Lecture 5 Geometrical Optics

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Lecture 5Geometrical Optics

• Generalization of Fermats Principle
• What is Geometrical Optics and when does it
  apply?
• Refraction at Aspherical Surfaces
• Refraction at Spherical Surfaces

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Generalization of Fermat Principle

• Fermat's principle of optics, in its historical
  form states
• The actual path between two points taken by
  a beam of light is the one which is traversed in
  the least time.
• The modern, full version of Fermat's Principle
  states
• the optical path length must be extremal,
  which means that it can be either minimal,
  maximal or a point of inflection (a saddle point

Optical Path Length
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Stationary Optical Path Length

• Examples when O.P.L is not minimal
• Point Source at focus of Ellipsoid
• Spherical Interface
• Lens

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Fermats PrincipleEllipsoid

• Ellipsoid with observer at F2 and point source at
  F1
• Distance F1-gtx-gtF2 constant for all paths that
  reflect from ellipsoid surface
• No path that is minimal!!!
• OPL for all of these paths are the same
• Each path is equally probable!!!

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Fermats PrincipleRefraction at Spherical
interface for paraxial rays

• Plane interface where there was only one path
  that minimized time
• At spherical interface there are multiple paths
  between A and B that have same OPL
• These paths are equally probable

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Fermats PrincipleLens

• All trajectories for light to travel from A to B
  are equally probable

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Geometrical Optics

• The study of how images form in optical systems.
• Uses the concept of rays
• Rays have direction and position but no phase
  information.
• Optics is an approximation of how Electromagnetic
  Radiation behaves in an optical system
• This model works well when the smallest dimension
  of the optical system is much larger than the
  wavelength of the incident EM Radiation.
• Physical Optics must be used when the wavelength
  is large
• An object is viewed as a collection of many
  pin-point sources that produce bundles of rays.
• These rays are traced through an optical system
  to determine what image will be formed.

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Geometrical Optics

• Objects are point sources of Rays .

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Geometrical Optics

• If rays emerging from point S are imaged into
  same focal point P
• S and P are conjugate focal points
• Sytem is stigmatic for points S and P

Point source of rays
Optical System
Focal point
S
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Geometrical OpticsLimitation
If system accepts only segments of wave fronts ?
Waves will be diffracted Diffraction Limited As
l ? 0 no diffraction limit (More on this later)
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Refraction at Aspherical Surfaces

• Refer to figure 5.3 of text
• Consider a point source at S
• Consider a material with index nt
• When nt is greater than ni , the wave slows
  when entering medium
• Extremeties of wavefront will overtake the
  midregion
• If interface is properly shaped the spherical
  wavefront can be bent into a plane wave
• Required shape can be found by finding shape
  where any point A on the surface where the path
  SAD will have the same phase where D lies on a
  plane surface in the medium (draw this on
  board).
• Same phase ? same number of wavelengths for
  whichever path from S to DD is taken
  F1A/liAD/lt is constant
• constant is the equation
  for a hyperbola with eccentricity
• Reversible
• http//www-optics.unine.ch/education/optics_tutori
  als/aspherical_surface.html

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Aspherical Surfaces

• To obtain object and image points that lie
  outside of transparent medium use a second
  interface with similar shape

• Apheric Optical Elements
• With one or both surfaces neither planar nor
  spherical
• Difficult to manufacture ? Expensive
• Surface Quality inferior to spherical optics

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Refraction at Spherical Surfaces
Fermat?

• Refer to figure 5.6 of text
• Derive relationship between
  on board
• Paraxial Optics
• For rays that arrive at shallow angles w.r.t
  optical axis

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Geometrical Optics Java Applets

• http//www.phy.ntnu.edu.tw/oldjava/optics/mirror_e
  .html
• http//www.phy.ntnu.edu.tw/ntnujava/main.php?t65
• http//www.kamikawas.com/physics/thicklens/thicl_e
  .htm