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I would like to use laser beam deflection to measure a small movement (100s nm) in a cantilever system as shown in the diagram, but I am unsure of the calculations. This may be something simple to do and I am making it more complicated in my head.

I would like to be able to calculate how the movement of the laser point on the detector/screen (PSD in the diagram) relates to the displacements I expect of the cantilever. So I can work out a rough set up and to where to place the equipment to achieve sufficient sensitivity.

Any advice or help would be appreciated, whether calculations or setup as this is not an area I am not familiar with.

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  • $\begingroup$ Is your cantilever such that you are guaranteed a nice specular reflection? That might complicate the actual experiment greatly. $\endgroup$ – Jon Custer Aug 17 '18 at 14:47
  • $\begingroup$ My cantilever should have a mirror finish across the entire length. $\endgroup$ – Dave Aug 26 '18 at 12:11
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The change in $D$ will be negligible (unless the angle of incidence is very small) because the deflection is very small. The angle of incidence of the light onto the cantilever mirror equals the angle of reflection. It appears that your cantilever is tilted relative to the laser beam and the screen, so you need to do your trigonometry in 3D. In your calculation, just assume that D does not change and that the mirror tilts by a small angle. However, the angle of tilt as a function of the distance the cantilever is deflected actually will depend on the structure of the cantilever. If I were handed this problem as a homework assignment, I would ask for more specifics about the cantilever -- or would state in my answer that I'm assuming a specific structure (e.g., a uniform rectangular cross section "diving board").

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  • $\begingroup$ This was just a diagram I found for the type of set up, if my cantilever is not tilted relative to the laser bean and the screen, I can reduce the problem to 2D? The structure of the cantilever will be a uniform rectangular cross section, 1 mm in width, 5 mm in length with some backing to clamp it in place. I am hoping to implement a simple set up and calculation. I am having trouble visualising how the change in angle relates to the beam's deflection. $\endgroup$ – Dave Aug 26 '18 at 12:09
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    $\begingroup$ When possible, it's a good idea to do a physical experiment to build your intuition. Yes, by aligning the cantilever and the laser you can reduce the problem to 2D. Try building a large-scale model. You could use a laser pointer and a microscope slide (light will reflect off the unmirrored surface of the microscope slide when incident at an angle). Cantilever calculations are found here: [web.ncyu.edu.tw/~lanjc/lesson/C3/class/Chap05-A.pdf] and here [engineersedge.com/beam_bending/beam_bending9.htm] and here [me.berkeley.edu/~lwlin/me128/BeamDeflection.pdf] . $\endgroup$ – S. McGrew Aug 26 '18 at 12:54

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