Grating Challenge

A grating (or more properly, a diffraction grating) is an object with a patterned surface that causes light to diffract. For visible light, these structures are microscopic, but tall buildings in a city can act like a diffraction grating for radio waves. Gratings can be made by creating variations in the thickness of an object (physically etching a surface), by creating variations in the material composition or density, or by creating variations in the opacity (transmission or tint) of a surface. Your grating has a surface that is textured with small grooves that change the thickness ever so slightly.

The image at the left is the surface of one of our gratings under 80x magnification, taken by Physics Department Lab Technician, Zach Eridani. The image shows many microscopic grooves. The variations in thickness of the surface intoduce a spatial modulation of the phase of the wave, which results in a bending of the wavefront (the direction of propagation) of light, where the magnitude of the effect depends on wavelength.

The obvious way to use the grating is look through the grating at a light source. If you look at a white light, you will see a rainbow pattern to the sides of the light source.

The fluorescent lights in Bowers hall show clear rainbow patterns, with surprisingly sharp lines between the colors (why is this?).

An EXIT sign viewed from below creates a line source of red light. The diffraction pattern also only shows red light.

A good place to start with a new puzzle is to play around with it and learn a thing or two. You can start by looking at a light source (not the sun and nothing very bright) through the grating and changing the setup to infer how it behaves.

d sin(θ) = λ,

where d is the spacing between the grating lines, θ is the angle at which the light bends (measured relative to the straight-forward direction), and λ is the wavelength of light. If you are familiar with this equation, you may notice that the right-hand side of this equation is often written as n λ, where n is a positive or negative integer (including zero). While that is entirely correct, we are going to limit this analysis to the case where n=1, which is referred to as the first-order diffraction band. If you put the grating close to your eye you should be able to another band further to the side that is fainter - this is the the second order (or n=2) band, but it is not useful for the purposes of this challenge.

While holding the grating in front of your eye is a good demonstration of the effect of the grating, it does not work particularly well for measuring an angle. If passing light through the grating toward our eyes is not the path to the solution, it must be that the path to a solution requires that we turn the system around and let the light head away from our eyes.

To determine the angle from the grating to the observation point you will need to make some measurements. A good way to do this is to make some marks on a paper and then measure the positions afterward. If you aren't sure how to do this, take a look at our setup under the previous hint and consider doing something similar.

Unless you have a calibrated light source, this part takes a little guesswork or some further research. Very roughly, the visible spectrum spans 400 nm (violet) to 750 nm (red). Below that is the UV spectrum and above that is the IR spectrum. To start, you can mark off the boundaries of your spectrum. If you don't see violet, but see something more of a blue color, then you may want to adjust the short wavelength limit to be about 475 nm (blue is considered to be in the range of about 450-500 nm). The greens are roughly in the range of 500 to 570 nm. If you have normal vision (not color blind), then you can mark off the green region as well and use that for additional data points.

The flashlight on your cell phone likely uses a single white LED that produces a continuous spectrum of light of different color. Using this light means that we have to make some educated guesses about the wavelengths we see. While decent, they are nonetheless guesses.

Alternatively, if you were to use the light from a white image on the screen of your cell phone, you would see that the diffraction pattern has three distinct color bands in the red, green, and blue. This is because color on our screens is produced by a combination of red, green, and blue pixels. When these pixels are small enough, they blend together to create what appears to us to be a uniform color.
Image credit: The Exploratorium

You could do some research on the wavelengths of these LEDs (they are pretty standard) and try to use the light from a white screen to make more precise measurements.

The simplest place to start is to calculate d for each color using the angle and wavelength data. There are more sophisticated ways to do this kind of analysis that attempt to eliminate systematic errors (bias), and one could include additional data points, but this is good enough for a start.

Solving the grating equation for the groove spacing gives d = λ / sin(θ). Recalling that sin(θ) ≈ x / L for small angles, an approximate solution combining all measured terms is d ≈ λ L / x.

According to the manufacturer, these gratings have 500 lines per mm, equivalent to 2.0 μm between lines.

Hopefully you learned something new as you worked on this project. This kind of thing is representative of what we aim to teach you in the Physics program at SUNY Cortland. There are a few important "morals" to this story:

1. Seemingly simple things get deep very quickly, especially if one has ambition to do something precise and quantitative. This is why scientists and engineers get paid good money to do what they do.
2. Success in a project like this requires a combination of (1) theoretical understanding that allows us to see the relationships between different things and allows us to simplify models, and (2) physical intuition that allows us to do real things in the real world.
3. Don't expect your first go at something to lead you to instant success. Physics is best approached as an iterative process. Patience and perseverance are the real keys to success, in physics and life more generally.