December 15, 1996 / Vol. 21, No. 24 / OPTICS LETTERS
Nonreciprocity of natural rotatory power
P. J. Bennett, S. Dhanjal, Yu. P. Svirko,* and N. I. Zheludev
Department of Physics, University of Southampton, Southampton SO17 1BJ, UK
Received June 27, 1996
We have observed that a light beam that passed through an optically active crystal of Bi12SiO20 and
that was then reflected exactly back through it did not recover its initial polarization orientation.
nonreciprocal component of the rotation was of the order of 2 3 1023of the reciprocal, single-pass rotation.
This nonreciprocity is unambiguous evidence of broken reversality of the light–matter interaction process.
1996 Optical Society of America
It has commonly been believed that polarization ro-
tation in nonmagnetic optically active media is recip-
rocal:The rotation does not depend on the direction
of propagation and is compensated for when the light
beam is reflected back through the sample.
procity was seen as a direct consequence of the time
reversality of the light–matter interaction process in
nonmagnetic media on the microscopic level. This re-
versality was recently questioned for the transmission
of light through the border with a chiral media,1for
the propagation of light in zinc-blend crystals,2for re-
flection from an antiferromagnetic material with zero
net magnetization,3,4and, more generally, for the in-
teraction of light with media lacking an inversion cen-
Here we report what we believe to be the first
experiment showing broken reversality for the funda-
mental optical effect of natural optical activity.
The optical properties of Bi12SiO20, which is in the
scribed by the constitutive equation Di? eij?v?Ej 1
gijm?v?=mEj.This crystal symmetry does not sup-
port birefringence, eij?v? ? e?v?dij, but supports
conventional optical activity, described by the anti-
symmetric part of the nonlocality tensor ga
With a diagonal dielectric
tensor and antisymmetric nonlocality tensor, the light-
polarization rotation is reciprocal.
the 23-crystal symmetry also supports the symmetric
been a conventional view that the microscopic time
reversality of the light–matter interaction demands
that the symmetric part be zero in nonmagnetic crys-
Although we do not doubt that the restriction on
the nonlocality tensor is valid when the light–matter
interaction is reversible, recently we showed that the
microscopic reversality of the interaction itself might
In fact, light–matter interaction Hamil-
tonians obey symmetry with respect to simultaneous
inversion in time and space (PT), where P and T are
space and time inversion operators rather than just in-
version in time. We make a clear distinction between
the indisputable reversality of the Hamiltonians of the
crystals and the nonreversality of the interaction of
light with the crystals.The latter is a consequence of
the fact that the propagating electromagnetic wave
does not have a definite parity with respect to space
inversion or a definite symmetry with respect to time
jim ? gijm 2 gjim.
As a matter of fact
jim? gijm 1 gjim.However, it has
reversal (see, for example, Ref. 7):
in the argument of an oscillatory propagating function
f?vt 2 kr? yields not only reversal of the direction of
propagation but also a phase lag, which is important
in nonlocal light–matter interactions with molecules
of finite size, interactions that are phenomenologically
related to the second, gradient term in the constitutive
equation above.Reversality of the interaction can
be violated only in restricted situations.
appear in electric-dipole interactions.
of the symmetric component of the nonlocality tensor
should be supported by the crystal symmetry point
group and in particular requires that the crystal
have no inversion center.According to Ref. 5 nothing
to which Bi12SiO20 crystal belongs.
along the  direction rotates nonreciprocally.
spatial evolution of the field E of the wave is governed
We solve Eq. (1) by substituting E ? A exp?i?v?c?nz?,
where n ? e1/2and A is a slowly varying amplitude,
so jdA?dzj , , jvA?cj.
polarized at an angle f0with respect to the  direc-
tion, i.e., Ax?0? ? A0cos f0and Ay?0? ? A0sin f0, the
Cartesian components of A for the transmitted light
2 A0ns6 na
reversal of time t
jimin crystals of the 23-point group,
In the presence
ijmthe plane of polarization of a wave propagating
If the incident wave is linearly
Here ns,a? ?v2?4c2??gxyz6 gyxz?, m2? ns22 na2.
the approximation that the symmetric part of the
nonlocality tensor is a small fraction of the symmetric
one, that is, jnsj , , jnaj, we have m ? ina, m??na1 ns? ?
i?1 2 ns?na?, and for the polarization azimuth f! of
the forward-propagating wave and f√of the backward
f-? ?f0,!6 fOA? 2 Re?ns?nasin?naL?
3 cos?2f0,!6 naL??,
1996 Optical Society of America
OPTICS LETTERS / Vol. 21, No. 24 / December 15, 1996
where fOA? ?v2?4c2?Re?gxyz 2 gyxz?L, which is the
angleof conventionaloptical activitypolarization-plane
rotation in a single pass through the crystal of length
L.Here we used the prescription for the treatment
of symmetric and antisymmetric components of the
nonlocality tensor gijm outlined in Refs. 9–14.
antisymmetric components of the nonlocality tensor
should be treated as i tensors, and the symmetric
components as time nonreversal c tensors:
do not change their sign with reversal of the direction
of propagation; c tensors do.
expression for f√as an initial condition gives us the
equation for the change df of the polarization azimuth
for a linearly polarized light beam that propagates in
the crystal and then is reflected back through it:
Substituting f!into the
df ? f√2 f0? 2 2ch?hOA?cos?2f01 fOA?
where hOA? Im?naL? is the ellipticity that is due to
conventional circular dichroism.
our measurements hOA, 2 3 1022rad, so we modify
Eq. (4) to
In accordance with
df ? 22fTNOA
Here fTNOA? ?v2?4c2?Re?gxyz1 gyxz?L is a figure of
merit for the dissipative nonreciprocal effect, which
should be seen only in the spectral range of noticeable
Although the optical properties of Bi12SiO20, and its
optical cavity in particular, have been studied compre-
hensively,13,15,16there has not been any attempt to ex-
amine the reciprocity of its rotatory power.
set up an experiment to detect this nonreciprocal ro-
tation in an unambiguous, retroreflecting, double-pass
configuration (see Fig. 1).
the crystal, polarization-plane rotation is due to both
conventional optical activity ?ga
ciprocal rotation ?gs
reflected exactly back through the sample, the conven-
tional optical activity is compensated for, but the non-
reciprocal rotation accumulates.
laser and a He–Ne laser we were able to measure the
nonreciprocal rotation at various wavelengths (514.5,
501.7, 496.5, 476.5, 457.9, and 632.8 nm) at the edge of
the transparency range. The measurement technique
described in Ref. 17 permitted polarization measure-
ments with an accuracy better than 5 3 1024rad, with
a deviation between the initial and the reflected beams
of less than 1023rad. This deviation ensures compen-
sation of the reciprocal component of rotation of bet-
ter than 1026of the single-pass rotation, or in absolute
terms better than ?6 3 1027rad.
high-grade, virtually stress-free sample of Bi12SiO20
with a thickness L of approximately 0.64 mm and di-
mensions of 5 mm 3 5 mm, cut and polished perpendic-
ular to the  crystal axis.
We measured the nonreciprocal rotation as a
function of the crystal orientation with respect to
the crystal around the  direction (see Fig. 2).
On the first pass through
When the light beam is
jim? and nonre-
With an argon-ion
We investigated a
For clarity the experimental points are presented
for only l ? 457.9 nm; measurements at all other
wavelengths are represented by best-fit curves.
characteristic oscillatory orientational dependence of
the polarization-plane rotation, predicted by Eq. (5), is
the most distinctive feature of the nonreciprocal effect
observed. We estimated that the effect cannot be
explained by the optical Faraday effect in a scattered
magnetic field in the laboratory.
fections of elements of the polarimeter could potentially
change the polarization state of light, this could not
contribute to the oscillatory dependence observed, be-
causeonly the Bi12SiO20sample was rotated during the
The phenomenological theory predicts a cos?2?f 1
fOA?2?? dependence for the nonreciprocal rotation.
That is, measurements of this dependence on f (see
Fig. 2) are expected to have an offset h ? fOA?2 with
respect to a 2cos?2f0?function (dotted curve in Fig. 2).
Figure 3 shows the experimentally measured depen-
dence of h onfOA. Itcorrelates very well with the pre-
diction that h ? fOA?2 for a nonreciprocal effect.
Although the imper-
of nonreciprocal polarization rotation.
modulator and the birefringent polarizing prism are parts
of the polarization-measurement system.
Schematic of the experiment for measurement
the polarization azimuth rotation df on the polarization
azimuth f0of the incident light wave with respect to the
crystal axes at different wavelengths in Bi12SiO20.
Dependence of the nonreciprocal component of
December 15, 1996 / Vol. 21, No. 24 / OPTICS LETTERS
single-pass optical activity fOAat different wavelengths in
Dependence of the orientation offset h on the
a single pass the light acquired some ellipticity, which
increased with absorption.
explained by small accidental crystal birefringence be-
cause it had no detectable dependence on the crystal
orientation.It was attributed solely to conventional
circular dichroism owing to Im?gxyz2 gyxz? that accom-
panies natural optical rotation in the absorption range.
We measured the dependence of the nonreciprocal ro-
tation on the wavelength (see Fig. 2).
we calculated Re?gyxz 1 gxyz? ? 4.9 3 10212cm and
Re?gyxz2 gxyz? ? 2.1 3 1029cm.
the nonreciprocal effect, which is due to the dissipative
nonlocal susceptibility Re?gyxz1 gxyz?, increases as ab-
sorption and conventional optical activity increase.
Although the crystal symmetry does not support
birefringence and linear dichroism, we must exclude
any contributions to the rotation that could appear
as a result of imperfections of the sample, such as
residual stress. Stress would have created an inho-
mogeneous birefringence–dichroism pattern across the
sample, so the magnitude and the sign of polariza-
tion disturbances associated with the stress would
depend on where the beam propagated through the
sample. We measured the nonreciprocal rotation at
different points of the crystal over the whole crys-
tal surface, scanning it with a beam of approximately
300 mm diameter. No significant change in the non-
reciprocal rotation effect was seen.
The laser light intensity was attenuated so that the
power of the beam that was incident upon the sample
was 1 mW for all the wavelengths.
photochromic coloring of Bi12SiO20, which is known to
happen in this material, was seen after several hours of
exposure to more-intense blue light, the nonreciprocal
component of rotation did not depend on the coloring.
This experiment allows us to conclude that the non-
reciprocal polarization rotation detected in Bi12SiO20
is due to the presence of the symmetric part of the
The ellipticity could not be
At l ? 457.9 nm
As we anticipated,
nonlocality tensor gs
ated with the spin-orbit coupling terms in the noncen-
trosymmetric crystal potential2,5and (or) distortion of
the optical electrons’ wave functions resulting from the
noncentrosymmetric of the crystal lattice.2
sider our first experimental observations of nonrecip-
rocal polarization rotation in an optically active crystal
to be an unambiguous and positive indication of broken
reversality of the microscopic light–matter interaction
jim. It is likely to be associ-
The authors thank S. V. Popov for experimental
assistance and discussions.
by the UK Engineering and Physical Sciences Research
Council (grants GT?J26854 and GR?K14230) and the
International Science Foundation (grant N4V300).
*Permanent address, General Physics Institute, 38
Valilov Street, Moscow, Russia 117942.
This study was supported
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