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Abstract
A mixed multi-order vortex beam generator, based on a Reuleaux triangle core fiber chiral grating (RCFG), is proposed. The triangular perturbation and off-axis effects induced by core shape, result in the simultaneous coupling of the core mode with the 1st- and 3rd-order vortex modes. To the best of our knowledge, this is the first time that a mixed vortex beam was generated in a single chiral fiber. The phase matching conditions required for the co-coupling of multi-order vortex beams are analyzed based on the coupled mode theory. Additionally, a cladding shrinkage method is proposed to flexibly adjust the co-coupling wavelength. We found that the key to co-coupling lies in balancing the different order perturbations of the Reuleaux triangle core fiber (RTF). The proposed method offers a new approach for the design of mixed multi-order vortex beam generators, with potential applications in fields such as fiber OAM communications, optical tweezers, and super-resolution imaging.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent decades, vortex beam has attracted significant attention from researchers due to their unique helical phase and annular intensity distribution. The phase wavefront of a vortex beam takes the form $\textrm{exp}({il\theta } )$ [1], where $\theta $ and l represent the azimuthal angle and topological charge, respectively. Unlike spin angular momentum, which represents the polarization of the beam, the orbital angular momentum (OAM), indexed by the any integer topological charge l, of the beam is theoretically infinite. The positive and negative topological charges represent the right-hand and left-hand rotation of the beam phase vortex, respectively. Fractional-order vortex beams are considered as multi-order vortex beams mixed in specific proportions and phases. The orthogonality of OAM makes vortex beams highly promising in optical communication [2] and quantum information [3]. Research on OAM multiplexing communication in free space and optical fiber systems is rapidly advancing [4]. Additionally, the singularity of the central phase of the vortex beam causes its intensity distribution to display a dark hollow ring. This property makes vortex beams valuable in nonlinear optics [5], microphotography [6], optical tweezers [7], and holography [8] applications.
So far, a variety of methods for generating vortex beams have been developed, some of which are based on space beam transformation, such as using helical phase plates [9], q-plate [10], Spatial light modulators [11], etc. Other methods focus on generating vortex beams in fiber-optic systems, such as mode selective coupler [12], long-period fiber grating [13], photon lantern [14],fiber-end spiral zone plate [15], etc. The selective coupler is limited by the degeneracy of the fiber vector mode, thus lacking the flexibility to control the polarization state of the vortex beam. The long-period fiber grating couples the fundamental mode to higher-order modes, and a polarization controller is used to adjust the phase difference between the two degenerate modes and convert them into vortex modes. In the case of the photonic lantern, the generation of the vortex beam requires precise phase control for each input port. The spiral zone plate at the fiber-end is used to collimate and shape the Gaussian beam emitted from the fiber core, thereby generating a vortex beam. However, this method requires expensive and complex fabricating equipment.
In particular, the vortex beam generator based on chiral fiber grating offers high mode conversion efficiency, low insertion loss, and compact structure. In 2004, Kopp et. al fabricated the chiral fiber gratings first [16], which have since attracted significant attention from researchers as vortex beam generators. Table. 1 illustrates the different order chiral fiber gratings that have been reported for generating different order vortex beams. Twisted off-axis single mode fiber (SMF) [17,18] or few mode fibers (FMF) [19] with 1-fold rotational symmetry can excite 1st-order vortex beams. Similarly, ellipse [20,21], Four-leaf clover [22], Hexagram core [23], and photonic crystal fibers (PCF) [24] generated vortex beams of the same order as their respective rotational symmetry. And the high-order diffractive chiral fiber gratings [25] with harmonic perturbation have also been utilized to excite high-order vortex beams. In addition, the special fibers containing inherent n-fold rotational symmetry excited vortex beams with better purity and polarization independent features. However, these methods are limited to exciting vortex beams with a single vortex order at a single wavelength. The generation of mixed multi-order vortex beams remains an important and unresolved issue.
Table 1. Theoretical and experimental work on different order chiral fiber gratings
In this work, an all-fiber mixed multi-order vortex beam generator based on RCFG is proposed, to generate mixed 1st- and 3rd-order vortex beams. In the Principle and simulation section, fiber vortex mode and coupled mode theory are introduced. The phase matching conditions for co-coupling of different order vortex modes are investigated in detail. In the Results section, we introduce the fabrication of RTF and the fiber-heated twisting device. The RCFG transmission spectra recorded during fabrication are presented. Using an interference pattern observation system, mixed, 1st- and 3rd-order vortex beams are observed. In the Discussion section, a cladding shrinkage method for exciting mixed vortex beams at arbitrary wavelengths is presented. Finally, we discuss the influence of the mode overlap integral and the core shape on the energy balance of different order vortex beams.
This fabrication technique can also fabricate square or higher-order shaped core fibers that can excite higher-order mixed and fractional-order vortex beams. Additionally, the method of co-exciting the OAM mode also allows for the generation of a specialized structured light field at the fiber end. It has potential applications in all-fiber communication systems with OAM mode division multiplexing. With further research, more innovative designs can be developed for multi-order vortex beam generation using such fibers.
2. Principle and simulation
2.1 RCFG with triangular and offaxis perturbation
In order to generate 1st- and 3rd-order vortex beams simultaneously, we designed and fabricated a fiber with a Reuleaux triangular core, which was then offaxis twisted into the RCFG, as shown in Fig. 1(a). The RCFG converts a core mode with Gaussian distribution into a mixed vortex beam consisting of 1st- and 3rd-order vortex components. This coupling process entails two distinct mechanisms that simultaneously coupled the core mode to vortex modes. The triangular perturbation of the core dominates the coupling of the 3rd-order vortex mode, while the off-axis effect caused by fiber twisting couples the core modes with the 1st-order vortex mode. Additionally, phase matching is another crucial factor that influences the feasibility of mode coupling. The periodic twist abolishes the orthogonality of the modes, allowing the core mode to couple with other modes. By selecting a suitable twist pitch P, the phase matching conditions of the 1st- and 3rd-order vortex modes are simultaneously satisfied, resulting in the generation of the mixed vortex beam.
Fig. 1. (a) Schematic diagram of the RCFG generates mixed multi-order vortex beam. (b) The cross-section of RTF with slight offaxis. (c) Reuleaux triangular core is formed by overlapping three circles with the same radius.
Figure 1(b) is a cross-sectional detail of the RTF fiber with a slightly off-axis core. The core shape is formed by overlapping three circles with the same radius, resulting in 3-fold rotational symmetry as shown in Fig. 1(c). The distance from the center of the circle (point O1) to the fiber center (point O) is 7 µm, while the distance from the intersection point (vertex of the triangle, point a) to the center (point O) is 5 µm. The core boundary of the twisted RTF remains unchanged and is only rotated with the z-coordinate.
2.2 Vortex mode and coupled mode analysis
The vortex modes in fiber are formed by the linear combination of quasi-degenerated vector eigenmodes with ${\pm} {\pi / 2}$ phase shift [13]:
Since the orthogonal polarization modes in optical fibers are quasi-degenerate, the OAM mode does not exhibit polarization dispersion at the grating scale (< 10 cm), so the polarization state of the OAM mode is no longer marked.
Due to the rotational symmetry of RCFG, the fiber core mode can be coupled to the cladding vortex modes. We assume that the modes (with time-harmonic factor ${e^{ - j\omega t}}$) propagate along the z-axis, and the coupled mode equation expressing its energy transfer between different modes is [16]:
The nco and ncl are the core and cladding refractive index, r0 is the core radius of the reference circular fiber, rm is the Fourier series of the core boundary (as perturbation term), m is an integer, ${k_u} = {P / {2\pi }}$, $\delta $ is the impulse function. Different orders of perturbations excite their respective vortex modes, for example, the off-axis perturbation r1 excites the 1st-order vortex modes, and the triangular perturbation r3 excites the 3rd-order vortex modes. The simplified coupling coefficients yield two matching conditions for mode coupling [17]:
2.3 Simulated RCFG transmission spectra
Considering the relation between the fiber mode propagation constant and effective refractive index (neff), $\beta (\lambda ) = {k_0}{n_{eff}}(\lambda )$, the expression of the coupled possibility mode curve (CMC) is obtained:
where, ${n_{eff,OAM01}}(\lambda )$ is the neff of the core mode. A simulation model of the reference circular fiber is established, which is used to calculate the fiber mode neff curve. The reference circular fiber has a core radius of r0 = 4 µm and a cladding radius of 60 µm, with a triangular perturbation of r3 = 0.25 µm and an off-axis perturbation of r1 = 0.07 µm. The numerical aperture is 0.1082. Figure 2(a) shows the simulated fiber core mode neff curve (black), 1st-order vortex mode CMC, and its intersection point with the 1st-order vortex mode neff curve (blue). The corresponding twist pitch is 894 µm, 850 µm, and 743 µm. Figure 2(b) shows the 3rd-order vortex mode CMC and its intersection point with the 3rd-order vortex mode neff curve (green). When the neff curve of the vortex mode crosses the CMC, the mode resonance coupling occurs at the corresponding wavelength. Figure 2(c-d) shows the simulated transmission spectra of the RCFG at the twist pitch are 894 µm, 850 µm, and 743 µm, respectively. The resonance coupling points with markers correspond to the intersection points of the curves in Fig. 2(a) and (b). Two resonant coupling points (brown inverted triangles) of OAM3,8 modes are shown in Fig. 2(e), consistent with experimental results. The appearance of two resonance points of the same mode results from the occurrence of two intersections between the OAM3,8 mode neff curve and the CMC.Fig. 2. (a) The neff curves of fiber core mode (black), 1st-order vortex modes (blue), and CMC. (b) The neff curves of 3rd-order vortex modes (green) and CMC. The simulated RCFG transmission spectra at different twist pitches. (c) 894 µm. (d) 850 µm. (e) 743 µm.
Figure 2 illustrates that different twist pitches of RCFG correspond to their respective CMC, allowing for the excitation of a specific mode at different wavelengths or the simultaneous excitation of multi-order vortex modes at the same wavelength. When the core boundary consists of multi-order perturbations, it produces multiple CMCs corresponding to different orders of vortex mode. Moreover, with a suitable twist pitch, the CMC and the neff curve of the different vortex mode can intersect at the same wavelength, which means that the RCFG simultaneously excite a multi-order mixed vortex beam.
3. Result
3.1 Fabrication of Reuleaux triangular core fiber
Figure 3(a) illustrates the fabricated steps of RTF. Firstly, a standard SMF preform is prepared using modified chemical vapor deposition (MCVD). Next, ultrasonic drilling technology is used to create three air holes around the core of the preform in a circumferential arrangement. Finally, the preform is vacuum-pumped during the fiber-drawing process, the temperature and air pressure are controlled to close the holes. Since the fusion fiber needs to fill the three air holes, the circular core is forced to stretch and transform into a Reuleaux triangular core. The measurement results of the fiber refractive index are given by the refractive index profiler (Photon Kinetics, S14). Figure 3(b) and (c) are the refractive index distributions of a straight line through the core and the RTF cross-section, respectively. Since the core shape is not much different from the original single-mode preform, the fusion loss of RTF and single-mode fiber (SMF, Corning, SM-28e) is 0.071 dB.
Fig. 3. (a) The RTF fabrication process. (b) The Refractive index distribution of a straight line through the core. (c) The Refractive index distribution of the RTF cross-section.
3.2 Fiber twisted device and vortex beam observation
The RCFG is fabricated from RTF using a homemade fiber-heated twisting device, as shown in Fig. 4(a). The device consists of two stacked motorized stages, a hollow rotating motor, two fiber clamps, and four electrode rods. One clamp is connected to the rotating motor shaft, while the other clamp is fixed at the end of the upper motorized stages. During fabrication, the RTF is placed in the plane formed by the electrode rods, and a discharge is induced between the paired electrode rods, heating the RTF to a fusion state. The rotating motor speed is 60 degrees/s, with the upper stage moving to the right at a speed of 12 µm/s to provide tension, and the lower stage moving to the left at a speed of P/6 µm/s. The sample length is proportional to fabrication time. The tension results in a slight reduction in the RCFG outer diameter. During fiber twisting, the transmission spectra were continuously monitored in real time using a supercontinuum light source (Yslphotonics, SC-5) and an optical spectrum analyzer (Yokogawa, AQ6370C).
Fig. 4. (a) Schematic diagram of the fiber-heated twisting device. (b) Schematic diagram of the pattern observation system. PBS, polarized beam splitter; NPBS, Non-polarized beam splitter; HWP, half-wave plate.
We built a pattern detection system based on a Mach-Zehnder interferometer, as shown in Fig. 4(b). The beam of the wavelength tunable laser (Santec, TSL-550-C, 1480-1630 nm) is divided into two beams after being collimated through a 20x objective lens. One beam is input into the RCFG to excite the mixed vortex mode, while the other beam is expanded and collimated into a Gaussian beam after passing through a SMF, and thus serves as a reference beam. PBS and NPBS are respectively positioned at the intersecting points of the two beams. To adjust the power ratio of the two beams, a half-wave plate (HWP1) is placed in front of the PBS, and the other two half-wave plates (HWP2 and HWP3) are employed to adjust the polarization state. The intensity profile and interference pattern of the generated mode were observed using an infrared camera (Hamamatsu, C12741-03).
3.3 Transmission spectra and vortex mode observation
Figure 5(a) shows the transmission spectra of different sample lengths collected during RCFG fabrication, with a pitch of 900 µm. There are two resonance dips OAM1,2&OAM3,7 (1503 nm, -18 dB) and OAM1,3 (1640 nm, -16 dB). The mode intensity profile observed through the camera at 1503 nm is shown in Fig. 5(b). The first row shows the simulated mixed vortex beam profile, the Gaussian beam, and the interference pattern. The second row shows the observed experimental result. The mixed vortex mode exhibits a multi-layer symmetrical double-crescent shape due to the coherent enhancement of OAM1,2 and OAM3,7 along the ring path.
Fig. 5. (a) Transmission spectra of different RCFG lengths with a twist pitch of 900 µm. (b) The mixed vortex mode intensity profiles, Gaussian beam, interference intensity patterns, and their corresponding simulation results.
Figure 6(a) shows the transmission spectra of different sample lengths collected during RCFG fabrication, with a pitch of 850 µm. There are two resonance dips of the 1st-order vortex mode, OAM1,2 (1460 nm, -7 dB) and OAM1,3 (1580 nm, -18 dB). The mode intensity profile observed at 1580 nm is shown in Fig. 6(b). The first row shows the simulated 1st-order vortex mode profile, the Gaussian beam, and the interference pattern. The second row shows the observed experimental result. The 1st-order vortex beam is in the form of a multi-layer ring, and the interference pattern with the Gaussian beam is a single spiral.
Fig. 6. (a) Transmission spectra of different RCFG lengths with a twist pitch of 850 µm. (b) The 1st-order vortex mode intensity profiles, Gaussian beam, interference intensity patterns, and their corresponding simulation results.
Figure 7(a) shows the transmission spectra of different sample lengths collected during RCFG fabrication, with a pitch of 745 µm. There are two resonance dips of the 3rd-order vortex mode OAM3,8-left (1323 nm, -21.1 dB) and OAM3,8-right (1610 nm, -23.6 dB). The mode intensity profile observed at 1615.5 nm is shown in Fig. 7(b). The first row shows the simulated 3rd-order vortex beam profile, the Gaussian beam, and the interference pattern. The second row shows the observed experimental result. The generated 3rd-order vortex beam is in the form of a multi-layer ring, and the interference pattern with the Gaussian beam is a triple spiral. The slight difference between the experimental spectra and the above simulations may be due to without consideration of the material dispersion and the effects of internal stresses generated during RTF fabrication.
Fig. 7. (a) Transmission spectra of different RCFG lengths with a twist pitch of 745 µm. (b) The 3rd-order vortex mode intensity profiles, Gaussian beam, interference intensity patterns, and their corresponding simulation results.
4. Discussion
4.1 Adjusting co-coupling wavelength
The above simulation results reveal that adjusting the pitch of the RCFG enables simultaneous coupling of the core mode to the 1st- and 3rd-order vortex modes. However, this outcome is only achieved for specific wavelengths and pitches. To address this limitation, we propose a cladding shrinkage method that provides flexible adjustment of the co-coupling wavelength. Figure 8(a) illustrates the relation between the cladding shrinkage and the matching pitch of OAM1,n, and OAM3,n modes at 1503 nm, according to the phase matching conditions. As the cladding shrinks, the matching pitch of the 3rd-order mode gradually decreases, while the pitch matching the 1st-order mode increases, multiple co-coupling intersections between the two modes emerges. The star points denote the shrinkage rate and pitch of the mixed 1st- and 3rd-order vortex beam generated by the RCFG. This method allows calculation of the corresponding cladding shrinkage and matching pitch for any wavelength, enabling simultaneous excitation of 1st- and 3rd-order vortex modes.
Fig. 8. (a) The 1st and 3rd order mode matching pitch versus cladding shrinkage in 1503 nm. (b) Overlap integrals of paired 1st- and 3rd-order modes with core mode.
In order to excite more OAM modes at the same wavelength, their respective phase matching conditions needs to be meet. For co-excitation of two OAM modes, we adjusted the cladding diameter and twisted pitch. For co-excitation of more OAM modes, we need more freedom degrees means to adjust respective neff, which is difficult for general optical fibers. Therefore, the simultaneous excitation of multiple OAM modes can only be achieved by designing fiber refractive index distribution and the cladding vortex mode neff.
4.2 Mode overlap integrals
Besides, the magnitude of the coupling coefficient is also an important factor in determining coupling. According to Eq. (5), the coupling coefficient of the mode is not only related to the perturbation quantity, but also to the mode overlap integral. Let's first focus on the mode overlap integral. Figure 8(b) shows the normalized overlap integral of the 1st- and 3rd-order vortex modes with the core mode, whose mode pairs are determined by the intersection of Fig. 8(a). As the radial order of the mode increases, the coupling coefficients of the 1st- and 3rd-order vortex modes also increase, with the 1st-order vortex mode displaying a higher growth rate. Although the mode overlap integrals of OAM1,1 and OAM3,6 are balanced, their values are too small, resulting in grating lengths more than 100 mm. In the experiment of Fig. 5, the mixed vortex beam consists of OAM1,2 and OAM3,7 modes. The mode overlap integral of the former is larger than that of the latter, necessitating a larger 3rd-order perturbation in the RCFG as compared to the 1st-order perturbation. Only balanced coupling coefficients can generate modes of different orders simultaneously. Otherwise, only modes of a single order are generated.
In addition to the co-excitation of two OAM modes, it is also possible to selectively excite a specific OAM mode. When the targeted mode fulfills the phase and OAM matching conditions, adjusting the twisted pitch allows for control the radial order n of the OAM mode. Furthermore, by designing the core shape with an l-order rotational symmetry perturbation, the angular order l of the excited vortex mode can be adjusted. Increasing the angular order l and decreasing the radial order n will result in a reduction in the mode overlap integral. Reducing the diameter of the fiber cladding can enhance the mode overlap integral and decrease the required length of RCFG.
4.3 Perturbations of core boundaries
The perturbations of RCFG originate from the non-rotationally symmetric components of the core boundary, which can be decomposed into a superposition of the reference circular fiber and harmonic perturbations of each order:
Any chiral fiber grating contains multi-order perturbations. Figure 9 shows the core boundary Fourier series expansion results of off-axis SMF and off-axis RTF fiber and their perturbation proportion pie chart. As shown in Fig. 9(a), the 1st- and 2nd-order perturbations of off-axis SMF are large enough, allowing it to generate 1st- and 2nd-order vortex beams [21]. However, the 3rd-order and higher-order components are extremely low. For comparison, as shown in Fig. 9(b), off-axis RTF with large enough 1st- and 3rd-order perturbations to generate 1st- and 3rd-order vortex beams.
Fig. 9. Fourier series for different core boundary and their perturbation proportion pie charts. (a) Offaxis SMF. (b) Off-axis RTF.
However, there is a difference in the proportion of perturbation between the two fiber cores. In off-axis SMF, the 1st-order component accounts for 87% of the total perturbation, while the proportion of the 3rd-order component is close to 0. On the other hand, in off-axis RTF fiber, the 1st-order component only accounts for 18%, while the 3rd-order component accounts for 63%. The relation between the both is inversely proportional to the above mode overlap integral, indicating that the coupling coefficients for the 1st- and 3rd-order vortex modes are approximately equal. Upon comparison, it can be observed that off-axis SMF chiral grating cannot simultaneously excite vortex modes of different orders. This is because any high-order perturbation is extremely lower than the 1st-order perturbation, resulting in much smaller coupling coefficients for the former. Consequently, the coupling phenomenon only occurs between the core mode and the single order vortex mode.
As an example, Fig. 10 illustrates the coupling process between the core mode and vortex mode in different fiber chiral gratings. All involved modes meet the phase matching conditions. Figure 10(a) shows the coupling process of core mode OAM0,1 and vortex mode OAM1,2&OAM3,7 in an off-axis SMF chiral grating. Due to the extremely low 3rd-order perturbation, most of the energy couples to the OAM1,2 mode. On the other hand, Fig. 10(b) shows the coupling process in off-axis RTF chiral fiber grating. Since the coupling coefficients of OAM1,2 and OAM3,7 are approximately equal, the energy of the core mode simultaneously converts to two vortex modes of different orders. Although the two vortex modes are not equal in energy, this can be improved by increasing off-axis perturbation. Precisely controlling the amount of perturbation requires more advanced optical fiber preparation processes. In following work, we plan to employ the preform stacking technology to fabricate RTF, similar to the fabrication of photonic crystal fibers. In addition, more optical fibers with special core will also be designed and fabricated, such as square core or hexagonal core fibers.
Fig. 10. The coupling process of core mode and vortex modes in different chiral fiber grating. (a) Offaxis SMF. (b) Offaxis RTF.
5. Conclusion
In summary, a mixed multi-order vortex beam generator based on RCFG is proposed and fabricated. Due to the 3rd-order perturbation and off-axis effects contained in the twisted core boundary, the core mode is simultaneously coupled into the 1st-order and 3rd-order vortex modes, resulting in the generation of a mixed vortex beam. By calculating the neff of the vortex modes, the phase matching conditions for simultaneous coupling of different order vortex modes are investigated. Using a homemade fiber-heated twisting device, RCFG with different twisting pitches were experimentally fabricated. The results of mode interference observation confirmed the excitation of mixed vortex beams, as well as independent 1st- and 3rd-order vortex beams.
In order to flexibly adjust the excitation wavelength of the mixed vortex beam, the cladding shrinkage method is proposed to achieve phase matching of different order vortex modes at any wavelength. In addition, we investigated the effects of the overlap integral and the core shape on the coupling coefficient. As the radial order of the mode increases, the overlap integral of the 1st-order mode is much higher than that of the 3rd-order mode. This indicates that to achieve mode power balance in the mixed beam, the 3rd-order component of the perturbation in the chiral fiber grating needs to be much larger than the 1st-order component. We found that the conventional off-axis SMF chiral grating is incapable of exciting mixed multi-order vortex beams, whereas the RCFG, containing inherent 3rd-order perturbation, successfully achieves excitation. We believe that the proposed mixed multi-order vortex beam generator will play a crucial role in fiber communication systems based on OAM multiplexing, optical tweezers, and super-resolution imaging.
Funding
National Natural Science Foundation of China (61827819); Bagui Scholars Program of Guangxi Zhuang Autonomous Region (2019A38).
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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