Geometric Optics

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Geometric Optics

• December 28, 2014

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Areas of Optics

• Geometric Optics
• Light as a ray.
• Physical Optics
• Light as a wave.
• Quantum Optics
• Light as a particle.

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Mirrors
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Reflection

• Reflection occurs when light bounces off a
  surface.
• There are two types of reflection
• Regular reflection, off a smooth surface
• Diffuse reflection, off a rough surface

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Types of Mirrors
Plane Concave (converging) Convex (diverging)

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Ray Diagrams

• Ray tracing is a method of constructing an image
  using the model of light as a ray.
• We use ray tracing to construct optical images
  produced by mirrors and lenses.
• Ray tracing lets us describe what happens to the
  light as it interacts with a medium.

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Light Rays

• Inherently, rays do not bend. However, if they
  encounter a different medium, they will react.

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Plane Mirrors
Incident ray
A Normal Line is perpendicular to the mirrors
surface drawn at the point of contact.
Normal Line
Plane Mirror
Reflected ray
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Law of Reflection

• The angle of incidence of reflected light equals
  the angle of reflection.
• Note that angles are measured relative to a
  normal line.

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Describing Images

• Nature
• real (converging rays)
• virtual (diverging rays)
• Orientation
• upright
• Inverted
• Size
• true
• enlarged
• reduced

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Ray Tracing
Extend Reflected Rays Behind the Mirror
Reflected Rays Diverge
Identify the Image Virtual, Upright, True
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Spherical Mirrors
Concave (Converging)
Convex (Diverging)
Positive Focal Length
Negative Focal Length
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Parts of a Spherical Mirror

• These are the main parts of a spherical concave
  mirror.
• The focal length is half of the radius of
  curvature.
• The focal length is positive for this type of
  mirror.
• R 2f

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Focus

• Rays parallel to the principal axis all pass
  through the focus for a spherical concave mirror.

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Ray Tracing Spherical Concave Mirrors

• The three principal rays to construct an image
  for a spherical concave mirror are
• the p-ray, which travels parallel to the
  principal axis, then reflects through focus.
• the f-ray, which travels through focus, then
  reflects back parallel to the principal axis.
• the c-ray, which travels through center, then
  reflects back through center.
• You must draw two of the three principal rays to
  construct an image.

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Ray Tracing Spherical Concave Mirrors

• Construct the image for an object located
  outside the center of curvature.
• It is only necessary to draw 2 of the three
  principal rays
• Identify the Image Real, Inverted, Reduced

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Ray Tracing Spherical Concave Mirrors

• Construct the image for an object located at the
  center of curvature.
• Identify the Image Real, Inverted, True

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Ray Tracing Spherical Concave Mirrors

• Construct the image for an object located between
  the center of curvature and the focus.
• Identify the Image Real, Inverted, Enlarged

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Ray Tracing Spherical Concave Mirrors

• Construct the image for an object located at the
  focus.
• Identify the Image No Image

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Ray Tracing Spherical Concave Mirrors

• Construct the image for an object located inside
  the focus.
• Identify the Image Virtual, Upright, Enlarged

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Mirror / Lens Equation

• si image distance
• so object distance
• f focal length

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Magnification Equation

• si image distance
• so object distance
• hi image height
• ho object height
• M magnification

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Sign Conventions

• Focal length (f)
• Positive for CONCAVE mirrors
• Negative for CONVEX mirrors
• Magnification (M)
• Positive for UPRIGHT images
• Negative for INVERTED images
• ENLARGED when M gt 1
• REDUCED when M lt 1
• Image Distance
• si is POSITIVE for real images
• si is NEGATIVE for virtual images

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Sample Problem

• A spherical concave mirror, focal length 20 cm,
  has a 5-cm high object placed 30 cm from it.
• a) Draw a ray diagram and construct the image.
• b) Identify the image
• c) Mathematically verify your results

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Parts of a Spherical Convex Mirror

• These are the main parts of a spherical convex
  mirror.
• The focal length is half of the radius of
  curvature, and both are on the dark side of the
  mirror.
• The focal length is negative for this type of
  mirror.

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Spherical Convex Mirror

• Construct the image for an object located outside
  a spherical convex mirror.
• Identify the image Virtual, Upright, Reduced
• All Diverging Mirrors (and Lenses) create an
  image with the same identity.

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Sample Problem

• A spherical convex mirror, focal length 15 cm,
  has a 4-cm high object placed 10 cm from it.
• a) Draw a ray diagram and construct the image.
• b) Identify the image
• c) Mathematically verify your results

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Mirror Summary

• Concave
• Image is real when object is outside focus
• Image is virtual when object is inside focus
• Focal length f is positive

• Convex
• Image is always virtual
• Focal length f is negative

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Refraction
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Refraction

• Refraction is the movement of light from one
  medium into another medium.
• Refraction cause a change in speed of light as it
  moves from one medium to another.
• Refraction can cause bending of the light at the
  interface between media.

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Index of Refraction

• n index of refraction
• c speed of light (3 x 108 m/s)
• v velocity of light in the medium

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Snells Law

• n1 index of refraction of incident medium
• ?1 angle of incidence
• n2 index of refraction of refracting medium
• ?2 angle of refraction

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Snells Law
When the index of refraction increases, light
bends toward the normal.
?1
n1
n2 gt n1
n2
?2
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Snells Law
When the index of refraction increases, light
bends toward the normal.
?1
n1
n1 gt n2
n2
?2
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Sample Problem

• Light enters an oil from the air at an angle of
  50 with the normal, and the refracted beam makes
  an angle of 33 with the normal.
• a) Draw this situation.
• b) Calculate the index of refraction of the oil.
• c) Calculate the speed of light in the oil

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Prism Problem

• Light in air enters a 30-60-90 prism
  perpendicular to the long side and passes through
  the prism. If the refractive index of the glass
  is 1.55, calculate the angle of refraction when
  it leaves the prism.

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Critical Angle
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Critical Angle

• If light passes into a medium with a greater
  refractive index than the original medium, it
  bends away from the normal and the angle of
  refraction is greater than the angle of
  incidence.
• If the angle of refraction is gt 90, the light
  cannot leave the medium.
• The smallest angle of incidence for which light
  cannot leave a medium is called the critical
  angle of incidence.

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Calculating the Critical Angle
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Sample Problem

• What is the critical angle of incidence for a
  gemstone with refractive index 2.45 if it is in
  air?

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Lenses
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Lenses
Converging
Diverging
Positive Focal Length
Negative Focal Length
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Focus

• All rays parallel to the principal axis refract
  through the focus of a converging lens.

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Ray Tracing

• Ray tracing is also used for lenses. We use the
  same principal rays we used for mirrors.
• the p-ray, which travels parallel to the
  principal axis, then refracts through focus.
• the f-ray, which travels through focus, then
  refracts parallel to the principal axis.
• the c-ray, which travels through center and
  continues without bending.
• You must draw 2 of the 3 principal rays.

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Principle Rays for Lenses

• Construct the image for an object located outside
  2F.
• It is only necessary to draw 2 of the three
  principal rays

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Mirror / Lens Equation

• si image distance
• so object distance
• f focal length

• si image distance
• so object distance
• hi image height
• ho object height
• M magnification

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Sample Problem

• A converging lens, focal length 20 cm, has a 5-cm
  high object placed 30 cm from it.
• a) Draw a ray diagram and construct the image.
• b) Mathematically verify your ray diagram.
• c) Identify the image

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Sample Problem

• A diverging lens, focal length -15 cm, has a 4-cm
  high object placed 10 cm from it.
• a) Draw a ray diagram and construct the image.
• b) Mathematically verify your ray diagram.
• c) Identify the image

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Summary

• Converging Lens
• f is positive
• so is positive
• si is positive for real images and negative for
  virtual images
• M is negative for real images and positive for
  virtual images
• hi is negative for real images and positive for
  virtual images

• Diverging Lens
• f is negative
• so is positive
• si is negative
• M is positive and lt 1
• hi is positive and lt ho

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Multiple Lenses / Mirrors

• When drawing ray diagrams for a combination of
  lenses/mirrors, use the image from the first
  lens/mirror as the object for the second.
• When appropriate, apply the p-ray, f-ray, and
  c-ray rules to the second lens/mirror.
• To Identify the image, the result is compared to
  the original object.

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Sample Problem