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Abstract
A vector beam is a type of topological beam in which the polarization direction of light rotates around a singularity on the wavefront. This paper proposes a method to generate a vector beam by tightly focusing a pump beam in the crystalline direction such that the second-order nonlinear optical effect is forbidden. The directional dependence of the effective nonlinearity in zincblende crystals, such as ZnTe, was analytically investigated. Two types of nonlinear polarization singularities were found in [111] and [100] directions. Their polarization topological charge ℓ was +1 and −1, respectively. To experimentally demonstrate the proposed method, a (111) cut ZnTe crystal was selected as the nonlinear crystal. The polarization state of the generated terahertz (THz) beams was measured with a custom-built THz spectroscopic polarization imaging system. Radially polarized distributions were observed within the entire generated spectral region. Such a broadband feature of the generated vector beam is likely due to the topological nature of the focused pump beam, where the wavevectors are winding once about the optical axis. This simple method for generating THz vector beams will accelerate its applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A cylindrical vector beam, in which the polarization is distributed by rotating around the optical axis, has a singularity where the polarization direction cannot be defined. In that singularity, a dark spot is formed [1]. The rotation number $\ell$ around the singularity is referred to as the winding number, which characterizes the topological nature of the vector beam. The rotation numbers of typical radial and azimuthal vector beams are unity. When beams of $\ell =+1$ are focused, the electric or magnetic fields oscillate longitudinally about the propagation direction at the focal point. In contrast, a typical electromagnetic wave in air propagates as a transverse wave [2,3]. Longitudinal fields have been used in applications that include a super focusing probe for a scanning near-field optical microscope (SNOM) [4,5], a probe for transient magnetic phenomena in material science [6], and charged particle acceleration [7]. Due to the narrower nature of the singularity’s dark spot when compared to the diffraction limit size, cylindrical vector beams have also been proposed for use in a super-resolution imaging technique [8].
In the terahertz (THz) regime, methods to generate cylindrical vector beams have been proposed. Examples include using a specially designed photo-conductive antenna [9], putting THz beam through a spatially distributed phase plate, called q-plate [10], using a specially designed Fresnel rhomb [11], combining a spiral phase plate with spatially distributed polarizer sheets [12], and exciting a device combined with nonlinear crystal segments cut in special directions [13]. These methods are necessary for devices that require high-quality fabrication techniques or fine-tuning when aligning special optics that generate vector beams. Although there has been one report of terahertz vector beams generation utilizing the down conversion processes of an optical vector beam [14], the study has remained a theoretical investigation for a long time. Recently, a vector beam generation based on such down conversion techniques has been experimentally demonstrated [15]. A simple method of combining a nonlinearity, regarding the epsilon-near-zero effect of a thin metallic film, and an axicon lens to generate a Bessel radial beam was also reported [16]. However, a high-power laser system, such as a regenerative amplifier, was required to utilize the nonlinearity of the boundary between metallic and dielectric materials. This paper theoretically proposes a simple way to generate a THz vector beam based on the nonlinear effect, the crystalline symmetry, and the topology of the wavevector distribution in real space. The radially polarized THz wave generation was then demonstrated experimentally using a ZnTe crystal for proof of concept.
2. Concept
The topological nature of vector beams provides a hint for their generation principle. The concept presented in this paper is depicted in Fig. 1. First, consider THz pulse generation through a difference frequency generation (DFG) process using a femtosecond laser focused on a nonlinear crystal. The focused pump pulse has wavevector components whose directions deviate from the optical axis and distribute by winding once around the same axis. The wavevector’s winding was interpreted as the topological nature of the wavevector distribution and if it could be projected onto the generated THz beam according to the following procedure, it lends a vectorial nature to the THz beam. Note that this projection to the polarization is, in a way, analogous to the topological nature of the moiré pattern projected onto the phase of the terahertz beam through meta-surfaces, which has previously been shown [17,18].
Fig. 1. (a) THz vector beam generation in nonlinear crystals concept. A focused pump beam including different wavevectors excites a nonlinear crystal. A THz wave is generated through a second-order nonlinear process. When the surface direction of the crystal coincides with the high-symmetry direction, in which the nonlinear process is forbidden, the THz wave becomes a vector beam. (b) The relationship between the crystalline frame and the laboratory frame. The x-axis in laboratory frame is parallel to the wavevector of the incident beam.
When the optical axis of the pump pulse corresponds to a highly symmetric direction in the crystal, the DFG process may be forbidden. However, since the focused pump includes the wavevector components that deviate from the forbidden direction, THz generation is allowed, except for at the center of the focusing beam. Consequently, the generated THz beam has a dark point at its center.
Additionally, when the circularly polarized pulse is incident to the forbidden direction, because of high symmetry, a characteristic direction of the nonlinear polarization cannot be defined; in other words it is isotropic. Conversely, in the case where the incident direction deviates from the forbidden direction since the deviation breaks the symmetry, the crystal behaves in the nonlinear polarization as an anisotropic material with the anisotropic axis along the deviation direction. In such a case, the anisotropic axis may also rotate around the highly symmetric direction accompanying the nonlinear polarization rotation. This can be the origin of the polarization rotation of the generated THz beam around the forbidden direction.
Hence, the intrinsic topological nature of the focusing wavevectors will be transferred to the THz beam with a dark singularity at the beam’s center and polarization rotation around the same. These features correspond to those of vector beams. Since this topological nature is expected to be robust during a frequency change, this methodology has the potential to generate a vector beam in the broadband THz region.
3. Theory
To reveal the symmetric direction and the polarization distribution around it, the nonlinear polarization was calculated for the case where a circularly polarized pump beam is incident to the arbitral direction of a nonlinear crystal. In the second-order DFG process, the nonlinear polarization in the crystallographic frame $P^{({\rm cry})}_{I}$ can be written as
If the nonlinear crystal satisfies Kleinman’s symmetry and is a zincblende crystal, such as ZnTe, which has a crystal point group of $\bar {4}3m$, the susceptibility components corresponding to $d_{14}$ ($= d_{25}= d_{36}$), where the suffixes are represented by contracted notation, are nonzero, but others are zero [20]. The same behavior is expected for crystals classified in the crystal point group of $23$.
ZnTe crystals have been utilized to generate THz pulses [21,22]. One of the main reasons is that ZnTe crystals satisfy the collinear phase matching conditions for pump lasers in the near-infrared region including the oscillation wavelength of femtosecond Ti:sapphire lasers [23]. Since this crystal behaves as an isotropic material for the linear response, the collinear phase matching condition can be satisfied in the arbitral incident directions of the pump beam. This is an advantage for the proposed method because the complicated phase matching conditions, as well as the work-off effect in the nonlinear interaction among incident and generated beams, can be neglected. In contrast, these effects must be considered in the case of general axial crystals [14].
Furthermore, a circularly polarized pump beam, described as ${\boldsymbol E} = E_0(0, 1, \pm i)^T$, was assumed to be incident to the nonlinear crystal, where the double sign indicates the direction of circular polarization rotation. For this case, the generated nonlinear polarization was derived analytically in all solid angles of the crystallographic frame as
Note that the rotation direction of circular polarization did not affect the nonlinear polarization.
The nonlinear polarization calculated from Eq. (3) is visualized in Fig. 2. In the figure, the nonlinear polarization vectors are drawn on the surface of (a) the one-eighth sphere around the [111] direction and (b) the half sphere around the [100] direction of the crystallographic coordinates. The vector is the polarization direction of the generated THz wave when the circularly polarized pump pulse is incident to the direction indicated by the latitude and longitude on the spherical surface. In this figure the nonlinear polarization component perpendicular to the spherical surface, namely the longitudinal wave component, is not depicted because such a component does not contribute to the generation of the propagation wave in air.
Fig. 2. Distributions of polarization directions obtained from Eq. (3) are mapped on (a) the one-eighth sphere around the [111] direction and (b) the half sphere around the [100] direction on the crystallographic frame. Polarization distributions for $\ell =+1, -1$ around a singularity are indicated as insets.
The singularities of the polarization direction were found in [111], [100] directions and their equivalent directions, in which the polarization direction cannot be defined and hence, THz wave generation via a circularly polarized pump is forbidden. Around the [111] direction, the polarization direction radially rotates +1 time, whereas that of [100] rotates −1 time, as illustrated in the inset of Fig. 2. These polarization distributions are known as the radial mode and the quadrupole mode distributions, respectively.
To investigate analytical formula when the optical axis is parallel to the symmetric directions, an arbitral coordinate frame obtained by rotating the crystalline coordinates $\theta$ around the y-axis and $\phi$ around the z-axis is introduced as shown in Fig. 3(a). The z-axis of the arbitral coordinate frame is parallel to the optical axis. The direction of the wavevector on the arbitral coordinate is described with the polar and azimuth angle of $(\Theta, \Phi )$ as shown in Fig. 3(b). The rotation tensors for transforming the base from the arbitral frame to the crystalline frame and from the laboratory frame to the arbitral frame are described as
Fig. 3. Definition of (a) the arbitral coordinate on the crystalline coordinate and (b) the laboratory coordinate on the arbitral coordinate. (c) The incident pump beam distribution assumed a Gaussian distribution according to our experimental conditions. Calculated THz beam distribution in (d) [111] and (e) [100] directions with considering a geometric optics. The arrow symbols represent the polarization direction at that point.
To obtain the nonlinear polarization around the [111] direction, the polar and azimuth angles of the arbitral coordinate should be $\theta =\arcsin \sqrt {2/3}$ and $\phi = \pi /4$. In that case, the nonlinear polarization can be derived as
Since the z direction of ${\boldsymbol P}_{[111]}^\textrm {(arb)}$ is corresponding to the [111] direction, x- and y-components indicate the polarization of the propagation wave. In the paraxial approximation, namely $\Theta \ll 1$, the nonlinear polarization can be approximately described as
In the case of the [100] direction, the nonlinear polarization can be obtained by substituting $\theta =\pi /2$ and $\phi = 0$ to Eq. (4) as
Note that due to $\bar {4}$-symmetry, this formula is equivalent to Eq. (3) by replacing of $\Theta \rightarrow \theta$ and $\Phi \rightarrow \phi$, except that the sign is reversed. Since the complex expression can be written as $P_x+iP_y \propto \sin \Phi + i\cos \Phi = ie^{-i\Phi }$, the nonlinear polarization is perpendicular to the radial direction at $\Phi =0$ and rotates $-1$ revolution in each azimuthal direction. The amplitude also has a hole when $\Theta =0$.
The paraxial intensity distribution can be predicted from Eqs. (7) and (9). We assumed a geometric optics that a Gaussian beam of 4 mm in diameter was focused using a lens with a focal length of 25.4 mm according to our experimental condition mentioned after. The distribution was shown in Fig. 3(c). In this figure, $\Theta$ and $\Phi$ are the same with the definition in Fig. 3(b). The expected THz beam distributions for the [111] and [100] directions are shown in Fig. 3(d) and (e), respectively. Both distributions are donut-shaped with a hole in the center. The arrow symbols on the donut rings represent the polarization direction at each location. Around [111] ([100]) direction, the arrows revolve $+1$ ($-1$) time with increasing azimuthal angle up to $2\pi$. The very slight triangular shape distortion of the intensity distribution in Fig. 3(d) is attributed to the three-fold symmetry around the [111]-axis neglected in the paraxial approximation of Eq. (8). It also makes the polarization direction slightly deviate from the pure radial distribution.
Hence, the THz vector beams with polarization topological charge $\ell =+1$ and $-1$ will be generated by focusing a circularly polarized pump pulse tightly onto the highly symmetric directions of [111] and [100], respectively, in the ZnTe crystal.
4. Experiments
In this section, the concept is demonstrated thorough the experimental observation of a radial vector beam ($\ell =+1$) using a (111) cut ZnTe crystal. The experimental setup was based on a traditional THz time domain spectroscopic (THz-TDS) system as shown in Fig. 4. A femtosecond pulse from an oscillator type Ti: Sapphire laser system was divided into pump and probe arms. The pump pulse passed through an optical delay. The polarization state of the pump pulse was converted to circular polarization via a quarter wave plate (QWP). The pulse duration and the average power of the pump beam were 100 fs and 800 mW, respectively. The beam width of the pump pulse was approximately 4 mm and was tightly focused on the (111) cut ZnTe crystal using lens 1(f = 25.4 mm) with slightly deviating from the focal point to avoid the optical damage to the crystal surface. The thickness of the crystal was 0.5 mm. The [111] direction is forbidden to generate a THz wave for a circularly polarized pump as shown in the previous section. However, the wavevector components that deviated from the [111] direction were included in the focused beam and became a source of the vector beam. The generated THz-wave was collimated with lens 2 (f = 17.4 mm) and then put through a low pass filter made of a Teflon film to cut the collinearly propagating pump beam. At the focal point of lens 3 (f = 40 mm), the THz beam forms the same image as that on the ZnTe crystal. The THz beam passed through a wire gird (WG) polarizer used to measure mutually orthogonally polarized components and was introduced to a photo-conductive antenna (PCA, BATOP Company) in which a bow-tie type antenna was fabricated on a low temperature growth GaAs substrate. At the front of the PCA’s lens tube, lens 4 (f = 50 mm) coupled the THz wave from the focal point. The PCA, together with lens 4, was mounted on an XY-auto-stage to measure the spatial distribution of the polarization. Note that lenses 2, 3, and 4 (Edmund optics) were aspheric lenses made of a cyclo-olefin polymer with high transparency for the THz wave.
Fig. 4. Experimental setup to observe the polarization distribution of generated THz waves from a ZnTe crystal (thickness: 0.5 mm) whose incident surface is directed to [111]. The pulse width of the laser system was 100 fs. The average power of pump and probe pulses was 800 and 4 mW, respectively. The pump beam diameter at the front of a focusing lens 1 is 4 mm. The focal length of lenses 1, 2, 3, and 4 are 25.4, 17.4, 40, and 50 mm, respectively. QWP: quarter wave plate, WG: wire grid analyzer, PCA: photo-conductive antenna, LPF: low pass filter.
The PCA was coupled with an optical fiber so that the optical path length did not change even if the PCA was moved. The average power of the coupled probe beam was 4 mW. To compensate for the group velocity dispersion within the optical fiber [24], the probe pulse ahead passed through a dispersion compensator consisting of a Treacy grating pair [25].
THz waveforms were acquired at each PCA position during a scan. After scanning, the WG was rotated 90 degrees to measure orthogonally polarized components. The PCA position scan was performed again. The optimum measurement angles of the transparent axis of the WG were tilted by −32.5 and 57.5 degrees from the horizontal axis due to the individual specificity of our PCA, and we labeled the electric fields directed to those directions as $E_{\rm X}$ and $E_{\rm Y}$, respectively. The THz polarization distribution was reconstructed by combining two independent polarization distributions.
5. Results and discussion
First, the waveform distributions in the horizontal and vertical directions through the beam center were acquired for the $E_{\rm X}$ and $E_{\rm Y}$ polarization components. Figures 5(a) and (b) show the $E_{\rm X}$ and $E_{\rm Y}$ components of the horizontal waveform distribution, respectively. In these panels, the horizontal axis represents the spatial position, the vertical axis represents the delay time, and the colors represent the waveform signal. On these figures, THz waves are shown propagating from below to above. The corresponding Fourier intensity distribution is shown below each panel. The vertical axis of these is the frequency and the color represents the intensity. Similarly, the vertical distributions are shown in Fig. 5(e), (f), (g) and (h). Schematic diagrams of the polarization and the scanning direction on the beam are depicted as insets on the respective panels.
At all frequencies, there was a clear reduction in intensity at the center of the beam. The waveform was inverted in phase on either side of the reduction. Comparing the intensity of the $E_{\rm X}$ and $E_{\rm Y}$ components, $E_{\rm X}$ was stronger in horizontal scanning, while $E_{\rm Y}$ was stronger in vertical scanning. This is consistent with the assumption that the polarization was radially polarized, while taking into account that the $E_{\rm X}$ and $E_{\rm Y}$ components have different inclinations from the horizontal axis as mentioned above. These behaviors suggest that the generated beam was radially polarized. The waveform distribution of the $E_{\rm Y}$ component was not completely zero at the center. Considering that linearly polarized THz beam can be generated at the center of the beam when a linearly polarized pump is incident to the [111] direction of the ZnTe crystal [19], the nonzero waveform at the center could be caused by an unintended linearly polarized component in the pump light.
Fig. 5. Experimentally observed (a), (b), (e), and (f) THz wavefront and (c), (d), (g), and (h) the corresponding spectral distribution. The left- (right-) side panels are $E_{\rm X}$ ($E_{\rm Y}$) components. The top four panels (a), (b), (c), and (d) represent the results of the horizontal scan and the bottom four panels (e), (f), (g), and (h) represent the results of the vertical scan. The scanning areas on the beam and measured polarization components are schematically shown in the insets.
Next, the PCA was scanned both horizontally and vertically to measure the two-dimensional distributions of the polarization and the intensity. The reconstructed THz polarization and intensity distributions at 0.5 THz are shown in Fig. 6(a). The color of each pixel indicates the normalized intensity. The linear or elliptical symbol in each pixel indicates the polarization state of that point, and the size of the symbol is indicating the amplitude of the normalized signal.
The color of the symbol reflects the Stokes parameter $S_3$, the index of circular polarization. $S_{3}$ can be derived from the orthogonal electric field components of $E_{{\rm X}\omega }$ and $E_{{\rm Y}\omega }$ where the suffix of $\omega$ represents frequency components in the THz spectrum Fourier-transformed from a THz waveform. $E_{{\rm X}\omega }$ and $E_{\rm Y\omega }$ can be described in the amplitude and phase terms as $E_{{\rm X}\omega }=E_{{\rm X}\omega 0} \exp [i\delta _X]$ and $E_{{\rm Y}\omega }=E_{{\rm Y}\omega 0} \exp [i\delta _Y]$, respectively. $S_3$ is given by
where $\Delta =\delta _X-\delta _Y$ is the relative phase between $E_{{\rm X}\omega }$ and $E_{{\rm Y}\omega }$[26].Fig. 6. Polarization and intensity distribution of (a) 0.5 THz component. The color and the linear /elliptic symbol in each pixel, indicate the normalized intensity and the polarization state, respectively. The color of the elliptic symbol shows Stokes parameter $S_3$. Distributions in the 0.33, 0.42, 0.75, and 1.0 THz components are shown in (b), (c), (d), and (e), respectively. (f) A THz spectrum of the total intensity integrated on the imaging area. Labeled frequencies on the graph are corresponding to the figures above. Color bars for (b) and (d) are common with (c) and (d), respectively.
A circular intensity distribution with a hole in the center was observed in Fig. 6(a). Moreover, the polarization was radially distributed surrounding the central hole. Such donut-like intensity and radially polarized distribution correspond to the features of the $\ell =+1$ vector beam predicted in Fig. 3(d). The slight triangular distortion of the intensity distribution seen in Fig. 3(d) could not be read off from the obtained distribution. There was elliptic polarization in some pixels. The cause of this is still under investigation, but one possible explanation is that the imaginary part of the susceptibility may not be zero in the ZnTe crystals used in the experiment.
The other frequency components of 0.33, 0.42, 0.75, and 1.0 THz are depicted in Fig. 6(b), (c), (d), and (e), respectively. The THz spectrum of the total intensity integrated on the imaging area was shown in Fig. 6(f) on which the labels correspond to Fig. 6(a), (b), (c), (d), and (e). Similar radial polarized distributions were also observed in this range where the signal-to-noise ratio was high enough to discriminate the polarization states in this THz-TDS system even though at the edge of this frequency range (0.33 - 1 THz), the intensity distributions slightly collapsed from the donut shape. This was attributed to the measurement noise. The broadband generation of the vector beam was reflected in the topological nature of the wavevectors included in the focused pump pulse. Namely, the wavevectors projected onto the crystal surface rotated once around the optical axis regardless of the pump beam frequency. The observed beam size increased as a function of the frequency. This is because the longer the wavelength, the larger the spot size at the focal point of lens 3. In the high frequency region, a ring-like fringe was observed on the outer part rather than the center of the donut. This fringe was assigned to the higher radial mode of $p=1$ because its phase was opposite of the inner donut, as determined in the phase analysis. When the THz radiation area on the nonlinear crystal was narrower than the diffraction limit of the generated THz wave, the component of the higher radial mode was nonzero. Similar results were obtained when the ZnTe crystal was excited by an oppositely rotating circular polarized pulse as predicted in Eq. (3).
The total power of the generated THz vector beam by the presenting method can be estimated as a quarter of the optimum condition to generate a usual Gaussian beam using a (110)-cut ZnTe crystal [19] (see Supplement 1). In the current method, efficiency has been sacrificed in favor of simplicity. Specifically, the energy at the center of the pump beam does not contribute to terahertz generation. To improve this point, an axicon lens could be introduced to the optical axis of the pump beam to efficiently contribute the energy of the pump beam to the vector beam generation. The advantage of using the axicon lens that, with the proper choice of lens system, the energy density of the pump beam can be kept lower than without, since the beam shape at the focal point is ring, not spot [27]. This means that more excitation energy can be injected below the damage threshold of the crystal, which is expected to generate a more powerful vector beam.
It is worth noting that the optics required to realize the proposed method for generating THz vector beams are only a QWP for generating a circularly polarized pulse, a lens with a short focal length, and a ZnTe crystal, which was readily available commercially. Another advantage of this method is that an oscillator-type femtosecond laser system can be used as a light source. Preparing these optics and the light source is not difficult for researchers familiar with a THz-TDS system. Hence, the proposed method bolsters interest and investigations on THz vector beams, which will accelerate their applications.
6. Conclusion
Via calculation, the [111] and [100] directions in the ZnTe crystals were determined to be the best candidates for special directions that can be used to generate THz vector beams of $\ell =+1$ and $-1$, respectively. THz radial vector beam generation was demonstrated using the proposed method in which a circularly polarized pulse was focused on the [111] direction. This method can be easily replicated in a laboratory using a general THz-TDS without any complicated techniques or devices. The authors believe the proposed method will contribute to accelerating the applications of the THz vector beam.
Funding
Fusion Oriented REsearch for disruptive Science and Technology (JPMJFR2036); Japan Society for the Promotion of Science (JP21H01018, JP22H01253, JP22H01980, JP22K04940).
Acknowledgments
The authors would like to thank Mr. Makoto Saitou in Tohoku University for his cooperation in maintaining the experimental facilities.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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