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Refraction at plane surfaces• Speed of light in air
– Slight digression on what is light
– Fizeau’s measurement
• Index of refraction
– speed of light in transparent materials
– dispersion
– wave picture of refraction
– ray picture, Snell’s law
• Plane refractive surfaces
– single surfaces, images – plane-parallel plate
• Critical angle
• Prisms
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What is light?• Carries energy
• Exerts force on electrical charges• Wave similar to sound or water wave
– Frequency related to color (high
frequency=blue, low frequency = red)
– Wavelength also related to color (long
wavelength=red, short wavelength=blue)
– Other electromagnetic radiation behaves
similarly (radio, microwave, xrays,
gamma rays)
• Wave fronts perpendicular to rays
rays
wavefronts
Wavefront – surface of
peaks of waves
Wave
length,
λ amplitude
Wave speed, V
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How fast does light travel
• Light travels in a straight line – How fast does it travel?
• Galileo’s method (1600)
– Lantern’s on neighboring hills
– One person unblocks his lantern
– When the other sees the lantern, he unblocks his
– First person sees light from the second lantern, giving
round trip time
– Doesn’t work, light too fast
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Measurement of speed of light in air • Olaf Romer (1675)
used eclipses of
Jupiter’s moons
estimatedc=2.967x1010cm/sec
• Fizeau (1849) firstearthbound
measurement
• Value in vacuum – 3x108m/sec=3.0x1010cm/sec
– 30 cm/nsec≅1 ft/nsec
Speed of light is constant
Independent of color (wavelength)
Independent of motion of source
Symbol, c (Latin, celere=speedy)
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Speed of light in transparent media
• First measured byFoucault in 1850
– helped to settle
controversy aboutwave/particle nature of
light
– light travels slower indenser media
• Be careful about the
meaning of the word
“density”
mediuminspeed
in vacuumspeed refractionof index =
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Dispersion
• Index of refraction (speed) changes with wavelength
• Decreases for longer wavelengths
• Small, but important effect, note y-axis doesn’t start at zero
Index of refraction of Quartz
1.454
1.456
1.458
1.460
1.462
1.464
1.466
1.468
1.470
1.472
350 400 450 500 550 600 650 700
Wavelength (nm)
R e f r a c t i v e I n d
e x
Blue Green Red
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Refractive index also varies
widely between materials
Index of refraction of variousmaterials at 589 nm (d light)
Material IndexWater 1.333
Air 1.000293BK7 glass
(crown) 1.517
BaSF6 glass
(flint)1.668
FDS9 glass 1.847Quartz 1.458
Diamond 2.419
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Refraction of a wavefront at plane surface
• Wavelength in higher index
material is shorter
– velocity slower but frequencyunchanged
• Wavefronts are continuous across
surface• Wave bends towards the normal
when going to a higher index
material
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Graphical construction for refraction
n1
n1
n2
A
n1
n2
A
B
n2
n1
parallel
• Set a compass to radius n1 (units don’t matter!!!)
– Use this to mark off a distance n1 along input ray from point of intersection with surface
– This is point A on the diagram
• Set compass to n2
– With point at A draw arc that intersects surface normal
– Intersection point is B in diagram
• Refracted ray is parallel to line AB – Transfer a parallel line to the intersection point
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Principle of reversibility• If a ray is reversed in direction:
– output angle Θ2 becomes input angle – final index, n2 becomes initial index
– angle which satisfies Snell’s law for the
final angle is then Θ1, the original inputangle
– therefore, the ray retraces its path
• The law of reflection is symmetricalwith respect to its initial and finaldirections also
Θ1 Θ2
Θ1Θ2
n2
n2
n1
)sin()sin( 1
2
12 Θ=Θ
n
n
n1 In geometrical optics, if any ray is
reversed it will retrace its original
path. This holds for all rays.)sin()sin( 21
2
1 Θ=Θ n
n
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Refraction of rays from a point source
• Rays at a higher angle bend more
• From in the high index material,light appears to come from an
image point farther from the
surface than the object – if n2<n1 then the image is closer to
the surface
• The image is virtualn2
n1
R f i f f i
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Refraction of rays from a point source
h
h’
R
L
R’
ΘΘ
Θ’
Θ’
Object
Image n’
n
Snell’s law:
In this approximation hn
n
h
'
' =
)sin()sin( Θ′′
=Θn
n
)cos()cos( Θ
′=Θ′
′ h
n
nh
For small angles this is
approximately
(paraxial approximation
angle in radians)
Θ′′=Θn
n
Exact result Not a perfect
image, location
depends on rayangle!
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Rays out of plane give different image from in
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Rays out of plane give different image from in
plane rays – astigmatism
• Observer will see both sets of rays, therefore blurryimage
• Only meridional rays appeared on previous diagram
object point
image point
for meridional rays
object point
image pointfor sagital rays
R f i h h l ll l l b
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Refraction through a plane-parallel slab
• final ray parallel to the initial ray – not true if faces are not parallel
– displacement, d, proportional to
thickness, t, increases with largerindex
t
φ
φ2
φ
d
φ
φ2
β=φ1-φ2
A
BC
Dh
)cos(
)sin()sin(2
2121
φ
φ φ φ φ −=−= t hd
n2
n1
Geometry gives:
This can be expressed in different ways using trigonometry
−
−=
−=
−=
)(sin
)cos(1)sin(
)cos(
)cos(1)sin(
)cos(
)cos()sin()sin(
1
2
2
1
2
2
11
22
111
2
121
φ
φ φ
φ
φ φ
φ
φ φ φ
n
n
t n
nt t d
Last form doesn’t require calculation of φ2
I i f i h h ll l l
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Imaging or focusing through a parallel plate
• Imaging – object locatedat O, appears to be
at O’
– Application, microscope
imaging through
cover glass
• Focusing – light to left
of plate is focusing towards O’
Actually focuses at O
– Application, machining laser focused through a window, for
example to prevent debris from getting on lens
• Imaging or focusing depending on direction of travel – reversibility
O’, location of focus
in absence of plate
E
n
n
t
1E
−=
t
O
n
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Prisms
• Deflections at the two surfacesdon’t cancel
• Deflection angle depends on apex
angle of prism, α, not angles of base
• From geometry δ=φ1
+φ2
’-α – can be considered a function of φ1
since φ2 depends on φ1 and α is fixedfor a given prism
– depends on index of refraction andtherefore also on wavelength
α
φ1
φ1’ φ2
φ2’δ
G hi l t ti f
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Graphical construction for
deviation by a prism
• Begin by finding the refraction at the first surface as before
– Before setting compass for n2, draw an arc of radius n1 centered
on A
• Through the point B draw a line perpendicular to the
second surface – the point where this line intersects the n1 circle drawn previously
is C
• The final ray exiting the prism is parallel to AC
n2
n1α
AB
C
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30
35
40
45
50
55
60
65
30 40 50 60 70 80 90
Incidence angle (degrees)
D e f l e c t i o n a
n g l e
( d e g r e e
Deflection angle of an
equilateral prism with
an index of 1.517
minimum deflection angle
Deflection of a prism as a function of
incidence angle
• Minimum
deviation depends
only on index and prism apex angle
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Minimum deviation formulas
• Formulae derived simply by setting φ1=φ2’ and φ1’=φ2
• Obviously, φ1’ can be found from φ1 from Snell’s law
• The details of the derivation are not too important, but
not too difficult either
Final results VALID ONLY FOR MINIMUM DEVIATION
To find minimumdeviation angle,
given index and
apex angle
To find input angle atminimum deviation angle
given index and apex
angle
= −
2sinsin 1
1
Anφ A−= 1min 2φ δ
To find index, givenminimum deviation
angle and apex angle
+
=
2sin
2sin min
A
A
n
δ
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Critical angle, total internal reflection
• A ray traveling from a higher indexto a lower index bends away from the
normal
– at an angle called the critical angle, the
refracted angle is 90°, the ray travels
along the surface
– for larger incidence angles, Snell’s law
says the sine of the refracted angle is
greater than one, this is impossible
• For incident angles larger than the
critical angle, the light is completely
reflected, there is no refracted ray
n
n
critical
'
)sin( =Θ
Θn
n’
C i i t t l i t l
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Common prisms using total internal
reflection