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A novel approach to adaptively control the beam profile in a few-mode fiber is experimentally demonstrated. We stress the fiber through an electric-controlled polarization controller, whose driven voltage depends on the current and target modal content difference obtained with the real-time mode decomposition. We have achieved selective excitations of LP01 and LP11 modes, as well as significant improvement of the beam quality factor, which may play crucial roles for high-power fiber lasers, fiber based telecommunication systems and other fundamental researches and applications.
© 2015 Optical Society of America
1. Introduction
Single-mode (SM) operation (normally referring to the fundamental mode) of the optical fiber is desirable in many applications. The most straightforward example is in present optical fiber communications, where the SM fiber has been employed as the main communication channel for long range links. The SM operation could enable high accuracy in case of high data rate after long distance propagation, thanks to the lack of the intermodal dispersion among multi-mode and low attenuation. In addition, the power scaling of SM beam with excellent beam quality (with M2 ≈1) has been beneficial widely for applications in laser community and other areas [1]. However, one could find that the selective excitation of a specific transverse mode and of a mixture modes with proper fractions in few-mode fibers (FMF) instead of the pure fundamental mode, as a key component of the mode division multiplexing (MDM) technique, have also attracted much attention in recent years. The MDM technique opens a promising route to overcome the capacity limit of the current fiber communication systems based on SM channel only [2]. Besides, the laser beam with specific spatial profile of optical field distribution will help to improve the processing speed and quality for materials processing purposes [3]. Thus, the adaptable control on the beam profile in high-power level would be beneficial to these applications.
Several approaches have been proposed to control the mode content within a FMF until now. Flamm et al. adjust the encoded phase pattern on a spatial light modulator (SLM) to shape the profile of a laser beam before being coupled into a FMF [4]. Moreover, Jung et al. have built a fiber oscillator with the SLM as one of the cavity feedback mirrors [5], known as the digital laser [6]. Thanks to highly mode-dependent phase pattern, selective transverse mode excitation can be achieved by inducing feedback of specific mode when applying specific phase pattern to the SLM. Further, Brüning et al. couple the laser beam from a solid-state digital laser into a FMF to create single transverse mode with high fidelity [7]. In addition, Daniel et al. have also realized adaptable mode excitation by using volume Bragg grating [8] or acousto-optic tunable-filter [9] to select the Bragg resonance wavelengths of different transverse modes induced by multi-mode fiber Bragg gratings (FBG) [10]. However, it is worth noting that the systems mentioned above are relatively sensitive to external perturbations with the free-space optics and require pre-calibration. To overcome those limitations, Lu et al. have proposed an adaptive-optics-based approach to select individual transverse mode, where the feedback signals can be provided by either the correlation between the experimentally measured intensity distribution and the desired mode profiles [11], or the coupling ratio of a directional coupler [12], or the reflection peak of a FBG [13]. Besides, the SLM, divided into several phase blocks (at least 5 × 5), has also been used to convert the beam mode into a FMF in their demonstrations. The phase blocks of the SLM enable adaptive control of the wavefront of the incident beam according to the feedback signal. To optimize the large number of parameters (the phase shift of each individual block) normally needs long processing time (over 100 seconds).
In this work, we experimentally demonstrate for the first time a novel all-fiber system to control the waveguide modes within a FMF adaptively where the feedback signal is provided by real-time mode decomposition (MD) technique. Compared with the aforementioned approaches, the system exhibits a significantly higher process speed with less optimization parameters and the all-fiber configuration contributes to a more compact and efficient system. Conceptually, we illustrate the versatility of the incorporation of the adaptive mode control with the real-time MD, which is essential for further development of the MDM-based fiber communications and high-power applications of complex beam shapes.
2. Experimental setup
The scheme of the experimental setup is shown in Fig. 1. The laser source is a SM narrow linewidth laser diode at 1073 nm (VSLL-1073, Connect Laser Corp.) with SM fiber pig-tailed. The SM beam from laser source then is free-space coupled into an all-fiber polarization controller (PolaRITEIII, General Photonics Corp.), whose delivery fiber at the input and output ports are both SMF-28 compatible step-index fiber with a core with 8.2 μm diameter and 0.14 NA. Since the pig-tailed fiber of the polarization controller (PC) can be considered as weakly guiding, the optical fields within the fiber can be described to good approximation by the linearly polarized (LP) mode [14]. The V-value is 3.36 at 1073 nm, thus only supporting the LP01 and LP11e,o modes, where “e” denotes “even” mode and “o” denotes “odd” mode. By varying the free-space coupling condition of the input port of the PC, one could excite different input mode contents within the pig-tailed fiber. Once the certain input mode content is achieved by adjustment, it will remain fixed during the experiment process. After propagating through the PC, the mode content would be modified, leading to the output mode content different from the input. For the laser beam emitting into free space from the output fiber end, a 4f imaging system with the magnification factor of 37.5, consisting of 2 lenses whose focal lengths are 8 mm and 300 mm respectively, is used to expand the beam. Between the two lenses, a half-wave plate and polarization beam splitter (PBS) are placed to change and choose the state of polarization (SOP) separately. Finally a CMOS camera (Camera 1, Firefly MV FMVU-03MTM/C, Point Grey Research Inc.) is located at the focal point of the second lens to capture the laser beam profile, which is the key component to perform MD to be described below. In addition, in front of Camera 1, a non-polarization beam splitter (nPBS) is used accompanied with another camera (Camera 2, SP620U laser beam analyzer, Ophir-Spiricon Inc.) to monitor and record the beam profile during adaptive control process. Between the second lens and the nPBS, a neutral density filter is employed to adjust the power of the beam within the linear response range of the cameras.
Fig. 1 Experimental set-up of adaptive mode control system (SMF, single-mode fiber; PC, polarization controller; DAQ, data acquisition device; L, lens; HWP, half-wave plate; PBS, polarization beam splitter; nPBS, non-polarization beam splitter; NDF, neutral density filter;f1/f2, focal length of L1/L2;). The SMF-28 fiber is FMF at 1073 nm and the dashed lines represent electrical connections.
Generally, the PolaRITE PC is used to transform the SOP of input beam in a SM fiber to an arbitrary output SOP. The feasibility is realized by 4 fiber squeezers driven by four piezoelectric actuators respectively, while the fiber squeezing produces linear birefringence, effectively creating a fiber wave plate whose retardation varies with the squeezing [15]. In our previous publication [16], we have demonstrated that the fiber squeezing of the PC will mainly induce the variation of both the SOP of LP01 and LP11e,o instead of mode coupling. Since the PBS allows transmission of only one orthogonal polarization, the mode content of the transmitted laser beam with certain SOP depends on the fiber squeezing in the PC. The four channels of the PC, corresponding to the four squeezers, are controlled by 0-5 V analog electrical signal individually. Here we only drive one channel of the PC whose electrical signal voltage is provided by a data acquisition (DAQ) device (NI USB-6009, National Instrument Inc.).
The real-time MD technique is applied to analyze the modal content based on the image of the near-field beam captured by the Camera 1. Then the electrical control signal will be generated according to the MD result to drive the PC through the DAQ device. Finally, the closed control loop is built to achieve the adaptive mode control, consisting of the camera, the PC driven by the DAQ device and the computer. The details on the MD and the adaptive control will be described below.
3. Real-time mode decomposition
For an arbitrary propagating field U(r,φ) in the fiber, one can express it as a superposition of the eigenmodes ψn(r,φ):
where ρn and θn are the modal amplitude and phase respectively. In addition, the beam intensity is defined as .The total number of the eigenmodes is N and all of them form an orthonormal basis set. The normalized propagating field leads to and |ρn|2 stands for the mode weight of the nth mode. In general, the coefficients of eigenmodes are theoretically calculated according to the prior known parameters of the fiber and the scaled parameters [17]. In principle, the MD method enables to achieve the modal amplitudes and phases of all the eigenmodes.Compared with several MD approaches developed previously, such as the spatially and spectrally resolved imaging [18] and all-optically correlation filter method [19], the MD method we applied here is based on numerical analysis, i.e. the mode coefficients are extracted by numerical analysis to minimize the difference between the measured beam profile and the reconstructed one achieved by Eq. (1), leading to the advantage of fast speed, simplicity and low hardware requirement [20]. There are some algorithms used for numerical MD, such as Gerchberg-Saxton algorithm [20], line-search algorithm [21], or simplex-search algorithm [22]. However, the algorithms mentioned-above suffer from long computation time or high sensitivity to the initial value. To address these limitations, we have proposed and demonstrated the numerical MD based on stochastic parallel gradient descent (SPGD) algorithm [23, 24]. Thanks to the superior properties, we have further demonstrated the real-time MD whose details can be referred to our previous publication [16]. It should be noted that the quality metric of SPGD algorithm to be optimized during the MD process is the cross correlation function between reconstructed and measured intensity distributions respectively:
where with j = re, me (respectively denoted the reconstructed and measured intensities) and is the corresponding mean value of the intensity distribution. J has the maximum of 1 in the case of the reconstructed pattern matches the measured one perfectly.Furthermore, one can derive additional beam characteristics based on the optical field reconstructed by the MD, such as the beam quality factor [25] and wavefront [26]. In this paper, the beam quality parameter M2 is directly calculated in real-time based on the decomposed mode weights and phases by using the derived equations in [27], which has been demonstrated in [16].
It is worth noting that the uniqueness of the MD result is an essential issue for the numerical MD method based on the measured intensity profile [21]. As demonstrated by Brüning et al. [21], there is only the uncertainty of the phase sign for two-mode fiber even through only the near-field beam profile is used to perform MD, which is the case of current paper. However, it will not affect the accuracy of the decomposed mode weights, as well as the calculation of M2 according to the formulas in [27].
4. Adaptive control process
The ultimate target of the adaptive control system could be the realization of a pure specific mode, or the combination of the certain modes, or other mode-related parameters such as beam quality or wave-front. For example, to obtain a pure LP01 mode, the quality metric of the close control loop can be expressed as the power content of the LP01 mode. Then the difference between current LP01 mode weight provided by real-time MD and the ideal value (quantity of 1 for pure LP01), serves as the feedback signal to adjust the driven voltage of the PC. Similarly, one could also define the beam quality parameter M2 as the quality metric based on the decomposed mode contents and phases.
For the sake of simplicity, the algorithm to optimize the quality metric by adjusting the driven voltage of the PC is the SPGD algorithm, which is the numerical method we use for the real-time MD [16]. The SPGD algorithm described in [28] is customized below for adaptive mode control purpose:
- (1) Generate statistically independent random perturbation δV of the control voltage V which is small value;
- (2) Apply the positive perturbation V+ = V + δV and perform MD to evaluate the quality metric J+ then apply the negative perturbation V- = V-δV and evaluate the quality metric J-;
- (3) Calculate the difference between the two quality metrics δJ = J+-J-;
- (4) Update the variable V = V + γδJδV where γ is the corresponding update gain. In addition, γ >0 for maximizing J while γ <0 for minimizing J. Iterate the cycle until the performance metric meets the requirements.
5. Results and discussion
With the experimental setup and the optimization algorithm described above, we have adaptively controlled the output beam to achieve optimizations of three different quality metrics. In the first and second case, almost pure LP01 and LP11o mode are adaptively realized respectively. Note that only the LP11o mode, separated from its degenerate LP11e mode, is selectively excited in the second case. In the third case, the target is to minimize the M2. In addition, the adaptive control begins at the 10th step in those cases.
Figure 2 depicts the mode weight of LP01 mode and the corresponding control signal during the process of the optimization in the first case. In the first 10 steps, the system works with the open loop without the feedback signal, where the mode content varies randomly. Once the feedback signal is applied at the 10th step leading to the closed loop, the LP01 mode weight starts to increase under the adaptive control and finally reaches up to 96% over the whole mode content with less than 10 convergence steps. The weight of LP01 mode saturates around 96% and stays constant under 50 iteration steps. The image of the laser beam after the optimization as shown in Fig. 2 also certifies the high agreement with the target compared with the initial beam profile. Besides, the Visualization 1 presents the beam profile recorded by Camera 2 during the optimization process, indicating an excellent evolution of the LP01 mode content.
Fig. 2 The evolution of the LP01 content and the control voltage during the experimental process. The insets show the initial beam and the optimized beam (at 50th step) respectively. Visualization 1 shows the video of the process optimizing the beam to almost pure LP01.
Figure 3 shows the beam profiles “before” and the “after” optimization with four different initial beam profiles which are randomly selected. The results show that the capacity to convert the beam to LP01 with high purity regardless of the initial beam profiles.
Fig. 3 Beam profiles “before” and “after” optimization of four different optimization processes and the corresponding LP01 weight.
In the second case, the quality metric to be maximized is the mode weight of LP11o mode. Figure 4 shows the evolution curves of the LP11o content and the corresponding control voltage on the left and right coordinates separately. The LP11o mode becomes dominant once the closed loop applied. The number of convergence step is about 8 and the weight of LP11o mode finally raises up to 86%. The image of the optimized optical field implies an efficient conversion to the target from the initial state. The evolution of the LP11o mode monitored by Camera 2 is depicted in Visualization 2.
Fig. 4 The evolution of the LP11o content and the corresponding control voltage during the process of selective excitation of the LP11o mode. The insets show the initial beam and the optimized beam (at 40th step) respectively. Visualization 2 shows the process optimizing the initial beam to almost pure LP11o.
Figure 5 shows the trend of the M2 and the applied control signal in the closed loop on the left and right axes respectively. The result encourages the feasibility of the system on beam quality improvement. The value of M2 decreases from about 1.55 to 1.02 (corresponding to the 2nd example shown in Table 1) with 13 convergence steps. Visualization 3 shows the beam profile evolution. Furthermore, Table 1 shows the initial and the optimized values of M2 with the corresponding mode contents of five examples. In these cases, the M2 converges to values less than 1.1 by using adaptive control with the initial values ranging from 1.32 to 1.98. As shown in Table 1, excellent M2 can be achieved even though there are some HOM contents. Additionally, a higher LP01 content associated with a lower M2 also convinces the feasibility of the method according to Table 1.
Fig. 5 The evolution of the M2 and the corresponding control voltage during the process of minimizing M2. The insets show the initial beam and the optimized beam (at 30th step) respectively. Visualization 3 shows the process of optimizing the initial beam to the beam with M2~1.02.
Table 1. Initial and optimized M2 with the corresponding mode contents of five examples
Several different targets and the corresponding processes deserve further comments. First of all, the time for one MD is about 0.1 second [16] and the whole process of each optimization involving 50 iteration steps shown above takes about 20 seconds, while the convergence time converting the initial beam to the target is only about 4 seconds averaged by a number of experiments. It has been dramatically improved compared with the other approaches in [11–13] thanks to the single optimization parameter. The speed is mainly limited by the MD at the present stage, which could be replaced by optimizing the cross correlation defined by Eq. (2) as the feedback similar in [11]. However, the absence of mode information will lead to failure to derive and optimize the beam characteristics (e.g. the beam quality or wavefront). Therefore, for selective mode excitation, one could apply the cross correlation as the feedback signal to speed up conversion. Otherwise the MD should be used to optimize the derived beam characteristics.
It should be noted that the discussion above is only limited to the two-mode fiber. When the number of supporting eigenmodes is more than two, another kind of ambiguity induced by modal interference will inevitably appear [21]. To address this ambiguity, not only the near-field beam intensity but also the far-field one should be utilized for MD, which requires some improvements of the system. For example, another camera recording the far-field beam intensity should be added into the experimental set-up, which should be synchronized with the Camera 1. Besides, the algorithm should be able to calculate the far-field propagation. As a consequence, the real-time ability of MD and the optimization rate of the adaptive control will be dropped down. Additionally, the accurate mode control based on fiber squeezing could remain applicable for the FMF supporting a limited number of modes (e.g. 4 or 6) on the cost of longer optimization time accordingly. But the multi-mode fiber supporting a large number of modes will make a challenge to the proposed technique in this paper because of the highly complicated variation of SOP of the modes induced by fiber squeezing while SLM-based adaptive mode control technique [11–13] is a promising way in such case.
Additionally, the pig-tailed fiber of the laser source can be spliced with the input fiber of the PC, instead of free-space launching. In that case, the feedback signal can be the coupling ratio of a directional coupler [12] or the reflection peak of a FBG [13], indicating the whole system could be potentially further fiberized. As described, the proposed adaptive mode control system in this paper may be potentially alternative to the selective mode excitation by using phase plates [29] in application of mode multiplexing. Besides, wavelength division multiplexing can also be acquired by replacing the narrow linewidth input laser source with a wavelength multiplexing source in the system. Finally, the PC and the PBS, the essential elements for adaptive mode control, will change the SOP of the input beam, so the multiplexing of polarization should not be performed before the selective mode excitation.
Owing to the enormous demand of the high-power fiber lasers with excellent M2, many efforts have been made to suppress the HOM with bad beam quality in the FMF, such as the use of distributed mode filtering through bend loss [30] or inducing delocalization of HOM via special fiber design [31]. However, according to the simulation results of Wielandy [32] and Flamm et al. [25], the quantity of M2 could still be close to one even though some contents of HOM exist in the output beam, which is also verified by the present paper. Therefore, adaptively optimizing the M2 through current system may be a novel way to pursue excellent M2 instead of merely suppressing the HOM.
6. Conclusion
In conclusion, we presented the experimental demonstration of rapid and adaptive control of the modes in a FMF based on a numerical real-time MD technique for the first time. As a proof of concept, the real-time MD is applied to characterize the mode content difference between the measured and target values, which further acts as the feedback signal to the PC for the fiber squeezing control. Not only the pure LP01/LP11 mode can be selectively excited, but also other complex parameters corresponding to the beam characteristics, such as the beam quality, can also be adaptively controlled. Generalization to the effective all-fiber adaptive approach allows a novel level of flexibility for selective mode excitation favoring applications in FMF-based fiber communications and advanced laser processing.
Acknowledgments
This work was supported by National Natural Science Foundation of China (NSFC) (No.61322505).
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