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We propose an improvement to the electrostatic membrane deformable mirror technique introducing push-pull capability that increases the performance in the correction of optical aberrations. The push-pull effect is achieved by the addition of some transparent electrodes on the top of the device. The transparent electrode is an indium-tin-oxide coated glass. The improvement of the mirror in generating surfaces is demonstrated by the comparison with a pull membrane mirror. The control is carried out in open loop by the knowledge of the response of each single electrode. An effective iterative strategy for the clipping management is presented. The performances are evaluated both in terms of Zernike polynomials generation and in terms of aberrations compensation based on the statistics of human eyes.
©2006 Optical Society of America
1. Introduction
Membrane deformable mirrors have a widespread diffusion in several applications. Comparing to other common devices for adaptive optics, such as liquid crystal modulators, bimorph mirrors, thermal mirrors, they have a lot of advantages: low cost, large dynamic behavior, achromaticity, no hysteresis, relatively high optical load, good performance in aberrations generation, low power consumption. The drawback of these devices is the limited amount of maximum stroke and the high correlation within the electrodes. Silicon nitrite membrane mirrors proposed by Vdovin and Sarro [1] have good properties in terms of initial flatness and maximum stroke. The application of deformable mirrors play an important role in visual optics for the correction of the eye aberrations. Literature shows that membrane mirrors can be used in visual optics obtaining good results [2, 3]. Dalimier and Dainty [4] showed a comparison within adaptive optics mirrors from different technologies. The results show that membrane mirrors, despite their advantages, do not have enough optical power to completely compensate eye aberrations. The improvement of membrane mirrors is still an open challenge. Recently Kurcinski [5, 6] has obtained higher strokes decreasing the membrane stress and applying a big transparent electrode on the top of the membrane. These important results still need of improvements in the fabrication techniques in order to reduce the stress on the membrane during the mounting that affects significantly the initial flatness. Furthermore new technologies are under development in membrane deformable optics. For example liquid mirrors based on electrocapillary or electromagnetic membrane deformable mirrors seem to be promising techniques for push-pull motion of the electrodes [7]. We propose an improvement in the electrostatic membrane mirror fabrication to obtain push-pull capabilities by the application of additional electrodes over the upper side of the membrane, where the light is coming in. Ten electrodes were added in the prototype here presented, one of these is realized with an indium-tin-oxide (ITO) coated glass to guarantee high transparency and electric conduction. Hence, the membrane is deformed by the non-transparent electrodes placed on its bottom side and, on the top side, by an external ring of nine electrodes and by the transparent one in order to add more stroke, improve the aberrations generation and to allow the use of the mirror without biasing the membrane [6]. The performances of this push-pull mirror are compared to those obtained using a normal membrane mirror with only pull capability [8]. The mirror design is studied by the solution of the Poisson equation [9]. The comparison has been carried out using the approach of Dalimier and Dainty for the test of new mirrors4 observing both the capabilities in the generation of Zernike polynomials and, for visual optics, in the generation of 100 aberrated eye following the statistics published by Castejon-Mochon [10]. The generation was carried out in open loop by the measures of the displacement of each single actuator.
2. Push-Pull mirror device
The device is composed by a nitrocellulose silver-coated membrane tensioned over a circular frame with a 25-mm diameter. The thickness of the membrane is 5 µm and the diameter of the useful area of the mirror is 10 mm. A schematic of the device is shown in Fig. 1.
Thirty-seven non-transparent electrodes are placed on the bottom side of the membrane and shaped as circular sectors. The electrodes are placed in three complete circular rings: one in the inner, six in the second, twelve in the third and eighteen in the outer for a total of 37 actuators, as shown in Fig. 2(a). Each of the electrodes has the same geometrical area. The distance between the membrane and the printed circuit board is about 70µm. The spacer is calibrated in order to obtain a maximum deflection of about 6µm. The electrodes are deposited on a diameter of 15 mm.
The presence of electrodes on only one side of the membrane gives a deformation capability in one direction, i.e. the distance between membrane and electrodes decreases applying voltages to the electrodes themselves. Such a device is then defined pull mirror.
Push-pull capability is obtained placing additional pads over the top side of the membrane. Nine non transparent electrodes are faced to the outer ring of the lower electrodes and a circular transparent electrode is placed on the center of the membrane and faces the first three inner rings of the lower electrodes see (Fig. 2). The electrical connection of the ITO-coated glass with the external amplifier is obtained by gluing a 5-µm Al-coated Mylar film. The ITO layer is deposited on a float soda lime 1-mm-thick glass. The thickness of ITO is 150 nm with a surface resistance of 12 WΩ/□ and a transmittance of 89%.
Fig. 2. Geometry of the actuators: a) non-transparent electrodes placed under the bottom side of the membrane; b) transparent actuators placed over the top side of the membrane. The red line indicates the size of the membrane. The ITO coated glass is the dashed blue pattern.
Figure 3 shows the interferogram on the active area obtained with no voltages applied on the actuators. The rms deviation is 29.8 nm, that is about λ/20 @633 nm, that is mainly caused by the surface flatness of the glass disc.
The configuration was chosen to obtain a deformation of the membrane on the edge of the active area thanks to the faced electrodes of the outer ring. Applying voltages to the upper electrodes the membrane moves towards the incoming beam and towards the back of the device. The possibility of push and pull the membrane allows to use the deformable mirror without biasing the membrane [2, 3] and to obtain better performance in terms of spatial resolution, Zernike polynomials amplitude and compensation of eye aberrations. These items will be analyzed in details in the next sections. The results will be compared with those obtained by the pull membrane mirrors realized in our laboratory and described in detail elsewhere [8]. The pull mirror uses a 18mm diameter membrane that has the optimal active area of about 10mm [8, 9]. The typical rms deviation from a flat surface is less than 60nm. The maximum stroke obtained pulling all the electrodes at the maximum voltage of 230 V, is 3.5 µm in the active region of 10.5 mm.
3. Push-Pull mirror design
We developed a mathematical model for the prediction of the mirror deformation based on the analytical solution of the Poisson equation proposed by Clafin [9]. The model gives the possibility of design the mirror geometry and to optimize the generation of the first four radial orders of the Zernike polynomials. The evaluation of the reproduction is computed by the peak-to-valley amplitude and a term that we define Purity. Literature reports the ability in Zernike generation in terms of rms residual of the actual shape with the desired one [2]. We propose to quantify the mirror performances computing the Purity:
where the terms Di are the projection of the normalized shape M(x,y) over the Zernike orthonormal base versor zi:
It is clear that Di=1 if the M(x,y) is parallel to zˆi, that it means that they have the same shape and M(x,y) has no contribution from other non desired Zernike terms. Hence, the quantity Pi represents how the generated shape is similar to the desired shape Zi and gives a measure of the contribution of different Zernike polynomials. We chose to use this representation because respect to the RMS residual, is independent from the peak-to-valley value, giving a more direct interpretation of the quality of the reconstruction.
The design of the mirror was carried out to obtain the maximum peak-to-valley values for the first three orders of the Zernike polynomials at the maximum values of Purity. Figure 3 reports the peak-to-valley values and the correspondent Purity as functions of the membrane normalised radius. The simulation has been performed using 37 lower electrodes and 9 upper electrodes faced to outer ring of the lower ones and the ITO coated glass disc.
The mirror optimal Active Region was obtained defining the Influence functions matrix as:
where Ai are columns containing the influence functions relative to the i-th electrodes.
The vector of the electrostatic pressure p is obtained, under the hypothesis of linearity, as:
where Z(x,y) is the vector containing the desired shape.
The calculation of A-1 was carried out using the non-negative constraints least square (NNLS, Lawson and Hanson algorithm) [9] of the electrostatic pressures.
The maximum peak-to-valley value is obtained for a normalised radius of 0.4, suggesting that for a membrane of 25 mm the optimal active region is 10 mm that corresponds to the whole area covered by the ITO glass. The minima in the peak-to-valley values correspond to the area closes to the places where the first derivative of the influence functions relatives to the actuators of the second rings are nulls (about in the centers of the electrodes). The maxima are placed about where the derivatives have their maxima which happens close to the separation between the second and the third rings. This configuration allows an optimum Zernike’s reproduction for normalised radius in the 0.3 to 0.7 range.
Fig. 3. (a). peak-to-valley amplitude obtained by the NNLS fitting of the first three terms of Zernike polynomials over the active region. The horizontal red lines represent the pad positions of the first, second and third ring. (b) Purity of the first three terms of the Zernike polynomials over the active region.
4. Influence functions
In order to generate the desired shapes in open loop, the influence functions were measured. We used a Twyman-Green interferometer (Zygo) to measure the deformation caused by a single actuator which is called influence function Aj. Figure 4 shows some measurements of the influence functions. The maximum displacement for the lower actuators is 0.8µm peak-to-valley relative to the active area. This low value is due to the excessive thickness of the spacer. By using a thinner spacer, the quality of the results can be further increased. The positive displacement due to the transparent electrode is easily recognizable in the upper actuators influence functions, with a 2.5µm peak-to-valley value. The external ring of upper actuators gives a maximum peak-to-valley displacement of 1.3µm and a minimum of 0.8µm.
This asymmetry, clearly visible in the number of fringes in the interferograms, is due to imperfections in the realization of the calibrated frame. The maximum voltage applied to all the upper actuators (230V) gives a displacement of about 3µm. Since the edge of the membrane is covered by the electrodes, it was not possible to measure the value of the pedestal of each influence function. The results here presented are obtained using the pedestal values from the simulations.
Fig. 4. Examples of measured influence functions. Surface and interferograms are shown for the electrodes 2, 8, 20, 38, 42.
5. Performances
Literature reports several publications that relates the capabilities of deformable mirrors in visual optics, with the generation of Zernike polymomials. As suggested by Dalimier and Dainty [4], this analysis is not completed because, as the Zernike terms are orthogonal, problems of maximum mirror deflection and electrodes clipping can reduce the capability of generating aberrations. We address the evaluation of the performance of the push-pull mirror using both the Zernike polynomial fitting and the reproduction of the ocular wavefronts. The results are compared with a pull membrane mirror with an equal active region.
The tests here presented are the generation of the first four radial order aberrations and of 100 aberrations based on the statistics of well-corrected eyes [10].
Several methods were published [2, 3, 11, 12, 13] on the techniques for the generation of a desired shape of deformable mirrors. We follow the open loop approach [9]. Literature reports iterative algorithms in closed loop for the optimization of the mirror controls; these methods give better results because they take into account the non-linearities in the mirror response [2].
5.1. Zernike Polynomials
We evaluate the mirror performance using as target the first four radial order of the Zernike polynomials. In order to full exploit the mirror capabilities the values exceeding the maximum pressure (pmax) are clipped as described in the following paragraph.
At first we calculate the vector p using (4). Than, we define the vector p′={pi∈p, pi>p max} and we set p′i=p max∀p′i∈p′. We also define the sub-matrix A′ of A according to: A′={Ai|pi>p max}, the sub-set of elements p″={pi∈p, pi≤p max}, and the matrix A″={Ai|pi≤p max}.
The vector p” can be computed as:
So, the new vector of electrostatic pressure is p=p′∪p″.
The algorithm has to be repeated until the vector p” does not contain any saturated electrode. This iterative strategy is applied to increase the peak-to-valley amplitude of the desired Zernike terms until the RMS residual is lower than 10%, to finally obtain the voltage vector to be applied to the mirror. The interferograms of the results are shown in Fig. 5 and Tab. 1. The mirror response is better for the lower orders, as attested by the purity and the good peak-tovalley amplitude. The worst response is related to the spherical aberration and the polynomial of order (4, 2). This lack of performance is explained by to the values of the pedestal of the influence functions, as discussed previously, that weights more in the higher order aberrations for their limited peak-to-valley amplitudes. The ratio between the measured RMS residual and the peak-to-valley does not exceed 18% as for the case of the (4, 0).
The peak-to-valley results of the simulation compared to the ones obtained in the measures are shown in Fig. 6(a). The results asset that the linear model is in good agreement with the measurements.
Table 1. results of the aberrations produced with the Push-Pull mirror.
Fig. 6. (a). histogram of the Peak to Valley measurements of the main Zernike Polynomials obtained from the model (green) and measured (red). b) comparison between the peak to valley measurements of the Pull mirror (red) and the Push-Pull mirror (green).
As a comparison with the pull membrane mirrors, we report the measurements of the generation of the first four order aberrations with one of the pull mirrors described above. The same strategy for aberration generations and measurements was followed. In this case, the membrane is biased at half of its optical power. The Fig. 6(b) reports the performance obtained with the two different systems. It is clear that, in general, the strokes induced by the push-pull mirror are higher than the ones obtained with the pull mirror.
6. Simulation of performance in visual optics
We followed the approach suggested by Dalimier [4]. We generated a random family of 100 aberrations following the statistics presented by Castejon-Mochon [10]. The amplitude of the aberrations was computed as normal distributed random variables around their averages, with their variances. The aberrations rotations was treated as white noise within [0, 2π].
The reproduction is carried out using the least square method explained in the previous section. In order to full exploit the dynamics of both mirrors according to the statistics reported by Castejon-Mochon, we apply a bias to the optical system relative to the push-pull mirror at the same amplitude of the average value of defocus. As usual, the pull mirror was used biasing the membrane to its half power [1, 2, 3].
The average RMS values of the family is about 1µm. The residual of the corrected wavefronts is 0.3µm for the pull mirror and 0.1µm for the push-pull one. We underline that the RMS residual for the pull mirror is in good agreement with the value obtained by Dalimier. The improvement given by the use of a push-pull mirror in the optical system is about a factor 3.
Fig. 7. Residual wavefront rms error after fitting with Pull mirror and Push-Pull mirror over a 10mm pupil.
7. Conclusion
We have designed and realized a push-pull membrane deformable mirror by adding some electrodes over the top side of the membrane. The performances for its application in visual optics are evaluated by the capability of generating Zernike polynomials and ocular wavefronts. The results are compared with a pull mirror with performances similar to commercial membrane mirrors. The results asset that a definitive improvement in the peak-to-valley amplitude and RMS residual are obtained.
References and links
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