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Simulating interference and diffraction in instructional
laboratories
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P A P E R Swww.iop.org/journals/physed
Simulating interference anddiffraction in
instructionallaboratoriesL Maurer
Department of Physics, University of Wisconsin-Madison, USA
E-mail: [email protected]
AbstractStudies have shown that standard lectures and
instructional laboratoryexperiments are not effective at teaching
interference and diffraction. Inresponse, the author created an
interactive computer program that simulatesinterference and
diffraction effects using the finite difference time domainmethod.
The software allows students to easily control, visualize
andquantitatively measure the effects. Students collected data from
simulationsas part of their laboratory exercise, and they performed
well on a subsequentquiz, showing promise for this approach.
Introduction
Often, interference and diffraction are taught byshowing the
mathematics in lectures and thenperforming laser interference and
diffraction ex-periments in an instructional laboratory.
However,studies show that this teaching method is lacking.
Consider the following problem (seefigure 1), which should be
straightforward fora student who understands the concepts. Twopoint
sources, 2.5 wavelengths (λ) apart, generatewaves in phase. Is
there constructive interference,destructive interference or neither
at points A, Band C?
A University of Washington study asked thisand other questions
of ≈1200 undergraduatesin their standard calculus-based
introductoryphysics course. Only ≈35% answered correctlyfor both
points A and B, and only ≈5% answeredcorrectly for point C.
Graduate students alsoperformed poorly; only≈25% answered
correctlyfor point C [1]. These mistakes were primarilydue to
fundamental misunderstandings; post-quiz
Figure 1. The diagram for the interference quiz.
student interviews included responses like ‘Isuppose that 2.5λ
(the slit separation) is smallcompared to 400λ and 300λ, so the
sources hereact like a single source’ [2]. Moreover, the
typicalinstruction method did not help; the scores forpoint C
before and after lecture and laboratorywere basically unchanged [1,
2].
0031-9120/13/020227+06$33.00 c© 2013 IOP Publishing Ltd P H Y S
I C S E D U C A T I O N 48 (2) 227
http://www.iop.org/journals/physedmailto:[email protected]
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L Maurer
Part of the problem is that the wave natureof light is only
visible indirectly—through effectslike interference and
diffraction. Light’s peaksand valleys cannot be viewed directly.
Hands-on experience seeing and controlling the waves’interactions
is useful for student understanding,so a better wave medium is
needed. Rippletanks provide this, but there are significant
trade-offs between ease of use, cost and measurementaccuracy.
For example, Wosilait et al provided studentswith simple ripple
tanks—pans, dowels andsponges. They are cheap and easy to use but
donot allow quantitative measurements. Still, theyproved effective
as part of an intensive tutorialsystem that raised scores for point
C to≈45% [1].
Measurements can be performed with aripple tank—at the price of
increased cost andcomplexity. They require a strobe device and
aconsistent wave source—not a student rolling adowel back and
forth. The strobe can be either alight [3] (easier but more
expensive) or a spinningslotted disc [4] (cheaper but more
complicated).Either makes the waves appear frozen, makingtheir
measurement easier.
Computer simulations can also displaywaves, and they have
potential advantages overripple tanks. Simulations can allow finer
controland more accurate measurements. They areavailable for free.
They can be used outside of thelaboratory. They also allow better
visualizationby avoiding unwanted reflections, being easilypaused
and having superior contrast.
Dozens of simulations are available1. Somedisplay what would be
seen in a laserexperiment—without showing the waves interact-ing
[5]. Others are non-interactive animations [6].Some have
interactive displays [7]. A fewallow measurement of the
instantaneous waveheight/field strength [8].
These simulations are primarily used forqualitative
understanding. However, I wanted asimulation that could answer
questions like theaforementioned one—albeit on a smaller scale—and
could thereby test interference conditions.This requires easily
measured amplitudes at anyuser desired grid point. No simulations
combinedthat with interactive, easy to control, animatedwaves.
1 Search www.merlot.org/.
Figure 2. The single narrow slit simulation.
Similar simulations have been proposed,but their use in
instructional laboratories hasnot been reported. Frances et al
proposed theuse of a finite difference time domain
(FDTD)electromagnetics simulation to show interferenceand
diffraction effects, but they have only reportedits use in lecture
demonstrations [9]. Werleyet al also suggested using FDTD
simulations inlaboratories, but they instead used
pre-recordedvideos of actual propagating radiation [10].
Therefore, I wrote an FDTD simulation thatallows easy and
accurate measurements, and Iconstructed a laboratory exercise that
uses thesimulation for quantitative measurements, not
justqualitative understanding.
The programThe FDTD technique solves differential equationsby
discretizing them in both space and time. Ithas proved popular and
effective for simulatingelectromagnetics, and there are many
fineworks on the technique [11–13]. Therefore,the equations are not
reproduced here2; thissection summarizes aspects of the simulation
andinterface relevant to its use.
Figure 2 shows the program’s interface,which has five plots. The
two large plots showEz (upper plot) and EzRMS (lower plot). For
both,
2 For reference, the program simulates a TMz wave usingthe
standard Yee lattice and second-order-accurate finitedifference
approximations.
228 P H Y S I C S E D U C A T I O N March 2013
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Simulating interference and diffraction in instructional
laboratories
black represents the smallest value and whiterepresents the
largest value, with shades of greyin between. The three smaller
plots show Ezand ±
√2EzRMS—an envelope for Ez—along the
horizontal and vertical dashed lines through thetwo larger
plots. These lines can be moved withthe keyboard or mouse, and x,
y, Ez and EzRMS atthe intersection of the lines are displayed in
thecentre right area between the two vertical plots.Knowledge of
EzRMS at that point allows users tohome in on extrema.
A plane wave, with a wavelength of 20 gridcells, enters from the
left. It is not simulatedbut calculated analytically. At the start
of thesimulation, the wave’s magnitude is ramped upgradually to
avoid potentially unstable high-frequency components.
The barrier—the red line visible in both largeplots—is a perfect
conductor, and the openingsin the barrier are hard sources that
inject theincoming wave into the FDTD domain—the areato the right
of the barrier. Openings in the barriercan be added, removed and
modified using thebarrier control frame at the right of the
interface.
Split-field perfectly matched layers [12, 14]terminate the other
three sides of the FDTDdomain. These boundaries reduce
reflectionsto imperceptible levels, effectively giving
thesimulation open boundaries.
When the simulation starts or the barrier ismodified, a timer
appears over the EzRMS plot,counting down until a steady state is
reached.Afterwards, EzRMS is reset to remove transients,and another
countdown appears for one wave timeperiod. The steady state EzRMS
is calculated byaveraging over that time.
Among its other features, the simulation alsohas a fast-forward
mode, which saves time by notupdating the plots. In that mode, the
simulationruns until the current countdown is completed.
The laboratory’s computers, running Win-dows 7 with Intel Core 2
Duo processors, take≈55 ms per timestep, resulting in a
smoothanimation.
The software is written in Python usingNumPy for the
calculations, TkInter for theinterface and the Python Imaging
Library for theplots. These libraries are available for Windows,OS
X and Linux. Executables, source code and
program information are available online3; theprogram’s source
code is available under the GNUPublic License version 2.
The simulationsThe laboratory was tailored for a particular
class,aimed at future physics majors, but tweaking ofthe exercises
for other courses should be straight-forward. Besides simulations,
the laboratory alsoincluded pen and paper work and short
laser/slitexperiments4. However, four simulations are atthe
laboratory’s heart—narrow single, double andtriple slits, and a
wide single slit.
Narrow single slit
See figure 2. This simulation provides a baselinefor comparison.
It shows that a single narrowopening is not responsible for the
interferencepatterns seen in the following simulations.
The course’s lectures included a mathemat-ical description of
point sources in 3D, so thissimulation introduced students to the
asymmetricsources used in the laboratory. To acquaintstudents with
the simulation’s controls, I hadthem measure roughly how fast the
amplitude de-creased with distance from the opening; becausethis is
a 2D simulation, the amplitude falls off lessquickly than in
3D.
Narrow double slits
See figure 3. This is the key simulation; it letsstudents
discover the conditions for constructiveand destructive
interference. Students were askedto find four maxima and four
minima of EzRMS,in either x̂ or ŷ and in the right half of the
domain,calculate |d0−d1|
λfor each, where d0 and d1 are the
distances from the extrema to the openings, andthen find the
pattern in those numbers.
In principle, the students already knew theconditions, but they
seemed new to many. Further-more, most students seemed to be
comprehendingthe conditions for the first time5. Pen and paper
3 http://lnmaurer.github.com/Interference-Inference-Interface/.4
The worksheet is available at
https://github.com/lnmaurer/Interference-Diffraction-Worksheet.5
Discretization in the program can slightly alter theconditions.
However, |d0 − d1| was always within one.For example, the point
where the dashed lines crosses is a
maximum in figure 3, and |d0−d1|λ=|235−296|
λ=
6120 = 3.05.
These small errors were not problematic for students.
March 2013 P H Y S I C S E D U C A T I O N 229
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L Maurer
Figure 3. The double narrow slit simulation, with thedashed
lines intersecting at an EzRMS maximum.
Figure 4. The triple narrow slit simulation, with thedashed
lines intersecting at an EzRMS minimum. Itis located 15.9λ, 16.5λ
and 18.25λ from the threeopenings.
work followed to reinforce their understanding.This included
drawing a couple of waves thatresulted in these conditions (using a
simplifiedversion of [15, 16]) and mathematically verifyingthat the
conditions lead to waves in or out ofphase.
Narrow triple slits
See figure 4. The additional slit makes studentsconsider the
logic behind the extrema conditionsclosely.
Figure 5. The wide single slit simulation with thedashed lines
intersecting at the first minimum. Thisoccurs at θ = arctan(
150−42599−100 ) = 0.213, whereas the
far-field limit has θ = 20100 = 0.2.
Here, the students were asked to find pointsthat were
simultaneously extrema in both x̂ andŷ. This is straightforward
with the simulation;however, since this requires fine control in
bothx̂ and ŷ, it would be difficult using the standardexperimental
setup.
The condition for constructive interferencestill holds. To get a
maximum, all pairs should bemaximized.
However, the minimization condition is morecomplicated [17]. The
two-slit condition causedthe two waves to roughly cancel. That
wouldleave the third wave undiminished. Additionally,it is
mathematically impossible for all differencesin the distances to be
half integer numbers ofwavelengths.
This is an instructive example that is missingfrom laboratories
that do not use simulations forquantitative measurements.
Wide single slit
See figure 5. The right side of the domain getsclose to the
far-field limit. While the domainis too small to truly display
far-field effects,a limitation common to many ripple tanks,
thesimulation successfully shows diffraction effectsarising from
waves. The simulation could be usedto investigate near-field
effects, which is difficultto achieve with normal instructional
laboratoryequipment.
230 P H Y S I C S E D U C A T I O N March 2013
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Simulating interference and diffraction in instructional
laboratories
Table 1. Scores for point C. The first three columns of results
are from Wosilait et al and are rounded to thenearest 5% [1].
Old UWash (%) UWash grad (%) New UWash method (%) My 15 (%)
Right method 10 55 70 73And right answer 5 25 45 60
to point
W
λ
W/2
to point to point
Figure 6. The figure for the second quiz question.
ResultsOnly 15 students used the final version ofthe laboratory,
so the only conclusion I candraw with confidence is that students
enjoyedit: I collected anonymous feedback, and it wasoverwhelmingly
positive. Still, for a rough gaugeof the method’s effectiveness, I
quizzed thestudents with questions from the University ofWashington
study. The scores for point C aregiven in table 1.
They performed well on all parts of thequiz, relative to others.
Sixty per cent of thestudents answered correctly for point C.
Eightyper cent correctly identified the behaviour at bothpoints A
and B—students in the old Universityof Washington system scored 35%
[1]. The quizincluded another question from the University
ofWashington study. See figure 6. What would beobserved at the far
away point, first with just theleft two slits and then with all
three slits? Eightyper cent got both parts right.
ConclusionInterference and diffraction are tricky subjectsto
teach, but this approach shows promisein making them more
accessible to students.The simulation allows them to easily
control,visualize and quantitatively measure interference
and diffraction effects. That makes the software apowerful
tool.
The progression of simulations describedhere appears to be
effective. First, a single narrowslit familiarized students with
the simulation andshowed that more was needed for
interference.Then, a double slit simulation showed
Young’sexperiment in a new light, with visible waves andeasy
measurements. Next, a triple slit experimentmade students
reconsider the constructive anddestructive interference
conditions—one held butthe other failed. Finally, a diffraction
simulationshowed that similar effects could arise from onewide
slit.
While the sample size here was small, theresults are
encouraging. I hope, eventually, to trythis approach with more
students in more classes.
It would be difficult to experimentallymimic some of the
simulations with instructionallaboratory equipment. The simulation
couldalso easily demonstrate more advanced topicswith minimal
modification. The permittivity,permeability and conductivity are
definable atevery point in the simulation domain. Changingthose
parameters would allow the simulationto demonstrate waveguides,
reflection due tochanges in dielectric constants, etc.
Simulationsof a modified two-slit experiment, where theupper and
lower halves of the domain havedifferent indices of refraction,
could illustratemore conceptual topics, like the differencebetween
physical and optical path lengths [18].
AcknowledgmentsI would like to thank Professor Susan Hagnessfor
suggesting this simulation as a project andfor giving feedback on
it, Dr James Reardonfor encouragement during this laboratory’s
devel-opment, and Professors Daniel Chung and LisaEverett for
letting me try this approach in theircourse, where I was the
teaching assistant.
Received 1 October 2012, in final form 2 October
2012doi:10.1088/0031-9120/48/2/227
March 2013 P H Y S I C S E D U C A T I O N 231
http://dx.doi.org/10.1088/0031-9120/48/2/227
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L Maurer
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Leon Maurer is a graduate student at theUniversity of
Wisconsin-Madison. Hispersonal interest in physics
educationdeveloped during several semesters as ateaching assistant.
He is currently aresearch assistant modelling thermaltransport
through silicon nanowires.
232 P H Y S I C S E D U C A T I O N March 2013
http://dx.doi.org/10.1119/1.19083http://dx.doi.org/10.1119/1.19083http://dx.doi.org/10.1119/1.19210http://dx.doi.org/10.1119/1.19210http://www.nuffieldfoundation.org/practical-physics/measuring-waves-ripple
tankhttp://www.nuffieldfoundation.org/practical-physics/measuring-waves-ripple
tankhttp://www.nuffieldfoundation.org/practical-physics/measuring-waves-ripple
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tankhttp://www.nuffieldfoundation.org/practical-physics/measuring-waves-ripple
tankhttp://www.nuffieldfoundation.org/practical-physics/measuring-waves-ripple
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tankhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://vsg.quasihome.com/interfer.htmhttp://dx.doi.org/10.1119/1.1631622http://dx.doi.org/10.1119/1.1631622http://dx.doi.org/10.1119/1.1526622http://dx.doi.org/10.1119/1.1526622http://dx.doi.org/10.1016/j.compedu.2005.06.008http://dx.doi.org/10.1016/j.compedu.2005.06.008http://dx.doi.org/10.1109/TE.2011.2150750http://dx.doi.org/10.1109/TE.2011.2150750http://dx.doi.org/10.1119/1.3652698http://dx.doi.org/10.1119/1.3652698http://dx.doi.org/10.1109/TAP.1966.1138693http://dx.doi.org/10.1109/TAP.1966.1138693http://dx.doi.org/10.1119/1.2835056http://dx.doi.org/10.1119/1.2835056http://dx.doi.org/10.1006/jcph.1994.1159http://dx.doi.org/10.1006/jcph.1994.1159http://dx.doi.org/10.1119/1.1407132http://dx.doi.org/10.1119/1.1407132http://dx.doi.org/10.1119/1.3098213http://dx.doi.org/10.1119/1.3098213http://dx.doi.org/10.1016/S0030-4018(02)02305-2http://dx.doi.org/10.1016/S0030-4018(02)02305-2http://dx.doi.org/10.1119/1.2120373http://dx.doi.org/10.1119/1.2120373
Simulating interference and diffraction in instructional
laboratoriesIntroductionThe programThe simulationsNarrow single
slitNarrow double slitsNarrow triple slitsWide single slit
ResultsConclusionAcknowledgmentsReferences