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Abstract
We propose and demonstrate a method for the adaptive phase correction of dynamic multimode fiber beams. The phase of incident beam is reconstructed in real-time based on the complete modal information, which obtained by using the modal decomposition of correlation filter method. For the proof of principle, both of the modal decomposition and the phase correction are implemented using the same computer-generated hologram, which was encoded into a phase-only spatial light modulator. We demonstrate the phase correction of dynamic multimode beam at a rate of 5 Hz and achieve a 1.73-fold improvement on the average power-in-the-bucket. The experimental results indicate the feasibility of the real-time phase correction for the large mode area fiber laser by adaptive optics.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Large mode area (LMA) fiber plays an essential role on the suppression of nonlinear effects in the high power fiber laser (HPFL) by enlarging the fundamental mode (FM) field area and lowering the power density in the core [1,2]. As the core diameter increases, the number of supported propagating higher-order modes (HOM) in the fiber core will accordingly increase, which will cause a multimode (MM) operation and generally degrade the beam quality [3,4]. Furthermore, the onset of transverse mode instabilities (TMI) will sharply degrade the beam quality when the output power exceeds a certain threshold, even if the initial beam quality is excellent and the HOM content is low [5,6]. Thus, it appears that the key problem for LMA fiber is to enlarge the mode field area (MFA) and simultaneously realize effective single-mode operation. To solve this, many mode controlling approaches have been proposed based on the principle of increasing the HOM loss while reducing the FM loss. Among these approaches, bending is the most direct way, but it is not capable to discriminate the HOMs when the core diameter becomes larger than 20 μm [7] and TMI still occurs at high-power level [8,9]. Novel fiber designs, such as large pitch fiber (LPF) [10], leakage channel fiber (LCF) [11], chirally- coupled-core (CCC) fiber [12], et al. have been presented to realize effective single-mode. However, these fibers should be critically designed, which greatly increases the difficulty of the manufacture technique. More importantly, none of these methods fully utilize the gain ability of the active fiber because HOMs are mostly lost, which indeed limits the output power. From another perspective, if the beam quality of the MM beam could be reformed, it would be beneficial to allow the presence of HOMs and to fully use the gain ability of the active fiber.
Actually, the phase correction of a static single HOM has been demonstrated by using the binary phase plate [13–16] or the interferometric element [17]. However, these rigid devices are not capable to correct the dynamic phase of the beam with more than one HOM. Comparing with that, the adaptive optics (AO) technique is more competent, which has successfully achieved the dynamic correction of the phase of the high power solid-state lasers [18–20]. Generally, in the AO systems, there are two ways for the close-loop control. One is based on the iterative optimization algorithm [21–25] and the other one is based on the phase conjugation [26]. However, both of them have some deficiencies in the application of MM fiber laser. For the first one, the iteration process of optimization algorithm often takes more than hundreds of steps and several minutes due to the large number of optimization parameters, thus the previous experimental demonstrations have only involved static MM beams [21–24]. For the second one, current wavefront sensors (WFS) (such as the Shack-Hartmann wavefront sensor (S-H WFS) and lateral shearing interferometer (LSI) [27]) are insufficient to accurately measure the phase steps in the phase of MM beams [25,28–30]. To the best of our knowledge, the dynamic correction of the MM beam has not been reported.
In this paper, we proposed a method for the phase correction of dynamic MM beam. In the method, the phase of incident beam is reconstructed based on the modal decomposition (MD) technique. The MD and phase correction are implemented with a set of transmission functions, which is encoded into a computer generated hologram (CGH). The effectiveness of this method is also experimentally verified by static and dynamic MM beams respectively. The rest of this paper is formatted as follows. Section 2 details the methods of the MD and phase correction. The experimental setup and results are presented in Section 3. Conclusions are given in Section 4.
2. Method
2.1 Modal decomposition and phase reconstruction
In our system, the phase measurement of MM beam is achieved by means of MD technique. With the prior information of eigenmode fields and the modal coefficients (weight and phase) determined by MD, the phase of incident beam could be precisely reconstructed. This phase measurement method cannot only accurately measure the phase steps, but also achieves the same rate as MD, which paves the way to measure the phase of MM fiber beam in real-time. In the past few years, several MD methods have been proposed, such as the numerical analysis method [31–34], spatially and spectrally resolved imaging (S2) technique [35,36], ring-resonators method [37], and wavefront analysis method [28,38]. By using numerical analysis method and S2 technique, the fastest speed of real-time MD is 9 Hz [31] and 1 Hz [39], respectively. There are a few reports of high-speed MD based on these two methods, but they are realized with off-line data processing [34,35,40,41]. For numerical analysis method, the speed is limited by the time of iterative optimization process. For S2 technique, the speed is mainly limited by the time of spatial scanning or wavelength scanning, and the time of intensive data processing. In this paper, correlation filter method (CFM) [42–45] is used for MD because it only requires straightforward algebraic calculation to obtain the modal components, which significantly increase the rate of in-line MD up to 30 Hz [46] (only limited by the frame rate of camera) and greatly benefits real-time operation.
In the CFM, a correlation filter encoded in the CGH is the most crucial element. The beam emitted from the end face of fiber is diffracted by the CGH and then Fourier transformed by a lens. The far-field on the optical axis is proportional to the correlation of the incident optical field and the transmission function of the CGH. In this way, the modal amplitudes and phases can be measured by designing the CGH with a set of specific transmission functions. The details regarding the designing of the transmission functions can be found in the [42,43]. The final transmission function for the MD can be expressed as follows
where are the transmission functions for the modal weight measurement; and are the transmission functions for the modal phase measurement. For the MD involving modes, transmission functions are required. These transmission functions are encoded in a CGH by means of angular multiplexing. Each transmission function is modulated onto a spatial carrier frequency and then superimposed together to obtain the final transmission function. In the Fourier plane, the diffraction patterns corresponding to their respective transmission functions are simultaneously arranged at different positions.Based on the measured modal amplitudes and phases , the complex amplitude of the optical field in the investigated polarization direction could be reconstructed. To evaluate the agreement of the actual field and the reconstructed field, the cross correlation function is proposed
where, is the mean value of the near-field intensity profile with denoting the reconstructed and measured one respectively. For an ideal MD and reconstructed field, the correlation coefficient is one; for an inaccurate result, it’s close to zero.2.2 Adaptive phase correction
From the former section, it is shown that the phase distribution of the optical field can be obtained from the complex amplitude reconstructed by MD. Thus, the MD can be used to measure the phase of the incident beam in the AO system. For a proof of concept demonstration, our aim is to verify the feasibility of real-time phase correction, rather than actually correct the entire optical field. Thus, it is no need to use an individual phase corrector, such as deformable mirrors (DM) or phase-only spatial light modulators (SLM). In our system, the corrector is substituted by a transmission function, which carries the conjugate phase of the incident optical field, as shown below
The diffraction pattern of this transmission function can also be observed on the Fourier plane, which is equivalent to the far-field intensity profile of the corrected beam. Besides, another mirror-like transmission function with a tilted phase only is added to monitor the original far-field intensity profile without correction, as shown belowThe transmission functions in Eqs. (3) and (4) act as a phase corrector and a beam splitter respectively. The KCorrect and KOri are the carrier spatial frequencies of these two transmission functions to separate the corrected and original far-field intensity profiles. This method not only simplifies the experimental configuration, but also allows the acquirement of the two far-field intensity profiles with and without correction in the same optical path and camera simultaneously.In the real-time correction procedure, modal information is measured with the kth CGH firstly, then the (k + 1)th CGH is generated with conjugate phase of the reconstructed optical field. The final transmission function of the CGH is a superposition of the stationary measurement transmission function and the dynamic correction transmission function, which can be expressed as
The terms without and with index k indicate the stationary and dynamic transmission functions, respectively. The ACorrect and AOri can be adjusted to obtain the proper illuminance of the far-field intensity profiles with and without correction, respectively. In the Fourier plane, the mode information and the far-field intensity profiles are acquired simultaneously by one camera. By selecting a series of carrier frequencies, the diffraction patterns can be spatially separated without crosstalk. To illustrate the arrangement of diffraction patterns, a simulation result generated with a typical MM beam is shown in Fig. 1. The + 1st rank diffraction pattern on the Fourier plane is shown in Fig. 1(a), the spots for modal weight analysis and modal phase analysis are in the red polygon and yellow polygon, respectively; the far-field intensity profiles are in the white box. The modal weight and the near-field intensity profile of the beam used in the simulation are shown in Figs. 1(b) and 1(c) respectively.
Fig. 1 Simulation of the diffraction pattern of a MM beam. (a) + 1st Rank diffraction pattern on the Fourier plane. Corr., the corrected far-field intensity profile; Ori., the original far-field intensity profile. (b) Modal weight of the incident beam. (c) Near-field intensity profile of the incident beam. The spots in the red polygon are for modal weight analysis; the spots in the yellow polygon are for modal phase analysis; the spots in the white box are for far-field monitoring.
Figure 2 shows the simulation results of the far-field intensity profiles without and with phase correction of the six eigenmodes of a 0.065-NA LMA fiber with core of 25 μm (the operation wavelength is 1064 nm; the near-field is expanded 125 times and the far-field is obtained by a focus lens of f = 175 mm). As expected, with the phase correction, each of the far-field intensity profile of MM beam transforms from a spot with multiple lobes to a Gaussian-like spot. The power-in-the-bucket (PIB) of far-field intensity profile and beam propagation ratio M2-parameter are used to evaluate the effectiveness of the phase correction respectively [47]. The diameter of the bucket is 97 μm, which is defined at the 1/e2 of the peak intensity of the FM. The M2x and M2y are the M2-parameters in the x and y orientations, respectively. They are calculated with the beam waist widths and the beam divergence angles of each mode, according to ISO 11146-1 [48]. With the correction, the PIBs of LP11, LP21 and LP02 modes are increased from 50.9%, 23.6% and 26.3% to 74.5%, 68.6% and 83.1%, respectively. In contrast, the M2-parameter of each mode remains the same with the correction. This is because the far-field beam width defined by the second-order moment exaggerates the contribution of the outer wings of the intensity profile [47,49]. It also can be seen from the PIB curves shown in Fig. 2, the corrected curve is building up faster than the original one in the range of radius < 40 μm, and contains approximately 70% of the power. Since the AO system is mainly used to improve the far-field power density of the output laser, the PIB should be more appropriate to evaluate the effectiveness of phase correction in this paper.
Fig. 2 Simulated comparison of the far-field intensity profiles of six LP modes. The subgraphs in the first row indicate the original far-field intensity profiles without correction. The subgraphs in the second row indicate the corrected far-field intensity profiles. The white circle in each subgraph indicates a D = 97 μm bucket. Subgraphs in the third row indicate the PIB vs the bucket radius. The blue curve and red curve indicate the original and corrected PIB respectively.
3. Experiments and results
3.1 Experimental setup
The experimental setup as schematized in Fig. 3 consists of a beam source part to generate the MM beam with aberrated phase, and a phase correction part. The beam source part comprises two light sources, a single-mode fiber (SMF) pigtailed single-frequency laser at 1064 nm serves as a coherent source to excite mixed modes in the LMA fiber, and an ASE source at 1064 nm with 10 nm bandwidth serves as an incoherent source to align and calibrate the MD system. These two light sources, respectively, connect to a SMF with a fiber connector, which allows rapid switch between them. Then the beam from the SMF is free-space coupled into a step-index LMA fiber with core diameter of 25 μm and core NA of 0.065. The supported eigenmodes in this LMA fiber at operating wavelength of 1064 nm are LP01, LP11e, o, LP21e, o, and LP02 modes. By moving the output end face of the SMF with a 3-axis nano-position stage, the coupling condition between the SMF and the LMA fiber can be changed, thus exciting variable MM beams in the LMA fiber. For the output beam of the LMA fiber, a 4-f system with magnification factor of 125 is used to get an enlarged optical field of the fiber end face. The 4-f system consists of a microscopic objective and an achromatic doublet lens whose focal lengths are 4 mm and 500 mm respectively (500/4 = 125). This enlarged field images at a phase-only reflective SLM (1920 × 1080 pixels of 6.4 μm pitch) and a near-field camera (Camera 1, 1920 × 1200 pixels of 5.86 μm pitch) through a non-polarizing beam splitter (NPBS). The CGH introduced in Section 3 is displayed on the SLM for the MD and the phase correction. The field is diffracted by the CGH and then Fourier transformed by a lens (Lens 2, f = 175 mm). A camera (Camera 2, 1928 × 1448 pixels of 3.69 μm pitch) is placed at the Fourier plane of the SLM to acquire the + 1st rank diffraction pattern which contains the correlation information and the far-field intensity profiles with and without correction. Since the SLM has a specific working polarization direction, a polarizing beam splitter (PBS) is used to choose one state of polarization of the optical field. The MD and the phase correction were only proceeding in the state of polarization parallel to the macro-axis of the SLM.
Fig. 3 Experimental setup. SMF, single-mode fiber; FC, fiber connector; LMA fiber, large-mode-area fiber; L1, L2, lenses; MO, microscopic objective; NPBS, non-polarizing beam splitter; PBS, polarizing beam splitter; SLM, spatial light modulator; 1st Rank, + 1st rank diffraction beam of CGH.
To give a detailed illustration of the experiment workflow, a typical simulation result is given in Fig. 4 as a paradigm. The kth CGH (Fig. 4(a)) is firstly generated with the TCorrect of a tilt phase only and displayed on the SLM. By analyzing the diffraction patterns in the kth frame (Fig. 4(b)) with MD algorithm, the modal components (Fig. 4(c)) in the fiber could be obtained and then the original near field (Fig. 4(d)) could be reconstructed in real-time. Next, according to Eq. (5), the (k + 1)th CGH which contains the conjugated phase of the near field (Fig. 4(e)), will be generated and sent to SLM. After that, the corrected far-field intensity profile can be acquired from the (k + 1)th frame of Camera 2 (Fig. 4(f)).
Fig. 4 Workflow of the adaptive phase correction in the experiment. (a) The kth CGH displayed on the SLM; (b) the diffraction pattern in the kth frame of the Camera 2; (c) the modal components analyzed by MD; (d) the reconstructed near-field phase distribution; (e) the (k + 1)th CGH with conjugated phase of reconstructed field; (f) the far-field intensity profiles acquired from the diffraction pattern in the (k + 1)th frame of the Camera 2.
3.2 Results and discussion
The experimental setup described above was used to demonstrate the phase correction for the MM beam. Its phase correction capability was verified under static and dynamic conditions separately.
In the first experiment, we excited a series of static MM beams by randomly coupling the SMF and LMA fiber with slight misalignment. Here the details of a typical case is illustrated in Fig. 5. The measured near field intensity profile is composed of one main lobe and several side lobes (Fig. 5(b)). According to the modal weight spectrum obtained by MD (Fig. 5(a)), the beam mainly consists of 40.6% LP21 odd mode, 29.0% LP11,o mode and 12.2% LP01 mode.
Fig. 5 Typical example of the static correction. (a) The modal weight spectrum. (b) The measured near-field intensity profile. (c) The reconstructed near-field intensity profile. (d) The original far-field intensity profile. (e) The corrected far-field intensity profile. The white circle in each subgraph indicates a D = 97 μm bucket.
The reconstructed near-field intensity profile in Fig. 5(c) shows a high consistency with the measured one. The cross correlation C is 0.99, which suggests a high reliability of the MD. The far-field intensity profiles without and with correction are shown in Figs. 5(d) and 5(e). Due to the phase fluctuation caused by the HOMs, the original far-field intensity profile is divided into several lobes. With the correction, the far-field intensity profile transforms to a Gaussian like distribution. The PIB of the corrected beam increases from 33.0% to 71.4%, which is close to 87.8% of the FM.
Based on the experimental verification with static field, the adaptive phase correction based on MD was tested with dynamic field then. By randomly adjust the nano-position stage, the modal components in LMA fiber can be varied. We excited MM beam with ever changing modal components to verify the effectiveness of the correction under dynamic condition. The far-field PIB of the original beam and the corrected beam are shown in Fig. 6 by the blue and red curves respectively, and the modal weight of FM is represented by the black dashed curve. It can be found from Fig. 6 that for the beam without correction, the far-field PIB is mainly relative to the modal weight of FM. A lower FM weight leads to a sharp drop in the PIB. For the beam with correction, the average value of PIB increases from 40.1% to 69.3% and the minimum value of PIB is always above 54%, which suggests a significant improvement in the PIB value. In other words, in order to transfer equivalent power into a far-field bucket as defined above (D = 97 μm), the output power of the MM HPFL only needs to be 1.11 times that of the SM HPFL, considering that the PIB of the LP01 mode is 87.8% (87.8%/69.3% = 1.11). To highlight the real-time correction ability of our method, the correction process of Fig. 6 is presented in the Visualization 1.
Fig. 6 The far field intensity PIB vs time curves for the original beam without correction (blue) referred to the corrected beam (red). The black dashed curve represents the modal weight of FM. The yellow curve represents the PIB of the ideal corrected beam. The green curve represents the cross correlation between the reconstructed far-field intensity profile and the measured one. The near-field intensity profiles of correspond time are shown on the top of figure. Ori. PIB, the original far-field PIB; Corr. PIB, the corrected far-field PIB; Ideal Corr. PIB, the ideal corrected far-field PIB; FM weight, the modal weight of LP01 mode; FF CC, the far-field correlation coefficient.
The simulated result of ideal correction is obtained by substituting the phase of reconstructed field with a perfect plane phase, and the PIB of its far-field profile is represented by the yellow curve in Fig. 6, with an average value of 79.2%. It also can be seen from Fig. 6 that the experimental result of PIB (see the red curve in Fig. 6) is lower than the simulated ideal result, which is mainly caused by the aberrations and noise. Firstly, a few aberrations are introduced by the imperfection of Fourier lens L2. Then the quality of far-field diffraction pattern will be degraded by the aberrations, which introduces errors into the MD and the calculation of PIB. To eliminate the influence of these aberrations, the CGH should be superposed on a correction phase which could compensate these aberrations. Secondly, the signal-to-noise ratio (SNR) of diffraction pattern images is affected by noise. The useless 0th rank of diffraction pattern is much brighter than the + 1st rank which we measured. Thus both of the outer-wings and stray light of the 0th order diffraction pattern introduce background noise to the + 1st order diffraction pattern, which also causes the decrease of the PIB.
The correlation coefficient between the reconstructed far-field intensity profile and the measured one (see the green curve in Fig. 6) is mainly affected by the output power of the fiber and it approximately decreases with the reduction of the FM modal weight (see the black dashed curve in Fig. 6). Because the LMA fiber was bent in the experiment, a little bit higher loss was introduced into the HOMs. When the modal weight of HOMs increase, the power of the output beam decreases and the far field diffraction pattern becomes faint (see the subfigure of 34.1 s in Fig. 6). Since the exposure configuration of the Camera 2 is fixed in the experiment, the SNR of diffraction pattern is decreased, which leads to more errors of the reconstruction of optical field and thus reduces the effect of phase correction. Therefore, for some cases where the FM modal weight is extremely low and the HOM dominate the beam (30~40 s in Fig. 6), the far-field PIB of the corrected beam shows a dip. Besides, the saturation of the diffraction pattern image also introduces errors to the MD, which result in the drop of correlation coefficient at high FM modal weight (e.g. 15 s and 48 s in Fig. 6). Due to the limited dynamic range of Camera 2, the choice of exposure time requires a balance among different output power levels. In order to maintain the SNR for low output power level (e.g. 30~40 s in Fig. 6), the exposure time of the camera was set a little bit longer. When the FM content is very high, the high output power level makes the spot for the measurement of the LP01 modal weight saturated. Thus, the modal power of the HOMs are erroneously exaggerated, because the modal power is calculated according to the ratio of the on-axis value of different spots. However, this shortcoming could be overcome by adaptively controlling the exposure time and gain of camera to preserve the SNR of diffraction pattern.
In our current system, the rate of MD and phase reconstruction is 25 Hz, which is limited by the maximum frame rate of Camera 2. Due to the calculation time of the high-resolution CGH and the refresh time of SLM, the average delay of each control step is about 170 ms for our current system, which results in a close-loop control rate of 5 Hz. When the modes in the fiber change rapidly, the incident beam changes markedly during the delay time, so that it cannot be corrected well by the correct phase. In this case, the corrected PIB curve follows the original PIB curve and shows sharp drops (15~16 s and 49~50 s). However, the rate of MD could be further improved by replacing the camera with a fast photodiode array and implementing the MD algorithm via a field-programmable gate array (FPGA). In addition, substituting the SLM by a DM with dozens of actuators as the corrector could achieve a correction rate up to ~5 kHz [50]. This method suggests great potential for correcting the drastically fluctuating phase of the MM beam. This makes it possible to obtain a high far-field PIB in the high power LMA fiber laser even when TMI occurs.
4. Conclusion
In conclusion, we presented the adaptive phase correction in a LMA fiber based on the MD for the first time. Experimentally, we shown that the correlation filter method could decompose the optical field in a six-modes LMA fiber with high precision. The close-loop phase correction for the dynamic MM beam was demonstrated at a rate of 5 Hz. With the phase correction, the average value of the far-field PIB of the dynamic MM beam increased from 40% to about 70%, which suggests a significant improvement in the PIB value. Accordingly, we believe this adaptive phase correction method will be an effective tool to combat the PIB degradation caused by TMI in the high power LMA fiber laser. In order to increase the power contained in the main lobe of high power LMA fiber laser actually, we expect to construct an AO system with a deformable mirror in the future and correct the phase of the entire field.
Funding
National Natural Science Foundation of China (NSFC, No.11504423)
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