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Abstract
This paper describes an opto-mechanical concept of a deformable metal mirror membrane, which can shift the focus position over a large range by use of a single actuator. The core element of the mirror is a diamond turned tulip-shaped membrane, the design is optimized to correct astigmatic aberrations which arise from the use of a curved mirror under a deflection angle. For this purpose, the target mirror surface is biconic. The manufactured mirror was tested with a maximum central deflection of 28 µm and, when used in combination with a 200 mm focal lens, is capable of producing a focus shift of up to 17.9 mm with a resulting wavefront aberration of 238.7 nm RMSerror.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Active deformable mirrors are nowadays used in many applications, e.g. in astronomy and laser material processing. Their function as well as their design is tailored to the respective applications. In astronomy, multi-actuator active mirrors are often used to correct wavefront aberration [1,2]. These can be induced by thermal changes of the environmental conditions [3] or turbulent disturbances of the atmosphere [4]. Due to improved manufacturing processes, metal mirrors are increasingly used due to their good mechanical properties which have advantages in terms of good ductility, manufacturability and mechanical integration [5,6]. In [7] different manufacturing processes and design guidelines for active metal mirrors to correct wavefront aberrations are described.
In laser material processing, tip-tilt-mirrors in the field of X-Y-scanner-based beam shaping can be found. This 2D-beam shaping technology speeds up laser cutting as it influences the heat distribution in the work-piece by enabling a moving spot. The spot position is oscillated on the workpiece surface to generate a ring, line or eight-shaped pattern in a long-time sampling [8]. This enables both, an increase of processing speed [9] as well as an improvement of machining quality [10–12]. In the application of laser beam cutting, focal position shifts of several centimeters might be necessary to cut the material [13]. This focus shift along the optical Z-axes extends the X-Y-scanner-based beam modulation. So far movable lens systems are used to implement the focus shift [14]. By changing the distance between the lens and the focal plane, these allow the laser spot to be adjusted. A disadvantage of this method is that the lens system must be carried along to change the focus position.
An alternative approach for focus shifting is to use deformable mirrors. A spherical curvature can be achieved on such membrane-based mirrors using piezoelectric actuators [15–17] or pressure [18]. Unlike the inclusion of a new lens in a collimated beam, however, an additional mirror has significant consequences for the beam geometry. Not least so, because oblique focussing mirrors result in the optical aberration astigmatism [19]. This means that a focussing mirror should be integrated under an angle of 0° so that the beam reflects directly back on itself - necessitating additional optical elements for beam separation and complicating the system considerably. This could be avoided if the focussing mirror was designed to correct the astigmatism error that results from reflection under an angle greater than 0°. The strength of this astigmatism is dependent on both the angle of incidence on the mirror, and on the focussing power of the mirror. For a given angle of incidence, the astigmatism aberration is constantly proportional to the defocus applied. Therefore, the creation of a focus mirror for a given angle of incidence which fulfills the condition of an astigmatism correction which is always proportional to the applied defocus, would enable fast focal shifting and flexible integration in optical systems using a single component. In [20] the realization of a mirror for 90° beam deflection using an elliptical design is shown. This design is capable of producing a biconic surface shape under pressure load.
The approach of this paper was to develop a variable focus mirror for a large deflection angle by adapting the thickness distribution of a mirror membrane. This mirror membrane is designed to correct the astigmatism problem for the entire focusing range of the mirror with a single actuator mounted orthogonal to the mirror surface.
2. Working principle of a focus shifting mirror membrane
A collimated laser beam is reflected by a deformable mirror with an deflection angle $\alpha$ and then focussed by a focal lens. Figure 1 shows a schematic design of beam focussing in a 90° beam deflection system with a shift of the focus position.
The key element is a mirror membrane which can be deformed by a central displacement of the membrane relative to its rim. The displacement is provided with a piezo actuator. The mirror membrane is plane and the focal point of the mirror-lens-system is at the nominal focal distance of the focal lens if no displacement is initiated. The actuator activation induces a biconic deformation of the mirror membrane, creating a divergence in the beam and shifting the focus of the mirror-lens-system away from the nominal focal distance. These two extreme displacement positions define the working range of the entire system. Figure 1 shows the beam path of the nominal (solid) and the divergent, Z-shifted beam (dashed lines).
2.1 Theoretical consideration of the reflective surface
As a result of the beam deflection the laser beam has an elliptical footprint on the mirror surface with the half-axes $a$ and $b$. In order to focus the widened and reflected beam into a single plane, the mirror membrane must have different radii of curvature in the X- ($R_{x}$) and Y-directions ($R_{y}$) and thus a biconical shape. A curved mirror surface with only a constant radius of curvature would result in an elliptical distortion in the focal plane under a deflection. Figure 2 shows the relation between the elliptical footprint and the radii of curvature schematically.
The ratio of the elliptical half-axes $a$ and $b$ depends on the deflection angle $\alpha$ of the circular laser beam with the beam radius $a$.
The associated radii of curvature result from the formulas for calculating the arc of a circle: where $s$ is the sagitta or displacement of the arc. Inserting Eq. (1) into (2) and forming the ratio $R_x$ to $R_y$ provides for ${b \to \infty }$:
Fig. 3. Ratio of the radii of curvature ($R_{x}/R_{y}$) of the mirror surface in relation to the deflection angle.
2.2 Astigmatism at the coal plane of oblique mirrors
A spherical mirror at an angle of incidence greater than 0° has two focal points [19].
where $R$ is the radius of curvature of the spherical mirror. In the focal plane this results in an elliptical distortion of the spot. This distortion can be described by the Zernike coefficient of astigmatism (C4). Figure 4(a) shows the dependence of the astigmatism on the focus shift for a 90° beam deflection and the optical boundary conditions from section 3.1. The astigmatism increases linear from 0.22 µm for a focus shift of 2.5 mm to 1.64 µm at a focus shift of 20 mm. An biconically shaped mirror in the same setup can correct the astigmatism to 0 µm for the entire focus shift. Due to the increasing astigmatism, the RMS (Fig. 4(b)) of a spherical mirror also increases linearly from 84 nm at a focus shift of 2.5 mm to 638 nm at a focus shift of 20 mm. A perfectly biconically shaped mirror also corrects the RMS to 0 nm for the entire focus shift, which means that an aberration-free wavefront is achieved in the focal plane.3. Development of the opto-mechanical design
To determine the ideal mirror surface with varying focal length, an optical design (Fig. 5) is implemented in Zemax OpticStudio©. The optical boundary conditions are implemented in this model. The mirror surface is generated by optimising the optical system for the smallest possible spot size under changing focal distances. The sole variable parameters for the optimization are the radii of curvature $R_{x}$ and $R_{y}$ of the biconic mirror surface as shown in Fig. 1. By repeating the optimisation for different focus positions, the two radii of curvature ($R_{x}$, $R_{y}$) of the mirror surface are determined for each value of the focal shift.
3.1 Optical design and optical boundary conditions
The mirror surface is optimised for the 90°-deflection angle and a fixed focal position of the mirror. Figure 5 shows the simulation model of the beam path with a beam aperture set to 10 mm. Therefore, the radii of curvature $R_{x}$ and $R_{y}$ are released as variable parameters. The simulation is based on a Gaussian beam with a beam quality factor $M^{2}$ of 1 and a wavelength of 1064 nm. The working range depends on the central deflection of the surface as well as the nominal focal length of the lens. In this optical design the nominal focal length is 200 mm. The maximum centric stroke is 30 µm. The simulated ratio of the radii of curvature should be 0.5 ($R_{x}$/$R_{y}$) in case of a 90° deflection angle and confirms the theoretical consideration according to Eq. (4).
The wavefront of the beam, which is reflected by a spherical surface depends on the angle of incidence ($\alpha$/2). It is composed of defocus (C3) and astigmatism (C4), and hence their ratio is dependent on the deflection angle as well. Figure 6 shows this behaviour for the range around 90°. Using an ellipsoidal surface instead of a spherical one will reduce astigmatism in the focal plane and will finally improve the focus spot size. For an ideal mirror with a deflection angle of 90° a fixed ratio C3/C4 of 1.5 must be achieved.
Fig. 6. Relation between C3 (defokus) and C4 (astigmatism) of the mirror surface to the deflection angle.
3.2 Mechanical simulation and boundary conditions
In [7] a variable thickness distribution (VTD) of the mirror membrane’s rear surface is suggested for manufacturing of a spherical metal mirror (angle of incidence = 0°). It is shown that a non-continuous thickness distribution of the membrane is required for a spherical deformation when a uniform load or a central force is applied. Here, we extend to create an ellipsoidal deformation as required for a deflection angle. The stiffness of the membrane and accordingly its thickness, $t$, will change along the radial radii $r_{x}$ and $r_{y}$ of the membrane. To generate an ellipsoidal surface with the optical radii of curvature $R_{x}$ and $R_{y}$ of an elliptical substrate with the semi-diameter $a$ and $b$ fixed at the edge, the dimensionless shape curves $\tau _{x}$ and $\tau _{y}$ are given by the equations:
The thickness distribution of the membrane is the product of the shape curves ($\tau _{x}$ and $\tau _{y}$) and the initial thicknesses $t_{0x}$ and $t_{0y}$. The initial thicknesses are dependent on the radii of curvature ($R_{x}$ and $R_{y}$) and hence the dynamic range of the focus shift that has to be achieved. Further influencing variables for $t_{0x}$ and $t_{0y}$ are the quantity of the central introduced actuator force $F$ as well as the material parameters Young’s modulus $E$ and Poisson’s ratio $\nu$.
Fig. 7. Shape curves of an elliptical deformable mirror membrane (semi-diameters $a$ and $b$) with radius-dependent thickness distributions. With a centric axial force application, this serves to create an ellipsoidal surface.
3.3 Opto-mechanical design of the mirror membrane
To determine the ellipsoidal deformation, the semi-diameters $a$ and $b$ were iteratively optimized by a static-mechanical FE-analysis using ANSYS Workbench. The resulting optimised mirror surface from the optical simulations as well as the mechanical boundary conditions and theoretical preliminary considerations are included in this analysis.
For a real mechanical system, the ideal boundary conditions can, however, not be fully implemented into the mechanical design. Therefore, a flexure hinge with low bending stiffness is introduced for $r_{x} = a$ and $r_{y} = b$. Furthermore, the values at $r_{x}=0$ and $r_{y}=0$ are shifted by the factor $k_{x}$ and $k_{y}$ to avoid asymptotic behaviour and allow the connection to an actuator. The ratio of the resulting elliptical half-axes $k_{x}$ and $k_{y}$ corresponds to the ratio of $a/b$. The thickness of the mirror membrane was iteratively optimized until a minimum surface shape deviation ($RMS_{error}$) at a centric stroke of 30 µm was achieved. The criteria for optimization are the radii of curvature $R_{x}$ and $R_{y}$ and their ratio of 0.5. To suppress the edge effects due to the flexure hinges, the mirror surface is oversized ($r_{x}$ = 30 mm and $r_{y}$ = 60 mm). The resulting optimized design of the mirror membrane is shown in Fig. 8. The blue areas are hollowed out. To provide connection to the housing, bores are included around the mirror surface. The centric connection for an actuator to the membrane is elliptically designed with $k_{x}$ = 3 mm and $k_{y}$ = 4.3 mm according to the boundary conditions. The thickness $t_{0}$ for $r_{x,y}=0$ is 6 mm, the flexure hinge at the membrane edge has a thickness of 0.5 mm.
Fig. 8. Top view (a) and cross-sectional views of the mechanical design of the mirror membrane for the X-X axis (b) and Y-Y axis (c) showing the variable thickness distribution on the back of the surface.
For experimental validation, the opto-mechanical design of the mirror membrane has been manufactured. The tulip-shaped rear side of the membrane was produced with a conventional milling process. The mirror membrane front is manufactured by a sophisticated diamond turning process. Due to its good machinability aluminium RSA-6061 was used [21].
4. Methods of design evaluation
For comparability between the opto-mechanical design and the manufactured mirror membrane, the deformed surface is described with Zernike coefficients. By use of the static-mechanical analysis in ANSYS Workbench, discrete centric displacements are initiated. The exported opto-mechanical design was evaluated by performing a zernike decomposition of the deformed surfaces. The fidelity of the manufactured mirror with respect to the mechanical and optical design was also determined through zernike polynomial analysis of the deformed surfaces by use of a central actuator. The manufactured mirror was characterised using a Shack-Hartmann wavefront-sensor and SHS Works Software. The zernike coefficients were recorded for different central deflections. Subsequently, the resulting iso-Zernike coefficients for both the opto-mechanical design and the manufactured mirror membrane were fed back into the ZEMAX model as introduced in 3.1. The Zernike polynomials C3 to C16 with a norm radius of 10 mm are used in these evaluations. The coefficients C0 to C2 are not examined, since they serve to define the reference plane and play no role in the description of the surface. In contrast to the description of the surface via the radii of curvature ($R_{x}$, $R_{y}$), the description of the surface with Zernike coefficients has the advantage that higher order aberrations can be detected. As a result, statements about the focus position, the focus shift, the spot size and the aberrations of this spot are obtained over the entire adjustment range of the mirrors. The final rms wavefront errors were calculated in comparison to the ideal biconic design file.
5. Results and discussion
To determine the general working principle of the opto-mechanical design and the mirror membrane, the defocus term is first evaluated. This is done by comparing the opto-mechanical design in the form of an FEM analysis with the wavefront measurement of the manufactured mirror membrane. Figure 9(a) shows a linear increase of the defocus term with the focus shift. The maximum centric displacement of the opto-mechanical design in the FEM analysis is 30 µm and leads to a focus shift of 18.4 mm. The centric displacement of the manufactured mirror membrane is initiated by a linear actuator with discrete steps. The maximum focus shift achievable with this actuator is 17.9 mm with an actuator stroke of 28 µm.
Fig. 9. Defocus (C3) and astigmatism (C4) of the mirror surface in relation to the focus shift a). Ratio between defocus and astigmatism in the working range b).
Figure 9(b) shows the ratio of the Zernike coefficients C3 (defocus) and C4 (astigmatism) as a function of the focus shift. When looking at the simulation results, a constant factor of 1.71 was achieved across the entire range of the focus shift. In agreement, the ratio C3/C4 is almost constant when evaluating the wavefront measurements of the manufactured mirror membrane. The maximum deviation between the simulation and measurement results is 4.6 $\%$. The comparison with the curve from Fig. 9 shows that the opto-mechanical design with a ratio C3/C4 of 1.71 fits over the entire focus shift to a deflection angle of 84.6°. This corresponds to a ratio of the radii of curvature ($R_{x}/R_{y}$) of 0.54, which deviates from the target value of 0.5 for a 90° deflection angle.
5.1 Characterization of the simulated opto-mechanical design
First, the FEM simulated surface is deformed in 10 discrete steps. These are imported in the optical model of the mirror surface described in the section 3.1. Figure 10 shows the Zernike coefficients of the surface at a focus shift of 1.5 mm, 8.9 mm and 18.4 mm, respectively.
Fig. 10. Zernike coefficients C3 to C15 of the simulated surface a) and the manufactured membrane b) for different focus positions.
The plot is dominated by the desired coefficients, C3 and C4, representing defocus and astigmatism, which show the highest values. An unwanted aberration, C11, is also apparent. This represents 2nd order astigmatism and increases with increasing deformation of the membrane. C11 describes the surface shape deviation ($RMS_{error}$) away from an ideal biconic surface. Consequently, the $RMS_{error}$ increases with increasing focus shift and is 238.7 nm at a focus shift of 18.4 mm. Table 1 shows the $RMS_{error}$ calculated with the ZEMAX model for selected focus shifts (1.5 mm, 8.9 mm and 18.4 mm). The residual error ($RMS_{error}$) of 238.7 nm of the optimized opto-mechanical design with a maximum centric deflection of 30 µm is too large. According to the marechal criterion [22], the $RMS_{error}$ value of the wavefront aberration in a diffraction-limited optical system must be smaller than $\lambda /14$. The entire surface of the opto-mechanical design corresponds to $\lambda /4.5$ for the wavelength of 1064 nm.
Table 1. Focus shift and $RMS_{error}$
The remainder of the analysed Zernike coefficients have no significant influence on the description of the surface.
5.2 Characterization of the manufactured mirror membrane
To evaluate the optical quality, the surface of the mirror membrane is measured using an interferometer. The surface shape deviation in relation to an ideal plane surface of the entire mirror surface is 577.8 nm PV and 75.9 nm $RMS_{error}$. Referring to the norm radius of 10 mm, the surface shape deviation is 52.8 nm PV and 7.1 nm $RMS_{error}$. The entire surface of the manufactured mirror membrane corresponds to the marechal criterium for the wavelength of 1064 nm. The surface of the norm radius has an $RMS_{error}$ value of approximately $\lambda$/150. Figure 11 shows the surface plot of the entire mirror membrane as well as the surface plot of the norm radius.
Fig. 11. Surface heatmap of the entire mirror membrane a) and surface heatmap of the measured diameter b).
For evaluation of the membrane under deformation, the mirror membrane is moved in 10 discrete steps. Figure 10 shows the focus positions of the steps 1, 5 and 10, which represents a focus position of 1.9 mm, 8.6 mm and 17.9 mm, respectively.
Analogue to the Zernike coefficients of the simulated opto-mechanical design, C3 (defocus) and C4 (astigmatism) have the highest values. The term C11 (astigmatism 2nd order) also increases with larger focus shifts. In addition, the Zernike coefficient C7 shows up. This represents coma and also increases with larger focus shifts. The occurrence of C7 (coma) can be caused both by imperfections in the manufacturing process and by a non-centric initiation of the displacement i.e. an improperly positioned actuator. In particular, a displacement introduced not orthogonally to the mirror surface results in an asymmetrical deformation of the mirror surface.
C11, present in both the manufactured mirror and opto-mechanical design, can be explained by the influence of the boundary conditions. The opto-mechanical design provides for an elliptical offset ($k_{x}$ and $k_{y}$) to connect an actuator. This introduces additional material into the center of the membrane (see 8), which leads to a stiffening in this area. The displacement of the actuator causes stresses. The theoretical principles from [7] do not provide for this. The associated stiffening can be assigned to aberration C11.
5.3 Analysis of the spot diameter in the focal plane
To determine the effect on the laser spot, the simulated and measured Zernike coefficients are fed back into the optical simulation model (see Fig. 5).
To evaluate the spot diameters, the beam width is described with the full width at half maximum (FWHM). According to the definition, the spot diameter is defined as the drop in beam intensity to 1/2 of its maximum intensity. The spot diameter is calculated by use of the Zemax model for the corresponding focus shifts and normalized to 1.
For this analysis the optimization of the minimum spot diameters for a 84.6° deflection angle is performed. Figure 12(a) shows the analysis of the spot diameters of an ideal biconical surface at the focus positions of 1.5 mm, 8.9 mm and 18.4 mm. The diffraction-limited spot is clearly visible over the focus shift range of 18.4 mm. By using a beam aperture of 10 mm, the beam radius at a focus shift of 1.5 mm is 24 µm. Due to the increased focal length, the diffraction-limited beam radius increases to 25 µm with a focus shift of 18.4 mm.
Fig. 12. Spotdiameter at the focal plane for different focus positions of an ideal biconic surface (a), the opto-mechanical design (b), the manufactured mirror (c) and the meassured spot (d). The dotted circles indicate the full width of half maximum (FWHM) intensity.
Considering the spots of the opto-mechanical design, it is noticeable that with a focus shift of 1.5 mm and 8.9 mm, a diffraction-limited spot is also achieved with a spot diameter of 24 µm and 24.5 µm. Figure 12(b) shows that the spot diameter increases to 30.8 µm when increasing the focus shift to 18.4 mm. The Zernike coefficient with a significant influence is the 2nd degree astigmatism (C11). This increases with increasing surface deformation. Due to the symmetry properties of C11, the spot diameter increases symmetrically cross-shaped. From this we conclude that C11 needs to be drastically reduced in a future design that employs the Zernike decomposition as part of the FEM design. A remaining aberration below $\lambda /10$ should be targeted. An expansion of the beam diameter with large deformations of the surface can be attributed to the opto-mechanical design.
The effect of enlarging the minimum possible spot diameter is also evident in the measurements of the manufactured mirror membrane. Compared to the opto-mechanical design and the ideal biconical surface, the spot is diffraction-limited at a deflection of 1.9 mm. Figure 12(c) shows that in addition to the Zernike coefficient C11, the aberration coma (C7) also occurs. This leads to an asymmetric spot distortion and can be described as a drop-shaped distortion in the focal plane. The spot diameter at a focus shift of 8.6 mm increases to 33.5 µm and at a focus shift of 17.9 mm to 41.9 µm. The spot aberrations can be caused both by manufacturing tolerances and by a flaw in the measurement setup in which the actuator did not transmit the force orthogonally when initiating the deformation.
To confirm the simulated spots, an experimental setup was set up based on the optical design (see 5) to measure the spot in the focal plane. The beam source is a diode laser with a single-mode fiber and a wavelength of 1064 nm. To ensure comparability, the laser beam is collimated to an aperture of 10 mm by means of a collimator. The manufactured mirror membrane is installed as an 84.6° deflection mirror according to the ratio C3/C4 of 1.71. A convex lens with a focal length of 200 mm is used for focusing on a CCD-camera with a pixel size of 2.2 µm. Figure 12(d) shows the measured spot diameters for the focus shifts of 1.5 mm, 9.0 mm and 17.0 mm, respectively. The measurements show a diffraction limited spot at a focus shift of 1.5 mm. The enlargements of the spot diameters can, in relation to the simulated spotdiameters of the manufactured membrane, also be seen at a focus shift of 9 mm with 35.1 µm and at 17.0 mm with 44.0 µm.
6. Conclusion and outlook
The paper introduces a design routine for a biconic deformable metal mirror for focus shifting under a large beam deflection angle. To determine the target shape of the mirror surface, an optical analysis was executed using ZEMAX. A FEM analysis was used to iteratively achieve the mechanical design with surface shape deviations of 238 nm $RMS_{error}$ of the optically effective radius of 10 mm with a central deflection of 30 µm. The description of the surface using only the radii of curvature ($R_{X}$, $R_{Y}$) is not suitable for detecting the residual error ($RMS_{error}$). By describing the surface using Zernike coefficients, this circumstance can be eliminated.
The evaluation of the simulations and measurements of the opto-mechanical design and the manufactured mirror membrane has shown that focus shifts of up to 17.9 mm are possible when using a lens with a focal length of 200 mm.
Analyses of the opto-mechanical design and manufactured mirror have shown that the mirror fulfills a significant requirement which demonstrates it’s suitability for high performance focus shifting. A constant relationship of defocus to astigmatism (C3/C4) across the full range of actuator motion is demonstrated and results in a circular spot at the focal plane. For the manufactured mirror, this constant ratio could be achieved for the large deflection angle of 84.6°, which is close to the design value of 90°. Under small deformation, the optical quality of both the opto-mechanical design and the manufactured mirror was very close to the ideal design. An undesired 2nd order astigmatism aberration was found to be present in both the opto-mechanical design and the manufactured mirror. This arises from the clamping of the mirror substrate to its housing and contributes to a deterioration of the spot quality under large deformation of the mirror membrane. This will be a targeted optimisation factor in future mechanical designs. Furthermore, it will be investigated to which extent the optically usable area can be increased by reducing unwanted aberrations (C7, C11). Further important considerations are the integration of the mirror membrane into a housing, the connection of the actuator and the investigations regarding the dynamic properties of a fully integrated mirror.
Funding
Bundesministerium für Bildung und Forschung (03ZZ1028F).
Disclosures
The authors declare no conflicts of interest.
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