VII

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VII2 Basic Optical Elements and Instruments
2
Main Topics

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

3
Refraction I

• Another important basic optical effect is
  refraction. This time rays pass from one material
  to another. Transparent materials differ in, so
  called optical density, the more dense material
  the lower is the speed of light in it. We
  characterize this by the absolute refraction
  index n c/v where c is the speed of light in
  vacuum and v speed in the particular material.

4
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 and we optimize for the shortest time,
  taking into account that we run much faster than
  swim.

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Refraction III

• We use the general definition that the correct
  ray is a stationary one. In other words, if we
  take some close 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 neighbor 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
• n1sin?1 n2sin?2

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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?2 n1/n2 the maximum reflected 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 this critical angle it would not get through
  the boundary but rather be totally reflected.
• The effect of total (internal) reflection is used
  in fiber optics.

9
Dispersion I

• Transparent materials have an interesting
  property that the speed of light and thereby
  their refraction index depend on the wavelength
  of the applied light.
• This means that light of every wavelength of
  color is refracted under a (little) different
  angle.

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Dispersion II

• The effect of dispersion complicates design of
  optical systems.
• On the other hand it gives us the possibility to
  decompose the visible light and near IR and UV
  regions into different wavelengths which has a
  great impact for instance on studies of
  properties of matter by spectroscopic methods.

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

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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, which were
  described in the case of mirrors aspherical
  surfaces are approximated by spherical ones.
  Again they can be successfully used only in the
  paraxial region.

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Thin Lenses I

• Very important lenses are those which can be
  considered as thin.
• They can be characterized by a single parameter
  the focal length f. It is the distance from the
  center of the lens to the focal point F. It is
  the point in which the rays approaching the lens
  in parallel with the optical or principal axis
  meet behind it.

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Thin Lenses II

• To understand many optical instruments it is good
  to remember that also other parallel rays which
  fall at the lens at an angle focus in one point,
  which lays in the focal plane of the lens.
• 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 III

• The, so called, lensmakers equation can be
  derived which relates the focal distance of a
  thin length with the radii of its surfaces
• 1/f (n-1)(1/R1 1/R2)
• Appropriate sign conventions must be obeyed.
• Note that the focal length is the same from both
  sides even if the radii are different.

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Thin Lenses IV

• As it was the case of the mirrors, lenses can be
  converging and diverging and images can be real
  and virtual.
• To find an image of some object, we can again use
  two of three special rays. We can employ the
  properties of a focal point and the fact that the
  beam passing through the optical center is not
  deflected.

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Thin Lenses V

• 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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Combination of Lenses

• We start from the lens closest to the object.
• We display the object as if only this lens were
  present.
• 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.

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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 had 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 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 on retina.

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Telescopes

• Astronomical telescope are two lenses an
  objective (longer f) and an eyepiece which share
  the same focal plane. The angle magnification is
  ratio of the focal lengths.
• Important are reflecting telescopes
• Large mirrors are easier to produce and support
• Mirrors dont suffer from color aberration.

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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

• Chapters 33, 34
• 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!

29
Lens Equation

• .

30
Maxwells Equations I

• .