Chiral photonic crystals with four-fold symmetry: Band structure … · fold symmetry. 4-fold...

Chiral Photonic Crystals with 4-fold Symmetry:Band Structure and S-Parameters of Eight-Fold Intergrown

Gyroid Nets ∗

Matthias Saba*, Mark D. Turner†, Min Gu†, Klaus Mecke*, and Gerd E. Schröder-Turk*

*Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg,91058 Erlangen, Germany

†CUDOS & Centre for Micro-Photonics, Swinburne University of Technology,Victoria 3122, Australia

November 15, 2013

ABSTRACT

The Single Gyroid, or srs, nanostructure has attracted interest as a circular-polarisation sensitive photonic material. Wedevelop a group theoretical and scattering matrix method, applicable to any photonic crystal with symmetry I432, todemonstrate the remarkable chiral-optical properties of a generalised structure called 8-srs, obtained by intergrowth ofeight equal-handed srs nets. Exploiting the presence of four-fold rotations, Bloch modes corresponding to the irreduciblerepresentations E− and E+ are identified as the sole and non-interacting transmission channels for right- and left-circularlypolarised light, respectively. For plane waves incident on a finite slab of the 8-srs, the reflection rates for both circularpolarisations are identical for all frequencies and transmission rates are identical up to a critical frequency below whichscattering in the far field is restricted to zero grating order. Simulations show the optical activity of the lossless dielectric8-srs to be large, comparable to metallic metamaterials, demonstrating its potential as a nanofabricated photonic material.

Keywords: Group Theory, Optical Activity, Chirality, Metamaterials, Photonic Crystals

1 INTRODUCTION

Dielectric photonic crystals (PC) and metallic metamaterials with chiral nanostructures attract interest because of theirchiral-optical behaviour, including circular dichroism,3 negative refractive index21, 22, 31 and optically-induced torque.13 Aparticularly intricate design, inspired by its occurrence in butterfly wing scales,16, 26, 27 is the single Gyroid (SG) or srsnet.9 It forms in inorganic materials on various length scales,17, 28, 30 with several applications.8, 14, 29, 30 The prediction ofcircular dichroism for a srs PC23 has been experimentally verified.28, 29 The circular polarization discrimination observedin metallic srs nets8 is lower than expected from the helical nature.19

Circular birefringence or optical activity (OA) and circular dichroism (CD) are polarisation effects related to the chiralproperties of a light-transmitting medium. In the literature, OA and CD correspond to the difference in the absorptioncoefficients and refractive indices between left- (LCP) and right-circularly polarized (RCP) light of a homogeneous non-transparent material.7 Here, we adopt these terms for a slab of a lossless and inhomogeneous material, and relate OA

∗An extended version of this conference article with detailed proofs and explanations is published elsewhere.24

1

Micro/Nano Materials, Devices, and Systems, edited by James Friend, H. Hoe Tan, Proc. of SPIE Vol. 8923, 89233T · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2033820

Proc. of SPIE Vol. 8923 89233T-1

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1-srs: cubic I4132 (214) 2-srs: tetragonal P4222 (93) 4-srs: cubic P4232 (208) 8-srs: cubic I432 (211)

Figure 1: (Color Online) Construction of the 8-srs by three replication steps. In each step the number of srs nets isdoubled by generating translated copies (blue) of the already existing nets (green). All nets are identical and equal-handed.

and CD to the transmission and reflection amplitudes (t,r). We define OA as the phase difference and CD as the relativedifference in absolute values between the complex scattering amplitudes s± = t± or s± = r±, respectively, for a respectiveincoming LCP (+) or RCP (−) plane wave:

CDs =|s+|− |s−||s+|+ |s−|

; OAs =ϕ(s)+ −ϕ

(s)−

2; eıϕ(s)

± =s±|s±|

(1)

While theoretical conclusions of this article, based on group theory and scattering matrix treatment, are valid for anystructure with I432 symmetry (all nomenclature for symmetry groups as in6), we use a particularly interesting geometrycalled 8-srs for illustration. The 8-srs is a periodic structure consisting of eight identical, and hence equal-handed, inter-threaded copies of the srs net.10

2 THE 8-SRS GEOMETRY

Figure 1 shows that the 8-srs is obtained by arranging translated copies of the srs net, such that all eight networks remaindisjoint and to give body centered cubic symmetry I432 1. With a0 the lattice parameter of the 1-srs in its symmetrygroup I4132, adding a copy translated by a := a0/2 along [100] gives the 2-srs of tetragonal symmetry; translation along adistinct coordinate axis by a yields the 4-srs with simple cubic (SC) symmetry; translation by

√3a/2 along [111] the 8-srs

with body-centered cubic (BCC) symmetry. Note that the 4-srs and the 8-srs have the same lattice constant a = a0/2.Importantly, the 8-srs has both four-fold rotation and four-fold screw axes along its [100] direction, see Fig. 2, in contrastto the 1-srs with only screw-rotations.

We consider the 8-srs as a dielectric PC obtained by inflating all edges of the 8-srs to solid struts (rods) with permit-tivity ε , embedded in vacuum. For the simulations, we use ε = 5.76, close to high-refractive index Chalcogenide glass attelecommunication wavelengths18 or TiO2 at optical wavelengths;17 the solid volume fraction is φ ≈ 31.4%, correspondingto a rod diameter of d ≈ 0.23a. Finite size effects are obtained for a slab of size ∞×∞×Nza with [100] inclination ofthe PC 2. All analyses are for wave vectors k along that axis and assuming termination planes perpendicular to [100]. Theparameter t denotes the position of the termination plane in the unit cell.

3 GENERAL PROPERTIES OF THE BAND STRUCTURE AND SCATTERING PARAMETERS

By Bloch’s theorem, the translational symmetry, or periodicity, of a PC leads to a natural representation of theeigenmodes by a set of orthogonal basis modes, the Bloch modes. The Bloch wave vector k characterises the

1See also the reticular chemistry structure resource www.rscr.anu.edu.au20 for details, where the 2-srs, the 4-srs and the 8-srsare denoted srs-c2*, srs-c4 and srs-c8, respectively.

2The infinite size in x and y direction is achieved by use of periodic boundary conditions assuming a single lateral unit cell of the8-srs.

2



432

*

*

* *

t = 0

*

*

* *

t = a/4

H ∆

Γ

Brillouin zone

Figure 2: (Color Online) Left and center: Different cross-sections with [100] inclination through the 8-srs reveal its four-fold symmetry. 4-fold rotation and 42 screw axes are marked by the symbols and *, respectively. The grey squarerepresents the cross section of the cubic unit cell with vertices at the 432 (Schoenfliess O) symmetry point (). Right: TheBCC Brillouin zone is a rhombic dodecahedron. The crystallographic [100] directions are marked by thin black lines. Theblue ∆ line connects the high symmetry points Γ and H.

translational symmetry behaviour of a mode and the band structure is the dependence of the frequency ω on k.In analogy to the translational symmetries, we now classify the behaviour of the eigenmodes under point

symmetries (here rotational symmetries). Group and representation theory provides the formalism for this clas-sification, in the form of irreducible representations A, B, E± and T1/2 and their characters that uniquely definethese, see Tab. 1. We analytically obtain the following general relationships valid for any PC with I432 symme-try (including also the 8-srs, see Fig. 3a):(a) Three-fold degeneracy at the H-point: The 4 lowest eigenstates at the H point are 3-fold degenerate. Thereare two T1 and two T2 modes defined in Tab. 1) and classified by their respective point symmetry behaviour 3.(b) Degeneracy fully lifted on ∆: The degeneracy is fully lifted when going away from the high symmetrypoints onto the ∆ line. Each mode split is summarized with a compatibility relation T1 = A + E+ + E− orT2 = B+E++E− (Tab. 1).(c) Inversion symmetry and slope at Γ and H: Each band ωi(k) along ∆ is characterized by its irreduciblerepresentation i ∈ A,B,E+,E−. It has inversion symmetry ωA/B(−k) = ωA/B(k) and ωE±(−k) = ωE∓(k). Thebands ωA/B(k) hence approach the points T1 and T2 with zero slope and the bands ωE± with equal and oppositeslope.

Group theory combined with an analytic scattering matrix treatment yields the following three further gen-eral rules for photonic scattering at a finite slab of an I432 PC, inclined at [100] direction and hit by a plane waveat normal incidence (see Fig. 3b for simulations of the 8-srs). These include previous results1, 12, 15 as specialcases.(d) A,B,E+,E− correspond to non-interacting scattering channels, E− and E+ represent RCP and LCP,respectively, A and B are dark modes: Modes of distinct representation do not interact. Scattering takesplace in four independent channels characterized by the four representations A,B,E+,E−. For each channel,a well-defined scattering matrix relating the amplitudes of the outgoing plane waves to those of the incomingplane waves is found. Both A and B representations represent dark modes that do not couple to any plane waveat normal incidence. At normal incidence, any E+ mode couples only to RCP and any E− only to LCP planewaves. This implies that the channels corresponding to A and B do not contribute to the scattering process and

3The upper T1 mode is out of frequency boundaries in Fig. 3a.

3



O (C4) 1 6C4 3C2 8C3 6C′2

A1(A) 1 1 1 1 1A2(B) 1 −1 1 1 −1E (A,B) 2 0 2 −1 0

T1(A,E+,E−) 3 1 −1 0 −1T2(B,E+,E−) 3 −1 −1 0 1

C4 1 C41 C2 C43 TRA 1 1 1 1 (a)B 1 −1 1 −1 (a)

E+ 1 i −1 −i (b)E− 1 −i −1 i (b)

Table 1: Character tables for the O and C4 point groups relevant for the H (Γ) point and ∆ line (Fig. 2 right),respectively. In the conventional manner, the rows cover the irreducible representations and the columns thesymmetry operations. The C4 representations that are included in each O representation (the compatibilityrelations) are listed in brackets on the left. The time reversal symmetry type TR for the C4 group is further addedin the last column.

there is no polarisation conversion between LCP and RCP in transmission and reflection at any wave length.(e) No CD and OA in reflectance: The reflection matrices on both sides are identical for the E+ (LCP) and theE− (RCP) channel. For the reflection spectrum, CD and OA are hence strictly zero for all wave lengths.(f) No CD in transmission below a critical frequency Ωc: The matrix norm of the transmission matrices isidentical for E+ and E− channels. Henceforth, at low frequencies Ω := ωa/2πc < Ωc := 1, where the portion ofenergy that leaves the crystal in the (00) Bragg order Σ± = |t±|2+ |r±|2 is strictly 100%, CD is zero. The matrixnorm imposes no condition for circular dichroism above Ωc. Optical activity may be finite at any frequency.

4 CONSEQUENCES AND CONCLUSIONS

The appendix outlines the proofs of the above claims. We now provide interpretations for the 8-srs PC:Due to rules (d)-(f), an 8-srs slab of fixed thickness acts like an effective, optically active material for which the

Kramers-Kronig relations are not valid; in contrast to homogeneous optically active materials, rotary power is not caused bya difference in the refractive indices for LCP and RCP but by the microstructure at the same length-scale as the wavelengthof the light.

Rule (d) also provides an interpretation for our definition of OA and CD: If a linear polarised plane wave at normalincidence impinges on a finite slab, the perpendicularly scattered (zero Bragg order) wave is generally elliptically polarised.The principal axis of its polarisation ellipse is rotated by OA compared to the polarisation axis of the incoming wave andhas eccentricity e =

√1− (CD)2. For Ω < Ωc, the polarization plane is hence rotated without introducing any ellipticity.

All results are accurately reproduced by numerical calculations. Fig. 3a compares the transmission, which is the samefor LCP and RCP light, through a thick slab with Nz = 53 to the photonic band structure (PBS) of the infinite periodicPC. The PBS modes are colored according to their numerically determined irreducible representation24 and have a sizeproportional to the coupling constant β 23 that describes the ability of the Bloch mode to couple to an incident plane waveof the same frequency 4. The PBS shows the behaviour predicted by results (a)-(c): It is 3-fold degenerate at the H pointand splits into 3 separate bands of irreducible representation A (B), E+ and E−. The slopes near the H point are consistentwith result (c). The PBS is further in good agreement with the transmission spectrum. Notably, the transmission drop atthe frequency Ωg := 0.64 is fully consistent with our band structure results: Since the A band has a dark mode behaviouraccording to rule (d), the band structure exhibits a small pseudo-bandgap (with T → 0 for Nz→∞, cf. teal points in Fig. 3a)in the frequency range 0.637 < Ω < 0.643. Note that different choices of φ and ε give significantly larger bandgap width,see Fig. 4.

Fig. 3b shows OA and CD spectra. In agreement with (e), OAr = CDr = 0 to any numerical precision. Consistentwith (f), CDt is only present above Ωc where the (10) Bragg order is non-evanescent. OAt is present at all frequencies

4β has been slightly improved compared to23 by taking the field overlap integrals of the 4 component vector F = (H‖,E‖)t insteadof the three components of H to take impedance mismatch between the coupled fields into account. β is still only a rough estimate fortransmission especially when several (partly evanescent) Bloch modes are involved in the scattering process.

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

0 0.4 0.8Transmission

48152953

(a) Numerical band structure

-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

Σ

Frequency Ω=ωa/(2πc)

CDt

CDr

Σ+

Σ-

1 1.01 1.02 1.03 1.04

Ωc

-90.0

-60.0

-30.0

0.0

30.0

60.0

90.0

OA

(de

gre

es)

OAt

OAr

(b) Simulated OA and CD

Figure 3: (Color online) (a) Band structure and transmission spectrum of the 8-srs PC along ∆ (Fig. 2): (Left) Bandsare colored according to mode’s symmetry behaviour corresponding to the irreducible representations i ∈ A,B,E+,E−of the C4 point symmetry defined in Tab. 1. (Right) Transmission of light at normal incidence through slab of thicknessNz = 53 and termination t = 0. Transmission is the same for both LCP and RCP (thin light grey line). The black line isa convolution with a Gaussian with δΩ = 0.002 that eliminates sharp Fano and Fabry-Pérot resonances. Teal points markthe transmission minima at the pseudo-bandgap at Ω≈ 0.64 for slabs of thickness NZ = 4, . . . ,53. (b) Simulated CD, Σ andOA for the reflection and transmission of a plane wave at normal incidence: The 8-srs PC slab has termination t = 0.25aand 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)).

and particularly strong above the fundamental bands with a large slope in the spectrum. Despite the absence of CD andellipticity below Ωc, the rotation angle goes up to ≈ −8 at Ω ≈ 0.6 even within the fundamental bands. This non-optimized PC exhibits therefore roughly 1/3 of the optical rotation that can be achieved with metallic metamaterials4

operating at comparable wavelengths in the near-infrared. Transmission is almost 1 at those wavelengths and dominatedby a single mode process so that an effective medium approach is justified. The 8-srs operating in the upper fundamentalband frequency region is hence a promising candidate for an optically active and lossless metamaterial.

In conclusion, the 8-srs is a prototype for a lossless chiral PC material that provides strong optical rotary powercombined with zero ellipticity below a frequency threshold that is way above the fundamental band edges for reasonabledielectric contrast. This unique combination of desired chiro-optical behaviour makes it a good candidate for opticalrotators or circular polarization beam splitters.29 Further, the scattering process with a PC slab is highly non-linear so thatoptical activity changes rapidly at the frequencies where poles in the generalized Airy formula exist.2 It therefore also isan attractive metamaterial that could be used for optical switches which is fully scalable in contrast to e.g. liquid crystalsin a smectic nematic phase.

Note that 1-srs and 2-srs PCs have been manufactured on the micron-scale with a0 = 1.2µm by direct laser writing(DLW) methods.28, 29 The DLW fabrication of the 8-srs appears therefore feasible at a structure size such that it exhibitsall the basic features predicted by our theory in the near infrared.

While the 8-srs geometry is a particularly interesting design from a chiral-optical perspective, it is important to notethat results (a-f) hold for any PC structure with symmetry I432. Even more generally, rules for the scattering matrix (d-f)are valid for any chiral PC with a 4-fold rotational symmetry in propagation direction.

Finally, we have restricted our representation analysis to the H point of lowest vacuum frequency H(0) for the sakeof simplicity. The reduction procedure, however, is equivalent for all higher H or Γ points with O symmetry yielding

5



inverse 8-srs 8-srs

join

t nets

disjo

int n

ets

RelativeGapwidth(in %)

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

0

10

20

30

40

50

60

Volu

me 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

Norm

aliz

ed F

requency Ω

neff

Dielectric Contrast

Chalcogenide

Figure 4: (Color Online) Relative gap width δG = 2(ω2−ω1)/(ω2 +ω1) of the symmetry induced band gap of midgap frequency ΩG; ω1 is the maximum frequency of the two fundamental bands of character E+ and E−, respectively,and ω2 the minimum of the two air bands of same character. (Left) Color coded map of δG as a function of volumefraction Φ and dielectric contrast ψ = ε8-srs : εbackground on a logarithmic scale. The join frequency ΦJ ≈ 38% indicatesa topological change: For Φ < ΦJ , the individual srs nets are disconnected, for Φ > ΦJ they overlap to form a singleconnected component. The orange point ’x’ marks the choice of parameters for Fig. 3a. (Right) Gap map along theline indicated by the white arrow on the left including the x-point. The frequency is scaled by the effective index neff :=√

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

for example the irreducible representations 2A2 + 2E + 4T1 + 2T2 at Γ(1) and 2E + 2T1 + 2T2 at H(1). Furthermore, otherdirections or symmetries can be treated in the same way. Preliminary investigation for k along [111] (a 3-fold axis ingeneral) yields the same main features as for k on ∆.

5 ACKNOWLEDGEMENTS

We thank Stephen Hyde for the inspiration to analyse multiple inter-threaded network structures, and Nadav Gut-man for pointing out the potential of group theory. We thank Michael Fischer for comments on the manuscript.MS, KM and GST acknowledge funding by the German Science Foundation through the Cluster of ExcellenceEngineering of Advanced Materials. MT and MG acknowledge funding by the Australian Research Councilthrough the Centre for Ultrahigh-Bandwidth Devices for Optical Systems (project CE110001018).

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APPENDIX: OUTLINE OF PROOFS OF RESULTS (A)-(F)

Detailed proofs are provided in.24 Here we outline the basic ideas, borrowing the group theoretical notationfrom5, 25 and referring to11 for PC theory.

(a) The operator in Maxwell’s wave equation in the Bloch form that acts onto the eigen-fields commutes withall point symmetries. Group theory tells us that the solution of the wave equation hence transforms as a spe-cific irreducible representation of the group of the wave vector, i.e. the subgroup of the PC’s point group underwhose symmetry operations the k-point is transformed into an equivalent point k+G that is only translated bya reciprocal lattice vector. The group of the wave vector at the H point is the full O point group (Tab. I in themain manuscript). We group the plane wave components fG(r) := eı(k+G)·r of a Bloch mode ∑G cG fG(r) at theH-point by the length of k+G of a specific wave vector in equivalence classes [G] :=

fG′ : |k+G′|= |k+G|

.

The plane waves within a class [G] form the basis of a reducible representation within the O point symmetrygroup. We determine the characters for each point symmetry of the original representation and reduce it withinthe O point group yielding 2 T1 and 2 T2 modes for the [(1,0,0)] class corresponding to the lowest H point (akaH(0)) in vacuum with frequency ΩV = 1. Both of the irreducible T representations in O are 3-dimensional. Any

8



mode that does not transform as T1 or T2 henceforth has zero amplitudes within the [(1,0,0)] class. The classwith next higher vacuum frequency is [(1,1,1)] with ΩV =

√3 leading to significantly higher frequencies if the

dielectric contrast of the PC is in a reasonable regime / 13.11

(b) On the ∆ line, the group of the wave vector is C4. Henceforth, the T1 and T2 representations are re-ducible for eigenmodes on ∆. The reduction procedure yields the compatibility relations T1 = A+E+ +E−and T2 = B+E++E− that are an abbreviation for the statement that the 3-dimensional T representations of Oreduce into three 1-dimensional representations within the lower symmetry C4, respectively.

(c) The time inversion operation within the monochromatic field approach is a complex conjugation. Inversionsymmetry of the whole band structure holds for any lossless PC.11 A straightforward analysis of the charactersyields that any A/B mode has an A/B counterpart at the opposite side of the BZ (time inversion symmetry type(a), Tab. I in the main manuscript) and that the E± modes form an inversion pair (type (b)).

(d) The statement that scattering takes place in 4 independant channels, corresponding to the irreducible repre-sentation of the C4 point group is a direct result of the orthogonality of irreducible representations. Using theprojection operator of irreducible representations of C4, the projection of an LCP plane wave with wave vectorparallel to the 4-fold rotation axis onto A, B and E− yields zero. Similarly, and RCP wave does not project ontoA, B and E+. Therefore, an incident LCP/RCP plane wave couples only to the E+/E− channel.

(e) A scattering matrix can be well-defined in each scattering channel relating propagating Floquet modes thatleave a slab of 8-srs crystal to incident Floquet modes. The scattering matrix in each channel is shown to beunitary for a non-dissipative system using global energy conservation based on spatio-temporal integration overPoynting’s theorem. Time inversion invariance further yields a correlation between the E+ and the E− channel:The entries of the scattering matrix in the E+ channel are the complex conjugate of the inverted matrix in the E−channel. The combination of both results yields that the reflection matrix is identical in both channels.

(f) The previous point yields that forward transmission in the± channel is identical to backward transmission inthe ∓ channel. However, this statement corresponds to two distinct experiments with sources on opposite sidesof a PC slab and is of less practical relevance. Comparing the same direction in both channels, only the matrixnorm of the transmission matrix is shown to be equal.

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Chiral photonic crystals with four-fold symmetry: Band structure … · fold symmetry. 4-fold...

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