1. Introduction

Dynamic and multidimensional laser spot control in laser material processing machinery allows for rapid processing of complex architectures and is highly desirable as compared to motion of the element being processed. Lateral control of the beam spot using fast deflection mirrors is already industry standard [1]. However the fixed distance of the focal spot from the laser machine head due to static focusing lens optics poses a problem to the addition of a third dimension or strong lateral positions. Focus shifting can be accomplished either by shifting the focusing lens [2] along the beam direction or by beam manipulation with deformable mirrors [3,4]. In lens shifting systems, the focus position is only influenced by the position of the focus lens and its typically constant optical power. The dynamic motion of this lens provides a focus shift. An alternative to lens shifting systems are tunable lenses, which change their optical power when actuated [5,6]. In a deformable mirror system, the position and optical power of the focus lens are constant while the optical power of the mirror surface is variable. Thus, the focus position depends on the optical power combination of the deformable mirror and the focusing lens. There is a significant difference in the masses to be moved for focus shifting. While the entire lens must be moved in the case of the lens shifting system, only a displacement or deformation of the surface is necessary in the cases of the deformable mirror and the tunable lens. Active mirrors allow significantly higher operating frequencies and hence significantly higher processing speeds than lens shifting systems and tunable lenses. In [7,8] solutions for highly dynamic focus shifting with deformable mirrors for an angle of incidence of 0° are described. The deformed surface of these mirrors has a spherical shape. By using these mirrors at a beam incidence angle of greater than 0°, an elliptical distortion of the spot in the focal plane occurs [9], commonly described using the optical aberration astigmatism. Due to considerations relating to the power threshold of high power laser systems and overall compactness, implementation of a mirror under 0° poses its own challenges. The design of highly dynamic deformable mirrors which could provide focus shifting and furthermore correct the undesired elliptical distortion of the beam, allowing for implementation in existing laser machine heads without posing further technical challenges, would significantly advance the state of the art and it is this topic which is addressed in this paper. Of particular interest are metal-based deformable mirrors for beam incidence angles of 45°. In [10], a design is described for an angle of incidence of 0° for a simply supported metal-mirror and a central force application. In [11] this approach is extended to angles of incidence of 45°, though, the opto-mechanical design shows unwanted aberrations of the reflected wavefront in simulations and experiments, which lead to the enlargement of the spot. Using an analytical approach followed by an iterative finite element analysis (FEA), this paper describes a mirror design process for arbitrary angles of incidence. Subsequently, a diffraction limited opto-mechanical deformable mirror design for orthogonal beam deflection is presented. The paper concludes with dynamic investigations to show the suitability of the opto-mechanical design for highly dynamic beam oscillation applications.

This paper is organized as follows. Section 2. describes the baseline scenario for mirror-based focus shift with orthogonal beam deflection and optical boundary conditions. In Section 3. the design routine from the analytical solution up to the opto-mechanical design is explained. The evaluation of the opto-mechanical design in terms of the performance for focus shifting is discussed in Section 4. Finally, in Section 5. results of the optical and dynamic performance of the opto-mechanical design are presented.

2. Baseline scenario and boundary conditions

The baseline scenario in Fig. 1 is analogous to the proposed optical model of a biconic deformable mirror for focus focus shifting in [11].

 

Fig. 1. Baseline scenario without mirror deformation (solid line) and with mirror deformation (dashed line).

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It consists of a collimated beam, a deformable mirror for orthogonal beam deflection and a focal lens $f$ with a focal length of 200 mm. A laser beam with the following parameters is assumed as the source:

The optical components are at a fixed distance from each other. The focus position in the Z-direction is shifted by deforming the mirror surface. The working range spans between the minimum and maximum surface deformation.

3. Development of the opto-mechanical mirror design

Figure 2 shows the routine for generating the opto-mechanical design of the deformable mirror’s surface with its mechanical interfaces. The starting point is an analytical investigation. For this, a mathematical model, which predicts the surface bending, is selected and connected with the optical boundary conditions. A finite element model (FE model) is used to verify the analytical solution of the surface bending.

 

Fig. 2. Routine for development of opto-mechanical design

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This FE model is extended with geometric boundary conditions, such as the dimensions of the mirror body and the fixation of the substrate. The FE model is parameterized for the optimization process. Considering the target shape of the curved surface as well as the dynamic properties of the opto-mechanical design an iterative optimization is performed. In consequence, this method shows a routine to develop a metal-based mirror design for central focal plane shifting at arbitrary deflection angles. Because of its good machinability, high surface shape accuracy, high thermal load capacity and good dimensional stability aluminium has great properties for deformable mirror applications. In addition, aluminum is suitable for high connection rigidities to actuator and housing with regard to high eigenfrequencies [12]. In this work, the specific case of highly dynamic orthogonal beam deflection using an aluminium-based deformable mirror membrane is highlighted. For the development of the opto-mechanical design the surface shape accuracy as well as the eigenfrequency of the bendable surface each have a major significance. As has been reported in [11], an ideal biconical surface can be characterized by a ratio of the radii of curvature of 0.5 or the ratio of the Zernike coefficients defocus and astigmatism of 1.5. Due to the orthogonal beam deflection the collimated beam results in an elliptical footprint on the mirror. Figure 3 shows a biconically deformed surface with different radii of curvature in X- and Y-direction as well as the elliptical beam footprint. The error of the reflected wavefront of such a surface is null.

 

Fig. 3. Biconically deformed surface and elliptic footprint (dashed line) on the surface for orthogonal beam deflection.

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3.1 Optical and dynamical boundary conditions

The quality criteria for the optical performance of the deformable mirror in this paper will be the wavefront error and the Strehl ratio. André Maréchal defined the wavefront error ($RMS_{WF}$) of a diffraction limited optical system as follows [13]:

The wavefront error of a diffraction-limited reflecting surface ($RMS_{SE}$) depends on the angle of incidence ($\alpha /2$) of the beam and the Maréchal criterion [14]:

At a wavelength of 1064 nm, a diffraction-limited mirror for orthogonal beam deflection yields a permissible wavefront error of 76.0 nm and a permissible surface error of 53.7 nm. Another quality criterion is the Strehl ratio $S$ [15], which is approximated by

To describe the dynamic behaviour of the opto-mechanical design, the eigenfrequency is used. In the state of the art, deformable mirrors are postulated up to eigenfrequencies of 3.65 kHz [8]. This paper will demonstrate that this approach of an aluminium-based mirror allows eigenfrequencies up to 20 kHz. Table 1 summarizes the main optical and dynamic properties for developing the opto-mechanical design for orthogonal and dynamic beam deflection.

Table 1. Properties and target values of the opto-mechanical design

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3.2 Analytical consideration of variable thickness plates

The analytical approach is based on the theory of thin plates. Thin plates are structures whose thickness $d$ is small compared to the characteristic semi-diameter. In addition, loads on the plate occur perpendicular to the mean surface and due to edge loads. Furthermore, Hooke’s law is valid [16,17].

The bending energy $U$ depending on the thickness $d$, the Young’s modulus $E$, the Poisson’s ratio $\nu$ and the second derivatives $w_{xx}$ and $w_{yy}$ of the surface shape can be written as [18]:

In order to find the solution for the surface shape $w(x,y)$ and the thickness distribution $d(x,y)$ we have to minimize the bending energy U using the Euler-Lagrange algorithm. The corresponding differential equation reads

In the first step the desired surface form is inserted

Then the differential equation reduces to an equation for the thickness distribution

Inserting the ansatz function:

There is a non-proportional dependence of the radii of curvature $R_{x}$ and $R_{y}$ on the coefficients $a$ and $b$ as well as a material dependence via the Poisson’s ratio $\nu$. For a spherically deformed surface $R_{x}=R_{y}$ and therefore $a=b$. Thus, the y term of the ansatz function of Eq. (8) can be eliminated and the solution same as in [10] is obtained. However, oblique focusing mirrors with a spherically deformed surface lead to the optical aberration of astigmatism for angles of incidence greater than zero [9]. This can be corrected by a deformed surface with $R_{x} \neq R_{y}$. To achieve a single focal point in the sagittal and meridional planes (X and Y direction) of a deformed surface the ratio of the radii of curvature must be 0.5 for orthogonal beam deflection [11]. Figure 4 shows the dependency of the radii of curvature ratio $R_{x}/R_{y}$ with respect to the ratio of the coefficients $a$ and $b$ for aluminum with a Poisson’s ratio $\nu$ of 0.33. This leads to an approximated ratio of the coefficients $a/b$ of 1.18.

 

Fig. 4. Ratio of the radii of curvature $R_{x},R_{y}$ in relation to the coefficients-ratio $a/b$

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3.3 Verification of the analytical solution by FE analysis

A numerical FE analysis using ANSYS Workbench is performed to verify the analytical solution. The boundary conditions with respect to the solution of the plate differential equation are included in the FE model. The thickness function Eq. (8) provides the logarithm of 1 for $x=a$ with $y=0$, respectively for $y=b$ with $x=0$. This results in a thickness of zero in the edge areas. Moreover, the logarithm for ${x\to \ 0}$ and ${y\to \ 0}$ converges to infinity. In order to be able to perform an FE analysis, the domain is limited. Furthermore, the optical boundary condition due to the elliptical footprint of a circular beam on oblique surface is added. For surfaces of orthogonal beam deflection the relation of the elliptical semi-axes x and y applies.

The axis-dependent domains are specified as.

To perform a verification of the analytical model a set of geometrical parameters for an FE model was chosen. These parameters represent typical values for mirrors in laser material processing. The plate, which represents the mirror substrate is simply supported at the edge and deformed with a central displacement $u$ = 1 µm. To suppress edge effects, the plate is twice oversized compared to the collimated beam diameter $\omega$ = 10 mm. Equation 10 yields the outer contours of the elliptical plate with $x=2 \cdot \omega$ = 20 mm and $y$ = 28.28 mm.

The maximum thickness in the plate center is $d_{max}$ = 2.1 mm, taking into account the domain (Eq. (11)) as well as the elliptic outer contour ($x,y$) and the coefficients $a$ and $b$. The parameters describing the FE model are summarized in Table 2.

Table 2. Geometric parameters for FE model

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Figure 5 shows the FE model as well as the boundary conditions. To describe a biconical surface, the Zernike coefficients defocus ($C_{3}$) and astigmatism ($C_{4}$) in the Fringe notation can be used in addition to the radii of curvature. This type of characterization offers the possibility to describe the optical surface more precisely and to detect surface errors. According to Table 1 an ideal deformed surface for generating a focal point with orthogonal beam deflection must have a $C_{3}/C_{4}$-ratio (defocus/astigmatism) of 1.5.

 

Fig. 5. Geometry sketch of the analytical solution and boundary conditions.

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Figure 6 shows the simulated deformed surface a), a fit of this surface with the Zernike coefficients $C_{3}$ and $C_{4}$ b) and a fit with the Zernike coefficients $C_{3}$ to $C_{36}$ c). The Zernike coefficients $C_{0}$ (Piston), $C_{1}$ (Tip) and $C_{2}$ (Tilt) are not considered. Figures 6(d) and (e) show the residual errors when subtracting the fitted surface (b,c) from the simulated surface (a). In order to fully evaluate the surface irradiated by the beam, the norm radius $R_{norm}$ of the Zernike decomposition is set to $y/2$ = 7.07 mm.

 

Fig. 6. Simulated surface by FE analysis a). Fit of Zernike coefficients with defocus ($C_{3}$) and astigmatism ($C_{4}$) b). Fit of Zernike coefficients $C_{3}$ - $C_{36}$ c). Residual errors between simulation and fits d+e).

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The fits (Fig. 6(b) and (c)) of the simulated surfaces both have a $C_{3}/C_{4}$ ratio of 1.61. The plate theory is valid for thin plates ($d/x\ll 1$, $d/y\ll 1$), a homogeneous thickness distribution and isotropic material [17]. The geometry, which is investigated here, has a variable thickness distribution. This results in a deviation from the ideal biconical surface.

For the development of the opto-mechanical mirror design, the residual error of the surface shape after subtracting $C_{3}$ and $C_{4}$ (Fig. 6(d)) from the simulated surface is a crucial quantity for evaluating the optical performance. Subtraction of the $C_{3}$ - $C_{36}$ fitted surface (Fig. 6(c)) from the simulated surface yields a surface error of 0.13 nm (Fig. 6(d)). This shows that it is possible to describe the simulated surface (Fig. 6(a)) with a surface error of 0.026 % regarding the peak to valley of the deflected surface. The FE model confirms the analytical solution considering the optical boundary condition and the deviations of the thickness convention of the plate theory.

3.4 Advancement of the FE model

Decisive for the evaluation of the FE model (Fig. 5) is the minimization of the surface shape error and maximization of the eigenfrequency. To evaluate the optical performance, the surface is deformed by a central deflection $u$ of 1 µm and decomposed into Zernike coefficients with a norm radius $R_{norm}$ of 7.07 mm, as can be seen in Fig. 6. The first eigenfrequency is determined with modal analysis using ANSYS Workbench. To investigate the influence of the membrane thickness $d_{max}$, the proportional factor $d_{0}$ of Eq. (8) is varied.

Figure 7(a) shows the dependency of the membrane thickness to the $C_{3}/C_{4}$-ratio. The curve shows that the surface shape approaches the analytical solution (target value) for thinner membrane thicknesses. The $C_{3}/C_{4}$ value is 1.58 for a membrane thickness of 0.5 mm. The curve in Fig. 7(b) is calculated by comparing the FEA surface to the target surface with a $C_{3}/C_{4}$-ratio of 1.50. Greater membrane thicknesses which do not match the target $C_{3}/C_{4}$ value consequently also show an increase of the residual error. The residual error is determined for the elliptically illuminated footprint area at an introduced deflection of 1 µm. The minimal residual error is 5.1 nm µm⁻¹ at a membrane thickness of 0.5 mm. The curve in Fig. 7(c) shows a steep increase in eigenfrequency up to a thickness of 1.85 mm. After this optimum, the curve decreases significantly.

 

Fig. 7. The effects of varying the membrane thickness, d, on a), the Ratio $C_{3}/C_{4}$, b) the residual surface error and c) the eigenfrequency of the membrane.

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To achieve a manufacturable opto-mechanical design, geometric boundary conditions are introduced. The simply supported edge is replaced by a circumferential low-bending-moment flexure hinge. Therefore, the FE model is advanced by the variable parameters $a_{y}$ and $d_{z}$. The geometry parameters $L$, $a_{x}$, $d_{fh}$ and $R$ completely describe the flexure hinge. Table 3 summarizes the parameters mentioned. Figure 8 shows the advanced FE model. To design the flexure hinge, it is sufficient to optimize the gap-parameter $a_{y}$ and the thickness $d_{z}$. Therefore, an iterative FE analysis is performed to find values of $a_{y}$ and $d_{z}$ to obtain a deformed surface with a $C_{3}/C_{4}$ ratio of 1.5.

 

Fig. 8. Geometry sketch of the enhanced FE model with flexure hinge

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Table 3. Geometric parameters for design of the flexure hinge

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As a starting point, the connection of the flexure hinge to the FE model is set with $a_{x}=a_{y}$, so that a gap of 1 mm is obtained circumferentially. The flexure hinge is fixed. A FE-based parameter analysis is used to increase the thickness stepwise from 0.25 mm to 1 mm. Figure 9(a) shows a steep increase of the $C_{3}/C_{4}$ ratio with increasing thickness and thus a greater deviation from the target shape. With a thickness of 0.25 mm, the $C_{3}/C_{4}$ value is 1.75. Also in this context, the residual error (Fig. 9(b)) increases from 6.5 nm µm⁻¹ to 12.5 nm µm⁻¹. The eigenfrequency increases from 17.5 kHz at a thickness $d_{z}$ of 0.25 mm to 24.1 kHz at a thickness $d_{z}$ of 1 mm.

 

Fig. 9. The effects of varying the thickness of the flexure hinge, $d_{z}$ on a), the ratio $C_{3}/C_{4}$, b) the residual error and c) the eigenfrequency of the membrane.

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For further dimensioning, the thickness $d_{z}$ is set to 0.25 mm. The next step is to design the gap width $a_{y}$. Modification of the boundary condition from a simply supported edge to a flexure hinge has a large effect on the $C_{3}/C_{4}$ ratio. The target value of the ideal biconical surface of 1.5 is not reached. Therefore, a correction factor $c$ is introduced as follows:

It is gradually increased from 1 to 3. Figure 10(a) shows that a $C_{3}/C_{4}$ value of 1.5 is achieved with a correction factor $c$ of 1.58. The corresponding surface error is 6.3 nm µm⁻¹ (Fig. 10(b)) and the eigenfrequency is 16.4 kHz (Fig. 10(c)).

 

Fig. 10. Dependency of the Ratio $C_{3}/C_{4}$ a), the surface error b) and the eigenfrequency c) on the correction factor $c$.

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Finally, the residual surface error is minimized. This is achieved by optimizing the domain of the logarithm function and results in a thickness modification. The curve of Fig. 10(b) shows a minimum residual error of 3.9 nm µm⁻¹ m at a membrane thickness of 1.7 mm. The $C_{3}/C_{4}$-ratio of 1.5 (Fig. 11(a)) is preserved. The eigenfrequency of the opto-mechanical design is 20.4 kHz at a thickness of 1.7 mm (Fig. 11(c)).

 

Fig. 11. Dependency of the Ratio $C_{3}/C_{4}$ a), the surface error b) and the first eigenfrequency c) on the membrane thickness as a result of the domain of the logarithmic shape curve.

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4. Evaluation of the opto-mechanical design

The mirror design is evaluated by examining the actuator influence function. For this purpose, a stepwise displacement of the centre of the mirror membrane from from −50 µm to 50 µm in steps of 5 µm is performed using ANSYS Workbench. After each of these load steps, the surface deformation is described via Zernike decomposition using the norm radius of 7.07 mm. Due to the linear independence of the Zernike coefficients, their ratio can be directly inferred from the values of $C_{3}$ and $C_{4}$. The residual error of the surface is determined by the root mean square of the Zernike coefficients $C_{5}$ to $C_{36}$. By means of this Zernike description, characteristic parameters of the spot can be determined in addition to the determination of the optical performance. This is accomplished by feeding the Zernike coefficients into Zemax OpticStudio (see Fig. 5). The influence on optical quality of the mirror surface is isolated by combining it with an ideal paraxial lens and the resulting spot is obtained. The spot position depends on the deformation and thus, the optical power of the surface. Thus, a focus position can be assigned to each set of Zernike coefficients or mirror deflection amplitudes. The resulting focal length $f_{res}$ of the mirror-lens-system can be determined with the optical power of the mirror $D$ and the focal length of the focusing lens $f_{lens}$ [19]:

The spot diameter and the Strehl ratio can be determined via Huygens point spread function ($PSF$) analysis. For this purpose, the Gaussian beam diameter is used as a criterion, which by definition includes the decrease of the beam intensity to $1/e^{2}$. For dynamic characterization, modal and harmonic vibration analysis were performed using ANSYS Workbench.

In addition to the optical characterization, a dynamic characterization of the mirror membrane is performed. A modal and a harmonic analysis are performed using ANSYS Workbench. The modal analysis is used to simulate the eigenfrequencies and eigenmodes. Furthermore, the influence of the type of oscillation excitation is simulated with a harmonic analysis.

5. Results and discussion

Both optical and dynamic characterization is conducted to evaluate the performance of the opto-mechanical design. To evaluate the optical performance, the surface from the FE analysis is compared with an ideal surface described by the Zernike coefficients $C_{3}$ and $C_{4}$ alone and the residual error is determined. Considering the baseline scenario (see section 2), the surface described by the Zernike coefficients $C_{3}$ - $C_{36}$ can be fed into the Zemax OpticStudio simulation (see Fig. 1). This allows conclusions to be drawn about the focus shift, the spot diameter and the Strehl ratio as well as the optical surface error. A modal and a harmonic vibration analysis are performed with ANSYS Workbench to determine the first eigenfrequencies and eigenmodes. A parameter study of the rigidity of the interface between the initiation of the deformation and the mirror membrane will be used to show the influence of an actuator on the first eigenfrequencies.

5.1 Optical performance of the opto-mechanical design

The optical power $D$ of the deformed surface is determined by the radius of curvature $R_{x}$ of the elliptical semi-axis $a$ due to the mounting position of the mirror membrane at 45°.

Figure 12(a) shows a linear progression for the defocus $C_{3}$ and astigmatism $C_{4}$ terms with respect to central displacement of −50 µm to 50 µm with a constant $C_{3}/C_{4}$ ratio of 1.5. The central displacement produces an optical power change between −1.7 dpt and 1.7 dpt over this range. The opto-mechanical design allows both concave and convex curvatures. The plots in Fig. 12 are also called actuator influence functions.

 

Fig. 12. a) The relationship between central displacement and resulting optical power and focal position shift. b) Zernike terms $C_{3}$, defocus and $C_{4}$, astigmatism and their ratio with changing central displacement.

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Figure 13 shows the relation of the Zernike coefficients $C_{3}$ and $C_{4}$ to the coefficients $C_{5}$ to $C_{36}$ for ±0.7 dpt, ±1.2 dpt and ±1.7 dpt. As expected, the plot is dominated by defocus $C_{3}$ and astigmatism $C_{4}$. The influence of the unwanted coefficients $C_{5}$ to $C_{36}$ is shown in Fig. 13(b). Depending on the optical power, the surface error increases linearly. At ±1.2 dpt, the surface error reaches the diffraction limit. This means that up to an optical power of ±1.2 dpt the optical system is diffraction limited. This corresponds to a focus shift of −31.3 mm to +41.0 mm using a nominal focal length of 200 mm.

 

Fig. 13. a) Zernike coefficients $C_{3}$ to $C_{36}$ of the simulated deformed surface. b) Residual surface error with respect to the optical power.

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The $PSF$ is evaluated to analyze the spot. Figure 14(a) shows the normalized intensity in the spot with a flat surface at the nominal focal length of 200 mm. The Gaussian beam diameter is 38.6 µm. The ideal absolute diffraction limited spot size is proportional to the focal length $f$, according to the definition of optical resolution and thus changes over the focal shift range of the mirror [20]. A convex deformed surface with an optical power of 1.2 dpt (Fig. 14(b) results in a Gaussian beam diameter of 45.4 µm and a focus shift of 41.3 mm. With an optical power of −1.2 dpt (Fig. 14(d), the Gaussian beam diameter is 34.5 µm with a focus shift of −30.3 mm. Increasing the optical power to 1.7 dpt (Fig. 14(c) results in a Gaussian beam diameter of 48.2 µm and a focus shift of 61.4 mm, −1.7 dpt (Fig. 14(e) results in a Gaussian beam diameter of 34.1 µm and a focus shift of −41.9 mm. The profile of the PSF shows a decrease in maximum irradiance (Fig. 14(f) and (g)) with increasing optical power. The relative normalised intensity maximum of the PSF in relation to an ideal optical system is description of the Strehl ratio [21]. The surface error depends on the optical power (see Fig. 13(b)), so that at an optical power of 0 dpt the ideal Strehl ratio of 1 is achieved. The Strehl ratio at an optical power of ±1.2 dpt is 0.81. Within this range the system is thus diffraction limited [22]. At an optical power of ±1.7 dpt, the Strehl ratio drops to 0.62 and secondary maxima of the $PSF$ appear (Fig. 14(c and e)).

 

Fig. 14. Normalized point spread function (PSF) of the opto-mechanical design at different optical powers a-e). Cross-section of the PSF with absolute intensity at different optical powers f and g).

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5.2 Dynamic performance of the opto-mechanical design

Both modal and harmonic vibration analysis are performed using ANSYS Workbench to evaluate the dynamic characteristics of the opto-mechanical design [23]. The determination of the eigenfrequency in section 3.4 was carried out under the assumption of an ideal, infinite actuator rigidity. The corresponding first ten eigenfrequencies are shown in Table 4.

Table 4. Eigenmodes and eigenfrequencies of the mirror membrane with an infinite actuator rigidity

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Linear actuators are used to initiate a central displacement. Their rigidity influences the dynamic behavior of the actuator-mirror membrane system (active mirror).

Figure 15 shows both the dependency of the actuator rigidity on the first eigenfrequency and the evolution of the first eigenmode. At low rigidities up to 0.01 N µm⁻¹, the oscillation mode corresponds to the biconic deformation, which is given by the design. Between 0.01 N µm⁻¹ to 10 N µm⁻¹, there is a transition from this biconic mode of vibration to a twin-peak-like deformation with a minimum at the center of the mirror membrane. Above a rigidity of 10 N µm⁻¹, the biconical oscillation is completely suppressed and the twin-peak-like oscillation form dominates. The membrane rigidity is 2.11 N µm⁻¹. Typically, piezoelectric linear actuators have rigidities between 10 N µm⁻¹ and 250 N µm⁻¹ [24].

 

Fig. 15. Evolution of the first eigenfrequency in relation to the actuator rigidity.

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For further investigation, we assume an actuator rigidity of 100 N µm⁻¹. Figure 16 shows the first ten eigenfrequencies and eigenmodes by use of such an actuator.

 

Fig. 16. The first ten eigenmodes and eigenfrequencies of the mirror membrane with an actuator rigidity of 100 N µm⁻¹.

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By introducing a harmonic displacement of 1 µm at the actuator a frequency response analyses of the active mirror is performed in a range between 1 Hz and 65 kHz. Figure 17(a) and b) shows the geometry of the active mirror as well as the top and side view of the FE mesh, respectively. A damping ratio of 0.02 is assumed to represent the influence of damping. The magnitude responses were evaluated at the points P1 (17(c)) and P2 (17(d)) with and without damping. Point P1 is located in the center of the membrane, point P2 has a lateral offset of 7.0 mm in y-direction. As a result of the harmonic centric displacement, the symmetric eigenmodes 2, 5, and 10 are excited at 21.6 kHz, 39.0 kHz, and 54.4 kHz, respectively.

 

Fig. 17. Top view a) and side view b) of the FE mesh. Frequency response at point P1 c) and point P2 d). A harmonic displacement of 1 µm is introduced as a displacement.

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To determine the effective frequency range of the active mirror, the influence of the frequency on the ratio of the radii of curvature $R_{x}$ and $R_{y}$ is evaluated. With increasing frequency, the mentioned twin-peak mode is excited in addition to the biconical fundamental mode. A superposition of the fundamental and the twin-peak mode occurs and leads to aberrations of the mirror surface. The fundamental mode has a radius of curvature ratio of 0.5. The tolerance limit for a diffraction-limited surface was determined with Zemax OpticStudio and is at a ratio of 0.45.

Figure 18 shows the influence of the frequency on the radius of curvature ratio for beam diameters of 5 mm, 7.5 mm and 10 mm. This results in diffraction-limited frequency ranges up to 4.5 Khz, 6.6 kHz and 9.5 kHz for the beam diameters mentioned. In applications where diffraction limited optical surfaces are not required, the active mirror can also be used in higher frequency ranges.

 

Fig. 18. Frequency dependence of the ratio of the radii of curvature for beam diameters of 5.0 mm, 7.5 mm and 10 mm.

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6. Conclusion and outlook

This paper shows both an analytical solution and the development of an opto-mechanical design for a metal-based deformable mirror for focus shifting with orthogonal beam deflection. A FE analysis using ANSYS Workbench confirms both the analytical approach and shows the limitations of this method. In an iterative process, an opto-mechanical model of the mirror membrane has been developed that can yield diffraction-limited spots at optical powers of up to $\pm$1.2 dpt. The investigations using Zemax OpticStudio have revealed, that in combination with a focal length of 200 mm, this optical power range results in a focusing range from −30.3 mm to +41.3 mm.

The analysis on eigenfrequencies and eigenmodes shows the influence of the orthogonal central displacement on the eigenfrequency and eigenmode. The simulation clearly shows that the lowest excited eigenfrequency occurs at 21.6 kHz. Compared to the state of the art the achieved eigenfrequency is approximately six times higher. Control loops in the kHz-bandwidth [25] within the given focus shift ranges can lead to focussing speeds in the range of several 1000 m/sec which is significantly faster than typical lens-based focus shifting devices. Even higher eigenfrequencies are possible by designing the substrate with a larger thickness or smaller elliptical outer dimensions [26]. This is accompanied by a new dimensioning of the flexure hinge. The effects on the optical quality have to be investigated for the new geometry.

Furthermore, the influence of the rigidity at the interface of the introduced displacement on the shape and frequency of the fundamental mode could be shown. The investigation has shown that the first excited eigenfrequency of 21.6 kHz is achievable with a rigidity of 100 N µm⁻¹ and thus with commercially available piezoactuators. The diffraction-limited frequency range of the active mirror depends on the beam diameter. For a beam diameter of 5 mm, the active mirror can be used up to 9.5 kHz.

The development of an active mirror prototype and their optical and dynamic characterization is part of further research activities.