientations are both in the appropriate dire&on, that is, the isoclinics correspond to an orientation differing from the polarizer setting by half the natural rotation in the quart,z. Fig. 9 shows the type B, ~1 = 1, mode for a O-5 mm thick crystal. The isoclinic cross, whose arms should make an angle of 45’ with the polarizer axis, is misorientated by 7O as in the raw of Fig. 7. The radii of the zero-order isochromat rircles have been measured on photographs of the 0.5 mm thick specimen for type A and 23 modes, m = 0 to 8. In
On the photoelastic examination of vibrating bodies
37
every case the radii agreed with the theoretical values to well within the accuracy
of the measurements. In spite of this good agreement it is known from multiplebeam interference examination of the surface displacements normal to the plane of the disk that all of these modes are strongly coupled to flexural modes (TOIANSKY and WOOD 1958). However, these flexural displacements do not influence the longitudinal displacements and they do not distort the photoelastic patterns since the stresses involved are antisymmetric about the median plane of the disk and the optical changes due to the stresses in laminae at equal distances on opposite sides of this plane cancel out, to the first order, in a sufficiently thin disk. Fig. 10 shows the type B, m = 1, mode for a 3 mm thick crystal. For this thickness various second-order effects, both elastic and photoelastic, become significant and the stress pattern is correspondingly distorted. It is included, however, to demonstrate the increase of rotation for high stresses. In spite of the distortion it is seen that the isoclinic cross is misorientated by about 40” clockwise, the natural rotation being 78” and the crystal left-handed. Fig. 11 shows the crystal vibrating in the same mode at a much higher amplitude. Those parts of the isorlinic in between the zero-order isochromat have moved still further in the closewisc dircction indicating an increase in the effective rotation. It is. perhaps, surprising that the displaced isoclinic is so clearly defined, for it is actually moving to and fro at twice the frequency of vibration. The reason is that in the extreme position the isoclinic is momentarily at rest, the intensity of the rest of the field being then at a maximum, and the system acts stroboscopically. In the case of the 0.5 mm thick crystal there was no sign whatever of an increase in rotation with high stress ; a 1.0 mm thick crystal showed some signs of increased rotation at stresses just short of the breaking point. Thus, for thin quartz crystals of this type, the photoelastic stress patterns may be interpreted as though the quartz were isotropic and the stresses static except that the isoclinics correspond to a virtual polarizer setting differing by half the natural rotation in the quartz from the actual setting. 7.
COMMENTSON PHENOMENAOBSERVED
BY
EARLIER
WORKERS
(i) Experiments of MOENS and VERSCHAFFELT(192’7). A vibrating quartz bar was studied by placing it between polarizer and analyser, so that white light passed along the optic axis, the transmitted light being examined with a spectrometer. The dimensions of the bar were 8, 34 and 12 mm along the X, Y and Z-axes respectively and the bar was excited into longitudinal vibrations along the X-axis at a frequency of about 450 kc js by electrodes on the X-faces connected to a valve oscillator. Initially there was an extinction band in the spectrum of the transmitted light, due to the rotary dispersion of the quartz along the Z-axis, the position of which depended upon the precise relative setting of polarizer and analyser. Having moved this band to a suitable part of the spectrum the crystal was set into vibration and, as the amplitude was increased, the band was observed to move bodily towards the red end of the spectrum becoming less distinct as it did so. This occurred whatever the initial position of the band and indicated an increased rotation of some tens of degrees as shown by re-orientating the analyser to restore the band to its original position. The authors expressed surprise that the change in rotary power was in a fixed sense for an oscillatory stress. Moreover they stated that
A. F. Is. \VOOl)
38
the effect could be produced neither by a uniform compression perpendicular the Z-axis of a Z-cut specimen nor by m intense electric field. The change in rotation was clearly a manifestation of the effects discussed Sections
3 and 4 of this paper
rotation irrespective more precisely than
which always
result
of the sign of the st,rrss. this since the exprrimcntal
in an increase
to in
of the effective
It ih not possible to comment any conditions are not very well des-
cribed. The light admitted to the spectrometer for instance. was probably collected from a large area of the spec~imcn over which the magrutudc of the stress varied ~1s to their failure to product the effects by a statics tricc~hanical or considerably. electrical stress, this was probably due to their failure to appreciate thr enormous stresses present in a resonating piezoelectric crystal. They dimensions of the specimen IIMYI for the static experiments
do not mcntlon the but considering that
used for the dynamic cxpcriments, a force of about ~‘OOkg wt would be needed to produce rotations of the order of %-30” by uniform compression along thca S-axis, from the t3pcrimr~ntal results of Section 5. Using Po(.Kl~:I.s’ data for the electrooptic effect (1890) the same result, c~oultl b(a produc*ctl by a potential difference of several
h11ndrrd kilovoit,s
extremes
although.
suitable
specimen
(ii)
Experimerd
bc~twc~~rrtlic X-fac*c3.
as sun by
ita Section
simple
IGvitlcntly they did not go t,o these 5. the nr~ccssary stress is easily realized in 21
bcntfing.
of Pax ‘l’~~rr~~sc;
K.\o
(l!G:i).
It
was noticed
that
the high
order modes of a vibrating rectangular Z-cut quartz crystal often had the appearance of a regular arrav of red and green spots when the crystal was examinedin reflexion optical
by white plane polarized
TC’HENQ Kao
quartz
to principal
stress
from the directions the
isoclinic
colour,
that
Thus
colour
parameter
varictl
would be stopped
the general
appearance
of the high order
regular
array
differing
for each
by half the rotation axes.
wavelength,
Since
modes
and the c*omplemcntary
in the
white light
owing
point of t,fle crystal where there stress cluections and the rotation
one side and green on the other. many
tlirec,tions
of polarizchr and analyser
dispersion of quartz. ht any relation between the principal up.
Now this is the
used in thr Mark described in Sections 5 and 6 and so P.\s was observing, although hc did not realize it, the isoclinic patterns
corresponding used
light, usmg a crossed analyser.
arrangement
to the
was
rotary
was the correct for a particular
colour
would show
was t flat of diffuse isoclinics coloured red on Owing to the gridlike nat,ure of the isoclinics for
of a rcc*tarrgular
cLrysta1 the colours
appear
as a
of spots. ,~(‘KNoI~I.KI)(:~r~~~-,rs
The author encouragement Research
is grateful to Professor S. T~LANSKY in this work and to the Department
for a maintenance
for his keen interest and of Scientific and Industrial
grant. bFl
L)RUDE:, P.
1903
Love, A. E. H.
1927
MOENS,R. and
VERSCHAFFEL?‘. .J. E:.
1927
PAN TCNENG KAO
1!I35
POCKELS,F.
1xx9 1x90
PRESTON, T.
1901
TOLANSKY, S. and \Voon, A. F. R.
195X
(London : Longmans Green). (Cambridge : I-niversity Press). C.12. Acad. Sri. Pan’s, 184, 1615. C’.R. Acud. Sci. Paris, 200. 56X. ;Inn. Phys. Lpz. 28, 737. The Throry of Optics The .~~attmnutirul
S.
Theory of Elmlicity
Jahrb. J. Mitrer. Reil. 7, 221. The Theory oJLight (London : Macmillan). Phtpicfl 24, 308.