## Abstract

We demonstrate a passively mode-locked all-fiber laser generating cylindrical vector beams (CVBs) only using a symmetric two-mode fiber optical coupler (TMF-OC) for both high-order mode excitation and splitting. Theoretical analyses show that for a symmetric TMF-OC with appropriate taper diameter, the second-order mode can be excited and coupled into output tap with high purity due to the effective index difference of different modes. Based on the fabricated TMF-OC, the passively mode-locked fiber laser delivers pulsed CVBs at a center wavelength of 1564.4 nm with 3 dB linewidth of 11.2 nm, pulse duration of 2.552 ps, and repetition rate of 3.96 MHz. The purity of both radially and azimuthally polarized beams is estimated to be over 91%. Due to simple fabrication method of the TMF-OC and high purity of the generated CVBs, this mode-locked CVB fiber laser with all-fiber configuration has potential applications in optical trapping, optical communications, material processing, etc.

© 2019 Chinese Laser Press

## 1. INTRODUCTION

Due to their symmetrical properties in both amplitude field and polarization, cylindrical vector beams (CVBs) with radial or azimuthal polarization have recently aroused strong interest recently [1]. A variety of applications have been found in many areas, such as material processing [2], single molecule imaging [3], optical sensing [4], and nanoparticle manipulation [5]. There are many different techniques for generating CVBs, initially using spatial polarization selection components including dual conical prisms [6], birefringent components [7], spatial-light modulators (SLMs) [8], sub-wavelength gratings [9], and others. These conventional methods based on bulky components are incompatible with a fiber-based system. Subsequently, the concept of an all-fiber structure for generating CVBs was put forward with the advantages of flexibility, robustness, and compatibility with an optical-fiber communication system. Nowadays, there are two all-fiber methods for CVBs generation. Specifically, the excitation of ${\mathrm{TM}}_{01}$ and ${\mathrm{TE}}_{01}$ eigenmodes with cylindrical symmetry in few-mode fiber (FMF), has been investigated mainly through two kinds of techniques: mode convertors and selectors, as well as their combination [10]. The mode-selective coupler (MSC), as a typical mode convertor, has been used to convert the fundamental mode in the single-mode fiber (SMF) to the ${\mathrm{LP}}_{11}$ group in the FMF [11,12]. The principle of the MSC is to meet the phase-matching condition of the fundamental mode in the SMF and higher-order mode in two-mode fiber (TMF) and thus achieve fundamental mode conversion to higher-order modes [13,14]. However, due to the diameter mismatch between two different kinds of fibers, the MSC fabrication commonly involves a precise pre-tapering process [15]. Another commonly used mode excitation method is to include offset splicing between an SMF and an FMF to excite multiple higher-order modes, together with a few-mode fiber Bragg grating (FM-FBG) to serve as a mode selector [16–18]. Unfortunately, for this method, excess loss inevitably induced by offset splicing and the narrow bandwidth of the FM-FBG are serious shortcomings for practical applications. More recently, mode coupling between ${\mathrm{HE}}_{11}$ and ${\mathrm{HE}}_{21}$ in a modal interferometer constructed in TMF was proposed, but it involves sophisticated micro-waveguide writing in TMF using a femtosecond laser [19].

In optical communications, sensing, and fiber lasers, fused fiber optical couplers [20,21] made from two of the same fibers (e.g., single-mode and polarization-maintaining fibers) are widely used fiber components to split/combine optical signals between fibers. Recently, FMFs have been prosperous because of their potential application in mode-division multiplexing (MDM) transmission with information transmitted simultaneously on several spatial modes in a single fiber [22–24]. A dissimilar fiber coupler formed between an SMF and a TMF was demonstrated as mode-selective fiber coupler, where the SMF was pre-tapered or etched to achieve the desired phase-matching condition at the coupler waist. The first symmetric fused optical fiber couplers using identical TMFs were presented by Jung *et al.* [25] by choosing appropriate pulling length, which offers various optical functions including an ${\mathrm{LP}}_{11}$ mode filter, an ${\mathrm{LP}}_{11}$ mode tap coupler, and a 50:50 power splitter for both the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes. In the add/drop filter via a whispering-gallery-mode resonator proposed by Huang *et al.* [26], the mode conversion between the fundamental mode (${\mathrm{LP}}_{01}$) and the higher-order mode (${\mathrm{LP}}_{11}$) is in a multi-mode tapered fiber. It is still a challenge to conceive a simple fiber component that can meet the requirements of both higher-order mode excitation and splitting.

In this paper, our theoretical analyses show that for a symmetric two-mode fiber optical coupler (TMF-OC) with an appropriate taper diameter, the second-order mode can be excited and coupled into output tap with high purity. Guided by the theoretical analyses, the symmetric TMF-OC with functions of both higher-order mode excitation and splitting, has been fabricated and used for CVB generation from a passively mode-locked fiber laser. Mode-locked CVB pulses at a center wavelength of 1564.4 nm with a 3 dB linewidth of 11.2 nm and pulse duration of 2.552 ps at a repetition rate of 3.96 MHz have been obtained. The purity of switchable radially and azimuthally polarized beams is estimated to be about 91%. This method features a simple fabrication process of the symmetric TMF-OC and high purity of the generated CVB.

## 2. SIMULATION AND FABRICATION

Figure 1 represents the schematic of a symmetric TMF-OC, composed of two TMFs (core/cladding $\text{diameter}=19/125\text{\hspace{0.17em}}\mathrm{\mu m}$, ${n}_{\mathrm{core}}=1.4425$, ${n}_{\text{cladding}}=1.4375$). A pair of identical TMFs is stretched/fused together to form a common taper waist, in which light can be coupled from one fiber to the other. The principle of the utilization of single TMF-OC for CVB generation is to excite and couple the ${\mathrm{LP}}_{11}$ mode from the first to the second optical fiber and simultaneously avoid the ${\mathrm{LP}}_{01}$ mode coupling as far as possible. According to the coupling mode equation,

Because the light source is injected into the TMF-OC from the first optical fiber, the second optical fiber has no light source input at the beginning, i.e., when $z=0$, ${A}_{2}(0)=0$. With this precondition, we can integrate Eq. (2) and get Eq. (3) as follows:

Equation (3) shows that an optical wave field with amplitude ${A}_{2}(L)$ will be established after propagation distance*L*in the second optical fiber. Furthermore, when ${\beta}_{1}-{\beta}_{2}\ne 0$, ${e}^{j({\beta}_{1}-{\beta}_{2})z}$ is a high-speed oscillation factor. It is impossible to make Eq. (3) equal to an effective value within the coupling distance $L$. This is to say, only if ${\beta}_{1}-{\beta}_{2}=0$ can an effective value of ${A}_{2}(L)$ be obtained. There are two cases for ${\beta}_{1}-{\beta}_{2}=0$. The first case is the coupling between different modes, which has been recently exploited to enable mode coupling and conversion from the fundamental mode to higher-order modes [13]. The propagation constant $\beta $ can be calculated as ${n}_{\mathrm{neff}}{k}_{0}$, where ${k}_{0}$ is the propagation constant in vacuum, and ${n}_{\mathrm{neff}}$ is the mode effective index, which varies with the diameter of the fiber. In this case, in order to meet ${\beta}_{1}-{\beta}_{2}=0$, the diameters of the two fibers should be accurately adjusted to appropriate diameter ratio through the pre-tapering method. The second case is the coupling between same modes, i.e., the mode in the first fiber remains the same mode after coupling to the second fiber. For example, the conventional couplers composed of two SMFs bring about the coupling of the fundamental mode between two SMFs. Here the TMF-OC is designed for the coupling ${\mathrm{LP}}_{11}$ mode as well as the excitation of the ${\mathrm{LP}}_{11}$ mode [27].

The calculated relationship between the effective indices of the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes in the TMF and the fiber diameter by the finite element method (FEM) is shown in Fig. 2. According to the total reflection principle of geometrical optics, total reflection at the interface between the fiber core and cladding is the necessary condition for the light to be confined in the core and propagate forward. This condition can be represented by the mode effective index of optical fibers, which is $1.4375<{n}_{\mathrm{neff}}<1.4425$ (1.4375 and 1.4425 are the refractive indices of the fiber cladding and core, respectively). From Fig. 2, the mode effective index of ${\mathrm{LP}}_{11}$ mode decreases faster than that of the ${\mathrm{LP}}_{01}$ mode, which means the ${\mathrm{LP}}_{11}$ mode should first break away from the bond of the core and diffuse to the cladding. We use ${n}_{\text{cladding}}=1.4375$ as the yardstick, which can be seen from the inset of Fig. 2. In detail, the effective index of the ${\mathrm{LP}}_{11}$ mode first decreases to 1.4375 as the fiber diameter goes down to 66 μm, when the total reflection condition is no longer satisfied for the ${\mathrm{LP}}_{11}$ mode. At this point, since the ${\mathrm{LP}}_{11}$ mode begins to diffuse into the cladding, the coupling of the ${\mathrm{LP}}_{11}$ mode can occur in the two tapered fibers of the TMF-OC, while for the ${\mathrm{LP}}_{01}$ mode, the diameter of the tapered fiber when the total reflection condition is no longer satisfied is 26 μm. At this time, the ${\mathrm{LP}}_{01}$ mode also increases the possibility of coupling into the second optical fiber. It will reduce the output purity of the ${\mathrm{LP}}_{11}$ mode, which is undesirable for CVB generation. Therefore, in order to couple the ${\mathrm{LP}}_{01}$ mode to the second fiber as little as possible and thus improve the purity of the output ${\mathrm{LP}}_{11}$ mode, the diameter of taper waist should be approximately between 26 and 66 μm, according to our calculation. All in all, when the diameter of the tapered fiber is relatively large, the ${\mathrm{LP}}_{01}$ mode cannot be coupled from the input fiber to the other fiber because the mode profiles have no overlap, while the injected ${\mathrm{LP}}_{01}$ mode can transfer light energy to the high-order mode ${\mathrm{LP}}_{11}$ in the other fiber with the ${\mathrm{LP}}_{01}$ mode suppressed. When the diameter of the tapered fiber is reduced to a certain value, the ${\mathrm{LP}}_{01}$ mode starts to be coupled to the other fiber, and the purity of the ${\mathrm{LP}}_{11}$ mode reduces dramatically.

By taking the tapering diameter of $\sim 30\text{\hspace{0.17em}}\mathrm{\mu m}$ as an example, the coupling situation of the ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes by the beam propagation method (BPM) has been simulated through the $X\u2013Z$ coordinate plane as shown in Fig. 3. When a bundle of ${\mathrm{LP}}_{01}$ mode light is directly injected into the core of the first fiber, Figs. 3(a) and 3(b) show the ${\mathrm{LP}}_{01}$ mode intensity distribution and its power exchange in the coupling region. It can be clearly seen that the ${\mathrm{LP}}_{01}$ mode in the first fiber cannot be coupled to the second one at all. No matter the intensity distribution or the power exchange in the coupling region, the ${\mathrm{LP}}_{01}$ mode in the second optical fiber can hardly be found. For the coupling of the ${\mathrm{LP}}_{11}$ mode, with a bundle of ${\mathrm{LP}}_{11}$ mode light injected into the cladding of the first fiber, Figs. 3(c) and 3(d) present the ${\mathrm{LP}}_{11}$ mode intensity distribution and its power exchange in the coupling region. The simulation results show that the ${\mathrm{LP}}_{11}$ mode in the first fiber can be almost completely coupled to the second fiber at an appropriate coupling location. Thus, the results of the dynamic coupling process validate the calculations according to the total reflection principle.

In order to fabricate such a TMF-OC with an appropriate taper diameter for both ${\mathrm{LP}}_{11}$ mode excitation and splitting, it is particularly important to know the relationship between the pulled length and the diameter of the tapered fibers. As shown in the inset of Fig. 4(a), solid and dotted curves describe the relationship between pulled length and fiber diameter. The dotted curve shows the calculated values, and the solid curve shows the measured values under the microscope. The blue curve in Fig. 4(a) shows the relationship between the pulled length and coupling efficiency of the TMF-OC. It can be seen that when the pulled length is 5500 μm, the power of the ${\mathrm{LP}}_{11}$ mode in the first fiber begins to couple to the second one. At this point, the total diameter of the two tapered fibers is about 119.92 μm [Fig. 4(c)], so the diameter of one tapered fiber is about 59.96 μm. When the tapered fibers are furthermore stretched to the diameters shown in Fig. 4(d), i.e., the total diameters of the two tapered fibers are about 58.73 μm (the diameter of one tapered fiber is about 29.37 μm), the ${\mathrm{LP}}_{01}$ mode begins to appear at port 3. When the diameter of a single fiber is stretched thinner than 29.37 μm, the coupling efficiency curve can be seen to rise rapidly. This is because the ${\mathrm{LP}}_{01}$ mode in the first fiber is starting to couple to the second one. From the above experiments, it is found that the TMF-OC with diameter of a single tapered fiber ranging from 59.96 to 29.37 μm can directly serve as an ${\mathrm{LP}}_{11}$ mode exciter and splitter, which is basically consistent with the simulation results shown in Fig. 2. Figure 4(b) shows the output power from port 2 and port 3 of the TMF-OC changing with the pulled length. The initial input power of port 1 is 140 μW. In the process of fabrication, there are repetitive changes in power. However, when the tapered fiber is thinner, the overall power is smaller and the loss is larger, and this general trend remains unchanged. In combination with the coupling efficiency curve of Fig. 4(a), it is not that the thinner fiber is better. The diameter should be optimally chosen so as to obtain the appropriate splitting ratio and smaller loss.

For fabrication of the TMF-OC, two TMFs are aligned together and fused directly using the modified flame brushing technique. With a tunable laser source launched into the first TMF and the beam profile from the second TMF output port monitored by a CCD camera, the pulling process is terminated immediately as a suitable ${\mathrm{LP}}_{11}$ mode spot is observed. After simple packaging, the TMF-OC is ready for use. The mode intensity distributions at the TMF output port of a fabricated TMF-OC with typical tapering diameter of 30 μm are shown in Fig. 5 at different wavelengths. The purity of the ${\mathrm{LP}}_{11}$ mode is about 92% at a wavelength of 1550 nm, and it is over 90% within the 40 nm bandwidth from 1540 to 1580 nm. Note that in order to verify the spatially varying polarization of the CVBs, the polarization should be quantified at every point in the beam cross section, i.e., the polarization vectors across the beam cross section for the polarization purity should be determined [28].

## 3. EXPERIMENTAL SETUP AND RESULTS

The schematic of our proposed passively mode-locked all-fiber laser for CVB generation with a TMF-OC is illustrated in Fig. 6. Similar to conventional mode-locked fiber lasers, the current laser cavity comprises a 980/1550 nm wavelength-division multiplexer (WDM), a 40 cm erbium-doped fiber (EDF) with a dispersion parameter ${\beta}_{2}$ of $69\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$ pumped by a 980 nm laser diode (LD), a polarization-dependent isolator (PDI) for mode locking with a nonlinear polarization rotation technique, an SMF optical coupler (OC), and three polarization controllers (PCs). Exclusively, a TMF-OC with the functions of both higher-order mode excitation and coupling output is critical for CVB generation. The length of the TMF inserted into the cavity is about 1 m (the dispersion parameter ${\beta}_{2}$ of the TMF is $-26.9\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$). The other fibers in the cavity are all SMFs with a length of 49.1 m and a dispersion parameter ${\beta}_{2}$ of $-21.7\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$. Therefore, the net dispersion of the cavity is $-1.064\text{\hspace{0.17em}}{\mathrm{ps}}^{2}$. The two PCs (PC1 and PC2) on both sides of the PDI are used to adjust linear phase delay of the cavity to ensure that the laser is operating in the positive-feedback regime for mode locking. The output beam profiles are captured by a CCD camera placed at output port 2 through a fiber collimator. The optical spectrum, pulse train, and autocorrelation trace are monitored at output port 1.

When the pump power is increased to be above the mode-locking threshold of about 150 mW, the mode-locked operation is easily obtained by adjusting the PC1 and PC2 in the cavity as a mode-locked fiber laser with all SMFs. Figure 7(a) shows the optical spectrum of the mode-locked pulses under pump power of 150 mW measured with an optical spectrum analyzer (OSA). The central wavelength is 1564.4 nm with a 3 dB linewidth of 11.2 nm. The corresponding pulse train is shown in Fig. 7(b), with 25.25 ns alternation between two adjacent pulses.

Figure 8(a) shows the RF spectrum of the laser output. The repetition rate of the mode-locked pulse train shown in Fig. 8(a) is 3.96 MHz. It is indicated that the repetition rate is in agreement with the fundamental repetition rate calculated with cavity length. The electrical signal-to-noise ratio (SNR) measured with 1 Hz resolution bandwidth (RBW) is 46.5 dB in a 100 Hz span range. The inset of Fig. 8(a) is the RF spectrum without spectral modulation in a 1 GHz span range. The measured autocorrelation trace is shown in Fig. 8(b). It is known from the ${\mathrm{sech}}^{2}$ fitting curve that the pulse duration is about 2.552 ps.

Figure 9 shows the output power versus the pump power from two different output ports (black line for output 1 and red line for output 2). The slope efficiency of output 1 is about 0.79%, and the slope efficiency of output 2 is about 0.22%. The relatively low power slope efficiency is determined by the fabrication and design of the TMF-OC. As in the analysis in the previous simulations and fabrication of the TMF-OC, for the sake of generation of an ${\mathrm{LP}}_{11}$ mode with higher purity and avoiding ${\mathrm{LP}}_{01}$ mode coupling into the second fiber as much as possible, the diameter of tapered fiber has a definite range, which limits the coupling efficiency of the TMF-OC. Under the condition of guaranteeing the output of a high-purity ${\mathrm{LP}}_{11}$ mode, the ${\mathrm{LP}}_{11}$ coupling efficiency of the TMF-OC can reach a maximum value of about 20%. This is because the optical fiber cannot be pulled very thin, at this time the evanescent field of ${\mathrm{LP}}_{11}$ mode is not very strong, and the ${\mathrm{LP}}_{01}$ mode is bound to the core, resulting in weak optical power coupling. After reaching a higher coupling efficiency, the ${\mathrm{LP}}_{01}$ mode begins to couple to the second optical fiber because the stretching diameter of the optical fiber is not within the calculated range. At this time, although the coupling efficiency is improved, the increased coupling power is actually the power of the ${\mathrm{LP}}_{01}$ mode. Despite the limitation of coupling efficiency, the fabrication process of this TMF-OC is relatively simple, which is as same as that of single-mode couplers. It does not require pre-tapering like an MSC, and it is a very simple and reliable method to obtain a high-purity ${\mathrm{LP}}_{11}$ mode with an all-fiber configuration.

Another polarization controller (PC3) is used to refine the final output polarization state and separate the degenerate ${\mathrm{LP}}_{11}$ mode group so as to generate a high-purity ${\mathrm{TM}}_{01}$ (radially polarized beam) or ${\mathrm{TE}}_{01}$ (azimuthally polarized beam) mode as shown in Figs. 10(a) and 10(f). The radial polarization and azimuthal polarization of the output mode could be confirmed by recording the intensity distributions by rotating a linear polarizer inserted between the collimator and the CCD camera. Figures 10(b)–10(e) and 10(g)–10(j) show the intensity distributions of the radial and azimuthal polarization beams after passing a linear polarizer with different orientations, respectively. Based on the fact that higher-order modes in FMFs are particularly sensitive to fiber bending, the bending method is utilized to evaluate the purity of obtained radial and azimuthal polarization beams. It has been experimentally verified that the CVBs would completely dissipate with a bending radius less than 2 cm [17]. When the laser operates at an azimuthally/radially polarized mode, the output power is adjusted to ${P}_{1}$ without bending. Then the fiber is bent to curvature radius less than 2 cm, and the output power is ${P}_{2}$. At the same time, in order to eliminate the influence of the ${\mathrm{LP}}_{01}$ mode, the percentage of power loss $\beta $ when ${\mathrm{LP}}_{01}$ propagates in the same fiber at the same wavelength should be measured, since the ${\mathrm{LP}}_{01}$ mode also suffers a small amount of loss. By comparing the different bending loss percentage, the purity of the cylindrical vector beam can be estimated according to $1-[{P}_{2}/{P}_{1}(1-\beta )]$, which is over 91% for both radial and azimuthal polarization beams obtained from our laser with TMF-OC. The slight inhomogeneity of intensity distributions in Fig. 10 may be due to the unflatness of the cut fiber end face used for measurement.

## 4. CONCLUSION

In summary, a passively mode-locked all-fiber laser with a symmetric TMF-OC for CVB generation has been proposed and demonstrated. The symmetric TMF-OC with an appropriate taper diameter used for both higher-order mode excitation and splitting has the merits of a simple fabrication process without pre-tapering and high purity of higher-order mode coupling. Pulsed CVBs at the center wavelength of 1564.4 nm with a 3 dB bandwidth of 11.2 nm and picosecond pulse duration have been achieved from the passively mode-locked fiber laser with the fabricated TMF-OC based on nonlinear polarization rotation. The high purity of about 91% is available for both radially and azimuthally polarized beams. Thanks to the simple method of TMF-OC fabrication and the high purity of the generated CVB, this mode-locked CVB fiber laser may find great applications in optical trapping, optical communications, material processing, etc.

## Funding

National Natural Science Foundation of China (91950105); Natural Science Foundation of Jiangsu Province (BK20161521, BK20180742); Nanjing University of Posts and Telecommunications Talents (NY214002, NY215002); Distinguished Professor Project of Jiangsu (RK002STP14001); Six Talent Peaks Project in Jiangsu Province (2015-XCL-023); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0856).

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