Journal of Infrared, Millimeter, and Terahertz Waves

, 32:1157

Analysis of Dual Frequency Interaction in the Filament with the Purpose of Efficiency Control of THz Pulse Generation

Authors

    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Nikolay A. Panov
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Roman V. Volkov
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Vera A. Andreeva
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Aleksey V. Borodin
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Mikhail N. Esaulkov
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Yanping Chen
    • Centre d’Optique, Photonique et Laser (COPL) and Département de physique, de génie physique et d’optique, Université Laval
  • Claude Marceau
    • Centre d’Optique, Photonique et Laser (COPL) and Département de physique, de génie physique et d’optique, Université Laval
  • Vladimir A. Makarov
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Alexander P. Shkurinov
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • Andrey B. Savel’ev
    • International Laser Center & Faculty of PhysicsM.V. Lomonosov Moscow State University
  • See Leang Chin
    • Centre d’Optique, Photonique et Laser (COPL) and Département de physique, de génie physique et d’optique, Université Laval
Article

DOI: 10.1007/s10762-011-9820-7

Cite this article as:
Kosareva, O.G., Panov, N.A., Volkov, R.V. et al. J Infrared Milli Terahz Waves (2011) 32: 1157. doi:10.1007/s10762-011-9820-7

Abstract

Cross-guiding of the 400 nm second harmonic of the Ti:Sapphire laser in the femtosecond filament produced by an 800 nm pump in argon leads to the efficient terahertz generation along the longitudinally extended high intensity region. Based on the vectorial model of the dual pulse co-propagation we found that terahertz yield due to four-wave mixing in the filament maximizes for the same temporal delay between 400 nm and 800 nm pulses as the 400 nm signal after the analyzer crossed to its initially linear polarization direction. This optimum delay goes up with increasing geometrical focusing distance and leads to the maximum terahertz yield if the initial 800 nm pump and the second harmonic polarization directions are parallel to each other.

Keywords

Terahertz (THz) generationFemtosecond filamentFour-wave mixing (FWM)

The possibility of low frequency (1012-1013 Hz) electromagnetic pulse generation from femtosecond light filaments in gases has been proposed in [1, 2] and demonstrated experimentally in [3, 4]. Femtosecond plasma channels can be localized at a desired distance [5, 6], producing a remote source of low frequency radiation including the THz range. If the pump pulse at the fundamental 800 nm frequency (ω) of Ti:Sapphire laser amplification system co-propagates together with its second harmonic (2ω) in air or noble gases, the energy of the nonlinearly generated low frequency component might be enhanced at least by an order of magnitude [79] as compared with THz generation in a single-color scheme.

The dual-color filamentation was used as a source of pulsed radiation tunable from the ultraviolet [10] through the visible [11] to the far infrared and THz ranges [9, 1216]. The suggested explanations are based on either the FWM process [713] or a transient photocurrent induced by free electrons driven by the asymmetric field of the two pump pulses [1416]. The comparative analysis of the effects of transient photocurrent, radiation pressure force, ponderomotive force on the THz generation in the presence of the fundamental and the second harmonic are given in [17].

The origin of THz radiation in the course of filamentation is now actively discussed in the literature [13, 1518]. However, the experiment that could ultimately answer the question if the plasma or the neutral molecules are responsible for THz generation in a dual—color filament is nontrivial. Indeed, plasma always accompanies filamentation, since it stops the self-focusing collapse and sustains dynamic equilibrium between the nonlinear divergence and contraction of the radiation, allowing the long-range propagation [19, 20]. At the same time the plasma density in 1 atm gas is rather weak, typically 1016-1017 cm−3, therefore the nonlinearity of neutrals may contribute essentially to the generation of new frequencies. Independent of the origin of THz radiation, the length of the filament [4, 21] as well as proper synchronization of the 1st and the 2nd harmonic in space and time [9, 13, 16, 17, 22, 23] are the key issues for the efficient energy conversion from the visible and near infrared to THz and far infrared ranges.

In this paper we study co-propagation of the 800 nm Ti:Sapphire pump (ω), producing a filament in argon, together with its second harmonic (2ω) based on the developed vectorial model of nonlinear interaction in gases. The THz radiation energy generated due to FWM is calculated as the function of the angle between the initial polarization directions of the ω and 2ω light fields. The conversion efficiency to THz is optimized by the adjustment of the ω − 2ω delay and obtaining the maximum second harmonic signal after the analyzer crossed to the initial polarization of the 2ω pulse. The effect of the geometrical focusing distance increase on the THz generation is discussed.

Numerical simulations were performed for the case of collinear propagation of the fundamental and the second harmonic of the 50 fs Ti:Sapphire laser pulse. The schematic picture of the experimental setup, which we assumed when performing numerical simulations, is shown in Fig. 1a, b. A single filament was created by focusing a 1.1 mJ, linearly polarized (0°) 50 fs Ti:Sapphire pump pulse into a gas cell filled with argon. The fundamental beam was split into two beams, one of which was used to form a filament and the other one to produce the second harmonic pulse (100 fs, 1 μJ). At the output of the cell we numerically “analyzed” the polarization state of the second harmonic pulse as shown in Fig. 1a or imitated the collection of THz radiation from the filament by a parabolic mirror (Fig. 1b). According to [24], the forward emission is more than three orders stronger than that of the sideways radiation. The experimental schemes assumed in the simulations are close to the ones used in Refs. [22, 23]. The difference with the latter is the longer geometrical focusing distance allowing us to simulate extended filaments.
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Fig. 1

The schematic picture of the experimental setup assumed when performing numerical simulations. a Analysis of the second harmonic pulse polarization. b Collection of terahertz radiation from the filament.

A theoretical analysis of the filament-induced THz generation requires the inclusion of the full spatiotemporal dynamics of both the 1st and the 2nd harmonic pulses [25]. Our vectorial model is a straightforward extension of the scalar version used in the previous studies of filamentation [19, 26], where the ultrashort 800 nm pump pulse is affected by material dispersion, diffraction, Kerr nonlinearity and self-produced plasma. Together with the equations for the second harmonic, the system below represents the cross-interaction of the two pulses and allows one to choose arbitrary relation between the 1st and the 2nd harmonic pulse energy and the initial polarization directions. The light field complex amplitude E contains the two components Ex and Ey in the plane perpendicular to the propagation direction z. The transition to the circularly polarized basis E± is performed according to the expressions E± = Ex ± iEy, with the light field intensity expressed as \( I = c{n_0}\left( {{{\left| {{E_{ + }}} \right|}^2} + {{\left| {{E_{ - }}} \right|}^2}} \right)/\left( {16\pi } \right) \). The coordinate system moves with the group velocity vgω (τ = t – z/vgω), the subscripts ω and 2ω denote the values related to the pump and the second harmonic, respectively:
$$ 2i{k_{\omega }}{\partial_z}{E_{{\omega \pm }}} = {\Delta_{ \bot }}{E_{{\omega \pm }}} - {k_{\omega }}k_{\omega }^{{\prime \prime }}\partial_{{\tau \tau }}^2{E_{{\omega \pm }}} + 2k_{\omega }^2\Delta {n_{{\omega \pm }}}{E_{{\omega \pm }}} + 2k_{\omega }^2\delta {n_{{\omega \pm }}}{E_{{\omega \mp }}} - i{k_{\omega }}{\alpha_{\omega }}{E_{{\omega \pm }}}, $$
(1)
$$ 2i{k_{{2\omega }}}{\partial_z}{E_{{2\omega \pm }}} = {\Delta_{ \bot }}{E_{{2\omega \pm }}} - 2i{k_{{2\omega }}}\delta_v^{{ - 1}}{\partial_{\tau }}{E_{{\omega \pm }}} - {k_{{2\omega }}}k_{{2\omega }}^{{\prime \prime }}\partial_{{\tau \tau }}^2{E_{{2\omega \pm }}} + 2k_{{2\omega }}^2\Delta {n_{{2\omega \pm }}}{E_{{2\omega \pm }}} + 2k_{{2\omega }}^2\delta {n_{{2\omega \pm }}}{E_{{2\omega \mp }}} - i{k_{{2\omega }}}{\alpha_{{2\omega }}}{E_{{2\omega \pm }}}, $$
(2)
where \( {\Delta_{ \bot }} = \left( {1/r} \right)\partial /\partial r\left( {r\partial /\partial r} \right) \) is a Laplacian, \( {k_{\omega }},{k_{{2\omega }}},k_{\omega }^{{\prime \prime }},k_{{2\omega }}^{{\prime \prime }} \) are the wavenumbers and the second-order dispersion coefficients, \( \delta_v^{{ - 1}} = v_{{g2\omega }}^{{ - 1}} - v_{{g\omega }}^{{ - 1}} \) is the group velocity walk-off. The self- and cross-action terms in an isotropic medium with comparatively weak dispersion described in Ref. [27] in the visible range are given by:
$$ \Delta {n_{{\omega \pm }}} = {n_2}\left( {{{\left| {{E_{{\omega \pm }}}} \right|}^2} + 2{{\left| {{E_{{\omega \mp }}}} \right|}^2} + {{\left| {{E_{{2\omega \pm }}}} \right|}^2} + {{\left| {{E_{{2\omega \mp }}}} \right|}^2}} \right)/6 - 2\pi {e^2}{N_e}/\left( {{m_e}{\omega^2}} \right),\delta {n_{{\omega \pm }}} = {n_2}{E_{{2\omega \pm }}}E_{{2\omega \mp }}^{*}/6, $$
(3)
$$ \Delta {n_{{2\omega \pm }}} = {n_2}\left( {{{\left| {{E_{{2\omega \pm }}}} \right|}^2} + 2{{\left| {{E_{{2\omega \mp }}}} \right|}^2} + {{\left| {{E_{{\omega \pm }}}} \right|}^2} + {{\left| {{E_{{\omega \mp }}}} \right|}^2}} \right)/6 - \pi {e^2}{N_e}/\left( {2{m_e}{\omega^2}} \right),\delta {n_{{2\omega \pm }}} = {n_2}{E_{{\omega \pm }}}E_{{\omega \mp }}^{*}/6, $$
(4)
$$ {\partial_{\tau }}{N_e} = \left( {{R_{\omega }} + {R_{{2\omega }}}} \right)\left[ {{N_0} - {N_e}\left( \tau \right)} \right], $$
(5)
where we assumed that argon is an isotropic medium with the following relations for nonzero components of the third order susceptibility tensor χ(3) in the plane (x, y) perpendicular to the propagation direction z: \( \chi_{{xxyy}}^{{(3)}} = \chi_{{yyxx}}^{{(3)}} = \chi_{{xyxy}}^{{(3)}} = \chi_{{yxyx}}^{{(3)}} = \chi_{{xyyx}}^{{(3)}} = \chi_{{yxxy}}^{{(3)}} = \chi_{{xxxx}}^{{(3)}}/3 = \chi_{{yyyy}}^{{(3)}}/3 \). The Kerr nonlinear coefficient n2 in the Eqs. 1, 2 is given by \( {n_2} = 3\pi \chi_{{xxxx}}^{{(3)}}/{n_0} \). The choice of the circularly polarized basis is motivated by the fact that in this basis the nonlinear contributions to the nonlinear refractive index \( \Delta {n_{{\omega \pm }}} \), \( \Delta {n_{{2\omega \pm }}} \) depend on the intensity [28, 29], and the phase dependence arises due to ω-2ω cross-interaction only in the nonlinear terms \( \delta {n_{{\omega \pm }}} = {n_2}{E_{{2\omega \pm }}}E_{{2\omega \mp }}^{*}/6, \)\( \delta {n_{{2\omega \pm }}} = {n_2}{E_{{\omega \pm }}}E_{{\omega \mp }}^{*}/6 \) on the right-hand side of Eqs. 1, 2 (see also Eqs. 3, 4).

Free electron generation Ne is taken into account by the optical-field-induced ionization rates Rω and R2ω of argon calculated according to Ref. [30]. For the ionization rate calculations, polarization of both the pump and the second harmonic radiation was considered as linear. This assumption is reasonable since in the simulations the maximum value of the pump light field component orthogonal to the initial linear polarization direction of the pump radiation is less than a parallel component by a factor of 1000. The second harmonic pulse becomes strongly elliptical, at the same time its maximum intensity value remains below 3 × 1011 W/cm2 and the fractional contribution to the free electron density Ne remains below 10−5% of the pump pulse contribution.

For the numerical simulations we have considered co-propagation in argon of initially linearly polarized Gaussian input pump and second harmonic laser pulses with central wavelengths λω = 800 nm, λ = 400 nm, e−1 of the intensity level beam radii a = a02ω = 0.5 mm, full-width-half-maximum pulse durations τω = 45 fs, τ = 100 fs, initial energy Wω = 1 mJ, W = 1 μJ, respectively. The delay between the pulses was varied from -25 fs (the pump goes first) to 150 fs and the geometrical focusing distance was varied from 20 to 100 cm. The initial pump light field vector Eω was parallel to the x-axis, while the vector E constituted an angle ψ with the pump vector. The value of ψ in the counterclockwise direction ranged from 7.5° to 82.5° with a step of 7.5°. The medium parameters are \( k_{\omega }^{{\prime \prime }} \)≈ 1.4 × 10−31 s2/cm, \( k_{{2\omega }}^{{\prime \prime }} \) ≈ 3.1 × 10−31 s2/cm, \( \delta_v^{{ - 1}} \) ≈ 7.1 10−16 s/cm. The critical power for self-focusing of the pump pulse is Pcr_ω ≈ 9 GW and for the second harmonic pulse Pcr_ ≈ 2.25 GW, under the assumption n2_ω = n2_ = n2, and for n2 ≈ 10−19 cm2/W. Thus, the ratio Ppeak/Pcr_ω for the pump pulse is 2.7, i.e. enough for the propagation in a single filament regime, while for the second harmonic Ppeak/Pcr_ ≈ 0.004, that means no nonlinear effects can occur without the pump pulse. Thus, the simulation parameters correspond to the experimental ones, except that the delay and focusing geometry were varied to find the optimum conversion efficiency to THz.

The pump Eω= (Eωx, Eωy) and the second harmonic E= (E2ωx, E2ωy) light fields are obtained from the Eqs. 15 in each radial r and longitudinal position z as well as in the local time τ. According to the four-wave mixing process 0 = 2ω − ω − ω, the third-order polarization at the THz frequency \( {\mathbf{P}}_{{THz}}^{{(3)}} = \left( {P_{{xTHz}}^{{(3)}},P_{{yTHz}}^{{(3)}}} \right) \) can be in the isotropic medium expressed as [31]:
$$ P_{{xTHz}}^{{(3)}} = \left( {\chi_{{xxyy}}^{{(3)}} + \chi_{{xyxy}}^{{(3)}} + \chi_{{xyyx}}^{{(3)}}} \right){E_{{2\omega x}}}E{_{{\omega x}}^{*2}} + \left( {\chi_{{xyxy}}^{{(3)}} + \chi_{{xyyx}}^{{(3)}}} \right){E_{{2\omega y}}}E_{{\omega x}}^{*}E_{{\omega y}}^{*} + \chi_{{xxyy}}^{{(3)}}{E_{{2\omega y}}}E{_{{\omega y}}^{*2}}, $$
(6)
$$ P_{{yTHz}}^{{(3)}} = \left( {\chi_{{xxyy}}^{{(3)}} + \chi_{{xyxy}}^{{(3)}} + \chi_{{xyyx}}^{{(3)}}} \right){E_{{2\omega y}}}E{_{{\omega y}}^{*2}} + \left( {\chi_{{xyxy}}^{{(3)}} + \chi_{{xyyx}}^{{(3)}}} \right){E_{{2\omega x}}}E_{{\omega x}}^{*}E_{{\omega y}}^{*} + \chi_{{xxyy}}^{{(3)}}{E_{{2\omega y}}}E{_{{\omega x}}^{*2}}, $$
(7)
where
$$ {E_x} = \frac{1}{2}\left( {{E_{ + }} + {E_{ - }}} \right),{E_y} = \frac{i}{2}\left( {{E_{ - }} - {E_{ + }}} \right). $$
(8)
As shown in [17], a THz field is represented by the optical rectification term \( {E_{{2\omega }}}E_{\omega }^{*}E_{\omega }^{*} \) independently of the source, which can be both free and bound electrons. To our knowledge, the absolute values of \( {\chi^{{(3)}}}\left( {0;2\omega; - \omega; - \omega } \right) \) tensor components have not been measured experimentally for femtosecond radiation. The electron—neutral collision frequency, which defines the THz driven force (see Eq. 5 of Ref. [17]), varies not less than within an order of magnitude 1012-10131/s. Therefore, quantitative estimate of the partial contribution of neutrals and free electrons into THz radiation cannot be performed without an additional research. At the same time, the spatio-temporal overlap of the 1st and the 2nd harmonics remains of crucial importance for the efficient THz generation. Thus, in this paper we will focus on the ω-2ω propagation and interaction allowing us to optimize this overlap.
Further on by the THz radiation intensity we will assume the value
$$ {I_{{THz}}} = \frac{c}{{8\pi }}\left( {P{{_{{xTHz}}^{{(3)}}}^2} + P{{_{{yTHz}}^{{(3)}}}^2}} \right), $$
(9)
and by the THz energy WTHz, the value integrated over the transverse spatial domain and the pulse extension as:
$$ {W_{{THz}}} = 2\pi \int\limits_{{ - \infty }}^{\infty } {\int\limits_{{ 0 }}^{\infty } {{I_{{THz}}}rdrd\tau } }, $$
(10)
Equation 10 gives the estimate of THz energy emitted locally by the filament at each propagation position z. THz propagation effects are not accounted for in our simulations. As shown in [24], the phase mismatch effects play an important role in the angular distribution of THz radiation. However, the overall THz energy from the filamenting pulse in a given focusing geometry does not depend on the phase of electromagnetic field. Experimentally this fact has been confirmed in [4], where a THz signal irradiated by 40 multiple filaments in the forward direction increased by a factor of 40 as compared to a single filament case. In our simulations the energy given by Eq. 10 is summed up at each grid step along the propagation direction and normalized to the filament length.
The intensity in Fig. 2a and the fluence in Fig. 2 (b, c) show a single filament created by a pump pulse (Fig. 2b) and the corresponding distribution of the second harmonic radiation (Fig. 2c) arising due to cross-interaction with the filament. The geometrical focusing distance is 50 cm and the second harmonic is sent ≈ 55 fs ahead of the pump to compensate the group velocity walk-off and to provide us with the maximum overlap between the two pulses. With this optimum delay the peak intensity of the second harmonic is by a factor of six higher in the filament than in the case of geometrical focusing without the pump (compare the curves marked by squares and circles in Fig. 2a). Besides, the fluence channel elongates in the propagation direction and contracts in the transverse direction in the presence of the pump (Fig. 2c) as compared with the case of no pump (Fig. 2d). This is due to the cross-guiding of the second harmonic in the filament.
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Fig. 2

a The peak intensity of the second harmonic as the function of the propagation distance in the presence of the 800 nm pump (squares) and without the pump (circles); (b, c, d) fluence distribution of the pump (b), the second harmonic in the presence of the pump (c) and the second harmonic without the pump (d). (e, g) The on-axis (r = 0) THz pulse intensity at z = 37 cm (e) and z = 57 cm (g) obtained from the overlap of the fundamental and the second harmonic pulses at the corresponding propagation distance. The spatio-temporal distribution of the ω − 2ω overlap is shown at z = 37 cm (f) and z = 57 cm (h) by the narrower blue contour (2ω) over a wider red contour (ω). For all panels (a-h) the simulation results are presented for 50 fs, 1 mJ 800 nm pump and 100 fs, 1 μJ second harmonic, both focused geometrically into 1 bar argon with f = 50 cm lens, the second harmonic is sent ≈ 55 fs ahead of the pump, the angle between the initial linear polarization directions is 45 degrees.

The largest conversion efficiency to THz radiation is at the positions of the maximum overlap of the fundamental and the second harmonic pulses. This takes place at the beginning of the filament at z ≈ 35–40 cm (Fig. 2e, f), where the second harmonic pulse repeats both the high-intensity self-compressed part in the front of the pump pulse as well as the diverging rings at the trailing part of the pulse (Fig. 2f, τ = −18 fs, the narrower blue contour within the wider red one, corresponding to the pump). Further in the propagation the second harmonic delays due to the group velocity walk-off, and the THz radiation is generated at the trailing edge of the pump pulse (Fig. 2g, h). The extension of the second harmonic pulse is from −25 to 25 fs, however, the efficient conversion to THz occurs on the trailing edge only (5-25 fs), where the cross-guiding takes place (Fig. 2h, compare narrow and extended blue outer contour of the second harmonic with the two maxima in the local time range 5–25 fs resulting from the cross-guiding). The real part of the time-dependent nonlinear polarization at THz frequency given by the Eqs. 6, 7 is shown in Fig. 3a. The presented waveform (Fig. 3a) results from the overlap of the high-intense part of the spatio-temporal distribution of the fundamental and the second harmonic radiation in Fig. 2f. Appearance of a direct current component in the frequency spectrum of the nonlinear THz polarization (Fig. 3b) is associated with the temporal asymmetry of the complex amplitude of the interacting ω and 2ω pulses. The bandwidth of the source equal to 40THz (Fig. 3b) originates from 25 fs characteristic temporal width in Fig. 3a, as received experimentally in [15]. We note that a waveform in Fig. 3a is the local near-field contribution to the THz radiation emitted as a result of ω-2ω interaction. The actual on-axis THz electric field profile emerging from the filament is the result of interference of electromagnetic THz waves, similar to the one shown in Fig. 3a and generated from all the local sources along the channel. Therefore, the profile that could be registered experimentally may differ significantly from the local time-dependent nonlinear polarization at THz frequency. For the detailed comparison with the experiment a THz propagation equation should be included into the simulations, which is the purpose of our future work.
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Fig. 3

The THz waveform and the frequency spectrum corresponding to the intensity shown in Fig. 2(e).

The interaction length equal to the length of the high intensity filament region is a key parameter for the increase in the conversion efficiency to THz radiation. This is confirmed in Fig. 4a, where the energy of THz (Fig. 4a, squares) is growing fast in the high-intensity region (Fig. 4a, 37–50 cm, circles) and saturates for z > 100 cm with the decay of the filament. The amount of THz energy accumulated by the end of the filament in the simulations is 4 nJ in agreement (within an order of magnitude) with 10 nJ THz energy obtained in the experiment with the filter transmission cutoff at 300 THz [32]. The corresponding THz conversion efficiency in the simulations is 4nJ/1 mJ = 4 × 10−6. In the similar dual-color scheme with long focusing geometry the larger conversion efficiency (8 × 10−4) to the newly generated pulse was obtained. The energy of this emerging pulse was 1.5 μJ, while the initial fundamental and the second harmonic pulse energy was 1.8 mJ and 150 μJ, respectively [12]. Therefore, a factor of 100 increase in the conversion efficiency in [12] as compared with our simulation results and the results of Ref. [32] is most likely due to the larger second harmonic pulse energy (1 μJ in our simulations and 150 μJ in Ref. [12]) and broadband registration range from 1500 to 1 THz in [12]. Recently, Thomson et al. [33] used the dual-color scheme to obtain 360 nJ pulse with a bandwidth of 100 THz. The high THz conversion efficiency of 360 nJ/420 μJ = 8.6 × 10−4 was attained with tight focusing and an optimized geometry of the second-harmonic generation crystal, producing a 2ω-field detuned from the second-harmonic of the ω-field, which promotes both a high polarization bandwidth and optimal ω-2ω temporal overlap. Therefore, the conversion efficiency obtained in the simulations may be further increased by increasing the second-harmonic pulse energy and controlling temporal overlap between ω and 2ω pulses.
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Fig. 4

a Accumulation of THz energy along the propagation distance (squares) and the peak intensity of the fundamental pulse (circles) for the radiation parameters and focusing geometry shown in Fig. 2. b The energy of THz radiation accumulated by the end of the filament as the function of the delay between ω and 2ω pulses (stars) and the energy of the second harmonic after the analyzer crossed to the initial polarization direction of the 2ω light field (circles). Except for the delay time, all the other parameters of the radiation and the focusing geometry are the same as in Fig. 2.

The detection of THz radiation with comparatively low amplitude might not be a trivial task and requires special equipment. It is much easier to detect the 400 nm second harmonic propagating in the forward direction by blocking the pump with 800 nm mirror. Four-wave mixing in the filament leads to the appearance of the second harmonic signal polarized perpendicularly to its original polarization direction [25]. The detection of this signal can be performed by inserting the crossed analyzer after the 800 nm mirror. The remarkably good correlation between the second harmonic signal after the crossed analyzer and the overall THz yield (Fig. 4b) is due to the similar nonlinear origin of the two “outcomes” of four-wave mixing. The optimization curves (Fig. 4b) are shown for one chosen geometrical focusing distance of 50 cm. By studying the THz and the second harmonic yields after the crossed analyzer, we obtained the optimization curves for f = 20 cm, 30 cm and 100 cm focal lengths (Fig. 5). The optimum temporal delay grows up with growing f since the start of the filament moves away from the laser system output according to the Marburger formula [34]. The increase in the focal and the interaction length results in the increasing walk-off distance between the first and the second harmonics. Thus, the optimum delay is adjusted by using the second harmonic signal. Then, after the ω-2ω delay is settled, the terahertz radiation measurements can be performed. In the case of tight geometrical focusing resulting in the filament length decrease down to 1 cm for the same initial 800 nm pulse with the energy of the order of 1 mJ, the four-wave mixing will still take place in the filament, however, the nonlinearly generated component of the second harmonic registered after the crossed analyzer will be strongly diverged in the laser-produced plasma. The registration of the diverging second harmonic is more complicated in the tight focusing geometry with the plasma density ~1017 cm−3 than in the loose focusing geometry with the plasma density ~1016 cm−3.
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Fig. 5

Increase in the optimum (for THz yield) ω − 2ω delay with increasing geometrical focusing distance.

When employing the second harmonic signal after the crossed analyzer for the delay optimization, one should take into account that this signal depends on the mutual orientation of the initial linear polarizations of 800 nm and 400 nm pulses. It was experimentally registered and theoretically justified that the maximum second harmonic signal after the crossed analyzer is reached for the angle of 45° between ω-2ω initial polarization directions [25]. However, if we consider the yield of the second harmonic signal integrated over all polarization directions in the filament core, it maximizes for the parallel initial orientation of 800 nm and 400 nm polarization directions. Similarly, due to the four-wave mixing mechanism of THz generation, the maximum THz yield is attained for the parallel orientation of the initial ω-2ω polarization directions (Fig. 6). Therefore, after the optimum delay adjustment, the angle between the ω-2ω polarizations should be changed from 45° to 0° in order to collect the largest THz signal energy.
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Fig. 6

Energy of THz radiation accumulated by the end of the filament as the function of the angle between the initial ω − 2ω polarization directions. The ω − 2ω delay is the same for all angles and equal to 55 fs. The parameters of the radiation and the focusing geometry are the same as in Fig. 2.

In conclusion, we have found that cross-guiding of the second harmonic of the Ti:Sapphire laser in the filament produced by a powerful 800 nm pump leads to the THz generation due to four-wave mixing induced by the Kerr nonlinearity. The conversion to THz radiation becomes efficient if the delay between the fundamental and the second harmonic is optimized and provides the maximum overlap of the two pulses. Based on the vectorial model of the co-propagation of the pump and the second harmonic in the filament, we suggest the algorithm for THz yield optimization in the experiment. For the given geometrical focusing distance we first set the angle between the initial ω − 2ω linear polarization directions at 45° and adjust the ω − 2ω temporal delay at which the second harmonic signal is maximum after the analyzer crossed to the initial 2ω light field vector. No THz registration is needed at this step. Second, we rotate the initial second harmonic light field back to the pump and with the optimum ω − 2ω delay time collect the maximum available THz signal from the extended filament.

Acknowledgements

We acknowledge the support from Russian Foundation for Basic Research (grants #09-02-01200a, #09-02-01522a, #11-02-12061-ofi-m-2011), Russian Federal Agency for Science and Innovation (contract #02.740.11.0223), the Council of the President of the Russian Federation Grant for Support of Young Scientists MK-2213.2010.2, National Research and Scientific Council of Canada, the Defence Research and Development Canada –Valcartier, Canadian Institute for Photonics Innovation, CFI, FQRNT, and Canada Research Chair.

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