2.3 Cantilever Bending Method for Measurement of Changes in Surface …
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Fig. 2.9 Scheme for deriving the relationship between the total deflection angle α and the curvature
radius R or curvature κ for the cantilever bending [1, 24]. The incident light is parallel to the mirror
axis. Modified from [24], Copyright 2010, with permission from AIP Publishing
2.3.2 Optical Detection of Curvature
Popular method for measuring the curvature of cantilever plate is a laser beam optical
detection, which is based on the reflection of the incident light at the cantilever surface
(as a mirror). Figure 2.9 shows the scheme for deriving the relationship between the
deflection angle α and the curvature radius R or curvature κ for bending of the
cantilever [1, 24]. The incident light is reflected at a point M on the segment DE
(depicted as a thick solid curve in Fig. 2.9) of a convex spherical or cylindrical
(mirror) surface, which corresponds to the cantilever. The angle of incidence 2θ is
measured from a normal line drawn to the curved surface at the reflection point M .
The angle of reflection is equal to the angle of incidence, since the law of reflection
holds at each point of the curved surface. Consequently, the total angle α of deflection
of the light is
α = 4θ.
(2.15)
The normal line drawn to the curved surface at M intersects at a point C (i.e.,
center of the curvature) with a normal line (i.e., mirror (principal) axis) drawn to the
same curved surface at a point D. The mirror axis is parallel to the incident light. The
length MC or DC corresponds to the curvature radius R (reciprocal of the curvature
κ). The bending of the cantilever within elastic limit is very small, i.e., 2θ ≈ 0, so
that the distance L defined as a length DF in Fig. 2.9 can be derived [1, 24]:
51
Fig. 2.9 Scheme for deriving the relationship between the total deflection angle α and the curvature
radius R or curvature κ for the cantilever bending [1, 24]. The incident light is parallel to the mirror
axis. Modified from [24], Copyright 2010, with permission from AIP Publishing
2.3.2 Optical Detection of Curvature
Popular method for measuring the curvature of cantilever plate is a laser beam optical
detection, which is based on the reflection of the incident light at the cantilever surface
(as a mirror). Figure 2.9 shows the scheme for deriving the relationship between the
deflection angle α and the curvature radius R or curvature κ for bending of the
cantilever [1, 24]. The incident light is reflected at a point M on the segment DE
(depicted as a thick solid curve in Fig. 2.9) of a convex spherical or cylindrical
(mirror) surface, which corresponds to the cantilever. The angle of incidence 2θ is
measured from a normal line drawn to the curved surface at the reflection point M .
The angle of reflection is equal to the angle of incidence, since the law of reflection
holds at each point of the curved surface. Consequently, the total angle α of deflection
of the light is
α = 4θ.
(2.15)
The normal line drawn to the curved surface at M intersects at a point C (i.e.,
center of the curvature) with a normal line (i.e., mirror (principal) axis) drawn to the
same curved surface at a point D. The mirror axis is parallel to the incident light. The
length MC or DC corresponds to the curvature radius R (reciprocal of the curvature
κ). The bending of the cantilever within elastic limit is very small, i.e., 2θ ≈ 0, so
that the distance L defined as a length DF in Fig. 2.9 can be derived [1, 24]: