Download lecture 41

Geometric Optics
April 6, 2014
Chapter 32
1
Law of Refraction
!  The law of refraction can be expressed as
n1 sinθ1 = n2 sinθ 2
!  where n1 and θ1 are the index of refraction and angle from
the normal in the first medium and n2 and θ2 are the index
of refraction and angle from the normal in the second
medium
!  The law of refraction is also called Snell’s Law
!  We will always assume the index of refraction of air is one
!  The situation of light incident on various media from air is
common so we will write the formulas for refraction of light
incident on a surface with index of refraction nmedium as
sinθ air
nmedium =
sinθ medium
April 6, 2014
Chapter 32
2
Total Internal Reflection
!  Consider the situation where light travels across the
boundary between two media where the first medium has an
index of refraction greater than the second medium, n1 > n2
!  In this case the light is bent away from the normal
!  As one increases the angle of incidence from medium 1, θ1,
one can see that the angle of the transmitted light, θ2, can
approach 90°
!  When θ2 reaches 90°, total internal reflection takes place
instead of refraction
!  The critical angle θc for total internal reflection is
n2 sinθ1 sinθ c
n2
=
=
⇒ sinθ c =
n1 sinθ 2 sin90°
n1
April 6, 2014
Chapter 32
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Color
Different colors are associated with light of different wavelengths.
The longest wavelengths are perceived as red light and the shortest
as violet light. The table below is a brief summary of the visible
spectrum of light.
Dispersion
The slight variation of index of refraction with wavelength is
known as dispersion. Shown is the dispersion curves of two
common glasses. Notice that n is larger when the wavelength is
shorter, thus violet light refracts more than red light.
Consequences of Dispersion
Red light passing through a
prism
White light passing through a
prism
Each wavelength has a
different index of refraction
and so refracts at a different
angle
Prism
Index of refraction for blue light is larger than for red light,
so blue light bends more than red light
Rainbows
Red light from high in the sky
reaches your eye; violet light from
lower in the sky does the same
Water droplets
Sometimes there can be a double rainbow, when there are two internal
reflections, but the order of colors will be reversed.
Sprinkler rainbow conspiracy
April 6, 2014
Chapter 32
9
Lenses
!  When light is refracted crossing a curved boundary between
two different media, the light rays follow the law of
refraction at each point on the boundary
!  The angle at which the light rays cross the boundary is
different along the boundary, so the refracted angle is
different at different points along the boundary
!  A curved boundary between two optically transparent
media is called a lens
!  Light rays that are initially parallel before they strike the
boundary are refracted in different directions
depending on the part of the lens they strike
!  Depending of the shape of the lens, the light rays can be
focused or caused to diverge
April 6, 2014
Chapter 33
11
Lens Maker’s 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 (Derivation 33.1 in book)
⎛ 1 1⎞
1
= (n −1) ⎜ − ⎟
f
⎝ R1 R2 ⎠
!  The curvatures R1 and R2 have different signs depending on
whether their center point is on the same side as object
(negative) or on the opposite side (positive)
•  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
1 2(n −1)
=
f
R
April 6, 2014
Chapter 33
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Lens Maker’s Formula
April 6, 2014
Chapter 33
13
Lens Focal Lengths
!  Unlike mirrors, lenses have a focal length on both sides
!  Light can pass through lenses
while mirrors reflect the light
allowing no light on the
opposite side
!  The focal length of a convex
(converging) lens is defined to
be positive
!  The focal length of a concave
(diverging) lens is defined to
be negative
April 6, 2014
Chapter 33
14
Convex Lenses
!  A convex (converging) lens is shaped such that parallel rays
will be focused by refraction at the focal distance f from the
center of the lens
!  In the drawing on the right,
a light ray is incident
on a convex glass lens
!  At the surface of the lens, the
light ray is refracted toward the normal
!  When the ray leaves the lens, it is refracted away from the
normal
!  Let us now study the case of several horizontal light rays
incident on a convex lens
April 6, 2014
Chapter 33
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Thin Lenses
!  These rays will be focused to a point a distance f from the
center of the lens on the opposite side from the incident rays
!  The rays enter the lens, get refracted at the surface, traverse
the lens in a straight line, and get refracted when they exit
the lens
April 6, 2014
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Thin Lenses
!  Let’s represent the thin lens approximation by drawing
a black dotted line at the center of the lens
!  Instead of following the detailed trajectory of the light rays
inside the lens, we draw the incident rays to the centerline
and then on to the focal point
!  This real-life lens is a thick lens and there is displacement
between the refraction at the entrance and exit surfaces
!  We will treat all lenses as thin lenses and treat the lens as a
line at which refraction takes place
April 6, 2014
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Images with Convex Lenses
!  Convex lenses can be used to form images
!  We will show the geometric construction of the formation
of an image using a convex lens with focal length f
!  We place an object standing on the optical axis
!  This object has a height ho and is located a distance do from
the center of the lens such that do > f
April 6, 2014
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Images with Convex Lenses
!  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, which is focused through the
focal point on the other side of the lens
April 6, 2014
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Images with Convex Lenses
!  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
April 6, 2014
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Images with Convex Lenses
!  The three rays starting at the top of the object intersect,
locating the top of the image
!  Any two of these three rays from the top of the object can be
used to locate the top of the image
!  In this case, with do > f, a real, inverted, and enlarged image
with height hi < 0 is formed at a distance di > 0 from the
center of the lens
April 6, 2014
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Images with Convex Lenses
!  Now let us consider the image formed by an object with
height ho placed a distance do from the center of the lens
such that do < f
!  The first ray again is drawn from the bottom of the object
along the optical axis
!  The second ray is drawn from
the top of the object parallel
to the optical axis and is
focused through the focal
point on the opposite side
!  A third ray is drawn through
the center of the lens
April 6, 2014
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Images with Convex Lenses
!  A fourth line is drawn such that it originates from the focal
point on the same side of the lens and is then focused
parallel to the optical axis
!  A virtual image is formed on the same side of the lens as the
object by extrapolating the three rays from the top of the
object back until they intersect
!  Red-and-black dashed lines
represent the extrapolated rays
!  In this case, with do < f, a
virtual, upright, and enlarged
image is formed with
hi > 0 and di < 0
April 6, 2014
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Images formed with Concave Lenses
!  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
!  A second ray is then drawn from
the top of the object parallel to the
optical axis
April 6, 2014
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Images formed with Concave Lenses
!  This ray is refracted such that the 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 is formed
at the intersection of these
two rays
!  The image is virtual, upright, and reduced
April 6, 2014
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The Lens Equation
!  The images formed by lenses are described by the lens
equation
1 1 1
+ =
do di f
!  This equation is the same relationship between focal length,
image distance, and object distance that we had for mirrors
!  To treat all possible cases for lenses we must define
conventions for distances and heights
!  The focal length of a converging lens is positive
!  The focal length of a diverging lens is negative
!  The object distance and object height for a single lens are
positive
!  If the image is on the opposite side of the lens then the
image distance is is positive and the image is real
!  If the image is on the same side of the lens as the object, the
image distance is negative and the image is virtual
April 6, 2014
Chapter 33
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The Lens Equation
!  The magnification formula is the same for lenses as mirrors
hi
di
m= =−
ho
do
!  If the image is upright, then the image height and
magnification are positive
!  If the image is inverted, the the image height is negative and
the magnification is negative
!  For converging lenses where do > f, we always get a real,
inverted image on the opposite side
!  For converging lenses where do = f, we get an image at
infinity
!  For converging lenses where do < f, we always get a virtual,
upright, and enlarged image on the same side
April 6, 2014
Chapter 33
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The Lens Equation
!  The following table summarizes the results for all values of
do for a converging lens
!  For a diverging lens, the image is always virtual, upright,
and reduced
April 6, 2014
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Mirrors and Lenses
The Lens Equation
!  Instrument makers often describe lenses in terms of the
power of the lens rather than the focal length of the lens
!  The power of a lens is given in diopters
1
D=
f expressed in meters
f
1 1 1
+ =
do di f
!  Eyeglasses are usually categorized in diopters
!  Typical reading glasses have D = +1.50 diopters
!  The focal length of these lenses is
1
1
f= =
= 0.667 m
−1
D 1.50 m
!  More later in this chapter about vision corrections
April 6, 2014
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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, which changes with age
!  Several common vision defects result from incorrect focal
distances
!  In the case of myopia (nearsightedness), the image is
produced in front of the retina
!  In the case of hyperopia (farsightedness), the image would
be produced behind the retina if it could, but the retina
absorbs the light before an image can be formed
April 6, 2014
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The Human Eye
Normal-sighted
Nearsighted
Farsighted
!  Myopia can be corrected using diverging lenses
•  Negative focal length, negative power
!  Hyperopia can be corrected using converging lenses
•  Positive focal length, positive power
•  Reading glasses
April 6, 2014
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Corrective Lenses for Myopia
PROBLEM
!  What is the power of the corrective lens for a myopic (nearsighted) person whose uncorrected far point is 15 cm?
SOLUTION
!  The corrective lens must form a virtual, upright image 15
cm in front of the lens of an object located at infinity
!  The object distance is ∞ and the image distance is –15 cm
1
1
1
+
= = −6.7 diopters
∞ −0.15 m f
!  The required lens is a diverging lens with a power of
–6.7 diopter (a focal length of -0.15 m)
April 6, 2014
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Corrective Lenses for Hyperopia
PROBLEM
!  What is the power of the corrective lens for a hyperopic
(far-sighted) person whose uncorrected near point is 75 cm?
SOLUTION
!  The corrective lens must form a virtual, upright image at the
uncorrected near point
Virtual image at
uncorrected near point
Close object
!  The object distance is 25 cm, the image distance is –75 cm
1
1
1
+
= = +2.7 diopters
0.25 m −0.75 m f
!  The required lens is a converging lens with a power of
+2.7 diopter (a focal length of +0.37 m)
April 6, 2014
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The Microscope
!  The simplest microscope is a system of two lenses
!  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 so that do,1 ≈ fo
!  This arrangement allows the objective lens to form a real,
inverted, and enlarged image of the object
!  This image then becomes the object for the eyepiece lens
!  This intermediate image is placed just inside the focal length
fe of the eyepiece lens, so that the eyepiece lens produces a
virtual, upright, and enlarged image of the intermediate
image and do,2 ≈ fe
April 6, 2014
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April 6, 2014
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