Dispersion Relation in Chiral Media: Credibility of Drude-Born-Fedorov equations

0.11 MB·English

by  Kikuo Cho

Dispersion relation in chiral media :
Credibility of Drude-Born-Fedorov equations
KikuoCho
InstituteofLaser Engineering,OsakaUniversity,Suita565-0871,OsakaJapan
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1
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2 Abstract

r
a
Disperion relation of EM field in a chiral medium is discussed from the viewpoint of
M
constitutive equations to be used as a partner of Maxwell equations. The popular form

1 ofDrude-Born-Fedorov(DBF)constitutiveequationsiscriticizedviaacomparisonwith
3
the first-principles macroscopic constitutiveequations. The two sets of equations show

] adecisivedifferenceinthedispersioncurveintheresonant regionofchiral,left-handed
i
c character, in the form of presence or absence of linear crossing at k=0. DBF equations
s
- could be used at most only as a phenomenology in off-resonant region, while the first-
l
r principlesonescan beusedforbothphenomenologicaland microscopicanalyses.
t
m
.
t
a 1 Introduction
m
-
d Symmetry plays an important role in the electromagnetic (EM) response of matter. It
n
is revealed in the form of susceptibilities relating electric and magnetic polarization
o
c (P and M) with source EM field. In high symmetry case, P and M consist of (the
[
superpositions of) independent groups of excitations belonging to different irreducible

3 representations of the symmetry group in consideration. This allows us to treat electric
v
8 and magnetic properties of matter independently. When a medium lacks in certain mir-
7 ror symmetry, i.e., the case of chiral symmetry, however, some (or all the) components
0
of P and M cannot be distinguished, so that they can be induced by both electric and
1
0 magnetic source fields. In addition, there is also a mixing between electric dipole (E1)
.
1 and electricquadrupole(E2)transitions.
0
ThestudyofchiralsymmetryintheEMresponseofmatterhasalonghistory(Intro-
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1 ductionof [1]). Chiral substances havebeen considered as unconventionalmaterials for
:
v a long time, but now it is regarded as an important source of new materials and states,
i providinghottopicsinthestudiesofmetamaterials[2],multiferroics[3], andsupercon-
X
ductivity[4].
r
a
In spite of its long history, theoretical description of chirality does not seem to be
standardized. In the documents of IUPAP and IUPAC dealing with the standard defini-
tions of physical and chemical quantities [5, 6], there is no mentioning about the chiral
susceptibilities. Correspondingly, there are two or more different forms of phenomeno-
logical constitutive equations in use for macroscopic response. Though the effect of
chiralsymmetryisexpectedalsoinmicroscopicresponses,itsfirst-principlesthoeryhas
been made only very recently [7]. From the viewpoit that all the different forms of EM
response theories should belong to a single hierarchy with logical ranking, one should
beableto choosethemostappropriateform oftheconstitutiveequationsforthemacro-
scopicchiralresponseonthebasisofthemicroscopictheory.
A typicaleffect of chiralityis thedifferencein thephasevelocityof EMwaves with
right- and left-circular polarizations, which appears in the off-resonant region of sus-
1
ceptibilities. However, this is not the only aspect of our interest in discussing chirality.
In fact, the dispersion curves in the resonant region of susceptibilityshow a remarkable
behavior, bywhichwe can selectthecorrect constitutiveequations.
Macroscopic EM response of matter is usually calculated by the combination of
Maxwelland constitutiveequations. Thestandard formofthelatteris
D = ǫE , B = µH (1)
withthedielectricconstant(permittivity)ǫandpermeabilityµ. However,ifthemedium
inconsiderationhaschiralsymmetry,theseconstitutiveequationsneedtobegeneralized.
A popularform ofsuchan extentionis
D = ǫ(E +β E) , (2)
∇×
B = µ(H +β H), (3)
∇×
which is called Drude-Born-Fedorov equations (DBF eqs) [8, 9]. The parameter (chiral
admittance)βdescribesthechiralityofthemedium. Thisisaphnomenologyforuniform
and isotropicmedia.
However, this is not the only way of generalization. From the viewpoint that the
fundamental variables of EM field are E and B, both electric and magnetic polariza-
tions P and M shouldconsistboth of the E and B -induced components,so that the
{ }
definitionD = E +4πP, H = B 4πM leadsto theextension
−
D = ǫˆE +iξB , (4)
H = (1/µˆ)B +iηE , (5)
where theterms with ξ and η takecare ofthechirality. For laterconvenience, let us call
them chiral constitutive equations (ChC eqs). Though they are a result of phenomeno-
logical considerationon theonehand, afirst-principles calculationofmacroscopiccon-
stitutiveequationscan beputalso inthisform ontheotherhand [7].
As to thedifferenceor similarityofDBF and ChC eqs, thereis acontroversy. There
have been arguments in the metamaterials community that DBF and ChC eqs are es-
sentially same [10], and also it is argued that the former can be derived from the latter
by assuming the uniformity and isotropy of matter [1](Sec.4.4). But there is also other
group of people preferring ChC to DBF eqs. The purpose of this article is to show that
there is a clear difference between the two, and that ChC eqs should be preferred. In
view of the fact that DBF eqs are frequently used in metamaterials studies and also in
recent textbookof standard electromagnetism[9], it will be important to clarify the dif-
ference between DBF andChC eqs.
WefirstnotetherelationbetweentheparametersofDBFandChCeqs. Bymeansof
therelation
E = (iω/c)B , H = ( iω/c)D , (6)
∇× ∇× −
DBF eqscan berewrittenas
D = ǫE +(iω/c)ǫβB , (7)
H = (iω/c)ǫβE +(1/µ)[1 (ωβ/c)2ǫµ]B . (8)
−
2
If DBF and ChC eqs are equivalent, the DBF parameters can be written in terms of the
ChC parameters bycomparing(7)and (8)with(4)and (5)as
ǫˆ= ǫ , ξ = η = (ω/c)ǫβ , (1/µˆ) = (1/µ) (ωβ/c)2ǫ. (9)
−
This relation will be shown later to lead to contradiction, which disproves the equiva-
lenceofDBF and ChCeqs.
The first-principles derivation of micro- and macroscopic constitutive equations is
done in the following way [7]. We assume a general form of non-relativistic Hamil-
tonian (including relativistic correction terms, such as spin-orbit interaction and spin
Zeeman term, etc.) for a many particle system in an EM field, and calculate the micro-
scopiccurrentdensityinducedbytheEMfield,whichisingeneralgivenasafunctional
ofthetransverse(T)partofvectorpotentialA(T) andthelongitudinal(L)externalelec-
tric field E(L). The integral kernel of the functional is the microscopic susceptibility
ext
of a separable form with respect to position coordinates. When the relevant quantum
mechanicalstates havespatialextensionmuchlessthanthewavelengthoftheEMfield,
wemayapplylongwavelengthapproximationtothemicroscopiccurrentdensity,which
leadstothemacroscopicconstitutiveequationstobeusedformacroscopicMaxwelleqs.
Inthismacroscopicscheme,weneedonlyasingle3 3tensortorelateinducedcurrent
×
density and source EM field, covering all the electric, magnetic and chiral polarizations
ofmatter. Thismacroscopicconstitutiveequationis givenintheform[7]
J(k,ω) = χ (k,ω) A(T)(k,ω) (ic/ω)E(L)(k,ω) . (10)
em { − ext }
The internal L field does not appear in the source field, since it is taken into account
as the Coulomb potential in the matter Hamiltonian. The susceptibility χ is written
em
in terms of the quantum mechanical transition energies and the lower moments of the
correspondingtransitionmatrixelementsofcurrent densityoperator.
UsingtheidentityJ = iωP+ick M inFourierrepresentation,wecanrigorously
− ×
rewritetheconstitutiveequationintotheform
P = χ E +χ B , M = χ E +χ B (11)
eE eB mE mB
The four susceptibilities χ , χ , χ , χ are again written in terms of the quantum
eE eB mE mB
mechanical transition energies and lower transition moments of electric dipole (E1),
electric quadrupole (E2), and magnetic dipole (M1) characters. Details are given in
sec.3.1 of[7]. Thelowestordertermsofthemare
1
χ = g¯ J¯ J¯ +h¯ J¯ J¯ ,
eE ω2V ν 0ν ν0 ν ν0 0ν
Xν (cid:2) (cid:3)
1
χ = g¯ M¯ M¯ +h¯ M¯ M¯ ,
mB ν 0ν ν0 ν ν0 0ν
V
Xν (cid:2) (cid:3)
i
χ = g¯ J¯ M¯ +h¯ J¯ M¯ ,
eB ν 0ν ν0 ν ν0 0ν
ωV
Xν (cid:2) (cid:3)
i
χ = − g¯ M¯ J¯ +h¯ M¯ J¯ , (12)
mE ν 0ν ν0 ν ν0 0ν
ωV
Xν (cid:2) (cid:3)
1 1 1 1
g¯ = , h¯ = , (13)
ν ν
E h¯ω i0+ − E E +h¯ω +i0+ − E
ν0 ν0 ν0 ν0
− −
3
whereV isthevolumeofacellforperiodicboundaryconditiontodefinek,andJ¯ and
0ν
M¯ are,respectively,theE1andM1transitionmomentsofcurrentdensityand(orbital
0ν
and spin)magnetizationoperators between themattereigenstates 0 (ground state)and
ν with transition energy E between them. (E2 moments app|eair in the J¯ terms
ν0 µν
| i
in the next higher order.) Chiral symmetry allows the existence of the transitions with
mixed(E1and M1)or(E1 and E2)character, leadingtotheO(k1)terms inχ .
em
Though there appear four susceptibilities, the single susceptibility nature is intact,
sincetherewritingof(10)into(11)isreversible. Combiningthenewformofconstitutive
equationswiththedefinitionofD and H, weobtain
D = E +4πP = (1+4πχ )E +4πχ B , (14)
eE eB
H = B 4πM = (1 4πχ )B 4πχ E , (15)
mB mE
− − −
which isessentiallyequivalentto ChCeqs. Theparameters ofChCeqs are givenas
1
ˆǫ = 1+4πχ , iξ = 4πχ , iη = 4πχ , µˆ = . (16)
eE eB mE
− 1 4πχ
mB
−
all of which are tensors, with no assumption of isotropy and homogeneity as for DBF
eqs. It should also be noted that the poles of χ , i.e., the magnetictransition energies,
mB
are,notthepoles,butthezerosofµˆ. Thisisduetothedefinitionofχ ,M = χ B as
mB mB
requiredinthefirst-principlesapproach,incontrasttotheconventionaloneM = χ H.
m
At this stage, the ChC eqs are not a phenomenology, but a first-principles theory. In
contrast,thiskindoffirst-principlesderivationdoes notexistforDBF eqs.
2 Dispersion equation
InordertoshowthedifferencebetweenDBFandChCeqs,itissufficienttogiveasingle
example. For this purpose,we compare the dispersion relations obtained from DBF and
ChC eqs.
2.1 Case of DBF eqs
IfwesolveDBF eqs andeq.(6)for H and E, weobtain
∇× ∇×
H = aH +bE , (17)
∇×
E = dH +eE , (18)
∇×
where
a = e = ǫµβ/∆ , b = icǫ/ω∆, d = +icµ/ω∆, (19)
− −
and ∆ = ǫµβ2 c2/ω2. From eqs (6), (7), (8) and B = 0, both E and H are
− ∇ ·
transverse,so that
E = k2E, H = k2H (20)
∇×∇× ∇×∇×
for Fourier components. Then, by operating to (17) and (18), we obtain a set of
∇×
homogeneous linear equations of H,E. The condition for the existence of non-trivial
solutiongivesus thedispersionequation
det k21 2 = 0 , (21)
| −A |
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where isa2 2 matrixwiththecomponentsa,b,d,e
A ×
a b
= . (22)
A (cid:20) d e (cid:21)
Thisdispersionequationcan berewrittenas
det k1+ = 0 or det k1 = 0 , (23)
| A| | −A|
so thatthesolutionis
k = a √bd , (24)
± ±
with all the combinations of being allowed, which finally leads to a compact expres-
±
sion
ck √ǫµ
= . (25)
ω ±1 (ωβ/c)√ǫµ
±
This gives two branches of dispersion curve. In homogeneous isotropic media, the two
modes correspond to right and left circular polarizations. It should be noted that the
condition for the existence of real solution is ǫµ 0. This means that the left-handed
≥
medium is defined in the same way as in non-chiral medium, in contrast to the case of
ChC eqs.
2.2 Case of ChC eqs
A same way of solution is possible in this case, too. After eliminating D,B from the
ChCeqs,thesolutionfor H, E hasasameformastheone,wherea,b,d,eof
∇× ∇×
previoussectionarereplaced withthefollowingf,g,h,j,respectively
f = i(ω/c)ξµˆ , (26)
−
g = i(ω/c)(ǫˆ+µˆξη), (27)
−
h = i(ω/c)µˆ , (28)
−
j = i(ω/c)µˆη . (29)
−
Further transformation of the equations of H, E into a set of homogeneous
∇ × ∇ ×
equations of E,H allows us to obtain the dispersion equation, as the condition for the
existenceofnon-trivialsolution,
ck 1
= µˆ(η ξ) µˆ(η ξ) 2 +4ǫˆµˆ , (30)
ω ±2 − ± { − }
(cid:2) p (cid:3)
wherewetakeall thecombinationsof . Theconditionforreal solutionsis
±
µˆ(η ξ) 2 +4ǫˆµˆ 0 , (31)
{ − } ≥
which islessrestrictivethannon-chiralcase.
3 Discussions
Firstofall,wenotethat,forbothDBFandChCeqs,thewell-knownresultof(ck/ω)2 =
ǫµ is obtained in the absence of chirality (β = 0 and ξ = 0,η = 0). Also both of the
constitutive equations exhibit the typical behavior of chiral medium, i.e., the existence
of the two branches with polarization dependent refractive indices. In the non-resonant
region, both of them could be used to fit experimental results via appropriate choice of
parametervalues.
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3.1 Resonant region of left-handed chiral medium
Adecisivedifferenceappearsinresonantregion. Anexamplewillbetheleft-handedbe-
havioremergingintheneighborhoodofachiralresonancewithE1-M1mixedcharacter.
Such a case has been treated by the first-principles theory of macroscopic constitutive
equation [7] (Sec.3.8.1 and Sec.4.1.1). It shows a pair of dispersion curves for left and
right circularly polarized modes, which have a linear crossing at k = 0. It will be a
test for the phenomenologies whether such a linear crossing can be realized or not by
choosingparametervalues.
Thedispersionequationinthefirst-principlesmacroscopicformalismis
c2k2 4πc
0 = det 1 1+ χ(T)(k,ω) , (32)
| ω2 − ω2 em |
(cid:2) (cid:3)
where χ(T)(k,ω) is the T component of susceptibility tensor (sec.2.5 of [7]). Let us
em
chooseachiral formofsusceptibilitytensorχ(T) as
em
4πc 0 ib′k
1+ χ(T) = (ǫ +a′ +c′k2)1+ , (33)
ω2 em b ib′k 0
(cid:2) − (cid:3)
where the terms with a′,b′,c′ represents the contribution of a pole 1/(ω ω) with
0
∼ −
mixed (E1, M1) character, while ǫ is the background dielectgric constant due to all
b
the other resonances. We assume that the resonance with mixed E1 and M1 characters
occurs in the frquency region of ǫ < 0, i.e., a chiral version of left-handed medium.
b
Thedispersionequationis
ck ck
2 = ǫ¯µ¯ β¯µ¯ , (34)
ω ± ω
(cid:0) (cid:1)
and itssolutionisgivenas
ck 1
= β¯µ¯ + β¯2µ¯2 +4ǫ¯µ¯ , (35)
ω ±2 ± q
(cid:2) (cid:3)
wherewetakeall thecombinationsof , and
±
β¯ = ωb′/c , ǫ¯= ǫ +a′ µ¯ = 1/[1 (ω/c)2c′]. (36)
b
−
Notingthatb′ isachiralparametercorrespondingtoξ,η oftheChCeqs,weseethatthis
equation is the same type as eq.(30), but not as eq.(25). An example of this dispersion
relationis giveninFig.1. Thecharacteristicbehaviorofthedispersioncurves isalinear
crossingat k=0 (Fig.4.1of[7]).
In order to check whether the dispersion equations (25) and (30) have this typical
behavior of ”linear crossing at k = 0”, we focus on the behavior of the dispersion
equations near k = 0. Since both of the dispersion equations are given in the form
”ck/ω = F(ω)”,weonlyneedtoexaminehowthefunctionF(ω)approachestozero in
each case.
Themicroscopicmodelofleft-handedchiralmediumgivenaboveconsistsofamat-
ter excitation level with (E1, M1) mixed character in the frequency range of ǫ < 0.
b
This means that all of χ ,χ ,χ ,χ have a common pole (at ω = ω ) and χ is
eE eB mE mB 0 eE
largely negativein the frequency range of interest to make ǫ < 0. Namely, ǫˆ,1/µˆ,ξ,η
b
of ChC eqs have a common pole at ω = ω and ˆǫ = ǫ + a¯/(ω ω). It may appear
0 b 0
−
that ther.h.s. ofeq.(30)becoms zero for frequency satisfyingµˆ = 0 orǫˆ= 0. However,
6
Figure 1: Dispersion curves of a chiral left-handed medium for the model in the text.
Both ordinate and abscissa are normalized by the frequency of the pole as ω/ω and
0
ck/ω . Twohorizontallinesshowthefrequncies ofǫ = 0 and µ = 0.
0
as mentioned before, the zero of µˆ corresponds to the pole of M1 transition, which in
this case is common to the pole of ξ and η. Therefore the zero of µˆ is cancelled in the
product µˆ(ξ η). Thus, the only remaining possibility of zero arises from ǫˆ= 0. The
−
ω-dependenceofther.h.s. eq.(30)nearzero pointcan befoundbyrewritingitas
1 ǫˆµˆ
. (37)
± 2µˆ(ξ η) µˆ(ξ η) 2 +4ǫˆµˆ
− ± { − }
p
At the frequency satisfying ǫˆ= 0, i.e., ǫ +a′ = 0, all of µˆ,ξ,η remain finite, and one
b
of the combinations in the denominator remains finite, so that the whole expression
±
becomeszeroforthiscombination. Thisoccursforbothsignsof infrontofthewhole
expression. Fornegativeǫ andpositivenumeratorofthepole 1±/(ω ω),ǫ +a′ = 0
b 0 b
∼ −
occurs at ω = ω < ω and in its neighborhood ˆǫ (ω ω ). This showsthat ther.h.s.
z 0 z
∼ −
ofeq.(30)behaveslike (ω ω ),whichmeansthelinearcrossingofthetwobranches.
z
∼ −
Notealsothatω liesinsidethefrequency rangeofeq.(31).
z
Now we check whether the same behavioris obtained for DBF eqs by assuming eq.
(9), from whichweobtain
1 1 ξ2 ω2β2 µ
= + , ǫµ = 1 . (38)
µ µˆ ǫˆ c2 − µˆ
This shows that 1/µ has the same pole as 1/µˆ at ω = ω , so that the factor µ/µˆ on the
0
r.h.s. of the second equation does not have the pole at ω = ω via cancellation. There-
0
fore, there is no chance for the denominator of the r.h.s. of eq.(25) to diverge. Hence
the only possibility of its becoming zero comes from the factor √ǫµ on the numerator.
In view of the fact that the zeros of ǫ,µ occur at different ω’s, e.g., at ω and ω (ω
z1 z2 z1
> ω ), the ω-dependence of √ǫµ should be √ω ω or √ω ω in the neig-
z2 z1 z2
∼ − ∼ −
borhood of the zeros. Therefore, no linear crossing is possible in the DBF dispersion
curves. Thetwo zeros aretheboundariesoftheregionofleft-handed behavior.
7
Onemightarguethatothertypeofω-dependencethaneq. (9)couldleadtothelinear
crossing behavior. But onecannot freely givethe ω-dependence even as a phenomenol-
ogy. Linear susceptibilities should be a sum of single pole functions. In the absence of
the first-principles theory for DBF eqs, it would be quitedifficult to givean appropriate
modelon areliablebasis.
3.2 Conventionality vs. Logical Consistency
DBF eqs have been popularly used in the macroscopic argument of chiral systems, es-
pecially inthefield ofmetamaterialsresearch. As longas they areused fornonresonant
phenomena as a practical tool, there is not much to say against it, except for the dif-
ficulty in assigning microscopic meaning to the parameter β. However, the restriction
to the nonresonant phenomena does not seem to be widely recognized, to the knowl-
edge of the present author. In fact there are examples of its use for resonant phenom-
ena [11, 12]. (The constitutive equations used in [12] are not exactly DBF eqs, but
D = ǫE iξH , B = µH +iξE,different alsofrom ChCeqs.)
−
From the qualitativedifference of the two dispersion equations (25) and (30) in res-
onant region, and from the fact that DBF eqs have no support by microscopic theory in
contrast to ChC eqs, the use of DBF eqs for resonant phenomenais risky. As a conven-
tionalapproachwithalonghistory,DBFeqsmightbekeptinusefurther,butthevalidity
limit should be kept in mind. However, if we consider that ChC eqs can be handled as
easily as DBF eqs, and that they are consistentwith the microscopicallyderivedmacro-
scopic constitutive equationn, it is highly recommended to use ChC eqs. For problems
requiringseveredistinction,logicalconsistencyshouldbepreferred toconventionality.
4 Conclusion
The DBF eqs, popularly used as constitutive equations of chiral media, should be re-
gardedasaphenomenologicaltheoryapplicableonlyinnonresonantregion. Inresonant
region,itwouldleadtoaqualitativelyerroneousresult. Ontheotherhand,theChCeqs,
consistent with the first-principles microscopic constitutiveequations, can be used both
forresonant andnonresonantproblems.
Acknowledgment
ThisworkissupportedbyGrant-in-AidforscientificresearchonInnovativeAreasElec-
tromagneticMetamaterialsofMEXTJapan (Grant No. 22109001).
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