1. Introduction

Long-period gratings (LPGs) are periodic structures written in optical fibers that generate resonant coupling between different spatial modes, especially between the fundamental mode and cladding modes or higher-order core modes. Various methods of fabrication of LPGs have been reported, consisting of periodic modification of the refractive index by focused UV laser beams [1–3] or femtosecond pulses [4], modification of the fiber structure by a CO₂ laser beam [5,6] electric arc [7], inelastic twist [8,9], and microbends [10–13]. A special type of LPGs is a rocking filter (RF) that resonantly couples the fundamental modes of orthogonal polarizations because of the periodic rotation of the fiber polarization axes by a small angle [14]. The RFs were so far demonstrated in birefringent single modes fibers of different types, including birefringent fibers with stress-applying elements [15], elliptical-core fibers [16,17], telecommunication fibers with twist-induced birefringence [18], and birefringent microstructured fibers [14,19–21]. Various methods have been used for the fabrication of RFs, such as preform rotation [15], irradiation by UV laser beams [16,17], CO₂ laser beams [14,19,20], and fusion arc splicers [21].

Different types of LPGs, including rocking filters, have attracted significant attention because of their numerous applications, including filtering [1], sensing [2,19–21], multiplexing and demultiplexing [12,22], and the generation of phase or polarization vortex beams [5,6,9]. Polarization vortex beams (often called cylindrical vector beams – CVBs) are obtained by resonant coupling between the fundamental and higher-order modes involving acousto-optic mode converters [23], fiber couplers [24], CO₂ laser-written LPGs [25,26], or excitation of a few-mode fiber with Laguerre-Gaussian beams [27].

In this study, we demonstrate, for the first time to our knowledge, the RF for resonant coupling of different LP₁₁ modes propagating in a two-mode birefringent PANDA fiber. We show that such filters can resonantly couple the LP₁₁ modes of the same spatial structure and orthogonal polarizations as well as the modes of orthogonal spatial structure and the same polarizations. Numerical simulations also reveal that because of the small overlap coefficients at twisting points, the resonant couplings between the LP₁₁ modes simultaneously orthogonal with respect to polarization and spatial distribution (doubly orthogonal LP₁₁ modes) are practically unattainable.

Moreover, we demonstrate experimentally that by employing such RFs, it is possible to obtain at the output of a two-mode birefringent PANDA fiber the TE₀₁, TM₀₁, and HE₂₁ beams or their coherent combinations, which can be easily tuned by the phase shift between respective pairs of LP₁₁ input modes. This is achieved by exciting a pair of LP₁₁ modes with the same spatial distributions and orthogonal polarizations at the fiber input using the recently proposed Wollaston prism-based method [28] and converting one of these modes using RF to the LP₁₁ mode of orthogonal spatial distribution. Therefore, we obtained at the fiber output a superposition of the doubly orthogonal LP₁₁ modes, which depending on the phase shift between them, produces pure TE₀₁, TM₀₁, or HE₂₁ beams or their coherent combinations. Consequently, we can generate all types of first-order cylindrical vector beams at the same wavelength using only one RF and a method for pure excitation of the LP₁₁ modes, which is considerably simpler and more universal compared to other methods of cylindrical vector beams generation. Such fiber-based tunable sources of CVBs, especially those with radial or tangential polarizations, may find many applications, including high-resolution microscopy [29], STED imaging [30], optical tweezers [31], mode division multiplexing for optical communication [11,32], electron acceleration [33], and material processing [34].

Finally, in the Appendix, we present a broadband interferometric method used for measuring the wavelength dependence of the difference in the effective indices between all pairs of LP₁₁ modes, which is necessary to properly design the RF and control the spectral positions of the respective resonances.

2. Principle of operation of LP₁₁ rocking filter

For the fabrication of the LP₁₁ rocking filters, we used a commercially available polarization-maintaining PANDA fiber (Nufern PM-1300B-XP), whose cross section captured in a scanning electron microscope is shown in Fig. 1. The estimated cutoff wavelength for this fiber is 1.08 µm and the core size is 7.34 µm × 8.06 µm.

 

Fig. 1. SEM image of PANDA type Nufern PM-1300B-XP fiber used for fabrication of the LP₁₁ rocking filters.

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The crucial parameters required to control the spectral positions of the respective resonances are the differences in the effective refractive indices between the LP₁₁ modes. They were measured over a wide spectral range using a special broadband interferometric method, as described in detail in the Appendix. The rocking filters (RFs) that we considered were fabricated by periodically repeated rotations of a two-mode birefringent PANDA fiber by a constant angle in the same direction (Fig. 2(a)). Such periodic twists enable the LP₁₁ modes in successive fiber segments to partially overlap and seed resonant coupling between different pairs of LP₁₁ modes (Fig. 2(b)), including the polarization and spatial modes.

 

Fig. 2. (a) Principle of operation of the rocking filter for conversion of LP₁₁ modes. Purple and blue arrows indicate directions of slow and fast axes at inputs to successive segments of the RF, and coupling points are marked with the dashed lines. (b) Intensity distributions and polarization structure of the LP₁₁ eigenmodes of a birefringent fiber. The green/red arrows mark pairs of LP₁₁ modes that are strongly/weekly coupled providing the phase matching conditions are fulfilled.

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We assume that the fiber is divided into N − 1 identical sections, with the polarization axes rotated with respect to each other by a constant angle Δα and only the forward-propagating LP₁₁ modes are considered. In each straight fiber section, the LP₁₁ modes propagate independently with different propagation constants, whereas at discontinuous interfaces, they couple to each other according to the overlap integrals.

In the fiber section rotated by an angle Δα clockwise relative to the preceding section, the electric field amplitudes (provided that longitudinal components are disregarded) in respective LP₁₁’ modes can be expressed in the local Cartesian reference system (x’,y’) as follows:

The above relations can be expressed in a more compact way using matrix notation:

Consequently, the amplitudes of the linearly polarized fields in the respective LP₁₁ modes after passing through the rocking filter composed of N − 1 twisted fiber segments can be expressed as follows in the local reference system (x’, y’):

The differences in the effective indices for the LP₁₁ modes in the PANDA PM-1300B-XP fiber used in the experiment and simulations (Fig. 3(a)) were measured over a wide spectral range using the interferometric method described in the Appendix. In the case of rocking filters with successive twists in the same direction, resonant coupling of q-th order between respective LP₁₁ modes may occur for wavelengths fulfilling the phase-matching conditions:

 

Fig. 3. (a) Spectral dependences of the differences in effective indices for selected combinations of the LP₁₁ modes used in the rocking filter simulations and fabrication and (b) spectral dependence of intiger multiples of beat length for all possible pairs of LP₁₁ modes. In (b) the dotted black lines represent three considered twist periods and the circles – the wavelengths at which the phase matching condition necessary for resonant couplings occurs.

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To examine numerically the couplings between different LP₁₁ modes in birefringent PANDA fiber with spectral characteristics of the intermodal beat lengths presented in Fig. 3, we employed the following structural parameters of the RF: twist angle Δα = 5°, number of twists N = 18 and three period lengths Λ₁ = 1.8 mm, Λ₂ = 4.3 mm, and Λ₃ = 5.3 mm, respectively. Because the total twist angle in such RF is equal to 90°, it results in nearly complete power transfer between the coupled LP₁₁ modes [35]. For the selected twist periods, the phase-matching conditions between different pairs of LP₁₁ modes are fulfilled, thereby potentially allowing for resonant coupling between the LP₁₁ modes with the same spatial amplitude distributions but orthogonal polarizations, orthogonal spatial amplitude distributions but with the same polarizations, and doubly orthogonal LP₁₁ modes, that is, with orthogonal spatial amplitude distributions and orthogonal polarizations (such as LP₁₁^xo and LP₁₁^ye modes). In Fig. 3(b), the spectral positions of the expected resonances for the different twist periods are marked with circles.

The results of the simulations for the RFs of the three different periods are displayed in Fig. 4. Figure 4(a) shows the spectra in each mode for Λ₁ = 1.8 mm when LP₁₁^xo is excited and reveals a strong first-order resonant coupling between the modes LP₁₁^xo and LP₁₁^yo of orthogonal polarizations and the same spatial distributions at λ = 0.741 µm. This is the only possible coupling for LP₁₁^xo excitation predicted based on the phase-matching diagram shown in Fig. 3(b). For the RF with a twist period Λ₃ = 5.3 mm (Fig. 4(b)), we obtained two strong resonant couplings when the LP₁₁^xe mode was excited, that is, first-order coupling between different spatial modes LP₁₁^xe and LP₁₁^xo with the same polarization at λ = 0.852 µm and third-order coupling between orthogonal polarizations LP₁₁^xe and LP₁₁^ye at λ = 0.763 µm. According to the diagram in Fig. 3, the pair of doubly orthogonal modes (LP₁₁^xe – LP₁₁^yo) is phase-matched at λ = 0.705 µm; however, the coupling between these modes is considerably weak and barely visible in Fig. 4. For the RF with Λ₂ = 4.3 mm and LP₁₁^ye excitation (Fig. 4(c)), there is a visible strong first-order coupling between the LP₁₁^ye and LP₁₁^yo modes of orthogonal spatial distribution at λ = 0.753 µm, and second-order coupling between the modes LP₁₁^ye and LP₁₁^xe of orthogonal polarizations at λ = 0.940 µm. Similar to the previous case, for the doubly orthogonal LP₁₁^ye and LP₁₁^xo modes, despite the phase matching at λ = 0.852 µm, the coupling was weak and barely reached 10% efficiency. This effect is related to the different overlap coefficients for singly and doubly orthogonal modes, which according to Eq. (4) can be expressed as cos(Δα)sin(Δα) and sin²(Δα). Note that for the twist angle Δα = 5°, the overlap coefficient for doubly orthogonal modes is approximately 11.4 times smaller than that for singly orthogonal modes. Therefore, to obtain efficient coupling between doubly orthogonal LP₁₁ modes, a greater number of coupling points (greater total twist angle) is required than in the case of singly orthogonal modes.

 

Fig. 4. Spectral intensity distribution near resonant couplings between pairs of LP₁₁ modes of orthogonal polarizations or/and spatial distributions (a) Λ = 1.8 mm and LP₁₁^xo input mode, (b) Λ = 5.3 mm and LP₁₁^xe input mode, (c) Λ = 4.3 mm and LP₁₁^ye input mode.

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To illustrate this effect, we numerically examined the influence of the number of RF sections and total twist angle on the coupling efficiency for different pairs of LP₁₁ modes. The results of the simulations shown in Fig. 5(a) confirm that regardless of the twist step Δα, full energy conversion can be achieved for the couplings between modes of the same polarizations or spatial structures (singly orthogonal) when the total twist angle is equal to 90°. As shown in Fig. 5(b), to achieve the maximum coupling efficiency for doubly orthogonal modes, a greater total twist angle (over 1000°) is required, and off-resonance couplings are visible to other LP₁₁ modes that are not phase-matched. This indicates that the fabrication of an RF that effectively couples doubly orthogonal modes is impractical.

 

Fig. 5. Coupling efficiency versus total twist angle NΔα (Δα = 5°) for different resonant couplings: (a) between modes of the same polarization LP₁₁^xo and LP₁₁^xe for Λ = 5.3 mm at λ = 0.852 µm, (b) between modes of orthogonal polarizations and spatial distributions LP₁₁^xe and LP₁₁^yo for Λ = 5.3 mm at λ = 0.705 µm.

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3. Rocking filter fabrication and characterization

For the fabrication of the RFs, we used a Fujikura LZM-100 glass processing station with a CO₂ laser beam equipped with translation and rotation stages. To experimentally demonstrate the different types of resonances, we manufactured two RFs with Λ₁ = 1.8 mm and Λ₃ = 5.3 mm and nominal twist step Δα = 5°. Before obtaining the first coupling point, the fiber was pre-twisted by approximately 25° (over a length of approximately 13 cm) to induce an initial shear stress, which was partially released during the heating of the fiber by the focused CO₂ beam. Thereafter, the fiber was rotated by an additional 5° to preserve the initial pre-twist and ensure the repeatability of twisting at successive coupling points. According to the simulations, a full energy transfer between the singly orthogonal LP₁₁ modes occurred for NΔα = 90°. This result is obtained for discontinuous and identical twists at all coupling points, represented by the coupling matrix denoted by Eq. (4). However, the actual twists are continuous, with the experimentally determined length of the twisted section equal to approximately 300 µm, as shown in Fig. 6. The continuous transition between successive segments of the RF causes the effective coupling coefficients between the modes to be lower than those used in the simulations, as shown in Eq. (4), and therefore a total twist angle greater than 90° is required to obtain the maximum coupling between the phase-matched singly orthogonal modes. We experimentally verified that the maximum coupling efficiency in RFs fabricated using the described procedure is achieved for a total twist angle of 120−160°. For such an RF, the measured insertion loss is approximately 3 dB.

 

Fig. 6. Image of the twisted section of the PANDA fiber captured using a polarization microscope. The measured twist angle is 10 °±2°.

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We examined the performance of the fabricated RFs using the setup shown in Fig. 7. For pure excitation of the LP₁₁ modes, we applied a recently proposed method utilizing a Wollaston prism, a half-wave plate, and linear polarizers [28]. With this method, the input Gaussian light beam of linear polarization is focused on the fiber end-face into two light spots of linear polarization and opposite phases, which overlap well with the LP₁₁ modes. As detailed in [28], the method allows pure excitation of each LP₁₁ polarization mode as well as certain combinations of LP₁₁ modes. In Fig. 8, we show the spectra registered at the fiber output before RF fabrication for the purely excited LP₁₁^ye mode using OSA with and without a polarizer at the input and the same spectrum split by a diffraction grating and captured by the camera. The output end of the fiber was oriented such that the line of zero intensity in the monitored LP₁₁ mode was horizontal. The image captured by the camera proves the high modal purity of the first-order mode over the full spectral range. The measured polarization extinction ratio (PER) was approximately −23 dB in the spectral range of 0.7–1.0 µm. A similar PER was obtained for the other LP₁₁ input modes, which proves the high polarization purity of the Wollaston prism-based excitation method.

 

Fig. 7. Setup for characterization of the rocking filters in a PANDA fiber. SC – supercontinuum, MO – microscopic objectives, P – polarizers, WP – Wollaston prism, H – half-wave plate, IF – interference filter, DG – diffraction grating, OSA – optical spectrum analyzer.

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Fig. 8. (a) Spectra registered for the LP₁₁^ye input mode at the output of the PANDA fiber before fabrication of the rocking filter: without polarizer (yellow), with the polarizer P3 in transmission (blue) and extinction (red), and OSA noise (black, dashed), (b) the same spectrum split by a diffraction grating and captured by the camera.

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After the fabrication of the RFs, we conducted similar measurements of the output spectra with the transmission direction of the output polarizer P3 set parallel and orthogonal to the polarization azimuth of the excited LP₁₁ modes. Figure 9 shows the spectra for the RF of Λ₁ = 1.8 mm registered by the OSA and split by the diffraction grating for two input LP₁₁ modes with the polarizer P3 set in transmission and extinction. For the LP₁₁^xo input mode, we observed a single resonant coupling to the LP₁₁^yo mode of depth −13 dB at λ = 0.707 µm (corresponding to λ = 0.741 µm in the simulations), whereas for the LP₁₁^xe input mode, there is a clearly visible coupling to the LP₁₁^ye mode of depth −14 dB at λ = 0.740 µm (corresponding to λ = 0.777 µm in the simulations). These experimental results are in agreement with the simulations and confirm the high quality of the resonant couplings between LP₁₁ modes with the same spatial distributions and orthogonal polarizations.

 

Fig. 9. Transmission (blue) and extinction (red) spectra registered with OSA and captured by the camera after reflection from the diffraction grating for (a), (c) the LP₁₁^xo and (b), (d) LP₁₁^xe input modes for the rocking filter of Λ = 1.8 mm, N = 23, nominal twist angle Δα = 5°.

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In the next step, we experimentally investigated the coupling between modes of the same polarizations and orthogonal spatial distributions. Figure 10 shows the spectra and images of the output beam diffracted by the grating at the output of the RF of Λ₃ = 5.3 mm for the four input LP₁₁ modes and the output polarizer P3 set in transmission or extinction. To collect most of the light from the input mode and filter out the light from the modes of orthogonal spatial distribution, the light spot outcoming from the fiber was enlarged and shifted along one of its symmetry planes with respect to the OSA input aperture (Fig. 7). As predicted by the simulations, for the LP₁₁^ye or LP₁₁^yo input modes, only couplings to orthogonally polarized modes of the same spatial distributions were observed (Fig. 10(c),(d)). When the LP₁₁^xo mode was excited, three resonant couplings were identified, i.e., between the LP₁₁^xo and LP₁₁^yo modes of orthogonal polarizations of third and second order at λ = 0.714 µm and λ = 1.01 µm, respectively (corresponding to 0.728 µm and 1.034 µm in the simulations), and between the LP₁₁^xo and LP₁₁^xe modes of the same polarizations and orthogonal spatial distributions at λ = 0.848 µm (0.852 µm in the simulations). When the LP₁₁^xe mode was excited, in addition to the already identified spatial resonance with the LP₁₁^xo mode, the third-order polarization resonance with the LP₁₁^ye mode was visible at 0.748 µm (0.763 µm in the simulations).

 

Fig. 10. Transmission (blue) and extinction (red) spectra for (a), (e) LP₁₁^xo, (b), (f) LP₁₁^xe, (c), (g) LP₁₁^yo, (d), (h) LP₁₁^ye input modes registered with OSA at the output of the RF of Λ = 5.3 mm, N = 18 and captured by the camera after reflection from the diffraction grating at the first diffraction order with the use of the polarizer to visualize polarization couplings, and zero diffraction order with the use of bandpass interference filters to visualize the spatial couplings.

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The existence of various types of resonances is also confirmed by the spectra split by the diffraction grating and registered with the camera for the output polarizer P3 set in transmission and extinction (Fig. 10(e)–(h)). In the case of polarization resonances, the missing part of the spectrum in the excited mode appears in the orthogonal polarization mode and the same spatial distribution. However, spatial resonance manifests itself through a gradual transformation of the spectrum structure from a double line to a single line. The intensity modulation visible in each of the lines in the vicinity of the spatial resonance is caused by the interference of both spatial modes (excited and partially coupled) with the same polarization. The existence of spatial resonance at approximately 850 nm is clearly visible in the images of the zero-diffraction order registered with different bandpass interference filters.

The quality of higher-order resonances is susceptible to technologically unavoidable random deviations from the phase-matching condition in the successive RF sections [35]. This effect was visible in our experimental results. The third order polarization resonances appearing near 0.7–0.75 µm in Fig. 10(a)–(d) are only 8 dB deep, the second order polarization resonances appearing near 1.01 µm in Fig. 10(a),(c) are around 11 dB deep, while the first order polarization resonances visible in Fig. 9 near 0.7–0.75 µm are 14 dB deep. It is difficult to experimentally determine the quality of the first-order spatial resonance between the LP₁₁^xo and LP₁₁^xe modes, as shown in Fig. 10(a),(b). Therefore, we filtered the modes of orthogonal spatial distribution by placing the OSA input aperture off the output beam symmetry axes at the maximum intensity of the collected mode and the minimum intensity of the spatially orthogonal mode, as shown in Fig. 7. The depth of the spatial resonance between the LP₁₁^xo and LP₁₁^xe modes evaluated in this manner was at least 11 dB. The resonance depths for all experimentally observed couplings are gathered in Table 1.

Table 1. Measured characteristics of all experimentally observed resonant couplings in the RFs of different periods Λ.

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4. Coherent superposition of LP₁₁ modes of orthogonal polarization and spatial structure

The coherent superposition of two LP₁₁ modes of equal intensity with orthogonal polarization and spatial structure allows for the generation of radially or azimuthally polarized TE₀₁ or TM₀₁ modes, HE₂₁ modes [36], or CVBs of more complex polarization structures with elliptical (circular) polarization on diagonals, which depends on the phase difference between the input LP₁₁ modes. Because in a cylindrical core fiber, the LP₁₁ modes can be represented as the following superpositions of the TE₀₁, TM₀₁, and HE₂₁ modes:

The above equations show that for Δϕ = 0 or Δϕ = π at the fiber output, we obtain pure TE₀₁ or HE₂₁^odd modes when LP₁₁^xo and LP₁₁^ye are excited at the fiber input. Correspondingly, for the same phase shifts, pure TM₀₁ or HE₂₁^even modes were generated at the fiber output for the excited LP₁₁^xe and LP₁₁^yo modes. The superposition of doubly orthogonal LP₁₁ modes with other phase shifts Δϕ also produces CVBs of zero angular and spin momentum; however, with more complex spatial polarization distributions (with elliptical or circular polarization states on the diagonals of the reference system). To illustrate this effect, Fig. 11 shows the numerically calculated spatial distribution of the ellipticity and azimuth angles for the superposition of different pairs of LP₁₁ modes with phase differences of 0, π/2, π, and 3π/2.

 

Fig. 11. Calculated amplitude distributions as well as ellipticity and azimuth angles for coherent superposition of the LP₁₁ modes of the same intensity with orthogonal polarization and spatial structures.

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Although the generation of such exotic vector beams by the superposition of doubly orthogonal LP₁₁ modes seems to be an attractive approach, and has been demonstrated in free space [36], the simultaneous pure excitation of such modes in a birefringent fiber is difficult.

As discussed in [28], the excitation of doubly orthogonal LP₁₁ modes cannot be obtained using a Wollaston prism-based method. In addition, as explained in Section 3, obtaining two doubly orthogonal LP₁₁ modes by exciting only one of them at the fiber input and the RF-induced resonant partial coupling to the other mode is ineffective because of the small overlap coefficients between the doubly orthogonal modes. In the following paragraphs, we show that pure excitation of two doubly orthogonal LP₁₁ modes is possible by exploiting RF-driven effective coupling between the singly orthogonal LP₁₁ modes. For this purpose, we excite two LP₁₁ modes of orthogonal polarizations and the same spatial structure at the fiber input using the Wollaston prism-based method (relative intensities of these modes can be easily tuned by rotating the input polarization state); then, one of these modes is coupled by the RF to the mode of orthogonal spatial structure and the same polarization. We show that such an approach allows the excitation of LP₁₁ modes with orthogonal polarizations and spatial structures at a selected wavelength determined by the RF period. Finally, depending on the phase shifts between the doubly orthogonal LP₁₁ modes, pure TE_01, TM₀₁, and HE₂₁ modes or CVBs of more complex polarization structures can be obtained at the fiber output.

To excite pairs of LP₁₁^xo and LP₁₁^yo (or LP₁₁^xe and LP₁₁^ye) modes, we added a half-wave plate after polarizer P2 in the setup shown in Fig. 7, which sets the polarization azimuth of the input beam at 45° with respect to the fiber polarization axis. The resonant coupling between the LP₁₁ x-polarized spatial modes occurs at 0.848 µm for an RF of Λ₃ = 5.3 mm. Therefore, as the light source, we used a laser diode operating near this wavelength (0.850 µm) with a coherence length of approximately 1000 λ. Depending on the excitation, we obtained at the output of the fiber with RF different pairs of doubly orthogonal modes, that is LP₁₁^xo and LP₁₁^ye or LP₁₁^xe and LP₁₁^yo. As the light source was monochromatic, we removed the diffraction grating DF and interference filter IF from the experimental setup and examined the output mode superposition using a camera, rotatable linear polarizer, and a Babinet-Soleil compensator placed directly at the fiber output and used to tune the phase difference between the interfering LP₁₁ modes (which can also be achieved by stretching the birefringent fiber). In Fig. 12, we show that by adjusting the phase difference between the interfering modes, we can obtain pure TE₀₁ and TM₀₁ polarized beams, HE₂₁ beams, and their superposition with circular polarization states on the diagonals. The handedness of circular polarization was determined using a quarter-wave plate. Note that for such beams, the orbital angular momentum and spin angular momentum are equal to 0, regardless of the phase difference.

 

Fig. 12. Output beams arising as a superposition of the LP₁₁ modes of orthogonal polarizations and spatial structures with different phase difference, registered after passing through a polarizer with different transmission azimuth indicated by black arrows and without polarizer (NOP).

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5. Conclusions

To the best of our knowledge, this is the first study to demonstrate a rocking filter in a birefringent two-mode fiber. Our simulation and experimental results show that such a filter enables resonant couplings between orthogonally polarized LP₁₁ modes of the same spatial structure, similar to the rocking filter in single-mode fibers and also the couplings between modes of the same polarizations and orthogonal spatial distributions, providing that the respective phase-matching conditions are fulfilled. Our simulation results show that both types of couplings between singly orthogonal modes are driven by identical overlap coefficients between interacting modes, equal to cos(Δα)sin(Δα), where Δα is the angle of the discontinuous twist at the interface between the RF sections. Consequently, both types of coupling have the same strength, and both reach the maximum efficiency for a total twist angle of 90°. The couplings between doubly orthogonal LP₁₁ modes (simultaneously with orthogonal polarizations and spatial distributions) are driven by a smaller coupling coefficient equal to sin²(Δα); therefore, they are significantly less effective. As shown by our simulation results, the maximum coupling efficiency for doubly orthogonal modes is reached for a long RF with a high number of segments and a total twist angle exceeding 1000°, the exact value of which depends on the specific filter structure. Because of the fabrication difficulties of such long filters with repeatable twist angle Δα, the couplings of this type are experimentally unreachable.

We proposed an experimental procedure involving white-light interferometry and the cut-back method to measure the spectral dependence of the difference in phase effective indices (corresponding to intermodal beat lengths) for all combinations of LP₁₁ modes. These parameters are required to control the spectral positions of the respective resonances by selecting an appropriate RF segment length. RF with different periods were fabricated in a PANDA-type Nufern PM-1300B-XP two-mode fiber using a Fujikura LZM-100 glass processing station with a focused CO₂ laser beam. Next, the RFs were experimentally examined using a recently proposed method of selective excitation of LP₁₁ modes employing a Wollaston prism. The possibility of coupling between the LP₁₁ modes of orthogonal polarizations and spatial distributions was experimentally confirmed. The locations of the respective resonances were in relatively good agreement with the predictions based on intermodal beat length measurements. Furthermore, we observed the effect of the degradation of the coupling strength versus the resonance order related to the fabrication imperfections. The resonance depths were −15 dB, −10 dB, and −8 dB for the first-, second-, and third-order resonances, respectively. It is expected that the quality of the resonances can be further enhanced using a fiber with optimized intermodal dispersion characteristics, allowing better spectral separation of different resonances and improving the fabrication precision.

Moreover, we demonstrated a new application of such RF structures as easily tunable sources of first-order vector beams, such as TE₀₁, TM₀₁, HE₂₁, and cylindrical vector beams with more complex polarization structures, including those with circular polarization states on diagonals. This functionality is achieved by the simultaneous excitation of the two LP₁₁ modes with orthogonal polarizations and the same spatial structure using the Wollaston prism method, and the resonant coupling of one of these modes by the RF to the mode with the same polarization and orthogonal spatial distribution. Consequently, we obtained two co-propagating LP₁₁ modes of orthogonal polarizations and spatial distributions, whose coherent superposition produces at the fiber output different cylindrical vector beams depending on the phase shift between them. Such tunable sources of vector beams, especially radial or tangential polarizations, have many potential applications, including high-resolution microscopy, STED imaging, optical tweezers and mode division multiplexing for optical communication. They can also be applied in high power regime for electron acceleration and material processing. However, in this case, the potentially harmful nonlinear spectral broadening of the input pulses should be limited by minimizing the RF length and using a bandpass filter matching the resonance band at the RF output.

Appendix: Measurements of the differences in phase effective indices for all pairs of LP₁₁ modes

Knowledge of the differences in the phase effective indices $\Delta {n_{eff}}$ for all combinations of the LP₁₁ modes is required to predict the spectral positions of the respective resonant couplings. To determine the spectral dependence of $\Delta {n_{eff}}$ for all combinations of LP₁₁ modes in the PANDA fiber used for RF fabrication, we developed an experimental method consisting of three steps. First, we measured the birefringence of LP₁₁ modes with the same spatial structure and orthogonal polarizations using the well-known spectral interferometry method combined with the point-force method [37]. An appropriate combination of LP₁₁ modes in the tested fiber was excited using the Wollaston prism method [28]. The results of the measurements for the two pairs of LP₁₁ modes with different spatial distributions are shown in Fig. 13. The uncertainty in these results was estimated to be approximately 3%.

 

Fig. 13. Spectral dependence of birefringence measured for different pairs of the LP₁₁ modes of orthogonal polarizations.

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In the second step, we utilized the cutback method to determine the order of spectral fringes arising at the fiber output owing to interference of the LP₁₁ modes with the same polarization and orthogonal spatial structures. The phase difference between the modes propagating in the fiber of length L can be expressed as:

(10)$$\Delta {\phi _{\exp }}\, + 2\pi q = \frac{{2\pi }}{\lambda }L\Delta {n_{eff}},$$

where $\Delta {n_{eff}}$ is the difference in the effective indices of interfering modes, $\Delta {\phi _{\exp }}$ is the phase difference modulo 2π determined experimentally based on spectral interference fringes, and q is an unknown interference order, which cannot be determined from a single spectral interferogram. For the interference maxima/minima, $\Delta {\phi _{\exp }}$ is 0/π. To determine the interference order q at a particular wavelength, we cut the output end of the fiber piece by piece with a step less than the intermodal beat length, which allows to follow the displacement of the spectral interference fringes without losing interference order unambiguity and determine the phase shift change $\Delta (\Delta \phi )$ after each cut, as shown in Fig. 14(a). As shown in Fig. 14(b), by linearly fitting $\Delta (\Delta \phi )$ versus ΔL for the chosen wavelengths, we could eventually calculate the intermodal beat length for the interfering pair of modes according to the following relation:

(11)$${L_B} = 2\pi {\left[ {\frac{{\Delta (\Delta \phi )}}{{\Delta L}}} \right]^{ - 1}}.$$

Thereafter, we can find interference order q as the highest integer satisfying the inequality qL_B≤ L and determine the spectral dependence of the absolute value of Δn_eff from Eq. (10), as the sign of the phase shift cannot be recognized from the interference fringes (Fig. 14(a)). The uncertainty of the measurement of |Δn_eff | in this manner is approximately 5%.

 

Fig. 14. (a)–(c) Measurements of the difference in effective indices for LP₁₁^xe and LP₁₁^xo modes using the cutback method. (a) Normalized intermodal interference fringes registered with Ocean Optics USB4000 spectrometer for three fiber lengths. (b) Change in the phase difference versus change of the fiber length for wavelengths of 0.8, 0.85 and 0.9 µm. (c) Calculated absolute value of the difference in phase effective indices |Δn_eff^xo−xe|. (d) Zero-order spectral interference fringes registered for all first-order modes in Mach-Zehnder interferometer of the same reference arm length for the fiber length equal to 113.7 cm.

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To properly assign to each LP₁₁ mode the value of the phase effective index with an accuracy to a common constant, we need to know the absolute values of Δn_eff for each pair of LP₁₁ modes as well as their signs. Because the core ellipticity is only 9% (Fig. 1), it does not change the sign of the overall fiber birefringence induced by stress-applying elements; therefore, the x-polarized modes have a greater effective index than the y-polarized modes of the same spatial structure. To order all modes versus to increase the effective index, we need to know the sign of the difference n_eff^xo – n_eff^xe. It can be seen in Fig. 14(c) that |Δn_eff^xo−xe| is weakly dispersive at longer wavelengths and has an extremum at 0.89 µm. This allows us to claim that in this spectral region, the sign of the difference in the group effective indices N_eff^xo − N_eff^xe is the same as the sign of the difference in the phase effective indices n_eff^xo – n_eff^xe.

The value of N_eff^xo – N_eff^xe was determined based on measurements of the group optical path delay (gOPD) in a white-light Mach-Zehnder interferometer (MZI) with an adjustable reference arm [38,39] conducted for all LP₁₁ modes. Knowing that the zero-order spectral interference fringe arises for λ₀ at which gOPD is balanced in both interferometer arms, the group effective index of the particular mode can be expressed as

(12)$${N_{eff}}({\lambda _0}) = \frac{{\Delta {L_R}}}{{{L_F}}} + C,$$

where ΔL_R is the changeable length of the MZI reference arm, L_F is the length of the fiber, C is the common constant factor for all modes, depending on the alignment of the MZI. The zero-order interference fringes for the first-order modes for a constant ΔL_R are shown in Fig. 14(d). By measuring the dependence of ΔL_R on the spectral position of the zero-order interference fringe for all first-order modes, we could determine the sign of N_eff^xo − N_eff^xe, which was positive over the entire examined spectral range. Therefore, we conclude that n_eff^xo > n_eff^xe. The spectral dependencies of the phase effective indices for all LP₁₁ modes relative to n_eff^xo determined in this manner are shown in Fig. 3(a).

Funding

Narodowe Centrum Nauki (DEC-2016/22/A/ST7/00089, Maestro 8).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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