This Exercise Set has been submitted for peer review, but it has not yet been accepted for publication in the PICUP collection.

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Fresnel's Equations - A Brief Activity

Developed by Todd Zimmerman

This exercise set has students plot the reflectance of light off of a glass substrate using Fresnel’s equations and Snell’s law. A hands-on activity requires students to compare their observations to the graph and to interpret the graph of reflectance.
Subject Area Waves & Optics Beyond the First Year and Advanced IPython/Jupyter Notebook, Python, and Octave*/MATLAB * Plot a function (Exercises 1, 2 and 5) * Interpret a graph (Exercises 3 and 4) * Relate observations to a model (Exercises 3 and 4) 20 - 40 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

Exercise 1: Use Snell’s law ($n_i \cos(\theta_i) = n_t \cos(\theta_t)$)to plot the angle of transmission as a function of the angle of incidence for $0 \le \theta_i \le \pi/2$. Use $n_{air} \approx 1$ and $n_{glass} \approx 1.5$. A good question to ask students is why $\theta_t$ levels off near 0.73 radians. If you’ve discussed total internal reflection you can have them compare this angle to the critical angle for light going from glass into air. You can bring up the idea that light paths are time-reversible so we’d expect light going from air into glass to have a maximum angle of refraction corresponding to the critical angle. Exercise 2: Plot the reflectances $R_s$ and $R_p$ for light in air entering glass as a function of the angle of incidence for $0 \le \theta_i \le \pi/2$. The equations for the reflectance are $$R_s = \frac{(n_i \cos(\theta_i) - n_t \cos(\theta_t))^2}{(n_i \cos(\theta_i) + n_t \cos(\theta_t))^2)}$$ $$R_p = \frac{(n_t \cos(\theta_i) - n_i \cos(\theta_t))^2}{(n_i \cos(\theta_t) + n_t \cos(\theta_i))^2)}$$ where $n_i$ is the index of refraction in the incident medium, $\theta_i$ is the angle of incidence (measured with respect to the surface normal), $n_t$ is the index of refraction in the transmitted medium, and $\theta_t$ is the angle of transmission found from Snell’s law. Ask students what it means when $R_p$ goes to zero and how this might be used to polarize light. You can mention that by placing a piece of glass inside a laser cavity at Brewster’s angle, the p-polarized light we be transmitted through the glass but some of the s-polarized light is reflected out of the cavity. This gives the p-polarized light an advantage so it dominates the behavior of the laser. Exercise 3: Part a) Based on your graphs for $R_s$ and $R_p$, predict whether the light intensity will change for normally incident light ($\theta_i$ near $0^{\circ}$) as you rotate the polarizer while looking at the light reflecting off of the glass slide. Part b) Place the glass slide directly under the light source and look straight down at the slide to view the reflection. Go ahead and look at the reflection of the light source through the polarizer and rotate the polarizer - describe whether the light intensity changes as you rotate the polarizer. Explain how your result does or does not agree with your prediction. Part c) Look at your graphs and predict how reflected light incident at a $45^{\circ}$ angle will behave as viewed through a rotating polarizer. Part d) Align your glass slide so that light from the light source arrives at roughly a $45^{\circ}$ angle and look at the reflection through the polarizer. Describe what happens to the reflection as you rotate the polarizer. Explain how your result does or does not agree with your prediction. Exercise 4: Find Brewster’s angle experimentally. Change the incident angle of light until the reflected light is completed blocked when the polarizer axis is perpendicular to the surface of the glass. Estimate this angle and then compare your angle to the graph. Identify where Brewster’s angle occurs on the graph. Optional Exercise 5: The light you are seeing is actually light reflected off both the front surface AND the back surface of the glass but we only calculated the reflectance for the front surface. Do you think the results of including both surfaces will vary significantly from the results for just the front surface of glass? Before proceeding, predict what the reflectance at $\theta_i = 0$ will be when you include reflections from both the front and back surface of the glass slide. Plot the total reflectance (the sum of the reflectance off of the front surface of glass and the back surface of glass) vs. the angle of incidence. To find the reflectance from the back surface (the glass-to-air interface) you need to multiply the transmittance through the air-to-glass interface by the reflectance from the glass-to-air boundary. You will also need to use Snell’s law ($n_i \cos(\theta_i) = n_t \cos(\theta_t)$) to relate the angle of incidence to the angle of transmission. The equations for transmittance are $$T_s = \frac{n_t \cos(\theta_t)}{n_i \cos(\theta_i)} \frac{4 n_i^2 \cos^2(\theta_i)}{(n_i \cos(\theta_i) + n_t \cos(\theta_t))^2}$$ $$T_p = \frac{n_t \cos(\theta_t)}{n_i \cos(\theta_i)} \frac{4 n_i^2 \cos^2(\theta_i)}{(n_i \cos(\theta_t) + n_t \cos(\theta_i))^2}$$