# Magneto-Optical Faraday Effect in Quasicrystalline and Aperiodic Microresonator Structures

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Faraday, M. On the magnetization of light and the illumination of magnetic lines of force. Philos. Trans. R. Soc. Lond.
**1846**, 136, 1–20. [Google Scholar] - Kimel, A.; Zvezdin, A.; Sharma, S.; Shallcross, S.; De Sousa, N.; García-Martín, A.; Salvan, G.; Hamrle, J.; Stejskal, O.; McCord, J.; et al. The 2022 magneto-optics roadmap. J. Phys. Appl. Phys.
**2022**, 55, 463003. [Google Scholar] [CrossRef] - Karki, D.; El-Ganainy, R.; Levy, M. Toward high-performing topological edge-state optical isolators. Phys. Rev. Appl.
**2019**, 11, 034045. [Google Scholar] [CrossRef] - Silva, C.; Guedes, M.; Pereira, N.; Alvarenga, A. Assembly of a Faraday modulator for polarimetric measurements. In Proceedings of the Journal of Physics: Conference Series, Brasilia, Brazil, 23–27 June 2015; IOP Publishing: Bristol, UK, 2015; Volume 575, p. 012019. [Google Scholar]
- Ho, K.S.; Im, S.J.; Pae, J.S.; Ri, C.S.; Han, Y.H.; Herrmann, J. Switchable plasmonic routers controlled by external magnetic fields by using magneto-plasmonic waveguides. Sci. Rep.
**2018**, 8, 1–8. [Google Scholar] [CrossRef] - Maccaferri, N.; E Gregorczyk, K.; De Oliveira, T.V.; Kataja, M.; Van Dijken, S.; Pirzadeh, Z.; Dmitriev, A.; Åkerman, J.; Knez, M.; Vavassori, P. Ultrasensitive and label-free molecular-level detection enabled by light phase control in magnetoplasmonic nanoantennas. Nat. Commun.
**2015**, 6, 1–9. [Google Scholar] - Rizal, C.; Manera, M.G.; Ignatyeva, D.O.; Mejía-Salazar, J.R.; Rella, R.; Belotelov, V.I.; Pineider, F.; Maccaferri, N. Magnetophotonics for sensing and magnetometry toward industrial applications. J. Appl. Phys.
**2021**, 130, 230901. [Google Scholar] [CrossRef] - Fujikawa, R.; Tanizaki, K.; Baryshev, A.; Lim, P.B.; Shin, K.H.; Uchida, H.; Inoue, M. Magnetic field sensors using magnetophotonic crystals. In Proceedings of the Photonic Crystals and Photonic Crystal Fibers for Sensing Applications II, SPIE, 2006, Boston, MA, USA, 1–4 October 2006; Volume 6369, pp. 85–92. [Google Scholar]
- Ignatyeva, D.O.; Knyazev, G.A.; Kalish, A.N.; Chernov, A.I.; Belotelov, V.I. Vector magneto-optical magnetometer based on resonant all-dielectric gratings with highly anisotropic iron garnet films. J. Phys. Appl. Phys.
**2021**, 54, 295001. [Google Scholar] [CrossRef] - Lodewijks, K.; Maccaferri, N.; Pakizeh, T.; Dumas, R.K.; Zubritskaya, I.; Åkerman, J.; Vavassori, P.; Dmitriev, A. Magnetoplasmonic design rules for active magneto-optics. Nano Lett.
**2014**, 14, 7207–7214. [Google Scholar] [CrossRef] - Belotelov, V.; Zvezdin, A. Magnetooptics and extraordinary transmission of the perforated metallic films magnetized in polar geometry. J. Magn. Magn. Mater.
**2006**, 300, e260–e263. [Google Scholar] [CrossRef] - Lyubchanskii, I.; Dadoenkova, N.; Lyubchanskii, M.; Shapovalov, E.; Rasing, T. Magnetic photonic crystals. J. Phys. Appl. Phys.
**2003**, 36, R277. [Google Scholar] [CrossRef] - Inoue, M.; Baryshev, A.; Goto, T.; Baek, S.; Mito, S.; Takagi, H.; Lim, P. Magnetophotonic crystals: Experimental realization and applications. Magnetophotonics
**2013**, Vol. 178, 163–190. [Google Scholar] - Sakoda, K. Optical Properties of Photonic Crystals; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2004; Volume 80. [Google Scholar]
- Inoue, M.; Yamamoto, T.; Isamoto, K.; Fujii, T. Effect of structural irregularity on propagation properties of optical waves in discontinuous magneto-optical media with one-dimensional quasirandom aray structures. J. Appl. Phys.
**1996**, 79, 5988–5990. [Google Scholar] [CrossRef] - Steel, M.; Levy, M.; Osgood, R. High transmission enhanced Faraday rotation in one-dimensional photonic crystals with defects. IEEE Photonics Technol. Lett.
**2000**, 12, 1171–1173. [Google Scholar] [CrossRef] - Koerdt, C.; Rikken, G.; Petrov, E. Faraday effect of photonic crystals. Appl. Phys. Lett.
**2003**, 82, 1538–1540. [Google Scholar] [CrossRef] - Inoue, M.; Arai, K.; Fujii, T.; Abe, M. Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers. J. Appl. Phys.
**1998**, 83, 6768–6770. [Google Scholar] [CrossRef] - Inoue, M.; Fujikawa, R.; Baryshev, A.; Khanikaev, A.; Lim, P.; Uchida, H.; Aktsipetrov, O.; Fedyanin, A.; Murzina, T.; Granovsky, A. Magnetophotonic crystals. J. Phys. Appl. Phys.
**2006**, 39, R151. [Google Scholar] [CrossRef] - Kahl, S.; Grishin, A.M. Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal. Appl. Phys. Lett.
**2004**, 84, 1438–1440. [Google Scholar] [CrossRef] - Dermeche, N.; Bouras, M.; Abdi-Ghaleh, R.; Kahlouche, A.; Hocini, A. Existence of high Faraday rotation and transmittance in magneto photonic crystals made by silica matrix doped with magnetic nanoparticles. Optik
**2019**, 198, 163225. [Google Scholar] [CrossRef] - Lyubchanskii, I.; Dadoenkova, N.; Lyubchanskii, M.; Shapovalov, E.; Zabolotin, A.; Lee, Y.; Rasing, T. Response of two-defect magnetic photonic crystals to oblique incidence of light: Effect of defect layer variation. J. Appl. Phys.
**2006**, 100, 096110. [Google Scholar] [CrossRef] - Pankin, P.S.; Vetrov, S.Y.; Timofeev, I.V. Tunable hybrid Tamm-microcavity states. JOSA B
**2017**, 34, 2633–2639. [Google Scholar] [CrossRef] - Goto, T.; Dorofeenko, A.; Merzlikin, A.; Baryshev, A.; Vinogradov, A.; Inoue, M.; Lisyansky, A.; Granovsky, A. Optical Tamm states in one-dimensional magnetophotonic structures. Phys. Rev. Lett.
**2008**, 101, 113902. [Google Scholar] [CrossRef] [PubMed] - Goto, T.; Baryshev, A.; Inoue, M.; Dorofeenko, A.; Merzlikin, A.; Vinogradov, A.; Lisyansky, A.; Granovsky, A. Tailoring surfaces of one-dimensional magnetophotonic crystals: Optical Tamm state and Faraday rotation. Phys. Rev. B
**2009**, 79, 125103. [Google Scholar] [CrossRef] - Khanikaev, A.B.; Baryshev, A.V.; Inoue, M.; Kivshar, Y.S. One-way electromagnetic Tamm states in magnetophotonic structures. Appl. Phys. Lett.
**2009**, 95, 011101. [Google Scholar] [CrossRef] [Green Version] - Levine, D.; Steinhardt, P.J. Quasicrystals: A new class of ordered structures. Phys. Rev. Lett.
**1984**, 53, 2477. [Google Scholar] [CrossRef] - 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] - Vardeny, Z.V.; Nahata, A.; Agrawal, A. Optics of photonic quasicrystals. Nat. Photonics
**2013**, 7, 177–187. [Google Scholar] [CrossRef] - Fu, X.; Liu, Y.; Zhou, P.; Sritrakool, W. Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices. Phys. Rev. B
**1997**, 55, 2882. [Google Scholar] [CrossRef] - Beck, M.; Geoghegan, R. The Art of Proof: Basic Training for Deeper Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Xu, P.; Tian, H.; Ji, Y. One-dimensional fractal photonic crystal and its characteristics. JOSA B
**2010**, 27, 640–647. [Google Scholar] [CrossRef] - Segovia-Chaves, F.; Vinck-Posada, H. Transmittance spectrum in a 1D photonic crystal with a Fibonacci sequence composed of polymer materials. Optik
**2019**, 196, 163141. [Google Scholar] [CrossRef] - Gong, Y.; Liu, X.; Wang, L.; Lu, H.; Wang, G. Multiple responses of TPP-assisted near-perfect absorption in metal/Fibonacci quasiperiodic photonic crystal. Opt. Express
**2011**, 19, 9759–9769. [Google Scholar] [CrossRef] - Lin, Z.; Kubo, H.; Goda, M. Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals. Z. Für Phys. B Condens. Matter
**1995**, 98, 111–118. [Google Scholar] [CrossRef] - Liu, N.h. Propagation of light waves in Thue-Morse dielectric multilayers. Phys. Rev. B
**1997**, 55, 3543. [Google Scholar] [CrossRef] - Liviotti, E. A study of the structure factor of Thue-Morse and period-doubling chains by wavelet analysis. J. Phys. Condens. Matter
**1996**, 8, 5007. [Google Scholar] [CrossRef] - Steurer, W.; Sutter-Widmer, D. Photonic and phononic quasicrystals. J. Phys. D Appl. Phys.
**2007**, 40, R229. [Google Scholar] [CrossRef] - Mehdizadeh, F.; Alipour-Banaei, H. Bandgap management in two-dimensional photonic crystal thue-morse structures. J. Opt. Commun.
**2013**, 34, 61–65. [Google Scholar] [CrossRef] - Moretti, L.; Mocella, V. Two-dimensional photonic aperiodic crystals based on Thue-Morse sequence. Opt. Express
**2007**, 15, 15314–15323. [Google Scholar] [CrossRef] - Loiko, V.; Miskevich, A. Optical properties of structures composed of periodic, quasi-periodic, and aperiodic sequences of particulate monolayers. Opt. Spectrosc.
**2017**, 122, 16–24. [Google Scholar] [CrossRef] - Xavier, J.; Probst, J.; Becker, C. Deterministic composite nanophotonic lattices in large area for broadband applications. Sci. Rep.
**2016**, 6, 1–12. [Google Scholar] [CrossRef] - Kasture, S.; Ravishankar, A.P.; Yallapragada, V.; Patil, R.; Valappil, N.V.; Mulay, G.; Achanta, V.G. Plasmonic quasicrystals with broadband transmission enhancement. Sci. Rep.
**2014**, 4, 1–6. [Google Scholar] [CrossRef] - Ghorbani-Oranj, F.; Abdi-Ghaleh, R.; Roumi, B.; Jamshidi-Ghaleh, K.; Madani, A.; Zhou, Y. Comparative study of Faraday rotation spectra of periodic and Fibonacci multilayer structures containing graphene sheets. Phys. B Condens. Matter
**2022**, 636, 413835. [Google Scholar] [CrossRef] - da Silva, R.; Zanetti, F.; Lyra, M.; de Oliveira, I. Polarization rotation of localized modes in magneto-photonic Fibonacci structures containing nematic layers. Mol. Cryst. Liq. Cryst.
**2017**, 657, 11–20. [Google Scholar] [CrossRef] - Zamani, M.; Amanollahi, M.; Taraz, M. Octonacci magnetophotonic quasi-crystals. Opt. Mater.
**2019**, 88, 187–194. [Google Scholar] [CrossRef] - Guo, S.J.; Hu, C.X.; Zhang, H.F. A reconfigurable device based on the one-dimensional magnetized plasma photonic crystals nested with the Pell and Thue–Morse sequences. Opt. Quantum Electron.
**2020**, 52, 1–18. [Google Scholar] [CrossRef] - Kalish, A.N.; Komarov, R.S.; Kozhaev, M.A.; Achanta, V.G.; Dagesyan, S.A.; Shaposhnikov, A.N.; Prokopov, A.R.; Berzhansky, V.N.; Zvezdin, A.K.; Belotelov, V.I. Magnetoplasmonic quasicrystals: An approach for multiband magneto-optical response. Optica
**2018**, 5, 617–623. [Google Scholar] [CrossRef] - Dotsenko, A.A.; Kalish, A.N.; Kozhaev, M.A.; Ignatyeva, D.O.; Achanta, V.G.; Zvezdin, A.K.; Belotelov, V.I. Magneto-optical effects in 2D plasmonic gratings with various types of ordering. AIP Conf. Proc.
**2020**, 2300, 020026. [Google Scholar] - Wu, J.; Wang, Z.; Wu, B.; Shi, Z.; Wu, X. The giant enhancement of nonreciprocal radiation in Thue-morse aperiodic structures. Opt. Laser Technol.
**2022**, 152, 108138. [Google Scholar] [CrossRef] - Guo, S.; Hu, C.; Zhang, H. Ultra-wide unidirectional infrared absorber based on 1D gyromagnetic photonic crystals concatenated with general Fibonacci quasi-periodic structure in transverse magnetization. J. Opt.
**2020**, 22, 105101. [Google Scholar] [CrossRef] - Guo, S.; Mao, M.; Zhou, Z.; Zhang, D.; Zhang, H. The wide-angle broadband absorption and polarization separation in the one-dimensional magnetized ferrite photonic crystals arranged by the Dodecanacci sequence under the transverse magnetization configuration. J. Phys. D Appl. Phys.
**2020**, 54, 015004. [Google Scholar] [CrossRef] - Wang, H.; Yu, B.; Cai, D.; Chen, K. Magneto-Optical Goos- Hänchen Displacement in Quasiperiodic Gradient 1D Photonic Crystal. Phys. Status Solidi
**2022**, 259, 2100624. [Google Scholar] [CrossRef] - Moharam, M.; Grann, E.B.; Pommet, D.A.; Gaylord, T. Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. JOSA A
**1995**, 12, 1068–1076. [Google Scholar] [CrossRef] - Li, L. Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors. J. Opt. A Pure Appl. Opt.
**2003**, 5, 345. [Google Scholar] [CrossRef] - Zvezdin, A.K.; Kotov, V.A. Modern Magnetooptics and Magnetooptical Materials; CRC Press: Boca Raton, FL, USA, 1997. [Google Scholar]
- Ignatyeva, D.; Kapralov, P.; Knyazev, G.; Sekatskii, S.; Dietler, G.; Vasiliev, M.; Alameh, K.; Belotelov, V. High-Q surface modes in photonic crystal/iron garnet film heterostructures for sensor applications. JETP Lett.
**2016**, 104, 679–684. [Google Scholar] [CrossRef] - Ignatyeva, D.O.; Karki, D.; Voronov, A.A.; Kozhaev, M.A.; Krichevsky, D.M.; Chernov, A.I.; Levy, M.; Belotelov, V.I. All-dielectric magnetic metasurface for advanced light control in dual polarizations combined with high-Q resonances. Nat. Commun.
**2020**, 11, 1–8. [Google Scholar] [CrossRef] - Voronov, A.A.; Karki, D.; Ignatyeva, D.O.; Kozhaev, M.A.; Levy, M.; Belotelov, V.I. Magneto-optics of subwavelength all-dielectric gratings. Opt. Express
**2020**, 28, 17988–17996. [Google Scholar] [CrossRef] [PubMed] - Levy, M.; Borovkova, O.V.; Sheidler, C.; Blasiola, B.; Karki, D.; Jomard, F.; Kozhaev, M.A.; Popova, E.; Keller, N.; Belotelov, V.I. Faraday rotation in iron garnet films beyond elemental substitutions. Optica
**2019**, 6, 642–646. [Google Scholar] [CrossRef] - Belotelov, V.I.; Kalish, A.N.; Zvezdin, A.K.; Gopal, A.V.; Vengurlekar, A.S. Fabry–Perot plasmonic structures for nanophotonics. JOSA B
**2012**, 29, 294–299. [Google Scholar] [CrossRef] - Poddubny, A.; Pilozzi, L.; Voronov, M.; Ivchenko, E. Exciton-polaritonic quasicrystalline and aperiodic structures. Phys. Rev. B
**2009**, 80, 115314. [Google Scholar] [CrossRef] - Poddubny, A.; Ivchenko, E. Photonic quasicrystalline and aperiodic structures. Phys. E Low-Dimens. Syst. Nanostruct.
**2010**, 42, 1871–1895. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The scheme illustrating the considered configuration of a Faraday rotation in the microresonators formed by multilayered structures with different layer arrangements and a magnetic layer. The multilayers labeled as ${\left({A}_{n}{B}_{m}\right)}_{k}$ are formed by non-magnetic dielectric layers, denoted as A and B, correspondingly arranged in various sequences of periodic, quasicrystalline and aperiodic ordering (see Materials and Methods for the details and Table 1).

**Figure 2.**Fourier spectra $\tilde{\epsilon}\left(K\right)$ of the multilayer permittivity spatial distribution $\tilde{\epsilon}\left(z\right)=\epsilon \left(z\right)-\langle \epsilon \left(z\right)\rangle $ for (

**a**) photonic crystal; (

**b**) Fibonacci quasicrystal; (

**c**) Thue–Morse multilayer (see Table 1).

**Figure 3.**Optical transmittance vs. wavelength $\lambda $ (angle of incidence $\theta $ = 0 deg) spectra for multilayers and corresponding microresonators formed based on (

**a**) photonic crystal; (

**b**) Fibonacci quasicrystal; (

**c**) Thue–Morse structure; (

**d**) optical transmittance of a bare magnetic film. Violet lines labeled as ‘multilayer’ and red lines labeled as ’microresonator’ denote the transmittance of the corresponding multilayers and microresonators formed by a magnetic layer sandwiched between them.

**Figure 4.**The distribution of the electromagnetic field magnitude ${\left|E\right|}^{2}$ for microresonator modes excited in multilayers based on (

**a**) photonic crystal at the wavelength 800 nm; (

**b**) just a bare magnetic film at 800 nm; (

**c**,

**d**) Fibonacci quasicrystal at the wavelengths; (

**c**) 662 nm and (

**d**) 1010 nm; (

**e**,

**f**) Thue–Morse multilayer at the wavelengths (

**e**) 686 nm and (

**f**) 958 nm.

**Figure 5.**The Faraday rotation spectra normalized to a uniform film value $\varphi /{\varphi}_{0}$ for microresonators based on the (

**a**) photonic crystal; (

**b**) quasicrystal crystal; (

**c**) Thue–Morse structure; and (

**d**) bare magnetic film. Notice the y axis scale difference.

**Figure 6.**(

**a**) The relative Faraday effect and (

**b**) peak FOM enhancements at the peaks corresponding to microresonator modes in photonic crystal, quasicrystal and Thue–Morse structures (see the legend) as a function of the number of layers in the structure.

Type | Multilayer Composition |
---|---|

Photonic crystal | A B A B A B A B A B |

Quasicrystal | A B A A B A B A A B A A B |

Thue–Morse sequence | A B B A B A A B B A A B A B B A B A A B A B B A A B B A B A A B |

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**MDPI and ACS Style**

Ignatyeva, D.O.; Golovko, P.V.; Belotelov, V.I.
Magneto-Optical Faraday Effect in Quasicrystalline and Aperiodic Microresonator Structures. *Magnetochemistry* **2023**, *9*, 54.
https://doi.org/10.3390/magnetochemistry9020054

**AMA Style**

Ignatyeva DO, Golovko PV, Belotelov VI.
Magneto-Optical Faraday Effect in Quasicrystalline and Aperiodic Microresonator Structures. *Magnetochemistry*. 2023; 9(2):54.
https://doi.org/10.3390/magnetochemistry9020054

**Chicago/Turabian Style**

Ignatyeva, Daria O., Polina V. Golovko, and Vladimir I. Belotelov.
2023. "Magneto-Optical Faraday Effect in Quasicrystalline and Aperiodic Microresonator Structures" *Magnetochemistry* 9, no. 2: 54.
https://doi.org/10.3390/magnetochemistry9020054