HAL Id: jpa-00249361
https://hal.archives-ouvertes.fr/jpa-00249361
Submitted on 1 Jan 1995
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
On the Frequency-Dependence of the Chirality Pseudoscalar of a Chiral Medium
Frédéric Guérin, Akhlesh Lakhtakia
To cite this version:
Frédéric Guérin, Akhlesh Lakhtakia. On the Frequency-Dependence of the Chirality Pseu- doscalar of a Chiral Medium. Journal de Physique III, EDP Sciences, 1995, 5 (7), pp.913-918.
�10.1051/jp3:1995170�. �jpa-00249361�
Classification Physics Abstracts
41.10 41,10H 78.20E
Short Communication
On the Frequency-Dependence of the Chirality Pseudoscalar of a Chiral Medium
Frdddric Gudrin (~,~) and Akhlesh Lakhtakia (~)
(~) Thomson-CSF Laboratoire Central de Recherches, Domaine de Corbeville, 91404 Orsay
Cedex, France
(~) IRCOM, Universit4 de Limoges, 123 Avenue Albert Thomas, 87060 Limoges Cedex, France (~) The Pennsylvania State University, Department of Engineering Science and Mechanics, Uni-
versity Park, PA 16802-1401, USA
(Received I February1995, accepted 2 June 1995)
Rdsumd. En utilisant les relations de Kramers-Kroning appliqu4es au cas de l'activit4 op- tique, et la valeur statique rigoureusement d4termin4e du parambtre de chiralit4 d'un milieu
chiral (dans la repr4sentation de Tellegen), nous d4montrons que les parties r4elle et imaginaire
de ce parambtre de chiralit4 doivent changer au moins
une fois de signe lorsque la f4quence
varie de o I cc. Des r4sultats exp4rimentaux disponibles dans la litt4rature soutiennent cette conclusion
Abstract. Starting from the available Kramers-Kronig relations for natural optical activity and the rigorously determined static value of the chirality pseudoscalar (in the Tellegen repre- sentation) of
achiral medium,
weshow that the real and the imaginary parts of the chirality pseudoscalar must change signs at least once as the circular frequency increases from o to cc.
This conclusion is supported by experimental evidence.
1. Introduction
Natural optical activity came to be discovered in France at the beginning of the nineteenth cen- tury. It encompasses the dual phenomena of optical rotation and circular dichroism. Whereas the first phenomenon is the rotation of the vibration ellipse of a plane wave after passage through a natural optically active or chiral slab, the second is the concomitant change in the
eccentricity of the vibration ellipse. Left- and right- circularly polarized (LCP and RCP) plane
waves propagate with distinct wavenumbers (noted as kL and kR) in a chiral medium. In terms
of these two wavenumbers~ the optical rotation # and the circular dichroism 9 are given as
~j~~
~kilLd) kilLd)
j~~~
2
© Les Editions de Physique 1995
914 JOURNAL DE PHYSIQUE III N°7
g~~~
~kilLd) kilLd)
2
'j~~~
respectively, where kL(R)
"k[j~~ + ik[~~~, and the explicit dependences with respect to the circular frequency cJ delineate the dispersive characters of # and 9.
Natural optical activity is not limited to infrared, visible, or ultraviolet frequencies: it was
reported in the microwave frequency range for suspensions of miniature helices in an isotropic dielectric host medium, as early as 1920 [ii. Indeed, such a property is exhibited by any medium with a microstructure possessing handedness or chirality. Modern electromagnetic field theory provides a framework for quantitatively describing wave propagation in chiral media, either chemically pure or composite [2-4].
Until recently, it was commonly believed that the constitutive relations of the most general
linear isotropic medium could be written in the frequency domain as
D(Ld)
=61Ld)ElLd) + i~lLd)H(Ld), (2a)
B(Ld)
=JJ(Ld)H(Ld) ix(Ld)E(Ld). (2b)
Here, e(cJ) and ~(cJ) are the permittivity and the permeability scalars, respectively, while
~(cJ) and x(cJ) are the biisotropy pseudoscalars, in the Tellegen representation. Lakhtakia and
Weiglhofer [5], however, recently showed that
"/(Ld) % x(Ld), (2C)
which precisely defines a chiral medium. This result is a specific consequence of a general uniformity constraint obtained by Post using covariance arguments (Ref. [6], Eq. (6.18)), which
was reinforced by an independent derivation emanating from the argument of uniqueness [7].
We observe in passing that equation (2c) does not emerge from a reciprocity relation.
Using equations (2a-c), and writing ~(cJ)
=
~'(cJ) + i~"(cJ)~ we can show that
4(Ld)
=LdV(Ld)> (3a)
o(~u)
=~uq'(~u). (3b)
Thus, the chirality pseudoscalar ~(cJ) in the Tellegen representation is quite simply related to
optical rotation and circular dichroism.
The objective of this communication is to draw attention to a specific feature of the frequency- dependence of ~(cJ). Kramers-Kronig (K-K) relations are naturally the mathematical tool of choice for this purpose. We shall commence by recalling prior results on the application of K-K
relations to optical rotation and circular dichroism, and extend them by means of a short math- ematical manipulation. Then, we shall show that a simple result on the static value of ~(cJ)
leads to interesting consequences regarding the frequency variations of ~'(cJ) and ~"(cJ), ob- tained using minimum assumptions. The obtained results shall be exemplified by experimental
observations.
2. Integral Relations for #(cJ) and 9(cJ)
In a landmark paper, Emeis et al. [8] considered #(cJ) and 9(cJ) to be causally connected and, therefore, related by the Hilbert transform. After replacing the real-valued cJ by the complex-
valued fl, they made the assumptions that (I) the function [#(fl) +19(fl)] is analytic in the upper half of the n-plane, and (it) this function goes to zero in the limit fl
-[oo +10]. Furthermore,
as natural optical activity is invariant with respect to time-reversal, #(cJ) is an even, but 9(cJ)
is an odd, function of cJ. Thus, Emeis et al. were able to write the Krarners-Kronig relations
as (Ref. [8], Eqs. 25 and 26)
~~~°~ ~x~~ %~ i~~~~~~°~' ~~~~
9(cJ)
=~~° P.V. /~ ~~°~ ~ dcJs, (4b)
x o cJ~-cJ
where P.V. stands for the principal value. Explicit values of the integrals of several combinations of the functions w~', #(cJ) and 9(cJ) have been derived by Smith [9] and King [10] after assuming specific asymptotic characteristics of #(cJ) and 9(cJ). It is, however, instructive to return to an earlier paper of King [11] that dealt with the K-K relations for the complex refractive index of
a dielectric medium.
On replacing [n(cJ) ii and ~(cJ) by #(cJ) and 9(cJ), respectively, in reference [11], it follows
from (4a,b) that (Ref. [11], Eq. (11)):
/°'<(~J)d~J
=o. (5)
This result had been obtained earlier by Altarelli et al. [12], but King [11] clarified that the necessary requirement for establishing equation (5) is that the integrand on the right side of
equation (4a) must be summable for all (co( < oo, (cJs( < oo.
The second important result derivable from King (Ref. [11], Eq. (13) is that:
hich
that
the integrand on the right side of 4b) be summable for all (cJ( < oo,
(cJs ( < oo.
urthermore,in the icinity of cJ = 0,
3. Consequences for ~(cJ)
Equations (5) and (6) contain information on the frequency-dependence of ~(cJ). Rewriting equation (5) using equation (3a) yields
cJ~'(cJ)dcJ
=
0, (7)
~~
which shows that the real part of the chirality pseudoscalar in the Tellegen representation
must change sign at least once in the interval 0 < cJ < oo. This deduction can also be inferred from a result of Bokut et al. (Ref. [14], Eq. (15)). It was also obtained by. Serdyukov [15](~
after using ~"(0)
=
0, which condition has not been invoked here. Of course, equation (5)
demonstrates that #(cJ) must also change sign at least once in the interval 0 < cJ < oo.
The static (cJ
=0) Lagrangian of a causal medium must be a perfect differential, as Post
(ref. [6], Eq. (6.17)) has shown and Lakhtakia [16] has commented upon. From this fact as well as from (2c), it follows that ~(0)/~(0)
=0. As the static permeability cannot be infinite,
the logical result
~(0)
=0 (8)
(~)The first author (F.G.) was kindly apprised of the thesis of Serdyukov (unfortunately, available
only in the Russian language) by S-A- Tretyakov.
916 JOURNAL DE PHYSIQUE III N°7
is mandated by the structure of the Maxwell postulates and the form of the Lorentz force. We
emphasize that this last result has been obtained by a formal proof, and is therefore neither
merely an assertion nor a simplifying assumption based on empirical arguments.
Significantly, equation (8) leads to ~'(0)
=
0, which automatically means that #(0)
=
0.
Equation (6) therefore yields
/~ ~~~~~~
#
0j (9)
o ld
equivalently,
/
m~"(Ld)dLd
=0. (10)
o
Equation (9), incidentally, had been obtained by Emeis et al. [8] through a different set of assumptions.
Equation (10) leads to the significant result that f'(cJ) must change its sign at least once
over the range of positive cJ. This result could have been inferred easily from the literature-e.g., (Ref. [8], Eq. (27)) and (Ref. [10], Eq. (15))-but it has never been stated elsewhere to our
knowledge. Parenthetically, let us note that equation (9) mandates 9(cJ)-just like #(cJ)-must change sign at least once over the range of positive cJ.
4. Discussion
We have shown above that both ~'(cJ) and f'(cJ) must change sign at least once in the range 0 < cJ < oo. This can be regarded as a new result for the electromagnetic engineering com- munity, even though it appears from previous studies (that laid the mathematical foundations of this result) that it could have been discovered much earlier.
Our derivation is based on the (physically reasonable) assumption that the function fl~(fl)
is analytic in the upper half of the complex n-plane and goes to zero as fl goes to infinity on
the real axis. Our derivation utilizes the time-reversal invariance of natural optical activity.
This is expressed through the even dependence of #(cJ) and the odd dependence of 9(cJ) with respect to cJ; equivalently, ~'(cJ) and ~"(cJ) are odd and even functions of cJ, respectively. The
relation ~'(0)
=
0, which is instrumental in obtaining the result of equation (10) for ~"(cJ), has been made available by a rigorous proof that ~(0) e 0 (I.e., no assumption is needed).
The question arises whether the theoretical results on the frequency dispersion of ~'(cJ) and
~"(cJ) have been experimentally verified. The abundance of experimental data in the scientific literature on #(cJ) shows that such is indeed the case as far as the sign changes of f(cJ) are
concerned. Doubts can exist regarding the sign changes of f'(cJ). There is no reason for the zero-crossings of ~'(cJ) and ~"(cJ)-or of #(cJ) and 9(cJ)-to coincide; that they do not coincide is responsible for the Cotton effect. A formal explanation for this fact is also afforded by K-K
relations. From, reference [11] (Eq. (3)) we can show that
/
mLd#(Ld)9(Ld)dLd
#0j (II)
0
I-e-,
/
mcJ~f(cJ)f'(cJ)dcJ
=
