Connecting optical intensities and electric fields using a triple interferometer
aa r X i v : . [ phy s i c s . e d - ph ] J u l Connecting optical intensities and electric fields using a triple interferometer
David Collins ∗ and Justin Endicott † Department of Physical and Environmental Sciences,Colorado Mesa University, 1100 North Avenue, Grand Junction, CO 81501 (Dated: August 31, 2020)We consider the issue of validating the relationship between electric fields and optical intensityas proposed by the classical theory of electromagnetism. We describe an interference scenario inwhich this can be checked using only intensity measurements and without any other informationregarding the details of the arrangement of the associated fields. We implement this experimentallyusing a triple Michelson interferometer and the results strongly suggest that the method validatesthe classical relationship between optical intensity and the associated classical field.
I. INTRODUCTION
A central tenet of classical optics is that light can bedescribed via an associated electric field. The behaviorof the electric field is determined by the theory of elec-tromagnetism, which eventually relates the intensity of alight source to the associated electric field . The pro-cess of validating the electric field description of light iscomplicated by the apparent difficulty of measuring theassociated electric fields directly. The theory is usuallychecked indirectly via inferences based on intensity mea-surements.One such indirect inference involves light producedfrom two or more sources. The associated electric fieldsinterfere, producing a superposition field which deter-mines the observed intensity. The resulting interferencephenomena, especially those produced by single or mul-tiple slits or interferometers, and their relationships withelectric fields are familiar to most undergraduate physicsstudents . In general, a detailed analysis of such in-terference phenomena is not done purely in terms of in-tensities but also involves phase relationships betweenthe individual sources. This typically depends very deli-cately on the configuration of the sources and when manyof these are present a precise comparison between experi-mental results and theoretical predictions can be difficult.However, recently there has emerged a type of multi-path interference scenario which only uses intensity in-formation to explore and validate an analogous quan-tum theory model of light . This has been checked withexperiments involving superposition of light from morethan two sources , produced by special multiple slitand mask arrangements. These experiments appeared tovalidate the underlying theory, when applied in a sim-plified fashion. However, a more detailed analysis re-vealed a small discrepancy and these deviations havebeen checked experimentally . Similar theoretical andexperimental investigations into analogs of multipath in-terference have been conducted in situations that do notinvolve optical slits .In this article we describe an adaptation of suchmultipath experiments that checks the relationship be-tween classical electric fields and optical intensity and amethod for assessing interference when multiple sources are present. Rather than a multiple slit arrangement ,the experiment uses a triple interferometer. This has theadvantage of being much easier to manage than the com-parable multiple slit experiments and is within the abil-ities of undergraduate students. It would also introduceundergraduate students to situations in which inferencesare made via correlating results (in this case intensities)from various experimental settings and observing howthis technique can illuminate the underlying physics. Fi-nally it also appears to evade the immediate critique ofprevious multipath experiments.This article is organized as follows. Section II describesthe theoretical background and how interference is quan-tified. Section III describes the experimental setup andsection IV describes the results of the experiment. II. INTENSITY AND INTERFERENCE TERMS
In optics, the intensity (or irradiance) of light is definedto be the time averaged rate at which energy flows acrossa surface per unit area of that surface . In classicalphysical optics light is described as an electromagneticwave and the rate of energy propagation per unit area isdetermined by the time average of the Poynting vectorassociated with the electromagnetic field. The simplestcase to assess is that of a monochromatic electromag-netic field whose electric field at location r and time t is E = E cos ( k · r − ωt + φ ); here k is the wavenumbervector associated with the direction of propagation andwavelength of the wave, ω is the angular frequency of thewave, φ is the phase of the wave, and E is a vector thatis independent of location and time. It emerges that,in free space, the intensity of this wave is I = ǫ cE / ǫ is the permittivity of free space. In the com-plex formalism, this electromagnetic wave is describedvia E = E e i ( k · r − ωt + φ ) and then the intensity is I = ǫ c E · E ∗ . (1)Crucially the intensity is proportional to E · E ∗ .Now suppose that two sources, A and B, each producelight with the same wavenumber and the superposition isincident on a detector. In classical electromagnetism theelectric fields produced by various sources are superposedlinearly to form the field that will be detected by anydetector. The electric field arriving at the detector is E = E A + E B where E A is the electric field producedby source A and E B is that produced by source B. Theresulting intensity is I = I A + I B + ǫ c E A · E ∗ B + E B · E ∗ A ) , (2)where I A = ǫ c E A · E ∗ A / I B is defined similarly. If E = E A e i ( k · r − ωt + φ A ) where E A and φ A are constant anda similar expression applies to source B, then straight-forward analysis gives I AB = I A + I B + 2 p I A I B cos (∆ φ ) (3)where ∆ φ := φ A − φ B is the phase shift between thesources. The resulting interference between the sourcesis a key prediction that results from the underlying elec-tric field description and is at odds with a simplistic de-scription which would assume that the intensity of thecombination is the sum of the two intensities. However,checking a prediction of Eq. (3) requires knowing thephase shift between the sources. This can depend in acomplicated way on the configuration of the sources.This complication can be avoided by considering multi-path interference experiments which typically combinelight produced by multiple sources at a detector. We ini-tially develop the associated theory purely in terms ofintensities, ignoring whatever underlying theory may de-scribe these intensities.Suppose that there are three sources, labeled A, B andC. Each source can be turned on and off independentlyand at will and whenever any source is turned on it pro-duces light with a set constant intensity; thus wheneversource A is turned on it produces light with the sameintensity as whenever it had been turned on previously.The detector only measures the overall intensity of thelight that arrives at it resulting from all the sources. Weuse the following notation to describe the possible inten-sities recorded by the detector when various sources areare on or off. Let I A be the intensity recorded by the de-tector when source A is on and sources B and C are off.Let I B be the intensity recorded by the detector whensource B is on and sources A and C are off. Define I C similarly. Then let I AB be the intensity recorded by thedetector when sources A and B are on and source C isoff. Define I AC and I BC in a corresponding fashion. Let I ABC be the intensity recorded by the detector if all threesources are on and I be the intensity if none are turnedon. The central questions ask how I ABC is related to I AB , I BC , I AC , I A , I B , I C and I or how I AB is related to I A , I B and I .Such issues have been addressed in the context ofvarious probabilistic descriptions and measures withinquantum theory and can be adapted to classical optics.Without knowing any details of the underlying theorythat describes the intensity we could entertain variouspossibilities. For example, if the theory were such that the intensities superimposed linearly, then I AB = I A + I B .This motivates the definition of a second order interfer-ence term , ∆ ( A, B ) := I AB − I A − I B . (4)The quantities on the right can be measured experimen-tally regardless of the underlying theory that describesthe values of that on the left. Various theoretical modelscould then predict ∆ ( A, B ) and checked against valuecomputed via measurements. For example, if the inten-sities superimposed linearly then ∆ ( A, B ) = 0 . According to classical electromagnetism, Eq. (3) pre-dicts that ∆ ( A, B ) = 2 p I A I B cos (∆ φ ) (5)and the sources could always be arranged with a phaseshift such that ∆ ( A, B ) = 0. At this point given achoice between a theory in which the intensities super-impose linearly and one in which fields superimpose lin-early, measuring the intensities and computing ∆ ( A, B )would allow us to decide which of these two possibilitieswould be correct.However, if we cannot measure the electric fields di-rectly or if it is difficult to ascertain or control the phaseshift, then we cannot use intensity measurements to eas-ily check the predictions of classical electromagnetic the-ory. We therefore seek a comparable quantity which willallow us to check the predictions of classical electromag-netism only using intensity measurements.If all three sources are turned on then, E = E A + E B + E C and I ABC = ǫ c E A + E B + E C ) · ( E ∗ A + E ∗ B + E ∗ C )= I A + I B + I C + ǫ c E A E ∗ B + E ∗ A E B )+ ǫ c E A E ∗ C + E ∗ A E C + E B E ∗ C + E ∗ B E C )= I A + I B + I C + I AB − I A − I B + I AC − I A − I C + I BC − I B − I C = I AB + I AC + I BC − I A − I B − I C . (6)It follows that, regardless of the intensities of the indi-vidual sources or the phase relationship between the as-sociated electromagnetic waves it is always true that I ABC − I AB − I AC − I BC + I A + I B + I C = 0 . (7)We then define a third order interference term , alsocalled the Sorkin parameter,∆ ( A, B, C ) := I ABC − I AB − I AC − I BC + I A + I B + I C . (8)Then if, as classical electromagnetism predicts, the in-tensity is determined via Eq. (1) and the electric fieldssuperimpose linearly, then ∆ ( A, B, C ) = 0 but, in gen-eral, ∆ ( A, B ) = 0, ∆ ( B, C ) = 0 and ∆ ( A, C ) = 0.We briefly consider the possibility that all second orderinterference terms are zero. A second order interferenceterm is only zero if and only if the phase shift betweenthe two sources is an odd multiple of π/
2. However, ifthe phase shift between A and B is an odd multiple of π/ π/
2. Thus it is impossible that all three second orderinterference terms are zero.Thus if the predictions of classical electromagnetismare correct then ∆ ( A, B, C ) = 0 and at least one sec-ond order interference term is non-zero. Note that thismethod for checking the underlying electric field descrip-tion is insensitive to the intensities of the individualsources and the phase relationship between them.This third order interference term can be expressed interms of second order interference terms such as∆ ( AB, C ) = I ABC − I AB − I C (9)and it is easily seen that, for example,∆ ( A, B, C ) = ∆ ( AB, C ) − ∆ ( A, C ) − ∆ ( B, C ) . (10)It immediately follows that any theory for which the sec-ond order interference term is always zero implies thatthe third order interference term would also be zero; anexample would be a theory in which the intensities su-perimpose linearly. The converse is clearly not true; onecounterexample is classical electromagnetism and optics.This entire framework has been extended to arbitrar-ily high order interference terms and has the feature thatany theory in which the interference term at a given orderis zero automatically implies that higher order interfer-ence terms are zero. Finding the boundary between theinterference terms which are zero and those which are notthen delimits the possible theory. In the case of classicalelectromagnetism and optics the boundary is between thesecond and third order. The experiment aims to checkthis. III. EXPERIMENTAL SET-UP
Previous experiments which have investigated interfer-ence in optics have used multiple slits to act as the re-quired sources . These used a succession of single pho-tons, followed by photon counting to check intensity pre-dictions given via the Born rule. However, it emergedthat a detailed theoretical analysis of the intensities pro-duced by various slit arrangements yields a small non-zero third order interference term and thus the fieldsproduced in this way do not superimpose exactly as themodel that yields ∆ ( A, B, C ) = 0 predicts.Additionally these experiments require delicate manip-ulation of a closely spaced multiple slit arrangement andthe masks which open or close various slits as well as in-tricacies associated with generating and counting singlephotons. The experiment that we describe avoids thesetechnical issues but still illustrates how the hierarchy ofinterference terms can decide between various theories. Our experiment uses a triple Michelson interferometerto produce three sources. This interferometer consists ofa parent interferometer and two offspring interferometersconfigured as illustrated in Fig. 1.
A CBBS 1 BS 3BS 2 MirrorMirrorMirror
Laser DetectorFIG. 1. Triple Michelson interferometer. Light from a sourceis incident on beam splitter (BS 1), which forms the parentMichelson interferometer. The resulting transmitted and re-flected beams are incident on two other beam splitters (BS 2and BS 3), which initiate the offspring interferometers. Thereflected and transmitted beams from BS 3 are redirected viamirrors, eventually reaching the detector. Prior to the detec-tor they form source B (dotted) and source C (dashed). Thebeam transmitted through BS 2 is also redirected via a mir-ror, eventually forming source A (solid) prior to the detector.The horizontal beams from BS 2 are discarded.
This arrangement can effectively produce beams fromthree sources incident on the detector. Sources can beturned on and off by blocking the relevant arms withinthe interferometer. Note that BS 2 in Fig. 1 is not strictlynecessary for the production of source A. However, itdoes allow for a situation where the intensity of all threesources is approximately equal and where the appear-ance of the interference pattern at the detector is roughlyequally sensitive to an adjustment in either offspring in-terferometer.The beam generation was done using a Melles Griot 25-LHP-111-249 1 . ± ± FIG. 2. Experimental set up. The red box indicates the lasersource, the blue the detector, the green the beamsplitters, theyellow the fixed mirrors and the brown the flip mirrors.
The voltage output from the PD was acquired via aPASCO 550 Universal Interface with a PASCO voltagesensor. The PD voltage output was recorded as a func-tion of time and displayed using PASCO Capstone soft-ware.The experiment was conducted in a dark room butthis did not eliminate all intrusion of light produced fromsources other than the laser and the PD would providea small non-zero voltage reading even when all threesources were blocked. The associated background inten-sity I must be subtracted from every intensity that en-ters into Eq. (8). The result is a modified version of thethird order interference term,∆ ( A, B, C ) = I ABC − I AB − I AC − I BC + I A + I B + I C − I . (11) We then aim to verify whether∆ ( A, B, C ) = 0 (12)provided that the intensities are those measured by thePD. Additionally note that since the intensity of the lightincident on the PD is proportional to the PD output,converted into a voltage, we can replace the intensitiesin Eq. (11) by the associated voltages and will do so forthe remainder of this article.A single “setting” of the experiment consisted of thefollowing sequence.1. Position the detector at a fixed location along theinterference pattern.2. Allow all three beams to be incident and record thePD output. This gives I ABC .3. Block path C and record the PD output. This gives I AB .
4. Open path C and block B and record the PD out-put. This gives I AC .
5. Open path B and block A and record the PD out-put. This gives I BC .
6. Block B and C and record the PD output. Thisgives I A .
7. Block A and C and record the PD output. Thisgives I B .
8. Block A and B and record the PD output. Thisgives I C .
9. Block all three paths and record the PD output.This gives I . Five runs were done at each setting. A total of elevendifferent settings were used, corresponding to eleven dif-ferent detector positions along the interference pattern.This effectively samples eleven different phase relation-ships between the three beams.
IV. DATA AND RESULTS
Capstone recorded the voltage produced by the PD asa function of time continuously during each run of theexperiment. Typical examples are illustrated in Fig. 3,representative of a cleaner data set, and Fig. 4, repre-sentative of a noisier data set. In each run the steepvertical transitions and spikes indicate the moments dur-ing which the flip mirrors are moved so as to alter thebeam combination incident on the detector. These thendelineate intervals during which the intensity is producedby particular combinations of beams. Each figure showseight such intervals, each typically lasting for five to tenseconds. During each interval, the intensity should beconstant although the degree to which this occurred var-ied. A representative intensity for each interval was de-termined via the mean and standard deviation of all volt-ages spanning the period between transitions but exclud-ing buffer periods of approximately equal duration (oneor two seconds) before and after the transitions. bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb I i n [ V ] Time in [s] I ABC I AB I AC I BC I A I B I C I FIG. 3. PD output voltage versus time for one particularsingle run. The boxes indicate the moments during whichvarious paths were opened or closed. During these periodsthe intensity was more or less constant. The vertical linesand spikes appear while the flip mirrors are being adjusted totoggle between sources. bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb I i n [ V ] Time in [s] I ABC I AB I AC I BC I A I B I C I FIG. 4. PD output voltage versus time for one particularsingle run that yielded messier data. Symbols have the samemeaning as in Fig. 3.
The collection of eight such data points for onerun were substituted into Eq. (11) to determine∆ ( A, B, C ) , ∆ ( A, B ) , ∆ ( A, C ) and ∆ ( B, C ). Foreach setting, the data was used to determine a weightedaverage for all interference terms and these are displayedin Table I.
Setting ∆ ( A, B, C ) ∆ ( A, B ) ∆ ( A, C ) ∆ ( B, C )1 2 . ± . . ± . . ± . . ± . − ± . ± . . ± . . ± . − ±
10 mV 117 . ± . . ± . ± − . ± . − ±
10 mV − . ± . ±
11 mV5 10 ±
21 mV − ± − ± ±
12 mV6 − ± − . ± . − . ± . . ± . ±
10 mV − ± − . ± . . ± . ±
18 mV 136 ± . ± . − . ± . − ±
17 mV − . ± . ± − ± − . ± . − ± ± − ± − ±
18 mV 23 ± ± ± The data for 54 runs are displayed in Figs. 5–8 (thedata from one run had been inadvertently erased).Of the eleven settings, the third order interference termis within one standard deviation of 0 mV for seven. Forthree of the settings it is within one standard deviationand for one setting it is within two. This should becompared against the second order terms for each set-ting. According to classical electromagnetic theory andEq. (3) the second order term could be 0 mV whenever∆ φ is an odd half multiple of π/ . The experiment didnot attempt to control the relative phases and it wouldhave been possible for at least one of the second orderterms to be 0 mV. However, all three second order termscannot be 0 mV. In general our data shows that none ofthe second order terms is within five standard deviationsof 0 mV. Furthermore, considering each run there is al-ways at least one second order interference term whichis beyond 50 standard deviations from 0 mV (the low- est such maximum occurs for setting 3). This and thefact that the majority of the settings yielded a third or-der term within a standard deviation of 0 mV stronglysuggest that this experiment validates the predictions ofclassical electromagnetism with regard to the relationshipbetween optical intensities and electromagnetic fields.The primary source of error in these experiments ismost likely the ability to produce stable interference be-tween the beams that are incident on the detector. Fig-ures 3 and 4 show that when only one source is turnedon (giving I A , I B or I C ) the signal produced by the PDis fairly stable. On the other hand when two or threesources are turned on the signal becomes less stable;Fig. 4 illustrates this. This probably is a result of thefact that either five or six optical components are in-volved in the production of the signal that arrives at thesource. Fluctuations on the order of the wavelength ofthe light will clearly dramatically alter the resulting in- b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b − . − . − . − . . . . ∆ ( A , B , C ) i n [ V ] Run number
FIG. 5. Data for ∆ ( A, B, C ). The intensities are representedby the voltage readings produced by the PD in V. The whiteand color bands delineate runs with the same setting alongthe interference pattern. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b − . − . − . − . − . − . − . . . . . . . . ∆ ( A , B ) i n [ V ] Run number
FIG. 6. Data for ∆ ( A, B ) with a set-up similar to Fig. 5. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b − . − . − . − . − . . . . . . . ∆ ( A , C ) i n [ V ] Run number
FIG. 7. Data for ∆ ( A, C ) with a set-up similar to Fig. 5. terference pattern. Such fluctuations can easily resultfrom vibrations as the optical bench was not isolated orthermal drift, which would alter the index of refractionand therefore the relative phase shifts along the paths. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b − . − . − . − . − . − . . . . . . ∆ ( B , C ) i n [ V ] Run number
FIG. 8. Data for ∆ ( B, C ) with a set-up similar to Fig. 5.
In fact, we noticed that when aligning the optics thequalitative appearance of the interference pattern wasextremely sensitive to adjustments to mirrors or beam-splitters. In preliminary attempts to gain data we alsoobserved that the airflow provided by the room ventila-tion system created a noticeable drift in the visible in-terference pattern. This airflow was eliminated and thatdata was excluded from consideration but this illustratesthe difficulty of producing stable interference patterns inthis type of triple Michelson interferometer.Note that the analysis via interference terms does notrequire knowledge of the phase shifts between the threesources. These phase shifts depend on the precise align-ment of the mirrors and beamsplitters and the positionof the detector. We found that, during optical elementalignment, the visual appearance of the interference pat-tern was very sensitive to adjustments and we doubt thatwe could have predicted the associated phase shifts. Therelative strengths of the the electric fields produced bythe three sources also depends on the reflectivity andtransmittivity of the beamsplitters and mirrors. Againthese details are irrelevant for the analysis in terms ofthe interference terms. All that is required are the vari-ous intensities at the detector for each setting.
V. DISCUSSION AND CONCLUSION
We have presented an experiment to validate the de-scription of optical intensity via electromagnetic fields as-sociated with light. The experiment only relies on inten-sity measurements and does not require any informationabout the relative phases between multiple sources thatproduce the light that is subsequently detected. The ex-periment introduces measurable interference terms andrelies on these to validate whether the usual theory iscorrect. The resulting data strongly suggests that theexperiment has established the validity of this approach.A strength of this approach is that it does not relyon knowledge of the precise phase relationship betweenlight sources that superimpose. This knowledge is essen-tial with typical investigations of interference using mul-tiple slits or even interferometers. The relative phasesbetween sources are invariably very sensitive to adjust-ments and the elimination of this issue vastly simplifiesthe experiment.We believe that the experiment is easier to managethan comparable experiments involving multiple slit in-terference and the particular critique of those experi-ments whose conclusion is that the third order interfer-ence term is non-zero does not immediately apply to ourexperiment . Whether a comparable issue might arisefor our type of experiment is an open question.This experiment could be extended in various ways.First, it could be done with true single photon sourcesand use photon counting rather than intensity measure-ments. In this way the rules connecting quantum statesand probabilities could be checked; this was done inthe multiple slit experiments . This would entail thecost and management of single photon sources and pho-ton counting devices and would also introduce statisticalanalyses associated with dark counts and detector effi-ciencies. However, for undergraduate students, it wouldoffer the benefit of direct use of the foundations of quan-tum theory to predict the outcome of experiments.Second, the layout of the experiment allows for in-troduction of additional optical elements into individualbeam paths. This could be used to incorporate the effectsof polarization of the light sources. For example, if thepolarization state of one source could be rotated relativeto the others then the relationships that the interferenceterms satisfy would change. These could be explored inan experiment where the data gathering is no more dif-ficult than that which we have done. At the classical level, this would expose undergraduate students to thevector nature of the electric field and its relationship tointensity. At the quantum level, it would allow studentsto explore the quantum nature of the path possibilitiesalongside that of polarization. Such investigations wouldbe very difficult to conduct with the multiple slit andmask arrangement of previous experiments .This type of analysis allows for testing of candidatetheories beyond those of simple addition of intensity orthat resulting from classical electromagnetic fields. Forexample, perhaps a possible theory predicts that the in-tensity produced when three sources superimpose is acombination of the intensities for all twofold combina-tions and does not depend on the intensities when a singlesource is active. With energy conservation, such a the-ory might predict that I ABC = ( I AB + I BC + I AC ) / that compare predic-tions of quantum physics to a broad class of plausiblephysical theories. Rather than try to investigate theproperties of quantum states or competing alternativesdirectly, such experiments use probabilities and correla-tions, toggling between a variety of experimental settingsand ultimately combining the resulting probabilities is asensible way. Our experiment has the same flavor and wethink would be very instructive way for undergraduatestudents to explore similar types of indirect inferences. ∗ [email protected] † Current Affiliation: Blue Line Engineering, 525 E. Col-orado Avenue, Colorado Springs, CO 80903 G. Brooker,
Modern Classical Optics (Oxford UniversityPress, Oxford, United Kingdom, 2002). E. Hecht,
Optics (Addison Wesley, San Francisco, CA,2002). C. A. Bennett,
Principles of Physical Optics (Wiley, Hobo-ken, NJ, 2008). I. R. Kenyon,
The Light Fantastic (Oxford UniversityPress, Oxford, United Kingdom, 2008). R. Sorkin, Mod. Phys. Lett. A. , 3119 (1994). U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, andG. Weihs, Science , 418 (2010). I. S¨ollner, B. Gsch¨osser, P. Mai, B. Pressl, Z. V¨or¨os, andG. Weihs, Found. Phys. , 742 (2012). T. Kauten, R. Keil, T. Kaufmann, B. Pressl, ˇC. Brukner,and G. Weihs, New J. Phys. , 033017 (2017). H. De Raedt, K. Michielsen, and K. Hess,Phys. Rev. A , 012101 (2012). R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha,Phys. Rev. Lett. , 120406 (2014). J. Q. Quach, Phys. Rev. A , 042129 (2017). A. R. Barnea, O. Cheshnovsky, and U. Even, Phys. Rev. A , 023601 (2018). B.-S. K. Skagerstam, J. of Phys. Comm. , 125014 (2018). G. Rengaraj, U. Prathwiraj, S. N. Sahoo, R. Somashekhar,and U. Sinha, New J. Phys. , 063049 (2018). F. Jin, Y. Liu, J. Geng, P. Huang, W. Ma,M. Shi, C.-K. Duan, F. Shi, X. Rong, and J. Du,Phys. Rev. A , 012107 (2017). J. P. Cotter and R. P. Cameron,J.l of Phys. Comm. , 045012 (2019). K. S. Lee, Z. Zhuo, C. Couteau, D. Wilkowski, and T. Pa-terek, Phys. Rev. A , 052111 (2020). M.-O. Pleinert, J. von Zanthier, and E. Lutz,Phys. Rev. Research , 012051 (2020). N. D. Mermin, Am. J. Phys. , 940 (1981). A. Aspect, P. Grangier, and G. Roger,Phys. Rev. Lett. , 460 (1981). M. Giustina, M. A. M. Versteegh, S. Wengerowsky,J. Handsteiner, A. Hochrainer, K. Phelan, F. Stein-lechner, J. Kofler, J.-A. Larsson, C. Abell´an,W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer,T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam,T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger,Phys. Rev. Lett.115