近年來，螺旋二十四面體(Gyroid)吸引了許多科學家爭相研究，而螺旋二十四面體是一種三維重複性排列的螺旋結構。介電質材料單螺旋二十四面體(Single gyroid)為一種光子晶體，具有光子能隙，因此提供了一種全新的控制光路的方法。並且由於其螺旋結構，單螺旋二十四面體具有偏振態相關的光學特性。具有此特性之材料將是極有潛力做為一些立體顯示元件，因此本文將探討此結構之偏振相關特性。本研究將利用有限時域差分法(FDTD)進行電磁模擬，以分析電磁波在結構內的特殊行為。根據能帶結構、螺旋相關參數、藕合係數與反射頻譜分析，單螺旋二十四面體以及非對稱之雙螺旋二十四面體將具有偏振態相關特性，此特性將會使得右旋光與左旋光的穿透反射值有所不同。同時，不同結構與材料參數之螺旋二十四面體將會被探討特性變化行為。並且透過螺旋材料，螺旋二十四面體的光學特性將可以被近似，並且發現其偏振特性的產生源由。根據這些分析結果，此論文可以做為以螺旋二十四面體為材料之新穎元件的設計準則。

Gyroid is a type of three-dimensional chiral structures, which have attracted much research
attention recently. A dielectric single gyroid (SG) can be a candidate for providing new
means of guiding light because it has been shown to exhibit complete photonic band gaps.
Owing to the chiral nature, the SG exhibits circular polarization-dependent properties, leading
to new types of polarization-sensitive devices. In this work, studies are presented based
on finite-difference time-domain (FDTD) method for analyzing the polarization-dependent
characteristics of dielectric gyroids. The polarization-dependent properties in SG and asymmetric
double gyroid (DG) are investigated since they are not mentioned previously in the
literature. The corresponding band structures, CD indices, coupling indices and reflectance
spectra for right circularly polarized (RCP) light and left circularly polarized (LCP) light
are examined to verify the existence of a polarization band gap. Moreover, the correlation
between the structural or material parameters of dielectric gyroids and the frequency range
of polarization band gaps is also investigated. According to helix array approaching analysis,
a polarization band gap is found originating from the geometrical characteristics of the
gyroids. These results are crucial for the design of functional polarization-sensitive devices
based on dielectric gyroids.

Abstract i
Contents ii
1 Introduction 1
1.1 Development of chiral structures . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Gyroid networks with chiral morphologies . . . . . . . . . . . . . . . . . . . 4
1.3 Optical properties in gyroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Complete photonic band gap . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Polarization-dependent band gap . . . . . . . . . . . . . . . . . . . . 8
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Methods 10
2.1 Photonic band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Finite-Difference time-domain method . . . . . . . . . . . . . . . . . 10
2.1.2 The irreducible Brillouin zone . . . . . . . . . . . . . . . . . . . . . . 11
2.2 CD index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Coupling index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Directions of circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Polarization-dependent characteristics in gyroids 18
3.1 Polarization band gap verification . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Single gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Double gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2.1 Symmetric double gyroid . . . . . . . . . . . . . . . . . . . 21
3.1.2.2 Asymmetric double gyroid . . . . . . . . . . . . . . . . . . . 22
3.2 Effect of structural and material parameters on the polarization band gap . . 26
3.2.1 Volume fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1.1 Single gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1.2 Asymmetric double gyroid . . . . . . . . . . . . . . . . . . . 27
3.2.2 Index contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2.1 Single gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Discussions of the origin of polarization-dependent properties 31
4.1 Polarization-dependent properties in helix array . . . . . . . . . . . . . . . . 31
4.2 Helix array approaching analysis . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 Single gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1.1 Optical properties in the RH and the LH helix array . . . . 33
4.2.1.2 Optical properties in SG-like structures . . . . . . . . . . . . 37
4.2.2 Double gyroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2.1 Symmetric double gyroid . . . . . . . . . . . . . . . . . . . 38
4.2.2.2 Asymmetric double gyroid . . . . . . . . . . . . . . . . . . . 39
5 Conclusions 43
Bibliography 44

[1] M. Decker, R. Zhao, C. M. Soukoulis, S. Linden, and M. Wegener, “Twisted split-ringresonator
photonic metamaterial with huge optical activity,” Optics Letters 35(10),
1593–1595 (2010).
[2] J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials:
a numerical parameter study,” Optics Express 18(2), 1059–1069 (2010).
[3] W. J. Chen, J. C. W. Lee, J. W. Dong, C. W. Qiu, and H. Z. Wang, “Fano resonance of
three-dimensional spiral photonic crystals: Paradoxical transmission and polarization
gap,” Applied Physics Letters 98(8), 081116 (2011).
[4] M. Thiel, M. Decker, M. Deubel, M.Wegener, S. Linden, and G. Von Freymann, “Polarization
stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced
Materials 19(2), 207–210 (2007).
[5] J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. Von Freymann,
S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular
polarizer,” Science 325(5947), 1513–1515 (2009).
[6] K. Michielsen, H. De Raedt, and D. G. Stavenga, “Reflectivity of the gyroid biophotonic
crystals in the ventral wing scales of the Green Hairstreak butterfly, Callophrys rubi,”
Journal of The Royal Society Interface 7(46), 765–771 (2010).
[7] M. Maldovan, A. M. Urbas, N. Yufa, W. C. Carter, and E. L. Thomas, “Photonic properties
of bicontinuous cubic microphases,” Physical Review B 65(16), 165123 (2002).
[8] V. Babin, P. Garstecki, and R. Ho lyst, “Photonic properties of multicontinuous cubic
phases,” Physical Review B 66(23), 235120 (2002).
[9] M. Saba, M. Thiel, M. D. Turner, S. T. Hyde, M. Gu, K. Grosse-Brauckmann, D. N.
Neshev, K. Mecke, and G. E. Schrder-Turk, “Circular dichroism in biological photonic
crystals and cubic chiral nets,” Physical Review Letters 106(10), 103902 (2011).
[10] M. D. Turner, G. E. Schrder-Turk, and M. Gu, “Fabrication and characterization of
three-dimensional biomimetic chiral composites,” Optical Express 19(10), 10001–10008
(2011).
[11] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s
equations in isotropic media,” Antennas and Propagation, IEEE Transactions 14(3),
302–307 (1966).
[12] C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, Inc., 2005).
[13] L. Bouckaert, R. Smoluchowski, and E. Wigner, “Theory of Brillouin zones and symmetry
properties of wave functions in crystals,” Physical Review 50(1), 58–67 (1936).
[14] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (John Wiley & Sons, Inc.,
2007).
[15] L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljaˇci´c, “Weyl points and line nodes in gyroid
photonic crystals,” Nature Photonics 7(4), 294–299 (2013).
[16] J. C. W. Lee and C. T. Chan, “Polarization gaps in spiral photonic crystals,” Optics
Express 13(20), 8083–8088 (2005).
[17] A. Demetriadou, S. S. Oh, S. Wuestner, and O. Hess, “A tri-helical model for nanoplasmonic
gyroid metamaterials,” New Journal of Physics 14(8), 083032 (2012).
[18] S. S. Oh, A. Demetriadou, S. Wuestner, and O. Hess, “On the origin of chirality in
nanoplasmonic gyroid metamaterials,” Advanced Materials 25(4), 612–617 (2013).
[19] L. Rayleigh, “XVII. On the maintenance of vibrations by forces of double frequency,
and on the propagation of waves through a medium endowed with a periodic structure,”
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
24(147), 145–159 (1887).
[20] E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,”
Physical Review Letters 58(20), 2059 (1987).
[21] S. John, “Strong localization of photons in certain disordered dielectric superlattices,”
Physical Review Letters 58(23), 2486 (1987).
[22] P. V. Braun, S. A. Rinne, and F. Garc´ıa-Santamar´ıa, “Introducing defects in 3D photonic
crystals: state of the art,” Advanced Materials 18(20), 2665–2678 (2006).
[23] K. Ishizaki, M. Koumura, K. Suzuki, K. Gondaira, and S. Noda, “Realization of threedimensional
guiding of photons in photonic crystals,” Nature Photonics 7(2), 133–137
(2013).
[24] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals:
molding the
ow of light (Princeton university press, 2011).
[25] E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev,
“Metamaterial with negative index due to chirality,” Physical Review B 79(3),
035407 (2009).
[26] H. H. Huang and Y. C. Hung, “Designs of helix metamaterials for broadband and hightransmission
polarization rotation,” Photonics Journal, IEEE 6(5), 1–7 (2014).
[27] M. Thiel, H. Fischer, G. Von Freymann, and M. Wegener, “Three-dimensional chiral
photonic superlattices,” Optics Letters 35(2), 166–168 (2010).
[28] M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar
chiral magnetic metamaterials,” Optics Letters 32(7), 856–858 (2007).
[29] E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen, “Giant optical
gyrotropy due to electromagnetic coupling,” Applied Physics Letters 90(22), 223113
(2007).
[30] V. Saranathan, C. O. Osuji, S. G. J. Mochrie, H. Noh, S. Narayanan, A. Sandy, E. R.
Dufresne, and R. O. Prum, “Structure, function, and self-assembly of single network
gyroid (I4132) photonic crystals in butterfly wing scales,” Proceedings of the National
Academy of Sciences 107(26), 11676–11681 (2010).
[31] A. Schoen, Innite periodic minimal surfaces without self-intersections (National Aeronautics
and Space Administration Cambridge, 1970).
[32] M. Wohlgemuth, N. Yufa, J. Hoffman, and E. L. Thomas, “Triply periodic bicontinuous
cubic microdomain morphologies by symmetries,” Macromolecules 34(17), 6083–6089
(2001).
[33] H. Y. Hsueh, H. Y. Chen, M. S. She, C. K. Chen, R. M. Ho, S. G. Gwo, H. Hasegawa,
and E. L. Thomas, “Inorganic gyroid with exceptionally low refractive index from block
copolymer templating,” Nano Letters 10(12), 4994–5000 (2010).
[34] M. D. Turner, M. Saba, Q. Zhang, B. P. Cumming, G. E. Schrder-Turk, and M. Gu,
“Miniature chiral beamsplitter based on gyroid photonic crystals,” Nature Photonics
7(10), 801–805 (2013).
[35] Lumerical Solutions, Inc. http://www.lumerical.com/tcad-products/fdtd/.
[36] B. P. Cumming, M. D. Turner, G. E. Schr¨oder-Turk, S. Debbarma, B. Luther-Davies,
and M. Gu, “Adaptive optics enhanced direct laser writing of high refractive index
gyroid photonic crystals in chalcogenide glass,” Optics Express 22(1), 689–698 (2014).
[37] J. R. DeVore, “Refractive indices of rutile and sphalerite,” Journal of the Optical Society
of America 41(6), 416–417 (1951).
[38] L. Poladian, S. Wickham, K. Lee, and M. C. J. Large, “Iridescence from photonic
crystals and its suppression in butterfly scales,” Journal of The Royal Society Interface
6(Suppl 2), S233–S242 (2009).
[39] C. Mille, E. C. Tyrode, and R. W. Corkery, “3D titania photonic crystals replicated
from gyroid structures in butterfly wing scales: approaching full band gaps at visible
wavelengths,” RSC Advances 3(9), 3109–3117 (2013).
[40] H. Y. Hsueh, H. F. Ling, Y. C.and Wang, L. Y. C. Chien, Y. C. Hung, E. L. Thomas,
and R. M. Ho, “Shifting networks to achieve subgroup symmetry properties,” Advanced
Materials 26(20), 3225–3229 (2014).