Adaptive modal gain controlling for a high-efficiency cylindrical vector beam fiber laser

Wen-Tan Fang; Run-Xia Tao; Yi-Min Zhang; Hong-Xun Li; Pei-Jun Yao; Li-Xin Xu

doi:10.1364/OE.27.032649

1. Introduction

In the past decades, a renewed interest in the transverse mode output in optical fiber laser, including cylindrical vector beams (CVBs) [1–4] and optical vortex beams (OVBs) [5], was sparked owing to their unique optical properties and application. CVBs are laser beams with cylindrical polarization symmetry [1]. The CVBs featuring excellent polarization and spatially symmetry, have a wide range of application such as optical trapping [6], high-resolution metrology [7], surface plasmon excitation [8], electron acceleration [9] and material processing [10]. Recently, attention has been drawn to the high order CVBs due to their properties under high order azimuthal and radial nodes which have potential application in scaling the transmission capacity of optical communication [11] and high power amplifier system. D. Lin demonstrate that CVB can obtain more gain than fundamental mode in a master oscillator power amplifier (MOPA) system. This is mainly due to the significantly larger effective mode area of high-order mode which counteract variety of nonlinear process [12]. The high-order CVBs may be a significant step towards high power fiber amplifier.

Motivated by the demands of practical applications, several methods to generate CVBs in free space and optical fibers have been proposed, which can be generally classified into the passive and active categories. The former points to the conversion of an incoming Gaussian beam into CVBs outside laser cavity by various spatial-variant components such as spatial light modulator (SLM) [13], geometric phase plates (q-plates) [14] and meta-materials [15] are generally required. The active CVBs generation denotes direct oscillation of vector mode in a laser resonator forced by an intra-cavity mode converter including fiber offset splicing technique [16], long period grating [3–4], few mode fiber Bragg grating [17] and mode selective coupler [18–21] and so on. Recently, attention has been drawn to the amplifications and generations of the high order CVBs [22–26].

Careful control over mode-dependent gain (MDG) of multimode fiber amplifier is necessary to ensure all signal modes are launched with optimal power in mode-division multiplexing (MDM) transmission [27]. From another point of view, control of MDG provide an efficient and simple way to generate specific vector beams in a fiber laser. MDG is mainly determined by three factors: (a) the concentration profile of the active dopant ions, (b) the transverse intensity profile of the pump, and (c) the transverse intensity profiles of the signal [28]. Hence, it is possible to obtain specific CVBs output by controlling the mode content of the pump and concentration profile of the doped fiber. Z. Lin and M. Tang, published theoretical work on a fiber laser generating CVBs based on a doughnut-shaped doping profile few-mode fiber [29]. In 2018, T. Wang proposed a wavelength-division-multiplexing mode selective (WDM-MSC), which improve efficient spatial overlapping between pump and signal transverse modes and generate CVBs in the end [16]. However, generation of higher order CVBs with a general theoretical model of a fiber laser based on adaptive modal gain control have not been reported.

In this work, we demonstrate a fiber laser emitting high-efficiency 2^nd CVBs with the control of MDG. A self-designed Ytterbium-ring doped fiber (Yb-RDF) and high-order mode pump are employed to control the MDG for mode selection. Since the propagation modes have different intensity distribution over the cross-section of the Yb-RDF, we analyze the modes competition based on multimode rate equations presented in [30]. The numerical and experimental results demonstrate the feasibility of MDG control by Yb-RDF and LP₂₁ mode pump. The high purity of 2^nd CVB denotes that LP₀₁ and LP₁₁ was depressed well by mode competition within laser cavity. The experimental result also demonstrate an arbitrary single transverse mode fiber laser can be achieved when the suitable of pump mode and fiber design is selected based on the theoretical model.

2. Theory of adaptive modal gain controlling

In the paper, we designed and fabricated the Yb-RDF has the profile shown in Fig. 1, where the ytterbium-doped region is located in a ring-core fiber (RCF). The index profile of RCF (Fig. 1) has a characteristic high-index and low-index ring that serves to achieve vector mode splitting. The doped region is located in the high-index ring. The refractive index profile as a function of the distance r from the center of the core, is given by:

(1)$$n(r) = \left\{ \begin{array}{l} {n_0},0 \le r < {R_{in}}\\ {n_{high}},{R_{in}} \le r < {R_a}({\textrm{Yb}}\textrm{ - doped}{\textrm{Region}})\\ {n_{low}},{R_a} \le r < {R_b}\\ {n_0},{R_b} \le r < {R_{clad}} \end{array} \right.$$

Fig. 1. Ytterbium-ring doped fiber

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We assume a weakly guiding few mode fiber where the modes are well approximated by linearly polarized (LP) modes. The LP mode group includes spatially degenerate modes HE and EH mode. There are four non-degenerate eigen-modes of fibers (TE₀₁, TM₀₁, HE₂₁ odd and HE₂₁ even) representing the 1^st order CVB modes. The other four eigen-modes (EH₁₁ odd, EH₁₁ even, HE₃₁ odd and HE₃₁ even) represents the 2^nd order CVB modes. Thus, CVB modes can be obtained by adjusting the polarization state of a fiber laser emitting LP mode. In the paper, we have targeted the generation of 2^nd order CVB modes. The modal analysis of the fiber are performed by solving wave propagation equation using the finite element.

When analyzing the transverse mode competition in the multimode fiber lasers, it is necessary to present the gain distribution in the transverse direction of the fiber. Each transverse mode has individual field intensity distribution, which interacts with the population inversion and shares it. In the competition by sharing the population inversion, the transverse mode which obtains enough gains will be excited and output. The schematic of our fiber laser with a typical linear cavity is shown in Fig. 2. To obtain high-order vector beam output, we chose LP₂₁ mode as pump mode in the paper. In our research, we assume a two-level system where the amplified spontaneous emission (ASE), the cross-coupling between modes and loss are neglected for simplicity. Considering the interactions between the active dopant and each potential transverse mode, the population inversion distribution is not constant in the transverse direction any more. Based on the rate equations presented in [31], the model we developed here to describe the transverse mode competition in multimode fiber laser can be expressed by the following space-dependent and time dependent steady rate equations:

(2)$$\frac{{{N_2}(r,\varphi, z)}}{{{N_0}}} = \frac{{\sum\limits_i {\frac{{[P_{pi}^ + (z) + P_{pi}^ - (z)]{\sigma _{ap, i}}{\Gamma _{pi}}(r,\varphi )}}{{h{\nu _p}}}} + \sum\limits_j {\frac{{[P_{sj}^ + (z) + P_{sj}^ - (z)]{\sigma _{as, i}}{\Gamma _{sj}}(r,\varphi )}}{{h{\nu _s}}}} }}{{\sum\limits_i {\frac{{[P_{pi}^ + (z) + P_{pi}^ - (z)]{\sigma _{ep, i}}{\Gamma _{pi}}(r,\varphi )}}{{h{\nu _p}}}} + \frac{1}{\tau } + \sum\limits_j {\frac{{[P_{sj}^ + (z) + P_{sj}^ - (z)]{\sigma _{es, j}}{\Gamma _{sj}}(r,\varphi )}}{{h{\nu _s}}}} }}$$

(3)$$\pm \frac{{dP_{pi}^ \pm (z)}}{{dz}} = \left\{ {\int\limits_0^{2\pi } {\int\limits_{{R_{\textrm{i}n}}}^{{R_a}} {[{\sigma_{eq}}{N_2}(r,\varphi, z) - {\sigma_{ap}}{N_1}(r,\varphi, z)]{\Gamma _{pi}}(r,\varphi )rdrd\varphi } } } \right\}P_{pi}^ \pm (z) - {\alpha _{pi}}P_{pi}^ \pm (z)$$

(4)$$\pm \frac{{dP_{sj}^ \pm (z)}}{{dz}} = \left\{ {\int\limits_0^{2\pi } {\int\limits_{{R_{\textrm{i}n}}}^{{R_a}} {[{\sigma_{es}}{N_2}(r,\varphi, z) - {\sigma_{as}}{N_1}(r,\varphi, z)]{\Gamma _{si}}(r,\varphi )rdrd\varphi } } } \right\}P_{sj}^ \pm (z) - {\alpha _{sj}}P_{sj}^ \pm (z)$$

Fig. 2. Schematic of fiber laser

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Where $h$ is Plank constant; $\tau$ is the spontaneous lifetime of the upper lasing level; ${\nu _p}$ and ${\nu _s}$ are pump and signal frequencies; ${N_2}(r,\varphi, z)$ are population densities of the lower and upper lasing levels at the position; ${N_0}(r,\varphi )$ is the doping concentration distribution which is constant along the fiber axis and has a radial symmetry; $P_{pi}^ + (z)$ and $P_{pi}^ - (z)$ are the pump powers of the ${i^{th}}$ transverse mode in the forward and backward directions, respectively; $P_{sj}^ + (z)$ and $P_{sj}^ - (z)$ are the signal powers of the ${j^{th}}$ transverse mode in the forward and backward directions, respectively; ${\sigma _{ap}}$ (${\sigma _{ep}}$) and ${\sigma _{as}}({\sigma _{es}})$ are the pump absorption (emission) and signal absorption(emission) cross section, respectively; ${\alpha _{pi}}$ and ${\alpha _{sj}}$ are the pump and signal of the ${i^{th}} {j^{th}}$ mode loss factors; ${\Gamma _{pi}}(r,\varphi )$ and ${\Gamma _{si}}(r,\varphi )$ are the pump and signal of the ${i^{th}}$ ${j^{th}}$ mode power filling distributions which can be expressed as follows:

(5)$${\Gamma _{pi}}(r,\varphi ) = \frac{{{\psi _i}(r,\varphi )}}{{\int\limits_0^{2\pi } {\int\limits_{{R_{\textrm{i}n}}}^{{R_a}} {r{\psi _i}(r,\varphi )drd\varphi } } }}$$

(6)$${\Gamma _{si}}(r,\varphi ) = \frac{{{\psi _j}(r,\varphi )}}{{\int\limits_0^{2\pi } {\int\limits_{{R_{\textrm{i}n}}}^{{R_a}} {r{\psi _j}(r,\varphi )drd\varphi } } }}$$

Where ${\psi _{i, j}}(r,\varphi )$ denotes the ${i^{th}} {j^{th}}$ mode envelope;

The boundary and initial conditions in this model is given by

(7)$$P_{pi}^ + (z = 0) = {P_{pi}}$$

(8)$$P_{si}^ + (z = 0) = {R_1}({\lambda _i}){P^{- }}_{si}(\textrm{z} = 0)$$

(9)$$P_{si}^ - (z = L) = {R_2}{P^{+}}_{si}(\textrm{z} = L)$$

Where and ${R_2}$ are the reflectivity of the FM-FBG and broadband reflector, respectively. And L is the length of the YDF. Equations (2)–(4) can be solved by using the standard fourth-order Runge-Kutta method given initial conditions for pump and signal power [28]. Gains and noise figures for all signal modes may similarly be calculated.

The quality of the match between signal and pump intensity profiles which can be evaluated by the overlap integral:

(10)$${\eta _{pj, si}} = \int\limits_0^{2\pi } {\int\limits_{{R_{\textrm{i}n}}}^{{R_a}} {rdrd\varphi {\Gamma _{pi}}(r,\varphi ){\Gamma _{sj}}(r,\varphi )} }$$

Then, we optimized the parameters of the fiber, including ${n_{high}}$, ${n_{low}}$, ${R_{in}}$, ${R_a}$ and ${R_b}$, to guarantee the largest gain of LP₂₁ mode. The modal characteristics of the RCF is theoretically analyzed by full vector finite element method. The optimal refractive index distribution across the center of the ring-core fiber is shown in Fig. 3(a). Comparing with the cladding, the refractive index difference of the central region, ring-core, and bound layer are about 0, 0.025, and −0.004, respectively. The annular ring core have high refractive index and the center region have a low refractive. The bound layer with a low refractive index is designed to resist the perturbations and bending loss. At 976 nm and 1060 nm, the RCF supports four LP modes (LP₀₁, LP₁₁, LP₂₁, LP₃₁) with high effective index difference (>10⁻³), which including 14 vector modes (2 modes in LP₀₁, 4 modes LP₁₁ with high effective index difference (>10⁻⁴), 4 modes LP₂₁ with low effective index difference (<10⁻⁵), and LP31 mode not shown). The annular ring core with the n_high index is uniformly doped with Ytterbium as an active fiber. Figures 3(b)–3(d) shows the gain experienced by each signal mode when pumping in the LP_{01, p}, LP_{11, p} and LP_{21, p} modes. It can be seen that the gain distribution of LP_{01, s}, LP_{11, s} and LP_{21, s} mode are quite different due to different intensity distribution of pump modes in the transverse directions of the fiber. And the power LP_{21, p} was continually adjusted to maintain a 2 dB difference between the gains of LP_{21, s} and LP_{01, s}, LP_{11, s.}

Fig. 3. (a) The RIP of the ring-core fiber overlaid with the normalized signal intensity profile of supported modes. (b-d) Modal gain of each signal modes at 1060nm varied with 976 nm pump powers in (b) LP_{01, p} (c) LP_{11, p} (d) LP_{21, p}

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The model in the manuscript was built up to demonstrate the behaviors of a fiber laser with multimode competition. As is shown in Fig. 4, we summarize the three pump LP modes and their corresponding numerical calculation MDG for all LP mode in the fiber into three groups. The pump power was set as 200 mw that is same in the following experiment. As is shown in Fig. 4, the desired LP₂₁ mode will get the highest gain which come from two main factors: (a) the special designs of the active fiber, (b) the optimal selection of the transverse intensity of the pump. The difference value of MDG between LP₂₁ mode and the adjacent two mode was 3 dB. The LP₂₁ mode can extract the population inversion most effectively. The adjacent modes will see an unsaturated gain coefficient that is far below the peak gain. Thus, the gain of the two mode are not too far above threshold of a laser, they cannot oscillate. Then the only LP₂₁ mode will oscillate in the cavity due to the mode competition. By adjusting the PC in the laser cavity, the ideal polarization state of 2^nd CVBs can be obtained.

Fig. 4. The distribution of MDG of three LP signal modes with three LP pump modes.

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3. Experimental setup and results

According to the theory illustrated in section 2, an experiment setup to generate the 2^nd order CVB was proposed. The experimental setup is given in Fig. 5 schematically, where a 50-cm-long Yb-RDF is used as gain fiber. The measured absorption coefficient of the home-made active fiber is ∼31 dB/m at 976 nm. The pump source is fiber coupled 976 nm laser diode, and a mode converter based on a vortex phase plate (VPP, u7155E RPC Photonics) is used to transform the spatial mode of the pump into LP₂₁ mode. Then, the pump laser is coupled into the annular doped Yb-RDF fiber’s core with a coupling efficiency of about 60%. The focal length of L₁ and L₂ are 75 cm and 150 cm respectively. The distance between VPP and the lens L1 is 150 cm. The cavity of the fiber laser comprise a fiber cleavage plane and a few mode fiber Bragg gratings (FM-FBG) written on the home-mad RCF. The FMFBG fabricated in the RCF serve as the highly reflective mirror (reflectivity 93% around 1061.3 nm). The end of the Yb-RDF was cleaved to be flat, yielding a 4% Fresnel reflection coefficient. The laser cavity was formed between the FM-FBG and the end of the Yb-RDF. The FM-FBG are written on the ring-core few mode photosensitive fiber with initial periods of 361 nm, and the reflection spectra are shown in Fig. 6(a). The three peaks in the reflection spectra od the two FMFBGs indicate three kinds of reflection among different order modes. From the left peak to the right peak represents the LP₂₁, LP₁₁, LP₀₁ mode reflection respectively. The laser spectrum shows that the laser oscillate at 1061.3 nm with 3 dB bandwidth of 0.1 nm and a signal-to-background ratio of more than 60 dB. The center wavelength of the laser coincides with the self-coupling reflection peak of the LP₂₁ mode, which indicates the high-order LP₂₁ modes oscillation in the laser cavity. In the experiment, it can be observed that the fiber laser can still operate in LP₂₁ mode stably with fiber bending and perturbations.

Fig. 5. Schematic illustration of the fiber laser

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Fig. 6. (a) Measured spectra of the FM-FBG. The pink line are the reflection spectra of the FMFBG. The blue line shows that the optical spectrum of the output laser. (b) Optical output power versus absorbed pump power. The solid red line is a linear fit for the output over the threshold.

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After passing through a 2^nd order VPP, the pump light is converted into 2^nd order CVBs and coupled into RCF by a collimator. Then the pump light is converted to the LP₂₁mode in the fiber. The intensity profiles of the pump LP₂₁ mode are captured at the end of the RCF using the CCD camera when the spatial mode coupled into a passive RCF (as shown in Fig. 7(a)). The fiber laser can operate at vector fiber mode i.e. CVB by adjusting the intra-cavity PC. The doughnut-shaped pattern of CVB laser can maintain with the increase of pump power (as is shown in Figs. 7(b)–7(d). The output power versus absorbed pump power of the fiber laser is shown in Fig. 6(b). The threshold of laser is as low as 47.73mw. The output of CVBs increase linearly above threshold pump powers and the slope efficiency of the fiber laser is 79.61% respect to absorbed pump power. By using a dichromic mirror, we measured the residual power ratio compared to the fiber laser emission. The residual pump power is quite weak to be 3%. Thus, the absorbed is almost the same with the launched pump power. The maximum output power is 89.6mw at a pump power of 172mw. Owing to the high modal gain difference between LP₂₁ and other modes and passive mode selection of laser resonator, the output doughnut mode has negligible intensity at beam center and no concentric outside the beam, which reveals the high purity of the output CVBs. Using the method described in [11], the mode purities of CVBs with 2^nd order are measured to be 96.3%. The 2^nd order CVB mode purity was tested by the coupled energy of the detected beams into a two-mode fiber. When the laser operate at a stable 2^nd order CVB mode, we obtain 1.1mw output from the two-mode fiber with the launched power 30mw. There is a little coupling loss when the hybrid mode of LP₀₁ and LP₁₁ coupled into the two-mode fiber. Thus, we estimated the mode purity of the 2^nd order CVB to be 96.3%.

Fig. 7. (a) Intensity profiles of the pump LP21 modes captured at the end of the fiber. Intensity profiles of 2^nd order laser CVB output versus pump power (b) 40mw (c) 70mw (d) 180mw

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The CVBs has good modal symmetry that could be recognized from Fig. 8. The polarization states of the output CVB beams can be confirmed by inserting and rotating a linear polarizer between the collimator and the CCD camera. Intensity distributions of the beam after passing through the linear polarizer at four different orientations are shown in the Figs. 8(b)–8(e), revealing the cylindrical polarization distribution of the generated laser beams. More generally and interestingly, tuning the intra-cavity PC will enable generation of all the states on a high-order Poincaré sphere [32]. Comparing to the relevant previous research [33,34], the high-order mode pump allow the fiber laser have higher slope efficiency and high flexibility. In future work, we intend to generalize the present scheme to be applied to efficient generation of higher-order CVB laser and CVBs amplifier. Moreover, comparing to the mode select technique based on the polarization dependence of FMFBG, this operating mechanism is not sensitive to polarization. Thus, the fiber laser has high stability to the environment.

Fig. 8. (a) Intensity profiles of the 2^nd order CVBs without a polarizer and (b)–(e) after passing through a linear polarizer with four transmission axis orientations denoted by the white arrows.

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4. Conclusion

In summary, we experimentally demonstrate a Yb-fiber laser generating second-order CVBs with high efficiency and low threshold, in which a combination of high-order mode pump and annular doping was used to the oscillation of cylindrical vector modes with adaptive modal gain control. Benefiting from the pump mode and doping region of active fiber, the fiber laser power reached 172mw with a high slope efficiency of ∼79.61%. The attractiveness of such approach is linked to its ability to be scaled up for generating single higher order CVBs with high efficiency. Owing to the flexible controllability of modal gain inside the laser cavity, it is possible for using such setup to perform arbitrary CVBs. It is important to note that our setup only perform second-order CVBs due to the unique fiber design and pump mode (see section 2). A high-efficiency fiber laser emitting single high-order mode output can be achieved through adaptive design of fiber and selection of pump modes. Moreover, high-order mode pump modes potentially enable key application in high-efficiency CVBs fiber amplifier and high-dimension entanglement source. In general, the simple and efficient setup opens the opportunity to drastically reduce the energy consumption compared to classical CVBs fiber laser. These characteristics may enable the present technique to play a key role in the advent spatial division multiplexing (SDM) system and high power fiber laser system.

Funding

National Natural Science Foundation of China (61675188); National Key Research and Development Program of China (2016YFB0401901).

Acknowledgement

We would like to thank Prof. Baosen Shi, Dr. Zhiyuan Zhou, Dr. Yinhai Li, Dr. Shilong Liu, Dr. Shikai Liu and Dr. Chen Yang for helpful discussion. We also thank Rui Zhang and Jingqin Su in Jiangsu Hengtong Optical Fiber Technology Com. Ltd. for the fabrication of the fiber.

Disclosures

The authors declare no conflicts of interest.

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Adaptive modal gain controlling for a high-efficiency cylindrical vector beam fiber laser

Abstract

1. Introduction

2. Theory of adaptive modal gain controlling

3. Experimental setup and results

4. Conclusion

Funding

Acknowledgement

Disclosures

References

Cited By

Figures (8)

Equations (10)

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