Geometrical Optics (general theory)

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Chapter 7

• Geometrical Optics (general theory)
• (April 27 , 2005)

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A brief summary to the last lecture

• Capillarity and surface absorption Capillary
  tube in liquid, capillary phenomena, stemmed from
  (???) additional pressure under curved surface
  surface tension.
• Geometrical optics light rays, dielectrics,
  light speed, refractive index, laws of reflection
  and refraction (three angles), total internal
  reflection (from optically denser medium to
  optically thinner medium), and refraction of a
  spherical surface.

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Some questions for you to answer

1. What is capillary tube? (???)
2. What is capillarity? (???)
3. What is air embolism? (???)
4. What causes the capillary phenomena? (In other
   words, how is it formed?) ???
5. What is the surface-active agent? What is the
   non-surface-active agent? Which one could change
   the surface tension greatly?(???)

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1. Does light travel at the same speed in different
   medium? Give your definition on the refractive
   index? (???)
2. What are reflective and refractive laws? ???
3. What is the equation of single spherical
   surface?? ?
4. How to determine the signs of the object
   distance, image distances and curvature radius?
   ???
5. What is the first focal distance and what is the
   second focal distance? ? ?

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1. Converge (??,??), diverge (??)
2. Lens (??),circumference (??),virtual (??)
3. Negligible (????), unimportant
4. Graphical (???,???),dioptric (???,???, ??????),
   coaxial (???)
5. un-deviated (????),un-displaced
6. Diopter (???), Substitute (??(?))
7. Reversed (???), principal points (??)
8. Cardinal points (??) focus (??, focal point),
   in focus (??) out of focus (??) (phase, order,
   question, shape, touch with )

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7.2 (b) Focus and coaxial spherical system
3. Coaxial spherical system Usually there are
several refractive surfaces in an optical system.
For example, we could make a system of a few
lenses and each lens has two surfaces. If all
the centers of the spherical surfaces are in a
line, the system is called coaxial spherical
system and the line is called light axis.
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• If we want to calculate the image of the system,
  we can calculate the image of spherical surface
  one after another.
• The image of first surface is regarded as the
  object of the second surface and so on.
• Especially, pay attention to the signs of
  object distance, image distance and curvature
  radius of each spherical surface.

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Example 7-2 A glass ball has a radius of 10cm
and refractive index (???) of 1.5. Use paraxial
rays (????) formula to calculate the image of an
object that is 40cm from the ball.
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Solution for the first surface, we know
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This is the first step. I1 is the virtual (??)
image for the first lens and it is regarded as
the object of the second spherical surface. So
for the second surface, we have
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Substituting the above data to the paraxial ray
formula, we have
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7.3 Thin lenses

• A lens is a simple coaxial system and it is an
  optical system including two refracting surfaces.
• If the thickness of a lens is much smaller than
  curvature radius, the object and image distances,
  the thickness of the lens can be negligible in
  comparison with them. Such a lens is called thin
  lens.
• Lens can converge and diverge light.

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7.3.1 Some special rays for converging and
diverging lenses (1). Light rays parallel to the
axis of a converging lens are refracted through
the focal point on the opposite side of the lens.
First and second focal point of a lens.
Planes through the focal points of a lens are
called focal planes.
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(2). Light rays parallel to the axis of a
diverging lens are refracted so that their
backward continuations pass through the focal
point on the same side of the lens.
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(3) Definition of converging and diverging
lenses
A lens is converging if the glass is thinner
around the circumference than at the center and
diverging if the situation is reversed.
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(4). Graphical method A light ray through the
center of a thin lens continues un-deviated and
un-displaced. A light ray parallel to the axis
passes through the second focal point of a
converging lens
Real image
F
Virtual image
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n
7.3.2 The lens equation
n0
n0
According to the formula of single spherical
surface, we can use it one by one and finally
calculate the lens equation.
v
u
v1
u1 u, v1 -u2, v2 v,
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For the first surface, n1 n0, n2 n
For the second surface, n1 n, n2 n0
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The above two equations can be added together,
then we have
If lens is in the air, n0 1,
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first focal distance for thin lens
second focal distance for thin lens
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Therefore we have f1 f2 . we can suppose that
they are all equal to f and for air medium, we
have
Finally we obtain the lens equation,
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Pay attention to the symbol of curvature radius
(r2 lt 0 generally). 1/f is called lens power (or
focal power), also denoted by D, with unit of
diopter (1/m).
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7.3.3 Systes of two lenses Many optical
instruments include combinations of two or more
lenses. In systems of multiple lenses, the image
formed by one lens becomes the object for the
next lens (see the following figure).
f1
u1
v1
u2
f2
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However, same as before, sometimes the second
lens is placed between the first lens and the
image. The original image disappears, and the
second lens may or may not form a new image, but
the image that the first lens would have formed
still serves as the object for the second lens.
Because the first image is not actually formed
but still functions as the object for the second
lens, we call it a virtual object. The object
distance u of a virtual object is negative.
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The case of two connected thin lenses
For the first lens, u1 u
For the second lens, u2 -v1, v2 v,
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Suppose that the focal distance for the group of
thin lenses is f, we have
For the more lens systems, we could use this
method one by one to solve them. Written as
dioptric strength,
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Example 7.3 A converging lens of focal length 12
cm is placed 52 cm from another converging lens
of focal length 8 cm. Calculate the image
position of an object that is 16 cm in front of
the first lens.
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(No Transcript)
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Solution the image formed by the first lens can
be obtained by the given data and the thin length
equation. Substituting the first object distance
of 16 cm and the focal distance of first lens
into thin lens equation, we have
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As the distance between the two lenses is 52 cm,
the object distance for the second lens is 52
48 4 cm. Using the focal distance for the
second lens of 8 cm, the final image can be
obtained as
The image is virtual.
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7.4 Thick lens
Thick lenses are similar to the thin lenses and
they also contain two systems of coaxial
spherical surfaces. The difference between them
is that thickness of the thick lenses cannot be
negligible while the thickness of thin lenses can
be ignored. As before, such a system can be
solved by spherical surface, but it contain a lot
of trivial details especially for coaxial optical
system of more spherical surfaces.
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Cardinal points (??) 1. Two focal points (F1,
F2), 2. Two principal points ( H1, H2, first
principal plane B1H1A1 and second principal plane
B2H2A2), 3. Two nodes (N1, N2).
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• Cardinal points (??, ?????)
• Two focal points (F1, F2)they have the same
  definition as above.
• Two principal points ( H1, H2, first principal
  plane B1H1A1 and second principal plane B2H2A2)
  the extension of incident ray and backward
  extension of refracted ray (note that the
  refracted line is parallel to the light axis)
  meet at point A1. First principal plane
• Two nodes (N1, N2), through these points, the
  incident line and refractive line are parallel.

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Positions of the three pairs of cardinal points
are based on the specific conditions of the
refractive system. When the refractive system is
put in one medium, in the air for example, the
two focal lengths can be proved to be equal to
each other. f1 f2 f, N1 and H1 are at the
same position, N2 and H2 are at the same
position. The same equation for the thin lens can
be obtained.
Note that the object distance is from the first
principal plane and the image distance is from
the second principal plane, not from the surface
of lens.
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• Cornea (??), membrane (?, ??).
• Aqueous humor (???, n 1.336, ???)
• Crystalline lens (???, n 1.437)
• ligament (??)
• Ciliary(???) muscles (???) .
• Vitreous humor (????,?????)
• Retina (???)
• Yellow spot (??)

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• Ciliary muscle (???) can change the curvature of
  the lens in order to focus objects at different
  distances in front of the eye.
• Vitreous humor (????) with refractive index 1.336
  is a thin jelly which is behind the crystalline
  lens.
• Retina (???) is a delicate film of nerve fibers
  which is a large part of the inner surface of the
  eye.
• Yellow spot (??) is a slight depression in the
  retina. Its center is a minute region, about
  0.25mm in diameter and it is a most sensitive
  parts in the retina as it contains cones
  excessively.

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• myopia (??),myopic, nearsighted, shortsighted,
• Hyperopic eye (??), farsighted (???,????)
• Astigmatism (??), astigmatical (???)
• retinal image (?????,????)
• Glasses, spectacles (??),
• prescribe (??,???),
• magnifying glass (???)
• fibrescope (??), microscope (???)
• unaided eye (??),