1. INTRODUCTION

A chiral fiber grating (CFG) is an attractive candidate for controlling various spatial states of light, such as polarization, guided mode, and optical angular momentum (OAM). From the first report of a circular polarized light filtering behavior in CFG [1] with birefringent core, many experimental works have been reported, such as polarization-dependent filtering in single- and double-helix fiber [2]. To explain complex polarization behavior in the CFGs, theoretical studies have also been conducted [3,4]. In Ref. [4], the guided modes in elliptical core twisted fibers (double-helix) were intensively studied, and the coupling between core and cladding modes affects the transmission characteristics and the selection rule of the coupling was proposed. In these studies, only the guided mode of the twisted fiber was investigated, and the longitudinal evolution of the spatial state of light has hardly been discussed.

Recently, helically twisted photonic crystal fibers (PCFs) were proposed and fabricated [5] that have periodic dips in the transmission spectra originating from the coupling between core and cladding modes with OAM. After this report, many unique features of twisted PCF were discovered, such as an optical activity and a circular dichroism [68" target="_self" style="display: inline;">–8], higher-order OAM mode generation [9] at a wavelength near the dip, and a circular-polarized mode filtering in off-axis core twisted PCFs [10]. In PCFs, it is easy to incorporate the linear birefringence by changing the size of a part of air holes. The magnitude of the linear birefringence can also be easily tuned with the size of air holes. Therefore, twisted birefringent PCFs (TB-PCFs) can be viewed as a new kind of CFG and a good platform for controlling the spatial state of light.

In Ref. [11], a preliminary work on TB-PCF was presented. Formulating the light propagation in single-mode TB-PCF with a periodically inverted twisting based on a transfer-matrix approach, the theoretical analysis of the polarization state, group velocity dispersion, and adiabatic connection to usual fiber were demonstrated. However, important physical phenomena in twisted fibers, such as circular dichroism, discovered by full-vector modeling methods [68" target="_self" style="display: inline;">–8], were not demonstrated. Also, a uniform TB-PCF was not treated. Furthermore, the formulation was done with the two modes (x and y modes), and OAM generation with the topological charge >0, which is an important topic for twisted fibers, was not treated.

In this paper, the polarization (including OAM) state of TB-PCFs is thoroughly investigated with the full-vector analysis method. It is shown that a circular polarization (CP) component (S3 of the Stokes parameters) is periodically excited [12] in uniform TB-PCFs when usual linearly polarized (LP) modes of a PCF are launched. The periodic excitation is interpreted as an effect of a geometric phase [13] in TB-PCFs. The S3 excitation is larger for larger linear birefringence for a fixed twisting rate. If the linear birefringence is large enough, a CP filtering behavior can be seen in addition to the S3 excitation. From the analytical consideration of the sign of the geometric phase, the method for efficiently accumulating the phase difference between two CPs is discovered. Based on this idea, TB-PCFs with periodical inversion of the twisting are analyzed to generate an arbitrary polarization state on the Poincaré sphere. Next, the polarization state of the multimode TB-PCFs is investigated for the first time. An OAM state generation in multimode TB-PCFs is shown for higher-order LP mode input. By observing a far-field interference pattern (FFIP) from TB-PCF mixed with LP01 mode, a vortex associated with the OAM state can be seen. Similar to the single-mode case, with the periodical twisting inversion, efficient OAM generation is possible. These phenomena are analyzed by the recently developed rigorous full-vector beam propagation method (BPM) specially formulated for helicoidal waveguides [1416" target="_self" style="display: inline;">–16], which can explain the experimental transmission spectra and polarization states of helically twisted PCFs, both qualitatively and quantitatively. By using the helical BPM, it is possible to track the longitudinal evolution of the polarization state in the fiber with an arbitrary input condition by considering important physical phenomena, such as CP filtering, which is difficult for conventional analysis methods based on guided mode analysis. The results shown in this paper make the understanding of optical physics in twisted waveguides deeper and are useful for fiber-based polarization and OAM controlling devices [17–20" target="_self" style="display: inline;">–20], since it adds new degrees of freedom to control the spatial state of light.

2. BPM FOR HELICOIDAL WAVEGUIDES

In the theoretical analysis of helically twisted PCFs, a guided mode analysis based on the finite-element method (FEM) formulated for helical systems [2123" target="_self" style="display: inline;">–23] has been used. In this FEM, the concept of transformation optics [24] is used for the formulation, namely, the helically twisted structure is replaced with a straight waveguide with equivalent permittivity, ε, and permeability, μ, tensors. For the twisted waveguides, left CP and right CP (LCP and RCP) modes are obtained from this guided mode analysis and much information can be extracted. However, it is difficult to treat longitudinal evolution of the spatial state of light in a waveguide with arbitrary input conditions. For example, in most experiments, the input light seems not to be in the CP mode, but in the LP mode. Therefore, a BPM analysis is more suitable for investigating the longitudinal behavior of the spatial state of light in the twisted waveguides. Here, we briefly summarize the BPM for helicoidal waveguides.

We consider a helically TB-PCF, shown in Fig. 1. The origin is at the center of the fiber. The cladding is composed of hexagonally arranged air holes with diameter d and pitch Λ. There are six rings of air holes, and there is no air hole at the center of the array that forms the core. The diameters of two side holes adjacent to the core are dside. If d=dside, there is no linear birefringence. The computational window is a circle, and the outer region is a cylindrical perfectly matched layer (CPML) region with a thickness dPML. The fiber is helically twisted along the z direction with a twisting rate of α [rad/m]. The propagation direction is z, and xy is the transverse plane. The transformation between helical and modeling coordinates is given by x′=xcos(αz)+ysin(αz),(1a)y′=−xsin(αz)+ycos(αz),(1b)z′=z.(1c)Here, the notations x, y, and z denote the modeling coordinates, in which the numerical discretization is made, and x′, y′, z′ are the original coordinates. Due to the equivalence under the coordinate transformation of Maxwell’s equations, the following wave equation is satisfied for both coordinates: where E is the electric field vector, εr and μr are the relative permittivity and permeability tensors, and k0 is the free-space wavenumber. In the modeling coordinate, εr and μr are modified due to the coordinate transformation, and are given by Here, we assume an ordinary optical isotropic medium with a refractive index of n and a relative permeability of 1 in the original coordinate. Tmetric is a metric tensor and is given by where J is the Jacobian matrix between the helical and Cartesian systems. By using these tensors given by Eqs. (3) and (4), we can treat the twisted PCF as a simple straight waveguide. We consider a propagating field with reference index n0 in the positive z direction as By substituting Eq. (6) into Eq. (2) and discretizing the waveguide cross section with full-vector FEM, we can obtain marching equations used for BPM. Detailed formulation can be found in Refs. [14,15] together with qualitative and quantitative comparisons with experimental transmission spectra.

Fig. 1. Schematic of (left) three-dimensional structure and (right) cross section of TB-PCF. The number of rings in the three-dimensional sketch is reduced for clarity. The simulation is carried out for the cross section shown in the right panel. The dashed lines are a square with the side length of 2Λ. The inside area is used to calculate the Poynting vector in the core.

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3. TRANSMISSION CHARACTERISTICS OF TB-PCF

3.1 A. Basic Characteristics

We consider TB-PCF, shown in Fig. 1, with d=1.1μm and Λ=2.9μm. We added a CPML region with the thickness of dPML=2μm outside the analysis region (a circle with the radius of 20 μm). The air hole at the center of the fiber is missing to form the core, and diameters of the two holes adjacent to the core in the x direction are changed to dside. The refractive index of silica is calculated by the Sellmeier formula described in Ref. [25]. The top left panel of Fig. 2 shows the effective refractive index (neff) of PCF without twisting (α=0) as a function of dside. In this paper, the wavelength of the light is λ=1.2μm. The neff of the y-polarized mode (neff,y) is larger than that of the x-polarized mode (neff,x) because the air hole in the x direction is large. In the same figure, the linear birefringence of the PCF as a function of dside is shown. The linear birefringence is given by For larger values of dside, the absolute value of B becomes larger, and it is larger than 2×10−4 for dside>2μm. The top right panel of Fig. 2 shows a polarization beat length (Lpol) and polarization twisting rate (αpol) as functions of dside. Lpol and αpol are given by The effect of αpol and the twisting rate of the fiber α will be discussed later.

Fig. 2. (Top left) neff and B and (top right) Lpol and αpol as functions of dside, and (bottom) neff as a function of the twisting rate.

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The bottom panel of Fig. 2 shows neff of the fundamental modes of PCF as a function of α for different values of dside calculated by helical FEM [2123" target="_self" style="display: inline;">–23]. The circular birefringence [6] can be seen as a split of neff to upper and lower branches. One corresponds to RCP and the other corresponds to LCP. The linear birefringence affects the modes much more for lower values of α.

Fig. 3. Optical power in the core of TB-PCFs as a function of propagation distance for different values of dside. Input modes are (left) LCP and (right) RCP modes. α=0.006rad/μm.

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3.2 B. Polarization States in TB-PCF

Figure 3 shows the optical power in the core of TB-PCFs as a function of propagation distance for dside=1.5, 2, and 2.5 μm. The step size in the BPM simulation is 10 μm. The twisting rate is α=0.006rad/μm. Here, the positive value of α corresponds to counterclockwise rotation from the viewpoint of +z. The input light is LCP (Fig. 3, left) or RCP (Fig. 3, right) and obtained by summing orthogonal fundamental modes with a 90° phase difference, namely, where Ex and Ey correspond to the x- and y-polarized fundamental modes of non-twisted PCF calculated by full-vector FEM [26]. The plus sign is for RCP and the minus sign is for LCP. Here, the optical power in the core is defined as the Poynting vector surrounded by the dashed line in the right panel of Fig. 1 (square with the side length of 2Λ). For dside<2.0μm (small linear birefringence), the loss difference between LCP and RCP is <0.1dB after 30,000 μm propagation. For large linear birefringence with dside=2.5μm, the loss of the LCP mode is larger than that of the RCP mode, and the difference is large (1.7 dB after 30 mm propagation, or 57 dB/m), showing the emergence of circular dichroism [7] due to the twisting. The left panel of Fig. 4 shows Stokes parameters S1, S2, and S3 (normalized by S0) calculated by BPM as a function of the propagation distance for TB-PCF with dside=2.0μm and α=0.006rad/μm. The x-polarized fundamental mode (Ex mode) of non-twisted PCF is launched (Ex input). Stokes parameters [27] are evaluated by S0=(∫|Ex|dS)2+(∫|Ey|dS)2,(11a)S1=(∫|Ex|dS)2−(∫|Ey|dS)2,(11b)S2=2∫|Ex|dS∫|Ey|dScos(ϕy−ϕx),(11c)S3=2∫|Ex|dS∫|Ey|dSsin(ϕy−ϕx),(11d)where ϕy−ϕx is the phase difference between Ey and Ex at the center of the core. The integration is done over the cross section of the fiber. In this paper, S1, S2, and S3 are normalized by S0 at each propagation step. Note that these Stokes parameters can be used for the LP01 mode only, which has the maximum field at the center of the core. From Fig. 4, there are two distinct features in the Stokes parameters. First, S1 and S2 are periodically exchanged. The period corresponds to the period of fiber twisting, and the phenomenon can be explained as follows. At the start, S1=1 because Ex mode is input. After a 90° rotation, the polarization becomes y polarized (S1=−1) and after a 180° rotation, the polarization becomes x polarized again. This is so-called optical activity. Second, S3 is periodically excited. After a 90° rotation, the absolute value of S3 is at a maximum, and after 180° rotation, the value becomes 0 again. The right panel of Fig. 4 shows S3 as a function of propagation distance for PCFs with dside=1.1, 1.5, and 2.0 μm as well as α=0.006rad/μm. For dside=1.1μm, there are no excitations of S3, since there is no linear birefringence. For dside=1.5 and 2.0 μm, the periodic excitation of S3 can be seen, and the magnitude of excitation is larger for large linear birefringence. Since it is difficult to explain this complex polarization evolution (the periodic excitation of S3) with only the BPM results, we formulate a simple analytical beam propagation model based on a weak guiding approximation, which is summarized in the Appendix A. Key results are summarized as follows.

Fig. 4. (Left) Stokes parameters of TB-PCF for dside=2.0μm and α=0.006rad/μm and (right) S3 of TB-PCF for different values of dside calculated by BPM.

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We consider a twisted birefringent waveguide shown in Fig. 16 in the Appendix A. The waveguide is twisted along the z direction, and an azimuthal rotation angle is θ=αz. We assume that the reference indices for x- and y-polarized modes are nx and ny when θ=0. From the analytical results of the propagation equation for x- and y-polarized modes, two basis transformations are performed, namely, xy to eo basis and eo to RCP-LCP basis. Finally, we obtain the propagation equation as where ΨR and ΨL are complex fields of RCP and LCP components. Other symbols are defined in the Appendix A. Here, an additional phase given by e2jθ with the opposite sign in nondiagonal terms is called the “geometric phase.” By using these equations, the evolution of input light (CP or LP modes) in the rotated frame can be calculated easily. If Δz is sufficiently small, the beam propagation in the twisted waveguides can be treated by concatenating two segments, θ and θ+Δθ.

The left and right panels of Fig. 5 show the same data as in Fig. 4, but calculated by the analytical method (in the right panel of Fig. 5, the results for dside=2.5μm are added). From the left panel of Fig. 5, it is shown that the optical activity (periodic exchange of S1 and S2) and the periodic excitation of S3 can be reproduced. The way to understand the phenomena is the term of e2jθ contained in Eq. (13). If the value of θ=0, S3 excitation disappears, and therefore, the physical reason for the periodic excitation of S3 can be attributed to the geometric phase [13], existing in twisted waveguides. From Fig. 5, the magnitudes of excitation of S3 are consistent with BPM results and are larger for larger linear birefringence. However, the behavior is different from the analytical results for larger values of B. Figure 6 shows S3 as a function of propagation distance for PCF with dside=2.5μm with α=0.006rad/μm calculated by the analytical method and BPM. The polarization behavior becomes more complex compared with the analytical results shown in Fig. 5. Negative S3 (LCP) is first excited periodically as in the analytical results; however, around 5000 μm, the S3 leans to the positive (RCP) side. The effect can be explained by the circular dichroism shown in Fig. 3. Since the loss of LCP is larger than that of RCP in TB-PCF with dside=2.5μm, the LCP component contained in Ex mode is lost during propagation (the LP mode can be expressed by the sum of LCP and RCP modes). Due to the loss difference, the S3 gradually approaches the RCP side for long-distance propagation.

Fig. 5. (Left) Stokes parameters of TB-PCF for dside=2.0μm and α=0.006rad/μm and (right) S3 of TB-PCF for different values of dside calculated by the analytical method.

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Fig. 6. S3 as a function of propagation distance for dside=2.5μm with α=0.006rad/μm.

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Figure 7 shows S3 as a function of propagation distance for PCF with dside=1.5, 2.0, and 2.5 μm with α=0.006rad/μm for Ey mode input. For low linear birefringence (dside=1.5 and 2.0 μm), completely opposite polarization is excited compared with the Ex mode input. For large linear birefringence (dside=2.5μm), the polarization state rapidly approaches the RCP side because the LCP component is not excited for Ey mode input and the LCP components contained in Ey mode are lost due to the circular dichroism during propagation.

Fig. 7. S3 as a function of propagation distance for α=0.006rad/μm and different values of dside. Ey mode is launched.

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Summarizing the results so far, the polarization behavior in the TB-PCFs is very complex, depending on the magnitude of the linear birefringence. For the PCF with small linear birefringence, there are two features in their polarization state, namely, optical activity and periodical excitation of S3, originating from the geometric phase. For the PCF with large linear birefringence, in addition to the above features, the circular dichroism makes the polarization behavior more complex.

3.3 C. Effect of Twisting Rate

The left panel of Fig. 8 shows the optical power of TB-PCFs as a function of propagation distance for dside=2.5μm. The twisting rate is α=0.0024, 0.006, and 0.01 rad/μm. The input light is LCP or RCP mode. For a larger twisting rate, the circular dichroism is also larger, since, for a large twisting rate, many cladding modes are phase-matched to the core mode [28,29]. The right panel of Fig. 8 shows S3 as a function of propagation distance for PCF with dside=2.5μm for Ex mode input. The magnitude of periodical excitation of S3 is maximum for α=0.0024rad/μm and is smaller for the larger twisting rate. This is because αpol≃α=0.0024rad/μm, as shown in Fig. 2. For α=0.01rad/μm, the polarization state approaches the RCP side due to the large circular dichroism. From these results, the periodical excitation of S3 becomes large when the twisting rate of the fiber is similar to that of the polarization twisting rate. For the large twisting rate, due to the phase matching to the cladding mode, the circular dichroism becomes large and CP filtering behavior can be seen.

Fig. 8. (Left) Optical power in the core of TB-PCFs and (right) S3 as a function of propagation distance for different values of α. dside=2.5μm.

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3.4 D. Wavelength Dependence

The left panel of Fig. 9 shows the transmission spectra of TB-PCFs as a function of propagation distance for different values of dside and the twisting rate of α=0.006rad/μm. The input light is in the Ex mode, and the fiber length is 30,000 μm. For dside<2.0μm, there are several decibel losses in the shown wavelength range. For dside=2.5μm, the loss increases due to the circular dichroism, especially for the long wavelength side. The center panel of Fig. 9 shows S3 as a function of propagation distance for different wavelengths, and the right panel is the enlarged view. For the long wavelength side, the excitation of S3 in one period is larger. This is because αpol is close to α for the long wavelength side. For long-distance propagation, S3 leans toward the RCP side due to the large loss of the LCP mode. The slope for the RCP mode is larger for the long wavelength due to the larger LCP loss.

Fig. 9. (Left) Transmission spectra of TB-PCFs for α=0.006rad/μm and different values of dside. Ex mode is launched, and the fiber length is 30,000 μm. (Center) S3 as a function of propagation distance for different wavelength and (right) enlarged view of the center panel.

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3.5 E. Periodic Inversion of Twisting

From the results in Fig. 4, when the Ex mode is launched, the negative S3 is excited, and the absolute value of the S3 component is at a maximum at 90° rotation and then returned to zero at 180° rotation. Therefore, the accumulated phase shift between 0° and 90° is canceled out between 90° and 180° rotation. If the sign of the geometric phase can be inverted between 90° and 180° rotation, the accumulated phase shift is not canceled out and continuously accumulated, leading to a larger S3 excitation. The sign inversion of the geometric phase can be achieved by changing the direction of twisting (the sign of α) periodically, as shown in the inset of Fig. 10, where Λtwist=2π/α. Although the fabrication of the structure is not easy, it is interesting to see what happens if the structure can be fabricated. According to Ref. [5], a permanent twist can be produced with proper processing. Therefore, multiple splicing of inverted sections may be one of the fabrication methods. Also, the length of each region cannot be Λtwist/4. Since the excitation of S3 is periodic, the length should be NΛtwist+Λtwist/4 (N is an integer), and the length of each section can be lengthened.

Fig. 10. Stokes parameters of periodically inverted TB-PCF calculated by (left) the analytical method and (right) BPM. dside=2.0μm and α=0.0024rad/μm. Dashed vertical lines show the distance of integer multiples of Λtwist/4.

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The left panel of Fig. 10 shows Stokes parameters calculated by the analytical method for TB-PCF with dside=2.0μm and α=0.0024rad/μm. The Ex mode is launched, and the sign of α is periodically inverted at each 90° rotation (Λtwist/4=655μm). Vertical dashed lines are the distance corresponding to integer multiples of Λtwist/4. As shown in the figure, the S3 component is constructively accumulated at each Λtwist/4, and the generation of a perfect CP mode is possible. The right panel of Fig. 10 shows the same data, but calculated with BPM. Both results agree very well, and the correctness of the analytical results is confirmed by numerical full-vector simulation.

The left panel of Fig. 11 shows the Poincaré sphere plot of the polarization state of constantly twisted TB-PCF with dside=2.0μm and α=0.0024rad/μm. The Ex mode is launched (S1=1), and the length of the fiber is 2000 μm. In this case, S3 is periodically excited as in Fig. 4, and the trajectory on the Poincaré sphere is elliptical. The center panel of Fig. 11 shows the same plot, but with periodic α inversion. At the inversion point, the polarization state trajectory is not continuous, and at 2000 μm, an almost perfect LCP mode is reached (please see Fig. 10). The right panel of Fig. 11 shows the trajectory of the polarization state for α=0.006rad/μm (Λtwist/4=261μm). Since the length of one segment is short, the trajectory is more complex. At 2000 μm, an almost perfect LCP mode is obtained, as in the case of α=0.0024rad/μm. From these results, periodic inversion of the twisting adds a novel degree of freedom to control and generate the arbitrary polarization state.

Fig. 11. Poincaré sphere plots of the polarization state in TB-PCF with dside=2.0μm. (Left) α=0.0024rad/μm with constant twisting, (center) α=0.0024rad/μm with periodically inverted twisting, and (right) α=0.006rad/μm with periodically inverted twisting.

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In the above examples, the value of Λtwist/4 is some hundreds of micrometers, which is difficult to control if the fiber is cut and spliced. To show the accumulation of the geometric phase for long Λtwist, Fig. 12 shows Stokes parameters calculated by BPM for periodically inverted TB-PCF with dside=1.25μm and α=0.000314rad/μm (Λtwist=20,000μm). The S3 component is constructively accumulated at each Λtwist/4, as in the case of Fig. 10.

Fig. 12. Stokes parameters of periodically inverted TB-PCF calculated by BPM. dside=1.25μm and α=0.000314rad/μm. Dashed vertical lines show the distance of integer multiples of Λtwist/4.

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4. OAM STATE GENERATION IN MULTIMODE TB-PCF

From the results shown in Section 3.E, it is possible to generate an arbitrary polarization state in single-mode TB-PCFs by using periodic twisting inversion. The results lead to a natural extension to a multimode case. For higher-order LP mode, complex polarization evolution in TB-PCF may lead to generating an OAM state with the topological charge ≥1. Here, the generation of the OAM state is shown by launching the usual LP mode to multimode TB-PCF.

The left panel of Fig. 13 shows the cross section of multimode TB-PCF. Seven air holes at the center of the fiber are missing to form a multimode core, and the diameters of two holes are changed to dside. Here, we consider multimode TB-PCF with d=1.1μm, Λ=2.9μm, and dside=2.0μm. The twisting rate α=0.00314rad/μm. The right panels of Fig. 13 show the intensity distributions of the Ex modes (LP01, LP11a, LP11b, LP21a, and LP21b) of the untwisted PCF. There are also Ey modes with intensity patterns similar to Ex modes.

Fig. 13. Schematic of a multimode TB-PCF and its NFPs.

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4.4 A. Constant Twisting

Here, the polarization evolution in a uniform multimode TB-PCF is investigated. The blue solid line in the left panel of Fig. 14 shows the overlap between input and BPM-propagated fields as a function of propagation distance for LP11bx input. Here, the overlap power is defined as where Ei is the electric field calculated by BPM at the ith step. Hin is the magnetic field of input light. The value of overlap oscillates with an approximately 1000 μm period. Since the 180° rotation period is 1000 μm, the overlap is at a minimum at 90° rotation (500 μm) and increases again at 180° rotation. The periodic nature of the overlap is collapsed for increased propagation distance. This is due to the phase retardation originating from the geometric phase in the twisted PCF. The black dashed line in Fig. 14 shows the total power in the cross section calculated from the Poynting vector. The loss is small in the shown length scale.

Fig. 14. Overlap powers (blue solid lines) and total Poynting vectors (black dashed lines) as functions of propagation distance for (left) LP11bx and (right) LP21bx mode input. The inset shows mixed FFIP at the distance marked by red circles.

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At some propagation distances, FFIPs from TB-PCF mixed with the LP01x mode are shown. The FFIP is calculated by [30] where n is the effective index of the free space (in this case 1), ϕNFP is the near-field pattern (NFP), and R=(x−x0)2+(y−y0)2. The FFIP of the output light of TB-PCF with 1000 μm free-space propagation and the FFIP of input LP mode with 2000 μm free-space propagation are added. This is the mimic of the typical experiment for OAM observation. The FFIP evolution is very complex, and at some distances, a clear single vortex associated with the OAM state with the topological charge of 1 can be seen. There are both left- and right-handed vortices. This is probably because the propagation constants of all four LP11 modes are different, and the relative phase differences between these modes are accumulated and not constant. Therefore, at some point the vortex is left-handed, and at a different point, it is right-handed. It should be noted that to confirm the value of the OAM, numerical evaluation of the mode field is effective [31]. However, in this paper, the “propagated field” containing the components of multiple guided and radiated modes is evaluated and is not the true mode field. Therefore, the numerical evaluation of the OAM for the BPM field is not easy to use to determine the OAM order.

The blue solid line in the right panel of Fig. 14 shows the overlap between input and BPM-propagated fields as a function of propagation distance for LP21bx input. FFIPs at some distances are also shown. The evolution is more complex than that of LP11bx input, and at some distances, a double vortex associated with the OAM state with the topological charge of 2 can be seen. The black dashed line in the right panel of Fig. 14 shows the total power in the cross section calculated from the Poynting vector. The loss is larger compared with the LP11 mode input.

4.5 B. Periodic Inversion of Twisting

Here, the polarization evolution in TB-PCF with periodic twisting inversion is investigated. The green solid line in the top panel of Fig. 15 shows the overlap between input and BPM-propagated fields as a function of propagation distance for LP11bx input of periodically inverted TB-PCF, together with NFP and FFIP at 1000 and 2350 μm. The period of twisting inversion is 500 μm (90° rotation). The behavior of the overlap value is very complex compared with the constant twisting case (blue dashed line). At 1000 and 2350 μm, doughnut-shaped NFPs are obtained, and left- and right-handed vortex FFIPs are seen. In terms of the value of overlap, it oscillates with an approximately 1000 μm period (π rotation) for a constant twisting fiber. However, the period is slightly shifted from the integer multiples of 500 μm for long-distance propagation, and the value of the overlap becomes small. Since the propagation constants of all four LP11 modes are different, the relative phase differences among these modes are accumulated. Therefore, the initial state is not recovered after one rotation, and the difference from the initial state is accumulated during the propagation, resulting in different periods and small overlap. The situation is more complicated in the case of periodic inversion of twisting.

Fig. 15. Overlap powers (green and blue) and total Poynting vectors (black) as functions of periodically inverted multimode TB-PCF for (top) LP11bx and (bottom) LP21bx mode input. The right panels show NFP and mixed FFIP at the distances marked by red circles.

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The green solid and blue dashed lines in the bottom panel of Fig. 15 show the same one, but for LP21bx input. The period of twisting inversion is 250 μm (45° rotation). The overlap value seems to have no more periodicity. At 450 and 1800 μm, doughnut-shaped NFPs and double-vortex FFIPs are seen. The black dashed lines in Fig. 15 show the total powers in the cross section calculated from the Poynting vector. As in Fig. 14, the loss is larger for LP21 mode input compared with LP11 mode input in the shown length range. In addition to these losses, there are intrinsic losses in the real fiber, and they deteriorate the characteristics further.

From these results, although OAM states can be generated both in constant and periodically inverted TB-PCFs, a periodically inverted TB-PCF has an additional degree of freedom for controlling the spatial state of light, and there is a possibility that the length can be shortened for generating the OAM state.

5. CONCLUSION

The spatial state of the light (polarization and OAM) in TB-PCFs is theoretically analyzed in terms of the effect of linear birefringence and twisting rate. Recently developed rigorous BPM analysis revealed that the polarization state of light in the TB-PCF is very complex, and some features can be extracted from the analysis. In the TB-PCF with small linear birefringence, the optical activity and the periodic S3 excitation can be seen. These two features can be explained with the concept of geometric phase existing in the twisted waveguides. Furthermore, a periodically inverted TB-PCF is proposed to control the polarization. By accumulating the phase rotation originating from the geometric phase, an arbitrary polarization state can be generated. In the multimode TB-PCF, the technique may be useful to effectively generate an OAM state. These results indicate that the TB-PCF adds a new degree of freedom to control the spatial state of light and is useful for fiber-based light-controlling devices.

6 Acknowledgment

Acknowledgment. This work was partly supported by the Yazaki Memorial Foundation for Science and Technology.