## Abstract

The annular laser beam (ALB) is widely used in many fields, which could be affected by laser power and beam quality. To effectively and flexibly improve the beam quality of high-power large-aperture thin-wall ALB, a two-stage enlargement and adaptive correction configuration (TEACC) consisting of a novel outer-surface tubular deformable mirror (OTDM) and two extra prism groups (EPGs) is proposed in this paper. The correction principle and design principle of the TEACC are derived and analyzed. Based on the principle, a typical OTDM prototype and EPG structure are designed. Annular aberrations are compensated by applying the OTDM’s influence functions and the least-square algorithm in simulation. The results show that the TEACC could perfectly compensate the wavefront distortions described by the 2^{nd} to 36^{th} order Zernike annular aberrations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Due to the unique intensity distribution and propagation characteristics, the ALB has received much attention in recent years and found increasingly broad application prospects in laser lidar [1,2], free-space laser communication [3], particle guidance [4,5], and laser industrial machining [6–8]. For example, in the laser lidar or free-space laser communication system, the thin-wall ALB could increase the spatial resolution and communication accuracy of the target at a far distance, as the thin-wall ALB with plane wavefront would not cause additional beam spreading or irradiance loss for long propagation distances in atmospheric turbulence [2,3]. In the particle guidance, the ALB with central intensity null would be more suitable for the precision control and guidance of the atoms along the beam. In the industrial application, compared with the conventional laser welding and trepanning, the ALB could enhance the productivity and quality of machining a workpiece with a circular seam, as the rotation operation of the workpiece is removed.

It is well known that the application effect of the ALB would be affected by the beam quality and optical power. However, limited by severe thermal effects, it is quite hard to maintain good beam quality in the continuous-wave high-power lasers even the tube gain- medium [9,10] and unstable resonator [11,12] are adopted. Furthermore, optical aberration of the output coupling system, thermal distortion of the optical elements in the output system [13–15] and propagation turbulence [1–3] in the free space could aggravate the wavefront distortion of the ALB. In order to obtain better application effect of the ALB, it is extremely important to improve the beam quality of high-power ALB using extra-cavity distortion correction approach.

The adaptive optics (AO) system, consisting of the wavefront sensor, the deformable mirror (DM) and the control system, has been successfully used in varieties of laser system to improve the beam quality [16–19]. In the AO system, the DM could be controlled to generate the conjugated phase of the distorted wavefront. As we know, there are typical five types of conventional DMs, including the liquid crystal spatial light modulator (SLM) [20,21], the MEMS DM [22–24], the membrane DM [25–27], the bimorph/unimorph DM [28–30], and the piezoelectric (PZT) stacked array DM [11,18,31,32]. However, for the distortion correction of the large-aperture thin-wall and high-power ALB (i.e. the 53mm-diameter, 8mm-wall thickness and 1.4kW-power ALB generated by the inside pumped Nd: YAG tube laser [10]), the demand of a large aperture restricts the application of the MEMS DMs [24] and bimorph/unimorph DMs [30]. Limited by the damage threshold, the SLM [21] and membrane DMs [27] are not suitable for high-power ALB. The conventional PZT DM (CDM) is also not an ideal solution for the large-aperture thin-wall and high-power ALB, although they have been widely used in larger-aperture high-power laser systems (e.g. the laser used in the National Ignition Facility). Restricted by the structure and size of the PZT actuators, the distance between the adjacent PZT actuators is difficult to be set smaller than 1mm [32], which indicates that the effective actuators covered by the thin-wall ALB is very limited. To meet the requirements of high aperture, high actuators’ density and high damage threshold for the large-aperture thin-wall and high-power ALB, a PZT stacked array tubular DM (TDM) and a novel AO system was proposed in our previous study. Through a beam transformation and control system consisting of a prism and a TDM prototype, the number of the effective actuators covered by the ALB could be improved from 16 [i.e. the number of the effective actuators of the CDM covered by the vertical incident ALB, Fig. 1(b)] to 180, while the normalized residues of the 2^{nd} to 15^{th} order Zernike polynomials aberration and the power density on the TDM could be effectively controlled below 10% and 6.1%, respectively. Although the installation and adjustment of the TCM’s actuators are easily to realize, however, restricted by the processing technology (e.g. the polishing technique and the film coating technique) and the measurement technology of the tube mirror’s inner surface, generally, the maximal height of TDM could not be set longer than 50 mm, which means that the stretch effect of the wall thickness is limited [33].

In order to overcome the processing and measurement restrictions of the TDM’s inner surface, the TEACC consisting of two EPGs and a novel OTDM, whose outer cylindrical surface is set as the effective reflectance surface, is proposed in this paper. Comparing with the reported TDM [33], the proposed OTDM has distinct advantages in the design and manufacturing, as the optical polishing [34], optical coating [35] and the high-accuracy absolute measurement techniques [36] for the tube mirror’s outer cylindrical surface are generally available. Different from the reported adaptive optics configuration based on TDM, as the ALB converges onto the outer surface of the OTDM, the radius of the OTDM should be smaller than the radius of the input ALB. What’s more, the EPG is adopted in the TEACC [Fig. 2(b)] to expand the radius of the OTDM and to make the manufacturing flexible.

This paper is organized as follows. In section 2, the configuration and the principle of the TEACC based on the OTDM and two EPG are discussed. The coordinate transformation equations between the 2D ALB and the OTDM are analyzed based on the simple ray tracing and geometric relations. In section 3, the influence of the structure parameters on the circumferential magnification of the OTDM in practical application is discussed in detail. In section 4, designing principle of the OTDM is discussed and an OTDM prototype is designed. The distribution of the actuators in 3D and 2D are analyzed. In section 5, a finite element model is built to study the typical influence functions of the OTDM. According to the coordinate transformation equation, the effective influence functions characteristics on the ALB are analyzed. In section 6, based on the effective influence functions on the ALB, the least square algorithm and 2^{nd} to 36^{th} order Zernike polynomials over an annular pupil are adopted to verify the correction ability of the TEACC.

## 2. Configuration and principle

A typical near field intensity distribution of the large-aperture thin-wall ALB is illustrated in Figs. 1(a) and 1(b) in orange ring. As shown in Fig. 1(a), the intensity distribution of the ALB could be represented by radius ${r_{m\textrm{1}}}$ (i.e. ${{({r_o} + {r_i})} \mathord{\left/ {\vphantom {{({r_o} + {r_i})} 2}} \right.} 2}$) of the middle circle (in black dash line) and the wall thickness $\Delta r$ (i.e. ${r_o} - {r_i}$). In this paper, the outer radius ${r_o}$ and the inner radius ${r_i}$ of the ALB are set 25mm and 21mm, respectively. Correspondingly, the middle radius ${r_{m\textrm{1}}}$ and the wall thickness $\Delta r$ are 23mm and 4mm, respectively. As shown in Fig. 1(b), for the CDM, only one effective actuator could be covered by the vertical incident ALB along the radial direction, as the diameter (5mm) of the PZT actuators are larger than the wall thickness $\Delta r$ (4mm). Generally, in order to improve the correction ability of the AO system, the input beam is normally enlarged using a beam expander and a large-aperture high-density-actuator CDM is adopted. Figure 1(c) shows the 3.28-time expanded ALB and a 170mm-aperture 441-actuator CDM with 8mm spacing of two adjacent actuators. The middle radius $r_{m\textrm{1}}^{\prime}$ and the wall thickness $\Delta {r^{\prime}}$ of the expanded ALB are 75.44mm and 13.12mm, respectively. However, as shown in Fig. 1(c), even for the 3.28-time expanded ALB, the effective number of the actuators covered along the radial direction is not more than 3. Obviously, the actuators’ number of the CDM is insufficient for the distortion correction of the large-aperture thin-wall ALB, even a beam expander and a large-aperture high-density-actuator CDM are adopted.

#### 2.1 Configuration of the AO unit based on OTDM

In order to achieve superior distortion correction ability, a novel TEACC, consisting of circumference enlargement stage (stage 1, making up of prisms P1 and P2), wall-thickness enlargement and adaptive correction stage (stage 2, making up of prism P2 and OTDM), beam profile recovery optical path (making up of another EPG, which is placed symmetrically around the OTDM with stage 1), a splitter mirror M1, a Shack-Hartmann wavefront sensor (SHWS), a voltage driver and one PC, is proposed and depicted in Fig. 2(a). In this configuration, the two EPGs and the OTDM form the beam transformation and correction system (TCS), which is shown in Fig. 2(b).

As shown in Fig. 2(b), in prism P1, the vertical incident ALB from the plane end would output from the inward cone end at the angle ${\beta _\textrm{1}}$ (i.e. the angle between the output light and the Z direction). Then, the beam from prism P1 would incident into the outer conical surface of prism P2 with half conical angle ${\alpha _\textrm{1}}$, which is equal to the half conical angle of prism P1. As a result, the middle radius ${r_{m\textrm{1}}}$ of the input ALB would be expanded to ${r_{m\textrm{2}}}$, which could be called circumferential stretching effect of the stage 1 (CSE-s1), while the wall thickness of the expanded ALB is still equal to $\Delta r$. According to the geometrical optics principle, the CSE-s1 of the ALB in the first EPG could be influenced by primary parameters of prisms P1 and P2, including the refractive index *n*, the half conical angle ${\alpha _1}$ and the distance between prisms P1 and P2. Thus, an important advantage of the OTDM is that the CSE-s1 could be regulated by changing the distance between prisms P1 and P2 in practical application, as the refractive index and the half conical angle of the prism are always unchangeable.

In stage 2, the expanded ALB in prism P2 would output from the right cone end (half conical angle ${\alpha _\textrm{2}}$) at angle ${\beta _\textrm{2}}$ (i.e. the angle between the output light and the Z direction) to the OTDM. As a result, the annular plane wavefront of the ALB is spread to the cylindrical wavefront on the outer cylindrical surface of the OTDM. According to the geometrical optics principle, in stage 2, on the outer surface of the OTDM, the wall thickness of the ALB would be stretched, while the middle circle’s perimeter of the ALB would be shrunk. The stretching and shrinking effects are determined by primary parameters of prism P2 and OTDM, including the half conical angle ${\alpha _\textrm{2}}$, the refractive index *n* and the distance between prism P2 and the OTDM. Considering the CSE-s1, the radius of the OTDM could be larger than the middle radius of the initial input ALB, which means that the middle circle’s perimeter of the initial input ALB could be stretched from the prims P1 to the OTDM. As the outer cylindrical surface of the OTDM is covered by sufficient actuators, the cylindrical wavefront distortion could be compensated in high precision. As prisms P4 and P3 is placed symmetrically around the OTDM with the prisms P1 and P2, after being reflected by the OTDM and transmitting through the prisms P3 and P4, the expanded ALB would be transformed back into the profile of the original ALB. Note that the materials of the four prisms in the TCS are the same.

The output ALB from prism P4 would be reflected by the splitter mirror M1 and detected by the SHWS. Based on the detected wavefront distortion and the measured influence functions of the OTDM, a close-loop algorithm is carried out to acquire the driven voltages of each actuator. By applying the driven voltages on the actuators, the outer cylindrical surface of the OTDM could be deformed to compensate the distortion of the ALB. According to the stretching effect of the TCS, the effective actuators covered by the enlarged ALB is sufficient and high precision distortion correction could be achieved.

#### 2.2 Coordinate transformation in the TCS

Based on the description of the optical configuration above, the correction ability of the OTDM is determined by the size of the stretched cylindrical wavefront and the distribution of the actuators covered by the stretched cylindrical wavefront. As the wavefront of the input ALB is plane and the stretched wavefront is cylindrical, it is important to derive the coordinate relationship between the plane annular wavefront and the cylindrical wavefront.

According to the symmetry of the optical path, Fig. 3 illustrates the coordinate transformation from prism P1 to the OTDM (i.e. the stage 1 and stage 2) based on the cylindrical coordinate system. As shown in Fig. 3(a), the inner orange ring, the outer orange ring and the grey ring represent the circumferential sections of the ALB in P1/P4 (i.e. the input/output ALB), the ALB in P2/P3 and the OTDM, respectively. Note that the red circle on the grey ring [Fig. 3(a)] represents the $\rho OX$ section of the stretched cylindrical wavefront, while the $\rho OZ$ section of that is depicted by the red lines on the OTDM along *Z* direction in Fig. 3(b). Obviously, the perimeter of the input ALB would be enlarged to the perimeter of the ALB in prism P2 and sequentially be shrunk to the perimeter of the OTDM (i.e. the perimeter of the cylindrical wavefront), which is called the circumferential stretching effect of the two stages (CSE-2s) and could be quantitatively expressed as the circumferential magnification ${\Gamma _c}$ [Eq. (1)]. On the other side, the thin wall $\Delta r$ of the input ALB along the radial direction is stretched to the generatrix *L* of the outer cylindrical surface [i.e. the red lines on the OTDM along the Z direction in Fig. 3(b)], which is called the radial stretching effect (RSE) and could be quantitatively expressed as radial magnification ${\Gamma _r}$ [Eq. (1)].

In Fig. 3, a cylindrical coordinate system is established, setting the conical point of prism P2 as the origin *O*, the radius direction of the ALB as the $\rho $ axis direction, and the optics propagation direction in prism P1 as the *Z* axis direction, respectively. Due to the rotational symmetry, the angle $\theta $ between the *X* axis and the $\rho OZ$-section could be set an arbitrary value between 0 and $\textrm{2}\pi $. The green lines in Fig. 3(b) represent the light rays from the middle circle of the ALB. Points ${C_1}$ and ${C_2}$ are the intersection points between the green light ray and the outer cylindrical surface of the OTDM. As shown in Fig. 3(a), from prism P1 to prism P4, angle $\theta $ between the *X* axis and the $\rho OZ$-section is not changed. Thus, in the cylindrical coordinate system, the coordinate transformation from the input ring wavefront of the ALB to the cylindrical wavefront is the transformation from the radius $\rho $-coordinate to the generatrix *Z*-coordinate of the outer cylindrical surface.

In order to illustrate the coordinate transformation, an arbitrary interaction point ${A_1}$ between the input thin-wall ALB and the cone end of prism P1 is selected, whose $\rho $-coordinate is ${r_{A1}}$. As the red solid line shown in Fig. 3(b), by the refraction of prism P1, the optical ray from point ${A_1}$ would interact with the left cone end of prism P2 at point ${A_\textrm{2}}$, whose $\rho $-coordinate is ${r_{A\textrm{2}}}$. Owing to the refraction of the left cone end of prism P2, the light would propagate along the Z direction and arrive at point ${A_\textrm{3}}({r_{A\textrm{2}}},{z_{A3}})$. After that, the ray arrives at point ${A_4}({R_D},{z_A})$ on the outer surface of the OTDM.

According to the geometrical relationship of triangle ${A_\textrm{1}}{D_\textrm{2}}{A_\textrm{2}}$ and triangle ${D_\textrm{1}}{D_\textrm{2}}{A_\textrm{2}}$, the difference $\Delta {r_{12}}$ between ${r_{A\textrm{2}}}$ and ${r_{A1}}$ is equal to the length of line ${A_2}{D_2}$, while the distance between point ${A_1}$ and point ${D_1}$ (the point on the left cone end of prism P2) along the Z direction could be expressed by Eq. (2). As the cone end of prism P1 is parallel to the left cone end of prism P2, the distance between point ${A_1}$ and point ${D_1}$ is also equal to ${L_1}$, which is the distance between the conical point of prism P1 and the left conical point of prism P2. Thus, the difference $\Delta {r_{A12}}$ could be expressed as Eq. (3).

*n*based on the Snell law. Due to the arbitrariness of point ${A_1}$, in Eq. (3), radius ${r_{A\textrm{2}}}$ could change from the minimum value ${r_i} + \Delta {r_{12}}$ to the maximum value ${r_o} + \Delta {r_{12}}$, while the radius ${r_{A\textrm{1}}}$ changes from the inner radius ${r_i}$ to the outer radius ${r_o}$ of the input ALB. Specially, the radius ${r_{A\textrm{2}}}$ represents the middle radius ${r_{m2}}$ of the ALB in prism P2, while the radius ${r_{A\textrm{1}}}$ equals to the middle radius ${r_{m\textrm{1}}}$ of the ALB in prism P1. Therefore, ${r_{m2}} = {r_{m1}} + {L_1}/(\cot {\beta _1} - \cot {\alpha _1})$.

According to the geometrical relationship between point ${A_\textrm{3}}({r_{A\textrm{2}}},{z_{A3}})$ and point ${A_4}({R_D},{z_A})$, the coordinates ${z_{A3}}$ and ${z_A}$ could be expressed by Eqs. (4) and (5).

*Z*direction [Eq. (6)]. ${R_D}$ is the outer radius of the OTDM (i.e. the radius of the cylindrical wavefront), which could be expressed in Eq. (7).

*O*and point ${C_1}$ along the

*Z*direction.

In Eq. (5), the Z-coordinate ${z_A}$ of point ${A_\textrm{4}}$ could change from the minimum value $[{r_i} + \Delta {r_{12}}](\cot {\beta _2} - \tan \gamma ) - {R_D}\cot {\beta _2}$ to the maximum value $[{r_o} + \Delta {r_{12}}](\cot {\beta _2} - \tan \gamma ) - {R_D}\cot {\beta _2}$, while ${r_{A\textrm{2}}}$ changes from the minimum value ${r_i} + \Delta {r_{12}}$ to the maximum value ${r_o} + \Delta {r_{12}}$ (i.e. the radius ${r_{A\textrm{1}}}$ changes from the inner radius ${r_i}$ to the outer radius ${r_o}$ of the input ALB). The difference between the minimum and the maximum value of the ${z_A}$ of point ${A_\textrm{4}}$ represents the generatrix *L* of the cylindrical wavefront [i.e. the length of the red solid line on the OTDM in Fig. 3(b)].

The coordinate transformation from the 2D rectangular coordinate $({x_o},{y_o})$ of point ${A_1}$ on the input ALB to the 3D rectangular coordinate $({x_A},{y_A},{z_A})$ of point ${A_4}$ on the OTDM could be expressed in Eq. (8).

Considering the symmetry and the reversibility of the optical path, as the red dash line with one arrow shown in Fig. 3(b), the $\rho $-coordinate $r_{A1}^{\prime}$ of point ${A_{D\textrm{1}}}$ on the input ALB could be calculated by the coordinate $({R_D},{z_D})$ of the corresponding point ${A_D}$ on the OTDM and be expressed in Eq. (9).

## 3. Analysis of the parameters’ regulation effects on the two-stage enlargement

As analyzed in section 2, the radial magnification ${\Gamma _r}$ could be regulated by angle ${\alpha _2}$ and refractive index *n* of prism P2, which could be rewritten as Eq. (11) by substituting *L* into Eq. (1). Meanwhile, the circumferential magnification ${\Gamma _c}$ depends on angle ${\alpha _\textrm{1}}$ of prism P1, refractive index *n* of two EPGs, angle ${\alpha _2}$ of prism P2, distance ${L_\textrm{1}}$ between prism P1 and prism P2, and distance ${L_\textrm{2}}$ from prism P2 to the half-height of the OTDM, which could be rewritten as Eq. (12) by substituting ${R_D}$ into Eq. (1).

Based on the monotonicity of trigonometric functions, as shown in Eq. (11), the radial magnifications ${\Gamma _r}$ of the OTDM would increase with the increasing of the output conical angle ${\alpha _2}$. Equation (12) illustrates the complex relationship between the structural parameters and the circumferential magnifications ${\Gamma _{c12}}$/${\Gamma _c}$, which quantitatively show the CSE-s1/ the CSE-2s, respectively.

Distributions of the stage 1’s circumferential magnification ${\Gamma _{c12}}$ with different angle ${\alpha _\textrm{1}}$ (vertical axis) and distance ${L_\textrm{1}}$ (horizontal axis) are depicted in Figs. 4(a) (*n *= 1.4) and 4(b) (*n *= 1.8). According to Figs. 4(a) and 4(b), varies of parameters combinations (i.e. angle ${\alpha _\textrm{1}}$ and distance ${L_\textrm{1}}$) could be chosen to achieve the target ${\Gamma _{c12}}$. It means that, in practical application, angle ${\alpha _\textrm{1}}$ and distance ${L_\textrm{1}}$ could be comprehensively considered based on the available conical angle ${\alpha _\textrm{1}}$ and practical assembly space. The difference between Figs. 4(a) and 4(b) shows that the circumferential magnification ${\Gamma _{c12}}$ could increase with the increasing of refractive index *n*. Figure 4(c) shows the positive linear relationship between circumferential magnification ${\Gamma _{c12}}$ and distance ${L_\textrm{1}}$, while the inverse proportional relationship between ${\Gamma _{c12}}$ and angle ${\alpha _\textrm{1}}$ is illustrated in Fig. 4(d). It means that the deviation of the target ${\Gamma _{c12}}$ caused by the processing error of angle ${\alpha _\textrm{1}}$ could be compensated by the adjustment of distance ${L_\textrm{1}}$, which shows unique operation flexibility of the TEACC.

As expressed in Eq. (12), the circumferential magnification ${\Gamma _c}$ would be influenced by angle ${\alpha _\textrm{2}}$ and distance ${L_\textrm{2}}$ for a given ${\Gamma _{c12}}$. Figures 5(a) and 5(b) show the distributions of the circumferential magnification ${\Gamma _c}$ with different angle ${\alpha _\textrm{2}}$ (vertical axis) and distance ${L_\textrm{2}}$ (horizontal axis). Note that the given ${\Gamma _{c12}}$ values of Figs. 5(a) and 5(b) are both set 3.85. As shown in Fig. 5(c), the circumferential magnification ${\Gamma _c}$ would decrease with the increasing of distance ${L_\textrm{2}}$, while it would increase with the increasing of angle ${\alpha _\textrm{2}}$ [Fig. 5(d)]. It also indicates that the deviation of the target ${\Gamma _c}$ caused by the processing error of angle ${\alpha _\textrm{2}}$ could be compensated by adjusting the distance ${L_\textrm{2}}$.

However, in practical application, in order to acquire high radial magnification ${\Gamma _r}$ (e.g. 16), angle ${\alpha _\textrm{2}}$ could be set close to 90$^\circ $ (e.g. 85.58$^\circ $). As a result, the regulation effect of distance ${L_\textrm{2}}$ on the circumferential magnification ${\Gamma _c}$ is limited. For example, as shown in Fig. 5(c), when ${\alpha _\textrm{2}}$=85.58$^\circ $, the circumferential magnification ${\Gamma _c}$ only changes from 3.7 to 3.43, while distance ${L_\textrm{2}}$ varies from 50mm to 150mm.

In order to achieve the target circumferential magnification ${\Gamma _c}$ (3.57), the change curve of distance ${L_\textrm{2}}$ with different angle ${\alpha _\textrm{2}}$ is depicted in black line in Fig. 6(a), while other parameters are set as listed in Table 1. From the black line in Fig. 6(a), in order to achieve the target ${\Gamma _c}$, distance ${L_\textrm{2}}$ need to be changed from the designed 100mm to the optimized 75.93mm, if angle ${\alpha _\textrm{2}}$ deviates from the designed 85.58$^\circ $ to the practical 84.58$^\circ $ (i.e. 1$^\circ $ deviation from the designed value). It means that, in order to maintain the target ${\Gamma _c}$, the spacing between prism P2 and the OTDM should be greatly reduced to compensate the slight processing error of the angle. As the red line shown in Fig. 6(a), the target ${\Gamma _c}$ could also be realized by increasing the distance ${L_\textrm{1}}$ from the designed 100mm to 102.67mm, when angle ${\alpha _\textrm{2}}$ deviates from the designed 85.58$^\circ $ to the practical 84.58$^\circ $. Figure 6(b) shows the linear relationship between the circumferential magnification ${\Gamma _c}$ and distance ${L_\textrm{1}}$.

For an assembled TCS unit, parameters of the elements are unchangeable, including the refractive index and half conical angle of the prisms. Adjusting the distance ${L_\textrm{1}}$ or/and ${L_\textrm{2}}$ is the effective approaches to compensate the processing error. However, as distance ${L_\textrm{2}}$ is shorter than half-height of the OTDM in the TCS unit, in some practical situation, it is unachievable to compensate the processing error by only reducing distance ${L_\textrm{2}}$.

## 4. Prototype and design principle of the OTDM

In order to achieve the targets values of ${\Gamma _c}$ (3.57), ${\Gamma _{c12}}$ (3.85), and ${\Gamma _r}$ (16), an OTDM prototype and the related structural parameters of TCS are designed. Figure 7 shows the designed OTDM prototype with 324 PZT actuators (i.e. 36 lines in the circumferential direction, 9 lines in the generatrix direction) covering the outer cylindrical surface of 64-mm generatrix length (*L*) and 82-mm radius (${R_D}$). Detailed structural parameters are listed in Table 1.

As shown in Fig. 7, the OTDM prototype is essentially a stacked array PZT DM with discrete PZT actuators and a continuous tube mirror. The PZT actuator is assembled in the specific channel and connected to the mirror through a post [Fig. 7(c)]. Figure 7(e) shows the distribution of the actuators’ posts on the OTDM. The radial [Fig. 7(b)] and circumferential [Fig. 7(d)] cross sections of the OTDM show that the distribution of the actuators could be described by the generatrix spacing *h* and the circumferential spacing *l* of the adjacent actuators. Besides, the circumferential distribution of actuators could also be characterized by the circumferential angle ${\theta _0}$ between the adjacent actuators.

In practical application, in order to increase the density of the actuators and improve the correction ability of the OTDM, the spacing between two adjacent actuators should be set as small as possible. However, restricted by the diameter *d* of the actuator, the spacing has certain limitation [Fig. 8]. In order to meet the manufacture requirement, the bottom spacing between the adjacent PZT actuators along the circumferential direction (i.e. the green dash circle) should be larger than the diameter *d* of the actuator. Thus, the radius ${R_B}$ of the green dash circle formed by the bottoms of the actuators should satisfy Eq. (13). Taking the heights (${h_{PZT}} = $36mm and ${h_{post}} = $3mm) of the actuator and the post, the thickness (${h_g} = $2mm) of the tube mirror into consideration, the radius ${R_D}$ of the OTDM’s outer cylindrical surface could be expressed by Eq. (14). Thus, the circumferential spacing *l* of the adjacent actuators on the OTDM is equal to ${\theta _\textrm{0}}{R_D}$.

The unfolded outer cylindrical wavefront and the unfolded outer cylindrical surface of the OTDM along the circumferential direction are illustrated in Fig. 9. The unfold view clearly shows the matching relationship between outer surface of the OTDM and the enlarged cylindrical wavefront. As shown in Fig. 9(a), in order to ensure the correction ability of the wavefront edge, the generatrix (*H*) of the OTDM is set larger $h/2$ than that of the cylindrical wavefront (*L*) in both sides. Thus, in practical operation, the coordinate ${z_A}$ of the OTDM would vary from (${L_\textrm{2}} - H/2$) to (${L_\textrm{2}} + H/2$). The distribution of the actuators in the unfold view is in rectangle, while the actuators’ spacings along the circumferential and radial directions are $l$ (${\theta _0}{R_D}$) and $h$[Fig. 9(b)], respectively.

Figure 9(d) shows the equivalent effective area and actuators’ distribution of the OTDM covered by the ALB on the XOY plane (i.e. the original ALB plane). In Fig. 9(d), the grey ring from ${r_{Di}}$ to ${r_{Do}}$[Eq. (15)] is the effective aperture of the OTDM, while the orange ring from ${r_i}$ to ${r_o}$ is the aperture of input ALB. Equivalent effective distribution of actuators is shown in the partial enlarged view [Fig. 9(c)]. According to the beam transformation principle, the actuators’ spacing ${h_r}$ along the radial direction is a constant value $h/{\Gamma _r}$, while the circumferential spacing ${h_{ck}}$ is the function of the distance ${r_{ck}}$ between centers of the ${k^{th}}$ (along the radial direction from outside to inside) actuator and the ALB [Eq. (16)]. Obviously, in the 2D plane, the distribution of the actuators is radioactive [Fig. 9(d)], which means that the actuator’s density would increase with the increasing of the *k*.

## 5. Influence functions of the OTDM

The influence function (IF, i.e. the deformation of the surface shape when one actuator is activated by the driven voltage) is a critical parameter to evaluate the performance of the OTDM. Different from the CDM, the IF of the OTDM reflects the deformation of the outer cylindrical surface of the OTDM (IF-OCS), and would be equivalently projected to the 2D plane of the input ALB. Obviously, it is of vital significant to calculate the IF-OCS and the projected IFs on the annular wavefront (IF-AW). As shown in Figs. 7(a) and 9(d), considering the rotational symmetry of the OTDM, the IFs in the same circumference direction (36 circumference directions in the prototype) are identical. As a result, the IFs of all the 324 actuators could be presented by nine typical IFs [marked from 1 to 9 in Fig. 7(e) and Figs. 9(b)–9(c)]. Note that the actuators marked in the three figures are identical.

In order to calculate the nine typical IFs-OCS, a finite element model is built in COMSOL Multiphysics software [37,38]. In the model, as shown in Fig. 7, the OTDM consists of a continuous tubular BK7 mirror with 324 discrete cylindrical posts, 324 discrete PZT actuators, and a stainless-steel cylindrical base with a hole. Primary parameters of the structure and the materials in the simulation are listed in Table 1 and Table 2, respectively. The whole model is built using the solid stress interface from the structural mechanics module. In the simulation, the steel base is fixed and each actuator is set pushing the mirror along the axis of itself in the local coordinate system. The equivalent 38µm (the maximum stroke 38µm of the PZT actuator, P885.91, Physik Instrumente GmbH) displacement is set to the contact surface between the post and the actuator, providing a maximum voltage of 120V is applied on the actuator.

Due to the similar boundary condition of the central seven actuators (No. 2- No.8), the profiles of their IFs-OCS are almost same, which could be represented by the IF-OCS of No.5 actuator. Thus, three typical IFs-OCS of the nine typical actuators numbered 1, 5 and 9 in Fig. 7(e) are shown in Fig. 10. It could be seen that the maximum value (31µm) of the IFs-OCS [Fig. 10(b1)] are smaller than the displacements (38µm) applied on the posts, which could be explained by the stiffness effect of the tubular mirror. Due to the less constraint on the 1^{st} and 9^{th} actuators, the maximum value (46µm) of the IFs-OCS [Figs. 10(a1) and 10(c1)] are larger than the displacements (38µm) applied on the posts. The IFs-OCS could be regarded as the IFs of the CDM intersecting with the outer cylindrical surface of the OTDM. Thus, the deformation of the reflective surface of the OTDM is away from the centre line, as the outer surface of the tube is pushed by the PZT actuator.

In consideration of the oblique incidence on the OTDM, a modulation coefficient should be considered on the relationship between the stroke (i.e. the PV values of the IFs-OCS, $\Delta \omega $) of the OTDM and the effective stroke (i.e. the PV values of the IFs-AW, $\Delta \omega ^{\prime}$) applied on the output ALB wavefront [Eq. (17)] [33]. Due to the reflectance effects on the OTDM, the dynamic range of the OTDM could be represented by 2-time effective stroke ($\Delta \omega ^{\prime}$).

where ${\beta _\textrm{2}}$ is the angle between the incidence light of the OTDM and the Z-direction. As the value $\sin {\beta _\textrm{2}}$ is smaller than 1 in practice, the effective stroke applied on the ALB is always smaller than the real stroke supplied by the OTDM. In the designed TCS, ${\beta _\textrm{2}}$ is equal to 3.558° and the modulation coefficient is 0.062.By introducing the coordinate transformation [Eq. (10)] and the stroke modulation [Eq. (17)] into the IFs-OCS by using MATLAB software, the IFs-AW of all the actuators could be acquired. Correspondingly, the IFs-AW of three typical actuators numbered 1, 5 and 9 are shown in Figs. 10(a2)–10(c2), respectively. The partial enlarged views of the 2^{nd} row is depicted in the 3^{rd} row respectively. Considering the modulation of the oblique incidence, the effective stroke $\Delta \omega ^{\prime}$ just is about 2µm, which means that the dynamic range of each actuator in the OTDM is about 4µm. Note that the dynamic range could be improved with the increasement of the piezoelectric properties of the PZT actuators. According to the latest report, the electric field-induced strain of the newest PZT material is approximately 90% higher than the previous one [39], which indicates that the dynamic range of the OTDM could be improved to about 8µm by using the newest PZT material. As shown in Figs. 10(a3)–10(c3), different from the circular/square/hexagon shape of the IF in CDM, the profile of the OTDM’s IF-AW appears in the shape of an approximate crescent moon. The distance *r _{ck}* between the center of the actuators (i.e. the IF-AW) and the center of the ALB could be calculated by Eq. (16) and marked in Fig. 10.

## 6. Correction ability of the OTDM

Based on the IF-AW analyzed above, in the numerical simulation, the Zernike 2-36 order annular aberrations [40,41] and the least square algorithm are taken to verify the correction ability of the TCS/TEACC prototype. It should be noted that the diameter of the OTDM prototype is 164mm, which is 3.28-time of the outer diameter $2{r_o}$ (50mm) of the input ALB [Fig. 1(a)]. It is known that the beam expanding is a common and effective way of improving the effective actuators’ density and correction ability of the CDM. Thus, to verify the superiority in the correction ability of the TCS/TEACC prototype, the performance of the CDM without beam expanding (1-CDM), the CDM with 3.28-time beam expanding (3.28-CDM) are also studied in the simulation. The effective aperture and actuators of the 1-CDM and the 3.28-CDM are shown in Figs. 1(b) and 1(c), respectively.

To show the distribution characteristics of Zernike annular aberration and the correction residues based on different AO systems. The 2^{nd}, 4^{th}, 7^{th} and 11^{th} order Zernike annular aberration and the correction residues of the 1-CDM, the 3.28-CDM and the OTDM are shown in Fig. 11 from the 1^{st} to the 4^{th} columns, respectively. Note that all the PV values of the original Zernike annular aberrations are set 1µm.

As shown in the 1^{st} column of Fig. 11, Zernike annular aberrations could be divided into two types based on their radial distributions. The first type is the circumferentially varying wavefront (e.g. Z_{2}), while the second type is the circumferentially and radially varying wavefront (e.g. Z_{4}/Z_{7}).

From Fig. 11, comparing with the 3.28-CDM and the OTDM, it could be seen that the correction ability of the 1-CDM on the first-type aberrations are very limited. The distortion along the radial direction could not be well corrected and the PV values of the residues are larger than 0.4µm. Comparing with the residues of the 1-CDM and the 3.28-CDM, using the OTDM, the original severe distortion disappears essentially and only tiny uniform local residues are left. After compensation using the OTDM, the distortion is well corrected and the PV values of the correction residues could be controlled smaller than 0.1µm.

Due to only one actuator along the radial direction [Fig. 1(b)], the correction residues of the 1-CDM on the second-type aberrations are almost the same with the initial aberrations [the 1^{st} and 2^{nd} column of Fig. 11], which means that the 1-CDM almost has no compensation ability on the second-type aberrations. By adopting 3.28-time beam expanding system, as shown in Fig. 1(c), the number of the actuator covered by the ALB wall could be improved to 3. As a result, comparing with the 1-CDM, the correction residues of the 3.28-CDM are improved. However, as shown in Fig. 11, local large random distortions are still left in the residues of the second-type aberration (e.g. Z_{4}, Z_{7}, Z_{11}). On the contrast, the OTDM still has strong compensation ability on all of the second-type aberrations with PV values of the correction residues only about 0.05µm.

In order to clearly illustrate the improvement in the correction ability from the CDM to the OTDM, the PV values of the original distortion and the correction residues are summarized in Fig. 12(a). The normalized correction residues are shown in Fig. 12(b). As shown in Fig. 12, although the PV values of the 3.28-CDM’s correction residues are all smaller than that of 1-CDM, the normalized correction residues of the 1-CDM and 3.28-CDM fluctuate severely with the aberration distribution (i.e. the order of the Zernike). The fluctuation phenomenon indicates that the correction ability of the CDM on the ALB relies heavily on the distribution characteristics of aberrations, which is unacceptable in practical application for the randomness and unpredictability of laser system’s aberration. On the contrast, as the purple bars and line shown in Fig. 12, the OTDM’s normalized correction residues of the 2^{nd} to 36^{th} Zernike aberration could be well controlled smaller than 0.1µm.

The simulation results show that the aberration correction ability of the TEACC based on the OTDM prototype. By stretching the input ALB along the radial and circumferential directions at the same time, the correction ability could be improved remarkably compared with the 1-CDM and the 3.28-CDM.

## 7. Conclusion

In conclusion, a novel TEACC based on the OTDM to correct the wavefront distortion of the larger-aperture thin-wall high-power ALB is proposed and investigated theoretically. In this configuration, the thin-wall wavefront of the ALB is enlarged along the circumferential direction in stage 1 and then stretched along the radial direction based on the stretching effect of the oblique incidence in stage 2. The CSE-s1 relies on the refractive index, the cone angle of the prism and the distance between the two prisms, while the radial stretching effect and the CSE-2s are determined by the index and output cone angle of P2 and the distance between P2 and OTDM. The correction principle and the coordinates transformation equations of the AO configuration are investigated according to the ray tracing method. A finite element model is built to investigate the IF characteristics of OTDM. The correction ability of the OTDM is also simulated using the least square method. The normalized residues of all the 2^{nd}- 36^{th} order Zernike polynomials aberration could be controlled below 10% after correction. Comparing with the CDM, the OTDM has distinct compensation ability on the aberration of the larger-aperture thin-wall ALB.

## Appendix

## Funding

National Natural Science Foundation of China (61775112).

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