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Rainbows

Developed by E. Behringer

Easy JavaScript Simulation by Fremont Teng and Loo Kang Wee

This set of exercises guides the student in exploring primary and secondary rainbows. It requires the student to generate, observe, and describe plots of the deflection angles for light of different wavelengths and identify rainbow angles for different values of the relative index of refraction.

Subject Area	Waves & Optics
Levels	First Year and Beyond the First Year
Available Implementation	Python & Easy JavaScript Simulation
Learning Objectives	Students who complete this set of exercises will be able 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); and crudely estimate the intensity as a function of detection angle (Exercise 5).
Time to Complete	120 min

1instructorguide

This set of exercises may be used in an introductory course that covers optics or in an upper-level optics course. The first three computational exercises are not demanding, but collectively provide an opportunity to deduce physical consequences (e.g., why is it that the primary and secondary rainbows are bright in particular directions). The last exercise is intended purely as a crude, ‘zeroeth order’ estimate of the irradiance as a function of deflection angle, purposefully neglecting polarization effects. Additional computational exercises have been described by D.S. Amundsen et al., Am. J. Phys. 77 (9), 795-798 (2009). An experimental apparatus to explore rainbows up to the 6th order has been described by G. Casini and A. Covello, Am. J. Phys. 80 (11), 1027-1034 (2012).

2theory

For Exercises 3 and 4, the theoretical concepts required are the Law of Refraction and the Law of Reflection, which are standard topics in introductory courses and are usually revisited in intermediate or advanced optics courses.

For Exercise 5, the crude model is nothing more than summing the irradiances associated with different rays, which implicitly assumes that the rays are incoherent with one another. Depending on the course, this can be contrasted with the ideas underlying the derivation of far field diffraction patterns - namely, that one works with the electric fields and carefully accounts for the relative phases of contributions from different parts of an extended source.

Exercise 4

EXERCISE 4: WHERE IS THE SECONDARY (DOUBLE) RAINBOW?

Repeat the computation of Exercise 3, but now assume that the ray undergoes two internal reflections within the raindrop. In this case, show that the deflection angle is given by

δ = 2 (θ i - θ r) + 2 (π - 2 θ r) (2)

and plot the deflection angle versus the incident angle for rays of wavelength 400 nm and 650 nm. The two curves should have minima that are close to each other, but not identical. What are the values of the incident angle corresponding to these two minima? Assuming that these minima correspond to the rainbow direction, what direction do you have to look relative to the horizontal to see the bright red band of the secondary rainbow? Can a ground based observer see these rays? Do the rays that produce the secondary rainbow enter the top of the raindrop? What direction do you have to look relative to the horizontal to see the bright violet band? Which band appears higher in the sky?

Near the minimum of the deflection function (that is, the curve), are two rays with slightly different incident angles deflected into different directions? Regarding the brightness of the deflected light perceived by an observed, what is implied by your answer?

Rainbows_Exercise_4.py

# Rainbows_Exercise_4.py

# Plot the deflection function

# for the secondary (second-order) rainbow

# assuming a spherical water drop

# and a single wavelength,

# which implies a single index of refraction

# The refractive index of water is taken from

# Eq. 3 of P.D.T Huibers, Applied Optics,

# Vol. 36, No. 16, pp. 3785-3787 (1997).

# The refractive index of air is taken from

# Eq. 1 of P.E. Ciddor, Applied Optics,

# Vol. 35, No. 9, pp. 1566-1573 (1996).

# Written by:

# Ernest R. Behringer

# Department of Physics and Astronomy

# Eastern Michigan University

# Ypsilanti, MI 48197

# (734) 487-8799

# This email address is being protected from spambots. You need JavaScript enabled to view it.

# 20160109 by ERB

from __future__ import print_function

from pylab import figure,plot,xlim,xlabel,ylim,ylabel,grid,legend,title,show

from math import pi,asin

from numpy import linspace,sin,zeros,argmin

# Define the index of refraction function for water

def water_index(wavelength):

n_H2O = 1.31279 + 15.762/wavelength - 4382.0/(wavelength**2) + 1.1455e6/(wavelength**3)

return(n_H2O)

# Define the index of refraction function for air

# Note: wavelength is supposed to be in micrometers

def air_index_minus_one(wavelength):

term1 = 0.05792105/(238.0185 - 1.0e6/(wavelength**2))

term2 = 0.00167917/(57.362 - 1.0e6/(wavelength**2))

return(term1+term2)

# Inputs

wavelength_r = 650 # vacuum wavelength in nm ("red")

wavelength_v = 400 # vacuum wavelength in nm ("violet")

n_r = water_index(wavelength_r) # refractive index of water at wavelength_r

n_v = water_index(wavelength_v) # refractive index of water at wavelength_v

npts = 901

n_ar = 1.0 + air_index_minus_one(wavelength_r) # refractive index of air at wavelength_r

n_av = 1.0 + air_index_minus_one(wavelength_v) # refractive index of air at wavelength_v

theta_i_deg = linspace(0.0,90.0,npts) # array of incident angle values [deg]

theta_i = theta_i_deg*pi/180.0 # array of incident angle values [rad]

# Set up the arrays of deflection angles

theta_r = zeros(npts)

theta_v = zeros(npts)

# Calculate the refraction angle for each incident angle

# and for red and for violet light

for j in range(0,npts):

theta_r[j] = asin((n_ar/n_r)*sin(theta_i[j]))

theta_v[j] = asin((n_av/n_v)*sin(theta_i[j]))

# Calculate the deflection angle for each incident angle for red light

theta2r = 2.0*(theta_i - theta_r) + 2.0*(pi - 2.0*theta_r) # First deflection function

theta2r_deg = theta2r*180.0/pi

# Calculate the deflection angle for each incident angle for violet light

theta2v = 2.0*(theta_i - theta_v) + 2.0*(pi - 2.0*theta_v) # First deflection function

theta2v_deg = theta2v*180.0/pi

print("The secondary rainbow deflection angle for red light is ",min(theta2r_deg))

# The index of this value is

index_r = argmin(theta2r_deg)

print("This secondary rainbow deflection angle for red light corresponds to")

print("an incident angle of ",theta_i_deg[index_r]," deg.")

print("The red band of the secondary band appears at ",min(theta2r_deg)-180.0," above the horizontal.")

print(" ")

print("The secondary rainbow deflection angle for violet light is ",min(theta2v_deg))

# The index of this value is

index_v = argmin(theta2v_deg)

print("This secondary rainbow deflection angle for violet light corresponds to")

print("an incident angle of ",theta_i_deg[index_v]," deg.")

print("The violet band of the secondary band appears at ",min(theta2v_deg)-180.0," above the horizontal.")

#----------------------------------------------------------------------

# Start a new figure. This will be a plot of the deflection functions,

# i.e., deflection angle versus incident angle

figure()

# Set the limits of the horizontal axis

xlim(0,90)

# Label the horizontal axis

xlabel("$\\theta_i$ [deg]", size = 16)

# Set the limits of the vertical axis

ylim(220,360)

# Label the vertical axis

ylabel("$\\delta$ [deg]", size = 16)

# Draw a grid

grid(True)

# Plot the deflection functions

plot(theta_i_deg,theta2r_deg,"r-",label="$\lambda = 650 \\, {\\rm nm}$")

plot(theta_i_deg,theta2v_deg,"b-",label="$\lambda = 400 \\, {\\rm nm}$")

# Generate the legend

legend(loc=1)

# Generate the title

title("Deflection function for two internal reflections")

show()

3exercises

EXERCISE 1: OBTAIN AND USE INFORMATION FROM PEER-REVIEWED LITERATURE

Water and air are the two materials involved with producing rainbows we may see in the sky after a thunderstorm. To accurately predict where rainbows will appear, we need to have accurate information about the refractive indices of water and air.

(a) Obtain a copy of “Models for the wavelength dependence of the index of refraction of water”, Applied Optics 36 (16), 3785-3787 (1997) by Paul D.T. Huibers, and use Eq. (3) of that paper to generate a plot of the refractive index of water as a function of wavelength in the range from 400 to 650 nm.

(b) Obtain a copy of “Refractive index of air: new equations for the visible and near infrared”, Applied Optics 35(9), 1566-1573 (1996) by Philip E. Ciddor, and use Eq. (1) of that paper to generate a plot of $n_{a i r} - 1$ , the deviation of the refractive index of air from unity, as a function of wavelength in the range from 400 to 650 nm.

Which material has the larger change in refractive index over the wavelength range from 400 to 650 nm? Calculate the ratio of the larger change to the refractive index of the material for a wavelength of 400 nm, and comment on the magnitude of the ratio.

EXERCISE 2: DEFLECTION ANGLE FOR A LIGHT RAY ENTERING A SPHERICAL RAINDROP

Assume that a light ray incident on a spherical raindrop at an angle $θ_{1 i}$ measured with respect to the surface normal undergoes one internal reflection before leaving the raindrop, as shown below.

Alt Figure: Geometry of a ray incident on a spherical raindrop. `

Show that if $θ_{i} \equiv θ_{1 i}$ and $θ_{r} \equiv θ_{1 t}$ , then the deflection angle $δ$ of the light ray is (in radians):

\begin{matrix} (1) & δ = 2 (θ_{i} - θ_{r}) + (π - 2 θ_{r}) \end{matrix}

The deflection angle $δ$ is the angle between the incident ray and the outgoing ray, and $θ_{r}$ is the angle of refraction for the light ray incident from air and entering water. To compute the deflection angle, what quantities must be known?

EXERCISE 3: COMPUTE THE DEFLECTION ANGLE VERSUS INCIDENT ANGLE

Generate a plot of the deflection angle $δ$ as a function of incident angle $θ_{i}$ for light rays of wavelength 400 nm that experience one internal reflection in the raindrop. Compare, on the same plot, the deflection angle for light rays of wavelength 650 nm. The two curves should have minima that are close to each other, but not identical. What are the values of the incident angle corresponding to these two minima? Assuming that these minima correspond to the rainbow direction, what direction do you have to look relative to the horizontal to see the bright red band of the rainbow? What direction do you have to look relative to the horizontal to see the bright violet band? Which band appears higher in the sky?

Near the minimum of the deflection function (that is, the $δ (θ_{i})$ 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?

EXERCISE 4: WHERE IS THE SECONDARY (DOUBLE) RAINBOW?

Repeat the computation of Exercise 3, but now assume that the ray undergoes two internal reflections within the raindrop. In this case, show that the deflection angle is given by

\begin{matrix} (2) & δ = 2 (θ_{i} - θ_{r}) + 2 (π - 2 θ_{r}) \end{matrix}

EXERCISE 5: CRUDE ESTIMATE OF THE IRRADIANCE VERSUS DEFLECTION ANGLE FOR A SINGLE WAVELENGTH: PRIMARY RAINBOW

Assume that the each outgoing ray produces an irradiance that is equal to $I (θ) = I_{0} \exp [- (θ - δ)^{2}]$ where here the angles are assumed to be in degrees and the implied width of this distribution is $1^{\circ}$ . (For simplicity, we neglect the loss of intensity during refraction and internal reflection. This is a huge oversimplification, but it allows us to focus on adding up contributions from each ray.) Sum up the contributions from rays uniformly distributed over the scaled impact parameter $\tilde{b} = b / R$ to compute the overall irradiance as a function of deflection angle for light of wavelength 400 nm. Plot the resulting irradiance distribution versus deflection angle for a single wavelength. How does this plot help explain why the rainbow is bright?

4solutions

Exercise 1: Obtain and use information from peer-reviewed literature

The plot for the refractive index of water should look like:

Alt Figure: Geometry of a ray incident on a spherical raindrop. `

The plot for the deviation of the refractive index of air from unity should look like:

Alt Figure: Geometry of a ray incident on a spherical raindrop.

Exercise 2: Deflection angle for a light ray entering a spherical raindrop

To calculate the deflection angle, one needs to know the refractive index of the air and the water at the particular wavelength and also the angle of incidence of the incoming ray. The laws of refraction and reflection are then applied to obtain the angles of refraction and the angle of reflection for the internal reflection. Students should be able to show that the equalities shown in the figure illustrating the ray/raindrop geometry.

Exercise 3: Compute the deflection angle versus incident angle

The solution for Exercise 3 is:

Alt Figure: Geometry of a ray incident on a spherical raindrop.

The minimum of the deflection function for $λ = 400$ nm ( $λ = 650$ nm) is $δ = {139.3}^{\circ}$ $(δ = {137.6}^{\circ})$ . This means that you must look (with the sun at your back and the distant raindrops in front of you) at an angle of $180^{\circ} - {139.3}^{\circ} = {40.7}^{\circ}$ $(180^{\circ} - {137.6}^{\circ} = {42.4}^{\circ})$ relative to the horizontal to see the bright violet (red) band of the rainbow. So the red band appears above the violet band. (It really does!) It is worth noting that the rays producing the primary rainbow enter the top half of the raindrops.

Exercise 4: Where is the double (secondary) rainbow?

The solution for Exercise 4 is: Alt Figure: Geometry of a ray incident on a spherical raindrop.

The minimum of the deflection function for $λ = 400$ nm ( $λ = 650$ nm) is $δ = {233.3}^{\circ}$ ( $δ = {230.2}^{\circ}$ ). These rays are directed back up into the sky and the ground-based observer wonâ€™t see them. However, if you realize that you have just computed the deflection angles for rays entering the bottom half of the raindrop, you realize that you must look (with the sun at your back and the distant raindrops in front of you) at an angle of ${233.3}^{\circ} - 180^{\circ} = {53.3}^{\circ},, ({230.2}^{\circ} - 180^{\circ} = {50.2}^{\circ})$ relative to the horizontal to see the bright violet (red) band of the rainbow. So the red band of the secondary rainbow appears below the violet band. (It really does!). Thus the rays producing the secondary rainbow enter the bottom half of the raindrops and the order of the color bands is reversed relative to the primary rainbow. Comparing these angles to those from Exercise 3, we see that the secondary rainbow appears above the primary rainbow. The reduced brightness of the secondary rainbow is due to the additional internal reflection and the corresponding loss of irradiance during that reflection. It is posible to extend Exercises 3 and 4 to account for the change in intensity occurring for every refraction or internal reflection. For fun, it must be noted that fascination with the double rainbow became an Internet meme in 2010 when a person living on the border of Yosemite National Park recorded his reaction to a spectacular double rainbow. At the time of writing, the video can be seen at: https://www.youtube.com/watch?v=OQSNhk5ICTI).

Exercise 5: Crude estimate of the irradiance versus deflection angle for a single wavelength: Primary Rainbow

The solution for Exercise 5 is: Alt Figure: Geometry of a ray incident on a spherical raindrop.

Note that irradiance is distributed over a large range of detection angles, and that the irradiance has a peak near (but not at) the minimum of the deflection function for these chosen parameters. If we use an angular width smaller than $1^{\circ}$ for the assumed irradiance distribution of a ray, the irradiance peak approaches the minimum in the deflection angle and the “background irradiance” decreases. 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, and it is also observed in scattering events such as atomic/molecular collisions or ion collisions with ordered surfaces.

5references

Most introductory textbooks mention rainbows and the angles at which they can be observed, but do not necessarily explain why rainbows are bright. This set of exercises is intended to show how irradiance “piles up” around the minima of the deflection functions calculated in Exercises 3 and 4.

Translations

Code	Language		Translator	Run

Credits

Fremont Teng; Loo Kang Wee

This document provides a briefing on the "PICUP Where is the Secondary Rainbow JavaScript Simulation Applet HTML5" resource, focusing on its purpose, content, and key ideas related to primary and secondary rainbows. The resource is designed as a set of exercises utilizing a JavaScript simulation to guide students in exploring the physics behind rainbows, particularly the secondary rainbow. It emphasizes a computational approach, requiring students to analyze deflection angles of light rays passing through spherical raindrops for different wavelengths and numbers of internal reflections.

Main Themes and Important Ideas:

Exploring Primary and Secondary Rainbows through Simulation: The core of this resource is a set of exercises that use a JavaScript simulation applet to visualize and analyze the formation of primary and secondary rainbows. The "About" section explicitly states: "This set of exercises guides the student in exploring primary and secondary rainbows. It requires the student to generate, observe, and describe plots of the deflection angles for light of different wavelengths and identify rainbow angles for different values of the relative index of refraction." The embed link provided allows for direct integration of the simulation into web pages.
Computational Approach to Understanding Rainbow Formation: The exercises heavily rely on computational tasks. Students are expected to "Plot an equation over a range of values. Analyze and compare plots ( Exercises 1 and 2 )" and to "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 )." This hands-on approach aims to deepen understanding by linking theoretical concepts with visual and numerical results.
Deflection Angle and Rainbow Angles: A central concept explored is the deflection angle of light rays as they interact with raindrops. The resource guides students to "produce and describe graphs of deflection angle versus incident angle for light of different wavelengths ( Exercises 3 and 4 )" and to "identify rainbow angles for light of different wavelengths ( Exercises 3 and 4 )." The minima in the deflection angle curves are identified as corresponding to the rainbow directions, where a higher intensity of light is observed due to the "piling up" of rays deflected at similar angles.
Secondary Rainbow Specifics (Exercise 4): Exercise 4 specifically focuses on the secondary rainbow, where light undergoes two internal reflections within the raindrop. The theoretical deflection angle for the secondary rainbow is provided: "δ = 2 ( θ i − θ r ) + 2 ( π − 2 θ r ) (2)". Students are tasked with plotting this for different wavelengths (400 nm and 650 nm) and analyzing the minima to determine the viewing angle of the secondary rainbow.
The solution to Exercise 4 reveals key differences between primary and secondary rainbows:

The red band of the secondary rainbow appears below the violet band, a reversal of the order in the primary rainbow.
"The rays producing the secondary rainbow enter the bottom half of the raindrops."
"Comparing these angles to those from Exercise 3, we see that the secondary rainbow appears above the primary rainbow."
The secondary rainbow is fainter due to the additional internal reflection and the associated loss of irradiance.

Role of Refractive Index: Accurate refractive indices of water and air for different wavelengths are crucial for predicting rainbow angles. Exercise 1 requires students to obtain and use information from peer-reviewed literature ("Models for the wavelength dependence of the index of refraction of water" by Huibers and "Refractive index of air: new equations for the visible and near infrared" by Ciddor) to plot these values. This highlights the wavelength dependence of refraction and its role in the dispersion of light that creates the colors of the rainbow.
Brightness and Minimum Deflection Angle (Exercises 3, 4, and 5): Exercises 3, 4, and 5 address the brightness of rainbows. The text explains that near the minimum of the deflection function, "two rays with slightly different incident angles [are] deflected into different directions." However, the crucial point is that many rays with incident angles around the angle corresponding to the minimum deflection are all scattered in approximately the same direction. This "piling up" of light intensity at the rainbow angle is why rainbows are visible and bright. Exercise 5 provides a "crude estimate of the irradiance" to further illustrate this concept, stating, "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..."
Learning Objectives and Target Audience: The resource is designed for "First Year and Beyond the First Year" students in courses covering "Waves & Optics." The learning objectives include:

Obtaining and using information from peer-reviewed literature.
Plotting and analyzing equations and graphs.
Writing deflection angles in terms of the refractive index.
Producing and describing graphs of deflection angle versus incident angle.
Identifying rainbow angles for different wavelengths.
Crudely estimating intensity as a function of detection angle.

Theoretical Foundation: The "Theory" section explicitly mentions that "the theoretical concepts required are the Law of Refraction and the Law of Reflection," fundamental topics in introductory and advanced optics courses. Exercise 5 also touches upon the concept of incoherent superposition of light waves for a simplified intensity model, contrasting it with coherent superposition in diffraction.

Key Facts and Figures:

The deflection angle for a primary rainbow (one internal reflection) is given by: δ = 2 ( θ i − θ r ) + ( π − 2 θ r ) (1)
The deflection angle for a secondary rainbow (two internal reflections) is given by: δ = 2 ( θ i − θ r ) + 2 ( π − 2 θ r ) (2)
For the primary rainbow, the minimum deflection angles calculated are approximately 139.3° for violet light (400 nm) and 137.6° for red light (650 nm), leading to viewing angles of 40.7° and 42.4° above the horizontal, respectively.
For the secondary rainbow, the minimum deflection angles calculated are approximately 233.3° for violet light (400 nm) and 230.2° for red light (650 nm), leading to viewing angles of 53.3° and 50.2° above the horizontal, respectively (measured from 180°).

Conclusion:

The "PICUP Where is the Secondary Rainbow JavaScript Simulation Applet HTML5" resource provides a valuable, computationally-driven approach to understanding the formation of primary and secondary rainbows. By engaging students in plotting, analyzing, and interpreting deflection angles for different wavelengths and reflection scenarios, it fosters a deeper understanding of the underlying optical principles, including refraction, reflection, and the wavelength dependence of these phenomena. The resource effectively connects theoretical concepts to observable phenomena, explaining why rainbows appear at specific angles and why the secondary rainbow exhibits a reversed color order and reduced brightness.

Rainbow Formation Study Guide

Quiz

What are the two primary optical phenomena responsible for the formation of rainbows as discussed in the theory section?
Describe the path of light that leads to the formation of a primary rainbow within a spherical raindrop.
How does the number of internal reflections differ between a primary and a secondary rainbow?
Explain why different wavelengths of light are deflected at slightly different angles when passing through a raindrop.
According to Exercise 3, at what approximate angle relative to the horizontal does the bright red band of the primary rainbow appear to a ground-based observer? What about the violet band?
In Exercise 4, the minimum deflection angle for the secondary rainbow is calculated to be greater than 180 degrees. Explain the significance of this result for a ground-based observer.
Do the rays that produce the secondary rainbow enter the top or bottom of a raindrop? How does this compare to the primary rainbow?
How does the order of colors in a secondary rainbow differ from that in a primary rainbow? Explain why this reversal occurs.
What is meant by "rainbow scattering" as discussed in the solution to Exercise 5, and why does it lead to the brightness of rainbows?
According to the instructor guide, what is the main purpose of the set of exercises regarding rainbows?

Quiz Answer Key

The two primary optical phenomena responsible for rainbow formation are the Law of Refraction, which describes how light bends when entering a different medium, and the Law of Reflection, which describes how light bounces off a surface. These laws govern the behavior of sunlight as it interacts with raindrops.
In a primary rainbow, sunlight enters the top half of a spherical raindrop and is refracted. It then undergoes one internal reflection off the back surface of the raindrop and is refracted again as it exits the top half, directed towards an observer. The deflection angle depends on the wavelength.
A primary rainbow involves one internal reflection of light within the raindrop before exiting, whereas a secondary rainbow involves two internal reflections. This additional reflection alters the path and the observed angle of the light.
Different wavelengths of light are deflected at slightly different angles due to the wavelength dependence of the refractive index of water. This phenomenon, known as dispersion, causes the separation of white light into its constituent colors, creating the spectrum observed in a rainbow.
According to the solution for Exercise 3, the bright red band of the primary rainbow appears at approximately 42.4 degrees above the horizontal, while the bright violet band appears at approximately 40.7 degrees above the horizontal. Thus, the red band is higher in the sky.
A minimum deflection angle greater than 180 degrees for the secondary rainbow initially suggests that the rays are directed upwards. However, considering the light entering the bottom of the raindrop, these rays are seen by a ground observer at an angle above the horizontal (deflection angle - 180 degrees).
The rays producing the secondary rainbow enter the bottom half of the raindrop. In contrast, the rays that produce the primary rainbow enter the top half of the raindrops.
The order of colors in a secondary rainbow is reversed compared to a primary rainbow. In a primary rainbow, the order is red on top and violet on the bottom. In a secondary rainbow, the order is reversed, with violet on top and red on the bottom, due to the two internal reflections.
Rainbow scattering refers to the accumulation of irradiance around the minimum of the deflection function. This occurs because multiple rays with slightly different incident angles are deflected into nearly the same direction, leading to a higher intensity of light observed at that particular angle, thus making the rainbow appear bright.
The main purpose of the set of exercises is to demonstrate how the irradiance of sunlight "piles up" around the minima of the deflection functions for primary and secondary rainbows, thus explaining why rainbows are bright in particular directions.

Essay Format Questions

Discuss the role of refraction and reflection in the formation of both primary and secondary rainbows. Compare and contrast the light paths involved and explain how these paths lead to the distinct visual characteristics of each type of rainbow.
Explain the concept of the deflection angle for light passing through a spherical raindrop. How does the deflection angle vary with the incident angle and the wavelength of light for both primary and secondary rainbows, and what is the significance of the minimum deflection angle?
Analyze the wavelength dependence of the refractive index of water and air, and discuss how this dependence leads to the dispersion of sunlight and the separation of colors observed in rainbows. Be sure to address the order of colors in both primary and secondary rainbows in your explanation.
The exercises mention that the brightness of a rainbow is related to the minima of the deflection function. Elaborate on why the intensity of light is higher at these angles, connecting your explanation to the concept of "rainbow scattering" and the paths of individual light rays.
Consider the perspective of a ground-based observer viewing a primary and a secondary rainbow simultaneously. Describe the relative positions of the two rainbows in the sky, the order of colors in each, and the reason for any differences in their apparent brightness.

Glossary of Key Terms

Refraction: The bending of light as it passes from one transparent medium to another due to a change in speed.
Reflection: The bouncing back of light when it strikes a surface.
Angle of Incidence (θi): The angle between an incoming ray of light and the normal (a line perpendicular to the surface) at the point of incidence.
Angle of Refraction (θr): The angle between a refracted ray of light and the normal at the point of refraction.
Deflection Angle (δ): The angle between the incident ray and the outgoing ray of light after interacting with a raindrop.
Wavelength (λ): The distance between successive crests or troughs of a wave, such as a light wave. Different wavelengths of visible light correspond to different colors.
Refractive Index (n): A measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. It depends on the wavelength of light and the properties of the medium.
Dispersion: The phenomenon in which the phase velocity of a wave depends on its frequency (or wavelength). In optics, this is seen as the separation of white light into its constituent colors when refracted by a prism or raindrop because different colors of light have different refractive indices.
Internal Reflection: The reflection of light that occurs at the boundary between two media when the light is traveling from a medium with a higher refractive index to one with a lower refractive index, and the angle of incidence is greater than the critical angle. In rainbows, it refers to the reflection off the back surface of the raindrop.
Rainbow Angle: The angle of observation relative to the direction of the sun at which the intensity of the rainbow is highest, corresponding to the minimum of the deflection angle.
Irradiance: The power per unit area incident on a surface, often used to describe the intensity of light.
Rainbow Scattering: The phenomenon where multiple light rays with slightly different incident angles are deflected into nearly the same outgoing direction, leading to an increased intensity of light observed at that angle, particularly near the minimum of the deflection function.

Sample Learning Goals

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For Teachers

Where is the Secondary Rainbow JavaScript Simulation Applet HTML5

Instructions

Control Panel

Adjusting the field boxes will change their variables respectively.

Toggling Full Screen

Double clicking anywhere in the panel will toggle full screen.

Reset Button

Resets the simulation.

Research

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Video

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Version:

Other Resources

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Frequently Asked Questions about Rainbows

What are the fundamental optical principles behind the formation of rainbows?

Rainbows are formed through a combination of refraction and reflection of sunlight within spherical raindrops. When sunlight enters a raindrop, it is refracted (bent) at the air-water interface due to the change in the speed of light. Different wavelengths of light are refracted at slightly different angles, a phenomenon known as dispersion. 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.

What is the difference between a primary and a secondary rainbow in terms of light interaction with raindrops?

In a primary rainbow, sunlight undergoes one internal reflection within the raindrop. The deflection angle for the primary rainbow has a minimum that depends on the wavelength of light, leading to the separation of colors and the formation of the bow. In a secondary rainbow, the light undergoes two internal reflections within the raindrop. This additional reflection reverses the order of the colors compared to the primary rainbow (red is on the inside and violet on the outside) and results in a fainter bow that appears higher in the sky at a larger angle from the original direction of sunlight.

How does the deflection angle of light relate to the observation of rainbows?

The deflection angle is the angle between the incident ray of sunlight and the outgoing ray after interacting with the raindrop. Rainbows are observed at angles where the deflection angle is at a minimum (or maximum, depending on the number of internal reflections). Near these minimum/maximum deflection angles, many rays with slightly different incident angles are deflected into nearly the same direction. This "piling up" of light rays at specific angles leads to the increased brightness of the rainbow at those angles. The different minimum deflection angles for different wavelengths result in the separation of colors that we see in a rainbow.

Why are rainbows typically seen as arcs?

Rainbows appear as arcs because the condition for observing a specific color (i.e., the minimum deflection angle for that wavelength) is met by raindrops that lie on a circle in the sky, centered on the anti-solar point (the point directly opposite the sun from the observer's perspective). For a ground-based observer, the lower part of this circle may be obscured by the horizon, making the rainbow appear as an arc.

How does the refractive index of water and air affect the appearance of rainbows?

The refractive index of water determines the angle at which light is refracted when entering and exiting the raindrop. Since the refractive index of water varies slightly with the wavelength of light (dispersion), different colors are bent by different amounts, leading to the separation of colors in the rainbow. The refractive index of air also plays a role in the initial refraction, although its variation with wavelength is smaller than that of water. Accurate knowledge of these refractive indices for different wavelengths is crucial for predicting the exact angles at which rainbows will appear.

What is the significance of the minima in the deflection angle versus incident angle plots for rainbows?

The minima in the deflection angle versus incident angle plots correspond to the angles at which the intensity of the deflected light is highest. This is because, near these minima, a range of incident angles results in outgoing rays that are deflected at very similar angles. This concentration of light at specific angles is what creates the bright bands of color that we perceive as a rainbow. The different minima for different wavelengths explain why we see distinct color bands at slightly different angles.

Why is the secondary rainbow fainter than the primary rainbow, and why are its colors reversed?

The secondary rainbow is fainter because the light undergoes an additional internal reflection within the raindrop compared to the primary rainbow. At each reflection, some light is lost due to refraction out of the drop, so after two internal reflections, less light emerges to form the secondary rainbow. The colors are reversed because the second internal reflection causes a different sequence of angular separation for the colors. The deflection angle for the secondary rainbow is derived differently, leading to a different order of the wavelengths in the observed bow.

Can the intensity of a rainbow be estimated based on the deflection angle?

A crude estimate of the intensity of a rainbow can be made by considering how irradiance accumulates near the minimum of the deflection function. While neglecting polarization effects and losses due to refraction and reflection, a model can sum the contributions from rays with different incident angles. The resulting intensity distribution as a function of deflection angle shows a peak near the minimum deflection angle, indicating that the rainbow appears bright in that direction because multiple rays are deflected towards the observer at similar angles. This "rainbow scattering" effect explains the enhanced brightness of the rainbow.

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Details: Written by Fremont; Parent Category: 03 Waves; Category: 04 Light; Published: 11 June 2018; Created: 11 June 2018; Last Updated: 22 March 2025; Hits: 4098

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