LECTURE 30
TOTAL INTERNAL REFLECTION
DISPERSION
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the indices of refraction of different materials,
Determine Given
determine when total internal reflection occurs
Objectives
Relate
Deduce
Relate dispersion to the color separation of white light
as it travels through a prism at non-normal incidence
Deduce the speed of light in a medium from its
dispersion curve
θa
θr
na = 1.52
θb
nb = 1.0
Is it possible that all of the light can be reflected back
from the interface with none of it being transmitted?
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Total Internal Refection (TIR)
Occurs only when a ray in material 𝑎 is incident on a second material 𝑏 whose index of
refraction is smaller than that of material 𝑎.
θa
θcrit
θr
na = 1.52
θb
TIR occurs only if nb < na.
nb = 1.0
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Total Internal Refection (TIR)
TIR will occur if the angle of
incidence θa is larger than or
equal to θcrit.
When the angle of incidence reaches 𝜃𝑐𝑟𝑖𝑡 , the
angle of refraction 𝜃𝑏 is exactly 90°. Hence,
𝑛𝑎 sin 𝜃𝑐𝑟𝑖𝑡 = 𝑛𝑏 sin 90° = 𝑛𝑏
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TIR in real life
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TIR application: fiber optics
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Example
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Example
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Example
Light is incident along the normal on face of a glass prism of
refractive index 1.52, as shown in the figure. Find the largest value
the angle 𝛼 can have without any light refracted out of the prism
at face if (a) the prism is immersed in air and (b) the prism is
immersed in water.
check your answer
in the next slide
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Solution
No refracted ray implies TIR.
𝑛𝑎 sin𝜃𝑎 = 𝑛𝑏 sin𝜃𝑏
𝜃𝑎 = 𝜃𝑐𝑟𝑖𝑡 ; 𝜃𝑏 = 90∘
𝑛𝑏
−1
𝜃𝑐𝑟𝑖𝑡 = sin
𝑛𝑎
𝛼 = 90∘ − 𝜃𝑐𝑟𝑖𝑡
(a) The index of refraction of air is 1.00029.
LHS: 1.52 sin 𝜃𝑐𝑟𝑖𝑡 = 1.52 sin 90° − 𝛼 = 1.00029 sin 90°
1.00029
90° − 𝛼 = sin−1
= 41.154°
1.52
𝛼 = 48.846°
(b) The index of refraction of water is 1.333.
LHS: 1.52 sin 𝜃𝑐𝑟𝑖𝑡 = 1.52 sin 90° − 𝛼 = 1.333 sin 90°
1.333
90° − 𝛼 = sin−1
= 61.279°
1.52
𝛼 = 28.721°
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No refracted ray implies TIR.
𝑛𝑎 sin𝜃𝑎 = 𝑛𝑏 sin𝜃𝑏
𝜃𝑎 = 𝜃𝑐𝑟𝑖𝑡 ; 𝜃𝑏 = 90∘
𝑛𝑏
−1
𝜃𝑐𝑟𝑖𝑡 = sin
𝑛𝑎
𝛼 = 90∘ − 𝜃𝑐𝑟𝑖𝑡
Objectives
• Index of refraction of primary medium is
greater than than index of refraction of
secondary medium
𝑛𝑎 > 𝑛 𝑏
• Angle of incidence is equal to or greater
than critical angle
𝜃𝑎 ≥ 𝜃𝑐𝑟𝑖𝑡
Determine
Given the indices of refraction of different materials,
determine when total internal reflection occursReduce
the speed of light in a medium from its dispersion curve
White Light
Ordinary white light is a superposition of waves with wavelengths
extending throughout the visible spectrum.
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Dispersion
➢ dependence of wave speed and index
of refraction on wavelength
where:
λ: wavelength of light in a material
λo: wavelength of light in vacuum
n: index of refraction of the material
v: speed of light in the material
c: speed of light in vacuum
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The amount of dispersion depends on the
difference between the refractive indexes for
violet light and for red light.
Dispersion is greatest for silicate flint glass, for
which there is a larger difference in the value of n
between red and violet.
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Example
1. Which material has the
slowest speed at violet
light?
Since are 𝜆0 and 𝑓 are constant
with higher 𝑛, we have lower 𝑣
Silicate flint glass:
largest 𝑛 at 400 nm
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Example
2. What is the speed of a
700nm-light (𝜆0 ) traveling in
silicate crown glass?
𝑐
𝑐
𝑛= →𝑣=
𝑣
𝑛
From the plot, we have 𝑛 ≈ 1.5. Hence,
𝑣≈
3×108 m/s
1.5
= 2 × 108 m/s
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How rainbows form
Dispersion
Refraction
Reflection
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Exercise!
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Exercise!
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Relate
Relate dispersion to the color separation
of white light as it travels through a prism
at non-normal incidence
Objectives
Deduce the speed of light in a medium
Deduce from its dispersion curve