Diffraction through a square aperture


The setup

Diffraction pattern observations

Calculating the Fresnel number

Calculating the distance to the maxima

Discussion


Rachel and I briefly looked into the various diffraction patterns that arise from shining light through different shaped apertures: circular, square, and triangular, however our measurements and calculations focus on the square aperture.

Diffraction refers to the spreading of light around an obstacle. The resulting pattern is a description of the obstacle in terms of its spatial frequencies. For more information on diffraction and spatial frequencies, see my mini project from last summer, in which I studied pinhole diffraction and the resulting Airy pattern.


The setup

For our setup, we sent a red HeNe laser (λ=632nm) through various shaped apertures (i.e. circular, rectangular, and triangular). The square and triangular apertures were actually created for another LTC student’s project, David, who used them to identify the topological charge of optical vortices.

The circular pinhole had a diameter of 200 μm, the square had a side length of 1.4mm, and the equilateral triangle had a side length of 1.7mm. The pinhole created an Airy pattern, as I had studied for my mini project last summer, but we were more interested in seeing the patterns that arose from the rectangular and triangular apertures.

In order to have the beam fill these larger apertures, we used a Keplerian beam expander. The focal length of our front lens was 35mm and the second had a focal length of 125mm, which magnified the original beam diameter (about 1mm) 3.5 times.

We then turned off all of the lights and projected each aperture’s diffraction pattern across the lab to the far door, which was about 13 m away.


Diffraction Pattern Observations

It was pretty neat to see the square and triangular aperture patterns. The square aperture diffraction pattern had two-fold symmetry with a very bright square in the center and smaller less-intense squares coming off in a cross shape. The triangle aperture had a bright triangular shape in the center and two layers of three-fold symmetric arms coming off of it.

Kevin took pictures of each of the patterns with his high quality camera. [These will be included shortly.] However, it was still a little difficult to capture the full extent of the pattern, since the camera doesn’t have as large of a dynamic range as the human eye. The extra-bright center of the diffraction patterns made it hard for the camera to register the lesser-intense side lobes.


Calculating the Fresnel number

The Fresnel number is a dimensionless value that describes which type of diffraction is occurring at a certain distance L from an aperture with radius a. If this number is greater than or equal to 1, it means that we’re observing Fresnel diffraction, also known as near-field diffraction. If this number is much, much less than 1, than we’re observing Fraunhofer diffraction, or far-field diffraction.

In the Fraunhofer zone, the shape of the intensity pattern is independent of distance; it will continue to expand the farther you move away from the aperture, however the pattern itself will remain the same. For larger apertures, you have to be farther away in order to be in the far-field diffraction zone. To check that we were, we performed the following calculation:

Even though the rectangular aperture was relatively large, we only needed to be at least a meter away in order to observe its Fraunhofer diffraction pattern.


Calculating the distance to the maxima

While the diffraction pattern for the square aperture was projected across the room, we marked where the central maximum and first order maximum were on a sheet of paper and found that to be about 9mm. This measurement was just from marking the distance by eye, so there is some uncertainty there that we should account for if we were doing a further study on this; but since this is just a mini-exploration of diffraction, our measurements aren’t that precise.

We then calculated what this distance should be, based on the wavelength of light and size of the aperture, to compare with this experimental observation.

Note that in order to use the above equation, our m integer for the location of the first maximum is 3/2, as shown in the graph. (Originally, we had incorrectly multiplied our answer by 2, but that would be the location of the second minimum). Our calculation was therefore 8.55 mm – with significant figures, about 9mm!


Image Source: http://www.pstcc.edu/


Discussion

For the diffraction through the square aperture, our mathematical calculation and experimental observation of the distance from the central optical axis to the first maximum agree, within the (presumed) uncertainty of our measurements. We measured the location of the bright spots by eye, and only did so once. To report the SDM, we would need to go back and measure this distance multiple times.

Further study into the mathematics that describes the intensity pattern that arises from a rectangular aperture would be interesting as well.