1. Introduction

Cylindrical vector beams (CVBs) are laser beams with cylindrical polarization symmetry [1]. Due to the unique polarization properties, CVBs attract a lot of attention and have been extensively implemented in a wide range of applications, such as optical trapping [2–4], optical communications [5–7], laser fabrication imaging [8–10], and remote sensing [11]. Particularly, great interests have been focused on radially and azimuthally polarized beams, which are two typical CVBs. The radially polarized beams can be used to generate tighter focal spots since the existence of a strong and localized longitudinal field component in the focal region [12,13], which provide benefits to applications in high-resolution imaging, plasmonic focusing and nanoparticle manipulation [1]. On the other hand, the azimuthally polarized beams preserve pure-transverse focal fields even at very high numerical apertures due to the symmetry in a tangential polarization distribution, which advanced applications in material processing and confocal microscopy [13,14]. Moreover, higher order CVBs are recently preferred with the purpose of scaling the transmission capacity in optical communication [5].

Motivated by the demands of practical applications, various methods have been proposed to generate CVBs. In general, depending on whether the cylindrical vector (CV) beam modes are generated directly at source, it can be classified into the passive and active categories [1,23]. For passive CVBs generation, spatial homogeneous polarizations, i.e., linear and circular polarization, are converted into spatially inhomogeneous CV polarizations outside laser cavity [23]. Specially-designed optical elements [15], such as segment half-wave plates [16], space-variant subwavelength gratings [17], geometric phase (q-plates) and meta–materials [18] or optical fiber [19] are generally required. Limitations of this venture lies in that the introduction of extra optical components may lead to beam distortion [20] and increase the complexity of the system [21]. Comparatively, active method has drawn rising attention in terms of compactness, good beam quality and high polarization purity. The basic idea for active CVBs generation is to force the laser oscillating in higher order beam modes instead of fundamental beam mode and deliver the CVBs directly from the laser cavity output [1,22,23]. Whereas in fiber lasers to produce CVBs at source, a mode converter or a mode-selective coupler (MSC) is commonly used. In 2014, making use of the combination of a few-mode fiber and a single mode fiber, R. Ismaeel et al. proposed an MSC, which can produce CV beam modes with high efficiency and purity [24]. Recently, a passively mode-locked Erbium-doped fiber laser generating pulsed CVBs based on an all-fiber MSC has been reported [25]. However, the demonstrated methods basically involve fiber offset splicing technique to excite CV beam modes or few-mode fiber Bragg gratings (FM-FBGs) as MSC to produce CVBs. The multimode fiber is always required in such manners. In order to eliminate mode conversion loss in the fiber, the single mode fiber is preferred in the system rather than the multimode fiber. Moreover, when the pump power increases, not only the mode profile trends to higher order multimode, but also the difficulty of controlling the beam modes will increase. Consequently, a simple and robust single mode fiber laser to generate CVBs with arbitrary polarization orders are highly demanded.

Despite that CVB generation based on a mode-locked Yb-fiber laser without multi-mode fiber components has also been demonstrated [26], only 1st order CVB has been observed. Moreover, the laser is based on nonlinear polarization evolution (NPE). Compared with NPE, nonlinear amplifying loop mirror (NALM) is an ideal mode-locking mechanism for realizing both environmental stability and high output performance in the mode-locked fiber laser [27]. In this paper, we propose a novel scheme to generate arbitrary order cylindrical vector beams based on a single mode fiber laser. An all polarization maintaining (all-PM) nonlinear amplifying loop mirror (NALM) mode-locking Er-fiber laser has been used here. Benefitting from the all-PM fiber configuration and NALM mode locking scheme, the laser could emit a stable mode-locked pulse train with a repetition rate and spectral bandwidth at full-width at half-maximum (FWHM) of 74.9 MHz and 32 nm, respectively. Specially, ultrafast 1st CVBs with different polarization states can be obtained and switched by controlling geometric phases inside the laser cavity. Higher order CVBs are also observed through switching the mode converter in the proposed system. To the best of our knowledge, this is the first demonstration on pulsed CVBs based on Er-doped single-mode fiber laser using NALM as mode-locker. The results represent a significant step toward a compact, stable laser source, which could facilitate many subsequent applications.

2. Experimental setup and principle

The cavity structure of the proposed all-PM CVBs mode-locked laser is illustrated in Fig. 1(a). The laser cavity is designed in a linear configuration, where the NALM serves as one end-mirror. The NALM consists of a wavelength-division multiplexer (WDM), a piece of gain fiber, 2.35 m single-mode PM fiber and a fiber-pigtailed polarization beam splitter (PBS1). The PBS1 is configured that the slow axis of two PM fiber pigtails are aligned with the maximum transmission polarization states [28]. A 0.8 m long Erbium-doped fiber (EDF) (Liekki, Er80-4/125-HD-PM) with group velocity dispersion (GVD) of −29.3 ps/nm/km at 1570 nm is used as gain fiber. A 700 mW single mode laser diode operating at 976 nm is used as the pump. Two counter-circulating pulses in the NALM are combined on PBS1. The combined pulse is then coupled into free space path through a fiber collimator (COL). Note that the PM fibers in the proposed system guarantee the single-polarization operation along the slow axis of the fiber. A nonreciprocal phase shifter (NPS), which consists of a half-wave plate (HWP1), a Faraday rotator (FR) and a quarter-wave plate (QWP), is inserted into the cavity (as highlighted with a shaded panel in Fig. 1(a)). The combined counter-propagating pulses in the fiber loop will interfere at PBS2. With adequate adjust of NPS, the interference on PBS2 will introduce an intensity-dependent loss which acts as an artificial saturable absorber and realizes self-starting mode-locking. In this process, the NPS inserted into the cavity provides an additional phase bias between the counter-propagating pulses in clockwise (CW) and the counter-clockwise (CCW) directions. To this end, the accumulated nonlinear shift in the NALM realize equivalent fast absorber for mode-locking operation. Figure 1(b) exhibits the sequence of pulse propagation together with polarization state evolution before PBS2. For better understanding the working principle of the NALM, we connect two ends of the PBS1 to make it a fiber loop. And one PM fiber pigtail is twisted by 90 degree accordingly [28]. To this end, this process is equivalent to place an HWP inside the fiber loop. Finally, the vertically polarized pulses are coupled out by PBS2, while the horizontally polarized pulses pass through PBS2 and then inject into the CVB generation setup.

 

Fig. 1 (a) Experimental setup. COL: collimator; EDF: Erbium-doped fiber; WDM: wavelength division multiplexer; FR: Faraday rotator; HWP: half-wave plate; QWP: quarter-wave plate; PBS: polarization beam splitter; VWP: vortex wave-plate; M: dielectric mirror with an intensity transmission of 2%. (b) Polarization state evolution before PBS2, NPS, nonreciprocal phase shifter; PMF, polarization maintaining fiber, point and arrow presents horizontal polarization and vertical polarization, respectively.

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CVB management part includes a HWP and a 1^st order vortex wave-plate (VWP, Thorlabs WPV10L-1550). On demand 1^st order CVBs can be directly measured at the output2 after the end mirror with a 2% transmission. The net cavity dispersion is estimated to be −0.0018 ps², and the laser works in stretched-pulse regime. The introduction of HWP2 and the 1^st order VWP into the cavity is crucial to generate at will CVBs. The detailed principle of the CVBs generation is depicted in Fig. 2. As can be seen from Fig. 2(a), the fast axis of the 1^st order VWP rotates continuously over the area of the optic. Thus, each point on the VWP could serve as a half-wave retarder, a linear polarized Gaussian beam will transform into CVBs with doughnut intensity beam profile after passing through the VWP. Moreover, the VWP is polarization dependent. When the fast axis orientation of the retarder changes with respect to the input laser polarization direction, the transmitted pulses can acquire different polarization directions from radially to azimuthally polarization. To this end, the HWP2 placed before the VWP is used to fine adjust the polarization direction of the incident pulse. As evident from Fig. 2(a), four typical CVBs, including radially, azimuthally, clockwise and anticlockwise polarized vector beams, can be switched by adjusting the orientation of the HWP. For instance, when the polarization orientation of the incident light is horizontal polarization, a radially polarized CVB is observed. While an azimuthally polarized CVB is created when the incident light becomes vertically polarized.

 

Fig. 2 (a) Principle for the generation of CVBs. Radially, azimuthally, clockwise and anticlockwise polarized vector beams, can be switched by adjusting the orientation of the HWP. (b) The mathematical derivation for the CVB generation.

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Perfect spatial mode recurrence can be achieved when the Gaussian beam gets a round trip through the VWP. As is displayed in Fig. 2(b), the mode transmission process can be derived using Jones matrix. Table 1 exhibits the Jones matrices of the basic optical elements in our experiment. Since the incident pulses are passing through PBS2, it has horizontal polarization state. Hence, the Jones matrix of incident pulse can be expressed by $\left(\begin{array}{c}1\\ 0\end{array}\right)$. The expression of the laser beam can be expressed through successively multiplying the Jones matrices of each optical element. As evident from Fig. 2(b), the light beam could reproduce itself after each round trip. Besides, the polarization state of the output depends on the angle of the HWP and the VWP. In our experiment, the demanded CVBs are achieved by proper adjusting the angle of a half wave plate. By setting the angle of the HWP relative to fast axis of VWP to be 0°, 22.5°, 45°, 67.5°, the output CVBs can be expressed as $\left(\begin{array}{c}\cos \beta \\ \sin \beta \end{array}\right)$,$\frac{1}{\sqrt{2}}\left(\begin{array}{c}\cos \beta -\sin \beta \\ \cos \beta +\sin \beta \end{array}\right)$,$\left(\begin{array}{c}-\sin \beta \\ \cos \beta \end{array}\right)$,$-\frac{1}{\sqrt{2}}\left(\begin{array}{c}\cos \beta +\sin \beta \\ \cos \beta -\sin \beta \end{array}\right)$, corresponding to the radial, clockwise, azimuthal, anticlockwise polarization CVB. What’s more, it is noteworthy that the adjustment of the HWP2 doesn’t affect the passively mode-locked operation, due to the fact that the polarization will restore after a round trip. As a result, the combination of the HWP2 and VWP ensure that a complete reversible cycle is obtained inside a laser cavity, while the output represents a selected CV beam mode simultaneously. Higher order CVBs can be easily obtained by cascading the VWPs and half wave plates in the linear section of the cavity [29]. In addition, the high transmission efficiency of VWP (>98%) secures the converted CV beam will experience negligible degradation except for the mode conversion loss. Therefore, the proposed system represents a flexible way to directly deliver CVBs with arbitrary order.

Table 1. Jones Matrices of Optical Elements

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3. Results and discussion

3.1 First-order CVBs generation

As mentioned before, the NPS inserted into the cavity provides an additional phase bias between the counter-propagating pulses in clockwise (CW) and the counter-clockwise (CCW) directions. While the accumulated nonlinear shift in the NALM realize equivalent fast absorber for mode-locking operation, the phase bias plays a crucial role for the self-starting. In experiment, reliable self-starting mode-locking operation can be easily obtained with suitable wave-plate angles under certain pump power condition. Initially, multi-pulses operation will be generated in the cavity when the pump power is raised to 530 mW. The laser tends to single-pulse operation with gradually decreasing the pump power. Steady single-pulse operation is obtained when the pump power is above the threshold pump power, which is found to be 430 mW for our current system. Note that, once the mode-locking is established, it can sustain in a wide range of pump power. The laser can preserve the single-pulse output state even when the pump power is increased to be about 500 mW.

As a result of inserting VWP after PBS2, the pulses extracted from output2 are CV beam modes, while the pulses ejected at output1 present Gaussian intensity profile. Figure 3(a) exhibits the average output power of output2 (CVBs, blue) and output1 (Gaussian, gray) of the proposed system at different pump power. A maximum output power of 1.42 mW is obtained for the CVBs at a launched pump power of 500 mW. In comparison, the pulses from output1 realize a maximum output power of 21.2 mW. Different output powers for the Gaussian and CV beam modes can be attributed to the different output ratio for each mode. In addition, we characterize the long-term stability of the generated 1st order CVB. The output power is recorded with a commercial power meter within 30 min, as shown in Fig. 3(b). The root-mean-square (rms) of the output power fluctuation is 0.24%, which shows good environmental stability.

 

Fig. 3 (a) CVBs output power of the laser as a function of the pump power. Mode-locked operation starts at pump power of 430 mW. Inset shows the Gaussian output power of the laser as a function of the pump power. (b) The long-term stability of the generated 1st order CVBs.

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The output spectra for both Gaussian (gray line) and CV (blue line) modes at the maximum output power are displayed in Fig. 4(a). The spectra are measured by an optical spectrum analyzer (YOKOGAWA, AQ6370D) with 0.5 nm resolution. It clearly shows that the pulses delivered from output2 with CV beam mode has a cleaner spectrum than the pulses ejected at output1 with Gaussian beam intensity profile. This could be explained by the mode-locking mechanism and pulses outputting from different cavity locations present totally different properties, which is similar to the previous reported results [27]. The CVB arriving at maximum transmission is passed with relatively low loss, and hence the CVB has a better quality. In contrast, the output1 present the spectrum of two side lobe of the propagation pulses after each roundtrip. The front part and the tail part of the pulses may interfere, which cause the oscillation in the spectrum domain of the Gaussian. The central wavelength and spectral bandwidth at FWHM for CVBs are 1571 nm and 32 nm, respectively. Meanwhile, the measured spectrum of Gaussian beam mode presents a central wavelength of 1560 nm and a 10-dB bandwidth of 74.9 nm. In the following, we mainly characterize the pulses from output 2 since CV beam modes leading to more suitable for practical applications. To further claim the stable the mode-locking operation, the radio frequency (RF) spectrum of the repetition rate is recorded under a frequency window of 5 MHz span (300 Hz resolution) and a window of 1.5 GHz (10 kHz resolution) by a radio frequency spectrum analyzer (RIGO, DSA815) are shown in Fig. 4(b). Clearly, the fundamental repetition rate of the pulse train is measured to be 74.9 MHz, with a signal-to-noise ratio > 80 dB at 300 Hz resolution bandwidth (RBW). No extra frequency sidebands are observed, which confirms the high amplitude stability of the generated pulses.

 

Fig. 4 Typical output characteristics of the NALM-based mode-locked fiber laser. (a) Normalized optical spectra for output1 and output2. (b) RF spectrum over a 5 MHz span with 300 Hz resolution. Inset: RF spectrum over a 1.5 GHz span with 10 kHz resolution.

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Next, various CVB modes are excited with different VWP and HWP orientation settings. When keeping the orientation of 1^st order VWP unchanged and simply rotating the angle of HWP2, 1^st order CVBs with arbitrary polarization state can be obtained. The generated CV beam modes are captured by a commercial high-resolution CCD camera (CCD, Beamage-4M-IR, Gentec-EO Inc., Canada). Four typical CVB modes measured in the far field are plotted in Fig. 5(a), exhibiting doughnut intensity profiles. The polarization distributions are further verified by measuring transmitted intensity profiles after an oriented linear polarizer at the angles of 0, 45°, 90°, and 135°. For example, when the HWP2 is set to an angle of 0°, the donut-shaped vector beam, as shown in the first row in Fig. 5, shows a radially arranged polarization pattern. The measured two lobes are aligned with the orientation of the linear polarizer, indicating that the generated radially polarized beams possess a high polarization purity. In another case, as the HWP2 is set at 45°, the annular beam turns to be in azimuthally polarization mode, which is confirmed by the two-lobed structure being perpendicular to the orientation of the linear polarizer, as presented in the second row in Fig. 5. Furthermore, as plotted in the last two rows of Fig. 5, the clockwise and anticlockwise polarized output modes can also be obtained with adjustment of the HWP2. The resulted two-lobe structures with arrows indicating the transmission direction of the polarizer are also showed in Fig. 5. It has been proven that, by simply controlling the orientation angle of the HWP2, the output beams can be continuously switched among the above four typical CV beam modes. Note that the mode-locking conditions will not change with the optical intensity under different CVBs generation, the output beams have the same pulse properties from output2. Benefit from the active generation method, all CVBs obtained in our experiment exhibit a high mode purity. We characterize and estimate the mode purity of the CVBs to be >94.2% according to the method described in Ref [30]. Consequently, we believe such a switchable, flexible CV laser system could find potential applications in many fields requiring continuous mode switching.

 

Fig. 5 (a) Doughnut-shape intensity profiles for four typical cylindrical vector beams, arrows are used to indicate the polarization distributions. Figures 5(b)-5(e) Transmitted intensity distributions after a linear polarizer in different angles of 0, 45°, 90° and 135°, where the white double-ended arrows indicate the polarizer orientation.

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3.2 High-order CVBs generation

The proposed technique is not limited to 1^st order CVBs generation. Indeed, higher order CVBs could be realized by simply augmenting the order of VWP. The generation of 2^nd order CVBs is achieved just by replacing the 1^st order VWP to 2^nd order. CVBs with even higher orders are slightly complicated, where the 1^st and 2^nd order VWP are employed as the basic unit to compose higher order VWPs. As a proof-of-principle experiment to verify the capability of generating arbitrary-order CVBs, 3^rd order CVBs are obtained by a combination of the 1^st and 2^nd order VWP, together with an HWP positioned between two VWPs [29].

The output parameters of the laser mode-locked at different orders of CVBs are summarized in Table 2. The output spectra of 2^nd order and 3^rd order CVBs for both the Gaussian (gray line) and CV (blue line) modes at the maximum output power are displayed in Figs. 6(a) and 6(b). The central wavelengths for 2^nd order and 3rd order CVBs are 1554 nm and 1551 nm, respectively. Meanwhile, the FWHM of optical spectra of the 2^nd and 3^rd order CVBs are recorded to be 17 nm and 23 nm. The difference in central wavelength and the FHWM of optical spectrum with the 1^st order CVBs is mainly owing to different pump intensities and cavity losses. When the order of VWP increases, the mode conversion loss will increase, as a result the mode-locking pulses tend to operate at shorter wavelength in order to acquire enough gain [28]. Table 2 also shows the output power of output1 and output 2 when the order of CVBs increases. The difference of the pump power is due to the different mode locking states of the laser. Since the cavity losses are totally different when the laser outputs different CVBs, the angles of the HWP1 and QWP are adjusted to be different to fulfil the mode-locking operation. The ratio between the power of output2 and output1 is also presented in Table 2. The variation of the ratio of those two output ports can be also attributed to the different mode-locking states. Due to virtue of this simple and critical design, the proposed system satisfies the self-reproducing conditions, no matter what higher order modes are achieved.

Table 2. Laser Output Parameters at Different Orders of CVBs

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Fig. 6 Normalized optical spectra of high-order CVBs generation at output1 and output2. (a) 2nd-order CVBs generation at output1 and output2. (b) 3rd-order CVBs generation at output1 and output2.

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The intensity profiles of 2^nd and 3^rd order CVBs captured by the CCD also show a doughnut shape which are shown in Fig. 7. The angles of HWP2 are set to α = 0° and α = 22.5°, respectively. As evident from Fig. 7, the intensity distributions of the CVBs with different orders are similar, except for the radius ratios. Higher order CVBs correspond to larger radius ratio. Here, a third-order CVB is generated by inserting a HWP and a 1st order VWP after a 2nd order VWP. In this process, the orientation of VWPs should be accurately homocentric and regulated to be directed parallel to each other, however, it’s hard to realize in the experiment, which results in the non-symmetric central null for the 3rd order CVBs. To further confirm the polarization states of the obtained annular beams, the intensity distributions are recorded by rotating a liner polarizer inserted between the output2 and the CCD camera. The measured intensity distributions after the linear polarizer at four different orientations are shown in Figs. 7(b)-7(e), revealing the cylindrical polarization distribution of the generated pulses. It should be pointed out that the lobe shape remains nearly the same when changing the angle of the polarizer, indicating the highly polarized quality of the generated CVBs. What’s more, owing to passive mode selection of laser resonator, the resultant doughnut mode has negligible intensity at beam center and there exists no concentric ring outside the beam, such a high contrast of the field intensity reveals the high purity of the output CVBs.

 

Fig. 7 Experimentally-obtained 2nd-order CVBs and 3rd-order CVBs. (a) Doughnut-shape intensity profiles for 2nd-order CVBs and 3rd-order CVBs with the HWP2 in angles of 0° and 22.5°. Figures 7(b)-7(e) Petal intensity distributions after a linear polarizer in different angles of 0, 45°, 90° and 135°.

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4. Conclusion

In conclusion, we demonstrated an all-PM passively mode-locked Er-fiber laser using a nonlinear amplifying loop mirror, which is able to directly deliver ultrafast pulses with CV beam profiles at telecommunication wavelength band. HWP and VWP are incorporated in the laser cavity to control the polarization states, thus creating the desired CVBs from radially to azimuthally polarized beam. Benefiting from the cavity design, the polarization and beam mode at any position are controlled and repeated after every round trip. To this end, stable mode-locked operation in all CVBs is obtained at a repetition rate of 74.9 MHz, central wavelength of 1571 nm, and FWHM bandwidth of 32 nm. Moreover, we also show that our cavity has the capability to generate higher order CVBs. Due to the flexible controllability of the polarization states inside the laser cavity, it is possible for using such setup to perform arbitrary CV beam with continuous switching capability. In general, the proposed system is simple but efficient. Thanks to its characteristics of being on-demand and flexible, the proposed laser has great potential for various applications such as optical trapping and optical communication.