Download Chapter 23: Geometric Optics

Chapter 23
LECTURE NOTES
Geometric Optics
In this chapter we concentrate on the ray model of light, which pictures light as traveling in
straight line paths (rays) like a laser beam and ignores the wave nature of light.
Law of Reflection
i r
Measure the angle of incidence  i and angle of reflection  r
from the normal (or perpendicular to the reflecting surface)
to the incoming and outgoing rays, respectively.
Then the law of reflection states  i =  r . This is the basis of image formation by mirrors.
Plane Mirrors
Plane mirrors (the common looking-glass mirror) reflect light from objects in front of the mirror
and form virtual images that seem to be behind the mirror.
A virtual image is not really "there." There is no energy behind the mirror; a piece of film placed
there would not capture an image. It merely looks to our eyes as if there is an image behind the
mirror.
For plane mirrors reflecting objects a distance do in front of the mirror and height ho, it seems as
if there is a virtual image of the object located a distance di behind the mirror of height hi.
Taking distances behind the mirror as negative, we write
di = -do
hi = ho
The image is reversed left to right also.
Spherical Mirrors
These are sections of spherical surfaces of radius r. We call this the radius of curvature of the
mirror. If the reflecting surface is on the outside of the spherical surface we have a convex
mirror (like the mirrors in stores used to monitor customers). If the reflecting surface is on the
inside of the spherical surface we have a concave mirror (it is like we are looking into a cave).
For all spherical mirrors we find:
 from objects very far away, rays arrive essentially parallel.
 parallel rays are focused at different spots; the farther the ray is from the principal axis,
the closer to the mirror it focuses.
 if the mirror is small compared to the radius of curvature, most of the parallel rays focus
at a point (the focus F) midway between the center of curvature R and the center of the
mirror (known as the vertex V).
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 The distance from the mirror to the focus VF, the focal length f, is given by f = CV or
2
half the radius of curvature.
People, calculators, or computers can trace the path of incident and reflected rays from the mirror
surface. We note:



incident rays parallel to the principal axis reflect outward through the focus F.
incoming rays passing through the focus F, travel outward upon reflection parallel to the
principal axis.
incoming rays whose direction of travel extends through the center of curvature C reflect
back upon themselves, passing through C while outbound.
C
F
V
Graphically determine the image
distance and height.
It is convenient to be able to solve for image properties algebraically. Use the mirror equation:
h
 di
1
1 1

 and i 
 m where . . .
di do f
ho
do
hi
ho
di
do
the height of the image; positive if upright relative to object.
the height of the object, always taken as positive.
the image distance from the mirror vertex;
positive if on the reflecting side of the mirror, negative if "behind" the mirror.
the object distance from the mirror vertex;
positive if on the reflecting side of the mirror (almost always).
If di > 0 we have a real image; if di < 0, vertical.
If do > 0 we have a real object; if do < 0 (which is rare), virtual.
Concave mirrors have positive radius of curvature r and positive focal length f.
Convex mirrors have -r and -f.
If hi > 0, the image is upright relative to the object; if hi < 0, inverted.
m = magnification.
If |m| > 1, the image is enlarged.
If |m| = 1, same size image and object
If |m| < 1, the image is reduced.
If m is negative this tells us the image is inverted relative to the object.
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Convex Mirrors -- remember to use f, r negative.
These mirrors always produce virtual images that appear to be behind the mirror, smaller than
the object, and upright. The image is virtual, reduced, and erect.
Concave Mirrors
A variety of image properties are possible depending on object distance.
1.
2.
3.
4.
5.
6.
do = 
do > r
do = r
r > do > f
do = f
do < f
Image (real) is a point at focus so di = f, hi = 0, m = 0.
Image (real) is between C and F, inverted, reduced.
Image (real) is at C, inverted, same size as object.
Image (real) is beyond C, inverted, and enlarged.
No image (or say image at  , m =  , real, inverted)
Image (virtual) is behind mirror, erect, enlarged.
Example
A 5cm candle is burning 15cm in front of a concave spherical mirror with focal length 10cm.
Describe the image.
1
1 1
1
1
1

 becomes

 ; di = 30cm
di do f
d i 15 10
hi  di
h
 30
Then

becomes i 
; hi = -10cm; m = -2
ho
do
5
15
This means the image is 30cm in front of the mirror, real, enlarged, inverted, 10cm high. (Its
magnification was of magnitude 2 and radius of curvature of the mirror was 20cm).
Index of Refraction
The speed of light in vacuum c = 3 x 108 m/s. Light travels more slowly in material substances;
the denser the material, the slower the speed.
c
where c is the speed of light in the vacuum
v
and v is the speed of light in the material. Note, n is dimensionless.
Define the index of refraction of a material as n =
Example:
The index of refraction of water is 1.33. What is the speed of light in water?
c
c
3x10 8 m / s
Since n =
,v= =
= 2.25 x 108m/s.
v
n
1.33
The index of refraction of air is nearly that of a vacuum nair ≈ 1.
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Law of Refraction
1
When a ray of light (at angle θ1 to the normal) travels
from a medium with refractive index n1 into a denser
medium of refractive index n2 (where n2 > n1), the light
slows down, its wavelength shortens, its frequency
remains unchanged, and it bends toward the normal so
θ 2 < θ1 . This is described by:
n1
n2
θ2
Snell’s law -- n1 sin θ1 = n2 sin θ 2 . θ1 is the angle of incidence, θ 2 the angle of refraction. Both
are measured from the normal at the point of incidence.



Note: The paths of the rays are reversible. Light would retrace its steps if it went from
bottom to top of our last diagram.
Note:  2  n 2  1  n1
Note: A ray moving from higher n to lower n (more dense to less dense) material bends
away from the normal.
Example
A laser beam shines from air into water with angle of incidence 30°. Find the angle of refraction.
We take the medium where light originates in the problem as n1. Thus n1 sin θ1 = n2 sin θ 2
becomes 1 ۰ sin 30° = 1.33 sin θ 2 or θ 2 = 22°. Note the laser beam bent toward the normal as
expected.
Total Internal Reflection
θ 2 = 90°
n2
n1
θc
Since n1 sin θ1 = n2 sin θ 2 , sin θ c =
When light travels from a more dense to a less
dense material (as shown, n1 > n2), an angle θ1
exists such that θ 2 = 90°. We call this incident
angle the critical angle θ c . If θ1 > θ c , there is
no refracted ray; the surface acts as a perfect
mirror, reflecting all of the light back into
material n1. This is total internal reflection.
n2
. The critical angle for water into air is les than 49°.
n1
Thin Lenses
A converging lens focuses parallel rays to a point. It is thicker at the center than the edges. A
magnifying glass is an example.
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A diverging lens causes parallel rays to diverge, seeming to come from a focus on the object side
of the lens. It is thicker at the edges than the center. If you wear lenses to correct for near
sightedness, they are diverging lenses.
The distance from the center of the lens to the focus mentioned above is the focal length f. We
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often talk of the power P of the lens. P = , when f is in meters. The unit of /m is called a
f
diopter (D). A converging lens has +f and +P (it is sometimes called a positive lens). A
diverging lens has -f, -P, and is called a negative lens.
Note: f is the same on either side of a lens even if the surfaces have different curvatures. Taking
 1
1
1 
 .
the radius of curvature of a surface + if convex and - if concave we have:  (n  1)

f
 R1 R 2 
This is the lens-makers' equation.
We can solve problems involving lenses as easily as problems involving mirrors. We will
discuss ray tracing and the lens equation.
Ray Tracing




Draw all components to scale.
Incoming rays parallel to the principal or optical axis change direction at the mid-plane of
the lens and the outgoing rays pass through the focus F.
Incoming rays passing through the focus refract and travel outward parallel to the axis.
Incoming rays that pass through the center of the lens continue outward (un-refracted)
along the same direction.
The (Thin) Lens Equation
h
 di
1
1 1

 and i 
m
di do f
ho
do
f is + for a converging lens, - for diverging lens.
do + if on side of lens where light comes from.
di + if on side of lens opposite where light comes from.
di - if on same side of lens as light source; a virtual image.
hi + if upright relative to object, - if inverted.
ho always taken +.
if m > 0, upright image; m < 0, inverted image
if |m| > 1, enlarged image; |m| < 1, reduced image
The important difference between the lens and mirror equations is that light naturally
passes through a lens so an image "beyond" the lens is natural, real, and di is +.
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Example
A 5.00cm tall burning candle is 15.0cm in front of a converging lens of 10.0cm focal
length. Describe the image.
1
1 1
1
1
1

 becomes


so, di = 30cm. Since this is +, the image is
di do f
d i 15 10
real and on the other side of the lens compared to the candle.
hi  di
h
 30

becomes i 
 -2, so hi = (-)10cm. The - tells us the image in
ho
do
5
15
inverted (note, so does the magnification of -2x).
Thus the image is 10cm high, 30cm away from the center of the lens on the side opposite
the candle, is real, enlarged, and inverted.
Multiple Lenses and Mirrors.
Often the image from one lens or mirror can be considered the object of another lens or
mirror in the same optical system. This is how we occasionally have virtual objects with
do < 0. In such a system, the total magnification is the product of all the individual
magnifications.
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