# High-Speed Microscale Optical Tracking Using Digital Frequency-Domain Multiplexing

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 6, JUNE 2009 1991

High-Speed Microscale Optical Tracking UsingDigital Frequency-Domain Multiplexing

Robert A. MacLachlan, Member, IEEE, and Cameron N. Riviere, Member, IEEE

AbstractPosition-sensitive detectors (PSDs), or lateral-effectphotodiodes, are commonly used for high-speed, high-resolutionoptical position measurement. This paper describes the instru-ment design for multidimensional position and orientation mea-surement based on the simultaneous position measurement ofmultiple modulated sources using frequency-domain-multiplexed(FDM) PSDs. The important advantages of this optical config-uration in comparison with laser/mirror combinations are thatit has a large angular measurement range and allows the useof a probe that is small in comparison with the measurementvolume. We review PSD characteristics and quantitative reso-lution limits, consider the lock-in amplifier measurement sys-tem as a communication link, discuss the application of FDMto PSDs, and make comparisons with time-domain techniques.We consider the phase-sensitive detector as a multirate DSPproblem, explore parallels with Fourier spectral estimation andfilter banks, discuss how to choose the modulation frequenciesand sample rates that maximize channel isolation under designconstraints, and describe efficient digital implementation. Wealso discuss hardware design considerations, sensor calibration,probe construction and calibration, and 3-D measurement bytriangulation using two sensors. As an example, we character-ize the resolution, speed, and accuracy of an instrument thatmeasures the position and orientation of a 10 mm 5 mmprobe in 5 degrees of freedom (DOF) over a 30-mm cube with4-m peak-to-peak resolution at 1-kHz sampling.

Index TermsComputer-aided surgery, lock-in amplification,medical robotics, optical position measurement, phase-sensitivedetector, position-sensitive detectors (PSDs).

I. INTRODUCTION

POSITION-SENSITIVE detectors (PSDs), or lateral-effectphotodiodes, are commonly used for high-speed high-resolution optical position measurement. PSDs that measurein 2-D are particularly attractive for multidimensional positionmeasurement [1]. When more than 2 degrees of freedom (DOF)are to be measured, instruments generally have multiple PSDsand laser/mirror combinations with distinct optical paths sothat each PSD still measures one spot with 2 DOF [2]. Thisis difficult to arrange when the measurement probe must bemuch smaller than the measurement volume and the angular

Manuscript received October 16, 2007; revised January 22, 2008. Firstpublished October 24, 2008; current version published May 13, 2009. Thiswork was supported in part by the U.S. National Institutes of Health un-der Grant R21 EY016359, by the U.S. National Science Foundation underGrant EEC-9731748, and by the American Society for Laser Medicine andSurgery. The Associate Editor coordinating the review process for this paperwas Dr. Juha Kostamovaara.

The authors are with the Robotics Institute, Carnegie Mellon University,Pittsburgh, PA 15213 USA (e-mail: camr@ri.cmu.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2008.2006132

range is large. We address this problem by using a probe withmultiple modulated sources, where each PSD simultaneouslytracks multiple sources, enabled by frequency-domain multi-plexing (FDM).

Analog lock-in amplifiers have often been used with PSDsto measure the position of a modulated source in the presenceof ambient light [3]. Multiple lock-in amplifiers can be usedto implement FDM, but the large amount of hardware requiredhas deterred the use of this technique with PSDs. The lock-inconcept is widely used with other sensors, and many designershave found that when the modulation frequencies are lowenough, it is more economical and more flexible to implementthe phase-sensitive detector in software [4], [5]. However, dueto the assumption of static signals, the frequency responseof the phase-sensitive detector is often neglected. This paperdemonstrates how to efficiently implement a digital phase-sensitive detector with any desired frequency response andevaluate the response tradeoffs in the context of dynamic signalmeasurement.

In Section II, we review the PSD characteristics and quantita-tive resolution limits, consider the lock-in measurement systemas a communication link, discuss the application of FDM toPSDs, and compare this approach with alternative time-domaintechniques in widespread use. As a reference design, Section IIIdescribes an instrument that performs 5-DOF tracking of mi-crosurgical tools. We discuss in detail several difficulties thatare widely encountered in applications of PSDs. Section IVconsiders the lock-in phase-sensitive detector as a multiratedigital signal processing problem, explores the connectionswith Fourier spectral estimation and filter banks, describesthe optimal choice of sampling rate and modulation frequen-cies under design constraints, and discusses the tradeoffs inwindow design as relevant to PSD demodulation. Section Vexperimentally characterizes the performance of the referencedesign (Section III) in terms of the noise spectrum, resolution,bandwidth, drift, effects of multiplexing, workspace size, rel-ative error, probe error, and rotation/translation coupling. TheAppendix describes calibration procedures for the PSD cameraand probe.

II. BACKGROUND

A. PSDs

A PSD is a nonimaging analog optical position sensor. It isa large-area planar photodiode with high-resistivity contacts. Aspot of light creates a photocurrent, and the contact resistanceforms a current divider that splits the current between two

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1992 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 58, NO. 6, JUNE 2009

terminals, allowing the position of the spot to be determinedfrom the ratio of the currents. If the currents along an axis arex0 and x1, then the position normalized to the range [1, 1] is

x =x1 x0x1 + x0

. (1)

In a 2-D duo-lateral PSD, the front and back contacts arehigh-resistivity sheets, and the front and rear contact pairs areoriented at right angles to each other. This allows the positionof the spot to independently be measured in 2-D.

Because of the analog spatial averaging performed by theresistive contact, the PSD responds to the mean center ofillumination (or centroid). The size and shape of the spot havelittle effect on position measurement, so precise focusing is notrequired. The disadvantage is that, having no spatial selectivity,the PSD is easily disturbed by ambient lighting and unintendedreflections.

PSD resolution is limited by noise. The noise analysis iscomplex because some noise components are correlated be-tween x0 and x1, and so are either cancelled or doubled by (1).If Is is the detected RMS signal current, Rie is the PSDinterelectrode resistance, and Ib is the mean dc sum of the darkcurrent and uniform background illumination, then the standarddeviation of the normalized x position is

n

B

Is

16 kTRie

+2qIb3

and the signal-to-noise ratio (SNR) is 20 log10(2/n) dB [3].If l is the PSD size, then the standard deviation of the spotposition on the PSD is

ln2

.

We have neglected the contribution of transimpedance am-plifier noise, which can be made small in comparison to theRie thermal noise (see Section III-D). The shot noise in Is doesnot affect the measurement because fluctuations in the signalamplitude are normalized out. The Ib shot noise does degradethe resolution because it is distributed across the PSD and thusis not perfectly correlated at the two terminals.

Using typical values Is = 1.4 A RMS, Rie = 10 k, Ib =1 A, and B = 1 kHz, then n 29 106, and SNR =97 dB. If l = 10 mm, then 290 nm RMS, giving a 1.9-m99.9% confidence interval (equivalent resolution peak-to-peak).

The PSD bandwidth is established by Rie and the junctioncapacitance Cj . These are both distributed, so precise analy-sis requires a position-dependent transmission line model [6];however, a lower bound on the bandwidth is

f3dB 1/RieCj .For a typical 1-cm 2-D PSD with Rie = 10 k and Cj =

175 pF, f3dB 180 kHz. To use more than a fraction of thisbandwidth, it is necessary to consider or even exploit the effectsof position-dependent phase shifts [7].

The effect of PSD size on system performance is worthconsidering. If the system incorporates optical magnification,we can measure any size displacement with any size PSD, so the

Fig. 1. Lock-in amplification application.

resolution at the sensor () is relatively unimportant, and ourprimary concern is the SNR. Is is determined by the spot power,and PSD saturation limits the power density (spot brightness).If d is the spot diameter, then the SNR is proportional to d2 atthe saturation limit. A larger PSD will tend to allow a largerspot size, so the SNR limit also approximately increases as l2.However, Cj is also proportional to l2, whereas Rie remainsconstant (at best), so larger devices have a lower bandwidth.

B. Lock-In AmplificationThe function of the lock-in amplifier (or phase-sensitive

detector) can be described as a narrow tracking bandpass filterwith capability to measure amplitude or phase [8]. Althoughcorrect, this completely obscures the reason for using lock-in amplification to make measurements near dc, which is tosuppress the 1/f noise and other low-frequency interference.

To apply lock-in amplification to a measurement problem(Fig. 1), we must somehow incorporate a modulator in the sen-sor, either by modulating the measurand (e.g., with a chopper)or by using a modulating sensor. A ratiometric sensor like thePSD multiplies the measurand by the excitation amplitude, so itcan serve as a modulating sensor. In communications terminol-ogy, observe that we have arranged for amplitude modulation(AM) of a carrier fc by the measurand. If the measurand hasa fixed dc value, then the output of the sensor is the carrierscaled by this value. The demodulator recovers the measurandvalue by multiplying the sensor output by the same carrierreference that originally modulated the measurand and thenlow-pass filtering the result. The principle of operation of thedemodulator is that multiplying by the reference signal convertsthe AM signal spectrum centered at fc into two componentsat fc fc. The 2fc component is above the low-pass filtercutoff fo and so is suppressed. With a dc measurand, thedifference frequency is zero, so the dc level is reconstructedat the modulator output.

By viewing the lock-in measurement system as an AMcommunication link, we can see that the frequency responsefrom the measurand to the output is simply that of the low-pass filter in the demodulator; signals below fo pass throughunchanged. The dashed signal paths in Fig. 1 carry AM signals;interference on these paths is rejected unless it falls within thebandwidth fc fo. This means that the measurement is insen-sitive to offsets, drift, 1/f noise, or any other low-frequencyinterference introduced at these points.

MACLACHLAN AND RIVIERE: HIGH-SPEED MICROSCALE OPTICAL TRACKING USING DIGITAL FDM 1993

Fig. 2. FDM sensor.

Because these low-frequency disturbances almost invariablydo exist in dc measurements, it is somewhat surprising to realizethat if the noise were either white or increased at high frequency(> fo), then the signal-to-noise improvement would be at leastas good if we discarded the rest of the lock-in system andused the low-pass filter alone. So the removal of low-frequency(< fo) noise is the only reason to use lock-in amplificationfor dc measurements. Phase-sensitive detectors also see use ininherently ac applications (such as impedance measurement),where 1/f noise is generally not a problem, and the phasemeasurement capability is often valuable.

C. FDM PSDs

The communications concept of frequency domain multi-plexing (FDM) can be implemented by a bank of lock-in am-plifiers. We can then simultaneously transmit multiple signalsover the same medium as long as they are sufficiently separatedin frequency to avoid any significant interference. Fig. 2 showsthe concept of FDM as applied to PSD measurement, whichhas been done in practice [9], [10] and has theoretically beenanalyzed and experimentally validated [7], [11], but remainsa little-used techniqueperhaps undeservedly so. There isnegligible loss of accuracy or resolution, and the hardwarecomplexity is minimal when software demodulation is used.

Note that FDM assumes that the sensor output is the sumof the two modulated measurands. If there were nonlinearinteraction between the two carriers, then the two measure-ments would become coupled, degrading the accuracy of both.However, the PSD centroid measurement property implicitlyrequires a sum response, so we can expect the nonlinear interac-tions to be small. To characterize this problem in the frequencydomain, observe that any nonlinear interaction between carriersat frequencies fc1, fc2 will cause intermodulation distortion(IMD), creating output components at fc1 fc2. We haveexperimentally measured these IMD products and found thatthey are at least 80 dB below the carrier.

In any multiplexing scheme, if the SNR (or resolution) is tobe held constant, then the power per channel must also remainconstant as the multiplexing is increased (total power increasesproportional to multiplexing). As for the feasibility of doing sowith FDM PSDs, Qian et al. [11] are overly pessimistic. SincePSD saturation is a localized phenomenon, it is possible to add

nonoverlapping spots of the same brightness without causingsaturation. In addition, the power may be limited by the sourcebrightness rather than PSD saturation, in which case addingmore sources proportionally does add more power. In theory,there is also some resolution degradation due to the shot noiseof the other spots, but this is position dependent and likely tobe submerged in the Rie noise as long as all the spots are closetogether. The degree of multiplexing is ultimately limited bythe PSD bandwidth, but in the applications we are currentlyconsidering, the total bandwidth is less than 10 kHz, so thePSD bandwidth is not a limiting factor.

D. Time-Domain Techniques

An alternative way to achieve the interferencerejectionfunction of lock-in amplification, which has also been used withPSDs, is to modulate the signal on/off, and then, in the receiver,sample the signal during on and off periods, subtracting thetwo values

y(n) = x(2n + 1) x(2n). (2)

Variously known as dc restoration, correlated double sampling,or background subtraction, this approach has the advantageof conceptual and implementation simplicity and can be im-plemented using either analog hardware or a data acquisitionsystem and software.

Note that if we define a reference signal

r = {1,+1,1, . . .},

then (2) becomes

y(n) = r(2n + 1)x(2n + 1) + r(2n)x(2n).

This allows us to reinterpret background subtraction as adiscrete-time...

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