Research Report – McNair Program 2006
LightLight-Matter Interaction
Richard Wooten
Mentor: Dr. Cristian Bahrim
Abstract
In order to understand the complex nature of light and light-matter interaction, two sets of
experiments were set up using PASCO equipment. One set of experiments uses the
phenomenon of dispersion of light through a transparent prism (which is a dielectric
material transparent to visible light) in order to investigate the radiation released by a
glowing (hot) object. The blackbody object used as a test system is simulated by the
filament of a light bulb enclosed inside a black cavity. We have developed a technique
which uses only accurate information about the position of the maximum radiancy, from
which the surface temperature and luminosity of hot objects can be calculated. This
technique enlarges the domain of applicability of a light sensor and would allow
measurements of temperature and luminosity from the stars’ corona.
A second set of experiments examined the transverse character of the light wave using
two different methods of polarization: (1) transmission of light through a polarizing sheet
and (2) reflection of light by a dielectric surface. We have explored several applications
using the polarization of light, such as: (a) the characterization of the stellar atmosphere,
(b) the precise determination of the index of refraction, (c) storage of information through
dielectric materials, etc. A recent study regarding the polarization of light revealed new
applications in astronomy that use polarized light as a messenger of the atomic and
molecular structures formed in the interstellar medium or in planetary atmospheres.
Due to recent theoretical and technological achievements, it is now possible to slow down
and even stop the light in optically dense media. This could eventually lead to the storage
of information in such media. In this project we discuss the possibility to slow down light
waves in an optically dense medium using (1) a technique called “electromagnetically
induced transparency” (EIT) developed recently at Harvard University and (2) a
technique based on the creation of a spectral hole in a crystal by using metastable states,
developed recently at The Institute of Optics, University of Rochester. Further, we
investigate the possibility of recording information by using circularly left- and rightpolarized light waves. If such a technique would be successful, it would make possible to
replace the magnetic supports, now responsible for binary recording, with optical devices.
This could eventually create a revolution in information technology (i.e. for computer
systems).
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1. Introduction
Optics is a complex field, which requires knowledge of electricity, magnetism,
wave mechanics, quantum mechanics, atomic, and molecular physics. Optics offers a
large variety of applications, including optical communications and astronomical
measurements, and therefore, it is a leading field in physics and technology. For example,
the last recipients of the Nobel Prize for physics have been awarded for outstanding
research in the field of Optics (see the Nobel Prize winners on 2005).
“What is light?” it is a seemingly simple question but one that has not been
decisively answered in the whole history of physics. It can be said that light travels as an
electromagnetic wave but can only transfer energy to atoms and molecules in discrete
bundles, known as photons. Phenomena such as polarization, interference, and diffraction
cannot be understood unless the light is considered a wave. However, blackbody
radiation, photoelectric effect, and Compton scattering show that light does not distribute
energy continuously as a wave does, but rather in discrete bundles of energy, called
photons. Today, it is widely accepted that the light has a dual character: it is both a
particle and a wave, but experimentally it can only be observed as one or the other.
Any object having a temperature will emit radiation. But when analyzing the
radiation emitted by various objects, the radiation reflected by the surface of the object
should be subtracted. In order to study only the radiation which originates from inside the
object, a theoretical model called the blackbody (or black cavity) is used (Krane, 1996).
Typically, a black cavity is considered as being a hole in the walls of an empty metal box.
We note that the blackbody is the hole itself and not the box! Any radiation incident
through the hole will have a negligible chance to exit. The only radiation that one
observes is formed inside the black cavity and is due solely to the temperature of the
object. (Krane, 1996).
Light is a transverse electromagnetic wave which travels due to an electric field
and a magnetic field oscillating in phase and perpendicular to one another. These fields
do not exist independently but induce each other according to the Maxwell
electromagnetic theory of light (Hecht 2001). Both fields also oscillate perpendicularly to
the direction of motion, and therefore, the light is a transverse wave (see Figure 1.1). The
direction of propagation of an electromagnetic wave is given by the Poynting vector,
r 1 r r
S = E×B
(1.1)
µ
r
where µ represents the magnetic characteristic of the material. The average value of S is
typically called the “intensity of light.”
r
I= S
(1.2)
avg
In most media the optical effects are typically produced by the electric field
component only. Therefore, the analysis of the propagation of the electric component
through materials gives us deep insight about the interaction between light and matter.
3
X
Y
Z
Figure 1.1 The electric
and magnetic fields are
traveling in the positive
z_direction and oscillate
perpendicularly to each
other.
Due to the light wave’s transverse character it is possible to re-orient the direction
in which the electric field oscillates. This phenomenon is known as polarization of light.
Longitudinal waves (which are waves that oscillate parallel to the direction of
propagation), like the sound wave, cannot be polarized. There are four mechanisms by
which a polarizer can orient incoming light: (1) dichroism (selective absorption), (2)
reflection on the surface of a dielectric, (3) scattering, and (4) birefringence (double
refraction). For each one of these mechanisms a form of symmetry must be present
because a polarizer isolates a particular direction of the electric field oscillation and
eliminates the rest (Hecht, 2001).
In this research project we experimentally investigate the radiation emitted by a
glowing object and several mechanisms to polarize a light wave.
In recent years, there were many discussions about the possibility to slow down
and even to stop light in optically dense media. This was recently proven to be possible
by a group of experimentalists at Harvard University (Hau et al. 1999, Phillips et al.
2001, and Liu et al. 2001) and at The Institute of Optics, University of Rochester
(Bigelow et al. 2003). According to these papers, after a light wave was stopped, it was
possible to re-generate the same light wave preserving all its initial characteristics. This
fascinating subject could eventually lead to a new revolution in the recording and storage
of information using light, including a new technology for creating long-lasting storage
spaces for computers. Today the information is mostly stored on magnetic supports, using
the alignment of magnetized atoms. However, the alignment relaxation of atoms induced
by atomic collisions (Bahrim et al. 1997) erases the information from hard disks after a
certain time due to the disalignment of atoms. The alignment relaxation of magnetized
atoms increases strongly with the temperature (Seo et al. 2003, Khadilkar and Bahrim
2006). A completely new technology based on the storage of light could be a solution for
keeping the information unaltered for a longer time.
This project includes a careful theoretical investigation of the optical techniques
currently in use for slowing down and storing the light in dielectrics. Also, we investigate
a new technique to store optical information by using circularly-polarized light.
4
2. Electric Dipole Oscillator Model
2.1 Transpacency of Dielectrics
Matter is just a collection of atoms and molecules. In order to understand how
light interacts with matter we must understand how light interacts with the atoms and
molecules that compose it. The distribution of charge around atoms has a spherical
symmetry when the atom is isolated. However, when light is incident on a dielectric the
mere presence of the electric field causes the positive and negative charges of the atoms
to separate. The shape of the electronic cloud changes into an ellipse, with a positive end
(where the charge of the nuclei dominates) and a negative end (where there is a
concentration of the electronic charge). This separation of charges is indicated in figure
2.1. Unless the incident light matches one of the characteristic frequencies of the atoms,
the electronic cloud will simply oscillate with respect to the positive nucleus at the same
frequency as the incident light, or resonantly. At this point the energy of the light has
been converted into the vibrational energy of the electronic cloud. Next, the oscillatory
electric dipoles will induce another electric field which will propagate forward. In this
way each atom acts as an antenna and passes the information about the incident light to
the next atom (Hecht 2001). Figure 2.1 shows the interaction between the electric field
component of light and the electric dipole moment of the atoms or molecules of a
material. This process represents the propagation of light through a transparent material.
r r
E = Eo cos (kz − ωt )
r r
E = Eo cos (kz − ωt )
r r
p = po cos (kz − ωt )
Figure 2.1 Propagation of light through a transparent dielectric. Electric
r r
dipoles oscillating under an incident E = Eo cos(kz − ωt ) field. The
r r
oscillation creates an oscillating dipole moment, p = po cos(kz − ωt ) ,
r r
which in turn induces an identical electric field E = Eo cos(kz − ωt ) .
2.2 Light Always Propagates Forward
The electromagnetic waves produced by a collection of oscillating electric dipoles
interfere with each other. There is no constructive interference in the lateral direction
because the electric fields are essentially independent from one another (random phase).
However, in the forward direction all scattered waves will interfere constructively
because they are practically in-phase. In this way the propagation of light through a
5
material is a conversion of energy from light to elastic energy of oscillating atoms or
molecules and then back into energy of light traveling in the forward direction.
2.3 Dissipative Absorption
When the energy of light doesn’t match the energy necessary to excite an atom or
a molecule, then the light induces only resonant oscillations of the atomic electric dipole.
In this case non-dispersive light-matter interaction occurs. This situation corresponds to
the transparency of dielectrics. But if the light’s frequency ω matches one of the
characteristic transitions of the atom in the dielectric, then the light is absorbed (see
figure 2.2) by atoms and next, it is typically dissipated in all directions further in the
material. At this frequency the material is opaque.
|3>
|2>
ω 03
|1>
ω
Figure 2.2 Atomic
transitions from the
ground state |0>.
02
ω 01
|0>
We consider the atoms and molecules in optically dense dielectrics with a linear
response (such as glass), as being simple harmonic oscillators under the action of an
incident monochromatic light wave of frequency ω having an electric field component
r r
E = Eo cos(kz − ωt ) .
(2.1)
ω
This light wave propagates in an arbitrary z-direction with a velocity v =
(called the
k
c
“phase velocity”) through a medium of index of refraction n = , where c is the speed of
v
8
light in free space ( 3 × 10 m/s) and k is the wave number. The index of refraction
represents the optical response of the material to incident light. The index of refraction
(n) of a dielectric depends on the incident frequency ω as
 n 2 − 1  Nq e 2

=
 n 2 + 2  3ε m
0 e


∑ω
j
fj
2
0j
− ω 2 + iγ j ω
,
(2.2)
and is plotted in figure 2.3. In equation (2.2) ωoj represents the natural or resonant angular
frequencies of an atom (at which the atom absorbs light, as shown in figure 2.2), N is the
density of the atoms or molecules in the dielectric, qe and me are the charge and the mass
of the electron, ε 0 is the vacuum permittivity, and i γ j is a complex damping coefficient.
6
The damping factor is only significant near atomic resonances ωoj, and is negligible
2
compared with ω0j − ω 2 elsewhere. The f j terms are known as transition probabilities
and satisfy ∑ j f j = 1 (Hecht 2001).
Figure 2.3 The variation of the
index n versus ω from equation
(2.2). The absorption bands are
indicated near the characteristic
frequencies ω 01 , ω 02 , …(see
n
ω 01
ω 02
ω 03
ω
figure 2.2).
2
As the frequency of the incident light ω approaches ωoj, ( ω0j − ω 2 ) decreases
and n will gradually increase as the frequency increases. This is known as the normal
dispersion and is shown in figure 2.2. For normal dispersive materials, the blue light has
a higher index of refraction than the red light. However, when ωoj = ω in equation (2.2)
the damping term iγ jω will dominate. This corresponds to dissipative absorption in
dielectrics. Dissipative absorption occurs when part of the energy of the excited atom is
converted into thermal energy and is dissipated within the material. The excited atoms
collide with other atoms, a process which induces de-excitation, and finally results in
dissipation of energy in the form of heat.
In conclusion we can say that light-matter interaction means the conversion of
energy between either the electric field of a light wave and the electric dipoles in the
material or between the photons and individual atoms/molecules.
3. Blackbody Radiation
3.1 Historical Importance of the Subject
The blackbody model (which explains the radiation built-up within a glowing
object) is an important phenomenon in physics both historically and conceptually. The
problem of blackbody radiation was one of the first to be solved by treating light as
discrete bundles of energy called quanta (or photons), not as a classical wave with a
continuous distribution of energy. When the classical wave theory is applied, the
theoretical result only agrees with the experiment at very large wavelengths. But as it
approaches shorter wavelengths (i.e. the ultra-violet region, for objects glowing in the
visible range), a large deviation from the experiment is observed, a fact known in the
history of physics as the “ultraviolet catastrophe” (Longair, 2003). This problem was
fixed in the early 1900s by Max Planck, who asserted that the atoms inside a cavity can
only absorb and emit energy in discrete bundles or quanta of energy. These energies are
proportional to the frequency of the atomic transitions. Planck’s prediction agreed
perfectly with the experiment, but Planck did not see the implications immediately, for he
still held to the classical ideas of the wave optics and believed that this assertion was
merely a calculation tool. It was Albert Einstein who had the vision to see that light was
7
actually quantized. The triumph of the discrete quanta of energy over the classical
continuous distribution of energy in physics led to the development of a new and
revolutionary field in physics, quantum mechanics. This theory has become one of the
twentieth century’s most successful scientific theories (Krane, 1996). Today’s advanced
technologies pay heavy tribute to the original quantum ideas developed by Planck and
Einstein.
3.2 Theoretical Background
The radiation reflected by the surface of any object complicates the study of the
radiation emitted from inside the object (which is due to its temperature). Therefore, a
blackbody object at thermal equilibrium (which means having a well-defined
temperature) is preferred. It has the advantage that none of the radiation incident on the
object is reflected, and that all the radiation emitted is coming from inside the object and
is caused solely by the object’s temperature.
Figure 3.1 shows the radiancy of a blackbody object as a function of wavelength
at four different temperatures. The radiancy of an object can be defined as the energy
emitted per unit area, per wavelength, per second by a glowing object. Planck’s theory of
electromagnetic radiation based on the photon concept successfully reproduces these
curves. The Planck’s formula for the radiancy is
 c  8π
R(λ ) =   4
 4  λ

1
  hc 
   hc λkT
− 1
  λ  e
(3.1)
where c is the speed of light in free space, λ is the wavelength of radiation emitted, h is
Planck’s constant, k is the Boltzmann constant, and T is absolute temperature of the
glowing object. From figure 3.1 we see that the peak of the radiancy shifts toward smaller
wavelengths as the temperature increases. Wien generated an experimental relationship
between the wavelength of the maximum radiancy and the temperature of a glowing
object; known as Wien’s displacement law (Krane, 1996)
λmaxT = 2.898 × 10−3 mK .
(3.2)
This equation can also be found by taking the derivative of the Planck’s formula (3.1)
with respect to the wavelength.
In figure 3.1, the area below the curve of radiancy represents the flux density of
the light released by a glowing object at a certain temperature, and is related to the
temperature by a formula known as the Stefan-Boltzmann law.
F =σT4
(3.3)
where σ is the Stefan-Boltzmann constant ( 5.6705 × 10 −8 W / m 2 K 4 ). The flux density
is an average measurement of the total amount of energy emitted from each square meter
of a light source per second. If this is multiplied by the area of the light source we have
the luminosity of a glowing object, which is a useful quantity for astronomers, and also
provides one of the seven fundamental units of the international system of units, called
candela.
8
2.5
Theorectical Blackbody Curves
2
2800 K
Radiancy(relative units)
2600 K
1.5
1
2300 K
1900 K
0.5
Wavelength (nm)
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Figure 3.1
The theoretical radiancy emitted by
a blackbody versus the wavelength
at several temperatures.
Figure 3.2
Our experimental data for 2800K,
2600 K, 2300 K, and 1900 K. The
wavelength at λmax is indicated by
vertical dashed lines.
A more detailed treatment of the physics related to the Planck formula (3.1) can
be found in the textbook “Theoretical. Concepts in Physics.” by Longair (2003). There
have been many theoretical investigations regarding the accuracy of various models used
to derive Planck’s law (Irons, 2005). It is accepted that due to our evolution under the
Sun the curve of response of the human eye has a maximum in the range 500-560 nm
which corresponds to the Sun’s peak of radiancy. Overduin (2003) has investigated this
phenomenon and concluded that the eye developed in such a way that it maximizes the
amount of energy absorbed, which leads to a peak of sensitivity at 560 nm. The
blackbody radiation also has some important applications in biology.
3.3 Equipment and Experimental Procedure
The setup is presented in figure 3.3. An infrared light sensor (CI-6628), attached
to the light sensor arm is used to analyze light emitted from a blackbody-like light source
(OS-8542). This light first passes through a set of collimating slits and a lens before it
reaches a prism (OS-8543). The light is then dispersed by the prism, focused by a lens,
and passes through an aperture slit before it reaches the light sensor. A light sensor arm is
attached to the plate and it can be rotated so that the sensor collects the entire spectrum of
radiation dispersed by the prism. The rotation of the arm is recorded by a rotary motion
9
sensor (CI-6538) through a small pinion which is in contact with the degree plate. The
rotary motion sensor and the light sensor transfer the data to a laptop computer via a
PASCO Science Workshop Interface (Model 750). Finally, the data is analyzed by the
DataStudio™ software (IM-BB 1999) to create accurate plots of the radiancy.
Figure 3.3
The setup for the study of the radiation
emitted by a blackbody. The path of
the light is shown by two thick arrows.
After the equipment is set up, the DataStudio software is configured and the
detectors calibrated for proper analysis. Several spectra of the radiation emitted by a
blackbody light source at various temperatures were collected as shown in figure 3.2. By
plotting the radiancy versus wavelength (figure 3.2), we determined the wavelength λmax
at which the blackbody is emitting the most energetic radiation. The wavelength λmax is
a function of temperature as predicted by Wien’s displacement law in equation (3.2). The
temperature of the blackbody was varied by changing the voltage supplied across it. A
voltmeter and an ammeter were used to determine the temperature as a function of the
supplied voltage and the current going through the blackbody. Details about the
equipment used and the experimental procedure are given in Appendix A.
3.4 Results
A glowing object was examined at various temperatures. The glowing object is
actually a resistor (the filament of a lightbulb) connected to a power amplifier. For
various currents and voltages established by a power amplifier, the filament will glow at
different temperatures. The values of the wavelength for the maximum radiancy for these
various temperatures were measured using the DataStudio program and then used to
determine the temperature. This temperature was then compared to the temperature
provided by direct measurements of current and voltage across the bulb. This experiment
was very challenging since we could only produce a narrow range of temperatures by
adjusting the voltage. This resulted in a narrow range of wavelengths (1000-1500 nm)
which corresponds to an angular interval measured on the rotary table less than 1 degree.
Anyone can imagine that setting up an experiment with such a resolution is extremely
difficult. However, our developed calibration procedure (for details see Appendix A)
allows to measure values with an accuracy within 15 % for both the wavelength and
temperature.
10
If voltages in the range of 6-10 volts are used, then the theory-experiment
agreement is very good. We have observed consistent large errors at lower voltages, and
therefore, such measurements are not performed with the present equipment. Figure 3.2
gives experimental data for this range of voltages, which is equivalent with temperatures
between 1900 K and 2800 K. These curves successfully reproduce the theoretical curves
presented in figure 3.1 near the peak of radiancy and at lower wavelengths.
Figure 3.4 shows the linear dependence of λmax with the inverse of the
temperature for four different measurements according to equation (3.2). Using a linear
regression procedure to measure the slope of the line, we are able to measure a Wien’s
constant of 2.8 ± 0.2 × 10 −3 mK . The error with respect to the accepted theoretical value
of 2.898 × 10 −3 mK from equation (3.2) is about 3%! This excellent agreement between
theory and experiment allows us to measure with a high accuracy the temperature of any
glowing object (such as Sun, hot filaments, etc.) from the peak of their radiancy.
Figure 3.4. The linear
dependence between the
wavelength λmax and the
inverse of the temperature
1/T for a blackbody. The
slope represents the Wien’s
constant in (3.2).
Also, we were able to verify the Stefan-Boltzman law (3.3). The flux density, F,
is defined as the area under the curve of radiancy for a certain temperature, such as those
displayed in figure 3.1. However, due to the calibration of the setup our experimental
curves are only accurate near the peak, which corresponds to a range of wavelength of
about 300 nm around the peak as seen in figure 3.5. By using our experimental value of
temperature derived from the meter readings, and introducing it in the Planck’s law (3.1)
we are able to produce an accurate curve of radiancy which vanishes at large wavelengths
(shown as the light green curve in figure 3.5). The area under the (light green) adjusted
curve leads to an experimental value for the area of 5.4384 × 10 15 W / m 2 / nm for the
flux density for a particular temperature. Using a linear regression to analyze the areas for
a series of different temperatures leads to Stefan-Boltzmann constant of
5.54 ± 0.05 × 10 −8 W / m 2 K 4 which agrees with the theoretical value from equation
(3.3) within 2.5% precision. This linear dependence from equation (3.3) is shown in
figure 3.6.
This experimental procedure increases the range of validity of the light sensor and
allows us to know both the temperature and luminosity (which is the flux density times
11
the surface area of a glowing object) of an object just by the precise location of the peak
of radiancy regardless of the accuracy for the rest of the experimental curve of radiancy.
Wavelength
Figure 3.5. The curve of radiancy emitted by a glowing object as a function of
wavelength. The dark green curve indicates our experimental data, while the
light green curve indicates the radiancy derived from the blackbody theory,
calibrated to the experimental value of the peak of radiancy. The total shaded
area under the red curve represents the flux density from equation (3.3).
Figure 3.6. The linear dependence between the flux density F
and the temperature raised to the fourth, T 4 , for glowing
filament. The slope represents the Stefan-Boltzmann constant,
σ , in (3.3).
3.5 Applications
Our blackbody experiment involves determining the temperature of an object
from scanning the radiation emitted by it. It is not necessary to know neither the distance
to the object, nor its size, nor any other physical property of the object for finding its
temperature. It is needed to find only the wavelength at which the maximum radiancy is
emitted. The temperature is determined strictly from an optical analysis, without any need
to touch the object or even be near it. This aspect is extremely important for astronomy
because it allows us to measure the surface temperature of stars. Also, this method can be
applied to measure the temperature of any glowing objects (i.e. in metallurgy) and it was
12
tested by us using other hot objects which have characteristics of the surface different
than a standard blackbody object as discussed above. This shows the performance of the
present experimental set up. The temperature of an object is an important parameter
giving insight about the physical and chemical processes that are taking place within.
Figure 3.7. The HertzsprungRussel diagram.
Figure 3.8. The solar activity
at the surface of the Sun.
For example, such a measurement in stellar observations is a primary tool which
astronomers use to classify and study the dynamics of stars and takes the form of the
Hertzsprung-Russel diagram, figure 3.7. This diagram presents the luminosity of stars (or
their brightness) versus their surface temperature. A Hertzsprung-Russel diagram is based
on optically acquired data giving insight about the stars’ age and the dynamics of the star,
such as the thermo-nuclear reactions within. The experimental technique studied in this
blackbody experiment helps to learn more about our Universe.
4. Polarization of Light Waves
4.1 Background
We recall that a light wave is a combination of electric and magnetic fields which
oscillate perpendicularly to the direction of propagation as shown in figure 1.1. Many
light sources that we encounter are randomly polarized, meaning that the direction of the
electric field’s oscillation is constantly changing in a plane perpendicular to the direction
of propagation (Halliday et al. 2001). Any light source emits radiation through downward
atomic transitions (called de-excitations). All the transitions will produce polarized
photons because of the selection rules that any transition should obey. However, because
photons are constantly emitted by excited atoms, the overall polarization of light cannot
13
be specified. If a preferred direction of polarization cannot be identified, then the light is
naturally or randomly polarized (Hecht 2001). Polarizers are optical devices able to
polarize the transverse light wave. The light wave can be polarized linearly
r
E = iˆEox ± ˆjEoy cos(kz − ωt ) ,
(4.1)
(
)
Circularly,
r
E = Eo iˆ cos(kz − ωt ) ± ˆj sin (kz − ωt ) ,
[
]
(4.2)
or even elliptically, which has a more complex formula than in equation (4.2).
4.2 Polarization by Transmission
A linear polarizer is simply a material with all the atoms or molecules, lined up in
a single file row. When an electric field is incident on it, only the component of the
electric field parallel with the row of atoms will cause the atoms to oscillate. In this way
the electric field component is selectively absorbed (dichroism). The oscillations of these
atoms re-radiate the light in the forward direction as explained in section 2. Since all
components of light contribute to its intensity eliminating all but one component will
lessen the intensity given in equation (1.2). When unpolarized light travels through a
polarizer, only 50% passes as shown figure 4.1 (Halliday et al. 2001). When linearly
polarized light passes through a second polarizer (often called “analyzer”) even less light
is transmitted.
Figure 4.1 Unpolarized light
incident on two polarizers.
The electric field of the
incident light as well of the
light transmitted through the
first polarizer and the second
polarizer (analyzer) are
indicated.
Malus’s law relates the intensity of the polarized light transmitted through a linear
polarizer to the angle, θ , between the transmission axis of the linear polarizer and the
polarization axis of the incident light, as seen in figure 4.1:
I (θ ) = I (0) cos 2 θ
(4.3)
where I (θ ) is the intensity of light after it passes through the polarizer and I (0) is the
intensity of the light incident on the polarizer. In order to investigate the wave nature of
light, we set up an experiment based on the polarization of light by transmission with the
verification of Malus’s law. Appendix B gives more details about this experiment along
with some general considerations for light-matter interaction.
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4.3 Polarization by Reflection and the Fresnel Equations
It is also possible to polarize light by reflection on a dielectric surface. Reflection
is a kind of back scattering that will occur whenever light experiences a discontinuity in
the medium (Hecht 2001). The interface between two media is an example of a large
discontinuity. The basic equations of electrodynamics can be derived using the Maxwell
equations, which describe the propagation of electromagnetic waves through a medium:
∫S D • da = Qenc
∫ B • da = 0
(4.4.a)
(4.4.b)
S
∫
P
E • dl = −
∫P H • dl
d
dt
∫
S
= I enc +
B • da
(4.4.c)
d
dt
(4.4.d)
∫S D • da
where (4.4.a) and (4.4.b) are integrals over a closed surface S and (4.4.c) and (4.4.d) are
B
integrals over a closed loop P. Where D = ε E and H =
(ε and µ represent the
µ
electric and magnetic characteristic of the material).
The Fresnel equations arise from manipulating the Maxwell equations applied at
the interface between two different media. A complete derivation of the Fresnel equations
can be found in the Appendix C. Based on these equations we can calculate the intensities
of the parallel and perpendicular components of a wave reflected by a dielectric surface.
Similar equations can be found for the wave transmitted through a dielectric. Also, this
theoretical model allows the derivation of the law of reflection and the law of refraction
(known also as Snell’s law) and also it shows that the incident, reflected, and transmitted
beams will all be in the same plane (called “plane of incidence”) (for a detailed
discussion see Griffiths, 1999).
The basic idea in Fresnel’s theory is that the electric and magnetic field
components of electromagnetic radiation experience a discontinuity when the light
travels through two different media.
The boundary conditions that result from applying Maxwell equations (4.4.) are:
ε1 E1⊥ − ε 2 E2⊥ = 0
(4.5.a)
B1⊥ − B2⊥ = 0
(4.5.b)
E1 − E 2 = 0
(4.5.c)
1
µ1
B1
−
1
µ2
B2 = 0
(4.5.d)
15
The preceding equations correspond to the case when there is no free charge or
current in either medium near their boundary. Equations (4.5) are very powerful and will
be used to derive many fundamental laws within the theory of light-matter interaction.
When a monochromatic (single frequency) plane wave propagating in an arbitrary
z_direction is considered, then the equations (4.5) become
(~
~
ε1 E 0I + E 0R
)z
~
= ε 2 E 0T
(
)z
(4.6.a)
(B~ 0 I
~
+ B0R
) z = ( B~ 0T ) z
(4.6.b)
(E~ 0 I
~
+ E0 R
) x, y = (E~ 0T ) x, y
(4.6.c)
1 ~
~
B0I + B0R
µ1
(
) x, y = µ1 ( B~ 0T ) x, y
(4.6.d)
2
In Appendix C we show how the following fundamental relations in Optics can be
proven:
* the law of reflection:
θI = θR ,
(4.7.a)
* the law of refraction:
nI sin θ I = n T sin θT ,
(4.7.b)
* the Fresnel equations for the electric component of the reflected wave by a dielectric
material in a plane parallel with the plane of incidence
E0 R
 nI cos θ T − n T cos θ I
=
 nI cos θ T + n T cos θ I


 E0 I


(4.8)
and perpendicular on the plane of incidence:
 n I cos θ I − n T cos θ T
E0 R ⊥ = 
 n I cos θ T + n T cos θ I


 E0 I .
⊥


(4.9)
where E0 R , E0 I , E0 R ⊥ , and E0 I ⊥ represent the parallel and perpendicular components
of the reflected and incident electric fields, respectively. The index of refraction of the
two media, are given by nI (incident) and n T (transmitted), and the angles of incidence,
reflection, and transmission are θ I , θ R , and θ T (see figure 4.2).
16
Reflected Ray
Incident Ray
E0 I
θI
E0 R
θR
θ
nI
nT
θ
Figure 4.2 The principle of
polarization by reflection
considering only the parallel
component of an incident
light beam.
T
E0 T
We define the intensity of the light as being the square of the electric field’s
amplitude (see equation (1.2)). The ratio of the reflected and incident intensities give the
reflectance R.
The two components of the reflectance are
 nI cos θ T − n T cos θ I
R =
 nI cos θ T + n T cos θ I

2

 =  E0 R
E

 0I

2
and
 nI cos θ I − n T cos θ T 
 =  E0 R
R⊥ = 
E
 nI cos θ T + n T cos θ I 
 0I





2
(4.10)
2


⊥
(4.11)
Figure 4.3 gives the theoretical variation of the perpendicular and the parallel
component of the reflectance versus the incident angle for an acrylic material of index of
refraction 1.49 for incident red light of 650 nm (Hecht, 2001). We notice that the parallel
component shows a minimum, R = 0 at a particular angle of incidence which is known
as the Brewster angle, θ B . At this angle no light parallel to the plane of incidence is
reflected. This phenomenon is explained in detail below.
The angle of reflection is always equal to the angle of incidence no matter what
the polarization of the light is (see Appendix C for the derivation which leads to equation
(C.12)). However, for light polarized parallel to the plane of incidence (the plane which
includes the incident ray, the reflected ray and the transmitted ray), there is an incident
angle where the rule doesn’t hold. For this angle, called Brewster angle, the reflection
simply doesn’t occur. This phenomenon can be understood by using the Lorentz electric
dipole oscillator model (discussed in Section 2) and the Fresnel equations. Recall the
electric field distorts the electronic cloud such that the atoms in the medium become
electric dipoles. Sometimes the electric dipoles exist regardless the presence of an
external electric field and they are formed by chemical bonds in molecules (e.g. the
molecule of water H2O is a permanent electric dipole, with the negative end located at the
oxygen atom and the positive end located at the two hydrogen nuclei). The oscillation of
the external electric field will cause these electric dipoles to oscillate resonantly at the
17
frequency of the incident electric field. The electric dipoles in turn radiate part of this
energy in the form of the reflected wave while the other part is transmitted into the
dielectric, as the refracted wave.
n t = 1.49 @ λ = 650 nm
Figure 4.3 The theoretical
reflectance of the light
decomposed in a parallel,
R , and a perpendicular,
θB
R⊥
R⊥ , component versus the
R
angle of incidence of light on
a acrylic surface. The middle
line indicates the average
value of R⊥ and R .
Angle of Incidence
For the purpose of better understanding the Brewster effect, lets consider an
incident wave ( E0 I ) polarized parallel to the plane of incidence. Due to the polarization
of the incident wave the reflected wave ( E 0 R ) and the transmitted wave ( E0 T ) will also
be polarized parallel to the plane of incidence, as shown in figure 4.2. An electric dipole
cannot radiate energy parallel to the direction of its own axis of oscillation (along the
dashed line in figure 4.2). Since the electric dipoles always oscillate perpendicularly to
the electric field’s direction of travel, it is inferred that if the angle between the
transmitted ray and the reflected ray is 90˚ there will be no radiation emitted by the
electric dipole back into the initial medium. This can be seen from figure 4.2 when
θ T + θ R + 90° = 180° and θ I = θ R (law of reflection).
θ T = 90° − θ I .
When no light wave is reflected E0 R
(4.12)
= 0 . From the Fresnel equation for the
parallel component (equation 4.8) we see that this can happen either if there is no incident
electric field, E0 I = 0 (condition that should be discarded as long as we manipulate light
at the interface between the two media) or if
18
 nI cos θ T − n T cos θ I

 nI cos θ T + n T cos θ I


=0 .


(4.13)
This implies that the numerator vanishes:
nI cos θ T − n T cos θ I = 0
and therefore:
nT
nI
=
cos θ t
.
cos θ i
(4.14)
(4.15)
If we substitute equation (4.12) into (4.15), and note that the particular angle of
incidence at which the reflected component disappears as being the Brewster angle
θ B = θ I , then we find that:
tan θ B =
nT
(4.16)
nI
Brewster’s law (4.16) allows determination of the index of refraction of a material by
simply measuring the Brewster angle. At any angles other than θ B , the reflected light is
partially polarized (for more details see Hecht 2004, Chapter 8).
Using the polarization of light by reflection on a dielectric surface we
experimentally verify the Fresnel equations and the Brewster angle, and also we can
calculate the index of refraction of an acrylic lens for a certain wavelength incident on
the dielectric.
The experiments based on Malus’ law and the polarization of light by reflection
clearly show that light is a transverse electromagnetic wave.
4.4 Equipment and experimental procedure
This experiment will make use of the equipment designed by PASCO and shown
in figure 4.4. For this experiment, the light source is a diode laser (OS-8525A) of
wavelength 650 nm. First, we polarize the laser beam by using two polarizers having
their transmission axes oriented at 45 degrees to one another. Next, the polarized laser
beam is reflected by an acrylic semicircular lens (OS-8170). The reflected light passes
through a polarizer (OS-8170) and finally is detected by a high sensitivity light sensor
(CI-6604). The rotary motion sensor (CI-6538) measures the angle of incidence. The data
is transferred into a DataStudio program through a PASCO Science Workshop computer
interface.
With the setup described above, the reflectance of the laser beam parallel to the
surface and perpendicular to the surface of a dielectric material are measured. An acrylic
semicircular lens is used as the dielectric material. The intensities are measured over a
large range of incident angles (20 to 85 degrees). Three values are reported for each angle
of incidence: the intensity of the beam without a polarizer and the intensities of the
polarized light parallel and perpendicular to the surface of the dielectric. By analyzing
this data using a spreadsheet and DataStudio we can find Brewster’s angle, and verify the
19
Fresnel equations (4.8) and (4.9). The relationship between the Brewster angle and the
index of refraction of the acrylic lens is indicated in equation (4.16). From this equation,
we determine the index of refraction of the lens (IM-BA 2002). Details about the
experimental procedure can be found in Appendix D.
Figure 4.4 A top
view of the setup
used in polarization
by reflection on an
acrylic surface.
This experimental technique was first reported by P.J. Ouseph, Kevin Driver, and
John Conklin (2001), in a paper entitled “Polarization of light by reflection and the
Brewster angle”, which was our main source of inspiration in setting up the experiment.
The experimental procedure proposed by P.J. Ouseph et al. leads to an index of refraction
of 1.51 ± 0.06, which has an error of about 1%. We have successfully acquired data of
similar precision, as detailed in section 4.6.
4.5 Results using Malus’ Law
Using a setup formed by two polarizers with their transmission axes at an angle θ
apart, we are able to verify Malus’ law from equation (4.3). Figure 4.5 shows the
measured value for the intensity of light as a function of the angle θ , as well as the
theoretical curves based on equation (4.3). As one can see, the two curves are in perfect
agreement. This experiment shows a method to efficiently attenuate the intensity of the
light using a linear polarizer and conclusively shows that light is a transverse
electromagnetic wave.
Polarization by T ransmission
20
18
16
12
experiment
10
theory
8
6
4
2
Angle
18
0
1
60
14
0
12
0
80
10
0
6
0
40
20
0
0
Intensity
14
Figure 4.5 Comparison
between our experimental
data and the theoretical
values. The minimum
corresponds to 90 degrees
angle where the cos
function (in equation
(4.3)) vanishes.
20
4.6 Results for Polarization by Reflection at the Surface of an Acrylic Lens
We have examined the variation of the intensity of light reflected by an acrylic
lens for different angles of incidence. The interest in decomposing the light and analyzing
the behavior of these components is to gain a better understanding of the way light
interacts with the molecules on the dielectric’s surface.
n t = 1.51 @ λ = 650 nm
θB
R⊥
R
Figure 4.6 Comparison
between experimental data
points and theoretical curves
predicted by equations (4.10)
and (4.11) for the parallel and
perpendicular components of
light incident on an acrylic lens.
Figure 4.6 shows a perfect agreement between the theoretical and experimental
curves for parallel and perpendicular components of the reflectance. This technique
allows finding with a high precision, the angle at which the intensity of the parallel
component from equation (4.10) goes to zero, known as the Brewster angle. When light
of wavelength 650 nm was reflected by an acrylic surface, the Brewster angle, θ B was
found to be 56.4 degrees. From the Brewster angle we can now determine the index of
refraction of the lens using equation (4.16). Our measured value for the index of
refraction is 1.51 ± 0.02 and differs from the theoretical one of 1.5%!
4.7 Applications to Polarization
The phenomenon of polarization of light is important for many applications, such
as the transmission of information through fiber optics. Also we consider that polarized
light might be the best mean for imprinting information in optically dense media (see
section 5.5).
The polarization of light has applications in various fields such as art, astronomy,
engineering, geology, and physics. Polarized sunglasses work in much the same way as
the linear polarizer described above. In this way they are often used to limit glare from
objects that reflect sunlight. The sunlight can be partially polarized by the atoms and
molecules in the Earth’s atmosphere. The reflected sunlight on surfaces, such as the sea,
can also be polarized. The intensity of this polarized light can then be limited by a linear
polarizer, such as a pair of polarized sunglasses.
21
The technology used for liquid crystal displays (LCD) in laptops makes use the
polarization phenomenon. A liquid crystal can polarize light in variable amounts
depending on the alignment of its strands of molecules. Controlling the alignment of
these molecules and therefore the polarization via an applied electric field will limit the
intensity of the light incident on a particular pixel. Each pixel can be a combination of
red, blue, and green. If it wouldn’t be possible to polarize the light and therefore vary the
intensity of each primary color, there would only be 8 possible colors from the basic
combinations of red, blue, and green. The 8 colors are red, green, blue, magenta, cyan,
yellow, white, and black. Varying the intensity of each blue, green, and red component
allows for the subtle color variation utilized in laptop displays.
The antennas responsible for radio and television signals are intrinsically
polarized. In order to take advantage of this and avoid potential confusion all television
signals are polarized horizontally and radio signals are polarized vertically. As such, only
receivers polarized in the appropriate way can receive signal. For details see the report
“Antenna Polarization Application Note” by Joesph H. Reisert from Astron Wireless
Technologies, Inc. (www.astronwireless.com/polarizaton.html).
Polarization of light has new applications in astronomy. Because, the polarization
of a light beam is dependent on how it interacts with matter, the analysis of polarized
light teaches us about the environment through which the light passes, such as the
existence of molecules in the upper corona of a star or of a planet’s atmosphere. Due to
recent technological advancements, astronomers have been able to analyze polarized light
emitted from stellar objects, such as supernovas. This analysis was not possible until
recently, because of the limitation in the amount of polarized light we can detect and
analyze. The study of the degree of polarization of light received from a stellar object
gives information about the composition and evolution of stars. In particular, the degree
of the light’s polarization can discern subtle variations in the explosion of a supernova.
Insight into the shapes of these explosions leads to knowledge of their nature and the
mechanisms which cause them (Cowen, 2006).
By studying the polarization of the cosmic microwave background radiation
measured by the WMAP (Wilkinson’s Microwaves Anisotropy Probe) astronomers have
learned more about the Big Bang. After studying how the background radiation has been
polarized after hitting the first stars astronomers agreed that no star had formed until
about 400 million years after the Big Bang. This estimate is 200 million years later than
originally predicted by the probes data, but is more consistent with the currently accepted
theoretical models. This same data has also confirmed one of the cornerstones of the Big
Bang theory: inflation. Inflation claims that there was a period of rapid expansion in the
early universe which caused the fluctuation that ultimately led to the formation of stars
and galaxies. By subtracting out the now known effects of polarization, cosmologists
have found larger variations in temperature over larger patches of sky than smaller ones.
This is exactly what is predicted by inflationary model (Cowen, 2006).
As a follow up to this research project, we plan to investigate the possibility of
using polarization by reflection in order to measure the curve of dispersion for different
dielectrics. The precision achieved in the experiment described above, which is 1/1000
for the index of refraction, allows drawing accurate curves of dispersion such as those
presented in figure 4.7. This idea is attractive because it allows the study of dispersion
without a direct measurement of dispersion but instead through a measurement of
22
reflection at the surface. In addition, this technique would allow measuring the degree of
homogeneity and purity of a material through comparison between the curve of
dispersion derived from reflection at the surface and the curve produced by a direct
measurement of dispersion.
Figure 4.7 Curve of dispersion
for different dielectric materials.
5. Slowing Down and Storage of Light
5.1 Status of the Problem
Light is composed by photons which only exist at speed c (c = 3 x 108 m/s, the
speed of light in vacuum). So, the light as a photon cannot be slowed down. It either
travels at speed c or does not exist! As a wave, the discussion is more complex and it
requires the introduction of the concept of group velocity for a light pulse. The
phenomena of “slow” and “fast” light uses the concept of group velocity, which
represents the velocity of a group of monochromatic waves, such as the one given in
equation (2.1), which form a light pulse. In general, when dealing with light waves, each
individual wave travels with a phase velocity. For example, a linearly polarized light
traveling in a positive z-direction as defined in equation (4.1) travels with a phase
velocity
v=
ω
k
(5.1)
When many waves, of slightly different frequencies, are added together (or
superimposed) a “modulation signal” envelopes the net, or resultant, wave. The speed of
this feature (usually called “envelope”) is known as the group velocity and is defined as
vg =
dω
dk
(5.2)
23
and can be slower or faster than the phase velocity (Hecht, 2001). The relationship
between the group velocity, v g , and the phase velocity, v , can be shown as follows:
vg =
d
(vk ) = v + k dv
dk
dk
(5.3)
dv dν
dν dk
(5.4)
vg = v + k
where v =
ω
is the frequency. By introducing the definition of the wave number k =
2π
in equation (5.4), we get
vg = v + k
dv d  ck 
c dv dk
c dv
=v+k
 = v+k
dν dk  2π 
2π dν dk
2π dν
λ
2π
(5.5)
In equation (5.5) we have used the relationships between the phase velocity, v, the
angular frequency, ω, and the wave number, k, from equation (5.1) and between
2π
frequency, speed c, and wave number ( v
= c ).
k
The relationship between the group velocity and the index of refraction
vg =
c
dn 

ω
n +
dω 

(5.6)
can be proven using the phase wave concept. Based on Boyd’s ideas (2002), a short
derivation of the equation (5.6), using the assumption that the wave’s phase remains
constant in space and time for a light pulse is given in Appendix E.
If we define a group index of refraction as
ng =
c
vg
(5.7)
we find that
ng = n +
dn
ω
dω
(5.8)
The only time when the group velocity coincides with the phase velocity is for a nondispersive medium (such as vacuum) (Hecht, 2001). In this case, the index of refraction is
independent of the angular frequency, ω, and the 2nd term in equation (5.8) vanishes. In
consequence, the light waves of different angular frequencies travel at the same speed. In
all other media the index of refraction depends on the angular frequency, ω.
In the case where ωoj = ω in equation (2.2), the light is absorbed by atoms and is
transferred into atomic excitations (see figure 2.2). Next, this energy is re-emitted by
atoms but in random directions, and therefore we say that the light is dissipated in the
material. In a region very close to a resonance, ωoj, the slope of the index of refraction,
24
dn
is negative. This is called the region of anomalous dispersion. This situation leads to
dω
the absorption bands shown in figure 2.3.
If the driving frequency is greater than any of the characteristic frequencies ωoj,
then from equation (2.2) we see that n < 1. Because by definition n = c/v, if n < 1 then v >
c, which is an apparent contradiction to Einstein’s special theory of relativity (which
states that light cannot move faster than the speed of light in vacuum, c of 3 x 108 m/s).
But a closer examination of equation (5.8) shows that it is about the group index of
refraction, and implicitly, the group speed associated (as defined in equation (5.6)) can
exceed c (Hecht, 2001)1. This result doesn’t contradict Einstein’s theory of relativity
which refers to the phase velocity specifically.
We have established that in the area of anomalous dispersion, we have a large
negative slope for n ( dn dω < 0) and the group velocity can be faster than c. Contrary,
for slowing down the light (which means that we look for v g << v ), we need to be in a
region of large normal dispersion with a large positive slope for n ( dn dω > 0). To create
this situation Hau et al. (1999) have used a technique called “electromagnetically induced
transparency” (EIT) and Bigelow et al. (2003) have used a spectral hole created in a
crystal using metastable states. Both techniques will be presented in separate sections
later.
At atomic resonance, (which represents the case where the incident frequency
matches a characteristic frequency of the atom) the phase index of refraction n equals 1,
because the light is absorbed as a photon which travels only at speed c (so n =c/c =1). In
consequence, in equation (5.8) the 2nd term dominates because of the sharp variation of n
(the slope dn dω is very large), and n g strongly depends on ω. Close to atomic
resonances, there is a large dissipative absorption. This process results from the loss of
energy by excited atoms due to atomic collisions (effect of the thermal motion) before
they can radiate photons. In consequence, the medium is opaque near its resonance.
However, because the region near a resonance has a steep dependence on the frequency
(see figure 2.3), a light pulse of angular frequency near a resonance attains a very slow
group velocity. In a normal dispersive medium we have v g < v , while in anomalous
dispersion we have v g > v (Hecht 2001).
5.2 Storing and Stopping Light with EIT
The EIT technique was built-up for slowing down the light and for storage of light
using a monochromatic laser pulse by experimentalists at Harvard University, the
Harvard-Smithsonian Center for Astrophysics, and the Institute for Theoretical Physics
(ITAMP) (sees Hau et al. 1999, Liu et al. 2001, Phillips et al. 2001). It requires a threelevel atom (as shown in figure 5.2): two Zeeman (hyperfine) states for the ground state
1
Strictly speaking any real wave is finite in spatial extend, and therefore, it is actually a pulse, though it
could be rather long in certain cases. It is known that any pulse is a superposition of sinusoidal waves and
travels actually at a group velocity. So, the index of refraction in equation (5.8) is actually the group index
of refraction for a light pulse.
25
1 and 2 , and one excited state 3 , and a probe laser pulse (of frequency νo = ωo/(2π))
resonant to the transition 1 → 3 . A second coupling laser, of frequency νc (figure 5.1)
very close to νo, locks all the atoms on the 1 state, and forbids the transition 1 → 3
to occur. The 3 state is now a “dark state”, and the medium becomes transparent at the
frequency νo of the probe laser. Now, the optically dense medium behaves as a normal
dispersive material, with a sharp increase of the index of refraction in the vicinity of the
resonance frequency (figure 5.2). Because dn dω is positive and large, according to
equation (5.8) the group index of refraction ng associated to the laser pulse centered at
frequency νo will be also very large. In consequence, v g of the probe laser pulse will be
very slow.
Figure 5.1
A transition from |1> state to
the |3> state occurs in EIT.
Figure 5.2
The steep positive slope of dn/dν
that occurs during EIT.
When the coupling laser is on, all the atoms on the 1 state are locked, the
medium is perfectly transparent but with a very high index of refraction. Now the probe
laser pulse is slowing down, from about 3x108 m/s in air, to only a few meters per second
in a optically dense medium. This makes the probe laser to be practically squeezed in this
optically dense medium. When the coupling laser is turned off, the transition 1 → 3 is
re-opened, and the atoms can absorb the probe laser of frequency νo, which is now
imprinted in the alignment of the atomic spins. In consequence, the light is stopped in the
medium. The procedure is reversible, and the information stored in the atomic spins can
be transferred back to the light field, reconstructing the original light pulse. Indeed, Liu et
al. (2001) have observed that when the coupling laser is turned back on, the original light
pulse is reconstructed. With this EIT technique two effects are realized: (i) the probe
pulse of frequency νo is slowed down significantly because of the large normal dispersion
of the atoms near the resonant frequency, and (ii) the information in the probe laser beam
can be imprinted in the spin alignment of the atomic state 3 . This accomplishment has
brought an increased interest in using photons for transmission of information to and
from atomic systems as part of quantum communication schemes (Levi 2001). It is clear
now that it is possible to squeeze the light onto an ensemble of atoms, which is an
amazing achievement.
26
5.3 Details Regarding EIT Experiments Realized at Harvard University
In 1999, a group of experimentalists at Harvard University were able to slow
down light to a speed of 17 m/s. The experiment was performed in a cloud of sodium
atoms which was cooled to nanokelvin temperatures. The atomic density was increased
by cooling the atoms below the transition temperature for Bose-Einstein condensation
which allows for an even larger index of refraction (Hau et al., 1999). Due to the slow
group velocity, the leading edge of the envelope entering the medium was slowed while
the back edge was still retained its free space speed. This allowed for the back edge to
“catch up” to the leading edge resulting in a spatial compression of the light pulse by a
factor of roughly 6 million (from 600m to 100µm) (Dutton et al., 2004).
The next step was to stop the light altogether by taking the group velocity to zero.
Using the EIT technique this can be accomplished by turning off the coupling laser beam
while the probe pulse is acting upon the atoms. Two experimental groups accomplished
this result in 2000 using a similar method: One group was led by Lene Vestergaard Hau
from The Rowland Institute for Science and Harvard University and has used an ultra
cold cloud of sodium atoms (Liu et al., 2001). The experimental results are shown in
figure 5.3. The experiment was based on the technique for slowing down the light builtup by this group along with Steve Harris (Hau et al., 1999). The other group, based at
Harvard-Smithsonian Center for Astrophysics, was led by Ronald Waslworth and
Mikhail Lukin and used warm rubidium atoms (Phillips et al., 2001). The experimental
results are shown in figure 5.4. Both teams achieved similar results regarding the storage
of light.
When the coupling laser is turned off the medium will no longer be transparent to
the probe pulse (centered on the resonance, νo). At this point all the information contained
in the light (frequency, angular momentum, and amplitude) is imprinted in the atoms in
the form of coherent spin excitations. During the storage process, the information about
the amplitude of the probe pulse is stored in the “population amplitudes” that define the
atomic dark states, while information about the “mode vector” of the probe field is
contained in the relative phase between different atoms in the macroscopic sample (Liu,
2001). However, it would seem that without the EIT mechanism, the excitation energy of
the atoms would be lost into dissipative absorption. This is because the medium is no
longer transparent. However, it is possible to decrease the intensity of the coupling beam
to zero without inducing absorption because “…the bandwidth of the pulse will continue
to narrow in such a way that it always remains within the transparency window” (Levi,
2001). When the coupling field is turned back on, making the medium transparent, the
original probe pulse is regenerated via stimulated emission (Liu et al., 2001). Since the
spin excitations do not couple to the electronic excited states, the atoms are immune to
spontaneous emission, which makes the light storage non-destructive (Phillips et al.,
2001). Using cold sodium this team has verified experimentally that the probe pulse was
regenerated by stimulated emission. First, they put all the atoms in the 1 state (figure
5.2) and then turned on the coupling laser which was then absorbed completely (Liu,
2001). After the coupling field is turned back on, the probe pulse travels as if it had never
been turned off. At this point information has been converted from purely optical to
purely atomic and then back to purely optical. The group using cold atoms reported a
maximum storage time of 1 ms while the group with warm Rubidium reported 0.5 ms.
27
Although the pulse is regenerated, as the storage time is increased the intensity of the
regenerated pulse dissipates (see figures 5.3 and 5.4). The primary limiting factor for the
storage is the atomic coherence lifetime of the medium (Phillips et al., 2001).
Figure 5.3
The plots show the data taken by the group using
cold sodium atoms. The dashed curves correspond
to the intensity of the coupling laser. The open
circles show a reference probe pulse in the absence
of the atoms. The filled circles show a probe pulse
in the presence of the atoms. The first diagram
shows a storage time of 38 µs and the second
diagram gives a time of 833µs.
Figure 5.4
Data reported by the group
using warm rubidium. The
dotted line corresponds to the
input signal or probe pulse. The
dashed line corresponds to the
applied control or coupling
field.
5.4 Experiments using a Spectral Hole Realized at The Institute of Optics
There have also been observations of slow group velocities in solids. A group
based in the Institute of Optics (Bigelow et al., 2003) has observed a group velocity of
57.5 m/s in a ruby crystal at room temperature. This was accomplished not by EIT but by
the “creation of a spectral hole due to population oscillations” (Bigelow et al., 2003).
Another team at Massachusetts Institute of Technology was able to slow and store light in
a solid by using EIT (Turukhin et al., 2002). A group velocity of 45 m/s was recorded
and the storage time was about 200 µs (Turukhin et al., 2002).
By using a spectral hole technique the group led by Bigelow at the Institute of
Optics were able to obtain low group velocities in a solid ruby crystal of the same order
as those observed at Harvard University. The spectral hole results from exciting the
ground state on a short-lived state followed by de-excitation to a metastable state. This
28
creates a population inversion between the ground state and the metastable state which
actually makes the short-lived state in the chain of transitions to become a “dark state”
(which is a state invisible to radiation that matches the energy difference between the
ground state and the metastable state). Therefore the medium is transparent to the
frequencies near the atomic resonance. Thus, the creation of a dark state generates a
dn
sharp increase of the index of refraction with a large positive slope
, and therefore,
dω
according to equation (5.8) creates the premises to slowing down the light near the
resonant frequency. This effect is similar with EIT which accomplishes the same goal by
locking off the ground state so that the medium is transparent to the incoming pulse. In
either case the incoming pulse cannot excite the atom, and therefore, the medium is
transparent.
The experiment using a spectral hole has the particularity that slows the group
velocity using a single laser. They determined that a single cw probe laser can
sufficiently provide the necessary saturation required to change the group index of
refraction. A benefit of the spectral hole technique is that there is no need to use a second
laser in order to lock any particular transition from the ground state, as is the case when
the EIT technique is used. Equally important is that in the spectral hole technique, the
laser can also operate in different modes and all modes will experience the same delay.
Finally, this technique is of particular importance because it was done using a crystal at
room temperature. These factors make this technique a prime candidate for future
implementation into the field of information technology (Bigelow et al., 2003).
5.5 Hypothesis for Storage of Information using the “Spectral Hole” Technique
The EIT and “spectral hole” techniques are based on storage of a monochromatic
laser pulse (of single frequency). This is impractical for quantum computing where two
bits, 0 and 1, need to be recorded. Our original idea is to use two laser beams, one
polarized circularly-left and the other one circularly-right of same frequency ωo to
imprint the light information through manipulation of circularly-left polarized photons (of
angular momentum + h ) and a circularly-right polarized photons (of angular momentum
− h ). The conservation of angular momentum for the atom-light system will record the
information in the spin alignment of atoms. Thus, the atoms in an excited (metastable)
state will be either with the spin-up or with the spin-down, similarly with the binary
recording on magnetic supports. The reconstruction of the information will use the same
principle of conservation of angular momentum, and therefore, the two circularlypolarized photons can be recovered.
According to the recent EIT experiments with cold (Liu et al., 2001) and warm
(Phillips et al., 2001) atoms, the temperature will not be a factor for loosing the
information. Therefore, the light information can be stored a longer time than on
magnetic supports. This could lead to a new generation of computers. Also, it would be
interesting to study if a larger band of frequencies could be stored. This would be
important for the storage of complex information such as a text or music with a large
bandwidth.
29
6. Conclusions
This research project addresses the problem of the light-matter interaction from
both experimental and theoretical view points. Two distinctive set of experiments were
developed during this project: (1) One experiment analyzes the radiation emitted by
glowing objects and provides an effective method to determine the temperature and
luminosity of, or the flux density emitted by, an object. This blackbody model uses only
the light emitted by the object and can therefore be applied to the study of stellar objects
and metallurgy. Our developed technique for increasing the range of applicability of our
sensor would facilitate this study. (2) The second experiment studies polarization by
transmission and conclusively shows that light is a transverse wave. A linear polarizer
can be easily used as a selective attenuating device. Based on the Fresnel equations
derived from the Maxwell’s theory, we precisely determined the index of refraction for a
dielectric material by using polarized light reflected by its surface. The precision of our
measurement will allow finding accurate curves of dispersion for different materials by
using various wavelengths (i.e. using several lasers). Also, the study of polarized light is
attractive for astronomy in the characterization of stellar atmospheres and their dynamics.
Another subject explored in this research project is related to the understanding of
techniques and models currently used for slowing down and eventually storing light in
the spin alignment of atoms. So far, this subject has been explored in a few select peerinstitutions such as Harvard University, The Institute of Optics at Rochester University
and Massachusetts Institute of Technology. Our plan is to develop a technique based on
the creation of a dark state within an absorption band for storing information using
circularly-polarized light. The method is based on the “spectral hole” technique recently
developed at The Institute of Optics.
7. References
Bahrim, C., Kucal, H., Dulieu, O., and Masnou-Seeuws F. Journal of Physics B 30, L797
(1997).
Bigelow, M.S., Lepeshkin, N.N., and Boyd, R.W. Physical Review A 90, 113903 (2003).
Boyd, R. W. and Gauthier, D.J. “Slow” and “fast” light. published in “Progress in
Optics”, 43, edited by Emil Wolf, Elsevier Science B.V. (2002).
Bransden, B. and Joachain, C. Physics of Atoms and Molecules. John Wiley and Sons,
New York (1983).
Cowen R. Astronomy Gets Polarized. Science News, 170, 24 (July 2006).
Dutton, Z., Ginsberg, N. S., Slowe, C., and V. Hau L. The art of taming light: ultra-slow
and stopped light. Europhysics News, 35, 2, (2004).
Griffiths, D. Introduction to Electrodynamics. Prentice Hall; 3rd edition (1999).
30
Hecht, E. Optics. Addison Wesley, San Francisco (2001).
Levi, B.G. Researchers stop, store, and retrieve photons—or at least the information they
carry. Physics Today, 54, 17, (March 2001).
Halliday, D., Resnick, R., Walker, J. Fundamentals of Physics. JohnWiley and Sons,
New York (2001).
Hau, L.V., Harris, S.E., Dutton, A. and Behroozi, C.H. Nature 397, 594, (1999).
(IM-BB) Instruction Manual and Experiment Guide for the PASCO Scientific Model OS8542. “Blackbody light source for the OS-8539 educational spectrophotometer.”
PASCO Scientific (1999).
(IM-BA) Instruction Manual and Experiment Guide for the PASCO Scientific Model OS8170. “Brewster’s angle accessory.” PASCO Scientific (2002).
Irons, F.E. Canadian Journal of Physics 83, 617, (2005).
Khadilkar, V. and Bahrim, C. 109th Conference of Texas Academy of Science (March 46), P57 (2006).
Krane, K. Modern Physics. John Wiley and Sons, New York (1996).
Liu, C., Dutton, Z., Behroozi, C. H. and Hau, L. V. Nature vol. 409, 490 (2001).
Longair, M. Theoretical Concepts in Physics. Cambridge University Press, Cambridge,
(2003).
Nobel prize winners in physics on 2005 at
http://nobelprize.org/physics/laureates/2005/index.html
Ouseph, P.J., Driver, K. and Conklin, J. American Journal of Physics. 69, 1166, (2001).
Overduin, J.M. American Journal of Physics 71, 216, (2003).
Phillips, D. F., Fleishhauer A., Mair, A., Walsworth, R.L. and Lukin, M.D. Physical
Review A 86, 783, (2001).
Seo, M., Simamura, T., Furutani, T., Hasuo, M., Bahrim, C. and Fujimoto, T. Journal of
Physics B 36, 1885, (2003).
Turukhin, A.V., Sudarshanam, V.S., Shahriar, M.S., Musser, J.A., Ham, B.S. and
Hemmer, P.R. Physical Review A 88, 023602, (2002).
31
Appendix A
Determining the Temperature and the Flux Density
of a Glowing Object
In this experiment we determine the temperature and the flux density of a glowing
object by measuring the wavelength at which the maximum radiancy occurs. The
dependence of the temperature of the glowing object on the wavelength at the maximum
radiancy is predicted by Wien’s displacement law (equation 3.2). For this experiment, it
is convenient to express the Wien’s constant from equation (3.2) in units of nanometers:
2.898 × 10 6 n m K . The temperature derived from the Wien’s displacement law applied
to the experimental data given in Figure 3.1 helps to find the intensity of the light emitted
by the glowing object (or its luminosity) using the Stefan-Boltzmann law given in
equation (3.3). Temperature and luminosity form the basis for astronomical observations.
The goal of this experiment is to determine accurate experimental values for the
Wien’s constant (of theoretical value 2.898 × 10 6 n m ⋅ K ) and for the Stefan-Boltzmann’s
constant [ 5.67 × 10 −8 W (m 2 ⋅ K 4 ) ] in order to precisely measure the temperature and
the flux density of various glowing objects, including stars.
A.1 Details about the Experiment
A.1.1 Overview
The equipment and the experimental procedure are briefly described in section
3.4. Here we present more details about the setup and the experimental technique. In this
experiment a PASCO blackbody light source will be used. The radiation emitted by the
glowing object will pass through a set of collimating slits, which help to reduce the
background light. A collimating lens will then focus the radiation (light) onto a dispersive
prism.
Because glowing objects emit radiation at various wavelengths, it is necessary to
use a dispersive prism in order to separate them. Dispersion occurs because the index of
refraction of a dielectric material depends on the frequency of the light passing through it.
Each frequency travels a different geometric path through the same dielectric. This will
spread out the radiation. Consequently, when the light leaves the prism each component
of light will leave at a slightly different angle.
Next, the radiation will be focused through an aperture bracket by a focusing lens
(see figure 3.3) and onto an infrared light sensor (or broad spectrum sensor) which will
detect the radiancy (which is the energy per wavelength) of the light. The light sensor, we
are using, can detect small variations in the energy of light. For our experiment, it is best
32
to use a high-sensitivity infrared sensor (IR). The choice of the sensor is related to the
range of temperatures we need to explore (2000-3000 K). Both the IR light sensor and the
focusing lens are mounted to a light sensor arm which is attached to a degree plate (see
figure 3.3). The degree plate will rotate as the light sensor is rotated. A rotary motion
sensor is attached to the degree plate in order to record the angle at which the degree
plate is positioned. The wavelength will ultimately be determined from this angular
position.
The light sensor arm has a stop attached underneath. This ensures that the light
sensor begins collecting the experimental data from the same starting position every time.
This stop is crucial for calibrating the setup and acquiring data.
Scanning the spectrum of the radiation dispersed by the prism through the rotation
of the sensor makes possible to precisely locate the angle at which the maximum radiancy
occurs. The relationship between the angular position and the wavelength will be
developed in the next section A.1.2.
A.1.2 Relationship between
the Angular Position of the Prism and
the Wavelength of the Radiation Dispersed by the Prism
The blackbody radiation is incident on a 60 degree triangular prism (see figure A.1).
Figure A.1 The
path of the light
through the prism.
By using Snell’s law given in equation (4.7.b) at each face of the prism, it is possible to
get the following expression for the index of refraction
1
 2
n= 
sin θ + 
2
 3
2
+
3
4
(A.1)
33
The Cauchy equation gives a relationship between the index of refraction and λ:
n(λ ) =
A
λ2
+B
(A.2)
The coefficients A and B depend on the type of glass being used. For an acrylic prism the
coefficients are A=13,900 and B=1.689
Solving equation (A.2) for wavelength we get
λ2 =
A
n−B
(A.3)
We arrive at an expression for wavelength by plugging equation (A.1) into equation (A.3)
and solving for wavelength
λ=
13900
1
 2
sin θ + 

2
 3
2
(A.4)
3
+
− 1.689
4
Equation (A.4) will be used to compute the wavelength from the angle of
refraction on the prism. There is a problem with this formula for the range of wavelengths
we are interested in. As one can see from figure A.2, there is a very strong dependence of
λ with respect to θ. Very small changes in the angle of few hundredths of a radian
correspond to changes of hundreds of nanometers in the wavelength!
Figure A.2.a Experimental values of
the wavelength versus the angle θ.
The region between the dashed lines
indicates the range of interest for the
present setup (1000-1500nm).
Figure A.2.b An enlarged view of
the entire range of wavelengths used
in this experiment.
34
In fact the entire range of wavelengths (1000-1500) of interest for the study of the
present blackbody object corresponds to a change in θ of only 0.02 radians, or 1.15
degrees, on the rotary table!! The use of formula (A.4) demands an extremely precise
calibration and alignment of the setup.
A.1.3 Initial Angle and Angular Ratio
Some special considerations need to be made related to formula (A.4). The
argument of sinθ function from the theory is not the same with the angle measured by
the rotary motion sensor (see figures A.1 and A.3). There are two measurements needed
to successfully relate these two angles: the first measurement gives the value “angle
ratio”: when the degree plate is rotated, a small pinion attached to the rotary motion
sensor also rotates. Through this pinion the rotary motion sensor measures the angular
position of the light sensor attached to the degree plate. A degree of the pinion is much
smaller than a degree of the plate. It is therefore necessary to find the ratio between these
two angles
pin angle
Ratio =
(A.5)
plate angle
A procedure for determining the value for the angle ratio is outlined in section A.4.1.
Light
Sensor
θ
Angle
Stop
Initial
Angle
Initial
position
Prism
Incident
Light
Figure A.3 Diagram of
the angles used during the
experiment. Note that all
angles are measured with
respect to the stop.
35
From figure A.1 we see that the angle needed in equation (A.4) is measured from
the optical axis or the normal to the back side of the prism (where the light exits). From
the following diagram we see that “Angle” is the angular position measured by the rotary
motion sensor, “Initial angle” (or Initial for the DataStudio software) is the angle
between the initial position (the stop) and the light sensor, and θ is the angle between the
optical axis and the light sensor. We see that θ = (Initial Angle – Angle). But this
relationship would only hold if all the angles were measured with respect to the degree
plate. Instead, they are measured by the rotary motion sensor and therefore, the division
by the angular Ratio (from equation A.5) is necessary:
θ=
Initial Angle − Angle
Ratio
(A.6)
By substitution of equation (A.6) into equation (A.4) yields to the equation:
13900
λ=
 2
1
 Initial Angle − Angle 

sin
 + 
Ratio
2

 3 
(A.7)
2
+
3
− 1.689
4
The Initial Angle is a critical piece of information needed for this experiment.
As our study proved, the success of the entire experiment depends on the accuracy of the
initial angle. It is nearly impossible to perfectly measure the initial angle because the
theory assumes that the light is perfectly incident on the apex of the triangular prism,
which is not true because of the imprecision in manufacturing the prism-mount system. In
order to have a perfect measurement of the Initial Angle, we would need a precision that
should reach a hundredth of a radian for the rotary motion sensor. These difficulties were
overcome by using an ingenious calibration technique which will be discussed in detail in
a later section of this Appendix. The calibration technique uses two methods to determine
the Initial Angle (which are outlined in the sections A.3.2 and A.3.3). The first method (in
Section A.3.2) is an approximation and it will be used as a starting reference to determine
the proper initial angle using the 2nd method. The second method (in Section A.3.3)
involves using three known values: λmax , Anglemax, and Ratio to find the value for Initial
Angle. It is necessary to perform both procedures in order to get an accurate value for
Initial Angle and successfully calibrate the setup. This calibration will be done for one
particular voltage (7 volts) which will not be used later in data analysis. This voltage was
chosen because it is in the middle of the 4-10 volts range we wish to measure.
A.1.4 Verifying the Temperature of the Glowing Object
In order to verify the temperature predicted by Wien’s displacement law, we
measure the current and voltage across the filament (blackbody) with meters. The
resistance of the filament depends on temperature in the following way
R = R0 [1 + α (T − T0 )]
(A.8)
36
where R is the resistance and α is the temperature coefficient of resistivity. For this light
bulb α = 4.5 x 10 −3 K −1 . The subscript “0” labels the parameters at room temperature,
while parameters without subscript characterize the filament at a temperature which
corresponds to a certain voltage across. Solving equation (A.8) for final temperature
yields to
R
T = T0 +
R0
−1
(A.9)
α0
The resistance of the bulb at room temperature is R0 = 0.84 Ω . The resistance, R , can
be found using Ohm’s law ( V = I R ):
R=
V
I
(A.10)
Both the voltage and current are measured using external meters. Finally, the expression
for temperature is:
T = 300 K +
V
−1
0.84 I
4.5 × 10 −3 K −1
(A.11)
The temperature (A.11) is compared to the temperature predicted by Wien displacement
law from the λmax of the experimental curves.
A.1.6 Finding the Flux Density of a Glowing Object
Once a theoretical value for the temperature has been calculated, it will be
possible to verify the Stefan-Boltzmann law given in equation (3.3). In general, the flux
density is simply the integral of the radiancy, equation (3.1), over the range of
wavelengths or the area under the curve of radiancy. Unfortunately, due to the difficulty
in manipulating the formula (A.7) for wavelengths (as discussed in section A.1.2 in
relation to figure A.2) our experimental curves of radiancy are only accurate near the
peak. In consequence, we cannot simply take the area underneath our experimental
curves of radiancy (shown in figure 3.2) to find the flux density. Therefore, we will use
the temperature measured by the meters for each specific run and next, plot the
theoretical radiancy versus our experimental wavelengths (as in figure 3.1). Then the flux
density can be determined by taking the area under this curve. A detailed description of
this method is provided in section A.5.3.
In summary, in this experiment the main steps to follow are:
1) Determine the wavelength at the maximum radiancy for a specific temperature.
2) Find the temperature of the object by using Wien displacement law.
3) Derive an experimental value for the Wien’s constant, using a linear regression of
λmax versus T for various settings.
37
4) Find the flux density of the object, for each temperature of a glowing object.
5) Using a linear regression, derive an experimental value for the Stefan-Boltzmann
constant.
A.2 Technical Aspects about Setting up the Equipment
Note: The right and left side of the setup will be referenced as if one is looking in the
direction of the traveling light (from left to right in figure 3.3).
A.2.1. Optics Bench Setup
a. Mount the optics table to the far end of the optics bench, opposite of where the
light source will be. The angle indicator should be on the left side of the bench.
b. Mount the rotary motion sensor to the right side of the optics table. It should be
opposite of the angle indicator. The rotary motion sensor should also be mounted
such that the small radius of the pinion is against the degree plate.
c. Rotate the degree plate so that the 0 degree mark is lined up with the angle
indicator.
d. Attach the light sensor arm to the degree plate.
e. Attach the aperture bracket to the light sensor arm.
f. Place the infrared filter in front of the infrared light sensor and attach both to the
far end of the light sensor arm. This experiment requires an infrared filter.
g. Attach the stop underneath the light sensor arm so that the smooth corner will be
against the angle indicator.
h. Attach the prism mount to the center of the degree plate from underneath using a
wing-nut. Be sure to use a washer in between them because the prism has to
remain stationary while the degree plate is rotated. Take special care to align the
prism such that the back face is perpendicular to the optical axis.
i. Place the Collimating lens on the optics bench (the lens should be close to the
degree plate but not touching it, about 1-1.5 cm).
j. Place the collimating slits on the optics bench 10 cm from the collimating lens
(the slits should be facing the lens).
k. Put the blackbody light source as close as possible to the collimating slits. The
source will actually fit inside the mount for the collimating slits.
38
l. Place the focusing lens on the light sensor arm between the prism mount and the
light sensor. The back of the lens should touch the screw that attaches the light
sensor arm to the degree plate.
A.2.2. Electrical Setup
a. Attach wires that go from the signal generator to the blackbody light source.
b. Wire an ammeter in series to measure the current.
c. Wire a voltmeter in parallel to measure the voltage.
d. Connect the light sensor into channel A of the PASCO Science Interface.
e. Connect the signal generator from a power amplifier into channel B.
f. Connect the rotary motion sensor into channel 1 and 2 (yellow in 1 and black in
2).
A.2.3. DataStudio Sensor Setup
a. Open the DataStudio software.
b. Under setup upload the appropriate sensor converters in order to have the correct
input as described above.
c. Set the rotary motion sensor for measuring the angular position (RAD) with a
sampling rate of 10 Hz
d. Set the infrared light sensor to a sensitivity of 10x with a sampling rate of 20 Hz.
e. In the DataStudio’s calculator, enter formula (A.6) for theta:
theta = (initial-angle)/ratio
where the parameters “initial” and “ratio” will be measured and entered in
manually and “angle” will be the angular position read by the rotary motion
sensor.
f. In the DataStudio’s calculator, enter formula (A.7) for wavelength:
wavelength= filter(0,8000,(13900/(((1.1547*sin(theta)+0.5)^2+0.75)^0.51.689))^0.5)
(A.12)
39
where “theta” is the angular position read by the rotary motion sensor according
to equation (A.6).
A.3 Calibration
It is very important to be as precise as possible when determining the angular
“Ratio” and the “Initial angle”. If these values are off by only 1 degree, it can skew the
value for wavelength by as much as several hundreds of nanometers!!
A.3.1. Determining the Angular ”Ratio”
In the theoretical model the angle at which the light is dispersed by the prism is
measured with respect to the degree plate. However, the angle is measure by DataStudio
with respect to the pinion on the rotary motion sensor. In order to correlate the degree
plate to the rotary motion sensor, one needs to determine the ratio in between the two.
This can be done in one of two ways: (1) either by measuring the number of degrees the
pin rotates in a full rotation of the degree plate or (2) by comparing the angular
displacements between the pinion and degree plate then performing a statistical analysis.
The latter method is more accurate because the sensor will be less likely so “slip” and
give an inaccurate reading. A description of this method follows.
a. Open DataStudio and an Excel Worksheet (or any spreadsheet program that can
perform a linear regression).
b. In Excel, set up the following columns:
Degree Plate Angle (DEG) – this is the angle that will be recorded from the actual
degree plate;
Pin Angle (RAD) - the angle measured by the rotary motion sensor;
Pin Angle (DEG) - convert Pin Angle from radians to degrees.
(See figure A.4 )
c. Under the Degree Plate column enter angle values from 0 to 55 in increments of 5
degrees.
d. Bring up a digits display for the angular position in the DataStudio software.
Note: If DataStudio is not measuring the angle in radians, then go to setup and
click on the rotary motion sensor. Under setup choose Angular Position (RAD).
e. Now, we will compare the angle measured by the degree plate to the angle
measured by the pin.
f. Press Start in DataStudio to begin collecting Data. Do not stop collecting data
until you have reached the final angle. No back-and-forth rotation of the rotary
40
arm is allowed.
g. Rotate the degree plate slowly and steadily to the 5 degree mark. It is very
important that the degree plate is rotated smoothly, otherwise the pin could stick
or the degree plate could skip.
h. Record the angular position displayed by DataStudio on the spreadsheet.
i. Repeat this data acquisition for the remaining angles.
j. In order to perform the linear regression on Excel, go to Tools, next to Data
Analysis, and next to Linear Regression.
k. Your x-values will be the degree plate angle measurements and your y-values
will be your pin angle (DEG) values.
l. Set the constant to zero
m. Designate a cell to display the result. While the analysis is performed the value
for Ratio will be the slope (in Excel it will be displayed as the coefficient for the
x-variable). This value is highlighted in figure A.4.
Figure A.4 Sample calculation of ratio, which is highlighted.
A.3.2 Approximate Determination of the Initial Angle
Finding the Initial Angle is very important for measuring accurate values of the
wavelength. Note that the value Initial Angle accounts for the sensor’s initial position but
41
since the rotary motion sensor starts from zero at the beginning of every trial, it is not the
sensor’s initial position as measured by DataStudio (see Figure A.3). In figure A.3, the
dotted line indicates the initial position of the light sensor arm when resting against the
stop. The stop ensures that every run will have the same initial position. “Angle” will be
the angle with which the light sensor arm is displaced from its initial position (this is the
angle measured by DataStudio). Initial Angle is the angle in between the initial position
and the optical axis and theta will be the difference between these two angles (see
formula A.6).
First, we will approximate the initial angle using an experimental procedure and
examining the radiancy versus angular position graph.
a. In DataStudio, create a graph of the light intensity versus the angular position.
b. Turn on the blackbody source (the light bulb) using the signal generator dialog
box. The voltage across the light bulb is set up by the power amplifier.
c. Position the light sensor arm at the initial position. Now, the stop should be
touching the angle indicator.
d. Click on the Start button in order to begin collecting data.
e. Begin rotating the light sensor arm slowly and steadily in order to avoid sticking
or slipping.
f. Rotate the light sensor until it passes through the light that actually passes under
the prism. The angle indicator will read approximately zero.
g. Stop collecting data.
h. Use the Smart Tool to determine the angle at which the maximum is reached. Do
not find the angle at which the maximum is caused by the dispersed light (will be
around 14 radians). This is the peak of the radiancy and gives us no information
about the optical axis. Instead, find the angle where the maximum is caused by the
light passing under the prism (approximately 70 radians). See Figure A.3.
42
Figure A.5 The approximate determination of initial angle.
From the graph we see that initial angle is 73.356 rad
A.3.3. Precisely Determining the Initial Angle
Theoretically, the method detailed above should give the Initial Angle precisely,
but in practice this value does not consistently reproduce the expected wavelengths. This
is mainly because the wavelength’s formula (A.4) has a strong dependence on θ in the
region of experimental interest (as shown in figures A.2 and A.3).
To remedy this problem an additional calibration step was implemented to zero
the wavelength for a certain value. This was done by scanning the blackbody spectrum at
7 volts (approximately 2500 K). Using the temperature determined from the meter’s
reading, we determined what λmax should be. Next we record the angular position at
which the peak for 7 volts occurs. This peak is independent on the wavelength’s formula
(A.4) and therefore, it can be used as a constant along with the angle Ratio and the
predicted wavelength to determine the proper value for Initial Angle. Next, we adjust the
value of the Initial Angle until the value of wavelength at this angular position agrees
with our predicted value of within 1%.
Steps to follow:
a. In DataStudio set the voltage to 7 in the signal generator window.
b. Record the value of your supplied voltage on Excel worksheet (entitle it “Initial”).
c. Rotate the light sensor arm such that the stop is against the angle indicator. This is
your initial position.
d. Press the Start button and collect data.
43
e. Record the current and the voltage displayed by the meters on Excel worksheet
f. Calculate the temperature from these readings.
g. Tare the light sensor.
h. Rotate the light sensor arm clockwise slowly and steadily. It will take a few runs
before you get the feeling of the speed you need to go. Use the graph Light
intensity versus wavelength as a guide.
i. After the spectrum has been scanned stop collecting the data.
j. In DataStudio use the Smart Tool on the Light intensity versus Angular Position
graph and find the angle at which the maximum radiancy occurs, 14.735 from
figure A.6. Record this value in the worksheet titled “Init” under the column
“Alpha @ max Radiancy.”
Figure A.6 Light intensity
versus angular position. The
maximum indicates the value
for angle max, 14.735.
k. Calculate the predicted value of the wavelength based on the temperature
determined from the meters, 923.6 nm. Record it in the column “λ - from T” and
here
l.
On the Excel worksheet introduce values for “Initial Angle” until the wavelength given
by the wavelength formula (A.4) matches the predicted value. In this case initial is
74.484.
44
Figure A.5
The adjustment
of the initial
angle value in
order to get the
predicted
wavelength at a
specified
m. Check this value by plugging it into the calculator in the DataStudio software and
then use the Smart Tool on the graph of the light intensity versus wavelength to
see if the peak in intensity occurs at the predicted wavelength. This should be
accurate to within 10 nanometers.
A.4 Data Acquisition
A.4.1 Measurements of the Temperature
a. In the DataStudio software, set the voltage to 4 in the signal generator window.
b. Record the value of the supplied voltage on the Excel worksheet “Data-Wien
Constant.” An example is shown in figure A.7.
c. Rotate the light sensor arm such as the stop to be against the angle indicator. This
is your initial position.
d. Start collecting data.
e. Record the current and voltage displayed by the meters on the Excel worksheet
f. Tare the light sensor.
g. Rotate the light sensor arm clockwise slowly and steadily. It will take a few runs
before you get a feeling of the speed with which you need to go. Use the graph
Light intensity versus wavelength as a guide.
45
h. After the radiation has been scanned, stop collecting data.
i. Use the Smart Tool in the DataStudio software and determine the wavelength at
which the maximum intensity occurs.
j. Record this wavelength in Excel under “λ – from DS”.
k. Calculate the temperature from the meter readings; record it on Excel
l. Calculate the temperature from the experimental wavelength and record it as an
experimental temperature.
m. Do the error analysis comparing the experimental temperature to the temperature
provided by the meters. The error will be larger for the smallest temperatures.
n. Repeat these steps for supplied voltages of 6, 8, and 10
Figure A.7 The experimental data from the scans of blackbody object. The percent
errors between the temperature measured and the temperature predicted by our λ max .
The highlighted run indicates the calibration.
A.4.2 Measuring the Wien’s Constant
It is possible to determine an experimental value of the Wien’s constant in
formula (3.2) by performing a linear regression procedure. We need to have written the
Wien’s displacement law in the linear form: y = mx + b , where m is the slope. The Wien’s
displacement law can be written for wavelength as
λ=m
1
T
(A.13)
where m is the Wien’s constant. In order to determine the Wien’s constant, m, we need to
have the wavelength plotted as a function of the inverse temperature. For that we follow
the steps:
a. On the Excel worksheet “Data-Wien Constant” determine the inverse temperature
for each measurement, also displayed in figure A.8.
46
b. From the tools menu select data analysis, and then select the linear regression.
c. The y-value will be for the four values of the wavelength and the x-value will be
the four corresponding values of the inverse temperature. Run the linear
regression (by selecting the appropriate cell in order to display the regression) and
set the constant to zero.
d. Perform the linear regression.
e. The value for m will be displayed as the Coefficient for the X Variable, .
f. Record the value and its standard error. 2.8 ± .2 × 10 6 nm K
g. Determine the percent error between this and the accepted value of the Wien’s
constant of 2.898 × 10 6 n m K , in this case 3.38 %
h. Insert a chart which plots the wavelength versus the inverse temperature.
Figure A.8 The spreadsheet used for
determination of Wien’s constant. The
value of 2810983 shows the experimental
value of the constant with an error of 3.38
% with the accepted value of 2898000.
A.4.3 Flux Density or Luminosity of a Glowing Object
In order to calculate the flux density, it is necessary to take the area under the
curve of radiancy, figure (3.1), or to use the Stefan-Bolztmann law, equation (3.3). A
problem arises when we use our direct experimental curves from figure (3.2). These
curves are only accurate in the region near the peak (see figure 3.5) because at the larger
and smaller values of the radiancy the curves don’t correspond to accurate values of
wavelengths. This is due to formula (A.4) which is very sensitive to the angle θ , and
therefore, it becomes impossible to measure accurately. The formula only allows for a
range of 345-7400 nm. The range between 2000 nm and 7400 nm corresponds to a
change of only 0.3 degrees on the degree plate, while the range between 300 nm and
47
2000 nm corresponds to a change of 14 degrees. This means that remaining 70% of the
wavelengths are covered within 2% of the angles measured. The wavelengths’ dramatic
dependence on θ will lead the light sensor to measure the same radiancy for all
wavelengths in this range. This causes the curves to accurate only near the peak of the
radiancy.
In order to fix this problem we derive an experimental Stefan-Boltmann constant
which includes all the limitations in the measurement of the radiancy. Using Planck’s
formula (3.1) we plot R (λ , T ) at various fixed T and fit the experimental curve for the
same T.
a. In the DataStudio software, we enter the Planck’s formula (3.1) into the calculator
setting x to be the data measurement wavelength
Radiancy = (2*PI*6.626E-34*300000000^2)/((x*10^-9)^5)*
(1/(EXP(6.626E-34*300000000/1.381E-23/temp/(x*10^-9))-1))
(A.14)
b. The value temp in the formula will be designated as a constant, which will be
changed for every run. For instance, if the temperature is 2464 Kelvin, the
formula will be
c. Make a graph of the planck radiancy versus wavelength
d. Select the same 4 runs that were selected to find the Wien’s constant.
e. From the previous spreadsheet identity the temperature for a run (use run 8 for
example)
f. Enter this temperature into the calculator.
g. Take the area under the curve using the Selected Statistics option as seen in figure
A. 9 below.
Figure A.9 Calulation of the area under an accurate curve of
radiancy, yielding an area of 1.9622 × 1015 .
48
h. Record this area on an Excel worksheet, “Flux Density” under the column, “Area
from under curve” As shown in figure A. 10 below
Figure A.10 Data collected from taking the area under the
adjusted radiancy curves.
i. Repeat the procedure for the remaining runs.
A.4.4 Determining the Stefan-Boltzmann Constant
This procedure is similar to that used in finding the Wien’s constant. The only
difference is that we need to account for the way in which the DataStudio program
calculates the area. To understand this aspect, we need to look at the units of Planck’s
formula and the units of radiancy.
W 
F = σ T 4 in units of  2 
 m 
 c  8π
R(λ ) =   4
 4  λ

1
  hc 
   hc λkT

− 1
  λ  e
 m ⋅ J ⋅ s ⋅ m   m2 ⋅ J 
s=
s = W 
in units of  s
5
5

  m
  m 3 
m

 

As one can see there is an additional m (meter) in the denominator. In order to get the
flux density, the radiancy should be integrated over the range of wavelengths
F = σ T 4 = ∫ R(λ ) dλ
(A.15)
Therefore, the additional meter at the denominator is cancelled by dλ . The DataStudio
program does not run the actual integral, but it performs a summation (Riemann Series)
over each nanometer in the range.
F = ∑ R(λ ) ∆λ
(A.16)
It is necessary to multiply the area under the curve by ∆λ = 10 −9 m for consistency in
units. Perform the regression by following the procedure used to determine the Wien’s
constant in section A.4.2 and make the appropriate substitutions.
49
Figure A.8 The spreadsheet used for analysis of the flux density. The highlighted
value of 5.54329E-08 shows the experimental value of the Stefan-Boltzmann constant
which has an error of 2.34 % with the accepted value of 5.6705E-08
50
Appendix B
Polarization by Transmission and Malus’ Law
B.1 General Considerations for Light-Polarizer Interaction
Measuring polarization by transmission requires the setup from figure 4.1. The
measurable quantity is the energy transmitted through a polarizer or a system of
polarizers. This energy (or Poynting vector) is typically called “intensity of a light beam”
and is proportional with the square of the electric field component of light. In this
experiment, the reference of a polarizer is its transmission axis. This axis can be rotated
conveniently. The linear polarizer is a device with all of its molecules lined up in fixed
rows. This preferential alignment of molecules represents the transmission axis. The
electric field component of an incident light beam causes an oscillatory motion of the
molecules in a direction perpendicular to the incoming light beam. Since these molecules
oscillate resonantly (at the same frequency) as the incident electric field, they will reradiate light having the same frequency and direction of propagation as the incident light.
This forward propagation was discussed in section 2.2. Because, the polarizer’s
molecules are quasi-fixed, the light can only be re-radiated perpendicularly to the row
of the molecules.
In the schematic shown in figure 4.1 the incident light is natural (or randomly
polarized) light, which means that there is no preferred direction of polarization, and
therefore, it can be thought of as randomly oriented in all directions. When the light is
incident on a polarizer, only the component along the direction of the molecules will
induce oscillations and therefore, the molecules can re-radiate energy. The light which
passes through a polarizer can oscillate in one direction only, and therefore, the light is
considered linearly polarized.
Malus’ law can be observed by implementing a second polarizer (called
“analyzer” in figure 4.1). One might ask “If this light is already polarized how it can
cause any molecules to oscillate except those that are already in its direction of
polarization?” The answer lies in the fact that the polarization of a light wave is a vector
meaning that any net vector can be broken down into two mutually orthogonal
components. This means that although the whole beam can not cause the polarizer’s
molecules to oscillate, the component parallel with the alignment of the molecules
actually can do it. We will simply vary this component by rotating conveniently the
transmission axis of the analyzer. In this way, we can see that the intensity of a polarized
light beam depends on the angle between the initially polarized light and the transmission
axis. The fact that the intensity of the light is less than the incident light is intuitive, and it
can be understood from noting that the components of any vector always have a
magnitude less than the magnitude of the resultant. The limit case is when the electric
field is lined up with the transmission axis ( θ = 0 ), and therefore all the light eventually
passes through the polarizer.
51
B.2 Experimental Technique
We use PASCO equipment composed by two linear polarizers, a diode laser, a
high sensitivity light sensor, an aperture bracket, an optics bench, a rotary motion sensor,
a PASCO interface, and the DataStudio software with the appropriate settings.
The light emitted from a diode laser passes through the first linear polarizer,
which linearly polarizes light. Next, the light passes through a 2nd polarizer (called
“analyzer”). The 2nd polarizer is attached to a rotary motion sensor via a rubber band. The
rotary sensor allows us to determine the angle between the transmission axis of the two
polarizers. In order to measure the intensity of light, it passes through an aperture bracket
and goes into a high sensitivity light sensor. In order to plot the light intensity versus the
incident angle, the DataStudio software is used. From this graph a cosine-square
dependence is observed (see figure 4.5). A comparison with theory is now possible.
B.3 Setting up the Experiment
1. Attach a diode laser to the optics table.
2. Place a lens holder with two polarizers (one on each side of it) about 10 cm
from the diode laser.
3. Attach a rotary motion sensor to the optics table in such a way that it faces the
analyzer.
4. Connect a rubber band to the polarizer and the rotary motion sensor. It should
be tight enough so that the rotary motion sensor rotates as the polarizer is
manually rotated.
5. Place a high sensitivity light sensor in front of the analyzer.
B.4 Setting up the DataStudio Software
1. Connect a high sensitivity light sensor to channel A of the PASCO Science
interface.
2. Connect a rotary motion sensor into channel 1 and 2 (plug the yellow jack in 1
and the black jack in 2).
3. Open the DataStudio software.
52
4. Calibrate the two sensors: (a) set the gain of the light sensor to Med(10x), and
the sampling rate to 5 Hz and (b) set the sampling rate for the rotary motion
sensor to 5 Hz.
5. Set the rotary motion sensor to measure the angular position in degrees.
6. Create a graph for the “Intensity” versus “Angular position”.
7. Bring up a digits display for both the intensity and the angular position.
B.5 Calibration
We need to align the laser beam with the light sensor and the two polarizers.
1. Align the laser by using the horizontal and vertical controls located on the
back of it. Use the digits display in DataStudio to determine when you have
reached a maximum of intensity. At this position the laser should be aligned.
2. Align both polarizers to 0 degrees between their transmission axis and thus,
keeping their axis parallel.
B.6 Procedure
The procedure will be performed over a single data run. Once you have started
collecting data do not stop (or go back) until the experiment is finished. We are going
to record the intensity of the light at various angles and then compare these values with
the theoretical curve based on equation (4.5).
1. Make sure the polarizers are lined up correctly.
2. Start the data acquisition.
3. Record the value of the intensity from the digits display at θ = 0 . This is I(0)
in equation (4.3), and represents the maximum intensity.
4. Begin rotating the polarizer. If the values of the angular position are negative
then stop the data acquisition and either rotate in the opposite direction or
switch the yellow and black jacks that connect the rotary motion sensor to the
PASCO Science interface.
5. Using the digits display for angular position as a guide, we rotate the polarizer
in steps of 10 degrees and record the intensity. Continue this measurement
until you reach 180 degrees.
53
B.7 Data Analysis
Use an Excel spreadsheet to analyze the experimental data. It is necessary to
create 5 columns: (1) angle recorded by DataStudio, (2) the angle converted into radians,
(3) the measured intensity, (4) the theoretical intensity, and (5) the percent error between
the intensities. The worksheet should look similar to the one below. Don’t forget that I(0)
is necessary to calculate the theoretical intensity. When finished, plot both the theoretical
and the experimental intensity versus the angle between the transmission axis of the two
polarizers.
Figure B.1 The
experimental data
acquired from the
Malus’ law
experiment.
54
Appendix C
Derivation of the Fresnel Equations,
Law of Reflection and Law of Refraction
Continuing from the Maxwell equations (4.4) and the boundary conditions (4.5)
discussed in section 4.3, lets suppose that we have a monochromatic plane wave traveling
in an arbitrary r _ direction. Its electric and magnetic components are:
1 ˆ
~
~
~
~
E I (r, t ) = E 0 I e i (k I • r − ω t ) , B I (r, t ) =
k I × EI
v1
(
)
(C.1)
The wave is initially in a medium with an index of refraction nI and is incident on a
dielectric surface with an index of refraction n T . This reflected wave has the components
1 ˆ
~
~
~
~
E R (r, t ) = E 0 R e i (k R • r − ω t ) , B R (r, t ) =
k R × ER ,
v1
(
)
(C.2)
and the transmitted wave is
1 ˆ
~
~
~
~
ET (r, t ) = E 0T e i (k T • r − ω t ) , B T (r, t ) =
k T × ET .
v2
(
)
(C.3)
~
Where k is the propagation vector whose magnitude is the wave number, k. E0 I ,
~
~
E 0 R , and E 0T are the complex amplitudes of the incident, reflected, and transmitted
electric fields of the waves. v, t, and ω represent velocity, time, and angular frequency
respectively
Figure C.1 A wave
traveling in the positive
r _ direction incident on
the interface between
two media ( n I < n T )
.
55
Regardless of what medium the wave is traveling through, its frequency will
always remain the same. Therefore, the frequency of the reflected, incident, and
transmitted wave will all be the same angular frequency ( ω R = ω I = ω T = ω ). The
wave number and the velocity will change according to ω = k v . Therefore, both the
reflected and incident waves have the same velocity ( v R = v I ) in the same medium.
k I v I = k R v I = kT vT = ω ,
(C.4)
Or equivalently,
kI = kR =
vT
vI
kT =
nI
kT
nT
(C.5)
Lets suppose the wave is traveling in the positive r _ direction and is incident on
an interface lying in the x, y plane at z = 0 (see Figure C.1). Now, by looking at the
equations of the waves (C1-C3), we see that the only space or time dependence is in the
exponential term. Since the boundary conditions must hold at all points on the interface
(all values for x and y at z = 0) for all times, all the exponential terms must be equal.
(k I
• r − ω t ) = (k R • r − ω t ) = (k T • r − ω t )
(C.6)
Since all the frequencies equal, this will only occur if
k I • r = k R • r = kT • r
(at z = 0)
(C.7)
In general the dot product will be k • r = k x x + k y y + k z z . So equation (C.7) can
only be true if all the components of the wave vectors are equal
(k I ) y
= (k R ) y = (kT ) y
when x = 0
(C.8)
(k I )x
= (k R )x = (kT )x
when y = 0
(C.9)
This means that if our incident beam is in a particular plane, the reflected and
transmitted beams will also be in that same plane. This plane is known as the plane of
incidence and is of particular importance in describing the interaction of light with a
dielectric at an interface.
Since all the components are equal the magnitude, the dot products give
k I r cos θ I = k R r cos θ R = kT r cos θ T
(C.10)
where θ I , θ R , and θ T are the angles between the vectors k I , k R , and k T of
propagation and r , respectively. This relationship infers that
k I sin θ I = k R sin θ R = kT sin θ T .
(C.11)
Note that k I = k R from equation (C.4). By substituting this into (C.11) we have
sin θ I = sin θ R and therefore
θI = θR .
(C.12)
56
This is the law of reflection and clearly states that angle of reflection is always equal to
the angle of incidence.
We can also prove the law of refraction by combining k I =
nI
kT from
nT
equation (C.5) and k I sin θ I = kT sin θT from equation (C.10):
nI
kT sin θ I = kT sin θT ,
nT
(C.13)
or equivalently,
nI sin θ I = n T sin θT
(C.14)
When we apply the boundary conditions to the waves (C1-C3), all the exponential
terms cancel out and we get the following equations:
~
~
~
ε I E 0 I + E 0 R z = ε T E 0T z
(C.15.a)
(
(B~ 0 I
(E~ 0 I
)
µ1
)
~
+ B0R
) z = ( B~ 0T ) z
~
~
E 0 R ) x, y = (E 0T ) x, y
(C.15.b)
+
(C.15.c)
1 ~
~
B0I + B0R
(
(
) x, y = µ1 ( B~ 0T ) x, y
(C.15.d)
2
Figure C.2 Shows a
wave polarized parallel
to the plane of incidence
incident on the interface
between two media
( n I < n T ).
Suppose that we have a wave that is polarized in the x_direction parallel to the plane of
incidence, from applying the boundary conditions we get
~
~
~
ε I − E0 I sin θ I + E0 R sin θ R = ε T − E0T sin θ T
(C.16.a)
(
)
(
)
57
0 = 0 (since the magnetic fields have no z components)
~
~
~
E0 I cos θ I + E0 R cos θ R = E0T cos θT
1
µI
1 ~
1 ~ 
1 ~
 E0 I −
E0 R  =
E0T
vI
µT vT
 vI

(since B =
(C.16.b)
(C.16.c)
1
E)
v
(C.16.d)
~
~
In equation (C.16.a), the z component of E0 I and E0T must have a negative sign
in front, because they are directed in the negative z_direction; similarly, for the reflected
term in equation (C.16.d). The angles can be shown using geometry. By re-arranging
terms and noting from the laws of reflection and refraction that sin θ I = sin θ R and
nI
sin θ I = sin θ T , equation (C.16.a) becomes
nT
~
~
~ n ε
E0 I sin θ I − E0 R sin θ I = E0T I T sin θ I
nT ε I
c
1
=
we can show that
εµ n
From
εI =
(C.17)
n I2
(C.18)
µI c 2
and
εT =
nT2
(C.19)
µT c 2
Substituting equations (C.18) and (C.19) into equation (C.17) yields
µI n T ~
~
~
E0 I − E0 R =
E0T
µT nI
(C.20)
Also, equation (C.16.d) leads to equation (C.20).
Equation (C.16.c) becomes:
(E~0 I
~
+ E0 R
)
~ cos θT
= E0T
cos θ I
(C.21)
We can summarize the discussion from above into the following equations:
(
~
~
)
(
~
ε I − E0 I sin θ I + E0 R sin θ R = ε T − E0T sin θ T
~
~
~
E0 I cos θ I + E0 R cos θ R = E0T cos θT
)
µI n T ~
~
~
E0 I − E0 R =
E0T
µT nI
(E~0 I
~
+ E0 R
)
~ cos θT
= E0T
cos θ I
58
1
µI
1 ~
1 ~ 
1 ~
 E0 I −
E0 R  =
E0T
vI
µT vT
 vI

µI n T ~
~
~
E0 I − E0 R =
E0T
µT nI
= µT , and therefore the equation (C.20) becomes
Typically µ I ~
nT ~
~
~
E0 I − E0 R =
E0T
nI
(C.22)
We can arrive at a value for the reflected component by noting
~
~ cos θ T
~
E 0 R = E0T
− E0I
cos θ I
(C.23)
from equation (C.21) and
n ~
~
~
E0T = I E0 I − E0 R
(C.24).
nT
from equation (C.22). Then by substituting equation (C.24) into equation (C.23)
(
)
n cos θ T
~
~
~
~
E0 R = E0 I − E0 R I
− E0 I .
nT cos θ I
(
)
(C.25)
Algebraic manipulations lead us to
 n cos θ T
 ~
 n cos θ T
~
E0 R =  I
− 1 E0 I −  I
 nT cos θ I

 nT cos θ I

n cos θ T
1 + I
nT cos θ I

~
 E0 R

~
 n cos θ T
 ~
 E0 R =  I
− 1 E0 I

 nT cos θ I

 nT cos θ I + n I cos θ T

nT cos θ I

~
 n cos θ T − nT cos θ I
 E0 R =  I
nT cos θ I


 ~
 E0 I .

Since we are only interested in the real part, we can drop the complex notation.
E0 R
 nI cos θ T − n T cos θ I
=
 nI cos θ T + n T cos θ I


 E0 I


(C.26)
This is the Fresnel equation for the reflected component of a light wave polarized
~
~
parallel to the plane of incidence. By solving for E0T instead of E0 I one would find the
parallel component of the transmitted wave.
The perpendicular component can be found similarly by applying the boundary
conditions to an incident wave polarized perpendicular to the plane of incidence.
 n I cos θ I − n T cos θ T
E0 R ⊥ = 
 n I cos θ T + n T cos θ I


 E0 I
⊥


(C.27)
59
Appendix D
Finding the Index of Refraction of a Dielectric Material
by Polarizing Light at Reflection on its Surface
D.1 Details about the Setup and the Experimental Procedure
It is possible to experimentally determine the index of refraction of a material by
measuring the intensities of the reflected light incident on its surface. For that we analyze
the parallel and the perpendicular components of light incident on the surface at different
angles. From this measurement we can determine the Brewster angle by using the Fresnel
equations as discussed in Section 4.3. The Brewster angle depends on the index of
refraction of the material as indicated by equation (4.16).
As a test sample, we will use an acrylic lens with a D shape. Although the shape
of the lens is not relevant for the present experiment, it does ensure that no light reflected
from the back surface will enter the light sensor. We will examine the intensity of a
monochromatic laser beam reflected by the acrylic lens for various angles of incidence.
Because the light is monochromatic we prefer to talk about the intensity of light rather
than radiancy. The acrylic lens will be rotated so that the intensities of two rectangular
components of the reflected laser beam can be measured. For a certain angle of incidence
(called Brewster angle) the light is completely polarized in a direction perpendicular on
the surface and the parallel component will disappear completely (the intensity of the
parallel component will be exactly zero). For more details see section 4.3.
Section 4.4 outlines the basic features of the setup. Here, we give more details.
Throughout this description, the reader should look to figure 4.4. In order to optimize the
outcome of the experiment light of a particular intensity must be used. This cannot be
controlled from the laser; instead an attenuator must be used. From section 4.2, we
understand that a simple linear polarizer can reduce the intensity of the light. Therefore,
the (unpolarized) light emitted from a diode laser passes through two linear polarizers
(see the round polarizers in figure 4.4). The first polarizer acts as an attenuator (according
to Malus’s law) leaving a linear polarized light of intensity reduced by 50%. The second
polarizer orients the direction of the electric field at 45 degrees with respect to the surface
of the acrylic prism. This second polarizer ensures an equal amount of energy on two
orthogonal axes (horizontal and vertical). Throughout the experiment, these two
polarizers will not be modified once they have been set. Next, the light passes through a
set of collimating slits which limit the background light incident on an acrylic D shaped
lens (a dielectric type of material). The lens is placed on a platform which will be used to
vary the angle of incidence. A third linear polarizer will further polarize the reflected
light by the lens’ surface. The axis of transmission of the third polarizer will be modified
for specific data acquisition. Finally, after passing through an aperture bracket the
intensity of the reflected light is measured by a high sensitivity light sensor attached to a
light sensor arm mounted on the degree plate. This light sensor is able to detect very
small variations of light intensity. The degree plate of the light sensor rotates as the light
sensor rotates. A rotary motion sensor is attached to the degree plate in order to record
60
the angle at which the degree plate is positioned. In this way, the DataStudio software
records the angle at which the light sensor is positioned. The information recorded by
both the light sensor and the rotary motion sensor are processed by the DataStudio
software and displayed on a laptop.
Throughout the experiment the lens will be rotated at angles of incidence between
20 and 85 degrees. There will be three measurements for each angle of incidence: (1) for
the total light intensity reflected by the surface (no polarizer is located in the path of the
reflected beam), (2) for the parallel component (when the polarizer’s transmission axis is
aligned horizontally) and (3) for the perpendicular component (when the polarizer’s
transmission axis is aligned vertically). All these measurements are taken in one data
run. The experimental results are shown in figure 4.6 and excellent agreement with the
theoretical curves shown in figure 4.2 is observed. We can see that the parallel
component in the plane of incidence (figure 4.3) of the laser beam vanishes when the
light is incident at a Brewster angle. For acrylic the Brewster angle is found equal with 56
degrees. From this type of measurement, precise values of the index of refraction can be
found. Obviously, each material will have a different index of refraction and therefore, a
different Brewster angle.
D.2 Setting up the PASCO Equipment
Note: Throughout the description below, I will designate the right side and the left side
of the optical bench with respect to the direction of the traveling light.
a. Attach the optics table to the far end of an optics bench (at about 10 cm
from the end).
b. Attach a rotary motion sensor to the optics table on the left side of the
optics bench. Position the degree plate such that the 180 degree mark is
lined up on the angle indicator mark.
c. Attach the light sensor arm to the degree plate.
d. Attach the aperture bracket to the light sensor arm. Set the aperture
bracket to the opening number 4 for the best intensity of the incident light.
e. Attach the high sensitivity light sensor to the light sensor arm.
f. Attach the Brewster’s angle plate to the degree plate.
g. Ground the optics table by connecting a wire from the optics table’ wing
nut. This will eliminate any excess charge which may affect the optical
response of the lens by polarizing the electric dipoles.
h. The angular mark N of the Brewster’s angle accessory should be lined up
with the 0 degree mark on the degree plate. In consequence, the 90 degree
61
mark on the Brewster’s angle accessory is lined up with the 180 degree
mark on the degree plate.
i. Place the lens table on the Brewster’s angle plate such that the mark on the
elevated side, lines up with the 90 degree mark on the Brewster’s angle
accessory. The mark on the low side should be lined up with the mark
above the letter N in the word ANGLE on the Brewster’s angle accessory.
j. Now, place the D lens on the lens platform. The curved side of the D lens
should face the rotary motion sensor.
k. Place the collimating slits on the optics bench close to the degree plate but
without touching (at about 2 cm). Set the collimating slits to the number 4
slit.
l. Place two circular polarizers on the optics bench. Both of them can be
attached to the opposite sides of an “optics lens mount.” Place the
polarizers roughly 20 cm away from the collimating slits.
m. Orient the 2nd polarizer to 45 degrees with respect to the transmission axis
of the first polarizer.
n. Attach a diode laser to the optics bench at about 10 cm from the first
polarizer.
o. The alignment of the diode laser
1. Position the degree plate to 180 degrees so that the light sensor is in the
direct path of the laser and make sure that the lens is lined up properly.
2. Turn on the diode laser and use the vertical and horizontal controls to
center the laser on the aperture bracket.
3. In order to fine tuning the alignment open up the DataStudio software
and bring up a digits display for the light intensity.
4. Set the gain on the sensor to 1 ( light sensor sensitivity should also be
set to low (1x).
5. Continue to use the horizontal control of the laser in order to attain the
maximum light intensity. If the sensor is maxing out, rotate the first
polarizer and limit the intensity of the light which reaches the sensor. A
good value would be between 80%-90%. Do not rotate the degree arm
in order to get maximum signal, but instead use the controls on the
laser.
62
D.3 Data Acquisition and Analysis
1. Setup an Excel spreadsheet
In excel set up the following columns
a. Platform angle (this will primarily be used only as a reference
and won’t be used to analyze the data).
b. DataStudio angle (this is the actual measured angle at which
the maximum intensity occurs, this angle will be used in analyzing the
data).
c. Total intensity (no square polarizer).
d. Parallel intensity (transmission axis horizontal).
e. Perpendicular intensity (transmission axis vertical).
f. Parallel reflectance (the parallel intensity divided by the total
intensity).
g. Perpendicular reflectance (the perpendicular intensity divided by the total
intensity).
2. Data Acquisition
Throughout the data acquisition, the angle of incidence will be varied by
rotating the lens’ platform on which the lens sits. For each angle of incidence
there will be three intensities of the reflected beam to measure:
(1)
the total light total intensity (in this case, no polarizer should be
placed in the path of the reflected beam),
(2)
the parallel component (for which the polarizer’s transmission
axis is aligned horizontally), and
(3)
the perpendicular component (for which the polarizer’s
transmission axis is aligned vertically).
The degree plate must be aligned properly when data acquisition starts,
otherwise the angle’s of incidence is measured incorrectly.
After the data collection starts, the light sensor will be rotated to find the
angle at which the maximum intensity occurs. This angle should approximately
match the angle on the lens’ platform. Actually, it is the angle of reflection that is
measured not the angle of incidence, but we know that these two are equal. Let’s
keep in mind that the angle of incidence and reflection are measured with respect
to the normal.
63
When the maximum intensity of each component of the reflected light
beam is measured, some special considerations must be made:
- after repositioning the polarizer it is necessary to move the light sensor
very slightly in order to find the maximum intensity of each
component;
- Do not assume that the intensity read after changing the polarizer is
indeed the maximum intensity. This may change the value for the
angle of incidence slightly, but not significantly
Once the light intensities become very small it is necessary to switch the
gain of the light sensor to a higher value. Do not forget to divide by the
appropriate factor when recording these intensities. A step by step layout of
the procedure follows.
a. In DataStudio open the calculator, and add a new function, and under
definition type angle = (180-abs(x)/15)/2. Define x to be Angular
Position, Ch 1&2.
b. Bring up two digits displays one for light intensity, and the other one for
angle.
c. Make sure the degree plate is set to 180 degrees, and that the laser and
lens are aligned properly.
d. Start the data collection. Once the data collection has started do
NOT stop it until ALL data has been collected.
e. Rotate the lens platform counterclockwise to 85 degrees
f. Rotate the light sensor arm counterclockwise to find the maximum
intensity
g. Record the angle at which this occurs (this is the actual angle of
incidence the 85 degrees is only a reference)
h. Record the intensity in the total intensity column
i. Now place the square polarizer on light sensor arm (being careful not to
move it from its present angle).
j. Record this intensity in the parallel intensity column. At this point it is
necessary to adjust the light sensor arm slightly so that a maximum
intensity is attained. The angular value might change slightly, but the
intensity is much more sensitive than the angle is. No matter how careful
you are, there will be a slight fluctuation of the light sensor arm while
changing the polarizer.
64
k. Rotate the square polarizer 90 degrees so that the transmission axis is
vertical (but be careful not to move it from its present angle!). Again
adjust the light sensor arm slightly for reading the maximum intensity of
the incident angle.
l. Record this intensity in the perpendicular intensity column.
m. continue steps e-l for the following angles
80,75,70,65,60,59,58,57,56,55,50,45,40,35,30,25,20,15
n. Note that when the intensities of the components start to get lower, it
will be necessary to raise the gain of the light sensor. In order to keep the
same order of magnitude, the intensity will need to be divided by the
appropriate factor (either 10 or 100). Since this gain switch is on the actual
sensor care must be taken not to move the light sensor arm from its present
angle.
o. Remember that the goal is to find the angle at which the parallel
component reaches a minimum intensity (preferably is exactly zero), so it
may be necessary to use increments of angles smaller than 1 degree, when
you get close to the minimum.
3. Data Analysis
a. Report the Brewster’s angle on an Excel worksheet
b. Convert this angle to radians
c. Calculate the index of refraction of the lens using formula (4.16).
d. Compare it to the accepted value for acrylic (Brewster’s Angle = 56)
e. Calculate the percent error for both the Brewster’s angle and the index
of refraction
f. Plot the Parallel Reflectance versus angle, the Perpendicular Reflectance
versus angle, and the Average Reflectance versus angle on the same graph
(see figure 4.6).
65
Figure D.1
The experimental data for
determining the index of
refraction. Our value for the
Brewster angle, 56.4, led to
an index of 1.51 which only
has a percent error of 1.52
% with the accepted valued.
66
Appendix E
Derivation of the Group Index of Refraction
The relationship between frequency and group velocity is important in understanding the
techniques for storage of light (Boyd, 2002). Consider a monochromatic traveling plane
wave of angular frequency ω propagating in the positive z_direction through a medium
of refractive index n
E ( z , t ) = A cos (k z − ω t )
(E.1)
This wave has a phase of ϕ = k z − ω t which remains constant in space (z) and time (t).
Since the phase’s wave remain constant, then its derivative with respect to time must be
zero
dφ
= k dz − ω d t = 0
dt
(E.2)
dz
ω
=
=v
dt
k
(E.3)
where v is the phase velocity. It is the velocity with which all the points on the wave of
constant phase propagate in the z_direction.
When a pulse enters a medium it is composed of several constituent waves, each
having a slightly different frequency. The components must be in-phase for all of the
distance z where they propagate, otherwise the pulse would be distorted. By noting that
c
ω
n =
and
= v we can rewrite the phase as
v
k
ω
nω
φ = kz − ωt = z − ωt =
z −ωt
(E.4)
k
c
dφ
Since the pulse is a composed of different frequencies, we must set
= 0 in order to
dω
ensure constant phase over the range of frequencies.
dφ
dn z n z
=
ω +
−t = 0
dω
dω c
c
Note that v g =
(E.5)
z
and is the group velocity of the wave, so
t
z  dn

ω + n = t

c  dω

(E.6)
67
vg =
c
dn 

ω
n +
dω 

(E.7)
If we define a group index of refraction as
ng = c
vg
,
(E.8)
we have the following relationship between the group index of refraction and the
frequency
ng = n +
dn
ω
dω
(E.9)