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The role of stimulated emission in luminescence decay Alexandra Rapaport, Michael Bass* School of Optics and Center for Research and Education in Optics and Lasers (CREOL), University of Central Florida, Orlando, FL 32816-2700, USA Received 20 May 2001; received in revised form 14 November 2001; accepted 14 November 2001

Abstract The effect of stimulated emission on observed ﬂuorescence dynamics was demonstrated. Experimental results show that the decay of strong transitions in four level systems is signiﬁcantly changed by stimulated emission even under conditions where the upper excited level population density is not noticeably altered. A new method for measuring the stimulated emission cross-section of a transition results and is presented. r 2002 Elsevier Science B.V. All rights reserved. PACS: 78.47; 32.50; 32.70.C Keywords: Luminescence dynamics; Ampliﬁed spontaneous emission; Stimulated emission cross-section; Excited state lifetime

1. Introduction In spectroscopy, different measurements are used in order to learn how impurities interact with the host matrix and with neighboring ions. Analysis of the luminescence decay dynamics of doped materials, which is accepted as being a signature of the time-behavior of the population density in the emitting level, is often used to obtain insight into the processes involved in various interactions (e.g., energy transfer and cross-relaxation). It is therefore essential to be aware of all possible processes against which to test the experimental data before drawing conclusions regarding the responsible mechanism. The present investigation concentrates on systems having four energy levels such as sketched in Fig. 1 and the mechanisms that must be taken *Corresponding author. Tel.: +1-407-823-6977. E-mail address: [email protected] (M. Bass).

into account to properly interpret the observed decay dynamics. Four level systems are of special interest as they result in efﬁcient laser media. Since the spectroscopy and luminescence dynamics of these media are used to judge their potential as lasers, the results in this paper are of particular interest. In systems where level 2 is strongly coupled to the ground state 1 it is very easy to produce an inverted population between levels 3 and 2. As a result, we studied how stimulated emission contributes to the luminescence decay dynamics of such a four level system even with weak pumping. The effects of stimulated emission have been extensively studied during the past 40 years since it was necessary in lasing media to understand and predict the role of stimulated emission on intensity growth and pulse ampliﬁcation. The dynamics of the various processes involved were explored and the physics of laser oscillators and ampliﬁers were broadly analyzed. The properties of ampliﬁed

0022-2313/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 0 2 ) 0 0 2 2 2 - 3

A. Rapaport, M. Bass / Journal of Luminescence 97 (2002) 180–189

1

τ

Level (4): Pumping state ( 4F5/2 +2H 9/2)

non _ rad 4

γ

Level (3): Upper excited state (4 F3/2)

Α31

1

τ 2non

_ rad

Α32

Β32

Level (2): Lower excited state ( 4 I11/2) Level (1): Ground state (4 I9/2)

Fig. 1. Energy level diagram of the system considered for the model. In parenthesis are indicated the corresponding states for Nd3+ ions.

spontaneous emission (ASE) were uncovered [1,2]: the effects of stimulated emission on the dynamics of the upper excited state (lifetime shortening) and on optical gain [3–6] were studied at length in media having large emission cross-sections such as dyes [7–12] and semiconductor materials [13,14], and when using strong pumping, or long emitting samples as in a laser with feedback [15,16]. In this paper, we report on the effects of stimulated emission on the time decay of the signal observed in luminescence measurements in rare-earth-doped crystals with extremely low gain. This is the case of weak pumping or low stimulated emission cross-section or both and yet we are able in the present study to show that stimulated emission cannot be neglected. We consider conditions in which stimulated emission affects the radiation detected but does not signiﬁcantly alter the population densities in the levels involved. Our experimental results show that the observed luminescence decay does not necessarily follow the decay of the population of the emitting level. The theoretical results derived in the thesis work of one of us (A.R.) [17] including the effects of both spontaneous and stimulated emission in the rate equations for luminescence decay in four level systems are used to evaluate the experiments. The following equation is derived in Appendix A and also in Refs. [9,17] gives the time decay of the ﬂuorescence intensity measured perpendicular to a pump beam: Iðt; DÞ ¼

LnA32;n ½expðs32 ðn3 ðtÞ n2 ÞDÞ 1 ; cs32 ð1 n2 =n3 ðtÞÞ

ð1Þ

181

where L is a geometrical factor characteristic of the detection system, n is the index of refraction of the material, A32;u is the Einstein coefﬁcient for spontaneous transition probability from level 3 to level 2 at frequency n; s32 is the stimulated emission cross-section of the transition 3 to 2, n3 ðtÞ is the population density in the upper excited level 3, n2 is the population density in the lower excited level 2 (supposed to be in thermal equilibrium with the ground state so n2 is very small and assumed to be constant) and D is the diameter of the pump beam. Ref. [17] also pointed out that the polarization of the emitted light, its spectral bandwidth and the pump beam distribution would affect the dynamics observed. We will illustrate the effect of the emission polarization and of the pump beam distribution later in this paper, but it is worth noting that Eq. (1) applies at any emission frequency n: When performing an experiment, the detection system collects light over a determined spectral bandwidth. If the stimulated emission cross-section varies signiﬁcantly over this bandwidth, then the intensity detected at each frequency contributes to the signal. The result is a signal with dynamics determined by a mixture of many different decay rates and can obscure the effects described below. Eq. (1) shows that the ﬂuorescence signal is dependent on the stimulated emission cross-section. We now show how to use the dynamics of the decay process to obtain a numerical value for the stimulated emission cross-section by ﬁtting of the luminescence decay signal recorded to Eq. (1). This does not require lasing at the wavelength studied [18–20] and is an easy technique to use as it only requires an excitation source, a spectral ﬁlter such as a monochromator, and a detector sensitive at the wavelength of the transition studied. We will now demonstrate experimentally the theoretical ﬁndings in Ref. [17] in the case of neodymium-doped barium ﬂuorapatite (BFAP). The experiments are conﬁned to the study of events occurring in time frames from hundreds of nanoseconds to hundreds of microseconds. In solid state materials at room temperature, this eliminates coherence effects such as superﬂuorescence [1,21–24], and insures that the population densities are thermalized.

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2. Nd3+:BFAP

1100

1200

1300

1400

1500

π-polarization

0.1

2 0.0

-19

2

(10 cm )

Emission cross-section

Experiments were conducted on samples of barium ﬂuorapatite (Ba5(PO4)3F or BFAP) doped with different concentrations of neodymium ions in order to illustrate the different behaviors uncovered by including the effect of stimulated emission on ﬂuorescence dynamics. The crystal preparation method as well as the energy level diagram for Nd3+ doped BFAP can be found in Refs. [17,25]. Nd3+ ion concentrations in the crystals were determined by electron microprobe analysis and the effective neodymium distribution coefﬁcient in BFAP was found to be B0.2. Emission cross-section spectra were previously obtained and are shown Fig. 2 to illustrate the strength of the different emissions. The experimental set-up is sketched in Fig. 3. A Q-switched, tunable Cr3+:LiSAF laser operating around 804 nm was used to excite the Nd3+ ions into their 4F5/2 state. The excitation pulse duration was 150 ns and the maximum pulse energy absorbed was about 6 mJ. Fluorescence was detected with a photomultiplier (risetime of less than a nanosecond) which was connected to a 3 GHz bandwidth oscilloscope. A 27.5-cm focal length monochromator as well as ﬁlters were used to select the wavelength of the signal detected.

1000

850

900

950

0

σ-polarization

0.1

2 0.0

0 1000

1100

850

900

1200

950

1300

1400

1500

Wavelength (nm) Fig. 2. Emission cross-section of Nd3+:BFAP at room temperature.

Photodiode

Focusing lens

Laser source (Q-switch)

Polarizer

Beamsplitter

Turning mirrors Sample

Photomultiplier

Collecting lens

Monochromator

Fig. 3. Experimental set-up for measurement of the effects of stimulated emission on luminescence decay detection.

2.1. Upper excited state population dynamics To isolate the effects of stimulated emission on the observed luminescence decay, we ﬁrst have to determine the upper excited state population dynamics. For that purpose, the ﬂuorescence signal due to the 4F3/2–4I9/2 transition was monitored. This transition near 850 nm not only has a small emission cross-section but is to the ground state. Population inversion is therefore very difﬁcult to achieve and the signal detected is certain not to be affected by stimulated emission. Self-absorption of this radiation by the ground state is ignored for several reasons. First, the absorption coefﬁcient at this wavelength is quite low [17,25], so few of the photons emitted at 850 nm are reabsorbed by the material. In addition, the excitation beam in the experiment pumped the material very close to the face of the

sample near the detector so that the path length through the material in which self-absorption could take place was minimized. As a result, the ﬂuorescence dynamics at 850 nm should exactly follow the decay of the upper excited level population density. A sample of 0.4% Nd3+:BFAP (the dopant concentrations given in this paper are the concentrations in the crystals) 6.3 mm high, 5 mm long and more than 9 mm wide was used. The beam obtained from the Cr:LiSAF pump laser was not a pure single mode. It was focused with a 7.5 cmfocal-length lens so that its diameter was about 0.4 mm when it entered the crystal. A 500 mm slit at the input of the monochromator allowed sufﬁcient ﬂuorescence to be detected, while a long-pass ﬁlter reduced the signal due to scattered 804-nm pump light.

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Nd3+ ions in the 4F3/2 level are known to be subject to cross-relaxation with the ground state as well as Auger upconversion with neighboring excited ions [26] in different hosts. Barium ﬂuorapatite is not free of these processes and the luminescence decay dynamics observed are nonexponential. The decay signal at 850 nm was recorded for several pump energies adjusted by introducing neutral density ﬁlters in the laser beam. Little variation if any was observed between low and high level of excitation. This rules out Auger upconversion (which would increase as more neighboring ions populate the 4F3/2 level). The data were then ﬁt with the mathematical form characteristic of an electric dipole–electric dipole cross-relaxation as given by the Forster–Dexter model [27–29]: t pﬃﬃ IðtÞ ¼ I0 exp expðg t Þ; ð2Þ t where I0 is the signal detected at peak emission (when the upper level population density is maximum), t is the lifetime of the upper excited level, and g is the parameter that characterizes the dipole–dipole interaction strength between neighboring ions. The ﬁt was performed on the data for different pump energies and g was found to be 21 s0.5 with a lifetime t of 390 ms. 2.2. Effects of stimulated emission on a strong transition The ﬂuorescence signal detected at the peak emission wavelength (1055.4 nm) in p-polarization was recorded for the same pump energies as used when examining the 850 nm emission. In these measurements, a 50-mm slit was placed in the monochromator. The resolution was 0.4 nm, the same as that used when performing the spectroscopic measurements of Fig. 2. Note that the 1055.4 nm transition is between levels 3 and 2 of the four level system corresponding to Nd3+ ion emission. The decay of 1055.4 nm luminescence is shown in Fig. 4a for an absorbed pump energy of about 1.2 mJ and the same focus as above (e.g., 0.4 mm diameter at the entrance face of the sample). In Fig. 4a, we see that the decay dynamics of the signal detected at the peak emission

Fig. 4. Luminescence decay detected at 1055.4 nm in p-polarization observed from a 0.4% Nd3+ doped BFAP sample. The thick solid curves in a and b represent decay following the Forster–Dexter model with a value of the parameter g ¼ 21 s0.5 for a pump beam focused to a diameter of 0.4 mm. (a) The absorbed pump energy was 1.2 mJ. (b) The absorbed pump energy was 6.4 mJ.

wavelength when weakly excited is similar to the one observed for the weaker transition to the ground state. Fig. 4b shows the signal detected at 1055.4 nm when the absorbed energy was increased to about 6.4 mJ. A comparison with the ﬁt of the data to the Forster–Dexter model using g ¼ 21 s0.5 in Fig. 4b proves that the time behavior of the signal detected for emission from the 4F3/2 level depends on the wavelength monitored when the pump energy absorbed is increased. This effect can only be caused by stimulated emission.

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To illustrate the consequences that stimulated emission can have when interpreting an experiment, the Forster–Dexter model alone was used to ﬁt the data in Fig. 4b at the peak emission wavelength, letting the parameter g vary. This yields a value of g ¼ 36:5 s0.5. However, from the measurement of the ground state transition decay we know that the decay of the excited level population is actually described by g ¼ 21 s0.5. Ignoring the effects of stimulated emission when evaluating spectroscopic measurements can clearly lead to erroneous interpretation of the dynamics of luminescence decay. This possibility is also evidenced by looking at the times at which the luminescence intensity reaches 1=e of its initial value when different wavelength emission signals are recorded. 2.3. Role of the polarization of the emission detected Using the same 0.4% doped Nd3+:BFAP sample, the signal emitted at 1055.4 nm for low and high levels of excitation was recorded for the s-polarization. Light polarized in this direction has a smaller emission cross-section than p polarized light. The results for low level of excitation are similar to the ones in Fig. 4a (the decay follows the upper excited level population density dynamics). However, the ﬂuorescence detected for a high level of excitation is shown in Fig. 5, along with the signal emitted in p-polarization for the same pump energy, and the Forster– Dexter ﬁt describing the population dynamics in the upper excited level. It is clear from these results that the signal measured is not a signature of the upper level population since the observed signal depends upon exactly which radiation is monitored. Furthermore, even though the stimulated emission cross-section in s-polarization is smaller than in p-polarization, the effect of stimulated emission on the dynamics of the signal detected can already be seen in Fig. 5. 2.4. Role of the pump energy The pump beam cross-section was assumed to be uniform in deriving the results in Ref. [17]. It

Fig. 5. Luminescence decay detected at 1055.4 nm in s-polarization (thin solid curve) observed from a 0.4% Nd3+ doped BFAP sample. The absorbed pulse energy was 6.4 mJ. The thick solid curve represents a decay following the Forster–Dexter model with a value of the parameter g ¼ 21 s0.5 for a pump beam focused to a diameter of 0.4 mm. The thin dotted curve represents the emission in p-polarization at the same wavelength for the same level of excitation.

was also assumed that stimulated emission does not signiﬁcantly affect the excited level population density. As a result, the dynamics of the excited level population density follows the Forster– Dexter model and is described by Eq. (2) at any point in the pumped volume. We use such an expression for the upper level population in the equation giving the time dependence of the radiation detected (Eq. (1)) and, neglecting the population density in the lower excited level, we ﬁnd that the time behavior of a signal detected at a speciﬁc wavelength follows the equation: h t i pﬃﬃ IðtÞ ¼ K exp L exp expðg tÞ 1 : ð3Þ t Here, K is a coefﬁcient proportional to the response of the detector, the geometry of the experimental set-up, and quantities characteristic of the transition monitored. L is given in Eq. (1) by: L ¼ s32 n3 ð0ÞD;

ð4Þ

where s32 is the stimulated emission cross-section of the strong transition considered, n3 ð0Þ is the initial upper excited state population density (time 0 is taken at maximum excited state population

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density) and D is the length along the direction of propagation of the detected signal (400 mm in our case). Eq. (3) was ﬁt to the data in Fig. 6 for an absorbed pump energy of about 6.4 mJ. The same sample of 0.4% doped Nd3+:BFAP was used so we know that the cross-relaxation parameter g has a value of 21 s0.5 (found when measuring the transition to the ground state for the same pump focus). The accuracy of the ﬁt is quite good even at early times, and the value of the parameter L obtained is 0.65. With the numerical values for our case (for an absorbed energy of 6.4 mJ and a pump diameter of 400 mm, n3 ð0Þ ¼ 4:4 1019 cm3) and this value of L; we obtain s32 ¼ 3:7 1019 cm2 from Eq. (4), instead of 3 1019 cm2 obtained earlier, spectroscopic measurements [17]. During our experiments, we noticed that stimulated emission along the direction of pumping could get so strong that it could actually deplete the upper excited level population. Nevertheless, we still observed different dynamics for different transitions from the same level indicating that the effects we report are not just determined by changes in the upper state population. However, one might be concerned that ampliﬁed stimulated

185

emission in the direction of pump light propagation might scatter into the detection system. The geometry of the detection optics makes this possibility negligibly small. To minimize any chance of such stimulated emission we made sure that the samples used did not contain any parallel faces to avoid resonator effects. As a result the length of material pumped as well as the exact energy absorbed were not very accurately known. Also, the energy distribution of the pump laser at the focal point was not uniform, leading to more inaccuracy. These experimental issues could be easily improved by using a TEM00 beam as a pump, and by using polished and AR coated samples. These steps would enhance the accuracy of this technique as a new and valid method to measure stimulated emission cross-sections. To further check the model, the pump energy delivered was adjusted by introducing neutral density ﬁlters in the laser beam. The geometry of the detection system was not changed, so the coefﬁcient K in Eq. (3) was kept constant while ﬁtting Eq. (3) for several pump energies. The results obtained for the coefﬁcient L for different incident pump energies are plotted in Fig. 7. From the analytical model, we know that L can also be expressed by L ¼ Cs32

Eincident ; D

ð5Þ

where C¼

Fig. 6. Luminescence decay detected at 1055.4 nm in p-polarization observed from a 0.4% Nd3+ doped BFAP sample. The absorbed pulse energy was 6.4 mJ. The thick solid curve is the best ﬁt to Eq. (3) including stimulated emission as well as Forster–Dexter decay of the upper level population density with g ¼ 21 s0.5 for a pump beam focused to a diameter of 0.4 mm. The parameter L obtained from the ﬁt is L ¼ 0:65:

1 expðalÞ ; hnpump l

a is the absorption coefﬁcient at the pump wavelength (measured to be 1.8 cm1), l is the length of material pumped, h is Planck’s constant and npump is the frequency of the pump. As seen in Fig. 7, the coefﬁcient L appears to be a linear function of the incident pump energy as expected from the model in Ref. [17], with a slope of 69 J1. From this slope we ﬁnd the stimulated emission cross-section to be 3.7 1019 cm2, consistent with the result from ﬁtting Eq. (3) to the data to ﬁnd a value of L: We would like to emphasize that this value is limited by the spectral resolution of our detection system: in the analysis above, we consider the

A. Rapaport, M. Bass / Journal of Luminescence 97 (2002) 180–189

Coefficient L obtained from fit to Eq. 12

186 0.8

0.6

0.4

0.2

0.0 0

2

4

6

8

10

Incident pump energy (in mJ)

Fig. 7. Values of L obtained from the ﬁt of the ﬂuorescence intensity detected at 1055.4 nm in p-polarization observed from a 0.4% Nd3+ doped BFAP sample to the model including stimulated emission for varying pump energies. The pump beam was focused to a diameter of 0.4 mm.

stimulated emission cross-section as constant over the bandwidth detected (about 0.4 nm). However, a higher-resolution monochromator could have yielded a more pronounced effect and a value for the emission cross-section closer to its peak value. 2.5. Role of focusing the pump beam From Eq. (5), we see that the dynamics of luminescence should depend on the pump beam diameter. We tested this relation by performing experiments on a sample of 0.8% Nd3+ doped BFAP. The pump energy was kept constant and the focusing lens was translated in order to change the spot size on the sample. The absorbed pump energy was held at about 4.5 mJ. Fig. 8a and b show the luminescence decay for the transition from the 4F3/2 state to both the ground level (near 920 nm) and the 4I11/2 level (at 1055.4 nm) in p-polarization. In Fig. 8a, the pump beam was focused to a spot size of about 1.5 mm in diameter, whereas in Fig. 8b it was focused to a spot size of 0.4 mm. In each ﬁgure, a ﬁt to the model including stimulated emission was performed on the decay detected at 1055.4 nm in p-polarization. The cross-relaxation rate was ﬁxed to the value found using Forster–Dexter model to ﬁt the decay signal recorded for a ground state

Fig. 8. Luminescence decay detected near 920 nm and at 1055.4 nm in p-polarization observed from a 0.8% Nd3+ doped BFAP sample. The absorbed pulse energy was 4.5 mJ. (a) The pump beam diameter was 1.5 mm. Signal detected near 920 nm. A ﬁt to a Forster-Dexter model yields g ¼ 18:7 s0.5. Signal detected at 1055.4 nm in p-polarization. Best ﬁt to Eq. (3) including stimulating emission and a Forster– Dexter decay of the upper level population density with g ¼ 18:7 s0.5 yields a value for the coefﬁcient L of 0.51. (b) The pump beam diameter was 0.4 mm. Signal detected near 920 nm. A ﬁt to a Forster–Dexter model yields g ¼ 24:7 s0.5. Signal detected at 1055.4 nm in ppolarization. Best ﬁt to Eq. (3) including stimulating emission and a Forster-Dexter decay of the upper level population density with g ¼ 24:7 s0.5 yields a value for the coefﬁcient L of 0.85.

transition. The effect of the pump size on the luminescence decay clearly appears in those two ﬁgures. We could attempt to verify the inverse relationship between the coefﬁcient L and the pump diameter. However, the pump beam proﬁle available to us was far from being uniform, and

A. Rapaport, M. Bass / Journal of Luminescence 97 (2002) 180–189

the pumping geometry varied as the focusing lens was moved. Both these factors affect the results preventing proper numerical analysis.

3. Comment concerning experimental procedures When performing ﬂuorescence dynamics measurements, it is common to increase the signal-tonoise ratio by averaging several recorded signals. Mathematically, it does not matter what exact pump energy is incident on the sample: the average of several exponential decays having the same exponent is still an exponential of the same exponent. However, we have shown that when stimulated emission is properly taken into account, the pump energy appears in the exponent of the exponential. It is therefore critical to keep the pump energy stable when performing the experiment to maintain the mathematical form derived (Eq. (1)) after averaging several decays. The accuracy of this technique as a means to measure stimulated emission cross-sections could be improved by using a pump beam of homogeneous proﬁle, as is assumed when deriving Eq. (3). In Ref. [17], the case of a pump beam having a cos2 proﬁle is also considered. It was observed that the dynamics of the luminescence decay are different for the two pump proﬁles even though the pump energy was held constant. We therefore suspect that the numerical results obtained during our study might be slightly off, as no adequate pump source was available to us at the time. What is important is that we showed that stimulated emission alters luminescence decay dynamics as predicted, and that this phenomenon could be used to ﬁnd the stimulated emission cross-section of the transition studied. The change in the dynamics resulting from stimulated emission is determined by the value of the stimulated emission cross-section averaged over the detected bandwidth and polarization.

4. Conclusion In this paper, we explain the role of stimulated emission on luminescence dynamics when neither

187

stimulated nor spontaneous emission dominates the other. Stimulated emission alters the luminescence decay detected without signiﬁcantly modifying the upper excited level population density. As a result, the luminescence decay observed can no longer be considered a signature of the upper excited level population density. We use the model developed in Ref. [17] to show how to use the luminescence dynamics to determine the stimulated emission cross-section of the transition studied. Experiments were performed to demonstrate how various parameters connected with stimulated emission can alter luminescence dynamics. These studies showed the signiﬁcant difference in the decay that can be observed when measuring luminescence at different wavelengths and for different polarizations. The mathematical expression for the photon density time dependence (Eq. (1)) was validated by studying its dependence on pump energy and diameter. The critical point is that we have shown that the luminescence decay observed during dynamic spectroscopic measurements is not always a true signature of the upper excited level population density. Our work demonstrates that the luminescence decay dynamics can depend on stimulated emission even in unlikely conditions of pumping and in systems having small stimulated emission cross-sections. We also show that properly understanding luminescence dynamics enables one to determine the stimulated emission cross-section of the material studied.

Acknowledgements We would like to thank Prof. G. Loutts and Dr. M. Noginov from Norfolk State University who provided us with the BFAP crystals.

Appendix A We consider the case of a four level system as shown in Fig. 1 and develop the set of rate equations that describe its behavior. Assume that the pumped level 4 quickly decays non-radiatively to the upper excited level 3 and that level 2 is

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strongly non-radiatively coupled to the ground level 1. Then include in the rate equations the stimulated emission term for the strongest transition from level 3 to level 2. The rate equations describing such a system are n2 n’ 1 ¼ gn1 þ non-rad þ A31 n3 ; t2 n2 n’ 2 ¼ non-rad þ A32 n3 þ B32 UðnÞðn3 n2 Þ; t2 n4 n’ 3 ¼ non-rad A31 n3 A32 n3 B32 UðnÞðn3 n2 Þ; t4 n4 n’ 4 ¼ gn1 non-rad : ðA:1Þ t4 In these equations, ni is the population density of level i; g is the pumping rate, ti is the lifetime of level i; Ajk is the spontaneous transition probability from level j to level k; Bjk is the stimulated emission probability for the transition between levels j and k; and UðnÞ is the energy density per unit frequency at the frequency of the strongest transition. We can write [2] B32 UðnÞ as c=ns32 ðnÞF; where c is the speed of light, n is the index of refraction of the material, s32 ðnÞ is the stimulated emission cross-section of the transition 3 to 2 (we drop the notation s32 ðnÞ and just specify s32 ) and F is the photon density at the frequency associated with that transition considered over all directions in space. We can also take into consideration the fraction L of the 4p solid angle of the spontaneous emission that is collected by the lens. The photon density in that particular solid angle collected by the lens grows according to the differential equation: qf c ¼ LA32;n n3 þ s32 ðn3 n2 Þf: ðA:2Þ @t n Assuming the pump beam to be homogeneous over a surface of square cross-section (length of one side: D), the equation that describes the propagation of this photon density, starting from the slice of excited material furthest from the detector to the slice closest, is qfðt; zÞ nh ¼ LA32;n n3 ðt; zÞ qz c i c þ s32 fðt; zÞðn3 ðt; zÞ n2 ðt; zÞÞ : ðA:3Þ n

We now restrict ourselves to the case of typical spectroscopic measurements. Unless pumping is unusually strong or feedback is present allowing UðnÞ or F to build up, the effect of stimulated emission on the population densities of the energy levels is much lower than that of spontaneous emission. When this condition applies, n3 does not depend on z and the stimulated emission term can be dropped from the set of Eqs. (A.1). However, stimulated emission can still contribute to the detected photon density (see Eq. (A.2)). Under the assumption that the energy density as well as the photon emission rate vary slowly with respect to the time it takes for a photon to escape from the pumped region, we can solve Eq. (A.3) to obtain fðt; DÞ ¼

LnA32;n ½expðs32 ðn3 ðtÞ n2 ðtÞÞDÞ 1 : cs32 1 n2 ðtÞ=n3 ðtÞ ðA:4Þ

For the simple system described by Eqs. (A.1), we can assume that the decay from the pump level to the emitting level is fast. Under this assumption, after absorption of the pump pulse energy, the solution for the rate equations considered is n3 ðtÞEnexcited expðtðA31 þ A32 ÞÞ:

ðA:5Þ

Using Eqs. (A.4) and (A.5), we can now write the equation for the photon density detected and obtain the same time dependence as in the expression given in Ref. [9]: fðt; DÞ ¼ LnA32;n ½expðs32 ðnexcited expðt=t3 Þ n2 ÞDÞ 1 : cs32 ð1 n2 =nexcited expðt=t3 ÞÞ ðA:6Þ The last assumption made in this derivation is that the excited population density is not signiﬁcantly reduced by the stimulated emission. This assumption is certainly correct if the stimulated emission due to the photon density at its strongest (that is at the exit face of the sample) is still negligible compared to the total spontaneous emission. This assumption will have to be veriﬁed in each particular case studied, as it depends on the pumping energy density, the various strengths of emission, and the doping concentration.

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The solution obtained in Eq. (A.1) is the photon density at the exit face, which then propagates, is collected by the lens and produces the electric signal given by the detector. As can be observed fðt; zXDÞ is not similar to the time-variation of the upper excited level population density. This means that in spectroscopic measurements in which the signal emitted at the strongest emission line is monitored, the decay observed might not represent the actual dynamics of the population of the excited level studied.

[13]

[14]

[15]

[16]

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