## Abstract

Photonic bandgap fibers have a critical constraint determined by wavelength. The principle of scale invariance requires that features remain unchanged even as the scale of an object changes. This paper introduces a new concept for fractal photonic crystal fibers integrating these two. Our simulation confirmed single-mode transmission is possible for a fiber whose core diameter exceeds 35 times the wavelength.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The recent gains in laser power for fine processing and long distance energy transmission have been remarkable. A typical configuration is a fiber laser of 20 KW or 10 KW, with expanding use of pulsed lasers for fine processing. For pulsed lasers, the pulse energy per pulse time width becomes the peak output. For high-output lasers, peak output values range from 200 MW to 20 GW. High output pulsed lasers cannot be transmitted by fiber at this time; they rely on mirrors to transmit energy. Transmitting this laser energy requires fibers of large core diameter. However, the number of propagation modes of refractive index waveguide type fibers increases with core area. Photonic bandgap fibers generally require comparable structure and wavelength length scales. A core diameter not more than 10 times the wavelength is regarded as the upper limit for single mode transmission [1]. If the energy density per area of the fiber cross-section exceeds the threshold value, stimulated Raman scattering occurs, and the output becomes unstable. Thus, a high-power laser requires a core diameter 20 to 40 times the wavelength. This exceeds the design limit for photonic bandgap fibers. The double lattice structure has been examined as a potential solution [2], but higher order modes increase with increased area, leaving little design freedom with respect to core area.

There has been growing interest in universality and scale invariance as a field of physics. Scale invariance refers to characteristics that do not change when scales of length are multiplied by a common factor. The difficulty with photonic bandgap fibers associated with large core diameters is that the scale of wavelength is not comparable with that of a large core diameter. It may be possible to solve the problem by applying universal scaling principles. The potential for resolving this issue drawing on scale universality motivates our research. Known for their scale universality, fractal are forms that emerge when matter undergoes phase transitions. While two-dimensional periodic and fractal structures are relatively unfamiliar, lattice structures obtained by absorbing hydrogen in the metallic palladium of a face centered cubic crystal become a discrete fractal when projected onto a two-dimensional plane. Palladium, a hydrogen absorbing metal, undergoes a structural phase transition when the number of hydrogen atoms exceeds approximately 0.6 times the number of palladium atoms.

Figure 1(a) shows a projection lattice of a palladium crystal on a two-dimensional plane, as viewed from the (111) direction [3]. Surprisingly, helium hydrate, characterized by helium in water with a regular structure, has the same two-dimensional lattice structure as palladium hydride [Fig. 1(b)] [4]. The structural phase transition also occurs with increasing helium content. Because palladium hydride has a three-dimensional lattice structure and helium hydrate has a two-dimensional lattice structure, the structures of the high concentration phases differ [5]. This paper presents the results of our investigation of eigenstates using two-dimensional patterns with chirality common to palladium hydride and helium hydrate in the design of photonic fibers.

## 2. Calculation methods

Calculations were performed using optical simulation software from Synopsys, Inc.: Rsoft FemSIM, a generalized mode solver based on the Finite Element Method (FEM) and Rsoft BandSOLVE, a tool for calculations involving photographic band structures [6]. The Maxwell’s equations formulated free of explicit terms for charge and current are presented below. Here, E is electric field, H magnetic field, µ permeability, and ɛ permittivity.

**x**) indicates a dielectric constant that depends on the (x, y) coordinate. Here c and $\mathrm{\omega }$ are the speed of light and angular velocity in the medium. Permittivity and permeability have the following relationship $\textrm{n} = \sqrt {\frac{{\varepsilon \mu }}{{{e_0}{\mu _0}}}} $, and $\mathrm{\mu } = 1,{\; }$so effective index

*n*is the square root of eigenvalue $\textrm{k} ={\pm} \sqrt {{{\left( {\frac{\omega }{c}} \right)}^2}} $ times $\frac{{{c_0}}}{\omega }$ . Here c

_{eff}_{0}is the speed of light in vacuum.

*k*is a complex number and has an indefiniteness of ± by taking the root.

The mode solver for photonic fibers solves this equation in real space rather than in reciprocal lattice space to obtain the eigenstates. Since the mode solver assumes that the electromagnetic field does not depend on the z coordinate, the analysis is performed by separating light into **E _{z}** having an electric field z component and

**H**having a magnetic field z component.

_{z}For the band solver, we assume that changes in the refractive index of the cross-sectional structure of the photonic fiber are periodic in the (x, y) plane. Applying Bloch’s theorem, we write ${\mathbf H}({\mathbf x} )$ as$\; \textrm{follows}:{\; }$

## 3. Models

We modeled the structure shown in Fig. 2(a) as a fractal two-dimensional periodic structure. The red portion is an air hole, the medium is silica, and the respective refractive indices are 1 and 1.4498670626. The air hole diameter d is 8.64 µm; air hole spacing *Λ* is 10.8 µm. Hexagons—aggregates of air holes with radius *R *= 50 µm from the center—are arranged at 60 ° intervals from a position of 90 ° ±$\textrm{CO}{\textrm{S}^{ - 1}}\left( {\frac{7}{{2\sqrt {13} }}} \right)$(≒90 ° ± 13.89789 °) with respect to the horizontal arrangement of air holes. The outside arrangement is determined so that a large hexagon composed of the sides of the center-to-center distance *R* fills the plane without gaps. The core diameter is approximately 40 µm. (See Discussion for an in-depth discussion of this structure.) Fig. 2(b) shows the minimum lattice structure of the fractal. We performed an eigenstate analysis of a real space using Rsoft FemSim and a wavelength of 1.07 µm, grid size of 0.2 µm, and simulation region of approximately 4 × 10^{4} µm^{2}. A perfectly matched layer (PML), a non-reflecting boundary for all electromagnetic waves, is used as the boundary condition [7]. Figure 2(c) shows a lattice model of periodic boundary conditions used to calculate the band solver.

For comparison, we analyzed fibers with double lattice structure [8] shown in Fig. 3(a) and simple large structure shown in Fig. 4(a). The double lattice structure shape connects two lattices of different pitch in the two-dimensional plane. The diameter *d* of air hole is 10 µm; the inner lattice spacing *Λ _{in}* is 16.66 µm; and the outer lattice spacing

*Λ*is 25 µm. The core diameter is 72 µm. The inner and outer grids connect at positions corresponding to the least common multiple of the grid spacing. A grid connects the innermost and inner gratings, reducing the number of air holes on the outside by one. The Discussion addresses similarities between the fractal structure and the double lattice model. The air hole diameter of the simple large diameter structure is 37 µm. The pitch is 48 µm. Figures 3(b) and 4(b) show periodic models of the double lattice and the simple hole used for the band solver. The periodic boundary condition is given as a simple trigonal lattice and a unit lattice, which are equivalent to the hexagonal lattice. The calculations use 128, 64, and 256 points per unit lattice vector for the fractal, simple hole, and double lattice models, respectively. The propagation constant assumed for the z-direction, the longitudinal direction of the fiber, is 10, an estimated figure obtained from $\frac{{2\pi \cdot {n_{eff}}}}{\lambda }$. Please see the checklist in Section 8 that summarizes all of the style specifications.

_{out}## 4. Results

Figure 5(a) shows the eigenstates of the electric field of the fractal model of Fig. 2(a). Figure 5(b) shows the eigenstates of the fractal model of Fig. 2(b). The diameter of the e2 beam is about 39 µm for the primary modes of both models. The electric mode fields of the single winding fractal fiber can be categorized as Laguerre Gaussian beam with indexes of (*p,m*), where *p* is radial mode index and *m* is angular mode index. The effective refractive indices are tabulated in Tables 1 and 2, respectively. The propagation loss *P _{L}* in dB/km unit is obtained from

*n*, the imaginary part of the effective refractive index based on the following equation, where λ is wavelength in µm units:

_{i}The refractive index is a physically correct solution where the real part is negative and the complex part is positive, obtained by multiplying the complex index by minus one. Depending on the chirality of the fractal structure, it means that the Lager-Gaussian has either positive or negative angular momentum, and the opposite momentum is reflected. The fact that half of the eigenmodes in the (0,1) mode and the (0,3) mode do not have a negative refractive index needs further study.

Figure 6 shows the eigenmodes of the electric field for the double-lattice fiber. With mode search, we can obtain higher order modes up to (1,1) mode, which does not exist in the single winding fractal model. The beam diameter was 57 μm for the 1st order. The effective refractive indices are tabulated in Table 3. As is clear from the comparison of the complex part of the refractive index, the transmission loss of the fractal model is about seven digits smaller than that of the double lattice model. Figure 7 shows the eigenmodes of electric field for the simple large-diameter air-hole model (Fig. 4(a)). As well as the double lattice model, solutions up to (1,1) order mode exist. The effective refractive indices are tabulated in Table 4. The imaginary part of the simple large air hole model is five digits larger than that of the double lattice model. This is because the eigenmodes of a simple large diameter air hole model are not confined by the band gap.

Since the core diameter of the double lattice model is twice that of the other two models, there is no question that it is highly effective as a large core diameter fiber.

Figures 8(a), 8(b), and 8(c) show the band structures of the fractal model, double lattice model, and simple large core diameter air hole model. The structure constant a is 50, 173, and 48 µm for these models. The normalized frequency ($\omega a/2\pi c = a/\lambda $) at a wavelength of 1.07 µm ranges approximately from 50 to 180. When the structural constant is 50 µm, the number of states existing in the frequency range 1 is about 2,000. The number of states increases as the structure constant increases. Since it takes about 10 hours to calculate the eigenstate of 2,048 for one lattice point, it is, in practical terms, impossible to find a band gap suitable for a 1.07 µm wavelength, considering the propagation vector in the z direction. However, a comparison of the features of the band structure is useful. Figures 8(a), 8(b), and 8(c) show calculations of the band diagram of 128 eigenstates from the lower one. The band structure of the simple large core diameter air hole model in Fig. 8(c) shows no band gap, indicating the eigenstates appear continuous because wavelength resolution is sufficiently fine relative to the scale of the structure. In contrast, Figs. 8(a) and 8(b) show band gaps in scale regions where the possibility of band gaps had been ruled out. The band gap in the fractal model appears narrower than in the double lattice model, even after accounting for the difference in normalized frequency, which is about 3.5 times enlarged in the double lattice model.

## 5. Discussion

We tried to identify similarities between a double lattice fiber and fractal fiber using a one-dimensional model. A double lattice fiber is considered to be a superposition of two triangular gratings with slightly different interstitial distances. Figure 9(a) shows the change in refractive index with the period of the lattice structure in one dimension. The number of short period waves in the synthesized wave is believed to correspond to the number of higher order modes. To reduce the number of modes, it is necessary to combine two lattices with as large intervals as possible. The long period interval that forms the envelope corresponds to the fiber core diameter and is determined by the least common multiple of the two gratings.

A fractal model fiber is considered to be a direct product of a large lattice and a small lattice [Fig. 9(b)]. The relationship between the double lattice structure and the fractal model structure is similar to that of the sum to product formulas for the trigonometric identities:

For the double lattice model defined by the left side of the equation, *n* and *m* must be integers, since they refer to the numbers of air holes of each lattice. The difference between n and m is always 1. This is because the envelope shape does not become a trigonometric function when two or more different n and m are superimposed. Since the maximum value of the lattice spacing is more or less determined by wavelength, the degree of freedom in designing a large-area core diameter has two parameters, lattice pitch and number of gratings. Since the number of higher-order modes also increases in proportion to the number of gratings, increasing the core area is difficult. A large core diameter requires setting lattice spacing to the maximum possible value and n and m to the smallest possible values, leaving minimal design freedom. In the fractal model structure defined by the right side of the equation, the combination of the period of the envelope and the period of the short wavelength is arbitrary, thus design has two degrees of freedom, and the setting of the core area can be made flexible. In addition, the double winding fractal model structure does not show the higher order modes corresponding to the number of short period waves. The reason for this is discussed next.

Schemes for the closest packing of spheres in three-dimensional space are limited to two methods: face-centered cubic (fcc) and hexagonal close-packed (hcp). When viewed from the (111) plane direction for fcc and the (0001) plane direction for hcp, the spheres are arrayed in triangular lattices. Fcc is composed of three different triangular lattices, while hcp is composed of two different triangular lattices. In Fig. 10, three colors indicate three different three grids of fcc, which constitute the fractal pattern. The two types of gratings appearing in hcp cannot form a fractal-like pattern of outward convex shape. Because there are only two close-packed structures in three-dimensional space, there is no other way to fill a plane by switching triangular lattices in our world. Note that this fractal structure can be tiled with regular hexagons and equilateral triangles.

This fractal structure is chiral and appears at the phase transition of palladium or water, depending on the concentration of hydrogen or helium. Reference [9] gives hints concerning this structure. The electromagnetic field in the 2D plane may be described as a nonlinear sigma model (Gade model) [10]. The structure has chiral symmetry; adjacent lattices belong to two different sublattices. The total number of cluster sites per unit cell is three, an odd number. These satisfy the Hamiltonian conditions defined in Reference 9, which states that chiral symmetry under these conditions guarantees the existence of zero energy eigenstate(s), or zero mode(s). In the double winding fractal model, three different lattices are specified, so the eigenstates are the primary modes only. Since there is no third grid in the single fractal model, higher-order modes may appear.

The fact that the energy in the in-plane xy direction is zero means the wave vector of light is directed entirely in the z-axis direction. When the energy of the primary mode is zero, the energy of the higher mode becomes positive; that is, the in-plane higher order mode cannot exist as a binding state. These pictures are consistent with the conclusion that only the solution of the first order mode exists in the double winding fractal model and higher order modes cannot be obtained in our simulation. It is considered that the Gade’s condition was not satisfied in the single winding fractal model because the third lattice is not explicitly indicated.

It has recently come to light that when two sheets of graphene are stacked at specific angles, unique physical properties emerge, including superconductivity and the Mott insulator [11,12,13]. The moiré image of two graphene sheets, named twisted bilayer graphene, t-BLG, is shown in Fig. 11. This moiré image is of a magic angle 1.05 degree. We can find a triangular chiral region at the three hexagonal junctions. Since the purpose of this paper is not the analysis of t-BLG, the region division is approximate. By changing the twist angle, the size ratio of the “chiral triangle” and the “bright moiré regular hexagon” changes. We suspect that a superconducting state will occur when this ratio matches that of the switching fcc triangular mesh (=3:7). It is our hope that field theory experts will examine the physical laws underlying these physical phenomena.

## 6. Conclusion

We modeled the fractal structure appearing in natural phase transitions as a shape of photonic fiber with core diameter more than 35 times the wavelength and analyzed eigenstates using FemSIM and BandSOLVE from Rsoft. We found that only the primary mode exists as an eigenstate for the double winding fractal model. The single winding fractal model has shown the possibility of transmitting Laguerre Gaussian beams with a very small attenuation rate. We compared the eigenmode and band structure of a fractal model fiber to those of double lattice fiber and a simple large-diameter air-hole fiber and studied the expression mechanism of the band gap. The results suggest a close relationship between the peculiar eigenmodes of fractal model fiber and field theory.

## Acknowledgments

We wish to express our gratitude to Yoshiyuki Kawazoe, Professor Emeritus, Tohoku University, for allowing use of the figures provided in their interesting paper. Reprinted with permission from R.V. Belosludov, Y.Y. Bozhko O.S. Subbotin, V.R. Belosludov, H. Mizuseki, Y. Kawazoe, and V.M. Fomin, J. Phys. Chem. C 2014, 118, 5, 2587-2593. Copyright 2014 American Chemical Society.

The image of t-BLG was created by Dr. Jun-Ichi Iwata of Advance Soft Co., Ltd. using the method of https://github.com/cometscome/TightBinding/blob/master/2Dgraphene.ipynb, with the help of Watari’s son Tamao to adjust the resolution.

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **M. Fujita, M. Tanaka, S. Yamadori, A. Suzuki, S. Koyanagi, and T. Yamamoto, “Photoinic crystal fiber,” Mitsubishi Densen Kougyou Jihou , **99**, 1–9 (2002) [in Japanese].

**2. **M. Napierala, T. Nasilowski, E. Beres-Pawlik, P. Mergo, F. Berghmans, and H. Thienpont, “Large-mode-area photonic crystal fiber with double lattice constant structure and low bending loss,” Opt. Express **19**(23), 22628–22636 (2011). [CrossRef]

**3. **N. Watari, S. Ohnishi, and Y. Ishii, “Hydrogen storage in Pd clusters,” J. Phys.: Condens. Matter **12**(30), 6799–6823 (2000). [CrossRef]

**4. **R. V. Belosludov, Y. Y. Bozhko, O. S. Subbotin, V. R. Belosludov, H. Mizuseki, Y. Kawazoe, and V. M. Fomin, “Stability and Composition of Helium Hydrates Based on Ices Ih and II at Low Temperatures,” J. Phys. Chem. C **118**(5), 2587–2593 (2014). [CrossRef]

**5. **N. Watari and S. Ohnishi, “Cluster model study for a new PdH phase with superabundant vacancies,” J. Phys.: Condens. Matter **14**(4), 769–781 (2002). [CrossRef]

**6. **https://www.synopsys.com/optical-solutions/rsoft/rsoft-products.html

**7. **J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic,” J. Comput. Phys. **114**(2), 185–200 (1994). [CrossRef]

**8. **T. Matsui and K. Nakajima, “Photonic crystal fiber and high power optical transmission system,” JP WO2017/098878 A1 2017.6.15

**9. **Y. Asada, K. Slevin, and T. Ohtsuki, “The Chiral Symplectic Universality Class,” J. Phys. Soc. Jpn. **72**(Suppl. A), 145–146 (2003). [CrossRef]

**10. **R. Gade, “Anderson localization for sublattice models,” Nucl. Phys. B **398**(3), 499–515 (1993). [CrossRef]

**11. **Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, “Spectroscopic signatures of many-body correlations in magic angle twisted bilayer graphene,” Nature **572**(7767), 101–105 (2019). [CrossRef]

**12. **K. Uchida, S. Furuya, J. Iwata, and A. Oshiyama, “Atomic corrugation and electron localization due to Moiré patterns in twisted bilayer graphenes,” Phys. Rev. B **90**(15), 155451 (2014). [CrossRef]

**13. **A. O. Sboychakov, A. L. Rakhmanov, A. V. Rozhkov, and F. Nori, “Electronic spectrum of twisted bilayer graphene,” Phys. Rev. B **92**(7), 075402 (2015). [CrossRef]