Optics: Descartes' Rainbow

1. Descartes' Rainbow

The white ray light travels from air of index of refraction n1, through a water droplet of index of refraction n2, and emerges to the air toward an observer.
What are the colors could we see? What is the minimum deviation of the ray?. We will use the model of Descartes to answer these two questions. We will see the two rainbows: the primary (one internal reflection and two refractions) and the secondary (two internal reflections and two refractions).

   δ is the deviation between the incidence and emergence of the sun light ray.
δ = (i - r) + (2π - 2r) + (i - r)
= 2Ï€ + 2i - 4r

We have Δ = 2π - δ
Δ = 2π - (2π + 2i - 4r)

Δ = 4r - 2i    (1)

d(Δ)/di = d(4r - 2i)/di = 0
Then: dr/di = 1/2    (2)
From Snell's law, we have:
n1 sin i = n2 sin r ⇉ sin r = (n1/n2) sin i. The ratio n1/n2 for air-water is generally taken equal to 3/4. Therefore:
r = arcsin((n1/n2) sin i)
dr/di = [1 - (n1/n2)² sin² i]^-1/2 x (n1/n2) cos i    (3)
The relationship (2) becomes:
1/2 = [1 - (n1/n2)² sin² i]^-1/2 x (n1/n2) cos i
Or
(1/4) [1 - (n1/n2)² sin² i] = (n1/n2)² cos² i = (n1/n2)² (1 - sin² i)
Then:
(1/4) = (1/4) (n1/n2)² sin² i + (n1/n2)² (1 - sin² i) = (1/4) (n1/n2)² sin² i + (n1/n2)² - (n1/n2)² sin² i
Thus:
(1/4) - (n1/n2)² = - (3/4) (n1/n2)² sin² i
or
(n1/n2)² - (1/4) = (3/4) (n1/n2)² sin² i (4/3) [(n1/n2)² - 1/4] (n2/n1)² = sin² i (4/3) [1 - (1/4) (n2/n1)²] = sin² i
sin i = ((4/3) [1 - (1/4) (n2/n1)²])^1/2
sin i = ((1/3) [4 - (n2/n1)²])^1/2
i = arcsin [([4 - (n2/n1)²]/3)^1/2]    (4)

i = arcsin [([4 - (n2/n1)²]/3)^1/2]


for n2/n1 = 4/3, we have :
i = arcsin [((1/3) [4 - 16/9])^1/2] = arcsin((20/27)^1/2)= 59.40

i = 59.40 ^o

From Snell's law: n1 sin i = n2 sin r, we have:
r = arcsin[(n1/n2) sin i] = arcsin[(3/4) (20/27)^1/2] = arcsin[0.64] = 40.20 = 40.20 ^o

r = arcsin[(n1/n2) sin i]


The relationship (1) becomes:
Δ = 4r - 2i

Δ = 4r - 2i = 4(40.20)^o - 2(59.40)^o = 160.80^o - 118.80 ^o = 42^o.

Δ = 42^o
Then, the deviation is : δ = 2π - δ = 222 ^o
Note &delta is min, then Δ is max.

The relative index of refraction of a ray from the medium n1 to the medium n2 is:
n = n2/n1

Thus, Snell's law is rewritten, if i and r are the angles od incidence and refraction respectively, as:
sin i = n sin r → sin r = (1/n) sin i
Using the relationship (2): dr/di = 1/2, we get:
cos r (dr/di) = (1/2) cos r = (1/n) cos i
Thus cos i = (n/2) cos r = (n/2) [1 - sin²r]^1/2 = (n/2) [1 - (1/n)² sin²i]^1/2
= (n/2) [1 - (1/n)² (1 - cos²i)]^1/2 = (n/2) [1 - (1/n)² + (1/n)² cos ²i]^1/2
Then:
cos² i = (n/2)² [1 - (1/n)² + (1/n)² cos²i] =
(n/2)² - (1/4)+ (1/4) cos²i)
Or:
(3/4) cos²i = (n/2)² - (1/4). Hence
3 cos²i = n² - 1. Then cos i = [(n² - 1)/3]^1/2

cos i = [(n² - 1)/3]^1/2

2. Properties of Descartes' Rainbow

Here is a table that gives the color, the wavelength in vacuum, the related index of refraction in water, and the related Descartes rainbow max deviation.
The index of refraction of air is taken equal to 1.0003 for all these wavelengths of the white light ray spectrum.

3. Secondary rainbow

In this case have, we have two internal reflections and two refractions:
δ = (i - r) + (2π - 2r) + (i - r) + 2π - 2r

δ = 2i - 6r + 4π
The condition of minimum is written as:
dδ/di = 0 = 2 - 6 (dr/di). Hence:
dr/di = 1/3

Using Snell's law:
sin r = (1/n) sin i, we get:
cos r (dr/di) = (1/3) cos r = (1/n) cos i
Thus cos i = (n/3) cos r = (n/3) [1 - sin²r]^1/2 = (n/3) [1 - (1/n)² sin²i]^1/2
= (n/3) [1 - (1/n)² (1 - cos²i)]^1/2 = (n/3) [1 - (1/n)² + (1/n)² cos ²i]^1/2
Then:
cos² i = (n/3)² [1 - (1/n)² + (1/n)² cos²i] = (n/3)² - (1/9)+ (1/9) cos²i = 0. Or:
(8/9) cos²i = (n/3)² - (1/9). Hence
8 cos²i = n² - 1. Then cos i = [(n² - 1)/8]^1/2.

The condition of minimum is:
cos i = [(n² - 1)/8]^1/2

For the example of n = 4/3, we have:
i = arcos [(n² - 1)/8]^1/2 = arcos [(16/9 - 1)/8]^1/2 = 72.0 ^o

i = 72.0 ^o

r = arcsin [(1/n) sin i] = arcsin [(3/4) sin 72.0] = 45.45^o
r = arcsin [(1/n) sin i] = arcsin [(3/4) sin 72.0] = 45.45^o

r = 45.45^o

δ = 2i - 6r + 4π = 2 x 72.0 - 6 x 45.45 + 00 = 144 - 272.7 + 720 = 591.3 == 591.3 - 360 = 231.3

Δ = δ - π = 231.3 - 180 = 51.3^o

Δ = 51.3^o