## Abstract

We propose and demonstrate the adaptive conversion of a multimode beam into a near-diffraction-limited flattop beam in the near field based on a combination of dual-phase-only liquid-crystal spatial light modulators (LC-SLMs) and the stochastic parallel gradient descent (SPGD) algorithm. One phase-only LC-SLM redistributes the intensity of the multimode beam, and the other compensates the wavefront of the output beam. The SPGD algorithm adaptively optimizes the phase distributions of dual-phase-only LC-SLMs to reduce the variance between the actual beam shape and the target beam shape. The experimental results on a fiber multimode beam show that the system is capable of adaptively creating square and rectangle flattop beams with desired parameters. Beam quality can be greatly improved by this system. The power in the main lobe of the far-field spot is about 4 times larger than that of the input multimode beam.

©2010 Optical Society of America

## 1. Introduction

In recent years, fiber lasers have attracted more and more attention due to their compactness, high efficiency, and beam quality. However, the output power from a single-mode fiber laser is limited because of deleterious, highly nonlinear effects and optical damage [1,2]. Multimode fibers provide an interesting alternative to single-mode fibers for achieving high-power laser beams. The core diameter of a multimode fiber can be in the hundreds of micrometers, so the output power increases largely as a result. However, with the power increase the beam quality degrades greatly, which limits capability for many applications. Beam quality can be improved by coiling up the multimode fiber, by using tapered sections, and by gain filtering [3,4]. However, the techniques mentioned above are wasteful of laser light and can only translate the multimode beam into a fundamental mode beam.

In the fields of laser fusion, laser radars, and coherent combinations, a flattop beam with a plane-wavefront in the near field is often required to improve the energy efficiency of a system. Many techniques, which include amplitude filtering and phase filtering by refractive or diffractive optics, have been proposed to convert a fundamental mode beam into a flattop beam with a plane-wavefront in the near field [5,6]. The conversion of a single higher-order mode beam to a fundamental mode beam can be realized by using phase elements and planar interferometric elements [7,8]. Unfortunately, differing from a single high-order or low-order mode beam, a multimode beam does not have defined amplitude or phase distribution (the amplitude and phase distribution can be influenced by heat and pressure). The beam-shaping techniques mentioned above are based on the transformation of a specific input-to-output transformation, so the techniques are not suitable for multimode beam shaping.

To the best of our knowledge, the conversion of a multimode beam to a flattop beam with a plane-wavefront in the near field has never been reported. Deformable mirrors (DMs) are popular choices for beam controlling and have been used in laser focal spot shaping. There are two ways for controlling DMs. One is to measure the wavefront of the input beam exactly by using a Shack–Hartmann wavefront sensor, and the other is to employ some iterative strategies to control the DM based on a single, experimentally measurable variable. Compared with the first approach, the latter one has the benefit of ease of implementation and can avoid errors in wavefront measurement [9]. As one kind of diffractive optical elements, phase-only LC-SLMs provide an interesting alternative to conventional DMs because of their high spatial resolution and programmable controller. In our preceding work, we reported on the conversion of a quasi-Gaussian beam to a near-diffraction-limited flattop beam with phase-only LC-SLMs based on a refractive beam-shaping system [10]. The design principle can only work well for a central symmetric beam. In this paper, we demonstrate the adaptive conversion of a multimode beam emitting from a multimode fiber into a near-diffraction-limited flattop beam based on the combination of dual-phase-only LC-SLMs and the SPGD algorithm. The algorithm adaptively optimizes the phase distributions of dual-phase-only LC-SLMs to reduce the variance between the actual beam shape and the target beam shape. The SPGD algorithm is well suited for finding a global minimum or maximum of some objective error functions by optimizing their variables and has been used in astronomy and coherent beam combination [11,12]. One phase-only LC-SLM redistributes the intensity and the other one compensates for the wavefront of the output beam. The technique can adaptively convert a multimode beam to a flattop beam with the desired intensity distribution in parallel with improving the beam quality. It is to be noted that the purpose of this paper is not to shape a beam with an extremely high power, but to demonstrate the feasibility of adaptive conversion of a multimode beam to a near-diffraction-limited flattop beam.

## 2. Experimental analysis

The experimental setup is shown in Fig. 1
. The essential instruments are phase-only LC-SLMs (BNS company, LC-SLM1 with 512 × 512 15 × 15 um^{2} pixels, LC-SLM2 with 256 × 256 24 × 24 um^{2} pixels), a 1064 nm single-mode laser (50 mW), and a multimode fiber (50 um core, 0.22 NA @ 1550 nm). The normalized frequency of the multimode fiber is *V* = 64 at 1064 nm, so the 1064 nm single-mode laser can inspire a multimode laser in the multimode fiber. The multimode beam is collimated and expanded by the telescope before incident on LC-SLM1. Reflected by LC-SLM1, the beam is separated into two parts: the main beam is incident on LC-SLM2, and the other beam is incident on CCD1. The beam reflected by LC-SLM2 is also divided into two parts. One is focused on the detection surface of CCD2 by a plano-convex lens with a 670 mm focal length. CCD1 (Dolphin F-145B, 15 Hz) with 1392 × 1040 6.45 × 6.45 um^{2} pixels and CCD2 (Guppy F0808B, 30 Hz) with 1032 × 778 4.65 × 4.65 um^{2} pixels are used to diagnose the near-field and far-field intensity distribution of the output beam. CCD1 and CCD2 are directly interfaced using IEEE 1394 (Firewire) connections and are addressed under VC + + control by using the SPGD program, which also controls phase-only LC-SLMs. By monitoring and processing the images from CCD cameras, the system is capable of adjusting phase distributions of LC-SLM1 and LC-SLM2 adaptively to produce a target beam profile with a nearly plane wavefront.

As shown in Fig. 1, the closed control loop consisting of LC-SLM1 and CCD1 redistributes the intensity adaptively, and the closed control loop consisting of LC-SLM2 and CCD2 adaptively compensates for the wavefront of the output beam. According to the analysis of M. A. Vorontsov, maximum convergence speed can be achieved if Zernike polynomials are chosen as a set of influence functions, which is regarded as the modal control strategy in [11]. The phase distributions loaded on LC-SLM1 and LC-SLM2 can be represented as

where*Z*(

_{i}*r,θ*) is the

*i*order Zernike polynomial, and

^{th}*a*is the coefficient of the

_{i}*i*order Zernike polynomial, which is also the control signal.

^{th}*j*is the denotation, which indicates the phase distribution for either LC-SLM1 or LC-SLM2. The quality metric

*J = J*(

**) is a function of the control parameters**

*a*

*a**=*{

*a*

_{2},

*…*,

*a*}. In this paper,

_{n}*J*has two definitions,

*J*and

_{fiterror}*J*, which are shown in the following paragraphs. The SPGD algorithm is used to optimize the quality metric. The working principle for the SPGD algorithm can be briefly described as follows [12]:

_{compensation}- 1. Generate statistically independent random perturbations
*δa*_{2},*…*,*δa*, where all_{n}*δa*are small values that are typically chosen as statistically independent variables having zero mean and equal variances, <_{i}*δa*> = 0, <_{i}*δa*> =_{i}δa_{i}*σ*^{2}*δ*, where_{ij}*δ*is the Kronecker symbol._{ij} - 2. Apply the control signal with perturbations and get the metric function from the CCD camera,
*J*(_{+}= J*a*_{2}*+ δa*_{2},*…*,*a*), then apply the control signals with opposite perturbations and get the metric function,_{n}+ δa_{n}*J*(_{-}= J*a*_{2}*-δa*_{2},*…*,*a*). Calculate the difference,_{n}-δa_{n}*δJ = J*._{+}-J_{-} - 3. Update the control signals,
*a*,_{i}= a_{i}+ γδa_{i}δJ*i*= 1, 2,*…*,*n*, where*γ*is the update gain, and*γ>*0 and*γ<*0 are corresponding to the procedure of maximization and minimization respectively.

The quality metric of the closed control loop of the intensity redistribution is chosen as the fit error between the actual beam shape profile and the target beam profile and is given by

*I*(

_{target}*x, y*) is the target intensity profile, and

*I*(

_{camera}*x, y*) is the actual beam shape profile recorded by CCD1. In this paper, the target beam profile is chosen as the super-Gaussian profile and is given by

*a*and

*b*determine the beam width in the

*x*and

*y*directions,

*p*and

*q*are integers that specify the steepness of the beam sides, and

*x*and

_{o}*y*are the centered positions of the target beam spot. The phase error metric is chosen as the metric function of the wavefront compensation closed control loop and is given bywhere

_{o}*I*(

_{farfield}*x, y*) is the intensity distribution of the focal spot. The global maximum of

*J*corresponds to an undistorted wavefront. In a closed control loop of the intensity redistribution, the SPGD algorithm is used to minimize the fit error between the target profile and the actual profile. In a closed control loop of the wavefront compensation, the SPGD algorithm is used to maximize the phase error metric

_{compensation}*J*. The optical length between CCD1 and LC-SLM1 is the same as the optical length between LC-SLM2 and LC-SLM1, which ensure that the intensity recorded by CCD1 is the same as the one incident on LC-SLM2. In this paper, the distance is about 40 cm

_{compensation}*.*When the system is an open loop, the near-field and far-field intensity distributions, which are recorded by CCD1 and CCD2, are shown in Figs. 2(a) and 2(b).

According to Figs. 2(a) and (b), it can be found that the multimode beam is effectively excitated by using the 1064 nm single-mode laser and multimode fiber at 1550 nm. The extended multimode beam is distributed in a 6 × 6 mm^{2} region. The intensity distribution is asymmetry and the beam quality is poor. In the closed control loop of the intensity redistribution, the SPGD algorithm is performed for a modal wavefront corrector with Zernike polynomials {*Z _{i}*(

*r*,

*θ*)}(

*i*= 2,…, 160) as a set of influence functions. During the process, the SPGD algorithm adaptively optimizes the phase distribution of the input beam and converges to give a beam that approximately matches the target profile. In the experiment, the super-Gaussian beam with parameters

*a*= 0.0045,

*b*= 0.0045,

*p*= 6,

*q*= 6 is chosen as the target profile. The output beam profile is shown in Fig. 3(a) . The curve for the evolution of the relative fit error during SPGD algorithm execution is shown in Fig. 3(b). It can be found that after 600 iterations (less than 9 minutes), the experimentally measured beam shape is approximately in agreement with the target profile. There exist ripples in the evolution curve of the relative fit errors, which are mainly caused by the intensity variation of the input beam.

As shown in Fig. 1, one part of the output beam is focused by a lens with focal length of 670 mm to form a focal spot on the detection surface of CCD2 for monitoring the far-field intensity distribution. When the wavefront compensation system is in open loop, the far-field intensity distribution of the output beam is shown in Fig. 4 , and the energy density is low.

In the closed control loop of the wavefront compensation, the SPGD algorithm is also performed for a modal wavefront corrector with Zernike polynomials {*Z _{i}*(

*r*,

*θ*)} (

*i*= 2,…, 60) as a set of influence functions. The far-field intensity distribution and the curve for evolution of the phase error metric during the SPGD algorithm execution are shown in Figs. 5(a) and 5(b). It can be found that after 600 iterations (less than 5 minutes), the relative phase error metric approaches an approximately absolute maximum value. According to the intensity distribution shown in Fig. 3(a), we calculate the diffraction-limited far-field intensity distribution, which is shown in Fig. 5(a) (dashed line). The experimental far-field intensity distribution (solid line) is nearly coincident with the calculated diffraction-limited far-field intensity distribution, which indicates good wavefront flatness.

Based on Figs. 2(b), 4, and 5(a), we calculate the power in the bucket (PIB) curves of the far-field intensity distributions of the input multimode beam, output beam without being compensated by LC-SLM2, and output beam compensated by LC-SLM2. The results are shown in Fig. 6
. The PIB can be greatly improved by adaptive beam shaping. According to Fig. 6, it can be found that after adaptive beam shaping, the power in the main lobe (with a 0.2 mm semi-diameter in the *x* and *y* directions) is about 4 times larger than that of the multimode beam.

In many applications, a flattop beam with other intensity distributions is also needed. In this paper, the adaptive near-field beam shaping for conversion of a multimode beam to a rectangle flattop beam with desired parameters is also realized. The corresponding results are shown in Fig. 7
. The top row corresponds to results of flattop beam shaping with parameters *a =* 0.0035, *b =* 0.0055, *p =* 6, *q =* 6. The bottom row shows the result of flattop beam shaping with parameters *a =* 0.0055, *b =* 0.0035, *p =* 6, *q =* 6.

## 3. Conclusion

In summary, the adaptive conversion of a multimode beam into a flattop beam with nearly a plane wavefront in the near field by using dual-phase-only LC-SLMs and the SPGD algorithm has been demonstrated. Under the SPGD algorithm controller, one phase-only LC-SLM adaptively redistributes the intensity of the input multimode beam, and the other one adaptively compensates the wavefront. After adaptive beam shaping, the power in the main lobe of the far-field spot is 4 times larger than that of the multimode beam. The beam quality has been greatly improved. Though the power of the multimode beam is low in the experiment, the technique proposed and demonstrated in this paper can also be used for controlling dual high-damage threshold DMs with high spatial resolution for converting a high-power multimode beam into a near-diffraction-limited flattop beam. This approach can also correct phase aberration of the input beam and can be implemented in many applications.

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