1. Introduction

Optical orbital-angular-momentum (OAM) beams (also called optical vortex beams), characterized by the helical-type wave front, the phase singularity, the intrinsic orbital angular momentum, and the infinite but orthogonal eigenstates, have lately generated a lot of attention and found widespread applications in fields such as optical communication, optical manipulation, optical tweezer, quantum entanglement, optical metrology, microscopy and imaging, and biochemical detection, among others [1–5]. To realize the aforementioned OAM beam-based applications, the OAM beam/mode converter, which allows the fundamental mode to be converted to a particular OAM mode, becomes the most basic and indispensable component. Various methods for OAM mode generations/conversions have been developed to date, including those based on the use of cylindrical lenses, q-plates, integrated silicon devices, J-plates, and optical fibers, among others [1–23]. Twisted fiber-based OAM mode converters, including helical long-period fiber grating (HLPG)-based ones and preset twist long-period fiber grating (PTLPG)-based ones, have lately received substantial research attention as a novel type of fiber-based OAM converters [6–23]. These OAM converters have a number of outstanding characteristics, including small size, low insertion loss, high OAM mode conversion efficiency, and inherent compatibility with commonly used fibers. As the particular examples, the twisted fiber-based OAM mode converters allowing mode conversion of the first-, the second-, the third- or the fourth-order OAM [6–11], simultaneous mode conversion of the first- and second-order OAM [12], and simultaneous mode conversion of the second- and third-order OAM modes [13] have been experimentally demonstrated, which are realized by using the HLPGs written in either single-mode fibers (SMFs) or few-mode fibers (FMFs). Additionally, the fifth- and sixth-order OAM modes have been successfully achieved by using a HLPG formed in a microstructure fiber [14]. It has been shown that the fiber-based OAM mode converter can be constructed directly utilizing HLPG. Theoretically, the orthogonal eigenstates of OAM modes are infinite, providing boundless transmission capacity when employed in mode-division multiplexing (MDM) fiber communication systems [2]. However, current reports on fiber grating-based OAM mode converters have only achieved up to the sixth-order OAM modes [14], as higher order OAM modes pose practical challenges due to complex structural requirements and fabrication constraints for both the fiber itself and the grating [15]. Nevertheless, exploiting the OAM modes with identical azimuthal order indexes but with the different radial order indexes (which are also mutually orthogonal) can introduce a new degree of multiplexing freedom with potential to significantly and realizably enhance transmission capacity [25,26].

On the other hand, the majority of the HLPG-based OAM converters discussed above are restricted to those that have a sufficiently high conversion efficiency but a narrow wavelength range. In particular, HLPGs with a broad yet flat-topped spectrum have been hard to come by, despite being desired for a variety of practical optical vortex applications especially the femtosecond vortex lasers, the OAM amplifiers, and the OAM-MDM systems combined with the wavelength division multiplexing (WDM) technique as well [3,16,17]. Up to now, three primary ways for increasing the HLPG's bandwidth have been proposed. The first are based on the use of phase-modulated HLPGs, which include linearly and arbitrarily chirped HLPGs. However, because to the complexity and difficulty in fabrication, such HLPGs have seldom been experimentally demonstrated [18–20]. The second is based on a graded-index fiber-based HLPG. Such an OAM converter provides both flexibility and robustness, but obtaining a flat-top spectrum is challenging [21]. The third is based on the so-called dual-resonant peaks approach, in which the period of the used HLPG is specifically chosen so that the resonant coupling between the two specified modes occurs at or near its dispersion-turning-point (DTPs) [22,23]. However, since the DTP wavelength is merely determined by the geometric structure and refractive indices of the used fiber, it cannot be altered even by modifying the period of the grating. Furthermore, for most HLPGs written in frequently employed SMFs or FMFs, DTP wavelengths such as 1350 nm in [22] lie outside the primary bands (S, C, and L bands) of the fiber communication system.

To overcome the above issue, most recently, some researchers have proposed and demonstrated a simple method enabling to changing the DTP wavelength of one particular cladding mode for the fabricated HLPG, which were realized by the tapered HLPGs [20,24]. This method enables the generation of a broadband OAM mode converter with a tunable center wavelength. However, achieving precise control of the taper waist dimensions on the submicron scale, which is crucial for the proposed device, remains challenging in practical fabrication, thereby limiting the robustness and flexibility of the proposed scheme. Consequently, the radial-orders of the generated OAM modes are fix, and OAM modes with identical azimuthal order indexes but different radial order indexes have not been thoroughly realized as required. These modes are also orthogonal to each other, representing a novel degree of freedom with potential applications for increasing transmission capacity through multiplexing [25,26]. Moreover, during the tapering process, the core diameter decreases in accordance with the reduction of the fiber cladding diameter, making multimode interference difficult to thoroughly avoid, particularly when the fiber is severely tapered. Consequently, achieving OAM modes with both a lower radial-order and high purity may prove challenging.

In this study, a new flat-top wideband OAM mode converter (FTWOAMMC) is proposed and experimentally demonstrated, which is based on utilization of a cladding-etched helical long-period fiber grating (CEHLPG). The proposed FTWOAMMC provides an excellent flexibility and robustness in controlling both the radial order and the central wavelength of the resulting flat-top wideband OAM mode conversion, which may find potential applications in OAM-MDM systems combined with the WDM technique, femtosecond vortex lasers, all-fiber OAM tweezers and the all-fiber OAM sensors.

2. Principle and simulation results

On the assumption that the core mode is completely converted into a local cladding mode at the central resonant wavelength satisfying the phase-matching condition, the resonance bandwidth at -20 dB for a standard HLPG with uniform pitch distribution can be determined by [19]

Based on Eq. (2) and the algorithm described in [27], the parameter $\psi (\lambda _{res}^{})$ in Eq. (1) can be derived as

As can be found that $\varDelta {\lambda _B}$ is inversely proportional to $|{{{d\Lambda } / {d\lambda }}} |$; further, a broad yet flat-topped dip can be anticipated to occur once if the HLPG is operated close to the DTP wavelength, where the condition ${{d\Lambda } / {d\lambda }} = 0$ is satisfied. However, when inserting ${{d\Lambda } / {d\lambda }} = 0$ into Eq. (4), the magnitude of $\varDelta {\lambda _B}$ is infinite. It is not possible to utilize Eq. (4) to evaluate the bandwidth of the HLPG operating at the DTP. In this scenario, the second derivative term of the Taylor series for $\Delta {n_{eff}}(\lambda )$ should be retained, and expression of the bandwidth at -20 dB therefore be rewritten as [28]

It can be seen from Eq. (5) that, for a fix-length HLPG working at the DTP, its bandwidth $\Delta {\lambda _\textrm{B}}$ is inversely proportional to |d²Λ/dλ²| which is must not be zero due to the extremal property at the DTP wavelength [28].

Figure 1(a) shows the calculated results for pitch spectra of twelve different cladding modes, where the curves LP_1,1, LP_1,2, …, LP_1,12 represent the cases that the interactions take place between the mode LP₀₁ and the modes LP_1,1-LP_1,12, respectively. For clarity, the zoomed curves of the LP_1,8-LP_1,11 shown in Fig. 1(a) are especially depicted in Fig. 1(b), where the blue solid circles denote DTPs of the cladding modes LP_1,8, LP_1,9, LP_1.10, and LP_1,11, respectively. Without losing generality, a standard SMF, characterized by core and cladding diameters of 8.2 µm and 125.0 µm, respectively, is employed in the computation. Refractive indices (RIs) of core and cladding are 1.4580 and 1.4536, respectively. The surrounding material is air. Figure 1 clearly shows that DTPs exist exclusively for the modes LP_1,9, LP_1,10, and LP_1,11 within the wavelength range of 1260 nm to 1675 nm, i.e., from original- (O-) band to ultralong- (U-) wavelength band. With the aid of the Eq. (5), bandwidths of the HLPGs operated at DTPs of these the modes, respectively, then can be estimated numerically as long as the utilized fiber and length of the HLPG are determined in advance. However, the DTP is only determined by the utilized fiber itself (i.e., the change of the $\Delta {n_{eff}}(\lambda )$ in terms of the wavelength), thus both the radial mode and the central wavelength of the resonance dip are fixed when the HLPGs are operated at these DTPs, which cannot be altered even by adjusting the grating pitch. As a result, a broadband OAM mode but with different radial-order cannot be obtained by using the conventional SMF-based HLPG.

 

Fig. 1. (a) The calculated results for relationships between the grating pitch and the corresponding resonant wavelengths, where the curves LP_1,1, LP_1,2, …, LP_1,12 reflect the instances when the interactions take place between the mode LP₀₁ and the modes LP_1,1-LP_1,12, respectively. (b) The zoomed curves of LP_1,8-LP_1,11 shown in Fig. 1(a).

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To address the above issue, the CEHLPG has been proposed in this study. Principle of the proposed CEHLPG lies in that by largely decreasing the cladding diameter of the fabricated HLPG, not only the DTPs will be linearly shifted to the short wavelength but also the index the corresponding radial-order mode will be changed to the smaller one in accordance with the changes in cladding diameter. Specifically, the decrease in cladding thickness resulting from the fiber etching processing leads to a reduction in the ERIs of fiber modes due to changes in mode confinement and numerical aperture (NA). However, unlike the cladding modes which experience a significant rate of change, the core mode undergoes only negligible variations. This can be attributed to the dominant confinement of optical field within the core region and its rapid decay within the cladding. As long as the cladding diameter remains significantly larger than the core diameter, any fluctuations in cladding dimensions have an insignificant impact on the ERI of core mode. Conversely, for cladding modes that exhibit optical field distribution not only within both core and cladding regions but also outside them into their surroundings, alterations in cladding diameter have a substantial effect on field distribution and consequently alter ERI values for these modes. Therefore, as the cladding dimensions decrease, the difference of ERI $\Delta {n_{eff}}$ will undergo alterations, leading to corresponding shifts in phase matching curves that encompass mode DTPs [29].

As typical examples, Fig. 2(a) shows the calculation results for pitch of the CEHLPG vs. the resonant wavelengths of the LP_1,9 mode under the cases of different cladding diameters. Figure 2(b) shows the relationship between the cladding diameters and the DTP-wavelengths of the LP_1,9 mode, which are summarized directly from the data shown in Fig. 2(a). For reference, the linear fitting to those discrete data is also given in this figure, showing that there exists a highly linear relationship (R² = 0.999) between these two parameters. Figure 2(b) shows that, as predicted, the DTP of the LP_1,9 mode linearly moves to the shorter wavelength in accordance with the decrement in cladding diameter, which implicitly means that the central wavelength of the CEHLPG-based broadband OAM generators operated at its DTP can be flexibly adjusted just by using the thinned HLPG.

 

Fig. 2. (a) The calculated results for relationship between the pitch of the CEHLPG and the resonant wavelengths of the LP_1,9 mode under the cases of 7 different cladding diameters. (b) Relationship between the cladding diameter and the DTP-wavelengths, which is directly obtained from the data in Fig. 2(a).

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For more information, we have done the same calculations like the ones in Fig. 2 to the modes LP_1,7, LP_1,5 and LP_1,2, respectively. The results are displayed in turn in Figs. 3, 4, and 5. To compare all the results shown in Figs. 2–5 each other, it can be found that by suitably choosing the cladding diameters as 118.8 µm, 92.6 µm, 66.4 µm, and 25.2 µm, respectively for the modes LP_1,9, LP_1,7, LP_1,5, and LP_1,2, respectively, all these four modes have almost a same resonant wavelength around the 1500 nm. The above results implicitly indicate that by consecutively decreasing the cladding diameters from the 125 µm to 25.2 µm, the mode transition phenomenon would occur at wavelength of 1500 nm, i.e., at the operating wavelength of 1500 nm, the mode transitions LP_1,9, LP_1,7, LP_1,5 and LP_1,2 can be realized just by continuously changing the diameter of the fabricated HLPG. The observation of similar properties has also been noted in conventional SMF-based long-period fiber gratings [29].

 

Fig. 3. (a) The calculated results for relationship between the pitch of the CEHLPG and the resonant wavelengths of the LP_1,7 mode under the cases of 7 different cladding diameters. (b) Relationship between the cladding diameter and the DTP-wavelengths, which is directly obtained from the data in Fig. 3(a).

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Fig. 4. (a) The calculated results for relationship between the pitch of the CEHLPG and the resonant wavelengths of the LP_1,5 mode under the cases of 7 different cladding diameters. (b) Relationship between the cladding diameter and the DTP-wavelengths, which is directly obtained from the data in Fig. 4(a).

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Fig. 5. (a) The calculated results for relationship between the pitch of the CEHLPG and the resonant wavelengths of the LP_1,2 mode under the cases of 7 different cladding diameters. (b) Relationship between the cladding diameter and the DTP-wavelengths, which is directly obtained from the data in Fig. 5(a).

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To verify the above presumption, we conducted further computations to examine the connection between the cladding diameter and the radial-mode, the results are shown in Fig. 6(a), where for convenience, the DTP wavelengths of all the considered radial-modes are purposely arranged to be located at same place of 1500 nm. As can be observed, the cladding diameter and the order index of the radial mode have a good linear connection. R² factor for the linear fitting is 0.9999. In accordance with the results shown in Fig. 6(a), the relationship between the DTP pitches and the corresponding radial order of the mode has also been numerically investigated, the result is shown in Fig. 6(b), to our surprise, it is found that the periods for all the considered modes (from LP_1,2 to LP_1,9) nearly remain a constant of ∼235 µm, which once again means that at a fix DTP period of ∼235 µm (corresponding to the operating wavelength of 1500 nm), the mode transitions LP_1,9-LP_1,7-LP_1,5-LP_1,2 can simply be realized just by continuously changing the diameter of the fabricated HLPG from 118.8 µm to 25.2 µm. Next, the second order differential term |d²Λ/dλ²| for each mode was calculated, where the DTPs of modes are located at a same wavelength of 1500 nm. The outcomes are displayed in Fig. 6(c). The value of |d²Λ/dλ²| from LP_1,2 to LP_1,9 changes a little, as can be seen. According to the results shown in Fig. 6(a), Fig. 6(b), and Eq. (3), it can be inferred that bandwidths of the proposed FTWOAMMC for LP_1,2-LP_1,9 modes are with a similar value due to the similar periods and values of |d²Λ/dλ²| while the grating length remains unchanged.

 

Fig. 6. (a) Relation between cladding diameter and the radial-order of the mode. (b) The relations between the period and radial-order of the mode and (c) the |d²Λ/dλ²| versus the radial-order of mode, where the DTPs of all the considered radial modes are purposely arranged to be located at a same wavelength of 1500 nm.

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Figure 7 shows the calculated transmission spectra of the CEHLPGs working at or near DTPs of LP_1,2, where the cladding diameters 25.18 µm, 25.20 µm, 25.22 µm and 25.24 µm, respectively. length L and grating pitch remain unchanged (i.e., 21.6 mm and 235 µm). The inset in Fig. 7 represents the calculated intensity distribution of the created OAM-carried LP_1,2 mode at wavelength of 1500 nm. As can be seen that when the cladding diameter is adopted as 25.18 µm, the grating works at the DTP wavelength and then a broad resonance dip but with a sharp envelope can be obtained, as shown as the red curve of the Fig. 7. When the cladding diameter is adopted as 25.20 µm, a resonance dip with an ideal flat-top envelope can be obtained, as shown as the blue curve of the Fig. 7. The bandwidth at -20 dB of the obtained flat-top resonance dip is 96 nm. When the cladding diameter is adopted as a greater value such as 25.22 µm and 25.24 µm, a broader resonance band can be obtained due to the large separation of the dual-resonant peaks, but the flat-top envelope will be lost, as shown as the black and green curves of the Fig. 7. The reason for this phenomenon is that, since the mode coupling coefficient for a HLPG is a function of wavelength, the depth of the two separated resonance peaks is somewhat different. All the results mentioned above return indicate that by reasonably adopting the diameter of the CEHLPG, a FTWOAMMC working at the desired radial-order and central wavelength can be flexibly customized, which may support a variety of practical optical vortex applications.

 

Fig. 7. The calculated transmission spectra of the CEHLPGs working at or near DTPs of LP_1,2, where the cladding diameters are 25.18 µm, 25.20 µm, 25.22 µm and 25.24 µm, respectively. The inset represents the calculated intensity distribution of the created OAM-carried LP_1,2 mode at wavelength of 1500 nm.

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3. Fabrication and characterization

Fabrication of the proposed CEHLPG is divided into two steps: fiber twisting and fiber etching. Figure 8(a) depicts the self-built fabrication setup that supports both sapphire-tube-assisted fiber thermally twisting [21] and hydrofluoric-acid-based fiber etching. A SMF is passed through the sapphire tube and fixed in the motorized rotator and fiber clamp. The part of fiber located inside the sapphire tube is without coating. In the first step, a single CO₂ laser beam is directly focused onto the sapphire tube, which functions as a micro-stove capable of generating a homogeneous hot zone internally. [21]. The CO₂ laser employed operates at a high repetition rate of 5 kHz, delivering a maximum output power of 100 W, while allowing for adjustable duty-cycle ranging from 1% to 99%. Heat will soften the fiber inside the sapphire tube to the point where it is overall deformable. By driving the motorized rotator (MR), motorized stages MS₁ and MS_2, a periodically twisted fiber (i.e., the HLPG) can be obtained. Here, MS₁ and MS₂ move in the same direction but with a tiny speed difference which keeps the fiber straight throughout twisting processing. The twisting pitch can be determined by the ratio between fiber twisting speed (speed of the MR) and fiber moving speed (the speed of the MS₁ or MS₂). In the fabrication process, the fiber twisting speed is stabilized at 1440°/s, and the fiber movement velocity is ascertained based on the fixed twisting speed of 1440°/s and the predefined grating pitch. The grating helicity is determined by the traveling direction of the MR. In this case, the left end of the SMF to be processed is spliced with a fiber slip ring (FSR) that is positioned on the left side of the MR in order to monitor the output spectrum of the helical grating in real time during manufacturing. It should be noticed that the helical grating produced by the above-mentioned indirect-heating fiber twisting technique is noninvasive on the surface and thus has a low insertion loss.

 

Fig. 8. Self-built fabrication setup for the CEHLPG. (b) Schematic diagram of the groove holder for fiber etching. (c) Microscope images of the fabricated grating before and after etching processing.

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On the other hand, it is challenging to identify the etching zone when utilizing some ex-situ fiber etching methods [30], [31] since any damage and deformation cannot be seen on the surface of the original helical grating as seen in the upper portion of Fig. 8(c) (i.e., the one before etching). But in our system, operating the MS₃ allows the HLPG to be accurately moved to the groove holder for fiber etching. According to Fig. 8(b), the groove's two ends are each separately bonded with UV glue, which serves as liquid stoppers. There are two processes in the etching process: coarse etching and delicate etching. The written HLPG is submerged in a 25% HF solution during the coarse etching process. Such concentration can guarantee the etched fiber's flat surface as well as a comparatively high etching efficiency. A CCD camera mounted on top of the groove holder can be used to estimate the diameter of the etched fiber in this instance. When the diameter of the etched fiber approaches the target diameter, the concentration of the used HF solution is reduced to 5% for delicate etching. Such low concentration is very beneficial to carefully adjust the suitable diameter and dip-wavelengths. In the lower portion of Fig. 8(c), a microscope image of a manufactured CEHLPG following etching processing is displayed. The proposed on-demand FTWOAMMC can be obtained with high yielding-rate because the output spectrum of the CEHLPG can be monitored in real-time during both the grating writing and fiber etching processes.

Some CEHLPGs have been successfully produced by utilizing the self-built fabrication setup depicted in Fig. 8(a). First, we fabricated an original helical grating working at LP_1,9 mode (designated Sample A), with a pitch and length of ∼235 µm and 22 mm, respectively. In Fig. 9(a), the measured transmission spectrum of sample A is displayed. It is evident that there is just one resonance-dip, which is found at approximately 1350 nm in wavelength. The Sample A was then subjected to the etching procedure. Figure 9(b) depicts the evolution of the transmission spectra for Sample A working at the LP_1,9 mode during the etching process as the fiber diameter decreased. Another resonance-dip can be seen on the right side as fiber diameter lowers, and the two resonance-dips move towards their DTP till they overlap. The depth of the resonance-dips will then gradually decrease until they vanish after the resonance-dips entirely overlap (i.e., the central wavelengths of such two resonance-dips are all located at the DTP wavelength). As shown in Fig. 9(c), a new resonance-dip working at LP_1,8 mode first appeared in the observed wavelength range (it was located at the wavelength of ∼1275 nm) when the fiber diameter dropped to 117.0 µm. Subsequently, a sequence of spectrum-evolutions resembling those for LP_1,9 mode occurred, and the same is true of lower order modes, such as LP_1,7 mode, LP_1,6 mode_, etc. Figure 9(d) shows the transmission spectra of the Sample A working at LP_1,5 mode, where the black and red curves denote the result measured in air and HF solution, respectively. The fiber diameter is ∼65 µm. By comparing these two curves, it can be found that, due to the different of refractive index between HF solution and air, the spectra of Sample A are huge difference; therefore, in order to obtain a FTWOAMMC, the etching processing must be stopped when the two new resonance-dips do not overlap. The red curve in Fig. 9(d) shows that a successful flat-top resonance dip with a maximum depth of ∼20 dB has been achieved. By comparing Figs. 9(b), 9(c), and 9(d), it is clear that when the mode number decreases (i.e., the fiber diameter decreases), the DTP wavelength moves to a shorter wavelength. Such phenomenon may be caused by index change of the fiber cladding produced by the etching processing, which mildly affects dispersion curves of the cladding modes.

 

Fig. 9. (a) The measured transmission spectrum of the Sample A working at LP_1,9 (an original helical grating). The evolutions of transmission spectra with decreasing fiber diameter for the Sample A (b) working at LP_1,9 and (c) working at LP_1,8 during the etching processing. (d) The transmission spectra of the Sample A working at LP_1,5 mode, where the black and red curves denote the one with and without HF solution, respectively.

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Based on the same method, a FTWOAMMC working at LP_1,2 mode (designated Sample B) has been also fabricated, the result is shown in Fig. 10, where the fiber diameter, grating pitch and length of Sample B are ∼26 µm, ∼265 µm and 24 mm, respectively. As can be seen that a nearly ideal FTWOAMMC with a central wavelength of ∼1500 nm and maximum depth of 25 dB is achieved. The -10 dB and -20 dB bandwidth of the flat-top resonance dip are ∼150 nm and 95 nm, respectively. The FTWOAMMC resulting from the aforementioned experiment closely match the predictions made by theory.

 

Fig. 10. The transmission spectra of the fabricated CEHLPGs working at LP_1,2 (named Sample B), where the diameter, pitch and length of Sample B are 26 µm, 265 µm and 21.6 mm, respectively.

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Finally, verification of the OAM mode generated from the realized CEHLPGs has been executed by utilizing the commonly used interferometry. In this method, the OAM mode can be verified by observing the helical interference pattern between two optical beams: one is emitted from the test objective (i.e., CEHLPG) and collimated well; and the other one is emitted directly from a SMF but without collimation. The above two optical paths use the same light source, i.e., a tunable laser. It can be found the details about the detection system in [21]. The Sample B (i.e., the FTWOAMMC working at LP_1,2 mode) is adopted as the test objective. The mode intensity and helical interference patterns at different wavelengths have been measured, Fig. 11 shows the results. The left-handed helical interference patterns have been clearly observed at all the investigated wavelengths, which indicates that the OAM _+ 1 modes are have been successfully created. Furthermore, the measured modes intensity patterns verify that the OAM _+ 1 mode generated by Sample B is LP_1,2 mode_. The above results confirm that the proposed CEHLPG can work as a radial-order and operating wavelength on-demand FTWOAMMC well, and the constructed in-situ processing platform for fabricating the proposed CEHLPG is robust and reliable since such low radial-order FTWOAMMC has been perfectly realized.

 

Fig. 11. Measured mode intensity and helical interference patterns at different wavelengths for Sample B.

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4. Conclusion

In this article, a new FTWOAMMC is proposed and experimentally demonstrated, which is based on utilization of a CEHLPG. By appropriately selecting the grating period and precisely controlling the diameter of the CEHLPG in-situ, both the radial order and central wavelength of the flat-top band for the generated OAM mode can be flexibly tailored according to specific requirements. As typical examples, the first azimuthal order OAM modes with a flat-top bandwidth of 95 nm at -20 dB, a central operating wavelength of ∼1500 nm, and the radial-orders of 9, 8, 5, and 2, respectively, have been demonstrated consecutively. The proposed method provides an excellent flexibility and robustness in controlling both the radial order and the central wavelength of the resulting flat-top wideband OAM mode conversion, which may support a variety of practical optical vortex applications.