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Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. B
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Modeling, numerical approach, and power scaling of alkali vapor lasers in side-pumped configuration with flowing medium Zining Yang,* Hongyan Wang, Qisheng Lu, Yuandong Li, Weihong Hua, Xiaojun Xu, and Jinbao Chen College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China, 410073 *Corresponding author:
[email protected] Received January 6, 2011; revised April 1, 2011; accepted April 4, 2011; posted April 5, 2011 (Doc. ID 140662); published May 5, 2011 Diode alkali vapor lasers (DPALs) with a flowing medium provide a pathway to extremely high-power CW or quasi-CW laser operations. In this article, the model for end-pumped alkali lasers [Beach et al., J. Opt. Soc. Am. B 21, 2151(2004)] is expanded to model DPALs in a side-pumped configuration. The difference between our model and the published model [Komashko et al., Proc. SPIE 7581, 75810H-1 (2010)] is studied, and a comparison with other people’s experimental results [Zweiback et al., Proc. SPIE 7915, 791509-1 (2011)] is made, which demonstrates the validity of our model. Some important influencing factors are simulated and analyzed. A conceptual power-scaled design of a megawatt-class side-pumped flowing DPAL is made. The results demonstrate an optical-to-optical efficiency over 60% with all the other parameters reasonable and available in the near future. © 2011 Optical Society of America OCIS codes: 140.1340, 140.3480.
1. INTRODUCTION As the first kind of laser proposed [1], the optically pumped alkali vapor laser was first demonstrated using a Cs species and a lamp pumping source by Rabinowitz et al. in 1962 [2]. The first laser-pumped (Ti:sapphire) Rb laser with high efficiency was realized by Krupke et al. in 2003, and the conception of a diode-pumped alkali vapor laser (DPAL) was proposed then [3]. As a new class of optically pumped gas laser, DPAL offers a new way to high-power, high-efficiency, highbrightness, and compact laser systems due to its many significant advantages. The quantum efficiency is very high (95.3% for Cs, 98.1% for Rb, and 99.6% for K), which is important for increasing the overall efficiency and minimizing heating problems. In addition, the thermal problems can be reduced, since the gaseous gain medium can be flowed to remove the heat. Good thermal management as well as the stability at high pump intensity [4] provides no significant single aperture power-scaled limitation for DPALs. For the large cross section, each atom could cycle many millions of pump photons per second during the lasing process, which makes it promising to extract high power from small gain volumes. In recent years, a great number of DPAL experiments have demonstrated high efficiency and good beam quality [5–12]. At the first development stage of DPAL, people demonstrated laser operations by use of a static gain medium in an end-pumped configuration, which provides the benefits of sufficient pump absorption, total population inversion along the laser path, and easy operation. But with the increase of pump power, the thermal effect becomes obvious in such endpumped static DPALs. In Zhdanov et al.’s experiment on a Cs laser, the output power rolled over and even dropped when the CW pump power increased from 30 W to 100 W, while a linear increase of laser power can easily be kept in pulsed0740-3224/11/061353-12$15.00/0
pump mode with the same average pump power [8]. By introducing a cooling channel into the cell, Zweiback et al. have improved the stability of performance for a Rb laser with CW output power of 145 W [11]. In power scaling for end-pumped, static-gas, and conductive-cooling DPALs, a planar waveguide configuration has been proposed, which seems practical in multikilowatt laser operation, perhaps even as high as 100 kW [13]. But a further power scaling for such a configuration will be limited due to the fact that the transverse temperature gradient must be restricted to some maximum tolerable value to guarantee the high beam quality. To scale a highbrightness DPAL to the megawatt power level, the gain medium must be flowed to convectively remove the waste heat. In a DPAL with a flowing medium, both end-pumped and sidepumped configurations can be used. And a conceptual design for a 2 MW end-pumped axially flowed DPAL has been made [13]. As compared with end-pumped configuration, the sidepumped DPAL with transverse-flow direction (mutually orthogonal pump, laser, and flow directions) is much more attractive and scalable. In such a configuration, the separation between pump and laser beams can simplify the optics arrangement and significantly decrease the power burden on laser windows. The demand of pump brightness is reduced to less than 1=10 of that required in end-pumped configuration, which leads to easy pump structure design. In addition, the gain along the laser path can be much more uniform, and the required flow rate can be much lower as compared with an axially flowed structure. All these physical and engineering advantages show the great potential of side-pumped transverse-flow DPAL in extremely high-power laser operation. In the theoretical field, some models for end-pumped alkali lasers have been proposed [14,15]. For a side-pumped configuration, a physically based model with an ingenious numerical approach was proposed by Komashko and Zweiback [16]. In © 2011 Optical Society of America
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their model, nearly all the key points for DPAL modeling are considered, including the fine-structure mixing rate that represents the three-level nature of this kind of laser, the spatially varying population densities, the spectral dependence of pump light absorption, and the inner cavity losses. For fast calculation, the spectral dependence of pump light absorption was neatly simplified. On the other hand, this simplification will more or less affect the predicted results. Even so, this model is still a very useful and convenient tool for laser system trade study. In this article, the successful model proposed by Beach et al. [14] for an end-pumped DPAL is expanded for the side-pumped configuration, which has considered all the points mentioned above with no significant simplifications. In Section 2, the model and numerical approach are introduced. In Section 3, the difference between our model and Komashko and Zweiback’s model [16] is studied (Section 3.A ), a comparison with Zweiback and Komashko’s experimental results is made (Section 3.B) [17], and some influencing factors on laser characteristics are calculated and analyzed (Section 3.C). In Section 4, a conceptual power-scaled design of a megawatt-class side-pump flowing DPAL is made, and some important design considerations are discussed.
2. MODELING AND NUMERICAL APPROACH For a DPAL in a side-pumped configuration, we are only interested in high-power laser operation. The diode pump sources are thus arranged in columns, and each column contains multiple laser diode (LD) stacks. The alkali vapor cell is designed as a rectangular box (L × W × H) to match the pump beam shape (see Fig. 1). Pump light enters the cell along the y axis and is reflected by side reflector for a second pass. In the fast-axis direction (along the x axis), the pump beam can be collimated by beam-shaping optics. In the slow-axis direction (along the z axis), an approximate 10° divergence angle (full angle, 95% enclosed power) exists, which will induce pump light overlapping between neighboring columns of LD stacks. The degree of overlapping depends on the cell width and the distance between the cell and the pump source. Because the neighboring columns may have a small spatial separation between them due to the packaging structure and because the intensity in the edge of the beam is relatively weaker, the overlapping of pump lights can be used to benefit
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the uniformity of pump power distribution along the laser path (z axis) by proper pump structure design. As for the alkali atoms, they only respond to the total pump intensity at their location without discrimination as to pump propagating directions. So as long as the pump power distribution is relatively uniform along the laser path, the slow-axis divergence angle and the separation between LD stacks can be ignored. Another simplification is to assume that the pump intensity along the x−z plane is homogeneous. In our model, different methods are used in the transverse (y) and longitudinal (z) dimensions. In the transverse dimension, the gain medium is divided into small volume segments (y, y þ Δy). In each segment, the population densities are assumed to be constant along the y axis. The spectral dependence of the pump light absorption and the transverse population variation are considered. To solve for double-pass configuration, an iterative algorithm is proposed with a high degree of accuracy and a high speed of convergence. In the longitudinal dimension, the model proposed by Beach et al. [14] is used in each segment (y, y þ Δy), with some modifications to adjust the side-pumped configuration. A. Modeling in the Longitudinal Dimension In this dimension, the assumption of a longitudinal population average is made to eliminate the z dependence of population densities. Since the pump intensity is assumed to be homogeneous in the x−z plane, the population densities are only y-dependent, denoted as ni ðyÞ (i ¼ 1, 2, 3). In each volume segment (y, y þ Δy), the rate equations are described as follows [14]: dn1 ðyÞ n ðyÞ n3 ðyÞ ¼ −Γp ðyÞ þ Γl ðyÞ þ 2 þ ; dt τ D1 τ D2
ð1Þ
dn2 ðyÞ ΔE n ðyÞ ¼ −Γl ðyÞ þ γ 32 n3 ðyÞ − 2n2 ðyÞ exp − − 2 ; dt kT τ D1 ð2Þ dn3 ðyÞ ΔE n ðyÞ ¼ Γp ðyÞ − γ 32 n3 ðyÞ − 2n2 ðyÞ exp − − 3 ; dt kT τ D2
Fig. 1. (Color online) Schematic diagram of DPAL in single-side double-pass pump configuration.
ð3Þ
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where Γp ðyÞ and Γl ðyÞ represent the pump absorption and laser emission rates, respectively. γ 32 ¼ nmethane σ 32 vr is the finestructure mixing rate that relaxes the population from 2 P 3=2 to 2P 1=2 in order to get population inversion. A sufficient finestructure mixing rate is usually realized by adding some small hydrocarbons, for example, methane or ethane. Here we take methane as an example; nmethane is the number density, σ 32 the fine-structure mixing cross section and vr the rms thermally averaged relative velocity between alkali atoms and methane molecules. τD1 ¼ 1=ðA21 þ Q21 Þ is the lifetime of the 2 P 1=2 energy level, where A21 is the spontaneous emission rate and Q21 the collisional quenching rate introduced by hydrocarbons. Q21 ¼ nmethane σ q21 vr , where σ q21 is the collisional quenching cross section. The definition of τD2 is similar to τD1 . ΔE is the energy gap between the 2 P 1=2 and 2 P 3=2 levels, k the Boltzman constant, and T the absolute temperature. Figure 2 shows the intracavity pump and laser powers in each segment. The pump light is assumed to be in Gaussian spectral profile; the forward- and backward-propagating spectrally resolved pump powers at the y plane are given by P p ðy; λÞ
¼
g3 3 A31 λ2D2 g1 4π 2 ΔνD2
1
1þ
2 ΔνD2
c λ
− λDc
2 ;
ð6Þ
2
where gi (i ¼ 1, 2, 3) are the degeneracy factors of the energy levels. The ‘factor of 3,’ 3 , equals unity when the linearly polarized pump light interacts with alkali atoms that have degenerate transition 52 S 1=2 ↔52 P 3=2 [18]. ΔvD2 is the spectral bandwidth (FWHM) of pump absorption transition, which needs to be broadened by buffer gases, such as helium or methane, to match the relatively broad pump linewidth. P ΔνD2 ¼ 1=τD2 þ i γ i;D2 P i , where γ i;D2 is the linewidth broadening coefficient and P i the partial pressure of buffer gas species i. λD2 is the center wavelength of the D2 line, which will be P shifted when buffer gases exist. λD2 ¼ λD2 ðvacuumÞ þ i δi;D2 P i , where λD2 ðvacuumÞ is the absolute wavelength in a vacuum and δi;D2 the line shifting coefficient of buffer gas i. The expression of Γl ðyÞ is similar to that in [14], except for a minor modification on intracavity losses: P 2 ðyÞ − P 1 ðyÞ þ P 4 ðyÞ − P 3 ðyÞ hνl V l P laser ðyÞ Roc T w ¼ fexp½ðn2 ðyÞ − n1 ðyÞÞσ 21 L − 1g hνl V p ηmode 1 − Roc
Γl ðyÞ ¼
dP p ðyÞ ¼ dλ P p ðyÞ
σ 13 ðλÞ ¼
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rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 4 ln 2 c −4 ln 2c 1 1 2 − · exp ; π Δνp λ2 Δνp λ λp ð4Þ
where P p ðyÞ are the total pump powers, c the speed of light, Δνp the linewidth (FWHM) of the pump light, and λp the center pump wavelength. Γp ðyÞ is given by 1 − − ½P þ ðyÞ − P þ p ðy þ ΔyÞ þ P p ðy þ ΔyÞ − P p ðyÞ hνp V p p Z ∞ 1 ¼ P þ ðy; λÞ hνp V p 0 p 1 × 1 − exp − n1 ðyÞ − n3 ðyÞ σ 13 ðλÞΔy dλ 2 Z ∞ 1 þ P − ðy; λÞ hνp V p 0 p 1 × exp n1 ðyÞ − n3 ðyÞ σ 13 ðλÞΔy − 1 dλ; ð5Þ 2
Γp ðyÞ ¼
where hνp is the pump photon energy, and V p ¼ LHΔy the pump volume that is assumed to occupy the whole alkali vapor cell. σ 13 is the spectrally resolved absorption cross section that is given by
× f1 þ T 2w T 2s Rback exp½ðn2 ðyÞ − n1 ðyÞÞσ 21 Lg;
where P 1 ðyÞ ði ¼ 1; 2; 3; 4Þ are the circulating laser powers and P laser ðyÞ is the output laser power in a volume segment (y, y þ Δy). hνl is the laser photon energy. The alkali laser is assumed to be in a single frequency centered on laser transition, and the emission cross section is given by 3 A21 λ2D1 σ 21 ¼ 2 ; ð8Þ 4π ΔνD1 where λD1 and ΔvD1 are, respectively, the center wavelength and spectral bandwidth of laser transition, which are also shifted and broadened by buffer gases. The 3 still equals unity for interaction between a linearly polarized alkali laser and degenerate transition of 52 S 1=2 ↔52 P 1=2 . Rback and Roc are the reflectivities of the back reflector and the output coupler. T w is the transmission of the single-cell window, and T s represents all the other intracavity losses that are assumed to be located in front of the back reflector. V l is the laser volume, and ηmode ¼ V l =V p is the mode overlap factor that is used to describe the spatial energy extraction fraction. The absorbed pump power, P absorb , will be inverted into different channels, including the output laser power, P laser , the fluorescence power, P fluorescence , due to spontaneous emission, the thermal power, P thermal , due to quantum defect, and the lost laser power, P scatter , due to the residual reflectivity of the cell window and other scattered losses of intracavity optical components. The expressions of these inverted channels in each segment (y, y þ Δy) are described as follows: P fluorescence ðyÞ ¼ V l ½n2 ðyÞA21 E21 þ n3 ðyÞA31 E 31 ;
Fig. 2. (Color online) Schematic diagram of intracavity laser and pump powers in a volume segment.
ð7Þ
ð9Þ
ΔE ; ð10Þ P thermal ðyÞ ¼ V l γ 32 ΔE n3 ðyÞ − 2n2 ðyÞ exp − kT
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P scatter ðyÞ ¼
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P 1 ðyÞ ð1 − T w Þ þ P 2 ðyÞð1 − T w Þ Tw
ð0Þ
þ P 2 ðyÞT w ð1 − T s Þ þ P 2 ðyÞT w T s ð1 − Rback Þ þ P 2 ðyÞT w T s Rback ð1 − T s Þ þ
P 3 ðyÞ ð1 − T w Þ Tw
þ P 4 ðyÞð1 − T w Þ;
ð11Þ
distribution ni ðyÞ (y ¼ ½0; W , i ¼ 1, 2, 3), which is taken as the initial value of this iterative process. 2. Letting the incident pump light P þ p ð0; λÞ double pass the gain medium through the already calculated population distrið0Þ bution ni ðyÞ, we can obtain the unabsorbed pump power −ð0Þ P p ð0; λÞ at the incident plane. The propagation equation of backward pump light, P −p ðy; λÞ, that is reflected by the side reflector is given by
and the energy conservation law requires that P −p ðy; λÞ
P absorb ðyÞ ¼ P laser ðyÞ þ P fluorescence ðyÞ þ P thermal ðyÞ þ P scatter ðyÞ:
ð12Þ
B. Iterative Algorithm in the Transverse Dimension In the transverse dimension, we have proposed an iterative algorithm to solve for the single-side double-pass pump configuration. The calculation process demonstrates a high degree of accuracy and a high speed of convergence. The iterative process is described as follows (see Fig. 3): 1. First, we calculate the solution for single-pass pump configuration, that is, with no side reflector. In this case, P −p ðy; λÞ in the expression of Γp ðyÞ becomes zero. The propagation equation of P þ p ðy; λÞ is given by 1 þ Pþ p ðy þ Δy; λÞ ¼ P p ðy; λÞ exp − n1 ðyÞ − n3 ðyÞ σ 13 ðλÞΔy : 2 ð13Þ Taking P þ p ð0; λÞ as the initial value, together with Eq. (13) and the longitudinal model mentioned above (‘L’ method for short in Fig. 3), we can solve for the transverse population
¼
P −p ðy
1 þ Δy; λÞ exp − n1 ðyÞ − n3 ðyÞ σ 13 ðλÞΔy : 2 ð14Þ
3. Now the total pump power at the incident plane is totð1Þ −ð0Þ P p ð0; λÞ ¼ P þ p ð0; λÞ þ P p ð0; λÞ. Taking it as the new pump light, we can use the ‘L’ method to calculate for a new populað1Þ tion distribution, ni ðyÞ (y ¼ ½0; W, i ¼ 1, 2, 3). In the calcu−ð0Þ þ lation, P p ðy; λÞ and P p ðy; λÞ just correspond to the forward and backward pump powers in the expression of Γp ðyÞ. The propagation equation for P þ p ðy; λÞ is Eq. (13), and for −ð0Þ P p ðy; λÞ is given by P −p ðy þ ΔyÞ ¼ P −p ðyÞ exp ð1Þ
1 n1 ðyÞ − n3 ðyÞ σ 13 ðλÞΔy : ð15Þ 2 ð0Þ
ð1Þ
ð0Þ
4. Compare ni ðyÞ with ni ðyÞ. If ni ðyÞ ¼ ni ðyÞ ð1Þ (y ¼ ½0; W , i ¼ 1, 2, 3), then ni ðyÞ is the solution, and other laser information has been obtained in the calculation proð1Þ ð0Þ cess. If ni ðyÞ ≠ ni ðyÞ (y ¼ ½0; W , i ¼ 1, 2, 3), then substitute ð0Þ ð1Þ ni ðyÞ by ni ðyÞ, and repeat steps 2 through 4 to continue the iterative process and finally get the solution. This iterative algorithm can also be expanded to solve for the double-sided symmetrically pumped or even the foursided symmetrically pumped configurations, which will be presented in a later paper.
3. SIMULATION RESULTS AND DISCUSSION
Fig. 3. Process of iterative algorithm for single-side double-pass pump configuration.
A. Comparison with Komashko and Zweiback’s model In Komashko and Zweiback’s model (K’s model for short) [16], the spectral dependence of pump light absorption was neatly reduced to an effective cross section, σ eff , for simplification, which was calculated by the convolution of the atomic absorption Lorentzian line shape and the initial pump Gaussian spectral shape. In contrast, this cross section is calculated in our model by the convolution of the atomic absorption spectrum and the changing pump spectrum when propagating along the cell width. To study the difference between these two models, a comparison is made in Table 1. In the calculation, all the parameters are set the same as in K’s model: T ¼ 120 °C, P He ¼ 3 atm (20 °C), P methane ¼ 200 torr (20 °C), Δvp ¼ 0:2 nm, Roc ¼ 20%, Rback ¼ 99%, Rside ¼ 90%, T w ¼ 95%, T s ¼ 100%, ηmode ¼ 100%. Also, as in K’s model, we assume that helium and methane only contribute to linewidth broadening and fine-structure mixing, respectively, with the values listed in [16], and the concentration of alkali atoms is calculated by the formula of Rb87 [19]. Before discussion, some parameters are defined: ηabsorb is the pump absorption efficiency, ηopt-absorb is the conversion efficiency from absorbed pump power into laser power,
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and ηopt-opt ¼ ηabsorb ηopt-absorb is the optical-to-optical efficiency. ηfluorescenceðthermal;scatterÞ represent the fluorescence, thermal, and scattered efficiencies, respectively. V laserðfluorescence;thermalÞ describe the relative variation of the laser, fluorescence, and thermal powers along the cell width. As an example, V laser is defined by
V laser ¼
max½P laser ðyÞy∈½0;W − min½P laser ðyÞy∈½0;W : max½P laser ðyÞy∈½0;W
ð16Þ
From Table 1, we can see that the ηabsorb calculated by our model is lower than in K’s model. As a consequence, ηopt-opt is also decreased. This difference is probably due to the introduction of the effective cross section σ eff . In K’s model, σ eff is calculated by using the initial pump spectrum and is kept constant. But as pump light propagates, strong absorption will change the pump spectral shape dramatically (see Fig. 4). As a result, σ eff will not keep constant, but will decrease as the pump light propagates. So K’s model may overestimate the pump absorption efficiency. The higher laser power variation (V laser ) predicted by our model is another necessary result. The spectral dependence in our model can be described by a decreasing effective cross section along the forward pump propagating direction denoted as σ eff ðyÞ. With either decreasing σ eff ðyÞ or constant σ eff , the maximal laser power normally occurs in the first volume ð0; ΔyÞ, but in subsequent volumes the relatively lower σ eff ðyÞ will induce lower laser power, which will finally increase the laser variation along the cell width. By our model, we can also obtain some detailed intracavity information (see Fig. 5), which is useful for laser design. In a side-pumped DPAL, a relatively flat output profile [see Fig. 5(a)] is needed, which is better for beam quality and has lower requirements for the damage threshold of optical elements. And a relatively symmetrical beam shape (0:1 × 5 cm in this example) can benefit the beam operation and application. Figure 5(b) shows the population distribution along the cell width, which decides the distribution of fluorescence and thermal powers [see Fig. 5(c)] that is calculated by Eqs. (9) and (10). A relatively uniform distribution of fluorescence and thermal powers is required for easy and efficient heat management. For the approximate nature of the discretization in the y direction and its importance to the model, we have studied the influence of division numbers (W =Δy) on calculated results. From Fig. 6(a), we see that the efficiencies (ηabsorb , ηopt-opt , etc.) of the whole laser (not of a single volume segment) and variations (V laser , etc.) are not sensitive to division numbers. But for intracavity information (e.g., laser intensity),
Fig. 4. (Color online) Evolution of spectrally resolved pump intensities (cell dimension 0:1 × 5 × 8 cm).
different division numbers will lead to different results. As shown in Fig. 6(b), the laser intensity distribution for five subsections (triangles) deviates obviously from that calculated for 100 subsections (solid curve). But the total output laser P power, P laser ¼ W y¼0 P laser ðyÞΔy, is nearly the same for these two cases (the area covered by the dotted line for 5 subsections and the solid curve for 100 subsections), which can explain why the efficiencies and variations are not sensitive to division numbers. We can also see that for 50 or 100 subsections, the laser intensity distribution is nearly the same. So if we want to obtain only the macroscopical information, e.g., efficiencies or variations, we can set only a few division numbers for a fast and accurate calculation, and if we want to get the detailed intracavity information, a relatively greater division number is needed.
Table 1. Comparison with Calculated Results Quoted from Published Literature [16] H × W × L (cm)
0:1 × 5 × 4 cm
0:1 × 5 × 6 cm
0:1 × 5 × 8 cm
2
3
4
P pump (kW) Model ηabsorb (%) ηopt-opt (%) V laser (%)
K’s Model 76.3 55.7 16.6
Our Model 65.6 41.1 19.4
K’s Model 82.4 61.2 22.9
Our Model 74.7 50.2 31.8
K’s Model 85 63.5 26.5
Our Model 78.2 53.7 38.1
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Fig. 6. (Color online) Influence of division numbers on calculated results (cell dimensions 0:1 × 5 × 8 cm).
Fig. 5. (Color online) Result of some intracavity information (cell dimensions 0:1 × 5 × 8 cm). In (c), each step corresponds to the power in one volume segment.
B. Comparison with Zweiback and Komashko’s Experiment We have studied the main difference between our model and K’s model above. To make a much more effective validation of our model, a comparison with Zweiback and Komashko’s experimental results [17] is made in this section. The experiment has successfully demonstrated a high-energy side-pumped double-pass rubidium vapor laser. In addition, the pump source (Alexandrite laser) is in pulsed mode (330 ns pulses at 10 Hz), so that the alkali vapor laser can actually reach steady-state operation to represent a CW system without thermal load. Thus, our model is appropriate to use here.
We have set the parameters according to the experimental conditions: L ¼ 13 cm, W ¼ 7 cm, H ¼ 1 cm, T ¼ 135 °C, P methane ¼ 2:7 atm (20 °C), Rback ¼ 99%, Rside ¼ 90%, Roc ¼ 30%, Δvp ¼ 0:2 nm, pulse duration τp ¼ 330 ns, pulse energy, Ep E p , pump power P þ p ð0Þ ¼ τp , T w ¼ T s ¼ 96:55% to get a single-pass transmission, T 2w T s ¼ 90%. Some basic physical parameters for interaction between rubidium atoms and methane molecules are chosen from the published literature: fine-structure mixing cross section σ 32 ¼ 42 Å2 [20], shifting and linewidth broadening coefficients for the D2 line δCH4 ;D2 ¼ −7:00 MHz=torr, γ CH4 ;D2 ¼ 26:2 MHz=torr, for the D1 line δCH4 ;D1 ¼ −7:92 MHz=torr, γ CH4 ;D1 ¼ 29:1 MHz=torr [21]. Because these values are measured at 394 K, a temperature modification for the pffiffiffi ffi linewidth broadening coefficient is made by including a T factor [14], and the linewidth of the D2 line (similar to the D1 line) is ΔνD2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T ¼ þ 26:2 ðMHz=torrÞ τ D2 394 ðKÞ × P methane ð20 °CÞ
T : 293:15 ðKÞ
ð17Þ
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On the other hand, some deviations exist between the experimental data and the model’s predicted results. In Fig. 8, the calculated output energies are a little higher when methane pressure is below 1:5 atm, and the maximum output energy occurs at a methane pressure of 2:3 atm rather than the reported 2:5 atm in the experiment. As methane pressure increases further, the output energy will experience a slightly decrease due to the quenching effect. Some reasons exist for these deviations. First, the quenching cross sections for both 2 P 1=2 and 2 P 3=2 levels we have used is the upper boundary of the 2 P 1=2 energy level that measured 40 °C, which may differ from actual values in the experiment. Second, the mode overlap factor is not measured but inferred for a better fit between calculated and experimental results; maybe an adjustment of both the quenching cross sections and mode overlap Fig. 7. (Color online) Calculated result for output energy as a function of input energy [P methane ¼ 2:7 atm (20 °C)].
For shifting coefficients, no temperature modifications are made, and the pump center wavelength is set equal to the shifted D2 line center. The upper boundary of the methaneinduced quenching cross section for the 2 P 1=2 level is reported as σ q21 ≤ 0:019 Å2 [22], and we assume quenching cross sections for both 2 P 1=2 and 2 P 3=2 levels the same as this value: σ q21 ¼ σ q31 ¼ 0:019 Å2 . The mode overlap factor is not mentioned in the experiment; by adjusting this factor to get a better fit with experimental data, we set this value as ηmode ¼ 89%. For a relatively large natural abundance, we use Rb85’s formula to calculate for the rubidium vapor pressure [23]: log10 P v ðtorrÞ ¼ 2:881 þ 4:312 −
4040 : T
ð18Þ
By using all the parameters set above, we have calculated the output energy as a function of input energy (Fig. 7) and methane pressure (Fig. 8). As compared with the reported experimental results (the original Figs. 3 and 4 in [17]), we see that the modeling results have made a good agreement with the experimental data, which has greatly encouraged us to ensure the validity of our model.
Fig. 8. (Color online) Calculated result for output energy as a function of methane pressure (Ep ¼ 49 mJ).
Fig. 9. (Color online) Temperature influence on laser characteristics.
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factor can give a better predicted result. Third, in our model we assume that the intensity of the pump beam is uniform at the entrance plane, which may not be satisfied in the experiment. In addition, some uncertainty about the interaction parameters between rubidium and methane may also affect the calculated results. As a whole, the agreement between the calculated and experimental results by using the reported experimental conditions and basic physical parameters quoted from the published literature has demonstrated the validity of our model that can be used for the design of side-pumped DPALs. C. Factor Influence on Laser Characteristics For the validity demonstrated in Section 3.B, we can extrapolate our model to study the influence of some important factors that affect laser characteristics, including operation temperature, dimensions of alkali vapor cell, pump linewidth, buffer gas pressure, etc., which are important for the laser design. Taking rubidium as the gain medium, the parameters are set as follows: L ¼ 4 cm, W ¼ 3 cm, H ¼ 1 cm, T ¼ 120 °C, P methane ¼ 400 torr (20 °C), P He ¼ 2 atm (20 °C), Δλpump ¼ 0:2 nm, I pump ¼ 5 × 103 W =cm2 , Rback ¼ 99%, Rside ¼ 99%, Roc ¼ 30%, T w ¼ T s ¼ 98%, ηmode ¼ 95%. Here we assume the narrowed linewidth for multikilowatt LD stacks as
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0:2 nm, which will be very possible to realize in the near future by the fast development of volume Bragg grating and LD technologies. When a parameter is changed, others remain constant, as above. Figure 9 shows the influence of temperature on laser characteristics. It can be seen that as temperature increases, ηabsorb increases, but ηopt-absorb decreases, due to the significant increase of ηfluorescence [see Figs. 9(a) and 9(b)]. As a result, an optimal temperature exists that is similar to that in the end-pumped configuration. ηthermal is always kept below 2% due to low quantum defect. Figure 9(c) shows that V laser and V thermal increase dramatically as temperature increases. When the temperature exceeds a certain value (∼140 °C), some parts of the gain medium may be pumped below threshold and no laser may exist (V laser ¼ 100%). In the design of DPALs, ηopt-opt and V laser should be optimized synthetically. In the design of the dimensions of the alkali vapor cell, the length is mainly decided by the column numbers of LD stacks, each of which has a width of about 1 cm. For example, a cell with a length of 10 cm may contain about 10 columns of LD stacks, and the power (or height) of each column can be varied according to the required total power. By adjusting the cell height as well as the focusing optical system, the pump intensity can be varied at a constant total
Fig. 10. (Color online) Pump intensity influence on laser characteristics at optimal operating temperature.
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pump power. Figure 10 shows the influence of pump intensity on laser characteristics. All the results are calculated under the optimal temperature (T optimal ) to get a maximal ηopt-opt . We can see that a strong pump intensity is needed (>4 kW=cm2 ) to obtain high ηopt-opt (>60%) [see Fig. 10(a)]. As a three-level laser, a relatively high pump intensity is necessary both to realize population inversion and to make the stimulated emission process dominant over the fluorescence process [see Fig. 10(b)]. The optimal temperature increases as pump intensity increases [see Fig. 10(d)], because higher pump intensity needs more alkali concentration for sufficient absorption. A disadvantage is the large V laser and V thermal [see Fig. 10(c)], and a tradeoff should be made between efficiency and variation by adjusting the operating temperature. Another important design consideration is the width of the alkali vapor cell. Figure 11 shows the influence of the cell width on laser characteristics. We can see that as the cell width increases, ηopt-opt has a slight decrease, due to the increase of ηfluorescence [see Figs. 11(a) and 11(b)]. Other efficiencies (ηabsorb , ηthermal , ηscatter ) and variations (V laser , V fluorescence , V thermal ) stay nearly constant. Thus, at optimal temperature, the laser characteristics are not sensitive to cell width. As for the optimal temperature, we can see that it decreases ob-
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viously as the cell width increases [see Fig. 11(d)]. For a low temperature that will benefit the system’s lifetime (a high temperature may induce a reaction between hydrocarbons and alkali atoms), the cell width should be designed to ensure the optimal temperature safe for a longtime laser operation. A crucial factor for a highly efficient DPAL is to ensure the spectral matching between the diode pump source and the atomic absorption band. Figure 12 show the calculation results when the pump linewidth is set as 0:1 nm. We can see that about 1 atm helium (additional 400 torr methane which also contributes to the absorption linewidth broadening) is enough for the maximal ηopt-opt (∼70%) [see Fig. 12(a)]. When the helium pressure increases further, no obvious ηabsorb is obtained, but ηfluorescence dramatically increases [see Fig. 12(b)], and, as a result, ηopt-opt finally decreases. The reason is that at high helium pressure, although the absorption linewidth is broadened, the peak stimulated emission cross section, σ 21 , is reduced at the same time. Although ηabsorb remains high, the reduced σ 21 will weaken the capacity for fluorescence suppression, and will finally reduce ηopt-opt . At a higher pressure of helium, the optimal temperature increases dramatically [see Fig. 12(d)]. In addition, for chemical safety between alkali atoms and hydrocarbons, which requires a relatively lower temperature, the buffer gas pressure should not be too high.
Fig. 11. (Color online) Cell width influence on laser characteristics at optimal operating temperature.
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Fig. 12. (Color online) Helium pressure influence on laser characteristics at optimal operating temperature (Δvp ¼ 0:1 nm).
Furthermore, a relatively low buffer gas pressure can benefit the laser beam quality and flow structure design. Figure 13 shows the helium pressure influence at the optimal temperature for different pump linewidths. We can see
clearly that as pump linewidth increases, the maximal ηopt-opt decreases even at the optimal pressure and temperature. For a uniformly spaced increase of pump linewidth (0:1 nm → 0:2 nm → 0:3 nm), the optimal helium pressure also experiences a uniformly spaced increase (1:5 atm → 2:5 atm → 3:5 atm) for the optimal spectral matching. On the other hand, if the pump linewidth is too narrow, the stability of laser performance will also suffer by the center wavelength drifting of the diode pump source [24]. Until now, a 1:28 kW line-narrowed LD stack with a 0:35 nm linewidth (70% power in-band) is realized by use of VBG for rubidium laser pumping [10]. We believe that, with the fast development of LD and VBG technologies, high-power laser diodes with even narrower linewidth and more stable performance could be realized in the near future.
4. CONCEPTUAL POWER-SCALED DESIGN FOR A 1 MW SIDE-PUMPED RUBIDIUM DPAL WITH A FLOWING MEDIUM
Fig. 13. (Color online) Helium pressure influence on ηopt-opt for different pump linewidths at optimal operating temperature.
A DPAL with mutually orthogonal pump, laser, and flowing directions has many attractive features and can be scaled to extremely high power. For an ideal side-pumped flowing DPAL, some standards should be met: sufficient pump power
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Table 2. Power-Scaled Parameters for a 1 MW Side-Pumped DPAL with a Flowing Gain Medium Design Parameter Temperature Rubidium atom density Cell dimensions (L × H × W) Methane pressure (20 °C) Helium pressure (20 °C) D2 line absorption FWHM Pump power Pump intensity Pump center wavelength Pump FWHM Pump reflectivity for double pass Single-cell window transmission Other one-way scattered loss Output coupler reflectivity Laser back mirror reflectivity Mode overlap factor
Value
Simulation Result
Value
107 °C 9:34 × 1012 cm−3 50 × 5:5 × 5:5 cm3 400 torr 2:5 atm 0:13 nm 1 MW 3:7 kW=cm2 780:2 nm 0:2 nm 99% 99% 1% 30% 99% 96%
Output laser power Output laser intensity Optical-to-optical efficiency Pump absorption efficiency Fluorescence efficiency Thermal efficiency Scattered efficiency Laser power variation Fluorescence power variation Thermal power variation Gain medium flow velocity Maximum temperature increase Temperature variation Buffer gas heating power Reynolds number Dynamic temperature variation
0:61 MW 21 kW=cm2 61% 72.5% 7.9% 1.3% 2.3% 37.7% 1.1% 35.8% 1 m=s 6:1 °C 34.3% 6300 W 1717 9:6 × 10−5 °C
absorption and high optical-to-optical efficiency, high beam quality, relatively uniform laser, fluorescence, and thermal power distributions, relatively symmetrical output laser shape, proper flow rate to remove the waste heat efficiently while no turbulence is induced, etc. To study the possibility of a flowing DPAL that can meet all these requirements, a conceptual megawatt-class side-pumped DPAL with a flowing gain medium is designed and moderately optimized. The parameters are listed in Table 2. It is designed to use 50 columns of LD stacks to produce a total pump power of 1 MW. Each column provides 20 kW of power, which can be realized by using four 5 kW or two 10 kW vertical LD stacks. For an LD stack with a width of about 1 cm, the length of the alkali vapor cell is thus designed to be 50 cm. The height of a column is approximately 45 cm (for two 10 kW LD stacks, Lasertel, USA [25]), and, to obtain the required pump intensity, we design the cell height as 5:5 cm to get a moderate pump intensity of 3:7 kW=cm2 by using a focusing optical system with a demagnification factor of 8. To get a relatively symmetrical laser beam shape, the cell width is also designed to be the same as the cell height. The linewidth of the diode pump source is assumed to be 0:2 nm, which is promising in the near future. Although methane itself can provide both a sufficient fine-structure mixing rate and an even higher collisional broadening rate than helium (γ CH4 ;D2 ¼ 26:2 MHz=torr and γ He;D2 ¼ 20 MHz=torr [21]), we choose 400 torr methane and 2:5 atm helium as buffer gases instead of methane only. This is because 400 torr methane is already sufficient for the required fine-structure mixing rate, and a high pressure of hydrocarbon may cause a reaction with the alkali atoms and decrease the system’s lifetime. Under these conditions, the absorption linewidth is broadened to 0:13 nm (FWHM) at an operating temperature of 107 °C, which results in a relatively high pump absorption efficiency (72.5%). The pump absorption efficiency (also the optical-to-optical efficiency) can be further increased by increasing the temperature, but the laser as well as the thermal and temperature variations also become very large, which will increase the difficulty of heat management and subsequent laser beam operation. Thus, a tradeoff between pump absorption efficiency and laser variation (∼38%) is made for a synthetically optimized result. In addition, a relatively low operation temperature
(107 °C) is also necessary for the chemical safety of the gain medium. A large mode overlap factor is crucial for highly efficient DPALs, which can be obtained by using an unstable resonator; this is important not only for the overall efficiency, but also for the reduction of fluorescence power, which is partly produced by the unlasing pumped gain medium. The fluorescence will heat the surrounding material, which may affect system performance. Based on the symmetrical beam shape, the output laser intensity is calculated as 21 kW=cm2 , which can be endured by active-cooling mirrors. The steady temperature increase in the flowing process is roughly estimated by Eq. (19) [13]: ΔTðy; y þ ΔyÞ ¼
0:4ηQ−defect T þ · ½I þ p ðyÞ − I p ðy þ ΔyÞ P He v þ I −p ðy þ ΔyÞ − I −p ðyÞ;
ð19Þ
where ηQ−defect ¼ 1:9% is the quantum defect for Rb, P He the pressure of helium, and v the flow velocity. For a rough estimation, the pressure of methane is ignored. The results show a maximal temperature increase of 6:1 °C, with a temperature variation of 34.3%, which is nearly the same as the laser and thermal variations. In the cell structure design, if the cold finger of rubidium is set apart from the lasing volume, that is to say, the temperature increase will not affect the alkali concentration, such a degree of temperature increase can be endurable. For a flow velocity of 1 m=s, the Reynolds number is below 2000, which means no turbulence in gain medium. In addition, a relatively low-flow velocity will also reduce the burden on the gas heating system when a recycling system is considered. In this example, to heat the gas to the required temperature (107 °C), about 6300 W of heating power is needed, which is easily realized. The temperature increase is approximately linearly distributed (see Fig. 14), which will result in a linearly distributed index of refraction. With no turbulence, the phase shift of the output laser will be pure tilt, which can be easily corrected. The dynamic temperature variation and the dynamic density variation can be ignored, which is important for a good beam quality. As a result, a high-power-scaled side-pumped flowing DPAL with 0:61 MW output power and over 60% optical-to-optical efficiency can be designed theoretically with all the other
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Fig. 14. (Color online) Temperature increase along the width of flowing gain medium.
parameters reasonable and available in the near future, which demonstrates the great potential of the DPAL in extremely high-power laser operation.
5. CONCLUSIONS In this article, a model of DPAL in a single-side-pumped double-pass configuration is set up, which has considered nearly all the main physical factors of alkali lasers. In the longitudinal dimension, the successful model proposed by Beach et al. for end-pumped configuration is applied with some modifications. In the transverse dimension, an iterative algorithm is proposed with a high degree of accuracy and a fast speed of convergence. The difference between our model and Komashko and Zweiback’s model is studied. A comparison with Zweiback and Komashko’s experimental results is made, and the agreement between the model’s predicted results and the experimental data demonstrates the validity of our model. Some important influencing factors are studied, which are useful for laser design. Finally, a power-scaled design for a megawattclass side-pumped flowing DPAL is made; the simulation result shows an optical-to-optical efficiency over 60% with all the other parameters reasonable and available in the near future, which demonstrates the great potential for DPAL in extremely high-power laser operation.
11. 12. 13. 14.
15. 16. 17. 18. 19. 20.
21.
22.
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