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 TM01 and TE01 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 LP11 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 ^{[1618" target="_self" style="display: inline;">–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 HE11 and HE21 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 ^{[2224" target="_self" style="display: inline;">–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 LP11 mode filter, an LP11 mode tap coupler, and a 50:50 power splitter for both the LP01 and LP11 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 (LP01) and the higher-order mode (LP11) 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 diameter=19/125μm, ncore=1.4425, ncladding=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 LP11 mode from the first to the second optical fiber and simultaneously avoid the LP01 mode coupling as far as possible. According to the coupling mode equation, A1 and A2 are the modal field amplitudes in the two TMFs, β1 and β2 are the propagation constants, z is the distance along the coupling region of the coupler, and K12 and K21 are coupling coefficients.

Fig. 1. Schematic of the symmetric TMF-OC.

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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, A2(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 A2(L) will be established after propagation distance L in the second optical fiber. Furthermore, when β1−β2≠0, ej(β1−β2)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 β1−β2=0 can an effective value of A2(L) be obtained. There are two cases for β1−β2=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 β can be calculated as nneffk0, where k0 is the propagation constant in vacuum, and nneff is the mode effective index, which varies with the diameter of the fiber. In this case, in order to meet β1−β2=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 LP11 mode as well as the excitation of the LP11 mode ^[27].

The calculated relationship between the effective indices of the LP01 and LP11 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<nneff<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 LP11 mode decreases faster than that of the LP01 mode, which means the LP11 mode should first break away from the bond of the core and diffuse to the cladding. We use ncladding=1.4375 as the yardstick, which can be seen from the inset of Fig. 2. In detail, the effective index of the LP11 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 LP11 mode. At this point, since the LP11 mode begins to diffuse into the cladding, the coupling of the LP11 mode can occur in the two tapered fibers of the TMF-OC, while for the LP01 mode, the diameter of the tapered fiber when the total reflection condition is no longer satisfied is 26 μm. At this time, the LP01 mode also increases the possibility of coupling into the second optical fiber. It will reduce the output purity of the LP11 mode, which is undesirable for CVB generation. Therefore, in order to couple the LP01 mode to the second fiber as little as possible and thus improve the purity of the output LP11 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 LP01 mode cannot be coupled from the input fiber to the other fiber because the mode profiles have no overlap, while the injected LP01 mode can transfer light energy to the high-order mode LP11 in the other fiber with the LP01 mode suppressed. When the diameter of the tapered fiber is reduced to a certain value, the LP01 mode starts to be coupled to the other fiber, and the purity of the LP11 mode reduces dramatically.

Fig. 2. Mode effective indices of the LP01 and LP11 modes in the TMF versus different fiber diameters at the wavelength of 1550 nm; the inset represents the enlarged part with a tapered fiber diameter from 20 to 70 μm.

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By taking the tapering diameter of ∼30μm as an example, the coupling situation of the LP01 and LP11 modes by the beam propagation method (BPM) has been simulated through the X–Z coordinate plane as shown in Fig. 3. When a bundle of LP01 mode light is directly injected into the core of the first fiber, Figs. 3(a) and 3(b) show the LP01 mode intensity distribution and its power exchange in the coupling region. It can be clearly seen that the LP01 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 LP01 mode in the second optical fiber can hardly be found. For the coupling of the LP11 mode, with a bundle of LP11 mode light injected into the cladding of the first fiber, Figs. 3(c) and 3(d) present the LP11 mode intensity distribution and its power exchange in the coupling region. The simulation results show that the LP11 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.

Fig. 3. (a) LP01 mode intensity distribution and (b) its power exchange in the coupling region; (c) LP11 mode intensity distribution and (d) its power exchange in the coupling region.

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In order to fabricate such a TMF-OC with an appropriate taper diameter for both LP11 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 LP11 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 LP01 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 LP01 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 LP11 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.

Fig. 4. (a) Relationship between the pulled length and the coupling efficiency; the inset represents the relationship between the pulled length and the diameter of the tapered fiber. (b) Output power from port 2 and port 3 of the TMF-OC. Microscopic images of the TMF-OC at the coupling region: (c) at the beginning of coupling and (d) when LP01 mode is observed from port 3 by a CCD camera.

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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 LP11 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 LP11 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].

Fig. 5. CCD images of the LP11 mode excited in the second TMF at different launching wavelengths.

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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 β2 of 69ps2/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 β2 of the TMF is −26.9ps2/km). The other fibers in the cavity are all SMFs with a length of 49.1 m and a dispersion parameter β2 of −21.7ps2/km. Therefore, the net dispersion of the cavity is −1.064ps2. 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.

Fig. 6. Schematic of the passively mode-locked all-fiber laser with a TMF-OC for CVB generation.

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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.

Fig. 7. (a) Mode-locked output spectrum and (b) corresponding pulse train at pump power of 150 mW.

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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 sech2 fitting curve that the pulse duration is about 2.552 ps.

Fig. 8. (a) RF spectrum of the mode-locked laser output measured with 100 Hz frequency span and 1 Hz RBW; inset is RF spectrum with span range of 1 GHz. (b) Measured autocorrelation trace and its sech2 fitting.

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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 LP11 mode with higher purity and avoiding LP01 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 LP11 mode, the LP11 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 LP11 mode is not very strong, and the LP01 mode is bound to the core, resulting in weak optical power coupling. After reaching a higher coupling efficiency, the LP01 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 LP01 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 LP11 mode with an all-fiber configuration.

Fig. 9. Laser output power versus pump power (black curve from output 1 and red curve from output 2).

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Another polarization controller (PC3) is used to refine the final output polarization state and separate the degenerate LP11 mode group so as to generate a high-purity TM01 (radially polarized beam) or TE01 (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 P1 without bending. Then the fiber is bent to curvature radius less than 2 cm, and the output power is P2. At the same time, in order to eliminate the influence of the LP01 mode, the percentage of power loss β when LP01 propagates in the same fiber at the same wavelength should be measured, since the LP01 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−[P2/P1(1−β)], 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.

Fig. 10. Intensity distributions of (a) radial polarization beam and (f) azimuthal polarization beam without polarizer; (b)–(e) show the intensity distributions of the radial polarization beam after passing a linear polarizer; (g)–(j) show the intensity distributions of the azimuthal polarization beam after passing a linear polarizer. Arrows indicate the orientations of the linear polarizer.

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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.