1. Introduction

A multipass amplifier of laser drivers for inertial-confinement fusion [1, 2] is to use an optical switch to trap an optical pulse within a laser cavity and divert the pulse out of the cavity when it reaches the required energy. The optical switch consists of a plasma electrode Pockels cell (PEPC) and a polarizer. By rotating the polarization of the beam, the PEPC controls whether the beam is transmitted through or reflected from the polarizer.

The operating principle of PEPC is based on the electro-optic effect of the longitudinal Pockels cells using a KDP crystal plat that is intrinsically fast. A high voltage larger than half-wave voltage of KDP, which is equal to16.4kV at wavelength of 1.06μm, is applied across KDP plat through plasma electrodes which are produced by a high current discharge. The luminescence of the plasma generated by helium discharge mainly is in the range of visible light, and cannot cause any optical problems to the laser at the wavelength 1.06μm.

These are two operating modes in PEPC. One is Two-Pulses Process (TPP). In TPP, the first pulse produces the discharge plasma when a voltage is applied between the electrode pairs by a plasma pulse generator and a high discharge current raises the plasma density to about 10¹²cm^-3, and then the second pulse drives charged particles to the surfaces of KDP plat to form transparent switching electrodes. Another operating mode of PEPC is One-Pulse Process (OPP) that was been proposed by J. Gardelle et al recently [3]. In OPP, the discharge plasma production and the plasma electrodes forming processes are finished in one pulse.

In reference [4], C. D. Boley and M. A. Rhode had modeled the plasma behavior for TPP of PEPC. In their model, static velocities equations for electrons and ions, static continuity equation, and energy equation for electrons were solved. Their static model is appropriate for TPP of PEPC because that a stable gaseous discharge plasma with a constant electron temperature has been produced after the first pulse. In OPP, the discharge plasma evolving and charging on the surfaces of KDP crystal are time dependent and have not been modeled so far. This paper presented a modeling for the OPP of PEPC in the following sections. In section 2, we describe a one-dimensional simplified model, which is based on a set of fluid equations: electron motion and ion motion governed by the equations of continuity, momentum conservation, mean electron energy equation and electric charging equations of charged particles on the surfaces of KDP crystal. The electric field distribution in the discharge cell was obtained by Poisson’s equation. In section 3, the time-dependent discharge current, charging process on the KDP crystal and switch efficiency were simulated. Finally, in section 4, we present conclusions.

2. Model description

Figure 1 shows a sketch of a PEPC which sandwiches a KDP crystal between two helium cells with fused silica windows forming the external seals. Within each gas cell, there is an electrode (cathode or anode) for discharge. When the cell is applied a pulse, the helium gas filled in the cell breaks down, high electrons and ions densities in the cell produce transparent plasma electrodes on the KDP crystal. The switch is open while the voltage of the plasma electrodes reaches half-wave voltage of the KDP. If the voltage of the pulser drops to zero volt, the electron and ion densities decay exponentially through recombination, and the switch closes rapidly.

 

Fig. 1. Sketch of PEPC optical switch (in cross section) driven by one-pulse process

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Our model consists of the fluid model of continuity equations for electrons, helium ions. For simplicity, we consider a one-dimensional model of OPP that is very adequate for large-aperture optical switches because that diffusion of charged particles to the walls perpendicular to the direction of light passing can be neglected.

The transient continuity equations and momentum transfer equations for charged species are as following

where i = e,p are for electrons, helium ions, n_i, D_i, v_i, q_i and Si are number density, diffusivity, drift velocity, charge of charged particles and net production rate of electrons and ions, respectively. The production rates of electrons and ions are mainly determined by electrons collision that births new electrons and ions while the kinetic energy of the electrons are larger than the ionized potential of helium atom. The loss of electrons and ions comes mostly from recombination of electrons and ions in the discharge space. The net production rates of electrons and ions are given by

where n_e, v_e, and n_p are density of electrons, drift velocity of electron and density of ions, respectively. The Re = 2×10^-12cm³sec^-1 is the coefficient of electron recombination [5]. The α is the ionizing coefficient of electron, which is the number of ionization on one centimeter electron path along the direction of electric field, and it is given by

where P is the pressure of helium gas (in Torr), $\overline{\epsilon }$ is mean electron energy (in V) that can be obtained from the energy flux equation in one-dimensional case [6,7]

where Ui = 24.58V is the ionized potential and Ue = 21.45V is the excited potential of helium atom, δ_e = 0.5α is the coefficient for the production of excited species [7]. The energy losse in electron elastic colliding was ignored in Eq. (4) because that the elastic colliding loss is much smaller than inelastic colliding.

For the transport characteristics of electrons and ions, we use a simple local field approximation (LFA), in which the drift velocities are depended on the local value of |E|/n_g, where |E| is the magnitude of the local electric field and n_g is the density of helium gas that is in proportion to the gas pressure of P. The drift velocities of electrons and ions are given by [6]

where μ_e and μ_p are mobilities of electrons and ions, respectively. The diffusion coefficients of electrons and ions are estimated through Einstein’s relation, in which the ratio of diffusion coefficient to mobility is in direct proportion to temperature of charged particles.

On the other hand, the densities of charged species in the space influence the electric potential in the discharge cell, and the electric potential distribution should be solved by Possion’s equation

With the surface charge density, the potential normal derivative condition on the surfaces of KDP crystal is given by

The evolution equations for the surface charge density are obtained given by

where σ_e,σ_p, are electron and ion charge densities on the surfaces of KDP, respectively, v _de = 10sec^-1 [8] is the electron desorption frequency and γ = 0.01 is secondary electron coefficient of cathode surface.

The efficiency of the optical switch is defined by

where V_KDP is the voltage on the KDP crystal and V_SW is the voltage on the KDP and the plasma sheaths. C_KDP = 0.11nF is the capacitance of the 8×8×1cm KDP crystal, and C_sheath is the plasma sheath capacitance given by [1]

where S = 64cm² is the transverse area of KDP crystal and λ_D is the Debye length.

3. Simulation results and discussion

The transient continuity equations and momentum transfer equations are approximated with exponential differential scheme that had been successfully used in many glow discharge simulations [9]. The electric field distribution is solved by successive over-relaxation method at the same time. The electron energy equation and circuit equation are solved by forth-order Rounge-Kutta scheme. The discharge cell distance from cathode to anode is 3cm sandwiched a KDP crystal with a thickness of 1cm in the middle of the cell. The loss of charged particles in transverse direction is ignored in the one-dimension simulation for a large aperture switch. A optical switch with the aperture of 8×8 cm² and circuit impedances of Z₁ = Z₂ = 50Ω was simulated.

When voltage of the pulser is ±19kV, the switch takes on the best operating state. The simulation results are shown in Fig. 2–6. After a high current pulse followed helium gas breakdown, as shown in Fig. 2, the discharge current tends to 3A at which it maintains the charge density on the surface of KDP to a stable value. However, before the breakdown almost no current passes the cell, the gaseous cell works like a dielectric isolator although a high voltage is applied the cell. The electric field distributions in the discharge gap at several times are given in Fig. 3. Before the gaseous breakdown, the electric field in the cathode area is gradually increased at t = 1.5ns, 2.5ns, 3ns and after the breakdown (t = 4ns), the electric field is decrease distinctly and the discharge plasma appears a low impedance characteristics. Figure 4 shows the variation of the voltage on the KDP crystal that controls the switch state of open or close.

The switch efficiency is given in Fig. 5. When t > 27ns the KDP voltage reaches over 98% of the half-wave voltage of KDP and the switch efficiency could be larger than 99%. After the time past 40ns, the discharge tends to a stable glow discharge although the discharge path is obstructed from the KDP crystal and the current is very small. The electron and ion density distributions at the time of 40ns are shown in Fig. 6, which look like a plasma with same electron and ion density except near the cathode area [6].

 

Fig. 2. Discharge current

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Fig. 3. Distribution of electric field

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Fig. 4. Voltage on KDP crystal

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Fig. 5. Switch efficiency of the PEPC

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Fig. 6. Electron and ion density distributions

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4. Conclusion

One-dimensional model of the PEPC driven by one-pulse process is proposed in this paper. The discharge characteristics in PEPC are solved with continuity equations momentum equations coupled with Poisson’s equation, electron energy equation and external circuit equation. The model is very useful for the design of PEPC and prediction the behavior of the optical switch.