Characterization of Fiber Optics Devices for LASIC II

Morgan DixonDepartment of Physics and Astronomy, Carleton College, Northfield, Minnesota

AbstractInterferometers used for gravitational wave detection require highly stabilized lasers in order to detect

small changes in arm lengths. LASIC II uses a hyperfine transition of Iodine to stabilize a Nd:YAG laser tobe used aboard LISA. Two fiber optics devices important to the experiment are an Evanescent Optics

evanescent wave coupler (EWC) and a Jenoptik electro-optic phase modulator (EOM). It was determinedthat both are polarizing maintaining, following manufacturer specifications. The EWC has a mechanicalhysteresis altering the location of maximum evanescent wave output, which the manufacturer warns of.

The input beam to the EOM must be of the correct polarization to obtain expected insertion losses and itscharacteristic Vπ is found to vary with the frequency of modulation, however, the exact relationship

between the two quantities is unknown.

1. Introduction

Contrary to Newton describing gravity as an ”ac-tion at a distance” force, Albert Einstein’s The-ory of General Relativity imposes a speed limiton gravity, a consequence of which is gravita-tional waves [1]. Described as ripples in space-time, gravitational radiation applies stretchingand compressing tidal forces upon matter with-out significantly altering the gravitational wave.Since gravitational waves propagate nearly un-altered through space time they contain undis-turbed sources of information about their cre-ation, whether it be astrophysical or cosmolog-ical in origin. Observing gravitational waveswould lead to unprecedented knowledge of theastrophysical bodies which emit them, such asrotating binary neutron stars and binary blackholes, as well as the time period, t ≈ tdec, afterthe big bang when gravitons, the force carrierparticle of gravity, broke free from the primordialplasma, providing information about physics athigh energies as of yet experimentally obtainedon Earth [2].

Gravitational waves were first indirectly de-tected by Joseph Taylor and Russel Hulse, whoobserved the expected effects of gravitationalradiation on a pulsar in a binary orbit witha neutron star. Attempts have been made todirectly observe gravitational waves since the1960’s, when Joseph Weber first introduced whatare now known as Weber Bars, long aluminumbars constructed to resonate at the expected fre-quency of passing gravitational waves. While

Weber failed to directly observe gravitationalwaves, current experiments hope to observeslight changes in interferometer arm lengths dueto gravitational wave tidal forces. Two landbased projects include LIGO and VIRGO, how-ever, they are limited by seismic noise, so LISA,an interferometer set to operate in space, is inplanning [1]. In order to make precise measure-ments of the interferometer arm lengths and de-termine if a variation is due to a gravitationalwave, very stable lasers are necessary. Onemethod of stabilizing lasers is through a hyper-fine transition of iodine, as the LASIC projecthas achieved and the LASIC II project is hop-ing to improve upon and make space worthy forLISA [3].

One goal of LASIC II is to decrease thesize of the stabilization apparatus, which canbe achieved by using fiber optics to guide laserbeams as opposed to free beams. Two fiber op-tics devices to be used in the proposed LASICII apparatus are an Evanescent Wave Couplerand an Electro-Optic Modulator. Before thesedevices are used in the apparatus, it is impor-tant to know whether they work according tothe manufacturer’s specifications, which was thegoal of the experiments presented in this pa-per. First the polarization of the devices wasstudied, namely determining if the devices werepolarizing maintaining, that is, if a linearly po-larized beam is input into the device the out-put beam is also linearly polarized. Specific tothe Evanescent Wave Coupler, the coupling ra-

M. Dixon 1

tio of how the device splits an input laser beamwas studied. Finally, the characteristic Vπ of theElectro-Optic Modulator was studied to find howit changes with the frequency at which the deviceis modulated.

It was found that both the Evanescent WaveCoupler and Electro-Optic Modulator are in-deed polarizing maintaining, following manufac-turer specifications. Additionally, a non-fiberedAcousto-Optic Modulator was found to be polar-izing maintaining, which was necessary for theapparatus to study the Electro-Optic Modula-tor. The coupling ratio of the Evanescent WaveCoupler varied as expected, where a mechanicalhysteresis altering the location of the maximumevanescent wave power output makes the devicemore difficult to use, however, the manufacturerwarns of this complication. Finally, it was foundthat the characteristic Vπ changes with the mod-ulation frequency of the Elector-Optic Modula-tor, however, the exact relationship between thetwo quantities is as of yet unknown. Unexpect-edly high values of Vπ were found, some sur-passing 5 V, complicating the final results, asthe manufacturer specifies that at a modulationfrequency of 1 kHz Vπ = 3.2V, which does notmatch experimental results and no mention ismade that Vπ could depend on the frequency ofmodulation.

2. Background

2.1 Gravitational Waves

Gravity and it’s effects have fascinated humankind since antiquity. Greek philosophers, suchas Plato and his student Eudoxus, described themotion of heavenly bodies in terms of a seriesof perfect crystal sphere centered on the Earth,which was updated by Ptolemy in Alexandria,who added the ”primum mobile” or the pri-mary mover as the motivating force behind themovement of the stars. The Ptolemic modelwas criticized by Islamic scholars but it was notuntil Copernicus revised the Greek philosopherAristoxenus’ work and reintroduced a heliocen-tric model of the solar system and Kepler in-troduced elliptical orbits as opposed to circular

orbits, that the correct model of planetary mo-tion was constructed [4]. Galileo then discov-ered the constant of acceleration due to gravityand supported the Copernican model of the solarsystem. Newton followed with three fundamen-tal laws of motion and a mathematical formal-ism of gravity’s effects that successfully predictedKepler’s laws of planetary motion, the universallaw of gravitation. One important consequenceof Newton’s gravity which ”acts at a distance”is that the force propagates at an infinite speed.In the case of a rapidly changing gravitationalfield, for example, with rotating binary neutronstars, the field would instantaneously change andan observer could register those changes imme-diately. With no time delay there could be nogravitational waves as the gravitational energydoes not need to propagate. The idea of grav-ity acting at a distance was disquieting to somephysicists, amongst whom was Laplace. Laplaceattempted to resolve the problem by describinggravity as a fluid which moved from its source tothe body it affected at a finite speed, however,he was unable to observe the expected effects ofgravity moving at a finite speed, thus he con-cluded that the speed of gravity must be large,in fact, larger than the speed of light. Laplacehad the correct idea, however, once Einstein in-troduced the Theory of Special Relativity [5] in1905, the universal speed limit of c, the speedof light, was imposed, ruling out the possibilitythat gravity could move faster than the speed oflight. Then, in 1916 Einstein published the The-ory of General Relativity [6], which gives the fi-nite speed of c to gravity, consequently introduc-ing the possibility of gravitational waves, whichwere later mathematically formalized by teamsof physicists in the 1960’s and 1970’s [1].

Deriving from two main sources, cosmolog-ical origins and astrophysical bodies, gravita-tional radiation is expected to be observed asa superposition of the many sources of gravita-tional waves, called the stochastic gravitationalwave background (SGWB) [7], where each sourceemits gravitational radiation at unique frequen-cies. Two types of waves are longitudinal andtransverse waves, differing in their direction ofpropagation. Longitudinal waves, such as sound

M. Dixon 2

waves, propagate in the direction of motion,whereas transverse waves propagate in the planeperpendicular to that of the direction of motion,and include water waves, electromagnetic wavesas well as gravitational waves. However, differ-ent from water waves and electromagnetic waves,which propagate through water and by oscillat-ing electromagnetic fields respectively, gravita-tional waves propagate through space-time, thefour dimensional ”fabric” consisting of the famil-iar three dimensions of space where time is thefourth dimension [1]. How a gravitational wavepropagates through space-time can be imaginedas follows. A mass distorts the space-time inwhich it exists analogous to the distortion cre-ated by a tennis ball placed on a taut fabric.Once the ball is stationary the fabric too is sta-tionary. Even if the ball were spinning within thedivot it created in the fabric, the fabric would notbe moving [8]. This is similar to how a rotatingastrophysical body will not create gravitationalradiation whereas a body with a changing grav-itational field, such as a star collapsing in a su-pernova or a binary system of neutron stars orblack holes, creates gravitational radiation whichcan be observed as gravitational waves [1]. Thecosmological source of gravitational waves comesfrom the moments after the big bang when theuniverse was a high energy plasma of particles,and gravitons, the force carrier particle of grav-ity, were able to break from the high energyplasma and radiate through space-time [2].

We come to observe gravitational waves asthey propagate through space-time to Earth,similar to replacing the taut fabric from abovewith a still pool of water, where the disturbanceof the tennis ball creates waves which propa-gate out from the point of contact. As thewater waves move away from the point of ori-gin their amplitude decreases, which also occurswith gravitational waves, whose amplitude de-creases inversely with the distance traveled, 1

r ,where r is the distance traveled, as shown in Fig-ure 1 [9]. This means that gravitational wavestraveling long distances diminish in amplitudeand are weak by the time they reach Earth.While gravitational wave signals are weak, oneof their important properties is that they dis-

tort mass they may encounter as they travel butdo not significantly loose energy, meaning thatthe information they contain is not appreciablychanged from their creation to their observation,making them a critical tool to further study theirsources [10].

Figure 1: In this artists rendition of a gravitationalwave the gridded two dimensional surface representsthe truly four dimensional space-time, where the twoblack holes in a binary orbit create the disturbancein space-time which propagates away from the originin waves of decreasing amplitude.

For example, they provide direct access to themass and period of rotation of rotating neutronstars and even allow for detection of pulsars thathave not been found by radio detection. Detec-tion of gravitational waves will also further thestudy of black holes as the only information thatthey directly emit is gravitational radiation andnearly undetectable Hawkings radiation, improv-ing upon the current indirect methods of observ-ing black holes by the effects they impose upontheir surroundings [1]. Background gravitationalradiation will also provide cosmological informa-tion about the inflationary models, pre-Big-Bangmodels, electroweak phase transition, and cosmicstrings, allowing physicists to probe the earlieststates of the universe, for example, when it was

M. Dixon 3

less than one minute old, a feat as of yet not ac-complished experimentally [7] as the energy atthat time was higher than we are currently ca-pable of creating on Earth.

2.2 Detection of Gravitational Waves

Gravitational wave detection involves observinghow an incoming gravitational waves distortsa mass, for example as shown in Figure 2 [1],where a gravitational wave distorts a circle offree particles by changing the circle to an ellipseof the same area, corresponding to tidal stretch-ing and compressing of a solid mass, detectableby changes in length. However the change indistance of a test particle due to a gravitationalwave is predicted to be at largest, 10−18 m, whichis smaller than the diameter of a proton and thethermal motion of atoms in a test mass [1].

Figure 2: Each ring of dots represents a ring of freeparticles, where the top and the bottom lines of ringsare distorted into different ellipses as a gravitationalwave passes but are identical, perfect circles, whenthere is no gravitational wave or it is equidistant fromthe masses.

In the 1960’s Joseph Weber was the firstphysicist to attempt to observe gravitationalwaves using an apparatus now known as WeberBars. The original bar was two meters long andconstructed from aluminum with piezoelectricsensors attached to convert vibrations to elec-tric signals. The bar was constructed such thata gravitational wave traveling at a right angle tothe long axis of the bar would stretch and com-press the bar along this axis, and if vibrated atthe correct frequency, 1 kHz, determined by theexpected frequency of gravitational waves emit-ted from astrophysical sources such as coalesc-ing binary neutron stars, the bar would amplify

the vibrations to ease detection. By placing onebar in Maryland and a twin at Argonne NationalLaboratory, Weber was able to cancel seismicnoise and random thermal noise unique to eachbar and observe the signals common between thetwo, believed to be gravitational waves. How-ever, repetitions of his experiment elsewhere pro-duced not but the expected noise, providing in-conclusive results as to Weber’s observations andleaving the scientific community without a di-rect observation of gravitational waves. WhileWeber was unable to obtain conclusive results,some current experiments continue to use We-ber Bars, however, cooled to a temperature of0.1 K to reduce thermal noise [1].

While Weber failed to prove the existence ofgravitational waves, Joseph Taylor and RusselHulse in 1974 observed gravitational waves indi-rectly by their effects on a binary pulsar. Hulseobserved a pulsar with a variable pulsation pe-riod, the first of it’s kind, and it was determinedwith Taylor that the pulsar is in a binary orbitwith a neutron star, moving at highly relativis-tic speeds of nearly 0.1% the speed of light in anelliptical orbit. These qualities allowed the pairto observe a gravitational redshift in the changeof spin rate as the pulsar moved through it’s or-bit as well as the orbit shrinking as the gravi-tational radiation carries energy away from thesystem, two effects predicted by the existence ofgravitational radiation and providing an indirectdetection of gravitational waves [1].

Even though Taylor and Hulse have provedthe existence of gravitational waves it is desir-able to directly observe the radiation in order tostudy it’s source, leading physicists to the cur-rent gravitational wave detectors which use in-terferometers.

2.3 Interferometers

The basic principle of an interferometer is to splita light source into two or more partial beamsthen send each beam on a different path and re-combine the partial beams to create an interfer-ence pattern which changes depending upon thedifference in path lengths and or phases. Usingthe effects of interferometry it is possible to de-termine the path length difference between the

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split beams, the basic premise of gravitationalwave detectors, as a gravitational wave’s tidaleffects would cause stretching and compressionof the two paths. Two basic interferometers, theMichaelson, the Mach-Zhender Interferometers,as well as a Fabry-Perot Cavity, also called aFabry-Perot Interferometer, are described below.

Created by Albert Michaelson, the Michael-son Interferometer uses a beam splitter to splita monochromatic beam into two parts of equalintensities, where a two plane mirrors redirecteach beam back to the beam splitter and theinterfered beams are incident upon a detector,as shown in Figure 3. One advantage of thisinterferometer is that when the arms are ofequal length the noise and instabilities originat-ing from the laser are canceled as the two in-terfered beams. This configuration also allowsfor easy distance measurements since moving onemirror a distance equal to a mere quarter of awavelength of the light in the apparatus, λ

4, pro-

duces a change in the interference pattern at thedetector from constructive to destructive inter-ference [11].

Figure 3: The basic apparatus for a Michaelson In-terferometer. A monochromatic light source is splitby a beam splitter, BS, then each beam is reflectedback towards the BS by means of two plane mirrors,M1 and M2, where the beams are recombined intoone and sent to a detector, which detects the inter-ference pattern created by a change in arm length.

A modified version of the Michaelson Inter-ferometer, a Mach-Zhender Interferometer, al-lows for precise measurements of phase shift be-

tween the split beams, a basic apparatus forwhich is shown in Figure 4. Beginning simi-larly to a Michaelson Interferometer, a Mach-Zhender Interferometer splits a monochromaticlight source with a beam splitter, sending eachbeam on a separate path guided by a plane mir-ror. Instead of recombining the two beams bythe same beam splitter that split them, they arerecombined by a second beam splitter and theinterfered beam is sent to a detector. Reflec-tions off of the plane mirrors introduce phaseshifts and are different for the two beams as thebeam splitter introduces different phase shiftsfor the beams that are reflected and transmit-ted through the cube [12].

Figure 4: A Mach-Zhender Interferometer beginswith a monochromatic light source which is split bya beam splitter, BS, where the two beams are di-rected by two plane mirrors, M1 and M2, to a secondbeam splitter which recombines the two beams andtransmits the recombined beam to be detected.

A Fabry-Perot Cavity or Interferometer con-sists of two mirrors facing one another, eitherplane or spherical mirrors of equal radii, wherethe mirrors are positioned such that a beam in-cident parallel to the axis of the interferometerresonates within the cavity. A spherical cavityis shown in Figure 5 [12]. Such cavities are use-ful as they resonate most efficiently when thefrequency of the laser is an integer multiple ofthe quantity, c

2L , where L is the length of thecavity, and the maximum transmission occursat the same frequencies, creating a filter on thefrequency of the laser, narrowing it’s bandwidth[10].

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Figure 5: A spherical Fabry-Perot Cavity consistsof two spherical mirrors of equal radii, M1 and M2,aligned along the axis, shown in a dashed line, whereL is the distance between the two mirrors. When alaser beam is incident parallel to the axis of the cav-ity it resonates within the cavity, as traced by thearrows, then is transmitted.

2.4 Interferometers for GW Detection

Michaelson interferometers are used in gravita-tional wave detection as a passing gravitationalwave causes deformation of the arms, subse-quently changing the length of the arms. A dis-turbance due to gravitational waves is expectedto cause a change in arm length on the orderof 10−18 m, a distance much smaller than thetypical infrared wavelength of laser light used,meaning that the interferometer is highly sen-sitive to external vibrations as well as thermalnoise of the mirrors creating slight changes inarm length, which is circumvented by placing theexperiments in underground vacuums. The pho-tons from the laser also introduce a shot noisewhich is reduced by using a high intensity beam[1].

Current experiments using interferometry in-clude VIRGO and LIGO ( Laser InterferometerGravitational-wave Observatory), both of whichconsist of two underground Michelson interfer-ometers created from suspended mirrors witharms between two and four kilometers long [1].While these interferometers have yet to make adirect observation of a gravitational wave, collab-orations have been able to place an upper limiton the SGWB in the frequency band around 100Hz, ruling out some aspects of early universe

models [7].

Another source of noise inherent in landbased gravitational wave interferometers is grav-itational noise from the Earth. Since the de-tector is built to be sensitive to tidal gravita-tional forces it cannot distinguish between thosecreated on Earth, by matter interacting on orin the Earth, and weak signals originating inspace. This limits the sensitivity range of suchland detectors to high energy gravitational wavesources from space and for that reason, LISA(Laser Interferometer Space Antenna), was con-ceived [1]. Consisting of three space craft form-ing an equilateral triangle with arms 5 ∗ 106 kmlong, each satellite contains a test mass in freefall and emits two laser beams, connecting it tothe other two satellites, resulting in a total of sixlaser links, as shown in Figure 6. The configu-ration is to orbit the sun about 20 behind theEarth and rotate once a year, as shown in Fig-ure 7. LISA aims to detect gravitational wavesin the 10−4 − 1 Hz range, and achieve a sensitiv-ity of detecting a gravitational deformation as

small as ≈ 7 ∗ 10−21√Hz−1 around the frequency5 mHz. With this frequency range and sensitiv-ity LISA hopes to observe gravitational radiationfrom black holes, binary neutron stars as well asthe Big Bang [13].

LISA’s main challenges come from the abil-ity of the spacecraft to follow the free falling testmasses and of precise phase shift measurements.This later goal is made more difficult since theinterferometer is not a classical Michaelson In-terferometer which cancels out laser noise byrecombining beams originating from the samelaser, instead, LISA combines two beams origi-nating from two different lasers, requiring highlyfrequency stabilized lasers and numerical algo-rithms to cancel the noise. Many laser stabi-lization methods have been considered, such asusing a Fabry-Perot Cavity. This provides highstability over a short time period, however, thematerial of the cavity is subject to thermal andmechanical noise that is difficult to control, soa second solution was proposed, to stabilize thebeam on a hyperfine transition of iodine whichoccurs at λ = 532 nm, exactly the second

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Figure 6: The basic configuration of LISA, threesatellites are connected by six laser links in the for-mation of an equilateral triangle with arms around5 ∗ 106 km long.

harmonic of the Nd:YAG lasers emitting at λ =1064 nm used for the laser links of the satellites.This method is more effective than that of theFabry-Perot Cavity as the iodine will be excitedand emit at only one wavelength of light ratherthan introducing noise due to excess radiationbuilt up from the beam resonating in the cavity.The goal of LASIC (Frequency Stabilized Laserto Iodine Lines for Space Applications) was todevelop this method, and the goal of LASIC II,the apparatus for which is shown in Figure 8, isto improve upon the methods developed in thefirst project and bring the stabilization methodcloser to being space ready, namely, by makingthe stabilization apparatus compact enough tofit aboard a space craft.

3. Devices Characterized

Two important devices used in the LASIC IIexperiment, shown in Figure 8, are a fiberedEvanescent Wave Coupler, which acts as a vari-able beam splitter and a fibered Electro-Optic

Figure 7: The location of the LISA satellites fromFigure 6 in orbit around the Sun, approximately 20behind the Earth.

Modulator, which introduces a phase shift to oneof the split beams. Before the apparatus can becompleted it was necessary to characterize thesedevices to be sure they would function properlyin the experiment. While these two devices werethe main goals of characterization, a non-fiberedAcousto-Optic Modulator was also characterizedfor use in a heterodyne interferometry experi-ment to study the Electro-Optic Modulator.

3.1 Evanescent Wave Coupler (EWC)

An Evanescent Wave Coupler (EWC) is an op-tical waveguide that splits an input beam invarying ratios via the phenomenon of evanescentwaves. In general, a waveguide is a device whichguides the direction of propagation of waves, forexample, a fiber optics cable uses the property oftotal internal reflection to guide electromagneticradiation in the optical wavelength range. Totalinternal reflection occurs when light strikes a sur-face at an angle greater than the critical angle,as derived from Snell’s Law, Equation 9, wheren1 and θ1 are the index of refraction and angleof incidence of the light through the first mate-rial, and n2 and θ2 are the index of refractionand angle of transmission of the light throughthe second material [11], as shown in Figure 9.

n1 sin(θ1) = n2 sin(θ2) (1)

If light passes from a material of higher indexof refraction to a material with lower index of re-fraction, the light bends away from the normal,as shown in Figure 9, and if the refracted angle

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Figure 8: The proposed LASICII apparatus. A Nd-YAG laser emitting at λ = 1064 nm in the TEM00 modeis frequency doubled to λ = 532 nm by a fibered frequency doubling crystal. The beam is then split by anEvanescent Wave Coupler, where the evanescent beam is sent to the iodine cell to be stabilized by meansof the hyperfine transition of iodine at its 532 nm line and the non-evanescent beam is phase shifted by afibered Electro-Optic Modulator.The fiber optics devices allow for a compact apparatus, small enough to fitwithin a space craft.

is set to 90, so that the light is reflected insteadof refracted within the second material, Equa-tion 1 can be solved for the critical angle, θc,also known as Bruster’s angle, where the lightis totally internally reflected within the secondmaterial [11], given by Equation 2.

θc = arcsin(n2

n1

) (2)

When light is totally internally reflected itcannot be discontinuous at the boundary of themedium of transmission, rather a portion of thelight has an exponentially decaying probabilityto be transmitted instead of reflected, creating

a second, less powerful beam whose amplitudedecays exponentially as the distance from themedium of transmission is increased, called anevanescent wave. The electric field of the trans-mitted, evanescent wave, ET , is given by Equa-tion 3 [14], where the real exponential showsan exponential decay in the z direction. Anevanescent wave transmission is shown in Figure10 [15].

ET = Aze−βzei(ωt−kr sin(θ)

nr) (3)

An EWC utilizes evanescent waves to splitan input laser beam by varying the distance be-

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tween two monomode fiber optics cores, wherethe closer the cores the greater the evanescentwave transmission and as the cores are separatedthe evanescent wave transmission decreases.

The EWC studied in these experiments wasmade by Evanescent Optics Inc [16].

Figure 9: Light passing from a material with indexof refraction, n1, makes an angle of θ1 with the nor-mal, and is bent towards the normal at an angle of θ2when the index of refraction of the second material,n2, is larger than the first.

Figure 10: Example of an evanescent wave trans-mission. A beam is totally internally reflected withina fiber optics core, shown by the blue triangle, anda second fiber optics core, shown as a pink circle, re-ceives a weak beam when the cores are further apart,as on the right, and a stronger beam when the coresare closest, as shown on the left.

3.2 Acusto-Optic Modulator (AOM)

The photo-elastic effect describes how applyinga mechanical stress to a crystal changes it’s per-mittivity, thus how it refracts optical light, orit’s index of refraction. The acousto-optic effect

makes use of this phenomenon by inducing me-chanical stress on a crystal via a sound wave. Atypical application of the acousto-optic effect isshown in Figure 11, where it is used to diffractan input beam by the Bragg angle, θB, as givenin Equation 4, where λ is the wavelength of theinput beam and Λ is the wavelength of the soundwave applied to the crystal.

θB = sin( λ2Λ) (4)

Figure 11 shows an Acousto-Optic Modula-tor (AOM), which is used to shift the input beamin frequency, also called a doppler shift, withthe diffracted output beams, while the trans-mitted beam remains at the same frequency asthe input beam. When an AC signal is appliedto the Piezo-Electric Transducer, pressure wavesare created which deform the crystal periodically,subsequently changing the index of refraction ofthe crystal in a periodic manner. The change inrefractive index acts similar to a diffraction grat-ing, resulting in a normally transmitted beamand diffracted beams at multiple harmonics, eachseparated by the angle given in Equation 4 [17].

The AOM studied in these experiments wasmade by AA Opto-Electric [18].

Figure 11: Basic schematic of an Acusto-OpticModulator. An AC signal is applied to the Piezo-Electric Transducer, deforming the crystal and creat-ing pressure waves which in turn deform the quartzcrystal, periodically changing it’s index of refraction,causing an input optical laser beam to be diffracted.

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3.3 Electro-Optic Modulator

The purpose of the Electro-Optic Modulator(EOM) in the LASIC II experiment is to intro-duce a phase shift to the input beam, which isachieved by means of the linear electro-optic ef-fect. Similar to the acousto-optic effect which in-troduces a change in refractive index to a crystalby means of an acoustic wave, the electro-opticeffect introduces a change in refractive index toa crystal by means of an applied electric field.Quantitatively the electro-optic effect is given byEquation 5, where n is the index of refraction inthe material altered by the electro-optic effect,n0 is the index of refraction of the material be-fore an electric field is applied to the material,a is the linear electro-optic effect, and E0 is theapplied electric field.

n = n0 + aE0 + bE2

0 + ... (5)

The second order term in Equation 5, bE20

is called the Kerr Effect, whereas the first orderterm, aE0 is the linear electro-optic effect, theunderlying principle of how the EOM functions.

The electric field applied to the crystal is de-fined in terms of the voltage applied perpendicu-lar to the polarization vector of the input beam,Vs, by Equation 6,

E0 = Vs

d, (6)

which allows for the phase shift induced onan input beam, ∆φ, to be solved for in terms ofthe voltage applied to the device by Equation 7,

∆φ = 2πkaVs. (7)

An important characteristic of the EOM isthe half-wave voltage or Vπ, the applied voltageneeded to introduce a phase shift of π to the in-put beam [17]. It is also important to note thatthe input beam to the EOM must be of the cor-rect polarization, otherwise the device will notfunction properly. Since the voltage to create theelectro-optic effect must be applied perpendicu-lar to the polarization vector, the input beammust not only be linearly polarized but at thecorrect angle. Without the correct angle of po-larization insertion losses of around 20 dBm were

obtained, however, once the polarization anglewas changed by a λ

2waveplate, the polarization

vector of the input beam was perpendicular tothe applied voltage and insertion losses of 5 dBmwere obtained, better than the manufacturer’sclaims of 7 dBm.

The EOM used in these experiments wasmade by Jenoptik [19].

4. Polarization

Each of the three devices studied, the Evanes-cent Wave Coupler, the Acousto-Optic Modula-tor and the Electro-Optic Modulator, are polar-izing maintaining, according to their manufac-turers, and since interferometry requires consis-tently polarized beams it was necessary to verifythat each device maintains the linear polariza-tion of the input beam. While the device mustbe polarizing maintaining, it does not have tomaintain the angle of polarization as that is eas-ily controlled with a half-wave plate.

4.1 Background

Light, as a transverse electromagnetic wave, con-tains an electric and a magnetic field which prop-agate perpendicular to one another. The polar-ization of light is a property of the electric fieldvector’s orientation, such that by projecting theelectric field vector, E, onto the cartesian coordi-nates, x and y, the relative magnitude and anglebetween the x and y components determines thepolarization of light. For example, if the electricfield of an electromagnetic wave creating a laserbeam is described by Equation 8,

E = xEox cos(kz−ωt)+ yEoy cos(kz−ωt+θ), (8)where Eox and Eoy determine the amplitude

of the EM wave and θ is the phase shift betweenthe x and y components, then the polarizationof the laser is determined by θ and the relativemagnitudes of Eox and Eoy .

Light is considered linearly polarized if θ =0,π as there is no phase shift between the x andy components of the electric field, thus the twocomponents are oscillating about a line, as shownin Figure 12a. If, however, there is a phase shift

M. Dixon 10

(a)

(b)

Figure 12: The electric field components, Ex andEy, are of equal magnitude for both linearly and cir-cularly polarized light, however, there is a phase shiftof θ = 0 for linearly polarized light (a), and a phaseshift of θ = ±π

2 for circularly polarized light (b).

of θ = ±π2between the two components of the

electric field and each component has the samemagnitude, Eox = Eoy , then the light is consid-ered to be circularly polarized as the phase shiftcauses the electric field vector to trace out a cir-cle in the x-y plane, as shown in Figure 12b [20].

If the x and y components of the electric fieldhave any phase shift other than θ = 0,±π

2, and if

θ = ±π2but the amplitudes of the components are

not equal, such that Eox ≠ Eoy , the light is con-sidered to be elliptically polarized as the electricfield vector can trace an ellipse in the x-y planewith any major to minor axis ratio at any arbi-trary angle, a shown in Figure 13 [20].

4.2 Polarizers and Waveplates

Two useful tools in studying the polarization oflight are polarizers and wave-plates, also calledphase retarders.

(a)

(b)

Figure 13: The electric field components, Ex andEy for elliptically polarized light can be of any phaseshift other than θ = 0,π,±π

2 (a) and any relative am-plitude (b).

One means of producing polarized light is byreflection and transmission, since light reflectedfrom a smooth surface at a non-zero angle of in-cidence becomes partially polarized. The lightwhich is transmitted through the plane is calledπ light, as it’s electric field is parallel to that ofthe plane, and the reflected light is called σ light,as it’s electric field is perpendicular to the planeand cannot be transmitted. By stacking manysuch plates a plate polarizer is created, wherechanging the angle of incidence upon the platepolarizer changes how much light is transmit-ted and how much is reflected. Thus, changingthe angle can result in all π light, where thereis complete transmission through the polarizer,called a maximum of transmission, or in all σlight, where there is complete reflection off ofthe plate polarizer, called a minimum of trans-mission, or in partial transmission and partialreflection [20]. By studying the amount of lighttransmitted through a polarizer the polarizationof the light can determined.

M. Dixon 11

While polarizers allow the polarization of anincident beam to be studied, wave-plates ac-tively alter the polarization of an incident beam.Wave-plates are also known as phase retardersas they change the relative phase between com-ponents of the electric field. A typical methodfor creating a wave-plate is using quartz crystal,where the optical axis is positioned such thatthe incident polarized light is resolved into it’scomponents, thus creating a phase shift betweenthem when they exit the crystal. The phase shiftis proportional to the thickness of the plate, thusit is also dependent upon the wavelength of theincident light, such that a wave-plate that actsas a λ

4for λ = 1064 nm light will act as a λ

2for

λ = 532 nm light.The difference between thesetwo types of wave-plates is that a λ

4will not only

change the angle of polarization but turn ellipti-cally or circularly polarized light to linearly po-larized light, whereas a λ

2wave-plate will only

change the angle of polarization. Thus, by usingthe two plates in conjunction with one anotherit is possible to finely control the linear polar-ization of light and the angle at which that po-larization occurs when observed through a planepolarizer [17].

4.3 Experimental Methods

To verify that each device studied was indeed po-larizing maintaining, that is each device’s outputbeam preserved the linear polarization of the in-put beam, the basic apparatus shown in Figure14 was used. For the fibered devices, the EWCand the EOM, it was necessary to bring the beamfrom the fiber optics cable to open air in orderto use the plane polarizer, thus a mounted fiberoptics coupler was used. A plane mirror directedthe beam through a plane polarizer whose out-put was read on a photodiode. The informationfrom the photodiode was then sent to an oscil-loscope and read through a LabVIEW programrun from the computer to which the oscilloscopewas connected. To measure the polarization ofthe non-fibered devices, such as the AOM, thecoupler was not used and the output from thedevice was directly sent through the polarizer.

Figure 14: Basic apparatus used to measure thepolarization of fibered devices. The beam is broughtfrom the monomode fiber optics cable to open air byuse of a coupler, the free beam is then directed to aplane mirror, through a plane polarizer, whose out-put is sent to a photodiode, marked PD. The outputfrom the photodiode is then read by an oscilloscope.

To find the polarization of a beam incidentupon the plane polarizer, the angle marked onthe polarizer was changed in increments of 10or 20 and the photodiode recorded the subse-quent change in power at each angle. As poweris proportional to voltage, the voltage read bythe oscilloscope and recorded in LabVIEW re-flects the change in power as the polarizer angleis altered.

4.4 LabVIEW Program

The LabVIEW program used to collect data uti-lized the drivers provided by the manufacturer tocommunicate with the Tektronix AFG 3252 os-cilloscope. A spreadsheet was initialized outsideof a stacked sequence, where first a data point iscollected from three different channels or typesof measurements, then each of the three datapoints are added to the spreadsheet by chang-ing the data point type from double precisionto a string, loosing a small amount of precisionbut not significantly such that it added error tothe measurements. The block diagram of theprogram is shown in Figure 15. The main assetof using LabVIEW was to provide multiple ac-curate measurements of the power transmittedby the polarizer, allowing the average power to

M. Dixon 12

be measured for each polarizer angle, where thestandard deviation of the power measurementsgave an estimate of error.

Figure 15: Image of the block diagram of theLabVIEW program used to record three channels ofdata from a Tektronix AFG 3252 oscilloscope into aspreadsheet.

4.5 Data Analysis

Once the raw data was collected, the mean powervalue and standard deviation from the meanwere calculated for each angle of the polarizer,giving a value and it’s error for each point.

To determine if the beam studied was linearlypolarized Malus’ Law, Equation 9 [11], was used.

P = Po cos2(θ) (9)

Malus’ Law states that the power of a linearlypolarized beam transmitted through a polarizer,P , is equal to the power incident upon the polar-izer, Po, multiplied by the Cosine squared of thepolarizer angle, θ. However, the equation onlyapplies when the angle at which the maximumpower is transmitted is defined as θ = 0. This wasachieved by taking the raw data, as shown in bluein Figure 16, finding the first maximum angle,then shifting each angle by that amount, obtain-ing the same trigonometric relationship as be-fore, however shifted by a slight angle, as shownby the red curve in Figure 16.

A linear relationship between P and cos2(θ)demonstrates that the beam is linearly polarized,while the extinction ratio of the beam demon-strates the quality of the polarization. The ex-tinction ratio of an optical beam can be definedin many ways, where one common means is touse a decibel system and report the ratio in unitsof dBm. This method is defined in Equation 10,

Figure 16: Polarization data taken on unalteredLASIC II beam. The blue curve displays the dataat the arbitrary angles measured on the plane polar-izer. The red curve displays the data at the anglesshifted such that the maximum occurs at θ = 0.where Pmin is the minimum power transmittedthrough the polarizer and Pmax is the maximumpower transmitted through the polarizer. Thelarger the ratio the higher the quality of the lin-ear polarization as it means that the beam iswell extinguished at the minimum power trans-mission and well transmitted at the maximum.

Log10(Pmin

Pmax) (10)

4.6 Results

In order to determine if the various beams fol-lowed Malus’ Law and thus were linearly po-larized, the power detected at the photodiode,normalized by dividing by the maximum powertransmitted, was plotted against the cos2 of theshifted polarization angle, θ. A linear relation-ship conveys that the beam is linearly polarizedas it follows Malus’ Law, Equation 9, and isdisplayed in Figure 17 where the output beamsfrom the Evanescent Wave Coupler, the Acousto-Optic Modulator and the Electro-Optic Modula-tor, are each linearly polarized as each displaysa linear relationship following Malus’ Law.

The extinction ratio’s of the three deviceswere then calculated to be 17.5 dBm for theAOM, 30 dBm for the EOM, 30 dBm for thenon-evanescent beam of the EWC and a range

M. Dixon 13

of 25 - 33 dBm for the evanescent beam as thecoupling ratio is raised from the minimum to themaximum.

a

b

c

Figure 17: The linear relationship following Malus’law indicates that the output beams from a.) theEvanescent Wave Coupler, b.) the Acusto-OpticModulator and c.) the Electro-Optic Modulator arelinearly polarized.

5. Coupling Ratio of Evanescent

Wave Coupler

The purpose of the EWC in the LASIC II ex-periment, as shown in Figure 8, is to split thefrequency doubled beam originating from theNd:YAG laser. Due to the EOM’s high inser-tion loss, around 5 dBm, it is necessary to splitthe beam into unequal parts, with the highestintensity beam input to the EOM so that whenthe two beams are recombined they are of nearlyequal power. The best ratio of beam intensi-ties needed for the apparatus is not known soit is desirable to use a variable splitter to addflexibility to the apparatus design. One way tocharacterize how the coupler splits the beam isby the coupling ratio, given by Equation 11, asthe ratio of the intensity of the evanescent beamin comparison to the non-evanescent beam.

coupling ratio = IEvanescentINon-Evanescent

(11)

5.1 Experimental Methods

The experimental apparatus used to measure thecoupling ratio of the EWC is given in Figure18, where the beam to be split is input into theEWC, then each beam is brought from the fiberoptics cable to open air by a coupler. Two planemirrors then direct each open air beam to a sepa-rate photodiode which is read on an oscilloscope.The LabVIEW program used in the polarizationexperiment was again used to collect informationon the relative powers of the split beams.

A micrometer knob on the body of the EWCalters the position between the two fiber opticscores within the device, consequently changingthe coupling ratio, as the power of the Evanes-cent beam is at maximum when the two cores areclosest together and diminishes when the coresare brought further apart. By changing the po-sition of the micrometer knob and recording thesubsequent change in the two beam’s powers us-ing the LabVIEW program, the coupling ratiowas found, as shown in Figure 19.

M. Dixon 14

Figure 18: The basic apparatus to measure the cou-pling ratio of the fibered Evanescent Wave Coupler.The beam input to the blue, monomode fiber opticscable is split into an Evanescent and Non-Evanescentbeam, represented by the yellow fiber optics cables.The beams are brought to open air by two couplers,then are guided to photodiodes (PD) by plane mir-rors. The output from the photodiodes is then readon an oscilloscope. The grey knob on the EWC’sblack body represents the micrometer which variesthe coupling ratio.

5.2 Mechanical Hysteresis

The LabVIEW program collects multiple datapoints at each micrometer position and takingan average of all the original points collectedthen computing the coupling ration, Equation11, reveals the relationship between the couplingratio and micrometer position shown in Figure19. The blue curve corresponds to the conditionwhen the micrometer position is set to a pointbeyond the maximum evanescent wave output,beginning at the position defined as 1.5, then de-creasing the position to 1.1. The red curve cor-responds to the inverse condition, where the mi-crometer position is begun at a low position, suchas 1.1, and is increased past the maximum tothe position 1.5. Under these two conditions themicrometer position where the evanescent waveoutput is maximum changes, where for the de-creasing condition the maximum output occursat the position 1.32 and for the increasing condi-tion the maximum output occurs at the position1.38.

Figure 19: Graph displaying the mechanical hys-teresis of the EWC. The horizontal axis is the mi-crometer position, as defined by the numbers dis-played on the knob, and the vertical axis displays thecoupling ratio. Under different conditions of findingthe maximum evanescent wave output the microm-eter position where the maximum occurs changes,where if the position is increased the maximum occursat the micrometer position 1.38 and if the position isdecreased, the maximum occurs at 1.32.

The two maximum positions indicate thepresence of a mechanical hysteresis in the mi-crometer knob, meaning that the position of theknob cannot be relied upon to find the maximumouput, unless if the exact conditions presented inFigure 19 are repeated. Similarly, searching forthe maximum by increasing and decreasing themicrometer position between, for example, posi-tions 1.3 and 1.4, results in a typical hysteresiscurve, where the maximum position alternatesbetween the positions 1.32 and 1.38.

It is not unexpected for the EWC to dis-play such a hysteresis, as the manufacturer statesthat the micrometer position should not be re-lied upon to find the maximum evanescent waveoutput. However, it is important to note whensearching for the evanescent wave’s maximumoutput, as the location of the maximum changesonce it has been passed, making it necessary toapproach the maximum location and stop nearthe maximum output. The only potential prob-lem is that it appears the hysteresis does not re-sult in a steady maximum, since the maximumcoupling ratio for the maximum at 1.38 is largerthan for the maximum at 1.32. It is not ideal

M. Dixon 15

that the coupling ratio changes by the methodof finding the maximum evanescent wave out-put, however, now that it is known, it is sim-ple to find the maximum output ratio followingthe method of increasing the micrometer posi-tion from a distance far away from the maximumand stopping one the coupling ratio has reachedaround 0.0225.

6. Vπ of Electro-Optic Modulator

In the LASIC II experiment, the EOM intro-duces a phase shift to the non-Evanescent beamsplit from the original, frequency doubled beam,as shown in Figure 8. In order to understandhow the EOM performs this phase shift, an im-portant characteristic of the device is Vπ, thevoltage needed to introduce a phase shift of πto the input beam. The manufacturer claimsthat at around 1 kHz Vπ = 3.2 V, and the fol-lowing experiment set to test this and find howVπ changes with the frequency of modulation ofthe EOM. The goal of the experiment is to splitthe original beam with the EWC, then introducea phase shift with the EOM to one beam andfrequency shift the second beam with the AOM,then recombine the two beams and use the beatfrequencies produced by the heterodyne interfer-ometry to fit a model of the expected results,which depends upon Vπ.

6.1 Theoretical Model

The heterodyne interferometry apparatus is sim-ilar to the Mach-Zhender Interferometer de-scribed in Section 2.3, as it involves interferingtwo beams derived from the same laser, however,one beam has been frequency shifted, which pro-duces a beat frequency in the interfered beams.

Beats occur when two waves, such as soundlight waves, with slightly different frequencies areinterfered, creating periodic variations in the vol-ume, for sound waves, or intensity, for light, thatare called beats. Figure 20 shows the interferenceof two such waves in black, where the frequencyof the successive minima and maxima is the fre-quency difference of the source waves.

Figure 20: Example of beats, achieved by interfer-ing two waves with slightly different frequencies.

In the heterodyne interferometer used in thisexperiment the electric field of the beam phaseshifted by the EOM can be described by Equa-tion 12, where Eo is the amplitude of the electricfield, ω0 is the optical frequency of the beam, βis a constant called the modulation index, and Ωis the RF frequency at which the EOM is driven,thus the frequency which was varied throughoutthe experiment.

EEOM = Eoei(ω0t+β sin(Ωt)) (12)

The electric field, Equation 12, can then berewritten as Equation 13,

EEOM = Eoeiω0teiβ sin(Ωt)), (13)

where the second complex exponential por-tion of the electric field can be written as a sumof Bessel Functions, as given in Equation 14.

eiβ sin(Ωt)) = n=+∞n=−∞Jn(β)einΩt (14)

The electric field of the beam doppler shiftedby the AOM can then be described by Equation15, where ω is the frequency at which the AOMis modulated.

EAOM = Eoei(ω+ωo)t (15)

It is then known that the intensity of the sig-nal received at the photodiode, thus the inten-sity of the interfered phase and frequency shiftedbeams, is proportional to the sum of the two elec-tric fields by Equation 16.

IPD ∝ EEOM + EAOM 2 (16)

M. Dixon 16

Then by substituting the electric field of theEOM, Equation 13, with the Bessel Function ap-proximation, Equation 14, and the electric fieldof the AOM, Equation 15, it can be calculatedthat the intensity of light detected at the photo-diode is proportional to the Bessel Function ap-proximation to first order as given in Equation17.

IPD ∝ J0(β) cos(ωt)+J1(β)[cos((ω −Ω)t) − cos((ω +Ω)t)]... (17)

The first three Bessel Functions in Equation17 show that the first three beat frequencies oc-cur at ω for the zeroth order, J0, and ω ±Ω forthe first order, J1.

In order to relate Equation 17 to experimen-tal data, recall that the intensity of the light inci-dent upon the photodiode is related to the powerof the beam by Equation 18, where A is the areathe light covers.

I = P

A(18)

Then power is related to the voltage readby the oscilloscope by Equation 19, where R isimpedance of the load to the oscilloscope, in thiscase R = 50Ω, as specified by the photodiodeused in the apparatus.

P = V 2

2R(19)

Finally, the characteristic Vπ is related to theinterfered electric fields through the modulationindex, β, by Equation 20, where VRF is the am-plitude of the RF signal modulating the EOM.

β = πVRF

Vπ(20)

6.2 Experimental Methods

The experiment was performed using the appa-ratus shown in Figure 21. A Nd:YAG laser pro-duces a beam in the TEM00 mode at λ = 1064nm, which is frequency doubled using a non-fibered frequency doubling crystal. The freebeam is then guided to through a λ

2waveplate

set at an angle of θ = 86 to control the angleof polarization and allow the EOM to function

properly. A lens with a focal length of 100 mm,followed by an aperture and a second lens witha focal length of 50 mm focuses the beam at acoupler which brings the beam into a monomodefiber optics cable. The Evanescent Wave Cou-pler splits the beam from the monomode fiberoptics cable into a non-evanescent and evanes-cent beam. The non-evanescent beam is broughtto a fibered Electro-Optic Modulator, which in-troduces a phase shift to the beam at the samefrequency of the signal from the signal genera-tor driving the EOM. The amplitude of the sig-nal applied to the EOM determines the ampli-tude of the modulation the EOM provides tothe input beam. The evanescent beam is thenbrought to a coupler which brings the beam fromthe fiber optics cable to open air. Two wave-plates, a λ

4set at an angle of θ = 249 and a

λ2

set at an angle of θ = 302 control the po-larization of the beam. A plane mirror thenguides the free beam to a non-fibered Acusto-Optic Modulator, which introduces a frequencyshift to the beam equal to the frequency at whichthe function generator is driving the AOM. Inthis experiment, the AOM was always driven at80 MHz. The amplifier serves to amplify thedriving signal before it reaches the AOM in or-der to increase the intensity of the first diffractedbeam output by the AOM. An aperture then se-lects the first diffracted beam from the outputof the AOM, blocking the transmitted beam aswell as all other harmonics. Similarly, the out-put from the EOM is sent to a coupler, whichbrings the beam from the fiber optics cable toopen air, where it’s polarization is controlled bymeans of two waveplates, a λ

4set at an angle of

θ = 94 and a λ2set at an angle of θ = 14. The

beam is then directed to a beam splitting cubeby means of two mirrors, to be interfered withthe first diffracted beam output from the AOM,whose position is also controlled by two mirrors.The interfered beam exiting from the BS cubeis focused on a fast-photodiode by a lens with a15 mm focal length. The signal from the FPDis then amplified and read on a spectrum ana-lyzer, which captures the beats created by theinterference of the phase and frequency shiftedbeams.

M. Dixon 17

Figure 21: The apparatus used to perform heterodyne interferometry. The beam is created by a Nd:YAGlaser at λ = 1064 nm, which is doubled in frequency by a non-fibered frequency doubling crystal. The beamis then sent through a λ/2 waveplate to control the angle of polarization. Focusing lenses and an aperturebring the beam from a coupler to a monomode fiber optics cable. The EWC then splits the beam, wherethe non-evanescent beam is sent to the EOM and the evanescent beam is brought to the AOM. WaveplatesW2 through W5 ensure that the beams exiting from the AOM and EOM are both linearly polarized at thesame angle once they exit from the beam splitting cube. The two beams are directed to the BS cube byfour plane mirrors where they are interfered and one half of the recombined beam is sent through a smallfocusing lens to a Fast-Photodiode, whose signal is amplified and read by a Spectrum Analyzer.

In order to observe beat frequencies on thespectrum analyzer the beams from the AOMand EOM must be perfectly aligned, which wasachieved using a variety of methods. Aligningthe beams by eye sight gives a rough approxima-tion of their locations. Then reading the pho-todiode into an oscilloscope allows for the sig-nals amplitudes to be maximized, so that eachbeam is well aligned on the photodiode. Finally,a beam camera, connected to a computer andrun from the program, DataRay, provides a pre-cise location of the beam on it’s CCD captors,and placing both beams in the same location onthe CCD means that they are interfering well, asseen in Table 1. Once the beats are observed onthe spectrum analyzer, final adjustments can bemade to maximize the signal observed, thoughthese are usually slight. An example of beat fre-quencies observed is given in Table 2.

Once the apparatus was complete, data wastaken by selecting a frequency of EOM modula-tion and varying the RF level of the modulationsignal. The EOM can receive a signal of 5 V am-

plitude at maximum, thus each data set begun atthe maximum allowed voltage and decreased theRF level of modulation until the first harmonicbeats were reduced to the noise threshold of thephotodiode. Since the model to fit the data andfind Vπ requires only the first order approxima-tion of the Bessel functions, only the peak ampli-tudes of the principle peak, at ω = 80MHz, andthe first harmonic peaks at ω ±Ω were recorded.The peak amplitudes were measured using thepeak search function of the spectrum analyzerand five readings were taken for each peak ateach RF level of modulation. Taking the av-erage of these data points provided the mean,which was used in data analysis, and the stan-dard deviation from the mean of the raw dataprovided the error of each measurement.

6.3 Data Analysis

Once the mean and standard deviation of theraw data is calculated, it is important to checkthe quality of the data set which can be quicklyassessed by observing how stable the peaks are.

M. Dixon 18

(a) (b) (c)

Table 1: Screenshots of the DataRay program observing the position of the beam from the (a) EOM, the(b) AOM and (c) the interfered beam.

(a) (b)

Table 2: Example of beat frequencies obtained from the heterodyne interferometry experimental apparatusin Figure 21 where ω = 80 MHz and Ω = 40 kHz. a.) The EOM is driven at an RF amplitude of +10 dBmand the beat frequencies descend from the principal peak every 40 kHz. b.) The EOM is driven at an RFamplitude of -15 dBm where the first harmonic peaks are pictures at 79.96 MHz and 80.04 MHz.

Ideally, the principle peak is consistent in ampli-tude as the RF level of modulation is changedsince the two quantities are not supposed to beinterdependent. It is also important that the twofirst harmonic peaks have nearly the same am-plitude throughout a data set, as they theoret-ically should be of the same amplitude. Whenthese conditions are met the data set is goodand the average of the two first harmonic peakswas taken in order to be analyzed. An exampleof good peak stabilities is shown in Figure 22.

Each peak was measured on the spectrum an-alyzer in decibel units, dBm, which is related tothe power detected on the photodiode in mW,P , by Equation 21.

dBm = 10log10(P ) (21)

The power of the beam is then converted tovolts by Equation 22, rearranged from Equation19, where only the positive root is used as nega-tive voltage does not make physical sense for this

apparatus.

V =√2RP (22)

In addition to measuring the peak amplitudeson the spectrum analyzer, for each data set thenoise level due to the photodiode was measured,in addition to any extra noise at the central peaklocation when only the beam from the AOM wasincident upon the photodiode instead of the in-terfered beam. These values were also convertedto mV using Equations 21 and 22. The noiselevel was then subtracted from each of the threepeaks measured and the principle peak noise wassubtracted from the principle peak only. Next,the ratio of the first harmonic peak amplitudeto the amplitude of the principle peak was cal-culated. Plotting this on a log-log scale againstthe RF level of modulation of the EOM resultsin a linear relationship, as shown in Figure 23.The linear relationship was useful to compare theexperimental data to the theoretical model, also

M. Dixon 19

Figure 22: The peak amplitudes of the principleand first harmonic peaks and how they change withRF level of modulation when the EOM was driven ata frequency of 40 kHz. The blue curve represents theprinciple peak, the red curve represents the first har-monic peak at 79.96 MHz, the green curve representsthe first harmonic peak at 80.04 MHz, and the purplecurve represents the average of amplitude of the firstharmonic peaks.

Figure 23: The raw experimental data has beenconverted from dBm to mV, from which the noisethreshold of the photodiode was subtracted and theratio of the first harmonic peak to the principle peakwas calculated. The experimental data is comparedto the theoretical model on a log-log scale, revealinga linear relationship. The value of Vπ that best fitsthe experimental data when the EOM is driven at afrequency of 40 kHz is Vπ = 2.54 ± 0.1 V.

shown in Figure 23. The Bessel Functions werecalculated using the built-in functions of Excel,where a value of Vπ was selected by control-ling the modulation constant, β, and how wellthe theoretical model matched the experimen-

tal data was determined by eye-sight, observingwhen the two linear relationships were most sim-ilar. The error was determined by how much Vπ

could change while still convincingly matchingthe experimental data.

The experimental data fit to the theoreticalmodel for all data points taken is presented inAppendix A.

6.4 Results

By varying the frequency at which the EOMwas modulated and calculating the subsequentchange in Vπ, the relationship between the EOMfrequency of modulation and Vπ shown in Fig-ure 24 was found. The blue data points wereanalyzed in the method outlined in the previoussection whereas the red and green data pointswere analyzed using a χ2 fitting program in Gnu-Plot and the Vπ of each first harmonic peak wasfound separately, instead of finding the Vπ whichmatches the average of the two peaks. Whilethese two methods results in slightly differentvalues of Vπ and the values are not always thesame within error, it is important to note thatthey appear to follow the same trend, a poten-tially exponential curve as the frequency of EOMmodulation increases, meaning that the resultsfrom the two methods are consistent.

The dependance of Vπ upon the frequency ofEOM modulation is unexpected as the manufac-turers makes no mention of the possible relation-ship. Additionally, the manufacturer states thatat a modulation frequency of 1 kHz Vπ shouldbe 3.2 V, which is inconsistent with the exper-imental data as Vpi at low frequencies plateausaround 2.5 V, however, this inconsistency is not alarge concern as the voltages are relatively simi-lar and within the range of the EOM’s maximumapplied voltage. What is concerning is the largevalues of Vπ found at higher frequencies, specifi-cally around 300 MHz when Vπ jumps above 5 Vand rises exponentially to 13.5 V at 750 MHz. IfVπ actually rises this high for these frequenciesit indicates that a full phase modulation cannotbe achieved as it would require applying a volt-age of an amplitude beyond what is safe for thedevice. This is not ideal as the LASIC II exper-iment would like to modulate the EOM between

M. Dixon 20

the frequencies 40 kHz and 440 MHz, and cur-rently it appears that the EOM will not achievea full phase modulation at the higher end of thefrequency range.

Figure 24: The potential relationship between thefrequency at which the EOM is modulated and Vπ

as found by two different methods. The blue curvefollows the method outlined in section 6.3 whereasthe second method compared the experimental to thetheoretical model using a χ2 fitting program for eachfirst harmonic peak separately.

One potential source of error in the ex-perimental measurements could be the ampli-fiers needed to observe the signal from the fast-photodiode on the spectrum analyzer. For fre-quencies below 200 MHz, a Femto HVA -200M-40-B amplifier was used as it not allow higher fre-quencies to pass. For the data set this includedthe frequencies of modulation from 40 kHz to 150MHz. The amplifier functioned properly for eachdata set except at 150 MHz where the ω+Ω firstharmonic peak occurring at 230 MHz was cut inamplitude by the amplifier, resulting in the largedifference between the values of Vπ found for thetwo different peaks, as shown by the second fit-ting method in Figure 24. However, this doesnot appear to be a problem as the Vπ fit to thefirst harmonic peak at ω−Ω occurring at 70 MHzmatches the value of Vπ fit to the average of thetwo peaks. The data points taken at frequencieshigher than 200 MHz, namely between 300 MHzand 750 MHz, were taken with a Mini-Circuits

ZX80-30186-57 amplifier whose frequency rangewas between 20 MHz and 3 GHz, however, thevoltage supply to run the circuit cut frequenciesat 1 GHz, limiting the actual frequency range at1 GHz. This amplifier created more noise thanthe lower frequency range amplifier, possibly afactor in the increasing values of Vπ, however, asmooth curve can be drawn between the pointstaken with each amplifier, indicating that thisdoes not appear to be a large issue in the vary-ing values of Vπ. For the data points taken at 410MHz it is unknown as to why the two methodsof finding Vπ result in different values, however,the theoretical model does not fit the experimen-tal data as well at that frequency as it does forother frequencies, as shown in Table 3 f2 and g2.

In order to truly test if the use of the two dif-ferent amplifiers resulted in different values of Vπ

it would be necessary to take a data set withinthe range of both amplifiers and compare the re-sulting values of Vπ under the two conditions. Ifthe two amplifiers result in the same value withinerror, the amplifiers are not the source of prob-lems. There was insufficient time to perform thisexperiment, thus if this experiment were to becontinued taking two data sets at the same fre-quency with the two different amplifiers wouldhelp to confirm or exclude them as a source oferror in the measurements.

7. Conclusion

The goal of this experiment was to character-ize two fiber optics devices to be used in thelaser stabilization project, LASIC II, which isto be used aboard LISA. The two main devicescharacterized were an Evanescent-Wave Coupler(EWC) and an Electro-Optic Modulator (EOM),while a third device, a non-fibered Acousto-Optic Modulator (AOM), was characterized tobe used in an experiment to study the EOM.First the polarization of each devices was stud-ied, in order to determine wether or not they arepolarizing maintaining as specified by the manu-facturers. This was done by inputting a linearlypolarized beam to each of the devices and an-alyzing the data with Malus’ Law, Equation 9,to determine if the resulting output beam was

M. Dixon 21

also linearly polarized. It was found that each ofthe three devices is indeed polarizing maintain-ing, where the non-evanescent beam of the EWChas an extinction ratio of 30 dBm, the evanes-cent beam of the EWC has an extinction ratioof 25 to 33 dBm depending upon the couplingratio of the beam, the AOM has an extinctionratio of 17.5 dBm and the EOM has an extinc-tion ratio of 30 dBm. Second, the coupling ratioof the EWC was studied, as the device splits aninput beam in varying ratios. By changing theposition of a micrometer knob on the device andrecording the subsequent change in output powerof the evanescent beam the device was found tohave a mechanical hysteresis. This means thatthe position of the knob associated with the max-imum evanescent beam output depends on howthe maximum is found, namely if the positionof the knob is increased or decreased. Increas-ing the knob’s position results in a maximumat the position named 1.38 while decreasing theposition results in a maximum at the position1.32. This is not a large issue in the usage ofthe device as once a maximum is found it isstable, additionally, the manufacturer warns tonot rely upon the micrometer to find the maxi-mum evanescent power output. Finally, a hetero-dyne interferometry experiment was constructedto study the half-wave voltage, Vπ, of the EOMby introducing a phase shift to one beam withthe EOM and a frequency shift to the secondbeam with the AOM. Recording the amplitudesof the principle and first harmonic peaks as boththe frequency and amplitude of the modulationsignal to the EOM were altered allowed the ex-perimental data to be compared to a theoreticalmodel depending upon Vπ, where two differentfitting methods produced similar values of Vπ.It was found that Vπ changes with the frequencyof modulation, however the exact relationshipbetween the two quantities is unknown and themanufacturer does not warn of this dependence.Also unexpected are the large values of Vπ forfrequencies over 300 MHz, where Vπ exceeds the

maximum voltage amplitude that can be appliedto the EOM, presenting problems if the device isused in this frequency range, as is planned forthe LASIC II experiment. A source of error inthese measurements may be the amplifiers usedin the experimental apparatus, however this av-enue requires further study to be conclusive.

8. Acknowledgements

This research was conducted at the Astropartic-ule et Cosmologie Laboratory in Paris, France,and was made possible by the University ofFlorida and the National Science Foundation’sResearch Experience for Undergraduate Stu-dents program. I would like to thank my advi-sor, Hubert Halloin, for his guidance during thisproject and everyone at the APC who graciouslyhelped make finicky optical components functionproperly.

9. Appendix A: All Measurements of

Peak Stabilities and Vπ for the

EOM

This appendix contains all data used to find therelationship between the half-wave voltage, Vπ,of the EOM and the frequency at which the de-vice is modulated, excluding the data set at amodulation frequency of 40 kHz which is pre-sented in Section 6.3. The first column presentsthe peak stability data for each measurement,showing that the principle peak’s amplitude isstable and that the amplitudes of the first har-monic peaks follow one another, such that theaverage of the two peaks can be taken, and isgraphed on the same plot. The second columnpresents the experimental data fit to the theoret-ical model in order to find the value of Vπ whichbest fits the data. The frequency of each mea-surement and the value of Vπ which fit the datais presented in the title of each graph and thedata is compiled for comparison in Figure 24.

M. Dixon 22

(a1) (a2)

(b1) (b2)

(c1) (c2)

(d1) (d2)

M. Dixon 23

(e1) (e2)

(f1) (f2)

(g1) (g2)

(h1) (h2)

M. Dixon 24

Table 3: The peak stabilities and Vπ measurements for the following frequencies, a.) 500 kHz, b.) 50 MHz,c.) 150 MHz, d-e.) 300 MHz, f-g.) 410 MHz, h.) 750 MHz. The data sets for 40 kHz are presented inSection 6.3.

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