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Abstract
Diode-pumped alkali lasers (DPALs) have drawn much attention since they were proposed in 2001. The narrow-linewidth DPAL can be potentially applied in the fields of coherent communication, laser radar, and atomic spectroscopy. In this study, we propose a novel protocol to narrow the width of one kind of DPAL, diode-pumped rubidium vapor laser (DPRVL), by use of an injection locking technique. A kinetic model is first set up for an injection-locked DPRVL with the end-pumped configuration. The laser tunable duration is also analyzed for a continuous wave (CW) injection-locked DPRVL system. Then, the influences of the pump power, power of a master laser, and reflectance of an output coupler on the output performance are theoretically analyzed. The study should be useful for design of a narrow-linewidth DPAL with the relatively high output.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A diode pumped alkali laser (DPAL) has been greatly developed during the past decade [1–3]. As a new concept of the hybrid laser, the DPAL owns a number of attractive advantages compared to the other types of high-powered lasers. The quantum defect of a DPAL is extremely low (4.72% for Cs, 1.85% for Rb, and 0.44% for K), which makes it possible to construct a laser with both high optical-optical efficiency and easy thermal management [4,5]. Furthermore, the thermal problems can be reduced by flowing the gaseous medium and the buffer gases to remove the generated heat during the lasing operation [6, 7]. In addition, converting the pump light into a coherent output laser should be very quick for a DPAL due to the large optical cross sections of atomic alkalis. All these desirable merits make a DPAL become one of the most promising candidates with high power, good beam quality, and compact size.
As the wavelength of a DPAL is located in the near-infrared wavelength region, a narrow-linewidth DPAL, especially for the single-frequency type, can play an important role in numerous application fields such as the coherent communication, laser radar, metrology, and high-resolution spectroscopy of weak atomic. Generally, a high-powered laser with the extremely narrow linewidth can be obtained by following three approaches. The first one is to construct a high power oscillator with some wavelength control elements including the gratings, prisms, and etalons. However, these inserted elements might greatly increase the laser transmission loss inside the resonator and lead to the reduction in efficiency and output power. The second method is to amplify a low-powered seed laser with the master oscillator power amplification (MOPA) configuration. Such method has already been theoretically and experimentally investigated in several DPAL studies [8–10]. However, in most practical cases, the power of a narrow-linewidth seed laser is very low. In this case, it is very difficult to amplify the seed power to a relative high level through a one-stage amplifier as the length of atomic alkali medium could not be fabricated too long. In addition, the undesirable amplified spontaneous emission (ASE) will decrease the amplification efficiency by consuming the inversed populations in a DPAL system with high gain. The third alternative technique is injection locking which has been widely utilized to realize a narrow-linewidth light source in the excimer lasers (ELs) and diode pumped solid-state lasers (DPSSLs) [11–14]. In an injection-locked laser system, a stable low-powered signal (master laser) is imported into an oscillator as diagrammed in Fig. 1. When the wavelength of the master laser is close to that of the free-running laser, the outputted slave laser will be the same as the master laser in wavelength and be similar to the free-running laser in power. This kind of laser has such advantages as high extraction efficiency, simple structure, high reliability, and good laser beam quality [15, 16].
Until now, the highest outputted powers have respectively achieved as high as 1.5 kW and 571 W for alkali lasers with the oscillator and MOPA configurations [17]. However, the research of a rubidium laser with the extremely narrow linewidth has still been rare so far. In this study, a theoretical model is established to describe the laser kinetics of an injection-locked DPRVL. Based on the model, we first discuss the laser performance in time-domain and then evaluate the laser output power as well as the locked wavelength duration under different physical parameters. To the best of our knowledge, there have been no similar published literatures for an injection-locked DPAL.
2. Theoretical Analyses
Figure 2 shows the electronic energy levels and the main kinetic processes of an injection-locked rubidium laser. The D2 and D1 lines are the pump absorption and laser emission lines broadened by the buffer gas, Q and γ32 represent the quenching rate and fine-structure mixing rate, and A stands for the spontaneous emission rate, respectively. n1, n2, n3, n4, and n5 are respectively the alkali number densities at the 52S1/2, 52P1/2, 52P3/2, 52D5/2, 3/2 and 72S1/2, and the ionized levels. ΓP, ΓSL, and ΓFL are respectively the absorption rate of the pump light, emission rate of the slave laser, and emission rate of the free-running laser. As diagrammed in Fig. 2, n2 level is mainly populated by the fine-structure mixing between n2 and n3 levels. The deleterious processes of quenching, energy pooling, ionization, and recombination would affect the population distribution of the excited energy level. Thus, we have considered all these kinetic processes in our theoretical model to get the precise simulation results.
According to the kinetic processes, we can obtain the following rate equations that describe the population density distributions for five energy levels.
where λFL and λML are respectively the wavelengths of a free-running laser and a master laser, σ21(λFL) and σ21(λML) are respectively the collisionally broadened cross sections at the free-running laser and master laser wavelengths, σ31(λ) is the spectrally resolved pump-absorption cross section, σphotoionization is the photo-ionization transverse section, T is the cell temperature, kb is the Boltzmann constant, △E is the energy gap between the 52P3/2 and 52P1/2 levels, n0 is the total alkali number density in the vapor cell, Γphotoionization is the transition rate of photo-ionization, kEP2 and kEP3 are respectively the rate constants of the energy pooling of the 52P1/2 and 52P3/2 levels, kPI is the Penning ionization rate coefficients, and krecombination is the recombination rate constant, respectively. In this study, the values of the main parameters for a rubidium laser are σphotoionization = 2 × 10−17 cm2, kEP2 = 2.09 × 10−10 cm3/s, kEP3 = 1.5 × 10−9 cm3/s, kPI = 3.5 × 10−8 cm3/s, and krecombination = 3 × 10−9 × T-4.5 cm6/s, respectively [18–22].In an alkali laser, the buffer gas is often added in the vapor cell to collisionally broaden the pump absorption line (D2 line). The homogeneously broadened linewidth can be calculated by
where γ is the linewidth broadening coefficient, Pgas is the pressure of buffer gas, and Tγ is temperature when the broadening coefficient was measured, respectively. Figure 3 shows the simulated linewidths of the D2 line under different cell temperatures.From Fig. 3, we can see that the Doppler linewidth is relatively narrow and insensitive to the cell temperature. The linewidth broadened by typical buffer gases, which gently increases with the cell temperature, is much wider than the Doppler linewidth. In the model, we consider that the homogeneous broadening of the D2 line is much greater than the unhomogeneous Doppler broadening due to the existence of the addition buffer gases [23]. Therefore, the collisionally broadened cross sections σ21(λFL), σ21(λML), and σ31(λ) are respectively expressed by
where σ021 and σ031 are respectively the peak collisionally broadened cross sections of the D1 and D2 transitions, △νD1 and △νD2 are respectively the broadened linewidths (FWHM) of the laser emission and pump absorption lines, λD1 and λD2 are respectively the center wavelengths of the D1 and D2 lines, and c is the light velocity in vacuum. The values of σ021 and σ031 are respectively 1.08 × 10−9 cm and 1.94 × 10−9 cm [23].For mathematic simplification, we assume that the master laser perfectly matches with the slave laser inside the cavity. Then, for an injection-locked DPAL with the end-pumped configuration, ΓP(t), ΓFL(t), and ΓSL(t) in Eq. (1) can be respectively expressed by [24]
where ηdel is the pump delivery efficiency from the pump source to the alkali medium, ηmode represents the mode-filling efficiency, Pp(λ) is the spectrally resolved pump power with a Gaussian distribution profile, Rp is the reflectance of the pump light at the rear mirror of the laser cavity, and VL is the mode volume of the alkali laser, respectively. In the study, VL is calculated bywhere z1 and z2 are the distances from the two side surfaces to the laser beam waist position, ωML is the waist radii of the laser beam, M2 represents the beam propagation factor, respectively.ΨFL(t) and ΨSL(t) in Eqs. (7) and (8) are respectively the longitudinally averaged intensities at the free-running laser and slave laser wavelengths inside the cavity. The differential equations describing the time evolution of ΨFL(t) and ΨSL(t) yield
where TT is the window single-pass transmission efficiency at both the pump and the lasing wavelengths, Roc is the reflectance of an output coupler, Lg is the length of the gain medium, tRT is the round-trip time for the laser cavity, ΦFL(t) is the spontaneous emission density coupled with the free-running lasing mode, and ΦML(t) is the longitudinally average intensity of the master laser, respectively. According to Ref. 25, ΦFL(t), which acts as a seed for stimulated emission at the onset of the free-running laser generation, can be presented bywhere Lc is the length of the cavity, S is the area at the laser beam waist, and Loss is the resonator loss, respectively.To obtain the function of ΦML(t), we first depicture the powers of a master laser, PM1(t), PM2(t), PM3(t), and PM4(t), towards two opposite directions at different locations of the laser cavity as shown in Fig. 4. Then, the average master laser intensity can be given by [25]
where PM1(t) and PM3(t) yield where Pmaster(t) is power of the master laser in time domain.With combination of Eqs. (13)-(15), the longitudinally average intensity of the master laser can be expressed as a function of the power of a master laser:
In the same way, we can obtain the output power of the slave laser in time domain [26]:
3. Results and discussions
3.1 Tunable range of an injection-locked DPRVL
Generally, the lasing wavelength of an injection-locked DPAL can be successfully locked within a certain wavelength range. When the master laser wavelength is out of this range, the output laser will operate at the free-running wavelength. Such a locking wavelength range is defined as the tunable range in the study. To evaluate the tunable range, we first calculate the temporal characteristics of an unlocked DPRVL system as shown in Figs. 5(a)-5(c). During the calculation, the wavelength of the master laser is by 2 GHz larger than that of the free-running laser. The other parameters used in the simulation are: Pp = 10 W, Pmaster(t) = 50 mW, T = 390 K, Lg = 4 cm, and Roc = 40%. Before the laser injection, both the pump and the master lasers are thought to be at null statuses and all populations are assumed to be located at the 52S1/2 level. From Fig. 5(a), the output power of the slave laser first rapidly increases and then reaches a short-time steady plateau after the process of relaxation oscillation. In this initial period of time, the free-running laser is hardly observed as the master laser has extracted most of the inversion populations. After this period of time, the power of the free-running laser begins to increase and the slave laser power exhibits an opposite tendency until both lasers reach their steady states. The final output is a combination of the relatively high free-running laser and the relatively low slave laser which is considered as the amplified master laser. Therefore, to obtain a wavelength-locked output, the free-running laser must be suppressed all the time, especially when the slave laser arrives at the first steady status (defined as the first plateau in this study) [27]. In other words, the free-running laser should not be amplified inside the cavity at the first plateau so that the following equations must be satisfied:
where n2_fp and n1_fp are respectively the number density of the n1 and n2 levels at the first plateau.
Fig. 5 Output powers of both the slave laser and the free-running laser in time domain (a). Temporal population density at the ground and excited levels (b) as well as the inversed population density (c).
Figure 5(b) shows the population density at the ground and excited levels as well as the inversed population density between these two levels. Initially, the population at n1 level rapidly decreases while the population at the 52P1/2 level quickly increases. When the population at the 52P1/2 level exceeds that at the 52S1/2 level, the lasing can be observed as shown in Fig. 5(a). Then the variations of the populations at both levels become unnoticeable. As to the inversed population density, it first reaches the maximum value at the laser oscillation relaxation and then decreases a little when the free-running laser increases (referring to Fig. 5(a)). Figure 5(c) diagrams the gain coefficient in time domain. It is obvious that the gain coefficient of the free-running laser is always somewhat higher than that of the slave laser. As the gain coefficient is highly related to the inversed population, the both curves in Fig. 5(c) exhibit the similar tendency with the invested population in Fig. 5(b).
According to the physical rule mentioned above, we investigate the steady-state output laser under different master wavelengths for an injection-locked DPRVL. The input parameters are the same as those in the above paragraph. The results are presented in Fig. 6. We can see that the output power first increases and then falls as the master wavelength increases. The maximum laser power can be achieved when the master laser has the same wavelength with the free-running laser. Note that the outputted power changes quite small with the master wavelength. It means that the wavelength can be tuned in a certain range without the obvious power decrease for such an injection-locked DPRVL system. This is because the master laser just acts as a seed of the laser generation. The output power is still dominated by the resonator cavity and the pump power. It is worthy to point out that the power of the suppressed free-running laser is close to zero inside the tunable range.
Fig. 6 Slave laser power versus the frequency difference between the master laser and free-running laser.
3.2 Different pump powers
In this subsection, we analyze the output features of an injection-locked rubidium vapor laser under different pump powers. During the simulation, the following parameters are adopted: Pmaster(t) = 50 mW, T = 390 K, Lg = 4 cm, and Roc = 40%. The results diagrammed in Fig. 7(a) show that the injection-locked laser power increases linearly with the pump power. Such a variation regulation is similar to that of a conventional DPAL system. With respect to the tunable range, it decreases when the pump power rises. This is because the inversed populations generally increase with the pump power. As a result, when the pump power becomes high, the free-running laser can be only suppressed in a small wavelength range according to Eq. (18). If the free-running laser is not restrained, the system will operate at the free-running wavelength instead of the master laser wavelength, which means the failure of injection locking. Thus, one should carefully choose the power of the pump source if an output laser with both high power and large tunable range is desired.
Fig. 7 Output power and tunable range under different pump powers (a). Output power versus pump power for different cell length (b) and different cell length (c).
Figure 7(b) displays the output power versus the pump power for different cell temperatures. We can see that the output power linearly increases with the pump power for every cell temperature. Figure 7(c) presents the output power versus the pump power for different cell lengths. In Figs. 7(b) and 7(c), all the curves are linear. We can deduce that when the pump power is very high, the relatively high cell temperature and long cell length should be desired for a high outputted power
3.3 Different powers of a master laser
In Fig. 8(a), the output power and tunable range are expressed as functions of the power of a master laser when the cell temperature, pump power, cell length, and reflectance of an output coupler are assumed to be 390 K, 10 W, 4 cm, and 40%, respectively. It is clear that both the output power and the tunable range increase with the power of a master laser. However, the accelerative extent is very small for the output power but relatively large for the tunable range. The increased output power might be regarded as the amplified master laser coupled into the slave laser, which is relatively small by comparing with the power of the oscillated slave laser in the resonator. Thus, we can conclude that increasing the power of a master laser is an efficient procedure to realize a wide tunable range in an injection-locked DPAL system although such a method contributes little to the output power.
Fig. 8 Output power and tunable range under different powers of a master laser (a). Output power versus master laser power for different cell length (b) and different cell length (c).
Figures 8(b) and 8(c) express the output power changing with the master laser power for different input parameters. It is clear that the output power increases slightly with the master laser power for all the curves in both figures, which coincides with the conclusion in the last paragraph. In addition, when the cell temperature rises, the output power first increases and then decreases. Similarly, an optimal cell length also exists. Thus, when designing an injection-locked DPAL system, one should perform an overall optimization for a maximum output power
3.4 Different reflectances of an output coupler
The output power and tunable range under different reflectances of an output coupler are illustrated in Fig. 9. During the simulation, the pump power, power of a master laser, cell temperature, and cell length are set to 10 W, 50 mW, 390 K, and 4 cm, respectively. We can see that the smaller the coupler reflectance is, the higher the output power becomes due to the high gain of the alkali laser. The curve of the tunable range also exhibits a diminishing tendency when the reflectance of an output coupler becomes larger as shown in Fig. 9. The reason can be explained as that the remaining laser power inside the cavity is relatively low after a round-trip transmission as most laser energy exports from the oscillator when the reflectance of an output coupler turns to be small. Among the remained laser energy, the slave laser and free-running laser components can respectively get supplement from the injected master laser and the spontaneous emission, where the former one is much larger than the latter one in intensity. The large augment to the slave laser somewhat make up for such a shortcoming as the gain of the slave laser is smaller than that of the free-running laser especially when the wavelengths of two lasers are different. Thus, the output laser can be locked in a relatively large wavelength region by adopting low reflectance of an output coupler.
4. Conclusions
In this study, we propose a novel method to construct a narrow-frequency alkali laser by applying the injection locking technique. A corresponding kinetic model has been established to investigate the output characteristics of an injection-locked DPAL. Based on the model, we first theoretically analyze the tunable range of a continuous wave (CW) injection-locking DPRVL. Then, the output features of an injection-locked DPRVL are investigated for different physical conditions. According to the simulation results, a low pump power, a high master laser power, and a small output coupler reflectivity are desired to obtain a relatively large tunable range. To achieve a relatively high output power, one should employ a high pump power, a high master laser power, and a small output coupler reflectivity. In addition, one should carefully choose the master laser and design the slave laser cavity to acquire an outputted laser beam with both good spectral and spatial profiles. Note that the phase of the laser is not taken into account in this study. We will carry out the systematical study of a pulsed injection-locked DPAL with the consideration of the laser phase in our continuous study.
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