Introduction

The elegant patterns of complex polymer systems driven by self-assembly mechanisms [1] inspire the design and functionality of plasmonic nanostructures [2, 3]. In the latter field, the arrangement of metallic nanoparticles at a periodicity corresponding to the wavelength of plasmonic excitations leads to a collective surface lattice resonance (SLR) [4]. The improved optical performance of SLR modes has paved the way for their application in various fields, including sensing [5, 6], energy conversion [7], the facilitation of strong light-matter interactions [8], the control of emission properties [9], and the advancement of plasmonic laser technologies [10]. For the breakthrough of plasmonic technologies, scalability through colloidal self-assembly and the versatility of structure formation at a defined distance are crucial [11]. Volk et al.’s research group demonstrates this with hexagonally ordered arrays on the centimeter scale, where the hydrogel shells control the lattice spacing [12]. Due to the soft and deformable character of the polymer shells and certain drying conditions, honeycomb monolayer lattices are also possible [13]. However, to make SLR reliably accessible for other nanostructures, template-assisted self-assembly (TASA) has been established in recent decades. TASA is compatible with soft lithographic techniques such as nanoimprint lithography and is suitable for various materials, colloids, and solvents, as Scarabelli et al. summarized in their review article [14]. However, the combination of TASA with various 2D colloidal nanostructures, such as the Lieb lattice or the Kagome lattice, which also supports SLR, is a research field that is still in its infancy.

A key to complex 2D grating geometries is phase-engineered interference lithography (PEIL). The method extends laser interference lithography with a phase-engineered spatial light modulator to generate photonic lattices. Xavier and Joseph’s research group has demonstrated the generation of both periodic right- and left-handed chiral structures as well as photonic, transverse quasi-crystallographic chiral structures [15]. This method can, in principle, also be considered to produce 2D lattice geometries such as Lieb lattices, honeycomb structures, or Kagome lattices to use as plasmonic band gap materials [16]. The Lieb lattice, for example, is known for its flat bands and localized states [17], which can significantly change the light-matter interaction and thus improve the sensitivity and selectivity of sensors. Honeycomb structures, which resemble the geometry of graphene, exhibit unique electronic and optical properties due to their Dirac cones and edge states [18]. In their theoretical work, Ochiai and Onoda explore this relationship using topological phase transitions in photonic honeycomb lattice crystals and compare them to electronic edge states [19]. In a follow-up paper by Han et al., these topologies are investigated with plasmonic honeycomb lattices, and the Dirac spectra are shown for both dipole and quadrupole modes [20]. Kagome lattices, characterized by their corner triangles, offer intriguing band gap features and topological properties crucial for advanced light manipulation and control [21]. Proctor et al. demonstrate this theoretically on plasmonic Kagome lattices and determine the properties by the local density of photonic states to realize plasmonic metasurfaces [22]. The review article by Moon et al. provides a comprehensive overview of how the above structures can be fabricated defect-free as dielectric 3D photonic crystals using multi-beam interference lithography [21]. Once the target structures and the polydimethylsiloxane stamp are generated from this master structure, the plasmonic nanostructure can, in principle, be created with colloidal ink [23]. The available optical design and the possibility of scalable implementation with high localized precision can take plasmonic band gap engineering a decisive step forward.

In this work, we conceptualize PEIL-based fabrication approaches to address SLR-assisted complex gratings on a large scale and numerically study their optical properties by electromagnetic simulations. The possibility of recording of grating structures in photoresists can serve as a basis for template fabrication. Although not carried out in the present work, such templates hold the key towards arranging regularly spaced plasmonic nanoparticles by utilizing the TASA method. Therefore, this research mainly presents an outlook with a colloidal self-assembly approach for fabricating diverse and complex 2D plasmonic lattice geometries. Using electromagnetic modeling, the optical extinction characteristics of the targeted plasmonic lattices are investigated for different periodicities with a detailed study of the electric field distribution at single particle and lattice resonance conditions. Further, the optical band diagrams are also calculated to show the existence of all possible modes in such lattices. Finally, as an approach towards applicability, the sensing properties of the resonant lattice modes are analyzed and compared through the cover index variation.

Experimental section

MATLAB computation

An XYZ mesh grid is constructed within the MATLAB domain to realize the phase-controlled approach, where the interference of multiple beams with varying propagating vectors is calculated. The X and Y axes are visualized from − 1.5 to 1.5 μm with a step size of 10 nm, whereas the Z axis is constrained to 500 nm with a resolution of 5 nm to recreate patterning on photoresist thin film. The interfering wavelength is considered to be 405 nm. For the 3D intensity distribution visualization, patch-isosurface and patch-isocaps commands are used.

Finite-difference time-domain simulations

For the FDTD method, a commercial-grade electromagnetic solver from Ansys Lumerical [24] has been utilized under normal incidence illumination with a broadband source (400 to 1000 nm) in the Z and periodic boundary conditions in the X and Y directions. An auto non-uniform mesh with a minimum mesh step of 0.25 nm is selected for all the simulations with an additional mesh overlay of 2 nm around the gold nanoparticles (AuNPs). Data from Johnson and Christy [25] are incorporated for the optical properties of these AuNPs. The FDTD background index has been considered 1.45 to impose a symmetric cover-substrate condition. For the square basis, the FDTD X and Y spans are kept identical, whereas the hexagonal basis is constructed by keeping the Y span times the X span. Frequency-domain field and power monitors calculate the transmittance (T) of the various lattices from which the extinction characteristics are calculated following Ext = − ln T. Frequency-domain field profile monitors record the electric field distribution using single-wavelength source and an additional 2-nm mesh overlay throughout the cross-sectional lattice plane. Further, several modifications to the above methodologies are implemented to calculate the band structures of the plasmonic lattices. All possible modes are excited for each wave vector (kx and ky) using multiple randomly oriented broadband (400 to 1000 nm) dipoles using built-in “dipole cloud” analysis. The number of dipoles, as well as the number of randomly distributed time monitors, is set to 100. At wavelengths where resonant modes exist, the field propagates infinitely. The monitors at different locations ensure that all modes are captured. The simulation time is set to 500 fs to allow for the decaying fields to dissipate. For the visual representation of the Brillouin Zone, Gamma-X path was considered.

Result and discussions

Phase-engineered interference lithography method (PEIL)

Laser interference lithography (LIL), also known as holographic lithgography [26], has been utilized as an exposure technique to the existing TASA method to produce 1D/2D plasmonic lattices. To produce complex lattices like Lieb, honeycomb, and Kagome (Fig. 1a), using conventional LIL-based exposure, one might need a bulky optical setup and additional components to generate and steer multiple non-planar beams [21]. In contrast, PEIL as an alternate exposure technique (Fig. 1b) offers possibilities for generating complex exposure patterns. Following the analytic considerations discussed in the later part of this section (Eqs. 1–6), a digital phase image can be constructed mathematically to generate “N + 1” (or N after the DC blocking) diffracted beams [27]. These diffracted orders from the SLM can be collected at a higher interference angle by introducing a custom-designed mirror mount that has been previously reported for chiral helices [28], woodpile structures [29], and chiral helices with sub-micron axial pitch [30]. A dual-cone setup allowed the exploration of chiral woodpile structures [31] by using interfering beams from both the hemispheres of the recording plane. Further, the intrinsic effect of polarization on incorporating higher angles has also been explored theoretically to understand the scalability in these PEIL-based geometries [32].

Fig. 1
figure 1

a (i–vi) Various plasmonic lattices as theoretically proposed in this article through the phase engineering of the interfering beams. b Schematic illustrating the general concept of a conventional PEIL setup capable of interfering N beams with specific phase offsets for complex lattice formation. In our study, we exclusively focus on analytically determining these phase offsets rather than implementing them experimentally

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Therefore, depending on the digital phase mask, N beams can interfere on the recording plane with the resultant intensity pattern given by

$${I}_{s}\left(\overrightarrow{r}\right)=\sum _{p=0}^{N}{\left|{E}_{p}\right|}^{2}+\sum _{p=0}^{N}{E}_{p}\sum\nolimits _{\begin{array}{c}p=1\\ q\ne p\end{array}}^{N}{{E}_{q}}^{*}\text{exp}\left[i\left({\overrightarrow{k}}_{p}-{\overrightarrow{k}}_{q}\right).\overrightarrow{r}+i\left({\psi }_{p}-{\psi }_{q}\right)\right]$$
(1)

Here, \({E}_{p}={E}_{0}\text{exp}\left[i\left({\overrightarrow{k}}_{p}.\overrightarrow{r}\right)+i{\psi }_{p}\right]\) with unit amplitude (\({E}_{0}=1)\), \({\overrightarrow{k}}_{p}\) is the wave-vector and \({\psi }_{p}\) is the phase offset of the pth beam. These beams diverging out of the SLM along the surface of a cone can be described by

$$\overrightarrow{k_p}=k_0\left[\text{s}\text{i}\text{n}\theta_{\mathrm{int}}\text{c}\text{o}\text{s}\phi_p,\text{s}\text{i}\text{n}\theta_{\mathrm{int}}s\text{i}\text{n}\phi_p,\text{c}\text{o}\text{s}\theta_{\mathrm{int}}\right]$$
(2)

For the basic 2D configurations, a single exposure of four interfering beams can easily achieve the square or rotated square lattice. However, dual exposure is required for the Lieb lattice case, where the first exposure results in a regular square lattice (Fig. 1b, i), with the second exposure resulting in a rotated square lattice (Fig. 1b, ii). Further, the Lieb lattice can only be idealized when the rotated square lattice’s diagonal matches the square lattice’s arm (Fig. 1b, iii). Such stringent condition is maintained when sin θ1 = (1/\(\sqrt{2}\))* sin θ2 where θ1 and θ2 are interference angles (θint) for the first and second exposure, respectively. Thus, the beam vectors for the first exposure are as follows:

$$\begin{aligned}&k_1=k_0\lbrack\sin\theta_1\cos(2\pi/8\ast1),\sin\theta_1\sin(2\pi/8\ast1),\cos\theta_1\rbrack\;\psi_1=2\ast(2\pi/4)\\&k_3=k_0\lbrack\sin\theta_1\cos(2\pi/8\ast3),\sin\theta_1\sin(2\pi/8\ast3),\cos\theta_1\rbrack\;\psi_3=2\ast(2\pi/4)\\&k_5=k_0\lbrack\sin\theta_1\cos(2\pi/8\ast5),\sin\theta_1\sin(2\pi/8\ast5),\cos\theta_1\rbrack\;\psi_5=0\ast(2\pi/4)\\&k_7=k_0\lbrack\sin\theta_1\cos(2\pi/8\ast7),\sin\theta_1\sin(2\pi/8\ast7),\cos\theta_1\rbrack\;\psi_7=0\ast(2\pi/4) \end{aligned}$$
(3)

whereas for the second exposure

$$\begin{aligned}& k_2=k_0\lbrack\sin\theta_2\cos(2\pi/8\ast2),\sin\theta_2\sin(2\pi/8\ast2),\cos\theta_2\rbrack\;\psi_2=0\ast(2\pi/4)\\&k_4=k_0\lbrack\sin\theta_2\cos(2\pi/8\ast4),\sin\theta_2\sin(2\pi/8\ast4),\cos\theta_2\rbrack\;\psi_4=0\ast(2\pi/4)\\&k_6=k_0\lbrack\sin\theta_2\cos(2\pi/8\ast6),\sin\theta_2\sin(2\pi/8\ast6),\cos\theta_2\rbrack\;\psi_6=0\ast(2\pi/4)\\&k_8=k_0\lbrack\sin\theta_2\cos(2\pi/8\ast8),\sin\theta_2\sin(2\pi/8\ast8),\cos\theta_2\rbrack\;\psi_8=0\ast(2\pi/4) \end{aligned}$$
(4)
Fig. 2
figure 2

a Orientation of the interfering beams in a square lattice. (i) A total of 4 + 4 beams shown in the “umbrella” geometry in 3D view, (ii) projection of the beams on the XY plane, and (iii) schematic of the mirror mount to accommodate the 4 + 4 beams at two sets of incidence angles. b MATLAB-calculated intensity distribution for the interference of 4 beams for (i) square lattice, (ii) rotated square lattice with a different period, and (iii) interference of 8 beams for the formation of Lieb lattice as a combination of both. The nearest points of intensity maxima are connected via yellow lines to reveal the nature of the lattice. c A 3D view to demonstrate the developed negative photoresist master (intensity threshold of 0.9) for the different lattice configurations. The dashed squares denote a unit cell

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The beam orientations have been explicitly described in Fig. 2a (i–iii), whereas the resultant outputs are depicted in Fig. 2b and c. Figure 2b (i–iii) contains the intensity distribution for the square lattice, rotated square lattice, and Lieb lattice, respectively, by considering θ1 = 27° and θ2 = 40°. Such an intensity pattern can be developed from a negative photoresist layer coated on a glass substrate to create the PR master. On successful exposure and development, the resulting PR pillars are produced (Fig. 2c, i–iii), where the exposure dosage can control the diameter of the pillars. The current figure reflects MATLAB simulated intensity profile with a threshold of 0.9, leading to the smallest possible diameter concerned with the square lattice periodicity “a = λ0 /(2√2 sin1).” The specialty of the PEIL technique allows an easy choice of beams between the pairs (1, 3, 5, 7) and (2, 4, 6, 8) through the introduction of two separate phase masks without arranging or changing any additional components.

Moving onto the hexagonal basis, we propose three different structures formed by interference six beams: hexagonal, honeycomb, and Kagome lattices. Most importantly, these various lattice configurations can be achieved in a single exposure just by adjusting the phases of the individual beams. The general beam configuration for the azimuthally arranged six beams in a regular umbrella geometry is given in Fig. 3a (i–iii), which is typical for all of these lattice configurations and represented as

$$\begin{aligned}& k_1=k_0\lbrack\sin\theta\cos(2\pi/6\ast1),\sin\theta\sin(2\pi/6\ast1),\cos\theta\rbrack\\&k_2=k_0\lbrack\sin\theta\cos(2\pi/6\ast2),\sin\theta\sin(2\pi/6\ast2),\cos\theta\rbrack\\&k_3=k_0\lbrack\sin\theta\cos(2\pi/6\ast3),\sin\theta\sin(2\pi/6\ast3),\cos\theta\rbrack\\&k_4=k_0\lbrack\sin\theta\cos(2\pi/6\ast4),\sin\theta\sin(2\pi/6\ast4),\cos\theta\rbrack\\&k_5=k_0\lbrack\sin\theta \cos(2\pi/6\ast5),\sin\theta\sin(2\pi/6\ast5),\cos\theta\rbrack\\&k_6=k_0\lbrack\sin\theta\cos(2\pi/6\ast6),\sin\theta s\mathrm{in}(2\pi/6\ast6),\cos\theta\rbrack \end{aligned}$$
(5)

However, as mentioned, the choice of the phase-offsets plays an essential role along with the beam vectors that are accordingly given as

$$\begin{array}{ccc}\mathrm{Hexagonal}&\mathrm{Honeycomb}&\mathrm{Kagome}\\\psi_1=0\ast(2\pi/6)&\psi_1=1\ast(2\pi/6)+\pi&\psi_1=1\ast(2\pi/6)\\\psi_2=0\ast(2\pi/6)&\psi_2=2\ast(2\pi/6)&\psi_2=2\ast(2\pi/6)\\\psi_3=0\ast(2\pi/6)&\psi_3=3\ast(2\pi/6)&\psi_3=3\ast(2\pi/6)\\\psi_4=0\ast(2\pi/6)&\psi_4=1\ast(2\pi/6)&\psi_4=1\ast(2\pi/6)\\\psi_5=0\ast(2\pi/6)&\psi_5=2\ast(2\pi/6)&\psi_5=2\ast(2\pi/6)\\\psi_5=0\ast(2\pi/6)&\psi_6=3\ast(2\pi/6)&\psi_6=3\ast(2\pi/6)\end{array}$$
(6)
Fig. 3
figure 3

a Orientation of the interfering beams in a hexagonal lattice. (i) A total of 6 beams shown in the “umbrella” geometry in 3D view, (ii) projection of the beams on the XY plane, and (iii) schematic of the mirror mount to accommodate the 6 beams at a single incidence angle. b MATLAB-calculated intensity distribution for the interference of six beams for (i) hexagonal lattice, (ii) honeycomb lattice, and (iii) Kagome lattice. The nearest points of intensity maxima are connected via yellow lines to reveal the nature of the lattice. c (i–iii) A 3D view to demonstrate the developed negative photoresist master (intensity threshold of 0.9) for the different lattice configurations. The dashed rectangles denote a unit cell.

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Figure 3b (i–iii) provides the intensity distribution of these different lattices by considering \(\theta\) = 40°, where the periodicity is given by “a = 2λ0/(√3sin\(\theta\))”. The corresponding 3D profiles of the lattices to be formed out of the resist for a threshold of 0.9 are given in Fig. 3c (i–iii) for the hexagonal, honeycomb, and Kagome lattices cases, respectively. Thus, one can idealize the importance of the phases of these beams, which the PEIL technique can successfully manipulate.

Optical responses via electromagnetic modeling

To investigate the optical properties of the proposed lattice configurations, we have carried out numerical analysis of the different lattice resonance conditions using finite-difference time-domain (FDTD)–based simulations. While other noble metals can result in localized surface plasmonic resonances depending on the shape and size of the individual resonant blocks [33], the current simulation study is restricted only to the cases of AuNPs. In general, for colloidally grown spherical AuNPs, the particle diameter and the surrounding medium directly relate to the position of the resonant wavelength; the absorption and scattering occur maximum for a specific frequency depending on the size and the supported modes [34]. It has been found that as the particle size increases, the resulting extinction peak shifts to a higher wavelength with an increase in the effective absorption cross-sects [33, 35]. Because of the pronounced radiative damping exhibited by AuNPs, typical LSPR extinction peaks show broader line widths (FWHM, full-width half-maxima), particularly notable for particle sizes exceeding 100 nm in diameter [4]. Interestingly, when such AuNPs are arranged in an array and illuminated with light perpendicular to their lattice plane, they scatter light in various directions due to the photonic diffraction involved. Further, waves diffracted parallel to this lattice plane can couple with the LSPR of individual AuNPs, significantly narrowing the otherwise broad plasmon resonances to as little as 1–2 nm in spectral width, resulting in exceptionally high-quality (Q) factors [36, 37]. Through hybridization of plasmonic and photonic resonances, plasmonic arrays can yield higher field enhancements and longer lifetimes compared to typical plasmonic systems, finding applications in plasmonic linewidth engineering [38], lasing [39], sensing [40, 41], and others. However, it is important to note that achieving ultrasharp modes is not guaranteed solely by employing a plasmonic array 42, [43]. The periodicity of the lattice needs to be comparable to the resonant plasmon wavelength [44]; hence, for a fixed AuNP diameter, the interparticle distance for different lattice geometry needs to be controlled separately through the TASA process, as proposed in this article.

Hereby, we explore numerically the optical characteristics of complex plasmonic lattices to observe the SLR modes [45]. For this, we have calculated the transmittance of the lattice geometries; this transmittances can be further used to calculate the optical extinction spectra as reported in previous findings [38, 46]. The presence of a uniformly matched index (RI = 1.45) around the lattices is a prerequisite for generating high-quality SLR modes [38, 47]; however, we have shown previously that an asymmetric environment with smaller differences between the cover and substrate can still produce the lattice modes [46]. Starting with the square basis, Fig. 4 shows the interplay between the extinction peak and the Rayleigh anomaly (RA) wavelengths (diffraction orders propagating parallel to the lattice plane) for the square, rotated square, and Lieb lattices. In general, the position of the RA modes can be analytically identified from the diffraction equation involving a 2D non-orthogonal lattice [46]. For the square lattice, the spectral position of the RAs under normal incidence is given by λRA = ns [(m2 + n2)/P2]-1/2, where P is the periodicity along the two principal axes X and Y, m and n are the 2D diffraction orders, and ns is the refractive index of the surrounding medium onto which the diffracted orders are finally propagated. We have limited our study case with incident electric field polarization parallel to the X-axis for the following cases. By varying P from 300 to 600 nm (in discreet steps of 10 nm), the contour plot in Fig. 4a (i) captures the interaction of the RA of different orders with the broad LSPR peak of the single AuNP having an 80 nm diameter [48]. Because of the discretization in simulation steps, the extinction spectra as a contour plot also appear to be discretized from post-processing in Origin software. Figure 4a (ii) shows the extinction spectra corresponding to the three distinct cases of 300 nm, 450 nm, and 600 nm periodicities. For P = 300 nm, the RA (0,1) is non-interactive at 436 nm, while the single-particle broad LSPR peak lies separately at 563 nm. For P = 450 nm, the RA (1, 1) is individually observed at 463 nm, and the RA (0,1) interacts with the dipolar plasmon mode to produce a sharp SLR mode at 660 nm. Coming to the P = 600 nm case, both the contour and line plots show that the RAs (1,1) and (0,1) can produce lattice modes at 620 nm and 874 nm, respectively. However, both of the modes are relatively weaker because of the lattice spacing. Further, RA (0,2) is also observable at 438 nm. It should be noted that for such square lattice, the other RA orders like (1,0) and (2,0) are degenerate correspondingly with (0,1) and (0,2).

Fig. 4
figure 4

Optical responses of 2D lattices on a square basis. a (i) Extinction vs. periodicity as a color plot for square lattice (unit cell shown in inset). (ii) Extinction line spectra corresponding to three different periodicities (P = 300 nm, 450 nm, and 600 nm. SLR peaks with ultra/narrow line widths are shown via magnification with a wavelength span of 20 nm. (iii) Electric field distribution over an FDTD unit cell, recorded at distinct resonant wavelengths corresponding to P = 300 nm, P = 450 nm, and P = 600 nm. b, c Similar studies are carried out for the rotated-square and Lieb lattice

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Next, for the rotated square lattice, the positions of the RAs corresponding to different orders can be calculated as \({\lambda }_{RA}= \frac{{n}_{s}}{\sqrt{2}}{\left[\frac{{m}^{2}+{2n}^{2}-2mn}{{P}^{2}}\right]}^{-\nicefrac{1}{2}}\), following the previous notations. From the contour plot in Fig. 4b (i), RA corresponding to (0,1) and (1,1) orders are identified that are also degenerate with (2,1) and (1,0), respectively. Meanwhile, RA from the diffraction orders (0,2), (1,2), and (2,0) are not observed as they lie below the investigated wavelength range. Figure 4b (ii) shows the extinction line plots corresponding to the three distinct P = 300 nm, 450 nm, and 600 nm cases. For P = 300 nm, there is only a single AuNP-based LSPR peak at 556 nm without any diffraction modes. As the periodicity is increased to 450 nm, the RA (1,1) order becomes weakly coupled with the LSPR peak, making it slightly narrower while shifting it to 571 nm. The mode splitting is more visible for P = 600 nm, where the dipolar LSPR and a strongly coupled SLR mode coexist at 562 and 626 nm, respectively. The RA corresponding to (0,1) order at 435 nm is also visible. The electric field plots in Fig. 4b (iii) also confirm the nature of the coupling; the strong coupling is readily visible from the intensified distribution around the AuNPs while showing the diffracted (1,1) orders. Following such analyses, the Lieb lattice can be easily assessed as a combination of the abovementioned cases. The contour plot in Fig. 4c (i) shows the presence of RAs corresponding to (0,1), (1,1), and (0,2) orders much like the square lattice. However, the extinction peaks intensify due to more AuNPs within the unit cell. Figure 4c (ii) reveals the line plots where, for even P = 300 nm, a side lobe at 601 nm exists, apart from the dipolar LSPR at 552 nm. For P = 450 nm, the strongly coupled SLR mode exhibits a sharp peak at 681 nm due to the interaction of the RA (0,1). Further, for P = 600 nm, SLR peaks at 633 nm and 878 nm also exist due to plasmon-photon interaction of the RA (1,1) and (0,1) orders, respectively. The electric field distribution at these resonant wavelengths is shown in Fig. 4c (iii), where the strongest interaction throughout this square basis is identified for the Lieb SLR mode associated with P = 450 nm. The Lieb-SLR modes for P = 600 are also stronger compared to their square counterpart.

A similar investigation is carried out on a hexagonal basis, as shown in Fig. 5. For the hexagonal lattice, the spectral position of the RAs corresponding to the 2D diffraction orders can be calculated using the generalized equation [46] as \({\lambda }_{RA}= \frac{{n}_{s}\sqrt{3}}{2}{\left[\frac{{m}^{2}+{n}^{2}-mn}{{P}^{2}}\right]}^{-\nicefrac{1}{2}}\), where P is the periodicity along the X axis, only. Figure 5a (i) shows the extinction spectra as a function of period variation, where three specific periodicities are displayed as line plots in Fig. 5a (ii). For P = 300 nm, only the dipolar LSPR is present at 558 nm, whereas, for P = 450 nm, the RA (0,1) mode and a sharper LSPR mode exist at 568 nm and 597 nm, respectively. For P = 600 nm, the most promising case is shown where a strongly coupled SLR mode is observed at 758 nm, apart from the dipolar LSPR at 558 nm. The RA (1,2) order is also present for such a case; however, it remains non-interactive with the plasmonic mode and is barely visible. There also exist other orders; for example, (1,0) and (1,1) are degenerate orders corresponding to the highlighted (0,1). Also, (2,1) is degenerated with the highlighted (1,2). The resonant wavelengths of interest are observed under electric field distribution, as shown in Fig. 5a (iii). The SLR mode at 758 nm corresponding to the P = 600 nm shows the intensified field accumulation across the particles with the propagation of the (0,1) modes along the lattice plane. For the honeycomb lattice (Fig. 5b (i–ii)), the single particle LSPR modes (P = 300 nm: λ = 557 nm, P = 450 nm: λ = 582 nm, P = 600 nm: λ = 565 nm) are comparatively stronger than the hexagonal counterpart which becomes even stronger for the Kagome lattice (Fig. 5c (i–ii); P = 300 nm: λ = 549 nm, P = 450 nm: λ = 566 nm, P = 600 nm: λ = 570). This can be related to the increment of AuNPs within the unit cell and the resultant near-field coupling. The SLR modes for the honeycomb and Kagome lattices exist at P = 600 nm: λ = 754 nm and P = 600 nm: λ = 761 nm, respectively. The electric field plots (Fig. 5b (iii) and c (iii)) show a comparative enhancement in mode localization near the AuNP surfaces for the SLR conditions, in contrast to the single particle–based plasmonic modes. Thus, this section concludes with a detailed analysis of the resonant modes present in these various plasmonic lattices under normal incident excitation.

Fig. 5
figure 5

Optical responses of 2D lattices on a hexagonal basis. a (i) Extinction vs. periodicity as a color plot for hexagonal lattice (unit cell shown in inset). ii Extinction line spectra corresponding to three different periodicities (P = 300 nm, 450 nm, and 600 nm. SLR peaks with ultra/narrow line widths are shown via magnification with a wavelength span of 20 nm. (iii) Electric field distribution over an FDTD unit cell, recorded at distinct resonant wavelengths corresponding to P = 300 nm, P = 450 nm, and P = 600 nm. b, c Similar studies are carried out for the honeycomb and Kagome lattice

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Study of band dispersion in 2D plasmonic gratings under oblique incidence

Fig. 6
figure 6

Normalized dispersion diagram for the various 2D lattices. a Square basis, showing the wavelength vs. propagation vector for (i) square, (ii) rotated square, and (iii) Lieb lattices. b Similar calculations for the hexagonal basis consisting of (i) hexagonal, (ii) honeycomb, and (iii) Kagome lattices. For all of the cases, P = 600 nm

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To extend the analysis for oblique incidence and consider all possible modes, we have also calculated the band dispersion diagrams for these 2D plasmonic gratings in Fig. 6, considering P = 600 nm. All possible orientations are considered for oblique incidence by choosing randomly orientated dipole sources (details in the “Experimental section”). Instead of energy versus momentum, the band diagrams are plotted to wavelength to correlate the modes with our previous findings. A normalized value of the k-vector: \(k_{\mathrm{norm}}=\frac{ka}{2\pi}\), where \(a\) is the periodicity of a particular structure, is used and is, therefore, unitless. It should be noted that the previously analyzed extinction spectra only highlight the most prominent modes, whereas such operations can investigate all possible existing modes. For better understanding, we have plotted the photon numbers on a logarithmic scale to capture specific modes that are barely visible on a normal scale.

For the square lattice in Fig. 6a, the similarity of the band structures along the kx and ky directions is expected due to the symmetry of the lattice; however, the photon counts show slight differences that indicate the response of the lattice to oblique incidence. In contrast, for the hexagonal basis (Fig. 6b), the spectral positions of the classically (mostly linearly) and conically (parabolic) diffracted modes differ in the kx and ky cases. We have identified the diffraction orders corresponding to Figs. 4 and 5 for the square and hexagonal bases. When we examine the complex gratings—Lieb, honeycomb, and Kagome—we find that the diffraction patterns and the resulting band structures have additional modes that are not present in the simpler square and hexagonal bases. This complexity is evident in the cancellation of degeneracy in specific diffraction orders at oblique incidence, which is particularly pronounced in the square lattice, with the pairs (1,0) and (0,1) and (2,0) and (0,2) showing no degeneracy. Similarly, in the hexagonal base, the previously degenerate pairs (1,0) and (0,1) and (2,1) and (1,2) have transitioned to non-degenerate states. This transformation can be attributed to the asymmetry introduced by oblique illumination.

The dispersion diagrams also show the existence of band gaps at specific frequencies, which are a direct consequence of the lattice geometry. The complex gratings—Lieb, honeycomb, and Kagome gratings—exhibit band gap features that are not present in the simpler geometries. The Lieb lattice, for example, is characterized by flat bands and localized states that are likely to alter the light-matter interaction significantly. The honeycomb lattice, which is analogous to graphene in its structure and optical properties, has a Dirac cone at the edges of the Brillouin zone. These can be recognized by the typical X-shape of the intersection of two linear bands at around 650 nm at ky. The Kagome lattice exhibits fascinating band gaps and topological properties due to the triangles dividing at the corners. Lieb and honeycomb show signs of a band gap. However, this is most clearly recognizable for Kagome gratings at around 700 nm and kx = 0.35. These features are crucial for advanced light manipulation and control and emphasize the importance of complex 2D lattice geometries for creating novel plasmonic band gaps.

Refractive index sensitivity and quality analysis of 2D plasmonic lattices in FDTD-based sensing studies

Finally, we have performed FDTD-based sensing studies utilizing high-quality surface lattice modes. All six plasmonic 2D lattices with P = 600 nm are investigated by varying the FDTD background refractive index (RI) while considering a substrate of constant RI = 1.45. The background RI are varied through 1.319, 1.353, 1.389, and 1.425, representing 0%, 25%, 50%, and 75% wt.-percentage of glycerol-D2O mixture [44] before coming to the symmetric case of 1.45 (index-matching liquid). Only for the hexagonal lattice case is an additional RI of 1.47 considered to calculate the sensitivity. Figure 7a shows the effect of background index variation, starting with the asymmetric condition and slowly tending towards the symmetric condition. For the square basis, it can be seen that the SLR modes corresponding to RA (1, 1) diffraction orders are quite prominent even for a background index of 1.353, which shows an asymmetry of δn ~ 0.1 (δn; the difference between substrate and cover RI). However, almost no SLR peaks are formed by the (0,1) mode for the hexagonal basis until 1.425 RI is reached (δn ~ 0.025). Thus, plasmonic lattices in hexagonal bases can only operate on a shorter range of RI variation while showing significantly higher sensitivity on an average than square bases (Fig. 7b). Table 1 calculates the quality factor (Q), Figure of merit (FOM), and effective FOM for the different plasmonic lattices offering SLR modes. Remarkably, the hexagonal and honeycomb lattices with ~ 1 nm FWHM can provide promising results in bio-analyte sensing and promote other light-matter interactions under strongly coupled conditions.

Fig. 7
figure 7

Effect of refractive index variation. a Extinction spectra, calculated at different FDTD background indices for (i) square, (ii) rotated square, (iii) Lieb, (iv) hexagonal, (v) honeycomb, and (vi) Kagome lattices. b Sensitivity calculation of the (i) square and (ii) hexagonal basis

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The hexagonal lattice possesses a higher electric field enhancement at the SLR condition compared to other lattice resonances studied so far in Figs. 4 and 5. To our belief, the unique geometric arrangement in hexagonal and honeycomb lattices could have led to the maximization the electromagnetic interactions between nanoparticles, thus magnifying the sensitivity of the LSPR to refractive index variations. The remarkable precision of the hexagonal and honeycomb gratings, reflected in their FOM and effective FOM, underlines their potential to pave the way for the next generation of plasmonic sensors.

Table 1 Calculation of quality factor (Q), Figure of merit (FOM), and effective FOM for different lattices with P = 600 nm
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Conclusion

This work demonstrates that PEIL can theoretically produce simple nanostructures and more complex 2D geometries such as Lieb, honeycomb, and Kagome lattices. Central to the development of optical band structures, these structures represent an advancement over conventional line, rectangular, and hexagonal configurations. Our suggestive approach with the PEIL, complemented by electromagnetic simulations, has opened up new horizons for the large-scale fabrication of SLR-assisted complex gratings with sophisticated optical properties. In our systematic analysis, we have investigated these gratings’ optical extinction characteristics and electric field distribution and gained insights crucial for their potential applications. The theoretically proposed technique to use these gratings as templates for the TASA method may enable the fabrication of regularly spaced plasmonic nanoparticles, thus highlighting the versatility and effectiveness of our approach to nanostructuring.

In addition, our comparative study of SLR versus isolated particle LSPR modes and sensing properties underscored these structures’ potential for excitation with other photonic modes. The resulting interaction highlights the presence of both photonic and plasmonic modes, enhancing the light-matter interaction. This aspect is essential for understanding the effects on Dirac cones and band gaps, suggesting applications in photonic devices where controlling light propagation is crucial. Therefore, this research expands the scope of colloidal nanofabrication and sets the stage for applications in low-loss plasmonic devices.