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SNELL'S LAW
Introduction
In this and the following lab the light is viewed as a ray. A ray is a line that has an origin,
but does not have an end. Light is an electromagnetic disturbance, and as such is
described using Maxwell's equations, which expresses the relationship between the
electric and magnetic fields in an oscillating wave.
Light propagates as a wave, yet many optical phenomena can be explained by describing
light in terms of rays, which in the model for light travel in straight lines in a
homogeneous medium. This model is referred to as Geometric Optics and is a very
elementary theory. In this theory, light travels from its origin at a source in a straight line,
unless it encounters a boundary to the medium. Beyond this boundary may be another
medium which is distinguished by having a speed of light different from the original
medium. In addition, light may be reflected at the boundary back into the original
medium. A light ray that returns to the original medium is said to be "reflected". A ray
that passes into the other medium is said to be "refracted". In most interactions of light
with a boundary both reflection and refraction occur.
In order to frame laws that govern these phenomena we must establish some definitions
of terms. The boundary between two media is defined as a surface. The orientation of a
surface at any specific point is characterized by a line perpendicular to the surface. This
line is referred to as the normal. A ray may encounter a boundary at any arbitrary
incidence angle. The angle of incidence is measured with respect to the normal line. A
reflected ray will have an orientation with respect to the boundary that is delineated by an
angle of reflection that is also measured with respect to the normal. So too, the refracted
ray will be oriented with respect to the boundary by the angle of refraction measured
between the ray and the normal to the surface.
What distinguishes the two media is that the speed of light is different from one media to
the other. We define the index of refraction n to be the measure of how much different
the speed of light is in a certain media from that of light through a vacuum. Light travels
through a vacuum at 2.99792 x108 m/s. This speed is thought to be a universal constant
and the highest speed allowed in nature as postulated in Einstein’s theory of Special
Relativity. We use the symbol c to represent this speed.
The index of refraction is a characteristic of the media. It is the only thing that
distinguishes one media from another in geometric optics. It is defined as the ratio of the
speed of light in a vacuum to the speed in a particular medium of interest.
n1 ≡ vo/v1
or
n1 ≡ c/v1
Therefore, the value of the index of refraction is always greater than unity. Gasses have
an index of refraction close to I (nair= 1.00028), while for water the index is about 1.33,
and for plastic it is approximately 1.4. Depending on the type of glass the index of
refraction of glass can vary from 1.5 to 1.7.
Normally we would think that the index of refraction is a constant with varying color of
light where the color of light is an indication of the frequency of the light. The index of
refraction actually depends on the frequency of the light wave to a small degree and as
such is different for different colors of light.
The rules for reflection of light are:
a) The angle of incidence is equal to the angle of reflection.
θ1 = θ1’
where θ1 is the angle of incidence and θ1’ is the angle of the reflected ray that propagates
in the same medium.
(This is the commonly known rule, but this next rule is rarely stated though equally
important)
b) The incident ray, the reflected ray, and the normal to the surface, all lie in a
plane.
We will not formally investigate these rules in this lab although you will be able to
observe the phenomena of reflection as a side issue while performing this lab experiment.
The rules for refraction are not so obvious although they where well known to the
ancients. The first rule is often sited as Snell's Law. It is:
a)
sinθ1/sinθ2 = n2/n1
where θ2 is the angle of refraction of the ray that is transmitted into the second medium.
Similar to the second rule for reflection we can say that the second rule for refraction is:
b) The incident ray, the refracted ray and the normal to the surface, all lie in a
plane.
In general the path of a light ray is reversible in that if a light ray were to be reversed it
would follow the same path. A ray travelling from a low index of refraction to a high
index of refraction will experience a bending toward the normal. However a ray passing
from a high index of refraction to a lower index will experience a bending away from the
normal. The angle of refraction will be larger than the angle of incidence. So, what
happens when the angle of refraction is greater than 90o for a given incidence angle. In
this case light cannot be transmitted through the interface and as such it is reflected
totally. The efficiency for this reflection is 99.99% (as compared to 95% for a typical
silvered surface mirror) The smallest angle for which a ray will be reflected is the critical
angle. One can show that the sine of this angle is the inverse of the ratio of the index of
refraction of the first media to the index of the second media. If the second media is air (n
= 1.00009), the sine of the angle is effectively the inverse of the index of refraction of the
first material.
Experiment 1: Index of Refraction
In this experiment you will use a
Light Ray Box. It consists of a
light source and a Multi-slit Slide . The light source housing is mounted on a colored
plastic base in which it can slide back and forth. The utility of this feature will be
explained in a future experiment. The a Multi-slit Slide is a flat square piece of plastic
with notches cut into each of the four edges. The a Multi-slit Slide slips into a slot on the
end of the ray box to create rays of light. Choose the side with just one narrow notch and
place that side down as you slip the a
Multi-slit Slide into place. A single
narrow beam should be observed
emerging from the ray box. Place the
ray box on the end of a sheet of paper
provided, such that the ray is
projected onto the paper.
Locate the oddly shaped tetrahedron lucite block. Place the block on the paper with the
frosted side down. Aim the ray at one of the two parallel side of the block. Note that the
ray emerges from the opposite side of the block in a direction parallel to the original ray
but displaced slightly due to the refraction of the ray inside the block.
You will now draw lines representing the incident and refracted rays so that you can
measure the angle of incidence and refraction for several angles of incidence. Arrange the
block for a specific angle of incidence. You will repeat the following steps through a
series of five angles from a shallow angle to a steep angle of incidence.
Using a pencil or pen, mark the location of the incident incident light ray θ2
refracted light ray
ray at two points along the ray, such that its direction
can be recreated later. It is advisable that one of the
θ1
points be located at or near the incident surface. Next
draw a line along the incident surface. Lastly, draw a
point on the opposite side of the block where the ray
emerges. These points should be sufficient to recreate
the geometry of the refraction. Remove the block and draw lines representative of the
incident and refracted rays, and a line representing the refracting surface. Use a protractor
to measure the incident and refraction angles. Repeat the procedure for four more angles.
Order the results in a table in your lab book. Plot the results as a function of increasing
incident angle using the plotting program resident on the PC’s in the lab. This program is
called GAX (Graphical Analysis). Your lab instructor will show you how to use the
program, if you haven’t already used it in previous labs.
Observe if there are any systematic changes in the index of refraction. To do this, one
must have an idea of how much deviation in the index can be attributed to random
variations. One way to assess this is to find an average value for the index, and from that
determine a standard deviation. The standard deviation can be obtained from the
STATISTICS menu item in GAX. Note the result.
Arrange the lucite block so
Experiment 2: Total Internal Reflection that the single ray from the
ray box hits the squared off
end of the block. This is the side opposite the slanted side. Have the ray cross through the
block and exit the slanted side. Observe the ray
θ1
that exits this side as you turn the block to vary incident light ray
θ2
the angle that the ray strikes the inside surface
reflected ray
of the slanted side, the incident angle θ1. You
should also observe that there is a significant
amount of intensity in the ray reflected off the
inside surface. As you increase the incident
angle the angle of refraction θ2will also
refracted ray
increase. Since θ2 will always be larger than θ1,
there will come an angle of θ1 for which θ2 should exceed the maximum of 90o. At this
point total reflection occurs and you should observe a noticeable increase in the intensity
of the reflected ray.
Adjust the block so that the refracted ray just skims the slanted surface of the block. Note
any observations of interest as you do this operation. Decide on what position best
represents the critical angle for the incident ray. Now mark the point where the incident
ray enters the block, where the ray hits the slanted side and where the reflected ray
emerges. Also draw a line along the edge of the slanted surface. Now, remove the block
and using the points recreate the rays as they appeared inside the block and measure the
angle of incidence. Use this angle to calculate the index of refraction for the lucite block
and compare with the value you got in part one. Does it agree within the uncertainty of
the first measurement as determined from the standard deviation? Comment. Estimate
errors in the second measurement.
One of the more recent and ubiquitous
applications of total internal reflection is
the optical fiber. Fiber optics has revolutionized the communications industry. A fiber
consists of a long thin thread of glass called the core, surrounded with an outer shell or
cladding of a material with a lower index of refraction. Light entering the end of the core
of an optical fiber is transmitted or piped to the other end with very little loss in intensity
even though the fiber may be bent in a circular shape. The thinness of the fiber and the
lower index of the cladding ensures that light will always strike the fiber side at an angle
greater than the critical angle for total internal reflection, and be totally reflected back
into the fiber. Thus the light bounces off the inside as it caroms down the length of the
fiber, following every bend or twist.
Observation 1: Optical Fiber
Find the optical fiber among the items on your lab table. Be careful in handling the
fiber. It is, after all, a piece of glass and will break if bent too sharply. Align one end
of the fiber with the ray emerging from the ray box. Observe the light emerging from the
other end of the fiber. It will look like a bright pinpoint of light. Point this end of the fiber
vertically down onto a sheet of paper on the table with the end of the fiber held about
three centimeters above the table. You should see a bright spot on the paper where the
light rays coming from the fiber hit the paper. The light has been channeled through the
fiber even around the bends as the ray bounces off the inside surface of the fiber, possibly
many times. The light persists with little loss of intensity because of the great efficiency
of these reflections that are at such steep angles to the surface normal as to always be in
the realm of total reflection.
Now, carefully rotate the incident end of the fiber to introduce and angle between the
incident ray and the fiber. Observe that the spot on the paper has expanded to form a ring
of light.
Questions
Following is a list of questions intended to help you prepare for this
laboratory session. If you have read and understood this write-up,
you should be able to answer most of these questions. Some of these questions may be
asked in a quiz preceding the lab.
• In optics angles are always measured with respect to what?
• For geometric Optics what assumption is made about the nature of light?
• If two adjacent media have the same index of refraction n, can you observe the
phenomena of reflection or refraction?
• What is the law of reflection?
• What is Snell's Law?
• What are the two conditions that allow total internal reflection to take place?
• In the case of fiber optics do you expect the core or the cladding to have a greater value
for the index of refraction?
• Why are optical fibers immune to electrical noise?