Applying tiling and pattern theory in the design of hollow-core photonic crystal fibers for multi-wavelength beam guidance

We apply tiling and pattern theory in the design of hollow-core photonic crystal fibers for guiding light in multiple spectral bandgaps. By combining two different glass apexes in a single [36;32.4.3.4] 2-uniform tiling, transmission regions with fundamental, second and third harmonic wavelengths are supported. This cladding design may also be an excellent candidate for high power beam delivery of Er/Yb, Nd:YAG and Ti:Sapphire laser sources.

For the past two decades, hollow-core photonic crystal fibers (HC-PCFs) 1,2 have had a tremendous impact on nonlinear optics. HC-PCFs have allowed to reduce the threshold of many nonlinear processes by several orders of magnitude 3 . This dramatic threshold reduction was obtained by tight confinement of the fundamental mode (FM), and by increasing dramatically the light-gas interaction length. HC-PCFs that guide a FM within a certain range of frequencies and reject frequencies outside this range are termed hollow-core photonic bandgap fibers (HC-PBGFs). It was shown that the FM transmission loss within the bandgap is limited by scattering losses at the core boundary 4 . Many nonlinear processes, such as high harmonic generation, require multiple spectral bandgaps that will guide the laser pump signal and generated harmonics with low transmission losses. It was shown that PBGFs with interstitial holes 5 and high air-filling structures [6][7][8][9] can guide light in two separate bandgaps. Yet, the ratio between their central normalized frequencies at the air line ck z /w = 1 is not suitable for second harmonic (SH) and third harmonic (TH) guidance. Recently it was theoretically demonstrated that HC-PBGFs can guide light in two well separated bandgaps suitable for TH guidance 10 . Such cladding designs can guide the FM and TH Gaussian modes with low transmission losses; yet, it is unclear if phase-matching of these modes is feasible.
Several approaches have been proposed to phase-match second harmonic generation (SHG) and third harmonic generation (THG) in PCFs. In solid core PCFs, phase-matching a Gaussian total internal reflection mode with a Gaussian bandgap mode was theoretically proposed 11 and experimentally demonstrated 12 . In HC-PCFs, it was theoretically proposed to quasi phase-match high harmonic generation by modulating the phase of ionization electrons using a counter-propagating beam 13,14 . THG was experimentally demonstrated in an Ar-filled kagome fiber by counterbalancing the fiber dispersion and the gas dispersion with two different order modes 15 . SHG was experimentally demonstrated in a Xe-filled kagome fiber by applying an external DC field 16 . Kagome fibers 3 are an excellent platform for high harmonic generation since their transmission region is broadband.
Another class of hollow-core fiber that is attracting much interest are antiresonant fibers 17 . Recently it has been demonstrated that antiresonant fibers can guide light with very low transmission losses 18 . Obtaining such low transmission losses in a hollow-core fiber is quite remarkable; yet, it was demonstrated with a relatively large core diameter, namely ∼ 37 µm . Large core diameters are very common in antiresonant fibers since the antiresonant reflecting optical waveguide (ARROW) model, which is used for designing antiresonant fibers, breaks down at small core diameters 19 . Small core diameters are preferable for enhancing high harmonic generation, and therefore it is necessary to investigate new approaches for designing HC-PCFs with small core diameters and low transmission losses at the FM and its higher harmonics 10 . Such HC-PCFs could withstand high energy pulses that cannot be guided in solid core PCFs.

OPEN
School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 8410501, Israel. * email: zevm@post.bgu.ac.il Bandgap formation in HC-PCFs is associated with three different resonators in the high air-filling structure: (a) apexes, (b) struts and (c) air holes 22 . The triangular (Fig. 1b) fundamental bandgap lowest frequency is bound by the silica apexes and the fundamental bandgap highest frequency is bound by the silica struts. Within the fundamental bandgap there is also a specific frequency and effective index range that is associated with the air holes. Similarly, the square (Fig. 1a) fundamental bandgap is also bound by the apex and strut resonators 23 . Interestingly, in higher order bandgaps it was shown that the apex and strut resonators can interchange their bandgap bounding positions 24 . Resonators bound the HC-PCFs' bandgap; yet, the fundamental bandgap bandwidth originates from the properties of the single apex in the high air-filling structure 25 . The struts connecting the apexes reduce the fundamental bandgap bandwidth by bounding the bandgaps' highest frequency; thus the struts have a similar effect as a low-pass filter. Even though most of the high frequency range is blocked by the struts, there are situations where they can support higher order bandgaps 6,8,9 . The most common way to calculate accurately the frequency bandwidth of these higher order bandgaps is with full-vectorial frequency domain methods, such as the plane wave expansion method (PWE) 26 . Figure 2 shows the struts and apexes band diagrams of the square and triangular structures. Glass dielectric constant was set to 2.1, apex radius curvature and strut thickness were set to 0.15 and 0.01 , respectively. Fixing the dielectric constant is useful since it allows to find the band diagrams in a scalable system and scale the structure according to the desired nonlinear process of interest 27 . Fused silica low dispersion allows to preserve quite well the ratio between the central frequencies of the bandgaps during the scaling process. The structures' fundamental bandgap lowest normalized frequency is located near the apexes' fundamental bandgap lowest normalized frequency. The struts' band diagrams span the entire apexes' fundamental bandgap and are known to reduce the frequency bandwidth of the structures' bandgaps 8 . In the square structure, the struts slightly narrow the apex fundamental bandgap (Fig. 2a); yet, in the triangular structure (Fig. 2b) the struts causes the apex fundamental bandgap to be discontinuous with several bandgaps. In the triangular structure, only the first higher order bandgap has a substantial frequency bandwidth at the air line. Properties of the first higher order bandgap were investigated rigorously in 8 with different apex radius curvatures and strut thicknesses. Total bandwidth of the apexes' fundamental bandgap is reduced dramatically in the triangular structure compared with the square structure. Figure 2 demonstrates that while the apex band diagrams can approximate the lowest normalized  Fig. 3. High air-filling structures with a smaller apex separation have a fundamental bandgap at a higher normalized frequency. If somehow several properties of these high air-filling structures could be combined there would be two obvious approaches to guide the SH and TH. The first approach would be to combine the guiding properties of the hexagonal, square and triangular structures. The hexagonal fundamental bandgap would guide the FM, the square and triangular fundamental bandgaps would guide the SH and TH. This approach has two disadvantages: (a) supporting the FM and the first two harmonics requires a structure with three different apexes. Since each apex is expected to reduce the frequency bandwidth of the other apex bandgaps as a result of interference effects, reducing the number of different apexes in the structure is preferred for high harmonic guidance. (b) the FM would be guided at a low normalized frequency, thus would have a high transmission loss in the fundamental bandgap. This high transmission loss could be reduced by increasing the number of periods in the cladding 10 or by designing a customized core 28 ; yet, these methods would make the fabrication of the fiber much more difficult. The second approach would be to combine the guiding properties of the square and triangular properties and guide the FM in the square fundamental bandgap and use the triangular fundamental and first higher order bandgaps to guide the SH and TH. This approach eliminates the disadvantages mentioned previously: (a) the structure would have only two different apexes instead of three, which would reduce apex interference effects and make the fabrication of the fiber easier and (b) the FM would be guided in the fundamental square bandgap at a higher normalized frequency, thus reducing the transmission loss of the FM.
Since the fundamental bandgap is shifted to lower normalized frequencies as the apex separation increases, and since guiding with a low transmission loss is difficult when the fundamental bandgap is at a low normalized frequency, it is preferred to construct hybrid structures that only have hexagonal, square and triangular tilings and avoid more complicated tilings with larger apex separation such as dodecagons. It is also preferred to use tilings that are invariant for 60 • rotations (such as the p6m symmetry tilings) since they facilitate construction of a circular-shaped core. The square and triangular apexes are located at the center of the square and triangular tilings and therefore, obtaining the guiding properties of the square and triangular structure could be realized in a single structure by simply placing the center of the preform capillaries on the vertices of a hybrid structure with square and triangular tilings. The higher order bandgap of the triangular structure is generated with struts of length �/ √ 3 , thus enforcing another limitation on the desired hybrid structure. Out of all eleven uniform tilings, none have only square and triangular tilings with a p6m symmetry; yet, there may be a uniform tiling that supports the TH. The trihexagonal tiling (Fig. 4a)    www.nature.com/scientificreports/ be approximated by adding the confinement loss and scattering loss (Fig. 5g). Transmission loss at the FM, SH and TH is ∼ 2.8 dB/m , ∼ 0.059 dB/m and ∼ 0.18 dB/m , respectively. Transmission losses of the [3 6 ;3 2 .4.3.4] HC-PCF are also shown in Fig. 5g for apex curvatures of r=300 nm and r=600 nm. By inspecting carefully Fig. 4c it is clear that the high transmission regions centered at ∼ 1.1 µm and ∼ 1.45 µm are associated with the square apex fundamental bandgap and that the high transmission region that spans the wavelength range of ∼ 0.50-0.85 µm is associated with the triangular apex fundamental bandgap. Since there are 245 triangular apexes and only 86 square apexes in a quarter of the fiber (including the apexes on the x and y axes), confinement from the triangular apexes is much stronger compared with the confinement from the square apexes. In addition, the triangular apex fundamental bandgaps' central frequency is located at a higher normalized frequency compared with the square apex fundamental bandgaps' central frequency. Usually, a better transmission loss can be obtained when the fundamental bandgaps' central frequency is located at a higher normalized frequency 10 . Balance between the scattering loss, number of resonators in the finite cladding and location of the fundamental bandgaps' central frequency causes the transmission loss to be the lowest at the SH. Low transmission bands, which originate from the resonant character of the 30 nm core strut, are expected above 9 µm and below 200 nm. For example, low transmission bands for N = 0.5 (see equation (2) in 19 ) are centered at ∼ 9.9 µm , ∼ 124 nm and ∼ 82 nm. Discontinuities and oscillations in the transmission loss shown in Fig. 5g  Many n-uniform tilings have been reported 32 , this implies that there may be many more tilings that will allow to guide light at multiple wavelengths with HC-PCFs. For example, the 3-uniform [3 6 ;3 3 34 . By stacking A7 unit cells in the preform, a solid core Archimedean-like PCF was fabricated. In this report the authors neglected to point out that the Archimedean-like structure is actually the [3 6 ;3 2 .4.3.4] structure. Stacking A7 unit cells may be a promising method for fabricating HC-PCFs that contain the [3 6 ;3 2 .4.3.4] structure. Another method to fabricate the [3 6 ;3 2 .4.3.4] tiling is by preparing a PCF preform with ultrasonic drilling and estimating the air hole geometry change ratio with the capillary Navier-Stokes model 35 . The final air hole diameter according to this model is expressed with the following equations 36 : where h 10 is the initial air hole diameter, h 20 is the initial fiber outer diameter, h 2 is the final fiber outer diameter, γ is the surface tension, µ is the viscosity, p is the hole overpressure, W f and W d are the feed and draw speeds and L is the heating zone length. The final air hole diameter can be expressed more elegantly with the imaginary error function: Using Eq. (2), the geometry change ratio is expressed with the following equation: Air hole expansion requires that the geometry change ratio fulfill the condition C > 1 . In 35 the air hole diameter was defined as the diameter of a circle that has the same circumference as the total of all the inner hole perimeters D perimeter . This diameter definition had a better correlation with experimental results compared with other air hole diameter definitions, such as the diameter of a circle that has the same area as the total area of the inner holes D area . In the extended work of 35 (Roman Kostecki's thesis), he showed that the former and latter definitions have a good correlation with experimental results, with D perimeter giving slightly better results. The equations for the perimeters of the two different air holes in the [3 6 ;3 2 .4.3.4] tiling are (see Fig. 6e):  . 6e): www.nature.com/scientificreports/ If we rewrite the geometry change ratio in terms of the air hole area we obtain: Where A is the total final air hole area and A 10 is the total initial air hole area. Since A 10 may be constructed of different air holes in the preform, A can be expressed with the following equation: Where A d 1 and A d 2 are the initial preform air holes with diameters d 1 and d 2 , respectively. Assuming the preform air holes A d 1 and A d 2 change their area independently, they will transition according to C into final air holes A 1 and A 2 that are expressed with the following equations: According to equations (10) and (11) obtaining an identical r, t and in the final PCF is only possible if the ratio between the final air hole areas is equal to the square of the ratio between d 1 and d 2 : For the [3 6 ;3 2 .4.3.4] PCF shown in Fig. 6e with parameters: r = 0.15 , t = 0.01 and = 3 µm we obtain d 2 /d 1 ∼ 1.035 . As shown in 35 , the Navier-Stokes model will not work properly without applying pressure and temperature offsets p ǫ and T ǫ . Figure 6a,b show the calculated geometry change ratio for different pressures and temperatures (equation (3)). The following parameters were used in these calculations: p ǫ = −479.9 Pa , T ǫ = −147.5 • C , W f = 1.5 mm/min , β ∼ 8.63 , L = 0.039 m , γ = 0.3 N/m 37 , h 10 ∼ 0.005 m . Glass viscosity was calculated with the following equation 38 : Where R is the gas constant in units of J/K mol and T is the temperature in units of K. Even though equation (3) can predict the geometry change ratio of the PCF, it cannot predict the final apex curvature, strut thickness and center air hole spacing. For a specified C, h 20 , h 2 and preform air hole diameters d 1 and d 2 , there are many apex curvatures r, strut thicknesses t and center air hole spacing that have the same final air hole areas A 1 and A 2 . Therefore, finding the exact r, t and for a specified C, h 20 , h 2 and preform air hole diameters d 1 and d 2 must be done experimentally. Figure 6a,b show different combination of pressures and temperatures that have an identical C, each combination will have the same air hole area; yet, r, t and will be different. With equations (3), (6), (7), (10), (11) and (12) Fig. 5a can be realized with a two stage cane and sleeve process 39 . Figure 6f-h show the cross-section of the [3 6 ;3 2 .4.3.4] PCF during the fiber draw calculated with the Navier-Stokes model. In the first stage, the preform (Fig. 6f) was drawn to a cane (Fig. 6g) Fig. 5a. Figure 6 demonstrates that the Navier-Stokes model can help estimate the geometry change ratio of high air-filling PCFs during the stack and draw fabrication process.

Conclusions
In conclusion, we demonstrated how to apply tiling and pattern theory in the design of HC-PCFs for SH and TH guidance. Since the apexes of the hexagonal, square and triangular structures are located at the center of their tilings, it is possible to obtain a single cladding structure with multiple spectral bandgaps by placing capillaries on

Methods
Band diagrams were computed with the plane wave expansion method. The number of plane waves in these calculations was 2791. Finite element calculations were performed with COMSOL. The number of elements in these calculations was 3373305 and the perfectly matched layer (PML) extended 10 µm in the radial direction. Material dispersion was included with a three-term Sellmeier equation, absorption losses were included by adding to the silica regions a constant 10 −7 imaginary refractive index 40 .

Data availability
All data generated or analysed during this study are included in this published article (and its Supplementary Information files).