measurements of first-passage-time distributions in laser transients near threshold

Vol. 5, No. 5/May 1988/J. Opt. Soc. Am. B 1011

Measurements of first-passage-time distributions in lasertransients near threshold

M. R. Young* and Surendra Singh

Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701

Received September 15, 1987; accepted November 3, 1987

The first-passage-time (FPT) problem for the time evolution of the optical field produced by a Q-switched laser hasbeen investigated theoretically and experimentally in the neighborhood of the laser threshold. Experiments wereperformed on a He-Ne laser operating near threshold in its steady state. Measurements for several operatingpoints were made. Q switching was achieved with the help of an intracavity acousto-optic modulator. Themeasured FPT distributions are compared with the approximate analytic results for the FPT distribution and thenumerical solutions of the laser Fokker-Planck equation based on the van der Pol oscillator model of the laser.

The threshold instability in the single-mode laser is an ex-ample of a nonequilibrium phase transition in which laserfluctuations undergo a smooth transition from a thermalstate below threshold to a coherent state above threshold.These fluctuations are a manifestation of the microscopicquantum noise that is due to spontaneous emission. Onlyfully quantum-mechanical treatments or semiclassicaltreatments based on Langevin equations that include quan-tum noise provide an adequate theoretical framework fordescribing the transition. 2

-4 The predictions of these treat-

ments have been tested and confirmed in many experi-ments5-"1 devoted to the study of the fluctuations in thelaser. Some of the most sensitive tests of the theoreticalpredictions have come from the investigations of the tran-sient characteristics of the laser performed by Arecchi andco-workers 7 and by Meltzer and Mandel.9 These experi-ments investigated the variation of the photoelectric count-ing probability with time following the turn-on of the laserby measuring the light intensity at a fixed time after thelaser was turned on. A complementary approach to thisproblem is to study the time that it takes the light intensityto reach a certain reference intensity after the laser is turnedon. The time at which the first passage of the laser intensitythrough this reference value occurs is called the first-passagetime (FPT).12 This time is expected to fluctuate, reflectingthe underlying quantum noise. The FPT measurementsprovide important new information on the role of noise inthe dynamics of the laser. For example, if the quantumnoise is important only during the early stages of laser dy-namics, the FPT variance is expected to be flat as a functionof the reference intensity. 3 -15 This type of information isnot available from intensity-versus-time measurements. Ifthe noise is important throughout the entire evolution to-ward the steady state, no such behavior is to be expected.

It is easier to treat the transient FPT problem theoretical-ly for a laser that operates far above threshold in its steadystate. In this regime the concepts of the decay of an unsta-ble state, scaling, and asymptotic approximation (low-noiselimit) can be fruitfully employed to derive analytic resultsfor the FPT distribution.' 4"5 When the laser's steady statelies in the threshold region, the concept of the decay of an

unstable state is not useful. The so-called states of the laserare not well defined because of the presence of relativelystrong quantum noise. In this regime noise is importantthroughout the entire evolution, and it is more appropriateto consider the growth of the laser's electric field amplitudeas a random walk under the influence of quantum noise.Direct evidence for this random walk was provided recentlyby means of the FPT measurements.' 6 An analytic treat-ment of the FPT problem appears difficult to achieve in thethreshold region. Arecchi and co-workers' 3 have presentedan approach based on the moments of the FPT distributionthat can be expressed in terms of a series of nested integrals.In practice these expressions become increasingly difficultto evaluate as moments higher than the second are consid-ered. Arecchi and co-workers tested the expressions for thefirst two moments by the FPT measurements of an electron-ic oscillator. FPT measurements in the scaling regime havebeen carried out by Roy and co-workers' 7 on a dye laseroperating far above threshold in the presence of strong ex-ternal fluctuations. For a laser operating near thresholdthere have been no reported FPT measurements of its tran-sient characteristics. In this paper we report on such mea-surements. Since the laser threshold is likened to a second-order phase transition,' these dynamical studies also providean example of the critical dynamics of a nonequilibriumsystem.

We begin by outlining the theoretical approach to theFPT problem during the turn-on of a laser near threshold.We then discuss various analytic approximations to the FPTprobability density and a method to compute it numerically.These results are then compared with the FPT measure-ments of the transients in a He-Ne laser operating nearthreshold. The laser was Q switched and was controlled tohave its steady state in the region of threshold.

FIRST-PASSAGE-TIME PROBLEM

The scaled slowly varying complex electric field amplitudeE(t) of the single-mode laser obeys the following equation ofmotion

2-

4:

0740-3224/88/051011-09$02.00 ©9 1988 Optical Society of America

M. R. Young and S. Singh

1012 J. Opt. Soc. Am. B/Vol. 5, No. 5/May 1988

E = E(a - E12) (1)

on resonance. Here a is the so-called laser pump parameter,which in terms of the gain per pass A and the loss per pass Ccan be expressed as

a = C-) 'nO, (2)

with no being the mean number of photons inside the lasercavity at the threshold of oscillation. 8 This number is oforder 104. The parameter a is negative below and positiveabove threshold, which occurs at a = 0 (A = C). q(t) repre-sents quantum noise due to spontaneous emission. Equa-tion (1) holds at least for laser pump parameters a < 100. Inthe region of threshold, q(t) may be taken to be a delta-correlated Gaussian random process with zero mean,

(q(t)) 0,

(q*(tj)q(t2 )) = 46(t - t2 ). (3)

Because of the presence of quantum noise in the equation ofmotion, the complex electric field amplitude E(t) becomes arandom process. Instead of the nonlinear stochastic differ-ential Eq. (2), it is convenient to deal with the Fokker-Planck equation for the probability density P(E, t) for thefield amplitude E(t) to be characterized by E at time t.19

This equation is

P(E, t) =- E(a - E12),P(E, t) + 2 a (E, t) + c.c.

(4)In the FPT problem discussed here the phase of the elec-

tric field plays no role. The equation of motion for theprobability density p(I, t) of the light intensity I = E*E isthen obtained from Eq. (4) to be

p(I, t) = -2 a [I(a - I) + 2]p(I, t) + 4 d2 p(I, t). (5)ai au2

We are now ready to discuss the FPT problem for the laserturn-on.

Let T - T(Ir, Io) denote the time for the light intensity (t)to reach a certain reference value 1r for the first time startingat (t) = Io at t = to. Then T is a random variable whoseprobability density is related to the conditional probabilitydensity G(I, tIo, to) for the light intensity (t) to have a valuebetween I and I + dI at time t, given that it had a value Io attime to. G(I, tIo, to) satisfies the initial condition

means that b(Ir, t11o, to) is the probability of the FPT T > t -to, so the FPT probability density P(T11o) - P(Ir, T11o, 0) isgiven by

P(TV ) =T a Ir to + T110, to)

a IG= -~ j dIG(I, t+ T110, t). (9

Equation (9) expresses P(T) in terms of G(I, tIo, to). FromEqs. (6) and (7) we note that G(I, to, to) is the Greenfunction solution to Eq. (5). To find this solution we writethe general solution to Eq. (5) in the form

p(I, t) = 1 C. exp[-/ 2U(I)]g(I)exp(-Xnt),n=O

(10)

where CQ1 are constants to be determined by the initialcondition and R1n} and {gn, respectively, are the eigenvaluesand the eigenfunctions of a certain Sturm-Liouville problemto be specified shortly. The function U(I) is given by

U() = -1/2a + 1/4 I2, (11)

in terms of which the stationary-state solution ps(I) to Eq.(5) may be written as

PU) = const. X exp[-U(I)]. (12)

Substituting Eq. (10) into Eq. (5), we find the equation forgn(I) to be

d d F/1 2 \4 I g(I) + - (I (aI) + a- 2I)] (I) =0.dI d I L 4 u J(13)

The boundary condition on gn(I) follows from Eq. (7) to be

gn(Ir) = 0. (14)

Equations (13) and (14) then define a Sturm-Liouville prob-lem whose eigenfunctions {gn} form an orthonormal completeset. With g(I) normalized according to

(15)t o of gn(I)my)dI = nsa

the completeness f 19n) may be expressed as

(16)

G(I, tolIo, to) = B(I -I)

and the boundary condition

G(Ir, t1lo, to) = 0.

(6) From Eqs. (10), (14), and (16) we find that both conditions(6) and (7) will be satisfied by solution (10) if we choose

(7)

Boundary condition (7) is typical of nonstationary problemssuch as those involving the FPT.12 In terms of G(I, t1lo, to)the probability (Ir, t Io, to) that I(t) remains less than I attime t will be

'r to, to) = | G(I, tIo, t)dI. (8)

This is also the probability that a first passage of (t)through 1r has not occurred in the interval [t, to]. This

Cn = gn(I0)exp[/ 2U(I0 )]exp(Xt 0 ). (17)

The corresponding solution G(I, tIo, to), therefore, has theform

G(I, tIo, to) = [ t))] gn(I)gn(IO)exp[-Xn(t -to)]

t > to. (18)

From Eqs. (9) and (18) it follows that the probability densityfor the first passage of I(t) through I to occur between T andT + dT, given that it started initially at Io, is given by


(9)

gn U)gn W) = 6 U - F) -


g,(I0 )P(TIO) = E An 2

X exp(-XnT) fIO dILp(I)]"/2gn(I). (19)

For the turn-on of a laser the initial intensity I0 = 0, so theFPT probability density P(T) P(TI Io = 0) is given by

P(T) Xngn(O)exp(-XnT) J dI exp[-/2U(I)]gn(I),n=o

(20)

where we have used Eqs. (11) and (12). The eigenfunctionsgn and the eigenvalues Xn have to be evaluated numerically.The eigenvalue problem defined by Eqs. (13) and (14) can becast in a more convenient form by the change of variable I =

r2 followed by the substitution

gn(r 2) = r-'1 2 (r), (21)

where in(r) satisfies a one-dimensional Schr6dinger equa-tion

dr 2 + [Xn - V(r)]On(r) = 0, (22)

with

V(r) = a-2r2 + (a-r2)2 1 (23)4 4r2

The boundary condition on 4in(r) is

An(Ci = 0. (24)

In addition to this, note that 4'(0) = 0, so that g (0) is finite.The new eigenfunctions are normalized according to

¢,~r)¢ (r)dr (25)

Several conclusions can be drawn regarding the FPT proba-bility density without solving the full eigenvalue problem.For large values of T only the lowest few eigenvalues willcontribute, since the sequence of eigenvalues IX,,} is an as-cending sequence, X0 < X < X2 ... . The large T behavior ofP(T) will, therefore, require a determination of only the firstfew eigenvalues and eigenfunctions. For XT >> 1 a singleexponential will dominate, and we can write

P(T) [xogo(0) J dI exp[-1/ 2U(I)]g0(I)]exp(-XoT). (26)

For small values of T, however, the problem is much morecomplicated. In this case XT << 1, and, since {Xn} form anincreasing sequence, the series in Eq. (20) may not be arapidly convergent series. Furthermore, not all the termshave the same sign. This means that a large number ofeigenvalues and eigenfunctions must be computed withgreat accuracy in order to evaluate the small-time form ofP(T). In fact, attempts to evaluate Eq. (20) did not producesensible results. Fortunately, it is possible to derive ananalytic approximation to the short-time behavior of P(T)following the work of Young and Singh.15 This approxima-tion is based on the fact that the extrema of U(r) [U'(r) = 0, r

= 0, r = , a > 0] correspond to minima of V(r) at r = 0 and r_ va_ separated by a local maximum at r a. Then forlarge a we can approximate V(r), in the vicinity of r = 0, by

V(r) = a + '/ 4a2r 2

-r (27)

Note that this approximation is exact for the Ornstein-Uhlenbeck process.19 For the laser problem this is expectedto be a good approximation so long as we do not considergrowth of laser intensity to large reference values. With thepotential given by Eq. (27), the eigenfunctions and the ei-genvalues of Eq. (22) are20

An(r) = (ar)/2 exp(-'/ar2 )Ln(/ 2ar2),

Xn = (2n + 2)a.

(28)

(29)

Here Ln(x) are the Laguerre polynomials. Note that theboundary condition An(4lr) = 0 [Eq. (23)] is not strictlysatisfied. However, for large a and Ir a/3, Eqs. (28) and(29) are expected to provide good approximation to the exactsolution since in that case 4ln(Clr) - 0[exp(-a'/6)]. UsingEqs. (21), (28), and (29) in Eq. (20), and carrying out thesummation, we obtain

a2Irexp(-2aT) [ 1 aIr exp(-2aT) 1

PTr) = [1 - exp(-2aT)]2 exPL 2 1 - exp(-2aT)

(30)

This equation is a special case of the result derived in Ref.15. For short times 2aT << 1 we obtain

P(T) = exp(-) (31)

which is independent of a. This result, although derivedfrom an expression valid for large a, is nevertheless valid as ashort-time approximation even for small values of a or thelarge-noise limit. This means that it is also valid for laseroperation near threshold. From Eq. (31) we also find thatP(O) = if I, $- 0. This result is difficult to establishnumerically. Physically this result is to be expected sinceI(t) is a smooth process and it will take a finite amount oftime in order for it to reach a finite reference value. Weprovide experimental and numerical evidence for the valid-ity of the short-time approximation in the threshold region.

Although the eigenfunction expansion is useful in discuss-ing the long-time and the short-time behavior of P(T), itbecomes cumbersome when a large number of terms contrib-ute to the series. It is, however, possible to compute P(T)directly. To this end we note that P(TI 0) satisfies theKolmogorov equation'2

a a __

d-TP(TIIo) = {[210 (a - IO) + 4] + 41o d2 P(TIIO),

(32)

which is the adjoint of Eq. (5). This equation has to besolved subject to the boundary conditions

P(T1I0 = I,) = M,

aP(TIIo) = 0.

dI0 Io=0

(33)

(34)

The first of these conditions simply states that if the intensi-



ty starts at the reference value I, its first passage occursimmediately. The second condition expresses the fact thatthe intensity cannot take on negative values. Once itreaches the value zero it is reflected back to positive values.Equation (32) is solved backward from the final value I, ofthe light intensity to its initial value I0. For this reason thisequation is also known as the backward Fokker-Planckequation. For the laser turn-on the initial value Io = 0, andthis equation can be solved numerically by using an explicitfinite-difference method. 2

1 Equation (32) can also be usedto derive the expressions for the moments of P(T). Expres-sions for the mean FPT, (T), and its variance, ((AT) 2), canbe found in Ref. 16, where they were compared with experi-mental results. We will not reproduce them here.

Ir 1

7

[1T;-<L

START

STOP

FPT COUNT GATE

EXPERIMENTAL APPARATUS

The experiments were performed on a single-mode stand-ing-wave He-Ne laser operating at X = 633 nm close to itsthreshold in the steady state. (See Fig. 1.) The laser wascontained within a dust-free temperature-controlled enclo-sure. This enclosure was placed in a sandbox mounted on avibration isolated table. The cavity was a three-Invar-rodstructure, and mirror mounts had no movable parts. One ofthe mirrors was mounted on a piezoelectrically driven trans-ducer, which allowed cavity-mode frequency to be tuned tothe center of the atomic gain profile. The laser was found tobe naturally quite stable once thermal equilibrium had beenreached. The operating point of the laser was controlled bycomparing the output of a monitor photomultiplier tube(PMT), illuminated by a fixed fraction of the output lightintensity, with some preset voltage level inside an electronicamplifier. The difference signal was amplified and appliedto a piezoelectric transducer which controlled the movement

-of a sharp knife edge in and out of the beam inside the lasercavity. With this arrangement the operating point of thelaser could be held constant to better than 1% throughoutthe threshold region.

The laser was turned on and off by an intracavity acousto-optic modulator (AOM). The time required for the modula-tor to switch from a high-loss (80% diffraction efficiency) toa low-loss (<2% insertion loss) state was less than 60 nsec,

HE-NEK PLFILTER 1 POL. PLASMA TUBE TUNING

IF7, . .___ PZT-- I I|' IHEI ril . i RED II IFILTER CONTROLEI MONITORt I PZT

_-J GATI)iSCLFig. 1. Outline of the experimental setup. POL, polarizer; PZT,piezoelectric transducer; PET/CBM, PET microcomputer (Com-modore Business Machines).

Fig. 2. Light-intensity trajectories and the measurement se-quence. The vertical scale is arbitrary, and the horizontal scale is 50,usec/division.

which was negligible compared with the typical rise time ofthe laser (50 /Asec) near threshold. The laser was turnedoff for approximately 0.6 msec and on for about 6 msec. Asthe laser is turned on a trigger pulse marks the beginning ofthe turn-on time. This trigger pulse opens a gate that allowspulses from a clock to reach a scaler. The laser intensitybuilds up from spontaneous-emission noise. The growth ofthe light intensity is monitored by a low-gain PMT (rise time-4 nsec) whose output is proportional to the light intensityI(t). The output signal from the PMT is fed to a discrimina-tor, where it is compared with a preset threshold voltage.When the PMT signal crosses the discriminator thresholdan output pulse is produced that stops the gate. The dis-criminator level determines the reference intensity 'r, Thenumber of clock pulses stored in the scaler then gives thepassage time for the light intensity to grow from a value zeroto some reference value Ir in units of clock period. Thisnumber is transferred to a 246-channel memory inside acomputer, and the scaler is cleared. The process was repeat-ed 10,240 times, and a histogram of the passage times wasbuilt up that became a measure of the FPT probabilitydensity P(T).

The steady-state intensity of the laser is stabilized by thefeedback loop. This is to ensure that each time the laser isturned on it evolves toward the same steady state. In thesemeasurements the pumping in the discharge tube was keptconstant, so the population inversion was maintained in itssteady state. Also, since the rise time of the AOM wasnegligible compared with the typical growth time of the laserintensity, we observed the free evolution of the light intensi-ty determined only by the steady-state net gain or the pumpparameter. The steady-state pump parameter was variedfrom a = -1 to a = 8. It is important in these measurementsto turn the laser completely off before turning it on. Thiswas ensured by aligning the AOM inside the cavity for itsmaximum diffraction efficiency (80%) and by waiting (0.6msec) thousands of cavity decay times (10-7 sec) beforeturning the laser on. The steady-state losses per pass C ofthe laser near threshold were about 3% (1% mirror transmis-sion and <2% AOM inversion loss). Since near threshold A/C 1, we estimate from Eq. (2) that in the off state, with C 83%, the laser pump parameter was a -8000, with themean photon number (n) - 10-2. This means that with

l



good probability each time the laser was tuned on it startedwith zero intensity.

The data-collection sequence was controlled by a micro-computer. Data were taken for several different operatingpoints of the laser near threshold, and for each operatingpoint several different reference intensities were used.These results are compared with the theoretical predictionsin the following section.

8 a=1.42

"~~~~~~~ J I= 1

6

4 4

2

10

EXPERIMENTAL RESULTS

A typical laser-light intensity trajectory is shown in Fig. 2.The trajectory was obtained from an oscilloscope trace of thePMT output voltage as a function of time. Since the PMToutput voltage is proportional to the light intensity, thegrowth of the PMT voltage reflects the buildup of laserintensity after it is turned on. In order to compare the

12000 400 800 1200 400 800T(,usec) T(Qsec)

8 __ +

6 +

00 400 800 1200 400 800 1200

T(,sec) T(psec)Fig. 3. Variation of the FPT probability density with laser pump parameter a for a fixed reference intensity I, = 1. The solid curves aretheoretical obtained by solving Eq. (32) numerically. The dashed curves are derived from Eq. (30).



0 400 800 1200 400 800T(usec) T(,sec)

400 800 1200 400 800T(Psec) T(,sec)

Fig. 4. Variation of the FPT probability density with laser pump parameter a for a fixed reference intensity Ir = 4.

measured and the theoretically predicted FPT probabilitydensities, we need two scale factors, one that converts themeasured intensity (in volts) to the dimensionless intensityused in the theory and the other that converts the measuredvalues of T (in microseconds) to the natural dimensionlessvalues of the theory. These scale factors were determinedby plotting (T) versus the steady-state intensity (I) for onereference intensity on a double logarithmic graph. A changein the scale factor corresponds to a translation along thecorresponding axis. The time-scale factor was determined

to be 800 i 20 ,tsec. An independent determination of theintensity scale factor was also made by plotting the steady-state normalized intensity variance against the steady-statemean light intensity. For a laser operating at line centerthis provides a unique scale factor for the light intensity.This was found to be consistent with the value determinedby the time measurements.

Figure 3 shows the measured forms of P(T) for I = 1 forseveral different values of the pump parameter near thresh-old. Superimposed upon the experimental points are the

8

6

4

2

0

8

6

4

2

0

1200

12000



theoretical predictions. The dashed curves represent thepredictions of Eq. (30), and the continuous curves representthe numerical solution of Eq. (32). It will be seen that thecontinuous curves always agree well with the measured dis-tributions. The dashed curves reproduce the short-timebehavior quite well, but the peaks and the tails of the distri-bution are not reproduced by this approximate solution.Figure 4 illustrates the same for a higher reference intensity,Ir = 4. It is clear that Eq. (30) can be taken to represent onlythe short-time behavior near threshold. In general the fullFPT problem must be considered in order to reproduceexperimental results. Equation (30) can, however, be con-

8_

6

4

2 N

sidered a good short-time approximation. We have alsoplotted the short-time result [Eq. (31)] in Fig. 5. Again wefind that Eqs. (30) and (31) reproduce well the short-timebehavior dominated by noise.

From Figs. 3 and 4 we can also see a progressive shift of thedistribution to the left as the pump parameter is increasedfor a fixed reference intensity. This is illustrated moreclearly in Fig. 6, where we have shown P(T) distributions forseveral different values of the pump parameter and a fixedreference intensity Ir = 2. This type of behavior is to beexpected since as the pump parameter becomes large theimportance of noise decreases. At low pump parameters the

T(Iisec)

Fig. 5. An example of the FPT distribution below threshold. The top curve represents numerical solution to Eq. (32). The middle and thelower curves are derived from Eq. (30) and (31), respectively.

8

6

4

2

0T(,usec)

Fig. 6. Forms of the FPT probability density for a fixed reference intensity Ir = 2 and several different operating points of the laser. Solidcurves are numerical solution to Eq. (32).



8

6

4

2

0 0


T(1usec)

Fig. 7. Forms of the FPT probability density for a fixed pump parameter a = 3.8 and several different reference intensities.

time evolution of the light intensity is dominated by noisethat is rapidly fluctuating, and it takes a long time for (t) toreach a macroscopic value. At high pump parameters, oncenoise initiates the buildup, amplification takes over andcauses (t) to reach macroscopic values quickly. Figure 7shows the behavior of P(T) for a fixed pump parameter andseveral values of the reference intensity. In this case we findthat P(T) shifts to the right as the reference intensity israised. This shift occurs for two reasons. First, it takesprogressively longer for the laser to reach the reference in-tensity as its value is raised. Second, for sufficiently highreference intensity, systematic forces tend to oppose thegrowth of the light intensity to such values. Note that Ir = 4corresponds to a value beyond the steady-state mean inten-sity. Predictions of Eq. (33) are shown by continuous curvesand are found to be in good agreement with the data. In Fig.8 we illustrate the behavior of the mean FPT as a function of

the reference intensity for two pump parameter values. Themean values reported in Ref. 16 were found to be systemati-cally higher by about 15 sec. This was in part due to adelay between the laser turn-on and the beginning of theFPT count gate and in part due to an error in the calibrationof the zero channel of the FPT histogram. This correctionaffects only the mean FPT. It leaves the FPT variance (andhigher-order moments) unaffected, as shown in Fig. 9 as afunction of the reference intensity for two operating pointsof the laser. The values of the reference intensity are uncer-tain by about 2-3%. The curves represent theoretical pre-dictions derived from Ref. 16. The absence of a flat regionin the variance-versus-reference intensity shows that, unlikethe scaling regime,' 3 laser dynamics near threshold is affect-ed by quantum noise throughout its evolution and not just inthe early stages of its evolution. Once again we find goodagreement between theory and experiment.

6x

4

AA

V

Fig. 8. Variation of the mean FPT with reference intensity for a =3.8 and a = 7.6.

2x

I,

Fig. 9. Variation of the FPT variance with reference intensity for a= 3.8 and a = 7.6.

1 200


CONCLUSIONS

We have examined the FPT problem for laser transientsnear threshold using two different approaches. The short-time and long-time behavior of the FPT probability densityhave been discussed by an eigenfunction expansion method.Exact solutions have been computed numerically by usingthe backward Fokker-Planck equation. These predictionswere confirmed by the FPT measurements of the transientsin a Q-switched He-Ne laser operating near threshold.These measurements show that, near threshold, quantumnoise dominates the entire dynamics of the growth of laserradiation. From the discussion above it is clear that thetransient dynamics of the laser involves all the eigenvaluesand the eigenfunctions, not just the lowest few as is the casewith the steady-state operation of the laser. These FPTmeasurements of laser transients, therefore, provide a sensi-tive test of laser theory, and they complement the earlierintensity measurements of laser transients in the thresholdregion by Arecchi and DeGiorgio7 and by Meltzer and Man-del.9

ACKNOWLEDGMENTS

We are grateful to R. Gupta for the loan of some equipmentand to G. Broggi and L. A. Lugiato for sending us a computercode for solving one-dimensional Fokker-Planck equations.This research was supported by the National Science Foun-dation and by a Joseph H. DeFrees grant from ResearchCorporation.

* Present address, Department of Physics, King's College,The Strand, London, UK.

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measurements of first-passage-time distributions in laser transients near threshold

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