## Step 1: Write the Fresnel Equations The **Fresnel reflection coefficients** for the electric field amplitudes for s-polarization (\(r_s\)) and p-polarization (\(r_p\)) are: \[ r_s = \frac{n_i \cos\theta_i - n_t \cos\theta_t}{n_i \cos\theta_i + n_t \cos\theta_t} \] \[ r_p = \frac{n_t \cos\theta_i - n_i \cos\theta_t}{n_t \cos\theta_i + n_i \cos\theta_t} \] Where: - \(n_i\): Refractive index of incident medium - \(n_t\): Refractive index of transmitting medium - \(\theta_i\): Angle of incidence - \(\theta_t\): Angle of transmission, found using Snell's Law: \[ n_i \sin\theta_i = n_t \sin\theta_t \] **Explanation:** These equations give the amplitude reflection coefficients for light polarized perpendicular (s) and parallel (p) to the plane of incidence, as functions of the angle of incidence (\(\theta_i\)). To get the reflected intensity, take the magnitude squared (\(|r|^2\)). To get the phase, use the angle (\(\arg(r)\)). --- ## Step 2: Plot for Both Cases Let’s outline the approach and show example code (Python/Matplotlib) for plotting both the **magnitude** and **phase** of the reflection coefficients for the given cases. ### **Case 1:** \(n_i = 2.\), \(n_t = 1.\) ### **Case 2:** \(n_i = 1.5\), \(n_t = 2.\) ```python import numpy as np import matplotlib.pyplot as plt def fresnel_coefficients(n_i, n_t, theta_i_rad): # Snell's Law to get theta_t sin_theta_t = n_i / n_t * np.sin(theta_i_rad) # Handle total internal reflection theta_t_rad = np.where(np.abs(sin_theta_t) <= 1, np.arcsin(sin_theta_t), np.nan) cos_theta_i = np.cos(theta_i_rad) cos_theta_t = np.sqrt(1 - sin_theta_t**2, where=(sin_theta_t**2 <= 1), out=np.full_like(theta_i_rad, np.nan)) # Fresnel equations rs = (n_i * cos_theta_i - n_t * cos_theta_t) / (n_i * cos_theta_i + n_t * cos_theta_t) rp = (n_t * cos_theta_i - n_i * cos_theta_t) / (n_t * cos_theta_i + n_i * cos_theta_t) return rs, rp angles_deg = np.linspace(, 89.9, 100) angles_rad = np.deg2rad(angles_deg) # Case 1 rs1, rp1 = fresnel_coefficients(2., 1., angles_rad) # Case 2 rs2, rp2 = fresnel_coefficients(1.5, 2., angles_rad) plt.figure(figsize=(14, 6)) plt.subplot(1, 2, 1) plt.title("Reflection Coefficient Magnitude") plt.plot(angles_deg, np.abs(rs1), label="Case 1 $r_s$") plt.plot(angles_deg, np.abs(rp1), label="Case 1 $r_p$") plt.plot(angles_deg, np.abs(rs2), label="Case 2 $r_s$") plt.plot(angles_deg, np.abs(rp2), label="Case 2 $r_p$") plt.xlabel("Angle of Incidence (deg)") plt.ylabel("|r|") plt.legend() plt.grid() plt.subplot(1, 2, 2) plt.title("Reflection Coefficient Phase") plt.plot(angles_deg, np.angle(rs1), label="Case 1 $r_s$") plt.plot(angles_deg, np.angle(rp1), label="Case 1 $r_p$") plt.plot(angles_deg, np.angle(rs2), label="Case 2 $r_s$") plt.plot(angles_deg, np.angle(rp2), label="Case 2 $r_p$") plt.xlabel("Angle of Incidence (deg)") plt.ylabel("Phase (radians)") plt.legend() plt.grid() plt.tight_layout() plt.show() ``` **Explanation:** - We calculate the reflection coefficients for both cases using the Fresnel equations. - We then plot the magnitude and phase as a function of the angle of incidence. - For angles beyond the critical angle (in Case 1, where \( n_i > n_t \)), the coefficients become complex, resulting in a phase jump. --- ## **Final Answer** - **The plots above (when executed)** show how the **magnitude and phase of the Fresnel reflection coefficients** vary with the angle of incidence for both cases. - In **Case 1** (\( n_i > n_t \)), you'll see total internal reflection and a phase shift past the critical angle. - In **Case 2** (\( n_i < n_t \)), the reflection coefficients remain real and there is no total internal reflection; the phase stays at or π. Let me know if you need the plots or code output visualized!