picosecond optical limiting in reverse saturable absorbers: a theoretical and experimental study

94 J. Opt. Soc. Am. B/Vol. 19, No. 1 /January 2002 Lepkowicz et al.

Picosecond optical limiting in reversesaturable absorbers:

a theoretical and experimental study

Richard Lepkowicz, Andrey Kobyakov, David J. Hagan, and Eric W. Van Stryland

School of Optics/Center for Research and Education in Optics and Lasers, University of Central Florida, 4000Central Florida Boulevard, Orlando, Florida 32816-2700

Received February 23, 2001; revised manuscript received June 29, 2001

We theoretically and experimentally study absorption of picosecond laser pulses in materials described by afour-level system that exhibit reverse saturable absorption (RSA). Using an approximate solution to the rateequations, we derive, analyze, and verify, numerically and experimentally, a single dynamical equation for thespatial evolution of the pulse fluence that includes both the rate equations and the propagation equation.This analytical approach considerably simplifies the study of optical limiting with picosecond pulses and helpsto predict the behavior of the nonlinear transmittance, the level of output signal clamping, and a possible turn-over from RSA to saturable absorption that restricts the performance of optical limiters based on RSA. Theresults obtained can also be used to characterize RSA materials by the pump–probe technique. © 2002 Op-tical Society of America

OCIS codes: 190.4180, 140.3360, 190.4710.

1. INTRODUCTIONMaterials that exhibit reverse saturable absorption(RSA), i.e., those with an excited-state absorption crosssection that is larger than that of the ground state, havedrawn a great deal of interest over the past severalyears.1–7 One of the most promising applications of thesematerials is their potential use as passive optical limiters,i.e., as devices designed to protect sensitive optical ele-ments and eyes from laser-induced damage. For an over-view of passive optical limiting see, e.g., Refs. 8 and 9.The nonlinear absorption of RSA materials is exploited todecrease the transmittance of a material at high input en-ergy while the material is kept transparent at low inputenergy as an ideal optical limiter requires. RSA has beenobserved in many organic materials (for an overview see,e.g., Ref. 1), and efficient optical limiting has been dem-onstrated in several materials by use of various opticalgeometries.10–14

The key quantities that determine the absorption coef-ficient of a material are the time-dependent populationdensities of the ground and excited states, which, in turn,depend on the parameters of the nonlinear medium andthe laser pulse characteristics. The population dynamicsare described by multilevel rate equations.1,15–23 For pi-cosecond input pulses, a three-level system (singlet bandsonly) allows RSA to be modeled but often fails to matchdata taken at high input fluences. The addition of ahigher-lying fourth level7,18,22 allows for a better matchwith data for many materials. The four-level model thatwe use is shown in Fig. 1.

Finding a general analytical solution to the system ofrate and propagation equations represents a fairly diffi-cult problem. Thus the solution is usually obtained nu-merically for a particular set of parameters.16–19 How-ever, it is still difficult to gain a basic insight into the

0740-3224/2002/010094-08$15.00 ©

properties of the nonlinear absorber. For example, to de-termine the dependence of the sample transmittance onthe input fluence with all other parameters fixed, one hasto perform a full integration of the rate equations and thepropagation equation for each value of the input fluence.To study the effects of any other parameter (e.g., pulseshape, width, lifetimes, absorption cross sections, samplethickness), one has to repeat the entire set of calculations.Even with contemporary computing capabilities, this taskis highly time-consuming.

Alternatively, an assumption of so-called fast absorb-ers1 (for which the pulse length is much longer than thelifetime of each excited state) allows for an analytical so-lution of the problem. Mathematically, this means thatthe differential rate equations reduce to a set of algebraicequations that can easily be solved.16–18,22 However, thissteady-state, or stationary, solution is not always appli-cable because it assumes long pulses, so a triplet excitedstate has to be taken into account for manymaterials.1,2,6,15–17,19,20,21,23

Rather than using the steady-state approach,16–18,22 inthe research reported in this paper we performed a dy-namical analysis; we show here that the stationary solu-tion to the rate equations gives only a rough estimate forthe absorption of short pulses. We show that the behav-ior of the limiting curve (dependence of the normalizedtransmittance on input fluence) can be accurately pre-dicted based on the solution of a single ordinary differen-tial equation (ODE). The analytical results will allow usto understand the effect of each individual parameter onthe absorptive properties of the material and accuratelypredict various phenomena, e.g., turnover from RSA tosaturable absorption (SA).16–18,22

The paper is organized as follows. In Section 2 we in-troduce rate equations and useful normalizations. In

2002 Optical Society of America

Lepkowicz et al. Vol. 19, No. 1 /January 2002 /J. Opt. Soc. Am. B 95

Section 3 we solve the rate equations for a rectangular(flat-topped) input pulse. We then use this solution tostudy the optical limiting properties of several materials.Particular emphasis is given to the effect of the turnoverfrom RSA to SA and to a comparison of the dynamical so-lution with predictions of the steady-state analysis. InSection 4 experimental verification of the theoreticalanalysis is presented for two organic dyes, a carbocyaninedye 1,18,3,3,38,38-hexamethylindotricarbocyanine iodide(HITCI; Refs. 7 and 18) and a polymethine dye (PD3).3,4

The chemical structure of PD3 is shown in Fig. 2. Sec-tion 5 concludes the paper.

2. BASIC EQUATIONS ANDNORMALIZATIONSThe model that we use to describe RSA is the singlet statefour-level structure depicted in Fig. 1. In as much as weare interested in picosecond excitation we can ignore thetriplet states because the intersystem crossing times aretypically much longer (.1 ns) than the pulse width.1 Wealso neglect the effects that are due to the vibronic sub-levels that have subpicosecond relaxation times. In ad-dition, the lifetime t32 is assumed to be much less thenthe pulse width, which results in negligible population inlevel 3. As a result, the rate equations can be written asfollows18,22:

Fig. 1. Schematic diagram of the four-level model of RSA withpicosecond pulse illumination; 0 is the ground state and 1–3 areexcited singlet states.

Fig. 2. Chemical structure of dye PD3.

]N0

]t5 2

s01I

\vN0 1

N1

t10,

]N1

]t5

s01I

\vN0 2

N1

t102

s12I

\vN1 1

N2

t21,

]N2

]t5

s12I

\vN1 2

N2

t21, (1)

where N0,1,2 are the level population densities that arefunctions of time t and propagation distance z, N0 1 N11 N2 5 N, I(z, t) is the irradiance of the incident pulse,and s01,s12,s23 and t10,t21,t32 are the corresponding ab-sorption cross sections and lifetimes (Fig. 1). The popu-lation densities determine the spatial pulse evolution bymeans of the propagation equation1,16–23

]I

]z5 2I~ s01N0 1 s12 N1 1 s23 N2!. (2)

It is easier to deal with Eqs. (1) both numerically andanalytically if we recast them into a dimensionless form.First we represent the input irradiance as I(0, t)5 I0f(t), where I0 is the peak irradiance and f(t) is a di-mensionless function with unit amplitude that reflectsthe temporal shape of the pulse (e.g., flat-topped, Gauss-ian, hyperbolic secant). Next we introduce a dimension-less variable that is proportional to the total fluence of thepulse A 5 s01I0tp /\v and scale time with pulse widthtp , i.e., T 5 t/tp . We also introduce absorption cross-section ratios a 5 s12 /s01 and b 5 s23 /s01 and dimen-sionless ratios T10 5 tp /t10 and T21 5 tp /t21 . Finally,we normalize the level population densities by the totalpopulation, nj 5 Nj/N, where j goes from 0 to 2. Substi-tuting the above definitions into rate equations (1), we ar-rive at the following set of dimensionless rate equations:

]n0

]T5 2Fn0 1 T10n1 ,

]n1

]T5 Fn0 2 ~aF 1 T10!n1 1 T21n2 ,

]n2

]T5 aFn1 2 T21n2 , (3)

where F 5 Af(T). Further, we define a normalized vari-able that represents linear absorption Z 5 s01Nz,23 andwe express the propagation equation as

]F

]Z5 2F~n0 1 an1 1 bn2!. (4)

In Section 3 we approximately solve the system of partialdifferential equations (PDEs) (3) and (4) and verify the re-sults by direct numerical integration of those equations.

3. SOLUTION TO THE RATE EQUATIONSAND NONLINEAR ABSORPTION OFTHE RSA MATERIALTo solve the system of Eqs. (3) and (4) we follow the rea-soning of Refs. 1, 19–21 and treat the rate and propaga-tion equations separately. We can solve the former by as-


suming constant irradiances, and we substitute theirsolution into propagation equation (4). Subsequent inte-gration of both sides of the propagation equation with re-spect to time will result in the spatial evolution equationfor the pulse fluence, F 5 *2`

1` FdT.19–21

It can be expected that, as in the five-level model usedfor describing the propagation of nanosecond pulses inRSA materials,1 the functional dependence of the popula-tion dynamics on the input fluence does not dependstrongly on the pulse shape if the medium is excited byshort input pulses.23 Below, we shall show that the flat-topped pulse approximation I(t) 5 I0 , 0 , t , tp (Refs.1, 20, and 21), provides reasonable accuracy for propaga-tion of realistic laser pulses in many RSA materials, whileit greatly simplifies the analysis. In our normalizedunits, this pulse shape results in f(T) [ 1, 0 , T , 1,and Eqs. (3) become a system of ODEs with constant co-efficients that are easily solvable.

The standard procedure consists in equating the corre-sponding determinant of Eqs. (3) to zero to find the eigen-values. As the order of system (3) equals 3, we obtain athird-order polynomial whose roots can be found explic-itly. One of the roots is equal to zero, and the other twoare

l6 5 ~1/2!@2~a 1 e 1 q 1 1 ! 6 AD# , 0, (5)

where we have denoted

D 5 a2 1 2a~q 2 1 1 e! 1 ~q 2 1 2 e!2, (6)

e 5 T10 /A, and q 5 T21 /A.With eigenvalues known, one can easily find the corre-

sponding eigenvectors and, using the initial conditions forthe medium before excitation, n0 5 1, n1 5 0, and n25 0, obtain the solution to system (3) as

S n0

n1

n2

D 5 C3q 2

~l2 1 q !l1

AD~l2 1 1 !

~l1 1 q !l2

AD~l1 1 1 !

q

e2

~l2 1 q !l1

ADe

~l1 1 q !l2

ADe

a

e2

al1

ADe

al2

ADe

43F 1

exp~l2AT !

exp~l1AT !G , (7)

where

C 5~l1 1 1 !~l2 1 1 !

l1l2~q 2 1 !. (8)

The solution remains continuous with T21 5 A, i.e., q5 1, because both the numerator and the denominator inEq. (8) approach zero. In the limit q 5 1, Eq. (7) simpli-fies to

S n0

n1

n2

D 5 3e

a 1 e 1 1

e

~a 1 e 1 1 !~a 1 e!

a

a 1 e

1

a 1 e 1 12

1

a 1 e 1 10

a

a 1 e 1 1

a

~a 1 e 1 1 !~a 1 e!2

a

a 1 e

43 F 1

exp@2~a 1 e 1 1 !AT#exp~2AT !

G .

To study the spatial evolution of the pulse, we mustsubstitute solution (7) into the propagation equation [Eq.(4)]. The resultant equation is too cumbersome to workwith analytically. However, we can significantly simplifythe exact solution to rate equations (7) without any sub-stantial reduction in accuracy by examining typical RSAmaterial parameters. Indeed, for a typical RSA materialthe lifetime of the first excited state, t10 , is of the order ofa nanosecond, whereas that of t21 is severalpicoseconds.3,7,18 This means that the parameter e, 1022 is small compared with both q and a (a . 1 forRSA materials). Hence we can make a series expansionwith respect to e and write Eqs. (5), (6), and (8) as

l1 ' 21 1 e1 2 q

a 1 q 2 1,

l2 ' 2a 2 q 2 ea

a 1 q 2 1, (9)

AD ' a 1 q 2 1 1 ea 2 q 1 1

a 1 q 2 1, (10)

C 'e

a 1 q, (11)

respectively. Substituting relations (9)–(11) into Eq. (7)results in the approximate solution to the rate equations.Furthermore, we take advantage of the fact that the so-lution found for the population densities has two quitedifferent characteristic times, which are proportional tothe eigenvalues l1 and l2 . As can be recognized fromEq. (9), ul2u @ ul1u, so one can neglect the transient pro-cesses that vary as ;exp(l2AT). As a result the approxi-mate solution for the population densities can be ex-pressed in the form

n0~T ! ' exp~2AT !, (12)

n1~T ! 'q

a 1 q1

1 2 q

a 1 q 2 1exp~2AT !,

(13)

n2~T ! 'a

a 1 q2

a

a 1 q 2 1exp~2AT !. (14)

To check the accuracy of the approximate solutions to therate equations we compare formulas (12)–(14) with theexact solutions, Eqs. (5)–(8). The results are shown inFig. 3 for two RSA materials, the carbocyanine dye HITCI(Ref. 7) and the polymethine dye PD3.3 It can be seen


clearly from Fig. 3 that formulas (12)–(14) accurately de-scribe the population dynamics. The slight initial devia-tion for n1 and n2 is due to the discarded terms that areproportional to ;exp(l2AT). It will not have a signifi-cant effect on the propagation equation because excited-state populations n1 and n2 are small for T , 0.05.

To validate the use of a flat-topped pulse we examinethe propagation of a pulse through the two RSA materialsfrom Fig. 3 and compare the output fluence as a functionof propagation distance for the flat-topped and Gaussianshapes. From Fig. 4 we can conclude that attenuation oftotal fluence F is nearly the same for both flat-topped andGaussian pulses. Thus, using a flat-topped shape in solv-ing the rate equations is appropriate and accurately pre-dicts the transmitted fluence of the corresponding shortpulses.

Integrating Eq. (4) with formulas (12)–(14), we obtainthe propagation equation for the fluence:

dF

dZ5 Fa

~b 2 a!F

~a 2 1 !F 1 T211 a 2 1G ~1 2 e2F!

2 aF1 1~b 2 a!F

aF 1 T21GF 5 2seff~F !F, (15)

Fig. 3. Population dynamics of two RSA materials excited by atemporal flat-topped pulse with a pulse width of tp 5 25 ps(FWHM). The results were obtained by use of the exact analyti-cal solution to the rate equations, [Eq. (7); solid curves] and theapproximate analytical solution [formulas (12)–(14); dashedcurves]. The parameters are (a) for HITCI: A 5 2, a 5 32, b5 0.1, t10 5 1.7 ns, and t21 5 10 ps and (b) for PD3: A 5 0.7,a 5 200, b 5 2, t10 5 1 ns, t21 5 1.9 ps. Each material has alinear transmittance of 0.905 (Z 5 0.1).

where F 5 *2`1` FdT and seff(F) 5 (1/F)*0

F(n0 1 an11 bn2)dF is the effective fluence cross section.19,23 Notethat this single differential equation now includes boththe set of rate equations and the propagation equation.Equation (15) cannot be solved analytically, but its nu-merical solution represents a much easier task than solu-tion of the full system [Eqs. (3) and (4)] for every value ofinput fluence. The results of numerical integration ofEq. (15) are shown in Fig. 4 by the solid curves.

We also computed limiting curves (transmittance ver-sus input fluence) for HITCI and PD3. We compared thenumerical solution to Eq. (15) with the full set of rate andpropagation equations (3) and (4) for a flat-topped and aGaussian input pulse. The solution to Eq. (15) almost co-incides with the numerical solution to the original systemwith the flat-topped input, and it is also close to the nu-merical solution for a corresponding (equal total fluenceand peak irradiance) Gaussian pulse (Fig. 5).

The important feature to notice in the limiting curves isthe effect of turnover from RSA to SA.18,22 As can be seenfrom Fig. 5, the turnover predicted by Eq. (15) occurs atslightly lower fluences than that given by the numericalsolution to the original system of Eqs. (3) and (4). Thus,our solution predicts a lower limit of the input fluencewhen this turnover occurs for a more realistic pulseshape. An approximate condition for the turnover point

Fig. 4. Transmittance (Fout /F in) as a function of normalizedpropagation distance Z 5 s01Nz for the dyes in Fig. 3: (a)HITCI, (b) PD3. Dotted and dashed curves, the numerical solu-tions to Eqs. (3) and (4) for a Gaussian (FWHM, tp 5 23.5 ps)and a flat-topped pulse shape, respectively. Solid curves, nu-merical solution to ODE (15).


(Fto) in terms of the material parameters can be obtainedas a solution to dseff(F)/dF 5 0.22 This criterion providesgood accuracy only for low linear losses (less than 10%)and can be expressed in terms of the transcendental equa-tion

a~b 2 a!T21F2~aF 2 F 1 T21!2

1 T21F~aF 1 T21!2~1 2 e2F!~a2 2 ab!

1 @~F 1 1 !e2F 2 1#~aF 2 F 1 T21!~aF 1 T21!2

3 @~2a 2 ab 2 1 !F 2 ~a 2 1 !T21# 5 0, (16)

whose numerical solution Fto gives the turnover fluence.To complete our analysis, we compare our results with

those obtained previously18,22 that exploit the steady-state solution to rate equations (3). Using the normaliza-tion procedure from Section 2, we arrive at the followingpropagation equation, obtained with the stationary popu-lation densities22:

dA

dZ5

2A~V 1 aA 1 baA2!

V 1 A 1 aA2, (17)

where A 5 s01I0t21 /\v 5 A/T21 and V 5 t21 /t10 . InFig. 6 we have plotted the limiting curve for HITCI calcu-lated from the dynamical propagation [Eq. (15)] and from

Fig. 5. Transmittance (Fout /F in) as a function of normalized in-put fluence for the dyes in Fig. 3: (a) HITCI, (b) PD3. Solidcurves, limiting curves calculated by use of the numerical solu-tion to ODE (15). Open and solid circles, limiting curves calcu-lated from the full system of PDEs (3) and (4) for flat-topped andGaussian pulses, respectively.

the steady state [Eq. (17)]. It is clear that the latter pre-dicts much stronger limiting and an even earlier turnoverfrom RSA to SA than that predicted by the dynamicalmodel [Eq. (15)]. As can be inferred from Fig. 6, theshape of the limiting curve predicted by the steady-statemodel differs strongly from the results of the dynamicalanalysis. It can be expected that the steady-state modelwill predict a much lower transmittance at higher inputfluences, but what is surprising is that the value of thefluence at the minimum in transmittance (the turnoverpoint) is in most cases an order of magnitude less thanthat given by the dynamical model.

4. EXPERIMENT AND DISCUSSIONTo confirm the theoretical predictions developed in Sec-tion 4 we performed optical limiting experiments with anactive–passive mode-locked Nd:YAG laser that operatesat a 10-Hz repetition rate. The fundamental wave of theNd:YAG laser was doubled to 532 nm, and a single 25 ps(FWHM) pulse was switched out from the 56-ns-longpulse train. The beam waist was measured by thethin-sample Z-scan technique24 and checked with a CCDcamera to be approximately 20 mm (half-width at 1/e2

max). Each sample was placed at the focal plane deter-mined from an open-aperture Z-scan measurement of thesample. The input energy was varied from 1 nJ to 10 mJby a stepper motor attached to a half-wave plate in frontof a fixed polarizer.

In the previous sections of this paper we reported usingthe materials HITCI and PD3 to compare the theoreticaland numerical results. We have experimentally studiedthese two materials along with the phthalocynaine dyechloroaluminum phthalocynaine (CAP).6 With the ex-ception of HITCI,7 we could not find any references tomeasured values of the second excited-state cross sections23 . Therefore we performed the experiment on HITCI

Fig. 6. Limiting curves calculated for HITCI from the numericalsolution to Eq. (15) derived from the dynamical solution to therate equations (filled circles) and the numerical solution to Eq.(17) derived from the steady-state solution to the rate equations(open circles). Pulse width, (FWHM) 25 ps.


to compare with previous results7,18 and on PD3 and CAPto determine s23 from fitting the experimentally mea-sured limiting curve. Also, the experimentally deter-mined value of the fluence at the turnover point can becompared with that predicted by the dynamical andsteady-state solutions. It will be seen that HITCI andPD3 fit the model exploited in this paper quite well butthat CAP does not.

The results of the optical limiting experiments forHITCI and PD3 are shown in Figs. 7(a) and 7(b), respec-tively. The experimental results are plotted along withthe limiting curves produced by the numerical solution ofEqs. (3) and (4) for a Gaussian pulse and the solution tothe ODE.15 The best-fit parameters for HITCI( s01 5 2.5 3 10217 cm2, s12 5 87.5 3 10217 cm2, s23

Fig. 7. Experimentally measured limiting curve (triangles) for(a) HITCI and (b) PD3 with the best-fit material parameters usedin the numerical solution to Eq. (15) (solid curves) and the fullsystem of PDEs (3) and (4) for a Gaussian pulse (dashed curves).

5 1.5 3 10217 cm2, t10 5 1.7 3 1029, t21 5 2.2 3 10212,Z 5 0.091) and for PD3 ( s01 5 1.03 3 10218 cm2, s125 2.2 3 10216 cm2, s23 5 5.2 3 10220 cm2, t10 5 1.03 1029, t21 5 6.0 3 10212, Z 5 0.089) are in good agree-ment with previously reported results for HITCI (Ref. 18)and for (PD3)3,4 We then determined the fluence at theturnover point from Fig. 7 for the experiment and the twofitting curves. We also calculated the fluence at the turn-over point, using transcendental equation (16) and thenumerical solution to the steady-state propagation, Eq.(17). The values for the fluence at the turnover point forboth HITCI and PD3 obtained with all the methods de-scribed above are listed in Table 1. As can be seen fromFig. 7 and Table 1, all the predicted values for the fluenceat the turnover point are lower than those of the experi-mental results. It has been shown for HITCI (Refs. 7 and18) and for PD3 (Refs. 3 and 4) that to explain nonlinearabsorption in these materials one needs to use a more de-tailed model structure. For HITCI, Hughes andWherrett18 and Swatton et al.7 have shown that their ex-perimental data could be fitted accurately if they ac-counted for a finite intraband vibrational lifetime. As forPD3, Lim et al.3 and Przhonska et al.4 showed that an all-singlet state five-level model, which includes reorienta-tional processes in the first excited state, would accu-rately fit their experimental data. Even neglecting thesedetails, our dynamical solution approximately predictsthe fluence at the turnover point to the same order ofmagnitude. For the material CAP our model could not fitthe data (Fig. 8) and therefore does not make accuratepredictions of the fluence at the turnover point. Weiet al.6 had to invoke both two-photon absorption andexcited-state absorption to fit their data accurately. So inthis case the two-photon absorption is large enough thatit cannot be neglected.

Below, we briefly explain how the material parameterseffect the behavior of the limiting curve. First, we con-sider the normalized time parameters T21 and T10 . AsT21 becomes larger (longer pulses or shorter lifetime ofthe second excited state) the limiting curve becomesdeeper and the turnover from RSA to SA shifts to higherfluences. However, a change in T10 of as much as an or-der of magnitude barely affects the nonlinear absorptionof the material. As for the excited-state cross-section pa-rameters a and b, a larger a (larger first-excited-statecross section or smaller ground-state cross section) pro-duces a deeper limiting curve, but the turnover from RSAto SA does not shift to higher fluences, as was seen withan increase in T21 . Parameter b is the most sensitivematerial parameter for determining when the turnoverfrom RSA to SA will occur. As b increases, the limitingcurve becomes slightly deeper and the turnover point

Table 1. Turnover Fluence (mJ/cm2) Determined from Experimental and Numerical Results

Material

Method

Eq. (17) Eq. (16) Eq. (15) Eqs. (3), (4), Gaussian Input Experiment

HITCI 3 13 27 36 84PD3 55 76 155 232 414


shifts to higher input fluences. It is important to notethat all these tendencies rely on the assumption of suffi-cient linear absorption to populate the first excited level.

Our theoretical results can also be used to analyze theoutput fluence at high input fluence levels (F @ 1). Inthis case, an exponential term in Eq. (15) can be droppedand the asymptotic behavior of the output fluence can beobtained from the condition

u~Fout! ' u~F in!exp@2a~b 1 T21!Z#, (18)

where

u~F ! 5 ~F 2 1 !T211aS F 1T21

bD T21~a2b!/b

. (19)

For instance, the output becomes clamped at the maxi-mum allowable energy, e.g., 1 mJ can be achieved for25-ps pulses by use of the polymethine dye PD33,4 with80% linear transmittance and a 5 50 if b . bcr ' 28.

An important conclusion to make is that for high inputenergies with sufficient linear absorption the key param-eter that influences optical limiting is the ratio of the sec-ond excited-state absorption cross section to the ground-state absorption cross section. In the above example theoutput energy is merely changed, with the increase in afrom 50 to 200. Therefore, from the viewpoint of efficientpicosecond optical limiting, there is no advantage in hav-ing a large first-excited-state absorption cross sectionwith a small second-excited-state absorption cross sectionbecause the latter will act as a bottleneck to the opticallimiting process. Recently synthesized polymethine dyesthat have a large first-excited state absorption cross sec-tion are quite poor optical limiters in the picosecond re-gime because of a low value of b. This considerationneeds to be taken into account when one is synthesizingnew materials for optical limiting. The qualitative basisfor practical estimates can be gained from formula (18).

5. CONCLUSIONSIn the picosecond regime the steady-state solution to therate equations does not always provide a reasonably ac-

Fig. 8. Experimentally measured limiting curve (triangles) forCAP with best-fit material parameters used in the numerical so-lution to Eq. (15) (solid curves) and the full system of PDEs (3)and (4) for a Gaussian pulse (dashed curve).

curate description of the nonlinear absorption in RSA ma-terials, whereas numerical treatment of the full system ofPDEs remains time-consuming. We have developed anapproach that accounts for the dynamical behavior of thepopulation densities and optical propagation but requiresonly the numerical solution of a single ODE. We haveshown that the specific pulse shape has little effect on thenonlinear absorption of the pulse, so a flat-topped pulseshape can be used that greatly simplifies and expeditesmodeling of beam propagation through RSA materialswith picosecond pulses. We have obtained an approxi-mate criterion for the turnover from RSA to SA to rapidlycalculate the turnover point from the photophysical prop-erties of the material. The dynamical equation for flu-ence allows us to determine the critical values of materialparameters to achieve the output energy clamping at thedesired level. This approach can be straightforwardlyextended to the case of arbitrary initial conditions tostudy the absorption properties of the material by doublepump–probe measurements.25,26 Additionally, we canuse the results of picosecond optical limiting experimentsto determine material parameters that in turn can thenbe used to predict the outcome of nanosecond optical lim-iting.

ACKNOWLEDGMENTSThe authors gratefully acknowledge the support of theNational Science Foundation (grant 9970078), the U.S.Office of Naval Research (grant N00014-97-1-0936), andthe Naval Air Warfare Center Joint Service Agile Pro-gram (contract N00421-98-C-1327). The authors thankOlga V. Przhonska for providing the PD3 dye.

R. Lepkowicz’s e-mail address is [email protected].

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picosecond optical limiting in reverse saturable absorbers: a theoretical and experimental study

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