Optical metasurfaces can be used to realize various peculiar optical effects, and their mechanisms of the controlling optical phase can be roughly categorized into three types: resonant phase, geometric phase, and propagation phase, also known as the dynamic phase. Multiple mechanisms can be employed to manipulate the phase and amplitude of one metasurface. Therefore, discovering more profound and diverse methods can provide additional degrees of application freedom. This paper proposes a control principle based on electric multipole expansion. We found that for a unit structure formed by dual-metal bars on a metasurface, the radiation of its dipole is equivalent to the interference results of the dual-metal bars. Moreover, the radiation of the quadrupole enables independent control for amplitude and phase. Therefore, we used quadrupole radiation to manipulate the phase and amplitude of the light and even investigated some simple applications, including the realization of focusing light and anomalous refraction. Such a new mechanism of controlling light, combined with other methods, can provide significant insights into achieving challenging goals, like steganography and multifunctional metasurfaces.

## I. INTRODUCTION

Since the research of optical metamaterials has stretched to the field of optical metasurfaces, researchers have extensively investigated the applications of optical metasurfaces, resulting in a thorough understanding of their control mechanisms.^{1,2} Currently, the resonant phase is achieved by analyzing the modes of nanostructures^{3} and identifying the resonant peaks of different geometric structures, from which we can achieve the desired phase and amplitude control.^{4} This approach can generate generalized Snell’s law and vortex beams.^{5} The aforementioned control method is based on manipulating geometric parameters of nanostructures. Another well-known method is based on the geometric phase,^{6,7} and its fundamental principle is that the solid angle in the Poincaré sphere of photon polarization is equal to twice its Berry curvature. Its origin can be traced back to the Berry phase in quantum mechanics.^{8} In addition to geometric phase control, there is another way called dynamic phase control^{9} in quantum adiabatic processes. Therefore, the third method of control is the so-called propagation phase (dynamic phase) manipulation, such as the all-dielectric metasurfaces,^{10–13} Metal–Insulator–Metal (MIM) sandwich structures,^{14,15} which utilize the formation of Fabry–Pérot cavities in metasurface units or the propagation phase accumulation of light^{16–18} to achieve phase control. A summary of these three control mechanisms is shown in Fig. 1(a). These three fundamental methods for controlling optical metasurfaces have found wide-ranging applications,^{19} including holographic metasurfaces,^{20–23} metalenses,^{24–27} surface plasmon generation,^{28} optical forces generation,^{29,30} optoelectric effect,^{31} multifunctional reconfigurable metasurfaces,^{32,33} and more. Additionally, there are metasurface designs that simultaneously utilize different principles for control.^{34} From this, we found that the discovery of more control principles for metasurfaces may lead to further applications in developing multifunctional optical metasurfaces. In this study, we propose a new way of controlling light based on the theory of multipole expansion of metasurface units. After the multipole expansion of the meta unit composed of two nanoscale metal bars, we found that its dipole radiation’s amplitude and phase distribution are similar to its interference results.^{35} Therefore, it is time to further analyze the radiation of its quadrupole, and with theoretical analysis, and simulations, we have found that dipole radiation consists of two components: a trivial component (which cannot modulate phase and amplitude) and a nontrivial component (which could modulate phase and amplitude). The results indicate that when the nontrivial component completely fails to modulate under specific situations, the radiation of our electric quadrupole can still independently control the amplitude and phase, which undoubtedly provides new ideas for controlling electromagnetic waves with metasurfaces. After that, we utilized this fundamental principle to fabricate a metalens and realized generalized Snell’s law, which is validated through finite-difference time-domain (FDTD) simulations. It is necessary to note that metasurface is an advanced diffractive media and previous research mainly focuses on different orders of diffraction field, like using Fourier expansion to get the scattered field.^{36} However, in this study, the multipole expansion serves as an alternative approach to deal with the scattered field, and the consistency between the dipole radiation and the interference results of the dual-metal bars confirms the theory.

## II. THEORY OF RADIATION CONTROL WITH ELECTRIC QUADRUPOLES

^{35}Under the situation that two nano bars are perpendicular to each other ( $ \theta 1\u2212 \theta 2= \pi 2$), the term for phase control of dipole radiation is ineffective. In such a situation, its quadruples radiation will modulate the scattered field’s phase and amplitude independently. As shown in Fig. 2(a), we simulated the radiation intensities of the electric quadrupole and electric dipole using the Cosmol Multiphysics: $ I P= 2 \omega 4 3 c 3| P | 2$, $ I Q= \omega 6 5 c 5\u2211 | Q \alpha \beta | 2$.

It can be seen that the simulation results are consistent with the theoretical predictions, as shown in Fig. 2(a). It is worth noting that when $ \theta 1\u2212 \theta 2= \pi 2$, the dipole radiation does not reach zero. This is due to the presence of the trivial term in Eq. (7). On the other hand, when $ \theta 1\u2212 \theta 2=0$, the electric quadrupole radiation completely reaches zero. As shown in Fig. 2(b), we analyzed the scattering amplitude of two perpendicular metal rods with respect to wavelength. It can be observed that the scattering amplitude of the electric quadrupole is more significant than that of the electric dipole. In the subsequent applications, we consistently utilized 1064 nm left circularly polarized (LCP) light as the light source.

## III. APPLICATION

### A. Metalens

Now, in the case where the electric dipole is completely ineffective (i.e., $ \theta 1\u2212 \theta 2= \pi 2$ at which the dual-bar interference method fails), we can still produce metalens. By analyzing Eqs. (7) and (8), it can be concluded that under the conditions mentioned above, the electromagnetic waves radiated by the dipole only contain negligible trivial terms (without any phase discontinuities).

We focus on Eq. (8) to obtain a simplified phase distribution with metalens: $\varphi (x,y)=\u2212 ( \theta 1 + \theta 2 )= 2 \pi \lambda ( f \u2212 x 2 + y 2 + f 2 )$.

We designed a structure with units of two perpendicular rods, leading to a phase distribution of the electric quadrupole radiation field to be focused with a focal length of $f$, which satisfies the above equation and serves as metalens [as shown in Fig. 3(a)]. Through FDTD simulation, we can obtain the spatial distribution of the electric field in the z-direction near the focal spot, as depicted in Fig. 3(b). To acquire the polarization properties of the radiation field, which is dominated by the electric quadrupole in the metalens metasurface, we calculated the distribution of the $ E z$ of the electric field on the x–y plane (see Appendix B). It can be observed that the focal spot is composed of a combination of zero-order Bessel functions and other higher-order Bessel functions, which is consistent with Fig. 3(b).

### B. Anomalous refraction phenomenon

In this section, we aim to utilize Huygens’ principle to construct the wavefront of an electric quadrupole radiation and achieve an anomalous refraction phenomenon similar to Ref. 3. We still employ two perpendicular metallic nanorods to ensure that the radiation of electric quadrupoles dominates the metasurface. The length of the rods remains at $350\xd770\xd770 n m 3$, and the incident light is an LCP light with a wavelength of 1064 nm, which is perpendicular to the metasurface. To obtain pronounced effects, we design a phase gradient distribution as shown in Fig. 4(a), and the transmitting angle satisfies the following relation: $sin\u2061 ( \theta t ) n t\u2212sin\u2061 ( \theta i ) n i= \lambda o 2 \pi d \Phi d x$, where $ \lambda 0 2 \pi d \varphi d x= 1.064 2.1=0.507$, which can be used for the refraction angle, and after conducting FDTD simulations, we obtained the anomalous refracted phenomenon of the metasurface dominated by electric quadrupoles, as shown in Fig. 4(b).

## IV. CONCLUSION

The proposed metasurface based on electric quadrupoles also enables independent amplitude and phase control. In addition to the current dual nanobars model, our theoretical analysis method is also applicable to other models. The analysis steps are as follows: first, analyze the charge oscillation modes of the model under a certain polarization of electromagnetic waves, locate the modes that can simplify the integrals (1) and (2), and then calculate its electric multipole expansion. While the resonance loop is formed, the analysis will not be limited to electric multipole expansion but also magnetic multipole expansion. Subsequently, the reasonable metasurface to achieve the desired phase and amplitude distribution will be designed by using the relation between the amplitude and phase change of the radiated field concerning that of the incident light. Although the conversion efficiency of higher-order multipole moments will decrease, there exist various types of multipole moments, each offering different possibilities for amplitude and phase modulation. This provides a rich selection of pathways for controlling the metasurfaces. This method inspires holography, vector field manipulation, transformation optics, and integrated optics design. Interestingly, the amplitude-phase control of the dipole radiation calculated in this work is consistent with the results of dual nanorods’ interference field, demonstrating the theory’s self-consistency. The reason behind it is that metasurface is essentially a diffractive grating. Researchers employ various methods to achieve the desired effects by utilizing certain diffraction orders of diffractive fields. In this study, we propose a different approach with multipole expansion and get results similar to the Fourier expansion method, highlighting the self-consistency of the theory. In conclusion, combined with other manipulation principles, this method will provide new ideas for increasing the degrees of freedom in the metasurface design.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (No. 12174052).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jiawei Zhang:** Conceptualization (equal); Formal analysis (equal). **Weijie Shi:** Software (equal). **Andong Liu:** Formal analysis (equal); Writing – review & editing (equal). **Lili Tang:** Software (equal). **Shuyan Zhang:** Visualization (equal). **Zhenggao Dong:** Funding acquisition (equal); Investigation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: MULTIPOLE MODEL

In this form of solution, the first term represents the background field (incident field), which does not undergo any phase or amplitude changes. Therefore, our focus lies on the phase and amplitude distribution of the second term, which represents the scattered field. To solve the scattered field, we can employ the multipole decomposition method, which is very convenient to conduct.

Among them, $ x 1 +, y 1 +, x 1 \u2212, y 1 \u2212$ represent the $x$ and $y$ coordinates of the positive and negative point charges on the left bar, respectively. The second set of coordinates are likewise.

**E**represents the electric field’s magnitude along the bar’s longitudinal axis, as shown in Fig. 5(b). In the case of LCP (left circularly polarized) light incidence, we project the electric field components along the $ E x$ and $ E y$ directions onto the longitudinal axis of the bar, and the electric field along the bar can be acquired

### APPENDIX B: ELECTRIC FIELD DISTRIBUTION IN THE FOCAL PLANE

By utilizing the integral expression of the nth order Bessel function: $ J n(x)= 1 2 \pi \u222b \u2212 \pi \pi e i ( n \tau \u2212 x sin \u2061 \tau )d\tau $, the distribution of the electric field $ E z$ on the x–y plane of the focal point can be calculated, as shown in Fig. 3(b). This distribution includes a zeroth-order Bessel function and other higher-order Bessel functions. As a result, the central point appears bright, which is consistent with the theoretical expectation.

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