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Two-dimensional (2-D) transverse photocurrent generation is studied and applied to control and optimize terahertz energy and polarization in two-color, laser-produced air filaments. A full control of terahertz output is demonstrated and explained in the context of 2-D photocurrent model.
©2012 Optical Society of America
1. Introduction
Recently, terahertz (THz) generation via two-color laser mixing in air [1–17] has attracted a considerable amount of interest owing to its capability of producing broadband and high-power THz pulses [7–9]. In this scheme, an ultrashort pulsed laser’s fundamental (ω) and its second harmonic (2ω) pulses are co-focused in air to ionize air molecules. From the ionized air plasma, intense THz radiation is observed to emerge. The nonlinearity responsible for such THz generation originates from rapid tunneling ionization and subsequent electron motion in a symmetry-broken electric field [7–9]. This so-called plasma (or photo) current model [7–9] can explain the microscopic origin of the third order nonlinearity χ(3) (generally all high order nonlinearities χ(n)) in THz generation via two-color photoionization.
In our previous studies [7–9], the photocurrent model was applied for two-color laser pulses polarized in the same direction. This can be achieved by splitting off the orthogonally polarized second harmonic pulse (type-I phase matching) and rotating its polarization to be parallel to the fundamental pulse polarization with a half-wave plate, before they are combined for THz generation. This scheme was previously demonstrated in Ref [5] and is useful for individually controlling two-color polarizations. However, the scheme is vulnerable to phase instability due to mechanical vibration or air turbulence occurring in the split and combined beam branches. Any sub-μm vibration can cause a phase shift of <2π, which can greatly affect the THz output.
To avoid the phase instability problem, most two-color photoionization experiments adopt an all-in-line second harmonic and THz generation arrangement as shown in Fig. 1(a) . However, in this scheme, the BBO crystal needs to be azimuthally oriented such that the ω field has a component parallel to the 2ω field polarized in the extraordinary (ê) axis of the BBO crystal. In this case, the fundamental pulse becomes elliptically polarized after passing through the crystal whereas the second harmonic remains polarized along the extraordinary axis. This setting invokes two-dimensional (2-D) transverse electron currents for THz generation [13]. Previously, it is shown that the polarization of THz radiation is sensitive to the phase difference between the fundamental and second harmonic fields [13, 14].
Fig. 1 (a) Schematic of all-in-line second harmonic and THz generation. (b) Vector diagram for fundamental (Eω) and its second harmonic (E2ω) generation before the BBO crystal and at the front end of plasma filament. ô and ê represent the ordinary and extraordinary axes of the BBO crystal respectively. Eωo and Eωe, polarized along the ordinary (ô) and extraordinary (ê) axes of the crystal.
In this paper, we extend the 2-D plasma current model to examine full parameters controlling THz yields and polarization, ultimately finding the conditions for optimized THz generation. We also examine the 2-D model by comparing our experimental results with simulations.
2. Two dimensional (2-D) photocurrent model and simulation
We consider a 2-D photocurrent model by taking into account of transverse 2-D laser fields as shown in Fig. 1(b). When the fundamental laser pulse passes through a birefringent BBO crystal, its amplitude can be decomposed into two orthogonal waves, Eωo and Eωe, polarized along the ordinary (ô) and extraordinary (ê) axes of the crystal, respectively. Here the second harmonic pulse produced by type-I phase matching is polarized along the ê axis.
Overall, they are given by
where Eω and E2ω are the amplitudes of the fundamental (ω) and the second harmonic (2ω) fields, deff is the effective conversion coefficient of the BBO crystal from ω to 2ω, and α is the angle between the incoming ω polarization (horizontal axis) and the ê axis [see Fig. 1(b)]. Then the laser field at the focal plane is expressed asWithwhere φ is the phase retardation between Eωe(t) and Eωo(t) after propagation through the BBO crystal of thickness l, nωo and nωe is the refractive index of the BBO crystal along the ô and ê axis, respectively. Also, θ is the relative phase between Eωe(t) and E2ω(t), c is the speed of light in vacuum, d is the distance between plasma and BBO crystal, nω and n2ω is the refractive index of air at ω and 2ω frequency, respectively, and θ0 is the phase difference right after the BBO crystal. In practice, θ can be varied by changing d [7], and φ can be controlled by tilting the BBO crystal and thus varying the effective thickness of the crystal l (see Sec. 3 for details).From the 2-D laser field in Eq. (2), one can calculate the tunneling ionization rate [18] given by
where EL(t) = |EL| is the laser field amplitude, Ea = 5.14 × 109 V/cm is the atomic field, and ωa = 4.134 × 1016 s−1 is the atomic frequency. In practice, we used the Ammosov-Delone-Krainov (ADK) tunneling ionization rate [19]. Although the ADK model has been used for primarily for noble gases, it also works well for structureless atomic-like molecules such as neutral N2 (the primary constituent gas of atmospheric air). From the ionization rate equation, the time-varying electron density Ne(t) can be computed with multiple degrees of ionization taken into account [9]. Here, we ignore collisional ionization, plasma recombination, and electron attachment to neutral molecules because those events occur much slowly [20] compared to the pulse duration <50 fs considered here.Once the electron density is determined, the plasma current density can be obtained as
where dNe(t′) is the change of electron density in the interval between t′ and t′ + dt′, and ve(t, t′) is the velocity of those electrons at t. Here, we need to consider 2-D transverse current density given by J⊥ = Joô + Jeê, where Jo and Je is the current density component directed along the ô and ê axis, respectively. The current surge radiates an electromagnetic field, given by ETHz ∝ dJ⊥(t)/dt. Here, the two orthogonal current densities (Jo and Je) are always in phase as long as θ and φ do not change over the entire pulse duration, and this makes the far-field THz radiation linearly polarized. Here, the THz polarization angle is determined by the relative strength between Jo and Je.Figure 2 shows a simulation result of 2-D photoionization model for two simple cases. In the first case (a), both Eω and E2ω are linearly polarized with an angle of 45°. This occurs when φ = 0° and α = 45°. In this case, the total current density summed over one cycle is maximized at θ = 90˚. This means that maximum THz radiation occurs at θ = 90°, consistent with our previous 1-D model [7]. The inset in Fig. 2(a) shows the polarization diagrams of all three waves, Eω, E2ω, and ETHz. In the second case (b), Eω is elliptically polarized (φ = 210°, α = 55°) while E2ω is linearly polarized. In this case, the maximum THz yield occurs at θ = 20°. The inset in (b) shows the angle of THz polarization with respect to those of the other two waves Eω and E2ω. Here, we note that the elliptical field case (b) produces almost four times higher THz fields compared to the linear polarization case (a). This suggests that linearly-polarized (but not parallel) two-color fields do not necessarily yield the maximal THz output in the all-in-line THz generation configuration. Thus, one needs to find the optimal φ, α, and θ parameters which maximize the THz output.
Fig. 2 2-D photoionization current producing linear fundamental field case; (a) φ = 0˚ α = 45˚, and θ = 90˚ (left) and elliptical fundamental field case; (b) φ = 210˚, α = 55˚, and θ = 20˚ at peak laser intensities of Iω = 1014 W/cm2 at λω= 800 nm and I2ω = 0.8 × 1013 W/cm2 at λ2ω= 400 nm. Over one cycle of the fundamental field (-π < ωt < π) are shown the laser fields Eωe (red solid line), Eωo (red dashed line), E2ω (blue dotted line); ionization rate calculated by the total field (black dotted line) and electron drift velocities along the ô (red dashed line) and ê (red solid line) axes; plasma currents Jo(red dashed line) and Je (red solid line) along the ô and ê axis, respectively. The insets show the polarization diagrams of Eω (red line),E2ω (blue line), and ETHz (black line) for the two cases.
We have examined the 2-D electron current density squared, |J⊥|2 (or equivalently THz yield or intensity), by varying all three parameters (φ, θ, and α). Figure 3 (top) shows |J⊥|2 as a function of θ and α for four φ values (φ = 0°, 45°, 90°, and 210°). For each 2-D plot |J⊥|2 (α, θ), the line which has a local maximum is selected and plotted as a function of α for |Jo|2, |Je|2, and |J⊥|2 (bottom). The maximum THz yield occurs at φ = 210°, θ = 20°, and α = 55°. Here, we note that maximal THz does not occur at θ = 90° anymore because of 2-D plasma currents. In the case of φ = 210°, the THz polarization angle is ~60° with respect to the ê axis as shown in Fig. 2(b). With φ = 90°, the fundamental field Eω is circularly polarized and the resulting THz field is purely polarized along the ê axis [see Fig. 3 (bottom, third column)].
Fig. 3 (Top) Resulting |J⊥|2 (or THz yields) as a function of θ and α (degrees) for four different φ = 0°, 45°, 90°, and 210°. (Bottom) In each 2-D simulation, the line which yields the local |J⊥|2 maximum at a fixed θ is selected and plotted as a function of α for |Jo|2 (line with circles), |Je|2 (line with asterisks), and |J⊥|2 (solid line).
For any given φ value (0° ≤ φ < 360°), the optimal α and θ that maximize |Jo|2, |Je|2, and |J⊥|2 are computed and shown in Fig. 4 . For example, the total THz output (blue solid line) peaks at φ = 30°, 150°, 210°, and 330° (see the last graph). At those φ angles, the α and θ values maximizing the total yield are shown in the first and second graphs. For all four cases, the maximum THz yield occurs at α ~55°. This agrees well with many previous experimental reports employing all-in-line schemes [2,13]. Note that the optimal θ has a broad range of values depending on φ and whether the total or polarized (ô or ê) yield is calculated.
Fig. 4 Optimal α and θ values maximizing THz output for a given φ value. It considers three maximum types: |Jo|2 (red dashed line), |Je|2 (red solid line), and |J⊥|2 (blue solid line). This shows the total THz radiation (or |J⊥|2) peaks at φ = 30°, 150°, 210°, and 330° and at α ~55°.
3. Experiments and simulations
The experimental setup is shown in Fig. 5 . A Ti:sapphire laser system delivering 800 nm, ~6 mJ, 25 fs pulses at a 1 kHz repetition rate is used to generate THz radiation. The laser pulses are focused by a 250 mm lens and propagate through a 0.1-mm thick BBO crystal (type-I) which generates second harmonics. The fundamental and second harmonic pulses co-propagate and ionize atmospheric air at the focus, generating THz radiation. The optical pulses are blocked by a 5-mm thick silicon (Si) wafer, whereas the THz radiation produced at the focal spot is collected and focused into a pyroelectric detector (SPI-A-62THZ, Spectrum detector Inc.). For low (<3 THz) and high (<10 THz) frequency THz detection, an additional 1.5-mm thick Teflon, 1.5-mm thick high-density polyethylene (HDPE), 2-mm thick Sapphire, or 2-mm thick Ge filter is placed, in front of the detector. For low-frequency (<3 THz) THz polarization detection, an optional wire-grid polarizer (G50x20, Microtech instruments, contrast >95:2 below 3 THz) with 50 µm wire spacing and 20 µm wire diameter is placed in front of the pyroelectric detector.
Fig. 5 Schematic setup for THz generation and detection. The far-field THz yield and polarization is measured with a pyroelectric detector combined with a wire-grid polarizer and filters (Teflon, HDPE, Sapphire, or Ge). THz output is controlled by varying the distant from BBO to filament (d), azimuthal angle of BBO (α), tilt angle of BBO (β), and tit angle of lens (γ).
We control the relative amplitude, Eωe/E2ω, by azimuthally rotating the BBO angle α, relative phase θ by moving the BBO position d, and phase retardation φ by tilting the BBO angle β [see Eq. (3)]. Furthermore, we change the filament length by tilting the focusing lens angle γ (see Fig. 5).
3.1 Relative phase (θ or d) effect on THz yields
The first experimental results show the effect of the relative phase (θ) between the ω and 2ω fields. Figure 6(a) shows the THz yield as a function of the BBO-to-plasma distance d. It shows five lines obtained with five different filters (Si plus Ge, Sapphire, HDPE, or Teflon) placed in front of the pyroelectric detector. All five lines increase quadratically with decreasing d with additional oscillatory modulations with a period of ~2.5 cm. This quadratic and oscillatory trend is well known [7] and consistent with the air dispersion term shown in Eq. (3) [2]. However, unlike some previous electro-optic (EO) based measurements [2, 7, 13], the yield does not reach the bottom when the yields are minimal. This imperfect oscillatory behavior can be explained by the 2-D plasma current model described in Sec. 2. As shown in our simulation result in Fig. 6(b), the total THz wave can be considered as the superposition of two orthogonal THz waves polarized along the ô and ê axes. The total yield curve (blue line) in Fig. 6(b) well reproduces the features shown in the experimental data. Note that the individual yields, |ETHz,o|2 or |ETHz,e|2 follow the complete oscillatory θ dependence, reaching the bottom at local minima, consistent with the previous EO-based measurements. The inset in Fig. 6(a) shows three polarization maps obtained with three different d (or θ) values. It is seen that the angle of THz polarization rotates with varying θ, consistent with the 2-D plasma current model and the previous reports [13,14].
Fig. 6 (a) Measured THz yield dependence on the relative phase (φ) obtained with various THz filters: Si (black solid line), Ge (blue dotted line), Sapphire (magenta line with circles), HDPE (green line with x), and Teflon (red dashed line), in addition to the Si filter. Here, the Sapphire, HDPE, and Teflon signal is rescaled by a multiplication factor of 2, 4 and 8, respectively. (Inset) THz polarization map obtained at d = 10, 11 and 12 cm with using a grid polarizer combined with the Si and Teflon. (b) Simulated THz yields with varying θ for the total (blue line), ô-polarized (red dashed line), ê-polarized (green dotted line) THz yields. In the simulation, φ = 210° (or β = 1°) and α = 55° are assumed.
3.2 Azimuth (α) and tilt (β) angle effects on THz yields and polarization
The azimuth α and tilt β angles of BBO crystal also affects THz output yield and polarization. To quantify those effects, THz yields are measured with varying α and β.
Figure 7(a) shows the THz yield as a function of α at two tilt angles β = 0° (black line with + ) and β = 1° (dashed red line). It shows that β = 1° provides the higher signal (60% more), which suggests that one needs to optimize φ, or practically β, to maximize the THz output. Here, we estimate φ to be 268° and 210° at β = 0° and 1°, respectively, in our BBO crystal case. In Fig. 7(a), co-plotted is the second harmonic intensity (blue line) with varying α. It is linearly polarized along the ê axis and peaked at α = 90°. By contrast, the THz yield is maximal at α = 55°, consistent with the 2-D photocurrent simulation [13]. At β = 0° and 1°, the corresponding THz polarization (black line with circles) measured with the wire grid analyzer is plotted in (b) and (c), respectively. Here, α = 55° is selected. Coplotted in Figs. 7(b-c) are the fundamental and second harmonic polarizations.
Fig. 7 (a) THz yields as a function of α angle at β = 0° (black line with + ) and β = 1° (dashed red line). Co-plotted is the second harmonic intensity (blue line). (b-c) Fundamental (ω, red line), second harmonic (2ω, blue line with + ), and THz (black line with circles) polarization maps obtained with rotating the polarization analyzer at β = 0° and 1°. Unlike 2ω, ω polarization is dependent on the tilt angle β.
Figure 8 shows how φ changes with the effective BBO thickness l or β under our experimental condition. In general, controlling φ by titling the BBO angle (β) is practically limited as it simultaneously affects the phase-matching condition in second harmonic generation. For example, Fig. 8(b) shows the second harmonic intensity (green line) as a function of β. If β deviates more than ± 2°, the second harmonic conversion efficiency drops significantly, ultimately reducing THz output power. This limits the achievable φ value as π < φ < 2π (or equivalently 0 < φ < π), which is just good for a full control of φ but this naturally drops the THz output energy when the optimal β approaches ± 2°. This, however, can be overcome, if an optical waveplate is inserted just before the BBO crystal and adjusted to control the tilt angle β separately. In addition, Fig. 8(b) shows the initial value of φ is ~268° at β = 0° but it drops to φ = 220° at β = 1°, which is near the optimal condition φ = 210° obtained in the 2-D simulation.
Fig. 8 Simulated φ value (blue line) as a function of (a) the BBO effective thickness in μm and (b) β in degrees. Co-plotted in (b) is the second harmonic intensity (green dashed line) produced by type-I phase-matching as a function of β. At β = 1°, φ ~220°.
3.3 Plasma filament length effect on THz yields
The 2-D plasma current model described in Sec. 2 predicts linear THz polarization, but circular or elliptically polarized THz radiation is often observed experimentally. For instance, the polarization map shown in Fig. 7(c) indicates the THz wave is not perfectly linearly polarized but rather elliptically polarized. We believe this occurs because the transverse plasma current (thus THz polarization) direction gradually rotates with varying θ along the plasma filament (recall θ varies with d). Thus, the THz pulse produced at each point along the filament has different linear polarization and arrives at different times on the detection plane. Because the THz velocity is faster than the optical group velocity, the THz pulse produced at the entrance of filament arrives earlier than that produced at the end of filament [21]. In addition, there is an instantaneous cross Kerr effect between ω and 2ω pulses occurring along the filament, which also rotates the plasma current and THz radiation direction. These two effects can produce elliptically or circularly THz polarization. More details about this macroscopic propagation effect will be reported separately. Here, instead we study the total THz yield dependence on the filament length.
Figure 9 shows the total THz yield with increasing d for three different plasma lengths, 12 mm, 14 mm, and 16 mm with two different filter sets: (a) Ge (<10 THz) and (b) Teflon (0.1~3 THz) [22,23]. Here, the filament length is adjusted by titling the lens normal angle as shown in Fig. 5. At both high and low THz frequencies, the longer the filament is, the more intense THz generation is produced as shown in Figs. 9(a)(b). This implies that tilt-induced aberration possibly drops the laser intensity in the transverse direction, but its effect looks less significant compared to the plasma volume (length) effect on THz generation. Here, the maximum enhancement factor is ~2.8 for the high THz frequency case. However, in the low THz case, the enhancement factor is ~1.4, and the yield even drops when the filament length reaches ~16 mm [Fig. 9(b)]. This is attributed to THz absorption in the plasma filament. It was previously shown that low frequency THz components start to saturate earlier than the higher frequency components [8]. We also note that the θ dependence is more dramatic with the low THz frequency case. We speculate that this may be attributed to certain effects which are not included in our model. For simplicity, we ignore collective plasma oscillations, electron-ion and electron-neutral collisional effects, rescattering with parent ions, and any propagation effects [24] including self- and cross-phase modulations, spectral shifting and broadening, Kerr-induced polarization rotation, phase- and group velocity walk-offs between two-color fields. In particular, plasma oscillations and collisional effects greatly influence the low frequency components of THz radiation initially produced by the plasma currents [25].
Fig. 9 THz emission as a function of the BBO-to-plasma distance d with three different plasma lengths, 12 mm (line with + ), 14 mm (line with circles), and 16 mm (line with squares) with (a) the Ge filter for high frequency (<10 THz) detection and (b) the Teflon filter for low frequency (0.1 ~3 THz) detection.
4. Conclusion
We describe the mechanism of 2-D plasma current generation and optimization of the magnitude and polarization of THz radiation in two-color photoionization. This is done by controlling the relative phase θ, BBO azimuth angle α, and tilt angle β. We show that these parameters can control many laser-THz properties including laser (ω and 2ω) amplitudes, ellipticity of ω, phase retardation φ, phase delay θ, polarization and the intensity of THz, all consistent with our 2-D photocurrent model. Furthermore, we have measured the THz yield dependence on the filament length. All these results have verified the sensitivity of control parameters in generating intense THz radiation and will be guiding factors for further investigation. For a more general characterization of THz radiation, however, our semiclassical approach needs to be replaced by full quantum mechanical calculations, in particular when the quasi-static tunneling ionization regime becomes inapplicable [26].
Acknowledgments
The authors thank the U.S. Department of Energy and the Office of Naval Research.
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