1376 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 C.-C. Wang and F. Davidson

Optical spectral analysis using moving space-chargefield effects in photoconductive semiconductors

Chen-Chia Wang and Frederic Davidson

Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, Maryland 21218

Received September 7, 1995; revised manuscript received January 26, 1996

The interference of two mutually coherent optical fields with unequal center frequencies generates movingspace-charge electric fields inside photoconductive semiconductors that contain deep level traps. The motionof the internal space-charge field drifts free charge carriers out of the photoconductive sample and resultsin measurable photocurrents that consist of ac and dc components even in the absence of external biaselectric fields. The dc photocurrent can be expressed as a convolution integral between the baseband powerspectrum of the signal optical field and the characteristic impulse response of the photoconductive crystal,provided that the local oscillator has negligible optical linewidth. We have demonstrated experimentally anew kind of optical spectrum analyzer with a resolution bandwidth smaller than 15 kHz by sweeping thelocal-oscillator center frequency over the spectral region of interest and by subsequent deconvolution of therecorded photocurrent with the characteristic response of the crystal. The scanning range of the analyzer islimited only by the tuning range of the local oscillator. 1996 Optical Society of America

1. INTRODUCTIONAn optical fringe pattern is formed when two mutually co-herent optical fields are brought to interference inside aphotoconductive semiconductor material. The spatiallyinhomogeneous distribution of the optical intensity pat-tern creates an internal space-charge electric field owingto the photoexcitation of free charge carriers and theirsubsequent separation by means of diffusion or drift.1

The motion of the optical interference pattern caused, forexample, by the mismatch between the instantaneous fre-quencies of the two optical fields sets the space-chargefield in motion. This causes free charge carriers to driftout of the photoconductive sample and results in measur-able photocurrent. One can understand this phenome-non by considering the expression for the photocurrentdensity inside the photoconductive semiconductor:

jSsx, td ­ emensx, tdEsx, td 1 mekBT≠nsx, td

≠x

1 e0er≠Esx, td

≠t, (1)

where, for simplicity, electrons with mobility me are as-sumed to be the only species of free charge carriersand the current density has only a single component inthe direction of motion of the optical fields, which as-sumption can be justified for materials with small opti-cal absorption coefficients sa , 1 cm21d. Esx, td is thetotal internal electric field, and nsx, td is the concentra-tion distribution of free electrons. The photocurrent den-sity consists of three components: drift current that isdue to the electric field, diffusion current that is due tothe nonzero charge-carrier concentration gradient, andthe displacement current. The potential difference be-tween the end surfaces of the crystal is given by U sx ­Ld 2 U sx ­ 0d ­

RL0 Esx, tddx, where L denotes the in-

terelectrode spacing of the crystal. Following the argu-ment of Petrov et al,2 it is useful to find the conditions

0740-3224/96/071376-08$10.00

under which the spatially averaged photocurrent density,jSstd ; s1yLd

RL0 jSsx, tddx, is nonzero when the sample

is operated in a short-circuit condition, i.e., U sx ­ Ld ­U sx ­ 0d. Under uniform illumination conditions, freecharge carriers are generated uniformly throughout thesample such that nsx, td ­ constant and jSstd ­ 0. Thephotocurrent density would, of course, not be zero if an ex-ternal bias field were applied so that U sx ­ 0d fi U sx ­ Ld,and the displacement current would then contribute tojSstd. Under nonuniform illumination the free charge-carrier concentration nsx, td varies with respect to x, andit is possible to achieve a nonzero value for jSstd, pro-vided that the host material contains recombination cen-ters for free charge carriers so that charge neutralityis destroyed locally and the resultant internal electricfield Esx, td also exhibits x variation. It is customaryto impose periodic boundary conditions on the sample sothat nsx ­ 0, td ­ nsx ­ L, td and that the diffusion cur-rent term in Eq. (1) does not contribute to the spatiallyaveraged photocurrent density. The functional form ofEsx, td is determined from the simultaneous solutions tothe equation of continuity, Gauss’s law, and the equa-tion describing the generation and recombination of freecharge carriers.1 This phenomenon was first observedunder open-circuit operation in which nonuniform illu-mination produced a photoinduced emf, which appearedacross the material as the result of diffusion of photoex-cited charge carriers from brightly illuminated regionsinto the darker regions. This phenomenon is known asthe Dember effect.3

Petrov et al.2 showed that one can generate a sinu-soidal photocurrent by imposing a small-amplitude si-nusoidal phase modulation on a signal optical field thatoriginates from the same laser source as the local os-cillator. Stepanov et al. further demonstrated that onecan use such sinusoidal photocurrents to detect vibra-tions of reflective surfaces.4 Recently it was shown thatdc photocurrents are generated when the two interfer-

1996 Optical Society of America

C.-C. Wang and F. Davidson Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1377

ing monochromatic optical fields have unequal centerfrequencies.5 Such dc photocurrents exhibit an anti-symmetric dependence on the relative frequency off-set between the optical fields. This characteristic hasfound application in optical phase-locked loops6 and laservelocimetry.7

Most recently we presented a theoretical derivationthat showed how measurable photocurrents are gener-ated when the two optical fields have nonzero linewidth.8

The dc photocurrent can be expressed as a convolu-tion integral between the optical power spectrum ofthe signal optical field and a deterministic character-istic response of the crystal. We also presented pre-liminary data showing that the signal optical powerspectrum can be determined from measurements of thedc photocurrent as the local-oscillator frequency is sweptthrough the spectral range of interest and subsequentdeconvolution of the dc photocurrent with the character-istic response of the crystal.8 However, the experimentalsetup used earlier was unrealistic in that a heterodynephase-locked loop was used to offset the center frequen-cies of the two lasers and to maintain phase coherencebetween the optical fields. In this paper we demon-strate the operation of a prototype moving space-chargefield optical spectrum analyzer that does not rely onany heterodyne phase-locked loops. Detailed theoreticalderivations are presented in Section 2, followed by discus-sion of system characteristics as well as the digital signalprocessing employed to extract the optical signal powerspectrum from the measured photocurrents. Experi-mental procedures and performance results presented inSection 4.

2. THEORETICAL DERIVATIONSThe incident optical fields are assumed to be linearlypolarized and are given by

ELOsr, td ­ ELO exph2j fkLO ? r 2 vLOt 2 fLOstdgj ,

ES sr, td ­ ES0 exph2j fkS ? r 2 vSt 2 fSstdgj , (2)

where kLO skS d is the wave vector, vLO svS d is the centerfrequency, fLOstd ffSstdg is the phase, and ELO sES0d isthe amplitude of the local-oscillator (signal) optical field.The amplitudes of the optical fields are assumed for sim-plicity to be independent of time. This argument can bejustified when ELO and ES0 vary on a time scale muchlonger than the space-charge field formation time of thehost material or, as in spectral analysis applications, ona time scale longer than the time that it takes to fin-ish one frequency scan. When the optical fields inter-fere inside the photoconductive semiconductor with a half-angle u as shown in Fig. 1, an optical interference patternis formed with a spatially inhomogeneous intensity pro-file given by I sx, td ­ I0h1 1 s1y2dfmstdexpsjKgxd 1 c.c.gj,where I0 is the total incident optical intensity and Kg ­skLOdx 2 skS dx is the grating wave number. The spa-tial period of the optical intensity pattern is denotedL ­ sin uy2l0 ­ 2pyKg, where l0 is the wavelength ofthe optical fields in vacuum. The modulation index ofthe interference pattern is denoted mstd and is given by

mstd ­ m0 exps2jvD tdexpf jDfstdg , (3)

where Dfstd ; fSstd 2 fLOstd is the phase difference and

vD ­ vLO 2 vS is the frequency difference between thelocal-oscillator and the signal optical fields. Also, m0 ­2ELO

pES0ysjELOj2 1 jES0j2d is the complex amplitude ofthe modulation index.

In our experiments the energy of the incident photonis smaller than the band-gap energy of the photocon-ductive semiconductor. Free charge carriers are excitedinto the conduction or valence band or both from mid-band-gap donor levels when the incident photons areabsorbed. The spatially inhomogeneous optical inten-sity pattern results in a nonzero concentration gradientfor the free charge carriers, which then diffuse from thebrighter regions into darker regions of the interferencepattern where they can become trapped by recombin-ing with deep level ionized donor–acceptor sites. Asa result of charge separation, an internal space-chargeelectric field is formed. The expression for the space-charge field is obtained from several material equa-tions that describe the behavior of charge carriers andelectric fields inside the host material. In the pertur-bation approach it is assumed that jmstdj ,, 1. Thereal-valued space-charge field can be approximated byEstd ­ s1y2dfEscstdexpsiKgxd 1 c.c.g in the absence of anyexternally applied electric bias fields. The equation gov-erning the evolution of the fundamental component of thespace charge field is given by9

dEscstddt

1Escstd

tg­ 2j

ED

tM s1 1 EDyEM dmstd , (4)

where tg ­ tM s1 1 EDyEM dys1 1 EDyEqd is the space-charge field’s formation time. The various parametersare defined as follows: tM ­ e0erys0 is the dielectric re-laxation time, ED ­ kBTKgye is the diffusion field, EM ­s meteKgd21 is the Maxwell field, and Eq ­ eNAye0erKg isthe screening field. The dielectric constant of the mate-rial is denoted e0er, the photoconductivity s0, the mobil-ity–lifetime product mete, Boltzmann’s constant kB, theabsolute temperature T, and the effective deep level trapconcentration NA. It has been assumed for simplicitythat electrons are the only species of free charge carriersin the host material in the derivation of Eq. (4). Similarexpression can be obtained readily for bipolar materials.10

Fig. 1. Experimental setup for the moving space-charge fieldoptical spectrum analyzer: LWE 122, LWE 120, LightwaveElectronics Series 122 and 120 Nd:YAG lasers, respectively;B. S.’s, beam splitters; E-O, electro-optic.

1378 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 C.-C. Wang and F. Davidson

By taking the Fourier transform of mstd and Esc(t) wecan solve the space-charge field from Eq. (4) and rewriteit as

Escstd ­ 2jm0ED

1 1 EDyEq

12p

Z `

2`

Z `

2`

11 1 jvtg

exps2jvDt0d

3 expf jDfst0dgexpf jvst 2 t0dgdt0dv . (5)

By arranging the sample in the short-circuit configura-tion and assuming the applicability of periodic boundaryconditions as explained above, we find that only the driftcurrent term in Eq. (1) contributes to the net photocur-rent density, which is given by2

jSstd ­s0

4s1 1 EDyEM dfmstdEsc

pstd 1 c.c.g , (6)

where c.c. denotes the complex conjugate of the preced-ing term. The photocurrent jSstd is a stochastic processthat is due to the random nature of the phase differenceDfstd. In the following discussions, quantities enclosedin boldface brackets represent stochastic processes.

The dc component of jSstd can be defined as thetime average of the photocurrent: jS,dc ; limT!`1y2TRT

2T fff jSstdgggdt, which could still be a random process. Toproceed further, it is assumed that the random processfff jSstdggg is mean ergodic, which implies that its time aver-age is equivalent to the ensemble average.11 It is furtherassumed that the random processes fffexpf jfSstdgggg andfffexpf jfLOstdgggg are statistically independent and wide-sense stationary11 so that their autocorrelation functionscan be written as RS std ; PSkfffexph j ffSstd 2 fSst 2 tdgjgggland RLOstd ; PLOkfffexphj ffLOstd 2 fLOst 2 tdgjgggl, respec-tively, where angle brackets represent ensemble averagesand the assumption of negligible amplitude variation ofthe optical fields has been applied. The power spec-tral density of a wide-sense-stationary random process isgiven by the Fourier transform of its autocorrelation func-tion; hence PSsvd ­ F hRS stdj and PLOsvd ­ F hRLOstdjrepresent the baseband power spectral densities for thesignal and the local-oscillator optical fields, respectively.Based on these assumptions, the dc photocurrent densitycan be rewritten as

jS,dcsvD d ­2s0EDPLOysPLO 1 PS d2

s1 1 EDyEM ds1 1 EDyEqd

3Z `

2`

PSbeatsvd

svD 2 vdtg

1 1 svD 2 vd2tg2

dv , (7)

where PSbeatsvd ­ PSF hhhkexph j fDfstd 2 Dfst 2 tdjljjj is

the baseband power spectral density of the signal opti-cal field broadened by the finite-linewidth local oscilla-tor. Note that PS

beatsvd ­ PSsvd, which is the actualbaseband power spectrum of the signal optical field iffLOstd ­ fLOst 2 td and the local-oscillator optical fieldis monochromatic. Equation (7) is a convolution integraland can be written as jS,dcsvD d ­ PS

beatsvD d ≠ TCRsvD d,where ≠ represents convolution and TCRsvDd is the deter-ministic characteristic response of the crystal, given by

TCRsvDd ­2s0EDPLOysPLO 1 PSd2

s1 1 EDyEMds1 1 EDyEqdvD tg

1 1 svDtgd2. (8)

Based on Eq. (7), one can infer the base-band powerspectrum of the signal optical field by sweeping the local-oscillator frequency over the spectral range of interest and

deconvolving the measured dc photocurrent with the char-acteristic response of the crystal, TCRsvD d. It should benoted that typically there is no information available fordetermining the frequency offset vD precisely during theexperiments; hence the value of TCRsvDd cannot be known.However, as the local-oscillator frequency is scanned, thefrequency difference between the lasers can be written as

vD ­ DvLO 1 vLOc0 2 vS , (9)

where DvLO [ f2vmax, vmaxg is the deviation of the local-oscillator frequency from its center value, vLO

c0, and vmax

is the largest frequency deviation. Equation (7) can berewritten as

jS,dcsDvLOd ­ PSbeatfDvLO 2 svS 2 vLO

c0dg ≠ TCRsDvLOd .

(10)

It is clear from Eq. (10) that one can determine the signalpower spectrum downshifted by the amount 2vS 2 vLO

c0,i.e., PS

beatfDvLO 2 svS 2 vLOc0dg, by deconvolving the

measured photocurrent with the antisymmetric function,TCRsDvLOd for DvLO [ f2vmax, vmaxg without requiringany information on the relative frequency offset betweenthe lasers, vD. Note that, if the center frequencies of theoptical fields are identical, vLO

c0 ­ vS and the inferredspectrum is equivalent to the local-oscillator-broadenedsignal baseband power spectrum.

3. SIGNAL PROCESSING ANDSYSTEM CHARACTERISTICS

A. Signal ProcessingIt was shown in the discussion above that we can de-termine the baseband power spectrum of the signal op-tical field by computing the inverse Fourier transformof the ratio F h jS,dcsDvLOdjyF hTCRsDvLOdj, which canbe computed time efficiently with the use of the fastFourier-transform algorithm. In the presence of noise,however, additional filtering of the experimentally mea-sured data is necessary. Parametric Wiener filteringcan suppress high-frequency noise and is widely used indigital signal-processing applications.12 The estimatedsignal base-band power spectrum is then given by

PSbeatsvd ­ F 21hhhF h jS,dcsDvLOdjW sjdjjj , (11)

where W sjd ­ F hTCRsDvLOdjpyhhhjF hTCRsDvLOdjj2 1 b

fSPS beat sjdySNN sjdgjjj is the parametric Wiener filter. Theconstant b has a value in the range [0, 1]. The quantitySNN sjd is the noise power spectral density of the fluc-tuations in the dc photocurrent density and associatedamplifier noise, and SPS beat sjd is the squared magnitudeof the Fourier transform of PS

beatsvd expressed in thej domain. Theoretical discussions have been proposedthat describe the behavior of the noise power spectraldensity for photocurrents generated by sinusoidally vi-brating optical interference patterns.13 However, weknow of no experimental studies that substantiate thesedescriptions. In order not to make any unnecessary as-sumptions about the behavior of the noise power spectraldensity, we used the modified parametric Wiener filtergiven below to filter the measured photocurrents:

C.-C. Wang and F. Davidson Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1379

W sjd ­F hTCRsDvLOdjp

jF hTCRsDvLOdjj2 1 a maxhhhjF hTCRsDvLOdjj2jjj.

(12)

In Eq. (12), maxh j denotes the maximum value of thefunction enclosed within the boldface braces and 0 , a ,,

1 is a constant whose value has to be chosen such that themodified Wiener filter, Eq. (12), can suppress unwantedhigh-frequency noise yet still preserve the useful signalinformation. Note that, based on the inferred basebandsignal power spectrum, a reasonably good choice of a

should produce photocurrent estimates that closely matchthose measured experimentally.

B. Resolution Bandwidth and Space-ChargeField Formation TimeIn the absence of measurement noise, F h jS,dcsDvLOdj ­F hPS

beatsDvLOdjF hTCRsDvLOdj, where it has been as-sumed, for simplicity, that the optical fields have identicalcenter frequencies, vS ­ vLO

c0. The estimated basebandsignal power spectral density, Eq. (11), can be written as

PSbeatsDvLOd ­ F 21

(((F hPS

beatsDvLOdj

31

1 1 a exps2jjjytgd

))), (13)

where Eq. (12) and the expression for F hTCRsDvLOdj(Ref. 14) have been taken into consideration. It is ap-parent from Eq. (13) that the moving space-charge fieldoptical spectrum analyzer can be modeled as a low-passfilter, FLP sjd ; 1yf1 1 a exps2jjjytgdg, with the fullwidth at half-maximum (FWHM) bandwidth j

LPFFWHM ­

tgjln ajy2. Furthermore, PSbeatsDvLOd can be a good

estimate of PSbeatsDvLOd if the bandwidth of the sys-

tem’s low-pass filter is large enough to pass the incomingpower spectrum without much distortion. This sets alower bound on the spectral linewidth of PS

beatsDvLOdin the DvLO in domain, which can be defined as theresolution bandwidth of the moving space-charge fieldoptical spectrum analyzer. We can obtain an analyti-cal expression for the resolution bandwidth by consid-ering the Gaussian spectrum given by PS

beatsDvLOd ­PS0 expf2sDvLOy2pd2ysf

2g, where PS0 is the peak powerspectral density and sf is the 1ye spectral linewidth in theunits of Hertz. The Fourier transform of PS

beatsDvLOdhas a Gaussian line shape with the 1ye linewidthof jGaussian ­ 1ypsf . The moving space-charge fieldoptical spectrum analyzer can faithfully reproducePS

beatsDvLOd from the measured photocurrents whenj

LPFFWHM $ 2jGaussian, that is, for sf $ sf

min ; 4yptgjln aj.The resolution bandwidth of the moving space-chargefield optical spectrum analyzer can be defined as

Dfres ; 2sfmin ­

8ptgjln aj

. (14)

One can achieve enhancement of the resolution band-width either by reducing the value of a or by increasingthe value of space-charge field formation time tg. Reduc-ing the value of a can be easily achieved under programcontrol of the fast-Fourier-transform signal-processing al-gorithm. However, the noise level, which consists of

at least three sources: photocurrent shot noise, crystalthermal noise, and electronic circuit thermal noise, is in-dependent of a. As the bandwidth of the system’s low-pass filter, j

LPFFWHM, is increased because of the reduction

of a, more noise energy is present in the filtered signal,resulting in inaccurate estimation of the baseband sig-nal power spectrum. In contrast, one can also increasej

LPFFWHM by reducing the local-oscillator power, which in-

creases the space-charge field formation time becauseof the reduction of photoconductivity; and as a result,the electrical impedance RCR of the host photoconductivesemiconductor is increased. The resultant noise leveldrops as a result of the decrease of crystal thermal noise,which is inversely proportional to RCR. The magnitude ofthe resultant dc photocurrent, however, is hardly affectedunder the condition that PS ,, PLO. That is, from thepoint of view of signal-to-noise ratio, increasing the valueof tg is preferable for enhancing the resolution bandwidthof moving space-charge field optical spectrum analyzers.Note that RCR . 1 MV under the typical experimentalconditions reported here, and its value depends on theconductivity as well as on the cross-sectional area of thehost crystal.

Recently optical spectrum analyzers based on finesse-enhanced optical fiber ring resonator designs were re-ported to have resolution bandwidth as small as 17 kHz(Ref. 15) and 20 kHz.16 As a comparison, under the typi-cal experimental conditions reported in this paper, tg ­20 ms, a ­ 1y2000, and the corresponding resolutionbandwidth is Dfres ­ 16.75 kHz, which is comparablewith the values quoted above. Furthermore, with a sta-ble local oscillator one can achieve a resolution bandwidthof the order of 1 KHz simply by increasing the value of theproduct tgjln aj by a factor of 10.

C. Scanning RangeOptical spectral analysis with excellent resolution band-width s,1 Hzd can be achieved by electronic spectrumanalyzers if the optical fields are mixed by a conventionalphotodetector, which generates an intermediate frequencysignal from which the spectral content of the signal opti-cal field can be determined. On the one hand, the scan-ning range is limited by the availability of high-speedelectronics with electrical bandwidths comparable withthe scanning range. In moving space-charge field opti-cal spectrum analyzers, on the other hand, high-speedelectronics are unnecessary because the photoconductivesemiconductor functions both as a conventional photode-tector and as an electronic mixer in that high-frequencyoptical beat notes are directly downconverted into dc sig-nals, as shown in Eq. (7). As a result, the scanning rangeof a moving space-charge field optical spectrum analyzeris limited only by the tuning range of the local oscilla-tor, which is potentially much larger than that achievablewith electronic spectrum analyzers (gigahertz). With theNd:YAG local oscillator laser used in our experiments,which had a continuous tuning range of 12 GHz with-out mode hopping, we successfully detected spectral peaksspaced apart by 2 GHz with a transimpedance amplifierthat had an electronic bandwidth of only 20 kHz. Thescanning rate of a moving space-charge field optical spec-trum analyzer is limited by the space-charge field forma-

1380 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 C.-C. Wang and F. Davidson

tion time because only steady-state photocurrent valuesshould be sampled by the sampling circuitry.

4. EXPERIMENTS

A. Experimental SetupFigure 1 shows the experimental apparatus that we usedto demonstrate the operation of an optical spectrumanalyzer in which two independent Nd:YAG lasers (Light-wave Electronics Series 122 and 120) operating at thewavelength of l ­ 1.064 mm were used to generate thestrong local-oscillator and weak-signal optical fields, re-spectively. The frequency of the local oscillator wasswept through the spectral region of interest by applica-tion of a sawtoothed voltage signal to the laser’s piezoelec-tric port, which had a tuning coefficient of 4 MHzyV. Thefrequency scan rates were typically less than 1 kHzytg,so accurate estimates of the signal optical power spectrawere obtained. An integrated-optic phase modulator wasused to vary the signal optical power spectrum. The twooptical beams were brought to interference inside a short-circuited photoconductive CdTe:V:Mn sample, which hadthe dimensions 5 mm 3 5 mm 3 4 mm with an interelec-trode spacing of 4 mm. The surfaces were polished butnot antireflection coated. The small optical absorptioncoefficient of the crystal (1.18 cm21) caused only a smallportion of the incident optical energy to be absorbed bythe crystal. The photocurrent generated was convertedinto a voltage signal by a transimpedance amplifier witha feedback resistance of Rf ­ 108 VyA and an electronicbandwidth of 20 kHz. A simple RC filter with the 3-dBcutoff frequency of 1y2p RC was used as an antialias-ing filter. The filtered voltage signal was sampled by aTektronix 11402 digitizing oscilloscope and then input toa computer for signal processing. The antialiasing fil-ter served to alleviate the aliasing effects in digital fast-Fourier-transform signal processing as well as to filter outunwanted ac photocurrents, which could have resulted infalse information appearing in the inferred baseband sig-nal power spectrum. As a comparison, the spectrum ofthe beat note of the optical fields was also monitored witha conventional electronic spectrum analyzer.

B. Free-Running LasersWhen we used free-running lasers, no phase modulationwas applied to the signal optical field. The linewidthof the local-oscillator-broadened signal baseband powerspectrum should be of the order of 1 kHz because theNd:YAG lasers used in the experiments reported here hadvery narrow linewidth and slow center frequency driftrates s,50 MHzyhd, according to the specification sheetssupplied by the manufacturer. The photocurrents gen-erated as the center frequency of the local oscillator wasswept through the spectral range jDvLOy2pj # 2 MHzwere measured, and the inferred signal baseband powerspectrum is shown in Fig. 2. Detailed experimental con-ditions are described in the caption of Fig. 2. The rip-ples in the estimated signal baseband power spectrumwere caused by the insufficient resolution bandwidth ofthe moving space-charge field optical spectrum analyzer,as indicated by the fact that the FWHM linewidth ofPS

beatsDvLOd s,11.7 kHzd shown in Fig. 2 is smaller thanthe resolution bandwidth of Dfres ­ 32.7 kHz under the

described experimental conditions. Thus PSbeatsDvLOd

is a distorted version of PSbeatsDvLOd and exhibited a

smaller magnitude and a broader linewidth.The power spectrum of an optical field should be a

positive-valued function. However, in addition to the rip-ples with negative values caused by insufficient resolutionbandwidth, the presence of noise can also cause negativevalues in the inferred baseband signal power spectrum.For the discussions below it will be assumed for conve-nience that the inferred signal baseband power spectrumis given by jPS

beatsDvLOdj, a procedure adopted by otherresearchers.17 It will be shown that, under appropriateexperimental conditions, such a procedure is reasonablein that it produces theoretical dc photocurrent values thatagree well with those actually measured experimentally.

C. Resolution BandwidthTo determine experimentally the resolution bandwidthof the moving space-charge field optical spectrum ana-lyzer, we introduced discrete spectral peaks of the sig-nal optical power spectrum by applying a sinusoidalphase modulation signal, D cossvmtd, to the electro-opticphase modulator, which had a p-phase shift voltage of6 V. For the signal optical field oscillating with centerfrequency vS the modulated spectrum consists of dis-crete spectral peaks spaced apart by amounts equivalentto integral multiples of the modulation frequency ac-cording to the expression14 exph j fvSt 1 D cossvmtdgj ­P`

k­2` jkJksDdexpf j svS 1 kvmdtg, where Jk is theBessel function of the first kind of order k. ChoosingD ­ 1.44 rad resulted in approximately identical magni-tudes of the 0th and 61st-order components sk ­ 0, 61d ofthe signal power spectrum, which were much larger thanthose of the higher-order spectral components. Figure 3shows the experimentally measured photocurrents overthe spectral range jDvLOy2pj , 1 MHz and the corre-spondingly inferred signal baseband power spectrum forthe modulation frequency of vmy2p ­ 100 kHz. Theresolution bandwidth remained at Dfres ­ 32.7 kHz be-cause other experimental conditions were the same as

Fig. 2. Inferred baseband optical power spectrum of thesignal optical field in the absence of spectral manipulations.The grating spacing was L ­ 12.38 mm, and the incidentoptical power densities were PLO ­ 109.1 mWycm2 and PS ­0.236 mWycm2. The corresponding space-charge field formationtime was tg ­ 10.25 ms, and the theoretical resolution bandwidthwas Dfres ­ 32.7 kHz for the modified parametric Wiener filterconstant a ­ 1y2000.

C.-C. Wang and F. Davidson Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1381

Fig. 3. Experimentally measured photocurrent (top trace) andthe correspondingly inferred base-band power spectrum of thesignal optical field (bottom trace) in the presence of a sinusoidalphase modulation signal, Dfmstd ­ 1.44 cosf2ps105tdg. The val-ues for the grating spacing and optical power densities wereidentical to those in Fig. 2.

those described in Fig. 2. The inferred signal powerspectrum is not centered at DvLO ­ 0 because the twolasers had unequal center frequencies. Spectral peaks ofthe signal optical field corresponding to k ­ 0, 61, 62 areclearly resolved, with relative magnitudes and frequencyspacings in good agreement with theoretically predictedvalues. Minor variations in the magnitudes of the dis-crete spectral peaks could be caused by jittering of thelaser center frequencies s75 kHzysd because the scanningtime was 1 s. As discussed above, the false sidelobesappearing between the major spectral peaks were causedby insufficient resolution bandwidth of the moving space-charge field optical spectrum analyzer. Note that eachsharp transition in the detected photocurrent signifiesthe existence of a spectral component in the signal op-tical field at the corresponding frequency value. Thisexplains intuitively why the resolution bandwidth is in-versely proportional to the space-charge field formationtime of the photoconductive semiconductor. As tg is in-creased, the width of the transition region s, 1ytgd isreduced; hence it is easier to identify two closely spacedspectral peaks in the signal power spectrum, which causetwo sharp transitions in the measured photocurrent.

The frequency of the sinusoidal modulation signal usedto alter the power spectrum of the signal optical fieldwas further decreased, with all other experimental con-ditions remaining unchanged. The minimum value ofvm under which the three distinct spectral peaks couldbe resolved corresponded to the resolution bandwidth ofthe moving space-charge field optical spectrum analyzer.The dashed–dotted and the center solid curves in Fig. 4represent the inferred signal baseband power spectra formodulation frequencies fm ­ 40 kHz and fm ­ 15 kHz,respectively. They clearly show that the moving space-charge field optical spectrum analyzer based on theCdTe:V:Mn sample can resolve spectral peaks spacedapart by 40 kHz while failing to resolve those spacedapart by 15 kHz, in agreement with the theoretical valuefor the resolution bandwidth, Dfres ­ 32.7 kHz. Effectsof laser center frequency jittering are also obvious, ascan be seen from the slight variations in the peak mag-

nitudes and frequency spacings in the inferred signalspectral features.

Reducing the local-oscillator power further enhancedthe resolution bandwidth, and the inferred signal base-band power spectrum is shown as the right-hand solidcurve in Fig. 4. The figure clearly indicates that, withthe enhanced resolution bandwidth Dfres ­ 10.9 kHz, themoving space-charge field optical spectrum analyzer cannow resolve the discrete spectral features of the signaloptical field spaced apart by 15 kHz, which were previ-ously unresolvable at the larger resolution bandwidth.Note that it is apparent that the effects of laser center

Fig. 4. Inferred signal baseband power spectra for differentphase modulation frequencies and resolution bandwidths forthe moving space-charge field optical spectrum analyzer. Thedashed–dotted and the center solid curves corresponding toresolution of Dfres ­ 32.7 kHz and the phase modulation fre-quencies fm ­ 40 kHz and fm ­ 15 kHz, respectively. The solidcurve at the right represents the inferred signal baseband powerspectrum for the modulation frequency of fm ­ 15 kHz and theenhanced resolution bandwidth of Dfres ­ 10.9 kHz, which wereachieved with tg ­ 30.8 ms. Other experimental conditionswere identical to those in Fig. 2.

Fig. 5. Inferred signal baseband power spectrum when thesignal optical field was phase modulated by a sequence ofpseudorandom pulses. The experimental conditions were L ­12.38 mm, PLO ­ 65.2 mWycm2, PS ­ 0.794 mWycm2, tg ­17.2 ms, and a ­ 1y2000, and the resolution bandwidth wasDfres ­ 19.5 kHz. Also displayed is the power spectrumof the local-oscillator and signal optical fields as observedby a conventional photodiode–electronic spectrum analyzercombination (inset) with a 1-MHz scanning range. Theresolution bandwidth of the electronic spectrum analyzer was10 kHz.

1382 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 C.-C. Wang and F. Davidson

Fig. 6. Experimentally measured photocurrents (scattered dots)and theoretical values (solid curve) based on the inferred signalbaseband power spectrum shown in Fig. 5.

Fig. 7. Inferred signal baseband power spectrum in the pres-ence of a broadband signal (3-dB linewidth , 3 MHz) over ascanning range of 40 MHz. The experimental conditions aredescribed in the text. The spectrum of the beat note of the twointerfering optical fields as measured by the electronic spectrumanalyzer is also displayed (inset), and the agreement with thatinferred by the moving space-charge field optical spectrum ana-lyzer is apparent.

frequency jittering become important as the resolutionbandwidth is further enhanced. Laser center frequencyjittering appeared to be the dominant factor that limitedthe resolution enhancement achievable with the movingspace-charge field optical spectrum analyzers in these ex-periments.

D. Broadband SpectrumThe signal power spectrum was continuously broadenedby application of a pseudorandom pulse sequence to theelectro-optic phase modulator. Note that more signi-ficant ac photocurrents are generated in the presenceof a broadband signal power spectrum because of theincreased mutual interference among the various spec-tral components of the signal optical field. Figure 5shows the inferred signal baseband power spectrumwhen PS

beatsDvLOd had a spectral linewidth of ,300 kHz.The antialiasing filter had a 3-dB cutoff frequency of0.234 kHz. As a comparison, the power spectrum of thebeat note of the local-oscillator and signal optical fieldsmeasured (at a later time) with a conventional electronic

spectrum analyzer is also shown in Fig. 5 (inset). Theagreement between these two spectra is evident. Notethat the moving space-charge field optical spectrum ana-lyzer can also resolve the discrete spectral features causedby the pseudorandom phase modulation process.

As mentioned above, ripples in the inferred signal base-band power spectrum can be observed if the signal spec-tral linewidth is smaller than the resolution bandwidthof the moving space-charge field optical spectrum ana-lyzer. As a result, the theoretical photocurrent derivedfrom jS,dc

th ; PSbeatsDvLOd ≠ TCRsDvLOd differs somewhat

from that measured experimentally. In the presence ofthe broadband signal power spectrum depicted in Fig. 5,fewer ripples are present in PS

beatsDvLOd because theresolution bandwidth of the moving space-charge field op-tical spectrum analyzer is smaller than the spectral widthof the signal power spectrum. Indeed, theoretical valuesof photocurrent based on the inferred signal basebandpower spectrum shown in Fig. 5 are depicted in Fig. 6(solid curve) together with experimentally measured val-ues (scattered dots), and the agreement between the twois reasonably good.

E. Scanning RangeAdjusting the peak-to-peak magnitude and the bit rateof the pseudorandom sequence of pulses used to phasemodulate the signal beam enabled the signal spectrumto be broadened further to have a linewidth of ,3 MHz,as shown in Fig. 7 (inset), as measured with an elec-tronic spectrum analyzer with a resolution bandwidth of100 kHz. With a correspondingly increased scanningrange, the baseband signal power spectrum inferredfrom using the moving space-charge field optical spec-trum analyzer is displayed in Fig. 7, which agrees wellwith that determined with the electronic spectrum ana-lyzer. The two spectral spikes at jDvLOy2pj ø 20 MHzwere caused by the windowing effect in data sampling.The scanning range of 40 MHz was limited by themaximum signal output from the function generator usedto sweep the frequency of the local-oscillator laser, and byno means does it represent any kind of fundamental limiton the scanning range for moving space-charge field opti-cal spectrum analyzers. The data shown in Fig. 7 werecollected under the following experimental conditions:L ­ 40.1 mm, PLO ­ 150.5 mWycm2, PS ­ 1.07 mWycm2,tg ­ 2.1 ms, and a ­ 1y5000. Note that we reducedthe space-charge field formation time mainly by increas-ing the grating spacing so that the resolution band-width of the moving space-charge field optical spectrumanalyzer had a larger value of Dfres ­ 142.4 kHz, appro-priate for wide scanning ranges.

5. CONCLUSIONIn conclusion, we have demonstrated experimentally theoperation of a new kind of optical spectrum analyzerbased on moving space-charge field effects in photo-conductive semiconductors. The characteristics of themoving space-charge field optical spectrum analyzerwere identified, and preliminary performance resultsunder various experimental conditions were presented.A resolution bandwidth of 15 kHz was achieved. It wasdemonstrated that one can further reduce the resolution

C.-C. Wang and F. Davidson Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1383

bandwidth by increasing the space-charge field forma-tion time of the photoconductive semiconductor. Thescanning range of the moving space-charge field opticalspectrum analyzer is limited only by the tuning rangeof the local-oscillator laser. Moving space-charge fieldoptical spectrum analyzers have the advantages of scan-ning ranges as large as those typically found in scanningdiffraction-grating- or Fabry–Perot-type optical spectrumanalyzers as well as of resolution bandwidth as smallas those associated with electronic spectrum analyzers.Moving space-charge field optical spectrum analyzers arespatially adaptive and do not require the stringent spa-tial alignment conditions typically found in conventionalcoherent optical detection systems.

ACKNOWLEDGMENTSThe authors thank Sudhir Trivedi of Brimrose Corpo-ration for the CdTe:V:Mn crystal grown under BallisticMissile Defense Organization phase II program contractDASG60-93-C-0066. C.-C. Wang acknowledges fruitfuldiscussions with Mark Martin, Phillippe Pouliquen, JerryPrince, and Richard Joseph. The authors also thank thereviewers for their helpful suggestions.

REFERENCES1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin,

and V. L. Vinetskii, “Holographic storage in electrooptic crys-tals: I. Steady state,” Ferroelectrics 22, 949 (1979).

2. M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofi-mov, “Non-steady-state photo-electromotive-force induced bydynamic gratings in partially compensated photoconduc-tors,” J. Appl. Phys. 68, 2216 (1990).

3. R. Bube, Photoelectronic Properties of Semiconductors (Cam-bridge U. Press, Cambridge, 1992), p. 53.

4. S. I. Stepanov, I. A. Sokolov, G. S. Trofimov, V. I. Vlad,D. Popa, and I. Apostal, “Measuring vibration amplitudesin the picometer range using moving light gratings in pho-toconductive GaAs:Cr,” Opt. Lett. 15, 1239 (1990).

5. F. M. Davidson, C. C. Wang, C. T. Field, and S. Trivedi, “Pho-tocurrents in photoconductive semiconductors generated bya moving space-charge field,” Opt. Lett. 19, 478 (1994).

6. F. M. Davidson, C. C. Wang, and S. Trivedi, “Optical phase-lock loops with photoconductive semiconductor phase detec-tors,” Opt. Lett. 19, 778 (1994).

7. C. C. Wang, F. Davidson, and S. Trivedi, “A simple laservelocimeter that uses photoconductive semiconductors to de-tect optical frequency difference,” Appl. Opt. 34, 6496 (1995).

8. C. C. Wang and F. Davidson, “Optical spectrum analyzer ofa new design,” Opt. Lett. 20, 1036 (1995).

9. Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huig-nard, “Two-beam coupling in photorefractive Bi12SiO20 crys-tals with moving grating: theory and experiments,” J. Appl.Phys. 58, 45 (1985).

10. F. P. Strohkendl, J. M. C. Jonathan, and R. Hellwarth,“Hole–electron competition in photorefractive gratings,”Opt. Lett. 11, 312 (1986).

11. A. Papoulis, Probability, Random Variables, and StochasticProcesses (McGraw-Hill, New York, 1965).

12. G. Zelniker and F. J. Taylor, Advanced Digital SignalProcessing: Theory and Applications (Dekker, New York,1994).

13. S. I. Stepanov, “Sensitivity of non-steady-state photoelectro-motive force-based adaptive photodetectors and characteri-zation techniques,” Appl. Opt. 33, 915 (1994).

14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series,and Products (Academic, New York, 1980).

15. H. Okamura and K. Iwatsuki, “A finesse-enhanced Er-doped-fiber ring resonator,” J. Lightwave Technol. 9, 1554 (1991).

16. K. Kalli and D. A. Jackson, “Ring resonator optical spectrumanalyzer with 20-kHz resolution,” Opt. Lett. 17,1090 (1992).

17. D. D. Marcenac and J. E. Carroll, “Maximum-entropy opticalspectrum analyzer,” Opt. Lett. 20, 1074 (1995).