Today’s Lecture
Brewster’s Angle
Mirrors and Lenses
Reflection and refraction indices - sparkling diamonds
• For a light beam incident upon a boundary between two
transparent media at 90°:
I r Intensity of reflected beam (n 1 − n 2 )
=
=
2
I i Intensity of incident beam (n 1 + n 2 )
2
Ir
• Glass – n1 = 1.5,
= 0.04
Ii
• Diamond – n1 = 2.4, I r = 0.17
Ii
What happens if the light is not coming at the angle of 90°?
What happens if the light is not coming at the angle of 90°?
The light coming at 90° corresponds to the angle of incidence θ = 0.
As a rule, at greater angles of incidence more light gets reflected.
Look at your watch glass at a grazing angle! It will act as a pretty
good mirror!
Is there any dependence on polarization of light?
How is the reflected wave created in the first place?
How does it depend on the medium the wave is going to?
The primary source of the reflected wave are oscillating charges in
the medium the light is going to.
Therefore the direction of polarization corresponds to the direction
of oscillation of the charges.
EM Waves by an Antenna
• Two rods are connected to an ac source, charges oscillate
between the rods (a)
• As oscillations continue, the rods become less charged, the
field near the charges decreases and the field produced
at t = 0 moves away from the rod (b)
• The charges and field reverse (c)
• The oscillations continue (d)
How does that apply to the light waves?
At a particular angle of incidence, called the Brewster angle, the
light polarized in the plane on incidence-reflection-refraction does
not get reflected at all!
You do not need any special coating – just appropriate polarization.
Brewster angle corresponds to the
situation, when the reflected
wave is supposed to go in the
direction of polarization of the
wave inside glass (2nd medium).
The directions of the reflected and
refracted waves are set by laws of
reflection/refraction.
The charges inside glass oscillate
in the direction of polarization (E)
and cannot radiate the reflected
wave.
Brewster Angle
The Brewster angle corresponds to
the situation when the reflected
wave is supposed to go in the
direction of polarization of the
wave inside glass (2nd medium).
The charges inside glass oscillate in
the direction of polarization (E) and
cannot radiate the reflected wave.
Brewster angle can be found from the
condition that the refracted ray propagates at
90° to the reflected ray. Therefore,
angle of incidence + angle of refraction = 90°
Under what conditions
is the polarizing angle
smaller than 45º?
At a particular angle of incidence, called Brewster angle, the light
polarized in the plane on incidence-reflection-refraction does not
get reflected at all!
By the same token, if the incident
light is unpolarized, only the
component perpendicular to the
plane of incidence gets reflected.
Therefore, the reflected light is
perfectly polarized perpendicular
to the plane of incidence-reflectionrefraction
A shop window photographed at the Brewster angle without and
with polarization filter….
Practical conclusions:
If you need to view objects through glass (camera, telescope
etc), try to look at the Brewster angle for air-glass - 56°.
Do not forget a good polarization filter (polarizer)!
Mirror as an image forming device…
Image – a pattern of light that provides visual representation of reality…
Image - the optical counterpart of an object produced by an optical
device as a lens or mirror …
… a likeness of an object produced on a photographic material.
Image formation in a plane mirror.
The image is always there, in a well defined position, whether you
look at it or not.
The image is formed behind the mirror at a distance equal to the
distance from the object, and has size equal to the size of the object.
It is a “virtual” image, since there is no light in the image location.
The light only appears to come from there.
How exactly do we see
(images of the) objects in a
mirror?
Pretty much the same way
as when we are looking at
the objects directly!
And there are always many
more rays emanating from
an object than getting to our
eyes.
That’s how different people
can see the same
object/image
In the overhead view of the figure below, the image of the
stone seen by observer 1 is at C.
Where does observer 2 see the image – at A, at B, at C, at
D, at E, or not at all?
Position of an image is defined just as well as position of the object!
A little geometry demonstrates that its location is independent of
the position of the observer!
Looking into a mirror at yourself:
For an unobstructed, complete view you
only need a mirror, which is a half of
your height.
Question: What is going to happen to your image in the mirror if
you walk away from it?
Answer: Nothing other than it will appear to be further away,
twice the distance of that from you to the mirror.
Mirrors are known to turn left into
right, that is to make the image of
your left hand look as your right
hand.
It is this effect that gives rise to the expression
“A Mirror Image”
The mirrors actually do a very special transformation, known as
inversion, which cannot be reduced to translations and
rotations…
Maybe to turning inside out?...
Chiral objects and
chiral molicules…
Parabolic Mirrors
Properties of a parabolic mirror
Any ray parallel to the mirror axis reflects
through the focal point
A point source of light at the focus will emerge
from the mirror in a beam of parallel rays
They can be used to concentrate light to high intensities – solar energy
They can be used to design light sources, head lights, flashlights, etc.
Parabolic Mirrors Form Images
The image of the pig is so real it seems
that you could reach out and touch it!
Spherical Mirrors
Spherical mirrors are easier to make
Over a small region near the apex
they are the same as a parabolic mirror
Deviation from parabolic causes
“spherical aberration” (more later)
It can be shown, an exercise for the student, that the
distance to the center of curvature is twice the focal length
Spherical Mirrors Form Images
Light passing through the focal point reflects
parallel to the mirror axis and vice versa.
(a) Object beyond C, image is real, reduced, and inverted.
(b) Object between C and F, image is real, magnified, and inverted.
(c) Object inside F, image is virtual, magnified, and upright.
Question: Can a parabolic mirror form an image?
Answer: Of course! The above applies, replace C with 2F
Spherical Mirrors Form Images
1. Any ray parallel to mirror axis reflects through the focal point
2. Any ray that passes through the focal point reflects parallel to the axis
3. Any ray that strikes the center of the mirror reflects symmetrically
about the mirror axis
4. Any ray that that passes through the center of curvature returns on
itself
Any two of these rays is sufficient to locate an image
Spherical Mirrors Form Images
Object located at l
Object height is h
Image located at l’
Image height is h’
The shaded triangles are similar triangles
M is the magnification
Image height is negative if
image is inverted
The further from the mirror (object vs image) the larger the object vs image.
Spherical Mirrors Form Images
l
Object located at l
Object height is h
Image located at l’
Image height is h’
The shaded triangles are similar triangles
Mirror Equation
Spherical Mirrors Form Images
Object located at l
Object height is h
Image located at l’
Image height is h’
As it turns out the image may also be virtual. This occurs when object is closer
to the mirror than the focal length. The image is then upright and virtual.
Convex Mirrors Form Virtual Images
l
l’
Object located at l
Object height is h
Image located at l’
Image height is h’
For a convex mirror the focal length is
always negative, − f. From the mirror
equation:
This result implies:
Image is always:
virtual, upright, and reduced
Hubble Mirror
Mirror Equation
A technician stands 3.85m in front
of the Hubble mirror which has a
focal length of 5.52m.
(a) The location of the tech’s image is?
Why is the answer < 0?
The tech is standing inside
the focal length. The image
Is virtual!
Hubble Mirror
Mirror Equation
A technician stands 3.85m in front
of the Hubble mirror which has a
focal length of 5.52m.
(b) The magnification of tech’s
image is?
If the technician is 15m in front of the
mirror then:
The image is real, reduced, and inverted.
Do you need a mirror to form an image?
Not necessarily. You can do reasonably
well with a flat refracting surface.
• The image formed by a flat refracting
surface is on the same side of the
surface as the object
– The image is virtual
– The image forms between the object
and the surface
– The rays bend away from the normal
since n1 > n2
The image appears closer, I,
appears closer than the
object, O.
Depth Perception Looking into Water
The object is at O, the image is at I
Assume a light ray from the
object has an incident angle of
Assume light ray from the
Image has an incident angle of
If the depth of a pool is p=2m then from
Snell’s law
The depth of the pool appears to be 1.5m
Thin Lenses
A thin lens consists of a piece of glass or plastic, ground so that
each of its two refracting surfaces is a segment of either a sphere or
a plane (a sphere of infinite radius) …
A lens is thin when the radii of curvature are much bigger than its
thickness.
Lenses have two major applications: light collection and imaging
Diverging lenses are
Converging lenses are
thickest at the edges and
thickest in the middle and
have negative focal lengths
have positive focal lengths
Converging Lens and its Focal Length, f
Light collection:
The parallel rays pass through the
lens and converge at the focal point.
The parallel rays can come from the
left or right of the lens.
A thin lens has two focal points equidistance from the lens,
corresponding to parallel rays from the left and from the right.
A thin lens is one in which the distance between the surface of the lens
and the center of the lens is negligible compared to f.
Converging Lens and its Focal Length, f
The parallel rays diverge after
passing through the diverging
lens.
The focal point is the point
where the rays appear to have
originated.
Imaging
f
l
Three principal rays emanating from every point of the object:
1. A ray parallel to the axis of the lens on the front side passes
through the focal point on the back side.
2. A ray that passes through the center of the lens and does not
get refracted.
3. A ray that passes through the focal point on the front side
emerges parallel to the axis of the lens on the back side.
Ray Diagram for Converging Lens
l
f
If the object is further away from the lens than the front focal point,
l > f , => the image is real, inverted, and located behind the lens.
If the object is located at more than 2f, l > 22f, => the image is
reduced. If the object is located at f < l < 2f, the image is
magnified. WHY?
What is different between a concave mirror and a converging lens?
Ray Diagram for Converging Lens
f
l
If the object is closer to the lens than the front focal point,
l < f, => the image is virtual, upright, magnified, and in front of
the lens
The image can only be viewed from behind the lens!
Ray Diagram for Diverging Lens.
The situation is pretty much the same no matter where the
object is located.
l
f
The image is virtual, and can only be seen through the lens.
The image is upright and is always reduced in size compared with
the object.
Note the analogy with convex mirrors!
Lens Magnification
l
The two shaded triangles, blue and gold, are
right-angled and similar. Therefore, for the
lens magnification, M, we have:
In an exact analogy with mirrors, the “−” sign
is taken to signify the fact that the image is
inverted.
l’
Lens Equation
l
l’
Consider the similar right triangles defined by the angle θ. From
the magnification we know that –h’/h = l’/l. Since these triangles
are similar:
The “Lens equation” is the exact form as the mirror equation. l and l’ are
both considered positive for the case shown. Whereas l’ is positive in
the mirror equation when the image is in front of the mirror.
http://www.mtholyoke.edu/~mpeterso/classes/phys301/geomopti/lenses.html