Design and Development of Bimorph Deformable Mirrors for Defocus Correction Kavita Rawal, Vijayeta Gambhir and D.P Ghai Laser Science and Technology Center, Metcalfe House, Old sectt, Delhi-110054, India. Email: rawalk@rediffmail.com ABSTRACT A single element Bimorph Deformable mirror 1, 2 was designed and fabricated for defocus correction of the laser wavefront. The design was optimized in terms of mirror substrate material and thickness of the substrate, dimensions and piezoelectric properties of the PZT discs. The series bimorph deformable mirror discussed in this paper consists of 2.5mm thick, 50mm diameter silicon mirror. The PZT used for the design had d 33 coefficient of 500-600 pc/n. The surface deformation and Radius of curvature of the DM was studied by setting up a Michelson Interferometer in the laboratory and also with a commercial Shack Hartmann wavefront sensor. The maximum surface deformation of 5 microns was recorded at the actuation voltage of ± 400volts. The frequency response of the bimorph deformable mirror was also studied. The deformable mirror shows the first mechanical resonance at 900Hz. Keywords: Adaptive optics, wavefront aberrations, deformable mirrors, bimorph, unimorph, PZT. 1. INTRODUCTION Although bimorph mirrors were envisaged a long time ago 3, 4 (Kokorowski 1979; Steinhaus and Lipson 1979), the actual use of a bimorph mirror in adaptive optics was first demonstrated in 1994 in a system developed at the University of Hawaii for astronomical applications (Roddier et al. 1994) A bimorph mirror now equips the PUEO user AO system of the Canada-France Hawaii Telescope (Lai et al. 1995). Bimorph mirrors are also under test for the SUBARU AO system (Takami et al, 1995) and for the Anglo-Australian Telescope AO system (Bryant et al. 1995). The performance requirements of deformable mirrors vary according to applications, which include high energy laser focusing, compensated imagery through atmospheric turbulence, and laser cavity control. Compared to astronomical applications, requirement related to defense applications are often more demanding. In the recent past a lot of progress has taken place in the field of adaptive optics which can be attributed to advancements in electronics, processing, control systems, high speed sensors but it is also associated with more complex design of deformable mirrors with a large number of sub apertures or actuators. Astronomical adaptive optics systems with a large number of elements have been fabricated. These systems are inherently expensive and complex to build. However the largest low-order phase distortions are rather slow in comparison with higher order ones, and it is cost effective to use two adaptive mirrors one a Bimorph for lower order is modal mirror with a large range of surface deformations and a second stacked actuator continuous or segmented face sheet mirror for higher order aberrations. Bimorph deformable mirrors are basically Laplacian correctors 5, 6 are becoming popular because of their low cost and ease of fabrication and do not involve complexity of design. 2. THEORY A bimorph mirror consists of two piezoelectric wafers 7, 8 which are bonded together and are oppositely polarized (parallel to their axes), this in turn is bonded to quartz or a silicon substrate. There is a common electrode between the substrate and the PZT, and a set of discrete electrodes on the rear face of PZT to which control voltages can be applied. The front face of the substrate has a reflective coating. When voltage is applied across two ends of the piezo disc, one-wafer contracts and the opposite wafer expands, which produces a local bending. The local curvature being
proportional to voltage. A tricky characteristic of bimorph DMs is that they are controlled not in surface shape, but in surface curvature and are called curvature mirrors 9,10 it tends to expand in the radial direction but glued interface of the substrate opposes this expansion. This results in the bending of the substrate, thereby, generating a defocus effect or forming a convex or concave curvature depending on the polarity of the applied voltage. The deformation of the mirror surface is typically a few microns, and a wide range of wavefront profiles can be generated by giving different combination of voltages and these can be interpreted in terms of various atmospheric aberrations as well as aberrations in the laser due to medium inhomogeneites like defocus, astigmatism, coma, spherical aberration etc. The bimorph deformable mirror 11 can be treated as a thin plate and the deformation can be interpreted according to the bending of thin plates therefore when a voltage applied across the thin dimension of the plate causes an expansion or contraction in the plane of the plate and bends the substrate in a manner analogous to a bimetallic strip. The relative change in length induced on an electrode of size L is given by L/L = V d 31 / t (1) Where d 31 is the transverse piezoelectric coefficient and t is the thickness of the wafer. Neglecting the stiffness of the wafers and three-dimensional effects, and assuming the piezoceramic wafer thickness to be small as compared to the substrate, the curvature produced is given approximately by: R=t 2 /2V d 31 (2) Where d 31 is the transverse piezoceramic strain constant and t is the glass plate thickness. V is the applied voltage and, as can be seen from equation (2), that the local curvature is proportional to the locally applied voltage. For a spherical deformation over the diameter d, the bimorph sensitivity Sb expressed as the ratio of stroke/voltage is: S b = d 2 /8 R V =( d 2 /4t 2 ) d 31 (3) The static equation of state for an ideal bimorph mirror has the form (Kokorowski 1979; Roddier 1988) δ 2 (δ 2 W+ AV) =0 (4) Where δ 2 denotes the two-dimensional Laplacian, W (x, y) is the mirror surface deformation, V (x, y) is the voltage distribution on the wafer, and A=8d 31 /t 2. The equilibrium is reached when the mirror surface is the solution of a poisson equation with appropriate boundary conditions. Radial tilts at the edge provide the boundary conditions required to solve the Poisson equation. A simple way to control these tilts is to use an extra ring of electrodes and to limit the pupil to the inner part of their surfaces (Jagourel et al. 1990). Besides this displacement, the applied voltage causes opposite changes of the thickness of each wafer (Kokorowski 1979). Even if this effect cancels for the entire bimorph 12 it still produces a displacement of the top and bottom surfaces, which is added to the displacement caused by bending. From equation (4) the deformation due to thickness changes over an electrode may be written Wt (x, y) = - b V (x, y) d 31. This effect can be compared with the pure bending deformation over an electrode of diameter d given by equation (3). These effects are opposite and the resulting displacement is locally zero when the wafer diameter d becomes of the order of the thickness. Since the ratio of the diameter of the whole wafer to the thickness is limited by polishing considerations, the number of electrodes is limited by the bimorph diameter to thickness ratio. Typically a few tens of electrodes are used. Hence, bimorph mirrors are best suited for lower order compensation systems. Owing to the k -2 spectrum dependence bimorph mirrors have a sufficient stroke at low spatial frequencies to compensate for turbulence induced tip/tilt errors. 3. DESIGN AND FABRICATION Although the Bimorph deformable mirrors are now commercially available and being used in various prestigious projects related to astronomy, adaptive optics and for laser source correction. As a step towards indigenous development, we have chosen to develop single element series type bimorph deformable mirror design for defocus correction as well as optimizing our design in terms of substrate material, diameter, thickness and also for PZT d 33
coefficient, diameter, thickness and mounting procedures. The silicon facesheet material is selected as the optimum choice based upon our theoretical studies on substrate material selection of deformable mirror facesheets for adaptive optics. The silicon substrates of 1, 2, 3 diameter and each of 3.5mm, 2.5mm and 1.5mm thickness were acquired and The PZT discs of different d 33 in the range of 400 pc/n to 650 pc/n and of diameters of 9mm, 20mm, 22mm were used to facilitate our design. To begin with all the optically polished silicon substrate surfaces were monitored on ZYGO for initial surface flatness with and without mounting so as to ensure that there is no deformation induced due to the mechanical mounts of the deformable mirror. We have selected to fabricate the series type bimorph deformable mirror as shown in figure (1). In this configuration two PZT discs with conductive coatings on both the surfaces were glued together in series with the conducting silver epoxy which should not expand or contract upon hardening and cured in the oven for 2-3 hours at 80 degree Celsius, till the adhesive had completely hardened. These cured bimorph discs were then adhered to the Silicon substrates of thickness varying from 1.5mm to 3.5mm and diameters of 25, 50 and 76mm with the conducting silver epoxy, and again kept in the oven for 2-3 hours at 80 degree Celsius and then allowed to cool to remove the residual thermal deformations. In the case of unimorph deformable mirror instead of two PZT discs only a single disc was used and same procedure was followed for its fabrication too. The Electrical connections were then made to the electrodes for its performance evaluation.the mechanical mounting of a bimorph DM is delicate in the sense that on one hand, it must be left to deform, on the other hand it must be fixed in the optical system. The mechanical mounts were designed and fabricated for mounting the deformable mirrors of different diameters. Figure (1): Morphology of series type Bimorph deformable mirror 4. PERFORMANCE EVALUATION The static as well as dynamic characteristics of the mirrors were studied. The surface profiles of all the mirror substrates were observed on Zygo for its initial surface flatness etc. A Michaelson Interferometer was setup in the laboratory to study the performance of the bimorph mirrors for the deformation produced as shown figure (2). We have used He-Ne (0.632microns) laser for these measurements. The deformation was estimated from the displacement of the interference fringes at the center of the pattern when the voltage was varied from 400 volts to +400 volts shown in figure (3) as one fringe shift corresponds to half the wavelength of the laser beam used. The deformation produced by these deformable mirrors was also evaluated on class 2D Shack Hartmann Wave front sensor set up shown in figure (4), on which the peak to valley surface deformation has been recorded as the 3D phase plot as given in figure (5). The results from both Michelson setup as well as SHWFS were in close agreement. The frequency response of the bimorph DM was also studied by monitoring the variation of the amplitude at the output of the silicon photo detector on varying the frequency of the signal at a constant voltage given to drive the bimorph deformable mirror. A number of prototypes of such bimorph and unimorph deformable mirrors were
fabricated for different dimensions of the mirror substrate and PZT discs parameters. Finally each of them was tested for its deformation and frequency response. Figure (2): Experimental setup for the evaluation of bimorph DM. Figure (3): Interferograms of single actuator BDM.
Figure (4): Testing of Bimorph deformable mirror on a Shack-Hartmann wave front sensor Figure (5): 3D Phase plot for Bimorph dm 50mm dia, 2.5mm thick at 400volts as Recorded on SHWFS. 5. RESULTS AND DISCUSSION The results of all the experimental measurements for the surface deformation have recorded. The effects of the variation of parameters and dimensions of the mirror substrate & PZT discs are compared and the results are shown in the graphical plots as given in figure (6,7,8,9,10,). The radius of curvature of the bimorph DM was also calculated and its variation with stroke is plotted in figure (11). The frequency response of the Bimorph deformable mirror for
the silicon substrate diameter of 50mm and thickness 2.5mm has been shown in the figure (12). This deformable mirror exhibits a bandwidth of about 500 Hz and the first mechanical resonance occurred at around 900Hz. The maximum surface deformation of 5 microns was recorded for the actuation voltage of 400volts to + 400volts The comparative analysis of all these experimental data indicates that for a fixed substrate diameter, decreasing the substrate thickness results in an increase of surface deformation and increasing the diameter of the PZT disc for the same diameter and thickness of the substrate also leads to an increase in the stroke or the deformation. Also with the increase in the diameter of the substrate, keeping all other parameters same, deformation of the mirror increases. The deformation measured was almost 1.5 times to two times more in the case of bimorph as compared to a unimorph. When increased d 33 value PZT material was used for the study, it resulted in increased deformation at the same driving voltages. This is in accordance with the theoretical relationship for the calculation of deformation in Bimorph Deformable mirrors. The experimental results are in accordance with the theory discussed above. The success of the single element Bimorph deformable mirror has encouraged us to go for multi-element bimorph deformable mirror development for producing and interpreting other higher order aberrations like astigmatism, coma, etc, and the work in this direction is in progress. Figure (6): The variation of surface deformation with dc voltage.
Figure (7): Surface deformation versus substrate thickness for 25.4mm mirror diameter and PZT dia of 9mm for bimorph and unimorph deformable mirror. Figure (8): Surface deformation versus substrate thickness for 50mm mirror diameter and PZT dia of 9mm for bimorph and unimorph deformable mirror.
Figure (9): Surface deformation versus thickness for 50mm substrate diameter and PZT diameter of 20mm for bimorph and unimorph deformable mirror. Figure (10): Surface deformation versus thickness for 25.4mm substrate diameter and PZT diameter of 20mm for bimorph and unimorph deformable mirror
Figure (11): Variation of the Radius of curvature versus stroke or deformation. Figure (12): Frequency response of 50mm diameters, 2.5mm thick bimorph deformable mirror.
6. CONCLUSION The performance of the DM strongly depends upon the type of mirror substrate, its thickness, diameter and initial surface flatness. It also depends on PZT disc dimension and its piezoelectric coefficient. The PZT disc should be thick enough to generate required localized bending force. However, a thicker plate demands a higher voltage to be applied across it for required expansion. The bonding material should be such that it does not under go shrinkage after application. The coefficient of thermal expansion of the mirror material and the PZT must closely match. However we can not go on reducing the thickness of the substrate or increase its diameter as the ratio has to be maintained due to polishing considerations and reducing the thickness of the mirror substrate below one mm will introduce some distortions or initial curvature on mounting so a trade-off needs to be established between these parameters for an optimum design of deformable mirrors. ACKNOWLEDGEMENT We wish to express our sincere gratitude to Shri K. S. Jindal, Director, LASTEC, Delhi for his constant support and encouragement. We also thank Dr. C. P. Rana and Dr. O. P. Thakur, SSPL, Delhi for providing us the piezoelectric discs to carry out this work. REFERENCES: 1. Journal Article: M.A.Ealey et.al, Proc SPIE 2201,680 (1994). 2. Journal Article: J.Christopher Dainty et.al,applied Optics, Vol.37,No21,4663(1998). 3. Journal Article: N.T. Adelman,,Applied optics,16,3075(1977). 4. Spherical mirrors for piezoelectrically controlled curvature N.T Adelman. Applied Optics No. 16, 3075(1977). 5. Semi passive bimorph flexible mirrors for atmospheric adaptive optics applications A. Kudyashov, and V Shmalhausen, Optical Engineering, No.35, 3064 (1996). 6. Bimorph deformable mirror design, S. Lipson, E. Ribak, c. Schwartz, proc. SPIE,vol 2201,703(1994). 7. Bimorph PZT active mirror, F.F Forbes, Proc. SPIE, vol 1114, 146(1989). 8. Bimorph adaptive mirrors and curvature sensing, C Schwartz, E Ribak and S, Lipson, JOSA A 11(2),895(1994). 9. Experimental studies of a deformable adaptive optical system, E Pearson, S. Hansen, JOSA No. 67, 325(1977). 10. Analysis of adaptive optical elements made from piezoelectric bimorphs, S A Kokorowski, JOSA No. 69,181(1979). 11. Controllable bimorph optics and the principles according to which they can be developed further, A G Safronov, J. Opt. Technol No. 65,4,(1996). 12. Comparison of deformability between multilayered deformable mirrors with a monomorph or bimorph actuator, O Ikeda and T. Sato, apllied Optics, No. 25 4591(1986).