Mapping Nodes Using Speckle Pattern Interferometry

Alex Igl and Levi Schwartzberg

University of Minnesota - Twin Cities (Methods of Experimental Physics II)

Introduction

In this experiment, we built and tested a speckle pattern interferometer, with the end goal of using it to find the fundamental nodes of a vibraphone bar. Speckle interferometry, like other variations, uses phase differences to determine displacement. However, using speckle, we can observe large areas of an object instead of just a single point. Theoretically predicting where a node lies can be challenging if the object being observed isn't very simple. A portion of vibraphone bars are cut out underneath for tuning purposes, making the locations of the nodes difficult to derive.

Experimental Setup

Our setup is similar to a Michelson interferometer. The laser beam first travels through a filter, which allows us to control the intensity. The diverging lens widens the beam, allowing us to create the pattern. The iris shaves off the edges of the newly widened beam, which aids in the uniformity of the intensity. After arriving at the beamsplitter, the beam branches into two arms of equal intensity: the reference beam and the measuring beam. The reference beam bounces off of a plate that remains unchanged in distance or amount of deformation throughout a set of measurements, allowing it to be used as a comparison. The measuring beam reflects off of the object being observed, which is moving in some way to cause a phase difference in the light. The two beams are recombined using the same beamsplitter they were branched by (interfering with each other), and the beam is detected by a camera.

Figure 1: The experimental set up is shown above.

Theory

The speckle patterns themselves are the result of coherent, monochromatic light being reflected off of an uneven surface and projected onto a plane. The roughness of the surface causes individual "wavelets" to reflect at different angles, and thus, reach the plane at different phases due to the varying distances they must travel. The wavelets then interfere (either constructively or destructively), and create small areas of distinct intensity, or "speckles" [1]. An example speckle pattern can be seen in Figure 2.

Figure 2: A sample image of a speckle pattern.

To obtain a picture of the nodes or fringes, one needs two pictures: one of the object while it is undeformed and unmoving, and the other while it is undergoing some kind of displacement. The subtraction of these two images reveals the pattern. The subtraction can be modeled by a series of equations. The general laser intensity addition formula is given by

where I_b is the measuring arm intensity, I_r is the reference arm intensity, and ε is the phase. Moving the object by some distance causes a phase difference of

Adding this into the intensity addition equation gives the formula for the reference image,

If the object is vibrating with some frequency ω, the phase change equation is altered slightly:

Adding this value to the intensity addition equation and time averaging results in the following formula:

where J_o is the zero order Bessel function of the first kind. Subtracting I_ω from I₁ as we would in the image subtraction results in the final intensity formula:

Note: Derivations taken from [2].

Testing

To test the interferometer, we first used a plate identical to the one in the reference arm. By bending the plate, we hoped to see equally spaced fringes after image subtraction. We added a cut down the center of the plate so that it would bend in a more triangular fashion, making the geometry (and therefore our predictions) much more straightforward (Figure 3). We also used a translation stage and dial caliper combination so that we could measure the amount of bend in the plate. The plate was bent in increments of 5 x 10^-5 inches. The following pictures show how the number of fringes increased with plate bend:

Figure 3: A horizontal optical post attached to a translation stage was used to bend the metal plate.

Figure 4: When multiplied by some constant, the two subtracted images produce the interference fringes.

Figure 5: The above interference patterns reveal increased amount of fringes with increased plate bend.

Figure 6: The observed amount of fringes (per deformation length) closely matched the amount predicted by theory.

Results

By using time averaged image subtraction of a vibrating bar, the following image of the nodal lines for the fundamental mode was obtained. The bar was excited by striking it with a mallet, before the image was captured. Due to the relatively small size of the circular speckle pattern on the bar, the image below is actually a construction of several images taken across the span of the bar.

Figure 7: Upon striking the bar with a mallet, and using an increased exposure time for the image, the image subtraction produced the above image of the nodal lines.

According to the Euler-Bernoulli model for a uniform rectangular bar, the nodal positions for the fundamental mode, as a fraction of bar length, are given as follows [3]. The fractional nodal positions for the image above are given for comparison. Note that although the vibraphone bar was not totally uniform, the theoretical model for the uniform bar was evidently a good enough approximation.

Table 1: The observed nodal positions are relatively close to those predicted by theory for a uniform bar.

Conclusions

In the case of the bending plates, we have shown that the interferometry works, in that it gives us fringes based on displacing the object of interest. Additionally, we showed that the observed amount of fringes closely match those predicted by theory, further emphasizing the success of the interferometry. Lastly, we used the interferometry to map the nodes of a vibraphone bar in the fundamental mode. Again, these results matched those predicted by theory.

Based on the results we have achieved, it is not hard to envision further application by a similar experiment to observe the vibrational modes of more complicated objects.

References

[1] J. C. Dainty, “Laser Speckle and Related Phenomena”, Topics in Applied Physics, Vol. 9, 9-11 (1975).

[2] T. R. Moore, “A Simple Design for an Electronic Speckle Pattern Interferometer,” Am. J. Phys. 72, 1380, 2004.

[3] T.D. Rossing and N.H. Fletcher, Principles of Vibration and Sound, 2nd ed., (Springer, New York, 2004), pp. 54-60.

Acknowledgements

We would like to thank Kevin Booth and Kurt Wick for the generous amount of help they gave us throughout the experiment.