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Today a specific attention has been paid to look into the modal characteristics of the high-power laser beam. And the instantaneous monitoring and analyzing on modal content via the mode decomposition technique will provide a novel route. We implement the first-ever experimental investigation on the real-time mode decomposition technique for few-mode laser beam based on stochastic parallel gradient descent algorithm. It will reduce the cost and the complexity of the mode decomposition system. We have succeeded to decompose the mode spectra as well as calculating the beam quality factor at about 9 Hz monitoring rate, while the high agreement between the measured and reconstructed intensity profiles in each frame indicating the high accuracy and stability during the process. By employing a fiber-squeezing-based polarization controller, the modal content under test can be time-varying automatically.
© 2015 Optical Society of America
1. Introduction
Recent years have witnessed the rapid power scaling of fiber lasers whose gain can be provided by rare-earth ions [1,2] as well as nonlinear effects [3]. To achieve fiber lasers with higher brightness, the emerging mode decomposition (MD) technology renders it attractive as a novel diagnostic tool to characterize the light emitting from optical fiber [4–6]. With the MD techniques, one can decompose the propagating field into individual transverse mode, which will result in the mode spectrum and the relative phase difference between modes. As a consequence, the MD provides insight into the propagating field with the mode information, which will straightly derive the beam quality [7], or wave-front [8]. MD has been gradually applied in the fiber lasers research, such as measuring fiber-to-fiber coupling process [9], mode-resolved gain [4] or loss [10] and evaluating the specialty fiber designs [5,6]. Real-time MD will extend the potential functionality to characterize output mode dynamics of fiber lasers [11,12] or monitor manufacturing mode-resolved components for mode division multiplexing [13].
In the past few years, several MD approaches have already been developed, such as the spatially and spectrally resolved imaging [14], the use of ring-resonators [15] and low-coherence interferometry [16]. These methods can provide accurate mode spectrum, but they require consuming post-data processing or intense experimental measurements. Therefore, they are not able to perform real-time MD which requires monitoring the mode contents without significant delays between real mode generation and reconstructing the mode contents. The most promising technique for real-time MD is the correlation filter method (CFM) [17]. CFM performs optical correlation analysis all-optically to achieve the mode information without any additional numerical evaluation [17] and the rate of MD has been demonstrated to be hardware limited [18]. On-line real-time monitoring the modal contents by using CFM has been demonstrated [7,18]. Computer generated hologram (CGH) was firstly used for implementing optical correlation [17]. However, it has the disadvantages that generating an appropriate CGH is costly and one piece of CGH is only suitable for one specific fiber design and scale parameter. To overcome these shortcomings, the phase-modulation spatial light modulator was recently employed to serve as a correlation filter [19]. Nonetheless, the relatively large pixel size and insufficient refresh rate of the device limit the real-time ability, compared to CFM using CGH. The newly-developed frequency domain cross-correlated imaging has also shown potential real-time MD ability [20], but the measuring time and the process time is still long and only near-real-time analysis is demonstrated at the present stage. F. Stutzki et al. demonstrated real-time monitoring of the modal spectrum of a monolithic few-mode fiber lasers by using multimode fiber Bragg gratings to achieve a spectral mode separation between modes [11], whereas this approach is restricted to fiber oscillator both supporting a few modes and employing fiber Bragg grating as the laser reflector. High speed MD process has been demonstrated within numerical analysis to measure the mode instability [12] but it is off-line measurement [18].
Overall, the MD method utilizing numerical analysis is believed to be unable to realize on-line real-time MD [18] because of its consuming numerical process [21,22]. However, we have demonstrated fast and accurate MD by using the stochastic parallel gradient descent (SPGD) algorithm theoretically [23] and experimentally [24]. Here, we will further show that MD method utilizing numerical analysis can also perform real-time MD when adapting the SPGD algorithm. It will reduce the cost and the complexity of the MD system. To the best of our knowledge, this is the first demonstration of real-time MD via numerical method.
2. Mode decomposition technique
The propagating field of a fiber can be projected on the Eigen modes of the fiber. It can be mathematically expressed as
where U(r,φ) is an arbitrary propagating filed and ψn(r,φ) is the nth Eigen mode of the fiber with modal amplitude ρn and phase θn. The total number of the Eigen modes is N and all of them form an orthonormal basis set. Normally, the fields mentioned above are normalized unit thus leading to. In fact, the principle of the MD method is to achieve the modal amplitude and phase of individual Eigen mode. It should be noted that this paper assumes the case of few-mode step-index (SI) fiber whose Eigen modes can be described in terms of linearly polarized (LP) mode [25], but the MD method is not limited to SI fiber and can be applied to the fiber with accessible Eigen modes. Prior to MD, one should firstly calculate the theoretical Eigen modes according to the parameters of fiber under test (FUT) and the scaled parameters. In addition, the coordinate systems of the measured and reconstructed intensities should share the same origin [22]. The alignment is vital for the accuracy of the MD.The iteration scheme of the real-time MD is outlined in Fig. 1 and consists of three steps per frame, namely acquiring near or far-field image of the output beam, SPGD algorithm and illustrating the results separately. The key step is the SPGD algorithm [26] and it is briefly described as follows:
(1) Initialize the variables for the first iteration. To be specific, the variables of the algorithm are the amplitudes {ρi | i = 1,2,···N } corresponding to all the Eigen modes and phase items {θi | i = 1,2,···N-1} denoting the relative phase differences of high order modes with respect to the fundamental mode. The initial values of the variables are randomly assigned in the first decomposition frame. After the first frame, the initial values of the variables inherit that of last frame. This handling will speed up the convergence if the variation of the beam profile is not severe, compared to randomly assigning the initial values of variables.
(2) Generate statistically independent random perturbations δρ1, δρ2,···δρN, δθ1, δθ2,···δθN-1 which are all small values and update the variables in the form ρi ± = ρi ± δρi (i = 1,2,···N) and θi ± = θi ± δθi (i = 1,2,···N-1). Two temporary reconstructed fields can be achieved within two set of the updated amplitudes and relative phase differences according to Eq. (1), thus deriving two temporary reconstructed intensity distributions Ire ± .
(3) Evaluate the merit functions J ± = J(Ire ± ,Ime) and calculate the difference between them δJ = J+–J–. The merit function J employed in this paper is the cross correlation function
where ΔIj(r,φ) = Ij(r,φ)–j with j = re, me (respectively denoting the reconstructed and measured intensity) and j is the corresponding mean value of the intensity distribution. This merit function has lower noise sensitivity and can lead to better reconstruction than the residual intensity [22].(4) Update the variables ρi = ρi + γ1δJδρi (i = 1,2,···N) and θi = θi + γ2δJδθi (i = 1,2,···N-1) where γ1 and γ2 are the corresponding update gain. The convergence criterion of the algorithm is |Jk + 1–Jk|<0.0001 and the iteration steps of every frame are constant in this paper.
3. Experimental setup
The experimental scheme is shown in Fig. 2(a). The laser source is a pig-tailed narrow linewidth laser diode at 1073 nm (VSLL-1073, Connect Laser Corp.). The delivery fiber is single-mode at the laser wavelength. The optical beam then is coupled immediately into the delivery fiber of an all-fiber polarization controller (PolaRITETMIII, General Photonics Corp.). By varying the relative positions of the two fibers, it is possible to excite different mode contents in the fiber of the polarization controller (PC). The PC is normally employed to convert arbitrary input polarization state to desired output polarization state and the principle of the PC is based on fiber squeezing driven by a piezoelectric actuator. However, the aim of the PC in our setup is to change the mode contents continuously instead of controlling the state of polarization and the reason of changing mode contents will be discussed in the following section. In this experiment, we control the mode contents by driving the PC with a 0-5 V analog signal which is provided by an arbitrary function generator (AFG3102, Tektronix Inc.). To be specific, the pig-tailed fiber of the PC is SMF-28 compatible SI fiber and it has a core with 8.2 μm diameter and 0.14 NA. This SI fiber can be considered as weakly guiding and the Eigen-modes can be expressed in terms of LP modes. 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. Accordingly, the parameters of the algorithm are the mode amplitudes of three modes and the relative phase differences between LP11e,o and LP01. What’s more, the intensity distributions of the Eigen modes of the fiber are also depicted in Fig. 2(b). The end facet of the fiber is imaged on the CMOS camera (Firefly MV FMVU-03MTM/C, Point Grey Research Inc., 752 × 480 pixels with the size of 6μm) through a 4-f imaging system consisting of 2 lenses whose focal lengths are 8 mm and 300 mm respectively. Accordingly, the magnification factor of this lens set is 37.5. A polarization beam splitter is located between the lenses to select only one polarization component of the beam and the half-wave plate is to change the polarization. A neutral density filter is used to adjust the power of the beam within the response range of the camera.
Fig. 2 (a) Scheme of the experimental setup. SMF, single mode fiber; AFG, arbitrary function generator; PC, polarization controller; L, lens; HWP, half-wave plate; PBS, polarization beam splitter; NDF, neutral density filter;f1/2, focal length of L1/2; (b) The Eigen modes in the fiber.
4. Results and discussion
The initial mode content is determined by the relative position between the fibers which will remain invariable during the measurement. We set the shutter time of per frame to 10 ms and the gain to 0. To alter the modal contents continuously and periodically, a triangular wave is used to drive one channel of the PC. Excitation of different mode mixtures in the multimode fiber is achieved under the squeezing driven by a piezoelectric actuator. The frequency of the triangular wave is 75 mHz while the value of low-level and high-level voltage is set to 0 and 1 V respectively. With those parameters, the mode contents can be considered to be slowly varying.
To highlight the on-line real-time ability of our method, we present a measurement process in Media 1. The first frame of Media 1 is shown in Fig. 3. The two images on top are the measured (left) and reconstructed (right) near-field intensity of the beam, respectively, which are both normalized. The final calculated result of the cross correlation function of the reconstructed near-field beam image defined by Eq. (2) together with the calculated M2 value is shown at the top right. The evolution of merit function including current frame and overall are shown on bottom, while the modal weight of each mode shown at the bottom right. By using a conventional computer, we have achieved monitoring rate of about 9 Hz. In total, the media includes 100 frames of decomposition.
Fig. 3 Excerpt from Media 1, showing a measurement process. The measured and reconstructed near-field intensity, the merit function of per frame and the total, as well as the corresponding M2 value and correction between the measured and reconstructed beam intensities are presented.
Figure 3 also reveals the evolution of the merit function during the first frame. In the first frame, there is no information on the mode contents at the beginning so the initial values of the parameters in numerical calculation are randomly assigned. The algorithm takes a few iteration steps to reach the optimum. For the rest frames, the initial values of the parameters inherit the optimized result of last frame. In this experiment, the variance of mode spectrum between two adjacent frames is small. Therefore, this processing can reduce the iteration steps required to converge in a decomposition frame. On the other hand, the processing will have little effect on reducing the convergence steps when measuring fast-fluctuating mode contents such as the case of mode instability [12]. The number of iteration steps for an individual frame is 60 here. This can be reduced to improve the monitoring rate, but it may cause instability of the decomposition.
From the media, we could see that the measured beam images vary gradually because of fiber squeezing and the reconstructed ones match well with the measured ones almost in every frame. Figure 4 illustrates the evolution of the correction coefficient between the measured and the reconstructed intensity pattern defined by Eq. (2) of every frame in the media. Almost all of them reach over 99%, indicating high accuracy of the decomposition. This can also be verified by the high agreement between the measured and the corresponding reconstructed intensity patterns as shown in the insets of Fig. 4(b). What’s more, the excellent performance of the decomposition during the process also confirms the high stability of the algorithm.
Fig. 4 The correction between the measured and the reconstructed intensity pattern defined by Eq. (2) of every frame. The insets depict measured (up) and reconstructed (down) intensity of the frame denoted with letters.
Figure 5(a) shows the mode spectra and the voltage of the PC modulation signal as a function of frame (on bottom) and time (on top). The time scale is about 11.3 s, which is 84% period of the PC modulation signal as shown in the figure. The signal increases linearly from 0.08 V to 1 V and then decreases to 0.22 V. The highest amplitude locates at the 54th frame (6.08 s) where the signal is symmetrical. One could obtain that the modal content distribution is also symmetrical to the position where the peak of modulation signal locates. Remarkably, the modal weight of each mode also presents a periodic change with a period of approximately 0.55 V signal amplitude. This phenomenon indicates the periodic mode coupling among the three modes under the regular fiber squeezing. The maximum weights of LP01, LP11e and LP11o attain 0.89, 0.56 and 0.61 respectively, while the minimum weights of them can all be about 0. On the other hand, the symmetrical and periodic results also indicate fiber squeezing driven by a piezoelectric actuator is an effective way to precisely control the mode contents. What’s more, the relative phases of LP11e and LP11o with respect to the fundamental mode a function of frame are shown in Fig. 5(b). The evolution of the relative phase is symmetrical as the corresponding modal weight but not periodical. It is worth noting that there is uncertainty of the sign of the modal phases because only the near-field image of the few-mode beam is utilized and decomposed this paper [22,23].
Fig. 5 (a) The modal weights of LP01, LP11e and LP11o and the input voltage as a function of the frame or the time. The time scale is 11.3 s. (b) The relative phases of LP11e and LP11o with respect to the LP01 as a function of frame. The dash lines in both figures are to denote the symmetry point of the modal weights and voltage.
The distribution of the mode spectra in Fig. 5 can be attributed for mode coupling induced by core distortion [25] and polarization variation [27] under fiber squeezing. Schulze et al. also changed the mode contents by squeezing the fiber [28] but they eliminated the influence of polarization on the mode contents. In order to compare with the result presented in [28], we pay attentions to the polarization effect. Here, we integrate the recorded beam intensities in each frame to reveal the evolution of the relative power level. Note that the camera is operated below the saturation during the process, so the power can be regarded as proportional to the sum of the grayscales of all pixels in image of the near-field beam. It is more convenient to perform such simultaneous power measurement compared with employing a power meter. What’s more, the power of the beam before the PBS keeps constant during the operation of the device, which has been calibrated before the experiment. Figure 6 illustrates the calculated relative power level and the modal weight of fundamental mode as a function of frame. The relative power is normalized to unit. The power varies following the evolution of fundamental mode weight. This indicates the obvious influence of polarization effect on the modal contents, which is different from the case in [28].
Fig. 6 The normalized power of beam after the PBS and the modal weight of fundamental mode as a function of frame.
The calculation of the beam quality factor M2 is based on the theoretical procedure introduced by H. Yoda et al. [29]. The quantity of the M2 can be derived directly by using analytical formulas with the already-known mode spectra and phases. Figure 7 shows the calculated M2 at x- and y-direction as a function of frame. The value of M2 changes periodically owing to the fiber squeezing, while the trend is almost inverse with that of the fundamental mode weight. The minimum M2 is 1.02 (at 33rd frame) and 1.01 (at 29th frame) for x- and y-direction respectively and the maximum M2 is 2.06 (at 98th frame) and 2.17 (at 67th frame) for x- and y-direction respectively. However, it is worth noting that the calculation procedure is only well suited to analyze stigmatic and simple-astigmatic beams but not to general astigmatic beams as pointed out in [7]. Besides, the uncertainty of the sign of the modal phase will not influence the calculation according to the formulas in [29]. Finally, this theoretical procedure has the advantage of fast speed, thus providing additional valuable information on the beam propagation characteristics in the real-time measurement.
The monitoring rate of MD can be further improved. In this experiment, all the processes are operated in serial. The technique of parallel processing will enable to further improve the monitoring rate, such as parallel calculation of merit function with positive and negative perturbations or parallel operating image acquisition, mode decomposition and displaying results. In addition, on the aspect of numerical calculation, better computer processors and more effective coding can also be expected to improve the monitoring rate. To sum up, 9 Hz monitoring rate is not the limit of our method, which is merely hardware and software limited currently. However, it is still difficult to achieve real-time decomposition ability as CFM. In spite of that, the best advantage of this method is simple and cost-effective to real-time decomposing optical modes.
It is worth noting that the MD proposed in this paper can be applied to arbitrary waveguides in addition to SI fiber when Eigen modes are accessible, such as large-mode-area fiber [24], multi-trench fiber [6] and multi-core fiber [27]. If one would like to extend the real-time MD to beam consisting of more than 3 modes, far-field image should be employed to eliminate another kind of ambiguity induced by modal interference [22], which can also remove the uncertainty of the modal phases. Another camera should be employed to record the far-field image. To realize real-time decomposition in that case, a much more complicated system and SPGD algorithm are necessary. Note that the two cameras, which collect the near- and far-field beams separately, should be synchronized. In terms of the algorithm, much more calculation is involved with increasing mode number and additional far-field propagation. Furthermore, the techniques propose here can also accurately decompose the beam of the fiber supporting more than 3 modes when only 2 modes are excited.
5. Conclusion
We have demonstrated on-line real-time MD technique for few-mode beam by using numerical method for the first time as far as we know. The variance of mode contents is realized by an electric-driven PC, which is mainly caused by polarization effect and the deformation of the fiber core. The real-time measurement of mode spectra and simulated beam quality factor with about 9 Hz monitoring rate are achieved by using SPGD algorithm. The experiment shows that the decomposition is highly accurate and stable during the whole process. The MD technique proposed in this paper provides a compact way to further look into the dynamics of fiber lasers.
Acknowledgments
The authors wish to thank Dr. Yu Cao for providing useful advice in image acquisition and Dr. Shihai Sun, Dr. Xiaolin Wang and Miss Suhui Dong for providing the PC and supporting the work. This work was supported by National Natural Science Foundation of China (No.61340017).
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