Active feedback scheme for minimizationof helicity-dependent instrumental

asymmetries

M. I. Fabrikant,1,* K. W. Trantham,2 V. M. Andrianarijaona,3 and T. J. Gay1

1Behlen Laboratory of Physics, University of Nebraska, Lincoln, Nebraska 68588, USA2Fort Hays State University, Hays, Kansas 67601, USA3Pacific Union College, Angwin, California 94508, USA

*Corresponding author: ifayam@gmail.com

Received 10 December 2007; revised 29 March 2008; accepted 4 April 2008;posted 7 April 2008 (Doc. ID 90666); published 29 April 2008

Amethod for the active feedback reduction of optical instrumental intensity asymmetries is presented. Itis based on the fast chopping of two spatially separated beams of light with orthogonal linear polariza-tions that are recombined and passed through a quarter-wave plate to yield a single beam with rapidlyflipping helicity. Active electro-optic feedback has been successfully employed to maintain this asymme-try below 10−5. © 2008 Optical Society of America

OCIS codes: 350.4600, 120.4570, 120.4820, 120.2130, 230.5440.

1. Introduction

Of the available methods to produce polarized elec-trons, laser-induced photoemission from GaAs crys-tals is advantageous because it allows the rapidflipping of electron spin using optical means [1].Weaknesses of this method emerge if the experimen-tal signal is sensitive to spin; helicity reversal of thelaser beam is generally associated with both spatialand intensity variations, which are produced by theoptical elements responsible for helicity reversal. Ifthe GaAs crystal has a nonuniform quantum effi-ciency over its surface, helicity-related spatial move-ment of the laser beam may produce a helicity-dependent current asymmetry emerging from thepolarized electron source. Similarly a helicity-dependent laser intensity will produce a correspond-ing helicity-dependent current, even in the absenceof spatial variations. Polarized electron currents thatdepend on their spin direction will subsequentlycause an instrumental asymmetry (IA) mimicking

a true spin-dependent signal. In this paper we definephoton intensity or electron current asymmetry as

A ¼ I− − Iþ

I− þ Iþ; ð1Þ

whereIþ isthecurrentorintensityofpredominantlyfor-ward-spin electrons or photons and I− is the equivalentquantity for spin-backward particles.We present an ac-tive feedback scheme to correct this problem at theoptical level. In previous work [2] we have shown thatthe upper limit of IA with passively stabilized opticalspin reversal is ∼5 × 10−4. Active feedback to correctthis problem has been investigated by several groups[3–8], but to our knowledge this is the first scheme thatdoesnotmakeuse of electro-optic or electromechanicalfeedback operating at the helicity-reversal frequencyto force instrumental asymmetries to zero.

The work reported here was necessitated by the re-quirements of a larger experiment designed to ob-serve electron circular dichroism (ECD) [9,10]. Inthis experiment, polarized electrons with alternatelyforward and backward longitudinal spins traversea chiral vapor target. The goal is to observe a

0003-6935/08/132465-05$15.00/0© 2008 Optical Society of America

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spin-dependent asymmetry in the number of elec-trons that traverse the target. This asymmetry is ex-pected to be very small—of the order of 10−4 [11]. Inorder to observe such small values, it is importantthat the experiment’s IA be significantly smallerthan 10−4.

2. Apparatus

Previous versions of our ECD experiment, whichwere plagued by large (∼10−4) IAs, employed passivemeasures such as temperature stabilization and re-focusing optics to minimize these effects [2,3]. Thehelicity of the polarized light was flipped by usinga single photoelastic modulator (PEM) to vary the re-tardance [12]. The primary source of instrumentalasymmetries resulted from the spatial drifting ofthe light beam, which is an inherent consequenceof the PEM [2,13]. To overcome these problems wedevised an optical system that flips the helicity ofpolarized light mechanically rather than electro-optically and that uses slow active feedback to limittime-varying IAs.The optical system is shown schematically in Fig. 1.

It uses a 75mW, 785nm Power Technology diode la-ser (Model LDCU5/7873). This laser was speciallybuilt to have excellent polarization and intensity sta-bility with regard to temperature. Nevertheless weuse a linear polarizer (LP) to ensure the initial lightpolarization. Afterward the laser light passes througha crystalline half-waveplate (Meadowlark H1557) anda computer-controlled liquid crystal retarder (LCR:Meadowlark LVR-200). It then passes through a polar-izing beam splitter (Thorlabs PBS3) creating twoorthogonally polarized spatially separated beams oflight. When the half-waveplate is rotated, intensityis redistributed between these two beams. A mechan-ical chopper (Thorlabs MC1000A) is inserted so that,when it is positioned correctly, it allows only one beamto pass at a time. The purpose of the LCR will be ex-plained in Section 3. After the chopper, each beam is

reflected by a mirror into a second polarizing beamsplitter that acts to spatially recombine the twobeams.The effect of this setup is to separate the beams in timeusing the chopper, but recombining them in spacewiththe two beam splitters. The recombined beam thenpasses into diode 1. The photodiode signal can bemon-itored with an oscilloscope or a computer. Finally athird beam, generated by a He–Ne laser, is shotthrough the chopper into diode 2. This beam is usedto create a reference signal with which we comparethe signal that reaches diode 1. A synchronization out-put from the chopper controller was available, but wasfound not to remain in phase with the diode 1 signal.

3. Measuring Asymmetry and Positive Feedback

The problems associated with IAs lie not in their non-zerovalueperse,but inthefact thattheydrift intime.Ifan IA did not drift, it could be minimized by passivemeans, measured carefully, and subtracted from theECDsignal toyield trueasymmetries.Weconsiderfirstthe problem of IA measurement, and then the use ofactive feedback to eliminate IA drifting.

The signals that reach diodes 1 and 2 are modifiedsquare waves [see Figs. 2(a) and 2(b), respectively].Both diode signals run to Keithly 1645 picoammeters,and the analog voltage outputs of these meters aresampled with a National Instruments PCI-6143 dataacquisition card. The card samples data in “burstmode”bymakingmanymeasurementsat thesamplingfrequency, storing them in a buffer, and outputting all

Fig. 1. Schematic of the optical system. Light from the laser isseparated into its constituent polarizations (arrows) and recom-bined in space, but not in time, with the aid of a chopper. The in-tensity of each polarization is measured with diode 1 (using theHe–Ne signal from diode 2 as a reference) and the LCR is adjustedto minimize the intensity asymmetry. Helicity is flipped by thepassage of the temporally alternating linear polarization througha fixed quarter-wave plate.

Fig. 2. Raw signals in arbitrary units as read by the computerdata acquisition card: (a) recombined intensity, (b) trigger wave-form, (c) time derivative of the trigger signal with rejection thresh-olds indicated.

2466 APPLIED OPTICS / Vol. 47, No. 13 / 1 May 2008

resulting data to the computer after the desired num-ber of samples has been reached. The diode 1 signal re-presents the intensity of the recombined beams [seeFig.2(a)].Sincetheseparatedbeamintensitiesaregen-erally different, a modified square-wave results. Thespikes in the signal are the result of both beams beingpartially on simultaneously as the chopper switchesfrom one blade to the next. The diode 2 signal is asquare-wave thatmaintains a fixed phase relationshipto the diode 1 signal.Asymmetry A can be calculated using Iþð−Þ taken

from the upper (lower) flat regions of the square-wave in Fig. 2(a). To get an accurate value for Iþand I−, we must eliminate any sampling of data inthe vicinity of the spikes in the square wave. We dif-ferentiate the reference signal [see Fig. 2(c)] and re-ject all data points where the absolute slope of thetrigger signal is less than some threshold—typically0:01V=s. We then use the trigger signal in combina-tion with the chopping rate to bin all Iþ and I− pointsand take their respective averages. The region of thechopper cycle over which sampling occurs, as dic-tated by the reference beam, is shown in Fig. 2(c).The choice of chopper speed, sampling rate, and

sampling time is dictated by three criteria: (1) atleast one chopper waveform must be sampled, thatis, a high and low intensity are required to calculatean intensity asymmetry; (2) the data acquisition timetaken to measure an asymmetry value must be shortcompared to characteristic IA drift times; and (3) thechopping rate must not be faster than the responsetime of the diodes. A typical chopper frequency inthese experiments was 500Hz with a sample rateof 200kHz and a data acquisition time for one asym-metry value of 50ms.We cannot assign an uncertainty to this value be-

cause there is no way of knowing whether these re-sults obey a normal distribution or what the width ofthat distribution is. We therefore repeat this process1000 times and extract a mean and standard devia-tion of the mean to give an average asymmetry mea-surement with an uncertainty. This compoundmeasurement takes ∼1:2 s to make. The number ofdata points used to determine a single asymmetryvalue is not taken into account explicitly when calcu-lating an uncertainty for the composite asymmetry.Nevertheless as a consequence of the criteria above,this number influences the uncertainty and must besufficiently high to allow a small uncertainty to bereached in a reasonable amount of time.Themainsourceof systematic error inanyasymme-

try measurement is asymmetry drift. One way tominimize this error is to chop the beam very quicklyand sample many times in a given chopping period;the shorter the chopping period, the less opportunitythe asymmetry has to drift within a given measure-ment of A. This improves the accuracy of any givenasymmetry measurement but does not improve theproblem of overall drift. To correct long-term drift, ac-tive feedback to set the IA to zero must be employed.An example of such an asymmetry measurement ta-

ken for our optical system over an extended periodof time is shown in Fig. 3. Over the course of 1:5h,the asymmetry drifts drastically to values muchgreater than the expected experimental asymmetry.To eliminate this drift we employ active feedback.

In the simplest systemwewould have only the half-waveplate to orient the linear polarization precisely at45°, just upstreamof the first beamsplitter.Oneway tofeedback onasymmetrywould be tomake anasymme-trymeasurement and rotate the half-waveplate to con-tinually force the asymmetry to zero. We can analyzetheeffectsofvariouselements inouroptical trainusingthe technique of Mueller calculus [14]. In our case webegin with a vertically polarized state of light emittedby the laser:

~P ¼

2664

1−100

3775: ð2Þ

In this vector, P0 is the total intensity, P1 is the differ-ence between the intensity of horizontally and verti-cally polarized light, P2 is the intensity difference inthe 45° and 135° polarization directions, and P3 isthe difference in intensity between right- and left-handed circularly polarized light. The half-waveplatecan be represented by the following operator:

�λ2

�¼

26641 0 0 00 1 0 00 0 −1 00 0 0 −1

3775: ð3Þ

Toaccount for the rotation of thehalf-waveplate,we in-troduce the rotation operators

½R�ðaÞ� ¼

26641 0 0 00 cosð2aÞ ∓ sinð2aÞ 00 � sinð2aÞ cosð2aÞ 00 0 0 1

3775; ð4Þ

where a is the angle of rotation. Multiplying thesema-trices together gives us a final Stokes vector:

Fig. 3. Long-term behavior of asymmetry without feedback.

1 May 2008 / Vol. 47, No. 13 / APPLIED OPTICS 2467

~Pfinal ¼ ½Ra� · ½λ2� · ½R−a� ·~P. The polarizing beam split-tersmaybe regarded as fixedLPs. The intensity asym-metry is therefore given by Pfinal;1=Pfinal;0 ¼sinð4aÞ. Since Pfinal;0 is defined to be 1, we can simplyextract thePfinal;1 component to obtain the asymmetry.The goal is to maintain the asymmetry at A ¼ 0; themagnitude of the rate of asymmetry change versushalf-waveplate angle at these points is 4=rad. If wewish to control the asymmetry towithin 10−5, wemustcontrol the angle of the half-waveplate to∼0:5 arc sec.Since this is near the precision limit of commerciallyavailable nanorotators, the plane of polarization is in-stead fine-tuned electronically by changing the retar-dance of the LCR (see Fig. 1).We represent the LCR as a retarder with variable

retardance d that depends on a voltage v:

½LCR� ¼

26641 0 0 00 1 0 00 0 cosðdðvÞÞ sinðdðvÞÞ0 0 − sinðdðvÞÞ cosðdðvÞÞ

3775: ð5Þ

The fast axis of this retarder is fixed at 45°. Thereforethe final Stokes vector is given by ~Pfinal ¼ ½Ra� · ½λ2�·½R−a� · ½R45°� · ½LCR� · ½R−45°� ·~P. The resulting asym-metry is A ¼ cosð4aÞ cosðdðvÞÞ.The retardance–voltage curve provided by Mea-

dowlark Optics is shown in Fig. 4. Also shown inFig. 4 is our measurement of the LCR retardance–voltage curve obtained by placing the LCR between(imperfect) crossed polarizers and measuring thetransmitted intensity as a function of v. Taking intoaccount that the two polarizers have a value ofK incident ¼ 0:965 [15], we obtain the results of Fig. 4.Using these results and the Mueller analysis of thesystem in Fig. 1, we can predict the optical systemasymmetry as a function of LCR voltage. Both thepredicted and the measured results of this are shownin Fig. 5, where the half-waveplate angle is 0°(vertical).In a typical linear feedback scheme, we wouldmea-

sure the asymmetry and alter the system (i.e., outputa voltage to the LCR) in a way that would make the

asymmetry tend to zero. This would all take placeduring a discrete time interval i. Our feedbackvoltage was determined by viþ1 ¼ Aikþ vi, whereAi is the measured asymmetry and vi is the voltageoutputted during this time step. In the region nearAðv0Þ≡ 0, AðvÞ ¼ ðdAdvÞjv0ðv − v0Þ. Thus, for smallA, k ¼ −dv=dAjv0 .

The asymmetry measurement must be made byaveraging over many chopper cycles to avoid feedingback on noise. However, in the beginning of a feed-back cycle, when changes in asymmetry are large,we average over fewer cycles so that the time it takesto feed back is not much larger than the time it takesfor the asymmetry to change. We therefore use anadaptive method that alters the number of choppercycles used to make a measurement with regard tohow large the asymmetry is. We use only two cate-gories of chopper cycle integration; if the asymmetryis >0:001, we use 10 chopper cycles, but if it is<0:001, we use 100. We find that this is sufficientto feed back in a reasonable amount of time. Figure 6shows the short-term behavior of the asymmetrywhen this feedback system is applied.

4. Results

Figure 7 shows the long-term asymmetry behaviorwith feedback, excluding the first few minutes whenthe asymmetry was >10−4. As discussed in Section 3,when no feedback is applied, the asymmetry exhibits

Fig. 4. Retardance of LCR versus applied voltage: data providedby the manufacturer (monotonically decreasing curve) and mea-sured value of this work (curve with peak).

Fig. 5. Asymmetry versus LCR voltage.

Fig. 6. Short-term feedback behavior.

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significant drifting to the point where it would eclipseany expected chiral effect (see Fig. 3). Using thismeth-od we are able to maintain the asymmetry below3 × 10−4. Over 1:5h the accumulated asymmetry ofFig. 7 is 3:7 × 10−6 � 2:2 × 10−6, well below the knownvalues ofECDasymmetries [11]. Inearlier resultswitha factor of 3, worse LCR control voltage resolution lim-ited theabilityof this feedbackmethod tocontrol short-term asymmetries to better than∼10−3 and could onlyachieve average asymmetry values below 1:5 × 10−5 inperiods of time greater than 15h.

5. Conclusions

Wehavedemonstratedtheabilitytoobtainlong-termop-tical asymmetries well below 10−5. Our system has theadvantage that helicity-correlated spatial variations ofthe beamarenot causedby the optical elements respon-sible for helicity reversal. Active feedback to eliminatetemporal drifting of asymmetry can thus be accom-plished by elements not operating at the helicity-rever-sal frequency.Webelievethatthe improvedresolutionofthe LCR voltage control can improve even further theupper limit on short-term asymmetry values, meaningthat the upper limit on average asymmetry requiredby a given experiment can be reached in shorter andshorterperiodsof time.Thisopticalsystemis ideallysui-ted for producingGaAs photocurrent since itminimizesthedrift of the laser beamover the surface of the crystal.In addition the feedback system described in this papermay be applied in a similar fashion to photocurrent,thereby eliminating any remaining helicity asymmetry.

We thankJackMaseberg, JoshMachacek, andSteveDucharme for helpful discussions. This work was sup-ported by National Science Foundation grant PHY-0653379 and Undergraduate Creative Activities andResearch Experiences Research Funding throughthe University of Nebraska.

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Fig. 7. Long-term behavior of asymmetry with feedback.

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