Hair Diameter Measurement Using Laser Diffraction Patterns

Cris Carianna

Experimental Design

My project consists of the diffraction of laser light of 2 known wavelengths around the hair follicles of several Vassar students. Each student will provide three hairs and the diameters of each will be calculated and averaged together, using both colors of laser light in an effort to see which of the two provide more precise measurements. The laser will be fixed to a wooden mount and shine from the inside of one end of an originally 1’ x 2’ x 6” rectangular prism made of 3/4” thick medium density fiberboard, wood glue, and pocket-holed screws. It will pass around the item to be measured, which will be fixed level to the laser and 1” away from its tip by a small frame made of 5mm thick sheet metal held steady between two halves of a 2 x 4, and will project a diffraction pattern on a piece of 1/4” thick MDF plate at the other end of the box. This plate was positioned exactly perpendicular to the laser to ensure that the measurement of the diffraction pattern was not skewed by the angle from which the laser emitted.

The diffraction pattern will be measured with digital calipers and this measurement will be taken from the center of the pattern’s brightest point to the edge of the pattern’s first ‘dark patch’ in millimeters to the nearest sig fig.

Using this formula in each measurement trial, I will plug in the distance, which has been standardized by the fixing of the laser to the inside of the box, and the known wavelength of the laser, either 532 nm or 473 nm, to find the diameter of the hair. Theta in this equation represents the angle of diffraction, and because this angle is very small, I can use the small angle approximation formula, which is the distance measured to the first minimum (or band) from the pattern’s center divided by the distance from the hair to the projection surface. To make things even simpler, the numerator in my equation can simply be the laser’s wavelength because I am only measuring to the pattern’s first minimum, meaning that m is 1.

At the end of initial data taking, I was not impressed with my results. The numbers seemed random and the only clear pattern I could discern was in one student’s results; his average head hair thickness was significantly higher than other subjects perhaps due to his south Asian background.

Unfortunately I think there were more failures than successes with this initial experimental setup, enough that I felt the need to readdress the experimental design and retake the data. First, the distance between my hair-holding frame and the laser itself as well as the distance between the hair-holding frame and the MDF board on which the diffraction pattern was projected is too small. The distances I used were 10 and 35 centimeters respectively, meaning that the total distance between laser and MDF was 45 cm. I set it up this way based on the set up of a similar experiment that I found online which I scaled down to fit the fiberboard box that I built to house the experiment. This downscaling of the set-up created diffraction patterns that were quite a bit smaller than I would’ve liked, meaning that the accuracy with which I measured the space between the bands was unacceptable for taking accurate measurements with the digital calipers. I measured the bands using digital calipers set to metric units to the thousandth decimal place, a method that I thought would yield highly precise results, but the patterns presented with an almost fish-eye quality that I attribute to the shorter distance and which I believe contributed to my less than ideal results.

Another obstacle in my attempts at precise data taking was in the design of the experimental enclosure’s lid. I initially used a piece of MDF fit to the exact dimensions of the box top attached with 3/4” strap hinges. I hoped this set up would allow for customizable levels of darkness so as to improve my ability to see the precise edges of the diffraction pattern’s bands, and I eventually added a hole in the lid which corresponded to an identical hole in the box frame and cut a series of dowels varying in size so that the box’s lid could be suspended at differing heights depending on how illuminated I wanted the inside of the box to be. This apparatus eventually began to restrict the range of motion of my arm during diffraction pattern measurement using the calipers, and this problem was exacerbated by the lid interfering with my ability to extend the calipers to the appropriate width needed for precise measurement. After data taking, I used my router to inlay the piece of PVC board into the MDF rather than simply attaching it with screws like I had done initially, and this seemed to solve the problem of the full extension of the calipers after a little more tinkering. However, I still had problems with the box’s top getting in my way. To solve both of these problems, I simplified. I removed the back wall of the box entirely by cutting off the part of the box which housed the laser mount and hair-holding frame. I moved this now independent apparatus to the other end of the table and clamped it in place to assure repeatability across trials. I then took my newly three-sided box, which I’ll now fancily refer to as the laser amphitheater because of the likeness, and clamped it to the other end of the table and extended the projection surface a full 9 inches off the edge of the table before clamping. Despite the lid and opposite wall being removed, my projection surface remained dark enough to see distinct lines in the diffraction pattern, probably because I conducted the experiment in my basement. All this resulted in a hair to projection surface distance of 226 cm, which yielded much more regular looking and measurable diffraction patterns.

This photo shows my new and improved experimental setup. The hair-holding apparatus was screwed into the MDF platform that was cut off of the box and which holds the laser mount to ensure that the distance between the hair and the laser remained unchanged throughout the trials. Moving the hair-holder onto a platform also helped to center the beam along the y axis of the metal holder to improve the pattern’s image. The projection amphitheater is projecting off the left side of the table about nine inches and is clamped in place to ensure distance repeatability. The platform with the laser mount and hair mount is also clamped in place.

This picture shows the apparatus from the back, with the 473 nm laser and the digital calipers I used for measurement beside it.

Pictured above from left to right is the hair-holder mounted in the block for stability. Tightening those screws would keep it in place. The middle shows the new apparatus. The block is screwed into the MDF, which is in turn clamped to the table. The laser mount is also screwed to the MDF with pocket holes. The right photo shows the hair holder being loaded into the block. Because the only thing I had to do was remove the holder, put the next hair in, and put it back, I could guarantee that both the distance between the laser and the hair and the hair and the projection surface remained identical for all trials.

Above are some additional pictures, depicting the metal hair-holder with a framing square for scale, the experimental apparatus housing the blue laser, and the measurement of a green diffraction pattern using the back arms of my digital calipers, which are used for measuring internal spaces rather than the external bounds of an object.

Results

After all of this adjustment, my new results came in with a lot more regularity. The measurements had much less variation across a given individual’s three trials in both wavelengths of laser, and the overall experiment yielded much less variation across the individuals’ average diameters as well. The best example is found in Diana Howland’s trials, especially in the 532 nm data. In my initial experimental setup, the data to which can be found in my initial data submission, Diana’s 532 nm wavelength three head hair trials yielded measurements of 30, 61, and 67 microns. In my improved setup, her three measurements for the same wavelength of laser light used were 57, 60, and 63 microns. These results instill much more confidence that my method is precise, I believe due to the changes I made to the repeatability of the standardized numbers in my calculations (by way of adjusting the enclosure and distances therein).

In terms of the differences between the green laser trials and the blue laser trials, I couldn’t find much. Most of the individual averages were within ten microns of each other; Caroline’s two averages were 54 microns for the green and 57 for the blue. I would say that that at least proves the reliability of my measurement method. Generally speaking, the blue trials yielded less precise data in that the diameters within an individual varied across the three trials more than that of the green laser trials. A notable exception was Jacob, whose blue trial data was noticeably more uniform than his green data. I also noticed that in his green trials, his average was 10 microns higher than the next highest number, which I expected due to his south Asian genetics. This spike was not observed in the blue trials, however. I expect the general trend toward imprecision in the blue trials is due to the brightness of the laser; during measurement, I had noticeable trouble looking at the center of the blue diffraction pattern for too long and this made locating the edge of the band and the middle of the pattern more difficult than it had been during the green trials. Interestingly, after standard deviation and variance calculations, the averages for the blue trials actually showed more precision in that the spread was far smaller than the green. However, I believe this actually points to inaccuracy in the blue trials because these numbers seem too close based on both the variance of human hair widths (around 20-120 microns) and on the data for the green trials appearing much more precise within an individual’s trial.

Conclusion

Overall, I would say the experiment mostly taught me the importance of repeatability across all trials of an experiment. In the context of our class, deriving the formula that we were shown early on gave me a lot more insight into why it works so well. I struggled for a while about how to find the sine of theta because I was convinced that the small angle approximation just wouldn’t do the trick. I tested a bunch of ways of measuring that angle, but none seemed as accurate as simply approximating the angle. Looking back now that I have results, I see that my method isn’t accurate enough to warrant that type of precision anyway.

This method of measurement is used in physics all the time; it is incredibly accurate when standardized with expensive equipment of course. One would think that this is an awfully expensive way of accomplishing simple measurement, but you’d be surprised how often it applies to the actual world of experimental physics.

If I could go back and change something, I would have projected my patterns and measured the distance to my patterns’ first minimum using some disposable surface, paper even, and marking on the paper exactly the bounds of my measurement. I would then turn the laser off, and measure the distance between my two marks with the digital calipers. This method would have been especially helpful during the blue laser trials because I believe its brightness was a hinderance to the accuracy of my measurements.

If the project was to continue, I would do my best to get my hands on a micrometer. This way I could first measure the hair’s actual diameter and I’d have a real understanding based in numbers of how accurate my methods actually are. I also would measure some smaller items, such as particles in pond water, in order to yield and measure diffraction patterns which show bands both above and below the one line, meaning that the light is diffracting both to the left and right of the object and around the top and bottom of the object. I would love to see what kind of accuracy I could get in finding the shape of a very small object with only light to help me out.