Diffraction errors in micromirror-array based wavefront generation

Optics Communications 284 (2011) 2317–2322

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Diffraction errors in micromirror-array based wavefront generation Nadine Werth ⁎, Mathias S. Müller, Johann Meier, Alexander W. Koch Institute for Measurement Systems and Sensor Technology, Technische Universität München, Theresienstrasse, 90/N5, D-80333 München, Germany

a r t i c l e

i n f o

Article history: Received 18 March 2010 Received in revised form 19 November 2010 Accepted 19 November 2010 Available online 8 January 2011 Keywords: Micromirror array Speckle interferometry Laser-based shape measurement Two-wavelength technique Phase shifting Simulation Fresnel diffraction Fraunhofer diffraction

a b s t r a c t Using Laser-based Speckle-Interferometers, the shape of optically rough surfaces can be measured precisely and contactlessly from variable measuring distances even in regions of difficult access. This work is concerned with the integration of a micromirror array (MMA) into an electronic Speckle-Pattern-Interferometer. With the adaptive optics, it is intended to adapt the phasefront of a reference wave to critical surface areas of the measurement object. Yet, due to the topography of the MMA, diffraction effects occur which affect the phase and intensity of the generated wavefront. We demonstrate how these diffraction effects can be efficiently modelled by a Fraunhofer diffraction method. We compare the results of this model to theoretical data obtained by a numerical Fresnel diffraction model and to measurement data obtained from a measurement setup incorporating a multi mirror array. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Optical measurement techniques, in particular measurements based on speckle interferometry, are becoming increasingly important in the area of industrial quality control. Among the optical and contact-free measuring approaches, speckle interferometry offers significant advantages over white-light interferometry and triangulation method. Shape measurement based on speckle interferometry and two-wavelength techniques combined with phase shifting [1] allows for a variable measuring distance as well as an identical illumination and observation path. It is possible to adapt the diameter of the illuminating laser beam to the diameter of the measurement surface area. The complete measurement area can thus be captured by a single camera image. As sampling is not required, measurement time is exceptionally short. The analysis of shape measurement is carried out through so-called phase images, which show the surface structure of measurement objects in the form of fringes. These fringes are similar to the contour lines of a map. Up to now, shape measurement based on electronic speckle pattern interferometry (ESPI), two-wavelength-technique including phase shifting reached its limits when surface areas were smooth enough such that fringes too narrow to be resolved by a camera limited by finite pixel size. In order to overcome such restrictions, an electronic speckle pattern interferometer for shape measurement [2] was designed in such a way that the phase front of the reference wave could be variably configured. ⁎ Corresponding author. Tel.: + 49 89 289 23357; fax: + 49 89 289 23348. E-mail address: [email protected] (N. Werth). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.11.049

Today, there are already many fields which make use of spatial phase modulation. The compensation of light oscillation during the observation of stars which is caused by turbulence in the atmosphere, for example, is noteworthy [3]. In the area of life science, the laser beam phase can be manipulated in order to correct amblyopia or defective vision [4]. These phase differences can be generated in different ways. One possibility is by changing the optical light path. This is performed by varying the refractive index of a refractive medium using liquid crystal displays [5]. Another possibility is a variation of the geometric light path by means of deformable mirrors. Two classes of phase shifting devices exist: phase modulators which continuously influence the illuminating wavefront, and phase modulators which discretely manipulate the wavefront at a local point. The first class of phase modulators comprises a continuously deformable mirror (OKO Technologies) [6], whereby electrodes deform a coated membrane. Although such mirrors have the advantage of continuously influencing the wavefront, they also have considerable disadvantages. Due to the physical properties of a homogeneous mirror surface, the surface cannot be deformed at as many support points as might be required. Only a limited number of actuators which have a certain spatial distance from each other can be used. Each actuator is in turn influenced by neighboring actuators. Because of this, the range of an individual and flexibly configurable local phase modulation is limited. The second kind of technical device is known as a micromirror array. This system operates on the basis of piezo actuators [7]. These are essentially tilting micromirrors and micromirrors with a pistontype device. Texas Instruments has developed a device which allows

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for up to a ± 10∘ tilt angle, and which has a mirror spacing of only 14 μm [8]. This Digital Micromirror Device (DMD) is used in highresolution projection displays as well as in lithography [9]. If the phase modulator is illuminated at normal incidence with a parallel laser beam, it is more convenient to use a mirror array with a piston-type device to obtain a clear local spatial mapping between the active and reactive area. This can be implemented in different ways, e.g. through electrostatic, magnetic, thermal or piezoelectric procedures [10]. With regard to a given task, there are certain requirements before a micromirror array can be employed. A continuous, deformable mirror is not applicable because of its insufficient lateral resolution. It is desirable to obtain an array which consists of many small mirror elements, which are separately controlled and which can be deflected parallel to the optical axis. A higher resolution, a higher optical fill factor and a resulting high reflectivity, as well as a local and global evenness of the mirror guarantee high measurement results. For the above-mentioned reasons a micromirror array available from the Fraunhofer Institute for Photonic Microsystems (IPMS) was selected for phase modulation [11,12,16]. This quadratic array consists of 48, 000 individual mirror elements, each with an edge length of 40 μm. The detailed geometric layout is illustrated in Section 2. Due to the spatial configuration of the micromirror array, there are considerable diffraction effects during the reflection of the laser wavefront on its surface. These diffraction effects influence both the phase and the intensity of the wavefront. As the phase of the illuminating light can be specifically, locally corrected in the following measurement application, the resulting diffraction during reflection at the array, due to its geometrical configuration, has the effect of a disturbance variable. This paper therefore presents the modelling of the diffraction effects of the micromirror array. For this research, a simulation tool which calculates the diffracted amplitude of the electric field in the plane of the camera was developed. With this tool, it is now possible to model the phase and intensity of the reflected wavefront, according to the deflection of the individual mirror elements. The reflective properties of the micromirror array will be analysed in a precise manner. At the same time, this research also allows for discussion about the feasibility and success of the device. One of the aims of this paper is to calculate the field distribution with high efficiency and without requiring excessively high computing power. Accordingly, the precise but extraordinarily time-consuming model of Fresnel diffraction is simplified in order to make calculation easier. Through comparison of both the Fresnel diffraction model and a refined Fraunhofer diffraction model with the measurement results, the simplifications are shown to be permissible and legitimate. To check for validity of the model, the simulation is compared with real measurement results. This is done by comparing the intensity images of the model with the intensity images of camera shots from the micromirror reflections. In the following, Section 2 gives an overview of the experimental setup, including a detailed description of the surface of the micromirror array as well as a description of the measuring system. Section 3 discusses the theoretical foundations of both simulation models. A comparison of both models as well as a comparison of measurement results and simulated pictures is given in Section 4. A final evaluation as well as an outlook regarding the benefits of the theoretical expertise, in terms of the reflection characteristics of the micromirror array, is presented in Section 5.

control each individual pixel. To protect against external influences, a 3 mm thick glass sheet is covering the whole arrangement. The micromirror array consists of 48, 000 quadratic, piston-type (phase only) mirror elements in 200 rows and 240 columns. The size of each element is 40 μm × 40 μm, which results in a total mirror surface area of 8.0 mm × 9.6 mm. The height of all mirror elements can be independently configured. A small part of the mirror array is illustrated in Fig. 1. Each individual mirror element has a maximum deflection of 400 nm and can be positioned with an 8 bit resolution in step-by-step height with steps of no less than 1.6 nm. Even with a perpendicular illumination, it is essentially possible to shift the phase of a laser beam of maximum 800 nm by 2π [13]. δ=

2π 2π Δ= λ λ

ð1Þ

An individual mirror has a flatness of λ/25 (RMS) at an illuminating wavelength of 633 nm. The flatness of the total mirror is λ/5 (RMS) when the wavelength is likewise 633 nm. The optical fill factor lies around 80%, the reflectivity is 85% when the wavelength is between 400 nm and up to 800 nm. In Fig. 2, the schematic setup of a single mirror element is illustrated. Each mirror pixel is connected to 4 support posts by 4 brackets [14]. The width of each bracket is 2 μm; the distance between each bracket and the mirror surface is 1 μm. Thus, the distance between the actual mirror surfaces is 4 μm. The holes which result from the etching process have an edge length of 1.2 μm. The entire mirror element, brackets and support posts are made of aluminium, fabricated by means of surface micromachining [16]. If the mirror surface is in undeflected condition, the distance between the mirror element and the counter electrode is 2.1 μm. A cross-section of this is seen in Fig. 3. The deflection of the mirror works on the principle of electrostatic attraction [15]. The mirror element and the counter electrode function as a plate condenser. When a voltage is applied to the counter electrode, there is a deflection of the mirror element. Oxide stoppers were introduced to prevent electrical shortening and sticking of the mirror elements in the fully deflected state [16]. To analyze the influence of diffraction effects on the intensity distribution of the image more accurately, the micromirror array was integrated into the measurement setup, as can be seen in Fig. 4. In order to compare the attained measurement results with the measurements from the speckle interferometer, identical distances and positions of the micromirror array relative to the camera were chosen. The mirror surface was illuminated perpendicularly with an

2. Experimental setup The micromirror array used is made up of a mirror coating and an underlying integrated CMOS-electronic component, which is used to

Fig. 1. Part of the micromirror array.

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brackets

Fig. 4. Measurement setup.

For a quadratic pixel aperture a(x, y) = rect(x, y) the integral in Eq. (2) becomes

laser beam with a wavelength of 488 nm. The reflection was likewise perpendicularly observed with a camera. The distance between the laser beam and the beam splitter as well as the distance between beam splitter and micromirror array was 14 cm. The camera was positioned 20 cm away from the beam splitter.

3. Theory We are considering the diffraction of monochromatic light from the MMA array. We disregard polarization effects, such that the scalar wave equation can be used to describe the diffraction problem. To obtain the field distribution at an image plane at a distance R, the Fresnel diffraction integrals can thus be employed [17–19]. In the far field limit, the diffraction pattern of an individual MMA pixel may be obtained from the Fraunhofer diffraction ∞

EðX; Y Þ =

1 iðωt−kRÞ ikðXx ⋅e ⋅ ∫∫aðx; yÞ⋅e R −∞

+ YyÞ = R

dxdy;

ð2Þ

EðX; Y Þ =

1 iðωt−kRÞ 2 ⋅e ⋅d R

sin

    kdX kdY sin 2R 2R ; kdX kdY 2R 2R

ð3Þ

with kdX/2R = α being the diffraction angle. For the Fraunhofer diffraction it is assumed that the phasefront is plane, as expressed by the constant term exp(i(ωt − kR)). Setting the image plane closer to the diffracting aperture, the phase fronts become more and more bent, while the approximate intensity hull expressed by the sin(α)/α term changes little (the zero positions remain at the same angle).

1 0.8

| E(X,0) | / a.u.

Fig. 2. Schematic setup of a single mirror element.

Fresnel our method

0.6 0.4 0.2 0

−0.01

−0.005

0

0.005

0.01

position / m 0.3 0.2

Re(E(X,0)) / a.u.

where R denotes the distance from the pixel center to the plane of observation, ω is the angular frequency, k = 2π/λ is the wave vector with λ being the wavelength, a(x, y) is the aperture function of the MMA pixel with the aperture coordinates x and y, X and Y being the coordinates in the plane of observation. Eq. (2) represents a good approximation for the actual observed field when the Fresnel number F = d2/(Rλ) ≪ 1, with d being the diameter of the pixel. For the distances in the setup of Fig. 4 the Fresnel number is approximately 10− 2, which is not very low.

Fresnel our method

0.1 0 −0.1 −0.2 −2.8

−2.7

−2.6

−2.5

position / m

Fig. 3. Cross-section micromirror element.

−2.4

−2.3

−2.2 −3

x 10

Fig. 5. Comparison of the image field distributions of a single MMA pixel as found by computing the Fresnel integral and applying the presented method. Top: Absolute value of the field along the Y = 0 direction, Bottom: closeup of the real part of the field.

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1

Hence, the diffraction pattern of the MMA at time t = 0 is described by

| E(X,0) | / a.u.

0.8

N

Fresnel

−ikRðX;Y;x−xi ;y−yi Þ

⋅     kdðX−xi Þ kdðY−yi Þ sin sin d2 2R 2R ; ⋅ ⋅ ⋅ kdðX−xi Þ kdðY−yi Þ R 2R 2R i=1

our method

0.6

−i2kΔzi

EðX; Y Þ = ∑ e

0.4

⋅e

ð4Þ

0.2 0 −5

0

5 −3

position / m

x 10 Fresnel

0.3

Re(E(X,0)) / a.u.

our method 0.2 0.1 0 −0.1 −0.2 −0.3 −2.5

−2

−1.5

−1

position / m

−0.5

0 −4

x 10

Fig. 6. Comparison of the image field distributions of four MMA pixels in the corner of the real MMA. Top: Absolute value of the field along the Y = 0 direction, Bottom: closeup of the real part of the field.

We use this property to refine the approximation of Eq. (3) for an image plane closer to the diffraction plane. We introduce a positiondependent distance R(X, Y, x, y) in the phase term exp(i(ωt − kR)). This way, the phase front bending is accounted for, while neglecting the hull shape change of the intensity pattern. For modelling the complete micromirror array we have to consider a set of single pixels, all contributing to the diffraction pattern. All pixels lie in one plane, and their deflection does not exceed 0.4 μm. Their lateral dimensions are approximately 100 times that of the deflection in z-direction. We therefore model the complete diffraction plane by summarizing the diffraction field generated by the individual pixels. For the model we assume illumination with a plane wave with perpendicular angle of incidence. This way, every pixel with the same deflection shares the same phase of illuminating light. If a pixel is deflected by an amount Δz, the phase of the illuminating light will change by exp(ikΔz). Considering the plane of diffraction at the zero position of the MMA pixel, the field to be taken for computing the diffraction pattern possesses twice the phase factor, since the path has to be traveled back and forth.

where i counts over the number of pixels N, xi and yi are the center positions of the i-th pixel and Δzi is its deflection. The MMA device features a cover plate for protection of the micromechanics. This cover plate is made from a transparent dielectric material, forming a Fabry–Perot resonator. Since the glass plate is not perfectly aligned with the diffraction plane of the MMA pixels, stripes of higher and lower intensity are induced in the image plane. We model this effect by calculating the intensity modulation for a tilted cover plate. Eq. (2) represents a good approximation for the actual observed field if the Fresnel number F = d2/(Rλ) ≪ 1, with d being the diameter of the pixel. For the distances in the setup of Fig. 4 the Fresnel number is approximately 10− 2, which represents an intermediate value between very high and very low. The computation for solving the more precise Fresnel diffraction is considerably longer (several days compared to several minutes). Therefore it is preferable to be able to use the described method to obtain the diffraction pattern instead of the Fresnel integral. Hence, the applicability of the described method will be tested thoroughly against the results of the Fresnel formulation. 4. Results 4.1. Testing the Fraunhofer diffraction method As a test case for checking the validity of the diffraction modelling we compute the diffraction from two different aperture models. The first is that of a single MMA pixel. The field distributions are calculated at a distance of 10 cm which is the closest that could be realized in our experiment. For this distance the highest deviations of the two methods are to be expected, since the Fresnel number increases with decreasing distance. Fig. 5 illustrates the results of the single pixel calculation. On the top, the absolute value of the field is shown. Both curves correlate closely. The figure on the bottom shows a closeup of the real part of the field. The high frequency sinusoidal modulation of the real part is visible. Again both methods are in agreement. The agreement of both methods for a single MMA pixel is to be expected, as the Fresnel number of the small aperture of a single pixel is very low. If we increase the total aperture, deviations are expected to increase. Therefore we construct a second model, with four MMA pixels at the outermost corners of the MMA (− 4.78 mm; 3.98 mm), (4.78 mm; 3.98 mm), (4.78 mm; − 3.98 mm) and (− 4.78 mm; − 3.98 mm). Again, the diffraction patterns obtained by both

Fig. 7. Spherical deflection of micromirror surface (left); measurement result (center); simulation with cover glass (right).

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Fig. 8. Rectangular deflection of the micromirror surface (left); measurement result (center); simulation without cover glass (right).

methods are compared in Fig. 6. The top pictures the absolute value complex diffraction pattern. The figure on the bottom confirms the close agreement of both methods in this case, too. We performed several other comparisons, changing the distance to the MMA, all yielding the same close agreement. We therefore conclude that the method is applicable to the problem of finding the diffraction pattern of the MMA for the described application, allowing for a fast computation of the field distribution. Continuing, we will use exclusively the method described in this work to compute the diffracted fields. 4.2. Comparison of simulation results and measurement results The measurement results are based on two examples. The result, which is seen in Fig. 7 (center), was obtained through the following procedure. A rounded surface area with a radius of 48 mirror pixels was deflected in the middle of the micromirror array. This micromirror array setup is illustrated in Fig. 4. Fig. 7 (left) shows the pattern which was produced on the mirror surface. The total surface area corresponds to the entire micromirror surface area with its 240 × 200 mirror elements. The black area corresponds to undeflected mirror elements. In the internal white area, with a radius of 48 mirror pixels, all mirror elements were deflected by 400 nm. Fig. 7 (center) shows an intensity image of the camera. The ringshaped pattern can be traced back to the intensity modulation caused by diffraction effects on the micromirror. A second model of a micromirror can be seen in Fig. 8 (left). Here, 32 × 32 quadratic mirror elements were left in the center of the array in undeflected condition. The rest of the mirror elements were attracted to the counter electrode by a maximum of 400 nm. The intensity image in Fig. 8 (center) shows the resulting diffraction phenomena. Fig. 8 (right) shows the results of the simulation. On closer examination of the intensity images, modulations using higher frequency result in a slight tilt across the image. These modulations are a result of the safety quartz glass, which works as a Fabry–Perot-Interferometer. This effect was likewise considered as an additional disturbance variable during the setup. The simulated

intensity image in Fig. 7 (right) takes into account the Fabry–Perot effect, so that the higher frequency modulations are visible. In order to show clearly this discrepancy, the model of the cover glass was left out in the simulation. This can be seen in Fig. 8 (right). 4.3. Comparison of detailed modelling and rough modelling The geometric surface structure of the micromirror array was modelled in detail. The entire surface of the micromirror was modelled. This includes models of the brackets, support posts, the holes resulting from the etching process as well as the distance between the brackets (see Fig. 2). The effect of this accuracy to detail on the results of the simulation can be seen by a more precise observation of the camera intensity image in Fig. 7 (center). Fig. 9 (left) shows a part of the image in Fig. 7 (center). A kind of granular structure may be observed with a periodic modulation of the intensity distribution in both spatial directions. Fig. 9 (center) shows the corresponding detailed image of the simulation from Fig. 7 (right). By using the detailed modelling of the micromirror surface, the same kind of granular structure can also be found in the image. Fig. 9 (right) shows a different setup. Unlike the above mentioned structure, accuracy to detail was dispensed with in Fig. 9 (right) while generating the image. Instead, a rough model was chosen. The foundation was a simulation which is based on a simplified version of the micromirror surface. The brackets, support posts, the holes which result from the etching process as well as the distance between the individual mirror elements were ignored and not modelled. An element was modelled only as a homogeneous mirror surface area with an edge length of 40 μm by 40 μm. The result is that the granular structure is no longer visible, and that the minute details disappear. 5. Conclusion Within this study, the presented approach in terms of modelling the diffraction effects of a micromirror array by means of a Fraunhofer diffraction model shows an excellent agreement with both, the

Fig. 9. Measurement result, granular structure visible (left); simulation, detailed modelling, granular structure visible (center); simulation, rough modelling, granular structure invisible (right).

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Fresnel diffraction model, which represent a highly precise model and the measurement results. A simulation tool was developed to calculate the complex field distribution with high numerical efficiency. Due to the approach of modelling, the calculation can be executed on a conventional personal computer. Additionally the detail of the calculated diffraction data can be enhanced by increasing the degree of detail in the modelling. The additional details have to be paid by an increased duration of computation. By successively adding more detail to the MMA model, individual artefacts in the measured diffraction image may be correlated with certain elements of the MMA. This allows the refinement of the MMA design by identification of the strongest interfering elements and their elimination. References [1] J.C. Wyant, http://www.optics.arizona.edu/jcwyant/Optics513/ChapterNotes/ Chapter05/PrintedVersionPhaseShiftingInterferometry.pdf8Online Resource. [2] A. Purde, N. Werth, A. Meixner, A.W. Koch, SPIE — Optical Measurement Systems for Industrial Inspection IV, 2005, p. 5856. [3] H.W. Babcock, Publications of the Astronomical Society of the Pacific 65 (386) (1953) 229.

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