1. Introduction

The generation and detection of high-field, single-cycle terahertz (THz) pulses has opened up new fields of science and novel applications ranging from strong field-matter interaction to imaging in the THz frequency band [1–3]. Plasma-based THz emitters excel in many aspects due to their broad radiation bandwidth, high output pulse energy and immunity to high power damage [4–8]. The most popular and efficient laser-plasma scheme used for THz wave generation is the two-color excitation scheme, in which a laser pulse at the fundamental frequency is supplemented by its second harmonic that is obtained with the use of a nonlinear crystal [9–13]. Several methods have been proposed to improve the efficiency of THz waves generated by laser-induced air plasma; these methods include the use of different gas media, circularly polarized or cylindrically focused two-color laser pulses, abruptly autofocusing laser beams, and chirped laser pulses [14–19]. Nonetheless, attempts to further increase the THz peak field have relied on increasing the pump energy, which is hampered by saturation effects due to plasma defocusing as well as other detrimental nonlinear mechanisms [18–21].

Theoretical studies predict that more efficient THz radiation can be produced by near-infrared pump pulses than pumps centered at 800 nm [22,23]. More recently, a wavelength-scaling mechanism that scales as λ^4.6, where λ is wavelength of the fundamental beam, for THz wave generation was experimentally demonstrated for the first time in [24]. However, only the THz electric field strength and spectrum under the fixed input energy and focusing conditions for different wavelengths were investigated. The power dependence, polarization, and beam profile distribution of THz waves emitted from different excitation wavelengths were unknown.

In this paper, we demonstrate that THz energy dependence on the input laser energy increases more rapidly when a longer laser wavelength than 800 nm is used in the two-color laser-plasma generation scheme. The exponents of the power laws in THz energy as a function of fundamental wavelength vary for different optical powers. We verify this approach by means of a model that considers the wavelength-dependent ionization rate. Furthermore, the THz polarization undergoes a continuous rotation with changes in the laser wavelength, which enables all-optical control of THz wave polarization with application to imaging and polarimetry.

2. Theoretical analysis

To quantify the yield of THz signal, we performed calculations taking into account photoionization and subsequent electron motion using a temporal Gaussian two-color laser field. In the ionization rate model, the wavelength dependent Keldysh parameter $\text{γ}=\text{ω}/(\text{eE})\sqrt{2{\text{mU}}_{\text{i}}}$ plays a crucial role to determine the multiphoton or tunneling ionization is dominant, where $\text{ω}$ represents the frequency of the laser, ${\text{U}}_{\text{i}}$ is the ionization potential of the atom, $\text{e}$ and $\text{m}$ are the electron charge and mass, respectively. In our experiments, $\text{γ}$ is less than 1 due to the long laser wavelengths. Thus, the model of the ionization rate we are considering is valid in the tunneling regime and includes a wavelength dependence, which can be written as [25–28]:

Here, ${\text{E}}_{\text{0}}$ and $\text{E}$ represent the atomic field of the atom and the electric field amplitude of the input laser, ${\text{U}}_{\text{H}}$ and ${\text{ω}}_{\text{H}}$ are the ionization potential and atomic frequency of the hydrogen atom, respectively; $\Gamma$ is the gamma function, and ${\text{n}}^{\text{*}}$ and ${\text{l}}^{\text{*}}$ are the effective principal and orbital quantum numbers, respectively. With the electron density ${\text{N}}_{\text{e}}\left(\text{t}\right)$ obtained from the given ionization rate, the electron current generated when the bound electrons are stripped off by an asymmetric laser field can be computed as $\text{J}\left(\text{t}\right)={\displaystyle \sum }_{\text{i}}{\text{eN}}_{\text{e}}\left(\text{t}\right){\text{v}}_{\text{i}}\left(\text{t}\right)$, where $\stackrel{\rightharpoonup }{\text{v}}\left(\text{t}\right)$ represents the electron velocity under the combined field, which is obtained by solving the classical equations of motion. Consequently, the generated THz field is given by ${\stackrel{\rightharpoonup }{\text{E}}}_{\text{THz}}\left(\text{t}\right)\propto \text{dJ}\left(\text{t}\right)/\text{dt}$.

The results of our simulation for three representative wavelengths of 800 nm, 1600 nm and 2000 nm are shown in Fig. 1(a) for Gaussian-enveloped two-color laser fields with the same pulse duration of 50 fs (FWHM), at peak intensities of ${\text{I}}_{\text{ω}}={10}^{15}\text{W}/{\text{cm}}^2$ and ${\text{I}}_{2\text{ω}}=2\times {10}^{14}\text{W}/{\text{cm}}^2$, and with a relative phase $\text{π}/2$. The longest wavelength adopted in the calculation is 2000 nm due to a sudden drop in plasma density for wavelengths longer than this value [24]. Here, the target gas is atmospheric pressure nitrogen with the characteristic stepwise ionization per laser cycle. By increasing the laser wavelength, the slope of the normalized electron density becomes steeper with almost no change in the temporal width of the velocity curve. This leads to a narrower temporal width for the transverse photocurrent, and since THz waveform is proportional to the time derivative of the transverse photocurrent, wider spectrum bandwidth is obtained. This explained the experimental observation in [24]. The shift to higher central frequency of the THz wave emitted from the current surge driven by relatively longer wavelengths results in shorter THz pulses and improved focus ability, thus contributing to a higher THz field.

 

Fig. 1 (a) Normalized free electron density and electron current as a function of time at three fundamental wavelengths (800 nm, 1600 nm and 2000 nm) superposed with their second harmonics. (b) Simulated ionization rate as a function of the pump power for five fundamental wavelengths superposed with their second harmonics. (c) Simulated THz energy dependence on the fundamental laser wavelength for four optical powers.

Download Full Size | PDF

A time-varying electron current becomes asymmetric and a quasi-DC current arises after the laser pulse is gone. As this current surge occurs on the time scale of ionization, extremely broadband THz radiation is expected. With the use of longer wavelengths, the current surge can be driven to higher values, since the average kinetic energy of an electron scales with the square of the wavelength of the driving laser. It is obvious that the air plasma induced by the longer-wavelength laser generates a stronger DC electron current, and thus gives rise to more efficient THz radiation in the far field.

The ionization rate is dependent on the electric field for each excitation wavelength of the input laser. Figure 1(b) shows the simulated ionization rate as a function of the fundamental optical power considering a constant intensity ratio of 20% between the two-color fields. The wavelength-dependent second-harmonic conversion efficiency of the BBO crystal and the resulting slight reduction in the second-harmonic intensity are negligible. The selected long wavelengths of the lasers are 1000 nm, 1200 nm, 1600 nm and 2000 nm. A curve for an 800-nm-wavelength laser typically used for two-color laser-plasma THz generation is also included as a reference using the same fixed pulse duration. The ionization rate is obviously wavelength-dependent, and a laser with a longer wavelength induces a higher ionization rate at identical pump power. Figure 1(c) shows the simulated THz energy as a function of the fundamental wavelengths for four specific optical powers. A monotonic increase in THz energy is shown with increasing wavelength, which is in accordance with the observation in [24]. In particular, we can highlight that the exponents of the power laws in THz energy as a function of pump wavelength vary for different optical powers.

3. Experimental results and discussion

To verify the calculations, we carried out experiments considering the same conditions used in the model. The schematic of the setup is shown in Fig. 2. Optical pulses with tunable wavelengths in the range from 1200 nm to 1600 nm were delivered by a commercial optical parametric amplifier (TOPAS), which was pumped by a Ti:sapphire laser system (Spitfire, Spectra Physics) with a central wavelength of 800 nm and a repetition rate of 1 kHz. The long-wavelength pulse was focused together with its second harmonic generated by a 100-$\text{μm}$-thick type I $\beta$-barium borate (BBO) crystal by means of an off-axis parabolic mirror (PM) with a 6-inch equivalent focal length. Maximum THz wave generation efficiency occurs when the extraordinary axis of the BBO crystal is oriented at ~$55°$ with respect to the polarization of the fundamental beam along the horizontal direction for all the wavelengths. The total THz energy radiated from the air plasma source was measured after eliminating the pump radiation by a low-pass THz filter with a cut wavelength of 13 $\text{μm}$ (Tydex Ltd.). The THz energy was detected with the use of a Golay cell that was equipped with a 6-mm-diameter diamond input window (Microtech SN:220712-D), which exhibited a nearly flat response over a broad spectral range (0.1-150 THz).

 

Fig. 2 Schematic illustration of THz wave generation from air plasma induced by a two-color laser field with tunable excitation wavelengths. SWL + SH_S: The shorter wavelength laser superposed with its second harmonic. LWL + SH_L: The longer wavelength laser superposed with its second harmonic.

Download Full Size | PDF

Figure 3(a) depicts the simulated corresponding THz energy as a function of the pump power. This result predicts that the THz energy from lasers with longer wavelengths increases more rapidly than that from lasers with shorter wavelengths with increasing pump power. Figure 3(b) depicts the measured THz energy for various pump optical powers with the operating wavelengths ranged from 1200 nm to 1600 nm. In this experiment, the BBO-to-plasma distance was fixed at 25 mm for all the measurements. The simulations agree with the experimental results. We attribute the observable inconsistencies between the experiment and simulation to unavoidable dispersion in the BBO crystal and air, and consequently, the optimal phase difference between the fundamental and its second harmonic fields could not be applied in the whole wavelength range for a fixed BBO-to-plasma distance. Experimentally, the pulse duration with respect to different wavelengths output from TOPAS is slightly variable, which is not considered in the simulation. Moreover, we find the measured THz wave generation from 1200 nm excitation laser is lower than the simulation data, which is attributed to the less efficient second harmonic generation for 1200 nm comparing to other wavelengths since the effective wavelength range of the BBO crystal we used is from 1300 nm to 1800 nm.

 

Fig. 3 (a) Calculated and (b) measured THz energy as a function of the pump power. The result for the 800-nm-laser is also plotted for comparison. The measured data corresponds to the excitation wavelengths of 1200 nm, 1300 nm, 1500 nm and 1600 nm.

Download Full Size | PDF

The THz signal obtained at 1400 nm is not shown due to the presence of low-intensity THz energy generated from the plasma, which was not observed at the other wavelengths and was inconsistent with the reported wavelength dependent tendency [24]. We verify that this phenomenon results from the emergence of the low-intensity plasma induced by 1400 nm laser in contrast to other wavelengths. We speculate that this phenomenon is due to the unusual pulse duration of the laser at this wavelength. To further demonstrate it, we measured the pulse duration of the lasers with different wavelengths using collinear auto correlator. The autocorrelation patterns reveal that the lasers with all the wavelengths have the almost identical pulse duration except for 1400 nm. The irregular pulse duration may result from the hardware damage of the TOPAS. The method to nullify the error is not available thus far.

To capture the full dependence of the THz polarization on the wavelength of the excitation laser, we consider the phase retardation ($\phi$) of the fundamental laser fields between the extraordinary and ordinary axes that occurs during laser propagation through the birefringent BBO crystal [29]. The optical field at time $\text{t}$ can be written as

 

Fig. 4 (a) The calculated relative phase between the two-color fields as a function of the excitation wavelength. (b) Simulation of transmitted THz energy as a function of the THz polarizer angle and excitation wavelength. The THz energy is normalized at each wavelength. (c) Normalized THz transmission (dots) as a function of the THz polarizer angle with the sinusoidal fit (solid line) for the excitation wavelengths of 1200 nm (red) and 1600 nm (blue). (d) Experimental result corresponding to case (b).

Download Full Size | PDF

Here, we remark that the ellipticity is highly correlated with the fundamental laser wavelength. Numerically, both the $\widehat{x}$ and $\widehat{y}$ components of ${\text{E}}_{\text{L}}\left(\text{t}\right)$ are considered as the driving fields for the THz generation process. The direction and amplitude of the net electron current determines the polarization of the emitted THz field, and we calculate these parameters to illustrate the polarization dependence of the wavelength of the input laser [29,30]. The simulation illustrated in Fig. 4(b) indicates that the THz polarization continuously rotates as the wavelength changes.

Experimentally, we recorded the THz energy transmission through a polarizer while varying the wire-grid THz polarizer angle at each laser wavelength. It is worth noting that the transmitted THz energy is normalized at each excitation wavelength, since the THz wave generation efficiency is varied with the wavelength. For clarity, we plotted the normalized THz transmission as a function of the THz polarizer angle for the 1200 nm and 1600 nm wavelength lasers, as shown in Fig. 4(c). Via positioning the THz polarizer on a rotation stage, we varied the polarizer angle from $-70°$ to $250°$ in $10°$ increments. The linear polarization angle of the THz wave ${\text{P}}_{\text{T}}$ with respect to the horizontal can be determined by fitting the THz transmission as a function of the polarizer rotation angle ${\text{P}}_{\text{p}}$ to the expression $\text{T}={\cos }^2\left({\text{P}}_{\text{p}}-{\text{P}}_{\text{T}}\right)$. Figure 4(d) shows the corresponding measurement results of the transmitted THz energy as a function of the excitation wavelength and the THz polarizer angle. The wavelength ranges from 1200 nm to 1600 nm with an interval of 10 nm. The shift in the maximum THz energy transmitted through the THz polarizer as a function of the wavelength is observed. The experimental result is in good agreement with the simulation.

THz radiation emitted from laser-induced plasma depends strongly on the length and radius of the plasma [31–33]. As the plasma length increases, we need to consider the phase-matching of the THz radiation emitted from each point along the propagation axis. Consequently, to complete the characterization of the THz radiation, we investigated the emission pattern of the THz beam from the air plasma induced by two-color optical lasers at different excitation wavelengths with the same pulse energies and focal lengths. The forward radiation profile was acquired by raster-scanning the Golay cell with a 6-mm-diameter aperture across the THz beam at a plane 7 cm away from the air plasma. The measured cross-sectional images of the beam profiles for two wavelengths are indicated by the 2D plot shown in Figs. 5(a) and 5(b). The THz energy is normalized for each image. Figure 5(c) plots the central lines of the images. As the laser wavelength increases, the forward THz beam becomes less directional. The directionality of the far-field THz radiation is correlated with the plasma length and radius, which scale as pump wavelengths over the linearly focused beam Rayleigh range. Moreover, if we treat the plasma as an array of coherent THz emission sources, the positions of the beginning and end of the source changed with varying pump wavelengths. Figure 5(d) shows the calculated THz field contributed from plasma with modified length and beginning position for different wavelengths using the equation described in [31].

 

Fig. 5 Cross-sectional images of the THz beam profile for wavelengths of (a) 1300 nm and (b) 1500 nm. (c) Measured and (d) calculated profiles of the center lines corresponding to the images.

Download Full Size | PDF

In particular, a prominent dip appears at the center of the measured THz beam generated by the 1500-nm laser, and this dip is wider than that of the 1300-nm-wavelength case. This feature is less prominent at shorter input pulse wavelength. This phenomenon is mainly due to the phase walk-off of the THz radiation, which is most probably caused by the self-phase modulation of the optical beam within the plasma. With plasma induced by long-wavelength lasers, the self-phase modulation dominates before the Kerr effect is balanced by the dispersion properties of the plasma, thus leading to a positive refractive index change $\Delta \text{n}\left(\text{λ}\right)\propto \text{I}/\left[{\text{n}}_0{\left(\text{λ}\right)}^2{\text{cε}}_0\right]$ along the focus, where ${\text{n}}_0\left(\text{λ}\right)$ denotes the refractive index of air, which decreases with increase in the wavelength [31]. This imposes a greater phase delay between the THz and optical pulses for longer wavelengths; consequently, interference patterns in the THz radiation are more apparent in the far field due to this phase walk-off.

4. Conclusion

In conclusion, longer excitation wavelengths are utilized to generate enhanced THz radiation from two-color laser-induced air plasma, in comparison to normal pump pulses centered at 800 nm. THz energy from lasers with longer wavelengths increases more rapidly than that from lasers with shorter wavelengths with increasing pump power, which is predicted by the wavelength dependent ionization rate. Furthermore, the THz polarization remains linear and smoothly rotates with change in the laser wavelength, which provides a new means of coherently controlling the polarization of THz waves. We envision that scaling further with longer wavelength is certainly feasible and will pave the way for applications in nonlinear THz optics.