1. Introduction

Generation of high-energy vacuum ultra-violet (VUV) radiation continues to remain for several decades in focus of basic spectroscopy, laser science and technology [1], and we are currently involved in research and development of high-energy and high-efficiency sources of Lyman-α (L-α) radiation for hydrogen (121.568 nm) and muonium (122.09 nm) resonance lines. Hydrogen L-α radiation is required for efficient laser cooling via 1S-2P transition of hydrogen-like atoms [2–6]. Muonium L-α radiation can be used for generation of ultra-slow muons via the resonant excitation of muoniums by 122.09 nm radiation followed by the ionization of excited muoniums by 355 nm radiation. Produced ultra-slow positive muons can be used for local magnetic microprobes for nanoscale solid state physics, material and life sciences [7–12]. In particular, we develop a new generation of the ultra-slow muon microscope using L-α radiation [13].

Research and development for efficient generation of L-α radiation by laser wave mixing in gases has been continued over few decades generating many significant advances [14–23], and our present work is based on a few references that we outline in more detail below. First, in our work we use the effective method to produce VUV radiation [21,23]. Second, we use the work of Marangos et al. [16] for optimizing hydrogen L-α generation by finding an optimal Ar/Kr pressure for high pressure range (10³- 10⁴ Pa) and obtaining the efficiency of 5x10⁻⁴ regarding resonant 212.556 nm radiation under relatively low input laser intensities. Third, we use the work of Bakule et al. [22] made for significantly higher laser intensities and high Kr-Ar pressure enabling the effect of gas ionization by the electron avalanche. Finally, we consider optical discharge pathways by using Keldysh mechanism of multi-photon ionization [24,25] combined with the general approach to the optical discharge in gases [26] and with the mechanism of step-wise (2 + 1)-photon ionization of Kr via the resonant 212.556 nm 2-photon excitation of Kr followed by 1-photon ionization [27].

In our present work we develop the high-efficiency L-α generation operating under high input laser intensities and low pressure conditions allowing us to avoid the full-scale Kr-Ar ionization. Our experimental study giving the final efficiency of 1.7x10⁻³ is combined with the theoretical analysis which focuses on the main mechanisms of photoionization and allows us to reveal the background of our experimental progress.

2. Experimental setup and results

The process for hydrogen L-α radiation generation (λ _L-α = 121.568 nm) shown in Fig. 1 uses the two-photon resonance four-wave mixing in phase-matched low pressure Kr-Ar mixture with Δk = k_L-α-2k₁ + k₂ = 0 (where k_i = n_iω_i/c is the wave vector) to generate pulses of ≈1.7 ns duration with the repetition rate of 25-50 Hz. First wavelength for two-photon pumping at λ₁ = 212.556 nm is defined by the energy-level spacing between the ground (4s⁶) and excited (4p-5p [1/2,0]) levels of Kr equal to 2hc/λ₁ (h is Plank’s constant and c is the speed of light). Second pump wavelength used, λ₂ = 845.015 nm, allows one to tune the generated radiation to the hydrogen L-α resonance line. However, this process can be inhibited by different mechanisms of photoionization leading to the loss of Kr atoms, loss of phase matching (Δk≠0) and related inhibition of nonlinear conversion efficiency.

 

Fig. 1 Schematization of laser four-wave mixing complicated by multi-photon ionization.

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In our experimental setup (Fig. 2) we use all-solid-state and fiber laser techniques for developing coherent 212.556 nm and 845.015 nm light sources required for the generation of 121.568 nm radiation in Kr-Ar gas cell having L = 100 cm in length. First, the radiation distributed feedback laser precisely tuned to the fundamental wavelength is injected into the fiber amplifier. Fiber amplifier provided with an electro-optic modulator allows us to increase the output energy up to 0.1 mJ in pulsed operation. After that we use Nd:Y₃Ga_xAl_(5-x)O₁₂ (Nd:YGAG) crystal with the fluorescence spectrum blue-shifted by the increase in theconcentration of gallium. In particular, for x = 2 the spectrum peak of this crystal agrees with a wavelength of 1062.78 nm as found by Walsh et al. [28] and confirmed by Oishi et al. [29]. Therefore, we use Nd:Y₃Ga₂Al₃O₁₂ for efficient amplification of 1062.78 nm radiation and for generating high intensity coherent 212.556 nm light. The duration of the generated output pulses is ≈1.7 ns. The all-solid-state amplifier is composed of a regenerative amplifier and two laser diode side-pumped amplifiers [29]. For the amplification of 1062.78 nm radiation pulses in the regenerative amplifier we use 10 mm-long Nd:YGAG crystal having an input plane area of 4 × 4 mm. For the gain material in the side-pumped amplifier we use 4 mm in diameter and 80 mm long Nd:YGAG ceramic taking into account several advantages of ceramic laser materials: large-size, high quality and high transparency [30, 31]. By using this all-solid-state amplifier the pulse energy can be increased up to 100 mJ at a repetition rate of 25 Hz coinciding with the repetition rate in the regenerative amplifier.

 

Fig. 2 Schematic of the optical system used for generation of L-α radiation.

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The frequency c/(212.556 × 10⁻⁹ m) Hz is the fifth harmonic of the fundamental frequency c/(1062.78 × 10⁻⁹ m) Hz. This frequency is obtained by using a set of (i) 20 mm long, antireflection-coated LiB₃O₅ (LBO) crystal, and (ii) two 15 mm long, uncoated CsLiB₆O₁₀ (CLBO) crystals. The output energy at 531.39 nm emitted by LBO crystal via second harmonic generation (SHG) reaches approximately 60 mJ. After that we obtain coherent 212.556 nm light by fourth and fifth harmonic generation in the CLBOs using a half of the output energy at 531.39 nm. After SHG the fundamental radiation is divided from the second harmonic and is combined with the fourth harmonic to generate the fifth harmonic.

The rest of the output energy at 531.39 nm is used as pump energy for the generation of coherent 845.015 nm light. Coherent 845.015 nm light is generated by the optical parametric generator (OPG) and the optical parametric amplifier (OPA) using 15 mm long uncoated LBOs crystals. We use a broadband (≈230 GHz) diode laser as the seeder to determine the output wavelength as well as the spectral linewidth of the coherent 845.015 nm light source. In order to eliminate timing jitter between 212.556 and 845.015 nm radiation pulses we use a half of the output energy of the second harmonic as the pump radiation in OPG and OPA.

This system provides our experimental setup with the pulses of 212.556 nm and 845.015 nm radiation shown in Fig. 3 on the timescale of 10 minutes. Experimental generation of L-α radiation was done using 3.2 and 1.6 mJ laser pulses of 212.556 nm and 845.015 nm radiation, respectively. However, due to the optical losses in the transmission system the input laser pulse energies in Kr-Ar gas cell are Q₁ = 1.5 mJ and Q₂ = 0.8 mJ for 212.556 and 845.015 nm, respectively. The input laser pulses are synchronized by the optical delay system, followed by the coaxial input into Kr-Ar gas cell enabling us to perform the two-photon resonance four-wave mixing defined by ω_L-α = 2ω₁-ω₂. In our experiments we use a tightly focusing optical system with the focal radii of r₁≈30 μm for λ₁ = 212.556 nm and r₂≈50 μm for λ₂ = 845.015 nm, and with the following confocal parameters: b₁ = 2πr₁²/λ₁≈2.7 cm and b₂ = 2πr₂²/λ₂≈1.86 cm, respectively.

 

Fig. 3 Generated pulse energy of input radiations vs the operation time.

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Our experimental study is targeted to optimize the process of L-α generation for low Kr pressure range P_Kr = 100-500 Pa diluted by Ar. In particular, our experimental data given in Fig. 4 for P_Kr = 1-550 Pa and P_Ar/P_Kr = 5 show the output energy of the generated L-α radiation pulses versus Kr pressure. In this range of pressure the output energy shows an increase for P_Kr = 5-250 Pa and the saturation for P_Kr = 250-400 Pa followed by the sharp decrease. The maximum of L-α pulse energy corresponds to P_Kr≈365 Pa (total pressure P_Σ≈2.2 × 10³ Pa) with the generated 8.4 µJ in the gas cell, whereas the pulse energy measured outside is 3.6 µJ.

 

Fig. 4 Output energy of hydrogen L-α radiation vs Kr pressure.

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Therefore, the resulting output energy corresponds to the conversion efficiency of 3.6 × 10⁻³ (≈8.4 × 10⁻⁶ J / [1.5 × 10⁻³ J + 0.8 × 10⁻³ J]). To finalize this section we would like to stress out that the experimental efficiency taking into account all optical losses is 1.7x10⁻³. However, the efficiency of the non-linear conversion estimated regarding the resonant radiation (212.556 nm) without all transmittance losses gives the value of 8.4 × 10⁻⁶ J/1.5 × 10⁻³ J = 5.6x10⁻³ which is an order of magnitude higher as compared with the experimental value of 5x10⁻⁴ [16].

3. Calculations and discussion

Below we show that this efficiency is achieved due to the operation under high input laser intensities and low gas pressures allowing us to avoid the full-scale photoionization and discharge. First, we estimate the multi-photon ionization rate (MPI) by using Keldysh model [24,25] with the modification of [26]:

By using Eq. (1) for the maximal laser intensities$I_i=2Q_i/(\pi r_i^2{\tau }_p)$ we calculate MPI rates and the resulting densities of electrons, n_e≈W_mpiN_Kr-0τ_{p ,}which are given in Table 1 for the conditions corresponding to the maximal L-α radiation output obtained under P_Kr = 365 Pa and T = 290 K. Data given in Table 1 show that the maximal contribution to the ionization is due to the generated L-α radiation. Namely, L-α radiation with the intensity of I_L-α = 3.5x10¹² W/m² generates during the pulse n_e≈0.75x10¹⁴ cm⁻³, which is ≈24 times higher as compared with the effect of 212.556 nm radiation generating n_e≈3.2x10¹² cm⁻³ under I₁ = 6.2x10¹⁴ W/m². Electron generation by 845.015 nm radiation is negligibly small.

Table 1. Estimated multi-photon ionization data of Kr.

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Second, let us consider another pathway of Kr photoionization via the resonant 2-photon excitation of Kr by 212.556 nm radiation followed by 1-photon ionization [27]. Order of magnitude estimations show that for our experimental conditions the effects of electron-ion recombination and diffusion can be completely neglected, and the densities of exited Kr atoms (n*) and free electrons (n_e) can be estimated by:

The first term in Eq. (2) represents 2-photon excitation taking also into account that the initial Kr density can be depleted by Kr excitation with the density of (n*) and Kr ionization with the density similar to that of electrons n_ion≈n_e. The analytical solution of this set of equations is given by:

The calculations made for I₁ = 6.2x10¹⁴ W/m² and N_Kr-0 = 0.9x10¹⁷ cm⁻³ show that the ionization by this pathway leads to the ion-electron density of n_e≈1.52x10¹⁶ cm⁻³ which is ≈200 times higher compared with that provided by all contributions given by Eq. (1). This mechanism of ionization leads to 17% loss of Kr density by the end of the pulse. Taking into account that I_L-α∝(N_Kr-0-n_e)² we conclude that by the end of the pulse the efficiency of nonlinear conversion has a factor of 0.7 as compared with that at the pulse start. Additionally, we have to note that the rate of Ar ionization (ΔE_ion = 15.76 eV) W_mpi = 2.34x10⁴ 1/s is ≈20 times lower as compared with Kr under the generated intensity I_L-α = 3.5x10¹² W/m².

Let us now consider the energy of free electrons under involved laser waves by [26]:

First, by using the values of I₁6.24x10¹⁴ W/m² and I₂ = 1.2x10¹⁴ W/m² we obtain for the quiver energies ε_q1 = 2.21x10⁻⁴ eV and ε_q2 = 7.56x10⁻⁴ eV giving ε_qΣ≈10⁻³ eV. This estimate shows that the main contribution to the electron heating (≈80%) is caused by 845.015 nm radiation due to the frequency effect (${\epsilon }_{qi}\propto {\omega }_i^{-2}$). Second, the initial energy of free electrons after photoionization can be within the range of ε₀ = 3ħω₁-ΔE_ion≈3.5 eV and ε₀ = 2ħω_L-α-ΔE_ion≈6.2 eV giving the electron velocity V_e = (2ε_e/m_e)^1/2 = (1.2-1.6)x10⁸ cm/s. Third, the collision rate is within the range of ν_c = (7-28)x10⁹ s⁻¹ for the total operational pressure P_Σ = 1200-3600 Pa and related atomic density range N_Kr-0 = P_Σ/k_BT = (2.9-8.7)x10²³ m⁻³. Finally, during the pulse the free electrons can experience about ν_cτ_p = 12-48 collisions with Kr and Ar atoms leading to the energy gain of Δε_e≈2ε_qΣν_cτ_p ≈(2.4-9.6)x10⁻² eV. Therefore, in our experiments the maximal energy of free electrons generated, ε_e = ε₀ + Δε_e≈6.3 eV, remains below the ionization energy (14 eV for Kr) and the avalanche ionization remains ineffective.

Taking now into account that for present experiments the value of Δε_e <0.1 eV the increase of 845.015 nm radiation input looks a quite realistic way leading to the linear increase of I_L-α ∝I₂. That is, I₂ can be increased about 2 orders of magnitude without the onset of the electron avalanche as compared with that used in our present experiments: I₂ = 1.2x10¹⁴ W/m². However, we do not expect a 100-fold practical increase of the output L-α pulse energy because our analysis extended to this range of I₂ allows us to identify another strong limitation. That is, in considering the increase of 845.015 nm laser input we have to take also in account that the linear increase of I_L-α by I₂ can lead to a drastic increase of photoionization by Keldysh mechanism. In particular, by using Eq. (1) and Table 1 one can find that MPI rate by L-α radiation is $W_{mpi}\propto I_{L-\alpha }^{1.4}$. By assuming only 50-fold increase of I₂ and I_L-α one finds that the related increase of the electron density due to generated L-α radiation will achieve the value of n_e = 0.75x10¹⁴x50^1.4 = 1.8x10¹⁶ cm⁻³ which exceeds the value generated by (2 + 1)-photon 212.556 nm radiation n_e≈1.52x10¹⁶ cm⁻³. Hence, the total loss of Kr should be close to ≈50%. That is, for our range of pressures even 50-fold increase in I₂ will not lead to the avalanche ionization. However, in this case Kr should be almost completely ionized by the combined action of (2 + 1)-photoionization by 212.556 nm radiation and by generated L-α radiation. This allows us to suggest that the practically feasible increase of L-α pulse energy will be limited by the factor of ≈30, for which the electron and ion density by the generated L-α radiation, n_e = (0.7-0.9)x10¹⁶ cm⁻³, remains below that produced by (2 + 1)-photon ionization by 212.556 nm radiation. Thus, our analysis suggests ≈30-fold increase of 845.015 nm pulse energy can allow the increase of L-α radiation pulse energy towards the level of ≈100 μJ by using other operational parameters of the present work.

Additionally, we should note that the avalanche ionization can give a significant contribution to the photoionization under higher pressures. In particular, our analysis used for high pressure experiments allows us to suggest that this mechanism is the valid explanation of the inhibition of L-α radiation pulse generation taking place for Q₂ = 8.4 mJ (λ₂≈845 nm) under increase of the total pressure from P_Σ = 10⁴ Pa to P_Σ = 2x10⁴ Pa shown in [22]. In particular, for this case we can estimate the laser intensity on the beam axis as I₂ = 4Q₂/(b₂λ₂τ_p)≈3.3x10¹⁴ W/m², where b₂ = 1 cm and τ_p≈12 ns [22]. The electron quiver energy is ε_qΣ≈ε_q2≈2.15x10⁻³ eV. By taking for this case the mean electron energy of ≈10 eV we can find the electron velocity V_e = (2ε_e/m_e)^1/2≈2x10⁸ cm/s. Hence, the electron collision rate with the atoms is ν_c = N_Σσ_cV_e≈10¹¹ - 2x10¹¹ s⁻¹ for the range of P_Σ = 10⁴ - 2x10⁴ Pa and, therefore, the free electrons experience about ν_cτ_p≈1200-2400 collisions resulting in the energy gain of Δε_e≈2ε_qΣν_cτ_p≈5.16-10.32 eV, respectively. Taking into account the initial energy after ionization one finds that for P_Σ = 10⁴ Pa the electrons can have the energy of ε_e = ε₀ + Δε_e≈8.66-11.36 eV whereas for P_Σ = 2x10⁴ Pa the value of ε_e = ε₀ + Δε_e≈13.82-16.52 eV. Taking additionally into account that this type of ionization can proceed via the excitation of Kr to the 4p-5p level (with ΔE_exc = 11.67 eV) we conclude that for P_Σ = 10⁴ Pa the laser-gas interaction gets very close to the initiation of Kr ionization by the electron avalanche whereas for P_Σ = 2x10⁴ Pa the laser-gas interaction proceeds under the onset of ionization of Kr and Ar by the electron avalanche able to develop within the time scale of ≈0.5 ns [27].

Finally, our analysis suggests that in near optical discharge operation performed in our study the composition of Ar-Kr mixture, the phase-matching of the laser waves and the conversion efficiency are very sensitive to any change of the laser intensities. In particular, on the one hand Eqs. (4)-(5) show that the ionization rate of Kr is $d{n}_e/dt\propto I_1^3$ and, therefore, even a small decrease in the beam radius ($I_1\propto 1/r_1^2$) or increase in I₁ can lead to loss of efficiency ($I_{L-\alpha }\propto {(N_{Kr-0}-n_e)}^2{I}_1^2{I}_{2}F$) via photoionization of Kr atoms and also by decrease of phase matching factor F which depends on Ar/Kr ratio [1]. On the other hand a small increase in the beam radius r₁ or decrease in $I_1^{}$ leads to the significant loss of the efficiency. Hence, L-α generation performed near onset of optical discharge allows one to operate at the maximum efficiency only under a very tight control of all laser parameters.

4. Conclusions

To conclude, we have developed and tested the laser system allowing a high conversion efficiency of resonant four-wave mixing for generating ns pulsed hydrogen L-α radiation in operation made near the onset of optical discharge in Kr-Ar gas mixture. Theoretical analysis performed shows that under our experimental conditions Kr-Ar gas mixture experiences the loss of about 17% of Kr only by the end of the pulse enabling high-efficiency operation during the pulse. The photoionization is due to 212.556 nm radiation via the mechanism of 2-photon Kr excitation followed by 1-photon Kr ionization. However, after photoionization the energy of generated free electrons remains during the pulse below the level of impact ionization sufficient for the onset of avalanche ionization and full-scale discharge. Our analysis also suggests that about 30-fold increase of 845.015 nm pulse energy can allow the increase of L-α radiation pulse energy towards the level of ≈100 μJ by using other operational parameters of the present study without critical enhancement of photoionization.