Induced Emission on Transitions from Vibrational Excited Levels of the KrF Molecule

Abstract

The paper considers the possibility of extending the spectral region of the wavelength tuning of a discharge KrF amplifier due to induced transitions from the vibrational excited states of the electronic level B. The model of the KrF amplifier on a He/Kr/F₂ mixture is presented, in which the behavior of the vibrational level populations is consistent with the excitation conditions of the active medium. The simulation results show that the shift in the operating wavelength to the short-wavelength region is possible in excitation modes, when the birth rate of excimer molecules is greater than the rate of their relaxation from upper to lower vibrational levels. The theoretical dependences of gain on the wavelength for different pressures were obtained. They confirm the possibility of tuning the KrF amplifier wavelength in the range of up to 10 nm while maintaining a gain of at least 0.5 of its maximum value.

1. Introduction

Over the last 30 years, gas discharge excimer lasers and amplifiers based on noble gas halides have been widely used as UV light sources. One of the promising application areas for these lasers is spectroscopy and the detection of various species in the air and on surfaces. In these systems, excimer lasers serve as sources of probing pulses. Spectroscopic methods are highly sensitive to the wavelength of the probing radiation, and the ability to tune the light source within a specific wavelength range significantly enhances the efficiency of these methods [1].

Since, for excimer molecules, laser transitions end on repulsive or weakly coupled potential curves of ground state, their fluorescence fundamentally have broad spectra lines. In the works [2,3], it was shown that by the simple optical tuning of KrF and ArF lasers, it is possible to obtain generation in a band width of ~2 nm. Unfortunately, since the shift in the generation wavelength relative to the central one led to a significant decrease in the laser efficiency, with energies of ~100 mJ achieved by that time, these studies did not find practical applications. A new wave of interest in the problem of tuning the working length of excimer lasers has occurred in recent years, which is associated with the need for more accurate spectroscopic methods. In addition, the current state of excimer laser technology makes it quite easy to obtain energy of the order of 1 J per pulse. In previous work [4], we experimentally demonstrated the amplification of radiation at the UV edge of the KrF molecule spectrum under the excitation of a He/Kr/F₂ gas mixture by a discharge. The gain efficiency drop for the 246.8 nm wavelength relative to the efficiency at the central wavelength (248.3 nm) was only 50%, which is significantly less than previously reported. We associated this result with the high excitation rate of the active medium (the pump power density at the maximum was ~10 MW/cm³).

Theoretical analysis based on numerical simulations has shown that the generation of radiation in this spectral range is related to stimulated transitions from the vibrational excited states of the upper laser level. The presence of vibrational excited excimer molecules in the active medium is related to the peculiarities of their formation. In the discharge plasma, excimer molecules are formed either as a result of the interaction between two particles, one of which is in an electronically excited state, or through ion–ion recombination reactions. In both cases, the molecules are born in highly excited vibrational states [5,6,7]. The relaxation of the excimer molecules to the ground state, from which laser generation usually occurs, is facilitated by collisions with heavy particles and electrons. In the 1980s and 1990s, the problem of the vibrational relaxation of excimer molecules attracted significant attention, as this process greatly affects laser efficiency. Several theoretical models were proposed, with models where vibrational levels were treated as a single state with generalized rates describing the interaction between the ground and excited states [8,9]; models that included rate equations accounting for collisional transitions between vibrational levels [7,10,11]; and models that described vibrational relaxation as motion in the vibrational energy space [12].

At the same time, several studies have focused on theoretical methods for calculating the probabilities of stimulated optical transitions from the vibrational excited levels of excimer molecules [11,13].

Despite a large number of theoretical works, until now it has not been clear how the gain depends on the operating wavelength for the different modes of its excitation. Our goal was to develop a model that would self-consistently take into account the conditions of discharge excitation, the kinetic processes in it, and laser generation, including from vibrational excited states. We expected that the calculated results would allow us to determine the width of the possible wavelength tuning interval during the discharge excitation of the He/Kr/F₂ gas mixture.

2. Materials and Methods

A 1D approach was used for modeling the amplification and laser generation processes. We assumed the pump discharge to be homogeneous in its cross section. Along the optical axis, the discharge was split into layers (Figure 1). The electron energy distribution function (EEDF) and plasma particle concentrations were found for each layer. The development of the photon avalanche along the optical axis of the resonator was described by the wave equations:

$${\displaystyle \frac{\partial F_{\mathrm{R},\mathrm{L}}\left(l,t\right)}{\partial l}}\pm {\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial F_{\mathrm{R},\mathrm{L}}\left(l,t\right)}{\partial t}}=F_{\mathrm{R},\mathrm{L}}\left(l,t\right)\left({\sigma }^{*}{N}_{\mathrm{K}\mathrm{r}F}-{\sum }_{\mathrm{i}}{\sigma }_k{N}_{\mathrm{k}}\right)+\Omega N_{\mathrm{K}\mathrm{r}\mathrm{F}}/\tau$$

(1)

where F_L, F_R—the two counter flows of laser photons; l—the distance along the optical axis; t—time; N_KrF—the excimer molecule concentration; σ*, σ_k—the cross sections of induced emission and the absorption of laser photons by particles of type k; N_k—the absorbing particle concentration; τ—the spontaneous lifetime of the KrF; c—the speed of light; and Ω—part of spontaneous photons, the direction of the propagation of which coincides with the direction of the optical axis.

The substitution diagram of the electrical system of discharge excitation is shown in Figure 2. The equations system describing the operation of the pumping electric circuit were written in matrix form according to the “variable states “ method [14]. For each moment in time, the resistance of the discharge R_d was calculated as follows:

R_d = d/en_eµ_eS_d,

(2)

where e—the electron charge; S_d—the discharge cross section; d—the distance between the electrodes; n_e and µ_e—the concentration and mobility of electrons. The resistance R_d was calculated from the values of electron concentration and electron mobility, which were determined in the previous calculation step.

2.1. Boltzmann Kinetic Equation

The electron energy distribution function (EEDF) was found from the solution of the Boltzmann equation. A quasi-stationary approach was used. The changes in the particle concentration and external electric field were assumed to be slow enough for the EEDF distribution to settle on a time step of ~1 ns; therefore, for each nanosecond of discharge duration, the steady-state equation was solved [15]:

$${\displaystyle \frac{\partial J\left(\epsilon \right)}{\partial \epsilon }}=S\left(\epsilon \right),$$

(3)

where J—the electron flux in the energy space, S—the collision integral, ε—the electron energy.

Flow J was represented as a sum of J(ε) = J_f (ε) + J_m(ε) + J_ee(ε) + J_ee(ε), where J_f—the heating of electrons by the electric field; and J_m, J_ee, and J_ei—the contributions of elastic, electron–electron, and electron–ion collisions, respectively. The collision integral S included terms accounting for inelastic and superelastic processes: S_u—interactions; S_i—ionization and recombination; and S_a—attachment. S(ε) = S_u(ε) + S_i(ε) − S_a(ε). A more detailed description of the expressions in Equation (3) was presented in [16]. In determining the expressions J(ε) and S(ε), the energy dependences of the excitation and ionization cross sections were used from the following works: Ne—[17,18,19,20,21]; Kr—[22,23,24]; and He—[25,26,27].

Equation (3) was reduced to a system of algebraic equations by the Galerkin weighted residual method [28], for the solution of which iteration methods were used. To minimize the numerical error, in addition to the convergence of the solution, we controlled the conditions for J: J(E₁) = 0 and J(E₂) = 0, where E₁ and E₂ are the boundaries of the computational domain.

2.2. Model of Plasma–Chemical Processes in the Active Medium

The model allows the numerical simulation of the KrF amplifier operation at the discharge excitation of mixtures containing the gasses He, Ne, F₂, and Kr in different proportions. The kinetics model describes changes in the following: the concentrations of the electrons n_e, excited atoms He and Ne (separated states 3s and 3p), Kr (5s, 5p, and an effective level combining states 4d and 6s), and F (3s and 3p); and the concentrations of the vibrational excited molecules F₂ (v = 0, 1, 2, 3) and ions He⁺, Ne⁺, Kr⁺, F⁺, F⁻, Kr₂⁺, He₂⁺, Ne₂⁺, NeKr⁺, and HeKr⁺. A total of 570 reactions between 60 particles were included in the model. A list of kinetic reactions included in the model for Ne as a buffer gas was presented in reference [14]. For working the mixtures containing He, similar processes were taken into account, and the rate constants for them were taken from the works of [25,26,29].

We assumed that excimer molecules are born at the highest vibrational levels of states B (²Σ_1/2) и C (²Π_3/2). The occupation of lower vibrational levels is a consequence of vibrational relaxation in collisions with other particles [11,30]. The model included the main, upper, and 20 lower vibrational excited levels of state B, as well as the main and level combining all the vibrational excited levels of state C. Taking into account that the difference between neighboring vibrational levels is of the order of the thermal energy, transitions are possible only between neighboring levels j = i ± 1.

The rate constants of the collisional relaxation down the energy scale k_ij, electrons, and BC-exchange, which were used in the model, are presented in Table 1. The rates of reverse transitions k_ji were determined from the following condition:

$$k_{ji}=k_{ij}\exp \left(-{\displaystyle \frac{\Delta {\epsilon }_{ij}}{kT}}\right).$$

(4)

In Equation (4), k—the Boltzmann constant, T—the temperature, k_ij—the rate of the direct process, Δε_ij—the energy gap between vibrational levels.

Table 1. Rates of vibrational relaxation and BC-exchange processes.

	k, 1/cm3	Ref.
KrF(B, i) + (He,Ne,Kr) → KrF(C) + (He,Ne,Kr)	5.0 (−10)	[8]
KrF(C) + (He,Ne,Kr) → KrF(B, i) + (He,Ne,Kr)	3.5 (−10)	[8]
KrF(B, i) + Kr → KrF(B, i − 1) + Kr	8.0 (−11)	[8]
KrF(B, i) + (He,Ne) → KrF(B, i − 1) + (He,Ne)	1.0 (−11) (i ≤ 10) 5.0 (−11) (i > 10)	[10] [10]

Table 1. Rates of vibrational relaxation and BC-exchange processes.

	k, 1/cm3	Ref.
KrF(B, i) + (He,Ne,Kr) → KrF(C) + (He,Ne,Kr)	5.0 (−10)	[8]
KrF(C) + (He,Ne,Kr) → KrF(B, i) + (He,Ne,Kr)	3.5 (−10)	[8]
KrF(B, i) + Kr → KrF(B, i − 1) + Kr	8.0 (−11)	[8]
KrF(B, i) + (He,Ne) → KrF(B, i − 1) + (He,Ne)	1.0 (−11) (i ≤ 10) 5.0 (−11) (i > 10)	[10] [10]

2.3. Induced Emission Cross Section and Einstein Coefficient for Bound-Free B–X Transitions from Vibrational Levels of the KrF Molecule

Since the lower state X (²Σ_1/2) of the KrF molecule is essentially repulsive, optical transitions occur from the bound state to the continuum of the lower state. In calculating the Einstein coefficients A_i and induced emission cross sections σ_i for the transition from the vibrational level ${|\psi }_i⟩$, potential curves and the dependence of the transition dipole moment µ on the distance between the atoms r were used from the paper [31]. The values themselves were defined according to [32] as follows:

$$A_i={\displaystyle \frac{64{\pi }^4{\mathsf{\nu }}^3}{3h}}{|⟨{\psi }_i|\mu \left(r\right)|E_g⟩|}^2$$

(5)

$${\sigma }_i={\displaystyle \frac{8c{\pi }^3}{3}}\nu {|⟨{\psi }_i|\mu \left(r\right)|E_g⟩|}^2$$

(6)

Here, ν = ν₀ + E_i − E_g—the energy of the radiation quantum (in cm⁻¹); ν₀—the difference in energy between the vibrational unexcited upper state and lower state; E_i—the vibrational energy in the upper state; E_g—the lower state energy. The vibrational wave functions ψ_i were found from the solution of the Schrödinger equation using the method proposed in reference [33].

Figure 3 shows the calculated induced emission cross sections σ_i for the transitions from the main and first two vibrational levels of KrF(B) as functions of the wavelength. It is possible to draw several conclusions from this figure. First, narrow line radiation interacts with several vibrational levels. In this connection, when solving Equation (1), a gain for the operating wavelength λ_w was determined as a sum over the oscillatory levels i:

$${\sigma }^{*}{N}_{KrF}={\sum }_i{\sigma }_i{{(\lambda }_w)N}_{KrF}^i$$

(7)

where σ_i(λ_w)—the stimulated emission cross sections of the i-th vibrational level for the wavelength λ_w; and Nⁱ_KrF—the population of the i-th vibrational level of the KrF(B) molecule.

Second, the large peak values for the σ_i of the higher levels suggests that their populations are accessible to photons of wavelengths other than 248.3 nm (transition wavelength from the main level).

3. Results

When testing the model, we compared the simulated results with the experimental results for a KrF laser operating at a wavelength of λ = 248.3 nm, as well as with the experimental results obtained when amplifying radiation at a wavelength of 246.8 nm.

3.1. Simulation of Laser Emission from the Ground State B

Figure 4 shows the dependences of the laser energy (λ = 248.3 nm) on the charging voltage U₀ obtained experimentally and numerically. The experimental results were obtained using a discharge KrF laser of the ELF series with pulse energy of up to 1 J and pulse repetition rate of up to 100 Hz, developed at the HCEI SB RAS. A more complete description of this setup is presented in [16].

In our calculations, it was assumed that in all the excitation modes the discharge has the same width. The spark preionization was modeled by a homogeneous in volume initial concentration of electrons of 10⁸ cm⁻³. A gas mixture of Ne/Kr/F₂ = 3200/110/4 at a pressure of 3.4 atm was used. The resonator was two flat mirrors with reflection coefficients R_R and R_L of 98% and 7%. It can be seen from the figure that a good agreement of the calculated results with the experimental results within the 9% experimental confidence interval in the whole range of operating voltages is obtained.

3.2. Amplification of the 246.8 nm Wavelength Radiation

The experimental setup of the KrF amplifier is presented in detail in [4]. A probing pulse with 0.1 mJ energy and a pulse duration of 8 ns was injected into the active medium at different time moments relative to the formation of the discharge.

Figure 5 shows the observed emission spectrums when a λ = 246.8 nm probing beam passes through the KrF amplifier. The results were obtained for a He/Kr/F₂ = 2300/100/5 mixture at a full pressure of 2.4 atm. The probing beam was entered in the discharge chamber after Δt = 5, 10, and 30 ns from the moment of the breakdown of the discharge gap. It can be seen that the probe pulse wavelength is at the boundary of the fluorescence spectrum of the B–X transition of the KrF molecule. The maximum gain g = 0.053 cm⁻¹ was obtained at a charge voltage U₀ = 22 kV and time Δt = 30 ns in this series of experiments.

The intensity of the amplified signal depends on the time delay of its entry into the active medium relative to the beginning of the pump pulse Δt. Figure 6 shows the dependences of the output pulse intensity on Δt, simulated and obtained in the experiment. The delay Δt = 0 ns corresponds to the breakdown time of the discharge gap. The maximum gain occurs when the moment of entry of the probing beam into the active medium coincides with the pumping maximum (Δt = 30 ns). In addition, the figure shows a good coincidence between the calculated and experimental results. The difference in the dependences for Δt > 40 ns is due to the deterioration of the discharge homogeneity at the end of the excitation pulse, which is not reflected in the numerical calculation. The maximum value of the calculated gain g was 0.052 cm⁻¹. This is in good agreement with the value of 0.053 cm⁻¹ obtained experimentally.

4. Discussion

As noted above, the possibility of radiation amplification on KrF molecules with a wavelength shifted relative to the main wavelength (248.3 nm) into the spectrum short-wave region depends on the excited states’ population. Figure 7 shows the calculated dependences of the KrF(B) molecules population on the vibrational excitation level i. The data are presented for the maximum pump power. The dependence is presented in relative units normalized on the total concentration of the KrF(B) molecules over all the vibrational levels. The same figure shows the equilibrium (Boltzmann) distribution. At the equilibrium state, 99% of the KrF(B) molecules populate the first four vibrational levels, with 80% of all molecules being in the ground state i = 0. However, with pulsed pumping, all the vibrational system KrF(B) is populated. The shape of the distributions depends on the excitation conditions of the active medium, and more precisely on the difference between the rates of the excimer molecules’ birth and their collisional relaxation at vibrational levels. For the pump power density of ~10 MW/cm³ and pressure of ~1 atm, the population of the upper vibrational levels is higher than that of the ground state. As the pressure increases, due to the increase in the vibrational relaxation rate, the proportion of KrF molecules that pass to the ground level increases. Despite this, the equilibrium distribution is not achieved even at a pressure above 4 atm.

Figure 8 shows the gain factors at the pumping maximum, which correspond to the population distributions 2 and 3 from Figure 7. The gas mixture’s composition is He/Kr/F₂ = 2300/100/2.5 (full pressure 4 atm) and 1300/100/2.5 (1 atm). The rest of the active medium excitation conditions were identical.

According to the results obtained, at a pressure of ~1 atm, the gain of the active medium remains within 50% of its maximum value in the wavelength tuning range of ~10 nm. At the higher pressures of the gas mixture, a sharper drop of the gain at the transition of the radiation wavelength to the short-wave region of the spectrum is observed, which is associated with an increase in the rate of th evibrational relaxation of excimer molecules on the vibrational levels. Nevertheless, KrF amplifier tuning in the ~2 nm range is possible while keeping the gain at least 0.5 of the maximum.

5. Conclusions

This paper studies the problem of the generation of transitions from the vibrational states of the upper level of the KrF molecule, which allows shifting the operating wavelength of the amplifier to the UV region of the spectrum. The results were obtained using numerical simulation, since it is a convenient tool for analyzing the relationships between pumping conditions, processes in the active medium of the amplifier, and laser generation. It was shown that the high gain coefficient on the transitions from the four and five vibrational levels obtained earlier experimentally [4] when exciting a He/Kr/F₂ mixture with an electric discharge is provided by a high peak pump power (~10 MV/cm³). Under such conditions, the rate of the production of excimer molecules is higher than the rate of their relaxation from vibrational excited states to the ground state. This leads to a significant population of the vibrational states of the upper laser level. The obtained dependencies of the gain on the radiation wavelength show the width of the possible range of the tuning of the operating wavelength of 2–3 nm while maintaining the gain of at least 0.5 of the maximum. The range width can be extended by reducing the working mixture pressure. This fact can be useful for applications where high energy values are not required. We hope that the obtained theoretical results will be useful in the development of electric–discharge excimer systems with wavelength tuning.