# Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Aperiodic Structures

#### 2.1. One-Dimensional Aperiodic Structures

_{n}of the Thue–Morse sequence have the following forms: S

_{0}= {A}, S

_{1}= {AB}, S

_{2}= {ABBA}, S

_{3}= {ABBABAAB}, and so on. The Thue–Morse lattice is not quasiperiodic but deterministically aperiodic, which shows the properties of an intermediate between periodic and quasiperiodic lattices [19,20].

^{n}B and B→A [23], whereas the metallic mean sequences can be generated by the inflation rule A→AB

^{n}and B→A [24].

#### 2.2. Two-Dimensional Aperiodic Structures

_{A}: A→AB, B→A; f

_{B}: A→B, B→BA) [25].

#### 2.3. Fractal Patterns

## 3. Aperiodic-Order-Induced Multimode Effects in Photonic Micro/Nanostructures

#### 3.1. Multimode Effects Induced by Self-Similarity

_{n}as the amount of the resonant transmission mode around the central frequency (where n represents the number of the generation). Then we obtain:

_{3}= 1 and R

_{4}= 3. Finally, we have:

_{0}. According to Equation (2), the inner feature of the Thue–Morse structure determines the mode amount of resonant transmissions. That is to say, special positional correlation between two blocks in the Thue–Morse structure causes resonant transmission. Multiple PBGs can coexist at the same frequency range in these structures, which is intuitively shown in the photonic band structures in Figure 2b. Moreover, the number of PBGs can be increased by tuning the refractive index contrast. This theoretical analysis was verified by measuring Thue–Morse SiO

_{2}/TiO

_{2}multilayers in the range of visible and near-infrared frequencies [34].

_{2}O

_{5}and SiO

_{2}was constructed (Figure 2c). From the transmission spectra calculated by the transfer matrix method, three photonic modes could be observed in the dispersion map shown in Figure 2d, as demonstrated experimentally [37].

#### 3.2. Multimode Effects in Symmetric Aperiodic Structure

_{2}and TiO

_{2}were chosen as two elementary layers with the thickness of a quarter wavelength (λ

_{0}/4), and the transmission coefficient for the two different systems can be calculated by the transfer matrix method (Figure 3b,c), which shows that the transmission coefficient of the symmetric Fibonacci structure behaves rather differently from that of the Fibonacci structure [39]. As shown in Figure 3, the localization property of optical waves can be influenced by the symmetric internal structure in a quasiperiodic system, which is demonstrated by the sharp transmission peaks with transmission coefficients near unity. That is to say, with the help of symmetric internal structures in the quasiperiodic system, a perfect transmission of the optical wave can replace the initially poor transmission. This improvement is benefited from the positional correlations in the system. Moreover, the resonant transmission can be varied to a certain frequency by tuning the aperiodic structures (Figure 3d). For example, as shown in Figure 3e, the transmission coefficients vary in different symmetric multilayers with defects (SMD) [41].

## 4. Optoelectronic Devices Based on One-Dimensional Aperiodic Structures

#### 4.1. Optical Filters for the Wavelength Division Multiplexing (WDM) Systems

#### 4.2. Open Cavities for the Light–Matter Strong Coupling

_{2}/Ta

_{2}O

_{5}multilayers was chosen and J-aggregates on the top surface of structure offered excitons (Figure 5a) [37]. Figure 5b shows the measured transmission spectrum, where three peaks of different optical modes are recognized. The Rabi splitting and newly generated hybrid polariton bands can be verified from the dispersion map of the hybrid system, clearly showing successive coupling between the modes H, C, and L and the excitons (Figure 5c,d). By varying the substitution rule of the photonic quasicrystal, the open-cavity system can be optimized to provide the various photonic modes in need. By introducing this design, multimode photon–exciton strong couplings can be realized, which may inspire some potential applications, such as optical spectroscopy and multimode sensors.

#### 4.3. Multiband Waveguides for Trapping “Rainbow”

#### 4.4. Solar Cells with Multi-Intermediate Band Structures

_{0.49}Ga

_{0.51}P/GaAs superlattices can be split by introducing aperiodic order, such as that of the Thue–Morse sequence, the Fibonacci sequence, or even the random case (Figure 7c). This approach by introducing a multi-intermediate band structure may produce low-dimensional high-performance photovoltaic devices based on electronic band gap engineering, and can also be used in other ranges such as optoelectronics.

## 5. Optoelectronic Devices Based on Two-Dimensional Aperiodic Structures

#### 5.1. Aperiodic Plasmonic Aperture Arrays with Extraordinary Optical Transmission

#### 5.2. Solar Cell with a Plasmonic Fractal

_{2}antireflection layer, the quantum efficiency of the solar cell could be improved further. Therefore, these kinds of plasmonic fractal structures can be applied to design miniaturized compact photovoltaic devices with high performance.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett.
**1984**, 53, 1951. [Google Scholar] [CrossRef] - Merlin, R.; Bajema, K.; Clarke, R. Quasiperiodic GaAs-AlAs heterostructures. Phys. Rev. Lett.
**1985**, 55, 1768–1770. [Google Scholar] [CrossRef] - Todd, J.; Merlin, R.; Clarke, R.; Mohanty, K.M.; Axe, J.D. Synchrotron X-ray study of a Fibonacci superlattice. Phys. Rev. Lett.
**1986**, 57, 1157–1160. [Google Scholar] [CrossRef] - Hu, A.; Tien, C.; Li, X.J.; Wang, Y.H.; Feng, D. X-ray diffraction pattern of quasiperiodic (Fibonacci) Nb-Cu superlattices. Phys. Lett. A
**1986**, 119, 313–314. [Google Scholar] [CrossRef] - Kohmoto, M.; Sutherland, B.; Iguchi, K. Localization in optics: Quasiperiodic media. Phys. Rev. Lett.
**1987**, 58, 2436–2438. [Google Scholar] [CrossRef] - Dharma-wardana, M.W.C.; MacDonald, A.H.; Lockwood, D.J.; Baribeau, J.M.; Houghton, D.C. Raman scattering in Fibonacci superlattices. Phys. Rev. Lett.
**1987**, 58, 1761. [Google Scholar] [CrossRef] - Gellermann, W.; Kohmoto, M.; Sutherland, B.; Taylor, P.C. Localization of light waves in Fibonacci dielectric multilayers. Phys. Rev. Lett.
**1994**, 72, 633–636. [Google Scholar] [CrossRef] - Maciá, E. Exploiting aperiodic designs in nanophotonic devices. Rep. Prog. Phys.
**2012**, 75, 036502. [Google Scholar] [CrossRef] - Grimm, U. Aperiodic crystals and beyond. Acta Crystallogr. B
**2015**, 71, 258–274. [Google Scholar] [CrossRef] [Green Version] - Peng, R.W.; Hu, A.; Jiang, S.S. Study on quasiperiodic Ta/Al multilayer films by x-ray diffraction. Appl. Phys. Lett.
**1991**, 59, 2512. [Google Scholar] [CrossRef] - Peng, R.W.; Hu, A.; Jiang, S.S.; Zhang, C.S.; Feng, D. Structural characterization of 3-component Fibonacci Ta/Al multilayer films. Phys. Rev. B
**1992**, 46, 7816. [Google Scholar] [CrossRef] [PubMed] - Chakraborty, S.; Marshall, O.P.; Folland, T.G.; Kim, Y.J.; Grigorenko, A.N.; Novoselov, K.S. Gain modulation by graphene plasmons in aperiodic lattice lasers. Science
**2015**, 351, 246–248. [Google Scholar] [CrossRef] [PubMed] - Verslegers, L.; Catrysse, P.B.; Yu, Z.; Fan, S. Deep-Subwavelength Focusing and Steering of Light in an Aperiodic MetallicWaveguide Array. Phys. Rev. Lett.
**2009**, 103, 033902. [Google Scholar] [CrossRef] [PubMed] - Pala, R.A.; Liu, J.S.; Barnard, E.S.; Askarov, D.; Garnett, E.C.; Fan, S.; Brongersma, M.L. Optimization of non-periodic plasmonic light-trapping layers for thin-film solar cells. Nat. Commun.
**2013**, 4, 2095. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fan, J.A.; Yeo, W.H.; Su, Y.; Hattori, Y.; Lee, W.; Jung, S.Y.; Zhang, Y.; Liu, Z.; Cheng, H.; Falgout, L.; et al. Fractal design concepts for stretchable electronics. Nat. Commun.
**2014**, 5, 3266. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jiang, S.S.; Hu, A.; Peng, R.W.; Feng, D. Quasiperiodic metallic multilayers. J. Magn. Magn. Mater.
**1993**, 162, 82–88. [Google Scholar] [CrossRef] - Peng, R.W.; Wang, M.; Hu, A.; Jiang, S.S.; Jin, G.J.; Feng, D. Characterization of the diffraction spectra of one-dimensional k-component Fibonacci structures. Phys. Rev. B
**1995**, 52, 13310–13316. [Google Scholar] [CrossRef] - Ma, T.; Liang, C.; Wang, L.; Lin, H.Q. Electronic band gaps and transport in aperiodic graphene superlattices of Thue–Morse sequence. Appl. Phys. Lett.
**2012**, 100, 252402. [Google Scholar] [CrossRef] - Li, Y.; Peng, R.W.; Jin, G.J.; Wang, M.; Huang, X.Q.; Hu, A.; Jiang, S.S. Persistent currents in one-dimensional aperiodic mesoscopic rings. Eur. Phys. J. B
**2002**, 25, 497–503. [Google Scholar] [CrossRef] - Ryu, C.S.; Oh, G.Y.; Lee, M.H. Extended and critical wave functions in a Thue–Morse chain. Phys. Rev. B
**1992**, 46, 5162–5168. [Google Scholar] [CrossRef] - Kola, M.; Nori, F. Trace maps of general substitutional sequences. Phys. Rev. B
**1990**, 42, 1062–1065. [Google Scholar] [CrossRef] [PubMed] - Bellissard, J.; Bovier, A.; Ghez, J.M. Spectral properties of a tight binding Hamiltonian with Period doubling potential. Commun. Math. Phys.
**1991**, 135, 379–399. [Google Scholar] [CrossRef] - Birch, J.; Severin, M.; Wahlstrom, U.; Yamamoto, Y.; Radnoczi, G.; Riklund, R.; Sundgren, J.; Wallenberg, L.R. Structural characterization of precious-mean quasiperiodic Mo/V single-crystal superlattices grown by dual-target magnetron sputtering. Phys. Rev. B
**1990**, 41, 10398–10407. [Google Scholar] [CrossRef] [PubMed] - Dotera, T.; Bekku, S.; Ziherl, P. Bronze-mean hexagonal quasicrystal. Nat. Mater.
**2017**, 16, 987–992. [Google Scholar] [CrossRef] [PubMed] - Dallapiccola, R.; Gopinath, A.; Stellacci, F.; Negro, L.D. Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles. Opt. Express
**2008**, 16, 5544–5555. [Google Scholar] [CrossRef] [PubMed] - Pierro, V.; Galdi, V.; Castaldi, G.; Pinto, I.M.; Felsen, L.B. Radiation properties of planar antenna arrays based on certain categories of aperiodic tilings. IEEE Trans. Antennas Propag.
**2005**, 53, 635–644. [Google Scholar] [CrossRef] - Wang, K. Structural effects on light wave behavior in quasiperiodic regular and decagonal Penrose-tiling dielectric media: A comparative study. Phys. Rev. B
**2007**, 76, 085107. [Google Scholar] [CrossRef] - Lifshitz, R. The square Fibonacci tiling. J. Alloys Compd.
**2002**, 342, 186–190. [Google Scholar] [CrossRef] [Green Version] - Maciá, E. The role of aperiodic order in science and technology. Rep. Prog. Phys.
**2006**, 69, 397–441. [Google Scholar] [CrossRef] - Uozumi, J.; Kimura, H.; Asakura, T. Fraunhofer Diffraction by Koch Fractals. J. Mod. Opt.
**1990**, 37, 1011–1031. [Google Scholar] [CrossRef] - Gefen, Y.; Meir, Y.; Mandelbrot, B.B.; Aharony, A. Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Phys. Rev. Lett.
**1983**, 50, 145–148. [Google Scholar] [CrossRef] - Zhang, X.F.; Peng, R.W.; Kang, S.S.; Cao, L.S.; Zhang, R.L.; Wang, M.; Hu, A. Tunable High-frequency Magnetostatic Waves in Thue–Morse Antiferromagnetic Multilayers. J. Appl. Phys.
**2006**, 100, 063911. [Google Scholar] [CrossRef] - Qiu, F.; Peng, R.W.; Huang, X.Q.; Liu, Y.M.; Wang, M.; Hu, A.; Jiang, S.S. Resonant transmission and frequency trifurcation of light waves in Thue–Morse dielectric multilayers. Europhys. Lett.
**2003**, 63, 853–859. [Google Scholar] [CrossRef] - Qiu, F.; Peng, R.W.; Huang, X.Q.; Hu, X.F.; Wang, M.; Hu, A.; Jiang, S.S.; Feng, D. Omnidirectional reflection of electromagnetic waves on Thue–Morse dielectric multilayers. Europhys. Lett.
**2004**, 68, 658–663. [Google Scholar] [CrossRef] - Huang, X.; Wang, Y.; Gong, C. Numerical investigation of light-wave localization in optical Fibonacci superlattices with symmetric internal structure. J. Phys. Condens. Matter
**1999**, 11, 517–520. [Google Scholar] [CrossRef] - Peng, R.W.; Jin, G.J.; Wang, M.; Hu, A.; Jiang, S.S.; Feng, D. Interface optical phonons in k-component Fibonacci dielectric multilayers. Phys. Rev. B
**1999**, 59, 3599–3605. [Google Scholar] [CrossRef] - Zhang, K.; Xu, Y.; Chen, T.Y.; Jing, H.; Shi, W.B.; Xiong, B.; Peng, R.W.; Wang, M. Multimode photon-exciton coupling in an organic-dye-attached photonic quasicrystal. Opt. Lett.
**2016**, 41, 5740–5743. [Google Scholar] [CrossRef] - Hu, A.; Wen, Z.X.; Jiang, S.S.; Tong, W.T.; Peng, R.W.; Feng, D. One-dimensional k-component Fibonacci Structures. Phys. Rev. B
**1993**, 48, 829–835. [Google Scholar] [CrossRef] - Huang, X.Q.; Jiang, S.S.; Peng, R.W.; Hu, A. Perfect transmission and self-similar optical transmission spectra in symmetric Fibonacci-class multilayers. Phys. Rev. B
**2001**, 63, 245104. [Google Scholar] [CrossRef] - Peng, R.W.; Huang, X.Q.; Qiu, F.; Wang, M.; Hu, A.; Jiang, S.S.; Mazzer, M. Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers. Appl. Phys. Lett.
**2002**, 80, 3063–3065. [Google Scholar] [CrossRef] - Peng, R.W.; Liu, Y.M.; Huang, X.Q.; Qiu, F.; Wang, M.; Hu, A.; Jiang, S.S.; Feng, D.; Ouyang, L.Z.; Zou, J. Dimerlike positional correlation and resonant transmission of electromagnetic waves in aperiodic dielectric multilayers. Phys. Rev. B
**2004**, 69, 165109. [Google Scholar] [CrossRef] [Green Version] - Peng, R.W.; Wang, M.; Hu, A.; Jiang, S.S.; Jin, G.J.; Feng, D. Photonic localization in one-dimensional k-component Fibonacci structures. Phys. Rev. B
**1998**, 57, 1544. [Google Scholar] [CrossRef] - Hu, Q.; Zhao, J.Z.; Peng, R.W.; Gao, F.; Zhang, R.L.; Wang, M. “Rainbow” trapped in a self-similar coaxial optical waveguide. Appl. Phys. Lett.
**2010**, 96, 161101. [Google Scholar] [CrossRef] - Hu, Q.; Xu, D.H.; Peng, R.W.; Zhou, Y.; Yang, Q.L.; Wang, M. Tune the “rainbow” trapped in a multilayered waveguide. Europhys. Lett.
**2012**, 99, 57007. [Google Scholar] [CrossRef] - Peng, R.W.; Mazzer, M.; Barnham, K.W.J. Efficiency enhancement of ideal photovoltaic solar cells by photonic excitations in multi-intermediate band structures. Appl. Phys. Lett.
**2003**, 83, 770–772. [Google Scholar] [CrossRef] - Matsui, T.; Agrawal, A.; Nahata, A.; Vardeny, Z.V. Transmission resonances through aperiodic arrays of subwavelength apertures. Nature
**2007**, 446, 517–521. [Google Scholar] [CrossRef] - Bao, Y.J.; Zhang, B.; Wu, Z.; Si, J.W.; Wang, M.; Peng, R.W.; Lu, X.; Li, Z.F.; Hao, X.P.; Ming, N.B. Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array. Appl. Phys. Lett.
**2007**, 90, 251914. [Google Scholar] [CrossRef] - Bao, Y.J.; Li, H.M.; Chen, X.C.; Peng, R.W.; Wang, M.; Lu, X.; Shao, J.; Ming, N.B. Tailoring the resonances of surface plasmas on fractal-featured metal film by adjusting aperture configuration. Appl. Phys. Lett.
**2008**, 92, 151902. [Google Scholar] [CrossRef] - Zhu, L.H.; Shao, M.R.; Peng, R.W.; Fan, R.H.; Huang, X.R.; Wang, M. Broadband absorption and efficiency enhancement of an ultra-thin silicon solar cell with a plasmonic fractal. Opt. Express
**2013**, 21, 313–323. [Google Scholar] [CrossRef]

**Figure 1.**Schematic description of the typical aperiodic structures. (

**a**) Fibonacci structure [16]. (

**b**) Thue–Morse structure [18]. (

**c**) 2D Fibonacci structure [25]. (Inset: Inflation rules of the first two generations of a 2D Fibonacci sequence). (

**d**) Regular Penrose tiling [27]. (

**e**) Koch snowflake fractal [30]. (

**f**) Sierpinski carpet pattern [31].

**Figure 3.**Multimodes in symmetric aperiodic structures. (

**a**) Schematic of the symmetric Fibonacci-class (SFC(1)) multilayer structure. (

**b**) The transmission coefficient of Fibonacci-class (FC(1)) (13 layers) and (

**c**) SFC(1) (26 layers) systems. (

**d**) Symmetrical fifth-generation Fibonacci TiO

_{2}/SiO

_{2}multilayer film. (

**e**) The measured (upper row) and calculated (lower row) transmission coefficient T as a function of the wave number for the symmetric TiO

_{2}/SiO

_{2}mutilayers with defects in the central gap with different layers (SMD V

_{2}, SMD V

_{3}, and SMD V

_{4}from left to right). (Adapted from ref. [39] (a–c) and ref. [41] (d,e)).

**Figure 4.**Transmission coefficient T as a function of the phase with the different incommensurate intervals k in k-component Fibonacci structures. The number of layers N are as follows: (

**a**) N = 28,657; (

**b**) N = 27,201; (

**c**) N = 31,422; (

**d**) N = 29,244; (

**e**) N = 233; (

**f**) N = 277; (

**g**) N = 250; and (

**h**) N = 245, respectively. (Adapted from ref. [42]).

**Figure 5.**Multimode photon-exciton coupling. (

**a**) Schematic of a Fibonacci photonic quasicrystal with J-aggregates on the top surface. (

**b**) Experimentally measured transmission spectra of the photonic quasicrystal; the modes labeled C, L, and H correspond to three peaks. (

**c**) Transmission spectra of the sample under various incident angles. Polariton bands were traced by dashed lines. (

**d**) Dispersion map of the sample. Calculated dashed lines fit the polariton bands and Rabi splitting. (Adapted from ref. [37]).

**Figure 6.**Multiband waveguide. (

**a**) Structure of a self-similar dielectric waveguide (SDW), where a coaxial Thue–Morse multilayer consisting of two building blocks was employed to cover a hollow core. The lower figure manifests refractive-index distributions in the SDWs. (

**b**) Photonic bands and transmission modes in the SDW. (

**c**) The electric-field time-average energy density distribution in the SDW for different modes. (Adapted from ref. [43]).

**Figure 7.**Enhancement of solar cells caused by photon-induced transitions in multi-intermediate band structures. (

**a**) Various radiation transitions between intermediate multiband structures in the designed solar cell. (

**b**) Limiting efficiency η for three model solar cells with diverse intermediate multiband structures (k = 1,2,3). (

**c**) Electronic miniband structures of several periodic and aperiodic In

_{0.49}Ga

_{0.51}P/GaAs superlattices. The inset shows the band-edge diagram of the In

_{0.49}Ga

_{0.51}P/GaAs interface (at room temperature). (Adapted from ref. [45]).

**Figure 8.**Transmission through aperiodic aperture arrays. (

**a**) A Penrose quasicrystal constructed of thin and thick rhomb tiles. (

**b**) Transmission of the Penrose-type quasicrystal perforated films with different side lengths. (

**c**) Scanning electron micrograph of an aluminum film perforated with a Sierpinski carpet fractal-featured aperture array. (

**d**) Transmission of the metallic Sierpinski carpet structure. (a,b) are obtained experimentally; (c–e) are simulated results. (Adapted from ref. [46] (a,b) and ref. [47] (c,d)).

**Figure 9.**Solar cell with a plasmonic fractal to achieve broadband absorption. (

**a**) Schematic of the solar cell with a plasmonic fractal. Including a Ag fractal-like pattern, a Si absorbent layer, and a silver back reflector. (

**b**) Absorbance spectra of the solar cell with or without the antireflection coating (ARC). (

**c**) Quantum efficiencies of the 50 nm thick silicon solar cells: ref-1 (bare Si film); ref-2 (Si film with Ag back reflector); solar cells with base-periodicity patterns (P1 = 100 nm, P2 = 200 nm, P3 = 400 nm); solar cells with a plasmonic fractal and additional dielectric ARC. (Adapted from ref. [49]).

© 2019 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jing, H.; He, J.; Peng, R.-W.; Wang, M.
Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices. *Symmetry* **2019**, *11*, 1120.
https://doi.org/10.3390/sym11091120

**AMA Style**

Jing H, He J, Peng R-W, Wang M.
Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices. *Symmetry*. 2019; 11(9):1120.
https://doi.org/10.3390/sym11091120

**Chicago/Turabian Style**

Jing, Hao, Jie He, Ru-Wen Peng, and Mu Wang.
2019. "Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices" *Symmetry* 11, no. 9: 1120.
https://doi.org/10.3390/sym11091120