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9-1-1991

Laser Bandwidth-Induced Fluctuations in theIntensity Transmitted by a Fabry-PerotInterferometerGerhard KlimeckPurdue University - Main Campus, gekco@purdue.edu

Daniel ElliottPurdue University - Main Campus

M. HamiltonUniversity of Strathclyde, Glasgow, Scotland

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Klimeck, Gerhard; Elliott, Daniel; and Hamilton, M., "Laser Bandwidth-Induced Fluctuations in the Intensity Transmitted by a Fabry-Perot Interferometer" (1991). Birck and NCN Publications. Paper 786.http://dx.doi.org/10.1103/PhysRevA.44.3222

PHYSICAL REVIEW A VOLUME 44, NUMBER 5 1 SEPTEMBER 1991

Laser-bandwidth-induced Auctuations in the intensity transmitted bya Fabry-Perot interferometer

G. Klimeck and D. S. ElliottSchool ofElectrical Engineering, Purdue University, West Lafayette, Indiana 47907

M. W. HamiltonDepartment ofPhysics and Applied Physics, University ofStrathclyde, Glasgow, G4 OXG Scotland

(Received 4 March 1991)

We have measured the power spectrum of the intensity Auctuations of light transmitted by a Fabry-Perot interferometer when the input field is the real Gaussian field. The real Gaussian field is a fieldcharacterized by real, random (Gaussian) amplitude Auctuations. The bandwidth of the real Gaussianfield was varied, taking on values less than that of the interferometer, as well as greater. Comparisons ofthe measured spectra with calculated spectra are quite satisfactory. Of special interest is a feature in thespectra centered at the laser-interferometer detuning frequency.

INTRODUCTION

Laser-bandwidth effects on nonlinear-optical processeshave been of growing interest for the past two decades,but recently systems that are inherently linear in theirresponse to the electromagnetic wave have shown in-teresting laser-bandwidth effects as well. These linearsystems become interesting when higher-order correla-tion properties of the system are involved. An exampleof this is the fluctuations (rather than just the averagevalue) of fluorescence from a linear atomic system in-teracting with weak laser radiation. Observations [1,2] ofan unusual dependence of fluorescence fluctuations onlaser detuning from resonance with the atoms motivatedseveral theoretical works [3—5]. Recent measurementsof the variance [6] and of the spectrum [7] of these Auc-tuations when the atom is excited by a phase diffusinglaser field are in good agreement with theory. In this pa-per, we discuss measurements of the spectrum of Auctua-tions of laser light transmitted by a Fabry-Perot inter-ferometer when the input field undergoes random ampli-tude fluctuations. The field is described by the modelknown as the real Gaussian field. The Fabry-Perot inter-ferometer is a linear system for all intensities used in thiswork, and measurements are in excellent agreement withcalculations [8].

EXPERIMENTAL TECHNIQUE

Generation of the real Gaussian field has been dis-cussed in detail in a previous publication [9]. The field isof the form

E(t)=E(t)ewhere coL is a constant frequency. The amplitude s(t) isa Gaussian random process with an average value ofzero; i.e., it is positive as often as it is negative. This fieldis generated by modulating the output of highly stabilizedcw tunable dye laser using an acousto-optic modulator(AOM). The AOM is driven by a rf signal whose ampli-

tude is also a real Gaussian process. Laser bandwidthsfrom 1 to 14 MHz full width at half maximum (FWHM)are attainable. The lower limit is determined by thebandwidth of the stabilized dye laser (-200 kHz). Mea-surements of the power spectrum of the intensity Auctua-tions of the real Gaussian field intensity indicate that itfalls off with frequency slightly faster than a Lorentzian.For all laser bandwidths used in this experiment, themeasured intensity spectrum is about 3 dB below a realLorentzian at about 25 MHz, and about 7.5 dB lower at40 MHz. Three factors contribute to this sub-Lorentzianshape. First is the response of the AOM. The efficiencyof the AOM is at a maximum for a drive frequency of 200MHz, and falls off by about 3 dB when the drive signal is50 MHz to either side of 200 MHz. Second is the spatialcoherence of the beam diffracted by the AOM. When theAOM drive signal is modulated, the acoustic wave in themodulator is not uniform across the optical wave front.This leads to a variation of the amplitude of the realGaussian field across the beam. Since the intensity spec-trum is measured by projecting the entire beam onto aphotodiode, the average intensity is measured, so that thehigher-frequency components of the fluctuations tend toaverage out. Both of these processes are discussed inmore detail in Ref. [9]. An additional contribution to thedecrease in the high-frequency components is the fre-quency response of the detector itself. This is secondaryto the preceding two effects. Approximately 4% of theincident field intensity can be converted into the realGaussian field. Higher diffraction efFiciencies or largerbandwidths result in distortion of the signal, such thatE(t) deviates from a Gaussian process. This limit wasdetermined through measurements of the intensity auto-correlation function, which for the perfect real Gaussianfield with a Lorentzian power spectrum decreases ex-ponentially from an initial value of three to a long-termvalue of 1:

(I(t+r)I(t))»,I (I(t))' —1+28

3222 Q~1991 The American Physical Society

LASER-BANDWIDTH-INDUCED FLUCTUATIONS IN THE. . . 3223

I

100I

200 300t (ns)

400 500

3.0

i';2.5- l '''

O~~

2.0

1.5

(b)

1.0

0.5

0.00

l

25I

50I I I

75 100 125 150g (ns)

FICi. 1. (a) Typical time trace of the intensity of the realgaussian field, and (b) the intensity autocorrelation functioncomputed from 26 such traces. The bandwidth of the laser wasb/~=9. 0 MHz (FWHM) for this figure, leading to a correlationtime of the intensity fluctuations of 1!2b=18 nsec. In (b) themeasured autocorrelation (solid line) is shown, as well as the re-sult of a least-squares fitting procedure to a function of the form1+[Rl(0)—1]exp( 2b~) (dashed curve—). The dotted curves in-dicate the standard deviation of the measured autocorrelationfunction, as determined from the scatter of the data.

where 2b is the width (FWHM) of the Lorentzian laserline in angular frequency. In practice, initial values ofRl(0) are measured to be in the range 2.4—2. 8. Figure 1

shows a typical measurement of the intensity of the realGaussian field and the corresponding intensity autocorre-lation function computed from a series of these digitizedtraces.The Fabry-Perot interferometer is of near confocal

geometry with a free spectral range (FSR) of 4 GHz. Theinterferometer is a commercial model of an optical scan-ning spectrum analyzer, from which we have removedthe photodiode detector and modified the mechanism foradjusting the cavity length. The rear mirror of the inter-ferometer is mounted on a shaft whose position is adjust-ed using a 40 pitch screw, and locked in place by tighten-

ing a collet nut. This allows fine tuning of the cavitylength to within approximately 10 pm of confocalgeometry, as evidenced by measurements of the cavitytransmission peak width, to be discussed later.Mode matching of the Gaussian laser beam into the

confocal cavity was important in order to make interpre-tation of the results simpler. Calculations [8] of thebandwidth effects were carried out for excitation of a sin-gle transverse and longitudinal mode of the cavity. Thelaser beam was focused to the center of the cavity usingan f=22-cm focal length achromatic doublet. The ebeam radius before focusing was 0.61 mm. The coupling[10] into the TEM„(n +m even) modes was nearly per-fect, as confirmed by observing the mode pattern of thetransmitted beam visually, and by measurements of theintensity of the various transverse modes as the inter-ferometer was scanned. We estimate that while theTEMo o cavity mode dominates, a few other higher-ordereven modes were also excited, as observed by slightmisadjustment of the cavity from confocal separation. Acavity length differing from that of a confocal geometryby as little as 15 pm, for example, would result in a shiftbetween the even-order modes of about 1 MHz. Thiscauses the transmission peak to become asymmetric andsomewhat broader when higher-order even modes arealso excited. Our previous claim of the precision of theadjustment of the cavity length is based on the high de-gree of symmetry we observed. The peak height corre-sponding to the odd values of n +I was reduced to lessthat —,

' the peak height of the even modes. Overall modematching was, we believe, quite satisfactory, although theeffect of a mismatch is not well understood.In order to measure bandwidth effects, it was necessary

to control the detuning of the laser frequency from reso-nance with the cavity. This was accomplished using stan-dard frequency locking techniques, as shown in Fig. 2.The output of the dye laser was split into two parts, oneof which was modulated to produce the real Gaussianfield, while the second was used for locking the laser fre-quency to the cavity. We will refer to these beams as theGaussian and locking beams, respectively. The centerfrequency of the real Gaussian field was shifted by theAOM by vG =200 MHz. The locking beam was frequen-cy shifted by a second AOM which was driven by asingle-tone frequency modulated signal. The carrier ofthis drive signal was varied in frequency v& from 160 to240 MHz, and was frequency modulated at 10 kHz. Thedeviation frequency of the modulation was about 8 MHz.The polarization of the locking beam was rotated 90, andthe Gaussian and locking beams were recombined on apolarizing beamsplitter. The two beams passed throughthe interferometer collinearly, and were separated with asecond polarizing beamsplitting cube upon exiting the in-terferometer. Cross coupling between the two beams wasobserved to be negligible. An error signal was generatedfrom the locking beam photocurrent by demodulation ofthe 10-kHz dither in a lock-in amplifier. The lock-in out-put was integrated, amplified and applied to the frequen-cy tune control of the dye laser, thus keeping the lockingbeam resonant with the interferometer. Since the Gauss-ian and locking beams differed in frequency by

3224 G. KLIMECK, D. S. ELLIOTT, AND M. W. HAMILTON

Dye Laser . .

Real Gaussian FieldGenerator

'"""""I aoM I""""1.0-

WavePlate

"""I &OM ) ~BS

= 0.1-.2

vco

Dither SourceIDC Voltage, ~g io kH

iCJ

FPI

LaserTuner Integrator Lock-In

Amplifier

BS:.O.O1

-40l ' ' ' I ' ' ' t

-20 0 20Detuning (MHz)

Il

UNIX Digitizing Spectrum AnalyzerPCMainframe Camera or Oscilloscope

FIG. 2. Schematic diagram of the experiment. The output ofthe tunable dye laser is split into two beams, one which is ran-domly amplitude modulated to form the real Gaussian field, theother of which is single-tone frequency modulated to lock thelaser frequency to the peak of the interferometer transmissionpeak. The two beams pass through the interferometer (FPI) col-linearly, and are separated using a polarizing beam splitter cube(BS). The licking beam is detected, amplified in a lock-in detec-tor, and integrated to produce a feedback signal to correct thefrequency of the dye laser. The real Gaussian field intensity isspectrally analyzed, digitized, and stored on a personal comput-er (PC).

6=vG —v&, the detuning of the Gaussian beam from cav-ity resonance was also A. This scheme was used for mea-surement of the cavity finesse as well as for measurementsof laser bandwidth effects.For the finesse measurements, the noise modulation

electronics were replaced by a single frequency source sothat the "Gaussian" beam was narrow band, and theaverage transmitted intensity was recorded as a functionof A. These data are shown on a logarithmic scale in Fig.3. The circles represent the data points and the solid linerepresents the result of a least-squares fit of a Lornetziancurve to the data, yielding a bandwidth (FWHM) ofabout 11.5 MHz. This corresponds to a cavity finesse of350. The data and the fit agree very nicely for detunings6 greater than —10MHz. For larger negative detunings,the poor agreement is due, we believe, to misalignment ofthe locking beam from the axis of the interferometer. Insupport of this assertion, we note that the hump on thelow-frequency side of this curve is not evident when sim-ply scanning the interferometer with the laser frequencyfixed. Misalignment results from changing the carrierfrequency of the locking beam, since the diffraction anglein the AOM varies with the drive frequency. For drivefrequencies less than 190 MHz (detunings less than —10

FIG. 3. The average intensity transmitted by the Fabry-Perotinterferometer. The open circles represent the measured inten-sities, while the curve represents a fit to the data of a Lorentzianfunction. From these data, the width of the transmission peakis determined to be 11.5 MHz, corresponding to a finesse of 350.

MHz), it appears that the locking laser excites a differentmode of the cavity sufficiently to cause the system to lockto the peak of that mode. This shifts the detuning of theGaussian beam, such that detuning measurements below—10 MHz are not accurate. We therefore limited ourmeasurements to detunings outside this range.The measurements of the power spectrum of the inten-

sity noise transmitted by the interferometer are carriedout by projecting the transmitted Gaussian beam onto asilicon photodiode, whose photocurrent is measured on arf spectrum analyzer. The spectra are measured in therange of frequencies from 3 to 39 MHz, for detunings ofthe Gaussian laser from resonance of 0—35 MHz. Thelaser bandwidth was adjusted to 4.8, 90, or 14.0 MHz(FWHM). Up to 23 power spectra of the intensity Auc-tuations were measured in each data set, and two datasets were recorded for each value of laser bandwidth.The intensity autocorrelation was checked for each laserbandwidth as well, yielding initial values of Rl(0) =2.552.57, and 2.44, respectively. Results will be presented inthe following section, where we show very good agree-ment with calculated results.

RESULTS

A few examples of the power spectra of the intensityfluctuations transmitted by the Fabry-Perot interferome-ter are shown in Fig. 4. Figures 4(a), 4(b), and 4(c) corre-spond to laser bandwidths of 4.8, 9.0, and 14.0 MHz, re-spectively. For each diagram spectra are shown forlaser-resonator detunings 6 of 0, 10, 20, and 30 MHz,from top to bottom. Each diagram shows the measuredspectra (rough lines) as well as the calculated spectra(smooth lines). We will first discuss the calculatedcurves, which are of the form

3225FLUCTUATION S IN THE. . .IDTH-INDUCED FLASER-BAND WID

Sl(co)= g Ai =1,2 +(2b)

b+b, co, )+2 Im( B )(&—co)2Re(B, ) b+ co,2(b,—co ) +(b+bco, ) co +(2b,co, )

B )(b, +co)2Re(B, ); (b+bco, )+2Im(2+ ) +(b+bco,(b.+coCO

(4)

at half max-vit width half width aC

ual in our caimum, equa' a

—b~' —ib)~ 2b1 +lb)(I R 1 2W=~(I —Re '

—br' —i 5b~' —b~')—&—Re—~—Re' (e ' —e

11—R

b~')i5 —b~Re eB R2 2i51—R e

lex, the power spec-B's are complex, to gNote that aya srea an

irrors, an ise of the inter eb hco =—(I/

tpcavity ro

f the inten-al to col w.

r s ectrum oto t}1f th t ttsity

1 t 1 bse terms qua ita 'derstand these term

-30—(a)

-30—

40Nz

-50—E

QJ

-60-G

&U

-70—

40z

g -50-

QJ

~ -60-

-70—

-80— -80-

I 3020frequency ( M z

40 10I3020

frequency ( M z40

-30—

-50-

G -60—

QJ

-70—

I30req Hz )frequency ( MH

40100

) in the figure are (aa 4.8h fi th d }1 ) d

nin 6 of 0,11

-'nterferome eter detuning

G. 4. Power specctra of h f 5.0 MHz. The wid '

hfiof curves in eac gale ulationsa c ' smooth curvestop to botto m.30 MH fro10, 20, and

3226 G. KLIMECK, D. S. ELLIOTT, AND M. W. HAMILTON

picture of the transmitted field. The transmitted-fieldspectrum should be expected to exhibit two maxima, oneat the laser frequency, the other at the cavity resonantfrequency. The intensity spectrum, therefore, consists ofpeaks with maxima at zero frequency corresponding toeach of these peaks beating against themselves, and also apeak centered at the detuning frequency resulting fromthe two peaks of the field spectrum beating against eachother. These are illustrated in Fig. 5, where we have plot-ted each term in Eq. (3) for a laser of width 9.0 MHz, acavity of width 11.5 MHz, and a detuning frequency of30 MHz. The first and last terms in Eq. (3) are Lorentzi-an in shape, and have widths equal to twice the laserwidth and twice the cavity width, respectively. Theseterms correspond to curves b and c in Fig. 5. TheLorentzian curve (d) and dispersion-shaped curve (e)centered at 30 MHz in Fig. 5 are represented in Eq. (3) bythe terms involving Re(B; ) and Im(B; ), respectively.These terms represent the cross term between the twopeaks in the transmitted-field spectrum. The disperison-shaped curve was omitted from the results in Ref. [8], dueto an error made in evaluating the Fourier transform ofEq. (27) and (28) of that work. We also remark here thatEq. (27) and (28) of Ref. [8], while they are not incorrect,could have been represented in a much simpler form. Seethe Appendix for further discussion.Several features of the measured data are noteworthy.

The overall shape of the data curves are in very goodagreement with the calculated curves. Within eachfigure, the only adjustment made to the calculated curvesis an overall amplification, which on a logrithmic scale ofcourse corresponds to a vertical displacement. The rela-tive amplitudes of each of the curves within any of thefigures are not adjusted. The data curves tend to fall off

1.0-

a P4

O Mcr t

O

2O

with frequency slightly faster than the calculated results,which we attribute to the spatial averaging and the fre-quency response of the detection system. Figure 6 showsthe frequency response of the random-modulation sys-tem. This was measured by single-tone modulating thefield amplitude at a frequency v, suppressing the car-rier, removing the interferometer, and measuring the am-plitude of the current generated by the photodiode at thefrequency 2v . Using the frequency response shown inFig. 6 to adjust our data actually overcompensates athigh frequencies. That this correction does not applyperfectly is not surprising since no account is taken of theeffect of the Fabry-Perot interferometer on the transmis-sion of a field with imperfect spatial coherence. The spa-tial coherence of the real Gaussian field was discussedearlier in the context of the laser intensity band shape.Nevertheless, the correction factor is of the proper orderto explain the difference between the data and calculatedcurves. The data for a laser width of 4.8 MHz shows athigher frequencies a contribution from the noise level ofthe spectrum analyzer.In Fig. 7 we show the amplitude of the power spectrum

at zero frequency versus detuning for the data. Thecurves represent the calculated amplitudes. Each curveand data set has been normalized to a peak value of 1.The data points are obtained by extrapolating the mea-sured power spectra to zero frequency. Agreement isgood for all laser bandwidths used, with the exception ofthe 14.0-MHz data for detuning frequencies greater than20 MHz. In this range the amplitude of the Auctuationsis measured to be less than that calculated, probably dueto the sub-Lorentzian power spectrum of the laser forhigh frequencies.We have found poor agreement when comparing mea-

sured and calculated bandwidths of the intensity spectra.This is not surprising, however, when one considers thelimited frequency response of the measurements due tospatial coherence of the field and the detection system.

0.0-I

10

~W~' R' m ' m.

I ~ II ~l1lll ~&gy ggggllll ~ IIIII~ty~

~III~ llNllllll ~ IIIIII ~ II

60I I I I

20 30 40 50Frequency (MHz)

-108

FIG. 5. Components of the calculated spectrum of the inten-sity Quctuations. The laser width for these curves is 4.5 MHz,the interferometer width is 11.5 MHz, and the detuning 6 is 30MHz. The five curves in the figure represent the various termsof Eq. (3) in the text. They are (a) the total power spectrum, (b)the laser bandwidth Lorentzian, (c) the cavity bandwidthLorentzian, (d) the Lorentzian cross term, and (e) thedispersion-shaped cross term.

-150 10

I ' ' ' I

20 30Frequency (MHz)

I

40 50

FIG. 6. Frequency response of the system: The measured am-plitude of the sidebands of the laser when single-tone amplitudemodulated with a suppressed carrier.

LASER-BANDWIDTH-INDUCED FLUCTUATIONS IN THE. . . 3227

10

10

since the filter is linear, the amplitude is still a Gaussianvariable. Preservation of the Gaussian properties isnecessary for a subsequent step. The intensity correlationfunction RI (r) of the transmitted field is given by

10

10~eg~~g

tggkg~

i g el ~

&lo(t+r)IO(t) &RI(r) = & IE,(t+r)l'IE, (t)l'&

(Al)

10 I

10 15 20 25Detuning (MHz)

I

30I

35 40

FIG. 7. Normalized amplitude of the intensity power spectraat zero frequency as a function of the laser-interferometer de-tuning b. The points correspond to measurement for laserwidths of 4.8 MHz (6), 9.0 MHz (0), and 14.0 MHz (X),while the curves are derived from calculated power spectra, forthe same laser bandwidth values (dashed, dotted, and solid lines,respectively).

The "0" denotes the transmitted field or intensity. Wenormalize to the average input intensity in order to retaininformation on the magnitude of the Auctuations. In or-der to evaluate this autocorrelation function for the out-put intensity Auctuations, we need to relate the transmit-ted field to the incident field. This is done in the usualseries expansion

Eo(t) =Eo(t)e =(1—R )[E(t)+RE(t r')e—+R E(t—2r')e

(A2)

In conclusion, we have presented results of measure-ments of the power spectrum of the intensity fluctuationsof light transmitted by a Fabry-Perot interferometerwhen a real Gaussian field is incident upon it. Agreementof the measured spectra with those calculated previouslyis very good, particularly when examining relative ampli-tudes, the cross term "hump" at the detuning frequency,and the overall shape of the curves. These measurementsdisplay very nicely the effect of a linear system, the inter-ferometer, on the higher-order statistical properties ofthe field.

where each term represents one more round-trip of thefield through the cavity than the previous term. We thenwrite the intensity correlation function in terms of thetransmitted-field amplitude

RI'(r) = & e,(t+r)E,*(t+r)E,(t)s,*(t)&/& le(t)l'&' . (A3)

Since the amplitude of the transmitted field Eo(t) is aGaussian variable, this fourth-order correlation functioncan be decomposed into a sum of products of second-order correlation functions,

ACKNOWLEDGMENTS

This work was supported by a grant from the U.S.Department of Energy, Office of Basic Energy Sciences.Participation by M.W.H. was made possible by a Colla-borative Research Grant from the NATO ScientificAffairs Division. One of us (G.K.) is grateful to theFriedrich-Ebert Stiftung for their financial support. Wewould like to thank Bruce Ferguson for many useful dis-cussions during the course of this work.

R,'(r) = [& IE,(t) I'&'+ I & s,(t+r)E,(t) & I'

+l&E;(t+ )E,(t)& ']/&IE(t)l'&'. (A4)

The first term here is the square of the average transmit-ted intensity. The second term vanishes for the thermalfield since E(t) is complex and of random phase. There-fore, when the thermal field is incident on the interferom-eter, the autocorrelation function is

APPENDIXRi'th(r)=&1&'+ I&E,*(t+1)EO(t)&I /& IE(t)l &, (A5)

In this appendix we discuss a simplification of thederivation and the results of Ref. [8] for the intensitycorrelation function of the field transmitted by a Fabry-Perot interferometer when a thermal field or a realGaussian field is incident upon it. Each field is represent-ed by Eq. (1), where c,(t) is a real Gaussian variable forthe real Gaussian field, and a complex variable for thethermal field. In the latter case we represent e(t) ass'(t)+ iE"(t), where c'(t) and E",(t) are each real Gaussianprocesses, independent of each other but of the samespectral density. After being transmitted by the inter-ferometer, the spectral density of E(t) is modified, but

and when the real Gaussian field is incident,

RI,RGF(r) RI, th(r)+

I & E,(t+r)e, (t) & I'/& le(t) I'& . (A6)

Equation (A4) represents a significant simplification overthe procedure outlined in Ref. [8]. Each of the second-order correlation functions in (A4) involve a product oftwo infinite summations of the field given by (A2). Thissummation can be simplified by regrouping terms accord-ing to their order in R, and the input field amplitudecorrelation functions evaluated using &e'(t+r)s(t)&/

3228 G. KLIMECK, D. S. ELLIOTT, AND M. W. HAMILTON

~e(t)~ =e ~'. After contracting the resulting summa-tions we find the following expressions:

( Eo (t +r)Eo(t) )( IE(t) I')

—b7.(1—R)

(1 Rebr'+i5)(1 R br' ——i5)

(Re—b~'+i5)M+1 br+(1—Re ' )(1—R )

( Eo(t +v )Eo( t) )( l.(t) ~')

—b7-(1—R )2

( 1 Rebr' i5—)( 1 Re b~' —l5)

e e—b~' i5)—M+1 br

(R b~ —i'5)M+1 —b~

( 1 R bT i5)( 1 R 2e —2i5) (A8)

(Reb ~'+i 5 )M + 1 —br

(1—Re +' )(1—R )(A7)

M in these expressions is the integer value of r/~'. In theusual case of bandwidths much smaller than the FSR ofthe cavity, M can be substituted by ~/~', leading to thepower spectrum in the text, Eq. (3).

[1]C. Weiman and C. Tanner (private communication).[2] L. Hollberg (private communication).[3]Th. Haslwanter, H. Ritsch, J. Cooper, and P. Zoller, Phys.Rev. A 38, 5652 (1988).

[4] K. Rzazewski, B.Stone, and M. Wilkens, Phys. Rev. A 40,2788 (1989).

[5]H. Ritsch, P. Zoller, and J. Cooper, Phys. Rev. A 41,2653 (1990).

[6] M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D.S. Elliott, H. Ritsch, and P. Zoller, Phys. Rev. Lett. 64,

1346 (1990).[7] M. H. Anderson, R. D. Jones, J. Cooper, S. J. Smith, D.S. Elliott, H. Ritsch, and P. Zoller, Phys. Rev. A 42, 6690(1990).

[8]B. A. Ferguson and D. S. Elliott, Phys. Rev. A 41, 6183(1990).

[9] Cheng Xie, G. Klimeck, and D. S. Elliott, Phys. Rev. A41, 6376 (1990).

[10]J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice Hall,Englewood Cliffs, NJ, 1989), pp. 134 and 135.