Reflective Optics

1
Reflective Optics

• Chapter 25

2
Reflective Optics

• Wavefronts and Rays
• Law of Reflection
• Kinds of Reflection
• Image Formation
• Images and Flat Mirrors
• Images and Spherical Mirrors
• The Paraxial Approximation and Aberrations

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Wavefronts and Rays

• A wave is the propagation of a condition or
  disturbance.
• A wavefront is a surface over which the value of
  that condition is constant.

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Wavefronts and Rays

• The direction of motion is always locally normal
  to the wavefront.
• A line drawn in the direction of advance is
  called a ray.

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Wavefronts and Rays

• The directional distribution of these rays
  depends on the nature and geometry of the source
  of the waves.

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Wavefronts and Rays

• As distance from the point source increases, the
  radii of the spherical wavefronts becomes larger,
  until the wavefronts approximate planes. Waves
  from an infinitely-distant source are sometimes
  called plane waves.

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Law of Reflection

• When light encounters the surface of a material,
  three things happen
• reflection
• transmission
• absorption

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Law of Reflection

• In reflection, the light bounces off the
  surface.
• The bounce occurs according to the law of
  reflection

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Law of Reflection

• Notice that
• Both angles are measured from the surface normal
• The incident ray, the reflected ray, and the
  surface normal all lie in a single plane the
  plane of incidence

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Law of Reflection

• Notice that
• If the surface normal is rotated through an angle
  a within the plane of incidence, and the incident
  direction is constant, the reflected ray rotates
  through twice the angle (2a)
• If the plane of incidence rotates, the reflected
  ray rotates with it (one for one)

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Kinds of Reflection

• We distinguish between two sorts of reflection
• Specular (from smooth surfaces)
• mirror
• polished metal
• calm liquid
• Diffuse (from rough or irregular surfaces)
• white paper
• projection screen
• clouds or snow

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Kinds of Reflection Specular

• A surface producing specular reflection has a
  constant, or a well-behaved (slowly and
  continuously changing) normal direction.
• For a constant incident direction, the reflected
  direction is either constant or changes
  continuously organized.

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Kinds of Reflection Diffuse

• A surface producing diffuse reflection has random
  surface normal directions that change chaotically
  with location on the surface.
• The law of reflection is everywhere obeyed but
  with random results.

14
Image Formation

• Consider an object that either produces light, or
  that scatters light from its surroundings. Each
  point on its surface acts as a spherically-symmetr
  ic source (point source), sending out rays in
  many directions.

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Image Formation

• If something acts on some of the rays that
  originate at one point on the object, and causes
  them to converge at a point somewhere else, or to
  diverge from a point somewhere else, then it has
  formed an image of that object.

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Image Formation Two Kinds of Image

• Images may be sorted into two categories
• virtual images formed when the rays never
  physically come back to one point, but instead
  diverge as if they came from one point. The
  place they appear to have come from is the image.
• real images formed when the rays converge, so
  that they physically arrive at the same point.
  That point of physical reconvergence is the image.

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Image Formation Real Image

• The rays here physically converge real image.

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Image Formation Virtual Image

• The rays here diverge as if they came from an
  image point virtual image.

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Image Formation by a Flat Mirror

• The image formed by a flat mirror
• virtual
• upright
• same size
• same distance
• on the other
• side of the
• mirror

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Image Formation by a Flat Mirror

• Two flat mirrors the image formed by one mirror
  acts as an object for the second mirror.

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Image Formation by Spherical Mirrors

• A spherical mirror is one whose surface is a
  portion of a sphere.
• The radius of the sphere at the mirrors center
  is called the optical axis. The center of the
  sphere is the center of curvature.

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Image Formation by Spherical Mirrors

• Necessary terms

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Spherical Mirrors Special Rays

• Chief ray a ray striking the vertex reflects
  symmetrically about the axis.

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Spherical Mirrors Special Rays

• Axial ray a ray parallel to the axis passes
  through the focal point after being reflected (or
  appears to have)

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Spherical Mirrors Special Rays

• This means that the image (real or virtual) of an
  infinitely-distant object is formed at the focal
  point.

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Spherical Mirrors Special Rays

• A ray passing through the center of curvature
  passes through it again after reflection.

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Finding Images by Ray Tracing

• We can use these special ray properties to find
  the locations where images are formed.
• We can also find out
• the image size
• the image orientation
• whether the image is real or virtual

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Finding Images by Ray Tracing

• Example concave mirror, object outside the
  center of curvature
• Image real, inverted, between focal point and
  center of curvature

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Finding Images by Ray Tracing

• Concave mirror, object at the center of curvature
• Image real, inverted, at center of curvature

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Finding Images by Ray Tracing

• Object between center of curvature and focal
  point
• Image real, inverted, outside center of curvature

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Finding Images by Ray Tracing

• Object at the focal point
• Image real, inverted, located at infinity

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Finding Images by Ray Tracing

• Object inside the focal point
• Image virtual, upright

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Image Formation Mathematics

• Trace a single chief ray from object to image
• magnification
• do and di are the conjugate distances

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Image Formation Mathematics

• Trace a single axial ray

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Image Formation Mathematics
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Image Formation Mathematics

• Mirror equation
• The mirror equation relates the conjugate
  distances and the focal length. With the
  definition of magnification
• it can be used generally to characterize images
  formed by mirrors.

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Image Formation Mathematics

• In solving problems, we must keep a standard set
  of sign conventions in mind.
• In the picture, all dimensions shown are positive
  except for hi, which is negative.

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Image Formation Mathematics

• Sign convention summary
• f for a concave mirror - for a convex mirror
• conjugate distances (do and di) if object or
  image is in front of mirror - if behind
• magnification, m if image is upright - if
  image is inverted

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The Paraxial Approximation

• Did you notice a stolen base?
• f, which is the distance from the focal point to
  the vertex, isnt quite the base of the green
  triangle.

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The Paraxial Approximation

• The larger the height of the axial ray, the more
  difference there is between f and the length of
  the triangles base.

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The Paraxial Approximation

• The mirror equation is valid only as a paraxial
  approximation. It applies to a threadlike
  cylinder of infinitesimal diameter, centered on
  the axis.
• The difference between the paraxial approximation
  and the consequences of exact spherical geometry
  cause what is called spherical aberration.
• The larger the height of an axial ray, the closer
  to the vertex it passes through the axis.

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Spherical Aberration
43
Spherical Aberration
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The Paraxial Approximation

• So what good is the mirror equation, since it is
  only an approximation?
• Optical system design
• paraxial layout (approximation)
• computer modeling and optimization (exact)