Excitation of high-quality orbital angular momentum vortex beams in an adiabatically helical-twisted single-mode fiber

Kaili Ren; Kaili Ren; Liyong Ren; Liyong Ren; Liyong Ren; Jian Liang; Jian Liang; Li Yang; Li Yang; Jie Xu; Dongdong Han; Yongkai Wang; Yongkai Wang; Jihong Liu; Jun Dong; Hanyu He; Wenfei Zhang; Wenfei Zhang

doi:10.1364/OE.419668

1. Introduction

In recent years, considerable attention has been paid to orbital angular momentum (OAM) beams due to the unique properties of their helical phase. Light beams carrying OAM mode play an important role in various areas, such as optical communications [1–8], vector lasers [9,10], optical tweezers [11], nanoscale microscopy [12], and nonlinear optics [13]. With increasing recognition of OAM vortex beams [14–17], fiber modes carrying OAM have attracted increasing attention owing to their potential in fiber communications [18–23], imaging [12], and sensing applications [24,25]. Especially, the quantum entanglement characteristic of OAM is suitable for high-quality, long-distance, and security information transmission [6–8]. In the vortex optical information system, the functions of quantum encoding, and error correction can be realized through the vortex optical mode conversion, which provides an important method to significantly increase the communication capacity. This new method enables single-fiber mode division, multiplexing, and multi-channel transmission based on the OAM mode. In addition, it could expand the capacity of optical fiber communications, and help ensure that the future information network continues to expand. For long-distance/high-capacity optical communication systems [5,18,19,26], the all-fiber designed OAM modulation devices have the advantages of low insertion loss, high conversion efficiency, and easy integration [27,28]. Moreover, it would enable the generation and transmission of OAM vortex beams in a flexible and compact manner. Therefore, the fiber-based compact and efficient technology for OAM converters has attracted many researchers and this research area has developed rapidly [29–36]. Several fiber-based methods have been proposed for generating OAM optical vortex beams, including twisted multicore photonics crystal fibers (PCF) [23,32,33,37], mechanical-microbend-induced long-period fiber grating (LPFG) in few-mode fiber (FMF) [38], acoustically induced LPFG in two-mode fiber (TMF) [24], and fiber Bragg grating in an OAM-supporting fiber [39]. It is worth noting that the above-mentioned research has mostly been concerned with the implementation of OAM converters based on special optical fibers [40–42], and rarely involved single-mode fibers (SMFs) owing to the uncertain eccentricity of SMFs [34]. Furthermore, one of the greatest challenges is to accurately control the period of the chiral fiber grating structure experimentally to meet its phase matching conditions, and thus achieve vortex optical field generation [34,43]. In addition, few reports have combined theoretical analysis and experimental results to conduct detailed mechanistic studies of the OAM converters in twisted SMFs. Therefore, in this study, the conditions of the OAM mode converters will be theoretically and experimentally presented. Moreover, to address the varied quality of the OAM converters made using SMF [44], a computer-aided, adiabatic fabrication method is used to acoustically induce high-quality chiral long-period fiber gratings (CLPGs). These CLPGs can obviously improve the conversion efficiency of the OAM modes for application.

In this paper, a method to efficiently generate high-quality OAM vortex beams is proposed and demonstrated by computer-aided, adiabatically twisting the SMFs. Note that the mode couplings can be monitored and controlled in real-time during the twisting fabrication process. For the CLPGs fabricated by twisting the SMF, each order mode of conversion has a coupling efficiency of over 99.9%, which is a significant improvement over previously reported values. Moreover, to clarify the mechanism of this OAM beam, we demonstrate how the angular momentum matching condition in couplings determines the conversion of the OAM beams. Both the theoretical and experimental results demonstrate a dual-OAM beam converter with charge ±1 for the right- and left-handed CLPGs, respectively. The achieved forklike and spiral interference patterns confirm the generation of the OAM beams. Furthermore, good agreement is obtained between the theoretical predictions and experimental results. The high-quality OAM beams generated in SMF may find excellent applications in optical communications and OAM-based sensing.

2. Principle

As shown in Fig. 1, CLPG is an optical fiber possessing a refractive index (RI) profile, and is formed by periodically twisting SMFs. The RI profile of CLPG has helical branches and possesses 1-fold rotational symmetry. Therefore, CLPG can control the topological charge of the output beam. In general, because the charge of an OAM beam changes with the mode couplings in CLPG [31], we have described the angular momentum (AM) of the modes in fibers. To analyze the coupling of the OAM modes, we defined a mode basis set in terms of the circular polarized mode as follows:

(1)$$HE_{11}^ \pm{=} ({HE_{11}^x \pm jHE_{11}^y} )/\sqrt 2 \approx ({\hat{x} \pm j\hat{y}} )F_{0n}^{}/\sqrt 2$$

(2)$$HE_{2n}^ \pm{=} ({HE_{2n}^e \pm jHE_{2n}^o} )/\sqrt 2 \approx {e^{ {\pm} j\theta }}({\hat{x} \pm j\hat{y}} )F_{1n}^{}/\sqrt 2$$

(3)$$V_{0n}^ \pm{=} ({TM_{0n}^{} \mp jTE_{0n}^{}} )/\sqrt 2 \approx {e^{ {\mp} j\theta }}({\hat{x} \pm j\hat{y}} )F_{1n}^{}/\sqrt 2$$

where $HE_{11}^ \pm$, $HE_{2n}^ \pm$, and $V_{0n}^ \pm$ correspond to the linear superposition of hybrid HE₁₁, HE_2n, and TE_0n/TM_0n modes with a phase difference of π/2, respectively, which result in the OAM beams. $HE_{2n}^e$ and $HE_{2n}^o$ denote the even and odd vector modes of circular polarization, respectively. Next, through the weak guidance approximation in SMFs, the vector modes can be accurately expressed as linear combinations of LP modes; F_mn corresponds to the radial wave functions of the LP_mn modes. A general linear combination of the two l= ±1 OAM modes has a topological charge with an integer total OAM. With the mode definition [31,45,46], the relationship between the angular order and charges of the spin angular momentum (SAM), OAM, and total AM of the modes, denoted s, l, j, respectively, is shown in Table 1.

Fig. 1. Schematically shown generation of the optical vortex (OV) from the incident Gaussian beam (GB) by CLPG. Insets show intensity distribution of the corresponding fields.

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Table 1. Charges of the SAM, OAM, and AM of CLPGs

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In this CLPG, we can obtain the interaction of transmission light based on the conversion conditions of the linear momentum and AM, which can be expressed as the following phase- and AM-matching conditions:

(4)$${\beta _1} - {\beta _2} - {\kappa _n} = 0$$

(5)$${M_1} - {M_2} \pm 1 = 0$$

where β_1,2 and M_1,2 represent the propagation constants and AM charges of the core and cladding modes that propagate independently in SMF, respectively; κ_n is the twist ratio where κ_n=2π/Λ (where Λ is the twist pitch of the CLPG). In this study, we assume β₁ > β₂, and upper and lower signs + and – are for left- and right-handed CLPGs, respectively. Furthermore, in association with the interactions of OAM and SAM with CLPG, the AM-matching condition in Eq. (5) signifies the conversion of AM in mode couplings, as shown in Table 2.

Table 2. Effectively coupled cladding modes in core and cladding mode couplings in 1-CLPGs

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In the case of right-handed and left-handed CLPGs, it follow from AM and mode coupling selection role, as shown in Eq. (4) and Eq. (5), respectively. When excited by the core modes $HE_{11,co}^ \pm$, effectively coupled cladding modes, the SAM, OAM, and AM charges of CLPGs are summarized in Table 2. The coupling between the core mode and low-order cladding mode is strong compared to that between the core mode and high-order cladding mode; thus the latter can be ignored [45–47]. Based on the above discussions, for a right-hand CLPG, the LCP core mode ($H{E_{11,co}}$) is coupled to the cladding mode (HE_2n) via the AM selective rule. The HE_2n cladding mode belongs to the same almost-degenerate quartet LP_1n as that of TE_0n and TM_0n, and the coupling strength of the LCP core mode to the HE_2n is the same as the coupling strength of the RCP core mode to the TE_0n and TM_0n modes [46]. Therefore, the RCP core mode primarily (equal degree) interacts with the TE_0n/TM_0n modes, while the LCP core mode primarily interacts with the HE_2n modes. As seen in Table 2, OAM beams with a charge of ±1 can be generated in CLPG.

3. Experimental results and discussions

In this study, a continuously twisted standard SMF with inherent core-cladding eccentricity is used to fabricate CLPGs. Consequently, it could to inhibit scattering between these modes, which is extremely difficult to achieve in a high-quality CLPG. The process of fabricating CLPGs is shown in Fig. 2(a). To realize the generation of the vortex optical field, the chiral fiber grating structure parameters should be controlled precisely in the experiment to meet the phase-matching condition of the chiral grating structure. Therefore, we used a flexible technique to accurately fabricate CLPGs by continuously twisting an SMF. In addition, based on the mode coupling between the core mode and cladding mode, the generated optical vortex (OV) in CLPG is investigated. The concept and principle of fabricating CLPGs are shown in Fig. 2(b). Relying on the fact that there always exists some eccentricity between the fiber core and the cladding [34,44], the core follows a helical path inside the cladding; a helical index modulation can be obtained when an SMF is twisted [46]. Based on the above experimental parameters, we twisted an SMF (Corning, SMF-28e(R)) to fabricate the CLPG. Because the SMF is periodically twisted, the propagating core mode receives the perturbation resulting from the core-cladding eccentricity of the twisted fiber. In the experiment, we observe the relationship between the selection of preparation parameters and the quality of CLPG online, and select appropriate parameters to fabricate high-quality CLPG. Firstly, the appropriate laser power must be selected through computer software. Improper laser power will cause excessive loss or even destruction of CLPG. Secondly, study the influence of torsion speed and translation speed on CLPG quality, and select the appropriate speed ratio according to the computer control software of this LZM-100. Finally, according to the online monitoring technology, the appropriate grating twist length is selected on the basis of computer control to achieve high-quality CLPG preparation. The CO₂ laser power is selected to be approximately 6 W for softening and twisting the fiber. A fiber optic rotary joint is adopted to eliminate the accompanying twisting spiral of the monitoring part in the fabrication process, and accurately control the CLPG’s structural parameters. A broadband light source (NKT Phonics, Superk Extreme) is employed as the input light source. Moreover, an optical spectrum analyzer (OSA) (YOKOGAWE, AQ6370C) is used for online monitoring of the transmission spectrum of the CLPG manufacturing process. Therefore, CLPGs can be adiabatically fabricated; the optical fiber deformation damage is extremely small.

Fig. 2. (a) Schematic diagram of the computer-aided CLPG fabrication setup; (b) scheme for CLPG twisted from SMF.

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However, one disadvantage of fabricating CLPG from standard SMF is that because the eccentricity of the core cladding is inconsistent and unknown [34,44,45], the average value of LP_1n in the fiber κ_n cannot be determined before the grating is fabricated. Therefore, to achieve a higher extinction ratio and lower loss in each cladding mode, CLPGs with different eccentricities usually have different and uncertain full coupling lengths, which cannot be determined before fabrication. This is a crucial factor that limits the capacity of CLPGs to achieve light field control. In this study, based on the proposed computer-aided adiabatic manufacturing solution, the abovementioned problems can be resolved and high-quality CLPGs can be readily fabricated. The cladding modes of different orders (LP_1n) have different coupling coefficients and unknown κ_n values [34,44]. As shown in Fig. 2(a), the computer-aided, adiabatic technology can monitor the spectral changes of the grating in real-time, and thus enable the fabrication of high-quality CLPGs with high extinction ratios and low losses that work in different coupling modes. As the twisting length increased, coupling occurred at a wavelength of 1648.52 nm corresponding to LP₁₄ with a twisted length of 32 mm, 1527.48 nm corresponding to LP₁₃ with a twisted length of 38 mm, 1392.68 nm corresponding to LP₁₂ with a twisted length of 40 mm, and 1326.88 nm corresponding to LP₁₁ with a twisted length of 45 mm. The extinction ratio of the main coupling wavelength reached 30 dB. Due to the over-coupling between the core mode and the specific cladding mode, as the L (the length of the twisted SMF) is further increased, the transmission spectrum of CLPG is blue-shifted slightly, and the loss is increased [44].

Compared with the previous method, the proposed computer-aided scheme effectively controls the preparation parameters during the production process to achieve adiabatic preparation of CLPGs. At the same time, clean, stable, and high-precision heating sources ensure the fabrication of high-quality CLPGs. More importantly, the computer-aided scheme can monitor the twisting process, using real-time transmission spectrum observation, to select appropriate experimental preparation parameters and accurately control the twisting length of the grating. Based on the above principles and techniques, this method can easily produce high-quality CLPGs with a large extinction ratio, low insertion loss, and dependable repeatability. As shown in Fig. 3, based on this computer-aided, flexible, and high-precision fabrication technique, the transmission spectra of CLPGs, working in each order of the coupling cladding mode, can be accurately achieved. This is of great significance to sensor application and conversion of mode filtering, laser beam shaping, all-fiber laser shaping, and OAM beam modulation [1,19,20,43]. To further study the mode coupling in CLPGs, seven CLPG samples with different pitches were fabricated for analysis.

Fig. 3. Transmission spectra of four CLPGs with an extinction ratio over 30 dB and a pitch of 800 µm, the corresponding strong coupling at LP₁₄ mode with L₁, LP₁₃ mode with L₂, LP₁₂ mode with L₃, and LP₁₁ mode with L₄, respectively.

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In the fiber, the temperature distribution and the corresponding stress relaxation have a specific relationship. However, heating the fiber causes diffusion changes and stress relaxation; different heating sources and parameters have uncertain effects [34,44]. Therefore, the effective RI of the core and cladding modes change accordingly; the quantitative analysis of this effect is difficult and complicated [44,48]. Based on these factors, simulation and experimental results usually have similar trends, but it is difficult to match them completely; this also leads to difficulty in the preparation of high-quality CLPGs. However, through our new method, we have obtained a strict relationship between the cladding mode order, working wavelength, and pitch. Furthermore, we modified our designed program parameters in the wavelength range of 1100 to 1700nm; Fig. 4. shows the relationship between the resonance wavelength of CLPG (pitch 400 to 1000 µm) and the structural pitch. The points are the experimental data, and the solid lines are the fitting results. Therefore, the grating can be fabricated to satisfy the response wavelength. The second focus of this study was to investigate the use of CLPGs to achieve OAM modulation.

Fig. 4. Wavelengths of cladding-mode resonances versus pitch of the CLPGs.

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Theoretically, when the left-handed and right-handed CLPGs have the same grating period and length, their transmission spectra are identical. However, left-handed and right-handed CLPGs can generate +1 and -1-order OV, respectively [31,34]. In the second experiment, we twisted the SMF to produce left- and right-handed CLPGs. First, we designed a dual-OAM converter based on the couplings of the LCP core mode $HE_{11, co}^ +$ to the cladding modes $HE_{2n}^ +$. To couple resonantly with $HE_{2n}^ +$ at 1550 nm, the twist pitch of the converter was chosen as 715 µm. The length was 25 mm, which was determined by the total power transfer lengths of the couplings. The transmission spectrum of CLPGs with a pitch of 715 µm is shown in Fig. 5. In [44], it was reported that in the wavelength region 1100–1700nm, there only exists non-zero coupling between the core mode LP₀₁ and the hybrid cladding mode LP₁₄ (TE₀₄/TM₀₄/HE₂₄) for CLPG, when the period of grating is approximately 715 µm. Therefore, the cladding number v=2 will only be considered in our case. Based on the right-/left-handed CLPGs, we designed two converters with ±1-order OAM charges.

Fig. 5. Transmission spectrum of the left-handed and right-handed CLPGs with pitch of 715 µm.

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In addition, to verify the theoretical analysis, as shown in Fig. 6(a), a light field detection experimental system is built to measure the interference light field between the vortex light, spherical light, and plane light. To observe the phase of the beam, we investigate the near-mode field output with a near-infrared CCD camera (Raptor, OWL-640) to visualize the mode intensity profile of CLPG. A tunable laser (Amonics, OTWL-C-R) with a wavelength range from 1530 to 1565 nm is used as a single wavelength light source. For CLPGs, the images in Fig. 6(b) show the observed near-mode field at the resonant wavelength of 1550.88 nm. The first column of Fig. 6(b) shows the beam radiating from an untwisted SMF at a wavelength of 1550.88 nm, and the mode field is a Gaussian beam (GB). The second column of Fig. 6(b) shows the mode field of the beam after radiating from CLPG at the wavelength 1550.88 nm. This indicates that the observed resonant dip of CLPG corresponds to the coupling from the fundamental mode to the cladding LP₁₄ mode. However, note that the beam has a dark spot in the center of the mode field. Many theoretical studies have demonstrated that a CLPG with a twisted fiber core can convert the fundamental mode into an OV beam due to the controllable twist defect in the helical fiber, thus changing the OAM of the GB [19,27,31]. From the viewpoint of mode coupling, the generation and conversion of OAM and SAM charges are dependent. Based on the linear momentum and AM conservation in the interaction between light and CLPG (CLPG with 1-fold rotation symmetry in the cross section), the general mode-selection rule of resonant couplings can be obtained. This can be revealed by the following phase- and AM-matching conditions:

(6)$${\beta _{co}} - \beta _{cl}^{14} = {\raise0.7ex\hbox{${2\pi }$} \!\mathord{\left/ {\vphantom {{2\pi } \Lambda }} \right.}\!\lower0.7ex\hbox{$\Lambda $}}$$

(7)$${M_{co}} - M_{cl}^{14} = 1$$

where ${\beta _{co}}$, $\beta _{cl}^{14}$ and ${M_{co}}$, $M_{cl}^{14}$ are the propagation constants and AM charges of the LP₀₁ core mode and the cladding LP₁₄ mode that propagate independently in the SMF, respectively. The AM-matching condition in Eq. (7) signifies the conversion of the AM in mode couplings, which is associated with the interactions of both OAM and SAM with the 1-CLPG. According to the theoretical results [31], the change in the OAM charge is 1. Therefore, in this case, the dark spot in the center of the mode field in the second column of Fig. 6(b) implies that GB is transformed into OV in the helical core [19]. The forklike and spiral interference patterns, as shown in the third and fourth columns of Fig. 6, respectively, confirm the high-quality generation of the OAM beams. We experimentally and theoretically demonstrate the passage of GB through such a twisted SMF to generate OV, with good agreement. This result has significance for the study of beam OAM control and communication.

Fig. 6. (a) Experimental setup for detecting interference between vortex light and plane light. (b) The left-handed and right-handed CLPGs generate the beam profiles and interference patterns of the OAM ±1 mode at a wavelength of 1550.88 nm, respectively. The first two rows are simulation results, and the last two rows are experimental results.

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4. Conclusion

By analyzing the generation and conversion of the OAM beams in CLPGs based on the AM matching condition in mode couplings, we demonstrated that the OAM beams with charge ±1 can be generated by excitation of the fundamental core mode in CLPGs (with 1-fold rotational symmetry in the cross section). The interactions of SAM and OAM accompanied with mode couplings have been reported for single-helix structures, where the change in the SAM charge is determined by the polarization-selectivity of the circularly polarized mode coupling. Real-time monitoring technology is employed in this entire, novel fabrication scheme. Typically, standard SMFs can be used to make ideal CLPGs. According to this study’s theoretical and experimental results, we have successfully demonstrated a computer-aided, adiabatic method to solve the uncontrollable quality effect. As an example, the generation light carried OAM directly in a twisted SMF with more than 99.9% mode coupling efficiency. Therefore, it has significant potential in beam OAM control.

Funding

Xi'an University of Posts and Telecommunications (YJGJ201905); Natural Science Foundation of Shaanxi Provincial Department of Education (19JK0807); Natural Science Foundation of Shaanxi Province (2019JQ-862, 2019JQ-864, 2020GY-101, 2020GY-127); National Natural Science Foundation of China (61275086, 61275149, 61875165).

Acknowledgments

The authors would like to thank the help of Xi'an Scientific and Technological Projects (2020KJRC0013)

Disclosures

The authors declare no conflicts of interest.

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Modes	$H E_{11}^{+}$	$H E_{11}^{-}$	$H E_{2 n}^{+}$	$H E_{2 n}^{-}$	$V_{0 n}^{+}$	$V_{0 n}^{-}$
s	+1	−1	+1	−1	+1	−1
l	0	0	+1	−1	−1	+1
j	+1	−1	+2	−2	0	0

Core Modes	Cladding Modes
Core Modes	Left-Handed	Right-Handed
$H E_{11, c o}^{+}$ (s = 1, l = 0, j = 1)	$V_{o n}^{+}$ (s = 1, l = −1, j = 0)	$H E_{2 n}^{+}$ (s = 1, l = 1, j = 2)
$H E_{11, c o}^{-}$ (s = −1, l = 0, j = −1)	$H E_{2 n}^{-}$ (s = −1, l = −1, j = −2)	$V_{o n}^{-}$ (s = −1, l = 1, j = 0)

Modes	$H E_{11}^{+}$	$H E_{11}^{-}$	$H E_{2 n}^{+}$	$H E_{2 n}^{-}$	$V_{0 n}^{+}$	$V_{0 n}^{-}$
s	+1	−1	+1	−1	+1	−1
l	0	0	+1	−1	−1	+1
j	+1	−1	+2	−2	0	0

Core Modes	Cladding Modes
Core Modes	Left-Handed	Right-Handed
$H E_{11, c o}^{+}$ (s = 1, l = 0, j = 1)	$V_{o n}^{+}$ (s = 1, l = −1, j = 0)	$H E_{2 n}^{+}$ (s = 1, l = 1, j = 2)
$H E_{11, c o}^{-}$ (s = −1, l = 0, j = −1)	$H E_{2 n}^{-}$ (s = −1, l = −1, j = −2)	$V_{o n}^{-}$ (s = −1, l = 1, j = 0)

Excitation of high-quality orbital angular momentum vortex beams in an adiabatically helical-twisted single-mode fiber

Abstract

1. Introduction

2. Principle

3. Experimental results and discussions

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (6)

Tables (2)

Equations (7)

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