Bidirectional description of amplified spontaneous emission induced by three-photon absorption

Baev et al. Vol. 22, No. 2 /February 2005 /J. Opt. Soc. Am. B 385

Bidirectional description of amplified spontaneousemission induced by three-photon absorption

Alexander Baev

Institute for Lasers, Photonics and Biophotonics, The State University of New York at Buffalo,Buffalo, New York 14260-3000

Viktor Kimberg, Sergey Polyutov, Faris Gel’mukhanov, and Hans Agren

Theoretical Chemistry, Roslagstullsbacken 15, AlbaNova, Royal Institute of Technology, S-106 91Stockholm, Sweden

Received March 9, 2004; revised manuscript received June 11, 2004; accepted August 31, 2004

A semiclassical dynamic theory of the nonlinear propagation of a few interacting intense light pulses is appliedto study the nonlinear counterpropagation of amplified spontaneous emission (ASE) induced by three-photonabsorption of short intense laser pulses in a chromophore solution. Several important results from the mod-eling are reached for the ASE process developing in the regime of strong saturation. Accounting for ASE inboth forward and backward directions with respect to the pump pulse results in a smaller efficiency of nonlin-ear conversion for the forward ASE compared with the case in which forward emission is considered alone,something that results from the partial repump of the absorbed energy to the backward ASE component; theoverall efficiency is nevertheless higher than for the forward emission considered alone. The efficiency of non-linear conversion of the pump energy to the counterpropagating ASE pulses is strongly dependent on the con-centration of active molecules so that a particular combination of concentration versus cell length optimizesthe conversion coefficient. Under certain specified conditions, the ASE effect is found to be oscillatory; theorigin of oscillations is dynamical competition between stimulated emission and off-resonant absorption. Thisresult can be considered one of the possible explanations of the temporal fluctuations of the forward ASE pulse[Nature 415, 767 (2002)]. © 2005 Optical Society of America

OCIS codes: 020.4180, 320.7130, 160.4890.

1. INTRODUCTIONThree-photon-active materials have been studied exten-sively over the past few years owing to their potential ap-plications in the fields of telecommunications andbiophotonics.1–4 Two major advantages of thesematerials—longer excitation wavelengths and much bet-ter spatial confinement—make them attractive in com-parison with two-photon-absorption-based materials.5

One of the most important applications of three-photon-active materials is three-photon-pumped frequency-upconversion cavityless lasing.1,2 Short infrared (IR)pulses induce the amplified spontaneous emission (ASE)process via three-photon absorption followed by fast non-radiative decay to a long-lived state that collects popula-tion. Conventional experiments with a pulsed longitudi-nal pump2–4 show that stimulated emission occurs in bothforward and backward directions with respect to thepump pulse. In our recent papers6,7 we modeled a previ-ous experiment2 and accounted for the forward ASE only.However, it is our belief that the counterpropagating ASEfield can drastically change the spatial and temporal pic-ture of the forward ASE by influencing the population dis-tribution induced by the pump field. Therefore the aimof this paper is to investigate the evolution of counter-propagating ASE pulses in connection with the efficiencyof nonlinear conversion.

We stress here the dire need for use of the strict dy-namical theory for adequate modeling of the interaction ofthree strong electromagnetic pulses (pumping and two

0740-3224/2005/020385-09$15.00 ©

counterpropagating ASE pulses). First, the nonlinearpropagation of the strong pumping pulse under conditionsof strong saturation makes it impossible to introduce theconventional coherent three-photon-photoabsorptioncross section owing to the considerable change of thestate’s populations, which necessitates solving thecoupled density-matrix equations and Maxwell’s equa-tions strictly. Second, together with this problem, wehave to solve the problem of propagation of two stronglyinteracting ASE pulses through the spatially inhomoge-neous inverted medium evolving in time. Thus for thefirst time we applied our recently developed dynamicaltheory to modeling of strongly interacting counterpropa-gating ASE pulses.

The paper is organized as follows. We start by outlin-ing the density matrix and field equations (Section 2).The numerical simulations are described in Section 3.Different aspects of the propagation of the bidirectionalASE pulse are discussed and compared in Section 4 withthe experimental results of Prasad’s group2 and with theresults of our previous simulations.6 The oscillatory re-gime of the ASE is analyzed in Section 5, with the use of asimplified model. Our findings are summarized in Sec-tion 6.

2. THEORYA. Wave EquationsWe explore the propagation along the z axis of an ASEfield through a nonlinear many-level medium. The ASEfield, E, consists of two components,

2005 Optical Society of America

386 J. Opt. Soc. Am. B/Vol. 22, No. 2 /February 2005 Baev et al.

E 5ER

2exp~2ivRt 1 ikRz 2 iwR! 1

EL

2

3 exp~2ivLt 2 ikLz 2 iwL! 1 c.c., (1)

propagating, respectively, to the right and to the left (Rcomponent and L component below), along the z axis thatis parallel to the propagation direction of the pump field.The R component propagates in the same direction as thepump field. The strengths, Em , the frequencies, vm , andthe wave vectors, km , of the ASE fields are indexed by m5 R, L. Three-photon absorption of the pump field withthe forthcoming nonradiative decay to a long-lived stateresults in the population inversion leading to the ASE.In turn, the ASE field affects the population distributionwhen it becomes strong. This effect makes the propaga-tion of the ASE field nonlinear. The polarization oscillat-ing with the frequency of the ASE field has the followingstructure:

P 5 Tr~dr!

5 PR exp~2ivRt 1 ikRz 2 iwR!

1 PL exp~2ivLt 2 ikLz 2 iwL! 1 c.c., (2)

where the trace is taken over the energy levels. To knowthe polarization, we need the transition dipole moments,dba(t) 5 dba exp(ivabt), and the density matrix, rab , ofthe medium. Here vab 5 (Ea 2 Eb)/\ is the frequencyof the transition a → b.

We use here the slowly varying envelopeapproximation.8 The substitution of E [Eq. (1)] and P[Eq. (2)] in Maxwell’s equations and a selection of contri-butions with the frequency vm and momentum kR or 2kLresults in the following paraxial wave equations for theamplitudes of the R and L components of the ASE field:

S ]

]z1

1

c

]

]t2

i

2kRD'DER 5

ikR

«0PR,

S 2]

]z1

1

c

]

]t2

i

2kLD'DEL 5

ikL

«0PL, (3)

where the SI system of units is used. The phase, wm , ofthe ASE fields is assumed to be constant.

B. Density-Matrix EquationsThe density-matrix equation for a medium reads as

S ]

]t1 GD r 5

i

\@r, V#, Tr r 5 N, (4)

where V is the interaction between the molecules and theelectromagnetic field. The concentration of the absorb-ing molecules is denoted by N, and G is the relaxation ma-trix. We used this equation to derive a general formal-ism describing the interaction of a many-modeelectromagnetic field with a nonlinear medium.7 The ki-netic equations for populations, raa , and off-diagonal el-ements, rab , of the density matrix read as

S ]

]t1 GabD rab 5 da,b(

g.aGg

argg 1i

\(

g

3 ~ragVgb 2 Vagrgb!, (5)

where the rate Gga of decay transitions g → a, which is

large in organic molecules owing to the nonradiative con-version, almost completely coincides with the total decayrate Gaa . We neglected the space derivatives at the left-hand side of the kinetic equations [Eq. (5)] that are re-sponsible for the Doppler effect @exp(6ikz) → 1# becausethe Doppler broadening in the condensed absorber is neg-ligible compared with the dephasing rate, Gab .9

Making use of Eq. (5), we derived the density matrix re-lated to the ASE field6,7:

habm '

i(g

~ragGgbm 2 Gag

m rgb!

Gab 2 i~vm 2 vab!, m 5 L, R, (6)

where Ggbm 5 Em

–dab/2\ are the Rabi frequencies of theASE field and rag are the elements of the density matrixinduced by the pump field (see Refs. 6 and 7 for details).Apparently, the populations raa , created by the pumpfield, contribute mainly to the density matrix [expression(6)]. The off-diagonal elements rab are small because oftheir off-resonant character, though they are included ex-plicitly. In our case the main mechanism initiating redis-tribution of populations is three-photon absorption. Toensure this in the modeling, we applied special resonantconditions: v 5 v10/3 where v is the frequency of thevertical transition from the ground to the first excitedstate of the molecule. We neglected the time derivativein expression (6) because of the large value of the dephas-ing rate in solutions (see Refs. 6, 7, 9, and 10).

C. Final Wave Equations for the IntensitiesA substitution of the polarization, Pm 5 (badbahab

m , inwave equations (3) yields finally the following paraxialequations for the components of the ASE field:

S ]

]z1

1

c

]

]t2

i

2kRD'DER 5

ikR

«0(ba

dbahabR ,

S 2]

]z1

1

c

]

]t2

i

2kLD'DEL 5

ikL

«0(ba

dbahabL .

(7)

If the transverse inhomogeneity of the field is small, onecan directly write down the equations for the intensitiesIR 5 c«0uERu2/2 and IL 5 c«0uELu2/2:

S ]

]z1

1

c

]

]t D IR 5 vR ImF ~ER!* (ba

dbahabR G , (8)

S 2]

]z1

1

c

]

]t D IL 5 vL ImF ~EL!* (ba

dbahabL G ,

(9)

where we assume real transition dipole moments and am-plitudes of the electromagnetic fields and use the follow-


ing expressions for the field amplitudes, ER

5 (2IR/c«0)1/2, EL 5 (2IL/c«0)1/2. Here ER 5 eER, EL

5 eEL; e is the polarization vector of the ASE field.It is important to note that the one-dimensional wave

equations [Eq. (8)] used in our numerical simulations ig-nore some important nonlinear optical processes such asself-focusing. The main reason of going for a one-dimensional approximation was computational feasibility.

D. Averaging over OrientationsAs the molecules in solution have random orientations,we have to average the right-hand side of the field equa-tions over molecular orientations: Pm → ^Pm& or hab

m

→ ^habm &. We perform the orientational averaging over

the angle 0 < w < 2p between the molecular axis andthe polarization vector, i x, numerically.6 It is worth not-ing that our averaging procedure ignores rotations of mol-ecules because they are slow in comparison with the in-verse broadening of the spectral transitions.

E. Influence of the Amplified Spontaneous EmissionField on PopulationsThe strict equations for populations take into account thechange of populations by the ASE field:

S ]

]t1 GaaD raa 5 (

b.aGb

arbb 1 Wa 1 WaR 1 Wa

L .

(10)

The pump and ASE fields change the population of theath level with the probabilities per unit time:

Wa 5 2 Im (b

@Gabrba~1,0! 2 rab

~1,0!Gba#,

Wam 5 2 Im (

b@Gab

m hbam 2 hab

m Gbam #, (11)

respectively. The pump and ASE fields are assumed lin-early polarized with the same direction of the polarizationvector. Equation (10) is strict: The only approximationwe used is the density matrix, hab

m [expression (6)], in-duced by the ASE field.

When the system is long, the ASE field becomes highenough to change the populations, in accordance with Eq.(10). It leads to a decrease of the population inversion,r33 2 r22 , and hence to a ceasing of the amplification.

3. NUMERICAL SIMULATIONSA. Computation of Dipole Moments and ExcitationEnergiesWe applied the above theory to studies of upconvertedstimulated emission, based on three-photon absorption ofthe organic stilbene chromophore 4-[N-(2-hydroxyethyl)-N-(methyl)amino phenyl]-48-(6-hydroxyhexyl sulfonyl)(abbreviated as APSS) dissolved in dimethylsulphoxide.2,6 The influence of the solvent was modeledby the collisional dephasing rate, which is assumed to bethe same, 0.01 eV, for all transitions. For details of theelectronic structure calculations, see Ref. 6. The experi-mental data such as the concentration of molecules in thesolvent, length of the active medium (cuvette length), in-

put laser intensity, and pulse duration are taken fromRef. 2. These and other input parameters are collectedin our previous paper (see Tables 1 and 2 in Ref. 6).

It is worthwhile to stress that the L and R componentsof the ASE field have different frequencies in general.3

The experimental data2 show that vR 5 vL for the stud-ied APSS molecule. We used this fact in our simulations.

B. Five-State ModelHaving performed an analysis of the calculated transitiondipole moments of the APSS molecule, we conclude thatthe transitions among the singlets S0 , S1 , and S2 aremost essential for the nonlinear optical process we are in-terested in. Therefore we chose these three states (seeFig. 1) for our simulations.

As we learned from the previous experiment,2 the ASEand pump pulses are delayed with respect to each other.That is the reason we took into account the vibrationalstructure of the singlets S0 and S1 . We modeled thisstructure by including two pairs of vibrational levels,(4, 3) and (2, 1), assuming the one-mode approximation.

The use of the few-state model is common practice forcalculations of the nonlinear properties (see, for example,Ref. 11). The five-state model used in this paper (see Fig.1) is described in great detail in our previous paper.6

Here we point out the physical meaning of this model.Vibrational state 4 models the group of vibrational levelsnear the point of vertical photoabsorption transition.State 4 decays nonradiatively to state 3, giving rise to thepopulation inversion between states 3 and 2. The lasingtransition between these states corresponds to the verti-cal decay. Afterward, the molecule experiences nonradi-ative decay to the lowest vibrational state 1.

Fig. 1. Energy-level diagram.


4. RESULTS AND DISCUSSIONA. Efficiency of Nonlinear ConversionAn important question concerning induced upconvertedlasing in a nonlinear medium concerns how much of theabsorbed energy of a pump pulse is converted into theASE radiation. The parameter answering this questionis the net conversion coefficient, h 5 h8/(1 2 T), whereh8 is the ratio of the energy of the overall output ASE(both forward and backward) to the energy of the incidentpump pulse and T is the transmission of the pump pulse.To make a comparison between the R and the L compo-nents of the ASE, we calculated the partial coefficients,hm8 and hm . The overall coefficients can be found as asum of the partial ones.

The results of our calculations of partial hm are col-lected in Table 1 and visualized in Fig. 2. We used sixvalues of the concentration for comparison. As one cansee, for each value of the concentration the net conversioncoefficient, hR , is larger in the case in which the leftpropagation is not accounted for. The cause behind thisobservation is that a part of the absorbed energy is con-sumed by the left-propagating ASE pulse in the oppositecase.

For the concentration value of 3 3 1019 cm23 (0.05 mol/l), which is a bit lower than that used in the previousexperiment,2 the net conversion coefficient hR was foundto be equal to 4.20% without consideration of the L com-ponent. When counterpropagation is taken into account,the net conversion coefficient hR changes almost negligi-bly to 4.19%. The overall h equals the sum hR 1 hL5 5.95%, and the experimental value is 2.1%. This dif-ference between our simulations and the experiment canbe attributed to a smaller output ASE energy in the caseof a higher experimental concentration (0.06 mol/l) owingto reabsorption of the ASE pulses (see discussion below).Indeed, extrapolation of the conversion function (seeFig. 2) to the experimental concentration value 0.06mol/l gives 3.6%. Another reason is that the three-photon-absorption cross section is overestimated in oursimulations owing to numerical accuracy of ab initio com-putations of the dipole moments and of the excitation en-ergies. The higher-absorption cross section results in thelarger portion of the incident energy converted to the ASEfield.

As one can see from Table 1, the difference between hRfor the single R-component case and the two-componentcase grows as the concentration of molecules decreases.The reason for this is found in the temporal oscillationsof the ASE R component at the end of the activemedium when both R and L components are accountedfor. We checked numerically that the oscillations cannotbe attributed to the accuracy of simulations. The originof these oscillations needs a somewhat more elaborateinvestigation and will be described briefly in Section 5and more in detail elsewhere. We note here only thatthe period and amplitude of oscillations stronglydepend on concentration, as demonstrated inFig. 3(a). As the concentration decreases, the period ofoscillations increases and the peaks become wider. Asimilar behavior is observed for the L component,as demonstrated in Fig. 3(b). However, as we cansee in the figure, the amplitude of oscillations is larger

for the L component, which certainly reflects theasymmetry of the ASE propagation with respect to thepump. In fact, even in the case in which we consideronly the R component of the ASE we can observe suchoscillations at certain distances from the left edgeof the cell. For example, for N 5 3 3 1019 cm23, weobserve oscillations at distances shorter than 2 mm fromthe left edge. For a given concentration, the increase ofintensity of the ASE along the medium leads to the fastattenuation of oscillations. For N 5 3 3 1019 cm23, theoscillations are bright at 0.5 mm from the entrance to thecell when the ASE intensity is approximately 1 GW/cm2.When the ASE propagates to the right, its intensitygrows, and, at 3 mm, when the intensity equals 2.5GW/cm2, the oscillations disappear. So the slow growthof the intensity means slow attenuation of oscillations,which is the case of lower concentrations. We have tostress here that for concentrations lower than a certainvalue, when only the R component of the ASE is takeninto account, the oscillations vanish completely—we donot observe them at N 5 1 3 1018 cm23 and N 5 13 1017 cm23. Once the L component is accounted for,

the oscillations reappear.

Fig. 2. Conversion coefficient versus concentration. ‘‘Right’’means simulations for the one-component case and ‘‘overall’’means simulations for both forward and backward components.

Table 1. Basic Results of Simulationsa

N 3 1019, cm23 T

hm8, % hm , %

Rb R L Rb R L

0.01 0.988 0.31 0.24 0.10 26.64 20.71 8.200.1 0.888 3.99 3.55 0.51 35.81 31.92 4.560.5 0.561 12.27 11.85 0.93 27.94 26.97 2.111 0.346 12.33 12.11 1.12 18.87 18.52 1.722 0.181 6.67 6.63 1.47 8.15 8.10 1.803 0.122 3.67 3.68 1.55 4.20 4.19 1.76

a The length of the cell is 10 mm, the incident pump intensity is 190GW/cm2. R, right-propagating ASE component; L, left-propagating ASEcomponent. T is the transmission of the pump pulse.

b Only the R component is accounted for.


On the basis of these facts we can conclude that oscil-lations built up in a certain point of the medium areblurred as the ASE pulse propagates to the right with anincrease of intensity. The closer to the left edge the pulseemerges, the longer distance it travels through. Appar-ently, for higher concentrations of the absorbing mol-ecules the formation of the ASE pulse is faster in spaceand time, according to Eq. (8). The degree of delay of theleading edge of the R and L components depending onconcentration, seen in Fig. 3, proves this statement.

The temporal behavior of the single R component andof both the R and the L components of the ASE is shownin Fig. 4 for N 5 1 3 1018 cm23. We see that in the casewhen the L component is taken into account the R com-ponent experiences oscillations compared with the oppo-site case in which the ASE temporal profile goes downsmoothly. Owing to this circumstance, the net conver-sion coefficient, hR , is different.

Let us now pay attention to the fact that the efficiencyof the conversion is a nonlinear function of the concentra-tion of the active molecules (see Fig. 2). We know thatthe peak intensity of the ASE in an optically thin mediumis determined by the abruption of the population inver-sion that occurs when the ASE intensity approaches some

Fig. 3. Temporal distributions of the (A) R-component intensityat the end of the cell at different concentrations and (B)L-component intensity at the entrance to the cell at different con-centrations.

critical value. This value refers to the resonant satura-tion intensity of the lasing transition and is independentof concentration. In our case the saturation intensity ofthe 3 → 2 transition is approximately equal to 3MW/cm2. As the ASE pulse propagates, it gains inten-sity until some maximal value is reached. This maximalvalue is determined by the pump depletion; i.e., when thepump pulse intensity is less than the lasing threshold in-tensity, the ASE ceases to grow and experiences reabsorp-tion as it propagates through the medium. If the pumppulse is almost depleted near the entrance to the cell,which is the case for higher concentrations, the right-propagating ASE pulse starts to become reabsorbed farfrom the right edge of the cell. In the case of lower con-centrations, the ASE pulse is formed further from the leftedge of the cell, which means a shorter reabsorption pathfor the R component. At the given length of the gain me-dium (10 mm in our case), the output energy of the for-ward ASE pulse is maximal for a certain value of the con-centration. It then drops down as the concentrationdecreases because the ASE starts to be formed so close tothe right edge of the cell that it does not have time toreach the maximum. It is interesting that the percent-age of the incident energy transferred to the L componentof the ASE decreases monotonously with decreasing con-centration (see Table 1). One reason for this behavior isthat the L component is always formed near the entranceto the cell (;1 mm according to Figs. 7 and 8), where theinversion of populations can be shared by two compo-nents.

Figure 5 shows that the maximum level of the ASER-component intensity is approximately the same for allvalues of the concentration. But for the lowest value thismaximum is reached at the end of the medium. Thus adecrease of the concentration shifts the maximum of theintensity toward the end of the cell.

Our simulations also show a good agreement of the fullwidth at half-maximum (FWHM) of the ASE pulse(FWHM ' 22 ps) with the experimental number2 of 40 psand a worse agreement of the delay between the ASE andthe pump pulse (Dt ' 3 ps) with the experimentalnumber2 of 12 ps. The theoretical results were obtainedfor the experimental concentration (0.06 mol/l) and pumpintensity (190 GW/cm2).

Fig. 4. ASE intensity dynamics.


B. Formation of the Amplified SpontaneousEmission PulsesTo understand how the right- and left-propagating pulsesof the ASE are formed, let us first make a simplifying as-sumption. We consider for the moment that we have aninfinitely narrow source of photons, moving to the rightalong the z axis with the speed of light. Narrow meansthat the lasing threshold is suddenly overcome at a pointinside the medium at a certain time and then suppressedat the same time because the population inversion iseliminated by the emitted photons. This source is thegain. So at each point of the medium a bunch of photonsis emitted in equal numbers to the right and to the left(Fig. 6). This number depends on the value of the gain.We assume the emission is discrete, with the time lapsesDt. The photons emitted to the right at time instant t5 0 will then move further with the gain. At time Dt, a

Fig. 5. Three-dimensional dependencies of the R-component in-tensity under different concentrations: (A) 1 3 1019 cm23, (B)2 3 1019 cm23, and (C) 3 3 1019 cm23.

new portion of photons will be emitted. The gain will beat the point cDt as well as the photons emitted at a pre-ceding moment. So the R component of the ASE wouldbe infinitely narrow in this case and would be growingwith negative gradient as the gain is being depleted. Incontrast, the left-propagating ASE component would bebroad. The reason is the following. At time t 5 0 abunch of photons is emitted to the left. At time Dt thegain coordinate is cDt, and the previously emitted left-propagating photons are located at 2cDt. In a case ofonly two emission acts (for instance, the gain is thennegative), the temporal width of the left-propagating ASEcomponent is 2Dt. The pulse would look like a saw jag ifthe gain were decreasing. Let us stress once again thatsuch an asymmetry of the right- and left-propagatingcomponents of the ASE is owed to the right-propagatinglongitudinal pump.

Apparently, the picture of the formation of the ASEpulses given above is a naıve one. In reality the gain iscontinuously decreasing through the medium if observedat a certain instant of time (see Figs. 7 and 8 at t5 15 ps). The reason for this is the population of thelasing level 3 through the exponential decay of level 4.Such character of the gain is maintained until the inten-sity of the ASE components is smaller than the saturationintensity of the lasing transition. When the ASE inten-sity becomes comparable with the saturation intensity,the ASE starts to change the populations and hence thegain (Figs. 7 and 8). The broad dips of the gain areformed by the leading edges of the R and L components.The leading edges with high intensity cut off the gain ow-ing to an abrupt decrease of the population inversion.Because of the continuous gain, the R component of theASE becomes broad, with a steep front and a long, slowlydecreasing tail. The forward pulse is growing, chasingthe leading edge of the decreasing gain. This leadingedge of the gain looks like our model narrow source ofphotons—it supports the growth of the leading edge of theforward ASE pulse. The source amplitude decreases be-cause the pump pulse, creating the population inversion,is absorbed when it runs through the medium.

As mentioned earlier, the formation of the L componentof the ASE takes place at a very short distance from the

Fig. 6. Illustration of a narrow gain propagation—formation ofthe L and R components of the ASE.


entrance of the cell, approximately 1 mm. As we can seein Fig. 8, the peaks of the gain supplying photons to R andL components are located at the same distance from theleft edge of the cell at different time instants. The right-hand slope of the gain, spanning to the right, is ‘‘eatenaway’’ by the forward ASE pulse because its intensity atthis short distance of 1 mm reaches the saturation inten-sity of the lasing transition.

It is necessary to note that the output energies of the Rand L components of the ASE differ substantially owingto the asymmetry with respect to the pump.

5. OSCILLATORY MODELTo check the numerical stability of the temporal oscilla-tions of the ASE, we applied a simplified open model con-

Fig. 7. Snapshots of the (A) gain and (B) single R component atdifferent time instants.

Fig. 8. Snapshots of the (A) gain and (B) and (C) R and L com-ponents at different time instants.

sisting of two states, of a pump, and of a few decay chan-nels (see Fig. 9). The physics of the model can bedescribed by means of two rate equations for populationsand one wave equation:

S ]

]t1 G1D r1 5 GNN 2 pI~r1 2 r0!,

S ]

]t1 G0D r0 5 G1r1 1 pI~r1 2 r0!,

S ]

]z1

1

c

]

]t D I 5 @B~r1 2 r0! 2 a#I 5 gI.

(12)

Here p 5 d102 /G\2c«0 , B 5 2vmd10

2 /G\c«0 , d10 is the 1→ 0 transition matrix element, vm is the ASE frequency,G is the dephasing rate, and G1 and G0 are the decay ratesof populations 1 and 0, respectively. The pump is mod-eled by the constant concentration, N, and the pump rate,GN . The simulations of the ASE based on the model[Eqs. (12)] showed clearly that the oscillatory regime hasnothing to do with the numerical accuracy but ratherwith physics. It is important to note that sensitivity ofthis regime to the effects of propagation is rather weakaccording to our simulations. It is therefore justified toneglect the term ]I/]z in the wave equation in order tocalculate the frequency, V, of oscillations. To obtain anequation for V, we have to write down equations for thesmall deviations of populations, r1 exp(iVt) andr0 exp(iVt), and of the intensity, j exp(iVt), from the sta-tionary solution of Eqs. (12) occurring for zero gain, g5 0. A solution to the corresponding eigenvalue prob-lem results in a cubic equation for the frequency:

V3 2 iV2~G0 1 G1 1 2pI ! 2 V@G0G1 1 ~2a 1 G0!pI#

1 iapIG0 5 0, (13)

where a 5 ac. The solution of Eq. (13) is straightfor-ward, resulting in three complex roots. The last root V3with zero frequency Re V3 5 0 (see Fig. 12) has nothing todo with oscillations. The two other roots, V1 5 Re V11 i Im V1 and V2 5 2Re V1 1 i Im V1 , with the samedamping rate correspond to ASE oscillations with thesame frequency (see Fig. 10).

The most important parameter of our model [Eqs. (12)]is the off-resonant absorption coefficient, a, which was setto 1 3 103 cm21 in our simulations. The decay rates andtransition matrix element d10 were set equal to those

Fig. 9. Simplified model of the ASE.


used in the complex model. The physical meaning of thequantity pI is the rate of stimulated emission and can beparametrized as pI 5 G1I/2Isat , where Isat is the reso-nant saturation intensity. The dephasing rate, G, is oneof the parameters determining the value of both thestimulated emission rate, pI ; 1/G, and the off-resonantabsorption coefficient, a ; G. Therefore the balance be-tween these two quantities, which is responsible for theappearance of oscillations, is strongly influenced by G.

Figure 11 shows that the oscillation frequencies lie onequipotential lines in the (a, pI) plane. The area belowthe line with the frequency 1 3 1011 s21 in Fig. 11 is theregion of slow oscillations. The frequency of the ASE os-cillations in this area changes irregularly around thevalue of ;1 s21. As mentioned above, the roots of Eq.(13) are complex, where Re V determines the oscillationfrequency and Im V is the damping rate of oscillations.Figure 12 demonstrates the real and imaginary parts ofthe roots of Eq. (13) as functions of the stimulated emis-sion rate, pI, at given a that was set to 1 3 103 cm21.Let us have a closer look at a branch with positive valuesof the frequency. The points on this branch correspond tocertain values of the damping rate. For example, at pI5 7 3 1010 s21, the oscillation frequency equals Re V5 2 3 1012 s21, and the damping rate equals Im V 5 1.33 1011 s21. In this case the damping rate is muchsmaller than the frequency, and hence the oscillations arenicely visible as demonstrated in Fig. 10. We also haveto note here that our estimation of the off-resonant ab-sorption coefficient, a, for the five-state model gave thevalue of approximately 30 cm21, which is 30 timessmaller than the value used in the model [Eqs. (12)].Our simulations showed that this estimated value resultsin fast damping of the oscillations—with a damping rateequal to the oscillation frequency. Such a disagreementis owed to the fact that the model [Eqs. (12)] is rough com-pared with the running-pump-five-state model. Never-theless, the simplified model gives good qualitative agree-ment with the complex model and allows for a simpleevaluation of the oscillation frequencies.

Fig. 10. Model oscillations of the ASE at the end of the 10-mmcell for different values of concentration: (A) 1 3 1019 cm23, (B)5 3 1018 cm23, and (C) 1 3 1018 cm23.

The experimental data [Fig. 3(b), Ref. 2] demonstratetemporal fluctuations of the ASE intensity with the char-acteristic time scale (*5 ps), which is larger than the timeresolution of the high-speed streak camera used in the ex-periment (;2 ps). Our simulations (Fig. 3) show the os-cillations with a similar period (;2–10 ps), which allowsus to relate the experimental oscillations to those pre-dicted by the model.

6. SUMMARYOwing to recent successful synthesis of different three-photon-absorption-based materials, especially dyes, thereare many ongoing discussions on the potential of newthree-photon-absorption-based optical applications. Al-though such materials thus are in reach for the design ofparticular optimal applications, their speedy scanningstill poses an arduous undertaking in the laboratory. Itis therefore highly relevant to make a versatile modelingtoolbox accessible for laboratory simulations of prospec-tive materials under different experimental conditions.

Fig. 11. Equipotential lines of the frequency, Re V.

Fig. 12. Frequency of the (A) ASE oscillations and (B) dampingrate versus the stimulated emission rate at a given a.


The research in our group has taken the view to combineinto such a toolbox quantum-chemistry methods for pre-dicting basic electronic–conformational structures andmolecular properties with classical Maxwell’s theory toinvestigate the properties of materials with respect to theinteraction with electromagnetic fields of various wave-lengths and strengths. The details can be found in ourprevious papers.6,7 An important asset of this toolbox isthat it transcends the power-series approach in which thenonlinear polarization is expanded over powers of theelectric field; instead, we account for the coupling of mul-tiphoton processes in strong optical fields and go beyondthe so-called rotatory wave approximation, meaning thatoff-resonant, in addition to resonant, effects are consid-ered.

In this paper we have modeled and analyzed three-photon-pumped upconverted lasing in connection with re-cently conducted experimental demonstrations of thiseffect.2–4 Upconversion of near-IR (1.2–1.7-mm) radia-tion to shorter wavelengths of approximately 0.5–0.6 mmis of particular importance for optical telecommunicationand biophotonics. Compared with the wavelengths usedto pump two-photon active materials, the longer IR rangeis preferable owing to specific characteristics of optical fi-bers and to deeper penetration of biological tissues.

As opposed to our previous study,6 we accounted herefor both forward and backward components of the ASE,which essentially enhances the predictive capacity of ourmodel. The simulations showed that accounting for ASEin both forward and backward directions with respect tothe pump pulse results in a smaller efficiency of nonlinearconversion for the forward ASE compared with the case inwhich forward emission is considered alone. The reasonfor this can be found in the partial repump of the ab-sorbed energy to the backward ASE component. Theoverall conversion is, nevertheless, higher when bothcomponents are present.

We found that the reabsorption of the ASE radiationdecreases the efficiency of the nonlinear conversion forhigh concentrations of the active molecules in solution, sothat a proper combination of concentration versus celllength should gives the maximal conversion coefficient.

We observed an oscillatory regime of the ASE for therunning-pump-five-state model. To figure out the originof these oscillations, we developed a simplified model ofthe ASE and analyzed the domain of parameters allowingfor observation of oscillations. The oscillations predictedby our theory seem to be observed in the previousexperiment.2

Our future efforts will be put into further developmentof the oscillatory model, to find conditions for clear experi-mental observation of oscillations, which is important asan additional check of our toolbox. Simulations of thespectral distribution of the ASE intensity will also be con-

ducted to understand the blue shift of the forward ASE forsome compounds as reported in Ref. 3.

ACKNOWLEDGMENTSThe authors are grateful to Paras N. Prasad and toGuang S. He for stimulating discussions. The authorsacknowledge a grant from the photonics project runjointly by the Swedish Materiel Administration (FMV)and the Swedish Defense Research Agency (FOI). Theauthors also acknowledge support from the Swedish Re-search council (VR). A. Baev thanks the Swedish Re-search Council and the Swedish Foundation for Interna-tional Cooperation in Research and Higher Education fora postdoctoral fellowship.

A. Baev, the corresponding author, can be reached bye-mail at [email protected].

REFERENCES1. G. S. He and P. N. Prasad, ‘‘Three-photon absorbing mate-

rials: characterization and applications,’’ in Nonlinear Op-tical Transmission and Multiphoton Processes in Organics,A. T. Yeates, K. D. Belfield, F. Kajzar, and C. M. Lawson,eds., Proc. SPIE 5211, 1–12 (2003).

2. G. S. He, P. P. Markowicz, T.-C. Lin, and P. N. Prasad, ‘‘Ob-servation of stimulated emission by direct three-photon ex-citation,’’ Nature 415, 767–770 (2002).

3. G. S. He, J. Dai, T.-C. Lin, P. P. Markowicz, and P. N.Prasad, ‘‘Ultrashort 1.5-mm laser excited upconvertedstimulated emission based on simultaneous three-photonabsorption,’’ Opt. Lett. 28, 719–721 (2003).

4. G. S. He, R. Helgeson, T.-C. Lin, Q. Zheng, F. Wudl, and P.N. Prasad, ‘‘One-, two-, and three-photon pumped lasing ina novel liquid dye salt system,’’ IEEE J. Quantum Electron.39, 1003–1008 (2003).

5. B. A. Reinhardt, ‘‘Two-photon technology: new materialsand evolving applications,’’ Photonics Sci. News 4, 21–33(1999).

6. A. Baev, F. Gel’mukhanov, O. Rubio-Pons, P. Cronstrand,and H. Agren, ‘‘Upconverted lasing based on many-photonabsorption: an all dynamic description,’’ J. Opt. Soc. Am. B21, 384–396 (2004).

7. A. Baev, F. Gel’mukhanov, V. Kimberg, and H. Agren, ‘‘Non-linear propagation of strong multi-mode fields,’’ J. Phys. B36, 3761–3774 (2003).

8. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

9. F. Gel’mukhanov, A. Baev, P. Macak, Y. Luo, and H. Agren,‘‘Dynamics of two-photon absorption by molecules and solu-tions,’’ J. Opt. Soc. Am. B 19, 937–945 (2002).

10. A. Baev, F. Gel’mukhanov, P. Macak, Y. Luo, and H. Agren,‘‘General theory for pulse propagation in two-photon activemedia,’’ J. Chem. Phys. 117, 6214–6220 (2002).

11. M. Albota, D. Beljonne, J. L. Bredas, J. E. Ehrlich, J. Y. Fu,A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R.Marder, D. McCord-Maughon, J. W. Perry, H. Rockel, M.Rumi, C. Subramaniam, W. W. Webb, X. L. Wu, and C. Xu,‘‘Design of organic molecules with large two-photon absorp-tion cross sections,’’ Science 281, 1653–1656 (1998).

Bidirectional description of amplified spontaneous emission induced by three-photon absorption

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