Overview:

This document provides a review of the "PICUP Refractive Indices of Water and Air JavaScript Simulation Applet HTML5" resource. This resource offers a set of exercises designed to guide students in understanding the formation of primary and secondary rainbows. It utilizes computational tasks and analysis of peer-reviewed literature to explore the relationship between the refractive indices of water and air, the deflection of light through raindrops, and the resulting visual phenomenon of rainbows. The exercises are suitable for introductory and upper-level optics courses.

Main Themes and Important Ideas/Facts:

1. Refractive Indices of Water and Air are Wavelength Dependent: The initial exercises (Exercise 1) focus on obtaining and utilizing data from peer-reviewed publications to understand how the refractive indices of water and air vary with the wavelength of light. Students are required to:

• Obtain the paper "Models for the wavelength dependence of the index of refraction of water" by Paul D.T. Huibers and plot the refractive index of water as a function of wavelength (400-650 nm) using Equation (3) from the paper.
• Obtain the paper "Refractive index of air: new equations for the visible and near infrared" by Philip E. Ciddor and plot the deviation of the refractive index of air from unity (n_air - 1) as a function of wavelength (400-650 nm) using Equation (1) from the paper.
• Compare the change in refractive index for both materials over the specified wavelength range and comment on the magnitude of the ratio of the larger change to the refractive index at 400 nm.

1. This exercise emphasizes the fundamental role of wavelength-dependent refractive indices in the dispersion of light, which is crucial for the formation of rainbows. It also introduces students to using real scientific literature and performing basic data analysis.
2. Deflection Angle of Light Rays in Raindrops: Exercises 2, 3, and 4 delve into the geometry of light rays passing through spherical raindrops, focusing on the concept of the deflection angle.

• Exercise 2: Derives the formula for the deflection angle ($\delta$) of a light ray undergoing one internal reflection within a raindrop:
• $\delta = 2 ( \theta i − \theta r ) + ( \pi − 2 \theta r )$ where $\theta_i$ is the angle of incidence and $\theta_r$ is the angle of refraction. The exercise prompts students to identify the necessary quantities to compute this angle (refractive indices of air and water at the specific wavelength, and the angle of incidence). This reinforces the application of the Law of Refraction and the Law of Reflection.
• Exercise 3: Requires students to plot the deflection angle as a function of the incident angle for wavelengths of 400 nm (violet) and 650 nm (red), assuming one internal reflection (primary rainbow). They are asked to identify the minima of these curves, which correspond to the rainbow angles for each color, and to determine the apparent position of the red and violet bands in the sky relative to the horizontal. The solution reveals that the minimum deflection angles for violet and red light are approximately 139.3° and 137.6° respectively, leading to viewing angles of 40.7° (violet) and 42.4° (red) above the horizontal, with red appearing higher. This explains the color order of the primary rainbow.
• Exercise 4: Repeats the process for two internal reflections (secondary rainbow), with a modified deflection angle formula:
• $\delta = 2 ( \theta i − \theta r ) + 2 ( \pi − 2 \theta r )$ Students plot the deflection angle for 400 nm and 650 nm light and identify the minima. The solution indicates minimum deflection angles of 233.3° (violet) and 230.2° (red). Considering that these rays enter the bottom of the raindrop, the viewing angles are 53.3° (violet) and 50.2° (red) above the horizontal, with red appearing below violet. This explains the reversed color order of the secondary rainbow, which appears above the primary rainbow due to the larger viewing angles. The text also notes the reduced brightness of the secondary rainbow due to the additional reflection.

1. Rainbow Angles and Brightness: The exercises highlight that rainbows are observed at specific angles corresponding to the minima in the deflection angle functions. The text explains: "Near the minimum of the deflection function (that is, the $\delta(\theta_i)$ curve), are two rays with slightly different incident angles deflected into different directions? Regarding the brightness of the deflected light perceived by an observer, what is implied by your answer?" The answer implicitly suggests that near the minimum, many rays with slightly different incident angles are deflected into nearly the same direction, leading to a higher intensity of light observed at these angles. This is the fundamental reason why rainbows appear as bright arcs.
2. Crude Estimate of Irradiance: Exercise 5 attempts to provide a basic understanding of the brightness of the primary rainbow by modeling the irradiance of each outgoing ray as a Gaussian function centered at the deflection angle. Students are asked to sum the contributions from rays with uniformly distributed impact parameters to obtain the overall irradiance as a function of deflection angle for a single wavelength (400 nm). The solution states: "The main result is that irradiance accumulates in the direction specified by the minimum in the deflection function because several rays are deflected into essentially the same direction. This is known as rainbow scattering". This exercise, while simplified by neglecting intensity losses and polarization, demonstrates the concept of enhanced brightness at the rainbow angle due to the concentration of deflected light.

Learning Objectives:

The exercises aim to enable students to:

• "obtain and use information from peer-reviewed literature. Plot an equation over a range of values. Analyze and compare plots (Exercises 1 and 2);"
• "write the deflection angle in terms of the relative index of refraction to calculate the deflection angle of rays entering spherical drops (Exercises 3 and 4);"
• "produce and describe graphs of deflection angle versus incident angle for light of different wavelengths (Exercises 3 and 4)."
• "identify rainbow angles for light of different wavelengths (Exercises 3 and 4);"
• "crudely estimate the intensity as a function of detection angle (Exercise 5)."

Target Audience:

The resource is intended for students in:

• Introductory courses covering optics.
• Upper-level optics courses.

Computational Tools:

The exercises require students to perform calculations and generate plots, suggesting the use of tools like:

• Python (as indicated by the provided Rainbows_Exercise_1.py code, which plots refractive indices).
• Easy JavaScript Simulation (EjsS), as the applet is built using this platform.

Instructor Guidance:

The instructor guide suggests that the exercises provide an opportunity to deduce why primary and secondary rainbows are bright in particular directions. It also mentions additional computational exercises in the American Journal of Physics.

Conclusion:

The "PICUP Refractive Indices of Water and Air JavaScript Simulation Applet HTML5" offers a comprehensive and engaging way for students to explore the physics behind rainbows. By combining the use of scientific literature, mathematical derivations, and computational analysis, it provides a deeper understanding of the role of refractive indices, light deflection, and the formation of these captivating optical phenomena. The exercises progressively build upon fundamental concepts of optics, culminating in a basic understanding of the intensity distribution that leads to the bright arcs we observe.

Frequently Asked Questions on Rainbow Formation

What are the primary optical phenomena responsible for the creation of rainbows?

Rainbows are primarily formed through the refraction and reflection of sunlight within water droplets. As sunlight enters a spherical raindrop, it is first refracted (bent) at the air-water interface. The light then travels to the back of the raindrop, where it undergoes internal reflection. Finally, as the light exits the raindrop, it is refracted again.

Why do rainbows exhibit different colors, and in what order are they typically observed in a primary rainbow?

The separation of sunlight into its constituent colors occurs due to the phenomenon of dispersion during refraction. The refractive index of water varies slightly with the wavelength (and thus color) of light. Violet light has a slightly higher refractive index than red light, causing it to be bent more. In a primary rainbow, the colors are observed in the order of red on the outer arc, followed by orange, yellow, green, blue, indigo, and violet on the inner arc.

What is the deflection angle of light in a raindrop, and what factors determine its value?

The deflection angle is the angle between the incident ray of sunlight and the outgoing ray after interacting with the raindrop. For a light ray undergoing one internal reflection (as in a primary rainbow), the deflection angle (δ) is given by δ = 2(θᵢ - θ<0xE1><0xB5><0xA7>) + (π - 2θ<0xE1><0xB5><0xA7>), where θᵢ is the angle of incidence and θ<0xE1><0xB5><0xA7> is the angle of refraction. The deflection angle depends on the incident angle and the refractive index of water at the specific wavelength of light.

What is the significance of the minimum deflection angle in the context of rainbow formation?

The minimum in the deflection angle curve (when plotted against the incident angle) is crucial for the visibility and brightness of rainbows. Near this minimum angle, many parallel rays of sunlight incident on the raindrops at slightly different angles will be deflected into almost the same outgoing direction. This "piling up" of light rays at a specific angle leads to a higher intensity of light observed by an observer, creating the bright arc of the rainbow. This phenomenon is known as rainbow scattering.

How is a secondary (double) rainbow formed, and how does it differ from a primary rainbow in terms of internal reflections and color order?

A secondary rainbow is formed when sunlight undergoes two internal reflections within the raindrop before exiting. The deflection angle for a secondary rainbow is given by δ = 2(θᵢ - θ<0xE1><0xB5><0xA7>) + 2(π - 2θ<0xE1><0xB5><0xA7>). Due to the additional internal reflection, the order of colors in a secondary rainbow is reversed compared to the primary rainbow, with red on the inner arc and violet on the outer arc. The secondary rainbow is also typically fainter than the primary rainbow because of the additional light lost during the second internal reflection.

At what approximate angles relative to the horizontal are the primary and secondary rainbows typically observed?

The bright bands of the primary rainbow (corresponding to the minimum deflection angles) are observed at angles of approximately 40.7° (violet) to 42.4° (red) relative to the horizontal, with the sun at the observer's back. The secondary rainbow appears at a higher angle, approximately 50.2° (red) to 53.3° (violet) relative to the horizontal, and the order of colors is reversed.

Why are rainbows observed as arcs?

Rainbows appear as arcs because the condition for seeing the intensified light at the minimum deflection angle is met by raindrops that lie on a cone centered on the line extending from the sun to the observer's eye. Only the raindrops along the circumference of this cone will return the intensified, dispersed light to the observer, resulting in the arc shape.

How does the wavelength of light affect the refractive index of water and the resulting rainbow?

The refractive index of water is wavelength-dependent; it is higher for shorter wavelengths (violet light) and lower for longer wavelengths (red light). This variation in refractive index leads to different deflection angles for different colors of light. Consequently, the minima in the deflection angle curves for different wavelengths occur at slightly different angles of incidence, resulting in the angular separation of colors observed in a rainbow. For the primary rainbow, violet light is deflected more and appears at a smaller angle relative to the anti-solar point than red light.