## Abstract

We propose and demonstrate the full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical element in a beam control system. The transformation of square dark hollow beam with unsymmetrical and inhomogeneous intensity distribution into square dark hollow beam with homogeneous intensity distribution is chosen to prove the validity of the technique. Dual deformable mirrors (DMs) based on the stochastic parallel gradient descent (SPGD) controller are used to redistribute the intensity of input beam and generate homogeneous square dark hollow beam with near-diffraction-limited performance. The SPGD algorithm adaptively optimizes the coefficients of Lukosz-Zernike polynomials to form the phase distributions for dual DMs. Based on the finite element method, the thermal deformations of CaF_{2} half transparent and half reflecting mirror irradiated by high power laser beam before and after beam shaping are numerically simulated and compared. The thermal deformations of the mirror irradiated by the laser beam with different powers and the influences of thermal deformation on beam quality are also numerically studied. Results show that full-field beam shaping can greatly decrease and homogenize the thermal deformation of the mirror in the beam control system. The strehl ratios of the high power laser beams passing through the beam control system can be greatly improved by the full-field beam shaping. The technique presented in this paper can provide effective guidance for optimum design of high power laser cavity and beam shaping system.

©2011 Optical Society of America

## 1. Introduction

High power laser beams are required for directed energy weapon and nuclear fusion applications. Because of the restriction of laser cavity, nonuniform gain distribution and phase distortion introduced by optical elements, the intensity of high power laser beam usually exhibits nonuniform and nonsymmetrical distributions [1,2]. If the power of the beam with nonuniform intensity distribution is the same as the power of the beam with uniform intensity distribution, the nonuniform intensity distribution will affects the propagation of high power laser beam in the following aspects: generating serious thermal blooming, resulting in nonuniform thermal deformation of optical elements or even destroying optical elements in the beam control system, and triggering or enhancing adverse nonlinear effects like stimulated Brillouin scattering, Raman scattering or filamentation. The nonuniform thermal deformation comes from the absorbed laser energy by windows, transparent and reflecting mirrors. Although the absorption is very small, the nonuniform absorbed energy causes a nonuniform temperature field, which induces local thermal stress and distortion. When the laser beam passes through the beam control system, the thermal deformation will result in beam distortion. The high power laser system is made up of many optical elements. The accumulation of nonuniform thermal deformation of optical elements will seriously degrade the outgoing beam’s quality [3–11]. In practical applications, compared with the beam with nonuniform intensity distribution, the outgoing beam with uniform intensity distribution can greatly improve the beam quality.

One effective way to redistribute the intensity of high power laser beam is beam shaping [12–17]. The study of laser beam shaping has a long history. A number of techniques and systems, which include using amplitude and phase filtering elements, are developed to redistribute the intensity of input beam [14–17]. Compared with amplitude filtering method, the phase filtering method by using refractive or diffractive optical elements has high conversion efficiency and can be used for high power laser beam shaping. Due to the simple design procedure and high conversion efficiency, the refractive beam shaping system with dual aspheric lenses has been widely studied. However, most of the currently available designs of refractive beam shaper use lenses with rotational symmetry. In addition, the shaping systems mentioned above are based on the transformation of specific input and output beam profiles.

In Ref. [18], we reported the adaptive conversion of quasi-Gaussian beam into near-diffraction-limited flattop beam based on dual phase only liquid crystal spatial light modulators (LC-SLMs) and stochastic parallel gradient descent (SPGD) algorithm. In the experiment, the intensity distributions of input and target beams are quasi-symmetric. In this paper, we propose using adaptive full-field beam shaping on high power laser beam with unsymmetrical intensity distribution to improve its intensity uniformity for decreasing and homogenizing the thermal deformation of optical elements. One deformable mirror (DM) adaptively redistributes intensity of input beam to uniform distribution at the second DM plane, and the other DM adaptively compensated the wave front of the output beam. The SPGD algorithm adaptively optimizes the coefficients of Lukosz-Zernike polynomials to form the phase distributions for dual DMs. The phase distribution generated by this technique can be used to fabricate refractive and reflective optical elements. Based on the finite element method, the thermal deformations of CaF_{2} half transparent and half reflecting mirror irradiated by the high power laser beam before and after beam shaping are numerically simulated and compared. The thermal deformations of the mirror irradiated by the laser beam with different powers and the influences of thermal deformation on beam quality are also numerically studied. Results show that full-field beam shaping can greatly decrease and homogenize the thermal deformation of the mirror in the beam control system. The strehl ratio of the high power laser beam passing through the beam control system can be greatly improved by the full-field beam shaping. To the best of our knowledge, the technique proposed in this paper has never been reported.

This paper is organized as follows. In the second section, working principle of full-field beam shaping and corresponding unsymmetrical beam shaping results are given. The third section reports the theoretical analysis of the influences of full-field beam shaping on thermal deformation of optical elements and on beam quality. In the forth section, the conclusions are given.

## 2. Beam Shaping Principle and Numerical Analysis

The working principle is shown in Fig. 1 . In simulation, the ideal 512×512 pixels transmission deformable mirrors are chosen as the intensity redistribution deformable mirror (DM1) and wave front compensation deformable mirror (DM2) respectively. After passing through DM1, the input beam is separated into two parts. The main beam is incident on DM2 and the other is incident on CCD1. After passing through DM2, the beam is separated into two parts too and one of them is focused on CCD2. The CCD1 and CCD2 are used to monitor the near field and far field intensity distribution of the output beam respectively. Using the information from CCD1 and CCD2, the computer calculates the phase distributions for loading on DM1 and DM2 to produce the target beam with nearly plane wave front.

The whole system is controlled by the SPGD algorithm. As shown in Fig. 1. The first closed control loop consisting of DM1 and CCD1 is used to redistribute the intensity of input beam. The second closed control loop consisting of DM2 and CCD2 compensates the wave front of the output beam. According to the analysis of Vorontsov, the maximum convergence speed can be achieved if the Zernike polynomials are chosen as a set of influence functions, which is regarded as the modal control strategy in reference [19]. Compared with the Zernike polynomials, the Lukosz-Zernike functions are more suitable for representing the larger wave front aberrations. The phase distributions would be loaded on DM1 and DM2 can be represented as the combination of a series of *N* Lukosz-Zernike modes.

*L*(

_{i}*r,θ*) is the

*i*order Lukosz-Zernike mode,

^{th}*a*is the Lukosz-Zernike expansion coefficient, which is also the control signal. The functions were first derived by Lukosz and, later, independently by Braat. Like Zernike polynomials, the Lukosz-Zernike functions are each expressed as the product of a radial and azimuthal function using the same dual index and numbering scheme [20–22]. They can be defined as

_{i}*n*and

*m*are the radial and azimuthal indices, respectively, and where

The quality metric *J*=*J*(** a**) is a function of the coefficient

*a**=*{

*a*

_{1},

*a*

_{2},

*…*,

*a*}. The SPGD algorithm is used to optimize the quality metric. The steps for SPGD algorithm can be briefly described as follows [19]. Each iteration cycle works as follows:

_{n}- 1. Generate statistically independent random perturbations
*δa*_{1},*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*_{1}*+δa*_{1},*a*_{2}*+δa*_{2},*…*,*a*), then apply the control signals with the opposite perturbations and get the metric function,_{n}+δa_{n}*J*(_{-}= J*a*_{1}*-δa*_{1},*a*_{2}*-δa*_{2},*…*,*a*). Calculate the difference between two evaluations of the metric function_{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.*γ>*0 and*γ<*0 according 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 ${I}_{actual}\left(x,y\right)$ and the target beam shape${I}_{t\mathrm{arg}et}\left(x,y\right)$.

For simplicity and convenience in dealing directly with the wave front distortion, the phase error metric is chosen as the metric function of the wave front compensation closed control loop and is expressed as [23]

where*I*(

_{farfield}*x,y*) is the intensity distribution of the focal spot, which is recorded by CCD2. According to the analysis of Muller, the global maximum of

*J*corresponds to an undistorted wave front. In closed control loop of the intensity redistribution, the SPGD algorithm is used to minimize the

_{compensation}*J*between the target and the actual beam shape recorded by CCD1. In closed control loop of the wave front compensation, the SPGD algorithm is used to maximize the phase error metric

_{fiterror}*J*. In this paper, taking the transformation of square dark hollow beam with unsymmetrical and nonuniform intensity distribution into near-diffraction-limited square dark hollow beam with uniform intensity distribution as an example, we study the full-field shaping of the beam with unsymmetrical intensity distribution based on the SPGD controller. The intensity distribution of input beam is shown in Fig. 2(a) . The target beam is defined as

_{compensation}*a*

_{1},

*a*

_{2}and

*b*

_{1},

*b*

_{2}determine the beam width in

*x*and

*y*directions.

*p*

_{1},

*p*

_{2}and

*q*

_{1},

*q*

_{2}are integers that specify the steepness of the beam sides.

*x*and

_{o}*y*are the centered positions of the target beam spot.

_{o}In simulation, the optical length between DM1 and DM2 is the same as the optical length between DM1 and CCD1, which ensures the intensity recorded by CCD1 is the same as the one incident on DM2. The distance is about 10*m* and the wavelength of the input beam is 3.8*μm*. The dark hollow flattop beam defined in Eq. (7) with parameters *a*
_{1}=*b*
_{1}=28, *p*
_{1}=*q*
_{1}=30, *a*
_{2}=*b*
_{2}=70, *p*
_{2}=*q*
_{2}=15 is chosen as the target beam. In intensity redistribution closed loop, the SPGD algorithm is performed for a modal wave front corrector with Lukosz-Zernike function {*L _{i}*(

*r,θ*)} as a set of influence functions. During the process, we use the first 300 order Lukosz-Zernike polynomials excluding the first order Lukosz-Zernike polynomial (piston). The SPGD controller continuously updates the phase distribution and converges to give a beam that approximately matches the target profile. The results are shown in Figs. 3(a) and 3(b). The relative

*J*is defined as the ratio of

_{fiterror}*J*in the process to the original

_{fiterror}*J*. According to Figs. 3(a) and 3(b), it can be found that the dark hollow beam with near flattop intensity distribution is achieved after 10000 iterations with use of modal control. The generated phase distribution for intensity redistribution is shown in Fig. 4(a) . It is recognized that there exist errors between the output beam and the target beam, which is mainly caused by the difference between the generated phase distribution and the ideal phase distribution for intensity redistribution. The convergence speed and convergence precision are related with the number and distribution of influence functions used in the mode control strategy. The convergence speed of the SPGD algorithm drops rapidly as the control variables increases. However, in the present paper, the shaping error only can reach a minimum value at the condition that the number of the influence functions is larger than 300. If we choose other more efficient influence functions, the convergence speed and convergence precision of the system can be improved a lot.

_{fiterror}As shown in Fig. 1, one part of the output beam passing through DM2 is focused on CCD2 by a lens with focal length 60*m* for monitoring the far field intensity distribution. Wave front information of the output beam can be obtained by analyzing intensity distribution of the focal spot. The far field intensity distribution of the output beam before being compensated by DM2 is shown in Fig. 5(a)
. Because of the phase aberration introduced by intensity redistribution, the energy density is low in the far field. In closed control loop of wave front compensation, precompensation by using the complex conjugate of intensity redistribution phase distribution is used to improve the effect. After precompensation, the SPGD algorithm is also performed for a modal wavefront corrector with Lukosz-Zernike polynomials {*L _{i}*(

*r*,

*θ*)} (

*i*= 2,…, 200) as a set of influence functions. After phase compensation, the phase distribution and far field intensity distribution of the output beam are shown in Figs. 5(b) and 6(a) . It can be found that the output beam has been compensated successfully. The root-mean-square (

*rms*) value of the wave front within a square dark hollow region with inner side length 0.01

*m*and outer side length 0.046

*m*is 0.045λ, where λ is the wavelength of the input beam. According to the far field intensity distributions shown in Figs. 5(a) and 5(b), we calculate the power-in-the-bucket (PIB) curve of the far field intensity of the beam without compensation, with compensation, and ideal compensation, which are shown in Fig. 6(b). It can be found that the far field energy density of the output beam has been greatly improved after compensation. The power in the main lobe of the far field intensity distribution of the beam after compensation is about 102 times larger than that of the beam without compensation. There exists error between the PIB curves of the adaptive compensated output beam and ideal compensated output beam, which is mainly caused by the system itself. The phase distribution generated by the SPGD algorithm and Lukosz-Zernike polynomials for wave front compensation is not ideal.

## 3. Numerical Analysis of the Thermal Deformation

It is necessary to study the influence of full-field beam shaping on thermal deformation of mirrors. Within a circle solid half transparent and half reflecting mirror of radius *r*
_{0} and thickness *d* shown in Fig. 7
, the temperature deformation *T*(*r*, *φ*, *z*; *t*) is given by the following thermal conduction equation [24]

Considering the heat convection on the substrate and side surfaces, the boundary conditions and initial conditions can be given as

*α*=

*κ*/

*ρC*is the thermal diffusivity.

*κ*is the thermal conductivity.

*ρ*is the density of the medium.

*C*is the special heat. $\stackrel{\u2022}{q}$ is the heat generation rate per unit volume.

*h*is the convection coefficient in

*K*.

*T*

_{∞}is the ambient temperature.

*q*(

*r*,

*φ*;

*t*) is the heat flux load on the mirror’s surface. Considering the high transmittance, the CaF

_{2}materials are usually chosen as splitter, and optical windows of high energy lasers. In this paper, the half transparent and half reflecting mirror using CaF

_{2}material as substrate is chosen to study the influence of full-field beam shaping on thermal deformation. Some parameters of the CaF

_{2}mirror are listed in Table 1 . As shown in Fig. 7, the beam is incident on half transparent and half reflecting mirror with tilted angle

*θ*=

*π*/4. The thermal deformation is induced by both surface heat and volume heat source. The surface heat source is the absorption of incident laser by coating and volume heat source is the absorption of transmitted laser by CaF

_{2}mirror itself. Because the thickness of the coating is much smaller than that of the substrate, thermal deformation of the coating can be neglected. The thermal deformation of half transparent and half reflecting mirror can be calculated by thermo-elastic equations of the substrate material [25].

Considering the side of the mirror is constrained, the boundary condition can be given by

where*u*and

_{r}*u*are radial and axial thermal deformations, respectively.

_{z}*ν*is the Poisson’s ratio.

*α*is the linear thermal expansion coefficient.

_{l}*ε*is the thermal strain. Because only the thermal deformation of the mirror along

*z*direction has influences on beam quality, only the thermal deformation on the surface of mirror

*u*(

_{z}*r*,

*φ*,

*d*) is considered.

The intensity distributions of the incident laser beam after and before beam shaping are chosen as Fig. 2(a) and Fig. 3(a), respectively. The intensity distribution area of the beam after and before beam shaping is chosen as 10*cm*×10*cm*. Considering the oblique incident of the beam (tilted angle *θ*=*π*/4), the power densities of the 150kW incident laser beam after and before beam shaping on the front surface *S*
_{1} of the half transparent and half reflecting mirror are shown in Fig. 8
.

The corresponding temperature distribution and thermal deformation of the half transparent and half reflecting mirror, which is irradiated by the incident laser beam with 50kW, 100kW and 150kW power, are calculated by using ANSYS Mechanical APDL application. The thermal deformations (at 5seconds) are shown in Fig. 9 . According to Figs. 9(a) and 9(b), it can be found that the increase of the power of the incident laser beam results in the larger thermal deformation of the mirrors. The thermal deformation induced by the beam after beam shaping is much smaller than that of the beam before beam shaping. In addition, the uniformity of the former thermal deformation is better than that of the latter.

In order to give a quantitative analysis of the thermal deformation, we chose the nodes at position *a* (0.03*m*, 0*m*) and position *b* (0.0068*m*, 0.024*m*) for analyzing the thermal deformation along with the irradiation time. The corresponding relative thermal deformations are shown in Fig. 10
. In Figs. 10(a), 10(b) and 10(c), pink and blue curves represent the thermal deformations induced by the incident beam before and after beam shaping respectively. When the irradiation time is 3 seconds and 6 seconds, the difference between the thermal deformation at positions *a* and position *b* irradiated by the beam with 50kW power after beam shaping are lowered by 92.3% and 88.3%. The difference between the thermal deformation at positions *a* and position *b* irradiated by the beam with 100kW power after beam shaping are lowered by 91.7% and 88.5%. The difference between the thermal deformation at positions *a* and position *b* irradiated by the beam with 150kW power after beam shaping are lowered by 91.7% and 88.0%.

According to Figs. 8(a) and 8(b), beams’ powers are mainly distributed in a square dark hollow region (−0.067*m*≤ *x* ≤-0.021*m* & 0.025*m*≤ *x* ≤0.067*m* & −0.047*m*≤ *y* ≤-0.018*m* & 0.016*m*≤ y ≤0.046*m*), so beam quality is mainly affected by thermal deformation of mirror in this region. The *PV* and *rms* values of the thermal deformation in this region are defined as

*D*(

*n*),

*D*

_{max},

*D*

_{min}are the displacement of node

*n*, the node with maximum displacement, the node with minimum displacement in the selection region respectively.

*M*is the number of nodes in this region.

*PV*and

*rms*values of thermal deformation in this region are calculated and shown in Fig. 11 . It can be found that

*PV*and

*rms*values of thermal deformation induced by the beam after beam shaping are much smaller than that induced by the beam before beam shaping. When the irradiation time is 6 seconds, the corresponding

*PV*values of thermal deformation induced by the beam with 50kW, 100kW and 150kW power after beam shaping are decreased by 0.6150

*μm*, 1.229

*μm*, and 1.884

*μm*, respectively. The corresponding

*rms*values of thermal deformation induced by the beam with 50kW, 100kW and 150kW power after beam shaping are decreased by 0.1302

*μm*, 0.2598

*μm*, and 0.39

*μm*, respectively.

In the following analysis, we study the influences of thermal deformation on beam quality. In practical applications, the beam control system consists of many mirrors and each of them may degrade the beam quality. For simplicity, we just take the high power laser beam reflected by one half transparent and half reflecting mirror as an example. Only the strehl ratio is considered and the configuration is shown in Fig. 7. According to the Fourier optics, the phase shift $\Delta \varphi $ of the reflected beam caused by the thermal deformation of the half transparent and half reflecting mirror can be given by [26]

*θ*is tilted angle of the input beam. In the numerical simulation, the tilted angle

*θ*is chosen as π/4. The beam with intensity distribution shown in Fig. 2 is chosen as the beam without beam shaping. The beam with intensity distribution shown in Fig. 3(a) and phase distribution shown in Fig. 6(a) is chosen as the beam after beam shaping. The intensity distribution area of the beam after and before beam shaping is chosen as 10

*cm*×10

*cm*. The half transparent and half reflecting mirror irradiated by the high power laser beam without and with full-field beam shaping (the irradiation time is 6s) is chosen to study the influences. We use the lens with focal length 150

*m*to focus the beam reflected by the mirror. After being reflected by the half transparent and half reflecting mirror, the far field intensity distributions of 50kW power laser beam without and with beam shaping are shown in Figs. 12(a) and 12(b). The corresponding strehl ratios of the beam without and with beam shaping after being reflected by the mirror are 0.417 and 0.61. The strehl ratio of the beam with beam shaping is 1.46 times as large as that of the beam without beam shaping. The beam qualities of the 100kW power laser beam and 150kW power laser beam reflected by the half transparent and half reflecting mirror have also been studied. The strehl ratios of the 100kW and 150kW power laser beams with beam shaping are 1.32 and 1.45 times as large as that of the beams without beam shaping, respectively. It can be concluded that the beam quality of reflected beam can be greatly improved by full-field beam shaping. If the beam control system consists of more mirrors, the beam quality will get greater increase.

## 4. Conclusion

The full-field unsymmetrical beam shaping for decreasing and homogenizing the thermal deformation of optical elements in a beam control system has been proposed and demonstrated. The transformation of square dark hollow beam with unsymmetrical and nonuniform intensity distribution into near-diffraction-limited square dark hollow output beam with uniform intensity distribution is used to prove the validity of the technique. The phase distributions for dual DMs are adaptively generated by using the SPGD algorithm and Lukosz-Zernike polynomials. Based on the finite element method, the thermal deformations of half transparent and half reflecting mirror irradiated by the high power laser beam before and after full-field beam shaping are studied in detail. In addition, the influences of thermal deformation on beam quality have also been investigated. Results show that the full-field beam shaping can greatly decrease and homogenize the thermal deformation of the optical element. The *PV* and *rms* values of the thermal deformation of the mirrors have been greatly decreased. The strehl ratio of the high power laser beam after passing through the beam control system can be greatly improved by the full-field beam shaping.

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