S18_Speckle Interferometry (w/ digital camera) Observing Vibrational Nodes of Metal Plate
Introduction
The "speckle effect" is an optical interference phenomenon resulting from random scattering of laser light from a rough surface. The reflected light from the surface
appears as areas of high intensity in some locations due to constructive interference, and areas of low intensity appear due to destructive interference. These spots
of high and low intensity are spread randomly across the object's surface, creating a speckled pattern. An example of a speckle pattern is shown in Figure 1.
Figure 1: A speckle pattern on a rough, aluminum plate from this experiment.
Although speckle patterns are random, they can be utilized to measure small deformations in an object through the technique of speckle interferometry.
This technique involves splitting a coherent laser beam into a reference beam and an image beam. The latter is diverged onto an object to illuminate it with a speckle
pattern. The two beams are then recombined into the lens of a digital camera, which records the pattern. This process is then repeated on the same object after it is
given a deformation on the order of microns or smaller, which creates a new pattern with speckles in different locations for the same object. These two images are then
subtracted pixel-by-pixel, and the resulting fringe pattern may be used to study the deformation which occurred (Figure 2). In this experiment, a speckle interferometer
was built and used to analyze micron-order bend of a metal plate, as well as determine the vibrational patterns of a rectangular plate with free edges.
Figure 2: Two speckle patterns of an approximately 200 mm by 200 mm by 8 mm aluminum plate before
and after is is bent down the center by several microns are subtracted pixel-by-pixel, revealing a fringe pattern.
Theory
1. The Speckle Effect
Speckle patterns arise from path length differences acquired by initially coherent light rays incident on a rough surface. Consider two initially in-phase light rays of wavelength λ heading towards a rough object in Figure 3.
Figure 3: Two initially in-phase light rays incident on a rough surface.
The bottom light ray acquires a path length difference of +2d relative to the top due to the roughness of the surface. When these rays end up on a screen, they interfere destructively when
This is just the usual condition for interference. Of course, the light can also add constructively, or somewhere in between, so this results in a randomly distributed pattern of bright and dark speckles across the surface, like in Figure 1. This is the origin of the speckle effect.
2. Speckle Interferometry of a Bent Plate
Figure 4: A metal plate which is bent by an amount D in its center.
Consider a light ray with wavelength λ incident on a metal plate before and after it is bent into the shape of a triangle by an amount D in the center (Figure 4). The light ray that reflects off of the bent plate has the same phase as the light ray which reflected off of the unbent plate the displacement from the unbent plate d at that point is
where n is an integer. Thus, if we make a speckle pattern on the plate, we expect the speckle patterns between the bent and unbent plate to be the same at locations x where this relation is satisfied. If the distance form the plate edge to the center, where it is bent, is L, trigonometry allows us to express this condition as
Rearrange this as
Thus, after bending a plate by an amount D in the center and subtracting the speckle patterns from before and after, there will be dark fringes in the subtracted image with spacing
3. Vibrational Modes
A vibrating plate in the x-y plane is described by the following time-dependent equation:
Here w represents the displacement of the plate. Additionally, p and D are constants defined as the density and the "flexural rigidity" respectively. The flexural rigidity is determined by the material itself using
where E is Young's Modulus, h is the thickness of the plate, and v is Poisson's ratio. To solve this differential equation it is easiest to first make the displacement, w, time-independent. Then, using the boundary conditions for the free plate, represent w as a double cosine Fourier series as such
noting that a and b represent the width and the height of the plate respectively. Using this form of w and inserting it into the differential equation above yields a set of solutions for the natural frequencies that allow the complete set to equal zero as desired
Solving this for the natural frequency quickly gives multiple solutions as a function of only m and n as all other quantities are constants dependent on the plate itself. These values of m and n give the vibrational modes of the plate as such
and were used to compare results to ones found in literature.
Experimentation
A speckle interferometer was constructed to ultimately observe changes in speckle patterns resulting from small bends of a metal plate, and vibrational patterns of a rectangular, aluminum plate. A diagram of this interferometer is shown in Figure 5. A 632.8 nm wavelength HeNe laser is used as a source. This beam is split by a non-polarizing beam splitter into a reference beam, and an image beam which ends up on the metal plate. The image beam is spread onto the surface of the plate with a combination of a microscope objective lens and a diverging lens. The image beam is made less intense with a neutral-density filter before going through a diverging lens, which spreads the light onto a piece of diffuse glass, which serves to create a speckle pattern for the reference beam. The reference beam and image beam then are combined in another beam splitter, where they interfere. This light then is recorded by a Canon EOS Rebel XTI digital camera, which is operated from a computer. The camera settings used were 0.125 seconds for the exposure time, f/5.0 for the f-stop setting, and an IS0 of 400.
Figure 5: The speckle interferometer built in this experiment.
For plate bending, a large metal plate with a central grove was mounted between two posts and held loosely by one screw near its central groove. The plate was then displaced by turning a knob on the translational stage that the screw was attached to and the displacement was measured with the Ames 311 dial gauge with a graduation of 1/10,000". This was done for varying increments on the micron scale, and the images were subtracted from the unaltered plate using MATLAB.
Figure 6: The physical set up used for plate bending data taking.
A similar process was used for vibrating the metal plate. The plate was swapped for a thinner aluminum plate with a hole punched in the center. The plate was then held loosely by a screw through this hole and attached to a horizontal post. The plate was made to vibrate using two speakers connected via BNC cables to a function generator operating at a single frequency for each trial. Two images were taken of the plate vibrating with a longer exposure time (2s) on the camera and these images were then subtracted from each other to reveal the vibrational pattern.
Results
After image subtraction in MATLAB, the perspective of the resulting images were then corrected using MATLAB so that the plate appeared as rectangles. These corrected images were then enhanced using LightRoom and finally analyzed in ImageJ.
1. Plate Bending
For plate bending, a picture of a ruler was used to calibrate the size of the pixels in ImageJ. Doing this allowed for direct calculation of the fringe spacing for each given deformation amount. The images were rotated until the observed fringes were vertical, and then a small horizontal segment along the center of the image was sampled to analyze the intensity histogram as a function of length.
Figure 7: An example of the intensity histogram as a function of distance for
one of the trials for plate bending. The minima were used to determine
where the fringes were located.
From this histogram, a plot of fringe number versus distance was formed for each trial. A slope was calculated from this to determine fringe spacing. Finally, a plot of fringe spacing versus inverse deformation was plotted and compared to the prediction from above. As discussed in the theory section, this data should follow a line.
Figure 8: A plot of fringe spacing versus inverse deformation distance.
Note the model seems to be a much better fit at large deformations.
The observed slope was (4.2 ± 0.3) x10-2 mm-2 which was three sigma away from the slope of 3.2 x10-2 mm-2 predicted by the model. This difference can most likely be attributed in the assumptions made in the model. It is a rough approximation assuming the incoming waves are plane waves, which is most likely not the case. However, the results agree on the same order of magnitude and at higher deformations verifying the ability of the interferometer.
2. Plate Vibrations
After image subtraction and enhancement, the resulting vibrational patterns observed were compared to a previous speckle interferometry experiment done by Huang [1]. The observed frequencies were matched to their closest values predicted by the equation above. Then the values for m and n were matched with their corresponding patterns found in literature. The patterns are shown below.
Figure 9: The observed vibrational pattern (left) versus the predicted vibrational
pattern (right) for 220Hz. This frequency corresponded to m=2 and n=1. The
predicted image is a linear combination of m=2 and n=1 with m=1 and n=2.
The dark spot in the middle is where the plate is bolted.
Figure 10: The observed vibrational pattern (left) versus the predicted vibrational
pattern (right) for 280Hz. This frequency corresponded to m=3 and n=1.
Figure 11: The observed vibrational pattern (left) versus the mathematically
predicted vibrational pattern (right) for 580Hz. This frequency corresponded to
m=4 and n=2.
The first image was observed at 220 ± 5 Hz and was predicted to be found at 219.2Hz. A complication arose mainly in this image because the plate was no assumed to be bolted in the center in the prediction equations. This adds an extra boundary condition when solving the differential equation, but this was accounted for analytically. This issue seemed to have negligible effect at higher modes as seen. The second image was observed at 280 ± 5 Hz and was predicted to be found at 278.9 Hz. Finally, the last image was observed at 580 ± Hz and was predicted to be found at 618.8 Hz.
Conclusion
This experiment can be seen as a verification of speckle interferometry based on the results for both plate deformations and vibrations.
In the future plates of different materials, shapes, and boundary conditions could be tested to better verify the theory. In particular, a circular plate would make for an interesting experiment as it is less difficult to account for the boundary condition caused by the center bolt. The equation for plate deformation due to vibrations would then take the form of polar coordinates, and still produce predictable frequencies for vibrational modes.
Additional images from this experiment may be found in the "Additional Images" subpage.
References
[1] Chi-Hung Huang and Chien-Ching Ma. Experimental measurement of mode shapes and frequencies for vibration of plates by optical interferometry method. Journal of Vibration and Acoustics, 123 (2):276-280, 200.