Abstract

We show that in twisted microstructured optical fibers (MOFs) the coupling between the core and cladding modes can be obtained for helix pitch much greater than previously considered. We provide an analytical model describing scaling properties of the twisted MOFs, which relates coupling conditions to dimensionless ratios between the wavelength, the lattice pitch and the helix pitch of the twisted fiber. Furthermore, we verify our model using a rigorous numerical method based on the transformation optics formalism and study its limitations. 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 9 mm helix pitch, thus allowing for fabrication of coupling based devices using a less demanding method involving preform spinning.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Twisting an optical fiber provides additional opportunities for controlling its propagation characteristics. First helical core fibers were obtained by twisting a preform of a step-index fiber with an offset core during the drawing process [1]. Such fabrication method allows for obtaining a uniform twist at long distances but the helix pitch is limited to no less than single millimeters. Helical core fibers obtained in such a way can be used in current sensors [2] due to their high circular birefringence, and in fiber lasers [3] because of single mode propagation at high normalized frequencies. It was recently shown that similar fabrication method can be used for microstructured optical fibers (MOFs) without disrupting their complex structure and the resulting twist significantly reduces the polarization mode dispersion [4].

Advances in microfabrication led to obtaining much higher twist rates and novel twist-related properties. Local twisting of an already drawn fiber, as it passes through a miniature heat zone, allowed for reduction of the helix pitch to the order of tens of micrometers [5], with smallest helix pitch reported being 16 µm [6]. First strongly twisted structures obtained in this way using fibers with an off-axes or elliptical step-index core exhibited new phenomena related to resonant coupling between the core and cladding modes [5–7]. The coupling effects in twisted step-index fibers can be used for development of sensors, circular polarizers, couplers [5–7] and optical vortices generators [8,9]. Coupling phenomena in twisted MOFs observed for reduced helix pitch suggest their applications as polarization insensitive dispersion controllers, sensors, and spin-orbit couplers [10–14]. Recently, the polarization-selective phenomena were also reported for twisted MOFs [15]. Typical helix pitch required for the above mentioned applications is between few tens to few hundred micrometers. The only reported twisted fibers in which coupling phenomena were obtained for a few millimeter helix pitch are chirally-coupled-core fibers [16,17], which were fabricated by twisting the preform during the fiber drawing process. In this case, however, there is no coupling between the fundamental and cladding modes which is a key phenomenon in previously mentioned applications. Instead, the higher order modes in a very large central core are cut off due to the coupling with helical side-cores.

While advanced twisting methods [6,14] allow for very precise control of the helix, including twist gradient [6], they are more challenging and can provide only short sections of a twisted fiber. For the applications discussed above, it would be beneficial if the coupling between the core and cladding modes could also be obtained in the fibers with larger helix pitch fabricated by the preform spinning method, which allows for obtaining complex structures such as twisted MOFs [14] of lengths exceeding 100 m. In this way, a fabrication of the coupling-based devices would be cheaper, obtained fibers would be more durable as there would be no need to remove the coating prior to twisting. Moreover, twisted fibers of long lengths could be used for distributed sensing applications.

In this paper we show that the twisted MOFs are promising candidates for obtaining the coupling phenomena in structures with millimeter long pitches, which are feasible for a preform spinning fabrication method. Until now, the only reported and analyzed effect, which can be used to tune the coupling in twisted MOFs, is the linear relation between the resonance wavelength and the twist rate, which was first observed experimentally in [10] and explained in [11] as coupling between the fundamental modes and the orbital angular momentum (OAM) states arising for certain combinations of radius ρ, twist rate A and wavelength λres:

λres=2πlnSMρ2A,
where l is the OAM order, nSM is the refractive index of the fundamental space-filling mode and A is the twist rate expressed in rad/m. It should be noted, however, that although Eq. (1) provides the correct relation between the resonance wavelength and the twist rate, it does not correctly describe the role of OAM in the coupling phenomena in twisted MOFs, as recently shown in [15]. Equation (1) suggests that the resonance peaks in MOFs will be present for any twist rate A but their spectral position will be shifted correspondingly. In case of the reported twisted MOFs [11], the increase of the twist pitch to millimeter level will result in shifting of the resonance peaks to the ultraviolet region. This problem could be possibly circumvented by utilizing the scaling properties of the electro-magnetic wave equation, which were already used for MOFs [18,19], but which were not yet adopted to twisted fibers. In principle the same phenomena will be observed in a uniformly rescaled MOF, if the ratio between the wavelength and the lattice pitch is conserved and the material dispersion is not taken into account. Therefore, one can suspect that uniform enlarging of the weakly twisted MOF could again move the resonance peaks to the practically desirable visible or near infrared wavelength range.

In this paper we analyze the scaling properties of the twisted MOFs and provide analytical model which relates phase matching conditions between two coupling modes to dimensionless ratios between the wavelength, the lattice pitch and the helix pitch of the twisted fiber. Our model shows that the resonance position can be modified by independently changing the lattice pitch or the helix pitch. This allows for designing twisted MOFs in which the coupling phenomena are observed for desired combination of the twist rate and resonant wavelength. Furthermore, we provide numerical verification of our model, study its limitations and discuss the effect of geometrical parameters of the fiber on recently observed polarization dependent characteristics of twisted MOFs [15].

2. Scaling properties of photonic crystal fiber

The electromagnetic properties of any system in which there is no material dispersion will be conserved, if the size of its components and the wavelength are rescaled by the same factor. This principle can be supported analytically if one considers a rescaled waveguide, which is uniformly enlarged by a factor Q in coordinate system x = {x1, x2, x3}. Rescaled waveguide can be described in a different coordinate system x’ = {x1’, x2’, x3’} defined by:

xi'=xiQ,
in which its apparent size does not change. According to the transformation optics formalism, the change of the coordinate system from x = {x1, x2, x3} to x’ = {x1’, x2’, x3’} is equivalent to the change in the permittivity and permeability described by the following relations [20]:
εi'j'=|det(Λi'i)|1Λi'iΛj'jεij,
μi'j'=|det(Λi'i)|1Λi'iΛj'jμij,
where:
Λi'i=δxi'δxi.
In case of coordinate systems related by Eq. (2) one obtains the following relation between the permittivity and permeability tensors:
ε'ε=εr'εr=μ'μ=μr'μr=Q.
Propagation properties of the original system are determined by the wave equation:
×(1μr(r)×E(r))=(2πλ)2εr(r)E(r),
while the propagation properties of the rescaled system can be obtained using the wave equation in which the equivalent relative permittivity εr’ and relative permeability µr’ tensors given by Eq. (6) are used:
×(1Qμr(r)×E(r))=(2πλ')2Qεr(r)E(r).
Equation (8) can be rewritten as:
×(1μr(r)×E(r))=(2πQ1λ')2εr(r)E(r),
which is identical to Eq. (7) for:
λ'=Qλ.
Therefore in the rescaled optical system one obtains the same form of the wave equation and consequently the same propagation properties, if the size of the optical system and the wavelength are rescaled by the same factor and there is no material dispersion, i.e., εr’ and µr’ are wavelength independent.

Similar scaling effect was discussed in [9] for multihelicoidal fibers, in which the resonance wavelength and the resonance peak width increase Q times if the helix pitch and the core radius are simultaneously enlarged by a factor of Q. In case of MOFs, it was shown in [18,19] that if material dispersion is disregarded, the complex effective refractive index ñeff of a mode guided in the microstructure with a fixed number of air hole rings depends only on the dimensionless ratios between wavelength λ, the lattice pitch ΛL and the air hole diameter d:

n˜eff(λ,d,ΛL)nmat(λ)=const=n˜eff(λ/ΛL,d/ΛL),
which is related to the fact that ñeff itself is dimensionless. According to Eq. (11), the uniform rescaling of the cross-section of the MOF and the wavelength conserves the complex effective refractive index ñeff of a mode. It must be noted, however, that the above relation is valid only for fibers which are invariant along their axis and therefore will not hold for twisted MOFs. Similar generalized relation, which is a direct consequence of the scaling properties of the electromagnetic wave equation, can be obtained for uniformly twisted MOFs by taking into account one additional parameter – the helix pitch ΛH. In this case, the relation equivalent to Eq. (11), has the following form:
n˜eff(λ,d,ΛL,ΛH)nmat(λ)=const=n˜eff(λ/ΛL,d/ΛL,ΛH/ΛL),
which can be used to predict some of the coupling properties of the twisted MOFs.

Real part of ñeff (denoted as neff in the later part of this paper) determines the phase velocity of the mode along the waveguide and the phase matching condition required for coupling while its imaginary part determines the mode loss:

L=20ln102πλIm(n˜eff)[dBm].
Therefore, if the loss maximum related to coupling between the modes is observed for a given wavelength λ in the MOF characterized by (d, ΛL, ΛH), it will also be observed for rescaled wavelength in a MOF characterized by (Qd, QΛL, QΛH). Furthermore, if the coupling occurs, the phase matching between two modes in the helicoidal coordinate system must be conserved in the rescaled structures. As shown in [8], the effective refractive index in helicoidal coordinates n’eff is modified with respect to the index in the Cartesian coordinates neff in the following way:
n'eff=neff+JλΛH,
where J is the angular momentum of the mode. In twisted MOFs, the value of J can be fractional and is related to the harmonic composition of the mode [14,15], which is also conserved in a uniformly rescaled structure. Therefore J also depends only on the dimensionless ratios between the wavelength λ, the lattice pitch ΛL the air hole diameter d and the helix pitch ΛH:
J(λ,d,ΛL,ΛH)nmat(λ)=const=J(λ/ΛL,d/ΛL,ΛH/ΛL)
and consequently one obtains:
n'eff(λ,d,ΛL,ΛH)nmat(λ)=const=n'eff(λ/ΛL,d/ΛL,ΛH/ΛL),
which is equivalent to the phase matching conservation in the rescaled fiber structure.

This general rule implies that the coupling observed in strongly twisted MOFs (ΛL≈3 µm ΛL≈1 mm) studied in [11,14] could potentially be observed, albeit at greater λ, in uniformly enlarged fibers with much greater helix pitch. In real fiber structures, in which the material dispersion must be taken into account, the relation given by Eq. (16) is not strictly valid but the effect of coupling in rescaled systems can still be observed for wavelengths, which are close to those predicted by Eq. (16), as shown in the results of numerical simulations presented in the later section of this paper.

Applications of the described scaling effect alone are limited because significant enlarging of previously analyzed MOFs [11,14] to ΛH in a millimeter range will result in shifting the resonance peaks to far infrared. It does however suggest that similar coupling phenomena could be possibly obtained for terahertz radiation in appropriately scaled MOFs. On the other hand, if the scaling effect is combined with Eq. (1) implying that the increase in ΛH alone leads to the blueshift of the resonance peaks, one can expect that for some combinations of ΛL and ΛH, the coupling will be observed in a useful spectral range even in twisted MOFs with ΛH in a millimeter range.

To analyze the effect of simultaneous variation of ΛL and ΛH on coupling in twisted MOFs, we use the phase matching conditions between the core and cladding modes. In Cartesian coordinates, the effective index neff of the guided modes in the MOF characterized by the nondispersive material refractive index nM of the glass matrix can be expressed in the following way for small normalized wavelength (λL) [21,22]:

neffnM(λΛL)2q,
where q is a constant which depends on mode order, fiber geometry and nM. According to Eq. (14), the effective refractive index n’eff in helicoidal coordinates is modified with respect to the index in the Cartesian coordinates neff in the following way:
n'eff=neff+(λΛL)JΛLΛHnM(λΛL)2q+(λΛL)JΛLΛH
Therefore, the difference between n’eff of the fundamental (1) and cladding (2) mode is given by:
Δn'eff=n'eff,1n'eff,2(λΛL)2(q2q1)(λΛL)(J2J1)ΛLΛH
and the phase matching condition (Δn’eff = 0) necessary to observe the resonant coupling between the core and cladding modes is equivalent to the following relation:
λresΛL=2ΔJΔqΛLΛH.
According to the above equation, the normalized resonance wavelength is equal to the ratio of ΛLH multiplied by a constant which depends on the fiber parameters and mode orders. The same relation can be expressed in terms of λ and twist rate A = 2π/ΛH in the following way:
λres=1πΔJΔqΛL2A.
Equation (21) predicts linear scaling of the resonance wavelength with the twist rate A, which agrees with the experimental results from [11] and predictions of Eq. (1) but additionally it provides correct understanding of the role of OAM in the coupling phenomenon. Furthermore, it shows the possibility of scaling the resonance wavelength quadratically with ΛL, which was not considered previously. This effect also agrees with Eq. (1), if we assume that the radius ρ is directly proportional to ΛL. Equation (21) has important practical consequences. The couplings, which previously were observed only in strongly twisted fibers, can be obtained for the same wavelengths in larger fibers with much smaller twist rates, possibly with helix pitch ΛH even in a centimeter range.

For larger normalized wavelengths, the relation between neff and the λL given by Eq. (17) no longer holds and some deviations from resonance wavelength given by Eq. (21) can be expected. In case of dispersive nM, both J and q become wavelength dependent. Consequently, the spectral positions of resonance peaks can be different for the same normalized wavelength λ/Λ in rescaled waveguides. However, as will be shown in the following sections, this effect is visible only in the wavelength region, in which the resonant peaks related to coupling vanish in the non-resonant loss determined by the properties of the untwisted fiber. In the wavelength range in which resonant coupling is well distinguished, Eq. (21) agrees very well with the results of numerical simulations.

3. Fiber structure and simulation method

The analytically predicted properties of the twisted MOFs were verified numerically using the model with a cross-section shown in Fig. 1.

 figure: Fig. 1

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

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Selected geometrical parameters of the fiber were modified during simulations to determine their influence on the coupling between the core and cladding modes in a twisted MOF. We examined the fibers with 4 air hole rings and lattice pitch ΛL ranging from 3 µm to 15 µm, corresponding to standard and large mode area (LMA) fibers. The smallest analyzed helix pitch ΛH was 1/6 mm. The filling factor d/ΛL was set to 0.4 to ensure endlessly single mode propagation in the core. Spectral dependence of the refractive index of silica glass wasaccounted for using the following Sellmeier’s equation [23]:

nM(λ)=1+i=13Aiλ2λ2Zi2,
where wavelength λ is given in microns, while A1 = 0.6961663, A2 = 0.4079426, A3 = 0.8974794, Z1 = 0.0684043, Z2 = 0.1162414 and Z3 = 9.896161. In some cases the material dispersion was disregarded (nM = 1.45) to better demonstrate the effects related solely to fiber geometry. To obtain the propagation characteristics of twisted MOFs, we combined the transformation optics formalism [24] with the finite element method using COMSOL Multiphysics software. Twisted perfectly matching layers (PML) [25] were placed outside of the air holes at a distance of 4.5ΛL 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.

4. Results

We verified the validity of Eq. (21) and analyzed its limitations related to used approximations by calculating loss spectra of the circularly polarized HE1,1 fundamental mode (J = −1) with the same handedness as the twisted MOF and for the HE1,1+ (J = + 1) fundamental mode with the opposite handedness. In the first step, the simulations were conducted for nondispersive refractive index of silica matrix nM. Since the phase matching condition given by Eq. (19) does not explicitly depends on nM, the difference in the position of distinct resonant peaks caused by material dispersion is negligible. As it is shown in the later part of this section, material dispersion influences mostly the non-resonant loss observed in the long wavelength range.

First, we examined the twisted MOF characterized by ΛL = 3 µm and A changing from 4π mm−1 to 8π mm−1. The obtained loss spectra are presented in Fig. 2. Sharp loss maxima are caused by resonant coupling between the fundamental and cladding modes, which satisfy symmetry conditions related to their orbital angular momenta [15].

 figure: Fig. 2

Fig. 2 Waveguide loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with ΛL = 3 µm, d/ΛL = 0.4 and A = 4π mm−1 (green), A = 6π mm−1 (blue) or A = 8π mm−1 (red). Dashed line shows the loss of the fundamental modes in untwisted fiber. In (b) we show the loss for the same parameters versus rescaled wavelength λ′ = λ·(4π mm−1)/A. In this case, colors of dashed lines correspond to values of A.

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Results in Fig. 2(a) show that the increase in the twist rate leads to the expected redshift of the resonance peaks. However, as can be seen in Fig. 2(b), the redshift is not always linear - in some cases, the rescaled spectral positions of the peaks differ, while according to Eq. (21) they should be the same.

Furthermore, additional phenomena, which are not directly related to Eq. (21) can be observed. In particular, the loss of the HE1,1 and HE1,1+ fundamental modes differs significantly. Most of peaks in the loss spectra of both fundamental modes arise from couplingwith modes of the same polarization handedness, but there are also clearly visible polarization dependent peaks related to the couplings to the cladding modes of opposite handedness. The possibility of observing polarization selective coupling in twisted MOFs was recently reported in [15] but in the analyzed case, the polarization dependent resonances observed for long wavelength are characterized by a much greater polarization dependent loss ratio and spectral width, which suggests that such fibers may be used as broadband circular polarizers. Additionally, the wavelength range in which the distinct resonant loss peaks are visible is limited by a non-resonant long wavelength loss edge (dashed line in Fig. 2), which is present even in untwisted fiber. For long normalized wavelength, the fundamental and especially the cladding modes strongly leak from the limited microstructure of the MOF. As a result, the coupling loss is lowered due to reduced mode overlap and increasing loss mismatch, which leads to incomplete coupling [26]. Consequently, the resonance peaks vanish and the observed loss asymptotically reaches the non-resonant loss level related to leakage of the fundamental mode. In the short wavelength range, many cladding modes are guided and multiple high loss resonance peaks are visible. This effect is similar to the emergence of the short wavelength loss edge in bend MOFs, which is also related to resonant coupling with the cladding modes [27]. Lowest loss and well visible polarization dependent loss peaks are observed in the intermediate wavelength range between the two spectral regions with increased loss. For higher twist rates, the separation of the two high loss regions decreases and the coupling loss rises.

In Fig. 2(b) one can see that the positions of the resonance peaks far from the long wavelength loss edge scale approximately linearly with the twist rate as predicted by Eq. (21), while the peaks located at longer wavelengths are redshifted for higher twist rates. This unexpected redshift can be attributed to the changes in ∆J/∆q factor in Eq. (21), which were not taken into account before. Close to the long wavelength loss edge, the parameter ∆q varies due the increase in neff of the guided modes, which is not taken into account by Eq. (17) and ∆J is affected by changes in their field profiles and consequently their harmonic representation [14,15]. To better illustrate this phenomenon in Fig. 3, we show the amplitude of the electricfield of the cladding modes responsible for selected resonance peaks, clearly showing the leakage effect, which increases for longer resonance wavelengths.

 figure: Fig. 3

Fig. 3 Electric field amplitude |E| distribution in the cladding modes at selected resonances shown in Fig. 2.

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As it can be seen in Fig. 3, the field profile and the confinement of the mode responsible for linear scaling of the resonance peak at λ = 430 nm × A/(4π mm−1) do not change significantly with the twist rate. For the mode responsible for the resonance peak at around λ = 835 nm × A/(4π mm−1) characterized by a moderate nonlinear redshift, the changes in the field profile and increasing leakage from the microstructure are better visible. The strongest effect of twist rate on leakage is observed for the third mode responsible for the polarization dependent resonance peak at around λ = 1130 nm × A/(4π mm−1). In this case the resonance peak, which is characterized by relatively small loss related to coupling between modes of the opposite handedness, almost vanishes in the non-resonant loss edge for A = 8π mm−1 due to the significant reduction of the electric field amplitude of the cladding mode around the fiber core and consequently the spatial overlap between the interacting modes.

In the next step, we analyzed the effect of enlarging the lattice pitch ΛL on the position of the resonance peaks. The obtained loss spectra are shown in Fig. 4.

 figure: Fig. 4

Fig. 4 Loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1, d/ΛL = 0.4 and ΛL = 3 µm (green), ΛL = 3·1.50.5 µm (blue) or ΛL = 3·20.5 (red) (b). Dashed lines show the loss of the fundamental modes in untwisted fibers, colors of lines correspond to ΛL. In (b) we show the loss for the same parameters versus rescaled wavelength λ′′ = λ·(3 µm)2L.

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The lattice pitch was enlarged in such a way that the expected resonance wavelength shift predicted by Eq. (21) is the same as in Fig. 2. While the spectral position of resonance peaks is similar as in Fig. 2, the changes in loss level and the nonlinear increase in resonance wavelength are considerably smaller than in the previous case, where the twist rate was varied. This effect is related to the fact that even if the resulting wavelength is the same, the normalized wavelength (λL) is smaller in the fibers with greater lattice pitch, which reduces the effects related to the long normalized wavelength loss edge. Therefore the confinement of the cladding modes is better and their loss is smaller. Furthermore, for smaller λL, Eq. (17) provides better approximation of n’eff, which reduces the nonlinear redshift of the resonant peaks.

Finally, we have verified the possibility of obtaining resonant coupling between the fundamental and the cladding modes in visible light and near infrared using MOFs with large ΛL and A sufficiently small to allow for fiber fabrication by preform spinning during the drawing process. As suggested by Eq. (21), we have changed ΛL and A simultaneously in a way, which conserves the value of AΛL2 and consequently the resonance wavelength. In this case, similar resonance spectra can be obtained for the combination of small lattice pitch ΛL and high twist rate A or large lattice pitch ΛL and small twist rate A. Loss spectra for the fundamental modes in twisted MOFs, for which strong resonance peaks are observed at different wavelength ranges, are shown in Fig. 5.

 figure: Fig. 5

Fig. 5 Loss calculated versus wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOFs with d/ΛL = 0.4, ΛL = 3 µm (red), ΛL = 9 µm (blue) or ΛL = 18 µm (green). In (a) ΛH = 0.5 mm for ΛL = 3 µm, ΛH = 4.5 mm for ΛL = 9 µm and ΛH = 18 mm for ΛL = 18 µm. In (b) ΛH = 0.25 mm for ΛL = 3 µm, ΛH = 2.25 mm for ΛL = 9 µm and ΛH = 9 mm for ΛL = 18 µm. In (c) ΛH = 0.1(6) mm for ΛL = 3 µm, ΛH = 1.5 mm for ΛL = 9 µm and ΛH = 6 mm for ΛL = 18 µm.

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As it can be seen in Fig. 5, the resonance loss obtained in the fibers with greater ΛL is much smaller. For the lowest analyzed twist rates (Fig. 5(a)), the resonance loss obtained for twisted MOFs with ΛH in a millimeters range (ΛL = 9 µm, ΛH = 4.5 mm and ΛL = 18 µm, ΛH = 18 mm) is too small for practical applications. However, if the twist rate is slightly increased, the resonance peaks with the loss of hundreds or thousands dB/m can be obtained in a broad wavelength range already for the fibers with ΛL = 18 µm, ΛH = 9 mm (green line in Fig. 5(b)) or ΛL = 18 µm, ΛH = 6 mm (green line in Fig. 5(c)).

The calculated resonance peak positions are redshifted in MOFs with smaller ΛL. This effect is related to changes of the phase matching conditions occurring for longer normalized wavelength (λL), which are not taken into account by Eq. (21). The shapes of the resonance spectra obtained for twisted MOFs with ΛL = 9 µm and ΛL = 18 µm, for which λL is small, arevery similar for each twist rate. The shape of the loss curve obtained for ΛL = 3 µm changes for higher twist rates due to the deformation of the modes, but most of the resonance peaks can be directly related to those observed in the MOFs with larger ΛL.

Additionally, it can be seen that the polarization effects observed in fibers with smaller ΛL (and constant AΛL2) are much stronger. This effect can be attributed to the angle α between the optical path of the fundamental mode propagating along the fiber axis and the spiral path of the cladding mode, which is given by [11]:

α=sin1(Aρ1+A2ρ2)Aρ
at radius ρ. We can assume that the average radius of the mode is proportional to the lattice pitch ΛL and therefore:
αAΛL,
which can be rewritten as:
αAΛL2=const1ΛL,
because AΛL2 is constant. Modes of opposite polarization handedness are perfectly orthogonally polarized only if their directions of propagation are parallel. In fibers with smaller ΛL, the angle α between directions of propagation of the fundamental and the cladding mode is greater and therefore the overlap between modes of the opposite handedness, which affects a strength of the polarization sensitive couplings, is increased.

Finally, we have analyzed the effect of material dispersion on the analyzed phenomena. All of the results presented above were obtained for a dispersive silica glass matrix characterized by nM given by Eq. (22). However, because Eq. (21) was obtained with the assumption that nM is constant, one may expect that the material dispersion can modify the resonance position and be in part responsible for the differences between analytical and numerical predictions.

In Fig. 6(a) we present the loss spectra for a twisted MOF with A = 4π mm−1 and ΛL = 3 µm calculated using dispersive and nondispersive nM. As it can be seen, the resonance peaks overlap while the non-resonant loss observed for long wavelengths differs for dispersive and nondispersive nM. In Fig. 6(b) we show the imaginary part of neff of the fundamental modes, which is related to loss by Eq. (13), in the MOF with A = 8π mm−1 and ΛL = 3 µm and the MOF with A = 4π mm−1 and ΛL = 6 µm. If nM was nondispersive, the relation between Im(neff) and normalized wavelength λ/ΛL obtained for these fibers would have been the same according to Eq. (12). However, as ΛL differs by factor of 2 the same λL is equivalent to a twice longer wavelength, which significantly modifies dispersive nM. As it can be seen in Fig. 6(b), even in this case the results overlap for smaller normalized wavelengths, for which the distinct resonance peaks are observed. In general, the effect of material dispersion is negligible for the resonance peaks, which are far from the long wavelength non-resonant loss edge. In this case, the effective refractive index of the coupling modes can be accurately described by Eq. (17) and therefore the phase matching condition given by Eq. (19) does not explicitly depend on nM. On the other hand, phenomena related to the modes leakage, such as emergence of the long wavelength non-resonant loss edge and distortion of the resonant peaks in its vicinity, are strongly affected by the material dispersion of the glass outside the microstructure thus causing that Eq. (17) is no longer accurate.

 figure: Fig. 6

Fig. 6 In (a) we present the loss spectra for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1 and ΛL = 3µm calculated using dispersive nM (red) and nM = 1.45 (blue). Dashed lines show the loss of the fundamental modes in untwisted fibers. In (b) we present the imaginary part of neff versus normalized wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 8π mm−1 and ΛL = 3µm (red) or by A = 4π mm−1 and ΛL = 6µm (blue) calculated using dispersive nM.

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5. Conclusions

In this paper, we have shown that in twisted MOFs the coupling between the core and cladding modes, originally observed for sub-millimeter helix pitch, can also be obtained in appropriately rescaled structures with pitches in a millimeter range. In consequence, instead of using advanced fabrication methods required for large twist rates, which limit the length of the twisted fiber usually to a few centimeters, one could use the much less demanding preform spinning method, which can provide twisted MOFs of unlimited length. Therefore, using the approach proposed in this work, fabrication of coupling based devices could be greatly facilitated.

We have provided an analytical model describing scaling properties of the twisted MOFs, which relates the phase matching condition between the coupling modes to dimensionless ratios between the wavelength, the lattice pitch ΛL and the helix pitch ΛH of the twisted fiber. Our model confirms reported experimentally in [11] linear scaling of the resonance wavelength with the twist rate A = 2π/ΛH and quadratic scaling with the lattice pitch not analyzed previously. It allows for designing the twisted MOFs, in which the resonant couplings are observed for a desired twist rate and wavelength range.

Furthermore, we have provided rigorous numerical verification of our analytical model based on the transformation optics formalism, studied its limitations and discussed the effect of geometrical parameters of the fiber on recently observed polarization dependent properties of twisted MOFs. The obtained results show that the increase in the twist rate or lattice pitch leads to expected redshift of resonance peaks, which is not always linear, because of strong leakage of the cladding modes at long normalized wavelengths. Additionally, we have observed that the wavelength range, in which the distinct resonant loss peaks are visible, is limited by the non-resonant long wavelength loss edge related to the geometry of an untwisted fiber. Furthermore, we have shown that the polarization selective couplings in twisted MOFs can result in the resonance peaks of much larger polarization dependent loss ratio and spectral width than previously reported, which suggests that that twisted MOFs may be used as broadband circular polarizers. Significant polarization sensitive effects can be observed only in fibers with small ΛL and ΛH, because their strength is related to the angle between the fiber axis and the spiral path of the cladding mode.

Finally, we have verified the possibility of obtaining resonant couplings between the fundamental and cladding modes in the visible and near infrared spectral range using MOFs with ΛH in a millimeters range. The obtained results show that for appropriately designed twisted MOF multiple resonance peaks with the loss of hundreds or thousands dB/m can be obtained in a broad wavelength range spanning between 400 nm and 1600 nm for fibers with helix pitch as large as ΛH = 9 mm.

Funding

National Science Centre of Poland, grant Maestro 8, DEC- 2016/22/A/ST7/00089.

Acknowledgment

M. Napiorkowski acknowledges the support of the Foundation for Polish Science (FNP) in the frame of START 2017 program.

References and links

1. R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987). [CrossRef]  

2. V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006). [CrossRef]  

3. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006). [CrossRef]   [PubMed]  

4. M. Fuochi, J. Hayes, K. Furusawa, W. Belardi, J. Baggett, T. Monro, and D. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 12(9), 1972–1977 (2004). [CrossRef]   [PubMed]  

5. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004). [CrossRef]   [PubMed]  

6. V. I. Kopp, J. Park, M. Wlodawski, J. Singer, D. Neugroschl, and A. Z. Genack, “Chiral Fibers: Microformed Optical Waveguides for Polarization Control, Sensing, Coupling, Amplification, and Switching,” J. Lightwave Technol. 32(4), 605–613 (2014). [CrossRef]  

7. V. I. Kopp, V. M. Churikov, G. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24(10), A48–A52 (2007). [CrossRef]  

8. C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008). [CrossRef]  

9. C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013). [CrossRef]  

10. W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009). [CrossRef]  

11. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012). [CrossRef]   [PubMed]  

12. X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013). [CrossRef]   [PubMed]  

13. X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014). [CrossRef]  

14. P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017). [CrossRef]   [PubMed]  

15. M. Napiorkowski and W. Urbanczyk, “Role of symmetry in mode coupling in twisted microstructured optical fibers,” Opt. Lett. 43(3), 395–398 (2018). [CrossRef]   [PubMed]  

16. X. Ma, C. H. Liu, G. Chang, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally-coupled-core fibers,” Opt. Express 19(27), 26515–26528 (2011). [CrossRef]   [PubMed]  

17. X. Ma, C. Zhu, I. N. Hu, A. Kaplan, and A. Galvanauskas, “Single-mode chirally-coupled-core fibers with larger than 50 µm diameter cores,” Opt. Express 22(8), 9206–9219 (2014). [CrossRef]   [PubMed]  

18. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25(11), 790–792 (2000). [CrossRef]   [PubMed]  

19. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibers, 2nd ed. (Imperial College, 2012), Chap. 8.

20. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006). [CrossRef]   [PubMed]  

21. B. Kuhlmey, R. McPhedran, C. de Sterke, P. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10(22), 1285–1290 (2002). [CrossRef]   [PubMed]  

22. B. T. Kuhlmey, Theoretical and Numerical Investigation of the Physics of Microstructured Optical Fibres, PhD thesis, (Université Aix-Marseille III and University of Sydney, 2004).

23. M. Bass, Handbook of Optics, Vol. 4, 3rd ed. (McGraw-Hill, 2009).

24. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004). [CrossRef]  

25. A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007). [CrossRef]  

26. Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008). [CrossRef]   [PubMed]  

27. J. Olszewski, M. Szpulak, and W. Urbańczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13(16), 6015–6022 (2005). [CrossRef]   [PubMed]  

References

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  1. R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
    [Crossref]
  2. V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
    [Crossref]
  3. P. Wang, L. J. Cooper, J. K. Sahu, and W. A. Clarkson, “Efficient single-mode operation of a cladding-pumped ytterbium-doped helical-core fiber laser,” Opt. Lett. 31(2), 226–228 (2006).
    [Crossref] [PubMed]
  4. M. Fuochi, J. Hayes, K. Furusawa, W. Belardi, J. Baggett, T. Monro, and D. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 12(9), 1972–1977 (2004).
    [Crossref] [PubMed]
  5. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
    [Crossref] [PubMed]
  6. V. I. Kopp, J. Park, M. Wlodawski, J. Singer, D. Neugroschl, and A. Z. Genack, “Chiral Fibers: Microformed Optical Waveguides for Polarization Control, Sensing, Coupling, Amplification, and Switching,” J. Lightwave Technol. 32(4), 605–613 (2014).
    [Crossref]
  7. V. I. Kopp, V. M. Churikov, G. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24(10), A48–A52 (2007).
    [Crossref]
  8. C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
    [Crossref]
  9. C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
    [Crossref]
  10. W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
    [Crossref]
  11. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
    [Crossref] [PubMed]
  12. X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
    [Crossref] [PubMed]
  13. X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
    [Crossref]
  14. P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
    [Crossref] [PubMed]
  15. M. Napiorkowski and W. Urbanczyk, “Role of symmetry in mode coupling in twisted microstructured optical fibers,” Opt. Lett. 43(3), 395–398 (2018).
    [Crossref] [PubMed]
  16. X. Ma, C. H. Liu, G. Chang, and A. Galvanauskas, “Angular-momentum coupled optical waves in chirally-coupled-core fibers,” Opt. Express 19(27), 26515–26528 (2011).
    [Crossref] [PubMed]
  17. X. Ma, C. Zhu, I. N. Hu, A. Kaplan, and A. Galvanauskas, “Single-mode chirally-coupled-core fibers with larger than 50 µm diameter cores,” Opt. Express 22(8), 9206–9219 (2014).
    [Crossref] [PubMed]
  18. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25(11), 790–792 (2000).
    [Crossref] [PubMed]
  19. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, A. Argyros, and S. Leon-Saval, Foundations of Photonic Crystal Fibers, 2nd ed. (Imperial College, 2012), Chap. 8.
  20. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
    [Crossref] [PubMed]
  21. B. Kuhlmey, R. McPhedran, C. de Sterke, P. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10(22), 1285–1290 (2002).
    [Crossref] [PubMed]
  22. B. T. Kuhlmey, Theoretical and Numerical Investigation of the Physics of Microstructured Optical Fibres, PhD thesis, (Université Aix-Marseille III and University of Sydney, 2004).
  23. M. Bass, Handbook of Optics, Vol. 4, 3rd ed. (McGraw-Hill, 2009).
  24. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
    [Crossref]
  25. A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
    [Crossref]
  26. Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008).
    [Crossref] [PubMed]
  27. J. Olszewski, M. Szpulak, and W. Urbańczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13(16), 6015–6022 (2005).
    [Crossref] [PubMed]

2018 (1)

2017 (1)

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

2014 (3)

2013 (2)

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

2012 (1)

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (1)

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

2008 (2)

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

Z. Zhang, Y. Shi, B. Bian, and J. Lu, “Dependence of leaky mode coupling on loss in photonic crystal fiber with hybrid cladding,” Opt. Express 16(3), 1915–1922 (2008).
[Crossref] [PubMed]

2007 (2)

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

V. I. Kopp, V. M. Churikov, G. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24(10), A48–A52 (2007).
[Crossref]

2006 (3)

2005 (1)

2004 (3)

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

M. Fuochi, J. Hayes, K. Furusawa, W. Belardi, J. Baggett, T. Monro, and D. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 12(9), 1972–1977 (2004).
[Crossref] [PubMed]

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004).
[Crossref] [PubMed]

2002 (1)

2000 (1)

1987 (1)

R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Agha, Y. O.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

Ahmed, G.

Alexeyev, A. N.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Alexeyev, C. N.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

Andrés, P.

Babic, F.

Baggett, J.

Belardi, W.

Beravat, R.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

Bian, B.

Biancalana, F.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Birch, R. D.

R. D. Birch, “Fabrication and Characterization of Circularly Birefringent Helical Fibers,” Electron. Lett. 23(1), 50–52 (1987).
[Crossref]

Chamorovskii, Y.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Chang, G.

Chao, N.

Churikov, V. M.

Clarkson, W. A.

Conti, C.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Cooper, L. J.

de Sterke, C.

Draper, C. W.

Euser, T. G.

Ferrando, A.

Frosz, M. H.

Fuochi, M.

Furusawa, K.

Galvanauskas, A.

Genack, A. Z.

Gubin, V.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Guenneau, S.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Hayes, J.

Hu, I. N.

Isaev, V.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Jiang, X.

Kang, M. S.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Kaplan, A.

Kopp, V. I.

Kuhlmey, B.

Lapin, B. P.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Lee, H. W.

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Lee, Y. L.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Liu, C. H.

Lu, J.

Ma, X.

Maystre, D.

McPhedran, R.

Milione, G.

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
[Crossref]

Miret, J. J.

Monro, T.

Morshnev, S.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Napiorkowski, M.

Neugroschl, D.

Nicolet, A.

A. Nicolet, F. Zolla, Y. O. Agha, and S. Guenneau, “Leaky modes in twisted microstructured optical fibers,” Wave Random Complex 17(4), 559–570 (2007).
[Crossref]

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004).
[Crossref]

Noh, Y.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Oh, K.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Olszewski, J.

Park, J.

Pendry, J. B.

Renversez, G.

Richardson, D.

Robinson, P.

Russell, P. S. J.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
[Crossref]

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Sahu, J. K.

Sazonov, A.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Schurig, D.

Shi, Y.

Shin, W.

W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

Silvestre, E.

Singer, J.

Smith, D. R.

Starostin, N.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Szpulak, M.

Urbanczyk, W.

Usov, A.

V. Gubin, V. Isaev, S. Morshnev, A. Sazonov, N. Starostin, Y. Chamorovskii, and A. Usov, “Use of Spun optical fibres in current sensors,” Quantum Electron. 36(3), 287–291 (2006).
[Crossref]

Wang, P.

Weiss, T.

X. Xi, G. K. L. Wong, T. Weiss, and P. S. J. Russell, “Measuring mechanical strain and twist using helical photonic crystal fiber,” Opt. Lett. 38(24), 5401–5404 (2013).
[Crossref] [PubMed]

G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber,” Science 337(6093), 446–449 (2012).
[Crossref] [PubMed]

Wlodawski, M.

Wong, G. K. L.

P. S. J. Russell, R. Beravat, and G. K. L. Wong, “Helically twisted photonic crystal fibres,” Philos Trans A Math Phys Eng Sci 375(2087), 20150440 (2017).
[Crossref] [PubMed]

X. M. Xi, G. K. L. Wong, M. H. Frosz, F. Babic, G. Ahmed, X. Jiang, T. G. Euser, and P. S. J. Russell, “Orbital-angular-momentum-preserving helical Bloch modes in twisted photonic crystal fiber,” Optica 1(3), 165–169 (2014).
[Crossref]

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W. Shin, Y. L. Lee, B. Yu, Y. Noh, and K. Oh, “Spectral characterization of helicoidal long-period fiber gratings in photonic crystal fibers,” Opt. Commun. 282(17), 3456–3459 (2009).
[Crossref]

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Optica (1)

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C. N. Alexeyev and M. A. Yavorsky, “Generation and conversion of optical vortices in long-period helical core optical fibers,” Phys. Rev. A 78(4), 043828 (2008).
[Crossref]

C. N. Alexeyev, A. N. Alexeyev, B. P. Lapin, G. Milione, and M. A. Yavorsky, “Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers,” Phys. Rev. A 88(6), 063814 (2013).
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[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Cross-section of the analyzed twisted microstructured optical fiber. Arrow shows the twist direction.
Fig. 2
Fig. 2 Waveguide loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with ΛL = 3 µm, d/ΛL = 0.4 and A = 4π mm−1 (green), A = 6π mm−1 (blue) or A = 8π mm−1 (red). Dashed line shows the loss of the fundamental modes in untwisted fiber. In (b) we show the loss for the same parameters versus rescaled wavelength λ′ = λ·(4π mm−1)/A. In this case, colors of dashed lines correspond to values of A.
Fig. 3
Fig. 3 Electric field amplitude |E| distribution in the cladding modes at selected resonances shown in Fig. 2.
Fig. 4
Fig. 4 Loss calculated versus the wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1, d/ΛL = 0.4 and ΛL = 3 µm (green), ΛL = 3·1.50.5 µm (blue) or ΛL = 3·20.5 (red) (b). Dashed lines show the loss of the fundamental modes in untwisted fibers, colors of lines correspond to ΛL. In (b) we show the loss for the same parameters versus rescaled wavelength λ′′ = λ·(3 µm)2L.
Fig. 5
Fig. 5 Loss calculated versus wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOFs with d/ΛL = 0.4, ΛL = 3 µm (red), ΛL = 9 µm (blue) or ΛL = 18 µm (green). In (a) ΛH = 0.5 mm for ΛL = 3 µm, ΛH = 4.5 mm for ΛL = 9 µm and ΛH = 18 mm for ΛL = 18 µm. In (b) ΛH = 0.25 mm for ΛL = 3 µm, ΛH = 2.25 mm for ΛL = 9 µm and ΛH = 9 mm for ΛL = 18 µm. In (c) ΛH = 0.1(6) mm for ΛL = 3 µm, ΛH = 1.5 mm for ΛL = 9 µm and ΛH = 6 mm for ΛL = 18 µm.
Fig. 6
Fig. 6 In (a) we present the loss spectra for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 4π mm−1 and ΛL = 3µm calculated using dispersive nM (red) and nM = 1.45 (blue). Dashed lines show the loss of the fundamental modes in untwisted fibers. In (b) we present the imaginary part of neff versus normalized wavelength for HE1,1 (dotted) and HE1,1+ (solid) core modes in twisted MOF with A = 8π mm−1 and ΛL = 3µm (red) or by A = 4π mm−1 and ΛL = 6µm (blue) calculated using dispersive nM.

Equations (25)

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λ res = 2π l n SM ρ 2 A,
x i' = x i Q ,
ε i'j' = | det( Λ i' i ) | 1 Λ i' i Λ j' j ε ij ,
μ i'j' = | det( Λ i' i ) | 1 Λ i' i Λ j' j μ ij ,
Λ i' i = δ x i' δ x i .
ε' ε = ε r ' ε r = μ' μ = μ r ' μ r =Q.
×( 1 μ r ( r ) × E ( r ) )= ( 2π λ ) 2 ε r ( r ) E ( r ),
×( 1 Q μ r ( r ) × E ( r ) )= ( 2π λ' ) 2 Q ε r ( r ) E ( r ).
×( 1 μ r ( r ) × E ( r ) )= ( 2π Q 1 λ' ) 2 ε r ( r ) E ( r ),
λ'=Qλ.
n ˜ eff ( λ,d, Λ L ) n mat ( λ )=const = n ˜ eff ( λ/ Λ L ,d/ Λ L ),
n ˜ eff ( λ,d, Λ L , Λ H ) n mat ( λ )=const = n ˜ eff ( λ/ Λ L ,d/ Λ L , Λ H / Λ L ),
L= 20 ln10 2π λ Im( n ˜ eff )[ dB m ].
n ' eff = n eff +J λ Λ H ,
J ( λ,d, Λ L , Λ H ) n mat ( λ )=const =J( λ/ Λ L ,d/ Λ L , Λ H / Λ L )
n ' eff ( λ,d, Λ L , Λ H ) n mat ( λ )=const =n ' eff ( λ/ Λ L ,d/ Λ L , Λ H / Λ L ),
n eff n M ( λ Λ L ) 2 q,
n ' eff = n eff +( λ Λ L )J Λ L Λ H n M ( λ Λ L ) 2 q+( λ Λ L )J Λ L Λ H
Δn ' eff =n ' eff,1 n ' eff,2 ( λ Λ L ) 2 ( q 2 q 1 )( λ Λ L )( J 2 J 1 ) Λ L Λ H
λ res Λ L =2 ΔJ Δq Λ L Λ H .
λ res = 1 π ΔJ Δq Λ L 2 A.
n M ( λ )= 1+ i=1 3 A i λ 2 λ 2 Z i 2 ,
α= sin 1 ( Aρ 1+ A 2 ρ 2 )Aρ
αA Λ L ,
α A Λ L 2 =const 1 Λ L ,

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