Super strong wide TM Mie bandgaps tolerating disorders

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Also, it is shown that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{TE}}_{01}$$\end{document}TE01 Mie bandgap appears in metamaterials containing a high refractive index, and disappears in metamaterials with a lower refractive index such as silicon; in contrast, a new phenomenon of intense and broadband \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{TM}}_{01}$$\end{document}TM01, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{TM}}_{11}$$\end{document}TM11, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{TM}}_{21}$$\end{document}TM21 in metamaterials with a lower refractive index such as silicon appear. 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This strong tolerance of disordering of TM modes in tellurium, germanium, and silicon metamaterials opens a new way to design small, high-efficient, and feasible fabrication optical devices for optical integrated circuits.


Physical background
Mie and Fabry-Perot (FP) resonances govern the physical background of high-n all-dielectric MMs. Shining a plane wave on a high-n dielectric rod that has length L ≫ radius r causes a m l and b m l resonances (where m and l are integer mode numbers), which are caused by strong light-matter interactions. Transverse magnetic (TM) polarized illumination yields a m l , and transverse electric (TE) polarized illumination yields b m l . A transverse wavevector k l ⊥ = 2π l (where l is the Mie resonant wavelength for different l mode number) is created that is responsible for Mie resonances, and a longitudinal wavevector k m � = mπ L is created that is responsible for the FP resonance appears as a result of the length of the rod 42 .
When a long Ge rod ( L ≫ r ) is illuminated by a TM-polarized plane wave (Fig. 1a), the long L means that extraction of time and frequency (Fourier transform) responses of electric and magnetic fields requires 2D FDTD simulations. Simulation results of the scattering cross-section (SCS) show Mie resonance peaks at normalized frequency 2πr = 0.254, 0.579, and 0.9225, respectively, which correspond to a m 1 , a m 2 , and a m 3 (Fig. 1b). E-field distributions in the xy plane for a m 1 , a m 2 , and a m 3 show one, two, and four electric dipoles, respectively, which originate www.nature.com/scientificreports/ from collective polarization of particle materials in response to the incident electric field (along the long axis of the rod) ( Fig. 1c-e). The circulating magnetic fields inside the rod are created due to the oscillation of electric dipoles. The rod and the surrounding medium (air) are nonmagnetic, so the permeability µ r = 1 ; therefore, the magnetic current loops penetrate to the air ( Fig. 1f-h).
Calculations of the total, K-direction, and H-direction SCSs of the Ge rod ( Fig. 1b) show that a m 1 represents the higher amplitude of the SCS in the H direction and that a m 2 , and a m 2 reveal a stronger amplitude of SCS a m 3 in that direction. The higher amplitude of SCS along H direction than the propagation direction (K direction) is responsible for a H-field coupling between rods that are in a periodic arrangement in that direction.
In a periodic arrangement of Ge rods in air with r = 0.3a (where a is the lattice constant), excitation a m 1 , a m 2 , and a m 3 modes induce the development of H-field couplings between rods located along the H-field direction, as a consequence of the induced circulating magnetic field inside each rod. These couplings suppress the incident wave to penetrate the periodic structure and result in bandgaps in the photonic band structure. In the periodic arrangement, the Mie resonances of a m 1 , a m 2 , and a m 3 are responsible for creating Mie bandgaps of TM 01 , TM 11 , and TM 21 , respectively.
The intensity of the H-field couplings is proportional to the scattering along the H-field direction (Fig. 1b, H  direction), consequently, the strengths of the H-field couplings between rods descend in order a m 1 > a m 2 > a m 3 . These couplings are depicted by H-field distributions as well as normalized H-field along x = 0 and x = 0.5a lines (Fig. 2). In this figure, TM polarized plane waves which are located at the bottom of the structure propagate from −y to y direction. H-fields of Mie resonances of a m 1 , a m 2 , and a m 3 that are responsible for Mie bandgap modes of TM 01 , TM 11 , and TM 21 have different distributions (Fig. 2a, c, e), and different normalized magnetic fields along x = 0.5 (green dotted-dashed line) and x = 0 (solid-blue line) (Fig. 2b, d, f). The incident plane waves that propagate from −y to y, interact mostly with the first row (I) of the periodic Ge MMs rather than with the second row (II). These interactions can be shown quantitatively using normalized magnetic fields (Fig. 2b, d, f). The Mie scattering amplitude along the H-field direction is highest in a m 1 , so the H-field couplings between the first row of rods (I) (Fig. 2b, green broken line) descend in strength in the order a m 1 > a m 2 > a m 3 . The normalized magnetic fields in Fig. 2b, d, f at x = 0 (solid blue) for the first row of Fig. 2a, c, e, show one, two, and three peaks that are created due to the shape mode of a m 1 , a m 2 , and a m 3 , respectively. The created shape mode of magnetic fields inside each rod for TM 01 and TM 11 (Fig. 2a, c) are the same as a m 1 and a m 2 (Fig. 1f, g), respectively. Due to the H-field couplings between rods along the incident H-field, the shape mode inside each rod for TM 21 (Fig. 2e) mode differs slightly from the shape mode of a m 3 (Fig. 1h). As described earlier the H-field couplings between rods are responsible for the creation of TM 01 , TM 11 , and TM 21 bandgap modes; also, there are other ways that describe the creation of a photonic bandgap which are originated from solid-state physics. The creation of electronic bandgap in electronic crystals is explained by band or bond theory 43 . The band theory uses the Bloch theorem under perfect periodicity of the lattice structure which results in the creation of the Bragg bandgap. In contrast, bond theory explains the creation of the bandgap Position disordering. Mie bandgaps normally appear as dips in transmission diagrams of the periodic arrangement of high-n dielectric rods, because of a transition from PCs to all-dielectric MMs. This part demonstrates that the Ge MMs show TM 01 and TM 11 bandgaps that tolerate significant disordering of the rod position. The position of a rod in a periodic structure is defined as ( x i , y i ), where x i = x i 0 + σ p U x and y i = y i 0 + σ p U y . ( x i 0 , y i 0 ), is the origin position of the rod in the periodic structure, σ p is the strength of the position disordering, and U i (i : x, y), are random variables over interval [− 1, 1] along x and y directions with a uniform random distribution, respectively. The position-disordering parameter is defined as η p = σ p a , where a is the lattice constant 29 . Transmission spectra under illumination of TM-polarized plane waves (magnetic field is along x direction) were obtained for borosilicate crown glass (BK7) PCs in air (Fig. 3a) and Ge MMs in air (Fig. 3b). BK7 PCs were composed of cylindrical BK7 rods with n = 1.5 and Ge MMs were composed of Ge rods with n = 4 ; both types were set in a periodic-cubic pattern under rod-position disordering of η p = 0, 20, 40, or 50%. Incident plane waves are located at the bottom of the structures, propagate from −y to y direction, and are monitored at the top of the structures (Figs. 3c, d and 6c, d).
The BK7 PCs showed two week Bragg bandgaps of BG 1 and BG 2 over 0.1 < a < 1 that do not tolerate position disordering (Fig. 3a). At η p = 0 the Bragg bandgaps of BG 1 and BG 2 appear at a = 0.44 and 0.87, respectively and disappear with increase in η p . The Bragg bandgap obeys f ∝ (acos −1 (θ)) , where θ is the propagation angle. Increase in position disordering changes both a and θ , so the Bragg bandgaps diminish and disappear. Also, position disordering breaks the symmetric of the periodic structure which results in degradation of the Bragg bandgaps. H-field distributions of BG 2 show penetration of TM polarized plane wave at a = 0.833 which results in degradation of BG 2 (Fig. 3c).
The Ge structure showed three Mie bandgaps TM 01 , TM 11 , and TM 21 , of which TM 01 and TM 11 reveal strong position disordering that can tolerate η p = 50% with average T = −135 dB. The strong tolerance of the Mie bandgaps of TM 01 and TM 11 under position disordering occurs for two reasons. The first is that the Ge rods have high SCSs a m 1 and a m 2 along with incident H-field direction, so adjacent rods develop high H-field coupling. The second reason is that TM 01 and TM 11   www.nature.com/scientificreports/ distance between some rods along x and y directions which results in declining the coupling between the rods as well as weakens coupling between optical quasi-bond states which in turn narrows bandwidths of TM 01 and TM 11 .
Position-disorder-tolerant Ge narrow straight, L-shaped and crossing waveguides. Ge MMs tolerate intense disorder of η p = 50% for TM 01 and TM 11 Mie bandgaps under TM polarized plane wave (Fig. 3b). This high tolerance of position disordering is a result of Mie scattering by individual rods in the first row of the structure (Figs. 2, 3d); therefore, the first row (I) has a significant effect on suppression of incident TM polarized light, and also, localization of H-field inside each rod of the first row.
To exploit this effect, we suggest ultra-narrow straight, L-shaped, and crossing waveguides that contain 14, four or two rows of rods, and that tolerate a specific degree of position disordering. For this purpose, straight, L-shaped, and plus shaped defects were surrounded by 14 (Fig. 4a, d, g; structure ' A'), four (Fig. 4b, e, h; structure 'B'), or two ( Fig. 4c, f, i; structure 'C') rows of dielectric rods (either BK7 or Ge) in air. Creating straight, L-shaped, or plus-shaped defect inside the Ge MMs localizes guided modes within the Mie bandgaps of TM 01 , TM 11 , and TM 21 (Fig. S1) that are called donor modes. For the BK7 PCs, the line defect localizes guided modes inside the Bragg bandgaps of BG 1 and BG 2 (Fig. S1). The guided mode inside the TM 01 bandgap propagates inside the rods by H-field coupling, but for TM 11 this guided mode acts as a total-internal-reflection effect and propagates through the waveguides; therefore, TM 11 is exploited for Ge waveguides. Incident Gaussian waves located at the bottom of the structures in an orange bell-shaped, propagate from -y to y direction and are monitored at the top (straight and crossing waveguides) and right (L-shaped waveguides) parts of the structures (Fig. 4).
Transmission spectra for the Ge and BK7 waveguides are monitored for the localized mode inside the TM 11 and BG 2 , respectively. In the Ge straight, L-shaped and crossing waveguides, the transmissions are higher and show robustness at all η p than for BK7 PCs (Fig. 5). Ge-based A-type straight, L-shaped and crossing waveguides have high average normalized transmission amplitude ( T T 0 ) > 96, 90, and 80% , respectively at η p ≤ 20% ; also, show the small variation of the average T T 0 . Ge-based B-type straight, L-shaped and crossing waveguides have average T T 0 > 86, 70, and 70% at η p ≤ 20% and C-type straight, L-shaped and crossing waveguides show average transmission > 70, 40, and 46% at η p ≤ 10% (Fig. 5a-c). H-field distributions of B-type waveguide under η p = 20% show high coupling of incident light waves to the outputs as well as represent robustness of T T 0 to the sharp bend in a L-shaped waveguide and a horizontal defect in a crossing waveguide (Fig. S2a-c). Decreasing the number of rows of Ge rods from A-type waveguides surrounded by 14 rows, to B-type waveguides with four rows decreased the average transmission slightly (Fig. 5a-c). The decreases in average T T 0 at η p > 20% for B waveguides and at η p > 10% for C waveguides, occurs because some rods are separated by large distances, which result in H-field coupling degradation, fleeing electromagnetic waves, and a from guided mode to radiative mode; also the increase in distances decreases the coupling between quasi-bond states between rods and results in escaping of electromagnetic waves.  www.nature.com/scientificreports/ In the BK7 PCs straight, L-shaped and crossing waveguides, the average T T 0 is much less than the Ge MMs ( Fig. 5d-f), owning to the lower refractive index. BK7-based A-type straight and crossing waveguides have average T T 0 < 0.6 at η p ≤ 10% and the average T T 0 decreased with increase in position disordering. Reducing the number of BK7 rods in straight and crossing waveguides from 14 (type A) to four (type B) and two (type C) decreases the average T T 0 , owning to the dependence of defect mode inside the Bragg bandgap the is strongly dependent to the periodicity and the number of unit cell of the structure. Because of sharp bend in BK7 based L-shaped waveguides, the average T T 0 is about zero for A, B, and C-type waveguides. The guided mode in L-shaped waveguides turn into the radiation mode at the sharp bend due to the low refractive index of BK7 which is represented in H-field distribution of B-type waveguides at η p = 20% (Fig. S2d-f). Also, by increasing the position disordering, the average normalized transmission amplitudes decrease sharply due to the breaking symmetry in the structure. As obvious, the average T T 0 of L-shaped BK7 waveguides (Fig. 5e) is less than the L-shaped Ge-based waveguides (Fig. 5b) that implies the robustness of TM 11 in Ge-based to sharp bends.
Radius disordering. The effect of radius disordering can also be exploited in the transition from BK7 PCs to Ge MMs. Under radius disordering, the radius is R i = R 0 i + σ r U , where R 0 i is the origin radius, σ r is the strength of the radius disordering, and U is a random variable over the interval [− 1, 1]. The radius-disordering parameter is defined as η r = σ r /R 0 i . BK7 PCs show weak Bragg bandgaps of BG 1 and BG 2 with T = −50 and −25 dB, respectively, that are tolerant of η r = 20% , because the periodic lattice is retained under radius disordering (Fig. 6a). For Ge MMs under radius disordering, the radii of the rods change, so their localized mode changes; these changes degraded the TM 21 Mie bandgap but TM 11 and TM 01 tolerated η r = 20 and 34%, respectively (Fig. 6b), because the TM 01 and TM 11 mode shapes inside each rod do not change at less than a certain R i .
As η r increased, the width of the TM 01 and TM 11 Mie bandgaps decreased, as a result of light penetration through rods with small radii, and the penetrated light waves in turn experience interferences that result in high oscillation in transmission spectrum around TM 01 , TM 11 , and TM 21 Mie bandgaps (Fig. 6b). H-field distribution for BK7 PCs and Ge MMs at the Bragg bandgap BG 2 ( a = 0.833 ) (Fig. 6c) and TM 11 Mie bandgap ( a = 0.4 ) (Fig. 6d) show the leaking of H-field distribution for Bragg bandgap and suppression of the incident field for the TM 11 bandgap by the individual Ge rods of the first row.
Radius-disorder-tolerant Ge narrow straight, L-shaped and crossing waveguides. As a result of the high-n elements of Ge MMs, the TM 01 and TM 11 Mie bandgaps tolerate radius disordering of η r = 34 and 20%, respectively (Fig. 6b). We exploited this tolerance to design ultra-narrow Ge straight, L-shaped and crossing waveguides that can endure radius disordering to a certain level. In this section the Ge MMs waveguides that use 14 (Type A www.nature.com/scientificreports/ waveguide), four (Type B), or two (Type C) rows of Ge rods are designed and simulated. The three types of Ge waveguides were analyzed under radius disordering (Fig. 7). Creating straight, L-shaped or plus-shaped defect localizes guided modes inside the Mie and Bragg bandgaps (Fig. S1). The average T for the Ge waveguides increased as the number of rows of rods increased, but decreased as η r increased (Fig. 8). Waveguides A tolerated strong radius disorder of η r ≤ 20% with average T T 0 > 0.93, 0.76, and 0.8, but under η r > 20% , T T 0 decreased gradually. Due to the straight path for light waves for A-type straight and crossing waveguides, the T T 0 has small variation at η r < 20% . Waveguides B tolerated η r ≤ 13% with average T T 0 > 0.82, 0.67, and 0.76 for straight, L-shaped, and crossing waveguides, respectively. H-field distributions of B-type waveguides at η r = 13% represent high confinement of light waves in the waveguides and robustness of the confinement to the sharp bend and horizontal defect in L-shaped and crossing waveguides (Fig. S3a-c). Also, T T 0 has larger variation at η r < 20% , owning to the sharp bend in L-shaped waveguide A and B. Waveguide C show lower toleration to radius disordering. There are three reasons governing decreasing average T T 0 for Gebased waveguides under radius disordering. First, increasing radius disordering decreases some radii of rods that increases the distances between them and decreases H-field coupling between them and change the guided mode to radiation mode which result in decreasing the average T T 0 . The second reason is diminishing quasi-bond states couplings between rods which have larger distances due to the radius disordering; the weaker coupling results in escaping electromagnetic waves from the waveguides. The last reason originated from non-localization of TM 11 mode inside the rods due to the different radii.
BK7 straight, L-shaped, and crossing waveguides have much lower transmission than Ge waveguides ( Fig. 8d-f). The BK7 straight and crossing waveguides show higher transmission than L-shaped wavegudies. The low transmission of L-shaped wavegudies are originated from a sharp bend in the structure that can not confine light waves due to the low refractive index of BK7 as shown in Fig. S3d-e.

Comparison between TM and TE modes in Te, Ge, and Si
MMs. This part describes the existence of TM 01 , TM 11 , TM 21 , TM 02 , TM 12 and TM 01 Mie bandgaps in periodic and disordered structures contain dielectric rods (either Te, Ge, or Si) in air. Te, Ge, and Si are non-dispersive materials over 4 µ m < < 10 µ m, 2 µ m < < 10 µ m, and 2 µ m < < 10 µ m, respectively (Fig. S5). In contrast to the TM 01 Mie bandgap mode that appear in all-dielectric MMs contain high-refractive index elements 29,30 , TM 01 , TM 11 , TM 21 Mie bandgaps appear in all-dielectric MMs contain lower refractive index elements such as Si as numerically simulated in the following.
Te is an anisotropic material with refractive indices under TE and TM polarized waves of n 0 = 4.8 and n e = 6.2 over a wavelength range of 4 µ m < < 10 µ m (Fig. S5). The Te MMs' structure consist of 15 × 15 Te rods in air under TE polarized light reveals a TM 01 Mie mode which just tolerates a rod-position disordering of η p = 20% (Fig. 9a). The structure under the illumination of TM polarized waves shows five bandgaps of TM 01 , TM 11 , TM 21 , TM 02 , TM 12 in which TM 01 , TM 11 , TM 21 , and TM 02 reveal wide bandwidths and high robustness to the rod-position disordering of η p = 50% (Fig. 9b). There are a few reasons behind the high robustness of TM 01 , TM 11 , TM 21 , and TM 02 to position disordering. First, the SCSs of each Te rod along the H-field are high enough that guarantee the coupling of the H-field between adjacent rods results in position-disorder toleration. Also, the coupling is originated from quasi-bond states between rods. Second, the bands are pure Mie type and reveal robustness to position disordering, whereas TM 21 is a mixture of Mie and Bragg bandgaps which weakens the tolerance to position disordering. The structure under rod-radius disordering and under illumination of either TE and TM modes show the strong and broadband TM Mie bandgaps and a narrow-weak TM 01 Mie bandgap mode (Fig. S4a, b).  (Fig. 9c). In contrast, the structure under illumination of a TM plane wave shows three intense and wide Mie bandgaps of TM 01 ( 0.17 < a < 0.266 ), TM 11 ( 0.351 < a < 0.461 ), and TM 21 ( 0.53 < a < 0.63 ). TM 01 and TM 11 tolerate a position disordering η p = 50% and TM 21 tolerates a position disordering η p = 20% (Fig. 9d). As mentioned earlier, the strong robustness of TM 01 and TM 11 is originated from the fact that they are pure Mie bandgaps while TM 21 is a combination of Mie and bragg bandgaps that does not tolerate position disordering. The structure under radius disordering and illumination of either TE or TM modes show the strong and broadband TM Mie bandgaps and a narrow-weak TM 01 Mie bandgap mode (Fig. S4c, d).
The TM 01 Mie bandgap mode almost disappears in a MMs structure contains Si rods with n = 3.48 in air under illumination of TE polarized plane wave (Fig. 9e). In contrast, the Si MM structure under a TM polarized incident plane wave shows three strong and broadband of TM 01 , TM 11 , and TM 21 ; furthermore, TM 01 and TM 11 reveal tolerant to position disordering η p = 40 and 20% , respectively (Fig. 9f). TM 01 approximately disappears and show no robustness to radius disordering in Si MMs structure while three intense and broad bandgaps appear in which TM 01 and TM 11 show robustness to radius disordering η r = 34 and 20% , respectively (Fig. S4e,  f). Based on the results, TM 01 only appears in Ge MMs and disappears in Si MMs, that shows highly dependent of the TM 01 Mie mode to the refractive index; the dependent is due to the highly-dependence of TE Mie scattering to the refractive index. Intense and broadband TM 01 , TM 11 , and TM 21 appear in Ge and Si MMs, which imply a very low dependence of Mie scattering modes of a m 1 , a m 2 , and a m 3 to the refractive index.
Robustness of TM 01 , TM 11 , and TM 21 to rod-position and rod-radius disordering. To illustrate the robustness of Te, Ge, and Si MMs to rod-position and rod-radius disordering, the parameter γ = (R 0 −R)

Rd
( 2 − 1 ) , R 0 , and i (i : 1, 2) are average reflection over the bandgap, the reflection of the bandgap with zero disordering, and bandgap wavelengths at 10% of the transmission, respectively. The γ parameter for three Mie bandgaps of TM 01 , TM 11 , and TM 21 for Te, Ge, and Si MMs under rod-radius and rod-position disordering shows the transition of Mie to Mie + Bragg and Bragg bandgaps (Fig. 10). γ < 0.25 , 0.25 < γ < 0.3 , and γ > 0.3 are categorized as Mie, Mie + Bragg, and Bragg bandgaps, respectively. TM 01 , TM 11 , and TM 21 Mie bandgaps of Te MMs reveal the highest robustness to a rod-position disordering of η p = 50% (Fig. 10a, b). Although, the Mie bandgaps show rod-radius robustness of η r = 34% , the bandgaps' edges disappear except for TM 01 (Fig. 10b). TM 01 and TM 11 for Ge MMs show rod-position robustness of η p = 50% and TM 01 reveals rod- www.nature.com/scientificreports/ radius robustness of η r = 34% with the narrower bandwidth (Fig. 10c, d). As obvious, the TM 01 in Si-based MMs shows rod-position robustness of η p = 40% , and rod-radius disordering of η r = 34% with an inevitable cost of bandwidth narrowing (Fig. 10e, f). In conclusion, increasing the refractive index from Si to Ge and then Te increased the robustness of the MMs to rod-position and rod-radius disordering for TM Mie bandgap modes. It is worth mentioning that the MMs contain dielectric rods of either Te, Ge, or Si illustrate three-wide bandgaps of TM 01 , TM 11 , and TM 21 , while the TE counterpart shows just one TM 01 bandgap. TM 01 and TM 11 for Te and Ge MMs under rod-position disordering show pure Mie bandgaps; for Te and Ge MMs, TM 21 is Mie and Mie + Bragg bandgaps, respectively (Fig. 10a, c). Si MMs under rod-position disordering show Mie + Bragg bandgap of TM 01 and Bragg bandgap of TM 11 and TM 21 (Fig. 10e). TM 01 under rod-radius disordering show the transition from Mie to Mie + Bragg and to Bragg for Te, Ge, and Si MMs, respectively (Fig. 10b, d, f). TM 11 and TM 21 under rod-radius disordering depict Mie bandgaps with an disappear bandgaps' edges for Te MMs (Fig. 10b). TM 11 and TM 21 under rod-radius disordering reveal Bragg bandgaps (Fig. 10d, f).

Conclusion
The paper has revealed the tolerance of Ge MMs to disordering of the position and radius of the rods that constitute robustness of the MMs to fabrication imperfections. This tolerance is a result of directive scattering of light waves by Te, Ge, and Si rods in air; this scattering induces H-field couplings between adjacent rods. TM 01 , TM 11 , and TM 21 appear in Te MMs that tolerate position disordering η p = 50% and radius disordering η r = 34, 27%, and 20%, respectively. The TM 01 and TM 11 Mie bandgap modes of Ge structures tolerate position disordering η p = 50% and radius disordering η r = 34 and 20%, respectively. Rod-position and rod-radius disordering were exploited to design ultra-narrow straight, L-shaped, and crossing waveguides composed of 14, four, and two rows of Ge rods in air. These waveguides show average T T 0 > 0.96, 0.9, and 0.8 for type A and T T 0 > 0.86, 0.7, and 0.7 for type B at η p ≤ 20% . Also, These waveguides show average T T 0 > 0.93, 0.76, and 0.8% for type A at at η p ≤ 20% and T T 0 > 0.96, 0.9, and 0.8% for type B at η r ≤ 13% . In addition, intense TM modes emerge in lower n (Si) MMs that tolerate a certain level of disordering; in contrast, TM 01 fades in Si MMs. The tolerances to position and radius disordering of Te, Ge, and Si MMs provide a resource for mass production of photonic components that are ultra-narrow and robust to fabrication imperfections, for use in high-density OICs.

Data availability
All data generated or analysed during this study are included in this published article [and its supplementary information files].