Download Optics-Diffraction - The Wave Nature of Light

The Wave Nature of Light: Superposition,
Interference and Diffraction
A. Tomasch July, 2009
10.1 Introduction
A wave is a traveling disturbance which transports energy from one point in space to
another. Waves are among the most important phenomena in all of physics. Mechanical
and acoustic (sound) waves propagate through an elastic medium, for example waves on
a string. Sound waves propagate through the air (another elastic medium) and allow us to
communicate via audible speech.
If you have ever watched ripples interact on the surface of a pond, say from two stones
thrown into the water, you have already observed that waves interact, often in complex
ways. What is remarkable is that the fundamental rule governing how waves combine is
simple and a wide variety of wave interactions can be readily understood and calculated
from this simple principle.
In this lab we will study electromagnetic waves in the form of visible light. Unlike sound
waves or other elastic waves, which must propagate in an elastic medium (such as air,
water, solids or strings), electromagnetic waves can propagate in the vacuum of space
itself. Much of we what we know about the world around us comes to us in the form of
electromagnetic radiation—visible light observed directly with our eyes—or other
electromagnetic waves (radio, infrared, ultraviolet, x-rays and gamma-rays) observed
with specialized instruments sensitive to the specific wavelength of the electromagnetic
radiation being observed. This is particularly true of our knowledge of the universe,
where virtually all information about the cosmos arrives in the form of electromagnetic
radiation observed with telescopes or specialized instruments located on the Earth’s
surface or aboard satellites in earth orbit. Similarly, much of what is known about atomic
structure—the arrangement of electrons within atoms and their corresponding energy
levels—is deduced from the study of the light emitted when electrons within atoms
undergo energy level transitions, an experimental technique known as spectroscopy.
By the end of this lab you should understand in detail the meaning of the terms linear
superposition and interference. You should understand how to derive the condition for a
dark fringe in the interference pattern of a single slit and the condition for a bright
maximum in the diffraction pattern produced by a transmission diffraction grating. You
should understand how to use a diffraction grating to measure the wavelength of spectral
lines emitted by a glowing gas and understand the difference between this type of
spectrum and that produced by a hot filament. You should also understand how to
interpret the diffraction pattern produced when a LASER illuminates a slit or diffraction
grating to deduce the width of the slit and the spacing of the grating.
Reading and Key Concepts
Before starting this lab, you should review the following concepts in Cutnell & Johnson
Physics:
 Basics of wave phenomena: §16.1- 4
 The Principle of Linear Superposition, constructive and destructive interference:
§17.1- 2
 Diffraction: §17.3
 The Principle of Linear Superposition: §27.1- 2
 Diffraction and diffraction gratings: §27.5-7
 You may also find §27.8-9 interesting.
Apparatus
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Laser slide projector and screen
Incandescent lamp
Single slit slide
Diffraction grating slide and mount
Hydrogen discharge tube
DNA structure slide
10.2 Preliminaries
10.2.1 The Principle of Linear Superposition, Interference and
Diffraction
When two waves arrive at a single point in space the rule for combining the effects of the
two waves together is remarkably simple and is called the Principle of Liner
Superposition, which states that the amplitude of the resulting wave is the sum of the
amplitudes of the two arriving waves at every instant of time. For electromagnetic
radiation we can characterize the amplitude of a traveling electromagnetic wave as the
magnitude of the wave’s electric field. The Principle of Linear Superposition arises
because the differential wave equation governing all wave phenomena is linear. This
means that if two waves satisfy the wave equation individually, then their sum also
satisfies the wave equation. We therefore have a simple and powerful rule for combining
waves arriving at the same place from different sources. When two waves combine the
effect is called interference. The same rule applies when more than two waves arrive in
the same place: the resulting amplitude is always the sum of the amplitudes of the
arriving individual waves at each instant of time. When many waves combine in
accordance with the Principle of Linear Superposition, the effect is called diffraction.
This distinction is largely semantic—the physical process responsible for all of these
phenomena is interference due to the linear superposition of waves from multiple
sources. The terms interference and diffraction are often used interchangeably since
there is no fundamental difference between them, nor any strict dividing line that
separates the two effects.
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Interference phenomena can be used to probe structures in two ways. Spectroscopic
observations are made by observing the interference effects produced by a diffraction
grating with a known spacing. The arrangement of atoms within matter can be deduced
by studying the diffraction pattern which arises when monochromatic radiation
(electromagnetic radiation of a single frequency and hence a single wavelength) interacts
with matter which exhibits a regular, periodic atomic or molecular structure—typically a
crystal. The accompanying photograph shows one of the most important such patterns
ever observed—the diffraction pattern produced when crystalline DNA was exposed to
monochromatic X-rays by Rosalind Franklin in 1953. It was the interpretation of this
pattern which led Watson and Crick to propose the double helical structure of DNA. The
internal arrangements of the atoms in the many “ball and stick” crystal models displayed
in most chemistry departments were similarly determined by studying the diffraction
patterns generated by crystals exposed to X-rays—the science of X-Ray crystallography.
Figure 10-1: Early X-ray diffraction pattern from B-DNA made by Rosalind Franklin in
1952.
In this lab we will learn how interference effects can be used to demonstrate the wave
nature of light, to study the energy level structure of electrons in atoms (spectral lines)
and to probe structures too small to see with the naked eye.
10.2.2 Monochromatic Light: The LASER
To demonstrate the wave nature of light and study interference effects, we need a source
of monochromatic light—light of a single wavelength. In the early 1960’s physicists
developed a new and powerful way to generate monochromatic light using electron
transitions between atomic energy levels as a light source. This technique was named
Light Amplification through the Stimulated Emission of Radiation—the LASER. Since
its invention, the LASER has become an indispensable tool for science and industry.
LASERs read the digital data from CDs and DVDs, cut industrial parts with amazing
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precision, replace metal knives in a variety of surgeries, particularly eye surgery, scan the
bar codes on your groceries and serve as pointers for public speaking presentations. The
first LASERs employed a high-voltage electrical discharge in gases such as Helium
mixed with Neon. Subsequently, solid-state LASER diodes have been developed and are
currently used for most applications. For our experiments we will use LASER diode that
emits monochromatic green light at a wavelength of 530 nm (1 nm = 10-9 meters).
Within the bright LASER light spot we can regard the light as plane waves with a 530 nm
wavelength. This is a good approximation to true plane waves (which are infinite in
spatial extent) since the laser spot size is ~10,000 wavelengths in diameter so that the
wave fronts look infinite in all directions for objects ~10 wavelengths in size illuminated
by the laser spot.
The LASERs we are using produce a total power output 10 mW (1 mW= 10-3 Watts).
While they are not dangerous, they should nonetheless be treated with respect. Never
look directly into the beam of the LASER and be careful not to accidentally shine the
beam into someone’s eye. Turn the LASER off when it is not in use.
10.2.3 Interference and the Wave Nature of Light
Previously, we have approximated the optical behavior of light by tracing light rays,
which are assumed to travel in straight lines in air or vacuum. Implicit in this view of
light is that a slit in an opaque plate should cast a single shadow with precisely defined
edges as shown in Figure 10.2 a.
Because light is a traveling wave, what actually occurs is an interference pattern of
alternating light and dark lines or “fringes” as shown in Figure 10.2 b. This effect proves
conclusively that light is in fact a traveling wave. To see why this is so, let’s study the
formation of the first dark fringe in the interference pattern generated by a narrow slit.
Figure 10.2: Interference in the image of a narrow single slit.
In Figure 10.3 an idealized “plane wave” of monochromatic light with a wavelength λ is
incident on a narrow slit of width W from a source far to the left of the slit. To deduce
the resulting interference pattern, each of the points along the face of the slit is considered
to be source of waves which travel outward from the slit to a distant screen at an angle θ
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to the slit axis. We have labeled five of the points along the face to aid in our analysis.
This method for analyzing the propagation of waves from an extended source (the slit) is
an example of Huygens’ Principle, which states that each point on a wave front serves as
a source of spherical “wavelets” which travel outward at the wave speed. At any later
instant of time the resulting wave front is tangent to all the wavelets.
We can use Huygens’ Principle to predict the formation of the first dark fringe in the
interference pattern from a single slit as shown in Figure 10.3. For a specific angle θ, the
distance traveled by waves originating from point 3 is one half of a wavelength longer
than for waves originating at point 1 so that the waves originating from points 1 and 3
arrive at the screen exactly out of phase and cancel by linear superposition.
Figure 10.3: Formation of the first dark fringe from a single slit, in accordance with
Huygens’ Principle
Similarly, a wave originating just below point 1 will destructively interfere with a wave
originating just below point 3 and so on across the entire slit, ending with waves
originating at point 3 destructively interfering with waves originating at point 5. For each
pair of points, the distances traveled by the two waves always differ by one half of a
wavelength to produce destructive interference.
Thus, destructive interference results in a dark fringe on the screen at the specific angle θ,
set by the width of the slot W and the wavelength of the incident light λ, the hypotenuse
and base respectively of the shaded right triangle shown in Figure 10.3 b. This analysis
can be repeated for the case where the difference in propagation distance between waves
originating at points 1 and 5 (the short base of the shaded right triangle) is two
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wavelengths, three wavelengths, four wavelengths and so on to produce the general
condition for a dark fringe from a single narrow slit:
sin   m

W
m  1,2,3... [10.1]
In light of this discussion we now understand that a “narrow” slit is: one that is not very
many wavelengths of light wide, say ten or fewer.
Once the angle θ has been determined from the known distance to the screen and the
distance on the screen to the chosen dark fringe of order m, equation 10.1 can be used to
determine the width of the slit if the wavelength of the incident light is known or the
wavelength of the incident light can be determined if the slit width is known. This is a
general property of interference and diffraction: a known wavelength of light can be used
to probe the size of a structure or the known dimensions of a diffracting object (in this
case the width of the slit) can be used to deduce the wavelength of incident light. We
will employ both techniques to make diffraction-based measurements.
10.2.4 The Diffraction Grating
As we have seen for a narrow single slit, interference effects are due to the wave nature
of light and cannot be explained with “geometric” optics (ray tracing). We will now
further our study of “physical” optics (based on the wave nature of light) by applying
Huygens’ Principle to deduce the properties of a transmission diffraction grating shown
in Figure 10.4.
The grating shown is like an almost-closed Venetian blind, with many long parallel slits
perpendicular to the paper which are uniformly spaced by a distance d. A real diffraction
grating would be about 10,000 times smaller than shown in the figure. Real gratings are
traditionally made by inscribing closely-spaced lines on a glass or plastic plate with a
diamond-tipped cutting tool. The spaces between the opaque lines (scratches) serve as
the slits in the grating.
Figure 10.4: Formation of a bright maximum for a transmission diffraction grating.
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As for the single slit, consider the ideal case of monochromatic light of a single
wavelength λ incident from a source far to the left of the grating. The initial wave front is
a plane W1 perpendicular to the direction of the incoming light (the direction of travel for
the light when viewed as a “ray”) and therefore parallel to the grating. To the right of the
grating Huygens’ “wavelets” spread out from each slit in grating, in phase with the
incident wave front W1. The wavelets combine to form the exiting wave front W2 in
accordance with Huygens’ Principle. For an angle θ such that the successive differences
in path length (the small distances l1, l2, and l3 , for example) correspond to integer
numbers of wavelengths, the waves from all the slits will arrive in phase at the screen far
to the right of the grating and interfere constructively, producing a bright maximum. This
will occur when l1=λ, l1=2λ, l1=3λ … So the general relationship that must hold to form a
bright maximum at the screen is
sin  m  m

d
m  1,2,3... [10.2]
where m is an integer, again referred to as the order of the maximum. Thus incoming
parallel light (plane wave fronts) of a given wavelength λ will be constructively enhanced
in the specific directions θm . Note that if the light is not monochromatic, but rather
contains a mixture of wavelengths, each different wavelength will have a different set of
θm directions, since d remains fixed but λ changes in equation [10.2]. Thus, a
transmission grating will disperse the spectrum of white light, performing the same
optical function as a prism. To produce diffraction angles of a few degrees requires a
spacing d ≈ 10λ or about 5 x 10-4 cm (~2000 slits/cm) for visible light.
10.3 The Experiments
10.3.1 Interference from a Single Narrow Slit
Procedure
Without a slide mounted in the slide holder, turn on the LASER slide projector and aim
the LASER so that the spot falls just above the zero mark on the centimeter scale marked
on the projection screen and verify that the distance from the slide to the screen is 1.5
meters. Mount the single-slit slide in the projector with the slit oriented vertically and
position it so that the LASER spot falls centered on the slit. Observe the diffraction
pattern and record the distances for the first three dark fringes from the beam position.
Analysis
Calculate the slit width for each order fringe using [10.1] and compare for consistency.
What would happen to the position of the first dark fringe if the slit width were doubled?
Halved?
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10.3.2 Characterizing the Transmission Grating
Procedure
Next, mount the transmission grating slide in the slide projector so that the LASER spot
falls near the center of the grating. Arrange the slide so that the bright maxima fall
horizontally along the centimeter scale on the screen, again with the LASER beam spot
centered at zero. Measure the distance to the first bright maximum
Analysis
How are the slits in the grating oriented (horizontally or vertically)?
Calculate the grating spacing d. How many slits per centimeter does your value of d
correspond to?
10.3.3 A Continuous Spectrum
Procedure
A hot, glowing solid produces a continuous spectrum, which looks like a complete
rainbow and can be observed through the transmission grating. The general scheme for
viewing a light source through the transmission grating is shown in Figure 10.5.
Figure 10.5: Scheme for viewing a light source through the transmission grating
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Observe the light from the incandescent light bulb filament through the transmission
grating. For this application, the narrow filament of the light bulb serves as the slit
shown in the drawing. Measure the positions of the ends of the visible spectrum (the
positions of the red end and the purple end).
Analysis
Calculate the corresponding wavelengths using [10.2]. You need only do this for the
first-order spectrum. How do your measured wavelengths correspond to the published
values for the ends of the visible spectrum displayed on the wall chart? Does the
spectrum appear truly continuous (no breaks)? Does the brightness seem uniform?
10.3.4 The Spectrum of Hydrogen
Procedure
The viewing setup is similar to that used for the continuous spectrum. This time, the
narrow hydrogen discharge tube serves as the “slit” for the apparatus and the grating is
placed 50 cm from the discharge tube. In the first-order spectrum you should see a bright
red line. This spectral line is called the Hα of the Balmer spectral series of Hydrogen.
Astronomers map the distribution of hydrogen gas in the Galaxy by observing the
intensity of this line to both determine both the presence and amount of hydrogen in
ionized gas clouds. The prominent greenish-blue line is called Hβ. You should also be
able to see a faint violet line (Hγ) and you may be able to see a very faint line in the far
violet (Hδ). Measure and record the positions of these lines on both sides of the discharge
tube.
Analysis
Average your two position measurements for each line (what error does this correct for?)
and calculate the wavelength corresponding to each line. The wavelengths of the lines in
the Balmer series are given by the Rydberg formula, a special case of the Bohr formula:
1  R(1/ 22  1/ n2 ) n  3,4,5,6... [10.3]

Where n = 3 corresponds to Hα , n = 4 to Hβ and so on. The parameter R is called the
Rydberg constant. Use each of your measurements separately to estimate the value of R.
Do the wavelengths of your lines all predict the same value for the Rydberg constant?
Are your values consistent with the published value R = 1.097 x 105 cm-1?
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10.3.5 Probing Structure with Diffraction
Procedure
Tape the diffraction pattern page of your worksheet to the screen and align the LASER
slide projector so that the beam spot hits the target on the page. Mount the unknown
sample slide in the LASER slide projector and observe the resulting diffraction pattern on
the screen. Trace the diffraction pattern on your worksheet page, measure and record the
distance from the beam to one of the primary bright maxima on the page. Be sure the
distance from the slide to the screen is 1.5 m.
Analysis
The sample you are studying consists of a regular array of identical closed shapes. What
is the symmetry of the basic repeated shape? (Hint: How many primary bright maxima
surround the beam spot? How many sides does the regularly repeated “cell” of the
sample have?). Treating the sample as a transmission diffraction grating, use [10.2] to
estimate the spacing between grating elements and hence the size of the individual
elements making up the sample. Can you identify a biological sample (animal) that
exhibits this regularly repeated shape? View the sample under a microscope and see if
you can guess what it is.
Figure 10.6: Characteristic sizes for various objects and the corresponding forms of
electromagnetic radiation needed to probe them using interference and diffraction
techniques.
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