## Abstract

In this study, we take the pump rate into consideration for the first time to give a theoretical description of radiation trapping in three-level systems. We numerically verify that under strong pumping, the population of the ground state is depleted, which leads to saturation of the radiation trapping within the pumped region. This saturation inevitably clamps the lifetime lengthening that is experimentally verified on a 0.05 at% thin ruby crystal based on the axial pinhole method. Our model is confirmed to be valid in lifetime measurement when the ruby fluorescence is collected from both the pumped and the unpumped regions.

© 2012 OSA

## 1. Introduction

Radiation trapping is the phenomenon that a photon emitted by an excited atom or ion can be absorbed by another ground-state atom or ion of the same type and re-emits before the photon leaves the sample volume. This phenomenon is also called radiation reabsorption and is well known responsible for lengthening the measured lifetime beyond the intrinsic radiative lifetime of the individual emitter. It was theoretically addressed in gases as early as 1920s [1,2]. However, there has been a renewed interest on this issue in recent years because the trapping effect is evident in solid-state laser materials [3–5].

The trapping effect can be separated into two cases according to the sample size
_{$\ell $}. For _{$\alpha \ell $}>> 1, _{$\alpha $} being the absorption coefficient, the sample is
large enough to allow many events of emission-reabsorption and the lifetime lengthening can be
observed. For _{$\alpha \ell $}<< 1, a photon may not be reabsorbed in one
pass through the sample but it will be reabsorbed after many passes inside the sample due to
total internal reflection (TIR), thereby the lifetime lengthening can also be observed. Many
axially pumped solid-state laser crystals belong to the latter case; however, in this paper, we
focus on the systems with the intermediate radiation trapping which cannot be considered as
_{$\alpha \ell <<1$}.

In the assumption of uniform radiation transfer due to TIRs, Guy [3] modeled the radiation trapping by dividing the sample into two parts: the
pumped region and the unpumped region. He showed that the measured lifetime is related to the
coupling of the two regions so that it is dependent on the collecting arrangement. Afterwards,
Kühn et al. [4] simplified the problem by considering only
the pumped region with spatially homogeneous excitation but neglecting the fluorescence from the
unpumped region. In their experiment called the pinhole method, the intrinsic lifetime was
determined. Recently, Toci [5] indicated that the
emission-reabsorption couplings in Guy’s rate equations should be modified when the short-range
radiation transfer was considered because the probability of excitation of another ion or atom
is always much higher than on average in the proximity of the emission site. Under the
assumption of _{$\alpha \ell $}<< 1, the short-range coupling was formulated
as a linear dependence of _{$\alpha \ell $} [5]; however,
in the intermediate radiation trapping regime, the couplings are lack of study.

Furthermore, the previous studies on radiation trapping always assumed an unchanged
ground-state population density because it is much larger than that of the upper level. However,
under strong pump in three-level systems, the increase of the upper level population strengthens
the emission while the decrease of the ground state population weakens the reabsorption. This
situation may occur in either lightly or heavily doped three-level materials with high pump
rate. Therefore, how depletion of the ground-state population influences the radiation trapping
is worth studying. To our knowledge, no theoretical approach describing this effect has yet been
given. On the other hand, high pump will induce other serious effects such as phonon-assisted
radiative and nonradiative energy transfer in a heavily doped crystal. So, in this paper, we
choose a 0.05 at% ruby disk (_{$\alpha \ell \approx 0.24$}without pump) to study radiation trapping. We model
the radiation trapping by taking into account the pump rate in a three-level system whose energy
diagram is shown on the left side of Fig. 1
. The right side of Fig. 1 is the geometry of our
axial pinhole method in numerical simulation and experiment. The volume ratio of the pumped
region to the whole crystal is very small in our axially focused pump scheme. The experimental
results show that the measured lifetime of the pumped region decreases as the pump power
increases but the lifetime stops decreasing at high pumps. The theoretical analysis justifies
the experimental results with a narrow pumped ruby laser crystal. In section 2, we construct the
rate equations including five emission-reabsorption couplings in our model. In section 3, the
numerical results are demonstrated. In section 4, we show the experimental apparatus of our
pinhole method. In section 5, we present the experimental results on a ruby crystal. The
pump-dependent lifetime is verified. Finally, the conclusions are given in section 6.

## 2. Model

The evolution of the excited population density after the excitation can be described by the Holstein-Biberman (HB) equation [2,3]

_{$n(\overrightarrow{r},t)$}is the density of ions in the excited state at the position

_{$\overrightarrow{r}$}and the time t,

_{${\tau}_{21}$}is the upper-level lifetime, W

_{r}is the radiative decay rate,

_{$G(\overrightarrow{r}\text{'},\overrightarrow{r})$}is the probability that the photon is emitted at

_{$\overrightarrow{r}\text{'}$}and is absorbed at

_{$\overrightarrow{r}$}, which in general can be expressed as

_{D1}is the population density of the ground state for the pumped region. For a size of few millimeters sample with uniform lower concentration, the fluorescence light undergoes many TIRs before being reabsorbed. If the sample is divided into the direct pumped region D and the unpumped region U, it had been considered as long-range radiation transfer and non-correlation between the emission point and the reabsorption point. In this case, Eq. (2) or the kernel function

_{$G(\overrightarrow{r}\text{'},\overrightarrow{r})$}reduces to a constant [3] due to

_{$\alpha \ell $}>>1, thus, the excited population densities for the two regions are governed by the rate equations [3]:

_{D}, V

_{U}, and V

_{S}are the respective volume of region D, U, and the whole crystal. The second term at the right hand side in Eq. (3) represents the emission-reabsorption events within D, in which the emission is proportional to N

_{D}, the possible trapped photons is proportional to V

_{D}, and the photon is eventually reabsorbed within D. The unity in Eq. (3) is to emphasize the reabsorption probability eventually reaches 1 after many times reflections due to TIR. The final term in Eq. (3) describes the events of emission in U and reabsorbed in D. The similar two terms in Eq. (4) describe the events of U-U and D-U.

Under the assumption of W_{r} = 1/τ_{21} with narrow pump condition of
V_{D} << V_{S}, the reabsorption D-D term due to long-range radiation
transfer is small so the short-range radiation transfer should be considered. By introducing a
damping parameter _{$\ell $a} for the uniformly pumped region as in [4], the HB equation can be described as

_{D1}. Due to

_{$\sigma {N}_{D1}{\ell}_{a}$}= 0.24 at low pump is no longer much smaller than 1, nonlinear absorption

_{$\frac{\sigma {N}_{D}{}_{1}{\ell}_{a}}{\sigma {N}_{D}{}_{1}{\ell}_{a}+1}$}should be considered instead and it is used to replace the linear term of

_{$\sigma {N}_{D1}{\ell}_{a}$}in Eq. (20a) of Ref [5]. Moreover, the absorption cross section of many laser crystals is about 10

^{−20}~10

^{−18}cm

^{2}and the doping concentration more than 1.0 at% is usually seen, so the condition with

_{$\sigma {N}_{D1}{\ell}_{a}$> 1}is possible for low pump regime. Second, if the nonlinear coefficient

_{$\frac{\sigma {N}_{D}{}_{1}{\ell}_{a}}{\sigma {N}_{D}{}_{1}{\ell}_{a}+1}$}in Eq. (5) is replaced by

_{$\sigma {N}_{D1}{\ell}_{a}$}, the reabsorption term becomes larger than the spontaneous emission term as

_{$\sigma {\text{N}}_{\text{D1}}{\ell}_{a}$> 1}.

Next, in order to consider the long-range transfer from the U region, we set a parameter
_{$\ell $b} to achieve _{$\frac{\sigma {N}_{D1}{\ell}_{b}}{\sigma {N}_{D1}{\ell}_{b}+1}=\frac{f{V}_{U}}{Vs}$} because _{$0<\frac{f{V}_{U}}{Vs}\le 1$} under the narrow pump condition V_{U}
≈V_{S}. Accordingly, the final term in Eq.
(3) of long-range term can be replaced by the mathematical representation of the
short-range reabsorption term. Therefore, we can impose the pump dependence on the reabsorption
terms of U-D and similarly on the terms of U-U and D-U. To describe the pump-induced variation
of N_{D1}, a three-level system is assumed as in Fig.
1, where 1, 2, and 3 denote the ground state, the upper state for emitting fluorescence
and the highest state, respectively. When the stimulated emission and the nonradiative energy
transfer are neglected, the population rate equations, in consideration of the spontaneous
emission and the reabsorption, are

_{31}and τ

_{32}are the decay time from level 3 to level 1 and to level 2, respectively. Note that we have added in Eqs. (9) and (10) the short-range and the long-range reabsorption in use of

_{$\ell $c1}and

_{$\ell $c2}for the events of U-U. The pump rate R

_{p}, atoms or ions per unit time per unit volume, is assumed to be

_{$\frac{{P}_{inc}/h\nu}{\pi {w}_{p}{}^{2}\ell}\frac{{N}_{D1}}{{N}_{t}}$}, where P

_{inc}is the incident pump power, hν is the pumping photon energy, w

_{p}is the pump radius, and

_{$\ell $}is the sample thickness. We have simply multiplied N

_{D1}/N

_{t}for describing the pumping efficiency, where N

_{t}is the total population density.

We understand the physics of D-D short-range coupling coefficient in Eq. (7) as follows. The emission photons are
proportional to N_{D2} but the trapped probability is
_{$\frac{\sigma {N}_{D}{}_{1}{\ell}_{a}}{\sigma {N}_{D}{}_{1}{\ell}_{a}+1}$}. A small value of N_{D1} has a small trapped
probability, which means that some spontaneous emission photons flee the pumped domain rather
than reabsorbed in the short-range interaction. The rare ground state ions are unable to
reabsorb all the spontaneous emission effectively. A small trapped probability also occurs when
_{$\sigma {N}_{D}{}_{1}{\ell}_{a}$} is very small. We can imagine that many emission
photons flee the small pumped domain rather than reabsorbed in this case. When
_{$\sigma {N}_{D}{}_{1}{\ell}_{a}$}>>1, the trapped probability is unity that
means the photons eventually to be reabsorbed. This is exactly the case of the long-range
interaction due to TIR. Therefore, we can extend the representation for the short-range
reabsorption coefficient to the long-range interaction.

## 3. The numerical results

The rate Eqs. (6)-(10) can be rewritten as the difference equations
by_{$\partial /\partial t\to \Delta /\Delta t$}. Given a pumping density rate and the initial
population density in ground state N_{t}, the equations can be solved numerically by an
iteration process. After each pumping period, the pump rate was reset to be zero and thus the
time evolution of the population density was obtained. Subsequently, the ratio of the collected
light from D and U was determined according to the position and the radius of the pinhole.
Finally, the decay of the composite population density was fitted. In the numerical simulations,
we used the parameters of a 0.05 at% ruby crystal with N_{t} = 2.4 × 10^{19}
cm^{−3}, τ_{31} = 3.3 × 10^{−6} s, τ_{32} = 5 × 10^{−8}
s, τ_{21} = 2.8 × 10^{−3} s, and σ = 1 × 10^{−19} cm^{2} [6]. We set the parameters for the long-range radiation transfer
of _{$\ell $b} = _{$\ell $c2} = 4.25 mm and _{$\ell $d} = 4.25 × 10^{−3} mm due to
_{${V}_{U}/{V}_{S}\approx 1$} and f = 0.5 for a thin slab with refractive index of
1.8 [7]. The short-range parameter of
_{$\ell $c1} was chosen as the smallest dimension of the
unpumped volume of 1 mm [5]. Each step of time difference
_{$\Delta t$} was chosen 2 × 10^{−8} s and 2.5 ×
10^{6} iterations were performed.

Figure 2(a)
shows the time evolutions of N_{D2}, N_{U2}, and N_{D1} within
one period for the population inversion pump power P_{th}, where P_{th} makes
the stable population density N_{D1} = N_{D2} = N_{t}/2. At the
beginning, N_{D2} grows up quickly to N_{t}/2 and then decays nearly single
exponentially when the pump is terminated. N_{U2} is much smaller than N_{D2}
and it exhibits non-exponential decay. As expected, N_{D1} decreases from N_{t}
to N_{t}/2 and then recovers to its initial value. Note that a two-region model predicts
a double exponential decay of N_{D2} [3,5]. Using _{${A}_{1}\mathrm{exp}(-t/{\tau}_{1})+{A}_{2}\mathrm{exp}(-t/{\tau}_{2})$} to fit N_{D2} at P_{th}, we obtain
_{${\tau}_{1}=3.51$} ms and _{${\tau}_{2\text{}}=1.69$} ms. The fitted result is shown with pink curve in
Fig. 2(a). To investigate why the fitted short lifetime
is smaller than the intrinsic lifetime, we decrease the pump to 0.05P_{th} and find
_{${\tau}_{1}=3.67$} ms and _{${\tau}_{2}=3.06$} ms. This result matches with the previous study [3,5], which show that both of the lifetimes are larger than the intrinsic lifetime as _{$\alpha \ell <<1$}. We have also replaced the nonlinear coefficient by _{$\sigma {N}_{D1}{\ell}_{a}$}, but we found the fitted short lifetime is still less than the intrinsic lifetime. Therefore, we know the aforementioned fitted short lifetime comes from depletion of the ground state rather than nonlinear coupling. However, the fitted weighting ratio is _{${\text{A}}_{\text{2}}{\text{/A}}_{1}=0.12$} at P_{th}, so single exponential fit with green curve in Fig. 2(a) fits reasonably well to calculate the decay time _{$\tau \text{'}$} of N_{D2} which will be used in the following discussion.

In order to trace the effect of radiation trapping, we plot the time evolution of the dominant reabsorption coefficient _{$\sigma {\text{N}}_{\text{D}}{}_{1}{\ell}_{a}/(\sigma {\text{N}}_{\text{D}}{}_{1}{\ell}_{a}+1)$} for different pumps in Fig. 2(b). We see that a low pump of 0.1P_{th} has a large coefficient all the time that presents serious radiation trapping and will lead to a large lifetime lengthening. However, a strong pump of 50P_{th} nearly depletes the population density in the ground state, which appears a very small dominant factor. After the pump is terminated, the resumption of the dominant coefficient is very close to that of 10P_{th} so we call the phenomenon as saturation of radiation trapping. The saturation inevitably limits the lifetime lengthening and will lead to a small measured lifetime. Indeed, when we analyzed the decay time of N_{D2}, we found that _{$\tau \text{'}$} decreases when the pump rate increases but the decrease nearly stops at high pumps. This result is shown in Fig. 3(a)
, in which the pump is normalized by P_{th} and the label for the right vertical axis is the lifetime lengthening ratio. We see from Fig. 3(a) that the lengthening lifetime ratio decreases from 33% to 19% for _{$\ell $a} = 1.4 mm but it is only from 13% to 8% for _{$\ell $a} = 0.6 mm. Therefore, an apparent reduction of _{$\tau \text{'}$} is for a large _{$\ell $a}.

In addition, a linear relation between _{$\tau \text{'}$} and _{$\ell $a}, as shown in Fig. 3(b), is obtained when the incident pump rate is fixed. The solid triangles for the low pump of 0.1P_{th} show that _{$\tau \text{'}$} decreases to the intrinsic value of 2.8 ms when _{$\ell $a} reduces to zero. The high pump of 10P_{th} has a lower slope in the linear relation in Fig. 3(b). This linear relation is the experimental basis of the pinhole method in [4], in which the pump rate was not considered. This shows that our model and the numerical simulation with single exponential fit work well so far.

When both of the fluorescence from U and D are collected, the light power is proportional to η_{D}N_{D2}V_{Dc} + η_{U}N_{U2}V_{Uc}, where V_{Dc} and V_{Uc} are the collected volume; η_{D} and η_{U} are the escape ratio of the light [5]. Therefore, the experimental measured lifetime can be compared with the numerical lifetime of N_{D2} + βN_{U2}, where β is the collected light ratio that equals to η_{U}V_{Uc}/η_{D}V_{Dc}. Figure 4
shows the decay time of the composite population density increases approximately linear with the increase of collected light ratio. The increase of the collected light ratio can be experimentally achieved by tuning the position of the pinhole transversely or axially and this will be demonstrated in the next section.

## 4. Experimental setup

The experimental setup for our pinhole method is shown in Fig. 5 , which is arranged according to the theoretical consideration in Fig. 1. The ruby crystal of 0.05 at% with a thickness of 1 mm and diameter of 5 mm was purchased from Roditi Company. The entrance face of the crystal for pump beam had a dichroic coating of 3-mm diameter with reflection greater than 99.5% at 694.3 nm and transmission greater than 99.5% at the pump wavelength of 532 nm; the other surface had no coating. The pump beam source was a continuous-wave DPSS laser modulated by a square wave with period of 50 ms. The fall time of the pump edge was smaller than 10 μs. A collimating lens was added in front of the crystal so that we could tune the pump size. We set a pinhole together with a 3-mm aperture behind the crystal at 24 cm to collect the fluorescence. The surface of the pinhole facing the crystal was sprayed paints to avoid the reflectance of the pump beam. The distance between the crystal and the pinhole was 0.5 mm and the pinhole could be moved when necessary. After passing a laser line filter and a convergent lens, the ruby fluorescence was detected by a photomultiplier. Output from the detector was displayed on an oscilloscope from which the decay could be determined. Curve fitting were performed with single exponent and the presented lifetime was the average of ten sets measured data. All the experiments were performed at room temperature 23°C.

The configuration of the second pinhole method was similar to that of [4]. A pinhole was set in front of the crystal to limit the pump beam size and simultaneously to collect the fluorescence. In order to collect the light from the pumped region as much as possible, we arranged the angle between the pump beam and the optical axis of the collecting system to be 45°. Since the pinhole faced the pump beam, the coated surface of the crystal was on the opposite side of the pinhole. The pump source and the collecting system were the same as in Fig. 5.

## 5. Experimental results and discussion

Using our pinhole method when the pump size was about 40 μm and a 30-μm pinhole was set on the pumping axis, we collected the fluorescence only from D but not from U. The squares in Fig. 6
show that the experimentally measured lifetime, with single exponential fit, decreases with increasing the pump power. In Fig. 6, the ratio of the absorbed power to the saturation power is labeled as the top x axis for comparison with Fig. 3(a). In order to totally exclude the stimulated emission, we operated the second pinhole method to investigate the pump-dependent lifetime. Because it is difficult to measure the lifetime using a 30-_{$\mu m$}pinhole in the second method, the experiment was done using a 100_{$\mu m$}pinhole. The result is shown in Fig. 6 with circles. Both of our experimental results match with the numerical result in Fig. 3(a). In the first pinhole method, the measured lifetime is believed not to be influenced by the stimulated emission or amplified spontaneous emission (ASE) because the measured lifetime is nearly unchanged as P_{inc} > 200 mW. According to Ref [8], ASE bring the right hand side of Eq. (7) a negative term _{$-{\sigma}_{e}{\text{N}}^{2}{{}_{\text{D}}}_{2}{\ell}_{s}/{\tau}_{21}$} that is equivalent to a coupling coefficient _{${\chi}_{e}=-{\sigma}_{e}{\text{N}}_{\text{D}}{}_{2}{\ell}_{s}$}, where _{${\sigma}_{e}$} is the emission cross section and _{${\ell}_{s}$}is an average path length for the spontaneously emitted photon. If ASE plays a part, it will further reduce the measured lifetime at a larger pump power. For the maximal population inversion 2.4 × 10^{19} cm^{−3}, the one trip small signal gain is still small (< 1.06 for _{${\sigma}_{e}=2.5\times {10}^{-20}{\text{cm}}^{2}$}). Based on the same reason, the stimulated emission is still weak although the absorbed power can be raised up to thirteen times of the saturation power. So the larger lifetimes with the second method in Fig. 6 are ascribed to the larger pinhole.

As explained below, the influence of temperature on the lifetime was also excluded. The temperature rising rate for D and U are nearly the same and the final rises are only 0.9°C and 0.6°C respectively after our maximal pump power for 10 second. The accuracy of our temperature measurement is 0.1°C for the area of 1 mm diameter. We have also calculated the temperature difference between the pump center (r = 0) and the crystal boundary (r_{b} = 2.5 mm). According to [9], the temperature difference between r and r_{b} is

_{$\alpha $}is the absorption coefficient at the pump wavelength, P

_{ph}is the fraction of pump power that results in heating, K

_{c}is the thermal conductivity and r

_{b}is the radius of crystal. The used ruby parameters are

_{$\alpha $}= 223 m

^{−1}and K

_{c}= 41.9 Wm

^{−1}K

^{−1}. After substituting our experimental conditions of P

_{ph}= 40 mW and w

_{p}= 20 μm into the aforementioned expression, we obtain

_{$\Delta T$}≒ 0.2°C between the pump center and the crystal boundary for the highest pump power we employed. The data of [10] indicates a decrease of 0.25 ms lifetime needs a rise of ~20°C, so the lifetime reduction we studied does not come from temperature. Therefore, we conclude that the reduction of pump-dependent lifetime comes from the pump-induced reabsorption saturation. Furthermore, when we compare the data at the highest pump and their total reduction between the experiment and simulation, we obtain the intrinsic lifetime to be 2.8 ms.

Next, we used the first pinhole method to testify our numerical result of Fig. 4. By moving the pinhole in the transverse direction x from D to U, we collect less fluorescence from D and more fluorescence from U. Because the fluorescence of U is weaker than that of D, the total collected fluorescence power decreases as x increases, as shown in black hollow squares in Fig. 7
. The corresponding measured lifetime, with the black solid squares in Fig. 7, shows that the lifetime increases with the increase of x. This qualitatively agrees with the result in Fig. 4. However, a quantitative discrepancy exists in the result that the measured lifetime increases with the pinhole shift up to x ~80_{$\mu m$}and then maintains stationary, rather than the predicted pump beam diameter of 40_{$\mu m$}. It can be explained in virtue of four factors. First, the boundary between D and U was not so sharp because the real Gaussian pump profile of the pump light made its intensity gradual decrease with x. Second, the experimental pump size was not constant but varied with the penetration depth in crystal. Third, some of the pump light would be reflected on the uncoated face of the crystal due to Fresnel reflection but it has been ignored in our theoretical analysis. Fourth, the collected area by the light collection system was larger than the pinhole area because the pinhole was not against the crystal but 0.5 mm apart in experiment.

When the distance between the crystal and the pinhole was increased to 5 mm, the collected power and the measured lifetime as a function of x are also shown in Fig. 7 with red hollow circles and solid circles, respectively. We see the red solid curve is shallower and wider than the corresponding black one. This is understandable because at x = 0, the collected area is large enough to cover the unpumped region, and thus the collected light ratio is large. This renders the measured lifetime larger than the case of z = 0.5 mm. At x = 100 μm, the pinhole at z = 5 mm still collect some fluorescence from D so that the measured time is shorter than the other case. Furthermore, by moving the on-axis pinhole farther away from the crystal the z-dependent lifetime for x = 0 is confirmed directly, as plotted in Fig. 8 . The measured lifetime indeed increases when the distance between the crystal and the pinhole increases. When the distance is too large, a maximum is acquired because the collected light ratio achieves the maximum.

Although our numerical results were calculated by using the parameters of a 0.05 at% ruby thin slab, our model can be used for higher concentrations and for other materials. Since _{$\sigma {\text{N}}_{D1}{\ell}_{a}/(\sigma {\text{N}}_{D1}{\ell}_{a}+1)<1$}, we think our model is valid for any concentration. However, there are some points need to be mentioned. First, the lifetime lengthening is not obvious as _{$\sigma {\text{N}}_{D1}{\ell}_{a}<0.01$}. Contrarily, when _{$\sigma {\text{N}}_{D1}{\ell}_{a}>0.2$}, the double exponential fit is suggested for both the numerical solution and the experimental data. Second, saturation of the radiation trapping will take place also at higher concentrations as long as depletion of the ground state can be achieved. Third, some three-level and quasi-three level materials such as Yb:GSO always have less populations in the ground state of emission, so the reabsorption is weaker, not to say about their small overlap between the emission and absorption spectra [11]. Finally, when the concentration is high, the phonon-assisted radiative and non-radiative energy transfer must be taken in account. Other effects like the ASE will also be noticed when the pump is high.

## 6. Conclusions

In summary, we have modeled the radiation trapping in a three-level system consisting of the direct pumped region D and the unpumped region U, both of which are assumed uniform and five couplings between them are described. We have also taken into account the pump rate and found the phenomenon of reabsorption saturation, which prevents the lifetime from further lengthening. This effect is straightforward but was not studied before. We verify it by two experiments of pinhole method using a 0.05 at% ruby laser crystal with narrow pump. On comparison of the simulation and the experiments, the intrinsic lifetime of our ruby crystal is obtained to be 2.8 ms.

In addition, we have testified our model by moving the pinhole both transversely and axially to investigate the lifetime variation. The results show the validation of our model. Moreover, the experimental basis of the second pinhole method for measuring the intrinsic lifetime, the linear relation between the population decay time and the parameter _{$\ell $a}, was numerically obtained. This linear relation exhibits a lower slope for a fixed higher pump rate that may expand the usage of the pinhole method because one can use the high pump to obtain strong fluorescence and collect these light using smaller pinholes. It should be careful to control the pump rate when the second pinhole method is used since the measured lifetime is pump-dependent.

Our two-region model with rate equations is simple and the numerical simulation is easily performed. We can treat the couplings between the pumped and the unpumped regions rather than only one pumped region. As compared with previous two-region model in Refs [3,5], the improvement of ours is that the nonlinear coupling coefficients are used so that we can treat the larger coupling coefficient due to high concentration or thick sample. In particular, we find the numerical solutions with two fitted lifetimes but one of them is smaller than the intrinsic lifetime. This phenomenon due to depletion of ground state is interesting and suggested to be further studied.

## Acknowledgments

This work was supported by the Nation Science Council of the Republic of China under Grant NSC 99-2112-M-415-001-MY2. We thank Professor W F Hsieh for his helpful discussions.

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