1 Introduction

In spite of the fact that the first optically pumped alkali vapor laser was proposed by Schawlow and Townes in 1958 [1], a real interest to these lasers appeared only in the past 10 years due to the development of efficient and high-power diode laser pump sources [2–6]. As a new class of laser, diode-pumped alkali vapor lasers (DPALs) have the advantages of both solid and gas lasers, such as the high quantum efficiency, the good optical quality of gaseous gain medium and the reduced thermal problems, and are expected to achieve high output power with high quality beam in near-infrared spectrum. These lasers have potential applications in laser cooling, directional energy transmission, material processing, medical magnetic resonance imaging (MRI) systems and so on [6, 7].

A crucial factor for a highly efficient DPAL is to ensure the lineshape matching between the diode laser emission line and the alkali atomic absorption line. At present, there are two approaches for this matching that are explored. One is the use of high pressure buffer gas to broaden the alkali absorption line to about 1 nm and to match it to the pumping diode linewidth, and the other is narrowing of the diode laser linewidth to match it to the alkali vapor absorption line broadened by low pressure (about 1 atm) buffer gas [8]. So far series of experiments have been conducted to improve the lineshape matching coefficient of DPALs with very promising results [3, 9]; however, there is rare theoretical study on the quantitative description of the lineshape matching coefficient.

In this paper, considering the effect of the linewidth ratio between D2 absorption line and the pump diode laser arrays, as well as the center frequency shift caused by mode hopping of the laser diode arrays (LDAs) on the lineshape matching, we put forward a formula to describe the matching coefficient and then calculate the dependences of the matching coefficient and optical–optical efficiency on the linewidth ratio and the relative center frequency shift of the LDAs using the experimental data. The good agreement between the calculation and the literature results shows the validation of the definition of the lineshape matching coefficient, which provides a useful guide to further improving the output characteristics of DPALs.

2 Definition of the lineshape matching coefficient

According to the formula of the optical cross sections for absorption [10] and Beer’s law [11], the lineshapes of the pressure broadened D2 absorption line and the narrowed LDA emission line can be considered to have Lorentzian profiles as shown in Fig. 1, where the D2 absorption and diode laser linewidths (FWHM) are 12 and 30 GHz, respectively. We can see that they are well matched with the pump absorption and laser excitation source spectral profiles in [3], which use a FWHM D2 absorption linewidth of 12.7 GHz, approximately 2.4 × narrower than the 30-GHz-wide Ti:sapphire pump excitation laser. The lineshape matching coefficient is related to not only each area of the two lineshapes, but also the shaded area that represents the partial pump light absorbed by alkali gain medium. Therefore, it can be defined as

$$\eta = \frac{S}{{(S_{\text{P}} + S_{\text{D2}} )/2}}$$
(1)

where S is the overlapping area of the lineshapes of the pump LDA light and the alkali D2 absorption line, S P and S D2 are the areas of the lineshapes of pump light and D2 line, respectively. What is more, after the lineshapes are normalized, the coefficient can be reexpressed by an expression containing only the overlapping area as

$$\eta = S$$
(2)

We can know from the Eqs. (1) and (2) that the coefficient will equal zero when the two lineshapes have no common part, and unity when superpose completely. According to the above definition of the matching coefficient, we can conduct further analysis and discussion on the dependence of the matching coefficient on the linewidth ratio between the alkali D2 line and the pump LDA light and the center frequency shift of the pump LDA light.

Fig. 1
figure 1

Lineshapes of the pump LDA emission and Rb D2 absorption line

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3 Results and discussion

3.1 Dependences of the matching coefficient and optical–optical efficiency on the linewidth ratio between D2 absorption line and pump light

3.1.1 Modeling both the lineshapes of the Rb D2 absorption line and the narrowed LDA emission as Lorentzian profiles

By assuming the center frequency of the pump LDA is fixed at that of D2 absorption line, the alkali vapor density is the same as the photon density of pump light, and both of the lineshapes of the pressure broadened D2 absorption line and the narrowed LDA emission line are normalized Lorentzian profiles, we calculated the dependences of the matching coefficient and optical–optical efficiency on the linewidth ratio between Rb D2 line and the pump LDA, as shown in Fig. 2.

Fig. 2
figure 2

Dependences of the matching coefficient η and optical–optical efficiency Eo–o on the linewidth ratio between D2 absorption line and pump light. Where square line is the simulated results of the matching coefficient when Δv D2 = 10GHZ and Δv D2 ≤ v p , star line is the simulated results of the matching coefficient when Δv D2 = 10GHZ and Δv D2 ≥ v p , solid line is the optical–optical efficiency versus the ratio of linewidths in [3], circle line is the computation results of the optical–optical efficiency when Δv D2 ≤ v p , dot line is the computation results of the optical–optical efficiency when Δv D2 ≥ v p

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It is noteworthy that the model can be utilized to obtain the optical–optical efficiency using the formula below with a conditional constant co:

$$E_{o - o} = 1 - \exp ( - co \times \eta )$$
(3)

We can know from Fig. 2 that both the matching coefficient and the optical–optical efficiency increase with the increase of linewidth ratio for two conditions, the linewidth ratio of unity turns out to be the best with a maximum matching coefficient of unity and a highest optical–optical efficiency of ~0.6, which means the rubidium atoms have the highest absorption efficiency to the pump photons when Δv D2 = v p , that is consistent with the experimental results in literatures [12–14]. Furthermore, when Δv D2 ≤ v p and Δv D2 ≥ v p , the trends of the matching coefficients are almost the same. Under the same ratio, the coefficients for these two cases are basically equivalent to each other.

For formulizing the simulation to provide simple and direct theoretical calculation, we make a curve fitting with the formula below:

$$y = a - e^{ - b(x + c)}$$
(4)

where y is the matching coefficient η, x = Δv D2/Δv p when Δv D2 ≤ Δv p, or x = Δv p/Δv D2 when Δv D2 ≥ Δv p., with parameter a = 1.1, b = 2 and c = 0.1.

For definiteness, we consider a pump irradiance level of 40 kW/cm2 with a FWHM spectral extent of 2 nm, which is then delivered into the end of a gain cell with 90 % efficient, that means we can set the conditional constant co equal to or approximately equal to 0.9, which have included the influences of most of the factors that may have impact on the optical–optical efficiency, such as temperature and output coupler loss. The 2-nm FWHM spectral extent assumed for the diode array is also achievable for cw bars run at high average power. And the pump light is double passed through the cell, the output coupler reflectivity of the laser is held constant at 0.3, and the one-way cavity transmission excluding output coupler loss is held constant at 0.99. For the baseline 40-kW/cm2 pump irradiance under consideration, it is assumed that the cells are filled with He at room temperature and then sealed before being brought to the operational temperature (T = 376 K) of the laser [3]. We give the dot line and circle line predications in Fig. 2 for the Rb laser optical–optical efficiency versus the ratio of the linewidths and compare them with the experiments results in literature [3]. As can be seen in the figure all of the optical–optical efficiencies increase with the rise of linewidth ratio, being maximized when Δv D2 = Δv p. The good agreement between literature data and model evident in Fig. 2 is achieved without the use of other adjustable parameters except co, giving us confidence that our model is valuable and practical, and we believe a simple and useful way to calculate the optical–optical efficiency straightly has just been found.

3.1.2 Modeling the lineshape of the Rb D2 absorption line and the one of narrowed LDA emission as Voigt and Lorentzian profiles, respectively

More detailed consideration shows that D2 absorption line cannot be strictly represented by a single Lorentzian profile due to the contributions from each hyperfine transition in both Rb isotopes [10], so we model the broadened Rb D2 absorption line as normalized Voigt profile including the effects of hyperfine splitting, pressure broadening and collisional shift. With all else assumptions being the same as Sect. 3.1.1, the dependences of the matching coefficient and optical–optical efficiency on the linewidth ratio between Rb D2 line and the pump LDA are calculated and shown in Fig. 3.

Fig. 3
figure 3

Dependences of the matching coefficient η and optical–optical efficiency Eo–o on the linewidth ratio between D2 absorption line and pump light. Where square line is the simulated results of the matching coefficient when Δv D2 = 10 GHz and Δv D2 ≤ Δv p, star line is the simulated results of the matching coefficient when Δv D2 = 10 GHz and Δv D2 ≥ Δv p, solid line is the optical–optical efficiency versus the ratio of linewidths in [3], circle line is the computation results of the optical–optical efficiency when Δv D2 ≤ Δv p, dot line is the computation results of the optical–optical efficiency when Δv D2 ≥ Δv p

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On one hand, we can see that for the two cases of Δv D2 ≤ Δv p and Δv D2 ≥ Δv p, the trends of the matching coefficients are somewhat different, and when the linewidth ratio ranges from 0.1 to 1, the coefficient of Δv D2 ≥ Δv p is no less than that of Δv D2 ≤ Δv p because of the pump linewidth of the former one being closer to the Rb D2 linewidth than the latter one under the same ratio. On the other hand, we can also see that for these two cases, the coefficients tend to be a constant value but cannot reach unity when Δv D2 = Δv p due to the influence of different profiles of Rb D2 line and pump light, which reasonably conforms to the actual situation in experiments that the pump light cannot be absorbed completely by the alkali vapor on account of the scattering, radiative transition and thermal effects.

For Fig. 3, the values of a, b and c are 1.0, 2.5 and 0.1, respectively.

Figure 3 also depicts the optical–optical efficiencies of not only the experimental results in [3] but also the calculated results of Eq. (3) as functions of the linewidth ratio. It is interesting to note that both of the hypotheses that the broadened Rb D2 absorption line being Voigt profile and Lorentzian profile make the computed results of Eq. (3) be in agreement with the optical–optical efficiency curve in literature with the same conditional constant co being essentially roughly equal to 0.9, meaning the compatibility of Eq. (3) with the two hypothetical profiles of the Rb D2 absorption line.

3.2 Dependences of the matching coefficient and optical–optical efficiency on the center frequency shift of pump light caused by mode hopping

Not only the linewidth ratio between pump light and alkali atomic D2 absorption line but also the center frequency shift of pump light are key factors for improving the output performance of DPALs. The center frequency shift of the pump LDA sometime appears as shown in Fig. 4 due to the variation of working temperature and injection current, which will affect the output performance of DPAL in some extent. Therefore, it is necessary to analyze the influence of the center frequency shift of pump light on the matching coefficient.

Fig. 4
figure 4

Normalized lineshapes of Rb D2 line and pump light before and after the appearance of mode hop

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According to Eqs. (2) and (3), we calculated the lineshape matching coefficient and optical–optical efficiency as functions of the relative center frequency shift \(|{\Delta}\nu_{\text{P}} - {\Delta}\nu_{\text{D2}} |/({\Delta}\nu_{\text{P}} + {\Delta}\nu_{\text{D2}} )\) under conditions of Δv D2 held fixed at 10 GHz and different Δv p, the calculation results are shown in Fig. 5 (Δv D2 ≥ Δv p) and Fig. 6 (Δv D2 ≤ Δv p), respectively. It is clear that if \(|{\Delta}\nu_{\text{P}} - {\Delta}\nu_{\text{D2}} |\) is larger than (Δv p + Δv D2), i.e., the relative center frequency shift exceeds unity, the matching coefficient will be less than 0.3, while the optical–optical efficiency fall below 0.2, causing the very low output power. That is why we select the relative center frequency shift \(|{\Delta}\nu_{\text{P}} - {\Delta}\nu_{\text{D2}} |/({\Delta}\nu_{\text{P}} + {\Delta}\nu_{\text{D2}} )\) as abscissa, for making sure that the domains of the lineshape matching coefficient function of relative center frequency shift and linewidth ratio can simply be 0–1.

Fig. 5
figure 5

Dependences of the matching coefficient η and optical–optical efficiency Eo–o on the ratio of the center frequency shift of pump light to the sum of linewidths of Rb D2 absorption line and pump light. Where solid lines are the simulated results of the matching coefficient when Δv D2 = 10 GHz and Δv D2 ≥ Δv p, dash lines are the computation results of the optical–optical efficiency when Δv D2 = 10 GHz and Δv D2 ≥ Δv p

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Fig. 6
figure 6

Dependences of the matching coefficient η and optical–optical efficiency Eo–o on the ratio of the center frequency shift of pump light to the sum of linewidths of Rb D2 absorption line and pump light. Where solid lines are the simulated results of the matching coefficient when Δv D2 = 10 GHz and Δv D2 ≤ Δv p, dash lines are the computation results of the optical–optical efficiency when Δv D2 = 10 GHz and Δv D2 ≤ Δv p

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It can also be seen that for the same value of the relative center frequency shift, both the matching coefficient and optical–optical efficiency increase as the linewidth ratio increases (which was confirmed in Sect. 3.1). However, when the linewidth ratio is set as 1.0, 0.7, 0.5 and 0.3, respectively, they dramatically decrease with the increase of the relative center frequency shift, that means the center frequency shift of pump light has a great impact on the matching coefficient and optical–optical efficiency, which was verified in literatures [15–17].

When the center frequency shift is larger than half of the sum of the linewidths, the optical–optical efficiency will fall below 50 %, which would lead to bad beam quality and low output power, thus if we want to design a highly efficient DPAL, we should keep the center frequency of pump light be as close as possible to that of alkali D2 line for obtaining a higher lineshape matching coefficient.

4 Conclusion

In this work, by assuming the lineshapes of pump LDA and D2 line to be identical normalized Lorentzian profiles, a definition of the matching coefficient of the lineshapes of D2 line and pump light in DPALs was proposed, and it was used to analyze the situation that the lineshape of the Rb D2 absorption line and the one of narrowed LDA emission are Voigt and Lorentzian profiles, respectively. Dependences of the matching coefficient on the linewidth ratio between D2 line and pump light and the center frequency shift of pump light were calculated and analyzed. Results showed that the matching coefficient increases with the linewidth ratio from 0.1 to 1, and decreases with the rising center frequency shift of pump light, which is qualitatively demonstrated by experimental results. Additionally, we have given a simple and promising method to calculate the optical–optical efficiency using the lineshape matching coefficient and conditional constant, good agreement is found to exist between model and literature data. Therefore, the matching coefficient we defined provides an effective way to improve the pump efficiency and output characteristics of DPALs. What is noteworthy is that the definition and analysis of the lineshape matching coefficient are based on the use of the D2 absorption line of alkali atoms and the diode laser emission line; nonetheless, they can also be applied to any other similar laser systems.