Invited Paper
Modeling of static and flowing-gas diode pumped alkali lasers Boris D. Barmashenko, Ilya Auslender, Eyal Yacoby, Karol Waichman, Oren Sadot, Salman Rosenwaks Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel ABSTRACT Modeling of static and flowing-gas subsonic, transonic and supersonic Cs and K Ti:Sapphire and diode pumped alkali lasers (DPALs) is reported. A simple optical model applied to the static K and Cs lasers shows good agreement between the calculated and measured dependence of the laser power on the incident pump power. The model reproduces the observed threshold pump power in K DPAL which is much higher than that predicted by standard models of the DPAL. Scaling up flowing-gas DPALs to megawatt class power is studied using accurate three-dimensional computational fluid dynamics model, taking into account the effects of temperature rise and losses of alkali atoms due to ionization. Both the maximum achievable power and laser beam quality are estimated for Cs and K lasers. The performance of subsonic and, in particular, supersonic DPALs is compared with that of transonic, where supersonic nozzle and diffuser are spared and high power mechanical pump (needed for recovery of the gas total pressure which strongly drops in the diffuser), is not required for continuous closed cycle operation. For pumping by beams of the same rectangular cross section, comparison between end-pumping and transverse-pumping shows that the output power is not affected by the pump geometry, however, the intensity of the output laser beam in the case of transverse-pumped DPALs is strongly non-uniform in the laser beam cross section resulting in higher brightness and better beam quality in the far field for the end-pumping geometry where the intensity of the output beam is uniform. Keywords: gas lasers, diode pumping, gas flows
1. INTRODUCTION DPALs are currently extensively studied due to their great potential as high power lasers.1,2 They operate on the D1(n P1/2 → n 2S1/2) transition of alkali atoms (where n = 4, 5, 6 for K, Rb and Cs, respectively) at ~ 800 nm, pumped via the D2(n2S1/2 → n2P3/2) transition, followed by rapid relaxation (by buffer gas, helium and\or small hydrocarbon molecules) of the upper to the lower fine-structure level, n2P3/2 to n2P1/2. (designated as levels 3 and 2, respectively; the ground state n 2S1/2 is designated as 1).
2
Static DPALs use simple hardware where pump beams enter a sealed laser cell containing a mixture of alkali vapors and buffer gases through optical windows. These lasers show high efficiency at low pumping power, however, pumping them at high power results in considerable heat release, mainly due to the relaxation, and leads to temperature rise and consequently to a decrease of their power and efficiency. Obviously, the gas flow in the laser section enhances the heat transfer out of this section, impedes temperature rise and increases the power and efficiency of the laser. Demonstration of subsonic DPALs3,4 and modeling of both supersonic and subsonic devices5-15 taking into account fluid dynamics and kinetic processes in the lasing medium, show the positive influence of the gas flow on the laser performance, where the highest lasing power and optical-to-optical efficiency were predicted for supersonic DPALs. A simpler alternative to supersonic devices is transonic laser, where the Mach number is a little smaller than unity. In such a laser, complex hardware, including supersonic nozzle, diffuser and high power mechanical pump (needed for recovery of the gas total pressure which strongly drops in the diffuser), is not required for continuous closed cycle operation.6 In the present paper modeling of static and flowing-gas subsonic, transonic and supersonic Cs and K DPAL is reported. A simple optical model of such lasers, where Gaussian spatial shapes of the pump and laser intensities in any cross section of the beams are assumed, is developed. Unlike semi-analytical models of DPALs9,16 this model does contain an empirical value of the mode-matching efficiency introduced to take into account the fractional spatial overlap of the pump and laser beams in the laser cell. Rather, the experimentally measured spatial shapes of the pump and laser beams
[email protected] High Energy/Average Power Lasers and Intense Beam Applications VIII, edited by Steven J. Davis, Michael C. Heaven, J. Thomas Schriempf, Proc. of SPIE Vol. 9729, 972904 · © 2015 SPIE · CCC code: 0277-786X/15/$18 · doi: 10.1117/12.2214975 Proc. of SPIE Vol. 9729 972904-1
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appear in the equations for the populations of the electronic levels of the alkali atoms and for the intensities of the pump and laser power in the resonator. The model is applied to the static K and Cs DPALs studied experimentally in Refs. 1719 and shows good agreement between the calculated and measured dependence of the laser power on the incident pump power. As discussed in Refs. 1 and 2, flowing-gas DPALs are of interest due to their scalability potential to megawatt (MW) class power. Possibilities for scaling up of these lasers were studied in Refs. 7 and 20, where rough estimates of their operation parameters indicated that efficient MW class flowing-gas DPALs are feasible. In the present paper scaling up flowing-gas DPALs is studied using accurate three-dimensional computational fluid dynamics (3D CFD) model, taking into account the effects of temperature rise and losses of alkali atoms due to ionization. Both the maximum achievable power and laser beam quality are estimated for Cs and K lasers. The performance of subsonic and supersonic DPALs is compared with that of transonic to learn if megawatt laser power can be achieved in them. The flow cross section is assumed to be rather large, 5 x 5 cm2. The main conclusions of the comparison are valid also for lower pumping power (where the dimensions of the device are correspondingly smaller). Comparison between the output laser power Plase of different lasers is performed for optimal pressures, temperatures and buffer gas compositions in the laser section found in Refs. 5 and 6. Optimal conditions for operation of Cs and K DPALs, as well as their beam quality, are modeled and analyzed in the following sections.
2. MODELING OF POTASSIUM AND CESIUM STATIC DPALS: ANALYSIS OF THE EXPERIMENTAL RESULTS 2.1 Description of the optical model The model considers typical configuration of Ti:Sapphire or diode pumped laser with end-pump geometry shown in Fig. 1. A pump beam with total power Pp ,0 enters a cylindrical laser cell of length l through windows with transmission t. The lasing medium consists of a mixture of alkali vapor X and buffer gas M for broadening the D2 transition and mixing between the fine-structure levels 3 and 2. The walls of the cell are heated to the temperature T ~ 180-190 C and ~85-95 C for K and Cs lasers, respectively. The laser resonator of length L consists of a concave reflector of radius R and a plane output coupler with reflectivity r1 (close to 100%) and r2, respectively, located outside the laser cell. Both the pump and laser beams propagate along the optical axis Z of the resonator. Unlike our previous models9,15 the pump and laser intensities, I p and I l , respectively, are not assumed to be uniform in the beam cross section xy and their spatial distributions have the form: I p ( x, y, z, ) Pp ( z, ) f p ( x, y, z) , l
(1)
I ( x, y, z ) Pl ( z ) fl ( x, y, z ) ,
(2)
where Pp ( z, ) is the spectral density of the pump power per unite frequency, Pl ( z ) is the total laser power and
f p,l ( x, y, z ) are the normalized spatial distributions of the pump and laser intensity over the beam cross section xy, slightly dependent of z, so that
dxdyf p ,l ( x, y, z ) 1 .
(3)
x , y
The spectral distribution of the pump power Pp ( z, ) over the frequency 𝜈 was taken into account, since in some experiments broad band pumping is applied. The laser powers Pl ( z ) are assumed to be monochromatic with frequency l; the superscripts + and – indicate laser beams propagating in the opposite directions +z and –z. Only the pump beam propagating in the +z direction was included into the model since checking the power of the pump beam propagating in the -z direction we found it should be negligibly small. The reason is that due to almost complete absorption of the pump beam during the passage of the cell in the +z direction followed by the reflection from the concave mirror only small fraction of the initial pump energy returns to the cell. The rates of changes of Pp ( z, ) and Pl ( z ) with z are described by the Beer-Lambert law, where z = 0 is assumed to be located at the inner surface of the window closer to the output coupling mirror with the reflectivity r2:
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dPp ( z, )
( z, ) Pp ( z, ), dz dPl ( z ) ( z ) Pl ( z ), dz where
(4) (5)
( z, )
n dxdy D2 ( ) n1 3 f p ( x, y, z ) 2 - x , y
( z)
(6)
and - x , y
dxdy D1 ( l ) n2 n1 fl ( x, y, z )
(7)
are the absorption coefficient for the D2 transition and the gain coefficient for the D1 transition, respectively, averaged over the beam cross section xy, n1 , n2 and n3 are the number densities of X atoms in levels 1, 2 and 3, respectively, whereas D ( l ) D , at 1
1
D1 , n
(8)
D1
and
D ( ) D 2
D2 , n 2 , at
D2
/ 2
2
D2
2
p
/2
D2
(9)
2
are the stimulated emission and pump absorption cross sections for the D 1 and D2 transitions, respectively, D1 (D2 ), at l2( p ) / 2 are the atomic cross sections, l ( p ) - the wavelengths and D1 ( D2 ), n and D1 ( D2 ) - the natural and the pressure broadened widths, respectively, of these transitions. In Eq. (9), it is assumed that the line shape of the D2 transition is described by pressure broadened Lorentzian spectral distribution.6,7
Windows, t Pump beam
Laser intensity Laser beam
l
Pump
Mirror, r1
Y
intensity
Mirror, r2
Z
X Fig.1. Schematics of static DPAL with end-pumped geometry.
The values of n1 , n2 and n3 are found from the rate equations for the 3-level system presented in Ref. 21. As shown in Refs. 5, 6 and 22 the rates of the excitation of K atoms to higher electronic levels followed by their ionization are negligibly small for pump intensities 1 MW the saturation of the absorption on the pumping D2 transition results in a weaker inhomogeneity of the laser beam. 2.4
1.8 -
-End pump,transonic (M -0.9)
0.6 -
- -Transverse pump,transonic (M -0.9)
0 0
1
2
3
4
5
Pn. MW
a
o
xmm
b Figure 12. (a) Dependence of Plase on Pp for end- and transverse-pumping, where the K DPAL parameters correspond to case 2 in Table 3. (b) Spatial distribution of the intensity of the output laser beam for DPALs with end- and transversepumping at Pp = 0.15 MW. For the end-pumping the spatial intensity distribution in the beam cross section is homogeneous, whereas for the transverse-pumping the intensity varies significantly along the pumping axis y.
The laser beam quality strongly depends also on the spatial distribution of the phase of the electromagnetic field. The best beam quality is achieved for a plane wave front of the laser beam corresponding to uniform phase distribution over the xy plane at the output coupler. The most important parameter affecting the wave front distortions is the optical path difference, OPD(x,y), defined as31
OPD x, y L n x, y ‹n› ,
(16)
where L is the flow width , n(x,y) the refraction index of the gas given by
n( x, y) 1 ( x, y )KGD ,
(17)
( x, y) is the gas density averaged over z, ‹n› the average n x, y over the xy plane and KGD is the Gladstone-Dale constant: 0.196 cm3/g for He and 0.615 cm3/g for CH4.32,33 Uniform phase distribution over the beam cross section can be achieved when the OPD is considerably smaller than the laser beam wavelength λ. Fig 13 shows the dependence of the OPD on x for Cs and K end-pumped transonic DPALs. The cases shown in Fig. 13 are for Pp = 0.15 MW and other parameters corresponding to case 2 in Table 2 and case 2 in Table 3, respectively. One can see that the maximum OPD of the Cs DPAL is about 10 times larger than that of the K DPAL and that the constant phase assumption is valid only for the latter, where the maximum OPD is ~0.2λ. The reason for this significant
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difference is that on the one hand KGD of CH4 is much larger than that of helium (0.615 cm3/g and 0.196 cm3/g, respectively) and that on the other hand the density changes in Cs laser are much larger than in K. 2 1.5
Cs
OPD / λ
1
K
0.5 0
-0.5 -1
-1.5 -2 0
5
10 x, mm
15
20
Fig.13. Dependence of the optical path difference (OPD) on the distance x in the flow direction for Cs and K end-pumped transonic DPALs at Pp = 0.15 MW and other parameters corresponding to case 2 in Table 2 and case 2 in Table 3, respectively.
For directed energy applications, obtaining maximum energy density on a remote target in the far field might be not less important than getting maximum output lasing power. The energy on the target strongly depends on the output laser intensity distributions I(x, y) shown in Fig. 12b. We estimated the far field intensity via Fraunhofer diffraction approximation where the absolute value of the electromagnetic field amplitude |uf (x, y)| in the far field is the Fourier transform of |u(x, y)| which is proportional to I ( x, y) . The laser intensity in the far field (as a function of x and y) is given by
I f ( x, y ) Plase
u f ( x, y )
2 2
.
(18)
u f ( x, y ) dxdy
x , y
Such an approach is valid when the phase of the complex electromagnetic field u at the resonator exit is independent on x and y, and when the far field distance l is much larger than W2/ λ, where W is the width of the laser output beam aperture.34 As shown above, the first assumption is correct for K DPAL with He as a buffer gas. Fig. 14 shows the intensity distribution in the far field for the end-pump K transonic DPAL for l = 30 Km and the same parameters as in Fig. 12b. I f ( x, y) for the transverse-pump case looks almost the same. For both end- and transversepumping the variations of the output intensity I ( x, y) in the x direction are very weak, resulting in very close dependencies of I f on x. At the same time, the dependencies of I f on y averaged over x and shown in Fig. 15 are different for the cases of the end- and transverse-pumping. For the transverse-pumping I f is smaller in the central lobe and larger in the wings of the intensity distribution. In order to estimate the beam quality in the far field "the power in the bucket" method was used,35 where the beam dimensions Wx ,86 and Wy ,86 are defined as the dimensions of the area that contains 86% of the total power. The smaller are Wx ,86 and Wy ,86 the better is the beam quality. Calculations using the data shown in Figs. 14 and 15 predict the values of Wx ,86 and Wy ,86 for the two pumping geometries. It appears that Wx ,86 ~ 1.4 m are the same for the end- and transverse-pumping. However, for the transverse-pumping Wy ,86 ~ 1.1 m is
almost two times larger than that for the end-pumping case. Therefore, in terms of brightness and beam quality, endpumping geometry is preferred, although the output power and optical-to-optical efficiency are not affected by the pump geometry. Note that the difference between the laser beam widths in the far field for the two pumping geometries gets smaller as the pumping power gets higher. The reason is the aforementioned weaker inhomogeneity of the laser beam in the near field at Pp > 1 MW. It is also important to note that the present optical model based on the intensity calculations does not give accurate results for the intensity distribution in the far field, and we used it only for comparison between
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