Theoretical simulation on exciplex pumped Rb vapor laser

Jinghua Huang; Chunsheng Xia; Xingqi Xu; Chenyi Su; Bailiang Pan

doi:10.1364/OE.27.000132

1. Introduction

The exciplex pumped alkali laser (XPAL) is a new class of laser in which the excited alkali-rare gas molecules quickly dissociate to populate the n²P_3/2 state. In XPALs, the blue or red satellites of the atomic transition (n²S_1/2 → n²P_3/2), created by the naturally occurring alkali-rare collision pairs, broaden the absorption linewidth to about 5 nm, which offers a route to address the issue of lineshape matching [1]. Since it was first proposed in 2008, the XPAL has been realized in Cs-Ar, Cs-Ar-C₂H₆, Cs-Kr-C₂H₆, and Rb-Kr-C₂H₆ systems by Readle’s team [2–4]. The highest slope efficiency of 27% for Cs-Xe at 453 K was recently reported by Mironov et al [5]. Meanwhile, there have been many theoretical works explaining laser kinetics and predicting the potential of high output power of XPALs. The BLAZE-V multi-dimensional model introduced by Palla et al. was applied to the XPAL system, in which simulations and the preliminary results were in good agreement with the Cs-Ar XPAL experiments [6,7]. In Carroll et al’ s and Huang et al’ s works, detailed theoretical models of CW XPALs were used to predict the optical-to-optical efficiency and analyze the laser kinetics [8,9]. A physical model combining the laser kinetics and thermodynamic processes was presented by Xu et al. to calculate and analyze 2-D and 3-D distributions of temperature [10].

In this paper, a physical model is established to describe laser kinetics and thermodynamics in the CW Rb-Kr XPAL system. A set of optimal parameters are obtained for designing an efficient CW Rb-Rr XPAL in further experiments.

2. Description of theory

The schematic diagram of the energy levels and kinetic processes in the Rb-Kr XPAL system is shown in Fig. 1. It invokes four energy states: 5²S_1/2 (Rb), $X^{2} Σ_{1 / 2}^{+}$ (Rb-Kr), $B^{2} Σ_{1 / 2}^{+}$ (Rb*-Kr), and 5²P_3/2 (Rb^*). The population densities of these states are denoted by n₀, n₁, n₂, and n₃, respectively. E₀ and E₃, E₁ and E₂ are the energies corresponding to the lower and upper levels of the laser and pump lights, and g_i (i = 0, 1, 2, 3) is the degeneracy of energy levels.

Fig. 1 Energy levels and kinetic processes in the Rb-Kr system.

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According to the kinetic processes shown in Fig. 1, the rate equations can be expressed as follows:

\frac{d n_{0}}{d t} = - k_{x a} n_{0} + k_{x d} n_{1} + (n_{3} - 2 n_{0}) \frac{σ_{30} (I_{l}^{+} + I_{l}^{-})}{h v_{l}} + A_{31} n_{3},

\frac{d n_{1}}{d t} = - k_{x d} n_{1} + k_{x a} n_{0} - (n_{1} - n_{2}) \int_{0}^{\infty} \frac{σ_{12} (v) I_{p} (v)}{h v_{p}} d v,

\frac{d n_{2}}{d t} = (n_{1} - n_{2}) \int_{0}^{\infty} \frac{σ_{12} (v) I_{p} (v)}{h v_{p}} d v - k_{b d} n_{2} + k_{b a} n_{3},

\frac{d n_{3}}{d t} = k_{b d} n_{2} - k_{b a} n_{3} - (n_{3} - 2 n_{0}) \frac{σ_{30} (I_{l}^{+} + I_{l}^{-})}{h v_{l}} - A_{31} n_{3},

n_{R b} = n_{0} + n_{1} + n_{2} + n_{3},

where k_xa, k_xd, k_ba, and k_bd represent the association and dissociation rates of Rb-Kr and Rb*-Kr states, respectively. A₃₁ is the spontaneous emission rate determined by the lifetime of Rb* state [11]. Equation (5) means that Rb atoms obeys mass conservation in the whole processes, where n_Rb is the total number density of Rb atom and it can be derived from an empirical function between saturated vapor pressure and temperature [11]. In addition, based on the assumption of CW operation, the Eqs. (1)-(4) are set to zero.

Furthermore, we suppose that the Rb-Kr pair is generated immediately as alkali atoms collide with rare-gas atoms [4], the association rate k_xa, which is proportional to the collision rate, can be determined by

k_{x a} = π υ_{R b - K r} d_{R b - K r}^{2} M,

where d_Rb-Kr is the interatomic separation [12]. M is the number density of Kr, and υ_Rb-Kr is the average relative velocity between the Rb and Kr atoms.

The dissociation rate of Rb*-Kr molecules in excited state, denoted k_bd, can be estimated using [4]

k_{b d} = k_{x d} \frac{1}{2} \exp {[(E_{1} - E_{0}) - (E_{3} - E_{2})] / k_{b} T},

When there is no pump beam, the value of dissociation rate k_xd and association rate of Rb*-Kr molecules k_ba can be given by [4]

k_{x d} = k_{x a} \frac{n_{0}}{n_{1}},

k_{b a} = k_{b d} \frac{n_{2}}{n_{3}},

To our knowledge, equilibriums between states of Rb and Rb-Kr, Rb*-Kr and Rb* are thermally maintained so that the relations of population densities can be described by the equilibrium fractions f₁₀ and f₂₃.

f_{10} = \frac{n_{1}}{n_{0}} = \frac{g_{1}}{g_{0}} 4 π R_{0}^{2} Δ R e x p (\frac{- Δ E_{10}}{k_{b} T}) M,

f_{23} = \frac{n_{2}}{n_{3}} = \frac{g_{2}}{g_{3}} 4 π R_{0}^{2} Δ R e x p (\frac{- Δ E_{23}}{k_{b} T}) M,

where ∆E₁₀, ∆E₂₃ are energy differences between the Rb and Rb-Kr states, Rb*-Kr and Rb* states, respectively. R₀ is the optimal internuclear separation (4.5 Å for Rb-Kr) for the blue satellite, and ΔR is the range of interatomic distance over which the resonance absorption can take place (1 Å) [12]. The parameters k_xa, k_xd, k_ba and k_bd, listed in Table 1, was calculated by Eqs. (6)-(11). However, the equilibrium assumption will be false if the rates of pump and production of the Rb-Kr collision pair are in the same order. Hence, the pump intensity I_ne required for non-equilibrium can be calculated from the following equation,

Table 1. Main parameters for the Rb–Kr XPAL system

View Table

(n_{1} - n_{2}) \int_{0}^{\infty} \frac{σ_{12} (v) I_{n e} (v)}{h v_{p}} d v = k_{x a} n_{0},

After the deep understanding of the rates of the atomic association and molecular dissociation, the propagation of laser in the cell can be analyzed. As shown in Fig. 2, the incident pump lights enter the vapor cell along longitudinal direction, and the initial intensity, considered as Gaussian spectral profile, can be given by

I_{p} (ν) = I_{0} \sqrt{\frac{\ln 2}{π}} \frac{2}{Δ ν_{p}} \exp [- \frac{4 \ln 2 {(ν - ν_{p})}^{2}}{Δ ν_{p}^{2}}],

where I₀ indicates the intensity at the entrance of cell. ν_p and Δν_p are the central frequency and the FWHM of the pump lights, respectively.

Fig. 2 Schematic diagram of the end-pumped Rb-Kr XPAL system

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The pump lights would be absorbed as they travel through the gain medium and absorption cross section can be described as

σ_{p} {=k}_{a b s} \frac{n_{0}}{n_{1}} M,

where k_abs refers the reduced absorption coefficient [13]. The lineshape of blue satellite is supposed as Lorentzian, and then spectrally resolved stimulated absorption cross section can be rewritten as

σ_{12} (ν) = σ_{p} \frac{{(Δ ν_{a b s} / 2)}^{2}}{{(ν - ν_{a b s})}^{2} + {(Δ ν_{a b s} / 2)}^{2}},

where ν_abs is the central frequency of blue satellite and Δν_abs is the FWHM of absorption line.

The stimulated emission cross section σ₃₀ can be calculated by [11,14]

σ_{30} = \frac{g_{3}}{g_{0}} \frac{λ_{l}^{2}}{2 π} \frac{1}{2 π t_{3} Δ ν},

where λ_l is the wavelength of laser and t₃ is the lifetime of Rb* state. ∆ν indicates the spectral bandwidth of lasing, which is mainly caused by the broadening of buffer gas, and corresponding rate has been obtained in Ref. 15 [15].

The spatial propagating intensities of pump light and laser beam along optical axial direction can be expressed as

I_{p}^{\pm} (z + Δ z, ν) = I_{p}^{\pm} (z, ν) \exp {\mp [n_{1} (z) - n_{2} (z)] σ_{12} (ν) Δ z},

I_{l}^{\pm} (z + Δ z) = I_{l}^{\pm} (z) \exp {\pm [n_{3} (z) - 2 n_{0} (z)] σ_{30} Δ z},

The temperature inside the vapor cell is higher than that in the walls due to the heat accumulation caused by the absorbed pump energy. To calculate the temperature distribution, the vapor cell is divided into many coaxial cylindrical annuluses, as shown in Fig. 3.

Fig. 3 Lateral view of a vapor cell.

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Each annulus is a non-negligible heat source, and the heat released by energy difference can be calculated by

Ω_{k, j} = [(k_{x a} n_{0} (k, j) - k_{x d} n_{1} (k, j)) Δ E_{10} + (k_{b d} n_{2} (k, j) - k_{b a} n_{3} (k, j)) Δ E_{23}] \times Δ V,

where ∆V is the volume of cylindrical annulus, which is

Δ V = π (r_{k + 1}^{2} - r_{k}^{2}) Δ z

.

In the longitudinal direction, the heat is carried by the flow from one cylindrical annulus to the next, and it can be given by [16]

F_{k, j} = u \frac{S_{k} n (T_{k})}{N_{A}} \int_{T_{k, j - 1}}^{T_{k, j}} C_{P} (T) d T, 0 < z < L,

where C_p (T) is the molar heat capacity of Kr obtained by fitting to existing data in [17], u is flow velocity, n(T_k) is the population density of the total mixed gas, and S_k is the cross-sectional area of the (k, j) th annulus.

In the radial direction, the conducted heat from the (k, j) th cylindrical annulus to the (k + 1, j) th can be calculated by Fourier law:

Φ_{k, j} = - K_{K r} (2 π r_{k} Δ z) \frac{T_{k, j} - T_{k -1, j}}{r_{k} - r_{k -1}},

where the thermal conductivity K_Kr = 2.4810 × 10⁻⁵T + 0.0017, fitted from the measured data [17].

Equation (22) demonstrates the energy conservation law in every cylindrical annulus, left-hand represents the sum of generated heat Ω_k,j in the (k, j)th segment and the conducted heat Φ_k-1,j from the (k-1, j)th cylindrical annulus to the (k, j) th, and the right-hand side is the sum of the removed heat F_k,j from the (k, j)th cylindrical annulus to the (k, j + 1) th and the conducted heat Φ_k,j from the (k, j)th cylindrical annulus to the (k + 1, j) th. Therefore, the temperature of the (k + 1, j) th cylindrical annulus can be solved when the Eq. (22) holds.

Ω_{k, j} + Φ_{k -1, j} = F_{k, j} + Φ_{k, j} .

3. Results and discussion

Firstly, we need to ensure that the kinetic equilibrium between Rb and Rb-Kr states is valid. Calculating by Eq. (12), the pump intensity required for breaking this equilibrium is approximately 1.9 × 10¹² and 2.4 × 10¹² W/m² at temperatures of 475 and 575 K, respectively. In our calculation, the maximum of pump intensity is set to 1 × 10¹² W/m². Therefore, the non-equilibrium effect need not be considered.

3.1 Comparison with Carroll’s model

Carroll et al. have made a preliminary prediction of CW Rb-Kr laser performance. In their model, the population density of alkali atom was assumed to be uniform along the longitudinal, so the average pump intensity was used to represent the true intracavity intensity [8]. In this part, the distribution of population is calculated, and a parameter NON, defined as $N O N (n_{i}) = \frac{\max (n_{i} (z)) - \min (n_{i} (z))}{\bar{n_{i} (z)}}$ (i = 0, 1, 2, 3, z ∈(0, L)) [18], was introduced to describe the non-uniformity of population. The non-uniformity of four states and the predicted optical-to-optical efficiency as a function of temperature are plotted in Fig. 4. We can see that in the range of 420 K to 500 K, the non-uniformity is tiny, and the efficiencies of these two models predicted are similar. As the temperature increases, the efficiency has a small difference because the effect of non-uniformity of population density is more pronounced.

Fig. 4 The non-uniformity of four energy states (a) and simulated optical-to-optical efficiency (b) versus cell temperature at a pump intensity of 1 × 10¹¹ W/m².

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3.2 Influences of parameters with uniform temperature distribution

The pump intensity, temperature, reflectivity of output coupler (R_oc), number density of Kr, and heat accumulation, all seem to impact on the performance of CW laser system. We give some calculations to further describe the characteristics of Rb-Kr XPAL. Figure 5(a) shows the optical-to-optical efficiency of end-pumped Rb-Kr XPAL as a function of pump intensity at different temperatures, where the reflectivity of output coupler is 0.25 and the length of cell is 0.1 m. The optical-to-optical efficiency of about 41% can be achieved at temperature of 575K when the pump intensity reaches to 2 × 10¹¹ W/m². At higher temperature, the optical-to-optical efficiency changes faster with pump intensity increasing due to the intense absorption of pump light. Noted that there is an optimal pump intensity due to the saturation of pump transition, the dependences of optimum pump intensity and its efficiency on temperature are plotted in Fig. 5(b). Apparently, high optimal pump intensity is needed for high temperature, and the value increases rapidly when temperature is over 560 K. This occurs because the population density restricted by temperature is quickly increasing and the reduction of pump intensity along the longitudinal direction becomes more significant. The high temperature case exceeding 600 K is not considered because higher temperature and pump intensity will lead to harmful thermal effects in the laser vapor cell.

Fig. 5 Dependences of performances with some factors in Rb-Kr XPAL systems. Figure 5(a) shows the optical-to-optical efficiency versus pump intensity at different cell temperatures. Figure 5(b) shows optimal pump intensity and maximum optical-to-optical efficiency versus temperature.

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The influences of reflectivity of output coupler on optical-to-optical efficiency with pump intensity of 4 × 10¹⁰ W/m² at 475 K, 6 × 10¹⁰ W/m² at 525 K, and 2 × 10¹¹ W/m² at 575 K are plotted in Fig. 6. It shows that the optical-to-optical efficiency increases with the decrease of R_oc, and a better performance can be achieved if lower reflectivity of output coupler is used.

Fig. 6 Optical-to-optical efficiency versus the reflectivity of output coupler.

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To analyze the influence of Kr number density, we enhanced the pressure of Kr to 5 atm. Figure 7 shows the dependence of optical-to-optical efficiency and absorption efficiency on Kr gas number density, where the reflectivity of output coupler is 0.25. We can see that both of efficiency increase with increasing number density of Kr, and then gradually keep stable at maximum. To reach the maximal optical-to-optical efficiency, higher number density of Kr is needed for low temperature case.

Fig. 7 Dependences of the optical-to-optical efficiency and absorption efficiency on the number density of Kr at temperature of 475 K, 525 K, and 575 K.

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3.3 Influence of gas flow velocity

Finally, the effect of heat accumulation generated by energy differences is analyzed, and the fluid system, as an option to cool the vapor cell, is introduced. Figure 8(a) shows dependences of optical-to-optical efficiency on the pump intensity with different flow velocities of 50, 100, 150, 200, and 250 m/s, where the reflectivity of output coupler is 0.25, the temperature of cell walls is 525K, and the Kr concentration is 2.50 × 10²⁵ m⁻³. We can see that the optimal optical-to-optical efficiency increases with raising flow velocity effectively, however, the value is very low comparing with Fig. 5(a) at uniform temperature distribution. The reason is that portion of the absorbed pump energy will be converted to heat and lead to the increase of temperature. The temperature gradient inside the cell results in inhomogeneous of population density and lower absorption efficiency (Fig. 8(b)). The longitudinal and radial temperature distribution of the vapor cell with pump intensity I₀ = 6 × 10¹⁰ W/m² was plotted in Figs. 8 (c) and 8(d). When the flow velocity increases from 50 m/s to 250 m/s, the heat effect is effective reduced and the temperature distribution in vapor cell is improved. The maximal optical-to-optical efficiency is 5.7% at the condition of I₀ = 5.2 × 10¹⁰ W/m² and u = 250 m/s.

Fig. 8 The influence of flow velocity on Rb-Kr performances. Figures 8(a) and 8(b) show predicted optical-to-optical efficiency and absorption efficiency versus pump intensity with difference flow velocities. Figures 8(c) and 8(d) draw the longitudinal and radial temperature distributions with pump intensity I₀ = 6 × 10¹⁰ W/m².

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4. Conclusion

In this paper, a physical model for a CW end-pumped Rb-Kr XPAL was established and the effects of pump intensity, temperature, reflectivity of output coupler, number density of Kr, and heat accumulation were calculated and analyzed. Results show that high optimal pump intensity is required for high operating temperature, and low reflectivity of output coupler and high Kr number density can promote optical-to-optical efficiency. With the heat accumulation considered, the Rb-Kr XPAL has poor performances. When the flow velocity is increased to 250m/s, the maximal optical-to-optical efficiency of 5.7% is attained at the condition of pump intensity I₀ = 5.2 × 10¹⁰ W/m² and T = 525 K. Our calculation also demonstrates that fast gas flowing method for the CW Rb-Kr XPAL should be introduced to effectively reduce the thermal effect due to the high pump intensity. Thus, our model provides a theorical way to design an efficient Rb-Kr XPAL system in experiment.

Funding

Aerospace Science and Technology Found (Grant No. KM20170269).

References

1. J. D. Readle, C. J. Wagner, J. T. Verdeyen, T. M. Spinka, D. L. Carroll, and J. G. Eden, “Excimer-pumped alkali vapor lasers: A new class of photoassociation lasers,” Proc. SPIE 7581, 75810K (2010). [CrossRef]

2. J. D. Readle, C. J. Wagner, J. T. Verdeyen, D. L. Carroll, and J. G. Eden, “Lasing in Cs at 894.3 nm pumped by the dissociation of CsAr excimers,” Electron. Lett. 44(25), 1466–1467 (2008). [CrossRef]

3. J. D. Readle, J. T. Verdeyen, J. G. Eden, S. J. Davis, K. L. Gabally-Kinney, W. T. Rawlins, and W. J. Kessler, “Cs 894.3 nm laser pumped by photoassociation of Cs-Kr pairs: excitation of the Cs D(₂) blue and red satellites,” Opt. Lett. 34(23), 3638–3640 (2009). [CrossRef] [PubMed]

4. J. D. Readle, “Atomic alkali lasers pumped by the dissociation of photoexcited alkali-rare gas collision pairs,” University of Illinois at Urbana-Champaign, 2010.

5. A. E. Mironov, D. L. Carroll, J. W. Zimmerman, and J. G. Eden, “Cs D2 line laser (852.1 nm) pumped by the photoassociation of Cs-Ar, Cs-Kr, and Cs-Xe collision pairs: Impact of rare gas partner on threshold and efficiency,” Appl. Phys. Lett. 113(5), 051105 (2018). [CrossRef]

6. A. D. Palla, D. L. Carroll, J. T. Verdeyen, J. D. Readle, T. M. Spinka, C. J. Wagner, J. G. Eden, and M. C. Heaven, “Multi-dimensional modeling of the XPAL system,” Proc. SPIE 7581, 75810L (2010). [CrossRef]

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10. X. Xu, B. Shen, C. Xia, and B. Pan, “Modeling of Kinetic and Thermodynamic Processes in a Flowing Exciplex Pumped Alkali Vapor Laser,” IEEE J. Quantum Electron. 53(2), 1–7 (2017). [CrossRef]

11. D. A. Steck, “Rubidium 85 D Line Data,” Available: http://steck.us/alkalidata.

12. M. C. Heaven and A. V. Stolyarov, “Potential energy curves for alkali metal - Rare gas exciplex lasers,” Proc. 41st AIAA Conf. on Plasmadynamics and Lasers (Chicago, IL, June) AIAA Paper 2010–4877. [CrossRef]

13. S. J. Davis, W. T. Rawlins, K. L. Galbally-Kinney, and W. J. Kessler, “Spectroscopic and kinetic measurements of alkali atom-rare gas excimers,” Proc. 41st AIAA Conf. on Plasmadynamics and Lasers (Chicago, IL, June) AIAA Paper 2010–5044.

14. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

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Parameters	Parameters Description	Parameters Value /Ref
T	Cell temperature	525 K
P_Rb	Saturated pressure	10^{2.881 + 4.312-4040/T} torr [11]
M	Number density of Kr	2.50 × 10²⁵ m⁻³
λ_p	Wavelength of pump lights	758 nm [13]
∆λ_p	Linewidth of pump lights	1 nm
∆λ_abs	Linewidth of absorption	2 nm
k_abs	Reduced absorption cross-section	1.66 × 10⁻⁴⁶ m [13]
σ_p	stimulated absorption cross section	6.99 × 10⁻¹⁹ m²
t₃₀	Lifetime of the Rb* state	26.2348 ns [11]
d_Rb-Kr	the distance of Rb and Ar atom	5 × 10⁻¹⁰ m [12]
R₀	The internuclear distance	4.5 $Å$ [12]
∆E₁₀	Energy difference between Rb and Rb-Kr state	25 cm⁻¹ [12]
∆E₂₃	Energy difference between Rb-Kr and Rb state	444 cm⁻¹ [12]
k_xa	Association rate of Rb-Kr state	7.9 × 10⁹ s⁻¹
k_xd	Dissociation rate of Rb-Kr state	1.3 × 10¹² s⁻¹
k_ba	Association rate of Rb*-Kr state	2.3 × 10⁹ s⁻¹
k_bd	Dissociation rate of Rb*-Kr state	2.4 × 10¹² s⁻¹
λ_l	Wavelength of laser	780 nm [11]
σ₃₀	Stimulated emission cross section	7.34 × 10⁻¹⁷ m²
L	The length of vapor cell	0.1 m
T_l	Transmission for laser	0.99
T_p	Transmission for pump lights	0.99
T_s	Intracavity single-pass optical element losses	0.70
R_oc	Reflectivity of output coupler	0.25
R_p	Reflectivity of the back reflector	0.99

Theoretical simulation on exciplex pumped Rb vapor laser

Abstract

1. Introduction

2. Description of theory

3. Results and discussion

3.1 Comparison with Carroll’s model

3.2 Influences of parameters with uniform temperature distribution

3.3 Influence of gas flow velocity

4. Conclusion

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (22)

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