Multi-transverse mode operation of alkali vapor lasers: modeling and comparison with experiments

Ilya Auslender; Boris D. Barmashenko; Salman Rosenwaks

doi:10.1364/OE.25.019767

1. Introduction

Optically pumped alkali vapor lasers operate on the D₁(n ²P_1/2 → n ²S_1/2) transition of alkali atoms (where n = 4, 5, 6 for K, Rb and Cs, respectively) at frequency ν_l (wavelength λ_l ~800 nm), pumped via the D₂(n²S_1/2 → n²P_3/2) transition at frequency ν_p. The pumping is followed by rapid relaxation (by buffer gas, helium and\or small hydrocarbon molecules) of the upper to the lower fine-structure level, n²P_3/2 to n²P_1/2 (designated as levels 3 and 2, respectively; the ground state n ²S_1/2 is designated as 1). Diode pumped alkali lasers (DPALs) are of interest due to their great potential as high power, efficient lasers [1,2].

In most alkali vapor lasers, pumped by Ti:Sapphire or low power diode lasers with well collimated output beams, only the fundamental transverse mode of the stable resonator oscillates [3,4]. The reason is that the beams of such pump sources are focused to small spots with the radii close to those of the fundamental laser modes. However, in the case of pumping by high power diode laser arrays it is difficult to focus the pump beam to small spot and usually its radius in the laser cell is several times larger than that of the fundamental laser mode. In this case several high order transverse modes of the resonator participate in the lasing. In the present paper we report on a simple optical model for multi-transverse mode operation of alkali lasers with stable resonators. The model is based on the analysis of the multi-transverse mode lasing performed in [5] and [6] for the chemical oxygen-iodine laser (COIL) and solid state laser, respectively. The power of the pump beam and the powers of the transverse modes of the stable resonator propagating in the gain medium are calculated in the model. For the azimuthally symmetric pump beam, it is assumed that the laser transverse modes are the azimuthally-symmetric Laguerre-Gaussian modes. Note that another approach, based on wave optics modeling for calculating lasing in DPALs, was used in [7,8]. However, application of that approach to multi-transversal mode lasing in stable resonators revealed problems with convergence of the solution for the electromagnetic field amplitude [7].

The present model was applied to Ti:Sapphire and diode pumped cesium vapor lasers studied in [4] and [9], respectively. The number and intensities of the oscillating modes as a function of the pump beam power and radius are obtained. Beam quality of the multimode laser beam as a function of the pump power is also calculated for the two aforementioned alkali lasers.

2. Description of the model

The model considers typical configuration of an alkali laser with end-pump geometry studied in [4,9] and shown in Fig. 1. The cases of one-side and two-side pumping applied in [4] and [9], respectively, are considered. A pump beams with total power $P_{p, 0}$ enter a cylindrical laser cell of length l through windows with transmission t. The gaseous lasing medium is a mixture of Cs vapor and a buffer gas that consists of He and C₂H₆ for broadening the D₂ transition and for mixing between the fine-structure levels 3 and 2. The walls of the cell are heated to the temperature T ~100 C. The laser resonator of length L consists of a total reflector with curvature radius R₁ and reflectivity r₁ (close to 100%), and an output coupler mirror with curvature radius R₂ and reflectivity r₂, located outside the laser cell. Both the pump and laser beams propagate along the optical axis Z of the resonator.

Fig. 1 Schematics of an alkali laser with end-pump geometry. Solid arrow corresponds to one side pumping applied in [4], for two sides pumping applied in [5] the second pump beam is indicated by the dashed arrow.

Download Full Size | PDF

Azimuthally symmetric intensity distribution in the pump beam cross section is assumed:

I_{p}^{\pm} (r, z, ν) = P_{p}^{\pm} (z, ν) f_{p} (r, z),

where r is the distance from the optical axis and

P_{p}^{\pm} (z, ν)

is the total power spectral density at z. The superscripts + and – indicate pump beams propagating in the opposite directions + z and –z; for one-side pumping geometry we set

P_{p}^{-} = 0.

f_{p} (r, z)

is the normalized spatial distribution of the pump beam which is supposed to have a Gaussian shape:

f_{p} (r, z) = \frac{2}{π w_{p}^{2} (z)} \exp (- 2 \frac{r^{2}}{w_{p}^{2} (z)}),

where

w_{p} (z)

is the radius of the pump beam at

1 / e^{2}

of maximum intensity given by

w_{p} (z) = w_{0, p} {[1 + {(\frac{z - z_{0, p}}{z_{R, p}})}^{2}]}^{1 / 2},

_{$w_{0, p}$}and

z_{0, p}

are the radius and coordinate of the pump beam waist,

z_{R, p} = π w_{0, p}^{2} / λ_{p} M_{p}^{2}

is the pump Rayleigh length and

M_{p}^{2}

is the pump beam quality factor. The values of

w_{0, p},

z_{0, p}

and

M_{p}^{2}

are taken from the experimental measurements.

As mentioned in the introduction the treatment of the multimode lasing is based on the analysis performed in [5] and [6] for the chemical oxygen-iodine laser and solid state laser, respectively. For given transverse modes composition of the resonator, the intensity distribution of the laser beam can be calculated as incoherent superposition of the individual modes and hence is equal to the sum of the intensities of the resonator modes. Note that as shown in [10] for the azimuthally symmetric pump intensity the laser intensity should be also azimuthally symmetric. Then the laser intensity $I_{l}^{\pm} (r, z)$ is given by

I_{l}^{\pm} (r, z) = \sum_{i = 0}^{N - 1} I_{l, i}^{\pm} (r, z),

where

I_{l, i}^{\pm} (r, z) = P_{l, i}^{\pm} (z) f_{l, i} (r, z),

_{$P_{l, i}^{\pm} (z)$}is the power of transverse mode of order i propagating in the forward ( + ) and backward (-) directions, N is the number of modes participating in the lasing and

f_{l, i} (r, z)

is the normalized intensity distribution in the transverse mode of order i. Following [5,6] we assume that the set of eigenmodes of the resonator filled by the gain medium closely resembles the analytically derived Laguerre-Gaussian modes of the empty resonator. Then

f_{l, i} (r, z)

are given by the azimuthally symmetric Laguerre-Gaussian functions

f_{l, i} (r, z) = \frac{2}{π w_{l}^{2} (z)} {[L_{i} (2 \frac{r^{2}}{w_{l}^{2} (z)})]}^{2} \exp (- 2 \frac{r^{2}}{w_{l}^{2} (z)}),

where

L_{i} (x)

is the Laguerre polynomial of order i,

w_{l} (z)

is the laser beam fundamental mode radius (at

1 / e^{2}

of the maximum intensity) given by

w_{l} (z) = w_{0, l} {[1 + {(\frac{z - z_{0, l}}{z_{R, l}})}^{2}]}^{1 / 2},

_{$w_{0, l}$}and

z_{0, l}

are the radius and coordinate of the fundamental mode waist, and

z_{R} = π w_{0, l}^{2} / λ_{l}

is the Rayleigh length for the laser fundamental mode.

The rates of changes of $P_{p}^{\pm} (z, ν)$ and $P_{l, i}^{\pm} (z)$ with z are described by the Beer-Lambert law and found from the system of differential equations:

\begin{array}{l} \frac{d P_{p}^{\pm} (z, ν)}{d z} = \mp α (z, ν) P_{p}^{\pm} (z, ν); α (z, ν) = 2 π σ_{D_{2}} (ν) \int_{0 \leq r < \infty} d r \cdot r (n_{1} - \frac{n_{3}}{2}) f_{p} (r, z) \\ \frac{d P_{l, i}^{\pm} (z)}{d z} = \pm β_{i} (z) P_{l, i}^{\pm} (z); β_{i} (z) = 2 π σ_{D_{1}} (ν_{l}) \int_{0 \leq r < \infty} d r \cdot r (n_{2} - n_{1}) f_{l, i} (r, z) \end{array}

where

α (z, ν)

is the spectral absorption coefficient for the D₂ transition and

β_{i} (z)

is the gain coefficient of the ith mode for the D₁ transition, averaged over the beam transverse cross section;

σ_{D_{1}} (ν_{l})

and

σ_{D_{2}} (ν)

are the stimulated emission and pump absorption cross sections for the D₁ and D₂ transitions, respectively, calculated as described in [3] and

n_{1},

n_{2}

and

n_{3}

are the number densities of Cs atoms in levels 1, 2 and 3. The values of

n_{1},

n_{2}

and

n_{3}

are found analytically as the steady state solutions of the rate equations for the 3-level system presented in [11] where stimulated emission caused by different laser transverse modes is taken into account:

n_{1} = n_{0} \frac{Φ_{s} (1 + Y F_{s} + Y τ_{R}) + F_{s} (K + Y) + 1 + K τ_{R} + Y τ_{R}}{1 + K τ_{R} + Y τ_{R} + F_{s} (2 + 3 K + 3 Y) + Φ_{s} (4 Y F_{s} + 2 Y τ_{R} + 2 + K)},

n_{2} = n_{0} \frac{Φ_{s} (1 + Y F_{s} + Y τ_{R}) + 2 F_{s}}{1 + K τ_{R} + Y τ_{R} + F_{s} (2 + 3 K + 3 Y) + Φ_{s} (4 Y F_{s} + 2 Y τ_{R} + 2 + K)},

n_{3} = n_{0} \frac{Φ_{s} (2 Y F_{s} + K) + 2 F_{s} (Y + K)}{1 + K τ_{R} + Y τ_{R} + F_{s} (2 + 3 K + 3 Y) + Φ_{s} (4 Y F_{s} + 2 Y τ_{R} + 2 + K)},

where

n_{0} = n_{1} + n_{2} + n_{3}

is the total number density of alkali atoms equal to the number density of the saturated alkali vapor at temperature

T,

Φ_{s} = [I_{l}^{+} (r, z) + I_{l}^{-} (r, z)] σ_{D_{1}} (ν_{l}) τ_{D_{1}} / h ν_{l},

F_{s} = {\int_{0}^{\infty} σ_{D_{2}} (ν) [I_{p}^{+} (r, z, ν) + I_{p}^{-} (r, z, ν)] d ν} τ_{D_{2}} / h ν_{p},

_{$I_{p}^{\pm} (r, z, ν)$}and

I_{l}^{\pm} (r, z)

are found from Eqs. (1) - (7),

Y = 1 / γ_{32} τ_{D_{1}},

K = 2 \exp (- Δ E_{32} / k_{B} T),

τ_{R} = τ_{D_{1}} / τ_{D_{2}},

τ_{D_{1}}

and

τ_{D_{2}}

are radiative life times of the levels 2 and 3, respectively,

Δ E_{32} = h (ν_{p} - ν_{l})

is the energy difference between the levels 3 and 2,

γ_{32} = σ_{32} n_{C_{2} H_{6}} \sqrt{\frac{8 k_{B} T}{π} (\frac{1}{m_{Cs}} + \frac{1}{m_{C_{2} H_{6}}})},

σ_{32}

is the relaxation cross section of the level 3 to 2,

n_{C_{2} H_{6}}

is the number density of C₂H₆ molecules, whereas

m_{Cs}

and

m_{C_{2} H_{6}}

are the atomic masses of Cs atoms and C₂H₆ molecules, respectively.

Note that just as in [3,4] the processes of the excitation of Cs atoms to higher electronic levels followed by their ionization are assumed to be negligibly small. Also the gas heating caused by the heat release due to relaxation between the fine-structure levels of Cs atoms and quenching of these levels is neglected. These assumptions are correct for the CW laser [4] with $P_{p, 0}$ < 10 W and the pulsed laser [9] with $P_{p, 0}$ < 100 W [12].

The boundary conditions for the pump power are:

P_{p}^{+} (z = 0, ν) = P_{p}^{-} (z = l, ν) = η_{del} t P_{p, 0} g_{p} (ν) / 2

for two side pumping and

P_{p}^{+} (z = 0, ν) = η_{del} t P_{p, 0} g_{p} (ν)

for one side pumping.

η_{del}

is the delivery efficiency defined as the fraction of the pump power delivered from the pump excitation source to the laser cell inlet,

g_{p} (ν) = {(4 \ln 2 / π Δ ν_{p}^{2})}^{1 / 2} \exp [- 4 \ln 2 {(ν - ν_{p})}^{2} / Δ ν_{p}^{2}]

is the normalized spectral distribution of the pump laser and

Δ ν_{p}

is the spectral width (FWHM) of the pump radiation.

The boundary conditions for the two way laser powers $P_{l, i}^{+}$ and $P_{l, i}^{-}$ of the transverse mode of order i at z = 0, were calculated by ray tracing of the given output laser power $P_{l a s e, i}$ in this mode: $P_{l, i}^{-} (0) = P_{l a s e, i} / [t (1 - r_{2})]$ and $P_{l, i}^{+} (0) = P_{l, i}^{-} (0) t^{2} r_{2} .$

The solution for the output laser flux was found by iterating on $P_{l a s e, i}$ until one of the following conditions at z = l is fulfilled:

P_{l, i}^{-} (z = l) / P_{l, i}^{+} (z = l) = t^{2} r_{1}

for the oscillating modes with

P_{l a s e, i} > 0;

P_{l, i}^{-} (z = l) / P_{l, i}^{+} (z = l) > t^{2} r_{1}

for the modes with

P_{l a s e, i} \to 0

that do not oscillate.

To determine $P_{l a s e, i}$ (i = 0, . . ., N - 1) we have to solve the system of algebraic Eqs. (17) and inequalities (18) together with the differential Eqs. (8). To avoid problems with the inequalities we replaced the system (17) - (18) by the system of algebraic Eqs. (17) where i = 0, . . ., N - 1 which means that all N modes are assumed to oscillate. The system of N Eqs. (17) was solved using Matlab computer program. It was found that the first K roots $P_{l a s e, i}$ (i = 0, . . ., K – 1, K ≤ N) are always positive whereas the remaining N – K values of $P_{l a s e, i}$ (i = K, . . ., N - 1) are equal to zero. We checked that conditions (18) are met for the latter modes for $P_{l a s e, i} < < P_{p, 0}$ and found out that they indeed do not participate in lasing. Note that the number K of the modes participating in lasing depends on the pump power, the pump beam radius and the optical resonator geometry. Then the total lasing power $P_{l a s e}$ is given by

P_{l a s e} = \sum_{i = 0}^{N - 1} P_{l a s e, i} .

The solution for the mode intensities for certain parameters of the system is not necessarily unique and stable [6]; however, detailed study of the multimode lasing stability is beyond the scope of this paper. Comparison of the output powers corresponding to different mode compositions for given $P_{p, 0}$ showed that the difference in the total power is <5%.

3. Results and discussion

3.1. Modeling of Ti:Sapphire pumped laser [4]

Parameters used for the modeling of Ti:Sapphire pumped Cs laser [4] are presented in Table 1. One-side pumping scheme was employed. The resonator consisted of a flat total reflector and concave output coupler mirror, the average radius of the laser fundamental mode ${\bar{w}}_{l}^{}$ over the cell length (calculated using Eq. (7) for $w_{l} (z)$ ) being 0.15 mm. As shown in [4], Ti:Sapphire laser produced high quality TEM₀₀ output beam with $M_{p}^{2} = 1.$ Two cases of different average pump beam radii ${\bar{w}}_{p}$ over the laser cell length were analyzed: i) ${\bar{w}}_{p}$ = 0.238 mm, the maximum pump beam radius for which the measurements were performed in [4] and ii) ${\bar{w}}_{p}$ = 0.404 mm when the pump beam radius is much larger than ${\bar{w}}_{l}^{00}$ and many transverse modes participate in the lasing.

Table 1. Parameters used for modeling CW Ti:Sapphire pumped and pulsed diode pumped Cs lasers

View Table

Figure 2 shows the dependence of the calculated total laser power $P_{l a s e}$ on $P_{p, 0}$ for the two values of ${\bar{w}}_{p} .$ For ${\bar{w}}_{p}$ = 0.238 mm the calculated $P_{l a s e}$ at low $P_{p, 0}$ < 1 W is in agreement with the measured values [4]. The same figure shows also the dependence of $P_{l a s e}$ on $P_{p, 0}$ (indicated by dashed lines) calculated under the assumption of single mode lasing when only the fundamental transverse mode of zero order oscillates. Such an assumption was used in our previous models of the alkali vapor lasers [3,4]. It is seen that for smaller ${\bar{w}}_{p}$ = 0.238 mm the values of $P_{l a s e}$ in the single mode case are only a little lower than in the multimode case, the ratio of the powers being ~0.9 at $P_{p, 0}$ = 10 W. However, the difference becomes stronger with the increase of the pump beam radius: for ${\bar{w}}_{p}$ = 0.404 mm the value of $P_{l a s e}$ at $P_{p, 0}$ = 10 W in the single mode case is by 60% lower than that in the multimode case. Therefore, if the pump beam is wider than the laser fundamental transverse mode the multimode model should be applied instead of the simple single mode model reported in [3,4]. It is also seen that the threshold power increases with increasing ${\bar{w}}_{p},$ which is due to increase of the volume occupied by the excited Cs atoms contributing to the spontaneous emission losses.

Fig. 2 Calculated and measured [4] output power of Ti:Sapphire Cs laser as a function of the pump power for the cases of multimode and single mode lasing and different average pump beam radii indicated in the figure. Other laser parameters are presented in Table 1. The insert shows comparison between the calculated and measured values at low pump powers.

Download Full Size | PDF

Modal composition of the laser radiation is shown in Fig. 3. It is seen that for smaller ${\bar{w}}_{p}$ = 0.238 mm the maximum number of the modes participating in the lasing equals to three at $P_{p, 0}$ = 10 W (Fig. 3(a)), whereas for larger ${\bar{w}}_{p}$ = 0.404 mm the maximum number of modes increases to 5 (Fig. 3(b)). The number of modes increases with increasing $P_{p, 0} :$ at low $P_{p, 0},$ close to the threshold value, only one mode oscillates, whereas at large $P_{p, 0}$ = 10 W three and five modes oscillate for ${\bar{w}}_{p}$ = 0.238 mm and 0.404 mm, respectively. Note that as seen in Fig. 3(a) at low $P_{p, 0}$ < 1 W applied in [4] only the fundamental transverse mode oscillates which is in agreement with the conclusion drawn in [4].

Fig. 3 Modal composition of the Ti:Sapphire pumped Cs laser power as a function of the pump power for (a) ${\bar{w}}_{p}$ = 0.238 mm and (b) ${\bar{w}}_{p}$ = 0.404 mm. Other laser parameters are presented in Table 1. The total laser power $P_{l a s e}$ is indicated by the black line, whereas the powers $P_{l a s e, i}$ of the modes of order i by the colored lines with markers.

Download Full Size | PDF

Figure 4 shows the calculated and measured dependence of $P_{l a s e}$ on ${\bar{w}}_{p}$ at three values of the pump power $P_{p, 0}$ including $P_{p, 0}$ = 0.8 W applied in [4]. As explained in [4], for each value of $P_{p, 0}$ this dependence is non monotonic, the maximum calculated $P_{l a s e}$ being achieved at some optimal ${\bar{w}}_{p} .$ The optimal ${\bar{w}}_{p}$ increases with increasing $P_{p, 0}$ which is due to the fact that higher order modes with larger radii start to participate in the lasing at higher pump powers.

Fig. 4 Calculated and measured [4] dependence of the laser power on ${\bar{w}}_{p} .$ For better visualization $P_{l a s e}$ is multiplied by 5 at 0.8 W pump power. Other laser parameters are presented in Table 1.The optimal value of ${\bar{w}}_{p}$ increases with increasing pump power.

Download Full Size | PDF

3.2. Modeling of pulsed diode pumped Cs laser [9]

In this laser two-side pumping was employed, namely, the pump beams entered the laser cell from the two opposite ends. The parameters used for the modeling are presented in Table 1. The dependence of the size of each of the pump beams (along the x and y directions) on the distance z along the optical axis was measured in [13]. However, due to incomplete overlap of the pump beams the spatial distribution of the pump intensity in the laser cell is unknown. For simplicity it was assumed that this distribution is azimuthally symmetric in the xy plane and has Gaussian shape given by Eq. (2). Note that if the pump beam intensity distribution is not azimuthally symmetric, the laser oscillations may be Hermite-Gaussian rather than Laguerre-Gaussian modes with non-zero azimuthal index. However, as mentioned above, the total output powers corresponding to different mode compositions differ by < 5%. The dependence of the beam radius $w_{p}$ on z is given by Eq. (3) where it was assumed that the coordinate of the pump beam waist $z_{0, p}$ coincides with the cell center. The pump beam quality factor $M_{p}^{2}$ was determined by fitting the calculated ratio of the pump beam areas at points with coordinates z and z_0,_p to the measured values for different z [13]. The pump beam waist radius w₀_,p was found by fitting the threshold pump power calculated using our model to the measured value. Low delivery efficiency $η_{del} = 0.85$ is mainly due to dichroic mirrors installed in the cavity of the laser [9] and transmitting only part of the pump beams. As mentioned in section 2 the processes of Cs atom ionization and increase of the gas temperature can be neglected. Indeed, as shown in [12], once these processes were important the dependence of $P_{l a s e}$ on $P_{p, 0}$ wound not be linear.

Figure 5 shows the calculated and measured values of the total laser power $P_{l a s e}$ as a function of the pump power $P_{p, 0}$ which are in good agreement. Note that unlike the semi-analytical model [12], where good agreement between the calculated and measured values of $P_{l a s e}$ was obtained using mode-matching efficiency as a fitting parameter, the present model does not use this empirical parameter.

Fig. 5 Measured [9] and calculated dependence of the total laser power on the pump power for pulsed Cs DPAL with the parameters presented in Table 1.

Download Full Size | PDF

Figure 6 shows mode composition of the multimode laser beam. Just as in the previous section the number of oscillating modes increases with increasing pump power from 1 at the threshold to 6 at 100 W pump power.

Fig. 6 Modal composition of pulsed Cs DPAL power as a function of the pump power. The parameters of the laser [9] are presented in Table 1. The total laser power $P_{l a s e}$ is indicated by the black line, whereas the powers $P_{l a s e, i}$ of the modes of order i by the colored lines with markers.

Download Full Size | PDF

3.3. Mode matching efficiency in the case of the multimode lasing

As noted above the present model does not use the mode-matching efficiency $η_{mode},$ [14]. This empirical parameter characterizes the partial spatial overlap of the pump and laser beams and is often used in semi-analytical modeling of the DPALs [12, 14, 15] and solid state lasers [16,17]. In such models $η_{mode}$ is connected with the alkali vapor laser slope efficiency $η_{slope}$ by a simple relation [12]

η_{slope} = η_{q} η_{mode} η_{res} η_{abs} η_{del} t,

where

η_{q} \equiv ν_{l} / ν_{p}

is the quantum efficiency,

η_{res} = \frac{(1 - r_{2}) t}{(1 - t^{2} \sqrt{r_{1} r_{2}}) (1 + \sqrt{r_{2} / r_{1}})}

the extraction efficiency of the resonator/windows system and

η_{abs}

is the fraction of the pump power absorbed in the laser cell and found from the calculations described in section 2.

_{$η_{slope}$}for the Ti:Sapphire [4] and diode pumped [9] Cs lasers was found by linear fitting to the dependence of $P_{l a s e}$ on $P_{p, 0}$ shown in Figs. 2 and 5, respectively. Then $η_{mode}$ was calculated from Eq. (20) using the values of $η_{slope}$ As a result we got that for both the Ti:Sapphire [4] and diode pumped [9] Cs lasers the values of $η_{mode}$ are ~0.8 – 0.85 which means that oscillation of multiple transverse modes provides for almost complete overlap of the laser and pump beams. Note that in the case of the single mode lasing, for which $P_{l a s e} (P_{p, 0})$ is shown in Fig. 2 by dashed lines, the values of $η_{mode}$ are much lower than for the case of the multimode lasing. E.g. for ${\bar{w}}_{p}$ = 0.404 mm the value of $η_{mode}$ is only 0.5 showing that for the single mode lasing the pump-to-laser beam overlap is poor.

3.4. Beam quality in the multimode case

Spatial distribution of the laser beam intensity at the output coupler, $I_{l a s e} (r) = I_{l}^{-} (r, z = 0) t (1 - r_{2}),$ for the multimode lasing is strongly different from the Gaussian distribution in the single mode beam. Figure 7 shows $I_{l a s e} (r)$ for the Ti:Sapphire pumped laser [4] at $P_{p, 0} = 10 W$ when three modes participate in the lasing. For comparison Gaussian intensity distribution in the single mode output beam is also shown in the same figure. For Cs DPAL [9] studied in Section 3.2 the intensity distribution has a similar shape. The total width of the multimode laser beam, for which a considerable part of the energy is in the wings, is greater than the width of the Gaussian beam. A narrow intensity peak with a maximum at r = 0 in the multimode case is due to the contribution of high order azimuthally symmetric modes, each of which has a sharp maximum at the origin.

Fig. 7 Spatial distribution of the laser beam intensity $I_{l a s e} (r)$ at the output coupler, for multimode and single mode lasing for the Ti:Sapphire pumped laser [4] at $P_{p, 0} = 10 W$ when three modes participate in the lasing. A narrow intensity peak with a maximum at r = 0 in the multimode case is due to the contribution of high order azimuthally symmetric modes, each of which has a sharp maximum at the origin.

Download Full Size | PDF

The output multimode beam has a sharp central lobe surrounded by a wide pedestal containing substantial fraction of the laser power. Such intensity shape in the near field results in poor beam quality in the far field. The laser beam quality factor $M^{2}$ for a beam consisting of the azimuthally symmetric Laguerre-Gaussian modes is given by [18]

M^{2} = \sum_{i = 0}^{N - 1} (2 i + 1) \frac{P_{l a s e, i}}{P_{l a s e}} .

Figure 8 shows the dependence of $M^{2}$ on the pump power $P_{p, 0}$ for the Ti:Sapphire pumped laser [4] and Cs DPAL [9]. It is seen that $M^{2}$ increases from 1 at the threshold power to 5-6 at the maximum value of $P_{p, 0} .$ The reason for such behavior is the increase of the number of transverse modes participating in lasing with increasing $P_{p, 0}$ resulting in the worse beam quality. Note that for the case of the Cs DPAL the pump beam with very large $M_{p}^{2} = 434$ (see Table 1) and hence very bad beam quality generated by the diode laser is converted to the laser beam with much smaller $M^{2} < 7$ and much better beam quality.

Fig. 8 Beam quality factor $M^{2}$ as a function of the pump power $P_{p, 0}$ for the Ti:Sapphire pumped laser [4] and Cs DPAL [9].

Download Full Size | PDF

4. Summary

A simple optical model of multi-transverse mode operation of alkali vapor lasers is reported. The model is based on calculations of the pump and laser beam intensities in the gain medium, where the laser beam intensity is a linear combination of the azimuthally-symmetric Laguerre-Gaussian modes. It was applied to Ti:Sapphire and diode pumped cesium vapor lasers studied in [4] and [9], respectively. The model predicts that for low pump power and small pump beam radii applied in [4], only the fundamental lasing mode oscillates, just as shown experimentally in [4]. However, for higher pump powers and larger pump beam diameters several transverse modes participate in oscillation. The number and intensities of the oscillating modes as a function of the pump beam power and radius were found. The number of lasing modes increases with the increase of the pump beam radius when the latter becomes larger than the fundamental laser mode radius. It was shown that for a given pump power, there is an optimal pump beam radius corresponding to the maximum laser power and determined by the modal composition of the laser beam. The value of the optimal radius increases with increasing pump power due to the increase in the number of modes.

In order to check the validity of the model, it was applied also to pulsed static Cs DPAL [9] with maximum pump power of 100 W and the pump beam radius larger than that of the fundamental laser mode. It is shown that for such operating conditions, multimode lasing takes place. The model predicts linear dependence of the laser power on the pump power, the values of the former being in agreement with the experimental results.

For the single mode lasing when only the fundamental transverse mode of zero order oscillates and the pump beam radius is much larger than that of this laser mode, the lasing power is much lower than in the case of the multimode lasing. The reason is that the mode-matching efficiency $η_{mode}$ for the single mode lasing is lower than for the multi-transverse mode lasing. For the latter the values of $η_{mode}$ are ~0.8 – 0.85 which means that oscillation of multiple transverse modes provide for almost complete overlap of the laser and pump beams.

Beam quality of the multimode laser beam as a function of the pump power is also calculated for the two aforementioned alkali lasers. The laser beam quality factor $M^{2}$ increases with increasing pump power from 1 at the threshold power to 5-6 at maximum values of the pump power. For Cs DPAL [9] the pump beam with $M_{p}^{2} = 434$ and hence very bad beam quality generated by the diode laser is converted to a laser beam with much smaller $M^{2} < 7$ and much better beam quality.

Funding

Air Force Office of Scientific Research (AFOSR) under grant FA9550- 15-1-0489, Israel Science Foundation (ISF) under grant 893/15 and Office of Naval Research (ONR) under grant N62909-16-1-2213.

References and links

1. W. F. Krupke, “Diode pumped alkali lasers (DPALs) — A review (rev1),” J. Prog. Quantum Electron. 36(1), 4–28 (2012). [CrossRef]

2. B. V. Zhdanov and R. J. Knize, “Review of alkali laser research and development,” Opt. Eng. 52(2), 021010 (2012). [CrossRef]

3. I. Auslender, B. Barmashenko, S. Rosenwaks, B. Zhdanov, M. Rotondaro, and R. J. Knize, “Modeling of pulsed K diode pumped alkali laser: Analysis of the experimental results,” Opt. Express 23(16), 20986–20996 (2015). [CrossRef] [PubMed]

4. T. Cohen, E. Lebiush, I. Auslender, B. D. Barmashenko, and S. Rosenwaks, “Influence of the pump-to-laser beam overlap on the performance of optically pumped cesium vapor laser,” Opt. Express 24(13), 14374–14382 (2016). [CrossRef] [PubMed]

5. B. Barmashenko, D. Furman, and S. Rosenwaks, “Analysis of lasing in gas-flow lasers with stable resonators,” Appl. Opt. 37(24), 5697–5705 (1998). [CrossRef] [PubMed]

6. P. Wittmuess, S. Piehler, T. Dietrich, M. A. Ahmed, T. Graf, and O. Sawodny, “Numerical modeling of multimode laser resonators,” J. Opt. Soc. Am. B 33(11), 2278–2288 (2016). [CrossRef]

7. H. Shu, Y. Chen, M. Bass, J. F. Monjardin, and J. Deile, “Numerical modeling of alkali vapor lasers,” Opt. Express 19(21), 19875–19885 (2011). [CrossRef] [PubMed]

8. M. Endo, R. Nagaoka, H. Nagaoka, T. Nagai, and F. Wani, “Wave optics simulation of diode pumped alkali laser (DPAL),” Proc. SPIE 9829, 982908 (2016).

9. B. V. Zhdanov, J. Sell, and R. J. Knize, “Multiple laser diode array pumped Cs laser with 48W output power,” Electron. Lett. 44(9), 582–583 (2008). [CrossRef]

10. Yu. A. Ananev and S. G. Anikichev, “Steady multimode lasing in stable cavities with spherical mirrors,” Opt. Spectrosc. (USSR) 68(3), 408–410 (1989).

11. A. M. Komashko and J. Zweiback, “Modeling laser performance of scalable side-pumped alkali laser,” Proc. SPIE 8581, 85810H (2010).

12. B. D. Barmashenko and S. Rosenwaks, “Detailed analysis of kinetic and fluid dynamic processes in diode-pumped alkali lasers,” J. Opt. Soc. Am. B 30(5), 1118–1126 (2013). [CrossRef]

13. B. V. Zhdanov, US Air Force Academy, Laser and Optics Research Center, 2354 Fairchild Dr, Ste. 2A31, USAF Academy, Colorado 80840 (personal communication, 2016).

14. G. D. Hager and G. P. Perram, “A three-level analytic model for alkali metal vapor lasers: part I. Narrowband optical pumping,” Appl. Phys. B 101(1), 45–56 (2010). [CrossRef]

15. N. D. Zameroski, G. D. Hager, W. Rudolph, and D. A. Hostutler, “Experimental and numerical modeling studies of a pulsed rubidium optically pumped alkali metal vapor laser,” J. Opt. Soc. Am. B 28(5), 1088–1099 (2011). [CrossRef]

16. R. Kapoor, P. Mukhopadhyay, and J. George, “A new approach to compute overlap efficiency in axially pumped solid state lasers,” Opt. Express 5(6), 125–133 (1999). [CrossRef] [PubMed]

17. J. Han, Y. Wang, G. An, K. Rong, H. Yu, S. Wang, W. Zhang, H. Cai, L. Xue, H. Wang, and J. Zhou, “Analyses of mode filling factor of a laser end-pumped by a LD with high-order transverse modes,” Proc. SPIE 10183, 101831I (2017).

18. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]

Parameter	Description	Value
Parameter	Description	Ti:Sapphire pumped Cs laser [4]	Pulsed Cs DPAL [9]
p_He (Torr)	He partial pressure at room temperature	0	500
p_ethane (Torr)	C₂H₆ partial pressure at room temperature	600	100
T(C)	Temperature in the laser cell	98	94
l(cm)	Length of the laser cell	2	2
t	Transmission of the laser cell windows	0.94	0.94
r₁	Reflectivity of the total reflector	1	1
r₂	Reflectivity of the output coupler	0.1	0.2
L(cm)	Resonator length	14.4	40
R₁(cm)	Radius of curvature of the total reflector	∞	50
R₂(cm)	Radius of curvature of the output coupler	15	∞
${\bar{w}}_{l}$	Average radius of the laser fundamental transverse mode	0.152	0.438
w₀_,p (mm)	Pump beam waist radius (at 1/e² of maximal intensity)	0.22, 0.29	1
${\bar{w}}_{p}$ (mm)	Average radius of the pump beam	0.238, 0.404	1.2
$M_{p}^{2}$	Pump beam quality factor	1	434
Δυ_p (GHz)	Pump beam spectral width	16	10
$η_{del}$	Delivery efficiency	1	0.85

Multi-transverse mode operation of alkali vapor lasers: modeling and comparison with experiments

Abstract

1. Introduction

2. Description of the model

3. Results and discussion

3.1. Modeling of Ti:Sapphire pumped laser [4]

3.2. Modeling of pulsed diode pumped Cs laser [9]

3.3. Mode matching efficiency in the case of the multimode lasing

3.4. Beam quality in the multimode case

4. Summary

Funding

References and links

Cited By

Figures (8)

Tables (1)

Equations (22)

Optics Express