We present deformable mirrors for the intra-cavity use in high-power thin-disk laser resonators. The refractive power of these mirrors is continuously adaptable from −0.7 m−1 to 0.3 m−1, corresponding to radii of curvature ranging between 2.86 m (convex) and 6.67 m (concave). The optimized shape of the mirror membrane enables a very low peak-to-valley deviation from a paraboloid deformation over a large area. With the optical performance of our mirrors being equal to that of standard HR mirrors, we were able to demonstrate the tuning of the beam quality of a thin-disk laser in a range of M2 = 3 to M2 = 1 during laser operation at output powers as high as 1.1 kW.
© 2017 Optical Society of America
Thermally induced effects in the laser crystal have been one of the major challenges for the design and operation of solid-state lasers and laser systems and have been subject of intense research for many years (for references, see e.g. ). These pump-power dependent, often predominantly spherical wavefront-distortions, which in a first-order approximation are commonly referred to as a “thermal lens”, limit the stability of the resonator to a certain range of pump powers and lead to a pump-power dependent change of mode size in the resonator . The concept of the thin-disk laser  has led to significant improvement in this respect. As the disk-shaped laser crystal is cooled from the back face, the heat flows predominantly in parallel to the direction of propagation of the laser beam, leading to a decrease of thermal lensing by orders of magnitude. This has enabled high output powers, both in cw-operation , as well as for the generation [5,6] and amplification [7–9] of ultrashort pulses.
About 4 kW of output power at an almost diffraction-limited beam quality have been demonstrated [10,11] in cw-operation so far. However, starting from an M2 of about 5 at low powers and reaching diffraction-limited beam quality only near the operation point, the dependency of beam quality on the pump power imposed by thermal lensing is still significant. This dependency can be overcome by the use of intra-cavity deformable mirrors featuring an adaptable radius of curvature, which give some amount of control over the mode size in the resonator.
Due to the thin laser crystal, the round-trip gain is comparably small in thin-disk lasers. Hence, laser operation is considerably more sensitive to intra-cavity losses, requiring high-quality mirrors with a high reflectivity of >99.9%. Only few of the diverse existing concepts for deformable mirrors meet these requirements at reasonable cost and have already been demonstrated for the use as intra-cavity mirrors, namely micromachined membrane deformable mirrors [12–14], bimorph piezoelectric mirrors [15, 16], and deformable mirrors based on deformation by a distributed surface load, i.e. pneumatic or hydraulic pressure applied from the back of a thin mirror membrane. Offering only one degree of freedom, the latter type of mirrors can be used to compensate for one single effect only, however bearing the benefits of an overall reduction of complexity and hence cost. Such deformable mirrors have been proposed , and demonstrated  for the intra-cavity compensation of thermal lensing in rod-type solid-state lasers at a moderate output power of 25 W. Later on, this mirror concept has also been adapted for the use in CO2 laser resonators  as well as CO2 flatbed laser cutting machines .
In the context of thin-disk lasers, a spherically deformable mirror based on the deformation of a mirror membrane due to hydraulic pressure has been used for the compensation of thermal lensing in multimode-resonators comprising multiple laser crystals . In , the operation principle has been adapted for the generation of aspheric deformations in order to compensate exclusively for the non-spherical components of the phase distortions caused by thermal effects in thin-disk laser crystals. With the optical performance of these mirrors being equal to that of standard mirrors, the potential for their use as intra-cavity mirror in thin-disk laser resonators has thus been demonstrated.
In the present paper, we report on spherically deformable mirrors designed for the intra-cavity use in kilowatt-level thin-disk laser resonators. These mirrors are deformed by applying pneumatic pressure to the back side. In contrast to similar concepts [18, 21], where the shape of the deformed mirror is spherical only in a small area around the center of the mirror, the distribution of the thickness of the deformable glass membrane of the mirrors presented in the following was optimized in order to enlarge the diameter of spherical deformation to more than 80 % of the deformable membrane. The corresponding design considerations are discussed in the following section 2. The experimental characterization of the mirrors with respect to the achievable radius of curvature and refractive power, respectively, are presented in section 3. Section 4 finally reports on the tests of our mirrors used as end mirrors in a 1 kW thin-disk laser resonator, which allowed to control the working point of the resonator and therewith to tune the output beam quality in a range between M2 = 1 and M2 = 3 at all power levels.
2. Mirror design
The principle design of the deformable mirrors is based on the deformation of a membrane caused by an evenly distributed surface load (see Fig. 1), i.e. a pneumatic or hydraulic pressure applied to the membrane. The desired optical performance is achieved by fabricating the deformable membrane from a high-quality fused silica HR mirror by locally thinning the solid mirror body to the desired thickness. In order to support the membrane, an annular rim with the full thickness of the mirror substrate is left on the circumference, acting as a mounting interface in order to glue the mirror to a metal submount that features the necessary pressure connectors. Using the full-thickness rim as a mounting interface effectively minimizes distortions of the thin membrane caused by the mounting process, e.g. shrinkage of the glue or by uneven mounting surfaces.
The deformation of this type of mirror can be described using the approximations of Kirchhoff-Love thin-plate theory, since the ratio of diameter to thickness of the thin mirror membrane is expected to be sufficiently high (i.e. >10) . For a uniformly loaded circular plate of radius a which is rigidly clamped at its circumference, the deformation w(r, p) of the middle surface of the plate caused by a distributed surface load p can be described by the differential equation 
E denotes the modulus of elasticity of the material, ν is Poisson’s ratio and h the thickness of the plate (for the following considerations, typical values for fused silica of E=73000 N/mm2 and ν =0.2 have been used). Note that p(r) = p in our case is spatially constant and denotes the differential pressure, i.e. the difference of the input pressure and the ambient pressure
As introduced above, w(r, p) describes the deflection of the middle surface of the thin membrane. For the application considered here, however the deflection of the actual surface wsurface (r, p) of the mirror is of interest. Since wsurface (r, 0 bar) is assumed to be flat (i.e. constant over r) and the thickness h(r) of the membrane is not affected by the applied pressure, the deflection of the actual mirror surface wsurface (r, p) equals the deflection w(r, p) of the middle surface of the mirror. Starting from a flat surface at p = 0, the absolute deformation of the mirror surface will therefore take the form of the deflection of the mirror’s middle surface when a differential pressure is applied.
In order to take into account the circular symmetry of the problem as well as the rigid clamping of the edges of the deformed plate, the boundary conditionsEq. (1) yields the solution  Eq. (1) for the flexural rigidity F(r) which is needed to achieve a given deflection w(r, p), the corresponding thickness distribution h(r) can be determined directly from Eq. (2). Unfortunately, the second boundary condition formulated in Eq. (4) cannot be fulfilled by a strictly parabolic deformation, as the first derivative of the deflection curve will be non-zero for all r ≠ 0. This implies, that for a plate with clamped edges, it is not possible to achieve a spherical deformation throughout the full area.
However, by allowing for a transitional area where the given deflection curve is deliberately non-spherical, the area of spherical deformation can be maximized. As an example, in Figs. 2(a) and 2(b) the solution of Eq. (1) for the flexural rigidity and the corresponding mirror thickness, are shown for a desired deformation described asEq. (4) are met and the function wg (r, pg) is twice continuously differentiable at r = rt.
It can be seen, that the flexural rigidity in the spherical section of the mirror corresponds to a quadratic function in r, whereas in the non-spherical section, the flexural rigidity and the mirror thickness are quickly decreasing towards the edge of the membrane.
The manufacturing of a freeform-surface according to the thickness distribution given by the solution of Eq. (1) as seen in Fig. 2 is challenging. Furthermore, the undercut at the edge of the mirror membrane leads to a very thin residual cross-section, potentially weakening the mirror. However, as can be seen in Fig. 2(b), the ideal thickness distribution can be roughly approximated by a sphere, which can be manufactured using standard technologies such as ultrasonic lapping.
Figure 3(a) shows the deformation calculated for the optimized (solid line) as well as for the spherically approximated shape of the mirror membrane (dashed line). The ideal parabolic deformation is indicated by the dotted line. Figure 3(b) shows the deviation of the deformations of both the optimized mirror and the mirror with the spherically approximated membrane thickness from an ideal parabolic deformation. For both mirrors (either with optimized membrane shape h(r) or with a spherical approximation of h(r)), the deformation closely resembles the ideal parabolic shape within the central 8 mm in radius, with only minor deviations with a magnitude of well below 20 nm for the membrane with the spherical thickness distribution in this section. Therefore, we conclude that the spherical thickness distribution of the membrane is an excellent compromise between ideal deformation and manufacturability.
3. Characterization of deformable mirrors
The spherically deformable mirrors have been fabricated from HR-coated fused-silica mirror substrates. The parameters of the geometry applied to the mirror substrates by ultrasonic lapping are shown in Fig. 4(a). The mirrors have been glued to brass mounts featuring a pneumatic connector (see Fig. 4(b)). An electrically controlled proportional valve (FESTO VPPM) allowed for a precise adjustment of the pressure applied to the mirror. A venturi nozzle was used for the generation of pressures below ambient pressure in order to be able to generate concave deformations as well.
The deformation of the mirrors was measured at different pressures using a twyman-green interferometer. In Fig. 5, the measured deformations at +0.4 bar and −0.4 bar are shown. It can be seen that the surface deformation is parabolic as expected and the magnitudes of convex and concave deformation are essentially equal.
A cross-sectional view of the mirror deformations measured at different pressures ranging from −0.4 bar to 0.7 bar is shown in Fig. 6 along with the corresponding parabolic fit functions, which are in excellent agreement with the measured data. The radius of curvature and the corresponding refractive power D=2/R derived from the parabolic fit functions evaluated over the central 7 mm are shown in Figs. 7(a) and 7(b). It can be seen, that the refractive power of the mirror can be tuned continuously from about −0.7 to 0.3 diopters for the pressures applied, corresponding to radii of curvature of −2.85 m (convex) and 6.67 m (concave), respectively. The magnitude of the concave deformation is limited by the minimum pressure that can be generated by the venturi nozzle. However, this is no fundamental limitation of the mirror deformation itself. If perfect vacuum is assumed (i.e. p = pamb ≈ −1 bar) a maximum refractive power of about 0.78 dpt can be extrapolated from the measurements given in Fig. 7(b) for the concave deformations. The magnitude of the convex deformation could potentially be further increased, as it was derived from supplementary FEM simulations, that the fracture limit of the mirrors is presumably reached at a pressure of about 1.4 bar, which would correspond to a refractive power of about −1.2 diopters. However, this could not be validated experimentally, as it was not possible to reliably measure refractive powers of more than approximately 0.8 diopters with the interferometer used in our experiments.
In order to quantitatively assess the surface quality of the deformable mirrors, it is useful to compare the measured shape of the deformed surface to an idealized shape. Since parabolic deformations are desired for the application considered here, the measured surfaces are fitted with ideal parabolic surfaces using the least-squares method. The sum of the absolute values of the maximum positive and negative deviation of the measured surface from the least-squares-fitted parabolic surface, the so-called peak-to-valley-deviation, can be used as a metric in order to evaluate the optical quality of the deformed surface.
In Fig. 8, the peak-to-valley deviation of the measured deformation of the deformable mirror from the parabolic fits is shown for different diameters around the center of the mirror. It can be seen, that the deviation increases linearly both with refractive power and the diameter over which the measurements are evaluated. The overall peak-to-valley distortions are slightly larger than expected from the mirror design. This can presumably be attributed to manufacturing inaccuracies.
4. Intra-cavity performance
The mirrors have been tested in a thin-disk laser resonator with the set-up shown in Fig. 9. In order to demonstrate the possibility to control the working point of the resonator and hence the beam quality of the output beam using one of our deformable mirrors, the resonator was designed to generate a low-order multimode output beam. Furthermore, the approximately 120 µm thick 10 at.% Yb:LuAG disk was pumped at the zero-phonon-line at 969 nm in order to reduce the influence of thermal effects and aspherical phase distortions caused by the temperature distribution in the laser crystal.
For the nominal radius of curvature of 3.58 m of the disk used in our experiments, an 1/e2-radius of the fundamental mode w00,Disk of about 2.16 mm can be calculated from the ray-transfer matrix method . As a rule of thumb for thin-disk lasers, the width wMM of the incoherent superposition of higher-order modes oscillating in the laser resonator roughly equals the width of the pumped area wpump. Hence, with the beam width of any multimode beam relating to the width w00 of the associated fundamental mode by
For the pump spot radius wpump of 3.5 mm used in our experiments, this gives an expected M2 of about 2.6.
The deformable mirror was used as an end mirror in the resonator placed at a distance of 600 mm from the disk. As the beam is almost collimated in this section of the resonator, the size of the oscillating laser beam should be nearly identical to the size on the disk. For the resulting effective mode diameter of 7 mm on the mirror, a maximum peak-to-valley deviation of λ/20 limits the useable refractive power to about +/− 0.3 m−1 (see Fig. 8).
Although the thin-disk laser crystal is mounted on a diamond heatsink, the thermally induced bending of disk and diamond heatsink generally more than outbalances the very weak positive thermo-optical component of the spherical thermal lens in the thin-disk laser crystal, leading to a negative sign of the thermally induced refractive power. The effective radius of curvature of the concave disk is hence increasing with pump power. In , the change of refractive power with pump power density has been evaluated to amount to about −64·10−3 dpt/kW/cm2 for thin-disk laser crystals mounted on a CuW-heatsink. Hence, the useable range of refractive powers of +/− 0.3 dpt should be sufficient to compensate for this effect for pump power densities as high as about 9 kW/cm2. For the case of diamond-mounted disks, the expected thermally induced refractive powers are significantly lower due to the higher stiffness of diamond in comparison to CuW. In  and , a typical order of magnitude of about −4 to −8 · 10−3 dpt/kW/cm2 was experimentally observed for conventional pumping at 940 nm. Since in our experiments, the disk was pumped on the zero-phonon line, a slightly lower thermally induced refractive power of about −3 · 10−3 dpt/kW/cm2 can be assumed.
Since the pressure controller used in our experiments allowed for minimum changes of pressure of 0.01 bar, the resolution of the control of our mirrors in terms of refractive power is limited. As can be seen in Fig. 7(b), a sensitivity of about 0.8 dpt/bar can be achieved with the mirrors based on the design given in Fig. 4(a). Hence, for the convex deformations a minimum change of refractive power of about 8 · 10−3 dpt can be achieved, which is quite coarse in comparison to the order of magnitude of thermal lensing expected for the disk used in our experiments. When using the venturi nozzle in order to generate negative differential pressures, the resolution of the pressure control and hence the resolution of refractive power increases by a factor of about 12, giving a minimum step in refractive power of about 0.67 · 10−3 dpt. While due to the limited resolution, it is not possible to achieve a perfect compensation of the comparably weak thermal lensing effect in our disk, the controllable refractive power of our mirror still enables some amount of control over the operating point of the resonator.
In Fig. 10, the calculated fundamental-mode radius derived from the ray-transfer-matrix method is plotted against the effective radius of curvature of the disk and the refractive power of the deformable mirror for the given resonator geometry. It can be seen, that within the range of refractive powers which can be generated with the deformable mirror, the resonator can be tuned throughout its full stability range, reaching a radius of the fundamental mode on the disk of 3.5 mm close to the edges of the stability zone, which are indicated with the red dashed lines. Thus, with the assumption from Eq. (8), the deformable mirror should allow for a tuning of the resonator from M2=2.6 to M2=1 at any pump power.
The performance of the resonator has first been characterized using a plane HR mirror as the end mirror of the cavity instead of the deformable mirror. As can be seen from Fig. 11, more than 1 kW of output power could be extracted at a maximum overall optical efficiency of 47%. With increasing pump power, however no further increase of output power could be achieved, which can presumably be attributed to the home-built volume-bragg wavelength stabilization of the pump diodes used in our experiments, which exhibited a considerable amount (up to 15% at maximum pump power in our experiments) of pump radiation emitting off the zero-phonon absorption peak. The beam quality of the output beam was measured using an automated z-scan measurement method according to ISO11146. It can be seen from the inset in Fig. 11 that for the plane HR mirror, the values of (which is the geometric mean of the values of M2 in x (horizontal) and y (vertical) direction), are increasing from about 2.3 to 3.6 with increasing pump power and then decreasing again to 3.0 at higher pump powers. Despite the reduction of thermal effects which can be achieved by the zero-phonon-line pumping scheme when compared to pumping at 940 nm , this observation clearly hints to the presence of a spherical thermal lens: With increasing pump power, the radius of curvature of the disk increases, leading to a change of the mode size on the disk following the dotted blue line in Fig. 10. Hence, starting at low pump powers, the mode size decreases with pump power towards the minimum of 2.14 mm, leading to an increase of M2. With the pump power further increasing, the mode size on the disk is increasing as well, leading to a decreasing value of M2.
Subsequently, the plane end mirror was replaced by the deformable mirror. The deformable mirror was aligned to be perfectly centered with respect to the incident beam using a translation stage in order to avoid any angular misalignment of the resonator when changing the curvature of the mirror. At each power level, the pressure applied to the mirror was tuned in order to optimize the beam quality of the output beam. As can be seen from Fig. 11, the performance in terms of output power and efficiency remained almost unchanged when compared to the results achieved with the plane HR mirror. However, the inset shows that the beam quality could be kept almost constantly close to the diffraction limit over the full power range up to slightly more than 1 kW of output power. More data obtained from further experiments with similar performance are shown in Fig. 12. Hence, although the resolution of the refractive power of the mirrors is quite coarse in comparison to the resolution needed for a total compensation of thermal lensing effects, it is sufficient in order to achieve a constant beam quality of the output beam throughout the full range of pump powers used in our experiments.
Figure 13(a) shows the change of M2 and output power with different pressures applied to the DM at a pump power level of 970 W. It can be seen that the resonator can be continuously tuned to any point in the stability range, allowing to tune the M2 to a value anywhere in between about 2.6 and almost 1. With higher or lower pressures applied to the deformable mirror, the resonator was observed to become unstable, inhibiting laser operation. The evolution of the output power follows that of the value of M2, presumably due to the increasing overlap of the oscillating laser beam and the super-gaussian pump spot with increasing number and transverse order of the oscillating modes. The same behavior can be observed at a pump power level of 2560 W as shown by Fig. 13(b). It can be seen, that even at an output power level of above 1 kW, the DM can be used to fine-tune the resonator in order to get an almost diffraction-limited output beam (M2<1.3), with an insignificant drop of output power of only about 2% from about 1.1 kW to 1.078 kW.
In conclusion, we have successfully demonstrated pneumatically actuated spherically deformable mirrors for the intra-cavity control of the output beam quality of high-power thin-disk lasers. For output powers of up to 1 kW, the deformable mirrors were found to perform equal to standard HR mirrors in terms of efficiency, but additionally have the significant advantage to allow for the adaptation of their refractive power. Given the comparably small magnitude of the thermal lensing effect induced in the zero-phonon pumped disk used in our experiments, the resolution of the deformable mirrors in terms of refractive power was found to be too coarse in order to allow for a perfect compensation of thermal lensing. Nevertheless we were able to demonstrate constant and almost diffraction limited beam quality throughout the full output power range when using our mirrors.
The excellent performance of our mirrors can be attributed to the optimized mirror design, enabling a large area of paraboloid deformation with measured refractive powers in a range from −0.7 to +0.3 m−1 (and presumably up to −1.2 to 0.78 m−1 if the mirror deformation at the fracture limit is taken as a fundamental limitation for convex deformations and perfect vacuum for concave deformations, respectively).
Based on these results, we are confident that the mirrors demonstrated in this paper can also be used for a number of other applications, including the full compensation of both spherical and aspherical thermal lensing in thin-disk lasers when used in combination with aspherically deformable mirrors . The excellent optical performance of the deformable mirrors makes them applicable for the intra-cavity use in pulsed lasers too. As for mode-locked lasers, the mode size on intra-cavity elements such as SESAMs or inside Kerr-media plays a crucial role for the pulse generation mechanisms, the amount of control offered by deformable mirrors should lead to an enhancement of the scalability and reliability of such laser systems. Recent developments in the field of thin-disk-based multipass-amplifiers have pushed average power levels of ultrafast laser systems to the multi-kW-level . Therefore an extra-cavity compensation of thermal lensing by means of spherically deformable mirrors might prove necessary as well, e.g. for the compensation of thermally induced focal shifts in beam-delivery optics or the fine adjustment of the focal position and spot size on the workpiece.
DFG GR 3172/15-1 and SA 847/19-1
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