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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 2

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  measured with respect to the surface normal undergoes one internal reflection before leaving the raindrop, as shown below.

`

Show that if  and , then the deflection angle  of the light ray is (in radians):

The deflection angle  is the angle between the incident ray and the outgoing ray, and  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

EXERCISE 3: COMPUTE THE DEFLECTION ANGLE VERSUS INCIDENT ANGLE

Generate a plot of the deflection angle  as a function of incident angle  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  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?


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 nair1nair1, 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 θ1iθ1i 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 θiθ1iθiθ1i and θrθ1tθrθ1t, then the deflection angle δδ of the light ray is (in radians):

δ=2(θiθr)+(π2θr)(1)(1)δ=2(θiθr)+(π2θr)

The deflection angle δδ is the angle between the incident ray and the outgoing ray, and θrθr 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 δδ as a function of incident angle θiθi 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(θi)δ(θi) 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

δ=2(θiθr)+2(π2θr)(2)(2)δ=2(θiθr)+2(π2θr)

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 δ(θi)δ(θi) 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?

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)=I0exp[(θδ)2]I(θ)=I0exp[(θδ)2] where here the angles are assumed to be in degrees and the implied width of this distribution is 11. (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 b~=b/Rb~=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?

Code Rainbows_Exercise_3.py

#

# Primary_Rainbow_Deflection_Function.py

#

# Plot the deflection function

# for the primary (first-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

theta1r = 2.0*(theta_i - theta_r) + 1.0*(pi - 2.0*theta_r) # First deflection function

theta1r_deg = theta1r*180.0/pi

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

theta1v = 2.0*(theta_i - theta_v) + 1.0*(pi - 2.0*theta_v) # First deflection function

theta1v_deg = theta1v*180.0/pi

print("The primary rainbow deflection angle for red light is ",min(theta1r_deg))

# The index of this value is

index_r = argmin(theta1r_deg)

print("This primary 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 primary band appears at ",180.0-min(theta1r_deg)," above the horizontal.")

print(" ")

print("The primary rainbow deflection angle for violet light is ",min(theta1v_deg))

# The index of this value is

index_v = argmin(theta1v_deg)

print("This primary 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 primary band appears at ",180.0-min(theta1v_deg)," 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(135,185)

# Label the vertical axis

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

# Draw a grid

grid(True)

# Plot the deflection functions

plot(theta_i_deg,theta1r_deg,"r-",label="\(\lambda = 650 \\, {\\rm nm}\)")

plot(theta_i_deg,theta1v_deg,"b-",label="\(\lambda = 400 \\, {\\rm nm}\)")

# Generate the legend

legend(loc=1)

# Generate the title

title("Deflection function for one internal reflection")

show()

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=400λ=400 nm (λ=650λ=650 nm) is δ=139.3δ=139.3 (δ=137.6)(δ=137.6). This means that you must look (with the sun at your back and the distant raindrops in front of you) at an angle of 180139.3=40.7180139.3=40.7 (180137.6=42.4)(180137.6=42.4) 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=400λ=400 nm (λ=650λ=650 nm) is δ=233.3δ=233.3 (δ=230.2δ=230.2). These rays are directed back up into the sky and the ground-based observer won’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 233.3180=53.3,,(230.2180=50.2)233.3180=53.3,,(230.2180=50.2) 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 11 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

Source Physics @ Singapore. This resource consists of a set of exercises and a JavaScript simulation designed to help students explore the physics behind primary and secondary rainbows. The materials focus on understanding the deflection angles of light passing through spherical raindrops of different wavelengths and how these angles relate to the observation of rainbows.

Main Themes and Important Ideas/Facts:

  1. Educational Resource for Optics: The resource is designed as a learning module for students in introductory and upper-level optics courses. It aims to guide students in understanding the formation of rainbows through a combination of theoretical concepts, computational exercises, and a simulation applet. The "About" section states that the 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."
  2. Focus on Deflection Angles: A central theme is the calculation and analysis of the deflection angle of light rays as they interact with spherical raindrops. The resource provides equations for the deflection angle for both primary (one internal reflection) and secondary (two internal reflections) rainbows.
  • Primary Rainbow Deflection Angle (Exercise 2): The deflection angle \(\delta\) for a ray undergoing one internal reflection is given by:
  • δ = 2 ( θ i − θ r ) + ( π − 2 θ r )
  • where \(\theta_i\) is the angle of incidence and \(\theta_r\) is the angle of refraction. The exercise asks students to determine what quantities must be known to compute this angle, which includes the refractive indices of air and water at the specific wavelength and the incident angle.
  • Secondary Rainbow Deflection Angle (Exercise 4): For a ray undergoing two internal reflections, the deflection angle is:
  • δ = 2 ( θ i − θ r ) + 2 ( π − 2 θ r )
  1. Wavelength Dependence and Dispersion: The exercises emphasize the role of wavelength in the refractive index of water and air, leading to the dispersion of sunlight into different colors in a rainbow. Exercise 3 instructs students to "Generate a plot of the deflection angle \(\delta\) as a function of incident angle \(\theta i\) 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 solution reveals that the minima of the deflection angles for these wavelengths are slightly different, leading to the separation of colors.
  2. Rainbow Angle and Brightness: The resource explains that the bright bands of a rainbow occur at the minimum deflection angle. Near this minimum, many rays with slightly different incident angles are deflected in nearly the same direction, causing an accumulation of light intensity.
  • Regarding the primary rainbow (Exercise 3), the solution states: "The minimum of the deflection function for \(\lambda = 400\) nm (\(\lambda = 650\) nm) is \(\delta = 139.3^\circ\) (\(\delta = 137.6^\circ\)). 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."
  • Exercise 3 also poses the question: "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 implication is that near the minimum, rays are deflected in almost the same direction, leading to increased brightness.
  1. Secondary Rainbow Characteristics: Exercise 4 explores the secondary rainbow, involving two internal reflections. The solution indicates that the order of colors is reversed compared to the primary rainbow, and the secondary rainbow appears at a higher angle in the sky. "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 is attributed to the additional internal reflection.
  2. Irradiance and Rainbow Scattering: Exercise 5 provides a simplified model for estimating the irradiance as a function of deflection angle, neglecting polarization effects. The solution explains 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..."
  3. Integration of Peer-Reviewed Literature: Exercise 1 requires students to obtain and use information from specific papers in Applied Optics to plot the refractive indices of water and air as a function of wavelength. This emphasizes the importance of using accurate data for predicting rainbow phenomena.
  4. Available Implementation: The resource is available in Python and Easy JavaScript Simulation, providing flexibility for different learning environments. The provided Python code (Rainbows_Exercise_3.py) demonstrates the calculation and plotting of the deflection angle for primary rainbows for red and violet light.

Quotes:

  • "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." (About section)
  • "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." (Solution to Exercise 2)
  • "The minimum of the deflection function for \(\lambda = 400\) nm (\(\lambda = 650\) nm) is \(\delta = 139.3^\circ\) (\(\delta = 137.6^\circ\)). 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." (Solution to Exercise 3)
  • "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..." (Solution to Exercise 5)

Conclusion:

The PICUP Primary Rainbow Deflection Function JavaScript Simulation Applet HTML5 resource offers a comprehensive and interactive approach to understanding the physics of rainbows. By combining theoretical derivations, computational exercises, and a simulation, it allows students to explore the relationship between light's interaction with raindrops and the formation of primary and secondary rainbows. The emphasis on wavelength dependence, deflection angles, and the concept of rainbow scattering provides a deeper understanding of this optical phenomenon. The inclusion of exercises requiring the use of peer-reviewed literature also promotes scientific literacy.

 

 

Study Guide: Primary and Secondary Rainbow Formation

Key Concepts

  • Refraction: The bending of light as it passes from one medium to another due to a change in speed. This is governed by Snell's Law (Law of Refraction).
  • Reflection: The bouncing of light off a surface. The angle of incidence equals the angle of reflection (Law of Reflection).
  • Deflection Angle (δ): The angle between the incident ray of light and the outgoing ray after interacting with a raindrop.
  • Incident Angle (θi): The angle at which a light ray strikes a surface, measured with respect to the normal (a line perpendicular to the surface).
  • Refraction Angle (θr): The angle at which a light ray travels after passing through a surface, measured with respect to the normal.
  • Index of Refraction (n): A measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum. The relative index of refraction between two media is the ratio of their refractive indices.
  • Wavelength (λ): The distance between successive crests (or troughs) of a wave. Different wavelengths of visible light correspond to different colors.
  • Dispersion: The phenomenon where the refractive index of a material varies with the wavelength of light, causing different colors of light to be refracted at slightly different angles.
  • Primary Rainbow: A rainbow formed by light undergoing one internal reflection inside a raindrop.
  • Secondary Rainbow: A fainter and higher rainbow formed by light undergoing two internal reflections inside a raindrop. The order of colors is reversed compared to the primary rainbow.
  • Rainbow Angle: The angle of observation relative to the anti-solar point (the direction opposite the sun) where the intensity of the refracted and reflected light is highest, corresponding to the minimum deflection angle.
  • Irradiance: The power per unit area carried by electromagnetic radiation. In this context, it relates to the brightness of the rainbow.
  • Rainbow Scattering: The phenomenon where light accumulates in specific directions corresponding to the minima in the deflection function, resulting in the bright bands of a rainbow.

Quiz

  1. Describe the path of a light ray that creates a primary rainbow within a spherical raindrop. How many refractions and internal reflections occur?
  2. What is the deflection angle (δ), and what factors determine its value for a light ray entering a spherical raindrop and undergoing one internal reflection?
  3. Explain the role of the index of refraction of water and air in the formation of a rainbow. How does the index of refraction vary with the wavelength of light?
  4. Why do different colors of light appear at different angles in a rainbow? What phenomenon is responsible for this separation of colors?
  5. What is the significance of the minimum deflection angle in the context of rainbow formation? How does it relate to the observed position of the rainbow?
  6. Describe the key differences in the formation and appearance of a primary and a secondary rainbow, including the number of internal reflections and the order of colors.
  7. According to Exercise 3, at what approximate angles relative to the horizontal would you observe the bright red and violet bands of the primary rainbow? Which color appears higher in the sky?
  8. Based on Exercise 4, where does the secondary rainbow appear relative to the primary rainbow? Are the rays forming the secondary rainbow directed towards a ground-based observer after exiting the raindrop?
  9. Explain why rainbows appear as arcs in the sky rather than full circles. What determines the size of the visible arc?
  10. How does the concept of "rainbow scattering" explain the brightness of the rainbow at the angles corresponding to the minimum deflection?

Quiz Answer Key

  1. A light ray enters the spherical raindrop, refracts at the air-water interface, undergoes one internal reflection off the back surface of the raindrop, and then refracts again as it exits the raindrop back into the air. This process involves two refractions and one internal reflection.
  2. The deflection angle (δ) is the total angle through which the light ray has been deviated from its original path. It is determined by the incident angle (θi) and the angle of refraction (θr), which in turn depend on the refractive indices of air and water at the given wavelength through Snell's Law.
  3. The difference in refractive indices between air and water causes the initial refraction and the final refraction of light. The wavelength dependence of the refractive index (dispersion) causes different colors of light to bend at slightly different angles upon refraction.
  4. Different colors of light have different wavelengths, and the refractive index of water varies slightly with wavelength (dispersion). This causes each color to be refracted at a slightly different angle upon entering and exiting the raindrop, leading to the separation of colors in the rainbow.
  5. The minimum deflection angle corresponds to the direction where a large number of light rays with slightly different incident angles are deflected into a very small range of outgoing angles. This concentration of light intensity at a specific viewing angle results in the bright bands of the rainbow.
  6. A primary rainbow involves one internal reflection and has red on the outside and violet on the inside. A secondary rainbow involves two internal reflections, is fainter due to additional light loss, appears higher in the sky above the primary rainbow, and has the order of colors reversed (violet on the outside, red on the inside).
  7. According to Exercise 3, the bright red band appears at approximately 42.4° above the horizontal, and the bright violet band appears at approximately 40.7° above the horizontal. Therefore, the red band appears higher in the sky than the violet band in the primary rainbow.
  8. The secondary rainbow appears above the primary rainbow. The rays initially computed in Exercise 4 are directed back up into the sky and would not be seen by a ground-based observer directly. However, by considering rays entering the bottom of the raindrop, the secondary rainbow is observed at a higher angle than the primary.
  9. Rainbows appear as arcs because the condition for seeing a particular color at its rainbow angle is met by raindrops that lie on a cone whose apex is at the observer's eye and whose axis passes through the sun. Only the portion of this cone intersecting the region of falling rain is visible as the arc.
  10. Near the minimum of the deflection function, many rays with slightly different incident angles are deflected into almost the same outgoing direction. This "piling up" of irradiance in a specific direction is called rainbow scattering and explains why the rainbow is brightest at these angles.

Essay Format Questions

  1. Discuss the role of refraction, reflection, and dispersion in the formation of a primary rainbow. Explain how these optical phenomena interact to create the observed separation of colors and the arc shape.
  2. Compare and contrast the formation and appearance of primary and secondary rainbows. What differences in the path of light within the raindrops lead to the distinct characteristics of each type of rainbow?
  3. Explain the concept of the deflection angle for light interacting with a spherical raindrop. How does analyzing the deflection angle as a function of the incident angle help to understand the formation and brightness of rainbows?
  4. Critically evaluate the "crude estimate of irradiance" model presented in Exercise 5 for the primary rainbow. What does this simplified model reveal about the brightness of rainbows, and what are its key limitations? How could a more sophisticated model improve upon this estimate?
  5. Based on the provided materials, discuss the relationship between the wavelength of light and the observed angles of the primary and secondary rainbows. How does the wavelength dependence of the refractive index determine the order and position of the colors in each rainbow?

Glossary of Key Terms

  • Angle of Incidence: The angle made by a ray incident on a surface with the normal to the surface at the point of incidence.
  • Angle of Reflection: The angle made by a reflected ray with the normal to the reflecting surface.
  • Angle of Refraction: The angle made by a refracted ray with the normal to the refracting surface.
  • Anti-Solar Point: The point in the sky that is directly opposite the sun from the observer's perspective. The center of a full rainbow arc would be on the horizon at this point.
  • Deflection Function: A mathematical relationship describing how the deflection angle of light changes as a function of the incident angle on a raindrop.
  • Index of Refraction: The ratio of the speed of light in a vacuum to the speed of light in a medium.
  • Internal Reflection: The reflection of light that occurs when a ray traveling in a transparent medium strikes an interface with a less optically dense medium at an angle greater than the critical angle. In rainbow formation, it refers to reflection off the back surface of the raindrop.
  • Law of Reflection: States that the angle of incidence is equal to the angle of reflection, and that the incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
  • Law of Refraction (Snell's Law): States the relationship between the angles of incidence and refraction, and the refractive indices of the two media at the interface: n₁ sin θ₁ = n₂ sin θ₂.
  • Normal: A line perpendicular to a surface at a given point. Angles of incidence, reflection, and refraction are measured with respect to the normal.
  • Primary Rainbow: The brighter, lower rainbow in which light undergoes one internal reflection within each raindrop. The colors appear in the order of red (outer) to violet (inner).
  • Rainbow: An optical phenomenon caused by refraction, internal reflection, and dispersion of sunlight by water droplets, resulting in a spectrum of light appearing in the sky as arcs of color.
  • Rainbow Angle: The specific angle of viewing relative to the anti-solar point where the intensity of light is highest for a particular order of rainbow.
  • Rainbow Scattering: The enhancement of light intensity in specific directions due to the behavior of the deflection function having a minimum. Many rays with slightly different incident angles are scattered into a narrow range of angles near this minimum.
  • Refraction: The bending of a wave (such as light) as it passes from one medium to another due to a change in its speed.
  • Secondary Rainbow: The fainter, higher rainbow in which light undergoes two internal reflections within each raindrop. The colors appear in the reversed order of violet (outer) to red (inner).
  • Wavelength: The spatial period of a periodic wave—the distance over which the wave's shape repeats. For visible light, different wavelengths are perceived as different colors.

Sample Learning Goals

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For Teachers

 

Instructions

Control Panel

 
Inputting new numbers into the field boxes will change
their wavelengths and number of points respectively.
 

Toggling Full Screen

Double click anywhere to toggle full screen.
 

Reset Button

Resets the simulation.

Research

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Video

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 Version:

  1. https://www.compadre.org/PICUP/exercises/exercise.cfm?I=129&A=rainbows
  2. http://weelookang.blogspot.com/2018/06/primary-rainbow-deflection-function.html 

Other Resources

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Frequently Asked Questions: Rainbow Formation

  • What are the basic optical phenomena responsible for the formation of a primary rainbow? A primary rainbow is formed when sunlight enters a spherical raindrop and undergoes three main steps: first, refraction as it enters the water, then one internal reflection off the back surface of the raindrop, and finally, refraction again as it exits the raindrop back into the air.
  • How is the deflection angle of light related to the observation of a rainbow? The deflection angle is the angle between the incident sunlight ray and the outgoing ray after interacting with the raindrop. Rainbows are observed at angles where the deflection angle is at a minimum. At this minimum, many rays of light with slightly different incident angles are deflected in nearly the same direction, leading to a concentration of light intensity that we perceive as a bright arc.
  • Why are different colors observed in a rainbow, and in what order do they appear in a primary rainbow? Different colors of light have different wavelengths, and the refractive index of water varies slightly with wavelength (a phenomenon called dispersion). This means that each color of light is refracted and deflected at slightly different angles. In a primary rainbow, red light (longer wavelength) is deflected the least, appearing at the top of the arc, while violet light (shorter wavelength) is deflected the most, appearing at the bottom.
  • What is the formula for the deflection angle of light in a primary rainbow, and what parameters does it depend on? The deflection angle (δ) for a primary rainbow, where light undergoes one internal reflection, is given by the formula: δ = 2(θᵢ - θ<0xE1><0xB9><0x9B>) + (π - 2θ<0xE1><0xB9><0x9B>), where θᵢ is the angle of incidence and θ<0xE1><0xB9><0x9B> is the angle of refraction of the light ray as it enters the raindrop. To compute this, one must know the refractive index of air and water at the specific wavelength of light and the angle at which the light hits the raindrop.
  • How does the secondary rainbow form, and how does it differ from the primary rainbow in terms of the number of reflections and the order of colors? A secondary rainbow is formed by light that enters a raindrop, undergoes refraction, then two internal reflections, and finally refracts out. The formula for the deflection angle in this case is δ = 2(θᵢ - θ<0xE1><0xB9><0x9B>) + 2(π - 2θ<0xE1><0xB9><0x9B>). Due to the additional reflection, the order of colors in a secondary rainbow is reversed compared to the primary rainbow, with red appearing on the inside (top) and violet on the outside (bottom). The secondary rainbow is also fainter than the primary rainbow because more light is lost with each internal reflection.
  • At what approximate angles relative to the horizontal are the primary and secondary rainbows typically observed? The primary rainbow is typically seen at an angle of approximately 42 degrees above the horizon (with the red band appearing slightly higher, around 42.4 degrees, and the violet band slightly lower, around 40.7 degrees). The secondary rainbow, when visible, appears higher in the sky at an angle of approximately 50-53 degrees above the horizon, with the red band around 50.2 degrees and the violet band around 53.3 degrees.
  • Why is the light intensity enhanced at the rainbow angles? The brightness of a rainbow is due to an effect called "rainbow scattering." Near the minimum of the deflection angle function, many incident light rays with slightly different angles are deflected into a very narrow range of outgoing angles. This "piling up" of light intensity at these minimum deflection angles (which correspond to the rainbow angles) is what makes the rainbow a bright and observable phenomenon.
  • What role do the refractive indices of water and air play in determining the appearance of a rainbow? The refractive indices of water and air are crucial because they determine the amount of bending (refraction) of light as it enters and exits the raindrop. Since the refractive index of water depends on the wavelength of light (dispersion), different colors are bent by different amounts. The specific values of these refractive indices dictate the precise deflection angles for each color, and thus the angles at which we observe the different colored bands of the rainbow. Accurate knowledge of these indices as a function of wavelength is essential for predicting the appearance and position of rainbows.
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