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The Physics of Rainbows
Prof. Chad A. Middleton CMU Physics Seminar
September 13, 2012
Charles A. Bennett, Principles of Physical Optics, 1st ed., pps. 111-115.
Jearl D. Walker, “Multiple rainbows from single drops of water and other liquids”, American Journal of Physics Vol. 44, No. 5, May 1976.
John A. Adam, “The mathematical physics of rainbows and glories”,
Physics Reports 356 (2002), 229-365.
Quiz: The Physics of Rainbows
1. For the primary rainbow, what is the ‘exterior’ (largest observed angle) color? a) Violet b) Red c) Yellow d) Green
2. For the primary rainbow, what is the ‘interior’ (smallest observed angle) color?
a) Violet b) Red c) Yellow d) Green 3. For the secondary rainbow, what is the ‘exterior’ color?
a) Violet b) Red c) Yellow d) Green
4. For the secondary rainbow, what is the ‘interior’ color? a) Violet b) Red c) Yellow d) Green
5. Is the region between the primary and secondary rainbows dark or bright?
6. Is the interior region of the primary rainbow dark or bright?
What path should the lifeguard take to minimize her transit time?
vi
vt
vi > vt Lifeguard
Drowning swimmer
Smallest time interval?
vi
vt
vi > vt
Shortest distance = smallest time?
Smallest time interval?
vi
vt
vi > vt Shortest H2O distance
= smallest time ?
Smallest time interval?
vi
vt
vi > vt
Shortest land distance = smallest time?
Smallest time interval?
vi
vt
vi > vt Path of
smallest time!
Smallest time interval?
vi
vt
vi > vt
Smallest time interval?
vi
vt
vi > vt
a
c
b
Smallest time interval?
vi
vt
vi > vt
pa
2 + x
2
pb
2 + (c� x)2
x
a
c
b
c� x
Smallest time interval?
vi
vt
vi > vt
pa
2 + x
2
pb
2 + (c� x)2
=
pa
2 + x
2
vi+
pb
2 + (c� x)2
vt
x
a
b
c� x
To calculate the total time…
t(x) = ti + tt
Plot of t(x) vs x…
=
pa
2 + x
2
vi+
pb
2 + (c� x)2
vt
The total time is…
t(x) = ti + tt
0 0.2 0.4 0.6 0.8 1x
3.25
3.3
3.35
3.4
3.45
3.5
3.55
3.6
t!x"
Time vs Position
a, b, c = 1vi = 0.8vt = 0.6
Smallest time interval?
vi
vt
vi > vt
pa
2 + x
2
pb
2 + (c� x)2
x
a
b
c� x
To calculate the path of smallest time…
d
dx
t(x) = 0
1
vi
xpa
2 + x
2=
1
vt
(c� x)pb
2 + (c� x)2
✓i
✓t
yields
Smallest time interval?
ni
nt
ni < nt
The path of smallest time… where
✓i
✓t
ni sin ✓i = nt sin ✓t
n ⌘ c
v
Fermat’s Principle:
The actual path between two points taken by a beam of light is the one that is traversed in
the least time.
When light enters a new medium, it’s path obeys Snell’s Law:
ni sin ✓i = nt sin ✓t
§ is the deviation angle. § is the observed angle, measured from the anti-solar direction.
§ Rays of sunlight are nearly parallel to each other.
§ Suspended H2O droplets
are nearly spherical due to surface tension.
↵
Geometrical setup…
✓d↵
Sunlight
✓d↵
§ At certain observed angles, a particular color will dominate.
§ This angle forms a cone around the anti-solar direction. § All raindrops that lie on this cone can contribute to the
rainbow => The rainbow is circular!
Geometrical setup… Sunlight
↵↵
Primary Rainbow:
✓i
1st refraction… • transmitted angle given by
Snell’s Law
✓t
✓t = sin�1
✓sin ✓in
◆
where n ⌘ nt
ni
nair ' 1.00
nH2O ' 1.33
✓iLight is incident at angle …
Primary Rainbow:
✓i
1st refraction, • by geometry, same angle
✓t ✓t
Primary Rainbow:
✓i
1st refraction, 2nd reflection… • Law of Reflection, same angle
✓t ✓t✓t
Primary Rainbow:
✓i
1st refraction, 2nd reflection… • Law of Reflection, same angle • by geometry, same angle
✓t ✓t✓t
✓t
Primary Rainbow:
✓i
1st refraction, 2nd reflection, 3rd refraction… • by Snell’s Law, same
✓t ✓t✓t
✓t
✓i
✓i
Calculate the net deviation angle, …
✓i
✓t ✓t✓t
✓t
✓i
✓d
✓d = (✓i � ✓t)+
(✓i � ✓t)
Calculate the net deviation angle, …
✓i
✓t ✓t✓t
✓t
✓i
✓d
✓d = (✓i � ✓t)+
(✓i � ✓t)
(180� � 2✓t)
(180� � 2✓t)+
Calculate the net deviation angle, …
✓i
✓t ✓t✓t
✓t
✓i
✓d
✓d = (✓i � ✓t)+
(✓i � ✓t)
(✓i � ✓t)
(✓i � ✓t)
(180� � 2✓t)
(180� � 2✓t)+
The net deviation angle, , is ✓d
where we used Snell’s Law.
✓d = 180� + 2✓i � 4✓tor
✓d
↵ ↵ = 180� � ✓dobserved angle
↵
✓d(✓i) = 180� + 2✓i � 4 sin�1
✓sin ✓in
◆
An infinite number of parallel sunbeams hit the spherical raindrop, so which ones do we see?
Notice: The ‘Cartesian Ray’ is the ray that has the minimum deviation angle.
Plot of θd vs sin-1θi …
The net deviation angle is…
0 0.2 0.4 0.6 0.8 1sin!1!Θi"
140
150
160
170
180
Θd
Deviation angle vs sin!1!Θi"The backscattered rays cluster at the minimum deviation angle, yielding an enhanced brightness
✓d = 180� + 2✓i � 4 sin�1
✓sin ✓in
◆
Minimum deviation angle…
To calculate the minimum deviation angle…
d✓dd✓i
= 0 yields
sin ✓i =
r4� n2
3
0 0.2 0.4 0.6 0.8 1sin!1!Θi"
140
150
160
170
180
Θd
Deviation angle vs sin!1!Θi"
Dispersion… Different frequencies of light have different indices of refraction.
nt,r = 1.331
↵r = 42.4�
↵v = 40.5�
�↵ = 1.9�
↵(n) = �2 sin�1
r4� n2
3
!+ 4 sin�1
r4� n2
3n2
!
nt,v = 1.344
1.33 1.335 1.34 1.345 1.35n
40
40.5
41
41.5
42
42.5
Α
Observed angle vs index of refraction
Red
Violet
Notice: The ‘outside’ of the primary rainbow
is red, whereas the ‘inside’ is violet!
✓d = (✓i � ✓t)+
(✓i � ✓t)
(180� � 2✓t)+
✓i
✓t✓t
✓t
✓t
✓i
(✓i � ✓t)
(✓i � ✓t)(180� � 2✓t)
✓t
✓t
(180� � 2✓t)(180� � 2✓t)+
Secondary Rainbow…
The net deviation angle, , is… ✓d
where we again used Snell’s Law.
or
observed angle
✓d = 360� + 2✓i � 6✓t
↵ = ✓d � 180�
✓d(✓i) = 360� + 2✓i � 6 sin�1
✓sin ✓in
◆
✓d
↵
↵
Minimum deviation angle…
To calculate the minimum deviation angle…
d✓dd✓i
= 0 yields
sin ✓i =
r9� n2
8
0 0.2 0.4 0.6 0.8 1sin!1!Θi"
240
260
280
300
320
340
360
!Θ d"
Deviation angle vs sin!1!Θi"
Secondary Rainbow…
nt,v = 1.344nt,r = 1.331
↵r = 50.3�
↵v = 53.8�
�↵ = 3.5�
↵(n) = 180� + 2 sin�1
r9� n2
8
!� 6 sin�1
r9� n2
8n2
!1.33 1.335 1.34 1.345 1.35
n
50
51
52
53
54
55
Α
Observed angle vs index of refraction
Red
Violet
Notice: The ‘outside’ of the secondary rainbow
is violet, whereas the ‘inside’ is red!
Why is the interior region of the primary rainbow bright?
Your eye receives backscattered light of all wavelengths from the interior of the primary rainbow => Bright white light!
0 0.25 0.5 0.75 1 1.25 1.5Θi!rad"
0
10
20
30
40
Α!deg
"
Observed angle vs Incident angle
For the primary rainbow… one has backscattering for all angles in the regime:
0 ↵ 42.4�
‘Normal’ incidence
‘Grazing’ incidence
Primary
=> Alexander’s dark band!
Why is the region between the primary and secondary rainbows dark?
One has ZERO scattering from one or two reflections when
For the primary rainbow… one has backscattering when
0 ↵ 42.4�
0 0.25 0.5 0.75 1 1.25 1.5Θi!rad"
0
10
20
30
40
Α!deg
"
Observed angle vs Incident angle
Primary
0 0.25 0.5 0.75 1 1.25 1.5Θi!rad"
60
80
100
120
140
160
180
Α!deg
"Observed angle vs Incident angle
Secondary
For the secondary rainbow… one has backscattering when
50.3� ↵ 180�
42.4� < ↵ < 50.3�
Is a 3rd (or l th) rainbow theoretically possible?
After allowing for l internal reflections, the net deviation angle is…
✓d(✓i) = l(180�) + 2✓i � 2(l + 1) sin�1
✓sin ✓in
◆
d✓dd✓i
= 0 yields sin ✓i =
s(l + 1)2 � n2
l(l + 2)
To calculate the minimum deviation angle…
⇒ The first 13 rainbows of water have been observed from a drop suspended in a spectrometer!*
*Jearl D. Walker, “Multiple rainbows from single drops of water and other liquids”,
American Journal of Physics Vol. 44, No. 5, May 1976.
Why are two rainbows sometimes visible in the sky, but one never sees a third (or fourth)?
For the tertiary rainbow (l = 3)…
↵r = 137.5�
↵v = 144.3�
�↵ = 6.8�
So why don’t you see it? 1. Each successive Cartesian ray is at a greater incident angle,
therefore a reduction in intercepting cross-sectional area. 2. Larger for each successive rainbow. 3. Loss of light due at each successive reflection.
�↵
