1. Introduction

Helical twist provides an additional degree of freedom in shaping propagation characteristics of conventional and microstructured optical fibers and gives rise to new wave phenomena. In the first twisted conventional fibers fabricated by preform spinning during the drawing process, the existence of circular birefringence and single mode guidance even for large normalized frequency was demonstrated [1]. More recently, in the conventional fibers twisted locally with the helix pitch of the order of hundreds or tens of micrometers, the effect of coupling between core and cladding modes was observed and further studied for applications in novel fiber-based devices like sensors, circular polarizers, couplers [2,3], optical vortex generators [4,5], fiber lasers [6,7] and for enhancement of optical activity [8,9].

In general, properties of twisted conventional fibers are determined by their structural parameters and helix pitch. It is known that propagation and sensing characteristics in microstructured optical fibers (MOFs) can be engineered more freely than in conventional fibers. For example, properly designed non-twisted microstructured fibers show unique features like endlessly single mode propagation [10] or hollow core guidance [11]. Furthermore, optimization of microstructure geometry allows for a precise control of chromatic dispersion important for nonlinear applications [12,13], obtaining increased birefringence [14], extreme polarimetric sensitivity to hydrostatic pressure [15] or broad-band single polarization guidance [16]. It is therefore understood that twisted microstructured fibers have attracted recently much attention, as the helical twist allows for an additional control of their propagation and sensing characteristics. It was already shown that the circular birefringence can be induced in twisted MOFs [17,18] and maximized by choosing a proper filling factor of air holes [19]. Twisted MOFs have been also studied for sensors [19,20], polarizers [21,22], vortex guidance [23] and coreless propagation [24].

The effect of resonant coupling between core and cladding modes in twisted MOFs, utilized in most of the proposed applications, has been first reported in [25] and properly explained in [26]. Moreover, the scaling rules which relate the resonant coupling conditions to dimensionless ratios between the wavelength, the lattice pitch and the helix pitch of the twisted MOFs have been investigated in [21]. The obtained results show that for appropriately designed twisted MOFs, distinct, high loss resonance peaks can be obtained in a broad wavelength range already for the fiber with the millimeter helix pitch, thus allowing for fabrication of coupling based devices using a less demanding preform spinning method. Moreover, the effect of air hole diameter on resonant couplings between core and cladding modes was partially analyzed using the beam propagation method [27].

Nonetheless, the influence of all microstructure parameters on twist-induced resonant couplings in MOFs has not been yet fully explained. In this paper we analyze the effect of the number of air hole rings and the filling factor on the position and strength of the resonance loss peaks in twisted MOFs. The obtained results show that the number of air hole rings, which in non-twisted MOFs influences primarily the mode confinement and loss, has a decisive impact on the number of twist-induced resonant couplings, wavelength range in which the resonances arise, spectral width of resonance peaks and their polarization features. The results presented in this paper, together with the ones reported in [21,26], are sufficient to fully understand the role of all geometrical parameters of the hexagonal microstructure on the resonance loss spectrum in twisted MOFs.

2. Fiber structure and simulation method

The numerical simulations were performed for twisted MOFs with a cross-section shown in Fig. 1. The analyzed fiber is characterized by the fixed value of the lattice pitch Λ_L = 3µm and the right-handed helix pitch Λ_H = 500µm. As shown in [21], the chosen values of Λ_L and Λ_H allow for observation of many strong resonance peaks in the visible and near infrared wavelength range, including the polarization dependent peaks. In the first part of the simulations, the filling factor d/Λ_L = 0.4 was used to be near the critical ratio for endlessly single mode propagation in the core in non-twisted fibers [28], while the number of the air hole rings N_R was changed from N_R = 1 to N_R = 9. In the second part of the simulations, the filling factor was changed from d/Λ_L = 0.35 to d/Λ_L = 0.5 and N_R was fixed (we show the results obtained for N_R = 3 and 4). In case of N_R = 4 and d/Λ_L = 0.4, the analyzed fiber is identical to the one described in [21], for which the effect of changing Λ_L and Λ_H was analyzed in detail.

 

Fig. 1 Cross-section of the analyzed twisted microstructured optical fiber. Arrow shows the twist direction.

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Spectral dependence of the refractive index of silica glass was determined using the following Sellmeier equation [29]:

The propagation characteristics of the twisted MOFs were obtained by using the finite element method combined with the transformation optics formalism [30,31] implemented in COMSOL Multiphysics software. Twisted perfectly matching layers (PML) [31,32] were placed outside of the air holes at a distance of (N_R + 0.5)Λ_L from the fiber center to estimate propagation loss. Changing the distance of the PML beyond this value has little influence on the loss of the fundamental mode. Solution convergence was verified by a mesh refinement study.

3. Results

We have calculated the loss spectra of the circularly polarized HE_1,1⁺ (orbital angular momentum L = 0, spin σ = + 1, total angular momentum J = L + σ = + 1) and HE_1,1⁻ (L = 0, σ = −1, J = −1) fundamental modes in twisted MOFs to analyze the influence of the number of air hole rings N_R and the filling factor d/Λ_L on the resonant coupling between the core and cladding modes.

3.1 Effect of increasing number of air hole rings

In the first step, we analyzed the loss spectra of the HE_1,1⁺ and HE_1,1⁻ fundamental modes in the twisted MOFs characterized by Λ_L = 3µm, Λ_H = 500µm, d/Λ_L = 0.4 and N_R changing from 1 to 9 in the wavelength range from λ = 400 nm to λ = 1600 nm. We show that addition of successive air hole rings in the microstructured cladding has a strong influence on the number of observed resonance peaks as well as their loss and spectral width.

Small number of air hole rings in the twisted MOF yields simple resonance spectra. In Fig. 2 we show the loss spectra of the fundamental modes in twisted and non-twisted MOFs with N_R = 1 and 2. Each loss peak corresponds to the resonant coupling between the fundamental mode and one of the cladding modes, which satisfy the phase matching and the symmetry conditions discussed in details in [26]. The loss peaks observed in the transmission spectrum of the fundamental modes are irreversible as they are caused by leakage of the cladding modes. Distribution of the axial component of Poynting vector (S_z) and azimuthal component of the electric field amplitude (E_θ) of the cladding modes coupling to the HE_1,1⁻ fundamental mode at successive resonances are also shown in Fig. 2.

 

Fig. 2 Waveguide loss calculated versus wavelength for HE_1,1⁻ (blue dotted) and HE_1,1⁺ (red solid) fundamental modes in twisted MOF with Λ_L = 3 µm, Λ_H = 500µm, d/Λ_L = 0.4 and N_R = 1 or N_R = 2. Dashed curve shows the loss of the fundamental modes in non-twisted fiber. Distribution of the axial component of the Poynting vector (S_z) and azimuthal component of electric field amplitude (E_θ) are also shown for the cladding modes coupling with the HE_1,1⁻ fundamental mode at successive resonances in twisted MOF with N_R = 2. Angular symmetry order of E_θ determines a total angular momentum carried by dominating angular harmonic denoted as J_D.

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The loss obtained for N_R = 1 in the analyzed wavelength range exceeds 1000 dB/m even in the non-twisted fiber and is too high for the majority of applications. The loss is additionally enhanced by a twist, especially in the short wavelength range. Furthermore, for N_R = 1 there are no cladding modes confined in the microstructure, which could be phase matched to the fundamental modes in the considered wavelength range.

In fiber with N_R = 2, the confinement of the fundamental mode is much better, which leads to reduction of loss in the non-twisted fiber by orders of magnitude. In the twisted fiber, we observe two resonance peaks related to the coupling to two similar ring-type cladding modes located between the first and second air hole ring (denoted as a and b in Fig. 2). The resonant couplings arise in the short wavelength range and are responsible for gradual formation of the short wavelength loss edge in twisted MOFs [21]. It is worth to mention that similar effect (i.e. resonant coupling with cladding modes) is responsible for creation of the short wavelength loss edge in bent microstructured fibers [33]. The peak (a) obtained for both fundamental modes at λ = 460 nm is characterized by very high losses, while the peak (b) is much smaller and arises only for the HE_1,1⁻ fundamental mode. The two peaks differ significantly because they are related to coupling with two different types of cladding modes. As shown in [26], the cladding modes in twisted MOFs with 6-fold symmetry are composed of many angular harmonics, each carrying well-defined spin σ = ± 1 and a total angular momentum J = 6k + σ (where k is an integer number), which is directly related to the angular symmetry order of the azimuthal component of electric field amplitude (E_θ). As shown in Fig. 2, the cladding modes responsible for the resonance peaks (a) and (b) in the loss spectra of the HE_1,1⁻ fundamental mode (σ = −1) are dominated by the angular harmonic carrying J_D = 6−1 = 5. High loss resonance peaks observed for both polarizations are related to coupling with the cladding modes composed primarily of harmonics with the same polarization handedness as the fundamental mode (σ = σ_fund). Resonance peaks characterized by lower loss, which can be observed in loss spectra of only one of the fundamental modes (usually HE_1,1⁻) are related to coupling with the cladding modes composed primarily of harmonics with the opposite polarization handedness than the fundamental mode (σ = −σ_fund).

In Fig. 3 we present the loss spectra of the fundamental mode in twisted and non-twisted MOFs with an additional air hole ring (N_R = 3). We also show the field distributions for all the modes responsible for couplings visible in the analyzed wavelength range and identify their dominant harmonics (one or more). As can be seen in Fig. 3, in the microstructure with three rings of air holes (N_R = 3) there are couplings to the cladding modes of intensity profiles and symmetries similar to those observed for N_R = 2 (a and b in Fig. 2 are similar to a and d in Fig. 3) and to more complex cladding modes, which cannot be guided in the structure with less than 3 air hole rings. For N_R = 3 most of the observed modes can still be treated as ring-type modes. The ring-type modes for which a radius can be well defined are located between 2nd and 3rd air hole ring (c, e, f and g in Fig. 3). For such modes the azimuthal component of the electric field amplitude (E_θ) is dominated by a single angular harmonic of order J_D. However, in case of the modes located mainly between 1st and 2nd air hole rings (a and d in Fig. 3), we observe additional local maxima of intensity arising further from the core, which are created by the leaking amplitude from similar modes shown in Fig. 2 reflected back to the microstructure by the third air hole ring. Those additional maxima introduce higher-order harmonics of significant amplitude in the distribution of azimuthal component of the electric field amplitude (E_θ) carrying higher total angular momenta. As a result, it is not possible to identify the dominant harmonic in case of the mode (b), which has the complex amplitude distribution containing strong harmonics corresponding to J = −1, 5, 11.

 

Fig. 3 Waveguide loss calculated versus wavelength for the HE_1,1⁻ (blue dotted) and HE_1,1⁺ (red solid) fundamental modes in twisted MOF with Λ_L = 3 µm, Λ_H = 500µm, d/Λ_L = 0.4 and N_R = 3. Dashed curve shows the loss of the fundamental modes in non-twisted fiber. Distribution of the axial component of the Poynting vector (S_z) and azimuthal component of electric field amplitude (E_θ) are also shown for the cladding modes coupling with the HE_1,1⁻ fundamental mode at successive resonances. Angular symmetry order of E_θ determines the total angular momentum carried by dominating angular harmonics denoted as J_D. For the cladding mode responsible for the resonant peak b there is no dominant harmonic as the three harmonics with J_D = −1, 5, 11 are almost equally visible.

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From the results presented in Fig. 3 one can draw additional conclusions concerning the spectral position and loss of the resonance peaks. The resonance peak positions corresponding to ring-type cladding modes can be related to their radius ρ, dominating symmetry order J_D and polarization handedness σ. In case of pairs of the cladding modes with similar J_D and σ ([a, e] and [d, g] in Fig. 3), the mode with greater ρ couples with the fundamental mode at longer wavelength, which results in the appearance of the resonance peaks in broader wavelength range for MOFs with greater N_R. Furthermore, for pairs of the cladding modes with similar ρ and J_D ([a, d], [c, f] and [e, g] in Fig. 3), the mode with σ = + 1 couples with the fundamental mode at longer wavelength than the mode with σ = −1. These two effects lead to significant polarization dependent loss clearly visible in Fig. 3 for a broad wavelength range. The loss related to couplings with the ring-type cladding modes is much smaller for the modes with greater radius because of a lower overlap coefficient with the core modes. Therefore, the loss related to the coupling between HE_1,1⁻ fundamental mode (σ_fund = −1) and the cladding mode (d) with σ = −σ_fund and small radius is much higher than the loss related to the coupling to the modes (c) or (e), for which σ = σ_fund but the radius is larger. Finally, for pairs of the cladding modes with similar ρ and σ ([c, e] and [f, g] in Fig. 3), the mode with lower J_D couples with the fundamental mode at longer wavelength.

The relations identified in the previous paragraph can be explained using one of the conditions necessary for resonant coupling, which is the phase matching between the fundamental mode (f) and cladding mode (cl) in helicoidal coordinates:

where q_i is a constant which governs dispersion of a given mode in the short wavelength range and depends on mode order, fiber geometry and material refractive index [34]. Because in the analyzed case q_cl>q_f then for every λ below the resonance wavelength λ_res, which can be obtained from Eq. (4) for for n’_f − n’_cl = 0, the effective refractive index in helicoidal coordinates is higher for the cladding mode than for the fundamental mode:

Using the above relation, it is possible to determine the sequence of resonant couplings by analyzing the values of n′_cl. In Fig. 4(a) we show the numerically obtained values of n′_cl for ring shaped modes located between the last and the second-last air hole rings in MOFs with N_R = 2, 3 and 4, which have intensity profiles similar to modes (b in Fig. 2, g in Fig. 3 and k in Fig. 5) at λ = 1280 nm (spectral position of peak k in Fig. 5). In Fig. 4(b) we present the approximated values of effective indices of the cladding modes in Cartesian coordinates (n_cl) obtained by replacing in Eq. (3) the total angular momentum of the cladding modes J_cl by the momentum J_D carried by dominating angular harmonics.

 

Fig. 4 Effective refractive index in helicoidal coordinates n′_cl (a) and in Cartesian coordinates n_cl (b) calculated for λ = 1280 nm versus dominating orbital angular momentum L_D = J_D−σ for selected ring-type modes in MOFs with N_R = 2 (red), 3 (green) and 4 (blue). In (a) full dots correspond to modes with σ = + 1 while empty dots to modes with σ = −1. Distributions of axial component of the Poynting vector (S_z) for ring-type cladding modes in MOFs with various N_R shown next to corresponding dots.

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Fig. 5 Waveguide loss calculated versus wavelength for HE_1,1⁻ (dotted) and HE_1,1⁺ (solid) fundamental modes in twisted MOF with Λ_L = 3 µm, Λ_H = 500µm, d/Λ_L = 0.4 and N_R = 4. Dashed curve shows the loss of the fundamental modes in non-twisted fiber. Distribution of the axial component of the Poynting vector (S_z) and azimuthal component of electric field amplitude (E_θ) are also shown for the cladding modes coupling with the HE_1,1⁻ fundamental mode at successive resonances. Angular symmetry order of E_θ determines total angular momentum carried by dominating angular harmonics denoted as J_D.

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Numerically obtained relations between n′_cl and dominating orbital angular momentum L_D = J_D−σ cannot be accurately described using analytical formulas derived for the modes guided in similar but simpler structures, like thin cylindrical layers [35] or cylindrical arrays of weakly coupling waveguides [36,37]. However, similarly as in case of annular modes guided in thin cylindrical layers, the effective index of the cladding modes in microstructured fiber n_cl decreases for greater L and increases for greater ρ, regardless of λ. These relations are sufficient to explain the sequence of resonance peaks observed in Fig. 3.

For modes characterized by the same J_D and σ, the mode located further from the fiber axis has greater n′_cl for any λ and therefore is phase matched to the fundamental mode at longer wavelength. Similarly, for pairs of the cladding modes with the same ρ and J_D, the effective index n′_cl of the mode with σ = + 1 and L_D = J_D−1 is always greater than n′_cl of the mode with σ = −1 and L_D = J_D + 1. It should be emphasized that these rules are valid for ring shaped modes guided in MOFs with any number of hole rings N_R and moderate twist rates, which do not spoil the dispersion relations depicted in Fig. 4. As shown in [21] high twist rate increases leakage and changes dispersion relations of the cladding modes, especially those surrounded by small number of air hole rings, which can alter the order of appearance of selected loss peaks in transmission spectrum of the fundamental modes.

One can also see that for the MOFs with N_R = 2,3 and 4, which were considered in detail, the ring-type cladding modes of similar radius and polarization handedness have longer resonance wavelength for smaller J. However, this is not a universal rule because the value of L_D = J_D−σ for which n’_cl of ring-type mode reaches maximum increases with N_R as can be seen in Fig. 4(a).

In Fig. 5. we show the loss spectra of the fundamental mode and field distributions of all cladding modes responsible for couplings in the analyzed wavelength range in the twisted MOFs with 4 air hole rings (N_R = 4). This fiber structure is identical as the one analyzed in [21] for which the effect of changing Λ_L and Λ_H was analyzed in detail. For twisted MOFs with N_R≥4, in addition to ring-type modes, there are couplings with the cladding modes spanning over several air hole rings and covering a large part of the microstructure, for which single dominating harmonic cannot be identified (b, c, d and e in Fig. 5). Once again the modes for which the radius and symmetry order can be identified without any ambiguity are located between the last and the second-last air hole rings (k in Fig. 5). For modes which are ring-type in the MOFs with N_R = 3, additional radial and azimuthal oscillations appear in the E_θ distributions for their counterparts propagating in the MOF with N_R = 4 (a, f, g, h, i, j in Fig. 5 are similar to a, d, c, e, f, g in Fig. 3). Even if in case of such complex modes the dominating harmonic can still be identified, its contribution to the overall amplitude distribution is noticeably reduced.

Adding a fourth air hole ring significantly reduces non-resonant loss in a non-twisted MOF and consequently improves visibility of the polarization dependent peaks, which is strongly reduced by the long wavelength loss edge in the MOF with N_R = 3 (i, j in Fig. 5 and f, g in Fig. 3). On the other hand, the non-resonant loss in the MOF with N_R = 4 is still high enough to completely suppress the low loss polarization dependent resonance peaks related to the coupling between HE_1,1⁻ fundamental mode and most of the cladding modes located between 3rd and 4th air hole rings, which in the analyzed case are phase matched for λ>1700 nm. Similar suppression effect was previously observed and discussed in [21].

In Figs. 6(a)-6(c) we show the simulation results for selected MOFs with N_R>4. For such fibers there are much more resonance peaks associated to both the ring-type and the extended cladding modes, which arise in larger microstructured cladding. As it can be seen in Fig. 6, increasing N_R above 4 leads to effects similar to those observed for lower N_R. Non-resonant loss is reduced and more couplings are observed. Additional resonance peaks appear in the whole analyzed wavelength range but are easiest to identify for longer wavelengths because for shorter wavelengths they are close to much larger peaks existing already in more limited microstructure. In some cases, these emerging resonances disrupt broad ranges of polarization dependent loss bands observed for the MOFs with small N_R. In Fig. 6(d), it can also be seen that the resonance peaks are getting higher and spectrally narrower with increasing N_R (e.g. the peaks around λ = 550 nm and λ = 700 nm). This effect is related to improved confinement of the cladding modes and the reduced loss mismatch between coupling modes, which for low N_R can lead to incomplete coupling characterized by broad, low loss peaks [38]. Additionally, improved confinement of the cladding modes observed for increasing N_R gradually stabilizes their field profiles as well as effective refractive indices and consequently the spectral positions of the resonance peaks.

 

Fig. 6 Waveguide loss calculated versus wavelength for HE_1,1⁻ (dotted) and HE_1,1⁺ (solid) fundamental modes in twisted MOF with Λ_L = 3 µm, Λ_H = 500µm, d/Λ_L = 0.4 and N_R = 5 (a), N_R = 7 (b) and N_R = 9 (c). Dashed curve in (a) shows loss of the fundamental modes in non-twisted fiber. For N_r>5 the loss was too small to be precisely determined. Inset in (c) shows one of the polarization dependent loss peaks observed only for HE_1,1⁺ fundamental mode. In (d) we compare the loss calculated for HE_1,1⁺ fundamental modes in twisted MOFs with different number of hole rings N_R.

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Finally, for large N_R we observe another type of polarization dependent loss peaks. Typically, as it can be seen in Fig. 3 and Fig. 5, respectively for N_R = 3 and N_R = 4, higher loss in the polarization dependent couplings occurs for the HE_1,1⁻ fundamental mode. For N_R greater than 5, another type of polarization dependent loss peaks begins to appear, which is characterized by greater loss of the HE_1,1⁺ fundamental mode. One of such peaks is shown in the inset to Fig. 6(c) for N_R = 9. Peaks of this kind are related to coupling between the HE_1,1⁺ fundamental mode with σ = + 1 and the cladding modes composed primarily of harmonics with σ = −1 and are much weaker than their counterparts observed for HE_1,1⁻ fundamental mode. The difference in the coupling strength is related to σ of the cladding modes responsible for the polarization dependent peaks. According to Eq. (6) and the results presented in Fig. 4, out of the cladding modes with the same J and ρ, those with σ = −1 will be coupled at shorter wavelengths and therefore will be surrounded by more distinct polarization independent peaks with higher loss related to the modes with σ + 1 and, in most cases, similar or smaller ρ.

3.2 Effect of filling factor variation

We have also analyzed the effect of filling factor d/Λ_L on the resonance loss spectra of the HE_1,1⁺ and HE_1,1⁻ fundamental modes in the twisted MOFs with Λ_L = 3µm, Λ_H = 500µm and d/Λ_L changing from 0.35 to 0.5. In Fig. 7, we show the results obtained for the twisted MOFs with N_R = 3 and 4, for which we could observe all effects related to changes in d/Λ_L. For N_R = 3, the effect of variation of d/Λ_L can be easily observed and analyzed because for such fiber structure the resonance peaks are less numerous, well separated and distinct. In fibers with N_R≥4, it is more difficult to analyze the effect of d/Λ_L on the loss spectra, because of much greater number of modes supported by the microstructure, simultaneous couplings to few cladding modes and consequently the overlap of the resonance peaks in the short-wavelength range.

 

Fig. 7 Waveguide loss calculated versus wavelength for HE_1,1⁻ (dotted) and HE_1,1⁺ (solid) fundamental modes in twisted MOF with Λ_L = 3 µm, Λ_H = 500µm, N_R = 3 (a) or N_R = 4 (b) and d/Λ_L = 0.35 (blue), d/Λ_L = 0.4 (red), d/Λ_L = 0.45 (green) or d/Λ_L = 0.5 (black). Dashed curve shows the loss of the fundamental modes in non-twisted fiber.

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Variation of the filling factor d/Λ_L has a strong influence on confinement of the guided modes and consequently on the observed loss. For lower d/Λ_L, the confinement of the fundamental mode is weaker, which leads to higher loss in the non-twisted MOFs and greater overlap with the cladding modes, which increases the resonant coupling loss in twisted MOFs. However lower d/Λ_L also increases leakage and loss of the cladding modes, which has a detrimental effect on the resonance peaks observed in the vicinity of the non-resonant long wavelength loss edge (dashed curve in Fig. 7). As shown in [21], strong leakage of the cladding mode from limited microstructure to the unstructured glass matrix characterized by a higher refractive index causes additional redshift and spectral broadening of the resonance peaks. As a result, some of the resonance peaks visible for greater values of d/Λ_L vanish in the non-resonant loss edge for low d/Λ_L, even if the phase matching condition is satisfied. In the analyzed cases, this effect significantly reduces relative difference between the loss of the HE_1,1⁺ and HE_1,1⁻ fundamental modes and consequently reduces the wavelength range in which polarization dependent loss can be observed.

The mode leakage alone cannot explain the changes in the resonance peaks positions and the polarization dependent loss far from the long wavelength loss edge. These effects are related to the decrease of n_eff of the guided modes caused by the increase of d/Λ_L, which is greater for the cladding modes than for the fundamental modes [39]. Therefore, according to Eq. (7), for greater d/Λ_L the cladding mode will be phase-matched to the fundamental mode at shorter wavelength. Furthermore, variation of d/Λ_L affects the loss mismatch and the angle at which n_eff dispersion curve of the fundamental and the cladding mode cross, which leads to reduction of the spectral width of the resonance peaks for greater d/Λ_L accompanied by reduction of non-resonant loss between resonant loss peaks. This effect increases the relative difference between the loss of the HE_1,1⁺ and HE_1,1⁻ fundamental modes, especially in the spectral regions near the polarization dependent loss peaks.

Finally, for N_R≥4, the increase of d/Λ_L leads to the appearance of additional resonance peaks. Most of those peaks (e. g. the peaks observed at around λ = 470 nm and λ = 545 nm for d/Λ_L = 0.5 in Fig. 7(b)) are related to additional core and cladding modes guided only for d/Λ_L>0.4. However, some of those peaks (e. g. the peak in Fig. 7(b) observed around λ = 830 nm for d/Λ_L = 0.5 in Fig. 7(b)) are related to the couplings with modes guided even for small d/Λ_L but resulting in resonance loss peaks which are too small to be observed in the presence of high non-resonant loss. The simulation results presented in Fig. 7 lead to the conclusion that increasing d/Λ_L in twisted MOFs reduces non-resonant loss, yields narrower, more distinct resonance peaks and enhances polarization effects. Practical applications of these effects can be limited in MOFs with d/Λ_L>0.42 due to possible multimode propagation in the core. To fully and accurately define the useful wavelength range, N_R value, and d/Λ_L ratio for applications where the single mode regime is required, one needs to investigate carefully the loss and spatial transitions of the second order modes in twisted MOFs as done previously in straight fibers [28,31,34]. This worthy but long study is beyond the scope of the present work, knowing that the parameter space to investigate is no longer 2D (λ, d/Λ_L) as for the straight MOFs but 3D (λ, d/Λ_L, Λ_H), not mentioning about finite size effects related to N_R.

4. Conclusions

We have analyzed for the first time the effect of the number of air hole rings (N_R) and filling factor (d/Λ_L) on resonance loss spectrum in twisted MOFs. Our results show that the variations of N_R and d/Λ_L change the number of resonance peaks, their spectral width and the wavelength range in which they arise. The successive addition of an air hole ring in twisted MOF significantly increases the number of resonance loss peaks. This is because the microstructure with greater number of air-holes rings N_R guides more cladding modes, which can couple to the fundamental mode if the symmetry and phase matching conditions are satisfied [26]. In the MOFs with small N_R (1<N_R≤3), the cladding modes responsible for coupling are localized between two adjacent air hole rings. These modes are ring-shaped and have well-defined principal total angular momentum order J. In the MOFs with larger N_R, there are both the ring-type cladding modes and the extended cladding modes spanning over several air hole rings, for which a single dominating J order does not exist. Furthermore, spectral positions of the resonance loss peaks arising for limited microstructure gradually stabilize with increasing N_R along with the field profiles of the cladding modes and their effective indices. Additionally, improved confinement of the cladding modes observed for increasing N_R reduces the loss mismatch with the fundamental mode and consequently leads to spectrally narrow, better localized resonance loss peaks, which can be beneficial for sensing applications of twisted MOFs. Finally, our results show that the polarization sensitive couplings depend on the number of air hole rings. For moderate N_R (4 and 5), it is possible to obtain broadband polarization dependent loss related to couplings between HE_1,1⁻ fundamental mode (σ = −1) and the cladding modes of opposite polarization handedness (σ = + 1). For large N_R (≥7) we observed the appearance of additional polarization sensitive peaks arising due to couplings between HE_1,1⁺ fundamental mode (σ = + 1) and cladding modes with σ = −1, which were never before reported for the twisted MOFs.

Variation of the air-hole filling factor d/Λ_L has a lesser impact on the resonant loss spectra than the variation of N_R but nevertheless it can be used to control some of its important features. In particular, it was shown that the increase of d/Λ_L in twisted MOFs induces the blueshift of the loss peaks, reduces the resonance loss and enhances polarization effects. Furthermore, it can be used to obtain narrower, more distinct resonance peaks without variation of N_R. However, practical applications of these effects can be limited in the MOFs with d/Λ_L>0.42 due to multimode propagation in the core.

In addition to the numerical results, we have provided simple qualitative rules, which can be used to predict the sequence of resonant couplings of the fundamental mode with cladding modes, especially those of ring-type. These rules explain why the additional resonance peaks arising for the increasing number of N_R are located at greater wavelengths than in more limited microstructure and why the polarization effects are best visible for longer wavelengths.