VII2 Basic Optical Elements and Instruments

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VII2 Basic Optical Elements and Instruments
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Main Topics

• Refraction, Dispersion and Refraction Optics.
• Thin Lenses. Types and Properties.
• Combination of Lenses.
• Basic Optical Instruments
• Human Eye
• Magnifying Glass
• Telescope
• Microscope

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Refraction I

• Another important basic optical effect is
  refraction appearing when rays pass from one
  material to another. Transparent materials may
  differ in their optical density.
• The more dense material the lower is the speed of
  light in it. Optical density is characterized by
  the absolute refraction index n c/v
• c is the speed of light in vacuum
• v speed in the particular material.

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Refraction II

• We can again use the Fermats principle to find
  the law of refraction.
• To find which ray makes it first from S to P is a
  similar problem as if we want to safe a drowning
  person in the shortest time, taking into account
  that we run much faster than swim.

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Refraction III

• We use the more general definition that the
  correct ray is the stationary one. In other
  words, if we take some neighboring ray its time
  of flight will be (roughly) the same.
• Let the point
• S be in a space where the light travels with the
  speed v1 c/n1 and
• P in the space where the speed is v2 c/n2.

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Refraction IV

• Now, let the SCP be the correct ray for and the
  SXP some neighboring ray. Should the time of
  flight be the same EC/v1 XF/v2
• We use EC XCsin?1 and XF XCsin?2
  substitute for v1 and v2 and get the
• Snells law

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Refraction V

• We see that the higher is the optical density or
  the slower is the speed of light the smaller is
  the refraction angle.
• If the angle of incidence from the less dense
  material is 90 the refracted angle is given
• sin?c n1/n2 the maximum refracted angle or the
  critical angle.

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Refraction VI

• If the beam would try to pass from the optically
  dense material under an incident angle higher
  than the critical angle it would not get through
  the boundary but rather be totally reflected.
• The effect of total (internal) reflection is used
  for instance in fiber optics.

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Dispersion I

• Transparent materials have an important property
  that the speed of light and thereby their
  refraction index depends on the wavelength of the
  applied light.
• The higher energy (lower ?) the stronger
  interaction and thereby higher optical density
  and higher deflection from the original
  direction.
• This means that light of every wavelength or
  color is refracted under a (little) different
  angle.

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Dispersion II

• The effect of dispersion complicates design of
  optical systems and has to be compensated by
  using more lenses of different materials.
• On the other hand it gives us the possibility to
  decompose the visible light and near IR and UV
  regions into different wavelengths.
• That is important for instance for studies of
  properties of matter by spectroscopic methods.
  The matter can be very far away in the universe!

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Refraction Optics I

• The effect of refraction is used to build optical
  components and systems.
• If we have a point S in the medium n1 and the
  point P in the medium n2 gt n1 we may use the
  Fermats principle to find the shape of the
  boundary between the media so the points are
  conjugated or the optical system is stigmatic for
  them.

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Refraction Optics II

• If we compare a time of flight of some refracted
  ray with the one directly connecting both points
  we find a relation
• l1n1 l2n2 s1n1 s2n2
• We readily understand from here, why the
  optically denser media must be convex.
• The corresponding surface is of the fourth order,
  so called, Cartesian ovoid.

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Refraction Optics III

• If we move one of the points S or P into infinity
  the surface becomes second order, either
  elliptical or hyperbolical.
• This can be in principal used to construct lenses
  - optical components from some material, which
  allow that the object as well as the image are in
  the same media.

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Refraction Optics IV

• Ideal lenses are for instance double hyperbolic
  or planar-hyperbolic.
• Although, recently they can be, in principle,
  machined, for the same reasons, as in the case of
  mirrors aspherical surfaces are approximated by
  cheaper spherical ones.
• But they can be successfully used only in the
  paraxial region.

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Refraction Optics V

• Spherical surface can be shown to be stigmatic
  for points on the optical axis in the paraxial
  approximation.
• Let the the ray come from the point O in matter
  n1 under an angle ? and hit the spherical surface
  in the point P, which is seen from the curvature
  center C under an angle ? and deflects to the
  point I in material n2, where it arrives under an
  angle ?.
• ?1 and ?2 will be the incident and refracted
  angles.

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Refraction Optics VI

• From triangle PIC ? ? ?2 OPC ?1 ? ?
  
• In paraxial approximation the angles are small,
  so we can write n1?1 n2?2
• ? h/d0 ? h/R ? h/di where h is the
  height of the point P from the optical axis.
• We can show that the angle dependence vanishes

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Refraction Optics VII

• It is important to obey the following convention
  
• If C in on the same side as the light comes from,
  it is negative.
• If O in on the same side as the light comes from,
  it is positive.
• If I in on the same side as the light comes from,
  it is negative.
• We can see that the reciprocity principle is
  valid!

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Thin Lenses I

• Very important lenses are those which can be
  considered as thin.
• All their properties can be characterized by a
  single parameter the focal length f.
• It is the distance from the optical center to the
  focal points F.
• There is one focal point in front and one behind
  the lens, both equally distant from the center.

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Thin Lenses II

• The, so called, lensmakers equation can be
  derived which relates the focal distance of a
  thin lens with the radii of its spherical
  surfaces
• Sign conventions must be obeyed.
• Note that the focal length is the same on both
  sides even if the radii are different.

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Thin Lenses III

• It is possible to make converging lenses with
  positive focal length when the positive radius of
  curvature is smaller or diverging lenses with
  negative focal length when the negative radius of
  curvature is smaller.
• Optometrist and ophthalmologist use the power P
  1/f to specify lenses. Its unit is diopter (D),
  1D 1m-1.

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Thin Lenses IV

• To find an image of some point, we can again use
  two of three special rays.
• A ray passing in any direction through optical
  center is not deflected.
• A ray arriving in parallel with the optical axis
  will pass through the image focus if f is
  positive or appear to leave it, if f is negative.

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Thin Lenses V

• A ray passing through the object focus if f is
  positive or heading towards it, if f is negative,
  will continue in parallel with the optical axis
  on the other side of the lens.
• If the imaging is stigmatic (sharp) all other
  rays leaving the object point must appear in the
  image point as well. But they cant be used to
  find it.

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Thin Lenses VI

• The lens equation which relates the distances of
  the object and image with the focal distance can
  be easily derived
• 1/do 1/di 1/f
• and lateral magnification is defined as the ratio
  of the image height to the object height
• m ho/hi - di/do

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Thin Lenses VII

• To comprehend functioning of almost any optical
  instrument it is necessary to fully understand
  the importance of the focal planes of the lenses.
• For converging lens a bunch of parallel rays
  coming under some angle with the optical axis
  will pass through a point in the image focal
  plane, which is on the other side of the lens.

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Thin Lenses VII

• We can locate this point using the ray passing
  the optical center and the one passing through
  the object focus.
• Using the lens equation we can verify that an
  object producing an image in the focal plane (di
  f) must be in infinity.

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Thin Lenses VIII

• For diverging lens all beams heading towards a
  point in the object focal plane, which is now
  behind the lens, will run as a bunch of parallel
  rays after the lens.
• We can find their direction using the ray passing
  the optical center and the one coming in parallel
  with the optical axis.

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Thin Lenses IX

• We can again verify this using the lens equation.
  If the object is in the object focal plane (do
  f , both negative now) the image must be in
  infinity.
• We can produce parallel bunches of rays by both
  types of thin lenses if the object is in the
  object focal plane. For the diverging lens the
  object distance is, however, negative!

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Combination of Lenses

• We start from the lens closest to the object.
• We display the object by this lens only.
• The image of produced by the first lens will be
  the object for the second lens.
• Then we display the new object by the second lens
  only. And so on.
• The sign convention must be strictly obeyed since
  now object distance may be negative!

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The Human Eye I

• Most of the focusing (refraction) is done by the
  cornea (n 1.376). The lens does just the fine
  tuning.
• The quality of focusing and the depth of focus
  depends on the iris. The smaller the aperture the
  better.
• Normal eye has the near point at 25 cm and the
  far point in infinity.

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The Human Eye II

• In the case of nearsightedness (myopia) the far
  point is not infinity. This has to be corrected
  by a diverging lens.
• In the case of farsightedness (hyperopia or
  presbyopia developed by age) the eye cant
  focus on near objects. This has to be corrected
  by a converging lens.

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The Human Eye III

• The eye is relaxed if it watches the far point so
  eyepieces usually produce parallel rays.
• Some other optical instruments produce a virtual
  image in the conventional length equal to the
  standard near point at 25 cm.

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Magnifying Glass

• Magnifying glass is used
• either the object is in the focal plane and we
  watch it by relaxed eye.
• or the lens is close to the eye (Sherlock Holmes)
  and a virtual image is produced in the
  conventional distance.
• Magnification is the angle magnification we see
  objects as big as is the angle of their image on
  the retina.

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Telescopes I

• Astronomical refractive telescopes have two
  lenses an objective (with longer f) and an
  eyepiece, which share the same focal plane.
• The eyepiece can be either a converging lens or a
  diverging one, then the shared focal plane is
  behind the eyepiece.
• The angle magnification in both cases is minus
  the ratio of the focal lengths.

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Telescopes II

• Important are reflecting telescopes
• Large mirrors are easier to produce and support
• Mirrors dont suffer from color aberration.
• But we have to realize that although reflection
  is not influenced by dispersion, it is still a
  complicated process and reflectivity of any
  material isnt ideal .

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Compound Microscope

• The principle of a microscope can be shown also
  using two lenses. The objective (now with very
  short f) produces a real image. It is watched by
  the eyepiece, which usually produces the
  imaginary image in the conventional distance.
• Good microscopes are complicated since it is
  important to compensate aberrations.

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Homework

• The last homework is due tomorrow!

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Things to read and learn

• This lecture covers
• Chapters 33 5, 6, 7, 8 34
• Advance reading
• Chapters 35, 36
• Please, read and try to understand even the parts
  which were not dealt with in detail in the
  lecture. You should have far enough background
  knowledge to understand everything!

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Lens Equation

• .