Table-Top Tunable Chiral Photonic Emitter

Controlling the interplay of light and its interaction with matter has long been a fundamental objective in optics. In particular, chiral light has attracted substantial research interests, as the light polarization spiraling along the propagation direction provides a variety of opportunities for control and spectroscopy of matter. To name a few, chiral light is applied in subcycle optical gates such as angular-dependent dynamics studies Bloch et al. (2021); Chen et al. (2017), chiral molecule spectroscopy Tanaka et al. (2018); Richardson and Riehl (1977), mechanical separation of chiral objects Canaguier-Durand et al. (2013), and chiral quantum optics Lodahl et al. (2017).

Despite the tremendous achievements in chiral light related research, the generation and control of ultraviolet and near-infrared, chiral pulses have continued to present challenges for the following reasons. Firstly, although chiral objects are widespread in nature, such as DNA and certain proteins Mun et al. (2020), the chiral light-matter interaction is typically very weak due to the large mismatch between their atomic feature sizes and optical wavelengths Forbes (2022); Mun et al. (2020). Although one can use external magnetic fields to enhance the material chirality, the typically required magnetic field strengths are of a few Tesla Fan et al. (2021); Stanciu et al. (2007), leading to cumbersome and costly magnetic structures that are unsuited to large-scale manufacturing. On the other hand, metasurfaces and plasmonics Hendry et al. (2012); Zhang et al. (2022); Mun et al. (2020) can be utilized for manipulating chiral light. However, the necessary complex optical configurations require nano-scale fabrication with multiple lithographic processes Santos et al. (2015); Chen and Pepin (2001).

Generation and control of chiral light in the ultra-violet frequency range (800 THz-30 PHz) is even more difficult, due to the lack of suitable optical elements Borrego-Varillas et al. (2018); Brahms and Travers (2023). This impedes progress in many areas, from high-resolution spectroscopy Perkampus (2013) and photo-lithography del Barrio and Sánchez-Somolinos (2019), to controlling atomic/ molecular electronic transitions Ranitovic et al. (2014) and inactivation of microorganisms Li et al. (2009).

Control and manipulation of terahertz (0.1 THz-30 THz) chiral pulses, on the opposite end of the optical spectrum, have proven equally challenging Fan et al. (2021). Terahertz pulses can serve as an active gate to enhance the spatio-temporal resolution of spectroscopy Cocker et al. (2013), realize relativistic electron acceleration Hibberd et al. (2020), and be the key to 6G communication Rikkinen et al. (2020). Nevertheless, the terahertz frequency can not be accessed easily, because it is in between the standard electronic and photonic manipulation methods Vitiello and Tredicucci (2021); Pang et al. (2022). The difficulties in handling Terahertz pulses are mainly due to two reasons. One is the large diffraction owing to their long wavelength nature. The second aspect involves the absorption within solids, as this frequency range is in resonance with lattice phonons Kozina et al. (2019). Consequently, minimizing interactions with solid-state optical elements is imperative to circumvent these challenges. The above difficulties have been referred to as the "terahertz gap"  Vitiello and Tredicucci (2021); Pang et al. (2022).

In this work, we introduce a versatile tunable chiral emitter addressing the challenges in both the ultra-violet and terahertz spectral ranges at the same time. The proposed setup employs only two planar Weyl semimetal (WSM) slabs and a linearly polarized field driving high harmonic generation (HHG), as illustrated in Fig. 1a. Our approach utilizes the chirality of the high-harmonic generation process in the WSM planar structures, which presents the basis for tunable chiral ultraviolet emission from a table-top source. Elimination of complex optical elements simplifies the experimental realization of the proposed setup. The theoretical model used in our analysis solves the two-band equations for semimetals and Maxwell’s equations self consistently. The model applies to WSMs, Dirac semimetals, and 2D semimetals such as graphene.

WSMs are a newly discovered class of quantum materials with unique optical properties and an extremely strong nonlinearity Patankar et al. (2018); Wu et al. (2017). In the momentum space, the points where valence and conduction bands cross are Weyl nodes, around which electrons behave as massless Weyl fermions Yan and Felser (2017). The Weyl nodes appear in pairs with opposite chirality, making them the perfect source and detector for chiral light Ma et al. (2017); Yoshikawa et al. (2022). An inherent feature of the chiral properties in WSMs is the intrinsic coupling of different polarization states. In other words, by choosing the right combination of the driving field’s polarization direction with respect to the Weyl nodes separation direction, it is possible to achieve elliptically polarized emission using a purely linearly polarized driving field, in the absence of any external magnetic field Wang et al. (2023); Wang (2023); Avetissian et al. (2022); Gao et al. (2020).

When exposed to a strong laser field, the nonlinear currents arise from the movement of electrons within the WSM through two primary pathways: the motion within the individual bands (intra-band) and the polarization accumulation between bands (inter-band) Ghimire and Reis (2019); Wu et al. (2015); Vampa et al. (2014). The nonlinear current emits photons with energies that are integer multiples of the driving field. This process is known as high-harmonic generation in solids Ghimire and Reis (2019); Wu et al. (2015); Vampa et al. (2014). In momentum space, the Weyl points act like magnetic monopoles. The non-vanishing Berry curvature flows from the source $\chi=1$ to the sink $\chi=-1$. These magnetic monopoles create magnetic field equivalent effects. Under the influence of the magnetic field, the electrons move toward a direction perpendicular to the applied field and velocity, resulting in harmonic emissions with more than one polarization. Details can be found in supplementary material (SM) sec. III sup . The generated harmonic pulses exhibit excellent coherence properties (Salieres et al., 1997). Combining the elliptical emission from the WSM and the two-slab structure, the proposed scheme realizes easy tunability of the chiral pulses via interference along the propagation direction. This is similar to the principles of a Fabry–Pérot cavity. Moreover, since the harmonics are sensitively related to the properties of the driving field, the chiral/3D polarization states of the high-harmonic radiation can be fully controlled through the intense pulse laser.

Since the model we chose to describe WSM breaks only the time-reversal symmetry and preserves the inversion symmetry, only odd harmonics are present (Methods sections Eqs.2-4). The two-slab structure forms a Fabry–Pérot interference cavity. By alternating the distances between the two WSM slabs, the boundary conditions change for the electric field (i.e. superposition of the harmonics and the driving field) interacting with the structure, resulting in modified harmonics. Visible spectral amplitude modification can be observed (note that the color bar is in log scale). The harmonic spectra are expected to exhibit periodicity concerning $L$, with a period of $\sim\lambda_{0}/2$, where $\lambda_{0}=1\,\textmu$m is the center wavelength of the driving field. The spectra observed in reflection (Figs. 1b,c) and transmission (Figs. 1d,e) manifest distinctive characteristics. For the reflected wave, the dominant emission originates directly from the immediate reflection, i.e., the wave reflected directly from the surface. Consequently, the spectra in reflection demonstrate weaker nonlinear effects, specifically lower-order harmonics, in contrast to the transmitted case, where more pronounced harmonic orders are evident due to the interaction of the driving pulse through two WSM slabs. It can be seen that our compact chiral light source offers the potential for generating diverse polarization states across various harmonic orders (see Fig.1f-m). These exotic polarization states can trigger nuanced angle-resolved electron dynamics that will enhance our understanding of light-matter interaction with a sub-cycle temporal resolution Jiang et al. (2022).

To further characterize the harmonic pulses and their temporal evolution, we examine the polarization state of a given harmonic as a function of time. When the driving pulse is polarized along the $y$ direction, the emitted harmonics are exclusively along the $y$ and $z$ directions (SM Sec. IV sup ). The ellipticity of a pulse is determined by both the relative amplitude and phase relations of the two orthogonal polarizations. For instance, the electric field of a particular harmonic order, polarized along a given direction can be expressed as $E_{i}(t)=\{E_{0i}(t)\exp[i\phi_{i}(t)]+c.c\}/2$, where $i\in{y,z}$ and $E_{0i},\phi_{i}\in\Re$. For a linearly polarized pulse, it necessitates $\phi_{z}-\phi_{y}=n\pi$, where $n=0,\pm 1,\pm 2,...$. Similarly, a circularly polarized light demands $E_{0z}/E_{0y}=1$ and $\phi_{z}-\phi_{y}=\pi/2+n\pi$, where $n=0,\pm 1,\pm 2,...$. In Figs. 2a-c we depict the 1st, 3rd, and 5th harmonics, respectively, in the transmitted direction with an air gap distance $L=450\,$nm. Figures 1k,m and 2b,c represent the same electric fields. In the upper panel of Fig. 2, the solid (dashed) black curve represents the electric field envelope of $E_{0y}$ ($E_{0z}$). To quantify the pulse evolution dynamics, we define two parameters: the ratio of the amplitude $E_{0z}/E_{0y}$, and the phase difference $\phi_{z}-\phi_{y}$. These two parameters are indicated by blue and red curves on the lower panels, respectively. The temporal evolution of the harmonic fields displays distinctive and dynamic changes throughout the entire pulse duration. In other words, for a given harmonic order, the ellipticity varies with respect to time within the pulse. Since the multi-photon ionization is highly sensitive to the driving pulse polarisation and carrier-envelope phase, these unusual and exotic polarization changes of the generated harmonics across a single pulse can be harnessed for ultrafast electron dynamics control Eickhoff et al. (2021); Song et al. (2023); Li and He (2021). In addition, these time-varying polarisation states of the fields may bring new symmetry to the free electron wave packets via the photon electron ionization, resulting in a powerful tool for coherent control of the quantum process Kerbstadt et al. (2019).

The proposed scheme in Fig. 1a is also valid for the terahertz frequency range. Thus, we will not repeat the same calculations. Instead, we analyze how the waveform of the terahertz pulse varies as a function of the Fermi energy $\text{E}_{\text{f}}$. Since terahertz pulses are frequently applied to electron acceleration and communication, the capacity to tailor their waveform is very important. With increasing Fermi energy, the inter-band transition is suppressed, so that the intra-band transition starts to dominate. Since the inter- and intra-band transitions are not always in phase, the final induced current has a phase shift (see Figs. 3a,b). The peak phase differences of the current are defined as $\Delta=(t_{y}-t_{z})\omega_{0}$ where $t_{i}$ is the time corresponding to the peak of the electric field polarized along the $i$ direction with $i\in\{y,z\}$. The value of $\Delta$ as a function of $\text{E}_{\text{f}}$ is plotted as the dotted curve in Fig. 3a with the horizontal axis on top. The transmitted electric fields with $\text{E}_{\text{f}}=0$ and $\text{E}_{\text{f}}=500$ meV are presented in Fig. 3c and d, respectively. The two Weyl semimetal slabs (100 nm each) are separated by 2µm in the air.

The compact chiral light source we propose opens the door to various applications. For example, it creates a new dimension for information coding in optical communication Forbes (2022). Besides, since chiral molecules respond differently to chiral light, chiral light sources can be used to sort molecules and achieve all-optical chiral discrimination Genet (2022); Zhang et al. (2017). Furthermore, this easily tunable setup can be integrated with chemical experiments and serve as a sensitive trigger for photochemical reactions Rikken and Raupach (2000).

Our methodology also has potential applications in the realm of band structure spectroscopy of WSMs. For example, the chirality of the emitted harmonics is determined by the Weyl nodes separation $b$, given the fact that the emission is only linearly polarized if $b=0$ (see Method section and SM Sec. IV sup ). As a result, emissions observed along the $\hat{z}$ direction become a pivotal metric for deducing the Weyl nodes’ separation, $b$. Besides, since the nonlinearity of the emitted harmonics is proportional to the Fermi velocities Wang et al. (2022) as shown in Eqs. (2-4), the Fermi velocities of WSMs can be retrieved by obtaining the amplitude of a given harmonic.

The general form of the current can be found in SM Sec. III Eqs. S34-S36 sup . These expressions are highly comprehensive. Setting $b=0$ results in the expression for Dirac semimetals Lim et al. (2020). Based on this, by further eliminating the third dimension $z$, one obtains the expression for graphene Ishikawa (2010).

In our configuration, the Weyl nodes separation is along $\hat{x}$ and the driving field is linearly polarized along $\hat{y}$. Applying the frozen band approximation and accounting for k-space symmetry (refer to SM Sec. IV sup ), we simplify the analytical expressions for the current as the following:

	$\displaystyle j_{x}(t)=0$		(2)
	$\displaystyle j_{y}(t)=-2\chi\mathcal{M}_{0}\frac{\Omega_{0}}{\omega_{0}}\text{Im}\left[\sum_{\begin{subarray}{c}n,m=-\infty\\ n-m\in\text{ odd}\end{subarray}}^{\infty}\mathcal{C}_{2}(N,n){C}_{3}(N,m)^{*}\large\right]$		(3)
	$\displaystyle j_{z}(t)=\frac{\sin(\theta)}{\cos(\phi)}\mathcal{M}_{0}\sum_{\begin{subarray}{c}n,m=-\infty\\ n-m\in\text{ odd}\end{subarray}}^{\infty}\mathcal{C}_{1}(N,n)\mathcal{C}_{1}(N,m)^{*}$
	$\displaystyle-2vq\frac{\sin(\theta)^{2}}{\cos(\theta)}\text{Re}\left[\sum_{\begin{subarray}{c}n,m=-\infty\\ n-m\in\text{ odd}\end{subarray}}^{\infty}\mathcal{C}_{1}(N,n)\mathcal{C}_{3}(N,m)^{*}\right].$		(4)

The related variables are defined as

	$\displaystyle\mathcal{C}_{1}(N,n)=\frac{n\text{J}_{n}(N)\exp(i\omega_{0}n)}{n+\Omega_{0}/\omega_{0}}\exp(-\frac{t^{2}}{\tau^{2}}),$
	$\displaystyle\mathcal{C}_{2}(N,n)=\frac{\text{J}_{n}(N)\exp(i\omega_{0}n)}{n+\Omega_{0}/\omega_{0}}\exp(-\frac{t^{2}}{\tau^{2}}),$
	$\displaystyle\mathcal{C}_{3}(N,n)={\text{J}_{n}(N)\exp(i\omega_{0}n)},$
	$\displaystyle\mathcal{M}_{0}=qv\frac{\sin(\theta)[\cos(\phi)^{2}+\sin(\phi)^{2}\cos(\theta)^{2}]}{\cos(\phi)\cos(\theta)},$
	$\displaystyle N=\frac{-\chi q\exp(-{t}^{2}/\tau^{2})\cos{\phi}\cos{\theta}E_{0}}{({2\omega_{0}\hbar|k|})}$

where $\text{J}_{n}$ is the Bessel function of the first kind, $\Omega_{0}=4v|q|E_{0}/\hbar\pi\omega_{0}$, $\theta=\arccos(k_{z}/|k|)$, $|k|=\sqrt{(k_{x}-\chi b)^{2}+k_{y}^{2}+k_{z}^{2}}$, and $\phi=\arctan[k_{y}/(k_{x}-\chi b)]$. It can be seen that only odd harmonics survive (see SM Sec. IV for details). In particular, Eq.(4) is proportional to the Weyl nodes separation $b$. If the two Weyl nodes of the opposite chirality merge to one place in k space (i.e. $b=0$), $j_{z}(t)=0$ [see SM Sec. IV Eq.(S84)].

The planar structure of the WSM slabs is modeled via a customized Finite-Difference Time-Domain (FDTD) method. The emission current (we have used the general form as shown in Eqs. S34-S36) is integrated into the Maxwell equations. The numerical details are presented in SM Secs. VI and VII sup . The total current requires integration over the 3D $k$-space. The most standard mesh point selection would be an even-distance mesh, where up to 500 mesh points along each dimension are typically required Avetissian et al. (2022). We choose the parameters for one of the most standard Weyl semimetals i.e. TaAs. The lattice parameters at 300 K are $a_{0}=b_{0}=3.4\r{A}$ and $c_{0}=11.6\r{A}$ Lv et al. (2015). We choose to integrate over the entire Brillouin space with the integration range as $[-\pi/r,\pi/r]$, where $r\in\{a_{0},b_{0},c_{0}\}$. For very large $k$ values, there is almost no transition Ishikawa (2010). This can be understood intuitively. With a very large $k_{z}$ for example, in the $k_{x}-k_{y}$ plane, the Weyl semimetal behaves as a material with a very large band gap ($\sim 2\hbar v_{z}\lvert k_{z}\rvert$). Thus, the inter-band transitions are harder. Since the dominant inter-band transitions are around the Weyl nodes, in the entire Brillouin zone, many mesh points can be ignored, i.e., the current is almost zero in regions far from the Weyl nodes (SM Sec. V sup ). Consequently, we use non-equal-distance mesh sizes, which are denser around the cone (SM Sec. V sup Fig.S7). With this approach, we obtained comparable results as Avetissian et al. (2022) (details see Sec. VI Fig. S9) using a mesh size of $60\times 60\times 60$, which is $\sim 10^{3}$ times smaller than the conventional method Avetissian et al. (2022); Vampa et al. (2014). For our work, one single job of the high-harmonic generation via the 2 Weyl semimetal slabs (mesh size 60$\times 60\times 60$) requires a computational time of $\sim 12$ hours on a single node with 128-core CPU @ 2.6GHz.