arXiv:1508.04049v4 [physics.optics] 28 Jul 2016

12 Pages·2016·2.88 MB·English

Objects of maximum electromagnetic chirality
Ivan Fernandez-Corbaton
∗
Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
Martin Fruhnert
Institut fu¨r Theoretische Festko¨rperphysik, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
Carsten Rockstuhl
Institut fu¨r Theoretische Festko¨rperphysik, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany and
Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
6 We introduce a definition of the electromagnetic chirality of an object and show that it has an
1 upper bound. The upper bound is attained if and only if the object is transparent for all the
0 fieldsofonepolarizationhandedness(helicity). Additionally,electromagneticdualitysymmetry,i.e.
2 helicity preservation upon interaction, turns out to be a necessary condition for reciprocal objects
to attain the upper bound. We use these results to provide requirements for the design of such
r
a extremal objects. The requirements can be formulated as constraints on the polarizability tensors
M for dipolar objects or on the material constitutive relations for continuous media. We also outline
two applications for objects of maximum electromagnetic chirality: A twofold resonantly enhanced

0 andbackgroundfreecirculardichroismmeasurementsetup,andangleindependenthelicityfiltering
1 glasses. Finally, we use the theoretically obtained requirements to guide the design of a specific
structure, which we then analyze numerically and discuss its performance with respect to maximal

] electromagnetic chirality.
s
c
i
pt An object is chiral if it cannot be super-imposed onto objects meet extra requirements. These requirements al-
o its mirror image. This simple definition hides significant low to significantly narrow down the design parameter
. problems that arise when attempting to measure chiral- space.
s
c ity[1]. Quantifyinghow chiral an object isisthepurpose In this article, and in the spirit of [40], we shift the
i ofscalarmeasuresofchiralitywhichvanishonlyforachi-
s focus from a geometrical definition of chirality to a defi-
y ral objects and assign the same value to an object and nitionthatisbasedontheinteractionwiththefield. We
h its mirror image [2, 3]. There are many different scalar introduce a definition of the electromagnetic chirality of
p measures of chirality [3], but none of them allows to sort
anobjectbasedonhowitinteractswithfieldsofdifferent
[
generalobjectsaccordingtotheirchiralityortoestablish polarizationhandedness(helicity). Ourdefinitioncanbe
3 what a maximally chiral object is [4] in an unambiguous stated in the following way: An electromagnetically chi-
v way.
ral object is one for which all the information obtained
9 Independently of these measurement problems, chiral- fromexperimentsusingafixedincidenthelicitycannotbe
4
ityisentrenchedinnature: Fromthelackofmirrorsym- obtainedusingtheoppositeone. Thevariouselectromag-
0
4 metry of some interactions among fundamental particles netic chirality measures arising from this definition take
0 [5], to its ubiquitous presence in chemistry and biology. the form of relativistically invariant distances. We then
. Chirality is studied in very diverse scientific disciplines. select a particular measure, which can be singled out on
8
Oneofthemistheinteractionofchiralmatterwithelec- physicalgrounds. Weshowthattheelectromagneticchi-
0
5 tromagnetic fields, which started two centuries ago [6] rality of an object has an upper bound, and that the up-
1 andstillattractssignificantattentionfrombothitstheo- perboundisattainedifandonlyiftheobjectistranspar-
: reticalandpracticalsides(e.g. [7–39]). Thelackofupper enttoallfieldsofonehelicity. Theupperboundisequal
v
i bounds and unambiguous ranking for the magnitude of tothesquarerootofthetotalinteractioncrosssectionof
X chirality is a handicap for both theoretical and practical the object. Our definition allows the absolute ranking of
r developments. In particular, it is a handicap for the sys- objects according to their electromagnetic chirality. We
a tematicdesignofchiralstructuresforinteractionwiththe also show that any maximally electromagnetically chiral
electromagnetic field. These ambiguities leave us unable and reciprocal object must have electromagnetic duality
to compare different structures and without an extremal symmetry, i.e., interaction shall not change the helicity
reference to design towards. Additionally, it leaves us of the incident fields. We then particularize these results
with no other design guidelines besides chirality itself. to obtain the constraints that reciprocity plus maximum
We will show that, under a different definition of chiral- electromagneticchiralityimposeonmaterialconstitutive
ity, chirality upper bounds exist and are attained when relations, and on the polarizability tensor of an isolated
scatterer. These constraints are precise requirements for
the design of maximally electromagnetically chiral ob-
jects. Electromagnetic duality symmetry is one of them.
∗ [email protected] We then discuss two possible applications for maximally
2
electromagnetically chiral objects: A twofold resonantly
enhanced circular dichroism setup, and angle indepen-
dent helicity filtering glasses. Finally, we numerically
analyze the specific design of an object whose properties
come close to those of a maximally electromagnetically
chiral object in a narrow frequency band. The analy-
sis and results contained in this article apply to linear
interactions with finite cross sections.
I. SETTING
We start with a brief introduction of the setting, and
the mathematical tools and notation that we use. The
setting is depicted in Fig. 1, where an object inter-
acts with an incident field and produces a scattered field
and/or absorbs part of the incident energy. We assume
linearinteractionandfinitecrosssections. Thissettingis
convenientlytreatedintheframeworkoflinearoperators
in Hilbert spaces and Dirac’s “ bra” “ket ” notation1.
(cid:104) | | (cid:105)
There,thefieldsarevectorsintheHilbertspaceoftrans- FIG. 1. Interaction in the Hilbert space of transverse
verse solutions of Maxwell’s equations. The effect of the Maxwell’s fields. An incident field |Φin(cid:105) interacts with an
object is described by its interaction operator S, which object, characterized by its interaction operator S, and pro-
duces a scattered field |Φ (cid:105) = S|Φ (cid:105). A detector like the
we take to be the non-trivial part of the scattering op- out in
erator S˜ = I +S [41, Sec. 6.4]. The identity term in S˜ one in the bottom right corner of the figure obtains infor-
mation about the field scattered through the solid angle Ω:
accounts for the portion of the incident field that does
f((cid:104)Ω|Φ (cid:105))=f((cid:104)Ω|S|Φ (cid:105)), where (cid:104)Ψ|Γ(cid:105) is the scalar prod-
not interact with the object, and S is proportional to out in
uct of the two vectors |Ψ(cid:105) and |Γ(cid:105). The interaction operator
the system transfer operator T (a.k.a T-matrix [42], [43,
Salsodescribesabsorptionandnearfieldilluminationand/or
Eq. 2.7.20]): S = 2T. The interaction operator S con- measurement.
tains both scattering and absorption information. For
example, in Fig. 1, a far field detector at solid angu-
lar position Ω provides information about the projection 57],
of the scattered field on the corresponding plane wave,
i.e. the scattering coefficient ΩS Φ , where Ω is a J P
(cid:104) | | in(cid:105) (cid:104) | Λ= · . (1)
plane wave, and ΨΓ the scalar product of Ψ and Γ . P
(cid:104) | (cid:105) | (cid:105) | (cid:105) | |
Besides far field scattering, the interaction operator S
also models near field interactions [44–46], and absorp- For classical electromagnetic fields in the complex no-
tion by the object. It contains all the information that tation, helicity has two possible eigenvalues λ 1, 1 .
∈{ − }
can be obtained from the object by means of its interac- The eigenstates of helicity are the Riemann-Silberstein
tionwithtransverseelectromagneticfields. WetakeS as linear combinations [50, 51] G = 1 (E iZH), with
± √2 ±
the only relevant representation of the object and define Z the medium impedance, so that:
its electromagnetic chirality through the properties of S
with respect to the helicity of the fields. The fundamen- (E iZH) (E iZH)
ΛG =Λ ± = ± = G . (2)
tal properties of helicity make it suitable for discussing ± √2 ± √2 ± ±
chiral interactions, as is done in particle physics [47].
Equation(2)canbederived2fromMaxwell’scurlequa-
tions and the representation of the helicity operator for
II. USING HELICITY TO CHARACTERIZE monochromatic fields of frequency ω = kc, which reads
INTERACTIONS WITH CHIRAL MATTER
Λ . Equation (2) is valid in general, including in
≡ ∇k×
thenearfieldsaroundscatteringobjects. Thechiralchar-
The helicity operator is the projection of the total an- acter of near fields can be readily determined by means
gularmomentumvectoroperatorontothelinearmomen- of the two Riemann-Silberstein helicity eigenstates. We
tum vector operator direction [48, Sec. 8.4.1], [49, Eq. nowshowtheirconnectiontotheopticalchiralitydensity
[9, 53].
1 Diracintroducedthisnotationinquantummechanics. Itisalso
very convenient for other situations that can be treated in the
frameworkofHilbertspaces. 2 See[52,Eqs.(2)-(3)].
3
We start from a monochromatic electromagnetic field As an operator, helicity is the generator of the elec-
around some scattering object [Eω(r),Hω(r)]. Af- tromagnetic duality transformation5. The relationship
ter expressing it in the Riemann-Silberstein basis between helicity and duality is the same as, for example,
(cid:2) (cid:3)
[Eω(r),Hω(r)] Gω(r),Gω(r) , we consider the fol- angular momentum and rotations. A dual symmetric
lowing space-de→penden+t quant−ity scatterer preserves the helicity of the fields interacting
with it, i.e., it does not couple states of opposite polar-
κω(r)= Gω(r)2 Gω(r)2, (3)
| + | −| − | ization handedness. The conditions for duality symme-
tryofascattererinthemacroscopicMaxwell’sequations
thatis,thedifferencebetweenthepointwiseintensitiesof
the two helicity eigenvectors. The quantity κω(r) com- [58, 59] and in the dipolar approximation [59, 60] are
known. Theuseofhelicityanddualityforthestudyand
pletely determines the optical chirality density [9, 53],
engineering of light matter interactions is developed in
defined as
detail in [61].
(cid:15)ω
Cω(r)= Im Eω(r) Bω(r) , (4)
†
− 2 { }
where (cid:15) is the permittivity of the surrounding medium. III. ELECTROMAGNETIC CHIRALITY OF AN
It can be shown 3 that: OBJECT
ε
Cω(r)= ωκω(r). (5)
4c Let us consider the electromagnetic interaction oper-
ator S of an object. We choose a basis for transverse
The optical chirality density is nowadays widely em-
Maxwell fields η λ where helicity is used as the polar-
ployed to discuss the chiral character of the near fields | (cid:105)
ization label (λ 1, 1 ) and η is a collective index
around scatterers, and the coupling of chiral molecules ∈ { − }
containing the other three defining numbers6. We can
atinodnd(5ip)ocloesnfitormsuscthhenesauritfiaeblidlisty(eo.gf t[9h,e1h5e,l2ic1i,ty32fo])r.mEaqliusma- tEhaecnhcSoλ¯nsaicdtesrotnheinppaurttisatlatoepserηatλorsofSh++e,licSi+−ty,λS−+ an1d, S1−−,.
for describing chiral near field interactions. and proλduces output states |λ¯ η¯(cid:105)of helicity λ¯ ∈{1,−1}.
Helicitycanalsobeunderstoodinoperationaltermsin (cid:104) | ∈{ − }
We define the object to be electromagnetically achiral
the momentum (plane wave) representation. An electro-
if and only if there exist four unitary operators U , V ,
magneticfieldisaneigenstateofhelicitywitheigenvalue 1 1
U and V that commute with the helicity operator, and
+1( 1) if and only if all the plane waves in its decom- 2 2
− satisfy
position are left(right) handed polarized with respect to
their corresponding momentum vectors, in which case
G (G+) is zero at all points. The decomposition can (cid:104)+ η¯|S++|η +(cid:105)=(cid:104)− η¯|U1S−−V1†|η −(cid:105), (6)
co−nFtoarinmbaostshlespsrofipealgdast,intgheanhdeleicviatnyesocpeenrtaptolarnecowmamveust4e.s (cid:104)− η¯|S+−|η +(cid:105)=(cid:104)+ η¯|U2S−+V2†|η −(cid:105),
with all the transformations of the Poincar´e group, for all (η,η¯).
i.e. space and time translations, spatial rotations, and Conversely, we define the object to be electromagneti-
boosts. It is a relativistic invariant of the field. Addi- callychiralwhenitselectromagneticinteractionoperator
tionally, it commutes with the time inversion operator. never meets Eq. (6).
None of these operations flip the helicity eigenvalues of Anycompositionofboosts,rotations,translations,and
the states they act on. Crucially, helicity flips only with time inversion is an example of a helicity preserving uni-
spatial inversion transformations: λ λ after parity, tary operator.
→ −
mirror reflections, and rotation-reflections. Helicity is WepointoutthatEq.(6)saysthat,foranelectromag-
hence a spatial pseudoscalar in the Poincar´e group ex- netically achiral object, all the information which can be
tended with space and time inversion. obtained from experiments using only one input helicity
Thesepropertieshavealreadyallowedtodrawconnec-
tions between material chirality and optical helicity [9–
11, 16], and to establish the fundamental role of helicity
preservation in optical activity [54–56]. 5 The duality transformation acts on the initial (E,H) fields as:
[57,Eq. 6.151]
E =Ecosθ−ZHsinθ,ZH =Esinθ+ZHcosθ.
θ θ
6 EachvectorofabasisoftransverseMaxwellfields,bothpropa-
3 Equation(5)followsfromEq.(4)andthesesteps: gatingandevanescent,hasfournumbersthatidentifyit. These
−Im{E(ω)†B(ω)} = −Im{E(ω)†µH(ω)} = four numbers are the eigenvalues of four commuting operators.
For example, multipolar fields are eigenvectors of the angular
−Im{(cid:16)G+(ω)√+G−(ω)(cid:17)†µ(cid:16)G+(ω√)−G−(ω)(cid:17)} = momentumsquared,theangularmomentumalongoneaxis,the
2 2iZ
(cid:16) (cid:17) energy (frequency), and the parity operator. The latter fixes
21cIm{i |G+(ω)|2−|G−(ω)|2+2iIm{G+(ω)†G−(ω)} } = theirpolarization. Thehelicityversionsofmultipolarfieldsand
21c(cid:0)|G+(ω)|2−|G−(ω)|2(cid:1) Besselbeamsarethesumandsubtractionofthemorecommon
4 A plane wave of helicity +1(−1) is the sum(subtraction) of parityandTE/TMmodes[62,App. A].Planewavescanbecho-
TE and TM plane waves of equal momentum, irrespectively of senaseigenstatesofthethreecomponentsoflinearmomentum,
whetherallthemomentumcomponentsarerealornot. whichfixesthefrequency,andthehelicityoperator.
4
can also be obtained from experiments using the oppo- which can be represented by complex matrices of infi-
site helicity. This is not the case for electromagnetically nitedimension. TheinteractionoperatorS iscompletely
chiral objects. continuous. Our initial assumption of finite cross section
The common geometrical definition of chirality is a guarantees this property [64, Chap. 8.6].
particular case of our definition of electromagnetic chi- Consider the sub-matrices of coefficients
rality. The non-superimposability of an object with its
mirror image implies that after S is transformed7 by a Mλ¯ λ¯ η¯S η λ for all (η,η¯). (10)
λ ≡(cid:104) | | (cid:105)
mirror operator S MSM 1, no arbitrary sequence of
−
a rotation R and a→translation T can undo the change: Let us denote by σ(A) the column vector containing
the singular values of matrix A in decreasing order, and
MSM 1 =(TR)S(TR) 1 for all T, R. (7) define the column vectors
− −
(cid:54)
CToRnvsuercsheltyh,aftor an achiral object, there exist at least a v+ =(cid:20)σσ(cid:0)(cid:0)MM++−+(cid:1)(cid:1)(cid:21), v− =(cid:20)σσ(cid:0)(cid:0)MM−−+(cid:1)(cid:1)(cid:21), (11)
−
MSM 1 =(TR)S(TR) 1. (8) whichcontainthesingularvaluesofthetwosub-matrices
− −
corresponding to each input helicity.
It can be shown8 that Eq. (8) leads to a particular case The implication of Eq. (6) for em-achiral objects is
of Eq. (6) with Ui/Vi restricted to rotations and trans- that the singular values of M++ and M− are equal, and
lations. Besides rotations and translations, the proposed the singular values of M+− and M+ a−re equal9. This
definition of electromagnetic chirality allows for other is not the case for em-chiral object−s. The definition of
kinds of transformations as well. Notably, the relativis- Eq. (6) is hence equivalent to saying that an object is
tic invariance of electromagnetic helicity allows for Ui electromagneticachiralifandonlyifv+ =v . Ifv+ =v
and Vi to contain boosts. Consequently, our definition the object is electromagnetically chiral. In−light of(cid:54) thi−s,
of electromagnetically (a)chiral objects is relativistically anydefinitionofascalarem-chiralitymeasure χ should
invariant. Furthermore,thepossibilitythatUi andVi do be based on a distance function between v+ an|d|v
notrepresentthesameoperatorsisalsoallowed,andcan −
be interpreted in Eq. (6) as different input and measure- χ =d(v ,v ). (12)
+
ment basis changes. | | −
For the purpose of brevity we will often use the prefix The properties of distance functions ensure that χ is
| |
em- from now on. For example, we will write em-chiral real,nonnegative,andiszeroonlyforem-achiralobjects.
instead of electromagnetically chiral. It is also clear that χ is invariant under any transfor-
| |
mation by unitary matrices since the singular values re-
main invariant. The transformations include the matrix
A. Scalar electromagnetic chirality measures representations of translations, rotations, boosts, time
inversion, and also parity. The latter flips both the in-
The proposed definition has an implication which al- put and output helicities and therefore the two vectors
lowstheuseofthesingularvaluedecompositiontodefine v v , which, thanks to d(v+,v )=d(v ,v+), leaves
measures of em-chirality, i.e. measures of how em-chiral χ± →unch∓anged. Weconcludethat χ− isrelat−ivisticallyin-
| | | |
an object is. The singular value decomposition of a com- variant and that it behaves as a scalar chirality measure
plex matrix A always exists, meaning that A can always as defined e.g. in [2]. We will show in Sec. IV that χ is
| |
be written as: also normalizable to the interval [0,1].
A=BDC , (9)
†
IV. MAXIMALLY ELECTROMAGNETICALLY
where B and C are unitary matrices and D is a diag- CHIRAL OBJECTS
onal matrix made of real numbers d such that d 0
l l
≥
and d d d .... The same decomposition exists
1 2 3 There are many ways of defining the distance between
≥ ≥
for completely continuous operators [63, Chap. II, 2],
the two vectors in Eq. (12). We will now impose a re-
§
quirementwhichnarrowsdownthechoices. Therequire-
ment that we impose is that, given two em-chiral ob-
jects(A,B)whoseinteractionoperatorsareproportional
7 O→XOX−1 isthetransformationruleforanoperatorOupon S = αS , their em-chiralities must then be related as
theactionofoperatorX. Theruleforavectoris|Ψ(cid:105)→X|Ψ(cid:105). B A
8 From Eq. (8) we obtain S = M−1(TR)S(TR)−1M, and write
(TR)−1M = (XG)−1, where G acts only on the polarization
index and its action is to flip helicity, and X acts only on the
other three indices. Both G and X are unitary. It then follows 9 This follows because two matrices are related by unitary trans-
that: (cid:104)+η¯|S++|η +(cid:105)=(cid:104)−η¯|XS−−X†|η −(cid:105),and(cid:104)−η¯|S+−|η +(cid:105)= formations if and only if their singular values are equal [65, p.
(cid:104)+η¯|XS+X†|η −(cid:105),whichisaparticularcaseofEq.(6) 193].
−
5
χ = α χ . This property, often called homogene- is always greater or equal than zero. Therefore, given
B A
| | | || |
ity or scalability, is not necessarily met by a distance, C = 0, χ has its unique global maximum for ei-
int
(cid:54) | |
but is always met by a norm. Since a norm also has all ther(v =0, v =0)or(v =0, v =0). Thisimplies
+ +
the properties of a distance, imposing scalability boils transparency fo−r fi(cid:54) elds of helic(cid:54)ity +1−or 1, respectively,
down to imposing that the distance between v+ and v since, for example v = 0 S+ =−S− = 0: That
in Eq. (12) must be measured by a norm. − is, the object neither−absorbs⇐o⇒r scat−ters fie−lds of helicity
There is a physical reason for selecting a particular λ = 1. The attained maximum value is χ = √C ,
int
− | |
norm. When an incident state η λ interacts with an and hence:
| (cid:105)
object and a measurement of the scattering into a dif-
(cid:112)
ferent state λ¯ η¯ is made, the number of “clicks” or χ [0, Cint]. (17)
(cid:104) | | |∈
the intensities at the detector are proportional to the
Since a particular norm can be chosen on physical
square of the absolute value of the corresponding coeffi-
cient: λ¯ η¯S η λ 2. Let us consider the sum over all grounds,wecanestablishanabsoluteorderingofobjects
|(cid:104) | | (cid:105)| with respect to their em-chirality using Eq. (15).
possible incident and output states
(cid:88)(cid:88)
C = λ¯ η¯S η λ 2. (13)
int
|(cid:104) | | (cid:105)| V. RECIPROCAL OBJECTS OF MAXIMUM
λλ¯ ηη¯ ELECTROMAGNETIC CHIRALITY MUST BE
This quantity can be understood as the total interaction DUAL SYMMETRIC
cross section of the object. It is a measure of the over-
all coupling between the object and the electromagnetic Let us now consider a maximally em-chiral object in
field,includingbothscatteringandabsorption. Itcanbe the common case that its electromagnetic interaction is
shown10 that reciprocal. Wewillshowthatmaximumem-chiralityplus
reciprocity implies electromagnetic duality symmetry.
In the basis of plane waves with well defined momen-
C =(v )Tv +(v )Tv , (14)
int + + tum p and helicity λ, the reciprocity condition [66, Eq.
− −
where T means transposition. 2.22] results in the following relationships between input
and output states11:
We see that the total interaction cross section is the
sum of the squared Euclidean norms of v and v . We
+
hence select the Euclidean norm to compute |χ|: − λ¯ p¯ S p λ = λ pS p¯ λ¯ . (18)
(cid:104) | | (cid:105) (cid:104) − | |− (cid:105)
(cid:113) (cid:115)
(cid:88)
χ = (v v )T (v v )= (v (l) v (l))2. Let us say that the maximally em-chiral object in ques-
+ + +
| | − − − − − − tion is transparent to the positive helicity. In particu-
l
(15) lar S+− = 0. If the interaction is reciprocal, it follows
We now reach a notable result: With the definition immediately from Eq. (18) that S+ = 0 as well, which
of Eq. (15), the electromagnetic chirality of an object is holds independently of the choice−of basis. The condi-
upper bounded, and the upper bound is attained if and tion S+− = S+ = 0 is the definition of helicity preserva-
only if the object is transparent for fields of one helicity. tion and is e−quivalent to the statement that the object
In order to show this, let us fix a given C = 0, and has electromagnetic duality symmetry. We have reached
int
(cid:54)
then maximize χ2: the conclusion that all maximally em-chiral reciprocal
| |
objects are necessarily dual symmetric. Duality is hence
χ2 (cid:26)(v+)Tv++(v )Tv 2(v+)Tv (cid:27) a requirement/necessary condition for reciprocal objects
max | | = max − −− −
Cint ⇒ Cint to be maximally em-chiral objects.
(cid:26) (v )Tv (cid:27) Itisworthmentioningthatreciprocalinteractiondoes
+
= max 1 2 − not need to be lossless, and that when it is, time re-
⇒ − C
int versal invariance is automatically fulfilled. These results
= v =0,or v =0,
+ have a notable parallelism with portions of the chiral
⇒ −
(16) electroweak theory in the standard model of high energy
physics, where only left chiral fermions interact via the
where the last implication follows because the elements
of v are all real and non negative, so the term (v )Tv
+
± −
11 The reciprocity condition is given in [66, Eq. 2.22]:
10 Thetotalsumcanbecomputedbyaddingthefourpartialsums (cid:104)εf pf|S|pi εi(cid:105) = (cid:104)ε∗i −pi|S|−pf ε∗f(cid:105), where εf,i are general
polarization vectors. Equation (18) follows from using helic-
for each combination of incident and output helicities Cint =
(cid:80)λλ¯Ciλnλ¯t = (cid:80)λλ¯(cid:80)ηη¯|(cid:104)λ¯ η¯|S|η λ(cid:105)|2 = (cid:80)λλ¯trace(cid:16)Mλλ¯†Mλλ¯(cid:17). ittiyonpooflatrhizeaTtiEonavnedctToMrsεp(opla,λri)z,awtiohnichveacrteortsh,eansudmthaencdorsruebstproanc--
The final result Cint = (v+)Tv+ +(v−)Tv− is reached using dences with our notation |p ε(p,λ)(cid:105) ≡ |p λ(cid:105) and [51, Sec. 3]:
Eq.(11)andthepropertiesofthesingularvaluedecomposition. |−pε∗(p,λ)(cid:105)=|−p(cid:15)(−p,λ)(cid:105)≡|−pλ(cid:105).
6
maximally em-chiral object have with respect to their
interaction with fields of pure helicity.
We will now discuss reciprocity together with maxi-
mum em-chirality in the macroscopic equations and in
the dipolar approximation.
A. Constraints in polarizability tensors and
constitutive relations
(a)Generalobject. Interactswithandmixesbothhelicities. For an object embedded in an isotropic and ho-
mogeneous medium with permittivity and permeability
((cid:15) ,µ ), the conditions of transparency to one helicity
s s
and reciprocity restrict the constitutive relations of the
material of which the object is made
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
Z D (cid:15) χ E
s = , (19)
B γ µ Z H
s
through the relations
reciprocity:[67,Eq. 5.5-17] (cid:15)=(cid:15)T, µ=µT, χT = γ,
−
(b)Maximallyelectromagneticallychiralobject. Transparentto transparency to helicity +1: (cid:15)=iχ, µ= iγ,
−
onehelicity.
transparency to helicity 1: (cid:15)= iχ, µ=iγ,
− −
(20)
Each of the ((cid:15),µ,χ,γ) is a 3 3 tensor. The boxed equa-
×
tions for transparency to one helicity are readily reached
by changing the basis in Eq. (19) to the combinations
F = 1 (Z D iB) and G = 1 (E iZ H) and
± √2 s ± ± √2 ± s
nulling the appropriate column of 3 3 blocks for trans-
×
parency to the +1 or -1 helicity, namely
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
F 0 b G F a¯ 0 G
+ = + , or + = + . (21)
F 0 a G F ¯b 0 G
− − − −
(c)Maximallyelectromagneticallychiralreciprocalobject.
As expected, the first line in Eq. (20) plus any of the
Transparenttoonehelicity. Necessarilypreserveshelicityupon
interaction. other two imply duality symmetry [58, 59]: (cid:15) = µ, χ =
γ. Thisforcesband¯binEq. (21)tobeequaltozero. In
−
FIG.2. Interactionofageneralobject(a),amaximallyelec- the end, the only freedom left in a maximally em-chiral
tromagneticallychiralobject(b),andareciprocalmaximally reciprocal object is a symmetric three by three complex
em-chiral object (c) with fields of pure helicity ±1. Fields of tensor and the choice of transparency to the +1 (upper
helicity+1areblueandmarkedwitha“+”. Fieldsofhelicity
signs) or 1 (lower signs) helicity:
−1areredanmarkedwitha“−”. Incomingfieldsaredrawn −
asbullet-likeshapesandscatteredfieldsascloudssurrounding (cid:15)=(cid:15)T =µ= iχ= iγ. (22)
thescatterers. Ageneralobjectinteractswithandmixesboth ± ∓
helicities (a). A most electromagnetically chiral object (b) is
Inthefieldofmetamaterials,effectiveconstitutiverela-
transparent to one helicity. A reciprocal maximally electro-
tionsareobtainedfromthejointresponseofanensemble
magneticallychiralobject(c)must,besidesbeingtransparent
of electromagnetically small objects. The response of a
toonehelicity,alsobehelicitypreserving: Thescatteredfield
small enough object is approximately determined by its
shall have the same helicity as the incident field. The object
must hence have electromagnetic duality symmetry. induced electric (d) and magnetic (m) dipolar response
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
d α α E
= dE dH . (23)
m α α H
mE mH
weak force, the interaction is unitary (lossless) and time
reversal is a good symmetry [47, Sec. 3.3.1]. The same kind of analysis that lead us to Eq. (22)
Figure 2 depicts the different behavior that a general leads to a similar result. The reciprocity conditions for
object, a maximally em-chiral object and a reciprocal polarizability tensors have the same form as in Eq. (20)
7
[68]. Transparency to one helicity can be imposed by a 3 3 block structure like:
×
changing the fields as before and changing the dipoles to
(cid:20) (cid:21) (cid:20) (cid:21)
the combinations (d im/c)/√2. These combinations 0 0 a¯ 0
± , or , (25)
radiate fields of single helicity content [61, Sec. 2.4.3]. 0 a 0 0
Again,reciprocityplustransparencytoonehelicityimply
(dipolar)duality(α =(cid:15) α , α = α /µ ),and which ensures a null response to E iZH or E+iZH,
dE s mH mE − dH s −
the final result is respectively. Thenullresponseisindependentofwhether
the E and H fields belong to the near, intermediate, or
α =αT =(cid:15) α = iα /Z = iµ α /Z . far field zones of the exciting source.
dE dE s mH ± dH s ∓ s mE s We now sketch two concept proposals for applications
(24)
ofmaximallyem-chiralandreciprocalobjects: Enhanced
We note that the findings in [13], obtained for the par-
circular dichroism measurements of molecules and angle
ticular case of planar circuits, are consistent with our
independent helicity filtering glasses.
results.
The conditions in Eq. (24) describe maximally em-
chiral dipolar objects which do not couple to one of the
helicity components of the field G . This zero coupling A. Double resonantly enhanced circular dichroism
is independent of whether (E,H) a±re far fields in the ra- setup
diation zone or near fields around a scatterer.
Circulardichroism(CD)isusedtodistinguishbetween
thetwoenantiomericformsofchiralmolecules. Thisdis-
tinction is particularly important for synthetic drug pro-
VI. APPLICATIONS duction because the two enantiomers can have very dif-
ferent effects. The weak response of the molecules typi-
cally results in low sensitivity and/or long measurement
Before discussing two practical applications of maxi-
times. Geometrically chiral plasmonic structures featur-
mally em-chiral reciprocal objects, we highlight two re-
ing strong scattered near fields upon external illumina-
markable benefits of using helicity to treat the polariza-
tionarebeingstudiedforenhancingtheCDsignalofthe
tion of the field [61, Sec. 2.9], which we will exploit.
molecules in their vicinity. This design principle has two
First, helicity commutes with rotations and transla- importantdrawbacks. Oneisthatthenearfieldofagen-
tions. This means that after rotating and displacing a eral geometrically chiral structure is not of pure helicity,
helicity preserving object, it remains helicity preserving. evenwhentheexternalexcitationis(seeFig. 2(a)). The
This is not the case if one uses a different description of molecule is thus illuminated by a field of mixed hand-
the polarization. For example, an object with parity in- edness which blurs the CD measurement. The second
version symmetry, like a sphere, preserves the parity of drawback is that the plasmonic structure itself produces
the fields interacting with it when located in the origin a strong CD signal. We argue that a double resonantly
of coordinates. After a displacement, the multipoles of enhanced circular dichroism setup can be designed by
differentparitywillmixwitheachotheruponscattering. placing two resonant maximally em-chiral reciprocal ob-
Second, helicity preservation and transparency to one jectsofoppositehandednessclosetoeachother,andthat
helicity are properties which do not depend on whether this scheme avoids the two aforementioned problems.
one considers the near, intermediate, or far field zones. Let us start by considering two maximally em-chiral
At the root of this property lies the fact that, for a field reciprocal objects of opposite handedness O and O¯.
of pure helicity one of the two combinations E iZH in StraightforwardsymmetryargumentsshowthatifO isa
±
Eq.(2)isequaltozeroatallspacetimepoints. Forexam- maximally em-chiral reciprocal object with a resonance
ple, the field scattered off a dual symmetric object upon forhelicity+1,asuitableO¯canbeobtainedasthemirror
illumination with a general field of helicity λ (G = 0) imageofO,whichwillbeamaximallyem-chiralrecipro-
λ
(cid:54)
haszerocomponentofhelicity λ(G =0)inanyfield cal object with a resonance at the same frequency as O,
λ
zone. Thisisillustratedin[52,−Fig. 1]−forthenearfields. but for the opposite helicity.
Different dual symmetric objects are illuminated with a As previously discussed, if we place O and O¯ close
field of λ = +1. The numerical solution of Maxwell’s together,theyremainelectromagneticallyuncoupled,in-
equations show that the fields at a distance of about dependently of their relative orientation or separation.
1/30-th of the wavelength away from the objects have As a result, illuminating the pair with light of a given
zero intensity of the λ = 1 component. Similarly, an helicity does not produce any scattering of the opposite
−
otherwise arbitrarily complex near field of pure helicity helicity. In Fig. 3, a chiral molecule is in the vicinity
will not excite an object which is transparent to that of such a system. The three panels show a sequence of
helicity. This can be deduced from the constraints for events for illustration purposes. In Fig. 3(a) an exter-
transparency and duality symmetry in Sec. VA. When nal field of well defined helicity λ = 1 is incident on the
expressed in the helicity basis, the constitutive relations system. The resonance in O will illuminate the molecule
and polarizability tensors for maximal em-chirality have with a strong field of pure helicity λ=1. Assuming that
8
(a)Illuminationwithabeamof (b)(Zoomed)Themoleculeisilluminatedby (c)Themolecularfieldwithhelicity-1excites
helicity+1,whichexcitestheblue thenearfieldoftheexcitedobject,whichisof theredobjectontheright,whichproducesa
objectontheleft. helicity+1,andproducesaweakfieldwith strongfieldofhelicity-1. Thisfieldismeasured.
bothhelicities.
FIG. 3. Double resonant enhancement for circular dichroism measurements. Panel (a): Two resonant maximally electro-
magnetically chiral and reciprocal objects are placed close to each other. A chiral molecule is in their vicinity. The external
illuminationexcitesonlyoneoftheobjects,whoseresonanceilluminatesthemoleculewithastrongfieldofthesamehelicityas
theincidentbeam. Panel(b): Uponillumination,themoleculeproducesaweakscatteredfieldcontainingbothhelicities,which
excite the two resonant objects. Panel (c): The scattered field of helicity opposite to the initial one is measured. This field
exists due to the presence of the molecule. The other half of the circular dichroism measurement is obtained by interchanging
the input and measured helicities. The final difference features the two resonant enhancements of opposite helicity: One in
illumination and one in the amplification of the field scattered by the molecule.
themoleculeisnotdualsymmetric12,theinteractionwill themsuitableasfocusingandcollectinglensesinthepro-
result in a weak field containing both helicities, as de- posed measurement scheme.
picted in Fig. 3(b). The molecular field will excite both
structures. In particular, the portion with λ = 1 will
excite the resonance of O¯ producing a strong sc−attered
field of λ = 1 that can then be measured by an appa-
−
ratus which selects a single field handedness (Fig. 3(c)).
The measured power of the λ= 1 component depends
−
on the helicity flipping operator S+− of the molecule. We
notethatthemeasurementhasbeenenhancedbytwores- B. Angle independent helicity filtering glasses
onant interactions of opposite helicity, one in amplifying
theilluminationandoneinamplifyingthefieldscattered
by the molecule. The other half of the circular dichro- A second application is helicity filtering glasses. For
ism measurement is obtained by changing the helicity of this purpose we consider a slab of material contain-
the incident field and measurement apparatus. Chiral ing randomly arranged maximally em-chiral and recip-
omnoltehceulmesaghnaveteo-Se+−lec(cid:54)=triSc−+pawrtheorfeththeeirdpiofflearreinzacbeilditeypetnedns- rnoecsas/lppaarrttiicclleesdewnistihtyl/olsossesse.s,Fothrelasrlgaebewniolulgfihlteslraboutthiocnke-
sors. The difference between the two measurements will of the helicities by absorption. The other helicity will
featurethetwofoldenhancement. Theschemeissuitable pass straight through. This behavior is independent of
fordistinguishingbetweenthetwoenantiomericformsof the angle of incident of the field due to the orientation
a chiral molecule. independentcharacterofhelicitypreservationandtrans-
Finally, we note that the generation and measurement parency. Two of these slabs made with particles that
of pure helicity modes in the collimated regime at opti- are the mirror image of each other make suitable glasses
cal frequencies is straightforward and can be done with forviewing3Dprojectionswheretheimagesdestinedfor
polarizers and quarter wave-plates [62, 69], and that mi- each eye are encoded in the two circular polarizations
croscope objectives designed to meet the aplanatic ap- (see Fig. 4). The filtering ability of the glasses is inde-
proximation preserve helicity [62, App. C], which makes pendent of the relative orientation between the user and
the projector. This is in sharp contrast to designs based
on the paraxial optical paradigm of “quarter wave plate
12 Dualitysymmetryrequiresamoleculetohavecomparableelec- plus linear polarizer”, whose polarization discrimination
tricandmagneticresponses,whichisnotthecaseformostchiral degrades asthe angleof incidencedeviates fromthe nor-
molecules. mal.
9
of a single plane wave. The plane wave has a wavelength
of 200 µm and its momentum is perpendicular to the
helix axis. In this section, all the quantities are com-
puted from the interaction matrices and are implicitly
frequency (wavelength) dependent.
1
χ/√C
int
| | (cid:26)(cid:26)D
0.8 ∆
0.6
FIG. 4. Two slabs containing lossy maximally electromag-
neticallychiralandreciprocalobjectsofoppositehandedness.
For large enough slab thickness/particle density/losses, each 0.4
slab filters out one of the helicities by absorption. The other
onepassesrightthrough. Thisbehaviorisindependentofthe
angle of incidence. The slabs can be used to design glasses 0.2
forviewing3Dprojectionswheretheimagesdestinedforeach
eyeareencodedinthetwocircularpolarizations. Theglasses
wouldallowtoseethe3Deffectevenatlargeanglesfromthe 0
perpendicular of the projector. 300 250 200 150 100
Wavelength in vacuum (micrometers)
√
VII. NUMERICAL STUDY FIG.5. Normalizedem-chiralityχ/ C ,measureofduality
int
breaking(cid:26)D, and contrast of helicity cross sections ∆ for the
double turn silver helix shown in the inset. Large values of
In this final section, we study the em-chirality proper- √
χ/ C coincide with simultaneously large values of ∆ and
ties of the double turn silver helix depicted in the inset int
small values of(cid:26)D. The dimensions of the helix are: Major
of Fig. 5. The aim of the study is two fold. On the one
radius a = 6.48 µm, height b = 8.52 µm, and wire radius
handweusearealisticobjecttoillustratetwoimportant
c=0.8 µm.
ideas: Large em-chirality needs duality symmetry, and,
large em-chirality is possible in the presence of material
absorption. On the other hand, we show a realistic ob-
ject which, in a narrow frequency band, approaches the 1
C+
maximum em-chirality. int
Our choice of structure is motivated by the geometri- 0.1 C+ /CCi−+nt
cally optimized helical antennas for circular polarization abs int
[70]. Notablepropertiesofsimilarantennashavebeenre- Ca−bs/Ci−nt
cently studied [71–73]. Under some approximations, the 0.01
geometrically optimized helical antennas can be shown
to meet the dipolar duality condition at their resonant
0.001
frequency. Additionally, they present largely different
crosssectionstothetwopolarizationhandednessofplane
waves with momentum perpendicular to the helix axis. 0.0001
Instead of using common approximations like thin helix
wire or restriction to dipolar scattering, we obtain the
complete interaction matrix at each frequency by using 1e-05
the permittivity of silver from Ref. 74 and a technique 300 250 200 150 100
similar to the one described in [75, Sec. 4.1]. Exact nu- Wavelength in vacuum (micrometers)
merical solutions of the Maxwell’s equations based on a
finite-element method allow us to obtain the T-matrix
FIG. 6. Interaction cross sections and the ratio of absorp-
of the helix, which is related to the interaction operator
tion to interaction cross sections for each helicity ±. At 200
[43, Eq. 2.7.20] as S = 2T. To obtain the geometrical
µm, the maximum of em-chirality in Fig. (5), there is non-
parametersofthehelix,afirstinitialguessfollowingRef. negligibleabsorptionofthehelicitywithdominantinteraction
70ismade. Thisisfollowedbyalocaltuningofitsmajor cross section.
radius a and height b. The local tuning seeks to maxi-
mize the difference between the scattering cross sections We now define two parameters which we use in the
that the helix presents to the two circular polarizations analysis. Recalling that C = vTv +vTv , we define
int + +
− −
10
thecontrast∆betweentheinteractioncrosssectionscor- been used to design an object which exhibits optical ac-
responding to two input helicities as tivity in general scattering directions [55]. As can be
seen in [55, Fig. 2a], the restriction to dipolar duality
∆= v+Tv+−v−Tv−, (26) constraintsthechoiceofthespherespermittivityandra-
C dius to a narrow region in such parameter space.
int
and the total helicity change (duality breaking)(cid:26)(cid:26)D as13
VIII. CONCLUSION
(cid:26)(cid:26)D = σ(cid:0)M+−(cid:1)T σ(cid:0)M+−(cid:1)C+σ(cid:0)M−+(cid:1)T σ(cid:0)M−+(cid:1). (27) In summary, we have defined the electromagnetic chi-
int
rality of an object based on how it interacts with fields
The contrast ∆ ranges from 1 to 1 and is equal to zero ofdifferenthelicities(polarizationhandedness). Thedef-
−
when the object presents the same interaction cross sec- inition leads to relativistically invariant scalar measures
tion to both helicities. The measure of duality break- of electromagnetic chirality. Physical considerations al-
ing(cid:26)(cid:26)D ranges from 0 to 1. Zero means complete helicity low to choose a particular measure. We have shown that
preservation and 1 complete helicity flipping. the electromagnetic chirality of an object has an upper
Figure(5)showsthenormalizedem-chiralityχ/√C , bound,andthatthisupperboundisattainedifandonly
int
(cid:26)(cid:26)D,and∆ofthetwoturnhelixasafunctionofthewave- if the object is transparent to one of the two helicities.
length in vacuum. The maximum value of χ/√C = Theupperboundisequaltothesquarerootoftheinter-
int
0.92 is achieved near 200 µm. The figure illustrates the actioncrosssectionoftheobject. Whentheinteractionis
two conditions needed for large em-chirality: Large con- reciprocal,asismostcommonlythecase,anymaximally
trast between the two helicity interaction cross sections electromagneticallychiralobjectisnecessarilydualsym-
and small helicity change. metric, i.e. it does not change the helicity of the fields
Figure(6)showstheinteractioncrosssectionsandthe interactingwithit. Wehavederivedtherestrictionsthat
ratio of the absorption to the interaction cross sections these extremal objects must meet in two settings: The
for each input helicity14 . The figure shows that large dipolar approximation and the macroscopic Maxwell’s
values of em-chirality can be achieved in the presence of equations. Therestrictionsintheirpolarizabilitytensors
absorption losses. This is consistent with the fact that ormaterialconstitutiverelationsarepreciserequirements
the conditions in Eqs. (22) and (24) can be met by both forthedesignofmaximallyelectromagneticallychiralob-
lossy and lossless objects. This possibility originates in jects. Electromagnetic duality symmetry is one of them.
the use of reciprocity instead of time reversal invariance We have sketched two applications that show that these
inEq.(18), whichavoidshavingtorestricttheresultsto theoretical results also have practical value. Numerical
the lossless case. The principle of reciprocity has been analysisshowsthat,atleastinanarrowfrequencyband,
recently used to obtain a theory of circular dichroism in a realistic structure can come very close (92%) to being
planar systems [20]. maximally electromagnetically chiral even in the pres-
Thesamedesignprocedurethatwefollowedat200µm ence of losses. The analysis is also an example of how
producessignificantlylowerem-chiralityvaluesathigher the theoretically obtained requirements can be used to
frequencies. For example, at 20 µm the maximum of guide a practical design.
χ/C that we obtain is equal to 0.65. While the design
int
procedure that we used is not optimal, we take this as
an indication that a different strategy and/or materials ACKNOWLEDGMENTS
may be needed for maximizing em-chirality beyond the
near infrared. As far as we know, objects with the de- IFCwishestowarmlythankDr. MauroCirioforread-
sired properties are not yet available at optical or near ingthemanuscriptandprovidingvaluablefeedback, and
UV frequencies, where they would be relevant for the Ms. Magda Felo for her help with the figures. We
two applications sketched in Sec. VI. We hope that our also thank Dr. Tilo Arens for his comments on the
contribution increases the research in that direction. At manuscript. M.F. acknowledges support by the Karl-
optical frequencies, one may consider the fashioning of sruhe School of Optics & Photonics (KSOP). We also
structures out of high index dielectric spheres meeting gratefullyacknowledgefinancialsupportbytheDeutsche
the dipolar duality condition. This strategy has recently Forschungsgemeinschaft (DFG) through RO 3640/3-1.
13 Thisdefinitionof(cid:26)D isequivalenttotheonein[55,Eq.(2)]. inputhelicityλcanbecomputedas: trace(cid:16)L†L (cid:17),whereL =
14 Whentherearemateriallosses,thescatteringoperatorS˜=I+ λ λ λ
(cid:18)I+Mλ(cid:19)
S is not unitary. The total absorption cross section for each λ .
M−λ
λ