Today’s Lecture
Electromagnetic Waves
Polarization
Reflection-Refraction
http://www.phy.ntnu.edu.tw/java/emWave/emWave.html
Electromagnetic Waves
• The E and B fields are
perpendicular to each other
• Both fields are perpendicular
to the propagation direction
which is parallel to E X B.
– Therefore, EM waves are
transverse waves
• Propagate with the speed of
light, which does not depend on
either frequency or amplitude;
• Speed of light in vacuum – a
fundamental constant,
c = 3⋅108 m/s
E
B=
c
In every point in space, at each
moment of time – magnitude that is.
Electromagnetic Spectrum
Frequency of visible light: 4.3×1014 to 7.5×1014 Hz
Difference between a sound wave and a light wave?
One comes out of the radio the other is received by the radio.
fλ = c
Why a single wave phenomenon is called so many names?
EM waves of different frequencies/lengths originate from
different sources and interact with matter essentially
differently.
Polarization
is a property of electromagnetic waves that specifies the
electric field direction. Why do we pick the electric field
out of the two? Aren’t they on an equal footing?
They are on an equal footing, but it is mostly the electric field,
which interacts with matter – atoms, electrons, nuclei.
EM waves from a TV tower are perfectly polarized – the Electric
field has a very well defined direction, which stays always the same.
In contrast, the light coming from the Sun or from a light bulb is
unpolarized. What does it mean unpolarized? Doesn’t the electric
field have some direction?
It certainly does at every instant. BUT this direction does not stay
constant and changes very rapidly and randomly.
So, averaging over any reasonable time interval you do not find any
particular polarization!
polarized
unpolarized
The frequency of light is about 5×1014 Hz, which means 5×1014 wave
crests per second. If the polarization changes once every 500 crests it
will still be 1012 times per second. Too fast for us to detect!
Any way to make a polarized wave (light) out of unpolarized wave?
Yes, but it is going to cost us some intensity loss… (No free meals…)
We can use a polarizer - a piece of material, whose molecular or crystal
structure has a preferred direction called the transmission axis.
A polarizer “decomposes” the wave into a component with the electric
r
field, E , parallel to the transmission axis, which passes through, and a
component with the E field perpendicular to the transmission axis, which
gets totally absorbed.
r θ
r
E0
E
The magnitude of the right
component of the electric field:
z
E = E0 cos θ
Intensity of the wave is
proportional to the square of
the amplitude
S~E
2
S = S 0 E 2 / E02 = S 0 cos 2 θ
transmission axis
Law of Malus
In an unpolarized wave the angle θ is changing randomly.
Therefore, after passing through a polarizer the average intensity is
The light gets polarized, but it
looses 1/2 of its intensity...
If the axis of a polarizer is set at θ = 90° to the axis of polarization
cos θ = 0
no light is passing through!
A system of two crossed polarizers never lets any light through.
Whatever passes through the first one is blocked by the second.
What happens to the intensity, S, and direction of polarization of
unpolarized light upon passing trough three polarizers shown here?
E
S1 = S 0 / 2
S 2 = S1 cos 2 25°
S3 = S 2 cos 2 (70° − 25°)
S1
S2
S = S3
= S 2 cos 2 45°
S3 = S 0 ⋅1 / 2 ⋅ cos 25° ⋅ cos 45° = 0.205 ⋅ S 0
2
2
Without the second polarizer
S3 = S 0 ⋅1 / 2 ⋅ cos 70° = 0.058 ⋅ S 0
2
http://lectureonline.cl.msu.edu/~mmp/kap24/polarizers/Polarizer.htm
Geometrical optics – Ray approximation.
Light is an electromagnetic wave…and waves are difficult!
In many cases, though, difficulties can be avoided and geometrical
optics can be applied.
It is based on the suggestion that “Light travels in straight lines
called rays”. These rays are perpendicular to the wave front.
This is the ray approximation and it reduces optics to ray tracing
and geometry. Hence the name geometrical optics.
A ray a is a line in the direction along which light energy is flowing.
A laser beam (or a beam from your car’s headlight) is really a
bundle of many parallel rays.
Question: How come light waves can be reduced to
rays? Is it always valid?
Consider an unbounded plane wave of light.
All it takes to characterize it is its direction and intensity (which can be
thought of as density of rays). So, the ray approximation is OK.
After passing through an aperture the plane wave becomes a beam and
gets bounded. Does it keep going along a straight line?
Depends on whether or not the aperture size is much larger than the wave
length. If λ is on the order of the aperture, then diffraction occurs (later).
Ray Approximation – Waves on Water Surface
Waves propagate in straight lines
unless they hit something (a barrier
or an aperture) having a size
comparable with the wave length.
This causes diffraction which we
will study later.
In general all bounded light beams
in free space, including laser
beams, are somewhat expanding
and loosing their intensity (density
of rays).
They are expanding no matter how
hard you try to keep them narrow,
just because of the fact that they
are bounded!
What about spherical waves? Can we apply ray approximation to
them too? Sure thing!
http://www.people.vcu.edu/~rgowdy/mod/104/sphraymv.htm#1
The ray model (continued)
Light travels through a transparent medium in straight lines called
rays, at speeds v = c/n, where n is the index of refraction of the
medium.
Light rays do not interact with each other.
A light ray continues forever unless it has an interaction with matter
that causes it to change directions or be absorbed.
Light has four different ways in which it can interact with matter.
At an interface between two media, light can be reflected
or refracted.
Within a medium light can be scattered or absorbed.
• Light has four different ways in which it can interact with matter.
At an interface between two media, light can be reflected
or refracted.
Within a medium light can be scattered or absorbed.
Examples of using of the ray approximation:
A point source with a screen and 1D or 2D apertures between them.
What is the shape of the image on the screen? Image of what? What
does it size depend on?
The shape of the image
corresponds to the shape
of the aperture;
The image is an image of the
aperture;
Its size is proportional to the size
of the aperture and to the ratio of
distances
source to screen
source to aperture
Image by an aperture and sunlight:
It is always the image of the aperture corresponding to its profile.
It is formed everywhere in space past the aperture (an opening, a
window).
The image in a screen past the aperture is going to be rather sharp,
wherever you put the screen.
Examples of Using of the Ray Approximation:
1) How do you get a point source?
What does it mean, rays filling the
aperture? What happens if we
close half of the aperture?
2) Two point sources, an aperture
and a screen. What kind of a
pattern on the screen do they
produce? What about a line of point
sources?
3) A line of point sources and a very small (pinhole) aperture. What
kind of image will be on the screen? What happens if we cover half
of the aperture?
Congratulations!
We have just discovered the earliest model of photo camera,
Camera obscura.
What are the drawbacks
of the camera obscura?
The primary problem is that you need a bright source.
A method to observe a partial solar eclipse.
What kind of light is emitted by regular (not self-luminous) objects?
How do we see objects?
Our vision involves formation of images of different objects on the
retina of our eye.
Reflection of Light
• A ray of light, the incident ray, travels in
a medium
• When it encounters a boundary with a
second medium, part of the incident ray
is reflected back into the first medium
Angle of incidence = Angle of reflection,
θi = θr
θ
Corner Reflector
From the geometry of the corner
Reflector and the law of reflection
φ
An incident light ray reflects in an anti-parallel
direction independent of the incident direction!
Adding a third mirror at right angles forms a corner
cube which returns any beam from which it came
Reflecting cubes left on the moon allows for laser-based
measurements of the moon’s distance to within 15cm!
Law of Reflection
A single ray of light is incident upon a reflecting surface at an angle
θ1 with the normal to the surface.
It is reflected at angle θ’1 with the normal, such that θ’1 =
θ1.
The angle of reflection is always equal to the angle of incidence.
Law of Reflection
If the surface is flat and smooth, and the rays in the incident beam of light
are parallel to each other (plane wave), they will all be reflected at the
same angle. This is the case of specular reflection.
If the surface is rough, the law of reflection will still work for every ray and
every small element of surface. The law of reflection will not work for the
beam, however, because of random orientation of different surface
elements. Different rays constituting the beam will be reflected in random
directions, which is diffuse reflection.
Specular reflection:
Diffuse reflection:
an incident parallel beam is
reflected as a parallel beam
a parallel beam is reflected
as bunch of diverging rays
What type of reflection is more useful?
Well, it depends, what you actually want to get…
Which part of the figure below shows specular
reflection of light from the roadway?
With diffuse reflection you see the road;
With specular reflection you see the lights of the car reflected
off the road.
Mirrors are designed to give nice specular reflection.
Therefore the mirror surface is invisible by itself.
There are no point sources of light on mirror surface.
Therefore, looking into a mirror you see reflections of
objects, not the mirror itself.
Refraction of Water Waves…
Refraction – change in the direction of wave propagation at a
boundary.
Waves change direction, or refract,
as they enter shallow water,
because the waves slow down.
Refraction of Light
• When a ray of light traveling through a transparent medium
encounters a boundary leading into another transparent
medium, part of the ray is reflected and part of the ray enters
the second medium
• The ray that enters the second medium is bent at the
boundary
– This bending of the ray is called refraction!
Following (tracing) the Reflected and Refracted Rays
•
•
•
•
•
Ray 1 is the incident ray
Ray 2 is the reflected ray
Ray 3 is refracted into the lucite
Ray 4 is internally reflected in the lucite
Ray 5 is refracted as it enters the air from the lucite
Refraction of Light
• The incident ray, the
reflected ray, the refracted
ray, and the normal all lie
in the same plane
• The angle of refraction, θ2,
depends on the properties
of the two media and
speeds of light in them,
v1,2 = c/n1,2
• The angle of refraction depends
upon the two materials and the
angle of incidence
• The path of the light through the
refracting surface is reversible
• Refraction occurs because the
speeds of light, v1 and v2, are
different in the two media
• The index of refraction, n, of a
medium can be defined
Frequency and Wavelength at Refraction.
• As light travels from one medium to
another, its frequency does not
change
– Both the wave speed and the
wavelength do change
– The wavefronts do not pile up, nor
are created or destroyed at the
boundary, so ƒ must stay the
same
1
2
1
2
1
2
f = f
v
λ
=
v
λ
λ1 v1 c / n1 n2
= =
=
λ2 v2 c / n2 n1
So we see that