The proceedings of this workshop are available here.

Proceedings of 66Bi-isotropicstgS "
Workshop on novel microwave materials
Ari Sihvola (editor)
Eelsinki University of Technology
Faculty of Electrical Engineering
Electtomagnetics Laboratory
Otalaari 5 A, SF-02150Espoo,Finland
Report 137
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Proceedings
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Workshop on novel
microwave materials
Ari Sihvola(ed.itor)
Report 132
February19g3
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Table of contents
Introduction
Workshopprogramme
List of participants
7
q
Selectedbibliography
11
Ismo Lindell: Electromagneticsin novel isotropic media
2t
Ari Sihvola: Dictionary between different definitions of bi-isotropic material parameters
33
Fr6d6ric Maiotte: Theory of guided waves in chiral media: a step for measuring chiral
material parameters
47
Arto Hujanen: Measuring electrical, magnetic, and chiral material parameters
43
Luk R. Arnaut and .Lione/E. Davis: Electromagnetic properties of losslessn-turn helices
in the quasi-stationary approximation
57
Ismo Lindell and Ari Viitanen: Plane wave propagation in a uniaxial bianisotropic medium
61
Ari Viitanen and Ismo Lindell: Plane wavepropagation in a uniaxial bianisotropic medium
with an application to a polarization transformer
63
Ismo Lindell, Ari Viitanen, and Pdivi Koivisto: Plane-wavepropagation in a transversely
bianisotropic uniaxial medium
65
SergeiA. Tretyakov and Alexander A. Sochava:Novel uniaxial bianisotropic materials 67
Ari Sihvola: Macroscopicpredictions of material parameters for heterogeneousbi media
7T
FedorI. Fedorov: Covariant methods in the theory of electromagneticwaves
7s
Anatoly N. Serdyukov.' Fundamental principles restricting the macroscopicphenomenological description of matter
77
Igor V. SemcJrenko;Particular wavesin bi-isotropic media
gs
Introduction
Electromagneticsand microwavetechnology are progressingthese years with
considerable
pace. One of the frontiers at which major conquests are being made
is that of new
materials. The interaction of electric and magnetic fields with matter is determined
by
the physical phenomenataking place within media, but in the electromagnetic
description,
these effects are reduced to plain material parameters that expressthe magnitua.
of tt
responsebetween the exciting field and the correspondingpolarization. It
is only during"
the latest years that microwave community has recognizlJ the potentinl
of norn"l,mo.I
complicated material effectsin the design of new componentsani systems.
Complex materials have, indeed, attracted the attention of severalresearchgroups
working
in the electromagnetics,microwave,millimeter wave, and optical areas. The
classof these
media - upon which sometimesthe label "exotictt is attached - contains
several types
of materials that loosely can be said to have one thing in common: the
ability to
some special efect when excited by the electromagneticwave. As
"*tibit
examplesbe mentoined
chiral, nonreciprocal, nonlinear, gyrotropic, anisotropic, high-?i superconducting,
etc.,
materials, all of which promise applications in microwave
"r,girr"".irrg.
Bi-isotropic media are a subgroup of these novel materials. These are
isotropic, i.e. they
behavesimilarly regardlessthe direction of the vector force of the electric
or magnetic fieli.
But they are 6i-isotropic, meaning that there is electrically incited
magnetic polarization
and vice versa. one might divide bi-isotropic materials in five groups:
c dielectric media, consistingmicroscopicallyof electrically polarizable
entities, which
are induced by the electric field
c rnagnetic media, displaying analogously magnetic polarization
due to an external
magnetic field
r reciprocal chiral mediu that due to their inherent left- or
right-handednessexhibit
magnetically caused electric dipole moment density and vice versa;
these are also
called Pasteur media
o nonreciprocalmedia, where the magnetoelectric effect
is cophasal unlike in chiral
media; also called as Tellegen media
o media characterizedby any combination of these four efiects
In the beginning of February, 1993, Helsinki University of Technology
hosted a workshop
on novel microwave materials. Bi-isotropicsSS was the name
of tie four-day-and-nighi
workshop that attracted 17 participants from six countries:
Finland, Russia, BelorrrrJiu.
France, United Kingdom, and Germany. The focus of the talks was intended to be
the theory and applications of bi-isotropic materials in electromagneticsand microwave
engineering,but it may be the nature of this rapidly progressingfield a reasonfor the fact
that also results on more general,bianisotropic materials were discussed(not to mention
the informal interaction sessionswhere topics like scientific ethics and turmoils in global
political structures were reached). Perhaps even Bianisotropics'9? would be too narrow
a title for the next workshop to follow. - Someform of continuation io Bi-isotropics'9?
will certainly materialize; the desire is evident.
This report is a compilation of written material from the presentationsheld during the
ttbook of abstracts" are diverse: the contributions range
workshop. The pages of this
from full reports through short summariesto copiesof the viewgraphs shown in the oral
presentation. Without apologizing for this type of formal shortcomingsof my collection,
I hope that the information about researchresults - especiallyby workshop participants
from the former Soviet Union - will percolate on the pages of this report through the
electromagneticsand microwave communities of our world.
*
Bringing together participants from various countries, and especially from economically
unequally-equippedsocieties,requirescommitment in terms of money. I wish to acknowledge the financial assistancefrom the following three agencies:
o IEEE (The Institute of Electricaland ElectronicsEngineers)MTT (MicrowaveTheory
and Techniques)Society within Region 8
o URSI (InternationalUnion of Radio Science)Finnish National Committee
o The ElectromagneticsLaboratory of Eelsinki University of Technology
St. Valentine'sDay, 1993
Ari Sihvola, Workshop Organizer
Bi-isotropics
Elec_trom_agnetics of Novel Microwave
Materials
Workshop in Espoo, Finland, L_4 February lggg
F I N A LP R O G R A M M E
i
Monday, I February
9.00 Ari Sihvola (Helsinki University of Technology,Finland)
Welcome and opening of the workshop
9.30 Ismo Lindell (Helsinki University of Technology,Finland)
Introduction to electromagneticsin novel bi-isotropic med.ia
11 .00 Discussion
1 1 . 3 0L u n c h
13.00 Ari Sihvola (Eelsinki University of Technology,Finland)
Constitutive relations in bi-isotropic desciption
13.30 Fr6d6ricMariotte (C.E.S.T.A., France)
and Nader Engheta (University of pennsylvania, USA)
Theory of guided wavespropagation in chiral media:
a step for measuring chiral material parameters
14.30 Discussion
15.30 Arto Hujanen (State TechnicalResearchCentre,Finland)
Measuring electric, magnetic, and chiral materiar param"t..s
16'00 sergei rretyakov (st. petersburg state Technical
universitv. Russia)
Comment on bi-slab measurementpossibilities
16.30 Discussion
I\resday, 2 February
9.00 Luk Arnaut (Manchester,United Kingdom,l
Microwave chiral researchat the UMIST
9 .45 Discussion
10.15 Ismo Lindell (Eelsinki University of Technology,Finland)
Theory of wave propagation in uniaxial bianisotropicmedium
1 , t l E Discussion
1 1. 3 0 Ari Viitanen (Helsinki University of Technology,Finland)
Numerical examplesof wave propagation in uniaxiar bianisotropic
medium
7 2 . L 5Discussion
1 2 . 3 0Lunch
1 4 . 3 0Sergei Tretyakov (St. Petersburg State Technical
University, Russia)
Novel uniaxial pseudochiral materials
1 5. 3 0 Discussion
Wednesday, 3 February
9.00 SergeiTretyakov (St. PetersburgState TechnicalUniversity,Russia)
Chiral and bi-isotropic materials in waveguidingapplications
9 .45 Discussion
10.00 Ari Sihvola (Helsinki University of Technology,Finland)
Macroscopicpredictions of materials parameters
for heterogeneousbi media
10.45 Discussion
11.15 Fyodor Fyodorov (BelorussianAcademyof Sciences,Minsk)
Covariant methods in the theory of electromagneticwaves
12 .30 Discussion
13.00Lunch
14.00 Visit to the Telecommunications
Laboratory
of the State Technical ResearchCentre in Otaniemi
Thursdayr 4 February
10.30 Anatoly Serdyukov(Gomel State University,Belorussia)
Microwave and optical chiralitv researchat the Gomel State
University
1 1. 3 0 D i s c u s s i o n
11.45 Igor Semchenko(Gomel State University,Belorussia)
Particular wavesin bi-isotropic media
12 . 30 Discussion
13.00 Closingof the Workshop
Bi-isotropics'93
Electromagnetics
'Workshop
of Novel Microwave
in Espoo, Finland, l-4
Materials
February 1993
LISTOF PARTICIPANTS
I{ORKSHOP
ORGAIIISER:
Ari Sihvola
Helsinki University of Technologr, ElectromagneticsLaboratory
Otakasri 5 A, 02150Espoo, Finland
Tel. (358)-0-4 5 l-226 l, Fax ( 358)-0-a 5 7-2267,E-mail ari. [email protected]
Luk Arnaut
UniversityofManchesterInstitute ofScienceand Technology,DepartmentofElectrical Engineering
and Electronics,P.O. Box 88, ManchesterM60 lQD, United Kingdom
Tel. (aa)-61-236-331
l, Fax (aa)-61-228-7040
Murat Ermutlu
Helsinki University of Technologr, ElectromagneticsLaboratory
Otakaari 5 A, 02150Espoo, Finland
Tel. ( 358)-0-a51-2260,Fax (358)-0+ 5l-2267, D'mail [email protected]
Fedor I. Fedorov
ByelorussianAcademyof Sciences,
Institute of Physics,Skarynapt. 70,220602Minsk, Byelarus
Tel. (7)-(0t72)-346-870(home), 39&559 (office)
Arto Eujanen
Telecommunications
Laboratory, State TechnicalResearchCentre, Otakaari 7 B, 02150 Espoo,
Finla.nd
Tel. (358)-0-a56-5608,Far (358)-0-455-01
15
Arne F. Jacob
TechnischeUniversitit Braunschweig,Institut ltr Hochfrequenztechnil
Postfach3329,3300Braunschweig,Germany
Tel. (a9)-531-2469,
Far (49)-531-391-5841,
Dmail i7091001@dbstul
Piivi Koivisto
Eelsinki University of Technologr, ElectromagneticsLaboratory
Otakaari 5 A,02150 Espoo, Finland
Tel. (358)-0-45 1-2260,Far (358)-0-4 51-2267,Bmail [email protected]
Jukka Lempidinen
Helsinki Universityof Technology,ElectromagneticsLabotatory
Otakaari 5 A, 02150Espoo, Finland
1-2267
Fax (358)-0-45
Tel. (358)-0-{5r-2260,
Ismo Lindell
Laboratory
Helsinki Universityof Technologl,Electromagnetics
Otakaari 5 A, 02150Espoo,Finland
Tel. ( 358)-0-a 5l-2266,Fax ( 35s)-0-a 5l-2267,Bmail [email protected]
Fr6d6ric Mariotte
Commissariati I'llnergie Atomique, C.E.S.T.A.- ServiceSystlme
B.P. No, 2, 33114Le Barp, hance
Fax (33)-56-s85-121
Tel. (33)-56-684-147,
Keijo Nikoskinen
Helsinki University of Technologt, ElectromegneticsLaboretory
Otakeari 5 A, 02150Espoo,Finland
Fax (35S)-0-a5l-2267,Dmail [email protected]
Tel. (358)-0-457-2262,
Igor V. Semchenko
Gomel State University,Departmentof Physics,SovjetskayaStr. 104,246699Gomel, Byelarus
Tel. (?)-0232-576-657and 577-520,F-mail dej%[email protected]'eu.net
Anatoly N. Serdyukov
Gomel State University,Departmentof Physics,SovjetskayaStr. 104,246699Gomel, Byelarus
Bmail dej%[email protected]
Tel. (7)-0232-563-002,
Johan Sten
Helsinki Universityof Technologr,ElectromagneticsLaboratory
Otakaari 5 A, 02150Espoo,Finland
Tel. (358)-0-.451-2260,Fax ( 358)-0-a51-2267,Dmail [email protected]
Sergei Tretyakov
29,
St. PetersburgState TechnicalUuiversity, Department of Radiophysics,Polytechnicheskaya
Russie
195251St. Petersburg,
Dmail [email protected]
Far (7)-812-552-6086,
Tel. (7)-812-552-7685,
Ari Viitanen
Helsinki Universityof Technology,ElectromagneticsLaboratory
Otalaari 5 A, 02150Espoo,Finland
Tel. ( 358)-0--a51-2262,Far (358)-0+5 1-2267,Fmail [email protected]
Wei Zhang
Hclsinli Udversity of Technologr, Radio Laboratoty
Otatesri 5 A,02150 Espoo,Finland
Tel. (358)-0-45l-2255, Far (358)-0-asL2r52
l0
Selected bibliography
l{hat
folloes
about
the
j.s a list
topi.c
electromagnetics
exhaustive;
in
the
it
of
our
of
and micronave
only
Electromegnetics
publications
recent
llorkshop:
reflects
in
bianisotropic,
applications.
tbe
range
ald
journals
different
especially
The
domain
list
of
is
chiral
by
literature
no means
folloced
Laboratory.
Afonin, A.A., A.N. Godlevskaya, V.N. Kapshai, S.P. Kurlovich, and A.N. Serdyukov: Scattering of
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Spectrcscopg (USSR), Vol. 69, No. 2, p. 242-245, August 1990.
Agranovich, V.M., and V.L. Ginzburg: On the phenomenological electrodynamics of gyrotropic media,
Journal of Experimenlal and. Theoretical Physics (JETF, I/SS-R/, Vol. 63, No. 3, 1972.
Ali, S.M., Habashy, T.M., and Kong, J.A.: Spectral-domain dyadic Gteen's function in layered chiral
media, Journal of lhe Opticol Soceily of Americo,.4, Vol. 9, No. 3, p. 473-423, March 1992.
Applequist, J.: Optical activity: Biot's bequest, Americon Scienlist, Vol. 75, No. 1, p. 58-68, JanuaryFebruary 1987.
Babonas, G., S. Marcinkevidius, and A. Shileike: Structural model of optical activity in semiconductors,
Journal of Eleclrmognelic Watsesand Applicatioas, Vol. 6, No. 5/6, p. 573-586, 19g2.
Barron, L.D.: Molecular lighl scaltering and oplical acliaity, Cambridge University Press, 1982.
Bassiri, S., N. Engheta, and C.H. Papas: Dyadic Green's function end dipole radiation in chiral media,
Alla Frequenza, Vol. LV, No. 2, p. 83-88, March-April 1986.
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Bhattacharyya, A.K., E.K. Miller, and G.J. Burke: Circular loop antenna in unbounded chiral medium:
a moment methods solution, Journal of Electromagneiic Watses and, Applicotions, Vol. 6, No. 4, p.
431-44,7992.
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p. 458-462,1974.
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Bokut,8.V., and A.N. Serdyukov, On the phenomenological theory of optical activity, Journal of Experimentol ond. Theoreticol Physics (JETF, t/55.R/, Vol. 61, No. 5, 1971.
Bokut,B.V.,A.N.Serdyukov,andF.I.Fedotov:
Formofconstitutiveequationsinopticallyactivecrystals,
Optics ond Speclrcscopy (USSR), Vol. 37, No. 2, p. 166-168, August 1974.
Bokut, 8.V., A.N. Setdyukov, and V.V. Shepelevich: On the phenomenologicaltheory of absorbing
optically ective media, Oplics and Speclrcscopy (USSR), Vol. 37, No. 1, 1974.
Bokut, B.V., and A.N. Serdyulov: On the theory of optical activity of non-uniform media, Jorraol of
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Chambers, L.G.: Propagation in a gyrational medium, Quarterlg Journol of Mechonics and Applied
M a l h e m a l i c s , V o l . 9 , N o . 3 , p . 3 6 0 - 3 7 0 ,1 9 5 6 .
Chen, D. and L. Xue: Dipole radiation in the presence of a grounded chirai slab, Miupuare
Technologg Lellers, Vol. 5, No. 2, p. 65-68, February 1992.
and Oplical
Cheng, D. and L. Xue: Dipole radiation in the proximity ofa chiral slab, International Journal of Infiured.
and Millimeler Wates, Yol. 12, No. 12, 1991.
Cheng, D, W. Lin, and L. Xue: General approach for finding the characteristics of chiral layered media,
I n t e r n a l i o n a l J o u r n a l o f I n f i n r e d . o n d .M i l l i m e l e r W o o e s , Y o l . 1 3 , N o . 1 , p . 1 0 5 - 1 1 5 ,J a n u a r y 1 9 9 2 .
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Micrpuaae and Oplical Technologg Lellers, Vol. 5, No. 3, p. 135-138, March 1992.
Chien, M., Y. Kim, and H. Grebel: Mode conversion in optically active and isotropic waveguides, Oplics
L e t t . , Y o l . 1 4 , N o . 1 5 , p . 8 2 6 - 8 2 8 ,1 9 8 9 .
Chilo, N.A., and A.N. Serdyukov: Electromagretic wave transmission through a magnetized optically
active layer, Journal of Applied, Speclroscopg (USSR), Vol. 25, No.l, p. 169-171 1974 (in Russian).
Cloete, J.H. and A.G. Smith: The constitutive parameters of a lossy chiral slab by inversion of plane-wave
scattering coefficients, Mictware and Optical Technology Lellers, Vol. 5, No. 7, p. 303-306, June 1992.
Condon, E.U.: Theories of optical rotatory power, .Reoieus of Moilern Physics, Vol.
October 1937.
9, p.
432-457,
Cory, H., and I. Rosenhouse: Electromagnetic wave propagation along a chiral slab, IEE Proceed.ings,
Parl H, Vol. 138, No. 1, p. 5l-54, 1991.
Cory, H. and T. Tamir: Coupling processesin circular open chirowaveguides, IEE Proceed,ings,Parl E,
V o l . 1 3 9 , N o . 2 , p . 1 6 5 - 1 7 0 ,A p r i l 1 9 9 2 .
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N o . 3 , p . 3 5 1 - 3 5 9 ,1 9 8 9 .
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with an impedance wall, lEE Proceedings .f1,Vol. 135, No. l, p. 23-26, 1988.
Engheta, N., and A.R. Michelson: Transition radiation caused by a chiral plate, IEEE
Antennas and Prcpogalion, Vol. 30, No. 6, p. 12f3-1216,1982.
Itransaclions on
Engheta, N., and D.L. Jaggard: Electromagnetic chirality and its applications, IEEE Anlennas and
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Iight waves, Journal of Applied Physics, Vol. 66, No. 6, p. 2274-2277, t989.
Engheta, N., and P. Pelet: Orthogonality relations in chirowaveguide, IEEE llansactions on Microuaoe
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, o l . 3 8 , p . 1 6 3 1 - 1 6 3 4 ,1 9 9 0 .
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Engheta, N., and P. Pelet: Reduction of surface waves in chirostrip antennas, Eleclrcnics Letlers, Yol.
27, No. 1, p. 5-7, January 1991.
Dngheta, N., and P. Pelet: Surface waves in chiral layers, Optics Letters, Vol. 16, No. 10, p. 723-725,
May 1991.
Engheta, N. and P.G. Zablocky: Etrect ofchirality on the transient signal wave ftont, Optics Lellers,Yol.
1 6 , N o . 2 4 , p . 1 9 2 4 - 1 9 2 61, 9 9 1 .
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Tlvrsoclions on Anteanos ond Prcpogotion, Vol. 40, No. 4, p. 357-467, April 1992.
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1 4 6 , N o . 4 , p . 1 8 5 - 1 8 9 ,1 9 9 0 .
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T2
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Graglia, R.D., P.L.E. Uslenghi, and C.L. Yu: Electromagnetic oblique scattering by a cylinder coated with
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1990.
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media, Oplics and Spectrcscopy (USSR), Vol. 49, No. 6, p. 639-640, December lgg0.
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f-a,
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January
1 9 9 0 ,p . 1 0 8 - r 1 5 .
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with a grounded chiral slab, Jourtal of Electrcmagnetic Waoes and Applications, Vol. 6, No. 5/6, p.
751-770, 1592.
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applications to measurement techniques, Micrcwaoe ond Opticol Technology Letlers, Vol. 5, No. 4, p.
17+777, April 1992.
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M i c r p u a v e a n d O p l i c a l T e c h n o l o g yL e t t e r s , V o l . 6 , N o . 2 , p . 1 1 2 - 1 1 5 ,F e b r u a r y 1 9 9 3 .
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M. Aspnis, and P. Lindroos, editors), Abo Akademi (lniuersity, Deparlmenl of Phgsics, S4 pages, November 11,1991.
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1 0 , p . 2 0 1 - 2 1 1 1, 9 9 0 .
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J o r n t o l o l P h g s i c s , 4 , V o l . 2 0 , p . 2 6 4 9 - 2 6 5 0 ,1 9 8 7 .
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using chital composites, Jounrcl ol Wooe-Moteiol Intercction, Vol. 2, p. 71, 1987.
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Propagation in parallel-plate waveguide wholly filled
1B
with a chiral medium, Journol of Waae-Malerial Inleraction, Vol. 3, No.3, p. 267-272, 1988.
Varadan, V'V., Y. Ma, and V.K. Varadan: Dfects of chiral microstructure on em propagation in discrete
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.Olec-
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l9
ELECTROMAGNETICS
IN NOVEL
ISOTROPIC
MEDIA
I V Lindelll
ABSTRACT
Basic relations of electromagnetic fields in bi-isotropic media are briefly discussed. Such
media are more general than the ordinary isotropic dielectric and magnetic media because
they can couple electric and magnetic polarizations. Bi-isotropic media have interesting
possible applications and the main problem today lies in the fabrication, which has not yet
reached an industrial level. However, before starting making suc.hmedia, their usefulness
must be shown theoreticallv.
INTRODUCTION
Isotropic chiral media have raised interest in the last decade due to their extra medium
parameter which gives added freedom in designing microwave and millimeterwave devices.
In fact, promising applications in antennas (polarization rotating lenses,compactly packed
microstrip antennas),microwave devices(low-lossphase shifters) and radar engineering(reflectionlesssurfaces)have been suggested,all based on isotropic chiral media.
New applications have induced interest in electromagnetictheory for such media [1,2].
Actually, chiral media form a subset of the most general isotropic media, the bi-isotropic
(BI) media, which again form a subset of the most general linear media, the bianisotroprc
media. Electromagnetic wave propagation in and reflection from such media is consideredin
this paper. The basic phenomena, rotation of polarization, in propagation due to chirality
and in reflection due to nonreciprocity, of the BI medium, are discussed.Transmission-line
analogiesare given to treat layered structures of bi-isotropic media. As an example it is
shown that a twist polarizer can be simply constructed with just a layer of BI material on a
conducting plate.
THE BI MEDIUM
The medium is seenby the electromagneticfields through the constitutive equations, which
for time-harmonic fields (ept dependence)have the general form involving four medium
parameters [3, 4]
D=eE*{H,
(1)
B=pH+(E,
(2)
or in terms of other parameters[6, 7, 5], where n is the chirality parameterand 1, the
Tellegenparameter:
€: (x - jn)Jp"e",
( = (X+ jn)Jtr"e".
(3)
The relative chiral and Tellegen parameters
lHelsinki University ofTechnolog;r, Electromagnetics
Laboratory, Otalaari 5A, Espoo 02150, Finland
2L
Kr:
,t
-l
X' = sind
n
x)
with
":
Trt, : yE,e,
(4)
tl}t'o "o
Y
will also be applied to simplify the notation. The present form of the constitutive equations
is not the only possibleone and severalother notations are in use in the literature. Different
forms of the constitutive relations for BI media are discussedin [5].
It is quite easily shown that, for losslessmedia, the parameters x and 1 together with e and
- (p have real values, which with other symbols can be written as {
[3]. Also, it can be
parameter
the
Tellegen
if
and
only
if
is
reciprocal
medium
1 vanishes
shown that the BI
parameter.
Both
nonreciprocity
called
as
the
also
be
parameter
can
[10]. Thus, the Tellegen
physical
in
meaning
a
simple
parameter
have
rc
and
the
Tellegen
the chirality parameter
t
shortly.
propertiesr
be
discussed
to
terms of plane wave
A medium with nonzero chirality parameter ,r requires microscopic constituents without
mirror symmetry, e.g., small -glallic heliceswith the same handedness.Actually, first electromagnetic experimentson a man-made chiral medium were done in the 1910'swith copper
helices randomly embedded in cotton [8]. The Tellegen parameter X can be theoretically
realized by a medium whose microscopic constituents are permanent electric and magnetic
dipoles tied together to form similar pairs [9]. It is not known whether this kind of media
have yet been actually fabricated.
FIELDS IN BI MEDIUM
Let us consider time-harmonic electric and magnetic current sourcesJ, J- giving rise to an
electromagneticfield E, H in a homogeneousBI medium. The Maxwell equations
r z - l -l s
\l : & { X'' : 1 "
\
"'\"/
\/e\
- l\=rl l
- X , + r ^ , 1/ \l n; /- =
\ -rrr
/o -1 \/
J \
o i \l-i
(, ,o, , ,
can be written in uncoupled form for a homogeneousmedium by writing the electric and
magnetic fields in terms of uorse felds, also called self-duol electric felds 14, 6] E+, Edefined by
p* : -l(
cos u
"-iou
- jrt[),
e- : -1(
' eou+ irirr.1,
cos 1,
(6)
where 4 = lfUle.In fact, (5) can be written as the uncoupledequation pair
v x E - + , b - E -: i q J -,
YxE*-h*E*:-jqJ+,
(7)
with
r) = fr(cosrl
* r'),
k* = k'6F 1i
(8)
if the sourcesJ, J- are split into two parts J1 and J- as follows:
r* : =l ^(e*j'J+ lr-).
zcosu
(e)
Jq
The magnetic field can also be split into two parts, H = H+ * H-, which have a simple
relation to the electric field:
H+ = *1E+,
22
\+ = \erio,
(10)
and, similarly, the magnetic source has the decomposition
J-* : J-1+iar
+ r+jrJ-).
z cosl)
(11)
\\'ith these definitions, the Maxwell equations (7) can ia fact be written in the simple
isotrooicform
-J-+r
V x E1 : -jup.alla
VxHl-joe1E1fJ1.
(1 2 )
(13)
tt+t' fields
This means, that if the sourcesJ, J* are split into two parts, the Ja gives rise to
just like in a simple isotropic medium with the efective medium parameters
p+:
e + : e d o ( c o s r19r , ) ,
I l e - i o ( c o s r*g r , ) ,
(14)
and the minus part to minus fields in another isotropic medium with
- r,).
,t- : y,eio(cosd
,"-io("os t9 - r,),
,- :
(15)
This idea simplifies the analysis considerably,since at any stage we may find the efrect of a
BI medium by a cumulative effect of two diferent isotropic media'
PLANE
\MAVE IN BI MEDIUM
Plane wave propagation in a BI medium was first analyzed already in 1956 [12]. Let us
consider a plane wave propagating in the positive z axis direction in a BI medium. Let the
electric field at the plane z = 0 be written as E(0) = E+ * E-. Each of the two wave fields
E1 see their own isotropic media and propagate with the propagation factors, whence the
total field is
E(z) = Eae-ik+' +E-e-ih-",
kt:
f , ( c o s r l* r c ' ) .
uJiF+:
(16)
(1 7 )
The polarizations of the wave fields are obtained ftom (7), which outside the sourcesgives
- jw, x E*(0) + ,t+E*(0): 0.
(1 8 )
It is easy to seethat the field vectors are transversal to u" and circularly polarized, because
they satisfy E+ 'E+ - 0 and, moreover, the plus wave has the right-hand and, the minus
wave, the left-hand polarization. Thus, we may write
1
E(z) : !r-ih'co"o(trtol - JU, X E(o;]r-;*t' + tE(0)* ju, x E(0)]/-t")
"-ik'"-0lcos(r[z)71
Ir=I-u,u'
- sin(r,tz)7] .E(0),
J:urxI.
( le)
(20)
The dyadic in square brackets is a rotation dyadic which turns the vector E(0) by an angle
-n,kz - -xkoz in the right-hand direction when looking in the direction of propagation,
u,. Thus, the chirality parameter r has the simple physical meaning: it gives the rate of
polarization rotation of a propagating linearly polarized plane wave, relative to the rate of
23
The
phase changeof the wave in air: at a distance of )" the rotation angle equals 6:2trn.
function
exponential
with
a
dyadic
rotation dyadic can be written in the compact form
,=
cos(nk"z)I 1- sin(r,t"-)7
= s-nkozJ
(21)
The polarization rotation in the BI medium is seen to be due to the chirality parameter,
becausethe Tellegen parameter only affects the phase change of the wave. The effect is
similar to the Faraday rotation in magnetoplasmaor ferrite except that it is the same in all
directions of the wave becausethe BI medium is isotropic. This effect has many possible
applications in microwave engineering. For example, a polarization correction for a lens
antenna fed by a dipole, by introducing suitably distributed chirality in the lens material,
has been suggested[11].
The magnetic field of the plane wave can be written as
H ( z :) H + ( ,+) : f - - ( z ) :! n * 1 " 1 - !q-" - f " l
\+
j
n
'
h
z
- '."o+1"121.8(r).
(tefol - ju" x E(0)]e:
1 [ E ( 0 ) + j u , x E ( 0 ) ] e i K " k' ' z )
t
\'!
rl
tztzl
This can be interpreted so that the magnetic field of the plane wave is rotated by an angle
3 +T 12 with respectto the electric field. Thus, the Tellegenparameter X : n.sin rl is directly
related to the excessangle 19over the right angle of the electric and magnetic field.
1L1- " - i k ' " o " o
NORMAL
REFLECTION
tr'ROM AN INTERFACE
Let us considerthe plane wave incident from air to an interface of a BI medium at z : 0. In
normal incidence the problem is easily solved by writing the incident field as a sum of two
circularly polarized fields, which do not couple but changeplaces in the reflection. In fact,
becausethe field vector must be rotating in the same direction before ans after reflection,
the handednessis changed in reflection, whence the reflected wave seesanother isotropic
medium than the incident wave.
L e t t h e i n c i d e n t w a v e a r r i v e f r o m a i r z 1 0 t o t h e i n t e r f a c ea t z : 0
z > 0. Writing the incident field at the interface as
E':Ei*El,
of a BI mediumin
(23)
the reflected field is
E ' ( 0 )= . R + E +
l R - E ' -: E ' E ' ,
(24)
ft:R+u+u-*rt-u-u*,
( 2 5)
where the circularly polarized unit vectors are
,r*=;l{rr"-ruy), .,-:;l{,r"+ruy).
( 2 6)
Inserting
n
J L*+ =
l- .x - r l "
?t*1o'
24
(27)
the reflection dyadic (25) can be written as [6]
F = |ta* + ft-)7,- t;W* - R-)i - R"6i,
( 28)
where .R denotes the total reflection coefficient and {, the angle of rotation of the reflected
polarization,
R_
T2 - 2T\" cosri I ql
T 2l 2 q \ o c o s 8 1 r 7 l '
t a n g -- + + s i n r 9 :
(2e)
2x
(30)
e,-Fr
\'-n;
It is seen that the chirality parameter r does not affect the reflection at all. On the other
hand, the Tellegen parameter x causesrotation of polarization in reflection. flowever, +g0"
rotation is possible only for €, : p.. The reflection rotation is another quantity giving
physical meaning to the Tellegen parameter a.
The rotation of the reflectedfield is a demonstration of nonreciprocity of the BI medium with
nonzero Tellegen parameter. In fact, a field incident with the polarization of the reflected
field does not reflect with the polarization of the incident field but is rotated by another
angle /. This efect also exists for BI slabs and it may find important applications in the
future when media with nonzero Tellegenparameter can be reliably fabricated.
LAYERED
MEDIA,
NORMAL
INCIDENCE
The previous concepts can be applied to a piecewisehomogeneousBI med-ium where the
interfaces are planes perpendicular to the z axis. If a plane wave is incident in u, direction
to the layered structure, in each homogeneoussection the field can be expressed.in terms of
four waves: one plus and one minus wave in both directions. The plus wavesin *u, direction
are coupled only to minus wavesin -u, direction and vice versa. So the problem can be split
into two scalar problems which can be handled through transmission-linetheory except tlat
the properties of the transmissionline are diferent for wavestraveling in opposite directions.
Thus, we are motivated to study a more general transmission theory
[lB].
GENERALTZED
SCALAR
TRANSMIS
SION-LINE
THEORY
The generalizedtransmission-lineequations are written in matrix form as
W)':-,,("2,u
),
"l,u)(?,,;i
(31)
where prime denotes differentiation with respect to z. L is the distributed inductance and
C the capacitanceon the line. a and 6 are two new parameters of the transmission line and
they may have complex values, although for losslesslines they are real. To emphasizethe
connection to the field theory, we can write
o = *.Jid,
g = y,t/Td : sinoy/LC.
(32)
It can be easily shown that the propagation factors for waves propagating in the opposite
directions are analogousto those of the previous sectionsif the plus wave is p.oprgrliog io
the positive z direction
25
-
9t :IJLC:b2lua=
B ( c o s d *r c , ) ,
(33)
B:uJLC.
are
impedances
characteristic
The corresponding
nr
B+ a w(a + ib)
uC
uL
- 2"1i8,
B+au(a-jb)
I
- : r tr
o
,cl : T
(34)
The most general voltage function is, then, of the form
U(z) : g+
z [J- dP-',
+
"-iB+
(35)
and the correspondingcurrent function,
I(z) = 1+
z - y-g'"iBz I- dp-' - y+g+
+
"-i9+
"-iB+
= yl(J+ e-i(B+,-d)- g"i@-,-o)1.
z
(36)
Reflection and transmission at a junction
Let us consider a junction of two transmission lines 1 (, < 0) and 2 (z > 0) and a voltage
wave incident to the junction in line 1, Ur+. There arisesa reflected wave and a transmitted
wave with the reflection coefrcient -R and transmission coefficient? defined by
Ul : TU{
U, : RtJ{,
(37)
From continuity of voltage and current at the junction we can solve for the two coefrcients:
Zl(Zl - Z{)
n Yr* -Yr*
^:ffi:-iffi1fi=fr
Z{ - Z{
*fi"'jo"
T
' ==Y , + Y + _ Z l ( Z { + Z t ) _ 2 Z l " o ' 3 r " i 0 , .
V7*4,:27127*27: z; * ^
( 3 8)
(3e)
The last expressionsare only valid for a losslessline I with Z{ = /rs+i0'. The coefrcients
defined for voltage amplitudes appear simpler in terms of admittances. Definitions in terms
of current amplitudes are not the same because of. Z+ I Z- and, they appear simpler in
terms of impedances. Similar expressionsare valid for the reflection from a load admittance
Ys,when the admittance Yr+ is replaced by Y6
Z,(Zt - Z{)
ZiQt+ Zr)
Y r *- Y "
'^" Y, +Y1
(40)
Unlike for ordinary transmission lines, the magnitude of the reflection coefrcient is not
necessarilyunity for the casewhen all the power is reflected. For example, for open circuit
at z = 0, we have Yr. = 0 and
y.+
R-=$=e2io'
r1
instead of .R = 1. For short circuit Yr = *
(41)
we have
Rr":
as for the ordinary line.
26
-I
(42)
Input impedance
The reflection coefrcient at a point z ( 0 in line 1 can be written as
R(z):ffi:
: R(0)etze,"o"o.(43)
R(o)ei@++B-),
W#:
Aga.in,it is noteworthy that the parameter o, or r,, does not affect the reflection coefrcient.
The admittance at the point z < 0 of a line loaded by an admittance Y1,is
-Y-U-(z)
-Y- R(z)
+U+(z)
_
vtit\o)
. r , \ :_ I ( " ) _ Y__uIrTu_A_
: Y+
ue):
:t,
I + R(r)
,,Y7cos(Bd,cosd + rl) * jY sin(pd cosr9)
(44)
The correspondencewith the expressionof the ordinary line is seen immediately. In fact,
setting d : 0, the well-known expression
r,
ri^:
cosBd * jY sinBd
,,Y1
r
Y"""pd+ jY"t]"N
(45)
is obtained.
Impedance matching
The quarter-wavelengthmatching sectionof the ordinary transmissionline can be generalized
for the present transmissionline. Requiring that the imaginary part of the input admittance
(44) vanishes,gives us the following relation between the parameterc Bd,,$ and,YelY:
sin(Bd cos0)[Y] cos(Bdcost9 + 0) - Y'cos(Bd cos19- d)l = O.
(46)
Excluding the simple solutions sin(Bd cosd) : 0, for which we have Yn : Yr, there is another
set of solutions satisfying
=ffi
tan(Bdcosrg)tanr9
(47)
Apply*g this condition in (44) gives us the simple relation
-Y'
u
t n. -
(48)
yLt
which is the same as for the ordinary quarter-wavelenthline. Thus, if we wish to match the
load admittance Y7 to an admittance Yo, the line admittance must be chosenas
y:\ffi,.
(4e)
Flom (47) it can be seenthat, fot 3 I 0, the matching length of is changedfrom the ordinary
vralueof quarter wavelength and shorter matching sectionsare in fact possible.
I n t h e u l t i m a t e c a s e ,c h o o s i n g0 = r 1 2 f o t Y 1 ) Y a n d 0 : - r 1 2 I o t Y 7 1 Y , ( 4 7 ) b e c o m e s
po=l##,1,
or o= +;lTITFl
_
^2lY? -Y2l
(50)
This gives always d values smaller than )/4. Actually, the length depends on the mismatch
of the two impedances: for small mismatch (hlY" = 1) the required iine length is small,
27
whereasfor the most completemismatch (YrlY = 0 or YtlY = o), the length is )f 2r x
)/6. This means more broadbanded matching than with a quarter-wavelengthsection of a
conventional transmission line.
GENERAL
PLANE
\^/A\/ES IN LAYERED
BI MEDIA
Finally, we consider general plane-wavepropagation in a piecewisehomogeneouslayered BI
medium with plane-parallelinterfaces [14]. In this case,the plus and minus waves couple to
one another and the problem cannot be handled in terms of two separatescalar transmission
lines as for normal incidence. Also, in the simple isotropic case (x : 0, X : 0) with
obliquely incident plane waves, the TE and TM polarizations do not couple to each other,
which again leads to two scalar transmission lines. In the general BI problem the TE and
TM polarizations have no special importance.
Let us assumethat the plane waves are propagating normal to the c axis, which imposes
no restriction to generality. For a single plane wave excitation, wavesin all layers propagate
with the same velocity in the g direction, which implies that all wave vectors have the same
component &r. It turns out that in each layer there are four plane wave components with
the same &, component: two plus waves and two minus waves [15]:
'* E1 e-jf*''-1E(r):fr* s-ji1''* fr[- ,-;i- ',
"-;i,-;[-''
H(r):fr* s-ji+'* fr- .-;i-''* fi*
"-ji*'1 fi-
(orl
(52)
The vectors denoted by 9 correspondto r{'avespropagating in the positive z direction ani,
those denoted by E, in the negative z direction. It is assumedthat the propagation factors
Ba and p- defined by
k+:
u,9+ I ur,ty,
kr:
g_:1fk_-nu,
-ur0+ * tvky,
(53)
k+:k"(
(54)
n2-y2tn).
have a positive real part. Becauseof the coupling between scalar transmission lines, the
general plane-waveproblem leads to a concept which can be labeled as vector transmission
line.
VECTOR
TRANSMISSION-LINE
THEORY
In the vector transmission-line (VTL) theory, the transversal field components are represented through the tvector voltage' e and 'vector current'j defined by
e:1r.E
j:-u"xH=-J'H,
(00J
The vector voltage in a homogeneousregion can be written in terms of those corresponding to the four eigenwavesand the result presented in the compact form with two dyadic
propagation factors, one for each direction of propagation,
e ( z )= s - i i ' . - ( o )+ r ' F ' . ; ( o ) ,
+
R_R
=)++l++l
0: 9+ a*a* *6- B-8-,
28
(56)
e + t ^ F e l
a+a+ +p- a-a_
(0/J
The basis vectors C*, E* are the transversal projections of the circular polarizations of the
electric wave fields propagating in the respective *u, directions:
i+:
E+: u" I jcatr,
u" T rc+uy,
c + : c o s0 + : / x . l k + ,
(58)
J : u,' d1 x d-: j(c1+ c-).
(59)
and their reciprocalvectorsare
+t
*
1
d*: +ju,x ia,
The basis {d*,d-}
tst
i*:
,1
+ju,x i*=
and the reciprocalburis {d'1,d'-} satisfy the orthogonality relations
(60)
The two propagationdyadics
and simil6,llyfor the other basisvectors{f,*, f -}, {;;,;'-},.
€+
*
if 9+ and &1 are real (losslesscase).
are not independent, in fact, they are related by 0:9
For a lossy medium, the relation is a bit more complicated [15].
The dyadic characteristic admittance for the transmitted wave can be easily identified from
expressionsof the transmitted vector currents
i e)=i .i ("),
(61)
in the form
-u", i (i*;'* - ;-;'-),
\-l
\?*
F t
F
F t \
q
a-a_ I
/ a*a,
(62)
i:
I'=u,^Jlq-l
\?+
The relationbetweenthe two dyadicadmittancescan be written for losslessmediasimply
€ = -1",
I
/ F
where denotesthe transposeoperation.
as Y:Y
For a vector transmission line z ( 0 terminated al z :0
defined through the relation
j(0) :ip
with a dyadic admittance 7r,
.e(0),
(63)
a reflectiondyadici 1r; ."o be definedfor wavesreflectinginto -u, direction:
E(,):i (z).;e)
E1r;
".o
(64)
be expressedin terms of admittance dyadics as follows
(65)
for the input admitt"o.. fr," (z) is obtainedftom the reflectiondyafic
Finally,an expression
expression
in the form
i
+
-7
$
J
-
y;^ (") = ly .s-ta' . V - yt)-'-.2+:t.:+:+
e
.:
e
$
y .dp' . (y + yL)-').
. l e - t 0 '. 1 y _ y t ) - , + d p ' . ( y + r r ) - t ] - t .
(66)
Other forms can be found in [15]. After some algebra, the expression (66) can be seen to
reduce to the well-known isotropic formula for normal incidence when 7; is a multiple of
the unit dyadic 7,.
29
APPLICATIONS
OF VTL THEORY
there
determinant of the reflection dyadic is zero'
If we require that the two-dimensional
one
that
of the plane wave' -since this means
arises an equation for the angle of incidence
zero'
is
polarizations *hith do not changein reflection)
of the eigenpolarizations(transverse
an explicit expression
Brewster angle. After considerablealgebra'
the angle can be
""X.dt;:
in the form
i.t ti""S."*ster angle 0B can be obtained
(67)
tan'208:*ffi'
with
R_
(68)
| - Qln-)'1
| - (tln*)2
U+-
(n*-,t")(n - \")
(6e)
6TiJC+iJ
whence R is a real quantity' In this case'
:
\!,
For a losslessBI medium we have \+
isotropic interface we have, by setting
tn.'.iropl"
fo,
t,
solutio;r
angle
(67) has real
the well-knownsolutions
c+ :;- = JW, - L)lp,q and R = ('t T)lQt * q")'
(?o
t:n0",=l#.
=mr
tandel
it turns out that two real Brewster angles
There is a new phenomenon, however, because
may exist for certain media [16]'
Asanotherexample,letusapplytheVTLtheorytotheproblemofaBlslabbackedby
aconductingplane'f,'ft'"tppfi"ationsandtestsofthetheorycanbefoundin[15]'A
for the
correspondsto ?t'= *'whence the expression
perfectly conductingplane at z:0
reducesto
iopot ui-ittance (66), for normal incidence'
(Jr,l
rl)7,+ sinr97l,
i ,^ Q) = l1i .o. 0 cot(kzcos
dyadic has the form
rralid for z < 0. The reflection corresponding
R*i
i: (17,*7,,)-''r!7,-7..): R"'7,+
(72)
tto
4o
defined by the expressions
with the co and cross-polarizedreflection coefficients
d)
1 cos2r9+ (rt' 4l) sin2(/cdcos
@dcosrl)12-4lsin
Rn=
-@cosrl)]2-4lsin
/cdcosd)
2qt7os|mt9sin2(lcd cos 19)
r9sinz( lcdcosrl )
For losslesscase,these coefficientssatisfy In-lt + l8-lt
,
(73)
(74
= 1.
slab of BI medium, the co-polarizedreflection
If we wish to design a twist reflector with the
condition between the parameters lcd' r9
is required to vanish. Thi, h"ppeos if the following
and 4 is valid [17]:
30
si n(&d coso1 = J:22!--- .
\/\; - \"
( 7 5)
This has real solutions for losslessBI media only for
q l q " < l s i n r 9: l 1 1 ' 1 ,
(76)
whidr requires nonzero Tellegen parameter 1.
The twist reflector rotates the reflected field by 90" and gives a phase shift in reflection
defined bv
(77)
1b: cos-r(T/4" sin r9).
,
"i{
This effect can be obtained even for small values of 1if Ilrl"i" small enough to satisfy the
above condition, but, a broad-bandedoperation is obtained fot r7fq": 0.8 when 1, is chosen
slightly larger than TlTo, as is seenin the figure.
Rn =
o'91R.o
0.8
0.7
0.6
0.5
0.4
U.J
0.2
0.1
0
0
reflectioncocfficicntfor a plancwavc incidcntto a layerof bi-isotropic
Co-polarized
medium with a conductorbacking,as a function of normalizedthickness&d. The
n o r m a l i z e idm p e d a n c he a st h e v a l u e q fq . : 0 . 8 a n d t h e n o r m a l i z eTd e l l e g e np a r a m e t c r v a r i e sb e t w e e nX , : 0 . 7 . . . 1 . 0 . l t i s s e e nt h a t t h e c o - p o l a r i z erde f l e c t i o ni s s m a l l
for a wide band when 1, has a valueslightlylargerthan qf q".
References
[1] A. Lakhtakia, V.K. Varadan, V.V. Varadar, Time-Harmonic Electromognetic Fields in
Chircl Mediq Berlin: Springer, 1989.
[2] S. Bassiri, "Electromagnetic wavesin chiral media", in RecentAdt:ancesin Electromagnetic Theory,eds. H.N. Kritikos and D.L. Jaggard,New York: Springer,1990,pp.1-30.
[3] J.A. Kong, Electromagnetic Woae Theory. New York: Wiley, 1986.
31
,,Radiation and scattering in homogeneousgeneral bi.isotropic reSions'''
[4] J.C. Monzon,
pp'227-235' Febtuary 1990'
I E E E Trans. Antennas Propagat', vol'38, no'2'
constitutive relations", Microwaue and opt'
t5l A.II. sihvola, I.v. Lindell, "Bi-isotropic
Tech. Lett.,vol.4,no.8, pp'295-297,July 1991'
general bi-isotropic (nonViitanen, "Duality transformations for the
[6] I.V. Lindell, A.J.
reciprocalchiral)med.ium,,.IEEETrans.AntennasPropagat.,toappear,vol'40,no'1
JanuarY1992.
[ 7 ] I . v . L i n d e l l , A . H . S i h v o l a , A . J . V i i t a n e n , ' ' P l a n e - w aTech.
v e r eLett''
f l e c t vol'5,
i o n f r ono'2,
m a bpp'79-81'
i-isotropic
Opt.
(nonreciprocal ctrirJ) interface". Microuorse ond
FebruarY 1992.
[ S ] X . f , i n a m " o , t U b " ' e i n e d u r c h e i n i s o t r o p e s S y s t e m v o Wellent',
n s p i r a l f i Ann.
irmige
n R e svol'63,
onatore
Phys',
erzeugte Rot.tioorpoirrisation der elektromagnetischen
n o . 4 ,p P . 6 2 1 - 6 4 41,9 2 0 '
[9]B.D.E.Tellegen,,,TheGyrator'anewelectricnetworkelement'',PhilipsRes.Rep
v o l . 3 ,p P ' 8 1 - 1 0 11, 9 4 8 '
media"' Microuate and Opt' Tech' Lett''
l10l I.V. Lindell, "On the reciprocity of bi-isotropic
to appeartvol' 5, June 1992'
['--'
1 1 ] I ' v ' L i n d e l l , A . H . S i h v o l a , A . J . V i i t a n e n , S . A ' T r e t y a k ocorrection
v , ' ' G e o min
e t rinhomogeneous
icalopticsininh
*og"o"ous chiral media with application to polarization
pp.533-548,1990'
lenJantenn.sr,,J. Electro.WauesAppI.,vol.4, no.6,
medium"' Quart' Jour' Mech' Appl'
I12l t.G' Chambers, "Propagation in a gyrational
M o t h . , v o l . 9 ,n o . 3 ,P P . 3 6 0 - 3 7 01,9 5 6 '
[13]I.V.Lindell,M.E.Valtonen,A'H'sihvola,"Theoryofnonreciprocalandnonsymm
rictransmissionlines,,HelsinkiUniu.Tech.Electromagn.Lob.Reptll7,May1
Submitted lor IEEE Trans' Micro' Theory Tech'
isotropic and
Tretyakov, I.V. Lindell, "vector circuit theory for
[14] M.I. oksanen, S.A.
1990'
pp'613-643'
chiral slabs", J. Electromag'WauesAppI',vol'4' no'7'
''Vector transmission.lineand circuit theory
Tretyakov, M.I. oksanen,
[15] I'v. Lindell, S.A.
AppI', to appear'
for bi-isotropic layered structures", J. Electro' woues
of isotroprcsihvola, "Explicit expression for Brewster angles
[16] LV. Lindell, A.E.
pp'2163-2165'November1991'
biisotropicinterface", Electron' Lett', vol'27,no'23'
slab as twist
Tretyakov, M.I. Oksanen, "conductor-backed Tellegen
[17] LV. Lindell, s.A.
1992'
anuary
J
1282'
pp'28
polarizet," Electron.LetI', vol'28,no'3,
32
Dictionarv between different definitions of
bi-isotropic material parametres
Ari Sihvola
Helsinki University of Technology,Electromagnetic Laboratory
Otakaari 5 A, 02150Espoo, Finland
Due to the fact that there coexist different ways of expressingthe constitutive
relations of bi-isotropic media, and furthermore, that the quantities used in these
may carry the same name without obeying the same definition, it is essential to
write explicitely the interconnectionsbetween these. The present report servesthis
purposel.
As a practical rule of thumb (often neededin the Workshop's discussions),which
will be derivedin this report, among others,is the following:
Ir
you
s E E s o M E B o D y p R E S E N T I N GR E S U L T s o F c I t I R A L M E D I A , , l . N o s t t o ( n e )
TALKs ABour
c H I R A L I T v A D M T T T A N o E S( € " ) , A N D y o u
wISH To KNow rHE Dr-
M E N S I O N L E S SC I I I R A L I T Y V A L U E O F T H I S P I E C E O F M A T E R I A L , M U L T I P L E T H E A D M I T T A N C E F I G U R E B Y T H E F R E E S P A C EI M P E D A N C E , 1 O HAvE THE CHIRALITY PARAMETER r.
terials where p =
-
3 7 7 Q , N T T OY O U W I L L
This is approximate, and exact only for ma-
po, but the erperimental results so far show permeabilities not much
diferent from ps. Should you desire more accurate transformation formulae, please keep on
reading
:-s
The basic relations
Most of the presentationsgivenin the presentBz-uotropics'93workshopspeakabout
bi-isotropic media using the languageand terms of the following constitutive relarMuch of the material to
follow is common to that in [1].
33
tions:
D=eE +(x- jn)J-p,oe&
(1)
B:pE+(x+jK)JpoeoE
(2)
The relations give the effect of the electric (E) and magnetic (,F) field strengths
on the electric (D) and magnetic (B) flux densities.The effectscan be seento be
isotropic (i.e. the direction of the fields does not matter), becausethe coefficients
representing the linear relation between these are scalars. These parameters ate,
in addition to the normally encounteredpermittivity e and permeability pr of the
material, those that describethe magnetoelectriccoupling. The degreeof chirality
is contained in rc, a dimensionless(Pasteur) parameter, and 1 is the (also dimensionless)Tellegenparameter measuringthe nonreciprocity of this generalbiisotropic
material. €s and ps are the permittivity and permeability of the vacuum. For 1 - 0,
the medium is (nonreciprocal) chiral, and it can also be termed Pasteur medium.
The case when the chirality vanishes,r : 0, representsa nonreciprocal medium,
that can be called Tellegenmedium, becausethe form the constitutive relations that
Tellegenfirst proposed for a nonreciprocalmedium was of the form of (1), (2) with
rc:0.
It is worth noting that these relations implicitely assume sinusoidal time dependence (followed with the convention exp(ja.,t) in this paper), meaning that the
parameters are dispersive. The time dependencecomesthrough the inverse Fourier
transform, and in order to render real electromagneticfields (to representa physical
quantity), the parametershave to be real functions of jo. On the other hand, the
four material parametersof a losslessmedium haveto be real [2],leadingto the conclusion that e, ptryhave to be even functions of r.r, and rc an odd function. Hence,
there is no chirality in electro- or magnetostatics.
In the researchof chiral media, there are many different constitutive relations
in use (see,for example [3]) due to the possibilitiesin connectingthe electric and
magnetic quantities with another. In the following, the parameters in different sets
of constitutive relations are related with each other, This will help translate electromagneticresults derivedin one system to the other system. The most common
chirality representationswill be treated and their extensions to the nonreciprocal
chiral regime, i.e., the chiral constitutive relations that are consideredwill be generalized for bi-isotropic media.
The problem will be that in different systems,the same name has been given to
parameters that are not absolutely the same, for example permittivity and permeability. Therefore bookkeepingis essentialwhich system's permittivity is considered
as one speaks about permittivity. In the following, il relations are referred against
34
the constitutive relations (1), (2) and this permittivity is plain e and permeability
plain p in the following analysis. The correspondingparameters in other systems
are denoted by subindices.
Post relations
The Post [4] set of constitutive relations for chiral media, also derived phenomenologically by Jaggard, Mickelson, and Papas [5], is the following
D:eerE-ie"B
_t
H:
-
(3)
B-ie"E
(4)
Ppt
where the subindicesin e and /.r,now refer to the system, and the chirality comes
forth through the chirality admittance {". To completethe relations in order to cover
generalbi-isotropic media, a fourth scalarparameter describingthe nonreciprocity is
needed. In harmony with the chirality admittance (", a nonreciprocity susceptance
ry'^is here suggested,which also has the dimension of amperes/volt. The relations,
after the inclusion of nonreciprocity, look after that like
D:err0+rlr^B-je,B
_1
H-_B_rb^E_je"E
(5)
(6)
Uot
The connectionsbetweenthe parametersin (5), (6) and (1), (2) can be worked
out and are the following
Foeo, t
€pt=(-::::(y'+x')
p
a
Pps=P
(7)
(8)
'
JP""
9n= -X
Lt
(e)
'
-Jlt'eo
lc
lc =
lL
(1 0 )
and, the other way
€:€rr+ttrtb!'"+€l)
35
(i1)
F:
( 1 2)
Ppt
X: trr'{^lJPoeo
(1 3 )
n: p'r,LfuQseo
(14)
Condon-Tellegen relations
The constitutive relations of chiral media that have receivedtheir label after Condon
[6],look like the following, permitting an explicit time dependencein the fields:
D:e"E-X"T
(1 5 )
Ap
B:p"E+X.fr
(1 6 )
with the parameter 1" (now not a nonreciprocity parameter) measuring the material's magnitude of chirality (dimension sec2/meters). This is a reciprocal chiral
material.
On the other hand, a material that is not chiral but is nonreciprocal was
suggested by Tellegen [7] to obey the following constitutive relations
D=rrE*rE
(1 7 )
B:pril*tE
(18)
where 7r, dimensionally sec/meters, measuresthe nonreciprocity of the material.
These material relations can be combined for a set of bi-isotropic constitutive relations:
D:
e.,E *j.,8
- x.,*
B : rr..E* -t",E
+ x.,#
(19)
(20)
For time-harmonic field dependence,the connectionof these material parameters
t o t h o s ei n ( l ) , ( 2 ) i s
€ct=€
36
( 2 1)
(22)
Fcr = lt
7", :
y.r:
4
Drude-Born-Fedorov
(23)
Y1/1t'seo
(24)
n1/ffif u
relations
The relations emphasizingthe nonlocal characterof a chiral medium in its interaction
with the electromagneticfield,
E)
(25)
B: po"*(E
+ Bv x E)
(26)
D = eo"*(E* 0Y x
have been in much use by Lakhtakia, Varadan, and Varadan, and termed by them [8]
after Drude, Born, and Fedorov. (An advantagein these relations is that these are
not restricted to Fourier space,and the fact that from these, it can be directly seen
that chirality vanishesfor electro- and magnetostatics.) Here the chirality parameter
B has the dimension of length. One way of incorporating nonreciprocity in these
relations would be through the Tellegenparameter:
D:eo",(E+pVxE)+rE
(27)
B : p o " , ( E+ p v x E 1 +7 E
(28)
However, as often only time-harmonic field dependenciesare consideredin electromagnetic applications, it might be justified to tolerate complex quantities and
the appearanceof the imaginary unit j in these relations (like happens in relations
(1), (2) and (5), (6)). This suggestscompletingthe bi-isotropic DBF relations in
the form
D=eo",lE+(9+ja)vxE)
( 2e)
B=po".LE+(p-jo)Vx//l
( 3 0)
where now the nonreciprocity para-eter a has the same dimension of length as the
chirality parameter p.
37
The conversionof the material parametersbetweenthe sets (1), (2) and (29),
(30) is, for time-harmonicfields
r
po€o'l
€ o g r = e l l _ A ' l n 2 ' 1,- ,
TLE)
L
(31)
f.
, )
z'Poeol
posp:tLt-{X"+x'l;)
( 3 2)
o=
^
P:
xJtt'oeolu
rr 1r4 *'7r*
n Jltseolu
p€-A?'+K2)ttoeo
( 3 3)
(34)
and, inversely,
€oop
(35)
f f i
r-K;BF\a'+p')
p:
Itoar
t-kf;",(a2+02)
ulto"reo"raf
X_
K:
,Foeo
r-k2o"r@2+82)
upo"r€o"*9lJW"
t-kf;",(a2+92)
(36)
(37)
(38)
with
k'or* : a2 P'o"reo",
( 3e)
Discussion
A generalbi-isotropic medium requiresfour scalarmaterial parametersin its characterization, In this presentation,insteadof connectingthe electromagneticquantities
in a classicalway, through a matrix [9]
/D\
\B/
(,1\/E\
=\p
p)\H)
(40)
the effects (electric and magnetic polarization,
chiralityand reciprocity)are separated in the constitutiverelations( i ) , ( 2 ) :
(3)=( ,, * ,ilu',n
38
a- i4'/r';a ( t \
) E)
(41)
For a lossy material, all these four parameters can be complex for time-harmonic
excitation of the fields.
In the other systemsof constitutive relations that have been briefly discussedin
this communication, the parametershave been expressedas functions of the material
parametersof (1), (2). It is worth noting that in all these transation formulas, always
the permittivity, permeability, and nonreciprocity are even functions of chirality
(given in one of the other systems). This means that these three parameters are
equal for a material and its mirror image (which is a medium possessingr with a
changein sign with respect to the original medium). Also rc is an odd function of
a.ll chiralities of the other notations, which also agreeswith intuition.
Not so intuitive is that also nonreciprocity behavesin a similar way, a fact that
is reflected by the similar position of 1 compared to rc in the expressionsabove. In
other words, also the permittivity, permeability, and chirality are even functions of
nonreciprocity. And again in the translation formulae, nonreciprocity is itself always
an odd function of the nonreciprocity parameter of the other notations.
References
A.H. Sihvola, and I.V. Lindell: Bi-isotropic constitutive relations, Micrcwaue and
Optical TechnologyLetters, Vol.4, No. 8, p. 295-297,1991.
J.A. Kong: Electromagnetic wate theory, Joh.n Wiley & Sons, New York, 1986.
A, Lakhtakia, V.K. Varadan, and V.V. Varadan: Time-harmonic electromagnetic
fiekls in chiml media, Lecture Notes in Physics, 335, Springer-Verlag, Berlin 1989.
E.J. Post: Forrnol stntctutE of electrcmagnelics,North-Holland Publishing Company,
Arruterdam, 1962.
tDl D.L. Jaggard, A.R. Miclelson, rnd C.H. Papas: On electromagnetic waves in chiral
media, Applied Physics, Vol. 18, p. 211-216, 1979.
[6] E.U. Condon: Theories of optical rotatory porper, Reriews of Modern P/rysics, Vol.
9 , p . 4 3 2 - 4 5 7 ,1 9 3 7 .
B.D.F. Tellegen:The gyrator, a new electricnetwork element, PhilipsResearchReporls,Vol. 3, No. 2, p. 81-101,1948.
[8] A. takhtakia, V.K. Varadan, and V.V. Varadan: Dilute random distribution of small
chiral spheres,Applied,Optics, Vol.29, No. 25, p. 3627-3632,I September 1990.
[9] J.C. Monzon: R.adiation and scattering in homogeneous general biisotropic regions,
IEEE thnsactions on Antennos and Prcpagatioa, Vol. 38, No. 2,p,227-235,L990.
39
THEORY OF GUIDED WAVE IN CHIRAL MEDIA :
A STEP FOR MEASURING CHIRAL MATERIAL PARAMETERS
Fr6ddric Mariotte
CEA-CESTA
ServiceSystdme
BPNo2
33 I 14 l,,eBarp, France.
The constitutiverelationsof isotropiclossychiral compositeconsistingof
chiral objectsfor time-harmonicfields exp(-irot)havethe form D = sp + ii.B and
H = i E . E + B / p . w h e r ee . = € o ( t ' + i e " ) ,F c = [ r o ( F ' + i g " ) a n dE . = E 6+ i l i
representcomplex perminivity and permeability,and chirality admittance,which is
of the medium.Thesematerialspossesstwo bulk
a measureof the handedness
eigenmodesof propagation,a right circularly polarized(RCP) and a left circularly
polarized(LCP) plane rvavewith two differing complex wavenumbers.Recentll',
the guided wave propagationin chiral media has been the subjectof intense
research.Among thosestudieswe can mention the introductionby P. Peletand N.
Engheta llf of chirowaveguideswhich consist of cylindrical wave-guiding
structuresfilled with isotropiclosslesschiral materials.
In this talk, we review all the resultsof our recent theoreticalstudy on
parallel-platewaveguidepartially filled with isotropicchiral materials(loss or
lossless)[24]. First, we presentan overviewof the theory of chirowaveguides.
Secondly, we analysethe study of the canonical problem of reflection and
transmissionof guidedmodesat an air-chiral interfaceand at a losslesschiral slab
transverselylocatedin a parallel-platewaveguide.In thosetwo problems,the chiral
materials were assumedto be lossless;now we theoreticallystudy the effect of
lossy chiral materialson guidedmodesin suchwaveguidesand furthermorewe
analyze,first, the reflectionand transmissionof guidedmodesat a lossychiral slab
(case a) and secondlythe reflectionof a lossy chiral slab backedby a perfect
conductor(caseb).The motivationbehindthis study is the potentialapplicationof
this problem in the design of novel measurementtechniquesfor determining
of lossychiral composites.
materialcomplex parameters
So we have studiedthe effect of chirality and then the influenceof lossy
materialsin all theseproblems.The resultsare the following. For our analysison
reflection and transmissionof guided electromagneticwaves at an achiral-chiral
(lossless)interfacein a parallel-platewaveguide,it is found that in order to satisfy
4I
theboundaryconditionsattheachiral.chiralinterface,thereflectedandtransmitted
we first presentthe restrltsof our
wavesneedto be hybrrd' For lossy materials'
analysisonthedispersionrelations,Brillouindiagrams,cut_offfreqtrencies.
the modes lt must be notedthat for
propagationand attenuationcoefficientsof
lossychirowaveguide,thedispersioncun'esapproachthelinelqorkaccordingto
w e h a v e o b t a i n e dt h e r e f l e c t e da n d
t h e c h i r a t p a r a m e t e rssc a n d U c ' S e c o n d l y '
transmittedmodes(onlycaseb)andhaveevaluatedtheeffectofchiralityandthe
lossofthemedrumonpowerofreflectedandtransmittedwa\'es.Finallywe
in material
for the potentialapplications
addressrulesto solvethe inverseproblem
characterizations.Electromagneticshieldinganddesignofdevicesandcomponents
studies'
could be also potentialapplicationsof these
MicrorvaveChiralityResearchat
In this talk, we presentalsobriefly other
CEA-CESTA:First,modellingofheterogeneouschiralmaterialsbythecalculation
secondlythe design
scatteringof a chiralelement(thin helix) and
of electromagnetrc
[55]'
andfree spacemeasurements)
of chiral shields(preparationof chiral samples
References
tll
l2l
t3l
IEEE Trans'
P. Peletand N. Engheta,'TheTheory of Chirowaveguides"'
1990'
AntennasctndPropagarion,vol'38' pp 90-98'
and transmissionof guided
F. Mariotte and N' Engheta.''Reflection
electromagnetlcwavesatanair.chiralinterfaceandatachiralslabina
publication in IEEE Trans'
parallel-plate waveguide", acceptedfor
in 1993'
Microwave Theorv and technique't'to aPpear
material loss on guided
F. Marrotte and N' Engheta"Effect of
w a v e g u i d e " 'a c c e p t e df o r
e l e c t r o m a g n e t im
c o d e si n a p a r a l l e l - p l a t e
and Applications'lo
publicationin Journal of ElectromagneticWoves
aPPearin 1993.
l4l
lrom a lossychiral slab (with or
F. Mariotte and N. Engheta,'Reflection
submittedto Radkl
without metallicbacking)in a parallel-platewaveguide'',
Science,1992.
151
16l
f o r p u b l i c a t i o ni n R e v u e
F . M a r i o t t e , " C h i r a l i t e6 t f u r t i v i t 6 " ' a c c e p t e d
in 1993'
Scienrifqueer Techniquede kt Dlfense' to appear
"On the relationships
S. Zoudhi' A. Fourrier L:mer and F' Mariotte'
betweentheconstitutivesparametersofchira|materialsanddimensionsof
chiralobjects(helix)',EuropeanJournalofPhvsicsIII'Vol.2'No.2,pp.
337-343,199242
Measuring electrical, magnetic, and chiral
material parameters
Arto Hujanen
State Technical ResearchCentre, Telecommunications Laboratorv
Otakaari 7 B,02LS0 Espoo, Finland
43
Problem
- unknown plane slab
- what arethe electricalmaterial
parameters
t,l-[,K
Measuringtraditional"nonchiral.
material
- two complexparameters
g,lt
- measuringmethods:
- resonatormethods
-eorp
- low lossymaterials
- reflectionI transmission
methods
- transmissionline methods
- free spacemethods
ssion methods
Reflection/Transmi
(o
>
- measurement
of the reflected
and transmittedfield
- measurements
aredonewith
well calibratednetworkanalyzer
46
Transmissionline method
sampleinside the waveguide
- preparattonof the sample is
difticult
- gapsbetweensampleand
waveguidemakeresults
inaccurate
47
Free spacemethods
- purposeto measurereflectedand
transmittedsignalsin free space
- no preparationof smallsamPles
- problem:diffractionfrom the
edgesof the slab
48
- focusingantennas
- no edgediffractions
-calibrationwith TRl-method
- time domaingatingto avoid
multiblereflections
/,o
yorivii antenni
Liukukisko
fL
LZ
50
Calculatingmaterialparameters
trom measuredS-parameters
P
- measuredS11- and 521parameters
t..
2+
- e,s,s".'
ull
)1
1
2St,
r - K+/61
.r.
S ,, * S r , - f
1- (S,,+ Sr,)f
i
k - L(ln(7)+n2n)
'
rL L -= \0.
J. +
_ ^ .I- +2....
;e....
d
(.r
cr'
F,
l-f
k
1+f ko
l+f
k
1-I- ko
C 1 2 N 3 4 R1 2 O c t
----x- Rc ( 51 1 )
---€- Ra(52 1)
Kuva
2.1
tS98:
H 4z I HM T- t .21-t1
FREQUE|{CY
--{im( 51 1 )
- - - - : t - ! r n ( S 21 )
perusaineelle
Esinerkki
nitatuista
paksuus
(hiilen
osuus 4{;, naytteen
C 1 2 N 3 4 R1 2 O c t t 9 9 8 :
( GH: )
S-pa:aner-reisti
l-.2m) .
H 4 z t ,H H T = t . 2 H i
,\l
.\: ,rl
'r'')".'l
I
l. (€,
.88
---+--.F
R c( E P S )
R e( H Y )
-iF
-<-
f.i
.24
t4.88
E-;rnr
|nrln
ls-
r 8 .e B
lrn(EPS)
!rn( HY)
Kuva 2.2 Mitatuista
S-parar.retreista
. rnalla lasketut
suhteel]iset
- ja
permiabiliteettik;iyrdt
52
(kuva 2.1) EpsMy-ohjeI
p e r m i t t i i v i s y-y- . s _
(6.)
tp-l.
Measuringthe chiral material
- three complex parameterst,l,l
and t< = more measurements
ateneeded
- polartzattonof the wave ts
changedwhen the wave goes
throught the chiral slab
- electricalfield behindthe slab
E, :
\
E,(n,+ tan(-kodK)u, )
- lets measurethe polartzatton
propertiesof the transmittedtield
53
- transmitterandreceiverantennas
linearly pol artzed
,-a- - sr",- S, cos(u)
\-,
AR sin( t) - 7cos( t)
v
--
AR cos(t)+7sin(t)
S, sin(u)
, , - -s , , t - . s . , t( l + G : ; + i
r\
-?" q
11
t<t^l K2-l
I
k"'d= i(ln(f) + n}n)
fl = 0, +1, +2,...
54
--- i
c
(Jc
' c
r-t k.
1+f
ko
l+l
k"
l-l
k(,
Lt-:
K_
-(arctan(G) + mTt)
kod
rr ^nr - O
55
v)
1')
epsilon,myy ja kappa,[,evy 4, alfa=-5O
Suhteellinen
l1
Taajuus
/ GHz
Re(er)
Taajuus
i,00E+10
1,10E+10
1,20E+
i0
3,67
i , 30 E +l 0
iq5
i,40E+10
I snF+ln
I 6ntr+lo
'l
1,70E+10
dnR
I enF+ln
4lR
I OOF+IO
A a1
aR
401
Im(er)
-0,23
-0,22
-0,22
-0,26
-0,32
-0,31
-0,37
-0,33
-0,33
- 0 , 39
Re(pr)
0,87
0,88
0,89
0,89
0,89
0 , 88
0,87
0,86
0,86
0 , 86
cml440
56
Im(pr)
0,00 \
0,00
0,00
0,00
0,00
0,00
0,00
0,01
0,01
0,02
Re(x)
0,38
0,29
0,23
0,20
0 ,1 9
0 ,1 7
0 ,1 6
0 ,1 5
0 ,1 4
0 ,l 3
Im(r)
0,03
0,02
0,03
0,03
0,02
0,02
0,02
0,01
-0,01
-0,02
ElectromagneticPropertiesof Losslessn-Turn Heiicesin the
Quasi-Stationary Approximation
LRArnautandLEDavis
Department of Electrical Engineering and Electronics
The Liniversity of L'lanchesterInstitute of Scienceand Technology
PO Box 88, It[anchesterM60 LQD, UI{
Introduction
The study of electromagnetic(EM) propertiesof bi-isotropic(BI) media at UMIST is comparatively new. It is the subject of the first author's PhD Project in Applied Biectronics.
The project aims to be a '3M' approach:t mod.elling
of chiral and BI media: this includesthe fundamentalstudy of the physical
propertiesof generalBI media, as well as the propertiesof specificchiral structures
that mav be implemented at microwave frequencies;
c manufacturingol
chiral and BI media: advances in the tailoring and practical imple-
mentation of suspended chiral microstructures are sought;
. measurement of chiral and BI media at microwave frequencies: because of the com
piexity of the EM effects, new measurement techniques are considered.
Advances in all three areas have been made in the recent past. In this presentation, u'e
wiil discussone aspect,ie the modelling of practical chiral structures. The BN{ properties
of an n-turn helix will be derived in the quasi-stationaryapproximation,its performance
in terms of ENI activity wiil be predictedand comparedu'ith publisheddata.
2
Field Induced bv an n-Turn Helix
Chiral propertiesofideal, short-rvirehelices,shownin Fig 1, have been derivedby Jaggard
el a/. The helical structure under considerationhere is sketchedin Fig 2. The spring has
n turns, free length 2/, external diameter 2c, gauge diameter 26 and pitch p : ?1. It is
made of perfectiy conductingrvire and embeddedin a losslessmedium. It is assumedthat
2l )
2a so that fringing of the magneticfield ma1'be neglected.
57
An external plane wave,characterizedby {E*, lL}, it incident onto the heiix. After some
calculations,it is found that the inducedelectricfield Q is related to the induced current
I bv:-
E;
c(
c(
n
/
A1'\
n'\-ar)'
n a2 (
a;
0t;\
l-al/'
rla
(1)
rla
(2)
and is directed tangential to concentriccirclesaround the helical axis. Thus, the induced
electric field has a constant amplitude on concentriccircles around the helical axis, it
increaseslinearly with distanceinside the helix and decreaseru. t' outside the helix.
This inducedfield affectsthe fieid scatteredby the helix. In a compositemedium of u helix
vol pct springsembeddedin a dielectrichost, this causesthe interactionbetweenspringsto
be much larger than in a composite medium of the same vol pct u of nonchiral conducting
particles. This explainswhy the BNI propertiesof dielectricswith embeddedmicrosprings
changedrastically at a thresholdu : uo which is much lorverthan the classical10% limit
used in multiple scatteringtheory [2].
Chiral Admittance
and EM Rotation for an n-Turn
Helix
EM activitl' caused by chiral objects is basically an interaction between .E and H, ie a
coupling betrveen the capacitance Cn and inductance Ln of a collection of chiral objects.
In the quasi-static approximation,
the helices are ver)' small as seen by the incident wave,
hence the wave impedance of the chiral medium mav then be calculated from the lumped
impedances of the helices.
Compared to the idealized short helix model. inclination of the windings as well a,sinteraction between them needs to be accounted for in the model. This results in an expression
for the chiral admittance of a sinsle round-wire helix:-
f
-
t7
2l (ra)2
I_
u p( " - r ) r . s (
Zi
;M-')
(3)
.r\
in which e, p denote the permittivity and permeabilityof thc host medium, respectively.
\\Iith appropriateaveragingover the (random) orientationof the helicesand their concentration -lV, this allows the resonancefrequencl' I
5B
as rvell as the chiral admittance (. at
resonanceto be calculated [1]. The results are iisted in Table 1, in whicii the computed
EM rotation is compared with previously measureddata. The subscripts 1 and If refer to
a singlehelix and N helices/m3,respectively.
Qumtity
Helix Concentration N
L.
Unit
Ref 2
Ref 3
2.5 (at 1 0 G H z )
2.95 (at 7 GHz)
au7*'
139.8
36.7
vol To
3.2
0.8
nH/m
6.06
fF/m
93.7
89.4
I.=:]-
CHz
7.13
6.84
( € " )r
p S m3
9.13
9.43
@t(J = l.))r (computed)
p de8
.o74
.100
({")N
ps m3
(dr(/ = J"))w (computed)
(6r(l = I.))N (mecued) (approx)
12.88
deg
425.4
42.27
NA
d.g
Table 1: Comparison between Predicted and Measured Data
Given the fact that the information on the host material and helices is insufficient for
accuratecalculationsat /" (and, therefore,dr), and taking into accountthat the computed
vaiuesfor Q1serve as a lower limit due to multiple scattering,it may be concludedthat
the model is sufficientlyaccurateto provide a good estimate of the EM rotation by the
sample.
Finally, Fig 3 showsthe dependenceof the EN{ activity of a singlehelix on its dimensions,
rvhich have been normalizedwith respectto the helix length.
References
1 Jaggard D L. N'IickelsonA R, Papas C ll; On Electromagnetic Waues in Chiral lIedia; Appl
P h y s v o l 1 8 ( 1 9 7 9 ) ,p p 2 I L - 2 1 6
2 Guire T, Varadan V V and \"aradan Y K;EIJect ol Chimlity on Refection of Electromagnetic
V'a"*esby Planar Dielectric .9lcDs;Trans IEED-EIVIC vol 32 no 4 (Nov 1990), pp 300-303
3 Hollinger R D, Varadan V V and \/aradan Y K; Etperimental chcructerization ol isotropic
chiral compositesin circular war:eguides:
Radio Sciencevol 25 no 2 (Mar-Apr 1992), pp 161-168
59
2b
r3
rli
2!
r-
<i
I!
-l
N
-l
?
)l
/r'/;',( ) )
\\ \v /
n io-
a^
Figure 2
Figure1
LoglC
Single
(Resonance
r-Turn
HelLx
(GHz) )
Frequency
(21
=
1Om;
2bl2I
LoglO (chlral
for
=
slngl.e
(s n3) ) for
21 = 10m
Adnlttance
5-Turn
Heltx;
'01)
= 1
spaclng
contour
-l
-1
e
ts.
o
-2.
!
E
o
E
-3.
o
q
LogiC (Nomallzeo
Ilellx
Dlane!er
(-) )
Ad'nLttance (s h3) ) for
= '7' 2b/2'r =
Single n-Turn HeLjx;24l21
Logio
Loglo
i , o g 1 0{ C h l r a l
'AAC)
Single
{NortraIlzed
{chtral
n-Turn
Dlaneter
HeIlx
(s
Admlllance
tselLxi
2a/2i
=
'1'
E
F
n
E
////////t
Con:o!.
E
"--.-]3
I
E
-i
Conlour
E
SPacito _ l
ffi:j
c
Figure3
60
spacing
= 2
-2
n3) )
2b/21
(-)
)
fo!
=
0001
Laboratory
Repor! 128, Helsinki
Electromagnelics
- a w a iI n h l c o n r e n r r e s t f r o m the authors
University
of
Technology
Plane wave propagation in a uniaxial
bianisotropic medium
Ismo V. Lindell and Ari J. Viitanen
Report 128
November1992
Abstract
Uniaxial bianisotropic medium is a generalizationof the bi-isotropic and chiral media
which recently have been subject to intensive research. Such a medium results, for
example, when microscopicheliceswiih parallel axesare positioned in a host dielectric
in random locations. Plane wave propagationin sucha medium is studied and a simple
solution for the dispersionequation is found. Numerical examplesfor the wave number
surfacesof the medium are displaved.
I S B N9 5 1 - 2 2 - 1 3 0 3 - 6
ISSN0784-848X
TKK OFPSET
61
Laboratory
Electromagnetics
- available
on request
from
Report 132,
the authors
Helsinki
University
of
Technology
Plane wave propagation in a uniaxial
bianisotropic medium with an
application to a polarizatuorr
transformer
Ari J. Viitanen and Ismo V. Lindell
Report 132
January 1993
Abstract
Uniaxial bianisotropic mefium is a generalizationof the well-studied bi-isotropic and
chiral rnedia. It is obtained, for example, when microscopicheliceswith parallel axes
are positioned in a host dielectric in random locations. Plane wave propagation in
such a medium is studied and a simple solution for the dispersionequation and for the
eigenwavesare found. As a numerical glamplel polarization properties of a transverse
wave propagating in a uniaxial bianisotropic medium is considered.The results give a
simple possibility to construct a polarization transformer with a transverselyuniaxial
chiral medium for changing the polarization of a propagating plane wave.
I S B N9 5 1 - 2 2 - 1 3 8 7 - 7
ISSN0784-848X
TKK OFFSET
63
Report 133, Helsinki
Laboratory
Electromagnetics
- available
frorn the authors
on request
University
of
Technology
Plane-wavepropagation in a
transversely bianisotropic uniaxial
medium
Ismo V. Lindell, Ari J. Viitanen, Piiivi K. Koivisto
Report 133
January 1993
Abstract
Transverselybianisotropic uniaxial medium consideredin the presentpaper can be obtained, e.8., by mixing metal heliceswith an isotropic basemedium in such a way that
the axes of the helices are randomly oriented but perpendicular to a fixed direction in
space. The medium is a generalizationof the well-studied chiral medium and somewhat similar to the recently stufied *i.lly bianisotropic uniaxial medium, which has
interesting polarization properties. Plane wave propagation in the medium is studied
and the solution for the dispersionequation is given. Numerical examplesfor the wave
numbers correspondingto the two eigenwavesof the medium are displayed. It is seen
that, unlike in the axially bianisotropic uniaxial medium, there are no optical axes in
the present medium, in general.
ISBN951-22-1393-1
ISSN0784-848X
TKK OFFSET
65
Novel uniaxial bianisotropic materials
SergeiA. Thetyakov,Alexander A. Sochava
Radiophysics Department
St. Petersburg State Technical University
195251,Polytekhnicheskaya29, St. Petersburg,Russia
Recently, a novel concept of artificial composite materials with O-shaped metal
elementswas introduced in [1]. The idea originated from wide and intensive studies
of isotropic chiral media and their electromagneticproperties in microwave regime.
A typical chiral inclusion - a small wire helix - can be consideredas an electric
dipole connectedwith a magnetic dipole in such a fashion that high-frequencyelectric f.eld induces a magnetic field parallel to the original electric field component,
and uice uersa. This causesoptical or microwave activity - rotation of the polarization plane of a linearly polarized propagating wave. Such chiral structures most
effectively interact with circularly polarized waves, since the right and left circular
polarizations are eigenpolarizationsof wavesin unbounded biisotropic media. However, one may suggestother geometrical configurations which can provide stronger
wave-material interaction in other circumstanceswe deal with in microwave engineering. One of such modifications was introduced in [1], where it was suggestedto
use particles shaped like the capital Greek letter O instead of chiral helical particles
to ensure first-order efect on the propagation factor in a partially filled rectangular
waveguide. The material can have other interesting applications [2]. In a regular
microstructure with O-shapedconductiveinclusions,there exists additional interaction between electric and magnetic fields which lay in orthogonal planes, and the
material can be modelledby bianisotropicconstitutiveequations.
Eere we advance an idea of another modification of such microstructure configurations which can be better suited for use in plane non-reflecting coverings and
antenna radomes.
We focus the analysis on plane screenswhich are designedto interact with linearly polarized electromagneticwavesand suggesta modification which can provide
uniform operation for linearly polarized waveswith any electric field direction (or for
67
unpolarized plane waves). This can be achievedby introducing a secondensembleof
O- particlesinsidethe matrix. As a result,the structurewill interact more effectively
with linearly polarized wavesof any polarization direction or with unpolarized plane
lraves, Such a medium can be named as uniaxial omega-medium,becausethere exists only one particular direction - that one normal to the interfacesof a layer. The
size of O-shapedelementsis assumedto be smallerthan the wavelength,hencethe
medium can be describedby effective averagedmaterial parameters and the material is modelled by uniaxial bianisotropic constitutive relations which couple electric
and magnetic fields.
In the report we developgeneraltheory of plane wave propagation in novel media
and study reflection and transmissionin plane uniaxial bianisotropic layered structures. As an example interesting for applications, we consider in detail reflection
from a plane metal surface coveredwith a lossy layer. The example demonstrates
that with the additional O-shapedwire elementsabsorption can be enhancedin
a wide frequency range. The additional material parameter can help to manage
properties of anti-reflection coatings,in a way similar to the effect provided by the
chirality parameter of biisotropic materials. Another example is the reflection and
transmissionthrough a plane slab. Here it is seenthat the material is perspective
for potential use in antenna radomes since a nearly transparent and non-reflecting
covering can be designed.Also, lossy slabs can be designedto serveas non-reflecting
absorbers.
It appears that the input impedance of a lossy layer on an ideally conducting
surface or in free space can be matched with the free-spacewave impedance by
tuning the additional coupling parameter. The impedancematching condition is
frequency-independent(provided the material parameter values can be assumedto
be constants) and that suggestssuperwide frequency band for prospective antireflection coveringsand antenna radomes.
Designingthe material, one can compromisebetweenthe coupling parameter
value, the lossesand the slab thickness. If the impedancesmatch, the thickness
is essentialfor transmissionproperties,but not for the reflection. Comparing with
the requirements known for chiral low- reflection screens,it seemseasier to achieve
desiredeffectswith the O-composites,
becausein the chiral screensrather high degree
of chirality is required.
68
References
[1] Saadoun,M.M.L, and N. Engheta,"A reciprocalphaseshifter using novelpseudochiral or O medium", Microwaueand Opt. Tech.Lett., Vol.5, 184-188,1992.
[2] Tretyakov, S.A., "Thin pseudochirallayers: approximate boundary conditions
and potential applications", Microwaueand Opt. Tech.Lett., Vol. 6, 112-115,
1993.
69
Macroscopicpredictions of material
parameters for heterogeneousbi media
Ari H. Sihvola
Helsinki university of rechnology, Electromagnetics Laboratory
Otakaari b A, 021b0Espoo, Finland
In electromagnetics,we often treat materials as homogeneous,
i.e. their material
parameter functions are constantswith respect to space.
These material parameters
are macroscopic descriptions that cover all polarization and
loss phenomena that
take place on the microscopic physical level, and with
certain restrictions, these
quantities are sufficient in solving electromagneticwave problems
involving maiter.
rlowever, strictly taken, no media are homogeneous,at least
as one rooks at
them in sufficiently small scale. There are material boundaries,
grain walls, etc.
Media are random. Especially, as one tries to ,,seer'(and
seeingis measuring the
reflection or emission of electromagneticwaves) how
the medium looks like, using
a wavelength equal to the correlation length of the spatial
permittivity function of
the medium, the sceneryis kaleidoscopic:the wave behaves
totally difierently than
in a homogeneousmedium; it does not propagate along
straight lines as the opticar
wave in air but it scatters strongly, it is reflected and
refracted.
The macroscopic properties of dielectrically heterogeneous
media have been a
researchtopic for physicistsfor over a century, and studies
have resulted in a wealth
of mixing theories. But how about the specialguests
of this workshop: ,,exotic,,
materials, chiral and bi-isotropic media?
Compared to classicaldielectricor magnetic materials,
it is intuitively clearer
that chiral media are heterogeneous.
The conceptof chirality is so much connected
with geometry,the handednessthat is presentat
somescalewithin the microstructure of the rnaterial, that the first thought as one
seesa sample of chiral material
is to try to find the helical elements responsible
for the chirality. An inevitable
consequenceof the geometric structure is dielectric
inhomogeneity in the medium.
Much efrort in the chiroelectromagneticresearchof these
yearsis being expended
7I
on the electromagneticmodelling of a canonical chiral structure, the metal helix.
There are studies of what kind of current distribution will be induced on the helix
as a plane wave is incident upon it. This is not an easy problem.
In the presenttalk, however,I will not considerthe emergenceof chirality through
geometry. Rather, in a way of "analytically continuing" the classicaldielectric mixture rules and equations,I shall discusshow, given the chirality parametersi of the
constituentsof the mixture, to predict the macroscopicchirality (and also permittivity and permeability) of the whole. In other words, in the mixture under study
there are inclusionsthat are assumedto be of homogeneousbi-isotropicmaterial,
and thesedeterminethe larger-scaleparameters.
In this ana.lysis,
the starting point is to find the polarizabilitiesof simple-shaped
bi-isotropicparticles.Due to the magnetoelectriccouplingin the complexmedium,
a four-elementpolarizabilitymatrix, including cross-polarizability
terms, is needed.
After knowing this, the classical Maxwell-Garnett and other mixing rules can be
generalizedto cover bi-isotropicmedia. In appearance,theseexpand and inflate in
this process,and the formulas becomecoupled: for example, the permittivity of one
inclusion affects the nonreciprocity, chirality, and permeability of the mixture.
The results shown in the presentationare collectedfrom my studies on biisotropic mixtures, and many illustrationshave been publishedbefore. Most results
c a n b e f o u n di n t h e r e f e r e n c e[ 1
s,2,3,4,5].
some general conclusions-
The salient featuresof the mixing processcan be
seenalreadyfrom the simplesttwo-componentmixture: bi-isotropicspheresembedded in isotropic background medium. The coupling of all parameters was already
mentioned: one macroscopicparameteris a function of all six material parameter
quantities of the problem. Secondly,there exist wonderful dualities in the macroscopicmaterial formula expressions.For example,the way the permeability of the
inclusion affects the effective permeability, chirality, nonreciprocity, and permittivity, is the same as the way the permittivity of the inclusion affects the effective permittivity, chirality, nonreciprocity, and permeabiiity. AIso, the functional
dependenceof the macroscopicpermittivity2 on the inclusion chirality is exactly
the same as on the inclusion nonreciprocity.Put into a more nonredundantform:
€ ^ ( e ; , p ; , r c i r X i i € rp, n ) :
e ^ ( q , p ; , X ; t K ; l e n , p n ) . F u r t h e r ;t h e m a c r o s c o p i c h i r a l i t y
dependson inclusion nonreciprocity identically with macroscopicnonreciprocity derend also nonreciprocity (fully
biisotropic) parameters
2The same applies
for the macroscopic permeability.
72
pendenceon inclusion chirality.
An essential observation is the fact that the effective (:macroscopic) permittivity and permeability of a mixture are even functions of both the chirality and
nonreciprocity of the component material. Hence, firstly, the sign of handedness,
i.e. whether left or right handed, should not have effect on these parameters, and
they are true scalars, invariant of spatial inversion. This is obvious: samples of
media that are mirror images of one another should have the same permittivity and
the same permeability. But such is also the dependenceon the sign of the inclusion
noreciprocity parameter, although the intuitive support for this is not as evident.
Also the effective chirality is an odd function of the chirality of the inclusion material (and a pseudoscalar):changing the handednessof the component changesthe
handednessof the mixture. (And again, a similar conclusion holds for the effective
nonreciprocity.)
Furthermore, due to the fact that these functional dependencesare even, pe!turbation expansionsstart with the second-powerterm, and therefore the effect for
small chiralities (and nonreciprocities)is small. In other words, the chirality of the
inclusion phase has little effect on the macroscopicpermittivity and permeability,
at least for high permittivity and permeability contrasts between the inclusion and
background phases. On the other hand, naturally the chirality of the inclusion is
the dominant parameter defining the effective chirality of the mixture. Finally, the
effectivechirality of a mixture is decreasedby high permittivity and/or permeability
of the inclusion phase.
How, then, about inclusions of other shapes?The only other forms of inclusions
whose polarizabilities can be solvedin closedform are ellipsoids. By using chiral inclusionsof ellipsoidal shapes,a further range of mixture parameters can be tailored.
The effects vary: by using needle-shapedor disk-shapedinclusions, larger effective
parameters can be achieved,although the shape-effectdependson the dielectric and
magnetic contrast between the inclusion and host phases. One observation is, however, always valid: sphericalinclusions produce minimum effectsin the macroscopic
Properties,and each deviation from this extremum shape always increasesthe value
achievedby the spherical geometry.
References
[1] A.H. Sihvola, I.V. Lindell: Chiral Maxwell-Garnett mixing formula. Electronics
73
L e t t e r s ,V o l . 2 6 , N o . 2 , p . 1 1 8 - 1 1 9 1
, 8 t h J a n u a r y1 9 9 0 .
[2] A. H. Sihvola, I. V. Lindell: Polarisability and mixing formula for chiral ellipsoids.E/eclronicsLetters,Vol. 26, No. 14, p. 1007-1009,Sth July 1990.
[3] I. V. Lindell, A. H. Sihvola: Quasi-staticanalysisof scatteringfrom a chiral
sphere. Journal of Electromagnetic Waaesand Applications, Vol. 4, No. 12, p.
1223-1231,1990.
[a] A. Sihvola,I.V. Lindell: Analysis on chiral mixtures. Journal of Electromagnetic
Waaesand Applications,Vol. 6, No. 5/6, p. 553-572,1992.
[5] A. H. Sihvola: Bi-isotropic mixtures. IEEE Transactions on Antennas ond
Propagation,Vol. 40, No. 2, p. 188-197,February 1992.
Covariant methods in the theory of
electromagneticwaves
Fedor I. Fedorov
BelorussianAcademy of Sciences,Institute of Physics
Skarvna Str. 70. 220602Minsk. Bvelarus
This presentationfocusedon the history and presentstate of crystal optics, starting from the basic works of Pockels(1906), Drude (1912), and Born (1932). The
different crystal types were classifiedaccording to the symmetry and other properties of the correspondingpolarizability matrices. Optical activity and gyrotropy in
crystals has received considerableattention starting from late 1950tsin Belorussia
and the former Soviet Union in general. The talk discussedin detail the historical development of the constitutive relations of optically active media, as the early
suggestionsby Drude and Born were shown to be inconsistent with the energy conservation principle. The material equations in the final form are nowadays labeled
often Drade-Born-Fedorourelations in the Western literature. Special emphasisin
the talk was given to the coordinate-freecovariant representationof quantities that
appear in the electromagneticanalysis of anisotropic and bianisotropic media.
(fron
the uotebook oJ the l{orkshop organizer)
75
Fundamental principles resticting the
macroscopicphenomenologicaldescription of
matter
Anatoly N. Serdyukov
Gomel State University Department of Physics
SovjetskayaStr. t04,246699 Gomel, Byelarus
77
Ttae- Reqrro/ Syaac/2y
ht'
f 0rt.,q". - et/"'"
Fo,Xo
xa
=zKoeFe
ti--zEi^
Koe= ?o?eKeo
vl.rc
7o
= *r/ or
7o=-/
ope.olti* f G -- l),
h
uqaeeltba
TFe=
,ro'// f,ae -r€tn
?o Q
/t/o cro su ^ia ( L in ar €?ee*ro // n a,nt ix
= D,eE+dU
@=[V),v")
t?l
+
=
(K*) = It;*-)
LP J a.r€ /r2r.-.t6
-t?
K,t
aa
W = ED* 48
TE=E,
*l ,
7Lr,s=
e ,r, € r,
K,, o.' Er3= t!
'2t
€.r --4
K,, K
4,, = -fu
f,=
d,z=-lzr
n
ot,
lB ---fst
=
"f4
o{=
e
-A
4tt = -f,7,
t
=
S Knt K^g(i
wl,e.c2,,(r,...a,z ctt. or iet f ora^2ttrs,tzlattc
Koe(4)= ?^7e
fto(Tq)
| {{,>=G,b}
I f g =g , T L = - L
=7e,-Di
e(e,D
$(e,D=
f G;h)
*k.h\=-E'€-n
Lledia
Q = €E *t4
,otE=-6
ro*! =/
I
f'=Ur+\
0 =FE */ 4
(D t =
u9,
t'= UeU
E= U€,
^/
p/=4q0
t-
,/= ltJltJa7
t=ltlluFil
Ir t/=S isama#''r ol slnaep!,a.n
vr
rql
1t =s/s l
rdelt'on of rgsvh/:
lSl S"cS
p= ls F
18 lnn 52 dasseg
(ose): 3 ,f 'llreruQhna(' e/) lon aa{\.a'*
F.Fyodoror
q
clasc
4nn
E=-d
(ei = 'ar,;)
n2
rP:
9n*,
9+-9,
f=' +
- ,C
!rP
a + - e- t
8+ 8.
C
= -+, O
g- 9, 9' -9, e + c
ls1=-t
lsl =-t
lSl'+'t
cl
:.1
(q.9-9's)
0{cgs
42^
4-
og= &ft(q.Q*
Boih'04 4 tll
6'q)
*-
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lo 4,t
0l
Io o
0)
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at ol f oler,sa/iort-
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4I"n t l=144g =lla,*9.!9
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il=!
:
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= kar" sin0'sin19'
fransfurnay'tVa a+d e,he*'ca( Srxqdr/
!tta/,9
osa'l(ofort
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t%ay'tl
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al
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7"!'=ry
70
f:
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28
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: --
t
m // / ' u -
ce1
t. 2
?
9e
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N'
l.s/H'=P'+i!
3
;
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ml =-,(1-yf
e/m
r=
+eE
jyz - a2 - t'rt)
2 =€ *4nil4
g"gr
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llt = /+
rl
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'
tL" -asz -itu
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, F4;
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A = ,PE *,,4!
llo=- 4+ lz'r'rj/u
ff^
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4n.{ eg/n
t'l&)
E(e,?)=
E6e,-g)
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*F,
4*-4
E
D*0,9--Q
a (e&) =- i(e,-g,
N
\,...N
F,/ - ai'r,
r ,' 9 / n y ' '
=
4= ea
lr
[/''D': P;
lJ'nQ'=
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t'il
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\,
+
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se
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j,.n*--reg !
t{r-,r4>+
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d /ta1ae/,'c F,e//
on Ae /h{q.a( orl,za/ lcrty,lF
D = eE * ,'dl\
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*2"=//,',tr-',/liir\
€(U = € + tlhx
*ft) = & +; f ht
d({)
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{1, o e g t
lv
tA r*
c
eA rx
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r=f
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SYrrq
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€'=h (,t4r'l4'l- Lss'l)
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I=-4:
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--->
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->
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f
Particular wavesin bi-isotropic media
Igor V. Semchenko
Gomel State University, Department of Physics
SovjetskayaStr. 704,246699 Gomel, Byelarus
85
PARTICALAR WAVESIN
BI-ISOTROPIC MEDIA
I.V. SEMCHENKO
Investigationof the possibility of existenceand featuresof propagationof particular
waves (longitudinaland helicoidal)in the magneto-activeplasmaand metalsin the magnetic
field are reported in [1,2]. It is demonstrated[3] that similar waves can also exist in natural
gyrotropic media within certain frequencyrangesclose to the absorptionband. The present
paper results from the study of propagationand excitation of the particular waves in biisotropic media having optical properties describablewith the help of phenomenological
material equations[4-7].
D=eE+Q+i$fr
E=p,E+(i-igE
(l)
Here t, p and o are tensors of permittivity, magnetic permeability and natural optical
activity, x is the tensordescribingthe mediumoptical non-reciprocal,the sign "tilde" (-)
designatestransposition. Since properties of a medium in material equations (1) are
characterizedby the tensorst, lL, e, x they are applicableto describethe optical behaviour
of anisotropicopticallyactivenon-mutualmedia.
From Maxwell's equationsfor flat waves [4]
ni'E=8,
rfi E=0,
ri'H =-D,
tn D=o
(2)
and materialequations(l) the wave equationfor the electricfield in thebi-isotropicmedium
follows as
lirttfi -riz *2i u fi.' +e v -x2 - u2i E =0
and its solutionare right-handed
and left-handed
circularpolarizedwaveswith the refraction
index
n,=Tv-:?*a
(3)
Here fr=ntis the refractionvector,iis a unit vectorspecifyingthe directionof propagation
of waves, f is an antisymmetrictensordual to the vector fr, ttre point betweenthe vectors
indicatestheir direct (double)producr.
The permittivity within field frequenciescloseto the mediumabsorptionbandsmay approach
zero making it possible for particular waves in bi-isotropic media to exist. When the
relationship
86
eq=x',
(4)
field frequencyc,r,due to the
is fulhlled, which is true for a certainelectromagnetic
it followsfrom formula(3) that l+ : d'
of the mediumparameters,
frequencydispersion
n : - o. In this caseonly a singlecircularlypolarizedwavewith the refractionindex
in the bi-isotropicmedium:
n J lo I canpropagate
(5)
.u,
fr.ii
E,=8 :e*pl,(;,a
id r--co,r)l
Here?and f,are singlevectorsformingthe right trio with the vectorof the wavenormalf,
to the parametricsign d at the frequencyc,rr.
and the circular polarizationsign corresponds
as
(1) and (2) the magneticfield intensityvectorH is expressed
From e4uations
-l
H=:-lrft'E-(x-ia)El
p
Using it and relationship(a) for wave (5) it is obtained
-l
H ' p=-:- xE
E , = . i a E ,'
-[
D -- = - i s ^ l : E
\p
So, for the above wave it is obtainedthat H 10, still Umov-Pointing'svector transformsinto
zero and the wave doesnot convey energy. Identical, the so-calledspiral, waves can appear
also in the magneto-activepiasma [], yet they can propagateonly in the direction of an
external masnetic field. Now consider the condition
(6)
l"1r-f=o,
field closeto the
which can occur within a certainfrequencyc,r,of the electromagnetic
(6) is fulfilledrefractionindexes(3) acquirethe
band.Whenrelationship
mediumabsorption
form
(7)
n . ( r , r 2=)2 l c ( < , l r )| ,
n _ ( o r r=
)0 .
(8)
to refractionindex (7), it is
wave corresponds
A circularly polarizedelectromagnetic
by the followingvectors
characterized
87
.r,,
.?
-
E=g "1, expg(-2lalf, i_a,t)l
./)c-l
H=_:_(x+ia,)E,
P
-
D= -
-/ t'N*
( x + i a )E ,
l.r
E=-ziuE
where the circular polarization sign is determinedby the optical activity parameter sign
0(<,lt.
A wave with arbitrary elipticity ? correspondsto the zero refraction index
;
.- dtivE
L=6
,tt+
e
-io.r
'
-l
p=:_(iu_x)E,
l.r
E=D=0.
which has an infinite phase velocity and is independentof coordinates.The energy flux
density vector of the wave averagedin time has the form
q'Y* '
< i i >.. = - c
4n p(l*y2)
Llngitudinal waves of the plasma type can also exist within the frequency co, for which
condition (6) is fulfilled:
E = 8fi expli(K, n-r-- o, r) l
,
a-=D=0.
88
(e)
-1
H=_:_(x_ia)E,
p
The wave number K,, of the longitudinal waves can be determined by considering the
second-orderthree-dimensionaldispersiontaking into account that the permittivity tensor
dependsupon the wave number [1]:
-yo(<o))[[
e (o,fl = eo(o)+yo(r,r)t'+(yr(r,r)
(10)
Thenthe waveequationof the electricalfield intensityhasthe followingform:
' r f i + 2 i u r f i ' } E = 0,
{ e o p - x 2 - u 2 - ( l - F o pr i)z * l l * ( g r - F o ) p l r n
wherethe followingdesignations
are used:
go=vo9i,
(11)
n -., a2
Pt - It
c2
c-
From wave equation (11) a condition of the eiectrical field longitudinal componentfollows
( eo p- x 2- a 2* 0 # n 2 ) E rr = 0 ,
from which the lonsitudinal wave number is derived
Klt
(l)l
c\
q.2*x2-eov
(12)
ItF
In response
to the parameter
signp, thenon-attenuating
longitudinal
wavescanexisteither
within the frequencyrangein which
qot P,> o)
a2*x2 > eov
or within the frequencyrange in which
(dt pr < 0)
a 2 * x 2< e o v
Since the energy flux densityvector of waveswith refractionindexes(7) and (8) differs from
zero these waves can be excited by an electromagneticwave incident upon a bi-isotropic
medium from outside. One of mechanismsexciting a longitudinalwave (9) can be the effect
of Vavilov-Cherenkovwhich is discussedbelow. For this purposethe wave equationof the
electromagneticfield with the sourcein the bi-isotropic medium is written as:
89
, , o t l E =_ ! l y 9 f
\ r o t r o t + e p , - x z - c3z _ z t s 9
c2
ot2
c ot
c2 dt'
.
When second-orderthree dimensionaldispersion(10) is takeninto accountthe wave equation
acquires the form
A2
v^
A2
"t.-,t^
p:lv.v-(r *i },ji)v' *
{1r--r-::
c'
dt'
e o v - x 2 - qd2
c2
c'
( 13)
dt"
_ziag v,)r= _!lt_ 9 f
dt2
c ot
c2 at'
Where V is a symbolic vector differential operator. In order to take into account the
frequencydispersionof the medium properliesthe parameters to, F ,X, d,.yolr
ShOuld
- dependentoperators By writing down the
be treatedas differential argument
i-:
currentdensityfor a pin-pointed
:'
charge e moving with a constantvelocity t
?
j = e -v^ ', -6 ( f- i t ) ,
non-homogeneous
equation(13) is solvedfor the Fourier-componentof the longitudinalfield:
Et= -
4t peik
eik(/-it)
k 2 ( e o p x 2 - a 2* y J t k 2 )
Also,the solution of equation(13) with the zero right term should be taken lnto account
E, =E^E eiGt-ot)
Now a full solutionof equation
(13)
E=Er*Ey
90
satisfyingthe original condition
E(f, t) =0 ,
at t=0 acquiresthe form
(14)
I -'-'(tu-o)r 4,.v-eid
,i{Er-'t),
E 1 r , r l- 1rz eop-x2-o'*Yrvk'
whereall the parameterscharacterizinga mediumare functionsof the argumentH
Whenthe conditions
e o p - x 2 - u 2+ y , p f t 2= 0 ,
El= a
( 15)
are fulhlled field amplitude (14) increasesin time linearly:
=.- .
L\r,t ) =
4t ueikt
den(o)
k'v---:-
dG)
It has been assumedthat magnitudesx2,o2 and 7,k2
than eo.
are less frequency-dependent
From relationships(12), (15) it follows that the Cherenkovemissionof longitudinalwaves
is possibleproviding
v>G)
g . 2* x 2 - e o l t
the phasevelocityof longitudinalwavesin a
i.e., whenthe velocityof an electronexceeds
medium.
bi-isotropic
91
REFERENCES
Ginzburgv.L. Propagation
of electromagnetic
wavesin theplasma.Moscow,1960,
pp. 108-223(in Russ.)
2.
Konstantinov
o.v., Perelv.I. Possibilityof magnetic
wavespassage
througha metal
in a strongfreld.Journalof Exp. andTheor.physics,1960,Vol.38, pp. 16l-164
3.
BokutB.v., Gvozdevv.v., SerdyukovA.N. particularwavesin naturalgyrotropic
media.Journalof AppliedSpectroscopy,
1981,Vol.34,No4, pp. 701-706.
+.
A
FedorovF.I. Theoryof gyrotropy.Minsk. 1979,p.456(in Russ.)
5.
SihvolaA.H., LindellI.V. Bi-isotropic
constitutive
relations.MicrowaveandOnt.
Tech.Lett.V 4, Ne8, pp.295-297,
July,l991.
6.
KongJ.A. Electromagnetic
wavetheory.New-york:Wiley,19g6.
7.
MonzonJ.c. Radiation
andscattering
in homogeneous
generalbi-isotropic
regions.
IEEE Trans.Antennas
Propagat.,
Vol. 38, Ne2, pp.227-235,
Februa.yftfO.
92