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Intense infrared lasers for strong-field science

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Abstract

The advent of chirped-pulse amplification in the 1980s and femtosecond Ti:sapphire lasers in the 1990s enabled transformative advances in intense laser–matter interaction physics. Whereas most of experiments have been conducted in the limited near-infrared range of 0.8–1 μm, theories predict that many physical phenomena such as high harmonic generation in gases favor long laser wavelengths in terms of extending the high-energy cutoff. Significant progress has been made in developing few-cycle, carrier-envelope phase-stabilized, high-peak-power lasers in the 1.6–2 μm range that has laid the foundation for attosecond X ray sources in the water window. Even longer wavelength lasers are becoming available that are suitable to study light filamentation, high harmonic generation, and laser–plasma interaction in the relativistic regime. Long-wavelength lasers are suitable for sub-bandgap strong-field excitation of a wide range of solid materials, including semiconductors. In the strong-field limit, bulk crystals also produce high-order harmonics. In this review, we first introduce several important wavelength scaling laws in strong-field physics, then describe recent breakthroughs in short- (1.4–3 μm), mid- (3–8 μm), and long-wave (8–15 μm) infrared laser technology, and finally provide examples of strong-field applications of these novel lasers. Some of the broadband ultrafast infrared lasers will have profound effects on medicine, environmental protection, and national defense, because their wavelengths cover the water absorption band, the molecular fingerprint region, as well as the atmospheric infrared transparent window.

© 2022 Optica Publishing Group

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Figures (38)

Figure 1.
Figure 1. Experimental demonstration of extending the cutoff photon energy of HHG by increasing the driving laser wavelength [20]. Figure 5 reprinted with permission from Shan and Chang, Phys. Rev. A 65, 011804 (2001) [20], https://journals.aps.org/pra/abstract/10.1103/PhysRevA.65.011804. Copyright 2002 by the American Physical Society.
Figure 2.
Figure 2. High-order harmonic generation in solids. (a) Microscopic mechanism comprised of two channels: the emission from driven intraband current (left) and the emission from interband polarization (right). In the presence of the strong laser field, a small fraction of electrons from the valence band tunnel to the conduction band leaving behind holes in the valence band. Then, both the electron and the hole get driven by the laser field in their respective bands, reaching high energies and momenta. For strong enough fields, electrons can tunnel to higher lying conduction bands (not shown). (b) Measured high-harmonic spectra from ZnO crystals subjected to intense mid-IR laser fields [23]. The arrows at the 17th and 25th orders mark the locations of the high-energy cutoffs for the respective driving pulse energies, as shown in the legend. (c) The scaling of high-energy cutoff with the peak field strength. The scaling is linear, and it intercepts the vertical axis near the location of bandgap energy of the source material. Adapted with permission from [24].
Figure 3.
Figure 3. Energy-level diagrams of Tm3+ and Ho3+ in YLF. Adapted from Walsh et al. [44] with permission from AIP Publishing.
Figure 4.
Figure 4. Transmission of ambient air near the Ho:YLF emission wavelength.
Figure 5.
Figure 5. (a), (b) Absorption cross-section. (c), (d) Stimulated emission cross-sections of Ho:YLF at cryogenic temperature and RT [62]. (e), (f) Absorption and stimulated emission cross-sections of Tm:YLF at RT [63]. Reprinted from Lippert et al. [62] and Tamer et al. [63] with permission from SPIE.
Figure 6.
Figure 6. Boltzmann fractions of the 5I8 and 5I7 levels of Ho:YLF at two temperatures, 20°C and –55°C. Here N0, N1, and N2 are the ground state, lower, and upper laser levels, respectively, for amplifying signals at 2.05 μm. Adapted from Zhou et al. [64] © Optica Publishing Group.
Figure 7.
Figure 7. Calculated gain cross-section versus inverted fraction β for π polarization in Ho:YLF at RT [64]. Reprinted with permission from Kroetz et al. [65] © The Optical Society.
Figure 8.
Figure 8. Tm:fiber CPA laser layout. Reprinted with permission from Stutzki et al. [112] © The Optical Society.
Figure 9.
Figure 9. Schematics of coherent beam combination. Adapted from Klenke et al. [117] with permission.
Figure 10.
Figure 10. Coherent beam combination processes: (a) partially reflective surface and (b) polarization-dependent cube. Reprinted with permission from Klenke et al. [119] © The Optical Society.
Figure 11.
Figure 11. RT emission cross-section in arbitrary units for (a) Cr:ZnS, (b) Cr:ZnSe, (c) Cr:CdSe, (d) Fe:ZnS, (e) Fe:ZnSe, (f) Fe:CdSe, and (g) Fe:CdTe. Adapted from Mirov et al. [132] with permission.
Figure 12.
Figure 12. (a) Temperature dependences of SWIR/MWIR luminescence lifetimes in (i) Cr:ZnSe, (ii) Cr:ZnS, (iii) Fe:ZnSe, and (iv) Fe:ZnS crystals. (b) RT absorption cross-sections of (i) Cr:ZnS, (ii) Cr:ZnSe, (v) Fe:ZnS, and (vi) ZnSe samples; RT emission cross-sections of (iii) Cr:ZnS, (iv) Cr:ZnSe, (vii) Fe:ZnS, and (viii) ZnSe samples. Adapted from Mirov et al. [132] with permission.
Figure 13.
Figure 13. Cr:ZnSe and Cr:ZnS gain elements (ϕ50 × 7 mm) fabricated by the PGTD method at IPG Photonics Corp.
Figure 14.
Figure 14. (a) Design of Kerr-lens mode-locked polycrystalline Cr:ZnS(Se) oscillator (not to scale). P, pump laser at 1.5–2.1 µm; L, pump focusing lens; G, AR-coated polycrystalline gain element at normal incidence; HR, high-reflectivity mirrors; HRCC, concave high reflectivity mirrors; OC, output coupler; f, the main output of the oscillator and the fundamental wavelength; 2f, 3f, etc., optical harmonics of the main signal; *, optionally, the oscillator can be equipped with a dispersive prism (P) for the wavelength tuning. For simplicity, only the end mirrors HR and OC are shown in the resonator; additional HR mirrors can be used to fold the laser beams for footprint reduction. (b) The output of a Kerr-lens mode-locked polycrystalline oscillator. SWIR fundamental band and optical harmonics, generated directly in the laser medium via the RQPM process, are dispersed with a prism are visualized with a laser viewing card. Adapted from Mirov et al. [132] with permission.
Figure 15.
Figure 15. (a) Transmission of standard air in the SWIR. (b) Measured spectra of selected Kerr-lens mode-locked polycrystalline oscillators. The spectra are normalized to average power and correspond to: (i) Pave = 1.4 W, ${\varepsilon _p}$ = 17 nJ, fR = 79 MHz, τ(S) = 20 fs, τ(IAC) = 23 fs; (ii) Pave = 2.2 W, W = 4.1 nJ, fR = 540 MHz, τ(S) = 32 fs; (iii) Pave = 0.9 W, ${\varepsilon _p}$ = 1 nJ, fR = 900 MHz, τ(S) = 76 fs, τ(IAC) = 80 fs. Here Pave is average power, ${\varepsilon _p}$ is pulse energy, fR is pulse repetition rate, and τ(S) and τ(IAC) are pulse widths derived from the spectra and measured with an interferometric autocorrelator. Measured pulse widths τ(IAC) exceed the FT limit τ(S) due to the residual third-order dispersion (TOD; output pulses are stretched by the OC’s substrates and then re-compressed by a combination of plane-parallel plates).
Figure 16.
Figure 16. (a) Single-pass polycrystalline amplifier and supercontinuum generator. L, lenses; DM, dichroic mirror. For simplicity, only the transmissive imaging components (L) are shown; actual setups may also include reflective imaging components. (b) Supercontinuum spectrum and frequency comb referencing scheme (see the text).
Figure 17.
Figure 17. Measured spectra of single-pass amplifiers presented in log-reciprocal scale. Numbers near the spectra show pulse repetition frequency (fR) input and output pulse energy (Win/out), and amplifier's gain (G). All spectra are normalized to pulse energy. (a) Low-threshold SCG in Cr:ZnS amplifier at full repetition rate of the oscillator. Output pulses were re-compressed to τ(IAC) = 19 fs (at the FT limit τ(S) = 16 fs). CW EDFL pump light is converted to 2-cycle SWIR pulses with 18% efficiency. (b) SCG in the spinning-ring Cr:ZnSe amplifier at a full repetition rate of the oscillator. The FT limit of output pulses is τ(S) = 18 fs. CW TDFL pump light is converted into a 20-W SWIR pulse train with 25% efficiency. (c) High-gain kilohertz Cr:ZnSe single-pass amplifier pumped by a 4-mJ Q-switched Er:YAG laser. The FT limit of the output pulses is τ(S) = 9.7 fs. Adapted with permission from Vasilyev et al. [184] © The Optical Society.
Figure 18.
Figure 18. Layout of the femtosecond Fe:ZnSe CPA laser system. Reprinted with permission from Migal et al. [188] © The Optical Society.
Figure 19.
Figure 19. Measured spectra of optical parametric generation in ZGP crystal. The colors of the curves correspond to the different orientations of the ZGP crystal with respect to the input beam direction. The experiments were performed in collaboration with Konstantin Vodopyanov and Peter Schunemann.
Figure 20.
Figure 20. Rotational–vibrational level structure and nomenclature. (a) Laser-active vibrational levels of the CO2 molecule, vibrational laser transitions, and numbers used to denote them in the notation system. (b) Schematic illustration of rotational splitting of vibrational levels, rotational–vibrational transitions, and examples of their notation. Diagram (b) is not to scale, and only a small subset of rotational sub-levels is shown for each vibrational level.
Figure 21.
Figure 21. Gain spectra of CO2 lasers calculated using the spectroscopic data from the HITRAN database [216] for a typical CO2:N2:He = 1:1:18 composition of the gas mixture: (a) 0.1-bar total pressure typical for glow discharge CO2 lasers; (b) 1-bar TEA CO2 laser; (c) 10-bar TEMA CO2 laser with three isotopologues of CO2 and their statistical-equilibrium mixture. The same normalization factor is applied to all four curves in (c) to show the difference in the transition intensities. The arrow shows the position of the broadest and strongest peak optimal for the short-pulse amplification.
Figure 22.
Figure 22. Temporal pulse structure of multi-terawatt CO2 lasers. (a) A 15-TW, 10.6-µm laser at UCLA. An experimentally measured pulse is shown. Adapted with permission from Haberberger et al. [218] © The Optical Society. The peak in the experimental curve is longer than the actual pulse duration due to the limited resolution of the measurement method. A Gaussian pulse corresponding to the deconvoluted 3 ps duration is shown in a dashed line. (b) A 5-TW, 9.2-µm laser at BNL. An experimentally verified result of theoretical modeling is shown. Adapted with permission from Polyanskiy et al. [232] © The Optical Society. (c) Predicted performance of a 9.3-µm system based on the final amplifier of the BNL’s laser seeded with a 300-fs, 10-mJ pulse [222]. The difference in the modulation time period is related to (a) P-branch and (b) R-branch operation.
Figure 23.
Figure 23. Phase-matching schemes in uniaxial birefringent crystals.
Figure 24.
Figure 24. Concept of frequency-domain optical parametric amplification (FOPA). Adapted from Schmidt et al. [331] with permission.
Figure 25.
Figure 25. Scheme of (a) OPA, (b) OPCPA, and (c) DC-OPCPA Reprinted with permission from Zhang et al. [334], © The Optical Society.
Figure 26.
Figure 26. Spectral ranges of various ultrafast OPCPAs. The light blue areas indicate the transparent ranges of nonlinear crystals shown on the left. Black dots are the pump wavelength, and thick black bars are the demonstrated spectral ranges of OPCPAs.
Figure 27.
Figure 27. The calculated cutoff photon energy of phase matched HHG driving by lasers at different wavelengths: red line, 4.1 μm; green line, 2.5 μm; blue line, 1.6 μm; purple line, 0.8 μm. Reprinted from Adv. Atomic Mol. Opt. Phys. 69, Han et al., “Tabletop attosecond X-rays in the water window,” pp. 1–65, Copyright 2020, with permission from Elsevier.
Figure 28.
Figure 28. HHG spectra measured with a sensitivity-calibrated X ray spectrometer. Reprinted from van Mörbeck-Bock et al. [499] with permission of SPIE.
Figure 29.
Figure 29. High-energy cutoff as a function of laser intensity. High-energy cutoff scaling in gaseous media follows a straight line whose intercept is at around the ionization potential of the atom. Solid-state high-energy cutoff shows a complex scaling, and extending beyond the atomic barrier. There are distinct intensity thresholds for solid Ar and solid Kr, where a secondary plateau appears abruptly. Adapted from Ndabashimiye et al. [500] with permission from the authors.
Figure 30.
Figure 30. High-harmonic spectra of 2D materials: (a) measured HHG spectrum from graphene [508], (b) measured HHG spectrum from monolayer MoS2 [509], and (c) calculated harmonic spectrum from hBN by for out-of-plane excitation configuration [510], where the vertical read line is the cutoff from the semi-classical atomic picture. Adapted from Yoshikawa et al. [508], Liu et al. [509], and Tancogne-Dejean and Rubio [510] with permission from the authors.
Figure 31.
Figure 31. High-order harmonic generation from 3D TIs. (a) Observation of both the odd and even-order harmonics showing the contribution from the surface as it does not exhibit inversion symmetry. Inset shows the top view of the crystal structure, one of its mirror planes, and the direction of laser polarization. (b) Valence and conduction bands of the surface states of Bi2Se3 forming a single Dirac cone. (c) In and out-of-plane spin polarization of the surface band. (d) Calculated high-harmonic response from the surface and bulk bands as a function of drive-laser ellipticity. (e) Measured high-harmonic response as a function of ellipticity. Adapted from Baykusheva et al. [519] with permission from ACS.
Figure 32.
Figure 32. A microscopic mechanism for the generation of high-order harmonics can be employed to probe the structure and dynamics of the source material. This strength was first demonstrated in diatomic molecules [523]. High harmonics from bulk crystals were reported in 2011 [23]. The underlying microscopic dynamics were discussed in terms of intraband current and interband polarization and their coupled interactions. In recent years, HHG has also been studied in 2D crystals including graphene, 3D TIs, and Weyl semimetals. The prospects include the possibility of creating, controlling, and probing novel quantum mechanical states.
Figure 33.
Figure 33. Transverse dimensions of a slightly focused laser beam at (a) 0.8 and 10.6 μm and (b) laser peak intensity as a function of distance in the air for 200-fs Ti:sapphire and 5-ps CO2 laser pulses with a peak power of P0 = 5 $P_{cr}^{Kerr}$. (Courtesy of Y. Geints, Institute of Atmospheric Optics, Tomsk, Russia.)
Figure 34.
Figure 34. The results of measurements of the spatial evolution of the CO2 laser beam propagating in the air in the range of distances from z = 0 to z = 32 m (red dots). Data points for FWHM size of the laser beam were obtained by averaging over 5–10 shots. The blue line indicates diffraction-dominated propagation of the beam in vacuum. For a few fixed propagation distances z = 11.2 m, z = 22.4 m, and z = 32 m, 2D images of the laser beam profile were recorded at CO2 laser powers above the self-guiding threshold (P0≥871GW) are shown (see Fig. 2 in Ref. [535]). Adapted with permission from Tochitsky et al. [536].
Figure 35.
Figure 35. Experimental setup of the two-color MWIR laser mixing in air. MWIR pulses (3.9 μm, 120 fs, up to 5 mJ) are focused first on a second-harmonic generator in GaSe and air producing a ∼1-cm long plasma. Terahertz generation efficiency as a function of the driving laser wavelength. Adapted with permission from Jang et al. [557]. © The Optical Society.
Figure 36.
Figure 36. (a) Structure of the self-modulated plasma wakes. (b) Transverse electric field (Ex). (c) Longitudinal electric field (Ez). The laser is linearly polarized where $x$ is the direction of polarization and z is the direction of propagation of the laser pulse. (d) On-axis transverse and longitudinal fields from a 4-J, 2-ps CO2 laser–plasma interaction simulation. The plasma density is Ne = 5.0 × 1017 cm-3. Reprinted with permission from Kumar et al., Phys. Plasmas 28, 013102 (2021) [569]. Copyright 2021, AIP Publishing LLC.
Figure 37.
Figure 37. Accelerated electrons after exiting the plasma. Simulation parameters correspond to Fig. 36. Both wakefield and DLA contribute to acceleration. Maximum energy obtained by these electrons is close to 22 MeV. Reprinted with permission from Kumar et al., Phys. Plasmas 28, 013102 (2021) [569]. Copyright 2021, AIP Publishing LLC.
Figure 38.
Figure 38. CO2 laser-driven LWFA in the blowout regime. (a i) and (b i) Charge density showing plasma “bubble”. (a ii) and (b ii) Corresponding wakefield (red) and on-axis plasma density (blue). Column (a) is obtained with a0 = 2.4: P0= 20 TW, ${\tau _p} = 0.5$ ps, w0 = 117 μm and column (b) is obtained with a0 = 3.8 : P0= 20 TW, ${\tau _p} = 0.5$ ps, w0 = 74 μm. Plasma density Ne = 2.0× 1016 cm-3. Reprinted with permission from Kumar et al., Phys. Plasmas 28, 013102 (2021) [569]. Copyright 2021, AIP Publishing LLC.

Tables (4)

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Table 1. Sub-Division of IR Radiation

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Table 2. Optical and Spectral Properties of Five Tm- and Ho-Doped Media and Yb:YAG [5456]

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Table 3. Materials Properties of ZnSe and ZnS Semiconductors [131,133]

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Table 4. Effective Nonlinear Refractive Indices for Major Air Constituents Scaled to 1 atm

Equations (64)

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$$I(r,t) = {I_0}{e^{ - \frac{{{r^2}}}{{{w^2}}}}}{e^{ - 4\ln 2\frac{{{t^2}}}{{\tau {{{}_p}^2}}}}},$$
$$w(z) = {w_0}\sqrt {1 + {{\left( {\frac{z}{{{z_R}}}} \right)}^2}} .$$
$$I(\lambda ) = {I_0}{e^{ - 2\ln (2)\frac{{{{(\lambda - {\lambda _0})}^2}}}{{\Delta {\lambda ^2}}}}},$$
$${I_0} = 1.88\frac{{{\varepsilon _p}}}{{{\tau _p}\pi {w^2}}}.$$
$$E(t) = {E_0}{e^{ - 2\ln 2{{\left( {\frac{t}{{{\tau_p}}}} \right)}^2}}}\cos ({\omega _0}t + {\varphi _{CE}}),$$
$${I_0} = \frac{1}{2}{\varepsilon _0}c{E_0}^2,$$
$$\overrightarrow E (\vec{r},t) ={-} \frac{{\partial \overrightarrow A (\vec{r},t)}}{{\partial t}} - \nabla \phi (\vec{r},t).$$
$$A(t) \approx{-} {A_0}{e^{ - 2\ln 2{{\left( {\frac{t}{{{\tau_p}}}} \right)}^2}}}\sin ({\omega _0}t + {\varphi _{CE}}),$$
$${a_0} = \frac{{{q_e}{A_0}}}{{{m_e}c}} = \frac{{{q_e}{E_0}}}{{{m_e}c{\omega _0}}},$$
$$p(t = \infty ) ={-} {q_e}A({t_0}).$$
$${I_{rel}} = \frac{1}{2}{\varepsilon _0}{c^3}{\left( {\frac{{{m_e}}}{{{q_e}}}} \right)^2}{\omega _0}^2 = 2{\pi ^2}{\varepsilon _0}{c^5}{\left( {\frac{{{m_e}}}{{{q_e}}}} \right)^2}\frac{1}{{{\lambda _0}^2}},$$
$${I_{rel}}\;[\textrm{W/c}{\textrm{m}^2}] = 1.37 \times {10^{18}}/{\lambda _0}^2\;[\mathrm{\mu m}].$$
$${w_{PPT}}({E_0},{\lambda _0}) \propto {e^{ - \frac{{2{{(2{I_p})}^{3/2}}}}{{3{E_0}}}g(\gamma )}},$$
$$g(\gamma )= \frac{3}{{2\gamma }}\left[ {\left( {1 + \frac{1}{{2{\gamma^2}}}} \right){{\sinh }^{ - 1}}\gamma - \frac{{\sqrt {1 + {\gamma^2}} }}{{2\gamma }}} \right],$$
$$\gamma = \frac{{2\pi c}}{{{\lambda _0}{E_0}}}\sqrt {2I{}_p} .$$
$$\begin{aligned} &{\hbar {\omega _X}({t,{t_0}} )= {I_p} + \frac{1}{{2{m_e}}}p{{(t,{t_0})}^2} = {I_p} + \frac{{{q_e}^2}}{{2{m_e}}}{{[A(t) - A({t_0})]}^2}}\\ &= {I_p} + \frac{{{q_e}^2{E_0}^2}}{{2{m_e}{\omega _0}^2}}{{[\sin ({\omega _0}t) - \sin ({\omega _0}{t_0})]}^2}. \end{aligned}$$
$$\hbar \omega _c^{atom} = {I_p} + 3.17\frac{{{q_e}{E_0}^2}}{{4{m_e}{\omega _0}^2}} = {I_p} + 3.17{U_p}.$$
$${U_p}[\textrm{eV}] = 9.33 \times {10^{14}}{I_0}[\textrm{W/c}{\textrm{m}^2}]{\lambda _0}{[\mathrm{\mu m}]^2}.$$
$$\hbar \omega _c^{solid} \simeq {\Delta _{gap}} + n\hbar {\omega _B},$$
$$k(t) = k({t_0}) + \frac{{{q_e}}}{\hbar }\int_{{t_0}}^t {E(t^{\prime})dt^{\prime}} .$$
$$P_{cr}^{Kerr}({\lambda _0}) \approx 3.77\frac{{{\lambda _0}^2}}{{8\pi {n_0}({{\lambda_0}} ){n_2}({\lambda _0})}}, $$
$$LIDT({\tau _p},{\lambda _0}) = \sqrt {\frac{{{\tau _p}}}{{{\tau _{spec}}}}} LID{T_{spec}}, $$
$${G_0}{(\lambda )^{s.e.}} = {e^{{\sigma _{em}}(\lambda )\Delta NL}},$$
$${\tau _p} = \frac{{2\ln 2}}{{\pi c}}\frac{{\lambda {{{}_0}^2}}}{{\Delta \lambda {}_{FWHM}}} = \frac{{2\ln 2}}{\pi }{T_0}\frac{{{\lambda _0}}}{{\Delta \lambda {}_{FWHM}}}. $$
$$\Delta {\lambda _{FWHM}} = \Delta {\lambda _a}\sqrt {\frac{3}{{10{{\log }_{10}}G({{\lambda_0}} )- 3}}} , $$
$${f_{k,j}}(T) = \frac{{{g_j}}}{{{Z_k}(T)}}{e^{ - \frac{{{E_{k,j}} - E{}_{k,1}}}{{{k_B}T}}}}, $$
$${Z_k}(T) = \sum\limits_j {{g_j}} {e^{ - \frac{{{E_{k,j}} - E{}_{k,1}}}{{{k_B}T}}}}. $$
$${g_0}(T,r,z) = \{{{\sigma_{em}}(T)\beta (r,z) - {\sigma_{ab}}(T)[1 - \beta (r,z)]} \}{N_{tot}}, $$
$${G_{small}}(T,r = 0) = {e^{\int_0^L {{g_0}(T,r = 0,z)dz} }}. $$
$${\beta _{th}}(T) = \frac{{{\sigma _{ab}}(T)}}{{{\sigma _{em}}(T) + {\sigma _{ab}}(T)}}, $$
$${\alpha _0}({\lambda _p}) = \{{{\sigma_{ab}}({\lambda_p})[1 - \beta ] - {\sigma_{em}}({\lambda_p})\beta } \}{N_{tot}}. $$
$${\sigma _{em}}({\lambda ,T} )= \sum\limits_{i,j} {\frac{{{g_j}}}{{{Z_u}(T)}}} {e^{ - \frac{{{E_j} - {E_1}}}{{{k_B}T}}}}{\sigma _{ji}}(\lambda ){g_i}, $$
$${\sigma _{em}}({{\lambda_s},T} )= {\sigma _{ab}}({{\lambda_s},T} ){e^{ - \frac{{hc}}{{{k_B}T}}\left( {\frac{1}{{{\lambda_s}}} - \frac{1}{{{\lambda_u}(T)}}} \right)}}, $$
$${\lambda _u}(T) = \frac{{hc}}{{{k_B}T}}\frac{1}{{\ln \frac{{{Z_g}(T)}}{{{Z_u}(T)}}}}. $$
$${F_{s,sat}} = \frac{{hc}}{{[{\sigma _{em}}({\lambda _s}) + {\sigma _{ab}}({\lambda _s})]{\lambda _s}}}, $$
$$F{}_{out} = {F_{s,sat}} \cdot \ln [{1 + {e^{{g_0}}}({{e^{{F_{in}}/{F_{s,sat}}}} - 1} )} ]. $$
$${F_{p,sat}} = \frac{{hc}}{{[{\sigma _{em}}({\lambda _p}) + {\sigma _{ab}}({\lambda _p})]{\lambda _p}}}. $$
$$\Delta {\nu _{FWHM}}(\textrm{GHz}) = 5.71{P_{C{O_2}}} + 4.19{P_{{N_2}}} + 3.66{P_{He}}, $$
$${\tau _R}[{\textrm{ns}} ]= {({9.75{P_{C{O_2}}} + 9.0{P_{{N_2}}} + 4.5{P_{He}}} )^{ - 1}}. $$
$${\omega _i} = {\omega _p} - {\omega _s}, $$
$${\vec{E}_{i,s,p}}(z,t) = {\vec{A}_{i,s,p}}(z){e^{i({\omega _{i,s,p}}t - {k_{i,s,p}}z)}}, $$
$${\vec{\tilde{P}}_i}({\omega _i}) = {\varepsilon _0}{\chi ^{(2)}}({\omega _i}; - {\omega _s},{\omega _p}):{\vec{\tilde{E}}_s}{({\omega _s})^\ast }{\vec{\tilde{E}}_p}({\omega _p}). $$
$$\Delta \vec{k} = {\vec{k}_p} - {\vec{k}_s} - {\vec{k}_i} = 0, $$
$$\frac{{d{A_i}}}{{dz}} ={-} i\frac{{2{d_{eff}}\omega _i^2}}{{{k_i}{c^2}}}{A_s}^\ast {A_p}{e^{ - i\Delta kz}}, $$
$$\frac{{d{A_s}}}{{dz}} ={-} i\frac{{2{d_{eff}}\omega _s^2}}{{{k_i}{c^2}}}{A_i}^\ast {A_p}{e^{ - i\Delta kz}}, $$
$$\frac{{d{A_p}}}{{dz}} ={-} i\frac{{2{d_{eff}}\omega _p^2}}{{{k_p}{c^2}}}{A_i}{A_s}{e^{i\Delta kz}}, $$
$$G_0^{OPA}({{\omega_s}} )\cong \frac{1}{4}{e^{2\sqrt {{\Gamma ^2} - {{\left( {\frac{{\Delta k}}{2}} \right)}^2}} L}}, $$
$${\Gamma ^2} = \frac{{2{\omega _i}{\omega _s}}}{{{\varepsilon _0}{c^3}}}\frac{{{d_{eff}}^2}}{{{n_i}{n_s}{n_p}}}{I_p}. $$
$$\scalebox{0.92}{$\displaystyle\Delta k({{\omega_s}} )\approx \Delta k({{\omega_{s0}}} )+ \left. {\left( {\frac{{\partial {k_i}}}{{\partial {\omega_i}}} - \frac{{\partial {k_s}}}{{\partial {\omega_s}}}} \right)} \right|{}_{{\omega _{s0}}}({{\omega_s} - {\omega_{s0}}} )+ \frac{1}{2}{\left. {\left( {\frac{{{\partial^2}{k_s}}}{{\partial {\omega_s}^2}} + \frac{{{\partial^2}{k_i}}}{{\partial {\omega_i}^2}}} \right)} \right|_{{\omega _{s0}}}}{({{\omega_s} - {\omega_{s0}}} )^2} + \ldots$}$$
$$\left. {\frac{{{\partial^2}{k_s}}}{{\partial {\omega_s}^2}}} \right|{}_{{\omega _{s0}}} = \left. {\frac{{{\partial^2}{k_i}}}{{\partial {\omega_i}^2}}} \right|{}_{{\omega _{s0}}} = 0. $$
$${\varphi _{CE,i}} ={-} \frac{\pi }{2} + {\varphi _{CE,p}} - {\varphi _{CE,s}}. $$
$$I({\omega _X}) \propto {\omega _X}^4{|{FT[D(t)]} |^2}. $$
$$\vec{D}(t) = \left\langle {a{\psi_a} + b{\psi_b}} \right|\vec{r}(t)|{a{\psi_a} + b{\psi_b}} \rangle \approx \left\langle {a{\psi_a}} \right|\vec{r}(t)|{b{\psi_b}} \rangle . $$
$$\vec{D}(t) = i{\int_0^\infty {d\tau \left( {\frac{\pi }{{ \in{+} i\tau /2}}} \right)} ^{3/2}}b(t){\vec{d}^\ast }(t){e^{iS(t,\tau )}}a(t - \tau )\vec{E}(t - \tau )\vec{d}(t - \tau ), $$
$$I({\omega _X}) \propto 1/\lambda {{}_0^5}. $$
$${|{d({\omega_x})} |^2} \propto 1/{\omega _x}^{7/2}. $$
$${n_p}(\omega {}_0) \approx 1 - \frac{{q{{{}_e}^2}}}{{2{\varepsilon _0}{m_e}}}\frac{{{N_e}}}{{{\omega _0}^2}}. $$
$${n_a}({{\omega_0}} )\approx 1 + \frac{{{q_e}^2}}{{2{\varepsilon _0}{m_e}}}\frac{{{N_a}}}{{{\omega _r}^2 - {\omega _0}^2}}, $$
$${n_{IR}}({{\omega_0}} )\approx 1 + \frac{{{q_e}^2}}{{2{\varepsilon _0}{m_e}}}\left( {\frac{{{N_a}}}{{{\omega_r}^2 - {\omega_0}^2}} - \frac{{{N_e}}}{{{\omega_0}^2}}} \right) = 1 + \frac{{{q_e}^2N}}{{2{\varepsilon _0}{m_e}}}\left( {\frac{{1 - {b^2}}}{{{\omega_r}^2 - {\omega_0}^2}} - \frac{{{b^2}}}{{{\omega_0}^2}}} \right). $$
$${b^2} = {\left( {\frac{{{\omega_0}}}{{{\omega_r}}}} \right)^2} = \frac{{4{\pi ^2}{c^2}}}{{{\omega _r}^2{\lambda _0}^2}}, $$
$$\hbar \omega _c^{atom} = {I_p} + \frac{{0.5I_p^{3.5}{\lambda _0}^2}}{{{{\left[ {\ln \left( {\frac{{0.86{I_p}{3^{2{n^\ast } - 1}}{G_{lm}}C_{{n^\ast }{l^\ast }}^2}}{{ - \ln [{1 - {b^2}({{\lambda_0}} )} ]}}{\tau_p}} \right)} \right]}^2}}}. $$
$${\lambda _p} = 2\pi c/{\omega _p}, $$
$$P_{cr}^{rel} = 17{\left( {\frac{{{\lambda_p}}}{{{\lambda_0}}}} \right)^2}[{\textrm{GW}} ], $$
$${n_p} = \sqrt {1 - \frac{{{\omega _p}^2}}{{{\omega ^2}}}} . $$
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