Reflection and Refraction
The language of experiment is more authoritative than any reasoning: facts
can destroy our ratiocinations - not vice versa
Alessandro Volta (1745-1820)
OBJECTIVES
To observe reflection and refraction of light rays at an interface, and to compare the
reflected intensity with predictions.
THEORY
When a light beam strikes the boundary between two transparent, non-conducting,
materials it will be partially reflected and partially transmitted. The basic phenomenon was
doubtless recognized in antiquity, and was mathematically codified by W. Snell and others early
in the 17th century. By the end of the 17th century, Descartes, Huygens, Fresnel and others
found that the intensity as well as the direction of the rays could be calculated using the new
wave theory. You will verify some of their results in this laboratory exercise.
The basic geometry of reflection and refraction at a planar surface is shown in Fig. 1. The
three rays shown and the normal to the surface are all in the same plane. Measured relative to the
surface normal, the angle of the reflected ray is the same as the angle of the incident ray, while
the angle of the refracted ray is given by Snell's law:
(1)
n1 sin !1 = n 2 sin ! 2
Here n1 and n2 are the optical indices of refraction of the two materials. Interestingly, the
n1
n2
θ1 θ1
θ2
Fig. 1 Reflection and transmission at a planar interface between two transparent, isotropic
materials. As drawn, the index of refraction n1 < n2, so that !1 > !2.
transmitted ray is reversible, so that a beam in material 2 striking the interface at !2 will be
refracted and emerge at !1. The reflected ray will, of course, make an angle !2 with the normal
and remain in material 2.
For the special case of a ray incident from within the higher-index material angle of
refraction can reach 90º when the angle of incidence is less than 90º, causing the transmitted ray
to vanish. This is called total internal reflection, and is responsible for guiding light along fiberoptic cables.
Using the fact that light is a transverse electromagnetic wave, it is possible to
quantitatively calculate the intensity of the transmitted and reflected rays at the interface. The
!
interesting new feature is that light with E parallel to the plane of incidence reflects differently
!
than light with E perpendicular. The somewhat lengthy calculation is in your textbook, using the
notation of Fig. 2, with the results
2
2
# tan(! " ! ) &
i
t
R|| = %
(
tan
!
+
!
$ ( i
t )'
# sin(! " ! ) &
i
t
R) = %
(
sin
!
+
!
$ ( i
t )'
(2)
The angle of the transmitted ray, !t, can be found in terms of the incident angle !i using Eq. 1.
Et
Ei
n1
n2
θi
θr
θt
Er
!
Fig. 2 Angles for reflected intensity with E parallel to the plane of incidence. The perpendicular
!
case looks the same except with E normal to the page.
Physics 231 Reflection and Refraction
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EXPERIMENTAL PROCEDURE
The first exercise is to determine the index of refraction of PMMA plastic by measuring
the refraction of a beam of light through a semi-circular shape. The second is a measurement of
the reflectance as a function of incident angle for the flat surface of the semicircle. Since the
!
direction of E and the index of refraction are known, the measured reflectances can be compared
directly with the values calculated from Eq. 2.
1. General arrangements
The light source is a laser diode (laser pointer) which produces a fairly narrow beam. It is
powered from an external supply, to minimize changes in output amplitude, but it should be
allowed to stabilize for at least 10 minutes before making intensity measurements. The direction
of the electric field is set by a piece of Polaroid magnetically clamped to the front of the laser
mount.
Caution: The laser beam is quite bright, and may cause eye damage. Do not, under any
circumstances, look directly into the beam or a specular reflection.
The laser is mounted on a carriage which can slide around a graduated circle. Check that
the beam crosses the circle along a diameter by locating the beam spot at the center and at a point
180º from the laser position. Ask the instructor for assistance if there is significant misalignment.
Additional carriages are provided for angle and intensity measurements, as explained
below.
2. Index of refraction with Snell's law
Set up the laser and a carriage with screen on the graduated circle as sketched in Fig. 3.
Remove the polarizer to gain some extra intensity. The flat face of the plastic semicircle should
be placed against the small ledge on the rotating table at the center of the graduated circle.
object
screen
laser
graduated
circle
Fig. 3 Apparatus set-up for refraction measurements.
Physics 231 Reflection and Refraction
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If the center of the plastic semi-circle is at the center of the graduated circle, a light ray
hitting the flat side at !1 will be refracted at the interface and travel through the higher-index
material to the circular side. For any refraction angle, the ray will be perpendicular to the curved
face and will exit without further deviation. Measuring the angle of the transmitted ray therefore
gives the angle of refraction !2 at the flat face. Since the path is reversible, the same argument
holds for a ray entering the circular side along a radius.
To align the semicircle, set the laser and screen at equal angles from a convenient
reference point such as 0º on the graduated circle. Rotate the semicircle, keeping it against the
ledge, until the reflection from the flat face is centered on the screen. This means the normal to
the flat face is at the reference angle. Now position the laser at the reference angle and the screen
180º away. Without rotating the semicircle, slide it along the edge until the transmitted beam hits
the center of the screen. The laser beam must now be traveling along a radius of the semicircle.
Go back and forth a couple of times, to be sure you didn't disturb the angle while setting the
lateral position.
Measure the angle of refraction for several angles of incidence on the flat side. Repeat for
the beam incident on the circular side. Make plots of sin!2 vs sin!1 and obtain the index of
refraction from each data set. With care and proper alignment the values should agree to 1% or
so. You should also note the angle for total internal reflection and use it to get a third estimate of
n. This method is not very accurate for reasons you can explain in your report.
3. Reflected intensity
Light intensity is measured with a light-sensitive diode which passes current in direct
proportion to the incident intensity. Fig. 4 shows the circuit which converts the light-induced
current to a voltage, allowing for measurement with a voltmeter. The gain control should be set
so that the incident beam registers about 8 V or less, since the maximum output cannot exceed
the battery voltage regardless of intensity. The whole assembly is mounted to a sliding carriage
for alignment with the desired ray.
9V
10KΩ
+
-
50KΩ
V
Fig. 4 Circuit for converting photodiode current to voltage. The variable resistor serves as a gain
control.
Physics 231 Reflection and Refraction
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The Polaroid material is mounted in a circular holder which mounts magnetically to the
!
front of the laser holder. It preferentially transmits light with E along the diameter from 0º to
!
180º, so set 0º at the bottom to get E perpendicular to the scattering plane, and 90º at the bottom
!
to get E in the scattering plane.
Since the incident intensity depends on the orientation of the polarizer and the output of
the laser source, you need to measure it immediately before and after a series of reflection
measurements. With the polarizer set in the desired orientation, remove the object and measure
the intensity of the incident beam, adjusting the gain so that the voltmeter reads less than 8 V.
Now put the semicircle back in place, being sure that you did not rotate it, and take
measurements for several angles of incidence. (The semicircle does not have to be accurately
centered since you are only using the flat surface.) Measure the incident intensity afterwards, and
try again if it has changed substantially.
!
When you have data for E parallel and perpendicular, plot reflectance vs angle and
overlay the values calculated from Eq. 2 using your measured value of index of refraction.
Comment on the agreement, or lack thereof, with the theory.
REPORT
Your report should document your data and conclusions. Are your results consistent with
Snell's law and the Fresnel reflectance equations?
Physics 231 Reflection and Refraction
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