Group Theory of Circular-Polarisation Effects in Chiral Photonic Crystals withFour-Fold Rotation Axes, Applied to the Eight-Fold Intergrowth of Gyroid Nets

Matthias Saba,1, ∗ Mark D. Turner,2 Klaus Mecke,1 Min Gu,2 and Gerd E. Schroder-Turk1, †

1Theoretische Physik, Friedrich-Alexander Universitat Erlangen-Nurnberg, 91058 Erlangen, Germany2CUDOS & Centre for Micro-Photonics, Swinburne University of Technology, Victoria 3122, Australia

(Dated: November 28, 2016)

We use group or representation theory and scattering matrix calculations to derive analyticalresults for the band structure topology and the scattering parameters, applicable to any chiralphotonic crystal with body-centered cubic symmetry I432 for circularly-polarised incident light.We demonstrate in particular that all bands along the cubic [100] direction can be identified withthe irreducible representations E±, A and B of the C4 point group. E+ and E− modes representthe only transmission channels for plane waves with wave vector along the ∆ line, and can beidentified as non-interacting transmission channels for right- (E−) and left-circularly polarised light(E+), respectively. Scattering matrix calculations provide explicit relationships for the transmissionand reflectance amplitudes through a finite slab which guarantee equal transmission rates for bothpolarisations and vanishing ellipticity below a critical frequency, yet allowing for finite rotation of thepolarisation plane. All results are verified numerically for the so-called 8-srs geometry, consistingof eight interwoven equal-handed dielectric Gyroid networks embedded in air. The combination ofvanishing losses, vanishing ellipticity, near-perfect transmission and optical activity comparable tothat of metallic meta-materials makes this geometry an attractive design for nanofabricated photonicmaterials.

Optical properties, such as optical rotation or cir-cular dichroism, that are caused by a chiral structureof a light-transmitting medium or of its constituentmolecules, remain of great interest in many different con-texts. Circular dichroism spectroscopy of optically ac-tive molecules in solution is used in bio-chemistry whereleft-handed (LH) and right-handed (RH) molecular ar-chitectures cause different absorption properties for left-circularly polarised (LCP) and right-circularly polarised(RCP) light1. Optical activity of natural crystals suchas quartz (see e.g.2) and of liquid crystals, both in thetwisted nematic3 and the blue phases4,5, is well-known.Circularly polarised (CP) reflections of insect cuticleswere observed by Michelson a century ago6, with circular-polarisation effects an active topic in bio-photonics ofbeetles7,8, crustaceans9,10 and butterflies11, and alsothe plant kingdom12. Nano-fabrication technologynowadays allows for the fabrication of custom-designedchiral materials, both dielectric photonic crystals13–15

and metallic meta-materials16–18, which the potentialfor technological photonic devices. This ability tofabricate custom-designed structures has led to note-worthy chiral-optical behaviour, including strong cir-cular dichroism19, negative refractive index20 basedon Pendry’s prediction21, optically-induced torque22,handedness switching in meta-molecules23 and circular-polarised beam splitting15. Metallic or plasmonic meta-materials have been designed to give strong opticalactivity16,18,24,25 that is orders of magnitudes strongerthan in the natural materials.

This article makes a two-fold contribution to a deeperunderstanding of circular polarisation effects in chiralmaterials. First, we combine group theory and scatteringmatrix treatment of chiral photonic crystals (PCs) to pre-dict those properties of the band structure that are rele-

a a0 = 2a

FIG. 1. (Color Online) Construction of the 8-srs structure:(left) The BCC translational unit cell is obtained by plac-ing degree-three vertices at the mid-points of the hexagonalfacets of a Kelvin body. Edges connect the hexagon’s centerpoint to every second vertex, such that rotational symmetriesof the Kelvin body are maintained. (right) When repeatedperiodically, the 8-srs consists of 8 equal-handed interwovensrs nets; only one of the eight nets is shown for clarity.

vant for coupling to circularly-polarised light. This anal-ysis applies to all structures with cubic symmetry with atleast one point that has 2-, 3- and 4-fold point symmetry.Many of the ideas underlying this formalism are not spe-cific to this symmetry group and apply, upon suitable ad-justments, to structures with other symmetries. Second,we analyse in detail a specific chiral geometry that fulfilsthe symmetry requirements, namely the so-called 8-srsstructure consisting of eight interwoven non-overlappingGyroid (or srs) nets. Numeric data for its band structureand transmission coefficient, obtained by Finite elementsimulations and finite-difference time-domain methods,is in perfect agreement with our theoretical predictions.Further, these simulations demonstrate that the 8-srsgeometry exhibits a particularly strong chiral-optical re-

arX

iv:1

309.

2910

v1 [

phys

ics.

optic

s] 1

1 Se

p 20

13

2

t = 0 (”eyes”)

432

*

*

* *

t = a/4 (”dog bones”)

*

*

* *

FIG. 2. (Color Online) Different cross-sections with [100] in-clination through the 8-srs reveal its four-fold symmetry. Theparameter t denotes the position of the termination planewithin the cubic unit cell. 4-fold rotation and 42 screw axesare marked by the symbols and *, respectively. The greysquare represents the cross section of the cubic unit cell whosevertices are located at the 432 (Schoenfliess O) symmetrypoint marked by a blue symbol (same choice as in26).

sponse. Specifically, in parts of the frequency spectrum,the lossless dielectric 8-srs material with ε = 5.76 is fullytransparent for left- and right-circularly polarised (LCPand RCP) light, yet exhibits an optical activity that iscomparable to metallic meta-materials. This suggeststhat the use of multiply intertwined Gyroid nets, thatcan be realised with current nano-fabrication methods,is promising for custom-designed photonic materials.

Polarisation conversion measures the relative ratio ofLCP to RCP light transmitted through a photonic mate-rial if the incident light was purely LCP. We will here usethe term optical activity (OA) to denote the rotation ofthe polarisation plane as a linearly polarised plane wavetransmits through a chiral medium and Circular dichro-ism (CD) is utilized to describe the difference in trans-mission rates for LCP and RCP light. The definition ofOA and CD represent a slight deviation from the con-ventional nomenclature where both refer to a respectivephenomenological origin, i.e. a respective difference of therefractive indices and the absorption coefficients betweenLCP and RCP light.

Our use of these terms to describe differences in scat-tering parameters for CP light represents a natural adap-tion for a slab made of lossless but inhomogeneous ma-terial. With t and r denoting the scalar complex trans-mission and the reflection amplitudes, respectively, OAis the phase difference and CD as the relative differencein absolute values between the complex scattering am-plitudes s± ∈ t±, r± for an incoming left (LCP, +) orright (RCP, −) circularly polarized plane wave:

CDs =|s+| − |s−||s+|+ |s−|

; OAs =ϕ

(s)+ − ϕ

(s)−

2; eıϕ

(s)± =

s±|s±|

Group theoretic approaches to the optical propertiesof photonic crystals are motivated by the obvious im-portance of spatial symmetries in these systems. The

common textbook example for this is the strict clas-sification of discrete dispersion bands ωn(k) of a twodimensional PC into transverse-electric and transverse-magnetic modes if the wave vector k is restricted to theplane of periodicity27. In terms of circular polarisation,the role of four-fold rotational symmetries, and more gen-erally of (m ≥ 3)-fold rotations, for the transmission andreflectance coefficients has been recognised28–36. In par-ticular, it has recently been shown that any lossless struc-ture with a 4-fold symmetry axis inclined perpendicularto this axis shows no circular dichroism and polarizationconversion for normal incidence28. Our results on thescattering parameters for chiral PCs presented in this ar-ticle are generalizations of previous works that derivedrelated restrictions on reflection and transmission ampli-tudes for two-dimensional diffraction gratings made ofquasi-planar particles29–31,33 and also particles with 3Dsymmetry34,35. Group theory, or more precisely represen-tation theory, are the natural language to deal with theinfluence of spatial symmetry on physical properties ingeneral37,38 and with photonic properties in particular38.

A particularly intricate design for a chiral photonicmaterial is the so-called single Gyroid (SG) or srs net39,with edges inflated to solid tubes with a given volumefraction φ. This periodic network is composed of identicalthree-coordinated nodes and has cubic chiral symmetryI4132, without pure four-fold rotation axes40; the three-fold and four-fold screw axes correspond to three-fold andfour-fold helical structures along the [111] and [100] direc-tions, respectively. The theoretical prediction of circulardichroism for a dielectric photonic srs crystal11 has beenverified by nano-fabrication experiments14,15. Recently,the first prediction of fully three-dimensional Weyl pointshas been published using photonic designs based on theSG41. For metallic srs nets, discrimination of LCP andRCP modes is observed42,43, but its magnitude is lowerthan what might be expected from its helical nature42.The relevance of the SG geometry, which is inspired bythe biological PCs in wing scales of several butterflies44,as a chiral photonic material is increased by the abil-ity to generate this structure at different length scalesby molding from the self-assembly copolymer structurewith unit cell size a = 50nm45, replication of the but-terfly structures with a ≈ 300nm46, direct laser writingwith a & 1µm14,15 and cm-scale replica for micro-waveexperiments47.

This article investigates a related geometry called 8-srs consisting of eight identical equal-handed intertwinedcopies of the srs net48–50. As Fig. 1 of ref. 51 demon-strates, the 8-srs can be obtained by arranging trans-lated copies of the srs net, such that all 8 networks re-main disjoint and to yield body-centered cubic (BCC)symmetry I432; this is achieved by three orthogonalcopy-translations along the perpendicular [100] directionsof the original 1-srs net by a = a0/2 where a0 is the crys-tallographic lattice parameter of the 1-srs in its spacegroup I4132. Figure 1 illustrates an alternative construc-tion where the 8-srs is obtained by the decoration of the

3

hexagonal facets of a Kelvin body52 by a degree-three net-work, such that 4 (), 3 () and 2 () fold rotations ofthe Kelvin body are maintained. This provides the BCCunit cell of the uncolored 8-srs structures, with all com-ponents undistinguishable, whose lattice parameter a ishalf the size of the parameter a0 of the individual 1-srsnets. Note also the similar construction of the 8-srs as anembedding on a Schwarz periodic Primitive minimal sur-face in Fig. 12 of ref.50. In crystallographic notation, the8-srs has a single vertex at (1/4, 1/4, 1/4) at Wyckhoffsite 8c (symmetry 32) and a single edge with mid-point(1/4, 0, 1/2) at Wyckhoff position 12d (symmetry 222),in the body-centered cubic (BCC) space group I432 (no

211 in26) with lattice parameter a = a0/253. The 8-srs

has four-fold rotation axes and four-fold screw axes alongthe [100] lattice directions (see Fig. 2).

A photonic crystal geometry with finite volume frac-tion is obtained by inflating all edges of the 8-srs struc-ture to solid cylindrical struts with permittivity ε, em-bedded in air or vacuum. For the quantitative resultsin figures 354 and 955 we assume a value ε = 5.76,that closely resembles high-refractive index Chalcogenideglass at telecommunication wavelengths56 or titanium-dioxide at optical wave lengths46; the solid volume frac-tion is set to φ ≈ 31%, corresponding to a rod diameterof d ≈ 0.115; this value is well below the threshold wherethe distinct nets overlap to form a single connected com-ponent. We consider in particular an infinitely wide slabof size ∞×∞ × nza (nz = 4 in Fig. 9 and nz = 53 inFig. 3)57, with the crystallographic lattice vector [100]aligned with the z-axis; all analyses are for wave vectorsk along that axis. The two different terminations58 illus-trated in figure 2 for the 8-srs crystal are chosen for thetransmission simulation in Fig. 3.

The spontaneous formation of the 8-srs geometry byself-assembly appears as an uncertain long-term goal, de-spite progress in self-assembly of simpler poly-networkgeometries59–62. However, the 8-srs structure providesa design pattern suitable for nano-fabrication by directlaser writing technologies. Considering the realisationof 1-srs structures in low-dielectric polymer14,15 and inhigh-dielectric Chalcogenide glasses63, the fabrication ofthe 8-srs material appears feasible.

This article is organised as follows: Section I devel-ops a group theoretical treatment valid for any photoniccrystal with symmetry group I432 (we henceforth referto any such crystal as a I432 PC), including the 8-srsgeometry as a specific case, that can be readily extendedto symmetries P432 or F432. This theory predicts thetopology of the band structure at the Γ and H point andidentify LCP incident light with modes with irreduciblerepresentation E−, and RCP with E+. Numerical datafor the band structure of the 8-srs (Figs. 3 and 8) is inperfect agreement with these predictions, provided thatthe symmetry of the Fourier grid is equally high as thestructural symmetry (Fig. 5). A gap map providing thewidth and position of the band gap in [100] directionas a function of dielectric contrast and volume fraction

is provided (Fig. 7). Section II uses scattering matrixmethods to provide analytical results for the scatteringparameters, both transmission and reflectance, that arevalid for any structure with 4-fold symmetry that is in-clined normal to its symmetry axis. The predictions thatOA and CD are always zero in reflection, and that CDis zero below a threshold frequency in transmission areshown to be correctly reproduced by simulations of the8-srs (Fig. 9). The potential of the 8-srs as a designfor photonic materials and in particular the magnitudeof its optical activity relative to metallic meta-materials,is discussed in the conclusion section.

The paragraphs (a)-(f) that extend over chapter I andII correspond to the statements with the same numberingin51.

I. REPRESENTATION THEORY

A photonic band structure (PBS) can be seen as aclassification of the eigenmodes of an infinite PC by theirtransformation behaviour under its translational symme-try operations characterized by the Bloch wave vector k.This classification yields a deeper understanding of theunderlying physics and is also if practical use, the trans-verse dispersion for example yields a matching conditionat interfaces. In this context, k acts as a continuousquantum number.Here, we additionally classify the band structure modesin the crystallographic [100] direction by their transfor-mation behaviour under the PCs point symmetries (rota-tions, mirrors etc.) and introduce a corresponding seconddiscrete quantum number i.

The PBS eigenmodes are shown to belong to several or-thogonal classes characterized by their symmetry trans-formation behaviour. Group theory (or more specificallyrepresentation theory37,38) is used to show that thereare four such classes with respect to the 4-fold rotationaxis along the [100] direction that can be seen as non-interacting transmission channels. A circularly polarizedand normally incident plane wave decomposes into one ofthe 4 symmetry classes alone and therefore couples lightinto the respective transmission channel only.In the following, we use Dirac notation where the basisfunctions are denoted |αi〉 and any operator acting on el-ements of the corresponding Hilbert space is marked witha hat. The basis functions are ortho-normal, see theorem(iv) below.

In this notation, a point group is a mathematical (ingeneral non-Abelian) group with (unitary or length pre-

serving) point symmetry operations R as its elementsand the operator · that is defined by the action onto an

arbitrary function |f〉 via(R1 · R2

)|f〉 := R2

(R1|f〉

).

The operator R defines the operation of a point sym-metry R ∈ S where the point group S is a finite sub-group of the orthogonal group O(3,R) for I432 and anyother symmorphic space group. It can be represented by

4

0.6

0.65

0.7

0.75

0.4 0.6 0.8 1

Fre

qu

en

cy Ω

=ω

a/(

2π

c)

wave number ka/(2π)

A

B

E+

E-

(χ=1)

(χ=-1)

(χ=i)

(χ=-i)

Γ Δ H

T1

T2

T2

0.6

0.65

0.7

0.75

0.4 0.6 0.8 1

Fre

qu

en

cy Ω

=ω

a/(

2π

c)

wave number ka/(2π)

A

B

E+

E-

(χ=1)

(χ=-1)

(χ=i)

(χ=-i)

Γ Δ H

T1

T2

T2

0 0.4 0.8Transmission

48152953

"dog bones" "eyes"

0 0.4 0.8Transmission

48152953

"dog bones" "eyes"

FIG. 3. (Color online) Photonic band structure and trans-mission spectrum of the 8-srs PC: (Left) Band structure fork along ∆ (see Fig. 2). The bands are colored according totheir symmetry behaviour corresponding to the irreduciblerepresentations i ∈ A,B,E+, E− of the C4 point symmetry(cf. Tab. I). The E± bands that are able to couple to planewaves at normal incidence are further underlaid by dots ofsize proportional to the coupling constant β. Dots are smallerthan the linewidth for β / 0.1 and hence invisible. See dis-cussion in Sec. I B on page 7 for the meaning of the inset.(Right) Transmission of light at normal incidence through aquasi-infinite slab of thickness Nz = 53, [100] inclination andthe two terminations shown in Fig. 2: t = 0 or the eyes cutshown in black and T = 0.25a or the dog bones cut shown inbrown. The spectrum is the same for any polarization stateand is illustrated by the thin lines. The thick and more satu-rated lines represent a convolution with a Gaussian of widthδΩ = 0.002 eliminating the sharp Fano and Fabry-Perot res-onances. The teal points in the transmission spectrum markthe transmission minima at the pseudo-bandgap at roughlyΩg = 0.64 for a slab of thickness NZ = 4, . . . , 53, respectively.

complex square matrices D (R) so that the correspondingmap is linear, i.e. D (R1 ·R2) = D (R2)D (R1). A matrixrepresentation D (R) is called irreducible if no similar-

ity transformation D′(R) = D (R0)D (R)D −1(R0) with

R0 ∈ S exists that simultaneously transforms all D (R)into the same block form, i.e. the representation cannotbe split into representations of lower matrix dimension.Each representation has a set of basis functions |α〉 forwhich the symmetry operator can be replaced by the rep-resentation matrix R |α〉 =

∑β Dαβ(R)|β〉; with α and β

being partners of i and the corresponding indices denot-ing the rows and columns of D . In general, the matricesof an irreducible representation of dimension > 1 andthe corresponding basis functions are not unique due tosimilarity transformation gauge freedom. An irreduciblerepresentation i is uniquely characterized by the similar-ity transformation invariant trace of the respective ma-

trix χ(i)(R) =∑αD

(i)αα(R) that is also known as the

character64:(∀R ∈ S : χ(i)(R) = χ(j)(R)

)⇔ i = j.

A. General Theorems

For our photonic mode analysis below we use fourrepresentation theorems that are all implications of theWonderful Orthogonality Theorem37 and the length pre-serving nature of all point symmetry operations of I432:

(i) The eigenfunctions |n〉 of any operator ϑ that com-

mutes with all operations R of a point group S aregenerally given by the superposition of basis func-tions |iα〉 of one irreducible representation i of Sonly: (

∀R ∈ S : [ϑ, R] = 0)

⇒ |n〉 =∑α

c(i)α |iα〉 =: |ni〉.

(ii) The characters of an arbitrary representation of agroup S can be decomposed into the characters ofthe irreducible representations i by

χ(R) =∑i

di χ(i)(R)

with di =1

h

∑R

χ(i)(R)χ(R).

where z denotes the complex conjugation of a com-plex number z and h =

∑R the number of symme-

try operations in the point group.

(iii) An arbitrary function |f〉 can be expressed by thecomplete set of basis functions |iα〉 of the irre-ducible representations i:

|f〉 =∑i,α

f (i)α |iα〉 =

∑i,α

P (i)α |f〉

with the operator P(i)α that projects onto |iα〉 given

by

P (i)α =

lih

∑R

D(i)

αα(R)R

where li =∑α is the dimension of the irreducible

representation i.

(iv) The basis functions are orthogonal and can be nor-malized so that we assume for all representations iand j and partners α and β:

〈αi|βj〉 = δijδαβ

5

H

Γ

∆

FIG. 4. (Color Online) Brillouin zone of the BCC lattice: Arhombic dodecahedron whose edges are illustrated by yellowbars. The irreducible BZ (1/48 of the BZ due to 24 rotationsin O (Tab. I) and another 24 time-inverted rotations) is thepyramid framed in orange. The ∆ line marked in blue is oneof its edges and connects the Γ point in the origin with the Hpoint that is a 4-connected vertex of the BZ at 2π/a ·(1, 0, 0)T

Representation theorem (i) is used to classify the bandstructure by the symmetry behaviour. We first show that

the operator of the magnetic wave equation65 ϑk of a PCthat has a given point symmetry S commutes with anysymmetry element R that transforms the wave vector kinto an equivalent wave vector k + G that is translatedby a reciprocal lattice vector G only. The set of all thoseoperations R form a subgroup Sk ≤ S and is called thegroup of the wave vector.

In our work we consider all modes of a bulk I432 PCthat can couple to a normally incident plane wave at a(100) interface66. We choose a symmetric set of basis vec-tors a1 = (−d, d, d); a2 = (d,−d, d); a3 = (d, d,−d), withd = a/2, for the body centered cubic translational sym-metry. The Brillouin zone of the BCC lattice illustratedin figure 4 and the reciprocal lattice vectors are given byb1 = (0, b, b); b2 = (b, 0, b); b3 = (b, b, 0), with b = 2π/a.All coupling modes67 lie on a straight line within the Bril-louin zone parameterized by ks = s ∗ (b2 + b3 − b1), s ∈(−0.5, 0.5]. The points at s = 0 and s = 0.5 are denotedby Γ and H, respectively. At both points, the group ofthe wave vector is equal to the full point group that to-gether with the BCC translation gives the full I432 spacegroup: That is the O (Schoenfliess) or 432 (Hermann-Maugin) point group that includes the point symmetriesof a cube, i.e. 6 4-fold (C4) and 3 2-fold (C2) rotationsaround the [100] axes, 8 3-fold (C3) rotations around the

[111] axes and 6 2-fold (C′

2) rotations around the [110]axes (cf. Tab. I). On the ∆ line that connects Γ and H(s ∈ (0, 0.5)), the (Abelian) group of the wave vector isC4 that only contains the 3 rotations around the [100]axis where positive (C41

) and negative (C43) 4-fold rota-

tions fall into distinct classes68.

O 1 6C4 3C2 8C3 6C′2

A1 1 1 1 1 1

A2 1 −1 1 1 −1

E 2 0 2 −1 0

T1 3 1 −1 0 −1

T2 3 −1 −1 0 1

C4 1 C41 C2 C43 TR

A 1 1 1 1 (a)

B 1 −1 1 −1 (a)

E+ 1 i −1 −i (b)

E− 1 −i −1 i (b)

TABLE I. Character tables for the O and C4 point groupsrelevant for the H (Γ) point and ∆ line (Fig. 4), respectively.The time-reversal symmetry type TR of the irreducible rep-resentations of C4 are added in the last column.

The magnetic wave equation is given by

ϑkukn(r) := (ık +∇)×[

1

ε(r)(ık +∇)× ukn(r)

]=ω2kn

c2ukn(r)

with the periodic part of the Bloch field ukn(r), the peri-

odic dielectric function ε(r), the eigenfrequency ωkn, the

imaginary number ı :=√−1 and the velocity of light c.

We define the action of the operator R on a vector fieldby RF (r) = R F (R−1r). The matrices R are here thecommon 3×3 matrices that transform three-dimensionalvectors according to a point symmetry R, for examplegiven by Rodriguez formula in case of a rotation etc.In the 432 group these matrices are a suitable choicefor the irreducible representation T1 (Tab. I). The skewcan be understood in terms of an active transformationacting on the vector field itself whereas the change ofits position is achieved by a passive transformation thatchanges space itself. With that definition it is trivial thatR 1ε(r)F (r) = 1

ε(r) RF (r) for any dielectric function that

is invariant under R so that we are left to show that Rcommutes with the cross products in the wave equation.The proof is done in a Cartesian basis with Einstein con-vention and the abbreviation r

′= R−1r69:[

(ık +∇)× Ru(r)]i

= εijk(ıkj +∂

∂xj)Rklul(r

′)

= εijkRkl

ıkj +∂x′

m

∂xj︸ ︷︷ ︸(a)=Rjm

∂

∂x′m

ul(r′)

(a)= εojkRopRipRklRjm

(ıRqmkq +

∂

∂x′m

)ul(r

′)

(b)= ±Ripεpml

(ı(km −Gm) +

∂

∂x′m

)ul(r

′)

= ±[R (ı(k −G) +∇)× u

]i(r′)

= ±[R(ı(k −G) +∇)× u(r)

]i

6

Here we make use of the facts that (a) point symmetriesare length preserving and hence the matrix representa-tion is orthogonal, i.e. RijRkj = δik with the Kroneckersymbol δij , and that (b) the Levi-Civita symbol εijk is

invariant under proper (det(R ) = 1) rotations R(+) andchanges its sign under improper (det(R ) = −1) rota-

tions R(−), i.e. R(±)il R

(±)jmR

(±)kn εlmn = ±εijk. The identity

in (b) is evident through expansion of the Levi-Civitatensor into a triple product of Cartesian basis vectors (sothat R takes a basic form) and making use of (a).The change of sign for improper rotations cancels out by

the double cross product in the ϑk so that:

ϑkR uk,n(r) = (ık +∇)×[

1

ε(r)(ık +∇)× R uk,n(r)

]= R

(ık′ +∇)×

[1

ε(r)(ık′ +∇)× uk′,n(r)

]= R

ω2k′,n

c2uk′,n(r) = R

ω2k,n

c2uk,n(r) = R ϑk uk,n(r)

where we substitute k′ = k−G and use the invariance ofthe periodic Bloch function uk+G = uk and the eigenfre-quencies ωk,n = ω(k+G),n.

B. Topology and Symmetry Classification of theBandstructure

While the exact shape of the PBS strongly depends ongeometry and dielectric contrast, the topology, i.e. de-generacies and hence also band connections at high sym-metry points, is generally induced by symmetry alone.This fact combined with symmetry based selection rulesis what makes group theory a powerful tool. Here, wederive the topology of the band structure of an I432 PCand show that, for the 8-srs PC, these results are inperfect agreement with numerical results.

(a) Degeneracies at the lowest Γ and H points

To derive the irreducible representation of the eigen-modes at the high symmetry points in the BZ, it issuitable to group the plane wave components uGσ(r) =

uGσeı(k+G)·r of a Bloch mode ukni

(r) =∑Gσ uGσ(r)

with σ ∈ u, v denoting a linear polarization basis by thelength of k + G, i.e. its vacuum frequency, in equiva-

lence classes [k +G] :=uGσG

′ : |k +G| = |k +G′|

.

We sort the equivalence classes at the high symmetrypoints by ascending vacuum frequency and denote thecorresponding class by Γ(n) and H(n) (n ∈ N0), respec-tively. All elements within each equivalence class forma basis for a (generally reducible) representation of thegroup of the wave vector k by its definition. This repre-sentation can then be reduced into irreducible representa-

[k] Ω0 k R =

Γ(0) 0 (0, 0, 0)T T †1

H(0) 1 (b, 0, 0)T 2T1 + 2T2

Γ(1)√

2 (b, b, 0)T 2A2 + 2E + 4T1 + 2T2

H(1)√

3 (b, b, b)T 2E + 2T1 + 2T2

TABLE II. Summary of the results of section B.(a). Thevacuum frequency Ω0, a representative wave vector and thecompatibility relation are respectively listed for the 4 low-est points where the group of the wave vector exhibits 432symmetry. We use a dagger (†) at the T1 representation for

Γ(0) to indicate that the T1 representation does not split intoA + E+ + E− but only into E+ + E−. The static A modeexists at Γ(0) itself to give a full basis for a homogeneous 3Dvector field. However, it violates the divergence theorem fork ' 0.

tions of the group of the wave vector with representationtheorem (ii).

We first consider the H(0) point with |k + G| = b.There are 6 points in the group of the wave vector withk = ±bei (i ∈ x, y, z). Hence, there are 12 plane wavesthat form a closed set of basis functions for a reduciblerepresentation R

(H(0)

)within the 432 point group. The

character of this reducible representation is determinedby the corresponding representation matrix of a pointsymmetry R. First, we need to know how many of thesepoints are unchanged under R as we only count diagonalelements to obtain the trace, i.e. equivalent to the num-ber of points on the axis (or plane) of symmetry. Then wemultiply by the character of the point symmetry actingon a two-dimensional vector in the plane perpendicularto the axis of symmetry which is the contribution of thepolarization mode pair on each point: The character ofthe identity transformation 1 is 2, of the C4 rotation 0and of the C2 rotation −2. All other operations changeall k on the Hn or Γn points into k + G with G 6= 0and hence cannot contribute to the trace. The describedprocedure yields the following characters:

R 1 6C4 3C2 8C3 6C′

2

χR(H(0))(R) 12 0 −12 0 0

Note, that the number −12 for the C2 rotation alreadyincludes the number of elements in that class, i.e. 3. Ap-plication of representation theorem (ii) using the char-acters in Tab. I we obtain the compatibility relationR = 2T1 + 2T2 that tells us that the 12 dimensionalrepresentation of the plane waves at the high symmetrypoints is reduced into respectively 2 three-dimensionalrepresentations T1 and T2 that are irreducible within 432.This result is also numerically obtained and illustrated infigure 3.

7

R(O) A1 A2 E T1 T2

R(C4) A B A+B A+ E+ + E− B + E+ + E−

TABLE III. Compatibility relations obtained by reduction ofthe irreducible representations of O in C4.

(a) (b)

(c)

FIG. 5. (Color Online) Incompatibility of planar hexagonaland spatial BCC symmetry with the discrete Fourier grid usedin PBS plane-wave based frequency domain eigensolvers. Forany planar object with hexagonal symmetry, a discretized rep-resentation using a Fourier grid with two linearly-independentbasis vectors cannot maintain the full hexagonal symmetry, as(a) illustrates for a circle; a discretization by smaller Wigner-Seitz cells, as shown in (b) would alleviate this problem butcannot be implemented for the Fourier analysis. (c) The BCCcase represents a 3D analogon, with the same problem. TheBrillouin zone (solid cell) is a rhombic dodecahedron withfull 432 symmetry, yet the parallelepiped formed by the basisvectors of the Fourier grid representation (hollow cell) onlymaintains D3 symmetry with a single 3-fold axis (red) andthree 2-fold axes (cyan).

Analogously, the irreducible representations for allhigher order Γ(n) and H(n) points can be obtained. Forexample, Γ(1) =

[(b, b, 0)T ]

]has 24 elements yielding the

reducible representation with the following characters

R 1 6C4 3C2 8C3 6C′

2

χR(Γ(1))(R) 24 0 0 0 −24

and the compatibility relationR(Γ(1)) = 2A2+2E+4T1+2T2. H(1) =

[(b, b, b)T

]has 16 elements on a C3 axis.

The compatibility relation is R(H(1)) = 2E + 2T1 + 2T2.These results are summarized in table II.

(b) Degeneracy fully lifted on the ∆ line

Instead of also reducing an equivalent representationto R on the respective ∆ line we reduce the irreduciblerepresentations of 432 in the C4 point group to learn howthe degeneracies are lifted when going away from the highsymmetry points onto ∆. The characters are:

R 1 C41C2 C43

χA1(R) 1 1 1 1

χA2(R) 1 −1 1 −1

χE(R) 2 0 2 0

χT1(R) 3 1 −1 1

χT2(R) 3 −1 −1 −1

The same reduction procedure as above yields the com-patibility relations listed in table III. The degeneracy isin each case completely lifted.

For the 8-srs PC, the modes perfectly match in thepredicted manner in numerical calculations if the Fourierlattice maintains the full 432 symmetry (figures 8 and 5).

The eigenmodes of the band structure in Fig. 3 thatare coloured by their respective C4 irreducible represen-tation show the predicted behaviour. Each mode rep-resentation is obtained numerically by projecting thecorresponding normalized magnetic field |H〉 onto therespective basis function with representation theorem(iii). The representation is determined with the norm

Ni =∑α

⟨P

(i)α H|P (i)

α H⟩

70 of each projection onto one

of the 4 one-dimensional irreducible representations. Upto numerical accuracy, the projection onto the true ir-reducible representation yields Ni = 1 while all otherprojections vanish if a simple cubic lattice with full Osymmetry is used for the MPB calculations of the PBS,see Fig. 8. The representation algorithm for the bandstructure in Fig. 3 on the other hand only yields goodresults if the modes are separated in frequency, i.e. awayfrom band crossings and high symmetry points. Thiserror is caused by the fact that a BCC Fourier grid bydefinition is incompatible with the full 432 symmetry (il-lustrated in Fig. 5) and only has the D3 point symmetryof the parallelepiped spanned by the three (reciprocal)lattice vectors.

The Maxwell version of the Helmann-Feynman theo-rem yields directly that the band function ωni(k) is con-tinuously differentiable and has a slope given by the av-erage energy flow of the corresponding eigenmode whichcan be shown to be always less than the velocity oflight c71. Orthogonality of the irreducible representa-tions further implies that the function Hnik(r) does notchange its representation i if the k vector does not changeits point symmetry behaviour under the transformationk → k + δk. The argument also holds at acciden-tal degeneracies where the different representations arestill orthogonal and usually even at symmetry triggereddegeneracies72. Bands can therefore either cross (A and

8

B band in the lower half of Fig. 8 ”ii, SC”) or anti-cross(two red bands in Fig. 8 ”i, SC”)73.

A zoom into the band structure in Fig. 3 reveals (a)that bands do not cross on ∆ but are anti-crossing corre-lated with an interchange of characters which is visible inthe inset in Fig. 3 where the band structure around thepoint 0.65,0.68 is magnified74 and (b) that the 3-fold de-generacies at the high symmetry points are slightly lifted(Fig. 8). Both phenomena can be explained with a degen-erate perturbation theory model that in case (a) includesonly two modes of different representation. The diago-nal matrix is perturbed by numerical breaking of the 4-fold symmetry. Case (a) can be treated analytically withthe help of direct product selection rules. The pertur-bation matrix is orthogonal to the A representation andhence the perturbation matrix does not have diagonalelements. It is however still hermitian so that diagonal-ization results in a repulsive force between two closelyspaced bands. Any crossing becomes an anti-crossing.The modes mix in equal proportions (and with a phasefactor given by the argument of the secondary diagonalentries of the perturbation matrix) at the point of de-generate frequency without perturbation. The modes re-gain their original character when leaving the degeneracywhich can be observed in the inset of Fig. 3 at coordi-nates (0.65, 0.68) where the color is grey at the crossingpoints and changes to the original E− (B) color whenleaving the degeneracy and also in the fundamental bandsat (0.95, 0.62)75.

With representation theorem (iii) we can further ex-plain, why in case (b) the 3-fold degenerate modes T1

and T2 split into a 2-fold and a non-degenerate state asillustrated in Fig. 8 iiz and why the E mode does notsplit up under the symmetry break. The character tablefor the D3 point group (cf. Fig. 5) is

D3 1 2C3 3C ′2χA1

(R) 1 1 1

χA2(R) 1 1 −1

χE(R) 2 −1 0

so that we derive:

R(O) A1 A2 E T1 T2

R(D3) A1 A2 E A2 + E A1 + E

Note that the reduction of the T1 representation in C4

includes a trivial behaviour A whereas the reduction ofT2 includes a trivial behaviour A1 in D3. The black andblue colors of the bands in Fig. 8 ”iiz, BCC” are notunique, indicating that the irreducible representation ofC4 is not uniquely determined due to the fact that nosymmetry except the identity transformation of C4 is asymmetry of the discretization and hence the cannot beperfectly met by the numerics.

All modes have the same irreducible representationin the trivial group of the wave vector Sk = 1 on ∆.

A

B

E+

E−

plane wave

FIG. 6. (Color Online) Illustration of the four mode struc-tures of a monochromatic vector field that spatially transformaccording to the respective irreducible representation of C4

(cf. Tab. I). The field is shown at an arbitrary point in timeat four symmetry equivalent points. For the E± modes forwhich the characters are not all real, the transformation de-pends on the polarization state and we split each mode into ared RCP part with temporally right rotation as seen from thereceiver and a blue LCP part rotating opposite. The A andB profiles do not couple to a plane wave (center) as oppositecontributions with a phase shift of π cancel out. The RCPpart of an E+ mode and the LCP part of an E− mode cannotcouple to a plane wave for the same reason.

Modes of same irreducible representation cannot crosseach other, they are exposed to a ”fermionic” repulsion76.That is the reason why the upper two bands in Fig. 8”ii, BCC” cannot cross each other. Fig. 8 ”i, SC” pro-vides another palpable example for a symmetry induced”fermionic” anti-crossing.

(c) Time inversion symmetry and slope at Γ and H

Due to time-reversal symmetry it is sufficient to showthe band structure along the ∆ line (including Γ and H).To obtain the mode representations of the other half weexamine the action of the time-reversal operator T onthe modes and eigenfrequencies. T is defined by T f(t) =f(−t). The action on the (complex) spatial part f(r)of a monochromatic field f(r, t) = Ref(r) exp(−ıωt) isthen given by the antiunitary complex conjugation opera-tor, i.e. T f(r) = f(r), that obeys T 2 = E and transformsthe Bloch-Maxwell operator into one with opposite wavevector:

ϑ′k = T ϑkT−1 = T ϑkT = ϑ−k.

The action on an eigenfunction |ki〉 characterized by itswave vector k and representation i is generally given by

T |ki〉 = | − k, i′〉.

The positive definite27 nature of ϑk yields the frequency”degeneracy” ωi(k) = ωi′(−k)78. We distinguish two

9

inverse 8-srs 8-srs

join

t ne

tsd

isjoin

t nets

RelativeGapwidth(in %)

1:10 1:3 1:1 3:1 10:1Dielectric Contrast

0

10

20

30

40

50

60

Vo

lum

e F

ract

ion

(in

%)

0 2 4 6 8 10 12 14

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

2 4 6 8

No

rma

lize

d F

req

ue

ncy

Ω n

eff

Dielectric Contrast

Chalcogenide

FIG. 7. (Color Online) Relative gap width δG = 2(ω2 −ω1)/(ω2 + ω1) of the symmetry induced band gap of mid gapfrequency ΩG; ω1 is the maximum frequency of the two fun-damental bands of character E+ and E−, respectively, and ω2

the minimum of the two air bands of same character. (Left)Color coded map of the δG as a function of volume fraction Φand dielectric contrast ψ = ε8-srs : εbackground on a logarithmicscale. The join frequency ΦJ ≈ 38% indicates a topologicalchange: For Φ < ΦJ , the individual srs nets are disconnected,for Φ > ΦJ they overlap to form a single connected compo-nent. The orange point ’x’ marks the choice of parametersfor Fig. 3. (Right) Gap map along the line indicated bythe white arrow on the left including the x-point. The fre-quency is scaled by the effective index neff :=

√∑m Φmεm

with m ∈ 8-srs,background on the ordinate. The positionof the band gap hence does not depend on the choice of theabsolute value of dielectric constants. It turns out that the x-point is close to a band structure topology change at Φ ≈ 31%and ψ ≈ 6.5 where the T1 (black curve) and T2 (green curve)points at H are accidentally degenerate.

cases79(cf. references 37 and 80) :

(a) ∃R′ ∈ G : ∀R ∈ GD (R′)D (R)D (R′)−1 = D (R′)D (R)D (R′)−1,

(b) @R′ ∈ G : ∀R ∈ GD i(R) = D (R′)D i(R)D (R′)−1,

As time inversion T commutes with the point symmetryoperations R, i′ is equal to i in case (a). As a prooffor that statement we choose the representation whosematrix entries are all real so that the time-reversal statetransforms as a state of irreducible representation i:

R| − ki′α〉 = RT |kiα〉 =∑β

TD(i)αβ(R)|kiβ〉

(a)=∑β

D(i)αβ(R)T |kiβ〉

=∑β

D(i)αβ(R)| − ki′β〉

Contrary, i and i′ are different irreducible representa-tions (of same dimension) in case (b) that form a quasi-

degenerate pair so that

ω2i (k)/c2 = 〈ki|ϑkki〉 = 〈T 2ki|ϑkT 2ki〉

= 〈(T ϑkT )T ki|T ki〉 = 〈ϑ−k − k, i′| − k, i′〉= ω2

i′(−k)/c2.

This quasi-degenerate pair naturally meets at both the Γand the H point. The respective case for each irreduciblerepresentation can be determined by the Herring rules81.The pair of two fold rotations in O perpendicular to the[100] direction is responsible for a non-trivial behaviouralong ∆. The different cases are listed in table I. At theΓ and the H point, the irreducible representations of Oare all of time inversion type (a).

In accordance with the compatibility relations above,a pair of E+ and E− modes along ∆ hence always meetsat the H (Γ) point. Their group velocity in the vicinityis further of same magnitude and opposite sign. The Aand B representations on the other hand always havevanishing group velocity close to Γ and H.

(d) A,B,E+, E− correspond to non-interacting scatteringchannels

The mode structure of the C4 representations isstrongly connected to a plane wave propagating along theaxis of symmetry. We derive the symmetry behaviourof such a plane wave using a circular polarization ba-sis, i.e. we make use of representations theorem (iii) tocalculate the composition of an RCP/LCP wave. Themodulation factor expıkz is obviously unchanged by

any operation within C4 so that R is only acting on theJones vector for which we obtain: 1

∓ı0

=∑i

lih

∑R∈C4

χi(R)R

1

∓ı0

=∑i

1

4

χi(1)

1

∓ı0

+ χi(C41)

±ı10

+χi(C2)

−1

±ı0

+ χi(C43)

∓ı−1

0

=

1

∓ı0

, if i = E∓

0 , else

Therefore, any LCP/RCP mode transforms purely ac-cording to the irreducible representation E± respectively,there is strictly no contribution of an A or B representa-tion to any plane wave travelling along a C4 axis. As a

10

1

1.3

1.6

1.9

Γ 0.2 0.4 0.6 0.8 H

Ω neff

K

SC PBS

i

ii

iii

iv

i, SC

ii, SC

iiz

iiz, SC

iiz, BCC

iii, SC

iiiz

iiiz, SC

iiiz, BCC

iv, SC

δφ=10o

FIG. 8. (Color Online) Bandstructure of the inverse 8-srs PC for ε = 1 within the networks and ε = 5.76 outside and volumefraction Φ ≈ 41% (upper left quadrant in Fig. 7). The PBS is calculated using BCC and simple cubic (SC) symmetry. In theplots labelled ’SC’ the band structure of the simple cubic lattice is calculated for a grid discretization of M3 = 323 and unfoldedto the BCC ∆ path77. The bands are colored by their respective symmetry behaviour corresponding to i = A,B,E+, E−.Some of the interesting regions around H and Γ are magnified to demonstrate the excellent agreement between theory andnumerics in the SC calculations. Plots labelled ’BCC’ use a BCC Fourier grid that does not maintain all symmetries of theBCC symmetry, see Fig. 5. The BCC calculations illustrate that the violation of correct symmetry lifts the degeneracy of thebands at 3-fold degenerate points, even for a fine grid with M = 64.

result, the two distinct circular polarization states scat-ter in orthogonal and hence independent channels corre-sponding to E± because of representation theorem (iv).Polarization conversion vanishes for any scattering pro-cess at a 4-fold symmetric structure.

While the rigorous proof is provided above, an intuitiveunderstanding is easily obtained by going the reverse log-ical way starting from the four irreducible representationsand calculate the coupling strength with a plane wave.This idea is illustrated in Fig. 6.

We finally note that time inversion exchanges the rep-resentations E± that are defined respective to a staticCartesian coordinate system whereas it leaves the circu-lar polarization state that is defined respective to a righthanded coordinate system depending on the propagationdirection k that changes its sign under time inversionsymmetry.

II. GENERAL PREDICTIONS FORSCATTERING PARAMETERS

General features of the scattering matrix for a finiteslab of an I432 PC with (100) inclination can be derivedby its 4-fold symmetry alone. In the following, no en-ergy dissipation in any of the PC materials is assumed.

Kaschke et al. recently calculated in28 that there is nocircular dichroism in the reflectance and transmittancefor any lossless structure if any higher order scatteringis neglected. Here, we show more precisely that the re-flectivity matrix of the same system is identical for RCPand LCP light at all frequencies, i.e. circular dichroismand optical activity are zero. In the transmission spec-trum, optical activity is present and particularly strongabove the fundamental bands whereas circular dichroismis zero at frequencies below Ωc = 1. Above the criticalfrequency Ωc, the (10) Bragg order becomes leaky so thattransmission + reflection = 1 is no longer a valid state-ment if both are just measured within the (00) Braggorder (Fig. 9).

We first show that scattering matrix for any photoniccrystal slab with 4-fold symmetry normal to its clippingplanes can be well defined. We use representation theo-rem (iii) (a) to define a set of symmetry adapted Floquetbasis functions of the in-plane vector r = (x, y)T thattransform according to the irreducible representations ofthe C4 point group:

|(nσ)i〉 ≡ f (i)σ,n(r) =

1

2

∑R∈C4

χi(R)R eσeıGn·r

Those basis functions are characterized by their irre-

11

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

0.4 0.5 0.6

CD

and

T

Frequency ωa/(2πc)

CDT

CDR

T+

T-

1 1.01 1.02 1.03 1.04

Ωc

-90.0

-60.0

-30.0

0.0

30.0

60.0

90.0O

A (

de

gre

es)

OAT

OAR

FIG. 9. (Color Online) Simulated CD, OA and transmissivityT± in both transmission channels E± for the reflection andtransmission of a plane wave at normal incidence: The 8-srs PC slab has termination t = 0.25a and thickness Nz = 4.Since optical experiments cannot measure phase differences >±90, OA is wrapped onto the interval [−90, 90] by OA 7→arctan (tan(OA)).

ducible representation i ∈ A,B,E+, E− (Tab. I), po-larization σ ∈ s, p and a corresponding unit vector eσ

82

and Bragg order n = (n1, n2) (with n1 ∈ N1 and n2 ∈ N0)that defines the reciprocal grating vector Gn = (2π/a)n.To obtain a complete set for the normal incidence scat-tering problem, only two additional (00) Bragg order ba-

sis functions fE±x,(0,0)T

(r) are added to the above83. The

symmetry adapted basis functions are easily shown to beplane orthogonal using the orthogonality of plane waves,the Cartesian basis vectors and the wonderful orthogo-nality theorem of representation theory:

1

a2

∫ a/2

−a/2dx

∫ a/2

−a/2dy f

(i)

σ,n f(i′)σ′,n′ = δi,i′δσ,σ′δn,n′

We use the compact Dirac notation so that the electro-magnetic fields |F(nσ)i〉 in vacuum that are involved inthe scattering process are expressed by

|F(nσ)i〉 :=

(|E(nσ)i〉|H(nσ)i〉

)=

(1

H(nσ)i,qd

)|(nσ)i〉 ⊗ |qd〉

where d = ± denotes the sign ambiguity for |qd〉 ≡ eıqdz

with qd = d√ω2/c2 − |Gn|2 defined by the vacuum dis-

persion relation. The Bloch fields within a thin layer84

of thickness δ also transform as one irreducible represen-

tation i only and are hence given by

|Fαi〉 =

(|Eαi〉|Hαi〉

)=∑nσ

(E(α)

(nσ)i,qd

H(α)(nσ)i,qd

)|(nσ)i〉 ⊗ |qd〉

where within the layer |qd〉 ≡ eıqdz is again a planewave whose wave vector qd however has no analyticalexpression85 and d is hence chosen by sgn(Imq) ifImq 6= 0 and sgn(Req) else86. Because of the or-thogonality 〈ai|bj〉 = δabδij a scattering matrix treat-ment within each channel i analogous to the calculationdescribed by Whittaker and Culshaw87 can be performedand (with δ → 0) yields a well defined scattering matrixfor the whole structure that relates the (generally count-

ably infinite number of) amplitudes c(a)O 1/2 of the outgo-

ing symmetry adapted Floquet waves to the amplitudes

c(a)I 1/2 the corresponding incoming waves; 1/2 indicates thezmin/zmax surface of the slab respectively so that outgoingwaves are those with the sign d = ± and incoming waveswith d = ∓ at the 1/2 surface. With a vector c going overall indices a, the scattering matrix S is defined withineach 4-fold channel i ∈ A,B,E+, E− by:(

cO1

cO2

)=

(S

11S

12

S21

S22

)(cI1cI2

).

The definition of qd implies that non-zero entries in cI1/2are only physical for Ω ≥ |n|, such that the correspondingfields are propagating. The same condition applies for allentries in cO1/2 that contribute to the far-field. Further,if the incoming field is at normal incidence only the 4-fold channels that transform as E± are relevant as shownabove with the LCP/RCP Jones vectors. We thereforerestrict further arguments to the square submatrices S ±that relate the propagating Floquet modes in the respec-tive scattering channel corresponding to E±.

We now prove by energy conservation and time inver-sion symmetry that the reflectance spectrum (in the farfield) is exactly the same for RCP and LCP incident lightwhereas the transmittance spectrum has different phasesfor all frequencies and different magnitudes if Ω ≥ 1.These predictions are precisely reproduced for the 8-srsby a numerical calculation in Fig. 9.

In scattering theory, the scattering matrix is often as-sumed to be unitary. We prove that this is also truefor the particular problem. Averaging Poynting’s the-orem over time for the monochromatic fields yields thespatially local equation ∇ · Re

E ×H

= 0. We inte-

grate this equation over a box of size a × a × l that iscentered around the photonic crystal slab with l → ∞.Because of the mode structure of the 4-fold modes theparallel components of the Poynting vector vanish at eachpoint so that the divergence theorem yields the equationRe

∫AE‖ × H‖ = 0 where only the propagating fields

parallel to the surface are integrated over the top andbottom a× a surfaces.

12

(a) band structure (b) transmission LCP and RCP (c) reflection LCP and RCP

´ϕÓ`“

´ϕÒ´

´ϕÓ´ ϕÒ

`“

´ϕÓ´

ϕÒ´

ω0

ω

pLkqE`

E´

E´

E`

LCP

ϕÒ` ϕÓ

` ϕÒ`

tÒ,L` rÓÒ,L

`

tÒ,U` tÒ,U

`rÒÓ,U`

LCP LCP

LCP

air

air

PC

(L)ow

er

interf

(U)pp

er

interf

RCP

ϕÒ´ ϕÓ

´ ϕÒ´

tÒ,L´ rÓÒ,L

´

tÒ,U´ tÒ,U

´rÒÓ,U´

RCP RCP

RCP

LCP

tÓ,L`tÒ,L

` rÒÓ,L`

rÒÓ,U`

RCP RCP

ϕÒ` ϕÓ

`

RCP

tÓ,L´tÒ,L

´ rÒÓ,L´

rÒÓ,U´

LCP LCP

ϕÒ´ ϕÓ

´

FIG. 10. (Color Online) Diagrammatic representation of the processes that lead to different circular-polarisation properties ofreflected and transmitted light. Assuming that there is a single propagating mode in each channel and direction (a) at a givenfrequency ω0, transmission (b) and reflection (c) can be understood by considering positive, upwards pointing k (↑) and theircounterparts for negative k (↓). If the modes further have a leading Bragg order in propagation direction (i.e. interference maybe neglected) and the thickness of the slab L is an integer multiple of the lattice parameter a, the difference in k between twomodes δk correspond to an optical phase shift δϕ = δkL. The regions labelled upper (U) and lower (L) interface representthe interfacial two-dimensional planes between free space and the photonic crystal that are inflated to finite thickness forvisualization. The sign and coloration represents the E± channels in the same manner as in figures 3, 6, 8 and 9.

The orthogonality of the basis functions when averagedyields that all mixed contributions in different Bragg or-ders vanish. We show that for interface 2 and s polar-ization (p polarization is analogously done) so that theelectric field only has the following ϕ component:

Eϕ =1

2

[cOe

ıqz + cIe−ıqz]∑

R

χi(R)eıGRn·r

Since the z component of the magnetic field is irrelevanthere we only calculate the in-plane field that is purelyradial and obtained by Faraday’s law:

Hr =ıc

2ω∂zEϕ =

−cq2ω

[cOe

ıqz − cIe−ıqz]∑R

χi(R)eıGRn·r

The left side of the integrated Poynting’s theorem for spolarization is therefore given by:

−Re

∫A

EϕHr

= Re

cqa2

4ω

[cOe−ıqz + cIe

ıqz]

×[cOe

ıqz − cIe−ıqz]∑R

|χi(R)|2

=cqa2

ω

[|cO|2 − |cI |2

]The purely imaginary cross correlated contributions van-ish (i.e. they average out over time as intuitively ex-pected). The same result is obtained for surface 1 sothat for q > 0 Poynting’s theorem yields for each Braggorder in each 4-fold channel:

|cO1|2 + |c(a)02 |2 = |c(a)

I1 |2 + |c(a)

I2 |2

The scattering matrix is hence norm-preserving andtherefore unitary by Wigner’s theorem.

A correlation between the scattering matrices in dif-ferent channels can be derived by time inversion invari-ance. We act with T upon the equation that definesthe scattering matrix and note that the vectors satisfycI±(E+) = cO±(E−) (and also with I and O exchanged)because the basis functions |ai〉 stay unchanged by thecomplex conjugation except for a change in representa-tion χE±(R) → χE±(R) = χE∓(R)88 whereas the func-

tion of z changes its sign T |q+〉 = |q−〉. The complexconjugated scattering matrix in one channel is hence theinverse of the matrix in the other channel, i.e. S± = S−1

∓ .

(e) No CD and OA in reflectance

We combine both results and identify the diagonal en-tries S

iiwith the reflection matrices on either side to ob-

tainR(+) = R(−), i.e. the reflection matrices are identicaland hence OA and CD vanish to any numerical precisionfor all frequencies (cf. Fig. 9). We present an illustrativeinterpretation of this result in figure 10 based on the as-sumption that there is only a single mode per channeland propagation direction. If the assumptions made in10 are met as for example in the two fundamental bands,the Airy formula in terms of the interfacial scatteringparameters defined in 10 yields for the transmission and

reflection amplitudes with p↑↓± := expϕ↑↓±:

t = t↑,L± p↑±t↑,U±

∞∑n=0

(r↑↓,U± p↓±r

↓↑,L± p↑±

)nr = r↑↓,L± + t↑,L± p↑±r

↑↓,U± p↓±t

↓,L±

∞∑n=0

(r↑↓,U± p↓±r

↓↑,L± p↑±

)n

13

The first two contributions (n ≤ 1) of the infinite ge-ometric series are shown for transmission and the first(n = 0) for reflection. As the surface acts as a planargrating, we can use the previous results of29 that report

a special case of our more general treatment: r↑↓+ = r↑↓− ,

r↓↑+ = r↓↑− and t↓± = t↑∓ can be used. Hence OAR = 0,but there is no general restriction on OAT , because thenumber of passes through the structure is even for re-flectivity and each contribution to the sum and hence ton the lower interface and p come only in pairs so that

t↓,L+ t↑,L+ = t↓,L− t↑,L− and p↑+p↓+ = p↑−p

↓− because of time

inversion symmetry (see Fig. 10).

The phase shift at the surfaces caused by the respectivescattering amplitudes is usually negligible for frequenciesin the fundamental bands and therefore optical activityis essentially due to the difference in the wave vectors ofboth eigenmodes (cf. Fig. 10). This straightforward in-terpretation yields easy and fast numerical analysis of OAthat is comparable in magnitude to that of planar, metal-lic meta-materials16 and therefore makes the frequencyregion below the fundamental band edges (0.5 / Ω / 0.6in Fig. 9) particularly interesting for engineering of opti-cal devices such as beam splitters based on the 8-srs orsimilar structures with 432 point symmetry.

(f) No CD in transmission below a critical frequency

For the transmission matrices we derive analogously

S(+)

21= S(−)

12. This statement is however of less prac-

tical relevance because it identifies transmission in theE+ channel where the source is on one side of the struc-ture with transmission in the E− channel with the sourceon the other side and hence corresponds to two distinctexperiments. Comparing transmission matrices for thesame experiment we derive from the diagonal entries ofthe unitarity condition in both channels and the identityof the reflection matrices that the norm of transmissionT †T is the same in both channels. Therefore, CDT iszero below the critical frequency Ωc = 1 above which en-ergy is able to leak into the (10) Bragg order and henceenergy conservation within the (00) order with k perpen-dicular to the plane of interface is not valid. There is norestriction on OAT that can be finite for all frequencies(see Fig. 9).

Particularly interesting is the magnified region labellediv in Fig. 8. Transmission is governed by the respectivecalculated mode of the fundamental band in each chan-nel so that the situation well fits the one illustrated inFig. 10. If we further assume that the interfacial reflec-tions are sufficiently small (which is expected as trans-mission is almost 100% in the fundamental bands, seeFig. 9), optical activity is given by δϕ = δkL which isestimated to be around 10% at Ωneff = 1.1.

III. CONCLUSIONS AND OUTLOOK

We have provided consistent analytical and numericalresults that demonstrate the potential of the 8-srsgeometry as a dielectric photonic material that shouldbe realisable by current nano-fabrication technology.Analytic results, based on versatile group theory andscattering matrix treatment and applicable to anyphotonic crystal with I432 symmetry, are obtained fortransmission and reflection amplitudes, and for thetopology of the band structure. For the 8-srs PC, theseresults are in perfect agreement with numerical simula-tions that further demonstrate that the 8-srs exhibitsstrong optical activity in transmission, comparable tometallic meta-materials, yet without losses and withoutany ellipticity.

The potential of the 8-srs for applications as an opti-cally active nanofabricated material, say for the relevantcommunication wavelength of λ ≈ 1.5µm, can be gaugedfrom the following considerations.

First, if robustness of the optical activity with respectto slight changes in λ is desirable, the data in Fig. 9 sug-gests a realisation of the 8-srs with a lattice parametera = 0.55λ ≈ 0.83µm such that ωa/(2πc) = a/λ ≈= 0.55;the rotation of the polarisation plane through a layer offour unit cells (of total thickness 3.5µm) would then cor-respond to approximately −8o and close to 100% trans-mission for both RCP and LCP light and with zero ellip-ticity, assuming the same values for φ and ε as in Fig. 9.Second, if a strong dependence of OA on the value ofthe frequency is acceptable, or even desired, a realisa-tion of the 8-srs with a = 1.44µm, corresponding toωa/(2πc) ≈ 0.95, gives a very strong optical rotation of≈ 50o, in the optical communication window, again withclose to 100% transmission for both RCP and LCP lightand with zero ellipticity.

Importantly, Fig. 7 suggests that the parameters cho-sen here (corresponding to the cross on the white line inFig. 7) are not optimized to give the strongest photonicresponse. Specifically, the inverse structure, consisting ofhollow air channels along the edges of the 8-srs structureembedded in a dielectric matrix, has a wider band gap.While the E± frequency split that is desired for manyapplications is generally stronger for the original 8-srsstructure (cf. higher order bands in figures 3 and 8) dueto the light-guiding topology of the networks, it is foundrelatively weak in the fundamental bands here as thosecross close to the band edge (Fig. 3).

These values for the degree of optical activity should becompared to those in other systems. For quartz, the ro-tary power varies from 3.2o/mm at wavelength 1.42µm to776o/mm at 152nm89; for the liquid crystalline BPSmA1blue phase between 0 and ≈ 250o/mm over the visiblespectral range4; for a smectic phase SmCAPA a valueof 100 − 1000o/mm has been reported90. For a quasi-planar “twisted-cross” gold meta-material of thickness87nm, Decker et al. have reported ’strong’ rotations

14

of the polarisation plane of up to 4o, yet with signifi-cant losses16; for the split-ring-resonator meta-materialof thickness 205nm, ’huge’ rotations of up to 30o are ob-served with different transmittances for RCP and LCPlight of less than 50% (see Fig. 3 in91). Song et al.have reported even ’more gigantic’ polarisation rotationin a microwave experiment using a chiral composite ma-terial, yet also with the losses typical for metallic meta-materials. Note that the dependence on sample thickness(that can be varied in the 8-srs by varying the numberNz of unit cells) is non-linear, making a comparison ofpolarisation rotation normalised to the sample thicknessless meaningful.

Even for the likely non-optimal parameters chosenhere, the lossless dielectric 8-srs photonic crystalhence has a significant degree of optical activity, thatis combined with the complete absence of ellipticity,absence of losses and transmission rates of close to100%. Further, for applications such as beam-splitterswhere optical properties in various transmission direc-tions are important, the cubic symmetry of the 8-srsstructure is a further benefit to the uni-axial designs ofthe quasi-planar meta-materials. Given that a singlesrs net with a0 = 2a = 1.2µm has been realised15,it appears likely that future advances in direct laserwriting technology make a realisation of the 8-srs or itsinverse structure with a lattice size a = 1µm a genuinepossibility. Taking all of these considerations into aaccount, we believe that the 8-srs is a design for achiral-optical material that is worth further investigation.

Evidently, the validity of all theoretic predictions ofthis article is not limited to the 8-srs geometry, butapplies more generally to all geometries with symme-try I432. It is therefore worth exploring structuraldata bases for other designs with that symmetry; thismay include network structures such as the fcd net92,sphere packings such as Fischer’s 4/4/c14 or 3/4/c3packings93, minimal surface geometries such as Koch’sNO32 − c2 structure94, rod packings such as utz-b95

or woven filaments such as Evans’ P129RL(cosh−1(3/2))structure (see Fig. 8 of96).

Beyond the specific symmetry group I432, our grouptheoretic arguments can be adapted to apply more widelyto other crystalline chiral materials. For example, thesuppression of polarisation conversion is expected to bevalid for any chiral structure that has m-fold rotations(as opposed to screw rotations) with m ≥ 3 along the[100] propagation direction. These conditions are for ex-ample met for all cubic structures with symmetry groupsI432 (no. 211), F432 (no. 209) and P432 (no. 207); there-fore other network structures with these symmetries maybe alternative photonic designs with similar properties.From a perspective of photonic materials, particularlystructures with simple cubic P symmetries may be at-

tractive, because of their robustness with respect to theincident wave vector angle; for transmission along the[100] direction of a simple cubic crystal, the X point ofthe Brillouin zone represents the center of a facet, incontrast to the H point which represents a vertex of theBCC Brillouin zone. Therefore, a small variation in theincident angle will, to first order, not affect the distancebetween Γ and X in the SC case, again in contrast tothe distance between Γ and H in the BCC case. As isthe case for the I432 symmetry, advanced geometry andstructural data bases can provide suitable structure can-didates, such as dgn or fce97 or P118RL(cosh−1(

√6)) (see

Fig. 14 of96).

As a by-product of our exact group theoretic results forthe topology of the band structure, we have identified ageneral problem with numeric Fourier-based methods forthe band structure of structures with BCC (and hexago-nal and face centered cubic etc.) symmetry. The Fouriergrid for BCC structures can never maintain the full pointsymmetry (Fig. 5), and hence discretization artefacts areunavoidable and, as Fig. 8 shows, significant even for rel-atively large sizes of the Fourier grid. This problem, thatis not specific to our application, needs to be consideredfor all numerical Fourier methods.

In a broader context, our study of the 8-srs geometrydemonstrates the benefit of using advanced unconven-tional concepts of modern real-space geometry, such asthe maximally symmetric intergrowth of multiple mini-mal nets, for the informed design process of functionalphotonic materials. While numerical photonic methodsfor transmission parameters and band structures clearlyare indispensable tools to translate geometric structureinto photonic properties, we have here demonstrated theability of group theory to provide a firm theoretical un-derstanding of the relationship between photonic func-tionality and geometric properties for complex three-dimensional chiral photonic crystals.

ACKNOWLEDGMENTS

We are grateful to Stephen Hyde for the inspirationto investigate multiple entangled srs nets. We thankMichael Fischer for comments on the manuscript. MS,KM and GST gratefully acknowledge financial supportby the German Science Foundation through the excel-lence cluster ”Engineering of Advanced Materials” at theFriedrich-Alexander-University Erlangen-Nurnberg. MSis grateful to Nadav Gutman for inspiration and sugges-tion group theory as an approach to analyse symmetrybehaviour. MT and MG are grateful for financial supportby the Australian Research Council through the ”Centrefor Ultrahigh-Bandwidth Devices for Optical Systems”.

15

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16

52 The BCC Wigner-Seitz cell forms a Kelvin body that isa cube truncated along its [100] directions to give a poly-tope with 14 facets, eight of which are regular hexagonsand six of which are squares, also known as truncated oc-tahedron. The use of the term Kelvin body for this cellis motivated by it being Kelvin’s proposition for the cellwith least surface area (given fixed volume) that can tes-sellate space98.

53 See also the reticular chemistry structure resourcewww.rscr.anu.edu.au97 for details, where the 2-srs, the4-srs and the 8-srs are denoted srs-c2*, srs-c4 and srs-c8,respectively.

54 Transmission simulations are performed with the opensource finite difference time domain package MEEP99.Simulations are for a slab of 53 unit cells of 8-srs struc-ture. Periodic boundary conditions are assumed in allthree directions of the simulation box with a Yee grid of 64points per unit cell and a size of 1x1x61 unit cells. A com-bination of respectively 2 unit cells vacuo and perfectlymatched layers on each side are used with a Gaussiansource in vacuum on one side and energy flux measuredon the other. Band structure frequencies and eigenmodesare calculated with the open source plane wave frequencydomain eigensolver MPB100 with a 1283 structural resolu-tion and a 323 Fourier grid in the primitive body centeredcubic basis.

55 Simulations are performed with the commercial softwarepackage CST Microwave Studio. We used the finite el-ement frequency domain solver with periodic boundaryconditions in the transverse plane and a 3 [4] unit cellswide slab of the 8-srs structure. Perfectly matched layersare used for the z boundaries of the simulation box.

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57 The infinite size in x and y direction is achieved by useof periodic boundary conditions with a single unit cell ofthe 8-srs structure.

58 The termination determines the planes at which the infi-nite periodic crystal is clipped to give the slab of heightnz a. The termination is indicated by the vertical clip-ping position t in crystallographic coordinates, with t = 0corresponding to a clipping plane through the point withsymmetry 432 (the origin in the notation of ref.26).

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64 Note, that for compactness of notation we suppress thatα itself depends on i, i.e. α = αi.

65 The procedure for the electric field equation is analogous.We note however, that the character of any improper rota-tion of the E-field is the negative of the H-field characterwhich can be seen for example by using Faraday’s equa-tion. While any I432 PC does not exhibit any improperrotation, any dielectric PC has time inversion symmetry

that is improper and hence induces the same phase shiftbetween H and E-field.

66 This defines a set of all propagating modes in a semi-infinite structure. However, for any real scattering prob-lem a complete set of PC modes includes evanescentmodes. The impact of evanescent waves onto the scatter-ing process becomes dominant if there is no propagatingmode present in the transmission channel that has a fi-nite amount of energy in the 00 Bragg order to which theincident beam couples only. This makes it for exampleimpossible to get any useful information about transmis-sion through a finite slab in the frequency range betweenΩ := ωa/(2πc) = 0.66 and 0.71 from the band structurein figure 3 alone.

67 Conservation of the spatio-temporal modulation at thesurface implies conservation of frequency and parallel partof the wave vector where the PC acts as a grating and backscattering can occur into several Bragg orders, i.e. backscattered waves are given by a plane wave Floquet basis.

68 Technically, a class is a subset Sn of symmetry elementsthat is obtained by one selected member and all elementsthat are conjugate. As a result, for most purposes all el-ements of a class can be treated equally, i.e. they havesame character etc.

69 An alternative proof is provided in38.70 The sum over α is superstitious in this special case as allC4 representations are one-dimensional.

71 For any dielectric PC27.72 This claim can be proved with degenerate perturbation

theory. The irreducible representation of the point groupof the wave vector is reduced in the lower symmetry groupof δk and the matrix elements of the perturbation opera-tor ϑδk in this basis are diagonalized (similar to the k · panalog in quantum mechanics). As all matrix elements sat-

isfy the symmetry relation < ϑ−δk >= − < ϑδk >, thereis always a band with continuous slope passing throughthe point without changing its representation. The Γ0

point where E and H are decoupled and hence the groupvelocity is zero whereas finite in the vicinity of Γ is anexception.

73 The term anti-crossing is taken from reference 37 and isused to characterize two bands that come close to eachother and seem to interchange group velocities from leftto right without actually touching one another.

74 The points are all equal sized in the inset and colored bythe color of i with dominating Ni with intensity maximumat Ni = 1 and continuously decreasing to light grey atNi = 0.25.

75 The color code of the E± modes in the main plot is inthe common hexadecimal rgb scheme defined by RGB =Cb255(NE+ −NE−)c where C = 2562 if NE+ > NE− andC = 1 else.

76 Electromagnetic eigenmodes (unlike photons in a particlepicture) behave like Fermions: There can only be a singlemode in a state classified by all symmetries (or quantumnumbers) because the two modes are uniquely determinedexcept for a scalar coefficient and hence interchangeable.The symmetries in Maxwell theory are translation invari-ance in time and space (ω and k), time inversion (k ↔ −k)and point symmetries (character).

77 The main Bragg contribution G to the SC Bloch modeis determined and the band structure unfolded plotting1 − k and exchanging E+ ↔ E− if 2πG · ek/a yields an

17

odd number.78 We have adopted the term degeneracy although the states

have opposite Bloch wave vector and generally differentrepresentation.

79 Generally, a third case can occur where neither statement(a) nor (b) holds. However, this case only occurs in thepresence of a 4-fold improper rotation or a two-fold screwaxis that are both not present and generally cannot be inthe group of the wave vector within the Brillouin zone81.

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onto eσ is the identity transformation. Note that we onlyuse the polar system for the field vectors and not for theposition vector so that all spatial derivatives etc. are stillCartesian.

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= 0 and

f(i)

x,(0,0)T= f

(i)

y,(0,0)T.

84 So that the structure is assumed to be homogeneous alongz within that layer (i.e. ε(z) = ε(z + z0) if |z0| < δ/2).

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