Bidirectional description of amplified spontaneous emission induced ...

Baev et al.

Vol. 22, No. 2 / February 2005 / J. Opt. Soc. Am. B

385

Bidirectional description of amplified spontaneous emission 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

˚ gren Viktor Kimberg, Sergey Polyutov, Faris Gel’mukhanov, and Hans A Theoretical Chemistry, Roslagstullsbacken 15, AlbaNova, Royal Institute of Technology, S-106 91 Stockholm, 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 applied to study the nonlinear counterpropagation of amplified spontaneous emission (ASE) induced by three-photon absorption of short intense laser pulses in a chromophore solution. Several important results from the modeling are reached for the ASE process developing in the regime of strong saturation. Accounting for ASE in both forward and backward directions with respect to the pump pulse results in a smaller efficiency of nonlinear 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; the overall efficiency is nevertheless higher than for the forward emission considered alone. The efficiency of nonlinear conversion of the pump energy to the counterpropagating ASE pulses is strongly dependent on the concentration of active molecules so that a particular combination of concentration versus cell length optimizes the conversion coefficient. Under certain specified conditions, the ASE effect is found to be oscillatory; the origin of oscillations is dynamical competition between stimulated emission and off-resonant absorption. This result 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. INTRODUCTION Three-photon-active materials have been studied extensively over the past few years owing to their potential applications in the fields of telecommunications and biophotonics.1–4 Two major advantages of these materials—longer excitation wavelengths and much better spatial confinement—make them attractive in comparison with two-photon-absorption-based materials.5 One of the most important applications of three-photonactive materials is three-photon-pumped frequencyupconversion cavityless lasing.1,2 Short infrared (IR) pulses induce the amplified spontaneous emission (ASE) process via three-photon absorption followed by fast nonradiative decay to a long-lived state that collects population. Conventional experiments with a pulsed longitudinal pump2–4 show that stimulated emission occurs in both forward and backward directions with respect to the pump pulse. In our recent papers6,7 we modeled a previous experiment2 and accounted for the forward ASE only. However, it is our belief that the counterpropagating ASE field can drastically change the spatial and temporal picture of the forward ASE by influencing the population distribution induced by the pump field. Therefore the aim of this paper is to investigate the evolution of counterpropagating ASE pulses in connection with the efficiency of nonlinear conversion. We stress here the dire need for use of the strict dynamical theory for adequate modeling of the interaction of three strong electromagnetic pulses (pumping and two 0740-3224/2005/020385-09$15.00

counterpropagating ASE pulses). First, the nonlinear propagation of the strong pumping pulse under conditions of strong saturation makes it impossible to introduce the conventional coherent three-photon-photoabsorption cross section owing to the considerable change of the state’s populations, which necessitates solving the coupled density-matrix equations and Maxwell’s equations strictly. Second, together with this problem, we have to solve the problem of propagation of two strongly interacting ASE pulses through the spatially inhomogeneous inverted medium evolving in time. Thus for the first time we applied our recently developed dynamical theory to modeling of strongly interacting counterpropagating ASE pulses. The paper is organized as follows. We start by outlining the density matrix and field equations (Section 2). The numerical simulations are described in Section 3. Different aspects of the propagation of the bidirectional ASE pulse are discussed and compared in Section 4 with the experimental results of Prasad’s group2 and with the results of our previous simulations.6 The oscillatory regime of the ASE is analyzed in Section 5, with the use of a simplified model. Our findings are summarized in Section 6.

2. THEORY A. Wave Equations We explore the propagation along the z axis of an ASE field through a nonlinear many-level medium. The ASE field, E, consists of two components, © 2005 Optical Society of America

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J. Opt. Soc. Am. B / Vol. 22, No. 2 / February 2005

ER

E⫽

2

exp共 ⫺␫ ␻ R t ⫹ ␫ k R z ⫺ ␫ ␸ R 兲 ⫹

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冉

EL 2

⫻ exp共 ⫺␫ ␻ L t ⫺ ␫ k L z ⫺ ␫ ␸ L 兲 ⫹ c.c.,

P ⫽ Tr共 d␳ 兲 ⫽ PR exp共 ⫺␫ ␻ R t ⫹ ␫ k R z ⫺ ␫ ␸ R 兲 (2)

where the trace is taken over the energy levels. To know the polarization, we need the transition dipole moments, d␤␣ (t) ⫽ d␤␣ exp(␫␻␣␤t), and the density matrix, ␳ ␣␤ , of the medium. Here ␻ ␣␤ ⫽ (E ␣ ⫺ E ␤ )/ប is the frequency of the transition ␣ → ␤ . We use here the slowly varying envelope approximation.8 The substitution of E [Eq. (1)] and P [Eq. (2)] in Maxwell’s equations and a selection of contributions with the frequency ␻ ␮ and momentum kR or ⫺kL results in the following paraxial wave equations for the amplitudes of the R and L components of the ASE field:

冉冉

⳵ ⳵z

⫹

1 ⳵ c ⳵t

⫺

␫ 2k R

冊冊

⌬⬜ ER ⫽

␫kR ␧0

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

⳵ ⳵t

冊

⫹ ⌫ˆ ␳ ⫽

␫ ប

关␳, V兴,

Tr ␳ ⫽ N,

␥ ⬎␣

␫

⌫ ␥␣ ␳ ␥␥ ⫹

兺

ប

␥

⫻ 共 ␳ ␣␥ V ␥␤ ⫺ V ␣␥ ␳ ␥␤ 兲 ,

(5)

where the rate ⌫ ␥␣ of decay transitions ␥ → ␣ , which is large in organic molecules owing to the nonradiative conversion, almost completely coincides with the total decay rate ⌫ ␣␣ . We neglected the space derivatives at the lefthand side of the kinetic equations [Eq. (5)] that are responsible for the Doppler effect 关 exp(⫾␫kz) → 1兴 because the Doppler broadening in the condensed absorber is negligible compared with the dephasing rate, ⌫ ␣␤ . 9 Making use of Eq. (5), we derived the density matrix related to the ASE field6,7:

␫ ␮ ␩ ␣␤ ⬇

兺共r ␥

␮ ␣␥ G ␥␤

␮ ⫺ G ␣␥ r ␥␤ 兲

⌫ ␣␤ ⫺ ␫ 共 ␻ ␮ ⫺ ␻ ␣␤ 兲

␮ ⫽ L, R,

,

(6)

␮ where G ␥␤ ⫽ E␮ –d␣␤ /2ប are the Rabi frequencies of the ASE field and r ␣␥ are the elements of the density matrix induced by the pump field (see Refs. 6 and 7 for details). Apparently, the populations r ␣␣ , created by the pump field, contribute mainly to the density matrix [expression (6)]. The off-diagonal elements r ␣␤ are small because of their off-resonant character, though they are included explicitly. In our case the main mechanism initiating redistribution of populations is three-photon absorption. To ensure this in the modeling, we applied special resonant conditions: ␻ ⫽ ␻ 10 /3 where ␻ is the frequency of the vertical transition from the ground to the first excited state of the molecule. We neglected the time derivative in expression (6) because of the large value of the dephasing rate in solutions (see Refs. 6, 7, 9, and 10).

冉

(3)

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

冉

⳵t

兺

C. Final Wave Equations for the Intensities ␮ A substitution of the polarization, P␮ ⫽ 兺 ␤␣ d␤␣ ␩ ␣␤ , in wave equations (3) yields finally the following paraxial equations for the components of the ASE field:

PR ,

1 ⳵ ⳵ ␫ ␫kL ⫺ ⌬⬜ EL ⫽ PL , ⫹ ⫺ ⳵z c ⳵t 2k L ␧0

⫹ ⌫ ␣␤ ␳ ␣␤ ⫽ ␦ ␣ , ␤

(1)

propagating, respectively, to the right and to the left (R component and L component below), along the z axis that is parallel to the propagation direction of the pump field. The R component propagates in the same direction as the pump field. The strengths, E␮ , the frequencies, ␻ ␮ , and the wave vectors, k␮ , of the ASE fields are indexed by ␮ ⫽ R, L. Three-photon absorption of the pump field with the forthcoming nonradiative decay to a long-lived state results in the population inversion leading to the ASE. In turn, the ASE field affects the population distribution when it becomes strong. This effect makes the propagation of the ASE field nonlinear. The polarization oscillating with the frequency of the ASE field has the following structure:

⫹ PL exp共 ⫺␫ ␻ L t ⫺ ␫ k L z ⫺ ␫ ␸ L 兲 ⫹ c.c.,

冊

⳵

(4)

where V is the interaction between the molecules and the electromagnetic field. The concentration of the absorbing molecules is denoted by N, and ⌫ˆ is the relaxation matrix. We used this equation to derive a general formalism describing the interaction of a many-mode electromagnetic field with a nonlinear medium.7 The kinetic equations for populations, ␳ ␣␣ , and off-diagonal elements, ␳ ␣␤ , of the density matrix read as

冉

⳵

1 ⳵

␫

冊冊

␫kR ␧0

兺d

1 ⳵ ⳵ ␫ ␫kL ⫹ ⫺ ⫺ ⌬⬜ EL ⫽ ⳵z c ⳵t 2k L ␧0

兺d

⳵z

⫹

c ⳵t

⫺

2k R

⌬⬜ ER ⫽

␤␣

␤␣

R ␤␣ ␩ ␣␤

,

L ␤␣ ␩ ␣␤

. (7)

If the transverse inhomogeneity of the field is small, one can directly write down the equations for the intensities I R ⫽ c␧ 0 兩 ER 兩 2 /2 and I L ⫽ c␧ 0 兩 EL 兩 2 /2:

冉冉

⳵ ⳵z

⫹

1 ⳵ c ⳵t

冊冊

冋冋

I R ⫽ ␻ R Im 共 ER 兲 *

兺d ␤␣

册册

R ␤␣ ␩ ␣␤

,

(8)

⳵ 1 ⳵ L ⫺ I L ⫽ ␻ L Im 共 EL 兲 * d␤␣ ␩ ␣␤ , ⫹ ⳵z c ⳵t ␤␣

兺

(9)

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

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ing expressions for the field amplitudes, ER ⫽ (2I R /c␧ 0 ) 1/2 , EL ⫽ (2I L /c␧ 0 ) 1/2 . Here ER ⫽ eER , EL ⫽ 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 ignore some important nonlinear optical processes such as self-focusing. The main reason of going for a onedimensional approximation was computational feasibility. D. Averaging over Orientations As the molecules in solution have random orientations, we have to average the right-hand side of the field equa␮ tions over molecular orientations: P␮ → 具 P␮ 典 or ␩ ␣␤ ␮ → 具 ␩ ␣␤ . We perform the orientational averaging over 典 the angle 0 ⭐ ␸ ⭐ 2 ␲ between the molecular axis and the polarization vector, 储 x, numerically.6 It is worth noting that our averaging procedure ignores rotations of molecules because they are slow in comparison with the inverse broadening of the spectral transitions. E. Influence of the Amplified Spontaneous Emission Field on Populations The strict equations for populations take into account the change of populations by the ASE field:

冉

⳵ ⳵t

冊

⫹ ⌫ ␣␣ r ␣␣ ⫽

兺

␤ ⬎␣

⌫ ␤␣ r ␤␤ ⫹ W ␣ ⫹ W ␣R ⫹ W ␣L . (10)

The pump and ASE fields change the population of the ␣th level with the probabilities per unit time: W ␣ ⫽ 2 Im

兺关G

共 1,0兲 ␣␤ r ␤␣

W ␣␮ ⫽ 2 Im

兺关G

␮ ␮ ␣␤ ␩ ␤␣

␤

␤

1,0兲 ⫺ r 共␣␤ G ␤␣ 兴 ,

␮ ␮ ⫺ ␩ ␣␤ G ␤␣ 兴,

put laser intensity, and pulse duration are taken from Ref. 2. These and other input parameters are collected in our previous paper (see Tables 1 and 2 in Ref. 6). It is worthwhile to stress that the L and R components of the ASE field have different frequencies in general.3 The experimental data2 show that ␻ R ⫽ ␻ L for the studied APSS molecule. We used this fact in our simulations.

B. Five-State Model Having performed an analysis of the calculated transition dipole moments of the APSS molecule, we conclude that the transitions among the singlets S 0 , S 1 , and S 2 are most essential for the nonlinear optical process we are interested in. Therefore we chose these three states (see Fig. 1) for our simulations. As we learned from the previous experiment,2 the ASE and pump pulses are delayed with respect to each other. That is the reason we took into account the vibrational structure of the singlets S 0 and S 1 . We modeled this structure 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 for calculations 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 levels near the point of vertical photoabsorption transition. State 4 decays nonradiatively to state 3, giving rise to the population inversion between states 3 and 2. The lasing transition between these states corresponds to the vertical decay. Afterward, the molecule experiences nonradiative decay to the lowest vibrational state 1.

(11)

respectively. The pump and ASE fields are assumed linearly polarized with the same direction of the polarization vector. Equation (10) is strict: The only approximation ␮ we used is the density matrix, ␩ ␣␤ [expression (6)], induced by the ASE field. When the system is long, the ASE field becomes high enough to change the populations, in accordance with Eq. (10). It leads to a decrease of the population inversion, r 33 ⫺ r 22 , and hence to a ceasing of the amplification.

3. NUMERICAL SIMULATIONS A. Computation of Dipole Moments and Excitation Energies We applied the above theory to studies of upconverted stimulated emission, based on three-photon absorption of the organic stilbene chromophore 4-[N-(2-hydroxyethyl)N-(methyl)amino phenyl]-4⬘-(6-hydroxyhexyl sulfonyl) (abbreviated as APSS) dissolved in dimethyl sulphoxide.2,6 The influence of the solvent was modeled by the collisional dephasing rate, which is assumed to be the same, 0.01 eV, for all transitions. For details of the electronic structure calculations, see Ref. 6. The experimental data such as the concentration of molecules in the solvent, length of the active medium (cuvette length), in-

387

Fig. 1.

Energy-level diagram.

388


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Table 1. Basic Results of Simulationsa

4. RESULTS AND DISCUSSION A. Efficiency of Nonlinear Conversion An important question concerning induced upconverted lasing in a nonlinear medium concerns how much of the absorbed energy of a pump pulse is converted into the ASE radiation. The parameter answering this question is the net conversion coefficient, ␩ ⫽ ␩ ⬘ /(1 ⫺ T), where ␩⬘ is the ratio of the energy of the overall output ASE (both forward and backward) to the energy of the incident pump pulse and T is the transmission of the pump pulse. To make a comparison between the R and the L components of the ASE, we calculated the partial coefficients, ␩ ␮ ⬘ and ␩ ␮ . The overall coefficients can be found as a sum of the partial ones. The results of our calculations of partial ␩ ␮ are collected in Table 1 and visualized in Fig. 2. We used six values of the concentration for comparison. As one can see, for each value of the concentration the net conversion coefficient, ␩ R , is larger in the case in which the left propagation is not accounted for. The cause behind this observation is that a part of the absorbed energy is consumed by the left-propagating ASE pulse in the opposite case. For the concentration value of 3 ⫻ 1019 cm⫺3 (0.05 mol/ l), which is a bit lower than that used in the previous experiment,2 the net conversion coefficient ␩ R was found to be equal to 4.20% without consideration of the L component. When counterpropagation is taken into account, the net conversion coefficient ␩ R changes almost negligibly to 4.19%. The overall ␩ equals the sum ␩ R ⫹ ␩ L ⫽ 5.95%, and the experimental value is 2.1%. This difference between our simulations and the experiment can be attributed to a smaller output ASE energy in the case of a higher experimental concentration (0.06 mol/l) owing to reabsorption of the ASE pulses (see discussion below). Indeed, extrapolation of the conversion function (see Fig. 2) to the experimental concentration value 0.06 mol/l gives 3.6%. Another reason is that the threephoton-absorption cross section is overestimated in our simulations owing to numerical accuracy of ab initio computations of the dipole moments and of the excitation energies. The higher-absorption cross section results in the larger portion of the incident energy converted to the ASE field. As one can see from Table 1, the difference between ␩ R for the single R-component case and the two-component case grows as the concentration of molecules decreases. The reason for this is found in the temporal oscillations of the ASE R component at the end of the active medium when both R and L components are accounted for. We checked numerically that the oscillations cannot be attributed to the accuracy of simulations. The origin of these oscillations needs a somewhat more elaborate investigation and will be described briefly in Section 5 and more in detail elsewhere. We note here only that the period and amplitude of oscillations strongly depend on concentration, as demonstrated in Fig. 3(a). As the concentration decreases, the period of oscillations increases and the peaks become wider. A similar behavior is observed for the L component, as demonstrated in Fig. 3(b). However, as we can see in the figure, the amplitude of oscillations is larger

␩ ␮⬘, % N ⫻ 1019, cm⫺3 0.01 0.1 0.5 1 2 3

T

Rb

R

␩␮ , % L

0.988 0.31 0.24 0.10 0.888 3.99 3.55 0.51 0.561 12.27 11.85 0.93 0.346 12.33 12.11 1.12 0.181 6.67 6.63 1.47 0.122 3.67 3.68 1.55

Rb

R

L

26.64 35.81 27.94 18.87 8.15 4.20

20.71 31.92 26.97 18.52 8.10 4.19

8.20 4.56 2.11 1.72 1.80 1.76

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

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

for the L component, which certainly reflects the asymmetry of the ASE propagation with respect to the pump. In fact, even in the case in which we consider only the R component of the ASE we can observe such oscillations at certain distances from the left edge of the cell. For example, for N ⫽ 3 ⫻ 1019 cm⫺3 , we observe oscillations at distances shorter than 2 mm from the left edge. For a given concentration, the increase of intensity of the ASE along the medium leads to the fast attenuation of oscillations. For N ⫽ 3 ⫻ 1019 cm⫺3 , the oscillations are bright at 0.5 mm from the entrance to the cell when the ASE intensity is approximately 1 GW/cm2. When the ASE propagates to the right, its intensity grows, and, at 3 mm, when the intensity equals 2.5 GW/cm2, the oscillations disappear. So the slow growth of the intensity means slow attenuation of oscillations, which is the case of lower concentrations. We have to stress here that for concentrations lower than a certain value, when only the R component of the ASE is taken into account, the oscillations vanish completely—we do not observe them at N ⫽ 1 ⫻ 1018 cm⫺3 and N ⫽ 1 ⫻ 1017 cm⫺3 . Once the L component is accounted for, the oscillations reappear.

Baev et al.

On the basis of these facts we can conclude that oscillations built up in a certain point of the medium are blurred as the ASE pulse propagates to the right with an increase of intensity. The closer to the left edge the pulse emerges, the longer distance it travels through. Apparently, for higher concentrations of the absorbing molecules the formation of the ASE pulse is faster in space and time, according to Eq. (8). The degree of delay of the leading edge of the R and L components depending on concentration, seen in Fig. 3, proves this statement. The temporal behavior of the single R component and of both the R and the L components of the ASE is shown in Fig. 4 for N ⫽ 1 ⫻ 1018 cm⫺3 . We see that in the case when the L component is taken into account the R component experiences oscillations compared with the opposite case in which the ASE temporal profile goes down smoothly. Owing to this circumstance, the net conversion coefficient, ␩ R , is different. Let us now pay attention to the fact that the efficiency of the conversion is a nonlinear function of the concentration of the active molecules (see Fig. 2). We know that the peak intensity of the ASE in an optically thin medium is determined by the abruption of the population inversion that occurs when the ASE intensity approaches some

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


Fig. 4.

389

ASE intensity dynamics.

critical value. This value refers to the resonant saturation intensity of the lasing transition and is independent of concentration. In our case the saturation intensity of the 3 → 2 transition is approximately equal to 3 MW/cm2. As the ASE pulse propagates, it gains intensity until some maximal value is reached. This maximal value is determined by the pump depletion; i.e., when the pump pulse intensity is less than the lasing threshold intensity, the ASE ceases to grow and experiences reabsorption as it propagates through the medium. If the pump pulse is almost depleted near the entrance to the cell, which is the case for higher concentrations, the rightpropagating ASE pulse starts to become reabsorbed far from the right edge of the cell. In the case of lower concentrations, the ASE pulse is formed further from the left edge of the cell, which means a shorter reabsorption path for the R component. At the given length of the gain medium (10 mm in our case), the output energy of the forward ASE pulse is maximal for a certain value of the concentration. It then drops down as the concentration decreases because the ASE starts to be formed so close to the right edge of the cell that it does not have time to reach the maximum. It is interesting that the percentage of the incident energy transferred to the L component of the ASE decreases monotonously with decreasing concentration (see Table 1). One reason for this behavior is that the L component is always formed near the entrance to the cell (⬃1 mm according to Figs. 7 and 8), where the inversion of populations can be shared by two components. Figure 5 shows that the maximum level of the ASE R-component intensity is approximately the same for all values of the concentration. But for the lowest value this maximum is reached at the end of the medium. Thus a decrease of the concentration shifts the maximum of the intensity toward the end of the cell. Our simulations also show a good agreement of the full width at half-maximum (FWHM) of the ASE pulse (FWHM ⬇ 22 ps) with the experimental number2 of 40 ps and a worse agreement of the delay between the ASE and the pump pulse (⌬ ␶ ⬇ 3 ps) with the experimental number2 of 12 ps. The theoretical results were obtained for the experimental concentration (0.06 mol/l) and pump intensity (190 GW/cm2).

390


B. Formation of the Amplified Spontaneous Emission Pulses To understand how the right- and left-propagating pulses of the ASE are formed, let us first make a simplifying assumption. We consider for the moment that we have an infinitely narrow source of photons, moving to the right along the z axis with the speed of light. Narrow means that the lasing threshold is suddenly overcome at a point inside the medium at a certain time and then suppressed at the same time because the population inversion is eliminated by the emitted photons. This source is the gain. So at each point of the medium a bunch of photons is 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 lapses ⌬t. The photons emitted to the right at time instant t ⫽ 0 will then move further with the gain. At time ⌬t, a

Fig. 5. Three-dimensional dependencies of the R-component intensity under different concentrations: (A) 1 ⫻ 1019 cm⫺3 , (B) 2 ⫻ 1019 cm⫺3 , and (C) 3 ⫻ 1019 cm⫺3 .

Baev et al.

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

new portion of photons will be emitted. The gain will be at the point c⌬t as well as the photons emitted at a preceding moment. So the R component of the ASE would be infinitely narrow in this case and would be growing with negative gradient as the gain is being depleted. In contrast, the left-propagating ASE component would be broad. The reason is the following. At time t ⫽ 0 a bunch of photons is emitted to the left. At time ⌬t the gain coordinate is c⌬t, and the previously emitted leftpropagating photons are located at ⫺c⌬t. In a case of only two emission acts (for instance, the gain is then negative), the temporal width of the left-propagating ASE component is 2⌬t. The pulse would look like a saw jag if the gain were decreasing. Let us stress once again that such an asymmetry of the right- and left-propagating components of the ASE is owed to the right-propagating longitudinal pump. Apparently, the picture of the formation of the ASE pulses given above is a naı¨ve one. In reality the gain is continuously decreasing through the medium if observed at a certain instant of time (see Figs. 7 and 8 at t ⫽ 15 ps). The reason for this is the population of the lasing level 3 through the exponential decay of level 4. Such character of the gain is maintained until the intensity of the ASE components is smaller than the saturation intensity of the lasing transition. When the ASE intensity becomes comparable with the saturation intensity, the ASE starts to change the populations and hence the gain (Figs. 7 and 8). The broad dips of the gain are formed by the leading edges of the R and L components. The leading edges with high intensity cut off the gain owing to an abrupt decrease of the population inversion. Because of the continuous gain, the R component of the ASE becomes broad, with a steep front and a long, slowly decreasing tail. The forward pulse is growing, chasing the leading edge of the decreasing gain. This leading edge of the gain looks like our model narrow source of photons—it supports the growth of the leading edge of the forward ASE pulse. The source amplitude decreases because the pump pulse, creating the population inversion, is absorbed when it runs through the medium. As mentioned earlier, the formation of the L component of the ASE takes place at a very short distance from the

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391

entrance of the cell, approximately 1 mm. As we can see in Fig. 8, the peaks of the gain supplying photons to R and L components are located at the same distance from the left edge of the cell at different time instants. The righthand slope of the gain, spanning to the right, is ‘‘eaten away’’ by the forward ASE pulse because its intensity at this short distance of 1 mm reaches the saturation intensity of the lasing transition. It is necessary to note that the output energies of the R and L components of the ASE differ substantially owing to the asymmetry with respect to the pump. Fig. 9.

5. OSCILLATORY MODEL To check the numerical stability of the temporal oscillations of the ASE, we applied a simplified open model con-

Simplified model of the ASE.

sisting of two states, of a pump, and of a few decay channels (see Fig. 9). The physics of the model can be described by means of two rate equations for populations and one wave equation:

冉

冉冉

⳵ ⳵t ⳵ ⳵t

⳵ ⳵z

冊冊

⫹ ⌫ 1 ␳ 1 ⫽ ⌫ N N ⫺ pI 共 ␳ 1 ⫺ ␳ 0 兲 , ⫹ ⌫ 0 ␳ 0 ⫽ ⌫ 1 ␳ 1 ⫹ pI 共 ␳ 1 ⫺ ␳ 0 兲 , ⫹

1 ⳵ c ⳵t

冊

I ⫽ 关 B 共 ␳ 1 ⫺ ␳ 0 兲 ⫺ ␣ 兴 I ⫽ gI.

2 d 10 /⌫ប 2 c␧ 0 ,

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

(12) 2 2 ␻ ␮ d 10 /⌫បc␧ 0 ,

Here p ⫽ B⫽ d 10 is the 1 → 0 transition matrix element, ␻ ␮ is the ASE frequency, ⌫ is the dephasing rate, and ⌫ 1 and ⌫ 0 are the decay rates of populations 1 and 0, respectively. The pump is modeled by the constant concentration, N, and the pump rate, ⌫ N . The simulations of the ASE based on the model [Eqs. (12)] showed clearly that the oscillatory regime has nothing to do with the numerical accuracy but rather with physics. It is important to note that sensitivity of this regime to the effects of propagation is rather weak according to our simulations. It is therefore justified to neglect the term ⳵ I/ ⳵ z in the wave equation in order to calculate the frequency, ⍀, of oscillations. To obtain an equation for ⍀, we have to write down equations for the small deviations of populations, r 1 exp(␫⍀t) and r 0 exp(␫⍀t), and of the intensity, j exp(␫⍀t), from the stationary solution of Eqs. (12) occurring for zero gain, g ⫽ 0. A solution to the corresponding eigenvalue problem results in a cubic equation for the frequency: ⍀ 3 ⫺ ␫ ⍀ 2 共 ⌫ 0 ⫹ ⌫ 1 ⫹ 2pI 兲 ⫺ ⍀ 关 ⌫ 0 ⌫ 1 ⫹ 共 2 ¯␣ ⫹ ⌫ 0 兲 pI 兴 ¯ pI⌫ 0 ⫽ 0, ⫹ ␫␣

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

(13)

where ¯␣ ⫽ ␣ c. The solution of Eq. (13) is straightforward, resulting in three complex roots. The last root ⍀ 3 with zero frequency Re ⍀3 ⫽ 0 (see Fig. 12) has nothing to do with oscillations. The two other roots, ⍀ 1 ⫽ Re ⍀1 ⫹ ␫ Im ⍀1 and ⍀ 2 ⫽ ⫺Re ⍀1 ⫹ ␫ Im ⍀1 , with the same damping rate correspond to ASE oscillations with the same frequency (see Fig. 10). The most important parameter of our model [Eqs. (12)] is the off-resonant absorption coefficient, ␣, which was set to 1 ⫻ 103 cm⫺1 in our simulations. The decay rates and transition matrix element d 10 were set equal to those

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used in the complex model. The physical meaning of the quantity pI is the rate of stimulated emission and can be parametrized as pI ⫽ ⌫ 1 I/2I sat , where I sat is the resonant saturation intensity. The dephasing rate, ⌫, is one of the parameters determining the value of both the stimulated emission rate, pI ⬃ 1/⌫, and the off-resonant absorption coefficient, ␣ ⬃ ⌫. Therefore the balance between these two quantities, which is responsible for the appearance of oscillations, is strongly influenced by ⌫. Figure 11 shows that the oscillation frequencies lie on equipotential lines in the ( ␣ , pI) plane. The area below the line with the frequency 1 ⫻ 1011 s⫺1 in Fig. 11 is the region of slow oscillations. The frequency of the ASE oscillations in this area changes irregularly around the value of ⬃1 s⫺1. As mentioned above, the roots of Eq. (13) are complex, where Re ⍀ determines the oscillation frequency and Im ⍀ is the damping rate of oscillations. Figure 12 demonstrates the real and imaginary parts of the roots of Eq. (13) as functions of the stimulated emission rate, pI, at given ␣ that was set to 1 ⫻ 103 cm⫺1 . Let us have a closer look at a branch with positive values of the frequency. The points on this branch correspond to certain values of the damping rate. For example, at pI ⫽ 7 ⫻ 1010 s⫺1 , the oscillation frequency equals Re ⍀ ⫽ 2 ⫻ 1012 s⫺1 , and the damping rate equals Im ⍀ ⫽ 1.3 ⫻ 1011 s⫺1 . In this case the damping rate is much smaller than the frequency, and hence the oscillations are nicely visible as demonstrated in Fig. 10. We also have to note here that our estimation of the off-resonant absorption coefficient, ␣, for the five-state model gave the value of approximately 30 cm⫺1, which is 30 times smaller than the value used in the model [Eqs. (12)]. Our simulations showed that this estimated value results in fast damping of the oscillations—with a damping rate equal to the oscillation frequency. Such a disagreement is owed to the fact that the model [Eqs. (12)] is rough compared with the running-pump-five-state model. Nevertheless, the simplified model gives good qualitative agreement with the complex model and allows for a simple evaluation of the oscillation frequencies.

Baev et al.

Fig. 11.

Equipotential lines of the frequency, Re ⍀.

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

The experimental data [Fig. 3(b), Ref. 2] demonstrate temporal fluctuations of the ASE intensity with the characteristic time scale (ⲏ5 ps), which is larger than the time resolution of the high-speed streak camera used in the experiment (⬃2 ps). Our simulations (Fig. 3) show the oscillations with a similar period (⬃2–10 ps), which allows us to relate the experimental oscillations to those predicted by the model.

6. SUMMARY

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

Owing to recent successful synthesis of different threephoton-absorption-based materials, especially dyes, there are many ongoing discussions on the potential of new three-photon-absorption-based optical applications. Although such materials thus are in reach for the design of particular optimal applications, their speedy scanning still poses an arduous undertaking in the laboratory. It is therefore highly relevant to make a versatile modeling toolbox accessible for laboratory simulations of prospective materials under different experimental conditions.

Baev et al.

The research in our group has taken the view to combine into such a toolbox quantum-chemistry methods for predicting basic electronic–conformational structures and molecular properties with classical Maxwell’s theory to investigate the properties of materials with respect to the interaction with electromagnetic fields of various wavelengths and strengths. The details can be found in our previous papers.6,7 An important asset of this toolbox is that it transcends the power-series approach in which the nonlinear polarization is expanded over powers of the electric field; instead, we account for the coupling of multiphoton processes in strong optical fields and go beyond the so-called rotatory wave approximation, meaning that off-resonant, in addition to resonant, effects are considered. In this paper we have modeled and analyzed threephoton-pumped upconverted lasing in connection with recently conducted experimental demonstrations of this effect.2–4 Upconversion of near-IR (1.2–1.7-␮m) radiation to shorter wavelengths of approximately 0.5–0.6 ␮m is of particular importance for optical telecommunication and biophotonics. Compared with the wavelengths used to pump two-photon active materials, the longer IR range is preferable owing to specific characteristics of optical fibers and to deeper penetration of biological tissues. As opposed to our previous study,6 we accounted here for both forward and backward components of the ASE, which essentially enhances the predictive capacity of our model. The simulations showed that accounting for ASE in both forward and backward directions with respect to the pump pulse results in a smaller efficiency of nonlinear conversion for the forward ASE compared with the case in which forward emission is considered alone. The reason for this can be found in the partial repump of the absorbed energy to the backward ASE component. The overall conversion is, nevertheless, higher when both components are present. We found that the reabsorption of the ASE radiation decreases the efficiency of the nonlinear conversion for high concentrations of the active molecules in solution, so that a proper combination of concentration versus cell length should gives the maximal conversion coefficient. We observed an oscillatory regime of the ASE for the running-pump-five-state model. To figure out the origin of these oscillations, we developed a simplified model of the ASE and analyzed the domain of parameters allowing for observation of oscillations. The oscillations predicted by our theory seem to be observed in the previous experiment.2 Our future efforts will be put into further development of the oscillatory model, to find conditions for clear experimental observation of oscillations, which is important as an additional check of our toolbox. Simulations of the spectral distribution of the ASE intensity will also be con-


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

ACKNOWLEDGMENTS The authors are grateful to Paras N. Prasad and to Guang S. He for stimulating discussions. The authors acknowledge a grant from the photonics project run jointly by the Swedish Materiel Administration (FMV) and the Swedish Defense Research Agency (FOI). The authors also acknowledge support from the Swedish Research council (VR). A. Baev thanks the Swedish Research Council and the Swedish Foundation for International Cooperation in Research and Higher Education for a postdoctoral fellowship. A. Baev, the corresponding author, can be reached by e-mail at [email protected].

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