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Lecture 25
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Friday, November 13, 15
Recall: Focal lengths
converging lens
f
diverging lens
f
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Recall: Images with Convex Lenses (2)
§
We start with a ray along the optical axis of the lens that passes straight
through the lens that defines the bottom of the image.
§
A second ray is then drawn from the top of the object parallel to the optical
axis
•
This ray is focused through the focal point on the other side of the lens
§
A third ray is drawn through the center of the lens that is not refracted in the
thin lens approximation
§
A fourth ray is drawn from the top of the object through the focal point on the
same side of the lens that is then directed parallel to the optical axis
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Friday, November 13, 15
Now, let’s study concave Lenses (1)
§ A concave lens is shaped such that parallel rays will be caused to
diverge by refraction such that their extrapolation would intersect
at a focal distance from the center of the lens on the same side of
the lens as the rays are incident
§ Assume that a light ray parallel to
the optical axis is incident on a
concave glass lens
§ At the surface of the lens, the
light rays are refracted toward
the normal
§ When the rays leave the lens, they are refracted away from the
normal as shown
§ The extrapolated line shown as a red and black dashed line that
points to the focal point on the same side of the lens as the incident
ray
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Concave Lenses (2)
§ Let us now study several horizontal light rays incident on a concave
lens
§ After passing through the lens, the rays will diverge such that their
extrapolations intersect at a point a distance f from the center of
the lens on the same side of the lens as the incident rays
§ To the right is a concave lens with five parallel lines of light incident
in the surface of a concave lens from the left
§ In the second panel we have drawn
red lines representing light rays
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Friday, November 13, 15
Concave Lenses (3)
§ We can see that the light rays diverge after passing through the
lens
§ We have drawn red and black dashed lines to show the extrapolation
of the diverging rays
§ The extrapolated rays intersect a focal length away from the center
of the lens
§ In the third panel we draw the diverging rays using the thin lens
approximation where the incident rays are drawn to the center of
the lens
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Friday, November 13, 15
Images Formed with Concave Lenses (1)
§ Here we show the formation of an image using a concave lens
§ We place an object standing on the optical axis represented by
the green arrow
§ This object has a height ho and is located a distance do from the
center of the lens such that do > f
§ We again start with a ray along the optical axis of the lens that
passes straight through the lens that defines
the bottom of the image
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Friday, November 13, 15
Images Formed with Concave Lenses (2)
§ A second ray is then drawn from the top of the object parallel to
the optical axis
• This ray is refracted such that its extrapolation of the diverging ray passes
through the focal point on the other side of the lens
§ A third ray is drawn through the center of the lens that
is not refracted in the thin lens approximation
• This ray is extrapolated back along its original path
§ The image formed is virtual, upright, and reduced
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Friday, November 13, 15
The Lens Equation (1)
§ The images formed by lenses are described by the lens
equation
§ This equation is the same relationship between focal length,
image distance, and object distance that we had found for
mirrors
§ To treat all possible cases for lenses we must define some
conventions for distances and heights
• We define the focal length f of a convex lens to
be positive and the focal length of a concave lens
to be negative
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Power of Lenses
§ Often, the power of a lens is quoted rather than its
focal length
§ The power of a lens, D (diopters), is given by the equation
§ For example, common reading glasses have a power of
D = 1.5 diopters
§ The focal length of these glasses is
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Friday, November 13, 15
Lens Maker Formula
§ If the front surface of the lens is part of a sphere with
radius R1 and the back surface is part of a sphere with
radius R2, then we can calculate the focal length f of the
lens using the lens-makers formula
R1
R2
positive
negative
positive
negative
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Friday, November 13, 15
Lens Maker Formula (1)
§ If the front surface of the lens is part of a sphere with
radius R1 and the back surface is part of a sphere with
radius R2, then we can calculate the focal length f of the
lens using the lens-makers formula
R1
R2
positive
negative
positive
negative
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Friday, November 13, 15
Lens Maker Formula (2)
R1
R2
positive
negative
positive
negative
§ The curvatures R1 and R2 have different signs depending on whether
they point to the positive side (same side as object) or negative side
• For a convex lens, R1 is positive and R2 is negative
• For a concave lens, R1 is negative and R2 is positive
§ If we have a lens with the same radii on the front and back of the
lens so that R1 = R2 = R, we get
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Friday, November 13, 15
The Human Eye (1)
§ The human eye “sees” by absorbing light
§ Refraction at the cornea and lens surfaces produces a real image
on the retina of the eye
§ For an object to be seen clearly, the
cornea
image must be formed at the location of
retina
the retina as shown to the right
§ The shape of the eye cannot be changed
so shaping the lens must control the
distance of the image
§ The lens is held in place by ligaments that connect it to the ciliary
muscle that allows the lens to change shape and thus change the
focus of the lens
§ The index of refraction of the two fluids in the eye are close to that
of water with a value of 1.44; the index of refraction of the material
making up the lens is 1.34
§ Thus most of the refraction occurs at the air/cornea boundary.
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The Human Eye
The extremes over which distinct vision is possible are
called the far point and near point
• The far point of a normal eye is infinity
• The near point of a normal eye depends on the ability of the eye to
focus
§
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Friday, November 13, 15
The Human Eye (3)
§ Several common vision defects result from
incorrect focal distances
§ In the case of myopia (near-sightedness),
the image is produced in front of the retina
§ In the case of hypermetropia (far-sightedness),
the image is produced behind the retina
§
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Reading glasses
Myopia can be corrected using convex (converging) lenses
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Systems of Lenses
§ Now we will look at images formed by systems of lenses
rather a single lens
§ We use the first lens to image an object
§ We use the image of the first lens as the object for
the second lens
§ Thus we can produce various optical instruments with
combinations of lenses
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The Telescope
§ First we will discuss the refracting telescope and then reflecting
telescopes
§ The refracting telescope consists of two lenses
• The objective lens and the eyepiece
§ In our example we represent the telescope using two thin lenses
§ However, an actual refracting telescope will use more sophisticated
lenses
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Geometry of the Telescope
§ Because the object to be viewed is at a large distance, the incoming
light rays can be thought of as being parallel (the object is at infinity)
§ The objective lens forms a real image of the distance object at distance
fo
§ The eyepiece is placed so that the image formed by the objective is at
distance fe from the eyepiece
§ The eyepiece forms a virtual, magnified image of the image formed by
the objective
§ The image is at infinity, again producing parallel rays
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Problems with Refracting Telescopes
§ The objective lens of a refracting telescope is large and
heavy
• The 40-inch refractor at Yerkes weighed 500 pounds
§ Supporting a large glass lens is difficult
• Must be supported by its edges
§ Constructing large glass lenses is difficult
§ Glass lenses are thick and absorbed light
§ A glass lens has chromatic aberration
• Different focal lengths for different colors
§ Solution: Replace the objective lens with a mirror
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The Reflecting Telescope
§ Most large astronomical telescopes are reflecting telescopes
with the objective lens being replaced with a concave mirror
§ Large mirrors are easier to fabricate and position than large
lenses
§ The eyepiece is still a lens
§ Various types of reflecting telescopes have been developed
§ We will discuss three examples of the
geometries of reflecting telescope
• Reflector
• Newtonian
• Cassegrain
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Friday, November 13, 15
Basic Reflecting Telescope
§ Basic reflector
§ Replace the objective lens with a parabolic mirror
§ This design is impractical because the observer must be in
the line of the incident light
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Newtonian Reflecting Telescope
§ In 1670 Newton presented his design for a reflecting
telescope to the Royal Society
• The idea for a reflecting telescope came from James Gregory
§ Newton solved the observer problem by placing a small mirror that
reflect the light out to an eyepiece
§ This mirror is small compared with the objective mirror and causes
only a small loss of light from the image
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Friday, November 13, 15
Cassegrain Geometry for Reflecting
Telescope
§ A further improvement on the geometry of the reflecting
telescope is the Cassegrain geometry (named for the French
sculptor Sieur Guillaume Cassegrain) first proposed in 1672
§ Here a small mirror is used to reflect the image through a hole in
the center of the objective mirror
§ This design and many improvements to this basic idea are the basis
of modern astronomical telescopes
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The Microscope
§ Microscopes exist in many forms
§ The simplest microscope is a system of two lenses
Objective Lens
Eyepiece
Object
§ The first lens is a converging lens of short focal length, fo, called the
objective lens
§ The second lens is another converging lens of greater focal length, fe,
called the eyepiece
§ The object to be magnified is placed just outside the focal length of
the objective lens
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Friday, November 13, 15