Compensation of disorder for extraordinary optical transmission effect in nanopore arrays fabricated by microsphere photolithography

: The work considers the effect of extraordinary optical transmission (EOT) in polycrystalline arrays of nanopores fabricated via nanosphere photolithography (NPL). The use of samples with different qualities of polycrystalline structure allows us to reveal the role of disorder for EOT. We propose a phenomenological model which takes the disorder into account in numerical simulations and validate it using experimental data. Due to the NPL flexibility for the structure geometry control, we demonstrate the possiblity to partially compensate the disorder influence on EOT by the nanopore depth adjustments. The proposed experimental and theoretical results are promising to reveal the NPL limits for EOT-based devices and stimulate systematic studies of disorder compensation designs.

Along with periodical structures EOT exists in polycrystalline geometries, too. Such kind of structures naturally appears in colloidal self-assembly process, where a close-packed array of nano/micro spheres is deposited onto a substrate using one of numerous implementations of the Langmuir-Blodgett technique [18][19][20][21]. A number of works consider EOT-supporting plasmonic structures prepared via metal deposition into the interstices between colloidal particles [22,23] or by covering of a nanosphere mask with a thin metal film [24] with possible subsequent removal of particles [25,26].
The main advantage of colloidal self-assembly over conventional periodic structuration is its high throughput, compatibility with industrial production lines [19] and flexibility in the nanostructuration of curved and non-conventional surfaces [27][28][29]. For example, a plasmonic sensor was prepared on an optical fiber cleaved face using nanosphere photolithography [30]. On the other hand, the presence of disorder imposed by polycrystallinity affects the resonant optical behavior. It was numerically shown in [31] that randomly positioned perforations in a dielectric thin film possess a smooth broadband absorption, while amorphous pattern excites multiple resonant modes not observed in spectra of periodically arranged nanopores. Polycrystalline structures in [32] were used as advanced light absorbers. Both numerical simulations in [33] and experimental measurements in [25] showcase a lowering and flattening of transmission through thin gold films when the disorder increases. Along with perforations placement disorder, the rotational disorder [34] and nanopores size deviations [31] have also been considered.
In this paper we use an elaborated method of surface nanostructuration called nanosphere photolithography (NPL) [35]. To the best of our knowledge, this is the first work where NPL-fabricated samples were considered in the context of EOT. Despite the simplicity of the colloidal-based nanostructuring methods mentioned above, they however possess a lack of flexibility: diameter of deposited micro/micro spheres and, consequently, the period of nanotopography is often a single parameter which can be easily varied experimentally. Thus, the influence of colloidal monolayer disorder on sample optical response can hardly be mitigated. In contrast, NPL technology utilizes photonic nanojets generated by every colloidal particle to inhomogeneously expose the underlying photosensitive layer [35]. The control of exposure [36], development time [37] and angle of incidence [38] offers much more degrees of freedom in comparison with conventional colloidal approaches. In particular, we demonstrate that an increased nanopore depth can improve the quality of EOT in highly disordered nanopore arrays. Measured EOT spectra are accompanied by numerical simulations, which possess a good correspondence with experiments. In addition, we propose a phenomenological theory, adapted from the 1D gratings case, to take the disorder factor into account in numerical modeling. Obtained results are promising to reveal the NPL limits for EOT-based devices and pave the way toward plasmonic sample designs with disorder compensation.

Sample fabrication and characterization
Plasmonic structures used for EOT measurements were fabricated as sketched in Fig. 1. Microscope BK7 glass slides 3.7 × 2.5 cm were cleaned in a three-step wet bench procedure: ultrasonic acetone, ethanol bath and DI-water bath. Samples were then dried under nitrogen stream, and Shipley S1805 photoresist thin layer with a thickness of 600 nm for 1.1 µm microsphere diameter and 250 nm for 300 nm diameter was spin coated on a glass surface, see Fig. 1(a). The resist was soft-baked in an oven at 60°C for 1 min. Using a Boostream process developed in CEA-Liten [19], silica nano/micro particles with diameters σ 0 =1.1 µm or 300 nm are arranged in monolayers of different quality on the water surface and are transferred onto the photoresist surface, see Fig. 1(b). In the Boostream process the water flow plays the role of the barrier for the particle film compression. It permits to tune the degree of compression by the water flow speed and allows to perform continuous close-packed large colloidal film deposition for industry [39]. The higher monolayer quality means larger crystallites with close-packed spheres. The deposited micro spheres act as microlenses in NPL technique [35]; under UV irradiation they generate nanojets which expose the photoresist underneath. After the exposure during the time t exp colloidal particles are removed in an ultrasonic bath; the exposed resist is developed in MF-319 at 8°C during the time t dev , and an array of nanopores of certain depth appears, repeating the hexagonal spatial arrangement of initial particles, see Fig. 1(c). In order to observe plasmonic resonances a thin 20 nm continuous aluminum film was deposited on the structured resist via magnetron sputtering, see Fig. 1(d). Finally, to improve the EOT and create a symmetrical "Insulator-Metal-Insulator" (IMI) geometry a second 600 nm thickness dielectric resist cladding was spin coated on the top of the metal film, see Fig. 1(e).
For the characterization of samples prepared using 1.1 µm and 300 nm spheres diameter an optical microscope and SEM were used, respectively. Nanopore depth was measured by AFM. Transmission spectra of IMI structures were measured using the UV/Vis/NIR optical spectrophotometer Cary 5000.

Enhanced optical transmission measurements
We start with the study of EOT in structures possessing long-and short-range order of nanopores. Figures 2(a) and 2(b) show images of nanopore arrays of different quality obtained from optical microscopy. The "Quality 1" sample has a long-range nanopore spatial arrangement, while the "Quality 2" sample demonstrates a short-range order with clearly seen distinct grains of close-packed nanopores. At the second fabrication step (see Fig. 1(b)) the deposited silica microspheres with a diameter of σ 0 =1.1 µm were UV-exposed during the time t exp = 10 s, the development time at the third step ( Fig. 1(c)) is t dev = 4 s for both qualities. The nanopore average depth measured by AFM is 220 nm. Full-size 227 µm×170 µm photographs were used for the image processing-based statistical analysis [19], shown in Figs. 2(c) and 2(d). The comparison of two samples reveals a narrower distribution of interpore (center to center) distances in "Quality 1" around the value σ 0 =1.1 µm, and an almost isotropic distribution of crystallite orientations in the "Quality 2" sample. The orientation histogram for the "Quality 1" has two pronounced peaks which indicate the presence of two large grains in the photograph. In spectrophotometric measurements the incident collimated light beam has a spot of size 1 mm×3 mm much larger than the single image area. Thus, several photographs should be used to get an appropriate statistics for "Quality 1" sample; considering these photographs total crystallite orientations are uniform as in "Quality 2" sample.
Results of experimental measurements of transmission under normal incidence through the samples are shown in Fig. 3(a). EOT in IMI structures with periodically corrugated continuous metal films appears due to the excitation of Surface Plasmon-Polaritons (SPPs) on the one side of the metal layer via grating coupling, tunneling through the metal and decoupling to the transmitted wave at the other dielectric/metal interface [40]. The resonant wavelengths are determined by Enlarged sections of optical microscope images of nanopore arrays "Quality 1" and "Quality 2", realized with spheres of diameter 1.1 µm. Nanopore depth is ≈ 220 nm. In (b) grains of nanopores are much smaller than in (a); (c) and (d) statistical information for interpore distance and crystallites orientation for "Quality 1" and "Quality 2", respectively. Scale bars in (a) and (b) denote 20 µm.
grating diffraction orders (m,n), and at normal incidence can be estimated by the equation: in the case of a hexagonal lattice (see, for example, [41]); ε d and ε m are dielectric permittivities of dielectric and metallic layers, respectively, and σ 0 is the interpore distance defined by the diameter of the self-assembled particles. Strictly speaking, in thin metal films, SPPs excited at opposite surfaces hybridize into Short-Range SPP (SR-SPP) and Long-Range SPP (LR-SPP) [42]. In practical measurements, however, only the EOT from LR-SPPs is visible because of the high absorption losses of SR-SPPs in metal. The propagation constant of LR-SPPs is confined in a narrow range between the dielectric cladding refractive index and the propagation constant of SPPs on a single metal/dielectric interface. Therefore, the last value, which is used in Eq. (1), is a good approximation for determining EOT spectral positions [40]. Taking ε d ≈ 1.6 2 for the S1805 dielectric resist and ε m ≈ (1.5 + 15i) 2 in Eq. (1) for aluminum, we get the EOT wavelength λ EOT ≈ 1530 nm defined by 6 symmetrical first diffraction orders (m, n) = (±1, 0), (0, ±1), (±1, ∓1).
As expected, the sample "Quality 1" with a long-range order showcases a sharp plasmonmediated transmission at λ EOT . In contrast, the "Quality 2" sample has an almost negligible EOT because of the high disorder. Numerical calculations of EOT presented in Fig. 3(b) are in a good agreement with the experiment. The simulations for the "Quality 1" sample were performed using the ideal hexagonal lattice of nanopores, whereas those for the "Quality 2" sample take the disorder into account. The corresponding numerical approach is described in the following section.

Modelling of grating disorder via the inverse space approach
Experimental data in Fig. 3(a) clearly shows that the EOT signal is suppressed by the disorder in polycrystalline gratings prepared via NPL. The presence of grains and defects strongly affects the optical response of the final device, therefore a number of in-situ approaches exist to control the self-assembly process [19,43,44]. These approaches estimate the quality of the formed monolayers by their diffraction patterns. Here we construct a model to relate the quality of nanopore arrays with EOT efficiency.
Simulations of infinite ideal periodic crossed gratings were done by the Lumerical FDTD software and a proprietary GSMCC code [45]. In the first case of the FDTD method a grating period contains a vertical wall hole with a metallic layer covering flat regions and the hole walls. In the second case the hole walls were slightly slanted. Both methods produced similar spectra, which assures the attained numerical results. The material dispersion was taken into account, the wavelength-dependent refractive index of the resist S1805 was measured by ellipsometry.
For the normal incidence at nanopatterned surfaces with structural elements arranged in periodical hexagonal lattices their transmission is insensitive to incident light polarization due to the axis of 6-fold rotational symmetry [46]. In our calculations of ideal hexagonal arrays of nanopores we verified that transmission variations are well below 1% for any incident linear polarization, which underlines the simulation accuracy. Consequently, if we consider an IMI structure with a polycrystalline array of nanopores with grains large enough (the size of grains will be clarified later), it will have almost the same transmission spectrum as an ideal hexagonal lattice, though the number of illuminated crystallites can be big. We study the EOT caused by plasmonic excitations from the first grating diffraction orders. In the reciprocal space these diffraction orders, coming from numerous differently oriented grains, create a circle with the radius k c ≡ |b 1 | = |b 2 |, where b 1 and b 2 are basis reciprocal vectors for the hexagonal lattice format, see Fig. 4(a).
Thus, in our model the plasmon-mediated transmission from the ideal hexagonal nanoholes array is the same as from a polycrystalline structure with large grains, see Fig. 3(a); this assumption is based on experimental and numerical results and the argumentation above. If the grains become smaller, the disorder increases, the mentioned circle in reciprocal space spreads and its intensity diminishes in analogy with diffraction patterns of amorphous solids or liquids [47,48], see Fig. 4(b). To get a Fourier transform ρ k of experimental nanopore spatial distributions we consider a set of two-dimensional Dirac delta functions ρ (r) = ∑︁ N i=1 δ (r − r i ), which are zero except nanopore centers r = r i , index i numerates the nanopores [49]; the resulting Fourier transform is ρ k = ∑︁ N i=1 exp (ik · r i ). Although we consider a two-dimensional system of nanoholes, they are organized in multiple randomly oriented grains, so the structure has no preferred in-plane direction. Consequently, it is convenient to consider a profile of the Fourier spectrum along the circle radius. Figures 4(c) and 4(d) represent the radial-averaged static structure factor S(k), which is proportional to the squared modulus of Fourier amplitudes: S (k) = ⟨ρ k ρ −k ⟩/N [50]. IMI-structures under consideration support a set of quasiguided and plasmonic modes (see [7] for the analysis in one-dimensional IMI counterparts), which are excited via different momenta k in the vicinity of k c in Figs. 4(c) and 4(d). Thus, every momentum k gives a contribution α k · T k (λ) to the transmission T(λ) of the overall sample, with a scaling factor α k depending on the density of excited modes, and T k (λ) obtained from the simulation of perfect samples [51]. Due to the random polycrystallic arrangement of nanopores we assume a simple incoherent summation of all contributions T (λ) = ∑︁ k α k · T k (λ). The density of modes excited by the momentum k is proportional to the squared modulus of the corresponding Fourier harmonic amplitude [51,52], i.e. to the structure factor S(k) introduced above.
The radial distribution functions g(r) for samples "Quality 1" and "Quality 2" in Fig. 5(a) show the average number of nanopores separated by a distance r from the central nanopore, σ 0 =1.1 µm. Along with the information concerning the quality of short-and long-range orders, these curves allow calculating the structure factors S(k) via the Fourier transform, as shown in the statistical mechanics theory of liquids [53]. The first peaks of these functions S(k) presented in Fig. 5(b) are then least square fitted to a Gaussian and used in numerical simulation of disordered nanopore arrays using the phenomenological approach described above. A natural parameter of disorder in the context of our study is a dimensionless value p d ≡ FWHM/k c , see Figs. 4(c) and 4(d) for notations; for "Quality 1" and "Quality 2" samples p d ≈ 0.05 and p d ≈ 0.13, respectively. Let us notice that, along with p d , a number of other parameters have been proposed to quantify the disorder in two-dimensional systems: nanopore concentration and hole-to-hole spacing [54], impurity concentration [55], statistical control parameter [56]. The choice of the particular parameter p d in the present work is convenient for two reasons. Firstly, due to its dimensionless formulation it allows to compare the disorder in polycrystalline systems with different mean interparticle distances, see, for example, Fig. 7(a). Secondly, the parameter p d defines the width of the first Fourier peak used in the described numerical approach.

Compensation of disorder via nanopore depth in NPL-fabricated samples
NPL process offers much more flexibility over the structure geometry than conventional approaches at the cost of additional step of nanojet generation. Because of the fact that nanopore arrays repeat the spatial arrangement of self-assembled colloidal particles, the disorder reduces and smoothens the transmission spectrum as in standard colloidal mask techniques [25], see Fig. 3. However, within the NPL approach the disorder influence on EOT can be compensated due to the nanopore depth adjustment, which might be difficult to realize in other methods.
A "Quality 3" sample with highly disordered nanopores was fabricated via NPL, see Fig. 6(a), using the silica spheres of diameter σ 0 = 300 nm. By varying the exposure time t exp the nanopore depth is efficiently controlled; in order to keep all other geometrical parameters as constant as possible, nanopore arrays of different depths were fabricated on a single glass substrate by a sequential UV-irradiation of different sample regions through a mask. This mask opens only one particular area of the nanosphere-covered substrate for the UV-exposure at a time. The development time t dev = 4 s is equal for all zones of the sample, whereas exposure times were chosen to be 4 s, 5 s, 6 s and 7 s. Fabricated nanopore arrays for these exposure times have depths of 23 nm, 64 nm, 80 nm and 103 nm correspondingly, measured by AFM. Figure 7(b) shows the AFM-measured profiles for "Quality 1" and "Quality 3" samples. Photonic nanojets generated via smaller colloidal particles are less confined, leading to smoother nanopore profiles. In contrast, nanopores fabricated via micron-sized particles have more pronounced slanted walls.  6. (a) Enlarged section of the SEM photograph of "Quality 3" nanopore arrays with depth ≈ 103 nm. Scale bar denotes 3 µm; (b) statistical information for interpore distance and crystallites orientation for "Quality 3"; (c) radial distribution function g(r) and static structure factor S(k) for "Quality 3" sample; (d) comparison of disorder parameters p d for samples "Quality 1", "Quality 2" and "Quality 3".
The fact that all nanopore areas of different depths are located at the same substrate ensures an equal nanopore distribution statistics for them; this statistics is presented in Figs. 6(b) and 6(c). Calculated disorder parameter for "Quality 3" sample p d ≈ 0.16 is the highest among all structures considered in the work, see Fig. 7(a). Figure 8 shows the recorded transmission spectra under normal incidence of final IMI structures with the same disorder level, but with different nanopore depths.
We can notice that there is no EOT in Figs. 8(a) and 8(b). While in Fig. 8(a) it originates from the absence of metallic film corrugations and, therefore, the impossibility to excite plasmonic modes, in Fig. 8(b) with a nanopore depth of 23 nm the mechanism is different. It is known [7,40,57] that the undulation depth is responsible for the efficiency of SPP coupling, therefore in the limit of corrugations much smaller than their period the increase of depth improves the EOT. However, in case of Fig. 8(b) the value of EOT is not enough to overcome the disorder, enhanced transmission in partial contributions T k (λ) only gives rise to overall transmission level. It leads to the presence of a varying transmission due to a Fabry-Pérot effect in Figs. 8(a) and 8(b) caused by multiple reflections of light in dielectric claddings. The EOT efficiency grows with the growth of nanopore depth, the resonant peak appears at the wavelength λ EOT predicted by Eq. (1) for nanopores depth above 60 nm (Figs. 8(c)-8(e)). Consequently, the effect of disorder compensation via nanopore depth occurs. In addition, the EOT in the "Quality 3" sample occurs in the visible spectral range in contrast to near-IR for "Quality 1" and "Quality 2", where larger (a) Comparison of disorder parameters p d for samples "Quality 1", "Quality 2" and "Quality 3"; (b) AFM-measured profiles of nanopores for "Quality 1" (red curve) and "Quality 3" (blue curve) samples. Fig. 8. Measured (black curves) and simulated (green curves) transmission spectra under normal incidence for metallized arrays of "Quality 3" nanopores in IMI structure. Parameter of disorder p d ≈ 0.16 for all graphs (see statistical data in Fig. 6), whereas depths are different and are denoted above each graph in colors in accordance with the legend in (a). particles were used for the colloidal self-assembly. Numerical simulations shown in Fig. 8 are in a good correspondence with experimental data, thus proving the validity of the phenomenological approach described in the previous section.

Conclusion
In conclusion, NPL fabrication approach offers much more degrees of freedom in the structure design in comparison with conventional self-assembly methods. In this work we focused on the possible NPL applications for EOT devices, and considered the effect in the visible and near-IR range. To the best of our knowledge, the article presents the first results concerning the EOT in NPL-fabricated samples. The polycrystalline nanopore arrays with different disorder p d and interpore distance were examined experimentally, obtained EOT spectra were used for the validating of the proposed phenomenological approach which takes the disorder into account in numerical simulations. The dimensionless parameter of disorder p d is used to compare the quality of nanopore distributions with different mean interpore distance, and in the numerical modeling. We have demonstrated both experimentally and numerically the effect of disorder compensation via nanopore depth to improve the EOT signal, which might be difficult to realize using conventional fabrication methods. We believe that the work will stimulate a further research in NPL-based plasmonic devices and pave the way toward designs with an integrated disorder compensation.