A Scanning, All-Fiber Sagnac Interferometer for High Resolution Magneto-Optic Measurements at 820 nm
aa r X i v : . [ phy s i c s . op ti c s ] M a r A Scanning, All-Fiber Sagnac Interferometer for High ResolutionMagneto-Optic Measurements at 820 nm
Alexander Fried,
1, 2
Martin Fejer,
3, 4 and Aharon Kapitulnik
3, 1, 2 Department of Physics, Stanford University, Stanford, CA 94305 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 Department of Applied Physics, Stanford University, Stanford, CA 94305 Ginzton Laboratory, Stanford University, Stanford, CA 94305 (Dated: 5 October 2018)
The Sagnac Interferometer has historically been used for detecting non-reciprocal phenomena, such as rota-tion. We demonstrate an apparatus in which this technique is employed for high resolution measurements ofthe Magneto-Optical Polar Kerr effect—a direct indicator of magnetism. Previous designs have incorporatedfree-space components which are bulky and difficult to align. We improve upon this technique by using allfiber-optic coupled components and demonstrate operation at a new wavelength, 820 nm, with which we canachieve better than 1 µ rad resolution. Mounting the system on a piezo-electric scanner allows us to acquirediffraction limited images with 1.5 µ m spatial resolution. We also provide extensive discussion on the detailsand of the Sagnac Interferometer’s construction. I. INTRODUCTION
Magneto-optical (MO) effects are described withinquantum theory as the interaction of photons with elec-trons through the spin-orbit interaction . Macroscopi-cally, linearly polarized light that interacts with magne-tized media can exhibit both ellipticity and a rotationof the polarization state. The leading terms in any MOeffect are proportional to the antisymmetric off-diagonalpart of the ac conductivity: σ xy ( ω ) = σ ′ xy ( ω ) + iσ ′′ xy ( ω ) .If a magnetic field is applied to the material, σ xy ( ω, H )is small and proportional to the field. Its zero frequencylimit is the known Hall conductivity of the material. Ifthe material exhibits finite magnetization, σ xy ( ω, M ) isalso small and proportional to the magnetization, andits zero frequency limit is known as an example of theanomalous Hall effect. Other, less straightforward effectsof magnetism in a material will also lead to finite MOeffect. Thus, a finite MO effect measured in a mate-rial points to time-reversal symmetry breaking (TRSB)in that system.Magnetization normal to the surface of a reflectivesample can be measured optically via the Magneto-Optical Polar Kerr Effect. Specifically, polarization rota-tion is characterized by the Kerr angle, which, in thecircular basis, is half the difference in the phase ac-cumulated, upon normal incidence reflection, betweenright and left circularly polarized light. Provided non-linear optical effects are weak, the Kerr angle is propor-tional to the magnetization. Because optical interactionsare strongly dependent on electronic band structure andspin-orbit coupling parameters, the proportionality con-stant varies with temperature and optical frequency. Asan example, optical resonances, such as the Fermi-Edgesingularity in GaAs, can enhance or even switch the signof the Kerr angle even if the magnetization is constant. While the widely used cross-polarization techniquesoffer some advantages at measuring Kerr rotation ,the past couple decades have seen their angular resolu- tion limits bested by new strategies, as demonstrated bycavity-enhanced Kerr rotation for quantum dots, “opti-cal bridge photodetection,” and, in particular, a mod-ified Sagnac Interferometer (SI), first introduced by Spiel-man et al. , in order to optically search for anyons inhigh-temperature superconductors . While many of theabove techniques can achieve Kerr angle resolutions bet-ter than 1 µ rad/ √ Hz , only the SI technique allows for adirect high-resolution detection of the MO effect withoutany external modulation of an external field that cou-ples to the measured magnetic signal such as magneticfield, electric field, or current. This property is of utmostimportance when intrinsic TRSB effects need to be de-tected, such as in search for such effects in unconventionalsuperconductors. II. THE SAGNAC INTERFEROMETER
Complicating any accurate measurement of the Kerrangle are non-magnetic effects, such as linear birefrin-gence and linear dichroism, which can also affect the po-larization state of light. These effects can arise not onlyin the sample, but in the apparatus itself and prudencerequires consideration of their existence. A Sagnac in-terferometer, however, is sensitive only to non-reciprocaleffects as we will discuss in detail below, whichin materials can arise from TRSB phenomena, such asmagnetism . With this as a guiding principle, giventhat the optical components are reciprocal even if theyare misaligned or imperfect, any signals measured willbe strictly indicative of non-reciprocity arising from thematerial being examined. Nonlinear optical effects are anexception and they can cause nonreciprocal polarizationrotation as they are outside of the scope of reciprocity ar-guments that apply to only linear processes . However,as nonlinear effects are expected to have an intensity de-pendence, they can easily be identified. We discuss thisin more detail within Appendix C. DetectorSource EOM l /4 PlatesPolarizing Beam SplitterSamplePolarizer at 45°Beam Splitter +θ F -θ F FIG. 1. Simplified illustration of the original Sagnac inter-ferometer in the Faraday geometry. The first polarizer ori-ents the source polarization so that the beam splitter evenlycouples the light into the two time reversed equivalent andtherefore, reciprocal, illustrated by the inner (blue) and outer(red) channels in the loop. The signals interfere destructivelyat the beam splitter after a single traversal of the loop if thesample is non-reciprocal and the two beams yield differentphase shifts. The modulated intensity is measured by lock-indetection and the amplitudes of the harmonics are used to cal-culate the Faraday angle, θ F . This design can also be adaptedto the normal and oblique incidence Kerr geometries . In a Sagnac interferometer, such as the one used byDodge et al. or Spielman et al. polarized light from asource at wavelength λ is passed through a polarizer andlaunched into the slow axis of a single-mode, polarization-maintaining (PM) optical fiber. The light is then splitinto two beams by a loop coupler (a fiber beam-splitter).These two beams propagate in opposite directions aroundthe loop, which contains the sample as shown in Figure1. In general, if the loop is a reciprocal path (i.e. if itsoptical path length does not depend upon the direction ofpropagation), then the two beams will return to the loopcoupler precisely in phase and constructively interfere atthe detector. If elements within the loop are nonrecipro-cal, the two beams yield a phase difference. If the phaseshifts originate from a Faraday effect, this phase differ-ence will appear as destructive interference which will re-duce the detected intensity to a fraction [1 + cos(2 θ F )] / ± ∼ ˆ x ± i ˆ y withrespect to a fixed coordinate axis. Light passing throughthe sample must be in circular polarization states; op-posite circular polarizations, traveling in opposite direc-tions are a pair of time-reversed states of light, and willbe affected differently by the sample only if the samplebreaks TRS, which in materials, occurs if magnetism ispresent. As the light emerging from the fiber is in a lin-ear polarization state, a quarter-wave plate, with slowaxis oriented at 45 ◦ with respect to the fiber’s slow axis can be aligned so that it converts the linearly polarizedlight in the clockwise (counter-clockwise) mode into pos-itive (negative) circularly polarized light when it emergesfrom the fiber, and the second waveplate re-converts thebeams back to linear polarization states, but now rotated90 ◦ . Since the only light on the slow axis of the fiber andthe two beams must be in opposite circular polarizationsat the sample, the outputs of the two fibers must bealigned so that their slow axes are also 90 ◦ from eachother. Any remaining light that couples to the fast axisof the fiber is filtered out by the polarizer.To enable the measurement of 2 θ F to µ rad sensitivity,the system is actively biased by an electrooptic phasemodulator (EOM) in the loop, which sinusoidally mod-ulates the index of refraction of the nonlinear crystalwithin it. The modulator is operated at a frequency f with period corresponding to twice the optical tran-sit time of the full length of the loop. The time-varyingphase shifts induced on the two beams by the modulatorare equal and opposite, since they pass through the mod-ulator at times separated by half the modulator period.From Equation A1, to first order in 2 θ F , the signal fromthe detector will contain a second harmonic of f , withamplitude proportional to the throughput of the loop( I ), and a first harmonic with amplitude proportionalto 2 θ F × I , the quantity of interest. These harmonics aremeasured with lock-in amplifiers and divided to computethe Faraday angle.The Reciprocity Principle states that the results of ameasurement are identical to another performed underthe conditions of time being reversed and the positions ofthe optical source and the optical detector interchanged.There are various uses of this terminology in the litera-ture. A “reciprocal material” is one which acts on lightin the same fashion if only the source and detector posi-tions are exchanged. A pair of “reciprocal optical paths”refer to the polarization state and path of light taken be-tween a light source and detector, and the polarizationstate and path that would be traversed if time was re-versed (i.e. if the the source and detector were exchangedand the conjugate operation performed on the complexelectromagnetic field).A Faraday rotator is a nonreciprocal device because itis magnetic and breaks time reversal symmetry, whereasa quarter-wave plate is reciprocal, since its index ofrefraction is only dependent on the linear polarizationstate. Likewise, the two beam paths shown in Figures1and 2 which make up the Sagnac Interferometer are re-ciprocal because they are identical if the time reversaloperation is applied to one of them. Any differences be-tween the phases or amplitudes of these two beams, whenmeasured at the detector after traversing the loop, aresignatures of non-reciprocity. The Sagnac interferometeris sensitive only to the phase differences, and since thereare nominally no non-reciprocal effects in the fiber andthe other optical components, the only source of non-reciprocity will be at the sample.An improvement in design and performance of the ba-sic Sagnac interferometer was implemented by Xia etal. , by using both the fast and slow axes of a PM fiber asoptical paths, thus forming an SI with a zero-area-loop.Furthermore, this setup applied the technique to a mea-surement in the Kerr geometry. To introduce active bias-ing, free space optics direct light from the source througha polarization dependent phase modulator, before pro-ceeding to the fiber, a single quarter-wave plate and thesample. Because the loop area is zero, non-reciprocal ef-fects stemming from any mechanical rotation, includingthe Earth’s rotation, will not appear.We have further extended upon the work of Xia etal. by using a new all-fiber approach which dramaticallysimplifies the alignment, has a small footprint and can beconstructed quickly from off-the-shelf components. Likeits predecessor, this design allows for flexible tempera-ture capability and can be ported into a wide varietyof experimental designs. Our source is a superlumines-cent diode (SLED) centered at 818 nm, temperature con-trolled at 25 ◦ C, and possessing a bandwidth of about 22nm. Using both Michelson interferometry and spectralanalysis, we measure the coherence length to be about 8 µ m over a wide range of output power. We operate theSI with about 180 µ W of power incident on the sample,with only about 20 µ W returning to the detector whenthe sample is a gold mirror. The low throughput to thedetector, following reflection from the sample, is a conse-quence of inefficient coupling and focusing of light fromthe fiber to the sample. A custom lens would offer betterperformance, as would, larger, off-the-shelf fiber-couplingcomponents which are easier to align.
SLED Polarization Controller Isolator
EOM
Polarizer LP Ref
Signal
Ratio V f V V /V Lock-In Amplifier Lock-In Amplifier Photodiode PM Circulator Sample on Attocube Lens l /4 Plate slow TM to slow to X
TE to fast to Y _ + fast l /4 Plate Analog-Out
Digital-Out Function Generator Frequency Doubler Filter Power Splitter 1 2
1 2 -θ K +θ K AB FIG. 2. Schematic of the interferometer. The “A” (red) and“B” (blue) paths commensurate with the fiber componentsillustrate the two counter-propagating beams in the zero-arealoop that forms the interferometer. The counter-propagatingbeams illustrated in the upper part of the Sagnac loop aretraversing along the TM mode of the EOM and coupling toand from the slow axis of the 10m PM fiber and to the X-polarization state in free space. The lower part of the loopcarries describes light along the TE mode of the EOM whichcouples to the fast axis of the fiber and exits into the Y-polarization state.
As shown in Figure 2 the light from the superlumines-cent diode passes through a polarization controller, twoPM isolators and a PM circulator, all optimally function- ing for wavelengths centered at 820 nm. Back-reflectionsto the SLED are the single most significant source of slowdrifts in our measurements, so the two isolators and cir-culator provide 85dB of isolation from such effects. Atthis level of isolation, no back-scatter is measured, nordoes the addition of a third isolator gives makes no no-ticeable difference. Light emerging from the second portof the circulator, will have power primarily on the slowaxis of the fiber and any remaining light on fast axis isfiltered away with the PM fiber polarizer.The critical custom design specifications for the EOMare (1) on Port 1, the slow axis of the fiber is oriented at45 ◦ to the TM mode of the its Lithium Niobate waveguidemodulator and (2) the slow axis of the fiber is aligned tothe high-index, TM, mode of the modulator on Port 2.When the beam then proceeds to the custom EOM, thefirst requirement ensures light couples roughly equally tothe TM and TE modes. While the EOM modulates bothmodes, the modulation depth of the former is larger andthe “B” mode gains a net phase. The second require-ment helps suppress parasitic interferometric paths andis discussed in detail in the next section.The TM and the TE modes in the EOM couple tothe slow and fast modes, of a 10.01m long fiber and arethen focused onto the sample by an aspherical lens withanti-reflection coatings. The lens’ numerical aperture onone side is matched to the fiber (0.12) and is 0.43 onthe sample side. An anti-reflection coated quarter-waveplate, with its slow axis at 45 ◦ to the slow axis of thefiber, transforms the orthogonal, linear polarized modesemerging from the fiber to opposing circularly polarizedmodes. When reflecting off a magnetic sample, thesemodes will yield different phases; half their difference willbe the measured Kerr angle.After reflecting, the second pass through the quarter-wave plate returns the modes to linearly polarized states,but with their polarizations now interchanged from be-fore. Consequently, the axes in the fiber into which theycouple will also be exchanged, as shown in Figure 2. Thebeams return to the EOM where the “A” beam will re-ceive the dominant phase modulation. When both beamsre-couple to the slow axis of the fiber, they will interfere.Any remaining power on the fast-axis is, again, filteredby the polarizer and the beam returns to the circulatorand sends the light to a 125 MHz photodetector.We use two lock-in amplifiers to measure the first andsecond harmonics at 4.867 MHz and 9.734 MHz respec-tively and a low-pass filter is placed on the input of theformer to remove the large second harmonics. The lock-in amplifier integration time restricts the time resolution;we typically set it to about 1 second. The in-phase firstharmonic signal is normalized by the second harmonicmagnitude and the recorded output voltage of the lock-in is the quotient of the two. This quotient is used tocompute the Kerr angle, while the second harmonics areusually interpreted as the reflectivity. However, it re-ally corresponds to the power present in the two inter-fering modes. There are many scattering processes suchas imperfections in the equipment or birefringence of thesample which can scatter light out of these modes anddecrease the second harmonic magnitude. We review thedetails of this calculation and the specific alignment tech-niques in the Appendix A. III. COMPETING INTERFEROMETERS
Essential to the technique reported in Xia et al. isthe free-space electro-optical modulator (EOM). Thistype of EOM introduces Residual Amplitude Modulation(RAM) in the form of beam steering, back-scattering,and etalon effects from piezoelectric resonances. Weobserve that these generate drifting, spurious signals ofseveral µ rad on top of the desired signal and which de-pend exactly on the trajectory of the light beam throughthe EOM. Nominally, because of the π phase difference onthe EOM drive signal for the forward and return paths,RAM should cancel itself out to first order and spurioussignals should be reduced. In a free-space EOM, it isdifficult to perform an alignment that ensures that theforward and reverse paths not identical. Another dif-ficulty is that the resonant modulators used in previousversions require the fiber to be cut precisely to the lengththat corresponds to the resonant drive frequency.In an all-PM-fiber system there are only four propa-gating modes in both the fiber and the EOM. At a givenfrequency, there are two polarizations traveling in two di-rections, so no beam-steering can occur. Since the mod-ulator is non-resonant, the fiber length does not matteras the drive frequency can always be adjusted withoutsacrificing the performance of the EOM. Furthermore, ifnot terminated with 50 ohms, the inline EOM can oper-ate at hundreds of MHz, allowing the fiber length to beshortened . A detailed analysis of RAM is presented inAppendix B.The two fiber components used in lieu of free-space op-tics are a PM fiber-coupled-polarizer made by Oz-Opticsand a custom non-resonant EOM made by EOSPACEInc. The polarizer has extinction ratio 1:100 and is po-larized along the slow axis of the fiber. This EOM haslarge insertion loss; throughput to the sample is about45% as compared with 70% or more with the free-spaceoptics. EOSPACE guarantees that they can align thefiber axes to waveguide modulator axes with an accu-racy of less than 3%, which will unevenly distribute lightbetween the two modes in this component. By collimat-ing the output of the EOM and passing it through ahalf-wave plate on a rotation stage, a fixed 1:100000 ex-tinction ratio Glam-Thompson polarizer, and finally, toa photodetector, we measured the power difference onboth channels and find it to be less than 1%. The mod-ulators also have power limits of around 20 mW, abovewhich throughput decreases and performance will devi-ate from specifications. The EOM generates RAM andtransmission measurements reveal it to be sharply peakedat the driving frequency, with almost no spectral weight above it and some below it. As in the free-space design,all fiber connectors are angle polished at 8 ◦ to minimizeback-reflections. slow TM to slow to X
TE to fast to Y fast slow TM to slow to X
TE to fast to Y fast Polarizer PM Circulator slow
TM to slow to X
TE to fast to Y _ fast l /4 Plate a)b)c)d) slow TM to slow to X
TE to fast to Y fast ++_ FIG. 3. A comparison of possible optical paths in the inter-ferometer. (a) shows the two primary interfering paths alsoshown in Figure 2. If the polarizer or the circulator is imper-fect, then three additional pairs of reciprocal paths can exist,which will also interfere with the primary two. Depicted in(b) is one which shows light from the fast axes of the circu-lator, coupling to the EOM and returning to the slow axisof the fiber. Light can also couple through the system start-ing in the slow axis and ending in the fast axis, as well asstarting on the fast axis and ending back on the fast axis.Those paths illustrated in (c) will be incoherent with thosein (a) and not interfere. (d) There are points in the setup,where back-scattering will form optical paths with the samepath-length as in (a), and thus can interfere with the primarysignal. However as these points are not near scattering facets,such scattering should not be present.
Because of imperfections and misalignments, there willalways be modes of light which do not follow the idealoptical trajectories illustrated in Figure 2 and Figure 3a.Several possible mechanisms are known to result in lighttraveling along non-ideal paths. 1) An imperfect polar-izer will route some light along the fast axis of the EOMand, upon return, can couple light back into the fast orslow axes of the circulator as illustrated in Figure 3b.2) Following the EOM, cross-coupling can appear eitherthrough scattering or fiber connectors that are imper-fectly aligned. 3) The quarter-wave plate can be angu-larly misaligned, be of the wrong thickness, or in general,not functioning ideally as the incident light is not a planewave. 4) The sample could be linearly birefringent, so thepolarization eigenstates of reflection may not be circular.The last three result in trajectories illustrated in Figure3c. 5) Back-scattering at interfaces can reflect light back-wards leading to competing interferometers. As discussedabove, angle polished components prevent back scatter-ing, and no evidence of it has been observed, as there isno signal at the detector when the sample is removed.6) The alignment of the slow axis of the polarizer mightnot be perfectly aligned at 45 ◦ to the TM and TE modesof the EOM at port 1 and unequal power will be dis-tributed along the two modes. As the SI measures phasedifferences, differences in power along the two modes areunlikely to cause an effect, provided there are no non-linear interactions. In the same way, circular dichroismin the sample will also not contribute as it only affectsthe amplitude. As through out the entire apparatus thePM fiber ensures that there are only two polarizationchannels are allowed in a given propagation direction,a detailed analysis of how any imperfection within theinterferometer is accurately accomplished by using theJones Calculus, as described in Appendix A.Two powerful principles can be drawn upon to quicklyprovide accurate qualitative understanding of how phasesaccumulate on the various optical pathways. The firstprinciple, is that, by definition, only non-reciprocal phe-nomena can cause pairs of otherwise reciprocal opticalpaths to yield phase differences upon a complete traver-sal of the instrument. The fiber, the polarizer, the EOM,the wave plate and the rest of the optical componentswithin the loop are non-magnetic, so only the sample isexpected to be a source of non-reciprocity, and any devi-ation from the baseline Kerr signal is therefore expectedto be a positive identification of non-reciprocity in thesample. Reciprocity can also be used to understand thebehavior of individual components which make up theinterferometer. Phases delays, polarization rotation, andattenuation will be identical for reciprocal waves passingthrough a reciprocal component.The second simplifying principle is that with a broad-band source, interferometric fringes are visible only be-tween modes with optical path length difference less thanthe source’s coherence length . Because of the strongbirefringence of the PM fiber and EOM, most scatteredmodes will have path lengths which differ by many timesthe coherence length. Light following these “incorrect”paths will reach the detector, but won’t interfere coher-ently with the two main modes, and therefore can notgenerate a spurious signal. These modes will, however,shift power away from the interfering modes, and willappear as increased signal at the “DC” output of the de-tector. The ratio of the power on the DC part of thesignal to the power on the second harmonics is a goodmeasure of scattering as indicated by deviations from thetheoretical prediction of Equation A2. By this indicator,we note that the all-fiber interferometer possesses morescatter than the free-space SI.The analysis of the interferometer is considerably sim-plified by the spatial filtering provided by the polariza-tion maintaining fiber as it allows for a description of theelectromagnetic state by only two polarization states atall points in the instrument. It does not matter that thebeam incident at the sample is not a plane wave; there arestill two basis states of the field, both of which possess anapproximate Gaussian beam profile, have opposite spin Sampleλ/4 Plateλ/4 PlateFiberFiber a bc d
BirefringentSample | X, − k | X, k | Y, − k | Y, k | + , k | + , − k |− , k |− , − k | X, − k | X, k | Y, − k | Y, k | + , k | + , − k |− , k |− , − k | X, − k | X, k | Y, − k | Y, k | + , k | + , − k |− , k |− , − k | X, − k | X, k | Y, − k | Y, k | + , k | + , − k |− , k |− , − k FIG. 4. Illustration of all eight possible mutually coherentscattering modes, grouped in reciprocal pairs (red/blue), leav-ing the fiber and interacting with the sample. All of thesecould affect to the measurement of the Kerr signal, since thesum of the four amplitudes per channel, yields the total tran-sition amplitude which is then interfered. The intermediatepolarization and momentum basis states are written in the ketnotation. (a) and (b) are those modes that appear alone forthe two ideal angular positions of the the quarter-wave plate.c and d are those modes added when there is birefringence inthe sample. The ± defines the circular polarization in termsof the spin angular momentum state of the field angular momentum and which are respectively mode-matched to the two linear polarization states within thefiber. Even if the quarter-wave plate is imperfect and theoutgoing states are only elliptically polarized, becausethe incoming polarization states are both linear and or-thogonal, the expected value of spin angular momenta(not including power differences between the beams) ofeach mode leaving the quarter-wave plate will be equaland opposite.In Figure 4, we consider intermediate basis states ofthe two modes of the optical field as they pass throughthe various optical components and the reflecting sam-ple. At each transition, the basis has been defined sothat in a perfectly aligned system, only a single basis el-ement describes the optical state of each mode at everypoint in the fiber. In an imperfectly aligned system, su-perpositions of these modes must be considered. Ideally,the modes leave the fiber linearly polarized, | X/Y, − k i ,pass through the quarter-wave plate and become circu-larly polarized as |± , − k i . After reflecting off the sample( |± , k i ), the quarter-wave plate returns the states to thelinear polarization, | X/Y, k i , and couple into the fiber.Since only light that traverses from X to Y polarizationand vice versa will be coherent and interfere, only suchmodes have to be considered in the analysis.In the figure, the eight relevant paths are grouped inreciprocal pairs, so by the reciprocity theorem, the con-tribution to the full transition amplitudes from each ofthese pairs of paths will be identical if all the opticalcomponents are reciprocal. When the the quarter-waveplate is aligned perfectly, the modes illustrated in eitheronly Figure 4a or only in 4b will be occupied. Thus,there are two possible angular positions of the quarter-wave plate, 90 ◦ apart, for which the interference fringevisibility is optimized. Switching between these two ori-entations exchanges the phase-delays accumulated uponreflection between the two channels. As it is the phasedifference between the two modes that is measured, theKerr angle signal will change sign. When there are mis-alignments and imperfections in the quarter-wave plateso that the incident light is elliptically polarized, thenthose paths represented in both a and b will be presentand coherent at the detector.In general, we have h X, k | Y, − k i b ∝ h Y, k | X, − k i a ∝ e − iθ K and h X, k | Y, − k i a ∝ h Y, + k | X, − k i b ∝ e iθ K . Here,the constant of proportionality is real and we are abus-ing the bra-ket notation, with subscript (a,b,c,d) to de-note the product of transition amplitudes between eachbasis state constituting the respective path in Figure4. So, for each element of the transition matrix forpropagation outside the fiber, we have, for example, h X, k | Y, − k i = P u h X, k | Y, − k i u and h X, k | Y, − k i a = h X, k | + , k i×h + , k | + , − k i×h + , − k | Y, − k i . When there isbirefringence in the sample, then those paths illustratedin Figure 4c and 4d become relevant. From the figure, itis apparent that the sample’s contribution to each pairof amplitudes is the same. Likewise, will be the con-tribution from the quarter-wave plate and the fiber, asthey are reciprocal components. Consequently, the fullamplitudes for each pair will be identical regardless ofwhether the sample is reciprocal or not: h X, k | Y, − k i u = h Y, k | X, − k i u , for u = c, d .When some or all of these paths are present, they willinterfere coherently at the detector because the opticalpath length will be the same for all of them. Thus, ifthe sample is non-reciprocal, then the instrument will re-turn an incorrect reading for the Kerr angle. If the sam-ple is reciprocal, then all reciprocal paths will contributethe same amplitude to both channels and no Kerr signalwill be measured. Consequently, misalignments, imper-fections, birefringence or any other reciprocal perturba- tion to the system will not introduce spurious signals.When the QWP is removed entirely, light will mostlyfollow 3c, but any amount of scattering, especially at thefiber facet, will give way to optical modes following theideal paths of 3a, and be sensitive to magneto-opticaleffects. We measure the magnitude of the second har-monics when the quarter-wave plate is removed to be5 × − smaller than when it is present and correctlyaligned. Because the scattering that gives rise to raysfollowing path 3a with the quarter-wave plate removed isfunctioning as a quarter-wave plate, upon adding a realquarter-wave plate back into the system, these scatteringmodes will follow the paths indicated by 3c, and so willbe incoherent with the primary paths and not contributeto the Kerr signal.There are two points in the system, as illustrated inFigure 3d, where back-scatter into the same channelwill result in beams of light which have the same pathlength as the primary interfering modes, and will there-fore interfere. To minimize the possibility of scatteringat these points, the second customization requirementfor the inline EOM maximizes the path length differ-ence between the fast and slow modes upon exiting thefiber at the sample. This guarantees that the two back-scatter positions are as far away from terminating facetof the PM fiber as possible, thus avoiding cross-couplingfrom scattering at the fiber facet surface. As the pathlength difference between the modes leaving the fiber is∆ ℓ = L × ( n e − n o ) = 9 . mm , one of the back-scatterpoints is located about ∆ ℓ n e = 3 mm behind the fiber ter-minus and the other point is outside the fiber withinthe focusing lens, where back-scattering is unlikely tore-couple into the fiber due to mode-mismatch. Sinceany scattering surfaces or any other sources of cross cou-pling are located a distance greater than the 8 µ m co-herence length of the light from any of these back-scatterpositions, only Rayleigh scattering from impurities canpossibly generate the unwanted modes. IV. INITIAL TESTS OF THE APPARATUS
To demonstrate the accuracy of the instrument, wemeasure the Verdet constant of a 0.5 mm thick sampleof z-cut quartz. One side is coated with a reflective layerof aluminum, so the light makes two passes through thequartz before returning to the fiber. Figure 5 is the Fara-day rotation from the double pass measured as a functionof magnetic field which gives 0.50 µ rad G . For reference,we flipped the sample over and measure the Kerr ro-tation reflecting off just the Aluminum layer: 0.27 µ rad G .The Verdet constant is given by the difference 0.23 µ rad Gmm ,which is compares well with a previous measurement byRamaseshan of 0.26 µ rad Gmm . In both measurements therewas a − . µ rad offset at zero field, which is likely dueto RAM.Much of SI’s performance can be characterized bystudying how the measurement varies as the power varies. −60 −40 −20 0 20 40 60−30−20−100102030 Magnetic Field (Gauss) K e rr A ng l e ( µ r a d ) FIG. 5. Faraday rotation measurement through a .5 mm thicksample of quartz with one side coated with Aluminum. As thebeam passes through the sample twice, the effective width ofthe sample is 1 mm. M i nnu t e R unn i ng A ve r a g e o f K e rr ( µ r a d ) S ec ond H a r m on i cs ( m V r m s ) FIG. 6. Drift of the Kerr angle (a) and the Second Harmonics(b) of a gold mirror over time. Although there are drifts inthe Second Harmonic signal, the Kerr signal is unaffected.
We tune the power received by the detector by movingthe lens out of focus from the sample, rather than chang-ing the power through the SLED, as this can affect itsspectrum, and therefore change how the performance ofthe SI, which is nominally aligned only for a single opticalfrequency. Noise, as measured by the standard deviationin the Kerr signal is well described by thermal detectornoise, as can be seen in Figure 7b. At higher powers, theshot noise will dominate; the two noise sources are distin-guished by their power law behavior and this is discussedin detail in Appendix A.Another of the standard measures of performance forSIs is how much the signals drift over long periods of time.These drifts can enter from thermal fluctuations withinthe EOM, air currents and re-amplified back-reflectionsto the SLED. In Figure 6, the Kerr signal does not driftappreciatively over about 84 hours, although in practicethe drift does not change over the course of many months.That the Second Harmonic signal does exhibit drift sug-gests that these variations is successfully canceled out inthe division operation when the Kerr signal is calculated.Finite offsets to the Kerr signal are another possibleway in which the SI may give false readings, and al- though stray signals of up to a µ rad have been recorded,the present alignment, as shown in Figure 6a exhibits anoffset of ∼ . µ rad, measured over Gold. Gold is notexpected to yield a Kerr effect in the absense of an ap-plied field and the fact that the observed offset changesbetween various alignments of the system suggests thatthe source is instrumental.We have found that these anomalous, yet stable, off-sets originate from stray signals coupling to the electron-ics at the EOM drive frequency and from RAM. Whenit is the former, the offset signal on the first harmonicswill not depend on power; this results in a false finitemeasurement of the Kerr signal: ( φ nr × I + δ ) / I . As thepower decreases, the presence of the δ offset will generatea larger false value for the Kerr angle in the calculationas shown in Figure 7. RAM can be discerned from a Kerrsignal, as the sign of the measured value will not changewhen the quarter-wave plate is rotated 90 ◦ . We discussthis in detail within Appendices B and D. −1 Average Power at Detector ( µ W) Z e r o S i gn a l K e rr ( µ r a d ) −1 Average Power at Detector ( µ W) N o i se µ r a d / H z . Noise DataShot Noise LimitDetector Limit ab FIG. 7. (a) Spurious offset and (b) the rms noise of the Kerrsignal off a gold mirror as the power returning to the detectoris varied by defocussing the sample from the focal point. Theestimated noise figures are given by Equations A5 and A6. Doto the large power loss in the EOM the all-fiber interferometeris detector-noise limited.
V. THE SCANNER
A homebuilt 2D scanner and driving circuit was con-structed to affect translation of the optical fiber at cryo-genic temperatures.
Two pairs of piezo bimorph “S-benders” allow translation with minimal change in thesample-to-scanner distance over approximately a 1 mm area at room temperature. An Attocube nanopositionerallows for vertical motion of the sample to allow for easyfocusing of the beam. The scanner’s voltage response wascalibrated with a gold bar of width 8.2 µ m. To demon-strate the imaging capabilities of the all-fiber interferom-eter scans of a 320 um thick Bismuth Iron Garnet crystalfrom Integrated Photonics was examined at room tem-perature as shown in Figures 8a and 8b. The domainwidths are about 20 µ m wide and have uniform magne-tization with a sharp transition at the domain wall. Thesecond harmonics suggest that the sample was slantedslightly with respect to the beam and so moved slightlyout of focus, decreasing the amount of light re-coupledto the fiber at points in the lower right-hand corner ofFigure 6b. Dirt on the surface is clearly visible and itpossesses no magnetization. µ m µ m µ m µ m mVrms µ rad FIG. 8. Scan of Bismuth Iron Garnet at room temperature.(a) is a rightward moving trace of the Kerr signal and (b) is thecorresponding Second Harmonic amplitude, which is a proxyfor reflectivity, though birefringence should also decrease itsvalue . The values of the Kerr angle on the domains are ± µ rad. In Figure 9 we show the profile of the beam as mea-sured by scanning across the edge of a gold bar and takingthe difference of the second harmonic magnitude at adja-cent points. This shows that the spot size is about 1.5 µ min diameter, which agrees with diffraction limit. The lenshas a focal length of about 1.1 mm and the diffractionlimited spot size allows the system to be robust againstangular misalignment of the sample. However, this makesit sensitive to mechanical shocks which induce vibrationsin the z-direction. We find that these vibrations dampout quickly and primarily register on just the second har-monic signal, a tribute to robustness of the measurementagainst intensity fluctuations. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 Distance from the Center of the Beam ( µ m) E s t i m a t e d B ea m I n t e n s i t y FIG. 9. The beam profile is estimated by taking the discretederivative of the second harmonic trace with respect to thedisplacement of the scanner as the beam is scanned back andforth across an Au/GaAs interface many times (a subsequentAbel Transform is an over-refinement for profiles that are ap-proximately Gaussian). As the sign depends on the directionof motion, positive values of the derivative are from rightward-moving traces, while negative values are from leftward traces.There is hysteresis in the scanner so each trace above has beendisplaced so the maximum/minimum is at 0 µ m. VI. CONCLUSION
The SI is an elegant method to measure weak mag-netization. To further emphasize the flexible nature ofthis technique we conclude by mentioning several alter-native imaging modes. Replacing the quarter-wave platewith a π /4 Faraday rotator will rotate the orthogonal,linearly polarized modes from the fiber by 45 ◦ . Uponreflection from the sample, those modes will, again, berotated by 45 ◦ by the Faraday rotator–for a net rotationof 90 ◦ . In this way, polarization states within the fiberwill again interchanged and the optical paths will form asimilar reciprocal zero-area loop to the above. The onlynon-reciprocal phase shifts in this configuration will comefrom linear birefringence in the sample and would, essen-tially, measure anisotropy. Another possible applicationis that natural gyrotropy in conductive materials may besensed by taking advantage of the inverse circular pho-togalvanic effect, in which a current propagating alonga screw axis may generate a weak spin-polarized currentthat would appear as a magnetic signal with sign de-pendent upon the direction of the current. Finally, thesystem may be operated as a magnetometer by simplyimaging a reflective material with high Verdet-constantin close proximity to the sample . VII. ACKNOWLEDGMENTS
Very special thanks to Hovnatan Karapetyan, Eliza-beth Shemm, Jing Xia, Min Liu, Alan Fang, ZhanybekAlpichshev, Carsten Langrock, Jason Pelc, Clifford Hicksand Beena Kalisky. We also thank Alexander Palevski,Axel Hoffmann, Helmut Schultheiß, Robert Feigelson,and Mark Randles for providing samples for calibration.We are grateful for support from the Center for Probingthe Nanoscale, Grant NSF NSEC Grant PHY-0830228.
Appendix A: Extracting the Kerr Signal
As the function of most of the components in the sys-tem is to ensure propagation of the two modes containingthe magnetic phase information, without cross-coupling,the behavior of the apparatus is modeled by consideringcontributions to the phase delay from the EOM and themagnetic sample only. Without loss of generality, we canassume that the EOM’s phase-modulation action occursonly along the TM mode. Below, we expand upon similardiscussions in references.
At the detector, the opticalsignal is: I ( t ) = I| e iφ m sin( ωt )+ θ K ± e iφ m sin[ ω ( t + τ path )] − θ K | (A1)The two terms within the norm above correspond tothe phases accumulated on the two counter-propagatingmodes in Figure 2. I is the power. φ m is the phase mod-ulation depth of EOM as set by the driving voltage. ω is the driving frequency as set by the function generatorsuch that ωτ path is the phase delay introduced by thetransmission through a double pass through the ∼
10 mof fiber, and the θ K phase shifts are those generated bythe nonreciprocity within the sample. In general, thesephases are not equal and opposite, as the reciprocal com-ponents will add a fixed phase to both beams, but sincethe instrument measures only the difference in phases,this is irrelevant. As illustrated in Figure 4, there are twoangular positions of the quarter-wave-plate which alignthe system, and these determine if a factor of − π phase shift on the second pass through the mod-ulator: ωc [( n e + n o ) L + ( N EOMe + N EOMo ) L EOM ] = π ,where L ≈ m is the fiber length and n e/o ≈ .
458 arethe indices of refraction for the extraordinary and ordi-nary axes within the fiber (birefringence is 3 . × − ),and N EOMe/o and L EOM are the indices of refraction ofthe two axes in the EOM and it’s length. By the Jacobi- Anger expansion: e iz cos θ = ∞ X n = −∞ i n J n ( z ) e inθ e iz sin θ = ∞ X n = −∞ J n ( z ) e inθ Where J n ( z ) is the n th Bessel Function. Equation A1thus yields: I ( t ) = I (cid:0) (1 + J (2 φ m )) + sin(2 θ K ) J (2 φ m )sin( ωt ) (cid:1) . +cos(2 θ K ) J (2 φ m )cos(2 ωt ) + ... ) (A2)The Kerr angle is calculated from the harmonic coeffi-cients in Equation A2: θ K = 12 tan − (cid:20) J (2 φ m ) V ω J (2 φ m ) V ω (cid:21) (A3)Where V and V are the rms voltage outputs from theLock-Ins measuring the first and second harmonics re-spectively. Since these voltages above are divided againsteach other, variations in the power, either from a noisyfluctuating SLED or changes in the sample’s reflectivitywill not contribute to the measurement as all harmonicswill be affected equally. Strong decreases in reflectivitywill affect the signal noise, however.The noise is estimated from Equation A3 by relatingthe noise voltages, δV , to the noise power δ I for eachnoise source: θ K = 12 J (2 φ m ) δV ω J (2 φ m ) V ω = 12 J (2 φ m ) δ I ω J (2 φ m ) I ω (A4)The noise for this measurement comes from two pri-mary sources:Shot noise:∆ θ K = 12 J (2 φ m ) J (2 φ m ) p hc/λ ) × η × EBW × P dc s × P dc (A5)andDetector noise:∆ θ K = 12 J (2 φ m ) J (2 φ m ) N EPs × P dc (A6) η is the quantum efficiency of the detector, NEP is thenoise equivalent power of the detector, and EBW is theEffective noise Bandwidth of the lock in amplifier as setby its internal filter and the time constant. The constant, s ∼
1, is a measured proportionality factor that relatesthe voltage on the DC part of the signal to RMS voltageon the second harmonics.0The Sagnac interferometer is a direct probe of reci-procity because it interferes light propagating along thetwo reciprocal channels of in Figure 2. Direct applicationof the reciprocity theorem of classical electromagnetismto this system guarantees that only nonreciprocal effectswill cause the amplitude and phase of the two opticalpaths to be different after traversing the loop. Asidefrom the sample, and the EOM, which introduces a timedependent bias in the controlled manner shown above,all of the other optical components along the path arenon-magnetic, and thus, reciprocal. Provided that theprocedure below is followed, even in the presence of mis-alignments or imperfect components, the SI will measureonly non-reciprocal effects in the sample. The robust-ness of Equation A1 to imperfect components and othervariations in the model can be demonstrated explicitlyby employing the Jones calculus, however the principleof reciprocity provides a simple and accurate intuition.Strong magnetic fields applied to the fiber or the lensesand along the optical propagation direction will alsocause nonreciprocal phase shifts. In the case of a Fara-day effect within the fiber, the two interfering paths willtravel through the applied field in both directions. Thephase shift on a linear polarization state from a magneticfield can not depend upon the direction of propagation,so both beams will yield the same phase and there willbe no net phase difference. For this reason, it is impor-tant to place the quarter-wave plate between the lensand the sample. Otherwise, if there is Faraday rotationin the lens from an applied magnetic field, opposite cir-cularly polarization states will yield different phase de-lays and this will contribute to the Kerr signal. Whenthe quarter-wave plate is placed after the lens, then thelinearly polarized states will still exhibit Faraday rota-tion, but because the rotation will be identical for bothbeams after traversing the lens twice, there will be no netphase difference. Nonlinear magneto-optical effects willnot exhibit this cancellation and we observe reciprocityfor magnetic fields orthogonal to the propagation direc-tion. This could be a consequence of the CottonMoutoneffect amongst other magneto-optical effects .A more explicit understanding of the workings of theSI can be gained by application of the Jones Matrix for-malism. However, this must be used carefully in the caseof a broadband source, as it is necessary, especially whenmodeling misalignments and imperfections, to performthe sum in Equation A7 over all wavelengths, so as totake into account amplitudes which are incoherent andwill not interfere. In the ideal alignment, calculated be-low, the sum over optical frequencies, k, is superfluous,because there is no weight on amplitudes correspondingto the improper paths. For convenience, we supply therelevant Jones matrices for the optical components illus-trated in Figure 2 and they can be modified without dis-cretion to explicitly test for non-reciprocity when model-ing the effects of imperfections. Polarizer: P = (cid:20) (cid:21) Intensity: ρ k = (cid:20) I e ( k ) I o ( k ) (cid:21) Beam Splitter at Port 1of the EOM: B = √ (cid:20) − (cid:21) EOM: E ( t ) = (cid:20) e iφ m ( t )
00 1 (cid:21) λ Plate: Q = √ (cid:20) ii (cid:21) λ Plate (rotated 90 ◦ ): Q ′ = √ (cid:20) − i − i (cid:21) Fiber: F k = (cid:20) e in e kL e in o kL (cid:21) Sample: M = (cid:20) cos θ K sin θ K − sin θ K cos θ K (cid:21) The trace operation is a succinct method for calculat-ing the modulated intensity observed at the detector: I ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k Tr h ρ k × P × B × E ( t + τ ) × F k × Q × M × Q × F k E ( t ) × B × P i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A7)which yields Equation A1. Appendix B: Models and Measurements of ResidualAmplitude Modulation
The Sagnac Interferometer exhibits a finite offset tothe Kerr signal. One possible origin of this offset is fromResidual Amplitude Modulation from the EOM and itscontribution to a spurious Kerr signal can always be dis-tinguished from those originating from the non-reciprocalsignals as its sign will not change when the quarter-waveplate is rotated 90 ◦ . RAM can add spurious harmonicsto the signal in many ways, but because of the reciprocalnature of the interferometer, many of these are canceledout. Though we are unable to fully account for the sizeof the offset, we can model some of the more qualita-tive features and provide understanding for methods ofreducing spurious signals.In addition to the fiber-position dependence, men-tioned earlier, we provide some additional informationregarding its behaviors. The offset does not vary whendifferent samples are imaged. It also is not from an RFelectronic noise signal inadvertently coupling to the de-tector or the lock-in amplifiers as the spurious Kerr signaldoes not depend on intensity, nor is it present when thebeam is blocked at the sample. It is also not likely to befrom stray magnetic fields, as none have been detectednear the instrument with a gauss-meter. It is not froma spurious intensity modulation present on the SLED, as1might be induced from stray coupling from the functiongenerator, as no signal is detected when the driving fieldon the EOM is removed. Finally, there is a wavelengthdependence to the offset, which suggests that the opticalpath length is an important part of the effect. The ef-fect is also different when the system is dismantled andthen re-assembled. Specifically, if the fiber-connectorsbetween the EOM and the sample are slightly misaligned,the spurious amplitude modulation increases. The RAMalso depends upon the layout of the fiber: when it iscoiled on its spool there may be no offset, but when thewhole fiber is stretched out or twisted, stable offsets onorder of a 1 µ rad occasionally appear. The only wayto eliminate this is by moving the fiber into a positionwhere the offset is negligible. Some of these characteris-tics suggest that the non-ideal modes of Figure 3 mightbe contributing to the spurious signals, and that scatter-ing and variations in the index of refraction within thefiber are the source.We measure the size of the RAM at the lock-in fre-quency for light after a single pass through the inline fiberEOM to be about α = 10 − times the average power.The exact value is stable, but it changes every time theSLED power is adjusted. The variations are greater thanby no more than a factor of about 2. We also measure thephase of this signal with lock-in relative to some constantphase, though the exact value can be estimated from thecable length to the detector. While there appears to bea hysteretic effect on the RAM when SLED power is var-ied, the value of the corresponding phase measurementis independent of power. We also measure RAM at thesecond harmonic frequency, but and this is decreased inmagnitude by about a factor of 4 from that on the firstharmonic.
1. Harmonic Expansion
There are many optical trajectories in the systemif scattering and improper couplings are considered asshown in Figures 3 and 4 and any of these might con-tribute to spurious signals. As mentioned before, wefind that only those paths which include reflection offthe sample can possibly generate an offset, as no signalis observed when the sample is removed. The most gen-eral model for RAM included on the two primary pathsis: I ( t ) = I| A ( t ) e i Φ( t )+ iθ K + A ( t ) e i Φ( t + τ path ) − iθ K | Φ( t ) = φ m sin( ωt ) A ( t ) = (1 + α o ←− sin( ω ( t + τ path ))) × (1 + α e −→ sin( ωt ))) A ( t ) = (1 + α e ←− sin( ω ( t + τ path ))) × (1 + α o −→ sin( ωt )))(B1) α e and α o distinguish the modulation applied to thelight on the extraordinary and ordinary axes of the EOM when it is moving in the direction indicated by the un-derscored arrow.Consider the situation where there is no directional de-pendence to the EOM’s amplitude modulation. If τ path isoptimized to yield a π phase shift for a complete traver-sal of light around the Sagnac loop, then the amplitudeof odd harmonics of the modulation will flip sign aftera second pass through the EOM. Using this condition,we study the most general model of RAM where it hasspectral weight on all integral harmonics of the drive fre-quency. Let F n ( t ) and S n ( t ) be equal to the complexharmonic (i.e. F n ( t ) = F ∗− n ( t ) = a n exp( inωt + iθ n )) ofRAM with arbitrary phase and amplitude at frequency nω on the fast and slow modes of the EOM respectively. A ( t ) = ∞ X n = −∞ F n × ∞ X n ′ = −∞ ( − n ′ S n ′ A ( t ) = ∞ X n = −∞ ( − n F n × ∞ X n ′ = −∞ S n ′ Where the sums above are taken over the harmonics ofthe driving frequency. Expanding A1 with this included I ( t ) = I (cid:0) A ( t ) + A ( t ) + A ( t ) A ( t ) × J-A Expansion (cid:1)
So, A ( t ) + A ( t ) = X nn ′ n ′′ n ′′′ F ω F n ′ S n ′′ S n ′′′ (cid:16) ( − n + n ′ + ( − n ′′ + n ′′′ (cid:17) (B2) A ( t ) × A ( t ) = X nn ′ n ′′ n ′′′ F n F n ′ S n ′′ S n ′′′ ( − n + n ′′ (B3)A term will contribute to the n th harmonic of theseterms if | n + n ′ + n ′′ + n ′′′ | = b . If b is odd, then an oddnumber of mixing frequencies will be odd harmonics. Ofthe four frequencies above, dividing them into any twopairs will require one set to sum to an odd number, andthe other set to an even number. This implies that thereare no odd harmonics in both of the above terms. Whenthere is no Kerr signal, then there are only even har-monics in the Jacobi-Anger expanded interference terms,so any mixing between them and the RAM will produceonly even terms. To conclude, there must be a drivefrequency at which amplitude modulation introduced inthis fashion can not contribute to the first harmonics inan aligned system. Spurious even harmonics must be atmost at the level of α .If the frequency is misaligned then the spurious sig-nals will be on order δ × α , where δ = π f − f ideal f ideal . A slightmisalignment in the frequency approximately involves re-placing (-1) with exp( iπ − iδ ) ≈ ( − iδ ) in EquationsB2 and B3, and expanding to first order in δ . Spuriousharmonics with magnitude comparable to a 1 µ rad signalwill occur when δ × α ≈ − or when δ ≈ − , and2since the precision of the frequency set-point is on thescale of 10 − , misalignments in frequency are unlikely togenerate a noticeable offset.Despite precautions, such as angle polished fibers, ad-ditional, unintended modes can be present in the interfer-ometer having formed by scattering or reflections, suchas in Figure 3d. If two such modes have the same op-tical path length, they will form a competing interfer-ometer with signal superimposed on top of the primaryone 3a at the detector. With exception given to thosemodes shown in Figure 3d, It is highly unlikely that suchmodes comprising a spurious interferometer have a pathlength identical to the primary modes. Consequently,they are driven by the EOM at a frequency not equalto c Path Length and as mentioned above, this will gener-ate an offset from RAM. We have noticed an increase inRAM when fiber connectors are not well aligned. Suchmisalignments, as well as others, might introduce modeswhich form additional interferometers.
2. Directional Asymmetry in Traveling-Wave Modulators
The analysis above demonstrates that the effects of di-rectionally independent RAM should be tuned away withproper drive frequency alignment for the EOM, howeverwe still observe a finite offset on the order of a µ rad tothe Kerr signal. We consider the case that there is direc-tional dependence to the modulation coefficients in Equa-tion B1 and demonstrate that it, too, can not account forthe observed spurious amplitude modulation. Directionaldependence might be expected given that the waveguidemodulator operates by applying a propagating RF elec-tric field to the Lithium Niobate crystal that travels atthe same speed as the TM (extraordinary) optical po-larization component moving in the direction away fromthe circulator in Figure 2. The relative difference in thepropagation speeds of the two waves is less than 10% fromfactory specifications, however when the optical beam istraveling in the direction opposite to that of the RF field,the velocity mismatch will be greater. Because of the di-rectional dependence of the phase modulation, it is to beexpected that there will be a directional dependence tothe amplitude modulation as well. F ω and S ω will bedirectionally dependent and the spurious first harmonicswill not cancel in Equations B2 and B3.We consider a model of the traveling wave modulatorwhere the action of the modulator on the optical beamsnot only depends on polarization, but on the propagationdirection as well. As a function of position and time, thevoltage at any point in the traveling wave modulator isgiven by : V ( z, t ) = V sin( kN m z − ωt )where N m ≡ cv RF is the propagation constant within theelectrodes for the driving voltage signal, k is the freespace RF wave number and ω is the frequency of theRF signal. This equation assumes that the wave starts on one side of the Lithium Niobate crystal waveguide, oflength L, and is absorbed at the other end. As the mod-ulator used in these experiments is not terminated witha 50Ω resistor that would otherwise allow for impedancematching, part of the signal will be reflected and a cor-responding term for this traveling wave will need to beincluded in the above equation. We will consider per-fect impedance matching and extend the argument tothe generalized situation. If the optical signal is prop-agating with the RF wave then its position, z , is givenby kN EOMe/o z = ω ( t − t ), where N EOMe/o is the index ofrefraction for the extraordinary or ordinary optical polar-ization states, and t is the time when the light beam firstenters the EOM from either direction. The phase shift,as a function of position, will be proportional to the ap-plied voltage by β e/o , which depends on the electro-opticproperties of the crystal. A propagating optical beam willhave a phase shift per unit length within the modulatorgiven by:∆ φ e/o ( z ) = β e/o V sin( k ( N m − N EOMe/o ) z − ωt )and it’s integral over the entire modulator is: φ → e/o = Z L ∆ φ e/o ( z ) dz = β e/o V k Γ e/o sin (cid:16) k Γ e/o L (cid:17) sin (cid:16) k Γ e/o L − ωt (cid:17) = β e/o V L (1 + ǫ → )sin (cid:16) k Γ e/o L − ωt (cid:17) → − β e/o V L (1 + ǫ ← )sin (cid:16) kN EOMe/o L ωt (cid:17) (B4)Where we use the substitutions Γ e/o ≡ N m − N EOMe/o and let Σ e/o ≡ N m + N EOMe/o for here and below. In thelast two lines, terms higher than first order in the Tay-lor expansion of sin are kept in ǫ → and, below, in ǫ ← . Inthe last line, a constant phase factor, kN m L has been re-moved from consideration. If the optical signal is travel-ing against the RF wave, then kN EOMe/o ( L − z ) = ω ( t − t )then the corresponding voltage equation and its integralis∆ φ e/o ( z ) = β e/o V sin( k ( N m + N EOMe/o ) z − kN EOMe/o L − ωt ) φ ← e/o = Z L ∆ φ e/o ( z ) dz = β e/o V k Σ e/o sin (cid:16) k Σ e/o L (cid:17) sin (cid:16) k Γ e/o L − ωt (cid:17) = β e/o V L (1 + ǫ ← )sin (cid:16) k Γ e/o L − ωt (cid:17) → − β e/o V L (1 + ǫ ← )sin (cid:16) kN EOMe/o L ωt (cid:17) (B5)3Here, from the specification sheet for the EOM, Γ e/o ≈ N e/o ≈ . N EOMe = 2 .
17 and N EOMo = 2 . L = 7 . cm so at 5MHz drive frequency, k Γ e/o L ≈ × − . Since EOM drive voltage is adjusted such that op-timal phase modulation is . r and r electrooptic coefficients, describing the modula-tion of the TM and TE modes of the EOM, are of thesame order of magnitude, β e/o V L ≈ − less than that of thefirst order term. Each of these expressions for the mod-ulation describes waves in the voltage signal traveling ina single direction for each term. To include possible ef-fects from a reflected RF signal as created by the knownimpedance mismatch in the modulator, then the result-ing directionally dependent modulation amplitudes willbe linear combinations of the two terms above, and thiswill only effectively change the amplitudes and phases forthe modulation terms in the analogous form for EquationA1: I ( t ) = I| e iφ m sin( ωt + ϑ )+ θ K ± e iφ m sin[ ω ( t + τ path )+ ϑ ] − θ K | (B6)Expanding this expression can not lead to spuriousterms leading to an anomalous Kerr signal. However,the directional dependence for the phase modulations,implies, as previously mentioned, that there may be acorresponding directional dependence in the amplitudemodulation and a spurious signal, and in this way, RAMwill be introduced.Estimating the resulting size of RAM can not be ac-complished without a model for a specific mechanismthat generates it. One possible mechanism is amplitudemodulation occurring from modulated scattering at thesurfaces of the Lithium Niobate crystal. The contribu-tions to the amplitude modulation when the beam passesthrough the two facets of the EOM twice are given by: A ( t ) = (1 + α sin( ωt )) × (1 + α sin( ωt + φ e )) × (1 + α sin( ω ( t + τ path ) + φ e )) × (1 + α sin( ω ( t + τ path ) + φ e + φ o )) A ( t ) = (1 + α sin( ωt )) × (1 + α sin( ωt + φ o )) × (1 + α sin( ω ( t + τ path ) + φ o )) × (1 + α sin( ω ( t + τ path ) + φ e + φ o ))From Equations B 2 and B 2, the optimal choice of fre-quency is π = ωτ path + φ e + φ o . As φ e/o = N e/o kL ≈ × − rad, expanding the above expressions will resultin all terms containing first harmonics canceling out orcontributing negligibly .
3. RAM and Birefringence
If light is scattered at the sample by birefringence intothe wrong modes as in Figure 3c, then the two will notinterfere and their power signals will just add at the de-tector. Before, it was shown that the effects of RAM onthe primary modes will cancel when the modulation sig-nal is delayed by π rad on the second pass through theEOM, as compared to the first pass. When aligned in thisway, this condition will not be the case for modes trav-eling along the incorrect paths, because the path lengthsfor these two modes will not be equal to that of the pri-mary modes. If we consider that some fraction of thelight, ρ , is scattered at the sample into the incorrectpaths, then we can estimate the size of the resulting spu-rious signals for RAM, A ( t ) = 1 + α sin( ωt ), applied tothe signal on each pass through the EOM. As measuredbefore, α = 10 − and the phase difference between thetwo beams of light after reaching the sample is given by ǫ = ωL n e − n o c = π ∆ ℓL ∼ − . Because of the incoherenceof these two beams, the phase modulation does not needto be considered in the model below, so only amplitudemodulation from the first and second passes through theEOM for both beams is included: I ( t ) = ρ I × (cid:0) A ( t ) A ( t + Ln e c ) + A ( t ) A ( t + Ln o c ) (cid:1) = ρ I × (cid:0) α sin( ωt ))(2 − α sin( ωt + ǫ ) − α sin( ωt − ǫ ) (cid:1) ≈ ρ I × (cid:0) − α sin ( ωt ) (cid:1) + ρ I αǫ sin( ωt )Let ρ = 10 − , as found before by measuring the reduc-tion in the second harmonic after removing the quarter-wave plate. The expected contribution to the first har-monics is then down by a negligible factor of 10 − ,demonstrating that this model can not account for thespurious signals. However, in systems where the ampli-tude modulation and birefringence is greater, this effectmight be relevant. Appendix C: Nonlinear Effects
The nonlinear Kerr effect within the fiber or other com-ponents can be a source of nonreciprocity if the powerin the two counter-propagating channels is different orthe source is pulsed . Unlike RAM, because nonlineareffects will generate a phase difference within the twobeams, the resulting Kerr signals will change their sign ifthe quarter-wave plate is rotated 90 ◦ . Nonlinear effectswithin the fiber should not contribute because the nonlin-ear Kerr effect is coherent process. For each polarizationchannel within the fiber, the two counter-propagatingand broad-bandwidth beams will not be coherent andtherefore the nonlinear Kerr effect will average to zero.The only place where the nonlinearities may contributeto a spurious signal is where the beams are coherent,4such as in the EOM. As mentioned before, an intensitydependent Kerr signal is an indicator that a nonlineareffect is contributing to the measurement. The intensityshould be tuned in such a way as to avoid changing thespectrum of the SLED, which could potentially affect sizeof the offsets. We have not seen clear evidence for thiseffect, thus suggesting that nonlinear responses in the in-strumentation are not significant with power levels usedin this experiment.Nonlinear optical effects can potentially induce mag-netism in the sample: if such a sample is radiated withunequal power from + and − circularly polarized lightwill magnetize the sample, however the measured inten-sity difference between the two beams is low. Anotherpossibility is if the incident light significantly changes thesample’s temperature or changes some other macroscopicproperty. Appendix D: Alignment Procedure
Aligning the all-fiber SI is performed by cyclingthrough a series of steps until satisfactory performanceis achieved.1) Fix the optical wavelength of the source. Smallchanges in the center wavelength may affect the per-formance, however we find that the system is unaf-fected by shifts of ± µ rad. It is also essential that the optical compo-nents be designed such they are optimally functioningwithin the selected wavelength range.2) Assemble the components and ensure that there is suf-ficient throughput to the detector. Use a mirror in lieuof a magnetic sample and carefully ensure there are nostray magnetic fields present.3) Spurious first harmonics can enter the system in theform of electronic noise coupling to the detector andthe light source, so it is prudent to characterize thesesignals and eliminate them. One method of distin-guishing if the signal is originating from either theSLED or the detector is to measure the size of a spu-rious harmonic when illuminating the detector withthe SLED and comparing it to the size of the signalwhen illuminating with a small battery-powered lightsource, such as a laser pointer.4) Set the modulation voltage of the EOM to give a . π phase shift given the length andindex of refraction of the fiber.5) Still using a mirror as a sample: Maximize the secondharmonics by rotating the quarter-wave. 6) From Equation A1, the optimal value of ω is foundby adjusting the EOM’s driving frequency until thesecond harmonics are maximized. Alternatively, asalso prescribed by Equation A1, the second harmon-ics should vanish at the twice the optimal frequency.The optimal frequency should ideally coincide witha minimum signal from spurious first harmonics, butdepending on the noise source, this is not always thecase.7) On a homogeneous, strongly magnetic sample (i.e.CoPd multilayer film), maximize the first harmonicmagnitude by tuning the modulation depth. Thisguarantees that the value of the modulation depth isthe half the value of the argument of the J Besselfunction that gives the maximum, 0.92. This condi-tion also maximizes the size of the first harmonics,making it easier to measure.8) On a homogeneous, strongly magnetic sample, alignthe lock-in phase offset so that the first harmonic sig-nal is entirely in phase with the reference. When re-turning to a mirror, there should be no signal on thequadrature component of the first harmonics. Notethe phase of any spurious signals (i.e. from RAM)that are present after this alignment.9) The RF signals on the co-axial cables between the elec-tronics have transmission amplitudes and phases thatare strongly dependent on frequency and cable length.Once the driving frequency for the EOM has been se-lected, calibrate the transmission ratios for both har-monics going from the detector output to the lock-inamplifiers using a second RF source in place of thedetector and phase-locked to the main function gen-erator.10) Determine which of the two orientations the waveplate is in by measuring the Kerr angle with a sampleof known magnetization.11) Check if RAM is causing offsets by examining how theKerr signal varies as the quarter-wave plate is rotated.12) Test the system with birefringent samples and chi-ral samples and demonstrate that the measurementof the Kerr signal is immune to such perturbations.Misalign and then realign the quarter-wave plate todemonstrate immunity to this, as well. Measure theVerdet constant of a known sample to check the cali-bration.13) Adjust the fiber layout until any spurious signals fromRAM are minimized.Pay particular attention to the following issues:A) The alignment of the miniature lens/quarter-waveplate assembly at the termination of the fiber at thesample is difficult. We recommend carefully model-ing the propagation of the beam through this optic5using ray tracing software. It is often helpful to diag-nose problems and check components in the systemusing a fiber-coupled collimator, a free-space quarter-wave plate mounted on a rotation stage, and a mirrorplaced at the focal point of the beam.B) Because the fiber only supports a single Gaussianmode per polarization per direction, it is essentialthat the wavefront is specularly reflected from thesample. Any roughness or curvature on the surfacewill scatter light out of those optical modes that arematched to the fiber, substantially decreasing theamount of power returning to the detector.C) The Kerr angle should be unaffected by changes inthe total power through the whole system (as mightoccur when samples of different reflectivities are mea-sured). This would otherwise indicate the presence ofa spurious signal superposed on the desired one, butit could also indicate the presence of an intensity in-dependent spurious harmonic from electronic noise aswas described previously.D) When dividing the first harmonic signal by the sec-ond harmonic signal it is important to ensure thatdigital quantization errors are not large which wouldotherwise result in this observation. We avoid this byoperating the lock-in which measures first harmonicsin “ratio” mode which normalizes its measurement bythe output of the lock-in which measures the secondharmonics.E) The Kerr signal should be drift-free. Plenty of opti-cal isolation in front of the SLED will prevent back-reflections possessing RAM from being re-amplified.Drift from thermal fluctuations within the EOM canalso appear when the its drive frequency is not opti-mal for the given optical path length.F) The first harmonic quadrature should also be closeto zero; it, too, will likely not be exactly centeredat zero, because of the variety of spurious offsets. Itis often helpful to create a real-time phasor plot ofthe first harmonics. Once the alignment procedure isfinished, the signal on the quadrature should be driftfree.G) The effects of thermal fluctuations can be examinedby several tricks; touching the fiber, uncoiling it, orcarefully heating it with a hair dryer. These testscan be performed for the EOM as well, as it can alsobe sensitive to temperature changes. We house theall the components in Styrofoam casings to minimizethermal expansion of the components and eliminateair currents. 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