To decrease the cost of integrated circuits, optical lithographic processes has been improved over the past decades. Its wavelength is decreased to extreme ultraviolet (EUV) light (13.5 nm) to miniaturize circuits, and source power is increased in favor of production speed. EUV light is highly absorptive, requiring mirrors instead of lenses in lithographic optical systems. These mirrors have a reflectivity of maximum 70%. The combination of high source power and low reflectivity results in high absorption, causing the mirrors to increase in temperature and consequently to thermally deform causing wavefront errors (WFEs) [1]. WFEs of only a few nanometers already lead to deteriorated images.

Thermal errors occur in many high-precision applications. For example, in laser interferometer gravitational observatories (LIGOs), using power levels of several kilowatts for high sensitivity to weak gravitational waves [2]. Although the mirrors in LIGOs have high reflectivity, the smallest absorption of the high optical power still leads to thermal deformation of the mirrors, compromising their subnanometer precision.

To compensate for thermal distortion, an adaptive optics (AO) system is proposed [3], using an active mirror (AM) [3–5]. The AM consists of sequentially a mirror substrate, a deposited absorptive coating, and reflective coating, as shown in Fig. 1. Actuation is realized by exposing the absorptive coating to the actuation profile, which is a spatially distributed irradiance profile or heat flux. This actuation profile is provided by a projector, propagates through the mirror substrate, and gets absorbed by the absorptive coating.

 

Fig. 1. Active mirror (AM): the absorptive coating on top of the mirror substrate absorbs an actuation profile provided by a spatially controllable heat source. (a) Its compensation aspect realizes a uniform irradiance by exposing an actation profile that is opposite to the irradiance profile originating from the heat source. (b) Its correction aspect exposes an irradiance profile to thermally deform the AM for wavefront correction.

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Utilization of the AM has two aspects: compensation of its own deformation and correction of other mirrors in the optical system. First, the compensation aspect prevents mirror deformation by uniform irradiance, accomplished by an actuation profile opposing the distorting irradiance profile. Second, the correction aspect uses thermal expansion to deliberately deform the mirror.

This specific concept has advantages above conventional deformable mirrors (DMs). To manufacture subnanometer precise surfaces, thick substrates are needed, which is undesirable in terms of actuator forces and achievable spatial frequencies in conventional DMs. In this concept these issues inherently do not occur as actuation is at the reflective side of the mirror. Furthermore, mounting actuators in conventional DMs add surface inaccuracies due to print through of the actuator pattern, which is highly undesirable in subnanometer applications. This is prevented by solely adding an absorptive coating to the AM. Finally, in conventional DMs mechanical connections between force frame and DM, could transmit vibrations from the environment to the DM. This is avoided by omitting mechanical connection.

Most conventional mirrors are used by the zonal or modal approach [6]. The zonal approach induces local deformation by energizing single actuators. This could be mimicked by partitioning the AM into a grid of discrete actuators [7]. The modal approach decomposes a WFE into orthogonal shapes, for example Zernike shapes, to determine their contribution [8], and corrects accordingly. In contrast to most conventional DMs, the AM does not have a trivial relationship between actuation profile and deformation, because local absorption of light causes a global deformation due to thermal conduction and bimorph-like deformation. To employ either the modal or zonal approach, appropriate actuation profiles must be obtained.

This Letter demonstrates the application of the modal approach to this AM using Zernike shapes. Hence, a model is used, an approach that has also been shown to calibrate a conventional microelectro mechanical system DM [9]. To obtain actuation profiles that induce Zernike shapes, a finite element (FE) optimization procedure is used. The obtained actuation profiles are exposed to the AM, which is positioned in a Michelson interferometer to measure the deformation to validate the FE procedure. This procedure is carried out for 25 different Zernike shapes. Besides obtaining these actuation profiles, four randomly chosen actuation profiles are simultaneously exposed and decomposed in the contributions of the Zernike shapes.

The desired actuation profile is obtained with COMSOL Multiphysics FE software by applying an optimization procedure with the optimization toolbox. The thermal stress toolbox relates the actuation profile, ${\dot{Q}}^{\prime \prime }(x,y)$, which is a heat flux in $\frac{J}{s}\frac{1}{m^2}$ indicated by the dot and the apostrophes respectively, with a mirror deformation $w({\dot{Q}}^{\prime \prime })$. ${\dot{Q}}^{\prime \prime }(x,y)$ is the control variable of the optimization procedure, exposing the actuated area or exposed area (EA), shown in Fig. 2. The EA is typically larger than the aperture (A) used for correction, since it improves the formation of the desired shapes [10,11]. The meshing is manually adjusted for the individual Zernikes, so the symmetry axes and the tangential and radial orders correspond to the particular Zernike [12]. Furthermore, only a quarter of the mirror is modeled, and symmetry and antisymmetry are used to decrease computational time and memory. The actuation profiles of nonspherical pairs are obtained for one orientation and are rotated 90° for the paired orientation.

 

Fig. 2. The active mirror (AM) is divided into different areas for exposure and measurement.

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The optimization compares the mirror deformation $w({\dot{Q}}^{\prime \prime })$ with the desired deformation $w_{\mathrm{des}}(x,y)$, which is a Zernike polynomial expressed on the coordinates $x$, $y$. The integrated quadratic difference between these is used as optimization objective $J(w)$:

The so obtained actuation profiles are applied to the AM in the experimental setup shown in Fig. 3. Its measurement part consists of a Michelson interferometer, which has a measurement area of 40 mm employing 2 in. optics. It has 0.3 nm out-of-plane resolution, and a relative uncertainty of 0.6 nm. The actuation part consists of a BENQ 5000 ANSI-lumen video projector that projects the actuation profile to the AM. The AM is sequentially an $50\times 50\times 4\mathrm{mm}$ BK7 glass substrate, an absorptive and reflective coating, as explained in the introduction and in Fig. 1. The interferometer and the AM are placed in a vacuum chamber, which avoids free convection-induced measurement errors.

 

Fig. 3. The measurement setup consists of (a) the measurement part being a Michelson interferometer and (b) the actuation part.

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Initially, the AM is exposed to an off-set irradiance, which is 50% of the maximum irradiance used. This creates an off-set temperature which enables the mirror to be relatively cooled as well to be heated. Hereafter, a reference measurement is taken to determine the shape of the AM in its thermal equilibrium. The actuation profile is superimposed to the off-set irradiance and exposed to the AM and the induced deformation is measured. Afterward, the reference measurement is subtracted from the measured deformation, and forms the basis for further analysis. The actuated Zernike shapes are compared with the Zernike polynomials in two ways: the purity, which is the normalized cross correlation between the actuated and desired shape [13], and the RMS error are obtained to assess the magnitude of the error.

Figure 4 shows that high local irradiance corresponds to two phenomena: out-of-plane (OOP) expansion causes local OOP deformation and in-plane expansion causes local bending inducing global deformation. Furthermore, $Z_8$ and $Z_{19}$, which are both spherical shapes, contain lower-order spherical errors and the pair $Z_{20}$ and $Z_{21}$ contain astigmatism. A general trend is that Zernikes shapes with a lower order exhibit a higher amplitude. Figure 5 shows the purity of the other 21 Zernike shapes higher than 0.9 and an RMS error of approximately 2 nm.

 

Fig. 4. The measured Zernike shapes (meas.), their actuation profiles (act.), and the resulting errors. The amplitude of the Zernike shapes and the errors are normalized to the measured amplitude, which is shown in Fig. 5.

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Fig. 5. The amplitude (ampl.), error, and purity of the excited Zernikes as defined in Fig. 4. The amplitude of the Zernike is decreasing for higher Zernike orders. Most of the excited Zernike shapes have a purity above 0.9 and have an error of about 2 nm RMS.

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The high purity and low error of 21 measured shapes indicate the viability of the procedure. The decreasing amplitude of higher-order Zernike shapes can be explained by the increasing available energy per spatial period. The RMS error is a result of optimization, modeling misalignment, and measurement errors. Residual optimization errors could be reduced by adjusting the stop condition, and the modeling errors can be improved by using more elements in the mesh, at the expense of memory and calculation time. In addition, misalignment errors could be introduced by the reference mirror, which is slightly tilted compared to the AM. This tilt generates interference fringes, which are used to obtain the deformed shape.

Specifically, the pair $Z_{20}$ and $Z_{21}$ contain astigmatism. Their error is likely to be caused by the alignment of the interferometer, but it could well be that changing the FE mesh will improve the results. In contrast to the other actuation profiles, the shapes $Z_8$ and $Z_{19}$, have an average irradiance that is not zero. This changes the average temperature of the AM, which phenomenon has a longer time constant than the time between consecutive experiments. This increase goes together with global bending, showing up in the measurement error as $Z_1$, but not in the modeling error, as the models were solved for steady state. This error could be improved by adding the optimization condition to require an average irradiance of zero. Besides these proposed improvements, the inaccurately estimated actuation profiles could also be improved by performing a Zernike decomposition of the error encountered and adding the actuation profiles of the Zernike contributions found.

As a final test, this Letter shows that actuation profiles can be superimposed to obtain superimposed Zernike shapes. Therefore, the following procedure is applied: four randomly chosen actuation profiles with randomly chosen amplitudes are exposed to the AM and its resulting deformation is measured. Next, a Zernike decomposition is applied on the measured shape. This decomposition is compared to the actuated profiles and amplitudes.

The actuated magnitudes of the four randomly chosen Zernikes shapes are $Z_5=13\mathrm{nm}$, $Z_{11}=4.2\mathrm{nm}$, $Z_{12}=11\mathrm{nm}$, and $Z_{20}=4.9\mathrm{nm}$, as shown in Fig. 6. The resulting deformation shown in Fig. 7 is decomposed and shows resemblance for $Z_5$, $Z_{11}$, and $Z_{12}$, but is different for $Z_{20}$. The measured amplitude of this Zernike shape is 2.3 nm, but the plot also shows increased astigmatism $Z_2$. Although the determined actuation signal for $Z_{20}$ could be improved, the superposition is successful implying linearity of the actuation principle.

 

Fig. 6. Four excited Zernike shapes are $Z_5$, $Z_{11}$, $Z_{12}$, and $Z_{13}$. Three exited Zernike shapes emerge closely to the expected value, and one additional Zernike shape clearly emerged ($Z_2$).

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Fig. 7. Resulting shape of four superimposed heatloads.

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Apart from the exact amplitude of the excited shapes, the obtained actuation profiles are expected to be generic or at least comparable. This is expected for mirrors with linear material constants, and sufficient ratio between the thermal conductivity and the heat transfer to its environment (Biot number $\gg 1$). Because thermal conductivity $k$ determines the steady-state thermal profile, this constant only proportionally scales the amplitude of the 3D thermal profile. Furthermore, a different coefficient of thermal expansion of the substrate material scales the thermal expansion proportionally. Other thermal relevant material constants, i.e., material density and thermal capacity, mainly affect the time constant of the deformation, but have no impact on the steady-state deformation. Mechanical material constants play a small role since the stresses in the substrate material are low: local bending depends on thermal gradients rather than shear stress. Finally, the thermal conductivity $k$ and other relevant material constants are in the same order of magnitude for most optical materials, supporting the general applicability.

In conclusion, this Letter presents actuation profiles of 25 Zernike shapes. These can be used to control a thermally actuated active mirror (AM), to correct for thermal aberrations in applications with subnanometer precision. These actuation profiles were obtained by an FE procedure, and validated by experiments. These showed a purity of more than 0.9 for 21 Zernike shapes and an RMS error of about 2 nm. From these experimental results three main conclusions can be drawn. First, the presented method is suitable to find actuation profiles for specific deformed shapes. Therefore, it is useful to find actuation profiles for higher-order Zernike shapes as the 25 shown in this Letter and perhaps other useful orthogonal shapes for wavefront correction. Second, the presented actuation principle can induce precise 3D shapes to be used for wavefront correction. Third, because the actuation principle is linear, the obtained actuation profiles can be superimposed. Hence, this method can be used for modal wavefront correction with the AM.