November 15, 2007 / Vol. 32, No. 22 / OPTICS LETTERS 3355

Digitally enhanced heterodyne interferometry

Daniel A. ShaddockJet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA

Daniel.Shaddock@jpl.nasa.gov

Received June 22, 2007; revised August 15, 2007; accepted October 8, 2007;posted October 19, 2007 (Doc. ID 84411); published November 14, 2007

Combining conventional interferometry with digital modulation allows interferometric signals to be isolatedbased on their delay. This isolation capability can be exploited in two ways. First, it can improve measure-ment sensitivity by reducing contamination by spurious interference. Second, it allows multiple optical com-ponents to be measured using a single metrology system. Digitally enhanced interferometry employs a pseu-dorandom noise (PRN) code phase modulated onto the light source. Individual reflections are isolated basedon their respective delays by demodulation with the PRN code with a matching delay. The properties of thePRN code determine the degree of isolation while preserving the full interferometric sensitivity determinedby the optical wavelength. Analysis and simulation indicate that errors caused by spurious interference canbe reduced by a factor inversely proportional to the PRN code length. © 2007 Optical Society of America

OCIS codes: 120.3180, 120.3930, 120.3940.

Spurious interference limits the performance ofmany interferometric measurements [1]. This inter-ference can originate from several sources such asscattered light [2], polarization [3] or spatial [4] leak-age, and electronic cross talk in the data acquisitionsystem. These mechanisms are examples of multi-path interference, where signals originating from thesame source travel different paths to the measure-ment point. Spurious interference leads to measure-ment nonlinearity commonly known as cyclic error.

Digitally enhanced interferometry, proposed here,provides a way to isolate interferometric signalsbased on their delay. Interferometric signals are ef-fectively time tagged by phase modulating the lasersource with a pseudorandom noise (PRN) code [5].Digitally enhanced interferometry improves mea-surement sensitivity by exploiting the autocorrela-tion properties of the PRN to isolate only the signal ofinterest and reject spurious interference. The proper-ties of the PRN code determine the degree of isola-tion. In contrast to standard PRN-based measure-ments, digitally enhanced interferometry retains thefull interferometric sensitivity determined by the op-tical wavelength.

Another benefit of delay-based signal isolation isthe ability to measure multiple optical componentswith a single metrology system. This multiplexing ca-pability significantly simplifies measurements ofmultiple components, such as displacement monitor-ing of a train of optics [6] or segments of a multimir-ror telescope. A similar technique [7,8] used ampli-tude modulation with a PRN code to multiplex fiberoptic sensor arrays.

Figure 1 shows a digitally enhanced heterodyne in-terferometer used to measure changes in the separa-tion of three partially reflective mirrors. A beamsplitter divides the laser output into a local oscillatorand probe beam. The local oscillator is frequencyshifted by an acousto-optic modulator (AOM) to pro-vide a heterodyne signal at the photodetector with afrequency fh. A PRN code generator drives an electro-optic modulator (EOM) to produce a zero or � phase

shift on the probe beam before it is directed toward

0146-9592/07/223355-3/$15.00 ©

the three mirrors. The reflected light is recombinedwith the local oscillator, and the interference signal ismeasured by the photodetector.

The optical system differs from a conventional het-erodyne interferometer primarily by the PRN phasemodulation. With no PRN modulation the phase ofthe heterodyne signal at the photodetector is deter-mined by the vector sum of the reflections from allthree mirrors, and information about individual mir-ror positions is lost. When the PRN modulation ispresent each signal possesses a time-varying phaseshift unique to its time of flight from the EOM. Theprobe beam electric field at the photodetector can bewritten

EP = E1e−i�1c�t − �1� + E2e−i�2c�t − �2� + E3e−i�3c�t − �3�,

�1�

where Ei is the magnitude of the electric field contri-bution from mirror i and the PRN phase shift of 0 or

Fig. 1. (Color online) Digitally enhanced heterodyne inter-ferometer for monitoring the separation of mirrors M1, M2,and M3. Reflections from the different mirrors are isolatedby matching the decoding delays to the optical delays.EOM, electro-optic modulator; AOM, acousto-optic modula-

tor; PRN, pseudorandom noise.

2007 Optical Society of America

3356 OPTICS LETTERS / Vol. 32, No. 22 / November 15, 2007

� is represented by a multiplication by c�t�= ±1. Lowreflectivity mirrors are desirable to ensure that eachreflection is of comparable amplitude and to reduceambiguity arising from multiple reflections associ-ated with optical cavity effects. Reflections from mul-tiple round trips will have a different delay and re-duced amplitude and can be excluded from thisanalysis. The local oscillator electric field is

ELO = e−i�2�fht+�LO�. �2�

The PRN phase modulation randomly inverts theamplitude of the heterodyne signal measured by thephotodetector. The ac-coupled photodetector outputVd is proportional to Re�ELO

* EP�.

Vd�t� = E1 cos��1 − 2�fht − �LO�c�t − �1�

+ E2 cos��2 − 2�fht − �LO�c�t − �2�

+ E3 cos��3 − 2�fht − �LO�c�t − �3�. �3�

To recover the heterodyne signal the photodetectoroutput is multiplied by the same code �c�t�= ±1� de-layed by an amount equal to the total optical-electronic delay of the reflection of interest. Recallthat the code has values of either ±1, so c�t−�i�c�t−�i�=1 whereas c�t−�1�c�t−�2� remains a randomcode. For example, to isolate the mirror 1 (M1) reflec-tion we multiply the photodetector output by a �1 de-layed version of the code, VM1�t�=Vd�t��c�t−�1�.

VM1�t� = E1 cos��1 − 2�fht − �LO�

+ E2 cos��2 − 2�fht − �LO�c�t − �2�c�t − �1�

+ E3 cos��3 − 2�fht − �LO�c�t − �3�c�t − �1�.

�4�

With matched delays this multiplication reverses theinversion of the M1 signal and recovers a standardheterodyne signal. M1 information is said to be de-spread into a pure sinusoid at the heterodyne fre-quency. The phase of this heterodyne signal containsthe mirror displacement information. For reflectionswhere the modulation and demodulation delays dif-fer by more than one code chip these signals are notdespread but instead randomly reinverted and ap-pear as a broadband noise background to the mea-surement. This broadband noise can be strongly re-jected by appropriate averaging in the phasemeter.

Mirror separation information is obtained by dif-ferencing the phase of the decoded signals. For ex-ample, M1 and mirror 2 (M2) relative displacement isproportional to

��VM1� − ��VM2� = �1 − �2 + �1 − �2, �5�

where �i is the error in measuring �i due to limitedisolation from other signals [i.e., the last two terms inEq. (4)]. Note that the local oscillator phase shift �LOand the imposed heterodyne phase shift 2�fht arecommon to both measurements and cancel when thedifference is taken.

The isolation between interference signals is deter-

mined by the properties of the PRN modulation. The

code’s chip frequency fchip determines the minimumdelay difference needed for optimal isolation betweenreflections. The mirror separation �L should satisfy�L�c /2fchip. The last two terms in Eq. (4) show thecontamination of the M1 measurement by M2 andmirror 3 (M3) reflections. The time average of theseindividual terms is similar to the code’s autocorrela-tion with the exception that the code product isweighted by a time varying sinusoid. If the chip fre-quency and code frequency are not coherently relatedthen this modified autocorrelation is approximately2/ ���n� for averaging over n chips. Under these con-ditions, a displacement resolution of 1.1 nm rms wasdemonstrated [9] using a substantially simplified andminiaturized optical head.

Significantly better suppression is attainable if fhis an integer multiple of the chip frequency fchip. Inthis case, each chip is weighted equally and the aver-age is simply the code’s autocorrelation. This allowsfor improved suppression by using PRN codes withlow autocorrelation values, such as a maximal lengthsequence [5] whose autocorrelation is −1/n for a codeof n chips. In this case, the expected phase error inthe measurement of M1 due to contamination fromM2 is

��� �E2�

n�E1�rad. �6�

The optical arrangement in Fig. 1 was numericallysimulated. A maximal length sequence PRN codewith n=32767 chips and fchip=50 MChips/s was usedwith a heterodyne frequency fh=50 MHz. Mirrorseparations of 6 m (2 chip round-trip delay) and 3 m(1 chip round-trip delay) were chosen for M1–M2 andM2–M3 distances, respectively. All reflections wereequal in amplitude at the photodetector, E1=E2=E3,simulating low reflectivity mirrors, with returnphases chosen as �1=0, �2=� /2, and �3=3� /4.

Figure 2(a) shows a simulated spectrum of the pho-todetector output. Note that no sinusoidal signal ap-pears because the PRN modulation spreads all het-erodyne signals over a wide frequency range. Theripples in the spectrum are caused by the interfer-ence between the three spread spectrum signals. Fig-ure 2(b) shows the simulated spectra of the three re-flections after decoding with a code delay �1. Aspredicted by Eq. (4) the M1 signal is despread andappears as a single spectral line at the heterodynefrequency. The contributions of the differently de-layed reflections from M2 and M3 are not despreadand remain as broadband noise.

The phasemeter was modeled as the inverse tan-gent of the ratio of in-phase and quadrature demodu-lation products. The inverse tangent result was thenaveraged over exactly one code length �655 s�.Simulation showed a worst case contamination phaseerror of ±30.5 rad, which agrees with the predicted1/n [Eq (6)] with n=32,767. Assuming a laser wave-length �=1 m, the total cyclic error from other re-flections was reduced to ±10 pm with a measurement

update rate of 1.5 kHz. There is a trade-off between

c�t−

November 15, 2007 / Vol. 32, No. 22 / OPTICS LETTERS 3357

isolation and measurement rate; better suppressionrequires longer codes and therefore longer averaging.

Errors in the decoding delay do not directly pro-duce a phase error. Instead, decoding delay errors re-duce the amplitude of the decoded heterodyne signal.This reduced amplitude increases contamination byunwanted signals [Eq. (6)], so for optimal perfor-mance the demodulation delay error should be�1/ fchip.

This technique shares the same sensitivity to shotnoise and laser frequency noise as conventional inter-ferometry. For an interferometer of length L and la-ser frequency �, laser frequency noise � will producea displacement measurement error:

L = �

�L. �7�

This noise can be mitigated in the usual ways, for ex-ample, by stabilizing the laser frequency [10] or bymeasuring the difference between two equal lengtharms [11].

Digitally enhanced interferometry reassigns therejection of spurious interference from a hardwareresponsibility to a signal processing function. This re-laxes requirements on mirror loss or polarizer extinc-tion ratio compared to conventional heterodyne mea-surements. Isolation through digital signalprocessing is deterministic with no degradation overtime, and provides scalability as processing powercontinues to improve.

An important benefit of multiplexing signalsthrough a single measurement chain is common-mode rejection of several noise sources. Photodetec-tor phase delays and analog-to-digital converter(ADC) latency changes that can corrupt standardheterodyne systems are completely common to allmeasurements because a single detector and an ADCare used. Interchannel electronic cross talk and ex-

Fig. 2. (Color online) Simulated root power spectral densitector output and (b) contributions of each reflection to the

ternal electronic interference are also suppressed be-

cause signals remain encoded until they reach thedigital domain. The technique’s immunity to scat-tered light and electronic interference will allow mea-surements to approach more fundamental limitssuch as shot noise and electronic noise. This immu-nity should allow digitally enhanced interferometryto achieve laboratory class performance in real-worldenvironments.

The author thanks Brent Ware, Oliver Lay, andSerge Dubovitsky for many insightful discussions.This research was performed at the Jet PropulsionLaboratory, California Institute of Technology, undercontract with the National Aeronautics and SpaceAdministration (NASA).

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of the configuration depicted in Fig. 1 for (a) the photode-�1� decoded output, VM1.

ties

(1999).