Optimizing Motheye Nanostructures and Achieving 100% Transmission

Nanostructured Antireflections Layers (NALs) are a promising substitute for conventional antireflection layers. In the nature, such structures have long existed on insects’ eyes. Here, we have derived a general form of mathematical relation that generates a profile close to that of surface of a Moth’s eye. By properly designing NALs over dielectric, light has been efficiently transmitted through our high index nanostructures with minimal reflection. The parame-ters determining the transmission efficiency are the pitch and the profile of the periodic nanostructures. By optimizing the profile, we have designed NALs with transmission reaching 100%. Our periodic structures have minimal dependence on the pitch. This makes it possible to fabricate NALs with the very broadband transmission, without inducing diffraction to lower wavelengths of the transmitted light. Also, our periodic NALs make it possible to transmit laser lights without scattering.


Introduction
Conventional antireflections consist of one or more layers over each other laying over a substrate. These layers work based on destructive interference of light [1] [2] [3]. While one layer can only be used for one specific wavelength or angle of illumination, multi-layer structures partially solve this limitation [1] [3] [4].
However, still, layers have rather low bandwidth and angular tolerance. A major problem with this technology arises from the residual stress between these layers or layers and substrate [4]. This lowers the lifetime of layered antireflection coatings when they're used in environments with fluctuating temperatures. Also, when a high-power beam of light passes through them, these layers peel off the substrate [5] due to the difference in the expansion coefficient of different materials. Recently, nanotechnology and nanostructures [6] [7] [8] [9] have solved many problems in technology introducing materials [10] that exhibit special characteristics electrically, mechanically [11]- [15] and when they're under illuminations of light [6] [7] [16] [17]. Fabrication method of these nanomaterials [6] [18] [19] usually depends on the type of the materials and the physical characteristics they're designed for.
NALs are the new generation of anti-reflection coatings designed to deal with previous problems [6]. These nanostructured materials are designed based on a fill factor concept [1]. This means that the effective refractive index depends on the volume occupied by the nanostructure [17]. Such coatings are widely used for increasing light transmission to solar cells [20] [21], and light extraction from LED's [22] [23] and lasers [24] [25]. Sub-wavelength NAL structures are usually carved out of substrate for smoother transition of refractive index. NALs are used for different applications. For example, these structures can be used for increasing transmission in infrared detectors [26] or transmission of laser through different optical elements like lenses or polarizers [6] [17]. For solar cells and LEDs, the goal is to enhance scattering by eliminating total internal reflection or enhancing near-surface reflection by scattering light off normal [20] [21] [27].
Nanostructuring the surfaces of dielectric materials is often used to improve transmission at the silicon/air interface in IR applications. As previously mentioned, one of the main applications of NALs is enhancing transmission of light through optical elements used in high-power-laser applications [25] [28] [29] [30]. In such applications, it is also crucial to avoid scattering. Therefore, the randomly distributed NALs are replaced by orderly ones [25] [30] [31]. Although periodic structures don't impose scattering to the transmitted light, usually such structure's max efficiency happens at a certain pitch for every specific wavelength [17]. To the best of our knowledge, the researchers who have designed these nanostrucutres, first calculated refractive index of different 2D antireflection films laying over each other for maximal transmission. Afterwards, they related the calculated refractive indexes to the amount filled by a 3D dielectric disk using fill factor or Maxwell-Garnett approximation [17]. Although the 2D layers in some cases result in 100% transmission, due to the approximation, the 3D geometries associated with these 2D structures do not yield the same max transmission [17]. One way to deal with this problem is directly designing the NALs and evaluating the transmission of light through the 3D design using numerical methods.
In nature already nanostructures exist that maximize light transmission into the eyes of some insects. A series of mathematical equations are fitted to a motheye structure. We've seen that for high index materials like silicon, the trans-

Maximizing Transmission
In this section, we have investigated multiple profiles for periodic nanostructures to reach 100% transmission. In our previous work, we investigated profiles which generated over up to 99.1% for a height equal to π phase shift [6]. Here, we find the profiles for maximal transmission of light, even superior to the previous profiles [6]. The heights of NALs will generate the same phase shift in the transmitted electromagnetic waves for a fair comparison. We try different mathematical equations for the profile of NALs and evaluate each profile using RCWA. The equations we use have a general form of represents the fill factor and x p represents side length of cross section at each elevation normalized to pitch. We have chosen this mathematical equation to its resemblance to motheye. By swiping 1 n , we have gained series of mathematical relations resulting in 100% transmission. As the first step, we assume that 2 1 n = . We find the suitable pitch for maximizing transmission at roughly 12 μm numerically ( Pitch 0.64 m = µ ) using Bisection method [33]. At this pitch, if we assume 1 1 n = , the transmission will be maximum around 12 μm wavelength.
Then we keep the pitch constant and swipe the wavelength from 0.95 μm to 25 μm. The transmission is shown in Figure 1. The max transmission of 100% happens at 1 0.7 n = . However, the transmission drops rapidly for longer wavelengths compared to cases with higher 1 n values. Note that by changing 1 n the wavelength that maximum transmission happens deviates slightly.  Table 1. Further investigation has shown that the max transmission bandwidth happens for 1 0.75 n = . For 1 0.75 n = , we have swiped 2 n between 0.55 and 0.75.
In Figure 3, the transmission is shown for a wavelength span from 9.5 μm to 25 μm. We further reduce 2 n and increase the wavelength span to find how much the performance drops at wavelengths even longer than 25 μm. Keep in mind that in designing the NALs the height of the pillar is set to a π phase shift.
Our simulations show that for 1 0.65 n = , although one would not get a 100% transmission, the spectrum of transmission is better than the previous case. Figure 4(b) shows that when 2 0.45 n = , the transmission reduces to 99.96% for high wavelengths (above 50 μm).
The importance of the profiles introduced is that their max transmission is not dependant on pitch. This is a big advantage compared to our previous work [6], where the transmission peak for each wavelength happened at a specific  pitch, linearly related to the wavelength. In that case, the max transmission for each wavelength happened at a specific pitch, therefore one would need to change the pitch of NALs for maximizing different wavelengths. This is not desired since for gaining high transmission at longer wavelengths, one would need Journal of Applied Mathematics and Physics to linearly increase the pitch of the nanostructure. Then, if we want to use the nanostructure for both longer and shorter wavelengths, diffraction will be inevitable for shorter wavelengths due to higher pitch. The limit of pitch for complete elimination of diffraction is shown in Equation (1) ( inc λ is the wavelength of incidence).
The RCWA approach shows that our new profile is not pitch dependant to a very good extend. This is to our belief related to the way it distributes the E-field in space above the nanostructure. The spacial distribution of wave inside and out the NALs is subject of a subsequent publication which we are currently working on using finite element method. The refractive index of the nanostructures is shown in Figure 5. The refractive index is related to the profile at each elevation using fill-factor concept. The mathematical relation relating the two is shown in Equation (2)   ( )

FillFactor 1
Si Air In the equation, we have assumed that the volume is occupied by Si, but in general it can be any material that is transparent at the illumination wavelength.
Previously, we had found the proper refractive index at each elevation from substrate and assigned a geometric profile to the calculated index. Here, we first find a proper profile by mimicking nature and then related to a refractive index at each elevation.
In Equation (3), x is length or width of the cross-section of NAL. We have not yet been able to fabricate a 3-D NAL with square cross-section from tip to base. Therefore, we have fitted a geometry with circular cross-section, to the square cross-section. The side-view and 3D profile of the periodic structures are shown normalized to the pitch in Figure 6. The resulting structures resemble that of motheye surface and can be considered "Enhanced Motheye-NanoStructures".

Conclusion
NALs are a promising option for enhancing transmission of light from air into high index materials. By engineering the profile of the NALs, one can reach very high transmissions. These structures offer zero residual stress with the substrate when exposed to high power laser beams and are of significant importance for transmitting light energy into high-index substrates. Our nanostructures yield a 100% transmission. This high transmission at different wavelengths has negligible dependence on the pitch of the nanostructure. This is another achievement compared to our previous design. Previously for each wavelength, we needed to change the pitch of the nanostructures to maintain a high transmission. Therefore, NALs designed for longer wavelengths can have small pitches (even below 1 μm). This is important if one wants the nanostructures to function at lower wavelengths alongside the longer wavelengths. Our broadband nanostructures do not impose diffraction to the shorter wavelengths (due to very small pitch), while they highly enhance transmission for above 50 μm wavelengths. This makes our NALs highly desirable for transmitting laser lights at a broad span of wavelengths. The profiles of these nanostructures resemble that of a motheye's surface, and considering the capabilities and designing approach, they can be considered enhanced motheye nanostructures.