simulating interference and diffraction in instructional...

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].

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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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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.

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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.



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

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L Maurer

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McDermott L C 1999 Addressing studentdifficulties in applying a wave model to theinterference and diffraction of light Am. J.Phys. 67 S5–15

[2] Ambrose B S, Shaffer P S, Steinberg R N andMcDermott L C 1999 An investigation ofstudent understanding of single-slit diffractionand double-slit interference Am. J. Phys.67 146–55

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[10] Werley C A, Nelson K A and Ryan Tait C 2012Direct visualization of terahertzelectromagnetic waves in classic experimentalgeometries Am. J. Phys. 80 72–81

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[14] Berenger J-P 1994 A perfectly matched layer forthe absorption of electromagnetic wavesJ. Comput. Phys. 114 185–200

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[18] Newburgh R 2005 Optical path, phase, andinterference Phys. Teach. 43 496–8

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.


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Simulating interference and diffraction in instructional laboratoriesIntroductionThe programThe simulationsNarrow single slitNarrow double slitsNarrow triple slitsWide single slit

ResultsConclusionAcknowledgmentsReferences

simulating interference and diffraction in instructional...

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