Analytical model of a transversely diode-pumped alkali metal vapour laser

In recent years, diode-pumped alkali metal vapour lasers have been intensively investigated experimentally and theoretically [1–5]. A great interest in these lasers is reasonably associated with their ability to generate cw radiation of very high power, while demonstrating a high conversion efficiency of pump radiation into laser radiation.

The maximum output of an alkali metal vapour laser has been achieved in work [4], where the creation of a longitudinally diode-pumped caesium vapour laser with closed-cycle laser-active medium circulation is reported. In cw mode, the laser radiation power was 1 kW with a conversion efficiency of pump energy into laser energy amounting to 48%.

Obtaining significantly higher powers requires the scaling of the process. In our opinion, the geometry with transverse pumping is optimal for scaling. In this geometry, the power of the generated radiation increases in proportion to the length of the laser. The idea of a transversely diode-pumped alkali metal vapour laser was patented [6] and first implemented in [7]. A theoretical model of such a laser was first proposed by Komashko and Zweiback [8]. The laser operation was described in [8] by a rather complex system of differential equations, which was solved numerically in the approximation of the effective absorption cross section of pump radiation. Yang et al. [9] developed a somewhat different numerical model of the laser than that presented in work [8]. The equations in [9] were solved numerically on the assumption that the population levels of the atoms of the active medium do not depend on the coordinate along the resonator axis, i.e., they are constant along the propagation direction of the laser output.

In paper [10], we developed an analytical model of a transversely diode-pumped alkali metal vapour laser, which describes the operation of the laser in the practically important case of high radiation intensity but at a sufficiently high buffer gas pressure and in the absence of radiation loss in the resonator. The aim of the present paper is to expand the analytical model [10] for the possibility of describing the generation of a high-intensity laser at an almost arbitrary buffer gas pressure and taking into account radiation losses in the resonator.

An alkali metal vapour laser operates in accordance with a three-level scheme (Fig. 1). The laser cycle includes optical excitation of the D₂ (²S_1/2 – ²P_3/2) transition, the population of the ²P_1/2 level in collisions of alkali metal atoms at the ²P_3/2 level with the buffer gas particles, and laser generation at the D₁ (²P_1/2 – ²S_1/2) transition. In Fig. 1, the levels ²S_1/2, ²P_1/2 and ²P_3/2 are denoted by figures 1, 2 and 3, respectively.

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Let us consider the operation of a transversely diode-pumped alkali metal vapour laser. The scheme of the laser is shown in Fig. 2. To simplify the analysis, we assumed that a cell with alkali metal vapours and buffer gases has the form of a rectangular parallelepiped with edges z₀ (length), y₀ (width) and x₀ (height). Pump laser diodes are arranged on one side of the cell. Their radiation enters the cell in the xz plane and propagates in the y direction. For a more efficient utilisation of the pump energy, a plane mirror is mounted on the other side of the cell, which reflects the pump radiation passing through it (the reflection coefficient of the mirror, R_p) back to the cell. The resonator consists of two plane mirrors with reflection coefficients R₀ and R₁. The transmission coefficients of the cell windows are T₀ and T₁. The radiation energy losses in the resonator (due to the diffraction of light at the edges of the mirrors, due to the geometric imperfection of the resonator, due to the scattering of light in the medium by inhomogeneities) are taken into account by introducing the effective transmittance T_s. The value of T_s characterises the relative losses of the radiation energy in the resonator in one pass, excluding the transmission losses of the windows. It is assumed that the losses of T_s are localised in front of the rear mirror R₁. Laser radiation is coupled out of the cell in the xy plane through a semitransparent mirror with a reflection coefficient R₀ and propagates in the z direction. For simplicity, we assume that the intensity distribution of pump radiation is uniform in the xz plane (at the input to the cell). In this case, as a consequence, the intensity distribution of laser radiation is also uniform along the height of the cell (along the x axis).

Under steady-state conditions, absorption of pump radiation and amplification of laser radiation are described by the equations:

Here, and are the spectral densities of the pump radiation power at a frequency ω, propagating along the y axis and in the opposite direction (after reflection by the mirror), respectively; and are the intensities of laser radiation propagating along the z axis and in the opposite direction, respectively; N₁(y, z), N₂(y, z) and N₃(y, z) are the populations of levels 1, 2 and 3; σ_p(ω) is the absorption cross section of pump radiation; and σ_las(ω_las) is the absorption cross section of laser radiation with frequency ω_las. The absorption cross sections are found from the formulas:

where λ_p and λ_las are the centre wavelengths of pump radiation and laser radiation; A₃₁ and A₂₁ are the spontaneous emission rates (the first Einstein coefficients) for 3 – 1 and 2 – 1 transitions; Γ_p = A₃₁/2 + γ₃₁ and Γ_las = A₂₁/2 + γ₂₁ are the homogeneous half-widths of 3 – 1 and 2 – 1 transition lines, respectively; γ₃₁ and γ₂₁ are the collision half-widths of 3 – 1 and 2 – 1 transition lines; ω₃₁ and ω₂₁ are the frequencies of 3 – 1 and 2 – 1 transitions. Equations (1) are supplemented by boundary conditions describing the change in radiation intensities on the mirror surface in the course of reflection and propagation through the cell windows:

The populations of levels 1, 2 and 3 in equations (1) are found from the balance equations. In the steady-state case, these equations have the form:

Here, N is the total concentration of active atoms; collision frequencies ν₃₂ and ν₂₃ describe collisional mixing between levels 3 and 2; collision frequencies ν₃₁ and ν₂₁ describe inelastic collision transitions along the 3 → 1 and 2 → 1 channels; and w_p(y, z) and w_las(y, z) are the probabilities of induced transitions under the action of pump radiation and generated laser radiation, respectively. We assume that pump radiation has a spectrum of arbitrary width, and generated laser radiation is monochromatic. Then

where ω_p is the centre frequency of the pump radiation spectrum; and

is the total spectral power density of pump radiation inside the cell and the total intensity of laser radiation inside the cell, respectively. The collision frequencies ν₃₂ and ν₂₃, due to the principle of detailed balance, are related by the expressions

where ΔE = ħω₃₂ is the energy difference between levels 3 and 2; and T is the temperature of the gas mixture inside the cell.

From the system of algebraic equations (4) we find the populations of levels:

and population differences characterising laser generation and pump absorption:

where

Ã₃₁ and Ã₂₁ are the transition frequencies from levels 3 and 2 as a result of spontaneous emission and collisional transitions to the ground state; and Γ₃ and Γ₂ are the total transition frequencies from levels 3 and 2 as a result of spontaneous emission and collisions. The quantities ϰ_p ≡ ϰ_p(y, z) and ϰ_las ≡ ϰ_las(y, z) are defined as

They denote the saturation parameters, since each of them characterises the degree of population equalisation at the 3 – 1 or 2 – 1 transition in the absence of the second field.

Taking into account relations (9), differential equations (1), describing the operation of the laser, have the form:

As follows from the equations for in (12), lasing occurs when the condition aϰ_p > 1 is fulfilled. To ensure effective generation (aϰ_p ≫ 1), one must meet the conditions

The first condition in (13) is satisfied with a large margin at a sufficiently high pressure of the buffer gas (∼1 atm and higher). To fulfil the second condition, the spectral power density of pump radiation must be sufficiently high. Under these conditions, pumping produces an extremely large population inversion at the laser transition.

The system of differential equations (12) describing the laser operation can be solved only by numerical methods. Nevertheless, it is possible, without solving it, to obtain a practically important relationship between the integral characteristics of radiations.

We will show how this relationship directly follows from Eqns (12). Integrating the difference between the third (for ) and fourth (for ) equations in (12) over the cell volume (with respect to x, y, z), taking into account formulas (5) and (11) for w_las and ϰ_las, for the output power of laser radiation coupled out of the cell

we obtain an expression

Hereafter, we assume that in the considered geometry of the transversely pumped laser, laser radiation is generated in the entire cell volume V = x₀ y₀ z₀.

The intensity of laser radiation emerging from the resonator through a semitransparent mirror with a reflection coefficient R₀ is given by the expression

The output power of laser radiation emerging from the resonator through a semitransparent mirror is defined as the integral of the intensity over the beam cross section:

Let us find the relationship between the powers (17) and P_out (15). From the last two equations in (12) we obtain the expression

from which it follows that the product does not depend on z. In particular, the relation

is valid. The integrand in (14), taking into account (16), (19) and boundary conditions (3), has the form

where

Expressions (14), (15), (17) and (20) yield the relationship between the powers and P_out (15):

Let us write the difference between the second (for ) and first (for ) equations in (12) and integrate it with respect to the frequency ω. Taking into account formulas (5) and (11) for w_p and ϰ_p, we obtain an expression

where

are the total intensities of pump radiation propagating along the y axis (the superscript '+') and in the opposite direction (after reflection by the mirror, the superscript '–'). Integrating equation (23) over the cell volume, we obtain for the absorbed pump power

an expression

Below we will take into account that the total losses of the pump power for spontaneous emission and collisional quenching in the cell volume are given by a fairly obvious relationship:

Taking into account formulas (8) for the population levels, expression (27) has the form

From (22), (26), and (28) we obtain the expression

relating the output power of laser radiation coupled out from the resonator with the absorbed pump power P_abs and the pump power losses P_loss due to spontaneous emission and collisional quenching.

The pump power loss P_loss during the laser operation arises both due to the loss of spontaneous emission and collisional quenching and due to the unabsorbed pump power

where P_0p is the pump power at the entrance to the cell. Taking this relation into account, formula (29) for the laser radiation power can also be represented in the form

The ratio of the laser power to the pump power, , characterises the efficiency of the conversion of pump radiation into laser radiation, and the ratio of radiation frequencies ω_las/ω_p is the quantum efficiency of this conversion. For alkali metal vapours it is close to unity (95% for caesium, 98% for rubidium and 99.5% for potassium), which ensures a high efficiency of alkali metal vapour lasers. The conversion efficiency of pump radiation into laser radiation the higher, the smaller the relative total pump power losses (P_unabs + P_loss)/P_0p.

The system of differential equations (12) describing the laser operation can be solved in general only numerically. In our previous work [10], an analytical solution of Eqn (12) was obtained for the case of high laser radiation intensity and a sufficiently high buffer gas pressure, which is reduced to the fulfilment of conditions

The first condition means both a high intensity of laser radiation (a large saturation parameter ϰ_las, not only in comparison with unity, but also in comparison with a saturation parameter for pump radiation ϰ_p) and a high pressure ensuring a small parameter b such that bϰ_p ≪ 1. In order to satisfy the second condition in (32), a sufficiently large saturation parameter ϰ_las (in particular, for rubidium atoms 1/q ≈ 5) is also needed.

In paper [10], based on approximation (32), we left in equation (12) only ϰ_las in the denominator and discarded unity in the first equation of (12) in the numerator in comparison with qϰ_las. As a result, the equations were greatly simplified and an analytical solution was obtained.

In the present paper we remove the strict restriction bϰ_p ≪ 1 and find out the influence of the term bϰ_p ϰ_las in the denominators of equations (12) on the lasing characteristics at not very high buffer gas pressures. Instead of restrictions (32), we impose weaker restrictions:

When conditions (33) are satisfied, differential equations (12) have the form:

where the saturation parameter ϰ_p is independent of the coordinate z and is given, in accordance with (5), (6), (11), by the expression

Solving the equations for the laser radiation intensities in (34) with allowance for the boundary conditions (3), we obtain the expression for :

where I_las(y, z) is the total intensity of laser radiation inside the cell, determined by the formula

If the effective reflection coefficients of the mirrors, and , are close to unity, then, according to (37), the laser radiation intensity I_las inside the cell is almost constant along the z axis of the resonator. From formulas (16) and (36) we find the laser output intensity

where R is given by the formula in (21).

Next, to solve the problem, we need to find the saturation parameter ϰ_p(y), determined by the first two equations for the spectral power densities of pump radiation in (34). We will solve these equations by the method of successive approximations with respect to the small quantity bϰ_p. For b = 0 (zeroth approximation), the first two equations in (34) have the form [10]:

According to [10], the solution of equations (39), taking into account boundary conditions (3), is given by the expressions

In the next approximation, the first two equations in (34) have the form

where the initial approximate value of the saturation parameter is given by the expression

The solution of equations (41) with allowance for boundary conditions (3) is given by the expression:

As the final expression for the saturation parameter ϰ_p, we have the expression obtained on the basis of relation (43):

Thus, formulas (38) and (44) describe the operation of a diode-pumped alkali metal vapour laser in the practically important case of a sufficiently high intensity of laser radiation and a weak restriction from below on the buffer gas pressure (for bϰ_p ≲ 1).

To further define the calculations using the above formulas, it is necessary to specify the spectral power density I_0ωp(ω) of the pump diodes at the cell input. We assume that at the input to the cell the pump radiation spectrum is Gaussian:

where I_0p is the pump radiation intensity at the cell input; and Δω is the half-width (at the level 1/e) of the pump radiation spectrum.

From (43), taking into account (45) for the total spectral power density I_ωp(y, ω) of pump radiation inside the cell, we obtain the relation

The saturation parameter (42) that appears in the formula for the function φ(y) is given by the expression

Using (46), we find the total intensity of pump radiation inside the cell:

The saturation parameters ϰ_p(y) and ϰ_las(y, z) under the conditions in question are given by the expressions:

For the laser output power, from expression (17), using (38), we obtain

Note here the following important circumstance. Analysis shows that on the right-hand side of formula (50) the concentration N of active particles and cell width y₀ enter only in the combination Ny₀. This means that, under the condition Ny₀ = const, the change in the cell width does not affect the laser output power .

For the pump power losses due to spontaneous emission and collisional quenching in the cell volume, from (28), when conditions (33) are satisfied, we find the expression

For the absorbed pump power in the cell volume, from expression (26), when conditions (33) are satisfied, we obtain the expression

We will calculate the energy characteristics of the laser using the formulas given above. Let the active medium in the laser cell be rubidium atoms, and a mixture of helium and methane be used as a buffer gas. Methane is usually used for effective collisional mixing between excited levels 3 and 2 in alkali metal atoms [1]. Helium is added to increase the collisional broadening of the D₂ line in order to make more efficient use of broadband emission from pump diodes [1].

We specify the initial parameters required for the calculation. For rubidium atoms, according to the NIST database [11], the rates of radiative transitions are A₂₁ = 3.6 × 10⁷ s⁻¹, A₃₁ = 3.8 × 10⁷ s⁻¹, transition wavelengths are λ₂₁ = 794.8 nm, λ₃₁ = 780.0 nm and the energy difference ΔE between levels 3 and 2 is 237.6 cm⁻¹. Collisional broadening for rubidium atoms in a helium buffer gas is as follows [12]: γ_He(D1) = 9.45 MHz Torr⁻¹ for the D₁ line and γ_He(D2) = 10.0 MHz Torr⁻¹ for the D₂ line. For rubidium atoms in a methane buffer gas, collisional broadening is as follows [12]: γ_CH4(D1) = 14.55 MHz Torr⁻¹ for the D₁ line and γ_CH4(D2) = 13.1 MHz Torr⁻¹ for the D₂ line.

To find the collision frequency ν₃₂, we used the following cross sections for the collisional transitions between the fine components of the excited state of rubidium atoms: σ_32(He) = 0.103 × 10⁻¹⁶ cm² for rubidium atoms in helium [13] and σ_32(CH4) = 42 × 10⁻¹⁶ cm² for rubidium atoms in methane [14]. Because of the small cross section σ_32(He), a molecular gas is added to the buffer mixture.

For the cross sections σ_31(CH4) and σ_21(CH4) of the collisional quenching of excited levels 3 and 2 of rubidium atoms interacting with methane, Zameroski et al. [15] experimentally obtained values not exceeding 1.9 × 10⁻¹⁸ cm². In calculating collision frequencies ν₃₁ and ν₂₁, we assumed that σ_31(CH4) = σ_21(CH4) = 1.9 × 10⁻¹⁸ cm². For rubidium atoms colliding with helium atoms, the collisional quenching cross sections are extremely small (σ_31(He), σ_21(He) ⩽ 3 × 10⁻²⁰ cm² [16]), and therefore quenching due to interaction with helium can be neglected.

We assume below in the calculation that the centre frequency of the pump radiation spectrum and the laser radiation frequency coincide with the frequencies of 3 – 1 and 2 – 1 transitions, respectively: ω_p = ω₃₁, ω_las = ω₂₁. The cell should be designed in such a way that the alkali metal vapour enters it through the side branches so that the concentration N of active particles inside the cell is set by the temperature of these branches containing an alkali metal and is not related to the temperature T of the gas mixture inside the cell.

We first consider the effect of the correction associated with the term bϰ_p in the denominators of equations (34) on the lasing characteristics. This correction should be taken into account at not very high pressures of the buffer gas, when the parameter bϰ_p is not too small. In the case of a sufficiently high buffer gas pressure, the parameter bϰ_p is very small (bϰ_p ≪ 1) and can be neglected in Eqns (34) (we can assume b = 0). In this case, and the analytical formulas given above, which describe lasing, coincide with the formulas obtained previously in [10] (it should be noted that in [10] we neglected the energy losses in the resonator, which are taken into account in this paper by introducing the transmission coefficients T₀ and T₁ of the cell windows and the effective transmittance T_s, which differ from unity).

Figure 3 shows the results of calculations using formula (50) (with a correction for the parameter bϰ_p and without it) for the conversion efficiency of pump radiation into laser radiation (pump radiation power P_0p = x₀ z₀ I_0p) as a function of the helium buffer gas pressure p_He at different values of the parameter Ny₀ (the number of active atoms in the cell in a gas column of width y₀ with a unit cross section). The methane buffer gas pressure p_CH4 was assumed fixed and equal to 0.1 atm. Condition (33) of applicability of formula (50) with the parameters corresponding to Fig. 3 is met if the cell is sufficiently long (if z₀ ≫ y₀). Thus, for example, for the parameters corresponding to curve (1) in Fig. 3, as well as for z₀ = 10y₀ and p_He = 10 atm [a maximum on curve (1)], relations (1 + bϰ_p) ϰ_las ⩾ 14 (1 + ϰ_p), qϰ_las ⩾ 29, bϰ_p ⩽ 0.53 hold true. It is clear from Fig. 3 that the conversion efficiency depends monotonically on the pressure p_He and the parameter Ny₀ and reaches a maximum value at some optimal helium pressure and some optimal parameter (Ny₀)_opt. If the parameter Ny₀ deviates from the optimal value [it corresponds to curve (1) in Fig. 3] in the direction of its increase [curve (2)] or decrease [curve (3)], the laser efficiency decreases dramatically. As can be seen from Fig. 3, the allowance for the correction for the parameter bϰ_p can both reduce [cf. curves (1) and (1'), (3) and (3')] and increase [cf. curves (2) and (2')] the calculated values of . The effect of the correction on the calculated value of is small for the optimal value of the parameter Ny₀ [cf. curves (1) and (1')] and becomes significant when Ny₀ deviates from the optimum value [cf. curves (2) and (2'), (3) and (3')].

Let us consider in more detail the effect of the correction on the calculated value at the optimal value of the parameter Ny₀. At a helium pressure p_He = 10 atm, the calculated values of with correction for the parameter bϰ_p [curve (1)] and without correction [curve (1')] are 0.503 and 0.512, respectively, i.e., the relative change is 0.018. The parameter bϰ_p(y) is a decreasing function of y. For the parameters corresponding to curve (1) in Fig. 3, and at a pressure p_He = 10 atm, we have bϰ_p(0) = 0.530 and bϰ_p(y₀) = 0.268. Thus, with the optimal value of the parameter Ny₀, the correction for the parameter bϰ_p leads to a relative change in the conversion efficiency by an amount much smaller than the parameter bϰ_p itself. Let us explain this somewhat surprising fact on the basis of the dependences of the laser radiation intensity on the coordinate y. Then, the reason for a more significant difference of curves (2') and (3') from (2) and (3) in Fig. 3 becomes clear.

The correction for the parameter bϰ_p has a much stronger effect on the local laser radiation intensity. Figure 4 shows the laser radiation intensity as a function of the coordinate y along the cell width for the parameters corresponding to curves (1) and (1') in Fig. 3 and for a helium pressure p_He = 10 atm. It can be seen that taking the correction into account leads to a smoother dependence of : the intensity decreases from one side of the cell (at y = 0) and increases from the other side (at y = y₀). Relative changes in intensity near the two sides of the cell have different signs and turn out to be of the order of magnitude of bϰ_p. Note here the following circumstance. The laser radiation power is an integral characteristic (17), i.e., it is proportional to the area under curves (1) and (1') in Fig. 4. The difference in the areas under these curves characterises the effect of the correction for the parameter bϰ_p on the calculated value of the laser radiation power. As a result, we see that locally the influence of the parameter bϰ_p is quite appreciable, but for the integral characteristic it is strongly suppressed due to the proximity of the areas under the curves. In the case of pairs of curves (2) and (2'), (3) and (3') in Fig. 3, the situation is such that for the corresponding values of the parameter Ny₀, the dependences, similar to those in Fig. 4, give comparable differences between both local (with and without bϰ_p) and integral values of power.

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Methane is added to the buffer mixture for rapid collisional mixing between upper levels 3 and 2 in the alkali metal atoms so that at a not too high buffer gas pressure the first effective generation condition is satisfied in (13) (ν₃₂ ≫ Ã₂₁, Ã₃₁). The influence of the pressure of the methane buffer gas on the conversion efficiency of pump radiation into laser radiation at optimum values of the parameter Ny₀ is illustrated in Fig. 5. It can be seen that the addition of methane to the buffer mixture significantly improves the conversion efficiency and greatly reduces the optimum helium pressure , at which the efficiency reaches its maximum value. Thus, in the absence of methane in the buffer mixture, the maximum efficiency is attained at a high helium pressure [curve (1) in Fig. 5], while an increase in methane pressure p_CH4 to 0.5 atm increases the efficiency maximum (up to 0.58), attained at a lower helium pressure [curve (4) in Fig. 5].

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The conversion efficiency depends monotonically on the helium pressure p_He and the parameter Ny₀ and reaches a maximum value at some optimal helium pressure and some optimal parameter (Ny₀)_opt. Figure 6 shows the results of calculations of the quantities , and (Ny₀)_opt as functions of methane pressure p_CH4 for two values of the intensity I_0p and half-width Δω of the pump radiation spectrum chosen so that the spectral power density of pump radiation I_0p/Δω remained unchanged. One can see from Fig. 6a that the maximum conversion efficiency is achieved at an optimum methane pressure . Figure 6a also shows the calculated values of in approximation b = 0 [curves (1') and (2'), for which the initial calculation parameters are the same as for curves (1) and (2), respectively]. It can be seen from a comparison of curves (1) and (1'), (2) and (2') that at an optimum helium pressure and optimum parameter (Ny₀)_opt, the effect of correction for the parameter bϰ_p on the calculated values of is small.

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Consider the following important circumstance. Numerical analysis shows that for a given spectral power density of pump radiation I_0p/Δω and selected optimal values of , and (Ny₀)_opt at different Δω, the lasing efficiency is the same (Fig. 6a). For example, the spectral power density of pump radiation is the same for curves (1) and (2) in Fig. 6a (I_0p/[Δω/(2πc)] = 1 kW cm⁻¹), and therefore for these curves the maximum of the ratio is the same and equal to 0.58 [at p_CH4 ⩾ 0.5 atm, the difference between curves (1) and (2) is so slight that they merge and become indistinguishable by the naked eye].

It can be seen from Figs 6b and 6c that the optimum helium pressure and the optimal parameter (Ny₀)_opt decrease monotonically with increasing methane buffer gas pressure. At the same time, smaller values of the half-width Δω of the pump radiation spectrum correspond to smaller values of and (Ny₀)_opt, which is quite obvious.

Figure 7 illustrates the above statement that for the same spectral power density of pump radiation and optimum values of (Ny₀)_opt, and , the maximum conversion efficiency of pump radiation into laser radiation being the same. For curves (1), (2), (3), (4) and (5), the spectral output power density is the same (I_0p/[Δω/(2πc)] = 1 kW cm⁻¹) and the same are the maxima of the ratios (equal to 0.58), which are attained at p_He = 0.7, 2, 4.6, 7.3 and 10 atm, respectively. At these pressures, the ratio of the half-width Δω of the pump spectrum to the homogeneous half-width Γ_p of the 3 – 1 transition line (Δω/Γ_p = 1.5) and the ratio Ny₀/I_0p = 3.6 × 10¹³ kW⁻¹ are the same for all the curves.

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Figure 8 shows the intensities of laser radiation and pump radiation I_p(y)/I_p(0) as functions of coordinate y. The intensity profiles were calculated for such parameters of Ny₀ and helium and methane pressures, at which the conversion efficiency of pump radiation into laser radiation has the highest value. It can be seen that the laser radiation intensity decreases more rapidly with increasing y than the pump radiation intensity. The numerical analysis shows that for a given spectral power density of pump radiation I_0p/Δω but at different Δω and optimum values of the methane and helium pressures as well as the parameter Ny₀, the intensity profiles of laser radiation and pump radiation do not depend separately on Δω and I_0p.The spectral power density of pump radiation is the same for all the curves in Fig. 7 (I_0p/[Δω/(2πc)] = 1 kW cm⁻¹). At optimum helium pressures (at the points of the curve maxima in Fig. 7), the intensity profiles of laser radiation and pump radiation are described by curves (1) and (2) in Fig. 8. Each of these curves actually contains five curves with the same parameters as for Fig. 7 and the corresponding optimum helium pressures.

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Let us compare the results of calculations of the laser energy characteristics obtained numerically in [9] with the results of our calculations using analytical formulas. In considering the possibility of creating a megawatt rubidium vapour laser, Yang et al. [9] used the following parameters of the working medium and pump radiation: T = 380 K, N = 9.34 × 10¹² cm⁻³, z₀ = 50 cm, x₀ = y₀ = 5.5 cm, p_He = 3.24 atm, p_CH4 = 0.683 atm, I_0p = 3.7 kW cm⁻², Δω/(2πc) = 1.97 cm⁻¹, ω_p = ω₃₁, R_p = 0.99, R₀ = 0.3, R₁ = 0.99, T₀ = 0.99, T₁ = 0.99, T_s = 0.99. The influence of the spatial overlap of the laser radiation and pump fields on the lasing efficiency was taken into account in [9] by introducing the parameter η_mode = V_las/V = 0.96 (V_las is the laser volume in the cell). With these parameters, Yang et al. [9] obtained the conversion efficiency of pump radiation into laser radiation and the ratio of the absorbed pump power to the total pump power P_abs/P_0p = 0.725. The corresponding calculations by our analytical formulas for the same parameters of the working medium and pump radiation yield (after multiplying by η_mode = 0.96) , P_abs/P_0p = 0.730. The results of calculations of the laser energy characteristics by analytical formulas are in very good agreement with the results of numerical calculations [9]. Such a good agreement is achieved only when the correction is taken into account for the parameter bϰ_p. Calculation by simpler analytical formulas in the approximation b = 0 gives not so good agreement with the results of paper [9]: , P_abs/P_0p = 0.780. Note that in Ref. [9], not all the parameters of the working medium of a megawatt laser are optimal. According to our calculations, if the concentration of rubidium atoms is increased to 17.8 × 10¹² cm⁻³ and the helium pressure is raised to 5.25 atm with the other parameters unchanged, the conversion efficiency will increase to 0.66.

We have developed an analytical model of a transversely diode-pumped alkali metal vapour laser, which describes the laser generation in the practically important case of a sufficiently high intensity of laser radiation and with a weak restriction from below on the buffer gas pressure. This restriction is removed by taking into account the correction for the parameter bϰ_p. In the case of a sufficiently high buffer gas pressure, the parameter bϰ_p is very small (bϰ_p ≪ 1) and can be neglected (it can be assumed that b = 0) in the equations describing the laser operation. This case was considered earlier in our work [10]. The analytical solution of the differential equations that describe the laser generation makes it possible to determine exhaustively any energy characteristics of the laser and to find the optimum parameters of the working medium and pump radiation (temperature, buffer gas pressure and composition, intensity and width of the pump spectrum).

The conversion efficiency of pump radiation into laser radiation is described by an analytical formula in which the concentration N of active particles and the cell width y₀ enter only in the combination Ny₀. The efficiency depends nonmonotonically on the parameter Ny₀, as well as on the pressure of helium p_He and methane p_CH4 and reaches a maximum value at some optimum parameter (Ny₀)_opt and some optimal pressures and . It is important to note that at the given spectral power density of pump radiation I_0p/Δω and selected optimal values of (Ny₀)_opt, and for different Δω, the lasing efficiency is the same. Calculations using analytical formulas show that the conversion efficiency reaches 69% with the spectral power density of pump radiation I_0p/[Δω/(2πc)] = 2 kW cm⁻¹.

Analytical formulas for intensities of laser radiation and pump radiation I_p(y) are obtained as functions of the coordinate y along the width of the cell. The laser radiation intensity decreases with increasing y faster than the pump radiation intensity. At a given spectral power density of pump radiation I_0p/Δω, but for different Δω and corresponding optimum values of the methane and helium pressures, as well as the parameter Ny₀, the intensity profiles of laser radiation and pump radiation do not depend separately on Δω and I_0p.

Allowance for the correction for the parameter bϰ_p can both reduce and increase the calculated values of . The effect of the correction on the calculated value of is small for the optimal value of the parameter Ny₀ and becomes significant when Ny₀ deviates from the optimum value. With the optimal value of the parameter Ny₀, the correction for the parameter bϰ_p leads to a relative change in the conversion efficiency of pump radiation into laser radiation by a value that is much smaller than the parameter bϰ_p itself. At the same time, the correction for the parameter bϰ_p exerts a much stronger effect on the local value of the laser radiation intensity . Taking the correction into account leads to a smoother dependence of : the intensity decreases from one side of the cell (at y = 0) and increases from the other side (at y = y₀). The relative changes in intensity near the two sides of the cell have different signs and turn out to be of the order of magnitude of bϰ_p.

The work was supported by the integrated programme of fundamental research of the Siberian Branch of the Russian Academy of Sciences (RAS) (Integration and Development Programme No. II.2P as part of the RAS programme, Project No. II.2P/II.10-13) and RF President's Grants Council (State Support to Leading Scientific Schools Programme, Grant No. NSh-6898.2016.2).