Numpy Snell

Module 3: NumPy

import numpy as np

Snell’s Law

When light travels from one medium to another (e.g., from air to water), its path bends, a phenomenon called refraction. The relationship between the angle of incidence ($ _1 $) and the angle of refraction ($ _2 $) is given by Snell’s Law:

\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]

Where:

$ n_1 $ is the index of refraction of the first medium (e.g., air).
$ n_2 $ is the index of refraction of the second medium (e.g., water or an unknown material in our case).
$ _1 $ is the angle of incidence, i.e., the angle between the incident ray and the normal to the surface.
$ _2 $ is the angle of refraction, i.e., the angle between the refracted ray and the normal to the surface.

The index of refraction indicates how much a ray of light is bent, or refracted, when entering a material. For example, air typically has an index of refraction of about 1.0003, while water has an index around 1.33.

Exercise

You’ve been conducting optical experiments, directing light from air into an unknown material and measuring the angles of incidence and refraction. Your objective is to estimate the refractive index ($ n_2 $) of this unknown material using Snell’s Law.

Data

You have a list of pairs of measurements. Each pair consists of an angle of incidence and its corresponding angle of refraction, in degrees.

# [theta_1, theta_2]
angles = [
  [30, 18],
  [32, 19],
  [34, 21],
  [36, 22],
  [38, 23],
  [40, 24],
  [42, 25],
  [44, 26],
  [46, 27],
  [48, 28],
  [50, 29],
  [52, 30],
  [54, 31],
  [56, 32],
  [58, 33],
  [60, 33]]

Task A: Calculating Refraction Indices

Create a function index_of_refraction to estimate the relative index of refraction for various angles of incidence and refraction. The function tales two input parameters: - theta_1: An array of incidence angles in degrees. - theta_2: An array of refracted angles in degrees.

The function should return an array representing the ratio of the sine of the incidence angles to the sine of the refracted angles:

$\sin(\theta_1) / \sin(\theta_2)$

Example: For the given incidence and refracted angles:

$\theta_1 = [90, 45, 60]$

$\theta_2 = [90, 30, 45]$

The function should output: $[1, 1.41, 1.22]$.

Task B: Calculating Average and Standard Deviation

Create a function mean_and_std to compute the average and standard deviation of a given array of values. The function should format the results with two significant figures and return them as a string. The input is: - n: An array of numerical values.

And the output should be a string representing the average followed by the standard deviation in the format: average +/- standard deviation.

Example: For the input array:

$n = [1, 1.41, 1.22]$

The function should output: 1.21 +/- 0.17.

Task C: Estimating the Index of Refraction

Design a function named estimate_index to determine the estimated index of refraction based on given incidence and refracted angles. The input is:

angles_2d: A 2D array where each row consists of a pair of incidence and refracted angles, both in degrees.
n_1: A keyword parameter representing the refraction index of the first medium. It defaults to 1, which is the typical refraction index of air.

The function should perform the following steps:

Separate the 2D array angles_2d into two arrays:
- incidence_angles for the first values of each pair ($\theta_1$).
- refracted_angles for the second values of each pair ($\theta_1$).
Utilize the index_of_refraction function to obtain an array of values for $\sin{\theta_1} / \sin{\theta_2}$.
Compute the array n_2_values representing the refraction indices for the second medium using the formula: \[n_2 = n_1 \cdot \sin{\theta_1} / \sin{\theta_2}\]
Feed the n_2_values array into the mean_and_std function to determine the estimated index of refraction.
Display the result with a message like: “The estimated index of refraction is…”

Now put all work together and compute the refraction coefficient. What material were we studying?

If we were shooting our beams through water ($n_1 = 1.33)$ instead of air, and got the same refracted angles, what would the second material be?

# Use these functions as mock functions,
# until your team mates give you the good ones
def index_of_refraction(theta_1, theta_2):
  return np.zeros(len(theta_1))

def mean_and_std(x):
  return "1.00 +/- 0.00"

--- title: "Numpy Snell" subtitle: "Module 3: NumPy" format: html --- ```{python} import numpy as np ``` --- ## Snell's Law When light travels from one medium to another (e.g., from air to water), its path bends, a phenomenon called refraction. The relationship between the angle of incidence ($ \theta_1 $) and the angle of refraction ($ \theta_2 $) is given by Snell's Law: $$ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) $$ Where: - $ n_1 $ is the index of refraction of the first medium (e.g., air). - $ n_2 $ is the index of refraction of the second medium (e.g., water or an unknown material in our case). - $ \theta_1 $ is the angle of incidence, i.e., the angle between the incident ray and the normal to the surface. - $ \theta_2 $ is the angle of refraction, i.e., the angle between the refracted ray and the normal to the surface. The index of refraction indicates how much a ray of light is bent, or refracted, when entering a material. For example, air typically has an index of refraction of about 1.0003, while water has an index around 1.33. <p align="center"> <img src="https://media.geeksforgeeks.org/wp-content/uploads/20220514021809/snelllaw1.png" width="400" height="auto" /> </p> ### Exercise You've been conducting optical experiments, directing light from air into an unknown material and measuring the angles of incidence and refraction. Your objective is to estimate the refractive index ($ n_2 $) of this unknown material using Snell's Law. ### Data You have a list of pairs of measurements. Each pair consists of an angle of incidence and its corresponding angle of refraction, in degrees. ```{python} # [theta_1, theta_2] angles = [ [30, 18], [32, 19], [34, 21], [36, 22], [38, 23], [40, 24], [42, 25], [44, 26], [46, 27], [48, 28], [50, 29], [52, 30], [54, 31], [56, 32], [58, 33], [60, 33]] ``` ### Task A: Calculating Refraction Indices Create a function `index_of_refraction` to estimate the relative index of refraction for various angles of incidence and refraction. The function tales two input parameters: - `theta_1`: An array of incidence angles in degrees. - `theta_2`: An array of refracted angles in degrees. The function should return an array representing the ratio of the sine of the incidence angles to the sine of the refracted angles: $\sin(\theta_1) / \sin(\theta_2)$ **Example:** For the given incidence and refracted angles: $\theta_1 = [90, 45, 60]$ $\theta_2 = [90, 30, 45]$ The function should output: $[1, 1.41, 1.22]$. ```{python} #| eval: false ``` ### Task B: Calculating Average and Standard Deviation Create a function `mean_and_std` to compute the average and standard deviation of a given array of values. The function should format the results with two significant figures and return them as a string. The input is: - `n`: An array of numerical values. And the output should be a string representing the average followed by the standard deviation in the format: `average +/- standard deviation`. **Example:** For the input array: $n = [1, 1.41, 1.22]$ The function should output: `1.21 +/- 0.17`. ```{python} #| eval: false ``` ### Task C: Estimating the Index of Refraction Design a function named `estimate_index` to determine the estimated index of refraction based on given incidence and refracted angles. The input is: - `angles_2d`: A 2D array where each row consists of a pair of incidence and refracted angles, both in degrees. - `n_1`: A keyword parameter representing the refraction index of the first medium. It defaults to 1, which is the typical refraction index of air. The function should perform the following steps: 1. Separate the 2D array `angles_2d` into two arrays: - `incidence_angles` for the first values of each pair ($\theta_1$). - `refracted_angles` for the second values of each pair ($\theta_1$). 2. Utilize the `index_of_refraction` function to obtain an array of values for $\sin{\theta_1} / \sin{\theta_2}$. 3. Compute the array `n_2_values` representing the refraction indices for the second medium using the formula: $$n_2 = n_1 \cdot \sin{\theta_1} / \sin{\theta_2}$$ 4. Feed the `n_2_values` array into the `mean_and_std` function to determine the estimated index of refraction. 5. Display the result with a message like: "The estimated index of refraction is..." Now put all work together and compute the refraction coefficient. What material were we studying? If we were shooting our beams through water ($n_1 = 1.33)$ instead of air, and got the same refracted angles, what would the second material be? ```{python} # Use these functions as mock functions, # until your team mates give you the good ones def index_of_refraction(theta_1, theta_2): return np.zeros(len(theta_1)) def mean_and_std(x): return "1.00 +/- 0.00" ```