Objective The quantum coherent effect of electromagnetic induced transparency (EIT) is characterized by a narrow transmission peak in a broad absorption band. This phenomenon is associated with the slow light effect, which can be used in optical buffering, refractive index sensing, and other applications. And the development of the EIT effect has been limited due to the extremely difficult implementation conditions. Metamaterials are artificial composite materials made from natural materials with unique physical and chemical properties that natural materials do not have. Adjusting the resonant frequency of bright and dark modes to make them resonant at a close frequency and then combining them to produce an atom-like electromagnetic induced transparency phenomenon is how EIT-like behavior in metamaterial is generally realized. However, in some reported research, the passive modulation method is used in EIT-like metamaterial, which greatly limits the application of related devices. Therefore, studying EIT-like metamaterials in the terahertz band under active regulation is an important research topic.

Electronic control is now widely used due to its simple operating conditions. In general, electronic control can be achieved by incorporating electrically adjustable devices, such as variograms and other electrically tunable materials. The reported graphene structures are typically polarization- and incident-angle-sensitive, with the transmission peak disappearing under oblique incidence. Because of numerous possibilities for the polarization direction and angle of the incident wave in practical applications, polarization- and angle-insensitive metamaterial devices are more suitable for application requirements. The proposed structure shows excellent characteristics, such as polarization-independence, incident angle-insensitivity, obvious slow light effect, and high refractive index sensitivity.

Methods A novel polarization- and angle-insensitive metamaterial structure based on the cross and 4L-shaped graphene is designed. CST Microwave Studio software is used to run all numerical simulations. By splitting the structure and analyzing the surface current and electric field distributions at different frequencies, the physical mechanism is discussed. The effect of geometric parameters, such as cross and 4L-shaped graphene lengths on a transparent window are investigated. Finally, the characteristics of the proposed structure, such as polarization- and angle-insensitivity, tunability, slow light effect, and refractive index sensitivity are studied.

Results and Discussions The structure, which consists of a cross and four L-shaped graphene resonant units, has an obvious EIT-like effect, with a peak value of more than 0.8 (Fig. 2). The resonant dips of the structure with a single cross or 4L-shaped graphene are observed dips near 1.81 THz and 2.19 THz, respectively. The transparent peak lies between the resonance frequencies of the two isolated resonators (Fig. 2). At the frequency of 1.75 THz, strong electric fields are concentrated at both ends of the horizontal cross, and the surface current moves unidirectionally on the cross graphene (Fig. 3). Therefore, the cross surface experiences electric dipole resonance. The surface current and electric field generated along the arms of the 4L structure are the same at the 2.18 THz resonance frequency, which is also known as dipole resonance (Fig. 3). Both cross and 4L structures can be directly excited by incident waves as bright modes exhibiting electric dipole oscillations for the transmission peak near 1.94 THz. Surface currents are excited on the cross and 4L-shaped graphene structures simultaneously, and the directions are anti-parallel (Fig. 3). As a result of the destructive interference caused by the coupling of the two bright modes, a distinct transparency window is formed.

When it comes to structure parameters, the high-frequency transmission dip has a slight redshift as the length of 4L-shaped graphene increases (Fig. 4). As the length of the cross increases, the low-frequency transmission dip in the transmission spectra is slightly red-shifted (Fig. 5). To investigate the effect of the distance between the cross structure and the 4L structure on the transparent window, transmission spectra, and electric field intensity distributions under the peak at various distances are drawn. The transmission spectra and electric field intensity distributions show very little difference at different distances. This demonstrates that the metamaterial under consideration in this paper is fault tolerant (Fig. 6). As the Fermi energy of graphene increases, so do the transfer window is blue shifted (Fig. 7). The corresponding frequency modulation depth is approximately 0.286, with the Fermi level varying between 0.3 eV and 0.6 eV. The maximum amplitude modulation depth is 0.724, with graphene has Fermi level ranging from 0.4 eV to 0.5 eV. Because the structure has symmetry, the metamaterial structure proposed in this paper is polarization insensitive (Fig. 8). When the incident angle is less than 60°, there is no significant change in the transmission spectra for both TE polarization and TM polarization. This shows that the structure is insensitive to the incident angle of the incident wave (Fig. 9). When the Fermi level of graphene is 0.5 eV, the maximum delay of the transmission peak is 0.81 ps and can be controlled by tuning the Fermi energy (Fig. 10). The designed structure has a refractive index sensitivity of 395 GHz/RIU, which is obviously higher than many conventional EIT-like structures and can also be regulated by tuning the Fermi level (Fig. 11). Thus, it has great potentials in the field of refractive index sensing applications.

Conclusions Finally, this paper investigates metamaterials made up of the cross and 4L-shaped graphene resonant units. Numerical simulations show that bright-bright mode interference produces the EIT-like effect in the terahertz band. The studied metamaterial has great polarization- and angle-insensitivity characteristics. The transmission spectrum does not change significantly when the incident angle is less than 60°, and the transmission peak can be kept above 0.75. By tuning the Fermi level of the graphene, the frequency modulation depth of 0.286 and the amplitude modulation depth of 0.724 are achieved. In addition, the proposed structure has obvious slow light effect and high refractive index sensitivity. Meanwhile, the actively tuned group delay and refractive index sensitivity are achieved by changing the Fermi level of the graphene. These characteristics can be used in many applications, such as modulators, switches, light buffers, and sensors in the terahertz band.