Scaling Law of THz Yield from Two-Color Femtosecond Filament for Fixed Pump Power

: In 3 D + time numerical simulations, we study the wavelength scaling law for the energy of terahertz (THz) radiation emitted from a two-color femtosecond ﬁlament, which forms during cofocusing into air the fundamental and second harmonics of the laser pulse. In our simulations, the central wavelength of the fundamental harmonic varied from 0.8 to 8 µ m and the numerical aperture varied from 0.006 to 0.03. While the harmonics and supercontinuum development are not extreme, so the harmonics spectra are clearly separated, the energy of the generated THz radiation is proportional to the oscillation energy of the electrons, which grows as the squared pump wavelength, and the total number of free electrons in the ﬁlament, which decreases quasi-exponentially with the pump wavelength. As a result, the scaling law for the THz energy on the pump wavelength is nonmonotonic with the maximum at 1.6–4 µ m depending on the focusing conditions.

The ponderomotive potential of a free electron in an electromagnetic wave is proportional to the squared wavelength λ 2 0 .Therefore, one can expect that the energy W THz of the THz radiation emitted from the two-color filament is proportional to λ 2 0 as well.This idea was checked experimentally in [8,18].The measured dependencies of the THz yield on the pump wavelength λ 0 , usually referred as THz energy scaling laws, increase much faster than the squared wavelength λ 0 : the fit of the measured dependencies using the power function W THz ∝ λ α 0 provides α = 4.6 ± 0.5 [8] and α = 5.6-14.3[18].The 3D + time simulations [18] reproduced these faster than λ 2 0 -scaling laws and explained such dependencies by the variation of the relative phase ϕ between ω 0 -and 2ω 0 -pulses at the output of the frequency-doubling crystal, as well as the dependence of the laser beam size and duration at the output of the optical parametric amplifier on the generated wavelength λ 0 .
The λ α 0 -trend should not be treated as universal or global since the measured dependencies W THz (λ 0 ) are nonmonotonic [19] and saturate or even decrease significantly in the long-wavelength region [8,18].This decrease of the THz yield for a long-wavelength pump was explained by the decrease in the plasma density in [8].However, in [18], this effect was associated with the variation of the relative phase ϕ between the harmonics.The measurements of the THz yield [8] were performed in the range of the pump wavelengths λ 0 = 1.2-2 µm.The conjoint fit by the function λ α 0 of the experimental results [8] and the THz yield at λ 0 = 3.9 µm measured in [12] provides the dependence of the optics-to-THz conversion efficiency with α = 2.6 close to the squared pump wavelength scaling law.The 3D + time numerical simulations of the THz emission from the two-color filament with the central wavelength λ 0 varied in the wide range from 0.6 to 10.6 µm [20] showed the optimal efficiency of the conversion into THz radiation at λ 0 = 3-4 µm.For λ 0 > 4 µm, the conversion efficiency decreases monotonically.However, the simulations [20] were performed for the fixed ratio between the pulse peak power and the critical power for self-focusing (this results in the increase of the pulse energy from 0.69 mJ at 0.6 µm to 216 mJ at 10.6 µm) and do not account for the optimization of the phase ϕ [9,[21][22][23][24] between the harmonics.Thus, it is still debated if the THz energy scaling is proportional to the squared pump wavelength, strictly monotonic, or has the optimal wavelength after which, it tends to decrease.The physical mechanisms that could saturate the λ α 0 -scaling law of the THz yield from the two-color filament are still unknown as well.
In this work, we search for the scaling law for the energy of the THz emission from the two-color femtosecond filament formed under the focusing of the pulses with a fixed energy and duration into air.To determine the scaling law, we performed a comprehensive numerical scan over the central wavelengths λ 0 in the range from 0.8 to 8 µm and focusing conditions from the numerical aperture (NA) of 0.006 to 0.03.For each pair of λ 0 and the NA, we identified the maximal THz yield W (max) THz corresponding to the optimal phase ϕ 0 between the fundamental and second harmonics [9,[21][22][23] according to the algorithm proposed in [24,25].While the harmonics are clearly separated in the spectrum (in our conditions, for NA 0.015), the dependence of the THz energy W (max) THz on the pump wavelength λ 0 is proportional to two competing factors: the free electron energy, which increases with the pump wavelength as λ 2 0 , and the total number of free electrons in the filament Q e , which quasi-exponentially decreases with λ 0 .This results in the maximum of the dependence W (max) THz (λ 0 ) at the wavelength optimal for the given focusing conditions.

Model
We simulated the two-color pulse propagation and THz generation using the unidirectional pulse propagation equation (UPPE) ( [26]) for the spatio-temporal harmonic E(ω, k r , z) of the axially symmetric electric field E(t, r, z): where t and ω = 2πν are the time and angular frequency, ) and r (k r ) are the longitudinal and transverse coordinates (projections of the wave-vector), respectively, n(ω) is the refractive index of dry air, and c is the speed of light.
The material current J(t, r, z) = ∂P (3) /∂t + J free + J abs , corresponding to the harmonic J(ω, k r , z) accommodates the third-order polarization [27]: where χ (3) corresponds to the Kerr coefficient of air n 2 = 10 −19 cm 2 /W given by ab initio quantum calculations [28], which agrees with the measurements for different pump wavelengths [29,30], the transient photocurrent of free electrons J free (t) [21] defined as where e and m e are electron charge and mass, ν c ≈ 5 ps −1 is the electron-neutral collision rate, and the absorption current responsible for the energy loss due to nonlinear ionization [31]: where is the ionization potential of a component, and α is either O 2 or N 2 .The free e (t) is calculated according to the rate equations: where N 0 = 2.7 × 10 19 cm −3 is the neutral density under atmospheric pressure, η O 2 = 0.21, is the tunnel ionization rate: where W H = 13.6 eV is the ionization potential of the hydrogen atom, ω a = 41.3 fs −1 is the atomic frequency, and E a = 5.17 GV/cm is the atomic field.
In simulations, we "focus" into air the linearly polarized two-color pulse: where E 1 and E 2 are the amplitudes of the fundamental (ω 0 ) and second (2ω 0 ) harmonics; 2τ 1 = 85 fs and 2τ 2 = 125 fs are the pulse durations of ω 0 -and 2ω 0 -pulses, respectively; ϕ is the relative phase between them; the input beam diameter is 2a 0 = 3 mm.To find the wavelength scaling law for the energy of THz radiation W THz emitted from the two-color filament, we considered pulses with several different central wavelengths λ 0 = 2πc/ω 0 in the range 0.8-8 µm.The energies of ω 0 -and 2ω 0 -pulses were fixed at 1.4 mJ and 10 µJ, respectively, for all pump wavelengths λ 0 studied.We studied the THz generation for four different focusing conditions: focal lengths of f = 5, 10, 15, and 25 cm.The former cases are barely a quasi-collimated propagation, though the vectorial effects such as the longitudinal electric field can still be neglected.For this reason, the nonparaxial input conditions [32] in the form [33] were applied to describe the focusing effect.In order to reduce the computational time, we translated the initial conditions from z = 0 to z = z 0 = 4.5, 9, 12, and 20 cm for f = 5, 10, 15, and 25 cm, respectively (see the detailed description of the translation routine in [34]).
The crucial influence of the relative phase ϕ between ω 0 -and 2ω 0 -pulses on the energy of the THz radiation W THz is well known from both the experiments carried out with 0.8 + 0.4 µm pulses [9,[21][22][23] and the 0D + time photocurrent model [21]: where A, B, and ϕ 0 do not vary for the chosen experimental conditions.In the local photocurrent model, ϕ 0 = 90 • and A = B. UPPE simulations inherit the sine dependence [24], but as soon as they account for the pulse propagation in a nonlinear dispersive medium and integrate the THz yield over the focal volume, ϕ 0 and the A/B ratio cannot be elucidated prior to the simulations.
Equation ( 8) includes three unknown constants, A, B, and ϕ 0 .Therefore, to retrieve these parameters, one has to perform three UPPE runs with three different phases ϕ = ϕ (j) (here, j = 1, 2, 3) and estimate the THz energy W (j) THz at the end of the filament for these three phases.For the arbitrary relation between the phases ϕ (j) , the values of A, B, and ϕ 0 can be found from the system of three transcendental equations.In the case of the special choice of the phases between the fundamental and second harmonics ϕ (1) = 0 • , ϕ (2) = 60 • and ϕ (3) = 120 • , this system has the solution in the elementary functions [24,25]: Using the parameters A and B, one can estimate the maximal THz yield for the certain pair (λ 0 , f ) as W (max) THz = A + B.

Results
The simulated maximal energy W THz of the THz radiation is shown in Figure 1 as a function of the pump wavelength λ 0 for all focuses f studied.In the range λ 0 = 0.8-2 µm, the THz energies are almost the same for any f ; however, with the further increase in the wavelength λ 0 , the influence of the focusing on the dependence W (max) THz (λ 0 ) becomes significant.The THz energy increases monotonically in the range of λ 0 from 0.8 to 8 µm in the case of 5 cm focusing (NA = 0.03).For longer focuses, there is a maximum in the dependence W (max) THz (λ 0 ) at ∼4 µm for 10 cm (NA = 0.015) focusing and at 1.6-2 µm for 15 cm (NA = 0.01) and 25 cm (NA = 0.006) focusing.In the experiment [8] performed in relatively close conditions (NA ≈ 0.01, 2τ 1 ≈ 72 fs, pulse energy of 0.4 mJ), the maximum of the THz energy was at λ 0 = 1.8 µm.THz obtained from UPPE simulations versus the pump wavelength λ 0 for four focal distances: 5 cm (blue squares), 10 cm (red stars), 15 cm (violet circles), and 25 cm (green triangles).
The major mechanism of THz generation from two-color femtosecond filament in gases is the transient photocurrent of free electrons [21,35].Therefore, the resulting THz energy should depend on the overall amount of free electrons in the plasma channel: where N e (r, z) is the plasma density remaining after the pulse passes.In our case of the fixed pump pulse energy, for all focusing studied the dependence Q e (λ 0 ) decreases quasiexponentially with the increase in the pump wavelength λ 0 : Q e (λ 0 ) ∝ exp(−Γλ 0 ), where Γ depends on the focal distance f (see Figure 2).12)) on the pump wavelength λ 0 for four focal distances: 5 cm (blue squares), 10 cm (red stars), 15 cm (violet circles), and 25 cm (green triangles).Every pair (λ 0 , f ) is represented by three points corresponding to three UPPE runs with different phases ϕ, resulting in a slightly different amount of electrons.
For f ≥ 10 cm, the dependence W (max) THz (λ 0 ) is nonmonotonic (Figure 1).Let us assume W (max) THz (λ 0 ) = Q e (λ 0 ) × F (λ 0 ), where F (λ 0 ) is an unknown function.To provide the maximum in the dependence W (max) THz (λ 0 ), the function F (λ 0 ) should grow up with the increase in the wavelength λ 0 more slowly than the exponential function.Any power function F (λ 0 ) ∝ λ α 0 with α > 0 satisfies this condition; however, only the choice of α = 2 is physically justified, since for the fixed pulse intensity, the electron oscillation energy is proportional to λ 2 0 .Figure 3 shows the dependencies of the maximal energy W (max) THz of the generated THz radiation on the pump wavelength λ 0 in comparison with Q e (λ 0 )λ 2 0 for four focal distances.For all focuses studied except the shortest one ( f = 5 cm), these dependencies are in good agreement (cf. the filled and open markers in Figure 3), and the simulated dependencies see the solid curves in Figure 3a-c.The proposed fit for the maximal THz energy (13) explains why the λ 2 0 -scaling laws cannot be observed in experiments even if the pulse durations, radii, phases, etc., were not changed as the pump wavelength varies.According to Equation ( 13), the most efficient THz generation is achieved at λ opt = 2/Γ.Therefore, Equation ( 13) can be rewritten as W (max) THz ∝ λ 2 0 exp −2λ 0 /λ opt .The squared pump wavelength factor dominates for λ 0 λ opt /2.Since the maximum of the dependence W THz (λ 0 ) is reached at λ opt = 2-4 µm, the λ 2 0 -scaling of the THz yield could be expected for the pump in the ultraviolet part of spectrum, where the conversion efficiency into the THz range is quite low [36].Let us now return to the case of 5 cm focusing, for which the simulated dependence W (max) THz (λ 0 ) qualitatively differs from the rest (see Figure 3): it increases monotonically, despite the dependencies Q e (λ 0 ) and Q e (λ 0 )λ 2 0 being similar to the other cases; see the blue squares in Figure 2 and the maximum of open squares in Figure 3d at λ 0 ≈ 7 µm.The reason for the discrepancy between the dependencies W (max) THz (λ 0 ) and Q e (λ 0 )λ 2 0 could be associated with the features of the pulse field transformation during the propagation.Let us investigate more deeply the pulse spectra at the end of the filament (Figure 4) so as to reveal the reason for such behavior.In the case of λ 0 = 2 µm for both f = 5 and 10 cm, the pulse spectra are similar, except for slightly more broadened harmonics in the spectral domain in the case of 5 cm focusing; see Figure 4a.In contrast, for λ 0 = 8 µm, the difference between the spectra for the two focusing conditions is significant (Figure 4b).In the case of 5 cm focusing, the optical harmonics are indistinguishable against the supercontinuum background, and the THz generation process is no longer a purely two-color process.The explanation of the harmonics vanishing against the supercontinuum is the following.The µm filament the third harmonic only [37].In contrast, in the experiment [38] with the 3.9 µm femtosecond filament, the harmonics up to the ninth order (wavelength of ∼0.4 µm) were observed.The pulse with the same wavelength of 3.9 µm generates the third and fifth harmonics during collimated propagation without filamentation [39].However, while a harmonic is well distinguished, its width in the frequency domain is determined by the initial pulse spectral width and does not decrease while one preserves the ∼100 fs pulse duration.Therefore, the increase in the pulse wavelength results in "denser" localization of the harmonics in the frequency domain.When the pulse intensity reaches the ionization threshold, the harmonics (including the fundamental one) broaden and can overlap with each other, thus forming the multioctave supercontinuum.This effect was observed experimentally for 0.8 µm pump [40] and reproduced in numerical simulations for the 3.9 µm [41] and 10 µm [42] ones.The studies [40][41][42] were performed in a collimated (or quasi-collimated) geometry of propagation.Therefore, the pulse travels a long distance inside the weakly ionized air.This provides the significant broadening of the spectrum and the vanishing of the harmonics.In our case of 5 cm focusing, the filament's length is ∼6 mm and the comparable spectral broadening of the harmonics occurs due to a much higher ionization degree.For λ 0 = 6-8 µm, the total number of free electrons for f = 5 cm is 5-10-times higher than in the case of f = 10 cm (see Figure 2).This difference in the plasma density is enough to overlap the harmonics closely localized in spectrum and make them indistinguishable against the supercontinuum background; see Figure 4.

Conclusions
In conclusion, using the self-consistent non-paraxial propagation model with the nonlinear source, which includes the third-order response of bound electrons and the transient photocurrent, we performed a numerical scan over the central wavelengths in the range from 0.8 to 8 µm and focusing conditions from the numerical aperture of 0.006 to 0.03 to determine the scaling law for the energy of THz emission from the two-color femtosecond filament.When the spectral broadening of the optical harmonics is not very large and they are clearly seen above the supercontinuum background, the THz yield is proportional to the total number of free electrons in the filament and the electron oscillation energy.For the fixed two-color pulse parameters, the number of electrons rapidly decreases with the increase in the pulse central wavelength (our simulations showed an exponential decrease).The energy of an electron oscillating in the electromagnetic wave is scaled proportionally to the squared central wavelength.The joint effect of these two competing factors leads to the nonmonotonic scaling law for the THz energy with the maximum at the wavelength 1.6-4 µm, which depends on the focusing conditions.

Figure 1 .
Figure 1.The maximal energy of generated THz radiation W (max)

Figure 2 .
Figure 2. Dependences of the overall amount of free electrons in the filament Q e (see Equation (12)) on the pump wavelength λ 0 for four focal distances: 5 cm (blue squares), 10 cm (red stars), 15 cm (violet circles), and 25 cm (green triangles).Every pair (λ 0 , f ) is represented by three points corresponding to three UPPE runs with different phases ϕ, resulting in a slightly different amount of electrons.

Figure 3 .
Figure 3. Dependence of the maximal energy of generated THz radiation W (max) THz obtained from UPPE simulations on the pump wavelength λ 0 (filled markers) in comparison with Q e λ 2 0 (triplets of open markers) for four focal distances: (a, green) 25 cm, (b, violet) 15 cm, (c, red) 10 cm, and (d, blue) 5 cm.Solid colored curves show the fits (13).

Figure 4 .
Figure 4. Comparison of the spectra obtained in UPPE simulations at the end of the filament in the case of 5 cm (blue) and 10 cm (red) focusing of the pulses with the central wavelength of (a) 2 µm and (b) 8 µm.