In an anisotropic medium, such as crystals, the polarization state of an electromagnetic wave may change during the propagation [1]. In an isotropic medium in linear approximation, the change of the state of lightwave polarization in the process of propagation is well known for the optically active medium [2]. The nonlinearity of the medium allows us to observe polarization-sensitive effects, even in the isotropic centrosymmetric medium [3–5], and polarization nonlinear optics has become one of the most interesting parts of nonlinear optics [6–11]. For an isotropic nonlinear medium, the third-order susceptibility tensor has three independent components [1]. There is a classic effect related to ${\chi }_{1221}^{(3)}$ which is well known for the isotropic medium: the self-rotation of the polarization ellipse becomes stronger with the increase of the polarization ellipticity degree and intensity in the incident wave and completely disappears for the linearly polarized light. The history of this effect goes back to 1964 with the publication of Maker et al. [3].

In order to observe a nonlinear response to electromagnetic fields, high intensities are needed which seem unattainable in the terahertz frequency range. It is obvious that the field intensity in the medium is determined by its energy divided by the pulse duration and the square of illuminated area. In the terahertz frequency range, these three factors reduce the achievable intensity [12]: the pulse energy is considerably smaller than the characteristic values for higher optical frequencies, the pulses are longer, and the radiated area equals at least the square of the wavelength. However, it is predicted that in the terahertz range, much lower intensities are required to observe nonlinearities than for higher frequencies [13].

The effect of the rotation of a polarization ellipse of radiation when it is propagated in the isotropic medium is well known [3]. Its observation requires quite high intensities, which are impossible to achieve in the terahertz pulse radiation. In this Letter, we set the target to demonstrate that in the presence of strong waves of visible radiation coherently connected with the terahertz wave [14], the rotation effect of the polarization ellipse of a weak wave of terahertz radiation can be considerably increased and be observed experimentally.

In this Letter, as a source of terahertz radiation, we used optical breakdown in air in the “two-color” laser scheme [15]. In this scheme, a femtosecond laser pulse ($\omega$) passes through a frequency-doubling crystal to generate a second-harmonic pulse ($2\omega$); then these two pulses are mixed to produce gas plasma. After the laser beam waist radiations at $\omega$, $2\omega$, and ${\omega }_{\mathrm{THz}}$ frequencies propagate coaxially and synchronized in time. Although the diameter of the filament is much smaller than the wavelength of the terahertz wave, the terahertz pulse could propagate inside the filament, rather than undergo natural diffraction in air [16]. It can be assumed that with further propagation these waves can affect each other due to interaction. For the purpose of observing this interaction, right after the laser beam waist, we placed the medium, whose nonlinearity considerably exceeds the nonlinearity of air. We chose liquid nitrogen (LN) as the nonlinear medium. Besides the nonlinearity, LN possesses the absence of considerable absorption in the terahertz frequency range [17]. Negligible absorption in visible and NIR ranges, together with the features mentioned above provide a good contrast between LN and air, which is highly desirable for our experiments. LN used as a sample in the experiments described below was obtained by the use of an ordinary membrane system of nitrogen separation from compressed air. The system provides nitrogen purity of around 98% of the feed compressed air.

The experimental setup used in this Letter is presented in Fig. 1. We used a laser system based on a Ti:sapphire regenerative amplifier (Spitfire, Spectra Physics, Inc.) with the energy up to 2.5 mJ per pulse, pulse duration of 30 fs, a wavelength of 797 nm, and a 1 kHz repetition rate. The laser beam with the frequency $\omega$ is directed vertically from top to bottom with the help of a set of mirrors and is focused by a lens L1 with focal length $F=11\mathrm{cm}$. For the realization of a “two-color” scheme of terahertz wave generation, a 300 μm $\beta$-barium borate (BBO) crystal is placed in the way of the laser beam after the lens L1. The laser beams with frequencies $\omega$ and $2\omega$ transmit through a hole in an “off-axis” parabolic mirror (PM) and are focused into free-space air. A flat aluminum mirror M positioned 1 cm below the focus point reflects the terahertz radiation generated in the beam waist back toward the PM mirror. A hole of 1 mm diameter in M mirror passes through the optical beams, which prevents a destruction of the mirror by powerful optical radiation. Next, the PM mirror (with a focal length of 50 mm, a diameter of 25.4 mm, and “off-axis angle” is 90°) sends a collimated terahertz beam through focusing a polypropylene lens L2 with a focal length $F=10\mathrm{cm}$ to a 4.2 K silicon bolometer detector (Infrared Laboratories, Inc.). The IR cut-on long-pass and cold filters, which transmit radiation up to the frequency of 3 THz, are placed in front of the detector input window.

 

Fig. 1. Experimental setup. L, lens; BS, beam splitter; M, mirror; PM, parabolic mirror; BBO, $\beta$-barium borate (BBO) crystal.

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A Dewar vessel with LN is positioned into the scheme, as shown at the Fig. 1, to study nonlinear interaction between waves at $\omega$, $2\omega$, and ${\omega }_{\mathrm{THz}}$ frequencies propagating in the LN. The M mirror is dipped into LN in this case; however, the optical beam waist is situated above the surface of LN, i.e., the generation takes place in the plasma of the air optical breakdown, and terahertz signal passes partly through air and partly through LN.

The polarization plane of linearly polarized radiation coming from the laser system to our experimental setup is perpendicular to the surface of Fig. 1. Polarization properties of the optical radiation after the BBO crystal were analyzed by use of Glan–Taylor prism (not shown at the Fig. 1). For THz polarization measurements, a polyethylene wire-grid terahertz polarizer (Tydex, LLC) is placed in the collimated terahertz beam between PM and L2.

Kerr nonlinearity of the isotropic medium has a noticeable influence on the polarization state of terahertz radiation which appeared as a result of the process $\mathrm{\Omega }\approx 2\omega -\omega -\omega$ due to elliptically polarized waves at the fundamental and doubled frequencies continuing to propagate collinearly with the terahertz radiation along the axis $z$. Please refer to Fig. 2. The red lines on this figure depict the polarization ellipse of the radiation at fundamental frequency $\omega$ and the main axes of the ellipse after passing through the BBO crystal. The blue line shows the polarization of the radiation at the doubled frequency $2\omega$. The orange line demonstrates schematically the polarization ellipse of the radiation at terahertz frequency (${\omega }_{\mathrm{THz}}=\mathrm{\Omega }$). All angles of rotation of all polarization ellipses ${\mathrm{\Psi }}_{\omega }$, ${\mathrm{\Psi }}_{2\omega }$, ${\mathrm{\Psi }}_{\mathrm{\Omega }}$, are calculated in relation to the initial polarization of radiation at fundamental frequency $\omega$ which corresponds to the horizontal direction—the black $x$-axis on Fig. 2. In our experiments, fundamental and doubled frequencies $\omega$ and $2\omega$ correspond to $3.762*{10}^{14}\mathrm{Hz}$ (wavelength 797 nm) and $7.523*{10}^{14}\mathrm{Hz}$ (wavelength 398.5 nm), respectively, and ${\omega }_{\mathrm{THz}}$ corresponds to radiation detected in the terahertz range up to $3*{10}^{12}\mathrm{Hz}$ (wavelengths longer than 99.93 μm).

 

Fig. 2. System of coordinates for the polarization measurements.

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The degree of ellipticity of polarization of terahertz radiation can be expressed by

In both experiments, the BBO crystal was adjusted to maximize the terahertz yield from the plasma of the air optical breakdown. However, if the BBO crystal is rotated around the propagation axis, there is not only change in the efficiency of second-harmonic generation takes place, but also a transformation of the initial linearly polarized radiation at the fundamental frequency getting to the crystal into elliptically polarized radiation at the same frequency. In our experiments, after passing through the BBO crystal, the fundamental radiation has elliptical polarization with a degree of ellipticity $M_{\omega }=0.97$, and the main axis of the ellipse angle ${\mathrm{\Psi }}_{\omega }=17.0°$, second harmonic, was nearly linear polarized ($M_{2\omega }\approx 0$) at the angle ${\mathrm{\Psi }}_{2\omega }=132.8°$. The notations used to characterize the polarization states of radiations in the experiment are shown in Fig. 2.

The obtained results are shown at Fig. 3. The green dots and the graph depict thepolarization state of the terahertz beam propagated through air with no LN on its way. The terahertz beam has elliptical polarization with a degree of ellipticity $M_{\mathrm{\Omega }}=0.323$; the angle of rotation of main axis of the ellipse ${\mathrm{\Psi }}_{\mathrm{\Omega }}$ is directed at the 14.2° to the horizon. The orange triangles and the graph depict the polarization state of the terahertz beam propagated partially through air and partially through an LN layer of 13 mm thickness. After passing through the LN layer, the terahertz beam still holds the elliptical polarization, but the degree of ellipticity $M_{\mathrm{\Omega }}$ increases to 0.565. The angle of orientation of the main axis of the ellipse ${\mathrm{\Psi }}_{\mathrm{\Omega }}$ increases up to 33.2°, which is 19° large than in the case of free air-space propagation of the terahertz beam. We would like to note that in the preliminary set of experiments, we specially checked that there were no accumulation effects caused specifically by the high repetition frequency of laser pulses.

 

Fig. 3. Rotation of the ellipse of polarization of the terahertz radiation after propagation through the LN. These measurements were done at the pulse energy of 2.5 mJ and pulse duration of 30 fs.

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We have provided modeling with the use of Eqs. (9) and (10), taking into account the initial values of terahertz radiation ellipticity $M_0$ as 0.323 and the angle ${\mathrm{\Psi }}_0$ as 14.2° measured in our experiment. The results of numerical simulations have showed that obtained values $\mu L=1.191\pi (214.38°)$, $\mathrm{arctg}\nu =0.182\pi (32.76°)$, and $\alpha L=1.6\pi (288°)$ provide a perfect agreement with the values of polarization parameters of terahertz beams measured after transmission through an LN layer of 13 mm thickness. The values $M_{\mathrm{\Omega }}$ and ${\mathrm{\Psi }}_{\mathrm{\Omega }}$, which were calculated according to the above-mentioned theory, are 0.566° and 33.3°, respectively.

In this Letter, we observed for the first time, to the best of our knowledge, nonlinear “Terhune-like” “mutual” rotation of the main axis of a polarization ellipse of terahertz radiation and its transformation when it propagates in isotropic liquid. It was demonstrated experimentally that the terahertz wave generated in air does not have sufficient intensity to show the impact of self-action [7,8]. However, during its simultaneous propagation with the waves of visible radiation, we observe their interaction at terahertz frequency, which leads to nonlinear effects on terahertz frequency. The observation of the above-mentioned nonlinear optical effect shows that the nonlinearity of the medium can no longer be neglected when we describe the terahertz wave propagation. The phenomenological approach to the description of nonlinear optical properties of the medium makes use of the resonance and nonresonance electronic contributions into the susceptibility tensor. The LN absorption line is located at 1.5 THz [17,18]. In this case, there is a resonance to the $\mathrm{\Omega }={\omega }_1-{\omega }_2$, taking into account that the spectral shape of the femtosecond pulse is sufficiently wide. Under high intensities of laser radiation, the vibrations and rotations of LN molecules should be efficiently excited, and both contributions may lead to considerable nonlinearity and should be taken into account when describing the observed effect. The effects that lead to this enhancement have Raman nature and have been described in literature [19,20].

Nonlinear susceptibility of the third order, which is much higher in LN compared to air, enables us to carry out polarization control of terahertz radiation with the help of polarization states of other visible ultrashort light pulses. The possibility to observe polarization control in the process of generating terahertz radiation in the filament under dual-frequency interaction has already been demonstrated by us [21]. However, in this Letter, we showed for the first time, to the best of our knowledge, the possibility to observe the effects of polarization control due to coherent reciprocal action under their propagation exclusively in the nonlinear medium.