Measurement in Micro

Micro-electromechanical systems (MEMS) technology is poised to play an important role in next-generation astronomy instrumentation as, for example, micro-deformable mirrors for adaptive optics and programmable slit masks for multi-object spectroscopy. The characterization of such devices is a key issue in assessing the performance of the actual components and in providing reliable inputs for our models, leading to a better design and optimization of the architectures. Optical techniques provide a means for inspecting such devices.

Our group at the Laboratoire d'Astrophysique de Marseille (LAM; Marseille, France) is particularly involved in the design, realization, and characterization of micro-deformable mirrors, in collaboration with the Laboratoire d'Analyse et d'Architecture des Systèmes (LAAS; Toulouse, France), a micro-technology laboratory. We have developed a dedicated characterization bench for the complete analysis of building blocks and operational deformable mirrors. The micro- mirror characterization (MIMIC) system allows either a high in-plane resolution or a large field of view (FOV). Using phase-shifting interferometry (PSI), we can extract features such as optical quality or electromechanical behavior from these high precision, 3-D component maps. Two-wavelength PSI (TWPSI) increases the measurable vertical range; time- averaged interferometry provides dynamic data like vibration mode and cut-off frequency.

The MIMIC bench consists of a high-resolution Twyman-Green interferometer (see figure 1).¹ Laser sources introduce extraneous fringes because of their long coherence length, so we selected as the light source an incandescent lamp whose spectrum is controlled using a set of interference filters, for example, λ₀ = 650 nm and Δλ = 10 nm.

Figure 1. The interferometer is based on a Twyman-Green configuration. In the diagram, red lines represent the illumination beam and green lines represent the imagery conjugation. For phase-shifting interferometry, we translate the reference flat.

We chose a modular design for the instrument. A simple lens change allows us to switch between high in-plane resolution and large FOV, achieving, respectively, either very sharp (around 4 µm) resolution of the MEMS structure inside a small field (typically 1 mm), or a whole-device study at sizes of up to 40 mm with 100 µm resolution. In the high-resolution configuration, the instrument reaches diffraction-limited (N = 3) performance.

PSI involves moving the reference flat of the interferometer in fixed steps with a piezoelectric stage, which shifts the fringes, resulting in a series of interferograms.² Consider the intensity signal I_i for the i^th step

and where I₀ is the incident illumination, δ_i is the phase step, and M₀ the visibility of the fringes. The wavefront phase φ (x,y) is directly related to the component shape h (x,y), where λ₀ is the central wavelength.

Our system implements the PSI algorithm in which the reference mirror is displaced in five positions, δ_i, defined in equation 1.³ We can then calculate the phase and contrast as

Because the phase is only known between 0 and 2¼, the overall phase is unwrapped using a path-following algorithm. Phase unwrapping assumes that the wavefront is continuous; that is, the wavefront discontinuities are less than λ/4 between two adjacent pixels on the surface.

Error sources of PSI measurements such as phase-step error, air turbulence, vibration, or thermal drift are identified and minimized. The accuracy, limited by the optical quality of the reference mirror and the beamsplitter, depends on the observed FOV. In general, the smaller the FOV, the better the accuracy; for example, out-of-plane accuracy exceeds λ/10 in. for a 20-mm-diameter FOV. PSI coupled with a controlled environment can achieve an out-of-plane resolution of a few nanometers.

TWPSI provides improvements over basic PSI, increasing the measurable vertical range without compromising accuracy.⁴ If we subtract two phases measured independently with classical PSI at two wavelengths, we obtain a new equivalent phase corresponding to a longer equivalent wavelength λ_eq. Thus, the height-step condition for a continuous wavefront increases from λ_a/4 to λ_eq/4. The equivalent phase is then unwrapped to yield an intermediate height map without the λ_a/2 ambiguities. This intermediate result is less accurate than a single-wavelength PSI, however, and is used only for extracting the interference orders of the wrapped phases. The final result of the TWPSI consists of the wrapped phases at λ_a or λ_b adjusted with the interference orders.

Dynamic behavior such as vibration modes or cut-off frequency of the device under test can be analyzed by the evolution of contrast as the operating frequency increases. The movement of the device decreases fringe contrast because of the time-averaging effect introduced by the integration time of the CCD camera. In case of sinusoidal motion, we link fringe contrast to the motion amplitude a_ω for each pixel via a Bessel function.⁵ If we use equation 2 to calculate the contrast for a given frequency and the contrast of the fringe pattern at rest, the ratio of these two contrast maps leads to the Bessel term. The bijective part of the Bessel function yields the motion amplitude a_ω. The main drawback of this technique is its limited range of motion; nevertheless, the frequency limit of the measurement is only electrical and its implementation is easy.

A challenging part of designing such a bench is the fringe stabilization: We mounted the interferometer on a 3-m × 1.5-m damped table and enclosed it with a Plexiglas box. All the lenses are commercial 2-in. optics, the camera is a 1280 × 1024, 8-bit camera, and the piezoelectric stage operates in a closed loop with a capacitive sensor. These elements are driven by computers, network-linked for automating and optimizing the acquisition procedures. The recording and analyzing software is home-developed.

Figure 2. An interferogram (λ = 546 nm) of a 100 µm × 170 µm micro-steering mirror with 10-µm-wide torsion bars shows optical power. Fringes are resolved on the 4-µm-wide reflective coating on top of the torsion bar.

Chips designed by LAAS-LAM have been fabricated with the commercial multi-user MEMS processes. Structural layers are in polysilicon and sacrificial layers are in silicon oxide. A typical device on the chip consists of a steering mirror with a 170 µm × 100 µm plate with a 10-µm-wide, gold-coated torsion bar (see figure 2). Applying a voltage between the electrodes and the mobile plate actuates the mirror.

Using TWPSI at λ_a = 546 nm and λ_b = 600 nm, we captured a profile along the white line in figure 2 (see figure 3). We measured a 590 ± 20 nm step, revealing that the deposition of the bottom electrodes creates steps on the top layer by print-through. The data further reveals that the mirror has a radius of curvature of 3.0 ± 0.3 mm, induced by fabrication stresses. These results show that even for an optimized process, high curvature values are still present on the devices. Future developments must be made in order to minimize these fabrication effects.

Figure 3. A profile of the steering mirror plate (no voltage applied) obtained with TWPSI shows print-through step and curvature caused by stresses introduced during the fabrication process.

We used the system to characterize the device during actuation, producing a displacement map (the difference between the shape of the mirror with 19.0 V applied on the left electrode and the shape at rest). The motion is a rotation movement (15 arcmin) plus a small piston effect (100 nm) caused by low stiffness of the torsion bar. Deflection of the mirror with increase in driving voltage reveals that the actuation is mostly linear with the square of the applied voltage. As the gap between the mirror and the steady electrode decreases, however, the relationship becomes nonlinear.

To capture dynamic measurements, we excited the mirror sinusoidally and used time-averaged interferometry to assess motion behavior and scan the reasonance frequencies. The frequency response function of this device is close to a second-order system. We fitted a theoretical curve with ƒ_r = 31 kHz and Q = 0.94 to the measured data; changing the fitting parameters provides an error estimation of 3 kHz. The first vibration mode of this device is the rotation mode. If we discard the rotation mode by applying an identical voltage on both electrodes, a piston motion of the plate appears as a second resonant mode. A theoretical curve of a second-order system with ƒ_r = 63 kHz and Q = 0.8 has been fitted with an estimated error of 5 kHz.

In parallel, we have simulated this component using finite-element modeling; results on static shape, displacement map, and dynamic behavior fall within the error bars of the measurements. By modifying the geometry and the material, we can evaluate performances for new devices and optimize the design of building blocks as well as complex architectures.

In the framework of the next-generation adaptive optics research, we have inspected other components, including piston actuators in polysilicon and in polymer, and a home-designed micro-deformable mirror. This bench is not limited to the study of MEMS; larger components (up to 40 mm diameter) such as electrostatic or piezoelectric commercial deformable mirrors can be and have been characterized. By replacing the source by a laser, we can implement electronic speckle-pattern interferometry for observing deformation and vibration modes of nonreflective objects.

Several bench improvements are planned for increasing MIMIC's capabilities. For instance, white-light interferometry should enlarge the measurement range. Using stroboscopic illumination to freeze fringes during a periodic motion, dynamic behavior of the component could be analyzed without sign ambiguity amplitude limitation and with a better resolution.⁶ With such tools, we should be able to further advance our research. oe

1. A. Liotard et al., Proc. SPIE 5716, p. 207 (2005).

2. D. Malacara et al., Optical Shop Testing, Second Edition, John Wiley & Sons (1992).

3. P. Hariharan, Applied Optics 26, p. 2504 (1987).

4. K. Creath, Applied Optics 26, p. 2810 (1987).

5. S. Petitgrand et al., Proc. SPIE 4400, pg. 51 (2001).

6. M. Hart et al., Journal of Micro-electromechanical Systems, 9(4) (2000).