# How rainbows are created

Optics series, lecture – IV

This lecture was delivered on February 02, 2017

( All other optics series lectures ) Read other available lectures in optics series

## “How is a rainbow formed: primary and secondary rainbows”

Sunlight is white in color.

Newton would have greatly disliked this statement, do you know why? “Color is not the property of light” but a property of an agent making observations assisted by light. We need a separate article for that. Don’t we?

That means it comprises of 7 primary colors. VIBGYOR is an acronym for these basic colors: Violet, Indigo, Blue, Green, Yellow, Orange and Red. Each color of light corresponds to a different wavelength. violet has the shortest wavelength and red has the highest wavelength.

Accordingly violet has the highest intensity or consequently energy and red has the lowest intensity or energy. In other words red is the faintest color in the primary visible spectrum.

Different colors or wavelength of light have different refractive indices, this fact is known as dispersion, that is, different wavelengths of light would travel in different directions upon refraction at any optically denser or rarer media.

That means different wavelength or color component of light would travel at different speed and correspondingly different angles, upon incidence on a media whose refractive index differs from the medium from where incidence occurs.

The above phenomena is known as dispersion. Prism is an optical device which shows such dispersion of colors. We say that the angle of deviation of different colors of light is different at refracting interfaces of optical media. This is mathematically given by the Snell’s law of refraction which we write as follows: $\frac{\sin{\theta_1}}{\sin{\theta_2}}=\frac{v_1}{v_2}=\frac{\lambda_1}{\lambda_2}=\frac{n_2}{n_1}$.

Rainbows are caused by dispersion of light. Dispersion is an optical phenomena where wave velocity or phase velocity of light depends upon frequency or wavelength of light. Different colors of light emerge at different directions during refraction in a transparent medium such as water droplets during rainfall. When sunlight falls on small water droplets rainbow manifests as the result of dispersion of the incident rays.

Rainbow are usually seen during afternoon, in the aftermath of rain. This is so because sun is in the west and its light falls on receding clouds, that are moving eastward. During other times of day the rainbow aren’t visible from ground, since there is a certain small angular range, which is involved, for us to clearly see the effects at our level on ground.

Usually we see two types of rainbow. The primary and the secondary and we will discuss them in greater detail.

### Primary rainbow

Primary rainbow angle
Primary rainbow forms at an angle between 400 and 420 from anti-solar point. The anti-solar point is the region of horizon which is directly opposite to the sun, the sun is now at west.
Primary rainbow mechanism
There are two refraction and one internal reflection processes involved in the  Primary rainbow formation. We will discuss exactly what happens during these 3 processes.
Primary rainbow condition
If water droplets are bigger than 1 mm in diameter, red, green and violet components are bright, but blue color is very less in amount.
Primary rainbow condition
When droplets get smaller, red color weakens in intensity.
Primary rainbow condition
If very fine mist is present in atmosphere, all colors except brown disappear.
Primary rainbow condition
If droplets are smaller than 0.05 mm in diameter, these droplets produce white rainbow, these are known as fog bow.
Primary rainbow disappearance
Rainbows are not seen in midday, this is because the circle that represents 420 angle is below horizon. Entire circle of rainbow can be seen from airplanes as droplets are both above and below observer in the airplane.

How is a rainbow formed: primary rainbow formation, white light refracts twice and internally reflected once i a raindrop, making a 2 degree difference in the violet and red components of light. Photo Credit: mdashf.org

### Total internal reflection

Central to our understanding of rainbow as depicted above in the picture is thus the phenomenon of dispersion. But we see that there is one reflection which brings the incident ray after first refraction to inside of the water droplet so that it goes through a secondary refraction.

The reflection where the light instead of getting refracted away gets reflected back into the same medium is called as Total Internal Reflection. Lets try to understand this process in a bit more detail.

When light travels from an optically denser medium — that is one with higher refractive index, such as for water with n = 1.33, to an optically rarer medium, such as air, whose refractive index is n = 1, Snell’s law predicts that sine of angle of refraction is greater than unity, for angles of incidence that are larger than certain value.

But this is not possible, since sine of any angle can’t be larger than 1. Hence any refraction beyond this angle of incidence from denser to rarer medium, is curtailed. This maximum angle of incidence from denser to rarer medium for which refraction occurs no more, are determined by setting the angle of refraction to be 900, and this limiting angle of incidence is called critical angle.

For angles more than such angle of incidence for which angle of refraction is 900, all rays are reflected back into the denser medium. This is known as total internal reflection. We depict this interesting phenomena by the following diagram. Also lets apply Snell’s law to a situation where denser medium is water, and angle of incidence is 500, to see what happens.

As the calculations show, sine of angle of refraction is 1.021 and this is not possible. By setting angle of refraction to be maximum of 900 angle, we see that the critical angle of incidence of ray in water is 48.60. All rays above this value will be all reflected back and stay inside water, till there is another refraction, which is allowed only for angle of incidence smaller than 48.60 of angle, if they are to be refracted again, into air.

$\boxed{\sin {\theta_2}=\frac{n_1}{n_2}\,\sin{\theta_1}=\frac{1.333}{1.0}\,\sin{50^0}=1.021}$

500 angle or more, is therefore not possible as an angle of incidence, for any refraction to take part, if the ray is incident on water surface and is to refract into air. Here is how we evaluate the critical angle of incidence, in our example:

$\boxed{\theta_{\,critical}=\sin^{-1} \,\Big(\,\frac{n_2}{n_1}\,\sin{\theta_2}\,\Big)= \sin^{-1}\,\Big(\,\frac{n_2}{n_1}\,\Big)=48.6\,^0}$

How is a rainbow formed: total internal reflection for n1 > n2. This diagram depicts the critical angle and the process of total internal reflection. This diagram depicts the critical angle and the process of total internal reflection. Photo Credit: mdashf.org

### Secondary rainbow

Lets us now discuss the curious case of secondary rainbow. The secondary rainbow forms 10 degree farther than primary bow. The 2ndary bow is fainter and has its colors reversed.

How is a rainbow formed: An actual image of a secondary rainbow Photo Credit: muslimheritage dot com

Secondary rainbow
Secondary rainbow are formed 100 farther out, from anti-solar point, than that for primary rainbow.
Nature of secondary rainbow
The secondary rainbow are twice as wide and here the colors are reversed wrt primary rainbow.
Mechanism of secondary rainbow
Instead of two refraction and one total internal refraction now we have two refraction as well as two total internal reflections of the primary refracted rays.
Intensity of secondary rainbow
Light of secondary rainbow has an intensity of 1/10th that of primary rainbow.

The process of secondary rainbow is similar to that of primary rainbow, so we will depict the process in the following diagrams, instead of doing another round of discussion.

How is a rainbow formed: secondary rainbow, the process behind secondary rainbow; two refractions and two internal reflections Photo Credit: mdashf.org

### Higher order bows

3rd order bows are created at an angle of 40020′ around sun. No records exist of 4th order bow. 5th order bow is attributed to 19th CE scientist Mascart. It forms between primary and secondary, its 70 wide and the red color of 5th order rainbow overlaps the red of primary bow. Up-to 13th order bows can be created by LASER falling on water drops in laboratory.