Geometrical Optics

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• Geometrical Optics

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Geometrical Optics

• Optics is usually considered as the study of the
  behavior of visible light (although all
  electromagnetic radiation has the same behavior,
  and follows the same rules).
• The propagation of light can be described in two
  alternative views
• a) As electromagnetic waves
• b) As rays of light
• When the objects with which light interacts are
  larger than its wavelength, the light travels in
  straight lines called rays, and its wave nature
  can be ignored.
• This is the realm of geometrical optics.

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Geometrical Optics
Light can be described using geometrical optics,
as long as the objects with which it interacts,
are much larger than the wavelength of the light.
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Propagation of Light
Light propagates in straight lines (rays). This
is valid as long as the light does not change the
medium through which it propagates (air, water,
glass, plastic), or finds an obstacle
(interface). The velocity of light in air is c c
3x108 m/s The velocity of light in other
media may be different from c (less than c).
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Reflection and Transmission
Most materials reflect light (partially or
totally). For example, metals reflect light
(almost totally) because an incident oscillating
light beam causes the metals nearly free
electrons to oscillate, setting up another
(reflected) electromagnetic wave. Opaque
materials absorb light (by, say, moving electrons
into higher atomic orbitals). Transparent
materials transmit light. These are usually
insulators whose electrons are bound to atoms,
and which would require more energy to move to
higher orbitals than in materials which are
opaque.
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Geometrical Optics
Angles are measured with respect to the normal to
the surface
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Reflection

• The Law of Reflection
• Light reflected from a surface stays in the
  plane formed by the incident ray and the surface
  normal
• and
• the angle of reflection equals the angle of
  incidence (measured to the normal)

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Reflection
Specular and Diffuse Reflection

Smooth ? specular ? shiny Rough ? diffuse ? dull
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Two mirrors are placed at right angles as
shown. An incident ray of light makes an angle of
30 with the x axis. Find the angle the outgoing
ray makes with the x axis.
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Refraction
More generally, when light passes from one
transparent medium to another, part is reflected
and part is transmitted. The reflected ray obeys
q1 q1.
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Refraction
More generally, when light passes from one
transparent medium to another, part is reflected
and part is transmitted. The reflected ray obeys
q1 q1.
The transmitted ray obeys Snells Law of
Refraction It stays in the plane, and the angles
are related by n1sinq1 n2sinq2
Here n is the index of refraction of a medium.
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Refraction
q1 angle of incidence ?1 angle of
reflection q2 angle of refraction
Law of Reflection q1 ?1 Law of Refraction n1
sin?1 n2 sin?2
n ? index of refraction ni c / vi vi velocity
of light in medium i
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Index of Refraction
The speed of light depends on the medium trough
which it travels. The speed of light in a given
medium is determined by the mediums index of
refraction n.
Air, n 1.000293 Glass, 1.45 ? n ? 1.66 Water, n
1.33
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Refraction
l1v1T
The period T doesnt change, but the speed of
light can be different. in different materials.
Then the wavelengths l1 and l2 are unequal.
This also gives rise to refraction.
q1
1
q1
2
q2
l2v2T
The little shaded triangles have the same
hypoteneuse so l1/sinq1 l2/sinq2, or
v1/sinq1v2/sinq2
q2
Define the index of refraction nc/v. Then
Snells law is n1sinq1 n2sinq2
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Example air-water interface
If you shine a light at an incident angle of 40o
onto the surface of a pool 2m deep, where does
the beam hit the bottom?
Air n1.00 Water n1.33 (1.00)sin40
(1.33)sinq sinqsin40/1.33 so q28.9o Then
d/2tan28.9o which gives d1.1 m.
40
air
water
2m
q
d
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Example air-water interface
If you shine a light at an incident angle of 40o
onto the surface of a pool 2m deep, where does
the beam hit the bottom?
Air n1.00 Water n1.33 (1.00)sin40
(1.33)sinq sinqsin40/1.33 so q28.9o Then
d/2tan28.9o which gives d1.1 m.
40
air
water
2m
q
d
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Example air-water interface
If you shine a light at an incident angle of 40o
onto the surface of a pool 2m deep, where does
the beam hit the bottom?
Air n1.00 Water n1.33 (1.00) sin(40)
(1.33) sinq Sinq sin(40)/1.33 so q
28.9o Then d/2 tan(28.9o) which gives ? d1.1
m.
40
air
water
2m
q
d
Turn this around if you shine a light from the
bottom at this position it will look like its
coming from further right.
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Some common refraction effects
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(No Transcript)
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Air-water interface
Air n1 1.00 Water n2 1.33
n1 sin?1 n2 sin?2 n1/n2 sin?2 / sin?1
When the light travels from air to water (n1 lt
n2) the ray is bent towards the normal. When
the light travels from water to air (n2 gt n1) the
ray is bent away from the normal.
This is valid for any pair of materials with n1 lt
n2
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Total Internal Reflection
n1 gt n2
q2
n2
q2
q1
qc
q1
q1
n1
n2sin p/2 n1sin q1 ... sin q1 sin qc n2 / n1
Total internal reflection no light is refracted
Some light is refracted and some is reflected
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Total Internal Reflection

• The critical angle is when q2 p / 2,
• which gives qc sin-1(n2/n1).
• At angles bigger than this critical angle,
• the beam is totally reflected.

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• Find the critical angle for light traveling from
  glass (n 1.5) to
• Air (n 1.00)
• Water (n 1.33)

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Example Fiber Optics
An optical fiber consists of a core with index n1
surrounded by a cladding with index n2, with n1
gt n2. Light can be confined by total internal
reflection, even if the fiber is bent and
twisted.
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Example Fiber Optics
Find the minimum angle of incidence for guiding
in the fiber, for n1 1.7 and n2 1.6
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Example Fiber Optics
Find the minimum angle of incidence for guiding
in the fiber, for n1 1.7 and n2 1.6

• sin qC n2 / n1
• qC sin-1(n2 / n1)
• sin-1(1.6/1.7) ? 70o.
• (Need to graze at lt 20o)

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Reflection and Transmission at Normal Incidence
Geometrical optics cant tell how much is
reflected and how much transmitted at an
interface. This can be derived from Maxwells
equations. These are described in terms of
the reflection and transmission coefficients R
and T, which are, respectively, the fraction of
incident intensity reflected and transmitted. For
the case of normal incidence, one finds
Notice that when n1n2 (so that there is not
really any interface), R 0 and T 1.