Focusing of mid-infrared polaritons through patterned graphene on van der Waals crystals

3.1 Focusing performances of polaritons

To demonstrate the focusing characteristics of the proposed structure, we determined the spatial distributions of the real part of electric field component in the z-direction Re(E _z ) and the corresponding values of f for m ranging from 2 to 5, as shown in Figure 2A–D, respectively. The value of m is measured from the excitation source to the graphene–MoO₃ interface (rightmost edge of graphene ellipse) of the GCHM (proposed) structure. These simulations were conducted under specific parameters: angular frequency (ω) = 910 cm⁻¹, t = 150 nm, and E _f = 0.5 eV. Further, we included the results from a previous study [47], which adopted a plain graphene sheet covering half of an α-MoO₃ slab (GCHM), as shown in Figure 2E–H. The vertical dashed lines in Figure 2A–D mark the right edges of graphene ellipses, and the distance between these edges and the dipole source is mλ _x . Similarly, the vertical solid lines in Figure 2E–H indicate the interfaces between the left graphene-covered and right bare α-MoO₃ slabs, with the distance between the interface and the dipole source also equal to mλ _x . It was observed that exciting the polaritons on the patterned graphene initiates their propagation outward in various directions. Upon reaching the edges of the graphene ellipse, these polaritons undergo topological shifts from elliptical to hyperbolic IFCs. Consequently, their spreading becomes confined along the [100] direction, transitioning into the hyperbolic wavefront supported by the bare α-MoO₃ slab, thereby enhancing light focusing. Contrarily, the GCHM required a relatively long distance to converge the divergent wavefront to the focal point because a relatively broad wavefront was required at the interface.

Figure 2:

Re(E _z ) field distributions for the proposed structure at various “m” values, including (A) m = 2, (B) 3, (C) 4, and (D) 5. The corresponding Re(E _z ) field distributions for the structure in reference [47] are illustrated for (E) m = 2, (F) 3, (G) 4, and (H) 5. These distributions are obtained using consistent parameters, including ω = 910 cm⁻¹, α-MoO₃ slab t of 150 nm, and E _f of 0.5 eV. Normalized Re(E _z ) line plots are provided on the right-hand side along the y-direction of the focal length, f. (I) Normalized Re(E _z ) line fields for the proposed structure are depicted along the x-direction, starting from the graphene edge (x = 0) to the position of x = 5 μm. (J) Normalized Re(E _z ) line plots start from the interface to the position of x = 5 μm for the structure in Ref. [47]. The focal points for different m values are indicated by arrows.

Polaritons propagate within damping media; thus, a high f results in high energy loss. A stronger power intensity at the focal point enhances imaging quality and light–matter interaction. It is noteworthy that a smaller m correlates with a shorter f, as shown in Figure 2A–D, leading to a stronger power intensity due to propagation over a shorter distance in lossy media. Therefore, reducing the f is advantageous for preserving energy, improving imaging quality, and enhancing light–matter interaction. The proposed structure exhibited f values of 0.72 μm at m = 2 and 1.84 μm at m = 5, representing only 0.42 and 0.60 times the respective values of the GCHM. Remarkably, with increasing m towards infinity, the proposed structure gradually approaches the behavior of the GCHM, indicating that the GCHM was a special case of the proposed structure. Due to the need for a relatively broad wavefront at the interface, the GCHM necessitates an extended distance to converge the divergent wavefront to the focal point. Furthermore, the Re(E _z ) distributions at the values of f along the y-direction exhibited significant ripples near the main lobe for the GCHM, as depicted on the right side of Figure 2E–H. These ripples considerably impacted the focal quality, reducing its overall performance. In comparing the field profiles for various m values, we presented the normalized Re(E _z ) line plots of the proposed structure along the x-direction. These fields started at the graphene edge and extended to the position of x = 5 μm, as depicted in Figure 2I. Similarly, we illustrated the corresponding normalized Re(E _z ) line plots for the GCHM, which started at the interface and spanned to the position of x = 5 μm, as shown in Figure 2J. The arrows within these figures indicate the focal points for different m values. Notably, the field profiles shown in Figure 2I and J are normalized by the field amplitudes of m = 2 in their respective structures. In our proposed structure, the field amplitudes at the focal points were at their maximum values (see Figure 2I). However, the field amplitudes at the focal points were not the maximum (see Figure 2J) in the GCHM. This was because the energy loss due to the high f in the GCHM exceeded the energy enhancement achieved through the focusing process. In particular, we observed that the distances between the dipole source and the graphene edge coincidentally aligned with the values of f, revealing a mirror imaging effect.

To investigate the wavevector characteristics of our proposed structure, we conducted a Fourier transformation of the Re(E _z ) field profiles presented in Figure 2A–D. The resulting numerical IFC plots in the wavevector space (k _x , k _y ) are depicted in Figure S1A–D for m = 2–4, respectively, overlaid with the corresponding analytical IFCs. The IFCs for the Re(E _z ) fields within the graphene ellipse and the bare α-MoO₃ exhibited elliptical and hyperbolic profiles, respectively. Notably, the thickness of the brightest ellipse in the IFC diminished as m increased. This reduction was attributed to the small spatial extent of the graphene ellipse for low m values, resulting in a thick IFC in the wavevector domain. Additionally, the number of ellipses between the center and the brightest ellipse corresponded to the value of m, as explained by the presence of the same number of polariton wavelengths within the graphene ellipse. To quantitatively assess the waist diameter (WD) and light intensity, we plotted the line intensities of |E _z |² at the focal points for the proposed structure and GCHM. These plots are displayed in Figure 3A–D for different values of m ranging from 2 to 5, respectively. Remarkably, our observations revealed that the intensities in the proposed structure were approximately five times stronger than those in the GCHM. Additionally, the WD in our design achieved approximately 60 % of the values in the GCHM. For m = 2 (5), the WD corresponded to approximately 1/92 (1/54) or equivalently 1.09 % (1.85 %) of the free-space light wavelength of 10.99 μm (ω = 910 cm⁻¹). To observe the focusing characteristics, we plotted the f and WD as functions of m, as shown in Figure 3E, for both the proposed structure and GCHM. It showed that the f and WD exhibited linear dependencies on m, indicating that large graphene ellipses resulted in high values of f and WDs. These results indicated that the proposed design significantly enhanced focusing qualities. These included achieving a lower f, cleaner beam profile, higher power intensity, and lower WD than the previously published results in the GCHM. The realization of an ultrasmall spot in the proposed structure enabled deep subwavelength focusing, coupled with strong light–matter interaction for mid-IR polaritons.

Figure 3:

Profiles of the |E_z|², values of f, and waist diameters (WDs). Field intensities of the |E _z |² of the proposed structure and GCHM for (A) m = 2, (B) 3, (C) 4, and (D) 5 and the associated WDs, where the |E _z |² line plots are normalized by those of the proposed structure. (E) f and WD are depicted as functions of m for both structures.

3.2 Focusing characteristics’ dependence on Fermi level, operating frequency, and noninteger positive real number

We delved deeper into the dependencies of the focusing characteristics of our proposed structure on varying E _f . The Re(E _z ) distributions, corresponding to different values of E _f ranging from 0.3 to 0.6 eV, are shown in Figure 4A–D under the conditions of ω = 910 cm⁻¹, t = 150 nm, and m = 3. Note that the polariton wavelength increased as E _f increased, thereby increasing the geometric size of the graphene ellipse. We observed that the f increased by approximately one polariton wavelength (λ _pb = 0.40 μm) in the bare α-MoO₃ slab for every additional increment of 0.1 eV within the range of E _f from 0.3 to 0.6 eV. These results are consistent with the notion that a high m affords a high f. The f can be actively controlled in a range of 0.38–1.52 μm when varying E _f from 0.3 to 0.6 eV, respectively. Notably, the WDs remained at approximately 135 nm, with minimal variations as E _f changed.

Figure 4:

Re(E _z ) field distributions of the proposed structure for E _f = (A) 0.3, (B) 0.4, (C) 0.5, (D) 0.6 under the parameters of angular frequency ω = 910 cm⁻¹, t = 150 nm, and m = 3.

Considering the impact of the spectral response on focusing characteristics, we examined the Re(E _z ) fields across the frequency range from ω = 880 to 930 cm⁻¹ under the condition of E _f = 0.5 eV, t = 150 nm, and m = 3. The results are shown in Figure S2A–F. The polariton wavelength on the graphene ellipse increased as ω decreased, affording a large graphene ellipse. Additionally, the polariton wavelength increased, supported within the bare α-MoO₃ slab, as ω decreased, further augmenting the f. Consequently, the variation in the f while altering ω was more pronounced than the changes observed while varying the E _f . The f varied from 4.01 to 0.43 μm and WD varied from 176 to 116 nm as ω was adjusted across the range from 880 to 930 cm⁻¹, illustrating the significant influence of spectral variation on the focusing characteristics. The Re(E _z ) fields at m = 3–3.9, with increments of 0.1, are exhibited in Figure S3A–J, respectively. The field profiles within the graphene ellipses exhibited slightly different interference patterns when noninteger values of m were employed. This confirmed that the values of f could be subtly adjusted by increasing m in a noninteger manner, without the constraint of integer values.

3.3 Diffractionless propagation and directional steering

To further steer the focusing light beam, we employed a composite structure that combined two differently oriented α-MoO₃ slabs, each covered with distinct graphene patterns, as depicted in Figure 5A. The left portion of this composite structure is identical to that shown in Figure 1A. However, the right side features an oriented α-MoO₃ slab at an angle θ between the x-axis and the crystallographic direction [100] of the α-MoO₃, as illustrated in Figure 5B, and is covered with a semi-infinite graphene (SG) sheet. The permittivity tensor of the α-MoO₃ slab with an oriented angle θ could be obtained by conducting coordinate transformation with a rotation matrix (see Supplementary Material, §4). The graphene patterns were electrically gated with distinct values of E _f , denoted as E_f1 for the left ellipse and E_f2 for the right sheet. The separation between the edge of the ellipse and the interface of the two α-MoO₃ slabs was set as the f, indicated as f _m for different values of m. To fabricate a lateral heterojunction combined with two differently oriented α-MoO₃ flakes as shown in Figure 5A or B, step (2) would be replaced by the interfacial sliding approach proposed by Li et al. [59]. Initially, a vertical heterostructure is prepared on a polyimide (PI) substrate by stacking two α-MoO₃ flakes vertically. Then, by applying tensile strain to the flexible PI substrate via mechanical bending, the α-MoO₃ flakes can gradually slide apart or toward each other during the substrate stretching process. By controlling the tensile strain, the two contacted flakes will eventually separate and convert into two distinct flakes. In our previous work [60], the focusing performances under a plain graphene sheet covering the left α-MoO₃ slab, which had no orientation to the x-axis, were identified as needing substantial improvement, particularly due to significant ripples near the main lobe impacting the focal quality. As illustrated in Figure 5A of [60], a gap was set between both sides of the graphene sheet without graphene coverage, requiring precise tuning of its width. In contrast, in the current design, the left boundary of the right graphene sheet aligns directly with the focal point, facilitating perfect light canalizations. This is made possible because the patterned graphene achieves high-quality beam profiles and narrower waist diameters at the focal points, surpassing those obtained with the plain graphene sheet.

Figure 5:

Achieving diffractionless propagation and directional steering. (A) A composite structure featuring two graphene patterns: an elliptical ellipse with E_f1 and a semi-infinite graphene (SG) sheet with E_f2 placed on two differently oriented α-MoO₃ slabs. (B) The structure shown in part (A) without the SG, revealing the oriented α-MoO₃ slab at an angle θ, with f _m denoting the f for a specific m. Profiles of Re(E _z ) at m = (C) 2, (D) 3, and (E) 4 and |E| at m = (F) 2, (G) 3, and (H) 4. These profiles are obtained under the following parameters: ω = 910 cm⁻¹, t = 150 nm, E_f1 = 0.5 eV, and E_f2 = 0.22 eV. Notably, the red dashed lines highlight the interfaces between the two differently oriented α-MoO₃ slabs. Line plots of Re(E _z ) along the x-direction with and without the SG for m = (I) 2, (J) 3, and (K) 4. Re(E _z ) profiles with SG under the conditions of (L) m = 2 and θ = 40°, (M) m = 3 and θ = 20°, and (N) m = 4 and θ = −30°. Corresponding |E _z | profiles are shown under the conditions (O)–(Q).

The specific parameters employed for these simulations were as follows: ω = 910 cm⁻¹, t = 150 nm, E_f1 = 0.5 eV, E_f2 = 0.22 eV, θ = 0°, and various values of f _m (f₂ = 0.72 μm, f₃ = 1.19 μm, and f₄ = 1.52 μm). Re(E _z ) and |E| profiles are shown in Figure 5C–E and F–H, respectively, ranging from m = 2 to 4. Notably, the red dashed lines in these figures denote the interfaces between the two α-MoO₃ slabs. The WDs were well preserved as the focusing beams traversed the interfaces, thereby canalizing the polaritons and unlocking the fixed values of f. This contrasted the conventional diffraction exhibited by polaritons without the SG, as depicted in Figure 2A–C. Furthermore, to evaluate the field enhancement achieved by adding SG, the Re(E _z ) fields are depicted in Figure 5I–K for values of m = 2–4, respectively. These results revealed a significant reduction in the decay rate of field amplitude, transitioning from approximately e^−0.89x for the structure without SG to e^−0.57x for the structure with SG at m = 3. This outcome suggests that the proposed approach offers a versatile imaging method that does not impose restrictions based on fixed values of f. Furthermore, we steered the direction of the focused polaritons by adjusting the angle θ of the right α-MoO₃ slab. The Re(E _z ) profiles for three conditions, (1) m = 2 and θ = 40°, (2) m = 3 and θ = 20°, and (3) m = 4 and θ = −30°, are presented in Figure 5L–N, respectively, and their associated |E _z | profiles are illustrated in Figure 5O–Q. These results provide compelling evidence to effectively control the directional canalization of polaritons. The outcomes presented herein demonstrate that our proposed approach enables high-quality focusing and facilitates the directional waveguiding of mid-IR polaritons. This establishes a viable system for constructing a diverse range of photonic circuits and imaging systems.