Total internal reflection

Reflection and Refraction • In total internal reflection all of the light is 4.2 reflected at the interface between two media. • Total internal reflection occurs when the angle Total internal reflection of refraction is equal to or greater than 90 o Dispersion • Total internal reflection is important for optical communications. (light pipe) Rainbows

Total internal reflection Total Internal Reflection

Total Internal reflection

Critical Angle θ c

o n 1 sinθ c = n 2 sin(90 ) =n 2

n 2 must be less than n 1

Critical Angle Question What is the critical angle at the air water interface? At the critical angle n 2 n 2 =1.00 n 2 n sin θ c = 1 θ c n1 n 1 =1.333 θc n 2 < n 1

n 1.00 When θ > θ c , light is totally internally reflected 2 sinθc = = = 0.750 n1 1.333 o θc = 48.6

1 The fishes’ eye view of the outside world is limited by total internal reflection Optical Fiber ­Light Pipe An optical fiber (light pipe) confines the light inside the material by total internal reflection. If the refractive index of the fiber is 1.52 what is the smallest angle of incidence possible when the light pipe is in air.

n 2 =1.00 θ 2 = 90

θ 1 n 1 =1.52

n1 sinθ1 = n2 sin θ2 n sin90 (1.0)(1.0) sinθ = 2 = = 0.66 1 n 1.52 At angles greater than the critical angle the whale sees only reflected light 1 This illustrates the point that the path of light rays is the same in the forward o and reverse directions. θ1 = 41

Fiber Optics Light Pipe

Fiber optics are used A light pipe is a flexible fiber that confines a extensively in beam of light by total internal reflection communications. Telephone, Internet,

The high frequency of light (compared to microwaves) allows it to be switched rapidly and carry more information.

Fiber Optic are used in optical communication networks Diamond cut

The advantage of optical communication systems is due to the higher frequency of light f~10 15 Hz compared to radio or microwaves 8 10 f~10 ­10 Hz For a well cut diamond the incident light is totally internally reflected.

The high refractive index of the diamond allows total internal reflection over a wide range of angles.

2 Dispersion Dispersion of white light by a prism

• The index of refraction generally is different for different wavelengths of light. • This dispersion can be used to separate light into different wavelengths using a prism. Separation of different • Dispersion is a problem in fabricating lenses for wavelengths by refraction imaging – causing chromatic aberration, through a different images for different wavelengths of light Prism • Dispersion is responsible for rainbows. Incident White Light

The index of refraction varies with wavelength Example 35.6 A prism disperses white light into different colors. example for Crown Glass A 90 degree prism of crown glass refracts light normally incident on the long surface.. Find the largest difference in angles of refraction of violet and red light for different values of θ. violet 400 1.538 θ 1= θ green 500 1.526 n=1.0000 n sinθ = n sin θ θ 1 1 2 red 700 1.516 n sin θ = 1 sin θ 2 n maximum change θ 2 violet n 1 = 1.538 violet­red θ 1 red n 1 = 1.516 ∆ n ≈ 1.4% n violet light is refracted more than red n 1

θviolet − θred = arc sin(nviolet sinθ) − arc sin(nred sinθ )

Prism max difference when Rainbow θ ­> 90 o for violet 2 ∆θ plotted vs θ 2 θ=40.556 10 9

) 8 n=1.0000 s e

e 7 r g

e 6 d ( θ 5 a t

e 4 h t 3 ta l

e 2 d 1 θ 1 0 0 20 40 60 80 100 The ta vi ol e t (de gre e s) n 1 ∆θ 90 o o A rainbow is seen on a rainy day when the sun is to your back, ∆θ ~9.5 degrees at θ=40.556 o sinθ low in the horizon (less than 42 above the horizon) The large difference is because near 90 o A second rainbow is often seen with the order of colors (Maximum) sin θ is quite insensitive to θ θ reversed.

3 The shape of the rainbow is due to parallel beam of sunlight light The raindrop reflects and refracts reflected and refracted from raindrops at a special angle (rainbow angle 42 o) the incident sunlight. The colors of the rainbow are due to dispersion of the light.

Incident beam is refracted twice and internally reflected by the raindrop Deflection angle = 180­ϕ

Rainbow angle Effect of dispersion

Incident light is reflected at many different angles However much of the light ϕ is reflected near the o rainbow angle 42 because n ϕ vs θ goes through a maximum

45 larger n => smaller ϕ 40

35

30 ) s e e

r 25 g e

d Blue light has smaller ϕ (

20 i h P 15 Near the maximum there are a 10 Blue light is on the 5 large number of values of θ with the 0 inside of the arc 0 20 40 60 80 100 theta(degrees) values close to ϕ max Red light is on the outside

Colors of the rainbow

The primary rainbow has blue on the inside and red on the outside The secondary rainbow is due to light that is internally reflected twice the effect of refraction is to increase ϕ and the colors are reversed (see problem 60).

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