Transmission Matrix-Inspired Optimization for Mode Control in a 6 × 1 Photonic Lantern-Based Fiber Laser

Abstract

A photonic lantern is a coherent beam combination device that can increase the fiber laser brightness by adaptively controlling the input light properties, such as phase, intensity, and polarization. However, the control effect is closely related to the initial optical field, which affects the convergence speed to obtain the optimum solutions. In this work, we propose a novel control strategy using the prior structural information of the photonic lantern. Taking a 6 × 1 photonic lantern as an example, we calculate the transmission matrix of the photonic lantern. The initial optical field conditions, fed as the control inputs, for various mode outputs can be obtained. Compared with the random and equal amplitude control methods, the preset method from the transmission matrix presents a significant improvement of the desired mode content. Our optimization method is generally useful for adaptive control systems to improve their performance, taking advantage of their own structural information.

1. Introduction

The “photonic lantern” (PL) is an optical waveguide, which enables low-loss light transmission and mode coupling between single-mode fibers (SMFs) and multiple mode fibers (MMFs) [1,2,3,4]. In 2005, Leon-Saval et al. demonstrated such devices when fabricating a multi-mode filter, the performance of which matched the SMF performance [5]. When used for optical signal processing in large telescopes, PL-based filters can filter out the bright and narrow hydroxyl emissions between 1000 nm and 1800 nm to improve the visibility of desired signals [6]. In optical communication, a PL can be used as a spatial multiplexer with mode-selective ability and low insertion loss [7,8,9,10,11]. Compared with a traditional spatial multiplexer [12,13,14,15,16,17], PLs are all-fiber devices that can greatly reduce the loss caused by the space optical path. However, it must be pointed out that the multiple abilities of PLs are limited by the number of their input channels. In addition to application in the low-power regime, researchers explored PL to be used in fiber lasers. With the adaptive spatial mode control in 3 × 1 PL, the power output after scaling has reached 1.27 kW for the fundamental mode using a large-mode-area (LMA) fiber [18,19], revealing great potential for achieving high-power laser output with high beam quality.

LMA fiber is used to increase the threshold of nonlinear effects in the power growth of the laser by increasing the modal field area [20,21]. However, in high power systems, the refractive index of the fiber changes with the temperature, which enables the mode coupling to be instable at kilohertz or even faster [22,23]. This phenomenon is called transverse mode instability (TMI) [24] and it can lead to the degradation of the beam quality. The conventional method of fiber coiling can filter high-order mode (HOM) signals, but it will also lead to power loss. It is difficult to solve these problems with the laser itself [25,26]. Therefore, active control methods have been used to suppress the instability of the transverse modes. Usually, the stability control of the two lowest linearly polarized (LP) modes, LP₀₁ and LP₁₁ mode, can be reached when the optical intensity is used as the evaluation function [27,28,29,30]. For example, an approach had been attempted to mitigate TMI by a dynamic excitation of the fiber modes using an acousto-optic deflector [31]. However, they only considered the optical amplitude, neglecting the important phase information. Such a control scheme is difficult to achieve the stable output of HOMs.

To solve this challenge, we combine the PL technology with a LMA fiber. Using the linear correlation between the input and output of the PL, we can modulate the input conditions at the side of its SMFs in real time to realize the output of a specific mode in the LMA fiber. The applicability of this scheme has been demonstrated in the MIT experiments [18,19]. They firstly described the correspondence between the input and output of PL with the help of transmission matrix (TM). However, the 3 × 1 PL can only achieve stable control of three modes. Achieving HOM control requires higher-order PLs [7].

In this paper, we propose a mode control scheme based on a 6 × 1 PL. We use the stochastic parallel gradient descent (SPGD) algorithm to control the phase and amplitude at the input of the PL in real time, and preset the initial amplitude condition according to the inverse solution of the TM of PL. To enhance the control of HOMs, we also take the content of the relevant modes as an evaluation function instead of using the received power of the photodetector, which is commonly used in experiments. In this work, we first simulate two amplitude control methods, namely random amplitude control and equal amplitude control. We also compare their effects on controlling the fundamental mode and HOMs with the control scheme proposed in this paper, namely, transmission matrix-inspired control.

2. Materials and Methods

2.1. The Transmission Matrix of the Photonic Lantern

The PL is made by splicing a tapered SMF bundle with a MMF. The transverse size of the SMF bundle is continuously decreased, and the mode evolution occurs when the beam is transmitted in it. Since this size decrease is slow enough [32], there is a linear relationship between the input and output light fields of the PL. In general, the PL is considered as a linear optical device, which can be characterized mathematically as a matrix that maps the input vector to the output vector [18]:

$$A\cdot V=U$$

(1)

where V is a column vector containing the amplitude and phase inputs of the SMFs, and U is a column vector representing the complex amplitude coefficients of the supported modes. A is the TM of the PL. TM is a complex quantity used to describe the transmission characteristics of optical waveguides [33]. Much research revolves around the measurement, calculation and application of the TMs of special optical waveguides [34,35,36,37]. In this paper, we numerically calculate the TMs of PLs by simulating the mode evolution process in PLs based on the beam propagation method (BPM) [38,39,40]. Each input fiber of PL is injected into the fundamental mode and the optical field distribution obtained in the output fiber is calculated by BPM. Using the mode decomposition algorithm, we can obtain the complex amplitude coefficients of each supported LP mode. Finally, the ideal TM of PL is inversely calculated according to Formula (1). The numerical calculation results of the TM of the 6 × 1 PL are as follows:

$$\left[\begin{array}{cccccc}0.61& 0.35& 0.35& 0.35& 0.35& 0.35\\ 0& 0.63e^{i(-0.83\pi )}& 0.19e^{i(-0.83\pi )}& 0.51e^{i(0.17\pi )}& 0.51e^{i(0.17\pi )}& 0.19e^{i(-0.83\pi )}\\ 0& 0& 0.60e^{i(-0.83\pi )}& 0.37e^{i(-0.83\pi )}& 0.37e^{i(0.17\pi )}& 0.60e^{i(0.17\pi )}\\ 0& 0.63e^{i(-0.94\pi )}& 0.51e^{i(0.06\pi )}& 0.19e^{i(-0.94\pi )}& 0.19e^{i(-0.94\pi )}& 0.51e^{i(0.06\pi )}\\ 0& 0& 0.37e^{i(-0.94\pi )}& 0.60e^{i(0.06\pi )}& 0.60e^{i(-0.94\pi )}& 0.37e^{i(0.06\pi )}\\ 0.73e^{i(0.93\pi )}& 0.27e^{i(-0.07\pi )}& 0.27e^{i(-0.07\pi )}& 0.27e^{i(-0.07\pi )}& 0.27e^{i(-0.07\pi )}& 0.27e^{i(-0.07\pi )}\end{array}\right]$$

(2)

Experimentally, the TM of an actual PL can be measured using the method in reference [33]. Due to the limitation of manufacturing technology, the TM obtained by theoretical calculation will be slightly different from the TM measured by the experiment. Once the TM of the PL is calculated or measured, if a specific output light field is needed, the input condition can be calculated according to:

$$V=A^{-1}\cdot U$$

(3)

In the actual optical system, it is very easy to set the input amplitudes as the calculated results of Equation (3). However, due to environmental disturbance, optical path difference and other factors, the phase of each input path is constantly changing. Therefore, in order to realize the mode control by PL, it is necessary to control each input phase by means of adaptive optics.

2.2. Stochastic Parallel Gradient Descent Algorithm

In adaptive optics, according to the signal detected by the output end, the controller actively modulates the information of the input end. The control algorithm is key to achieve good control effects. In our system, stochastic parallel gradient descent (SPGD) is chosen to be the control algorithm. SPGD algorithm is simple and effective, so that it is widely adopted in the field of optics to achieve a phase-locked loop [41,42,43,44,45]. In the PL-based mode control system, we also use SPGD to control the phase of each input. In particular, we compared the mode control effect under three prefabricated amplitude conditions. Figure 1 shows the basic control loop of SPGD. Evaluation function J, random disturbance voltage δu and gain coefficient γ are the three main parameters in the SPGD algorithm. In the control of the SPGD algorithm, the system generates a set of voltages with opposite perturbations below a certain threshold, denoted as δu₊ and δu₋, respectively. The effect of the disturbances in both directions on the performance can be expressed as J₊ and J₋. We can update the input u following this formula in Equation (4). The iteration continues and the evaluation function J converges to a better solution. It can be seen that disturbance voltage δu and gain coefficient γ determine the stability and convergence speed of the control.

$$u_{n+1}=u_n+\gamma \times \frac{J_{+}-J_{-}}{J_{+}+J_{-}}\times \delta u_n$$

(4)

In our simulations, the evaluation function J is the mode content of the target mode. We make a mode decomposition of the output optical field in real time to the complex amplitude coefficients of every mode. Additionally, J is the real of the desired mode’s complex amplitude coefficients. Voltage u is applied to each phase modulator at each input channel. The final parameters are set as follows: the perturbation voltage for the phase control is 0.07 V and the gain coefficient is 200; the perturbation voltage for the amplitude control is 0.04 V and the gain coefficient is 150. In the actual optical system, the frequency of phase disturbance is below kHz, and the control bandwidth of SPGD can be up to 10 kHz, so the system can realize real-time mode control.

The focus of the following research is the control condition. We studied the stability and convergence speed of the SPGD algorithm in controlling the input phase under three different prefabricated input amplitude conditions of the PL.

3. Results and Discussion

3.1. The 6 × 1 Photonic Lantern

The 6 × 1 PL used in our system satisfies the structure requirements: mode matching, adiabatic cone tapering and reasonable arrangement [1]. Following these rules, the propagation loss of multiple modes can be minimal in forward and backward directions.

The “mode matching” means that the number of supported modes in the MMF needs to match the number of SMFs in the input channel. Six 10/130 (NA = 0.08) SMFs are selected as the single-mode input, and a 25/250 (NA = 0.065) MMF is used as the multimode output, as shown in Figure 2. The approximate value of the number of modes supported in a multimode fiber N_m can be derived from the following expressions:

$$N_{\mathrm{m}}\approx {\left(\frac{\pi dNA}{2\lambda }\right)}^2$$

(5)

where d is the diameter of the fiber core, λ is the laser wavelength of 1064 nm, and NA is the numerical aperture of the output fiber. N_m is calculated to be 6.024, which means the MMF supports 6 LP modes. Therefore, 6 × 1 PL in this paper satisfies the mode matching condition. To meet the reasonable arrangement of the 6 × 1 PL, the outer five fibers are arranged following a pentagon and the other fiber is in the geometric center. However, if all six fibers are of the same size, there will be more gaps between the outer fibers. To fill the gap, we can experimentally erode the cladding of the center fiber from 130 μm to 91 μm. After this treatment, the SMFs fit closely to each other and are not easily slipped during the tapering process. We accordingly change the diameter of the fiber cladding in the simulation. The last requirement is the “adiabatic taper tapering”. In the taper pulling process, the taper angle $\Omega$ < 0.01 rad in the taper region [19]. This means that for every 10 μm decrease in the fiber bundle, the taper length increases by at least 1 mm. After the taper pulling, the diameter of the 6 × 1 PL decreases from 280 μm to 25 μm. The minimum length of the tapered region can be calculated to be 32.5 mm. The PL is then integrated into a fiber laser system, the details of which can be found in ref. [18].

3.2. Random Amplitude

To set the initial values of the SMF amplitudes, one common strategy is to use the random number first before SPGD algorithm comes into play. We thus simulate 6 × 1 PL using first six modes (LP₀₁, LP_11e, LP_11o, LP_21e, LP_21o and LP₀₂). Following the initial amplitude setting, SPGD algorithm updates the values according to the evaluation function. Initially, there is a transition period when multiple modes couple to each other. Then, the desired mode content starts to rise rapidly and stabilize. However, the mode content cannot rise to unity as other modes co-exist. This cannot improve much as the system has fallen into a local optimal solution, so extending the time window does not help. As a result, the mode content for these modes are 78.9% (LP₀₁), 30.4% (LP_11e), 44.4% (LP_11o), 67.7% (LP_21e), 42.7% (LP_21o) and 22.9% (LP₀₂), respectively. The highest mode content is only near 80%. As LP₀₁ and LP₀₂ are so close in amplitude distribution, SPGD algorithm cannot separate them well. For example, in Figure 3f, we find that controlling LP₀₂ is rather difficult, reaching around 20% compared to 55% of LP₀₁ mode.

3.3. Preset the Equal Amplitude

Another usually adopted strategy to initialize the amplitude input is to have the equal amplitude. From the control effect shown in Figure 4, the final beam shape is more symmetrical than the previous ones. Additionally, the time period to stabilize is shorter. The final mode content is also higher. Especially in controlling LP₀₁ mode, the percentage is as high as 99.1%. However, in controlling LP_11e (70.0%), LP_11o (78.7%), LP_21e (65.0%) and LP_21o (59.5%) modes, the stabilized percentage is still not significantly improved. In controlling LP₀₂ mode, the LP₀₁ mode still has the largest percentage (30% for LP₀₂ and 69.3% for LP₀₁).

3.4. Preset Amplitude Based on Transmission Matrix

We propose a novel method to initialize the amplitude inputs using the TM of the PL. If the PL is well-fabricated, the optimal solution should be quite close to the values calculated from TM. The simulation results are shown in Figure 5. For all six modes, we find that the final mode content is more than 99% within 1 ms. The beam shape is almost identical to the control target. For LP₀₂ mode, the control effect is much better than the other methods.

We summarize our simulation results in

Table 1. The stabilized mode content using different methods for initializing amplitudes.

Mode Content	Random	Equal	Transmission Matrix-Inspired
LP01	78.9%	99.1%	99.7%
LP11e	30.4%	70.0%	99.8%
LP11o	44.4%	78.7%	99.6%
LP21e	67.7%	65.0%	99.8%
LP21o	42.7%	59.5%	99.2%
LP02	22.9%	30.0%	99.4%

. Although the equal amplitude control has advantages in LP₀₁ control, both schemes of random amplitude control and equal amplitude control have defects in mode control. The reason is that the optimization pathway of the 6 × 1 PL has many local optimal solutions, and the SPGD algorithm may fall into these local optimal solutions. Using the TM of the PL itself, we are able to achieve a much better control result compared with two other methods of setting amplitude, either random or equal.

4. Conclusions

We compare three methods to set the initial amplitude in SPGD algorithm for optimizing mode control in a 6 × 1 PL based fiber laser. The methods using random and equal amplitudes can achieve moderate beam contents, but cannot control LP₀₂ mode, as a result of falling into local optimum solutions. Inspired from the physical character of the PL, we can calculate the out amplitude input requirement for controlling certain modes. Using these PL-related initial amplitudes, the fiber laser system can achieve almost perfect mode content for all six modes within 1 ms. Our work demonstrates that knowing the physical properties of the optical system (such as TM of the PL) can significantly improve the optimization results, and should be applicable to other optimization problems in optical and photonic fields.