May 1, 1995 / Vol. 20, No. 9 / OPTICS LETTERS 1035

Optical spectrum analyzer of a new designChen-Chia Wang and Frederic Davidson

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

Received January 23, 1994

The interference between two optical fields with nonzero spectral linewidths forms multiple moving space-chargefields inside photoconductive semiconductors that contain deep-level donors and traps. The dc components ofthe photocurrents generated can be expressed in terms of a convolution of a function that is characteristic ofthe crystal and the signal optical power spectrum when the other optical field (local oscillator) has a negligiblelinewidth. This permits the recovery of the optical signal power spectrum.

Previous research has shown that steady-state mov-ing space-charge fields formed inside photoconductivesemiconductors generate dc photocurrents that canbe used to identify material characteristics1 and tophase lock independent lasers.2 However, it wasassumed that the laser linewidths were negligibleso that monochromatic optical-field approximationswere valid. The effects on the space-charge fieldsand the resultant photocurrents generated by theinterference of nonmonochromatic local-oscillatorand signal optical fields are presented here. Itwas found that the optical interference patterns ofnonmonochromatic optical fields generate multiplemoving space-charge fields inside the semiconduc-tor samples. The resultant photocurrents containboth dc and ac components. The dc component canbe expressed in the form of a convolution involvingthe power spectrum of the incoming signal opticalfield when the local-oscillator laser has negligibleoptical linewidth. The ac components are identicalto the non-steady-state holographic currents studiedpreviously.3

For the sake of simplicity it is assumed thatthe various frequency components of the signaloptical field propagate along the same directionso that the multiple space-charge fields estab-lished inside the crystal move along identical di-rections. We also assume that electrons are theonly species of photoexcited free charge carriers inthe host material. The optical fields are expressedas ELOsr, td ­ ELO exph2j fkLO ? r 2 vLOt 2 fLOstdgjand ESsr, td ­ ES0 exph2jfkS ? r 2 vSt 2 fSstdgj,where kLO skSd is the wave vector, fLOstd ffSstdg isthe phase, and vLOsvSd is the center frequency ofthe local-oscillator (signal) optical field. Amplitudemodulation effects have been neglected so that ELO

and ES0 are time independent. The optical-field in-tensity distribution inside the photoconductive semi-conductor is then given by

jEtotsr, tdj2 ­ jELOj2 1 jES0j2

1 sssELO*ES0 exphj fskLO 2 kS d ? r 2 vDtgj3 expfjDfstdg 1 c.c.ddd , (1)

where vD ­ vLO 2 vS is the center frequency differ-ence between the two optical fields, Dfstd ­ fSstd 2

fLOstd, and c.c. denotes the complex conjugate of the

0146-9592/95/091035-03$6.00/0

preceding term. We have assumed the applicabil-ity of the one-dimensional model as a result of thesmall optical absorption coefficients associated withsuch photoconductive semiconductors when the in-cident photon energies are smaller than the band-gap energy of the host material. The optical powerdensity distribution inside the crystal can be writtenas I sx, td ­ I0h1 1 s1y2d fmstdexpsiKgxd 1 c.c.gj, whereI0 ­ PLO 1 PS is the total optical power density inci-dent upon the crystal, Kg ­ skLOdx 2 skS dx ­ 2k0 sin u

is the grating wave number, k0 ­ 2pyl0 is the opti-cal wave number in free space, and 2u is the intersec-tion angle between the lasers. The modulation indexis given by mstd ­ m0 expf2jvDtgexpfjDfstdg, wherem0 ­ 2ELO*ES0ysjELOj2 1 jES0j

2d.Next, the truncated versions of the modulation in-

dex and the space-charge field are defined so thatmT std ; mstd and E sT d

sc std ; Escstd in the time inter-val 2T # t # T and mT std ­ 0 ­ ET

scstd for jtj . T .The Fourier transforms of these two functions aregiven by MT svd ; s1y2pd

RT2T mT stdexps2jvtd dt

and ETscsvd ; s1y2pd

RT2T ET

scstdexps2jvtd dt. ForjmT stdj ,, 1, the spatially varying space-charge fieldcan be written in terms of its first spatial frequencycomponent ET std ­ s1y2d fET

scstdexpsjKgxd 1 c.c.g.The function ET

scstd obeys the following equation:

dETscstd

dt1

ETscstdtg

­ 2jED

tM s1 1 EDyEM dmT std , (2)

where tg ­ tM s1 1 EDyEM dys1 1 EDyEqd is the gratingformation time and the parameters tM , ED , and EM

depend on physical parameters of the host material.1The expression for the short-circuit photocur-

rent generated inside photoconductive semiconduc-tor crystals is given by3jSstd ­ s0fmstdEsc*std 1c.c.gy4s1 1 EDyEM d, which can be written as

jTS std ­

j jm0j2s0ED

4s1 1 EDyEM d s1 1 EDyEqd1

2p

3

""" Z `

2`

dv

1 1 jsvD 2 vdtg

3

√√√ Z T

2Texpf2jvst 2 t0dg

3 exphj fDfstd 2 Dfst0dgj dt0

!!!2 c.c.

###, (3)

1995 Optical Society of America

1036 OPTICS LETTERS / Vol. 20, No. 9 / May 1, 1995

where the turn-on transient has been assumed to dieoff. Interference among the various moving space-charge fields generates the ac components of the pho-tocurrent, as signified by the expf2jvst 2 t0dg termon the right-hand side of Eq. (3).

The time average of Eq. (3) is defined as jTS,DC ;

s1y2T dRT

2T jTS std dt, which is a random process

since the factor hexpfjDfstdgj is random in na-ture. In the following discussions it will be as-sumed that the random processes hexpfjDfSstdgj andhexpfjDfLOstdgj are wide-sense stationary so that theprocess hexpfjDfstdgj is also wide-sense stationary.Thus the ensemble average of jT

S,DC is given by

kjTS,DCl ­

j jm0j2s0ED

4s1 1 EDyEM d s1 1 EDyEqd1

2p

3

0B@0B@0B@ Z `

2`

dv

1 1 jsvD 2 vdtg

3

8<: 12T

Z T

2T

Z T

2TRS st 2 t0dRLO*st 2 t0d

3 expf2jvst 2 t0dg dtdt0

9=; 2 c.c.

1CA1CA1CA , (4)

where RS st 2 t0d ­ kexphjffSstd 2 fSst0dgjl andRLOst 2 t0d ­ kexphjffLOstd 2 fLOst0dgjl. The anglebrackets represent ensemble averages, and it is alsoassumed that the random processes hexpfjfSstdgjand hexpfjfLOstdgj are statistically independent ofeach other.

kjTS,DCl is equivalent to the dc photocurrent density

in the limit T ! `. It can be shown4 that, for thetypical optical signal power spectra of interest (e.g.,Gaussian, Lorentzian, and white-noise), the dc pho-tocurrent density can be written as

jS,DC ­ limT!`

kjTS,DCl ­

2s0EDysPS 1 PLOd2

s1 1 EDyEM d s1 1 EDyEqd

3Z `

2`

SbeatsvdsvD 2 vdtg

1 1 svD 2 vd2tg2dv, (5)

where Sbeatsvd ; s1y2pdR`

2` PSPLORSstdRLO*stdexps2jvtddt is the power spectrum of the beat notebetween the two optical fields. If the local-oscillatorlaser has zero linewidth, RLOstd ­ 1 for all values oft, and hence Eq. (5) can be reduced to

jS,DCsvDd ­2s0EDPLOysPS 1 PLOd2

s1 1 EDyEM d s1 1 EDyEqd

3Z `

2`

SSsvdsvD 2 vdtg

1 1 svD 2 vd2tg2

dv, (6)

where SSsvd ; s1y2pdR

`

2` PSRS stdexps2jvtd dt isthe power spectrum of the signal optical field. Theoptical power spectrum can be determined by de-convolution of the measured dc photocurrents as afunction of vD with the function TCRsvd, defined as

TCRsvd ;2s0EDPLOysPS 1 PLOd2

s1 1 EDyEM d s1 1 EDyEqdvtg

1 1 svtgd2,

(7)

which can be determined unambiguously once thecharacteristics of the host material and experimentalconfiguration are known.

If both the signal and the local-oscillator opti-cal fields are monochromatic, RSstd ­ RLOstd ­ 1for all values of t, and hence SSsvd ­ PSdsvd.Recalling that jm0j

2 ­ 4PSPLOysPS 1 PLOd2, wethen simplify Eq. (6) to jS,DCsvD d ­ fjm0j

2s0EDy2s1 1 EDyEM d s1 1 EDyEqdg 3 vDtgyf1 1 svDtgd2g,which is identical to the result obtained in previ-ous studies.1 On the other hand, no dc photocur-rent results if either of the optical fields has awhite-noise-like spectrum since no steady-state mov-ing space-charge fields can be established. Indeed,jS,DCsvD d ­ 0 when RLOstd ­ dstd, as can be seenfrom Eq. (5).

If the signal spectrum consists of discretespectral components caused by, e.g., periodicphase modulations, the modulation index canbe written as mstd ­ m0exps2jvDtdexpfjfSstdg ­Simi expf2jsvD 2 vSidt], where svS 1 vSid is the cen-ter frequency of the ith frequency component of thesignal optical field. The steady-state photocurrent,Eq. (3), is then given by

jSstd ­ limT!`

jTS std ­

s0ED

2s1 1 EDyEM d s1 1 EDyEqd

3

8<:Xi

jmij2svD 2 vSidtg

1 1 fsvD 2 vSidtgj2

1j2

Xifij

mimj *expsjvij td

3

"1

1 1 jsvD 2 vSj dtg

21

1 2 jsvD 2 vSidtg

#9=; , (8)

where vij ­ vSi 2 vSj is the frequency differencebetween the ith and jth frequency components ofthe signal laser. We can consider the special caseof a sinusoidal phase modulation D cossvmtd ap-plied to the signal laser such that, for a suitablechoice of D, the signal spectrum consists of the threelowest-order contributions. The modulation indexof the optical interference pattern can be approxi-mated by mstd ­ m0 exps2jvDtdexpfjD cossvmtdg ø

Fig. 1. Experimental setup for the detection of photocur-rents generated by moving space-charge fields. B.S.’s,beam splitters; LWE 120, LWE 122, Lightwave Electron-ics Series 120 and 122 nonplanar Nd:YAG ring lasers;PLL, phase-locked loop; E-O, electro-optic.

May 1, 1995 / Vol. 20, No. 9 / OPTICS LETTERS 1037

Fig. 2. dc photocurrents versus carrier frequency offset,fLO 2 fS , for the signal spectrum shown in Fig. 3. Thegrating spacing was L ­ 44.7 mm, and the optical powerlevels were PLO ­ 56.9 mWycm2, PS ­ 36.9 mWycm2.The modulation frequency was fm ­ 496.1 kHz, and theamplitude was 2.39 rad.

Fig. 3. Deduced signal spectrum from the deconvolutionof the impulse response of the crystal and the experi-mental data shown in Fig. 2. The resolution bandwidth,which was approximately 8 kHz, was limited by the fre-quency spacing between two adjacent samples.

m0J0sDdexps2jvDtd 1 jm0J1sDdhexpf2jsvD 1 vmdtg1 expf2jsvD 2 vmdtgj, where Jn is the Bessel func-tion of the first kind of order n. If the center fre-quencies of the two lasers are kept identical suchthat vD ­ 0, Eq. (8) can be simplified to

jSstd ­jm0j

2s0EDJ0sDdJ1sDd2s1 1 EDyEM d s1 1 EDyEqd

3

24 jvmtg

1 2 jvmtgexps2jvmtd

2jvmtg

1 1 jvmtgexpsjvmtd

35 , (9)

which is sinusoidal and identical to the result ob-tained by other researchers3 when the approxima-tions of small-amplitude phase modulation, jDj ,, 1,and large saturation field, ED ,, Eq, made by thoseauthors are taken into consideration.

The experimental setup used to detect the pho-tocurrents generated by the interference betweenthe local oscillator and a nonmonochromatic sig-

nal laser is shown in Fig. 1. The center frequen-cies of two independent Nd:YAG lasers (LightwaveElectronics Series 120 and 122) operating at thewavelength of l ­ 1.064 mm were offset by a conven-tional heterodyne phase-locked loop. Light emissionfrom the Lightwave Electronics Series 120 laser wasphase modulated by an electro-optic phase modu-lator to expand its spectral width. The resultantsignal-field spectrum was monitored by an electronicspectrum analyzer and exhibited the expected be-havior. The two lasers were brought to interfer-ence inside a photoconductive GaAs:Cr sample, andthe dc photocurrents generated are shown in Fig. 2for the sinusoidal phase modulation fSstd ­ 2.39coss2p 3 496.1 3 103td. Data points at larger val-ues of frequency offsets were not collected becauseof the instability of the heterodyne phase-locked loopat higher frequencies. The distinctive transitions atfLO 2 fS ø 6fm, 62fm signify the presence of spectralcomponents in the signal power spectrum at the cor-responding frequencies. As a comparison, we alsodeconvolved the fitting curve based on the experi-mentally observed photocurrent data shown in Fig. 2with the characteristic response of the crystal to de-duce the power spectrum of the signal optical field.Figure 3 shows the deduced signal power spectrum,which is in excellent agreement with the actual powerspectrum of the signal optical field. The resolutionbandwidth was limited by the frequency step betweentwo adjacent sampling points and was approximately8 kHz.

This experiment demonstrates the validity of theuse of this technique to recover the optical powerspectrum of the signal beam. It is unrealistic onlyin the sense that the local-oscillator laser was phaselocked to the center frequency of the signal fieldbefore the phase modulation was imposed. The cen-ter frequencies of both lasers drift with tempera-ture, and this was the easiest practical way to sweepprecisely the frequency difference between the twolasers through the range of interest. A practical sys-tem would differ only in that it would require amechanism to tune precisely the frequency of thelocal-oscillator laser through the center frequency ofthe signal beam. At the point at which vLO ­ vS ,i.e., vD ­ 0, the origin of the jS,DCsvDd versus vD

plot can be found by use of the symmetry propertySS svd ­ SSs2vd. This new kind of optical spectrumanalyzer is spatially adaptive because the stringentoptical beam-alignment conditions found in conven-tional interferometric systems are not required.

References

1. F. M. Davidson, C. C. Wang, C. T. Field, and S. Trivedi,Opt. Lett. 19, 478 (1994).

2. F. M. Davidson, C. C. Wang, and S. Trivedi, Opt. Lett.19, 778 (1994).

3. M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S.Trofimov, J. Appl. Phys. 68, 2216 (1990).

4. A. Papoulis, Probability, Random Variables, and Sto-chastic Processes (McGraw-Hill, New York, 1965),Chap. 10, p. 344.