## Abstract

Based on the dual-resonance principle around the dispersion turning point (DTP), for the first time, an ultra-broadband orbital angular momentum (OAM) mode converter formed by the helical long-period fiber grating (HLPG) is proposed. The converter used for delivering the OAM operation with 3-dB bandwidth of 287 nm, which is about 7 times of general OAM converters and has only one mode. Furthermore, by chirping the HLPG working around DTP, a flat-top broadband OAM mode converter with bandwidth of ∼182 nm@3 dB is conveniently achieved. The flatness of spectrum can be increased by apodizing and optimizing the length of the chirped HLPG. Subsequently, we significantly developed a flat-top broadband rejection filter with >30 dB bandwidth of a high level of ∼100 nm@1 dB by double-cascading the HLPG. It is shown that the performances of the OAM mode converter and the flat-top broadband rejection filter can be remarkably improved by accomplishing the DTP in the mode phase-matching for the HLPG.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, large attentions have been drawn on orbital angular momentum (OAM) due the unique properties of its helical phase. An OAM beam is dependent on the field spiral distribution, which can be featured by a helical phase front of exp(*ilφ*). In this case, *l* denotes the topological charge number that any integer value is acceptable, and *φ* denotes the azimuthal angle [1–2]. Light beams carrying OAM mode play an important role in various areas, such as optical communications [1–3], atom manipulation [4,5], optical tweezers [6], nanoscale microscopy [7,8] and nonlinear optics [9]. One of the greatest challenges is how to generate a high-quality OAM beams efficiently and at low cost. To date, plenty of schemes for generating the OAM beams have been proposed not only in free space but also in the optical fiber system. As for the OAM generation in free space, several methods are demonstrated, which is mainly based on q-plates [10], spatial light modulators [11], spiral phase plates [12–13], metamaterials-based phase plates [14,15], cylindrical-lens mode converter [16]. However, such OAM mode converters are incompatible with the existing optical fiber communication systems, and additional coupling systems are required for coupling the OAM beams from free space to the optical fiber system, which may introduce undesired coupling loss and free-space transmission loss while also increase the complexity and cost of the system.

In contrast, due to the high flexibility, simplicity, compatibility and low cost, all-fiber generation of the OAM beam has attracted immense interests [17–22]. To date, many efforts have been devoted into the all-fiber generation of the OAM mode, mainly including long-period fiber gratings (LPFGs) [17,18], fiber coupler [19], fibers with offset splicing techniques [20], and all-fiber lasers [21,22]. As an all-fiber versatile device, LPFGs have attracted significant interests for the merits of high conversion efficiency in mode coupling, low insertion loss, easy to fabricate, and low cost [23,24]. As a new type of LPFG, helical long-period fiber gratings (HLPGs) can easily generate OAM modes from the coupling between fundamental core mode and cladding modes due to its special helical grating modulation. Recently, researchers have reported a number of schemes for generating the OAM mode by HLPGs based on different types of fiber, including the standard single mode fiber (SMF) [25], few-mode fiber [26], photonic crystal fiber [27], ring-core fiber [28,29]. For examples, Lang et al. proposed a method for flexible generation of OAM mode in a ring core fiber, and the OAM mode couplings are comprehensively analyzed based on OAM-mode coupled theory [28,29]. Wang et al. has fabricated a polarization-independent OAM generator in a few mode fiber which is insensitive to the polarization state of the input beam [26]. However, like conventional OAM mode converters, this HLPG-based OAM converter also has a narrow bandwidth of ∼25 nm at −10 dB, which has some certain limitations in the OAM mode division multiplexing applications. Thus, all-fiber broadband generation of the OAM mode is strongly desired. Therefore, it has demonstrated a broadband generation of the OAM modes based on the utilization of offset splicing and fiber rotating technique, realizing the generation of the OAM modes in a broad wavelength range of 1530-1566 nm [20]. Unfortunately, as for the offset splicing approach, it would be a challenge to precisely control and stabilize the splicing distant and thus will be technically challenging for practical applications.

In this paper, we proposed an ultra-broadband and flat OAM mode converter based on a HLPG with high-radial-order cladding mode dips near the dispersion turning point (DTP). The dual-resonance mechanism and chirp characteristics of the HLPG at DTP are theoretically investigated by cladding mode coupling theory. The simulation results demonstrate that modes coupling of the HLPG at DTP can significantly improve the bandwidth of the OAM converter with uniform mode purity. At the same time, the introduction of chirp in HLPG, which effectively solve the problem of flatness of the dips. It leads to the proposed flat-top broadband OAM mode converter enables to provide a wide bandwidth of ∼182 nm@3 dB. Finally, by cascading two identical length-apodized chirped HLPGs, a flat-top broadband rejection filter with an extinction ratio larger than 34 dB and a bandwidth of ∼100 nm@1 dB has been readily obtained.

## 2. Broadband generation of OAM mode at DTP

HLPGs with special helical phase modulation can introduce ±π/2 into two degenerated eigenmodes (i.e., ${\mathop{\rm HE}\nolimits} _{2,n}^{{\mathop{\rm even}\nolimits} }$ and ${\mathop{\rm HE}\nolimits} _{{\rm{2}},n}^{{\mathop{\rm odd}\nolimits} }$) automatically with different helical structure (right- or left-handed twist structure), which is considered as a key to generate the OAM mode. Next, for the fact that, through the weak guidance approximation in SMFs, the vector modes can be accurately expressed as linear combinations of LP modes. A general linear combination of the two *l*= ±1 OAM modes have a topological charge with integer total OAM [25,28]. Many experimental demonstrations for the generation of OAM mode in HLPGs with different types of fiber have been reported [25–28]. Figure 1(a) shows the structure of a right-handed HLPG. The HLPG is usually fabricated by periodically twisting an SMF and thus the core of the HLPG follows a helical path due to the existence of small inherent core-cladding eccentricity in the SMF. For the refractive index distribution of HLPG, it has helical branches and possess the 1-fold rotational symmetry. As a result, the HLPG has the ability to control the topological charge of the output beam. In general, for the charge of an OAM beam changes with mode couplings in HLPG, as in [28,29], it has described the angular momentum (AM) of the modes in fibers. Meanwhile, like the conventional LPFGs, mode coupling occurs between fundamental core mode and forward-propagating cladding modes under phase-matching condition [23,30]:

_{01}core mode and the LP

_{1,n}cladding mode, respectively. According to the phase-matching condition, we calculate the first fourteen resonant cladding modes. For clarity, the cladding mode orders

*m*= 1 to 7 and

*m*= 8 to 14 with resonance wavelength ranges of 1.3 µm to 2.0 µm and 1.0 µm to 2.0 µm are depicted in Fig. 1(b) and (c), respectively. In our simulation, the refractive indices of the core and cladding are chosen as 1.449 and 1.4432, respectively. The surrounding index is 1 (air). For the core-cladding eccentricity, which is chosen as a typical 1 µm. The diameters of the core and cladding are 8.3 µm and 125 µm, respectively.

As can be seen from Fig. 1(b), for the cladding mode orders *m *= 1 to 7, the slope of each curve always keeps positive. While the slopes for the cladding mode orders *m* = 8 to 14 will change their sign at special points marked by the black dots as seen in Fig. 1(c), which is called the DTP [31]. Obviously, each DTP corresponds to a maximum grating period for each resonant cladding mode. For a cladding mode of *m *≥ 8, when the grating period is below the DTP, one grating period will correspond to two resonance wavelengths. With the increasing of grating period, the two resonance wavelengths will get closer to each other simultaneously, and finally overlap in the DTP. In such case, the resonance dip at DTP is expected to be broader due to the combination of two resonance dips around DTP.

To verify the broadband transmission characteristic, we simulate the transmission spectrum of the HLPG operating at DTP. The LP_{1,10} cladding mode is considered because its DTP in the vicinity of the telecommunications band. At the DTP of this order cladding mode, the corresponding grating period is 197.3 µm working at the wavelength of 1.612 µm. In this case, the grating length is set as 9.4 mm for total power transfer from LP_{01} core mode to LP_{1,10} cladding mode and can be expressed as:

*C*is the coupling coefficient between LP

_{01}mode and LP

_{1,10}mode. The simulation result is shown in Fig. 2(a). For comparison, the coupling between LP

_{01}mode and LP

_{11}mode is also simulated at the same working wavelength with a grating length of 8.0 mm. The 3-dB bandwidth of the HLPG working at DTP exhibits a broad bandwidth of ∼287 nm, which is almost 7 times larger than that of a HLPG without working at DTP, achieving an ultra-broadband mode conversion. Meanwhile, for the central working wavelength, the extinction ratio of this OAM mode converter is over 60 dB, which suggests that the mode conversion efficiency is almost 100%.

Then we give the simulations of the intensity profile and interferogram of the converted OAM mode at the DTP, which is shown in Fig. 2(b)-(c). It is clearly seen that there is an annular intensity profile with a phase singularity at the center, which is considered as a feature of the OAM mode beam. For the right-handed HLPG, the phase of which changes from –π to π counterclockwise. Figure 2(c) display the interferogram of the converted OAM mode interfered with a plane reference light. The spiral pattern confirms the generation of the OAM_{+1} mode at DTP of the right-handed HLPG. Correspondingly, for the left-handed HLPG, it also has the symmetrical characteristics and results.

## 3. Flat-top response in chirped HLPG

In order to solve the problem that the extinction ratio of the resonance wavelength is not flat in the broadband range, as shown by the red line in Fig. 2(a). In this section, we further introduce a linear chirp into the HLPG which is working at the DTP of LP_{1,10} mode for achieving an ultra-broadband and flat conversion of the OAM mode. By linearly chirping the HLPG’s period, the grating period is no longer a constant but changes along the fiber axis linearly. For a chirped HLPG with length of *L*, the grating can be divided into M uniform sections, and each section is a uniform HLPG with length of *L*/M. The number of sections depends on the required accuracy and is typically taken as M ∼100 [32]. The period of each section is given by:

_{0}corresponds to the starting period, z represents the position along the fiber axis ranging from 0 to

*L*, and

*c*is the chirp coefficient, which is given by: where the ΔΛ is the total period change of chirped HLPG. Then the whole chirped HLPG can be simulated through the piecewise-uniform approach. According to the phase matching condition expressed in Eq. (1), each section corresponds to a different resonance wavelength, and all the resonance wavelength will be combined under the chirped HLPG, realizing broadband transmission in spectrum.

Generally, to achieve a broadband transmission in chirped LPFG, it also requires relatively wide range of grating period changes. In this case, grating with great length is needed for improving the coupling strength at each resonance position of the spectrum. In Ref. [33], a 3-dB bandwidth of 100 nm requires a grating period change of 41 µm, and the required grating length is 41.5 cm, which is inconvenient for its practical applications. In addition, there are multiple orders of coupling modes in this working band range, which is extremely disadvantageous to the mode conversion efficiency and purity of the OAM converter, and its performance and application are limited.

As for the chirped HLPG working in LP_{11} cladding mode, a broadband resonance wavelength range of 1.495 µm to 1.740 µm requires a large period change of 225 nm ranging from 520 µm to 745 µm as shown in Fig. 3(a). As such, according to Eq. (4), a long grating length is needed for reducing the chirp coefficient to improve the coupling strength of the grating. In addition, it can be seen in Fig. 3(a) that the LP_{12} mode will also participate in the coupling with LP_{01} mode under the same chirped grating period, as a result, the resonance wavelength ranges of the two cladding modes will overlap each other, resulting in a large overlapping band from 1.528 µm to 1.740 µm (between red dashed line and black solid line), which may seriously affect the quality of the generated OAM mode. In contrast, for the chirped HLPG operating around the DTP, the same resonance wavelength range can be obtained in a rather small grating period change of ∼1 µm as can be observed in Fig. 3(b). Due to the rapidly change of the resonance wavelength around the DTP, a short grating length is sufficient for forming a wide resonance wavelength range in such chirped HLPG. Furthermore, it is apparent from the Fig. 3(b) that there is only one cladding mode (LP_{1,10}) within the resonance wavelength range of 1.495 µm to 1.740 µm under 1 µm grating period change from 197.3 µm to 196.3 µm, ensuring the high quality of the generated OAM mode without interference from other undesired cladding modes. In the simulation, the total grating length of the chirped HLPG is set as 9.4 mm, which is the same as the non-chirped HLPG mentioned above. And the chirped HLPG is uniformly divided into 100 sections with a constant section length of 94 µm, and the final transmission spectrum of the whole grating can be obtained through the piecewise-uniform approach [32].

The chirped HLPG is schematically shown in Fig. 4(a), the grating period linearly decreases along the fiber axis. The variation of the refractive index of a fiber core along the *z* direction for the chirped HLPG is shown in Fig. 4(b). As mentioned before, the chirped could be uniformly divided into M sections (*z*_{1}, *z*_{2}, …, *z*_{M}) with a section length of *L/*M. In our simulation, the number of the grating section M is chosen as 100. Figure 4(c) compares the transmission spectra between the chirped and non-chirped HLPG operating around the DTP. The chirped HLPG with 1 µm grating period change ranging from 197.3 µm to 196.3 µm is adopted in this simulation, corresponding to a resonance wavelength range from 1.495 µm to 1.740 µm. After chirping the HLPG, a broad 3-dB bandwidth of ∼328 nm has been successfully obtained, which is slightly larger than that of the non-chirped HLPG. Meanwhile, it is worth noting that not only the bandwidth, but also the depth of the dip is tailored after chirping the HLPG, and a flat-top response is obtained by sacrificing the extinction ratio of this resonance dip. In general, the OAM mode generation with uniform purity in the working band does not seem to be an issue for the narrowband OAM applications, but would be significant for the broadband OAM applications, especially for the ultra-broadband OAM applications. Attributing to the resulted flat-top response of the chirped HLPG, the generated OAM mode would have a uniform mode purity in wide wavelength range from 1.534 µm to 1.716 µm. In fact, as we mentioned above, the broadband spectrum of the chirped HLPG is formed by combining all the different resonances, however, the varying grating period shortens the effective length of the HLPG for each grating period, which reduces the coupling efficiency of the grating. Therefore, the chirped HLPG will show a smoother and flatter transmission spectrum compared with that of the uniform HLPG, as shown in Fig. 4(c). Finally, a flat-top band is readily formed with a bandwidth of 182 nm@3 dB.

From Fig. 4(c), we can see that there is a small concavity in the middle of the flat-top band, which affects the flatness of the resulted spectrum. To further improve the flatness of the OAM mode converter, we shape the spectrum by apodizing the section lengths of the chirped HLPG with linear function and raised-cosine function. The structure of the length-apodized chirped HLPG is similar to that of the chirped HLPG but with varying grating section length. As well know that the coupling strength of the LPFG is closely related to the grating length, so, if the grating section lengths are apodized to better match with the coupling strength of each wavelength position, the flatness of the flat band will be improved accordingly. As shown by the blue line in Fig. 4(c), the small concavity of the spectrum indicates that the coupling strength of this wavelength position is slightly smaller than both sides of the band. Therefore, two apodization functions are utilized for improving the flatness of the resulted transmission spectrum. For the first one, a linear function is adopted for apodizing the section lengths, which can be expressed as:

where*L*is the total grating length,

*N*is the number of the divided sections,

*n*denotes the

*n*-th order grating section ranging from 1 to

*N*, and

*N*is the sum of the arithmetic progression from 1 to

_{sum}*N*with a common difference of 1. Then another raised-cosine function is also adopted for shaping the spectrum, and the length for the

*n*-th section can be expressed as can be expressed as: where

*F*(

*n*) is the raised-cosine function and is given by: where ${z_n} = L\frac{n}{N}$ is the longitudinal position of the

*n*-th section of the chirped HLPG ranging from 0 to

*L*. Note that after apodizing, both the linear and the raised-cosine apodization functions only rearrange the section lengths according to their own trends but without changing the total grating length of the chirped HLPG, i.e., the sum of all section lengths of both linear apodization and raised-cosine apodization keep in 9.4 mm unchanged. The variation of the section lengths versus the grating position is shown in Fig. 5(a). Obviously, both the linear apodization and raised-cosine apodization exhibit a declining trend. In this case, the grating section with a period close to the DTP possesses a longer coupling length while the grating sections with a period far away from the DTP have a shorter coupling length compared with the uniform section length we illustrated before. Therefore, the resonance wavelength position near the DTP will be more fully coupled, thus extinction ratio of the flat-top band will be more uniform, and the flatness can be accordingly improved.

The spectra of the chirped HLPG before and after apodizing are shown in Fig. 5(b). It can be obviously seen that after apodizing, the flatness of the loss dip has been improved compared with the original spectrum (green line). In addition, the extinction ratio has also been slightly improved after linear or raised-cosine apodization of this grating.

For the chirped HLPG operating around DTP, a single grating period corresponds to two different resonance wavelengths distributed on both sides of the DTP because of the dual-resonance mechanism. As such, the required coupling length for the two wavelength positions is different. However, the two resonance wavelengths correspond to only a single grating period, and a single period can only correspond to one section length, which means that it is impractical to design the coupling length of each dual-resonance wavelength separately. Then we adjust the total length of the chirped HLPG. Note that for convenience, only the chirped HLPG under raised-cosine apodization is chosen for the all following simulations. Figure 6(a) shows the transmission spectra of the chirped HLPG with grating lengths of 9.4 mm and 9.45 mm. By adjusting the total grating length to be 9.45 mm, the resulted spectrum shows an almost perfect symmetrical flat-top band with a depth of ∼17 dB (The efficiency of OAM conversion with 98%) and a bandwidth of ∼95 nm@0.5 dB. The reason for this symmetrical band could be attributed to the fact that after increasing the total length of the grating, the length of each section of the grating will be increased accordingly, thus for a wavelength position to the left of the DTP, the coupling strength of this wavelength position will be improved due to the increase of the section length. In contrast, for the wavelength position to the right of the DTP that coupled from the same grating section, as illustrated before, the coupling coefficient of such position is larger than the one to the left of the DTP, the coupling strength is decreased instead due to the over coupling under the increased section length. Therefore, a flat-top band with perfect symmetrical structure can be readily obtained.

Additionally, it is worth noting from Fig. 6 that the high flatness as well as the broad bandwidth of the transmission spectrum make the length-apodized chirped HLPG much suitable for operating as a flat-top broadband rejection filter, except for the relative low extinction ratio of −17 dB. For solving this problem, we used a cascading method to further improve the depth of the resulted dip. By cascading two identical length-apodized chirped HLPGs, the fundamental core mode will undergo double couplings with the LP_{1,10} mode, thus improving the extinction ratio of the dip. In order to prevent interference of the two gratings [34,35], in this simulation, we assume that there is 10 cm separation between the two chirped HLPGs equipped with the fiber coating. Therefore, the cladding mode resulted in the first grating will be fully absorbed by the fiber coating [30,34], the spectrum of the chirped HLPGs can be considered as a direct superposition of two transmission spectra that result from the two-individual chirped HLPGs. The spectrum of the cascaded chirped HLPG under raised-cosine apodization of the section lengths is plotted in Fig. 7(b), in which the depth of the dip has been efficiently improved to be ∼34 dB with a bandwidth of 95 nm@1 dB. As we all known that the higher radial order cladding mode corresponds to smaller fiber grating period. Therefore, for the fabrication scheme of the HLPG working at/near DTP, the speed and accuracy of the rotation motor should be high enough to produce such small grating period, and the heater that softens the fiber also needs to have the period size.

## 4. Conclusion

In conclusion, we have proposed an ultra-broadband OAM mode converter based on the HLPG operating at the DTP of the cladding mode. An OAM mode converter with 3-dB bandwidth of ∼287 nm is readily achieved, which is almost 7 times larger than that of the HLPG without operating at DTP. Moreover, based on the dual-resonance mechanism, a flat-top broadband OAM mode converter with bandwidth of 182 nm@3 dB is achieved by chirping the HLPG under a small period change, ensuring a uniform purity of the generated OAM mode. Furthermore, the flatness can be further optimized by apodizing the length of the grating and adjusting the total grating length according to the variation of the coupling coefficient. As a result, the OAM mode converter with a broadband of ∼100 nm@0.5 dB has been achieved. Finally, by cascading two identical length-apodized chirped HLPGs, a flat-top broadband rejection filter can be readily constructed with a broad bandwidth of 95 nm@1 dB.

## Funding

National Natural Science Foundation of China (61275086, 61275149, 61535015, 61875165); Natural Science Foundation of Shaanxi Province (2019JQ-862, 2019JQ-864); Natural Science Foundation of Shaanxi Provincial Department of Education (19JK0807).

## Acknowledgments

The authors would like to thank Chengfang Xu and Xudong Kong in Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Science for the help.

## Disclosures

The authors declare no conflicts of interest.

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