A Modified Approach for the Blumlein-line Laser Power Calculations : Electrical and Optical Power Waveforms

In this paper, a modified approach for output power calculations of the nitrogen laser system is reported. The power calculation is based on the distributed parameter model of the Blumlein-line circuit along with the decoupling approach of the laser rate equation from the electrical circuit equations. The general laser power assumption is considered in calculating the output optical power. The effect of the laser gap inductance on both the electrical and optical power waveforms is simulated and discussed. The theoretical work presented here is quite general and could be applied to many other fast discharges laser systems, such as CO2 and copper vapor lasers. Keywords— Blumlein-line; Fast discharge laser; Nitrogen laser; Power calculations.


I. INTRODUCTION
Nitrogen lasers are important because they can provide high-power short-duration pulses of ultraviolet radiation (λ = 337.1 nm).These lasers are widely used in spectroscopy and fluorescence studies, pumping of dye lasers and other research and industrial applications.
The performance of the nitrogen laser is basically determined by the type of electrical system used to create the discharge in the gas.Many excitation schemes were employed for pumping nitrogen lasers, however, the Blumlein transverse excitation method has become very popular, because of its low cost and ease of construction [1][2][3][4][5][6][7].
The Blumlein-line pulse-forming network consists of two parallel plate transmission lines (or coaxial cables) acting as energy storage capacitors, located at both sides of the cavity charged to high voltage Vo.When one side is short circuited, for instance using a spark gap, a transient voltage occurs across the laser cavity creating a gas discharge between the electrodes.
The spark gap and the laser gap are usually represented by resistances and inductances.Depending on the relevant time constants of the spark gap and laser gap, and the wave propagation time on the transmission line, two concepts can be used in the analysis of the Blumlein-line circuit, the lumped parameter model (LPM) and the distributed parameter model (DPM).From a theoretical point of view, LPM offers the advantage of a much simpler analysis over the DPM.However, it has the disadvantage of being valid only when the relevant time constants of the spark and laser gap are much larger than the wave propagation time on the transmission line [8].
In spite of extensive studies and investigations that have been made so far for understanding the performance of -lasers based Blumlein-line pulse forming network, still extra research work has to be made in this connection.This includes the selection of more accurate circuit models for simulating the laser system and also includes the effects of the electrical parameters on the overall laser performance.

The
Laser is a highly integrated electro-optical system and the prediction of the behavior of laser requires a complicated comprehensive theory.The theory must include the electric circuit parameters, the gas kinetic parameters and detailed mechanism of energy transfer of the three laser levels of the molecular nitrogen.A significant simplification of the calculation has been brought about by Fitzsimmons and others [9,10].Instead of estimating the electron temperature by solving the energy balance equation as a part of the coupled system, they used, in a decoupled procedure, another alternative approach to predict the electron temperature.Within this framework of the assumptions the electron temperature is determined in terms of the instantaneous value of ratio of the electric field between the electrodes of the laser channel divided by the pressure inside the laser channel.Hence this procedure makes it possible to handle the laser rate equations in separate way.This can be done after getting the required time development of the electric field from the solution of the electric circuit equations.Still most of the researchers' approaches treat the electrical circuit of the Blumlein-line as LPM in all of their calculations concerned with the electrical and optical powers.However, their approaches do not reflect the real waveforms especially when the relevant time constants of the spark and laser gap are comparable or much smaller than the wave propagation time on the transmission line [8].
This paper reports the theoretical analysis, power calculations and the effect of the laser gap inductance on the laser performance of the Blumlein-line laser circuit that based on the DPM and the general laser power assumption.

II. TRANSMISSION LINE EQUATIONS OF THE BLUMLEIN
AND THE BOUNDARY CONDITIONS A configuration of the Blumlein circuit is shown in Figure 1.The Blumlein line mathematical is divided into two sections, the right hand section and the left hand section with different zero coordinates.An applicable transmission line equations for the voltage V and current I on a section of the transmission line of length at any time t are given by [8], ( , and are the distributed inductance and capacitance per unit length respectively, and is the total length of line section.The initial conditions are: The boundary condition at the end of the spark-gap is: (5) ( stand for inductance and resistor of the spark gap, respectively.
is the zero coordinate of the right hand section transmission line of the Blumleinline circuit.
is the current passed in the spark gap inductance and is voltage across the spark gap.
is the total length of the right hand section transmission line.The boundary conditions at the end of the channel are: (6) (7) stand for inductance and resistance of the laser gap, respectively.
is the zero coordinate of the left hand section transmission line of the Blumlein-line circuit.
is the laser gap current and is the voltage at the left side edge of the laser channel.The boundary condition at the open end is: is the total length of the left hand section transmission line and is the current at the open end side.The laser gap impedance is not linear and must be treated as voltage dependent: as an open circuit before laser gap breakdown and represented by a combination of after laser gap breakdown.This nonlinear behavior of the channel impedance, makes it necessary, in the time development of solution of the partial differential Eq. ( 6) to distinguish between two time intervals.The first one is before laser gap breakdown, during which the state of only the righthand section is subject to the dynamical changes, whereas the left-hand section remains in its initial state.The second time interval is after laser gap breakdown, in which the states of both sides are time varying.The subscripts L and R in the current, voltage and lengths denoting for the left hand and the right hand sides of the Blumlein-line.

III. ELECTRON DENSITY EQUATION AND LASER RATE EQUATIONS
Using of Fitzsimmons' expression of point-by-point for relationship between the electron temperature , the instantaneous electric field E and the pressure P inside the laser tube [5], the electron density equation and the laser rate equations of the molecular nitrogen can be easily derived in term of (E/P).These equations are needed to obtain the states densities and the optical laser power.The Fitzsimmons' expression for the electron temperature, is given by [9]: K is the Boltzmann's constant.The electron density equation and the laser rate equations can then be written as: is the stimulated emission cross section ( ).Upon obtaining the electron density from Eq. ( 10), the above laser rate Eqs.( 11), ( 12) and (13) can then be solved.

IV. SELECTION OF THE SCHEMES AND COMPUTATIONAL PROCEDURE
To obtain voltage and current solutions across the laser channel, we applied numerical schemes that based on the finite difference method, the forward, backward and central difference schemes along with the Lax-Wendroff scheme to the partial differential equations of the Blumlein-line and the boundary conditions.Since Eqs. ( 1) and ( 2) represent a coupled system of linear hyperbolic equations, and in order to apply the above mentioned difference schemes, it is convenient to reduce these coupled systems to a normalized and uncoupled form and can be written as [8]: The system of Eqs. ( 1) and ( 2) under the application of the Lax-Wendroff scheme becomes: (16) = , and the different operators , and are defined as follows: The method and the schemes used were presented in detail in the previous work [8].Standard numerical techniques were used in solving the excitation rate equations and the electron density equation.

V. COMPUTATIONAL RESULTS AND DISCUSSIONS
Following the application of the numerical schemes mentioned in (IV) and simulation of the laser system, useful results were obtained.The values of circuit elements (shown in figure 1) that used in the simulation are: =128.88nF/m,= 0.288nH/m, = 0.1Ω, R S = 0.0366Ω, =14nH, (open end side) and (spark gap side).The parameter represents the characteristic impedance of the line and equals to , whereas the propagation time on the transmission lines is represented by and equals to .The chosen values of for the simulation are 3.2nH and 30nH.The electrodes separation used in the simulation is 1.5cm whereas their lengths and widths are 50cm and 0.5cm respectively.The peak value of (E/P) across the laser channel is taken to be in the range of 20 and 120 . The electrical power is obtained from the expression ( whereas the optical power is estimated from (13).In the waveform figures, the time is normalized by the propagation time on the transmission-line .
Figures 2 and 3 show the 3D voltage waveform variations at all points and at any time along the parallel plate Blumlein-line circuit using the presented DPM model.The low rates of the voltage variation in Figure 3 can be explained by the large time constant compared to the propagation time on the transmission-line .Similar 3D voltage waveforms simulations were carried previously [11].This effect on the voltage waveforms is clearly appeared especially during the after laser gap breakdown time.The channel current and the electrical power waveforms, and the effect of the laser-gap inductance on them are shown in figures 4 and 5.It is clear from figures 4 and 5 that the current and the electrical power waveforms are a highly dependent on the laser gap inductance.Figure 6 shows the waveform of the instantaneous (E/P) across the laser channel along with the influence of the laser-gap inductance on ∂ (E/P) ⁄∂t.Figures 7 and 8 show the waveforms of the density of the C-state, the B-state, and the optical power as well as the density difference between the C and B states in the laser channel.The density of the Cstate is appeared to be higher than that of the B-state for the first time span after the breakdown , then the density of the B-sates becomes higher for the rest of the time.This behavior, however, is due to both the short lifetime of the C-state and the long lifetime of the B-state, and the density difference curves show this behavior.The effect of the laser gap inductance on the optical power waveform appeared on the time of creation of the optical pulse as well as on its peak value.

VI. CONCLUSIONS
In this paper we present a full-distributed parameter model of the Blumlein-line N2 laser with the use of decoupling approach of the laser rate equations from the electrical circuit equations.The electrical equations were first simulated in separate to obtain the instantaneous (E/P) waveform and then the electrical power absorbed by the laser channel was calculated.The effect of the laser-gap inductance on the instantaneous (E/P) and the electrical power waveforms were carried out.The investigation results show that the peak value of the electrical power is a highly dependent on the laser gap inductance.Precisely, the peak of the electrical power decreases with the increase of the laser gap inductance.Also there will be a delay on the occurrence time of the peak power as well as an increase in the output pulse width due to the increase of the laser gap inductance.The optical power waveform was obtained by simulating the laser rate equations under the generalized condition in conjunction with the use of (E/P) values resulting from the simulation of electrical equations.It is clear that the peak value of the optical power and the time of occurrence are also affected by the laser gap inductance.Perhaps one of the most important results of the mathematical model the approach developed here that it could be used to optimize the nitrogen laser system and to improve its performance over a wide range of parametric variations such as the laser gap pressure, Blumlein characteristic impedance, spark gap electrodes separation and other parameters.This is to achieve a better system performance.I think the analysis presented here is quite general and could be applied to many other gas laser systems such as Cuvapor, TEA CO2 laser, etc.Further experimental research will be pursued aimed at determining the optimum operating condition of the Blumlein laser circuit.

E
photon lifetime inside the laser tube.is the electron mass ( g) and C is the speed of Light.
procedure in time and space should be applied to the problem equations before the application of the Lax-Wendroff and other finite difference schemes.It is quite standard to use the following notations for an approximate value of u( , ) at each point of the discretized problem: u( , ) = (15)

Figure 2 .
Figure 2. Simulated 3D-voltage waveform variations at all points and at any time along the parallel plate Blumlein-line N2 laser with = = 60cm and = 3.2nH.

Figure 3 .
Figure 3. Simulated 3D-voltage waveform variations at all points and at any time along the parallel plate Blumlein-line N2 laser with = = 60cm and = 30nH.

Figure 6 .
Figure 6.waveform versus normalized time showing the influence of spark inductance on .