# Unidirectional Wave Propagation in Low-Symmetric Colloidal Photonic-Crystal Heterostructures

## Abstract

**:**

## 1. Introduction

## 2. Description of the Photonic-Crystal Heterostructure

**a**

_{1}= (a, 0, 0) and

**a**

_{2}= (0, a, 0) [(001) crystallographic surface]. This means that we have a square lattice with lattice vectors

**R**

_{n}= n

_{1}

**a**

_{1}+ n

_{2}

**a**

_{2}, n

_{1}, n

_{2}= 0, ±1, ±2, ··· and corresponding reciprocal lattice vectors

**g**= m

_{1}

**b**

_{1}+ m

_{2}

**b**

_{2}, m

_{1}, m

_{2}= 0, ±1, ±2, ··· where

**b**

_{i}·

**a**

_{j}= 2πδ

_{ij}, i, j = 1, 2. A unit layer of the crystal consists of four non-primitive planes of spheres at (0, 0, 0), (0, 0, a/2), (0, 0, a), and (−0.3a, 0, 3a/2). All the spheres of the crystal of Figure 1 have the same radius S = 0.2a. The spheres of the first layer are of type A, of the second type B while the spheres at the third and fourth layers are of type C. The (n + 1)-th unit layer is obtained from the n-th layer by the primitive translation

**a**

_{3}= (−0.3a, 0, 2a). Such a monoclinic PC lattice has only one mirror plane [xz-plane, i.e., (010) crystallographic surface] and is therefore a lattice of low symmetry. We can thus expect that light transmission through a finite slab of such a monoclinic PC lattice would be different for incidence from the two opposite faces of the slab, if the slab is considered as a stack of planes parallel to a crystallographic direction other than the (010). In what follows, the different sphere types correspond to spheres made of different materials. In principle, one-way propagation can also be achieved by having spheres of different size or scatterers of dissimilar shape.

**Figure 1.**Unit cell of 3D PC with one-way photonic band gaps: monoclinic crystal consisting of four non-primitive planes of spheres parallel to the (001) surface at positions (0, 0, 0), (0, 0, a/2), (0, 0, a), and (−0.3a, 0, 3a/2).

## 3. Calculation Method

## 4. Results and Discussion

_{2}(ϵ = 2.1) spheres, as type-B polystyrene (ϵ = 2.6) spheres and as type-C silicon (ϵ = 11.9) (Si) spheres. All three types of materials have been the basis for the fabrication of colloidal photonic crystals of nanospheres (for a review see [36]).

**Figure 2.**Finite slab of the photonic crystal of Figure 2 consisting of eight unit layers.

^{+}refers to the transmittance of light incident from the left (001) face of the PC slab, i.e., a wave propagating from left to right whereas T

^{−}refers to the transmittance of light incident from the right (001) face of the PC slab, i.e., a wave propagating from right to left. Figure 3 shows T

^{+}and T

^{−}for light incident off-normally on a finite slab of the Figure 2 with wavevector

**k**

_{||}= (0.25, 0)π/a, for s- (a) and p- (c) polarization. The PC slab consists of eight unit layers where each layer contains four square lattices (planes) arranged as shown in Figure 2. Evidently, there are spectral regions where T

^{+}and T

^{−}differ significantly, such as, e.g., from ωa/c = 4.8 − 5.0 and from ωa/c = 5.2 − 5.8 for s-polarized light. For p-polarized light, T

^{+}and T

^{−}differ substantially from ωa/c= 5.9 − 6.1. Despite the important differences in light transmission for left and right incidence a clear case where there is a frequency gap for only one case of incidence direction (left or right) cannot be observed. By inspecting the corresponding frequency band structure (Figure 3b) we see that there exists a single band gap (from ωa/c = 5.0 − 5.2) for both polarizations and directions of incidence (left and right) as well as other gaps for only s- or p-polarized incident light. We also observe a definite asymmetry in the frequency bands relative to the k

_{z}= 0 which stems from the lack of reflection symmetry at this crystal direction.

**Figure 3.**Transmittance for s- (a) and p- (c) polarized light incident with

**k**

_{||}= (0.25, 0)π/a on a finite slab of the PC (see Figure 2) consisting of eight unit layers whereas type-A spheres we have considered silica SiO

_{2}(ϵ = 2.1) spheres, as type-B polystyrene (ϵ = 2.6) spheres and as type-C silicon (ϵ = 11.9) (Si) spheres. T

^{+}(T

^{−}) is the transmittance for light incident from the left (right) (001) faces of the slab. (b) Frequency band structure of the infinitely periodic PC of Figure 1 for

**k**

_{||}= (0.25, 0)π/a.

^{+}and T

^{−}practically coincide apart from two very narrow regions denoted by the arrows in Figure 4c. This is in accordance with the corresponding frequency band structure of Figure 4b where all frequency bands are symmetric with respect to the center k

_{z}= 0. Obviously, the inclusion of materials of high-index contrast in the considered photonic-crystal heterostructures does not favor the emergence of unidirectional propagation.

_{z}= 0 which is reflected in the corresponding transmission spectra of Figure 5a,c. Namely, we identify two regions of clear unidirectional frequency gaps. Namely, for both s- and p-polarized light, there exists a T

^{+}-frequency gap around ωa/c = 5.0 and a T

^{−}-frequency gap around ωa/c = 5.4. Asymmetric transmission (for left- and right- incidence) is also evidenced ωa/c = 5.8. We note that the emergence of the unidirectional frequency gaps is not generated by complete absence of frequency bands for one of the k

_{z}-directions (k

_{z}> 0 or k

_{z}< 0). There is a more complex mechanism which explains the emergence of unidirectional frequency gaps in terms of the matching of the group velocities of incident waves with that of the frequency bands (described analytically in [29]).

**Figure 4.**Transmittance for s- (

**a**) and p- (

**c**) polarized light incident with

**k**

_{||}= (0.25, 0)π/a on a finite slab of the photonic crystal (PC) (see Figure 2) consisting of eight unit layers whereas type-A spheres we have considered silica SiO

_{2}(ϵ = 2.1) spheres, as type-B germanium (ϵ = 16.2) spheres and as type-C silicon (ϵ = 11.9) (Si) spheres. T

^{+}(T

^{−}) is the transmittance for light incident from the left (right) (001) faces of the slab; (

**b**) Frequency band structure of the infinitely periodic PC of Figure 1 for

**k**

_{||}= (0.25, 0)π/a.

**Figure 5.**Transmittance for s- (

**a**) and p- (

**c**) polarized light incident with

**k**

_{||}= (0.25, 0)π/a on a finite slab of the PC of Figure 1 consisting of eight unit layers whereas type-A spheres we have considered silica SiO

_{2}(ϵ = 2.1) spheres, as type-B polystyrene (ϵ = 2.6) spheres and as type-C sapphire (ϵ = 3.13) (Si) spheres. T

^{+}(T

^{−}) is the transmittance for light incident from the left (right) (001) faces of the slab; (

**b**) Frequency band structure of the infinitely periodic PC of Figure 1 for

**k**

_{||}= (0.25, 0)π/a.

## 5. Conclusions

## Conflicts of Interest

## References

- Zolla, F.; Renversez, G.; Nicolet, A.; Kuhlmey, B.; Guenneau, S.; Felbacq, D. Foundations of Photonic Crystal Fibres; Imperial College Press: London, UK, 2005. [Google Scholar]
- Bozhevolnyi, S. Plasmonic Nanoguides and Circuits; Pan Stanford Publishing: Singapore, 2009. [Google Scholar]
- Haldane, F.D.M.; Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett.
**2008**, 100, 013904. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Chong, Y.D.; Joannopoulos, J.D.; Soljacˇic´, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett.
**2008**, 100, 013905. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Chong, Y.D.; Joannopoulos, J.D.; Soljacˇic´, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature
**2009**, 461, 772–775. [Google Scholar] [CrossRef] [PubMed] - Yu, Z.; Veronis, G.; Wang, Z.; Fan, S. One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal. Phys. Rev. Lett.
**2008**, 100, 023902. [Google Scholar] [CrossRef] [PubMed] - Ao, X.; Lin, Z.; Chan, C.T. Dirac spectra and edge states in honeycomb plasmonic lattices. Phys. Rev. B
**2009**, 80, 033105. [Google Scholar] [CrossRef] - Li, Y.; Zhang, Q.; Nurmikko, A.V.; Sun, S. Enhanced magnetooptical response in dumbbell-like Ag − CoFe
_{2}O_{4}nanoparticle pairs. Nano Lett.**2005**, 5, 1689–1692. [Google Scholar] [CrossRef] [PubMed] - González-Díaz, J.B.; García-Martín, A.; García-Martín, J.M.; Cebollada, A.; Armelles, G.; Sepúlveda, B.; Alaverdyan, Y.; Käll, M. Plasmonic Au/Co/Au nanosandwiches with enhanced magneto-optical activity. Small
**2008**, 4, 202–205. [Google Scholar] [CrossRef] [PubMed] - Jain, P.K.; Xiao, Y.; Walsworth, R.; Cohen, A.E. Surface plasmon resonance enhanced magneto-optics (SuPREMO): Faraday rotation enhancement in gold-coated iron oxide nanocrystals. Nano Lett.
**2009**, 9, 1644–1650. [Google Scholar] [CrossRef] [PubMed] - González-Díaz, J.B.; Sepúlveda, B.; García-Martín, A.; Armelles, G. Cobalt dependence of the magneto-optical response in magnetoplasmonic nanodisks. Appl. Phys. Lett.
**2010**, 97, 043114. [Google Scholar] [CrossRef] [Green Version] - Dani, R.K.; Wanh, H.; Bossmann, S.H.; Wysin, G.; Chikan, V. Faraday rotation enhancement of gold coated Fe2O3 nanoparticles: Comparison of experiment and theory. J. Chem. Phys.
**2011**, 135, 224502. [Google Scholar] [CrossRef] [PubMed] - Pakdel, S.; Miri, M. Faraday rotation and circular dichroism spectra of gold and silver nanoparticle aggregates. Phys. Rev. B
**2012**, 86, 235445. [Google Scholar] [CrossRef] - Armelles, G.; Cebollada, A.; García-Martín, A. ; Gonzá M.U. Magnetoplasmonics: Combining magnetic and plasmonic functionalities. Adv. Opt. Mater.
**2013**, 1, 10–35. [Google Scholar] [CrossRef] - Christofi, A.; Stefanou, N. Nonreciprocal optical response of helical periodic structures of plasma spheres in a static magnetic field. Phys. Rev. B.
**2013**, 87, 115125. [Google Scholar] [CrossRef] - Christofi, A.; Stefanou, N. Nonreciprocal photonic surface states in periodic structures of magnetized plasma nanospheres. Phys. Rev. B.
**2013**, 88, 125133. [Google Scholar] [CrossRef] - Nahal, A.; Talebi, R. Ellipticity-dependent laser-induced optical gyrotropy in AgCl thin films doped by silver nanoparticles. J. Nanopart. Res.
**2014**, 16. [Google Scholar] [CrossRef] - Soljacˇic´, M.; Luo, C.; Joannopoulos, J.D.; Fan, S. Nonlinear photonic crystal microdevices for optical integration. Opt. Lett.
**2003**, 28, 637–639. [Google Scholar] [CrossRef] [PubMed] - Scalora, M.; Dowling, J.P.; Bowden, C.M.; Bloemer, M.J. Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials. Phys. Rev. Lett
**1994**, 73, 1368–1371. [Google Scholar] [CrossRef] [PubMed] - Mingaleev, S.F.; Kivshar, Y.S. Nonlinear transmission and light localization in photonic-crystal waveguides. J. Opt. Soc. Am. B
**2002**, 19, 2241–2249. [Google Scholar] [CrossRef] - Yu, Z.; Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nat. Photon.
**2009**, 3, 91–94. [Google Scholar] [CrossRef] - Hwang, J.; Song, M.H.; Park, B.; Nishimura, S.; Toyooka, T.; Wu, J.W.; Takanishi, Y.; Ishikawa, K.; Takezoe, H. Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions. Nat. Mater.
**2005**, 4, 383–387. [Google Scholar] [CrossRef] [PubMed] - Lu, C.; Hu, X.; Zhang, Y.; Li, Z.; Xu, X.; Yang, H.; Gong, Q. Ultralow power all-optical diode in photonic crystal heterostructures with broken spatial inversion symmetry. Appl. Phys. Lett.
**2011**, 99, 051107. [Google Scholar] [CrossRef] - Wang, C.; Zhou, C.Z.; Li, Z.Y. On-chip optical diode based on silicon photonic crystal heterojunctions. Opt. Express
**2011**, 19, 26948–26955. [Google Scholar] [CrossRef] [PubMed] - Serebryannikov, A.E.; Nojima, S.; Ozbay, E. One-way absorption of terahertz waves in rod-type and multilayer structures containing polar dielectrics. Phys. Rev. B
**2014**, 90, 235126. [Google Scholar] [CrossRef] - Lockyear, M.J.; Hibbins, A.P.; White, K.R.; Sambles, J.R. One-way diffraction grating. Phys. Rev. E
**2006**, 74, 056611. [Google Scholar] [CrossRef] - Serebryannikov, A.E. One-way diffraction effects in photonic crystal gratings made of isotropic materials. Phys. Rev. B
**2009**, 80, 155117. [Google Scholar] [CrossRef] - Kuzmiak, V.; Maradudin, A.A. Asymmetric transmission of surface plasmon polaritons. Phys. Rev. A
**2012**, 86, 043805. [Google Scholar] - Yannopapas, V. One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry. Phys. Rev. A
**2013**, 88, 043837. [Google Scholar] [CrossRef] - Stefanou, N.; Karathanos, V.; Modinos, A. Scattering of electromagnetic waves by periodic structures. J. Phys. Condens. Matter
**1992**, 4, 7389–7400. [Google Scholar] [CrossRef] - Stefanou, N.; Yannopapas, V.; Modinos, A. Heterostructures of photonic crystals: Frequency bands and transmission coefficients. Comput. Phys. Commun.
**1998**, 113, 49–77. [Google Scholar] [CrossRef] - Stefanou, N.; Yannopapas, V.; Modinos, A. MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals. Comput. Phys. Commun.
**2000**, 132, 189–196. [Google Scholar] [CrossRef] - Gantzounis, G.; Stefanou, N.; Papanikolaou, N. Optical properties of periodic structures of metallic nanodisks. Phys. Rev. B
**2008**, 77, 035101. [Google Scholar] [CrossRef] - Yannopapas, V.; Vanakaras, A.G. Layer-multiple-scattering theory for metamaterials made from clusters of nanoparticles. Phys. Rev. B
**2011**, 84, 085119. [Google Scholar] [CrossRef] - Yannopapas, V. Layer-multiple-scattering method for photonic structures of general scatterers based on a discrete-dipole approximation/T-matrix point-matching method. J. Opt. Soc. Am. B
**2014**, 31, 631–636. [Google Scholar] [CrossRef] - Kim, S.H.; Lee, S.Y.; Yang, S.M.; Yi, G.R. Self-assembled colloidal structures for photonics. NPG Asia Mater.
**2011**, 3, 25–33. [Google Scholar] [CrossRef] - Ishii, M.; Kato, H.; Hashimoto, I.; Homma, Y. Synthesis of sapphire nanoparticles with graphite shells by hot-filament chemical vapor deposition. Mater. Express
**2014**, 4, 135–143. [Google Scholar] [CrossRef] - Istrate, E.; Sargent, E.H. Photonic crystal heterostructures and interfaces. Rev. Mod. Phys.
**2006**, 78, 455–481. [Google Scholar] [CrossRef] - Kedia, S.; Reddy, M.S.; Vijaya, R. Photonic crystal based direct and inverse heterostructures by colloidal self-assembly. Opt. Photon. J.
**2012**, 2, 242–248. [Google Scholar] [CrossRef] - Cai, Z.; Liu, Y.J.; Teng, J.; Lu, X. Fabrication of large domain crack-free colloidal crystal heterostructures with superposition bandgaps using hydrophobic polystyrene spheres. ACS Appl. Mater. Interface
**2012**, 4, 5562–5569. [Google Scholar] [CrossRef] - Liu, X.; Zhao, D.; Geng, C.; Zhang, L.; Tan, T.; Hu, M.; Yan, Q. Fabrication of colloidal photonic crystal heterostructures free of interface imperfection based on solvent vapor annealing. J. Colloid Interface Sci.
**2014**, 434, 98–103. [Google Scholar] [CrossRef] [PubMed]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yannopapas, V.
Unidirectional Wave Propagation in Low-Symmetric Colloidal Photonic-Crystal Heterostructures. *Nanomaterials* **2015**, *5*, 376-385.
https://doi.org/10.3390/nano5010376

**AMA Style**

Yannopapas V.
Unidirectional Wave Propagation in Low-Symmetric Colloidal Photonic-Crystal Heterostructures. *Nanomaterials*. 2015; 5(1):376-385.
https://doi.org/10.3390/nano5010376

**Chicago/Turabian Style**

Yannopapas, Vassilios.
2015. "Unidirectional Wave Propagation in Low-Symmetric Colloidal Photonic-Crystal Heterostructures" *Nanomaterials* 5, no. 1: 376-385.
https://doi.org/10.3390/nano5010376