The kinetic process of Cu+ UV laser in Ne-CuBr longitudinal pulsed discharge is analyzed and a comprehensive self-consistent physical model is developed. The temporal evolutions of discharge parameters, main particle densities, the electron temperature, and the laser pulse intensity are numerically calculated. The model results illustrate the process of population inversion and the lasing mechanism. The calculations on the influences of the tube radius and Br atoms on the laser output characteristic well explain the experimental results.
©2006 Optical Society of America
Lately, there has been sustained interest in the longitudinal pulsed discharge Cu+ ultraviolet (UV) lasers, particularly in the 248.6nm line, which is very close to the 248.5nm laser line of the KrF excimer laser. The UV Cu+ laser with 1W order average power can be used in microelectronics, microbiology and genetic engineering. Although the UV Cu+ laser in a hollow cathode discharge (HD) was firstly reported in 1976 , and eight Cu+ laser lines were obtained in the 248.6–270.3-nm spectral range in HD [2–5], this kind of laser has disadvantages of instability of discharge, serious cathode erosion, short lifetime of several hours , and lower output power. So it is hard to develop a practical laser device for real applications. An important progress was made for the first time when laser oscillation was obtained on four Cu+ lines, 248.6 nm, 252.9 nm, 260.0 nm, and 270.8 nm in Ne-CuBr with a nanosecond pulsed longitudinal discharge . This discharge tube is the same as the steadily operated CuBr laser, which has the long lifetime about 1000 hours and high output power . Based on the similar technical method, it is promising to develop a long lifetime and high power Cu+ UV laser in Ne-CuBr longitudinal discharge. Therefore, in recent years, many investigations have been conducted on the Cu+ UV lasers [9–12]. On the other hand, due to the very low gain of UV Cu+ laser medium, compared to the visible 578.2nm and 510.6nm of the atomic copper laser, the refined experimental conditions are needed to obtain laser oscillation. Although many experimental attempts have been made to improve the output characteristic of Cu+ UV laser in Ne-CuBr discharge, the comprehensive computer model for the Cu+ UV laser, as an integral part of experiment and theory hasn’t been developed up to now. There for, it is necessary to establish a computer model to study the kinetic process of this laser in order to provide the detailed information of the mechanism and operation parameters to the experiment.
Computer model has been used to simulate the temporal behaviors of the microscopic parameters, and provide the detailed information on kinetic processes in the discharge plasma where experimental techniques cannot be used [13–17]. The model of self-terminating copper and copper halide vapor laser has been well developed [15, 16, 18], and the results predicted by the model agree well with the experiment. Based on our previous model on copper vapor laser , this paper describes a self-consistent model to simulate the kinetic behavior of UV Cu+ laser in Ne-CuBr pulsed longitudinal discharge.
A description of the model, including the external excitation circuit, main particle species, the electron temperature, and the intracavity laser intensity is given in Section 2. In Section 3, the results predicted by the model are compared with the experimental data and discussed in detail. A summary of the results is given in Section 4.
2. Description of the model
The model consists of an excitation circuit and a comprehensive description of the UV laser kinetics in Ne-CuBr discharge plasma. We choose the interactive circuit (IC) as the discharge excitation circuit [7, 19], which has the characteristics of compressing the width of the discharge current pulse and improving the current rising rate. The assumption that the discharge tube is longitudinally homogeneous with respect to all particle densities was used, and Maxwellian distribution was used to describe the electron energy distribution function (EEDF) in calculating the rate coefficient according to the impact cross section.
The partial Cu, Cu+ energy levels and corresponding laser radiation, together with Ne atom energy level are shown in Fig. 1(a) and Fig. 1(b). Eleven copper atomic energy levels and ten ionic levels were considered in the model. From the ionic energy levels, one can see that the upper laser level 5s3D1 lies closely to the Ne+ ground state with a small energy difference (~0.2eV), so it can be easily populated
via charge exchange reaction between Cu and Ne+ ground states (Eq. (1)). Analogously, the lower laser levels 4p states can be populated via penning collisions between neon metastable level Nem and copper ground state Cu0 because of the close energy level (Eq. (2)). Meanwhile, these levels can be mainly depopulated by UV spontaneous emission to the 4s Cu+ metastable levels (Cu+m) with fast transition probability.
Three neon levels are included: the ground-state Ne, metastable state Nem, and ion ground-state Ne+. The kinetic process of laser oscillation is focused on the 248.6nm in this model for economy. Accordingly, the varying rate equations of the species populations are given below.
2.1 Copper atoms
2.2 Copper ions
where Cu+m, Cu+l, Cu+u are the population densities of the Cu+ metastable level(4s), the lower (4p 3) and the upper (5s 3D1) laser levels of 248.6nm, respectively. The electron density ne is inferred from the sum of the local ion densities assuming that the degree of ionization and space-charge fields are sufficiently high to ensure charge neutrality throughout the bulk of the gas.
The subscripts i and k refer to processes involving lower and higher energy levels in relation to the energy level of a given species, κ and γ are the respective rate coefficients for excitation (or ionization) and deexcitation, α is the recombination coefficient, Ai is the corresponding transition probability for radiative decay, and Kp 1, Kp 2 are the rate coefficients for Penning collisions. The first term in Eq. (5) represents the charge exchange reaction between Ne+ and Cu atom ground state with the rate coefficient KCT . Stimulated emission process of 248.6nm laser line is represented in the Eqs. (6) and (7) with the coefficient B, where Ip is the intracavity photon flux. Particle diffusion is represented throughout by the divergence of the radial particle flux Γ.
2.3 Intracavity laser photon flux Ip
The continuity equation for the photon flux at 248.6 nm is given by
Where R 1 and R 2 are the cavity mirror reflectivities, c is the speed of light, and Lc is the cavity length. The first term represents stimulated emission or absorption in the plasma with gain γ 0, the second term gives the losses at cavity mirrors, and the third term denotes the spontaneous emission from which the laser field builds up, where A 1 is the transition probability from the transition at 248.6 nm and dΩ/4π is the solid angle.
2.4 Energy conservation
The equation used to solve the electron temperature is given by
The first term is for the energy gained from the longitudinal electric field E, the second term represents heating of electron gas by superelastic collisions, and the third term takes into account heating by fast electrons liberated in Penning collisions, and the fourth term refers to heating via helium metastable-mestastable ionizing collisions whereby a fast electron is liberated with energy ε m , and the fifth term represents energy loss due to elastic collisions with the neutral-ions gas, while the sixth term denotes the energy losses associated with inelastic electron collisions with atoms or ionic species.
Equations (3)–(9) and the equation of electron density as well as circuit equations given in  and Cu atom rate equations described in  constitute a set of coupled first-order differential equations of the model. The standard initial conditions used in this model (Table 1) such as neon pressure, charge voltage, circuit parameters, and the density of ground state copper atoms are the same as those from a recent experimental study . The model of 56 first-order coupled ordinary differential equations is solved by the method of Backward Differentiation Formula (BDF). A set of initial conditions is chosen to begin the calculation after calculating for 10 excitation-afterglow cycles to reach the condition of temporal self-consistency. The rate coefficients used in the model are cited from [13–18]. Thus, the lasing process parameters including population densities, the discharge current, the lasing pulse, the electron density, and the electron temperature can be acquired.
3. Simulation results and discussion
3.1 Discharge current pulse and electron temperature
The discharge current pulses of calculation and experiment are shown in Fig. 2. The raise ratio of the front edge and the magnitude of current pulse mainly influence the electron temperature Te in the excitation period. As can be seen, the model result agrees well with the experiment, and the accuracy of current pulse can give an accuracy value of Te [15, 16]. The difference of current back edge comes from the complex plasma resistance expression due to the large number of particles after the excitation pulse, but it rarely influences the maximum of the electron temperature Te, and the difference will disappear in the afterglow. Figure 3 gives the evolution of the electron temperature Te. Here, the maximum value of Te reaches about 3.2eV in the excitation pulse, which is slightly lower than those in Ne-Cu discharge plasma  because of the existence of vast Br atoms and ions.
3.2 Kinetic process of UV Cu+ laser
During the early discharge pulse, the current pulse I and the electron temperature Te all reach its maximum, and neon and Cu atoms are rapidly ionized predominantly by stepwise electron collisions via the excited states. So the electron density and all ions reach a max value in the excitation pulse. With the end of the discharge, Te begins to decrease quickly and the deexcitation and recombination processes will predominantly occur, as a result, the densities of all particles have a decreasing process in the afterglow pulse. Figure 4 represents the calculated temporal behavior of the major ions and the electron density at tube center in the discharge period. It is emphasized that the density of Ne+ decreases evidently faster than other ions in the afterglow. This difference mainly attributes to the charge exchange reaction between Ne+ and the Cu ground state, which has the higher density in the afterglow. As has been described above, the upper laser level of 248.6nm is pumped mainly by the charge exchange reaction between Ne+ and Cu ground state in the discharge afterglow. As shown in Fig. 3 and Fig. 4, about 1µs later after the excitation pulse, the electron temperature has fallen to about 0.3–0.5eV, the excited state copper species begin to deexcitation and the copper ground state begin to rise to its initial density. With the increase of copper ground state, together with the sufficient high density of Ne+ and Neon metastable level Nem, the upper and the lower laser levels begin to be populated via the charge exchange reaction and Penning collision excitation in afterglow. Such as shown in Fig. 5, the upper and the lower laser levels reach a maximum density at about 1µs, and then begin to decay due to the spontaneous emission. Until about 3.5µs, the gain coefficient and the relative inversion reach maximum value and the laser pulse occurs. Between 3.5–13µs, the gain coefficient and inversion density hold on a steady value, as a result, the laser pulse duration can last about 10 µs. These modeling results agreed well with the experimental observation in . With the depletion of Ne+ due to charge exchange reaction, the laser pulse intensity begins to decrease till disappear.
3.3 Depopulation of the lower laser level
The lower laser level can be pumped by electron impact in early discharge pulse and by Penning collision via Nem in the afterglow. While the depopulation process of the lower level is mainly through the UV spontaneous emission to the Cu+ metastable levels Cu+m. Although this process is an efficient way to the deexcitation of lower laser level, the high density of Cu+m in the afterglow will lead to a remarkable radiative trapping effect between the lower laser level and Cu+m levels, which will increase the density of the lower laser level. As a result, the density of inversion population and the laser output power will decrease significantly. Only if the radiative trapping effect is quenched or destroyed effectively, both the laser output power and efficiency can be improved. A theoretical analysis on how to restrain the radiative trapping effect is given below.
According to Holstein’s theory , in a cylinder of plasma of radius R, the effective transition probability Aeff can be written by
Where k 0 is the absorption coefficient at line center. As discussed in , Aeff is strongly affected by the radius R of laser tube. In this case, the depopulation of lower laser level is mainly through UV spontaneous emission with the coefficient Aeff , so the laser output characteristic is influenced strongly by the laser tube radius. Figure 6 shows the dependence of the photon (corresponding to the laser output power) and Cu+m densities on different radius R. As can be seen, the maximum photon density almost doubled when the radius decreased from 8mm to 6mm. It also shows that the laser pulse delay decreased from 2µs to 1.5µs, although at which time the density of Cu+m increased about 50% from 5×1012 to 7.5×1012. These varieties in microcosmic parameters clearly show that the negative influence of radiative trapping on depopulating the lower laser level was effectively restrained with the diameter reduction. All these results of the model and the experiment in  show that the delay of the laser pulse will be shortened, and the laser output characteristic will be improved well with the decrease of tube radius R.
3.4 Influence of Br on the laser output characteristic
The copper atoms in CuBr Cu+ UV laser are produced mainly by electron impact dissociation of the CuBr molecules during the discharge period. In the high repetition frequency discharging, CuBr vapor is almost fully dissociated and hardly recombined in one discharge cycle due to its longer recombination time. Br atoms would reduce the maximum value of electron temperature Te (~3.5eV) in the Ne-CuBr plasma, compared to 6eV in the Ne-Cu plasma. However, the existence of Br atoms has the advantage of enhancing laser output power for UV Cu+ laser due to its own mechanism. As seen in , four Br+ levels are located closely to the Cu+ metastable levels. These levels can be populated mainly by Penning collision with neon metastable levels Nem due to the high excitation energy. A preferential condition is that the rate constant for Penning ionization of Br+ is about ten times as large as that of the lower laser and metastable Cu+ levels . As a result, this process leads to the reduction of Nem and Cu+m densities and the increase of laser power.
Figure 7 represents the modeling results of the influences of Br atoms on neon metastable and neon singly charged ion densities. One can see from Fig.7, the process of Penning ionization of Br+ can strongly decrease the Ne metastable density in the discharge afterglow, while the density of Ne+ scarcely changes due to the affection of Br. The reduction of Nem decreases slightly the density of lower laser level Cu+l, which is mainly pumped by Penning collision between Nem and Cu ground state. As a result, the inversion population density increases evidently and the laser power has a marked enhancement by about 50%, such as shown in Fig. 8.
Based on the physical mechanism of Cu+ UV laser, a self-consistent computer model was developed to simulate the discharge kinetics and Cu+ UV lasing characteristic in Ne-CuBr longitudinal pulsed discharge. The temporal evolution of population densities of main particles in Ne-CuBr discharge plasma was calculated. The model results agreed well with the experimental data by comparing the electrical I-V characteristic of discharge tube. The excitation mechanism of the upper laser level and the lasing process were clearly predicted by the calculation. It was shown that the charge exchange reaction between Cu ground state and Ne+ was the dominant mechanism for the excitation of the upper laser level. When Helium was used as buffer gas in experiment, Cu+ UV laser cannot be obtained . Due to the marked negative influence of radiative trapping on depopulating the lower laser level, the laser output power will increase with the reduction of tube radius. Model results also predicted the influence of tube diameter on the restraint of radiative trapping, and explained clearly the enhanced effect of Br atoms on the laser output power observed in . Thus, the model can provide an effective way to improve the Cu+ UV laser output characteristic in experiment.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 10374081, 10574111), the Inertial Confinement Fusion Research Foundation (Grant No. 2004AA84TS04), and China Postdoctoral Science Foundation (Grant No. 2004036482).
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