Electromagnetic scattering laws in Weyl systems

Wavelength determines the length scale of the cross section when electromagnetic waves are scattered by an electrically small object. The cross section diverges for resonant scattering, and diminishes for non-resonant scattering, when wavelength approaches infinity. This scattering law explains the colour of the sky as well as the strength of a mobile phone signal. We show that such wavelength scaling comes from the conical dispersion of free space at zero frequency. Emerging Weyl systems, offering similar dispersion at non-zero frequencies, lead to new laws of electromagnetic scattering that allow cross sections to be decoupled from the wavelength limit. Diverging and diminishing cross sections can be realized at any target wavelength in a Weyl system, providing the ability to tailor the strength of wave–matter interactions for radiofrequency and optical applications. Scattering characteristics are important optical properties but they depend strongly on the relative electromagnetic size and environment of a particle. Here, the authors study the frequency-dependence of the scattering cross section for a scatterer located inside a photonic Weyl system.

lectromagnetic scattering is a fundamental process that occurs when waves in a continuum interact with an electrically small scatterer. Scattering is weak under nonresonant conditions; an example is Rayleigh scattering, which is responsible for the colours of the sky. Conversely, scattering becomes much stronger with resonant scatterers, which have an internal structure supporting localized standing waves, such as antennas, optical nanoresonators, and quantum dots. Resonant scatterers have wide application because the resonance allows physically small scatterers to capture wave energy from a large area. As such, large electromagnetic cross sections, σ, are always desirable: a larger σ value means, for example, stronger mobile phone signals 1 and higher absorption rates for solar cells 2 .
The maximum cross section of resonant scattering is bounded by the fundamental limit of electrodynamics. One might be tempted to enlarge the scatterer to increase the cross section, but this strategy only works for non-resonant scattering, or electrically large scatterers. In resonant scattering, physical size only affects spectral bandwidth, while the limit of cross section is determined by the resonant wavelength λ as 3 : The directivity D describes the anisotropy of the scattering; D = 1 for isotropic scatterers. Equation 1 shows that an atom 4 can have a σ max similar to that of an optical antenna 5 , despite the subnanometre size of the atom. This also means that optical scatterers cannot attain resonant cross sections as large as those of radiofrequency (RF) antennas, due to the smaller wavelengths involved.  Fig. 1 The length scale of cross section and its relation to dispersion. a In free space, the resonant cross section scales according to σ $ λ 2 or $ 1=ω 2 . Large cross sections always favour low frequencies. For example, the cross section of an optical transition in an atom is small (~10 À12 m 2 ) because of the associated short wavelength (~μm). The cross section of an RF antenna is much larger (~10 À4 m 2 ) due to a much longer wavelength (~cm). Diverging cross sections are obtained around the DC point, which happens to be the apex of a conical dispersion. The double lines indicate double degeneracy due to polarization. b By embedding the resonant scatterer in a medium where the dispersion of the continuum exhibits conical dispersions located away from the DC point, diverging cross sections can be realized at high frequencies. The cross section scales according to $ 1=Δω 2 , where Δω is the relative detuning between the resonant frequency of the scatterer and the Weyl point. In both a and b, regions with stronger colour indicate larger resonant scattering cross section. c Schematic of extraordinarily large cross section created by an electrically small resonant scatterer (red dot) placed inside Weyl photonic crystal Overcoming the limit just described has far-reaching implications for RF and optoelectronic applications. Many efforts have been devoted to realizing this goal, including the use of enhanced directivity D 6-10 , degenerate resonances 11 , decreased dielectric constants 12 ϵ and materials with negative refractive index 13 . While these approaches exploit certain trade-offs to slightly increase the pre-factor in Eq. (1), the fundamental limit of λ 2 remains, which can be proven directly from Maxwell's equations without requiring specific scatterer details 5 . Until now, extremely large cross sections have only been obtained at long wavelengths near the DC frequency.
Here we show that the scattering laws in Weyl systems 14 allows the cross section to be decoupled from the wavelength limit. This opens a new path to realizing strong wave-matter interaction, providing potential benefits to RF and optoelectronic devices that rely on resonant scattering. Moreover, the scattering effect discovered here is equally applicable to acoustic or electronic waves [15][16][17] .

Results
Length scale of the cross section. In free space, the dispersion directly leads to the wavelength limit of the cross section, which is shown by Eq. (1). Specifically, the DC point, which is located at the apex of the conical dispersion relation as shown in Fig. 1a, gives rise to the diverging cross sections at low frequencies. The apex of the conic dispersion can be also realized at other spectral regimes, such as that shown in Fig. 1b. All the special scattering properties associated with the DC point can be reproduced, resulting in diverging cross sections and exceptionally strong light-matter interactions at high frequencies. Weyl points 14, 18-28 , the three-dimensional (3D) analogy of Dirac points, have recently been shown to exhibit such conical dispersion relations. The scattering laws in Weyl systems allows the cross section to be decoupled from the wavelength limit. Extraordinarily large cross section can be realized even for very small scatterers inside a Weyl photonic crystal, which is schematically illustrated in Fig. 1c. Unlike the lensing effect that can concentrate incident waves to a fixed focus, the scatterer concentrates incident waves to itself no matter where it is placed inside the Weyl photonic crystal.
Conservation law of resonant scattering. To illustrate the underlying physics of the relation between the dispersion and cross section, we will first show a conservation law of resonant scattering. We start by considering the resonant cross section of a dipole antenna (Fig. 2a). Without losing generality, we only discuss the scattering cross section, assuming zero absorption. Similar conclusions can be drawn for the absorptive case, with the maximum absorption cross section 1 4 that of the scattering cross section 29 .
The dipole antenna is anisotropic due to its elongated shape, so the cross section σðθ; φÞ depends on the incident direction of the wave. At a normal direction, when θ ¼ π=2, it reaches its maximum value 5 of σ max ¼ 3λ 2 =2π. Along the axial direction, when θ ¼ 0, the cross section vanishes. While it is straightforward to calculate the cross section of a dipole antenna, it is not immediately apparent why the cross section follows the λ 2 rule and diverges around the DC frequency. Figure 2b shows the real-space representation of σðθ; φÞ at the resonant frequency ω 0 . We can also represent σðθ; ϕÞ in momentum space as σðkÞ, with k located on the isosurface defined by ω ¼ ω 0 . As shown in Fig. 2c, the isosurface is a sphere with jkj ¼ ω 0 =c. To visualize the momentum-space representation, σðkÞ is indicated by the colour intensity on the isosurface in Fig. 2d. As we show in Supplementary Note 1, the resonant cross section satisfies the following conservation law: The integration is performed on the isosurface ω k ð Þ ¼ ω 0 . Our proof is based on quantum electrodynamics 30 , so it applies to classical scatterers such as antennas, as well as to quantum scatterers such as electronic transitions that absorb and emit light. More importantly, the continuum, in which the scatterer is embedded, does not need to be free space; it can be anisotropic materials, or even photonic crystals 31 , as long as a well-defined dispersion relation ω ¼ ωðkÞ exists. Equation 2 dictates the scaling of σ with respect to the resonant frequency ω 0 of the scatterer. As ω 0 decreases, the area of the isosurface shrinks. To conserve the value of the integration over a smaller isosurface, the cross section σ k ð Þ must increase accordingly. For example, Fig. 2e and f show the isosurfaces of dipole antennas with resonant frequencies at ω 0 =2 and ω 0 =3, respectively. The cross sections, indicated by colour intensity, must increase proportionally to maintain a constant integration over these smaller isosurfaces. This can also be seen by the average cross section in the momentum space: Here, S ∬ ds is the area of the isosurface. When approaching the apex of the conical dispersion relation, the isosurface diminishes, i.e., S ! 0. As a result, the cross section diverges around the DC point (Fig. 1a).
Resonant scattering in Weyl photonic crystal. Recent demonstrations of Weyl points in photonic crystals 18 show that the dispersion of a 3D continuum can exhibit conical dispersion at any designed frequency (Fig. 1b). The Hamiltonian for the continuum around the Weyl point is where σ x;y;z are Pauli matrices. The momentum q q x ; q y ; q z À Á ¼ k À k Weyl defines the distance to the Weyl point in the momentum space, and q = 0 at the Weyl point. The linear dispersion described by this Hamiltonian produces an ellipsoidal isosurface that encloses the Weyl point. The isosurface shrinks to a point at the Weyl frequency ω Weyl . Scattering properties associated with the DC point are carried to high frequencies within Weyl medium, resulting in exceptionally strong resonant scattering. As illustrated in Fig. 1b We now demonstrate a specific example of resonant scattering in a Weyl photonic crystal 22 . We consider a localized resonant scatterer in an infinitely large photonic crystal. The simulations are performed in two steps. First, we numerically calculate the eigenmodes of the resonant frequency in the Brillouin zone by using MIT photonic bands 32 (MPB). Next, we use each eigenmode as excitation and numerically calculate the scattering cross section by using the quantum scattering theory we developed recently 33 , which is described in detail in Supplementary Note 2.
The structure of the Weyl photonic crystal consists of two gyroids, as shown in Fig. 3a. The magenta gyroid is defined by the equation f r ð Þ>1:1, where f r ð Þ ¼ sinð2πx=aÞ cosð2πy=aÞ þ sinð2πy=aÞ cosð2πz=aÞ þ sinð2πz=aÞ cosð2πx=aÞ and a is the lattice constant. The yellow gyroid is the spatial inversion of the magenta one. The two gyroids are filled with a material with a dielectric constant of ϵ ¼ 13. To obtain Weyl points, we add four air spheres to the gyroids to break the inversion symmetry (Fig. 3a). These spheres are related by an S 4 ðzÞ transformation. The resulting band structure has four isolated Weyl points at the same frequency of ω Weyl ¼ 0:5645 ð2πc=aÞ. Figure 3b illustrates conical dispersion in the 2k z ¼ Àk x À k y plane.
The resonant scatterer is a quantum two-level system (TLS) embedded in the above photonic crystal. We numerically solve the scattering problem using quantum electrodynamics 33,34 . The Hamiltonian is ¼ H pc þ H TLS þ H I . The first two terms, H pc ¼ P q hω q c y q c q and H TLS ¼ hω 0 b y b, are the Hamiltonian of the photons and the TLS, respectively 35 . Here, h is the reduced Planck constant, b † and b are the raising and lowering operators for the quantum dot, respectively, and c y q and c q are the bosonic creation and annihilation operators of the photons, respectively. The Lamb shift 35 is incorporated into the resonant frequency ω 0 . The third term, H I ¼ i h P q g q ðc y q b À c q b y Þ, is the interaction between the TLS and the radiation. The coupling coefficient is , where d is the dipole moment, ϵ 0 is vacuum permittivity,Ê q is the unit polarization vector of the photons and L 3 is the quantization volume.
Scattering inside photonic crystals is much more complex than that in free space. The continuum is highly dispersive, anisotropic and non-uniform, and thus the scattering cross section depends strongly on the location of the scatterer and orientation within the photonic crystal.
As an example, we consider a TLS with a transition frequency ω 0 slightly below the Weyl frequency: ω 0 À ω Weyl ¼ À0:0005 ð2πc=aÞ. The calculated cross section σ q ð Þ (see Supplementary Note 2 for derivation) is plotted on the isosurface as shown by Fig. 3e, f. As expected, it strongly depends on the incident wavevector q. In addition, σ q ð Þ varies greatly at different locations, as shown by comparing Fig. 3e and Fig. 3f. Despite all these differences, when integrated over the isosurface, ∬ σ q ð Þds always results in the same constant: 16π 2 . We perform the integration for a TLS at 20 different locations, all with the same constant, as shown in Fig. 3d.
As the transition frequency of the TLS approaches the Weyl point, i.e., ω 0 ! ω Weyl , the isosurface shrinks in size, as illustrated by the insets of Fig. 4a. The conservation law leads to an increasing σðqÞ, as shown by stronger colours. Near the Weyl frequency (black dashed line), the average cross sections σ is enhanced by three orders of magnitude compared to that in free space, eventually diverging at the Weyl point (Fig. 4a). The analytical prediction from Eq. 2 and the area of the isosurface agree very well with predictions from numerical simulation (circles in Fig. 4a).
Frequency dependence of resonant scattering. A Weyl point greatly enhances the cross section at the resonant frequency. However, it comes at the price of suppressed cross section away from the resonant frequency. Next, we discuss the spectral feature of the average cross section σðωÞ for a given TLS. The spectral dependence is shown in Supplementary Note 3 as Here, } is a constant that depends on the local electric field at the position of the TLS, but does not vary significantly with frequency. In order to derive Eq. 4, we use the fact that the spontaneous decay rate is proportional to ω 0 À ω Weyl À Á 2 (Supplementary Eq. 32). At the resonance when ω 0 À ω ¼ 0; the average cross section scales as 1= ω 0 À ω Weyl À Á 2 , which increases as the resonant frequency moves closer to the Weyl point. However, away from the resonance when jω 0 À ωj ) jω 0 À ω Weyl j, Eq. 4 reduces to σ ω 0 ð Þ $ ω 0 À ω Weyl À Á 2 = ω 0 À ω ð Þ 2 , which shows that being close to the Weyl point suppresses the cross section. In Fig. 4b, we calculate the spectra for three different TLSs with their transition frequencies approaching the Weyl point (black dashed line). While the peak value of the cross section grows, the full width at half maximum of the spectrum decreases. The spectral integration of the cross section remains around a constant (Supplementary Note 3).
Rayleigh or non-resonant scattering. The non-resonant scattering of electrically small objects in free space follows the Rayleigh scattering law. In great contrast to resonant scattering, the cross section of Rayleigh scattering scales as σ $ ω 4 , and diminishes at the DC frequency, as shown in Fig. 5a. This property is also closely related to the isosurface and can be carried to high frequency at the Weyl point (see proof in Supplementary Note 5). While the resonant cross section diverges, the nonresonant cross section diminishes at Weyl points, as illustrated in Fig. 5b.
Using perturbation theory and the first-order Born approximation 36 , the Rayleigh scattering cross section can be shown as (details in Supplementary Note 5): Here, u k ðrÞ is the eigenmode of the Weyl photonic crystal associated with wavevector k. k s and k inc are the wavevectors of the scattered and incident eigenmodes, respectively. V is a tensor for the scattering potential of the Rayleigh scatterer. The integral is proportional to the area of the isosurface S. At the DC point, the isosurface shrinks to a point with S ¼ 0, and the cross section of Rayleigh scattering is zero. Similarly, around the Weyl point, the area of the isosurface S $ Δω 2 ¼ ω À ω Weyl À Á 2 . The cross section scales as σ $ ω 2 Δω 2 , and is zero at the Weyl point (Fig. 5b). To validate our theoretical prediction above, we numerically calculate the Rayleigh scattering cross section of a small dielectric sphere embedded in the same Weyl photonic crystal. We use MPB 32 to obtain the eigenmode of the Weyl photonic crystal, then numerically calculate the Rayleigh scattering cross section using the normal-mode expansion 37-39 (see more details in   Fig. 5c as red circles. They are normalized by σ R , the scattering cross section of the same scatterer in free space. The blue dashed line indicates the scaling law σ $ ω 2 Δω 2 . The calculated Rayleigh scattering cross section agrees with the theoretical prediction well, with a vanishing cross section observed at the Weyl point. For high-frequency devices, such as integrated waveguides and laser cavities, Rayleigh scattering caused by interface roughness degrades performance and increases noise. The combination of suppressed Rayleigh scattering and enhanced resonant scattering could make Weyl media attractive for these optoelectronic applications.

Discussion
Moreover, the conservation law of resonant scattering can also be extended to lower-dimensional space. In two-dimensional photonic crystals, diverging cross sections can be realized at Dirac points; an example is provided in Supplementary Note 4. To further generalize the findings in this paper, we may not necessarily need conical dispersion. Quadratic dispersion found around the band edges of photonic crystals also provides shrinking isosurfaces. However, this is less useful in practice because the zero group velocity at the band edge makes it difficult to obtain propagating waves 40 in the presence of disorders. In addition, coupling into such media is difficult due to the large impedance mismatch.
As a final remark, the transport properties of electrons around Dirac and Weyl points has also been studied in the past few years [15][16][17] . Some of the observations are consistent with the physics of photon scattering shown in this paper. Here, we explicitly show the general conservation law of cross section and its connection to the dispersion relation. We expect that similar conclusions can be drawn for both electrons and phonons. It provides useful insight to understand general scattering physics beyond Dirac and Weyl systems.
In conclusion, large resonant cross sections are of great practical importance. They are only achievable with long resonant wavelengths when the frequency approaches the DC point. As shown in this work, the dispersion, rather than the wavelength, is responsible for the cross section. As a result, Weyl points, which can achieve similar conic dispersion as that around the DC point, lead to the diverging resonant cross section at any desired frequency. The exceptionally strong resonant scattering is also  Fig. 5 Rayleigh scattering in free space and Weyl systems. a In free space, the Rayleigh scattering cross section scales as σ $ ω 4 and vanishes at the DC frequency. b In Weyl systems, the Rayleigh scattering cross section scales as σ $ ω 2 Δω 2 and vanishes at the Weyl frequency, which is indicated by the black dashed line. In both a and b, regions with stronger red colours indicate smaller Rayleigh scattering cross section. c Numerical calculation of Rayleigh scattering in a Weyl photonic crystal. The Rayleigh scatterer is a dielectric sphere with a radius of 0.01a and a dielectric constant of 2. The numerical results (red circles) are normalized by the Rayleigh scattering cross section in free space σ R , and fit well the predictedσ $ ω 2 Δω 2 scaling (blue dashed line). The black dashed line also indicates the Weyl frequency accompanied by diminishing non-resonant scattering, which is also similar to that around the DC point. Since Weyl points can be realized at any frequency, we can effectively decouple the cross section and the wavelength. It opens up possibilities for tailoring wave-matter interaction with extraordinary flexibility, which also can be extended to acoustic and electronic wave scattering.
Data availability. The data that support the finding of this study are available from the corresponding author upon reasonable request.