## Saturation of radiation trapping and lifetime measurements in three-level laser crystals |

Optics Express, Vol. 20, Issue 23, pp. 25613-25623 (2012)

http://dx.doi.org/10.1364/OE.20.025613

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### 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

1. E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc. **1**(1), 40–51 (1926). [CrossRef]

2. T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev. **72**(12), 1212–1233 (1947). [CrossRef]

3. S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B **73**(14), 144101 (2006). [CrossRef]

5. G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B **106**(1), 63–71 (2012). [CrossRef]

_{ℓ}. For

_{αℓ}>> 1,

_{α}being the absorption coefficient, the sample is large enough to allow many events of emission-reabsorption and the lifetime lengthening can be observed. For

_{αℓ}<< 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

_{αℓ<<1}.

_{αℓ≈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

2. T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev. **72**(12), 1212–1233 (1947). [CrossRef]

3. S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B **73**(14), 144101 (2006). [CrossRef]

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

_{r→}and the time t,

_{τ21}is the upper-level lifetime, W

_{r}is the radiative decay rate,

_{G(r→',r→)}is the probability that the photon is emitted at

_{r→'}and is absorbed at

_{r→}, which in general can be expressed aswhich is the same as that in Eq. (15) of [5

5. G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B **106**(1), 63–71 (2012). [CrossRef]

_{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(r→',r→)}reduces to a constant [3

3. S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B **73**(14), 144101 (2006). [CrossRef]

_{αℓ}>>1, thus, the excited population densities for the two regions are governed by the rate equations [3

**73**(14), 144101 (2006). [CrossRef]

_{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.

_{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

_{ℓa}for the uniformly pumped region as in [4

4. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. **32**(13), 1908–1910 (2007). [CrossRef] [PubMed]

4. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. **32**(13), 1908–1910 (2007). [CrossRef] [PubMed]

5. G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B **106**(1), 63–71 (2012). [CrossRef]

_{D1}. Due to

_{σND1ℓa}= 0.24 at low pump is no longer much smaller than 1, nonlinear absorption

_{σND1ℓaσND1ℓa+1}should be considered instead and it is used to replace the linear term of

_{σND1ℓa}in Eq. (20a) of Ref [5

**106**(1), 63–71 (2012). [CrossRef]

^{−20}~10

^{−18}cm

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

_{σND1ℓa> 1}is possible for low pump regime. Second, if the nonlinear coefficient

_{σND1ℓaσND1ℓa+1}in Eq. (5) is replaced by

_{σND1ℓa}, the reabsorption term becomes larger than the spontaneous emission term as

_{σND1ℓa> 1}.

_{ℓb}to achieve

_{σND1ℓbσND1ℓb+1=fVUVs}because

_{0<fVUVs≤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 where τ

_{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

_{ℓc1}and

_{ℓc2}for the events of U-U. The pump rate R

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

_{Pinc/hνπwp2ℓND1Nt}, where P

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

_{p}is the pump radius, and

_{ℓ}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.

_{D2}but the trapped probability is

_{σND1ℓaσND1ℓa+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

_{σND1ℓa}is very small. We can imagine that many emission photons flee the small pumped domain rather than reabsorbed in this case. When

_{σND1ℓa}>>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

_{∂/∂t→Δ/Δt}. 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

6. T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett. **4**(11), 564–566 (1960). [CrossRef]

_{ℓb}=

_{ℓc2}= 4.25 mm and

_{ℓd}= 4.25 × 10

^{−3}mm due to

_{VU/VS≈1}and f = 0.5 for a thin slab with refractive index of 1.8 [7

7. W. A. Shurcliff and R. C. Jones, “The trapping of fluorescence light produced within objects of high geometrical symmetry,” J. Opt. Soc. Am. **39**(11), 912–916 (1949). [CrossRef]

_{ℓc1}was chosen as the smallest dimension of the unpumped volume of 1 mm [5

**106**(1), 63–71 (2012). [CrossRef]

_{Δt}was chosen 2 × 10

^{−8}s and 2.5 × 10

^{6}iterations were performed.

_{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

**73**(14), 144101 (2006). [CrossRef]

**106**(1), 63–71 (2012). [CrossRef]

_{A1exp(-t/τ1)+A2exp(-t/τ2)}to fit N

_{D2}at P

_{th}, we obtain

_{τ1=3.51}ms and

_{τ2 =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

_{τ1=3.67}ms and

_{τ2=3.06}ms. This result matches with the previous study [3

**73**(14), 144101 (2006). [CrossRef]

**106**(1), 63–71 (2012). [CrossRef]

_{αℓ<<1}. We have also replaced the nonlinear coefficient by

_{σND1ℓa}, 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

_{A2/A1=0.12}at P

_{th}, so single exponential fit with green curve in Fig. 2(a) fits reasonably well to calculate the decay time

_{τ'}of N

_{D2}which will be used in the following discussion.

_{σND1ℓa/(σND1ℓa+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

_{τ'}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

_{ℓa}= 1.4 mm but it is only from 13% to 8% for

_{ℓa}= 0.6 mm. Therefore, an apparent reduction of

_{τ'}is for a large

_{ℓa}.

_{τ'}and

_{ℓa}, 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

_{τ'}decreases to the intrinsic value of 2.8 ms when

_{ℓa}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

4. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. **32**(13), 1908–1910 (2007). [CrossRef] [PubMed]

_{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

**106**(1), 63–71 (2012). [CrossRef]

_{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

**32**(13), 1908–1910 (2007). [CrossRef] [PubMed]

## 5. Experimental results and discussion

_{μm}pinhole in the second method, the experiment was done using a 100

_{μm}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

8. N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron. **35**(1), 101–109 (1999). [CrossRef]

_{−σeN2D2ℓs/τ21}that is equivalent to a coupling coefficient

_{χe=−σeND2ℓs}, where

_{σe}is the emission cross section and

_{ℓs}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

_{σe=2.5×10−20cm2}). 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.

_{b}= 2.5 mm). According to [9

9. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. **56**(19), 1831–1833 (1990). [CrossRef]

_{b}iswhere

_{α}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

_{α}= 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

_{ΔT}≒ 0.2°C between the pump center and the crystal boundary for the highest pump power we employed. The data of [10

10. Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr^{3+}-doped insulating crystals,” Phys. Rev. B Condens. Matter **48**(11), 7772–7778 (1993). [CrossRef] [PubMed]

_{μm}and then maintains stationary, rather than the predicted pump beam diameter of 40

_{μm}. 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.

_{σND1ℓa/(σND1ℓa+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

_{σND1ℓa<0.01}. Contrarily, when

_{σND1ℓa>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

11. W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett. **88**(22), 221117 (2006). [CrossRef]

## 6. Conclusions

_{ℓa}, 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.

**73**(14), 144101 (2006). [CrossRef]

**106**(1), 63–71 (2012). [CrossRef]

## Acknowledgments

## References and links

1. | E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc. |

2. | T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev. |

3. | S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B |

4. | H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. |

5. | G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B |

6. | T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett. |

7. | W. A. Shurcliff and R. C. Jones, “The trapping of fluorescence light produced within objects of high geometrical symmetry,” J. Opt. Soc. Am. |

8. | N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron. |

9. | M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. |

10. | Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr |

11. | W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett. |

**OCIS Codes**

(000.6800) General : Theoretical physics

(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing

(160.3380) Materials : Laser materials

(260.2510) Physical optics : Fluorescence

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: September 13, 2012

Revised Manuscript: October 22, 2012

Manuscript Accepted: October 23, 2012

Published: October 26, 2012

**Citation**

Ching-Hsu Chen, Yue-Heng Wu, Cheng-Ping Fan, and Tai-Hei Wei, "Saturation of radiation trapping and lifetime measurements in three-level laser crystals," Opt. Express **20**, 25613-25623 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25613

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### References

- E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc.1(1), 40–51 (1926). [CrossRef]
- T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev.72(12), 1212–1233 (1947). [CrossRef]
- S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B73(14), 144101 (2006). [CrossRef]
- H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett.32(13), 1908–1910 (2007). [CrossRef] [PubMed]
- G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B106(1), 63–71 (2012). [CrossRef]
- T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett.4(11), 564–566 (1960). [CrossRef]
- W. A. Shurcliff and R. C. Jones, “The trapping of fluorescence light produced within objects of high geometrical symmetry,” J. Opt. Soc. Am.39(11), 912–916 (1949). [CrossRef]
- N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999). [CrossRef]
- M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990). [CrossRef]
- Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993). [CrossRef] [PubMed]
- W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006). [CrossRef]

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