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Atomic force microscope cantilevers as encoders for real-time displacement measurements


Cantilevers of Atomic Force Microscopes (AFM) allow high resolution imaging of surfaces with their fine tip. The PTB has investigated the use of pairs of cantilevers for encoding displacement by using a well-defined optical grating to achieve high-resolution positioning.

If one AFM cantilever is used as encoder when paired with 1D sinusoidal/rectangular/trapezoidal grating with the pitch of P (nm) as the reference, the decoding principle is based on direct counting of integer periods I plus calculation of two fractional parts of such periods fs standing at the beginning and fe at the actual position in the encoded signal corresponding to a given path of displacement s [1].

                         S = [fs + l + fe] ⋅ P (nm)                                 (1)

Due to the noise and small deviation of single points on the artifact a filter was necessary to ensure accurate implementation of this decoding process. A cross-correlation technique has been employed to filter 1D grating encoded signal Y(n) in real time: A half sinusoidal waveform template is found very efficient and correct to filter any 1D waveform grating encoded signal by cross-correlating with it [2]. The real-time filtered signal R(n) as well as Y(n) are shown in figure 1(a), the corresponding real-time displacement is recorded in figure 1(b):


Figure 1: (a) Real-time 1D sinusoidal grating position encoded signal Y(n) scanned by one tuning-fork cantilever, and its cross-correlation signal R(n). (b) Real-time displacement decoding curve.

The encoder has been used to measure forward or backward displacement and to monitor the scanning stage, during the motion, to control its displacement. The decoding method can also be used for the measurement of in-plane rotation angle between1D grating orientation and moving direction of micro-moving stage within 90° range.

In the case of using two AFM cantilevers to encode a 1D sinusoidal grating pattern with the pitch of P (nm) as the reference, the distance between two cantilever tips has to be preset in such a way that the two 1D sinusoidal grating position-encoded signals Y1(n) and Y2(n) have a quadrature phase (90°) shift.
Typical results are plotted in figure 2(a) and expressed as Y1(n) = A2 ⋅ sin(P/2π ⋅ S(n)) and Y2(n) = A2 ⋅ cos(P/2π ⋅ S(n)).

By directly unwrapping the phase between two encoded signals Φ(n) = arctan[A2 ⋅ Y1(n)) / (A2 ⋅ Y2(n))], forward and backward displacements in the direction cross the 1D grating lines S(n) = [P / (2π)] ⋅ Φ(n) can be detected and measured in real time. Additional differentiation of the signals is necessary to remove effects due to tilt and offset [3]. The two encoded signals Y1(n) and Y2(n) are plotted in figure 2(a), the differentiations of the signals are shown in figure 2(b), the phase waveform is shown in figure 2(c) and unwrapped real-time forward and backward displacement is shown in figure 2(d).





Figure 2: (a) Two encoded signals Y1(n) and Y2(n) with approximate π/2 phase shift, (b) the differentiation of the signals D1(n) and D2(n), (c)  Phase waveform and (d) unwrapped real-time forward and backward displacement.

In the work a 1D grating of 300 nm pitch has been used as a first step, but gratings down to 80 nm have been investigated, too. A future goal of such AFM cantilever based encoders would be the use of atomic gratings of crystalline surfaces as base.

[1]  Chen X, Koenders L, Wolff H, Haertig F and Schilling M. Atomic force microscope cantilever as an encoding sensor for real-time displacement measurement Meas. Sci. Technol. 2010, 21  105205.  http://stacks.iop.org/0957-0233/21/105205      

[2]  Chen X, Koenders L and Haertig F. Real-time cross-correlation filtering of 1D grating position-encoded signal, Meas. Sci. Technol. 2011, 22  085105.  http://stacks.iop.org/0957-0233/22/085105

[3]  Chen X, Koenders L, Wolff H, Neddermeyer H and Haertig F. Atomic force microscope cantilevers as encoder for real time forward and backward displacement measurements, Meas. Sci. Technol. 2011, 22 094017 . http://stacks.iop.org/0957-0233/22/094017 

[4]  Chen, X., Thesis “Atomic Force Microscope (AFM) Cantilevers as Encoder for Real-Time Displacement Measurements”, Fakultät für Elektrotechnik und Informationstechnik der Technischen Universität Carolo-Wilhelmina zu Braunschweig, Nov. 2011



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