Chapter 9 Optics (Section 4)

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Chapter 9Optics(Section 4)
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9.4 Lenses and Images

• In Section 9.2, we described how curved mirrors
  are used in astronomical telescopes and other
  devices to redirect light rays in useful ways.
• Microscopes, binoculars, cameras, and many other
  optical instruments use specially shaped pieces
  of glass called lenses to alter the paths of
  light rays.
• As was the case with mirrors, the key to making
  glass or other transparent substances redirect
  light is the use of curved surfaces (interfaces)
  rather than flat (planar) ones.

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9.4 Lenses and Images

• Suppose we grind a block of glass so that one end
  takes the shape of a segment of a sphere as shown
  in cross section in the figure.

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9.4 Lenses and Images

• Let parallel rays strike the convex spherical
  surface at various points above and below the
  line of symmetry (called the optical axis) of the
  system.
• If one applies the law of refraction at each
  point to determine the angle of refraction of
  each ray, the results shown in the figure.
• In particular, rays traveling along the optical
  axis emerge from the interface undeviated.

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9.4 Lenses and Images

• Rays entering the glass at points successively
  above or below the optical axis are deviated ever
  more strongly toward the optical axis.
• The result is to cause the initially parallel
  bundle of rays to gradually converge togetherto
  become focusedinto a small region behind the
  interface.
• This point is called the focal point and is
  labeled F in the figure.

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9.4 Lenses and Images

• The figure shows the behavior of parallel rays
  refracted across a spherical interface that,
  instead of bowing outward, curves inward.
• In this case, the emergent rays diverge outward
  as though they had originated from a point F to
  the left of the interface.

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9.4 Lenses and Images

• The ability to either bring together or spread
  apart light rays is the basic characteristic of
  lenses, be they camera lenses, telescope lenses,
  or the lenses in human eyes.

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9.4 Lenses and Images

• In most devices the light rays must enter and
  then leave the optical element (lens) that
  redirects them.
• Common lenses have two refracting surfaces
  instead of one, with one surface typically in the
  shape of a segment of a sphere and the second
  either spherical as well or flat (planar).

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9.4 Lenses and Images

• The effect on parallel light rays passing through
  both surfaces is similar to that in the previous
  examples with one refracting surface.
• A converging lens causes parallel light rays to
  converge to a point, called the focal point of
  the lens.

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9.4 Lenses and Images

• The distance from the lens to the focal point is
  called the focal length of the lens.
• A more sharply curved lens has a shorter focal
  length.
• Conversely, if a tiny source of light is placed
  at the focal point, the rays that pass through
  the converging lens will emerge parallel to each
  other. This is the principle of reversibility
  again.

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9.4 Lenses and Images

• A diverging lens causes parallel light rays to
  diverge after passing through it.
• These emergent rays appear to be radiating from a
  point on the other side of the lens.
• This point is the focal point of the diverging
  lens.

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9.4 Lenses and Images

• The distance from the lens to the focal point is
  again called the focal length, but for a
  diverging lens it is given as a negative number,
  15 centimeters, for example.
• If we reverse the process and send rays
  converging toward the focal point into the lens,
  they emerge parallel.

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9.4 Lenses and Images

• For both types of lenses there are two focal
  points, one on each side.
• Clearly, if parallel light rays enter a
  converging lens from the right side in the
  figure, they will converge to the focal point to
  the left of the lens.

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9.4 Lenses and Images

• Whether a lens is diverging or converging can be
  determined quite easily
• If it is thicker at the center than at the edges,
  it is a converging lens
• if it is thinner at the center, it is a diverging
  lens.

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9.4 Lenses and ImagesImage Formation

• The main use of lenses is to form images of
  things.
• First, lets consider the basics of image
  formation when a symmetric converging lens is
  used.
• Our eyes, most cameras (both still and video),
  slide projectors, movie projectors, and overhead
  projectors all form images this way.

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9.4 Lenses and ImagesImage Formation

• The figure illustrates how light radiating from
  an arrow, called the object, forms an image on
  the other side of the lens.
• One practical way of demonstrating this would be
  to point a flashlight at the arrow so that light
  would reflect off the arrow and pass through the
  lens.
• The image could be projected onto a piece of
  white paper placed at the proper location to the
  right of the lens.

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9.4 Lenses and ImagesImage Formation

• Although each point on the object has countless
  light rays spreading out from it in all
  directions, it is simpler to consider only three
  particular rays from a single pointthe arrows
  tip.
• These rays are called the principal rays.
• 1. The ray that is initially parallel to the
  optical axis passes through the focal point (F)
  on the other side of the lens.

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9.4 Lenses and ImagesImage Formation

• 2. The ray that passes through the focal point
  (F) on the same side of the lens as the object
  emerges parallel to the optical axis.
• 3. The ray that goes exactly through the center
  of the lens is undeviated because the two
  interfaces it encounters are parallel.

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9.4 Lenses and ImagesImage Formation

• Note that the image is not at the focal point of
  the lens. Only parallel incident light rays
  converge to this point (assuming an ideal lens).

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9.4 Lenses and ImagesImage Formation

• We could draw principal rays from each point on
  the object, and they would converge to the
  corresponding point on the image.
• This kind of image formation occurs when you take
  a photograph or view a slide on a projection
  screen.

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9.4 Lenses and ImagesImage Formation

• In the latter case, light radiating from each
  point on the slide converges to a point on the
  image on the screen.
• Note that the image is inverted (upside down).
• Thats why you load slides in the tray or
  carousel upside down if you want their images to
  be right side up.

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9.4 Lenses and ImagesImage Formation

• The distance between the object and the lens is
  called the object distance, represented by s, and
  the distance between the image and the lens is
  called the image distance, p.
• By convention (with the light traveling from left
  to right), s is positive when the object is to
  the left of the lens, and p is positive when the
  image is to the right of the lens.

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9.4 Lenses and ImagesImage Formation

• If we place the object at a different point on
  the optical axis, the image would also be formed
  at a different point. In other words, if s
  changes, then so does p.
• Using a lens with a different focal length for
  fixed s would also cause p to change.
• For example, the image would be closer to the
  lens if the focal length were shorter.

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9.4 Lenses and ImagesImage Formation

• The following equation, known as the lens
  formula, relates the image distance p to the
  focal length f and the object distance s.

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9.4 Lenses and ImagesExample 9.3

• In a slide projector, a slide is positioned 0.102
  meter from a converging lens that has a focal
  length of 0.1 meter.
• At what distance from the lens must the screen be
  placed so that the image of the slide will be in
  focus?

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9.4 Lenses and ImagesExample 9.3

• The screen needs to be placed a distance p from
  the lens, where p is the image distance for the
  given focal length and object distance. So

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9.4 Lenses and Images Example 9.3

• If the slide-to-lens distance is increased to
  0.105 meter (about a 3 change), the distance to
  the screen (p) would have to be reduced to 2.1
  meters (more than a factor of 2.4 reduction in
  distance).
• If the lens is replaced by one that has a shorter
  focal length, the distance to the screen would
  have to be reduced as well.

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9.4 Lenses and ImagesImage Formation

• The images formed in the manner just described
  are called real images.
• Such images can be projected onto a screen.
• We see the image on the screen because the light
  striking the screen is diffusely reflected to our
  eyes.

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9.4 Lenses and ImagesImage Formation

• A simple magnifying glass is a converging lens,
  but the image that it forms under normal use is
  not a real imageit cant be projected onto a
  screen.
• We see the image by looking into the lens, just
  as we see a mirror image by looking into the
  mirror.
• This type of image is called a virtual image,
• similar to the image formed by a concave mirror

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9.4 Lenses and ImagesImage Formation

• The figure shows how an image is formed in a
  magnifying glass.
• In this case, the object is between the focal
  point F and the lens, so the object distance s
  is less than the focal length f of the lens.

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9.4 Lenses and ImagesImage Formation

• Note that the image is enlarged and that it is
  upright.
• It is also on the same side of the lens as the
  object, which means that p is negative.

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9.4 Lenses and ImagesExample 9.4

• A converging lens with focal length 10
  centimeters is used as a magnifying glass.
• When the object is a page of fine print 8
  centimeters from the lens, where is the image?

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9.4 Lenses and ImagesExample 9.4

• The negative value for p indicates that the image
  is on the same side of the lens as the object.
• Therefore, it is a virtual image and must be
  viewed by looking through the lens.

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9.4 Lenses and ImagesImage Formation

• A virtual image is also formed when you look at
  an object through a diverging lens.
• In this case, the image is smaller than the
  object, as it is with a convex mirror.

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9.4 Lenses and ImagesMagnification

• One of the most useful properties of lenses is
  they can be used to produce images that are
  enlarged (larger than the original object) or
  reduced (smaller than the original object).
• In either case, the magnification, M, of a
  particular configuration is the height of the
  image divided by the height of the object.

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9.4 Lenses and ImagesMagnification

• If the image is twice the height of the object,
  the magnification is 2.
• If the image is upright, the magnification is
  positive.
• If the image is inverted, the magnification is
  negative (because the image height is negative).

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9.4 Lenses and ImagesMagnification

• The magnification that one gets with a particular
  lens changes if the object distance is changed.
• Because of the simple geometry, the magnification
  also can be written in an alternate, equivalent
  form as minus the image distance divided by the
  object distance.

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9.4 Lenses and ImagesMagnification

• From this we can conclude that
• If p is positive (image is to the right of the
  lens and real), M is negative the image is
  inverted.

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9.4 Lenses and ImagesMagnification

• If p is negative (image is to the left of the
  lens and virtual), M is positive the image is
  upright.

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9.4 Lenses and ImagesExample 9.5

• Compute the magnification for the projector in
  Example 9.3 and for the magnifying glass in
  Example 9.4.
• In the first case in Example 9.3, s 0.102
  meters and p 5.1 meters.
• Therefore

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9.4 Lenses and ImagesExample 9.5

• The image is 50 times as tall as the object, but
  it is inverted (because M is negative).
• A slide that is 35 millimeters tall has an image
  on the screen that is 1,750 millimeters (1.75
  meters) tall.
• When s is 0.105 meters, p is 2.1 meters, and the
  magnification is 20.

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9.4 Lenses and ImagesMagnification

• In Example 9.4, s 8 cm and p 40 cm.
  Consequently
• The image of the print seen in the magnifying
  glass is five times as large as the original, and
  it is upright (because M is positive).

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9.4 Lenses and ImagesMagnification

• Telescopes and microscopes can be constructed by
  using two or more lenses together.
• The figure shows a simple telescope consisting of
  two converging lenses.

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9.4 Lenses and ImagesMagnification

• The real image formed by lens 1 becomes the
  object for lens 2.
• The light that could be projected onto a screen
  to form the image for lens 1 simply passes on
  into lens 2.

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9.4 Lenses and ImagesMagnification

• In essence, lens 2 acts as a magnifying glass and
  forms a virtual image of the object.
• In this telescope, the image is magnified but
  inverted.
• Replacing lens 2 with a diverging lens yields a
  telescope that produces an upright image.

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9.4 Lenses and ImagesAberrations

• In real life, lenses do not form perfect images.
• Suppose we carefully apply the law of refraction
  to a number of light rays, all initially parallel
  to the optical axis, as they pass through a real
  lens that has a surface shaped like a segment of
  a sphere.

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9.4 Lenses and ImagesAberrations

• We would find that the lens exhibits the same
  flaw we saw in Section 9.2 with spherically
  shaped curved mirrors spherical aberration.
• Rays striking the lens at different points do not
  cross the optical axis at the same place.

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9.4 Lenses and ImagesAberrations

• In other words, there is no single focal point.
• This causes images formed by such lenses to be
  somewhat blurred.
• Lens aberrations of this type can be corrected,
  but this process is complicated and often
  necessitates the use of several simple lenses in
  combination.

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9.4 Lenses and ImagesAberrations

• Another type of aberration shared by all simple
  lenses even when used under ideal conditions is
  chromatic aberration.
• A lens affected by chromatic aberration, when
  illuminated with white light, produces a sequence
  of more or less overlapping images, varying in
  size and color.

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9.4 Lenses and ImagesAberrations

• If the lens is focused in the yellow-green
  portion of the EM spectrum where the eye is most
  sensitive, then all the other colored images are
  superimposed and out of focus, giving rise to a
  whitish blur or fuzzy overlay.
• For a converging lens, the blue images would form
  closer to the lens than the yellow-green images,
  whereas the reddish images would be brought to a
  focus farther from the lens than the yellow-green
  ones.

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9.4 Lenses and ImagesAberrations

• The cause of chromatic aberration has its roots
  in the phenomenon of dispersion.
• The remedy for this problem, originally thought
  to be insoluble by none other than Newton
  himself, was discovered around 1733 by C. M. Hall
  and later (in 1758) developed and patented by
  John Dolland, a London optician.
• It involves using two different types of glass
  mounted in close proximity.

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9.4 Lenses and ImagesAberrations

• The figure shows a common configuration called a
  Fraunhofer cemented achromat (meaning not
  colored).
• The first lens is made of crown glass, the second
  of dense flint glass.
• These materials are chosen because they have
  nearly the same dispersion.

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9.4 Lenses and ImagesAberrations

• To the extent to which this is true, the excess
  convergence exhibited by the first lens at bluish
  wavelengths is compensated for by the excess
  divergence produced by the second lens at these
  same wavelengths.
• Similar effects occur at the other wavelengths in
  the visible spectrum, permitting cemented
  doublets of this type to correct more than 90
  percent of the chromatic aberration found in
  simple lenses.