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Laterally structured antireflective sub-wavelength structures show unique properties with respect to broadband performance, damage threshold and thermal stability. Thus they are superior to classical layer based antireflective coatings for a number of applications. Dependent on the selected fabrication technology the local topography of the periodic structure may deviate from the perfect repetition of a sub-wavelength unit cell. We used rigorous coupled-wave analysis (RCWA) to simulate the efficiency losses due to scattering effects based on height and displacement variations between the individual protuberances. In these simulations we chose conical and Super-Gaussian shapes to approximate the real profile of structures fabricated in fused silica. The simulation results are in accordance with the experimentally determined optical properties of sub-wavelength structures over a broad wavelength range. Especially the transmittance reduction in the deep-UV could be ascribed to these variations in the sub-wavelength structures.
© 2010 Optical Society of America
1. Introduction
Alternatives to classical multilayer based antireflective (AR) interfaces are laterally structured sub-wavelength antireflective surfaces which provide a graded transition between the refractive indices of the two interfacing media. Such antireflective structures are also found in nature on the corneal surfaces of night active insects [1, 2], so-called ’moth-eye structures’.
Different characteristics of sub-wavelength AR structures are advantageous in comparison to stacks of thin dielectric films. Whereas the optical performance of multilayer based AR coatings is limited to narrow wavelength ranges and normal incidence of light, nanostructured antireflective surfaces reduce the reflectance over a broad spectral bandwidth and tolerate an extended variation of the angle of incidence. In addition, layer systems may suffer from adhesion problems due to different thermal expansion coefficients of substrate and coating material. This is in particular a problem for high-power laser applications.
Different top-down techniques such as e-beam writing [3], mask-lithography [4] and interference lithography (IL) [5] have been applied to the fabrication of sub-wavelength gratings. Alternative approaches are based on bottom-up methods which allow for a fast and cost-efficient etching mask fabrication. Porous alumina membranes [6] and nanoparticle arrays prepared by block copolymer micelle lithography (BCML) were already employed for mask generation which - in combination with subsequent dry-etching - led to nanostructured surfaces with antireflective properties [7–9]. Recently, Lohmüller et al. demonstrated the fabrication of antireflective surfaces for deep-UV applications by using BCML and reactive ion etching (RIE) [10–12]. This technique possesses several advantages over standard lithography approaches as it allows the creation of very small lateral feature sizes below 200 nm and down to 20 nm with an appropriate structure height and is also suitable for non-planar substrates like lenses, especially with a small radius of curvature.
In general the broad diversity of manufacturing technologies is accompanied by a wide variety of different shapes, heights or aspect ratios on the local scale of the sub-wavelength unit cell. Moreover, the specific profile form and height of the structures as well as the position within the unit cell may show a variation over the extension of the structured area. The optimal profile shape and the theoretical performance of antireflection structures are already known [13, 14], but the influence on the performance of manufactured antireflection surfaces caused by the local imperfectness of the nanoscopic structure has not been studied until now. In this contribution we simulate the theoretical performance and the effect of height and position variations of the individual protuberances within a structured surface. The calculated optical efficiency of fabricated AR structures is compared to the measured reflectance and transmittance.
2. Simulation of different manufactured profile shapes
Different manufacturing technologies are associated with a wide variety of shapes of AR sub-wavelength structures. Fig. 1 shows cross-sectional scanning electron microscopy (SEM) images of two fused silica samples nanostructured by either IL [Fig. 1(a)] or BCML in combination with dry etching [Fig. 1(b)]. AR structures prepared by IL show protuberances with a conical shape whereas the unit cell of samples fabricated by BCML can be described by a Super-Gaussian function. The shapes of these two basic structure types provide the basis for the models used in our calculation.
Fig. 1 SEM images of fabricated antireflective sub-wavelength structures. (a) AR structure generated by interference lithography and reactive ion beam etching (lattice spacing: 139 nm). (b) AR structure manufactured by a combination of BCML and RIE according to Lohmüeller et al (lattice spacing: 80 nm). [12]
The conical profile is completely specified by the base radius, the lateral grating period and the height of the structure. The Super-Gaussian profile is described by
whereas x, y are the Cartesian lateral coordinates, Ω is the profile determining order parameter. The distribution is related to the half-width w of the protuberance byFor Ω = 1 the profile is equivalent to a simple Gaussian function. For large values of Ω the single structure is converging to a cylindrical pillar.
The simulation of the AR structures has been carried out with Unigit, a RCWA solver providing solutions for 1-dim and 2-dim gratings [16, 17]. The used 2-dim profile matrix of one single protuberance consisted of 256x256 data values to ensure sufficient accuracy of parameter dependence. The height of the structure was approximated with 256 layers. To investigate the convergence of the RCWA method, we used a conical shaped AR structure with a lateral period of 150 nm. The working wavelength was set to 325 nm and the incidence angle was perpendicular to the surface. The height of the protuberances was varied from 0 nm to 500 nm in steps of 10 nm and the calculation has been carried out with different numbers of Rayleigh orders (Fig. 2). To use only four Rayleigh orders was found to be a good compromise between calculation time and accuracy of the simulation. The difference between the calculated values with four and higher counts of used Rayleigh orders was less than 0.01 % in the complete parameter space, which is much smaller than the accuracy of the comparable measurement.
Fig. 2 Calculated reflectance for a conical shaped AR structure in dependency of structure height and number of Rayleigh orders. The wavelength of light was 325 nm (perpendicular incidence). The simulated lateral grating period was exemplary set to 150 nm.
For supporting the manufacturing process of the AR-structures by providing specifications and evaluating the optical performance of the realistic profile shapes we simulated the reflectance as a function of the structure height as well as of the profile geometry at a wavelength of 325 nm (normal incidence). In accordance with the lateral spacing of structures fabricated with BCML (Fig. 1) the dimension of the lateral unit cell was chosen to be 80x80 nm2. Because the lattice spacing has minor influence on the performance of perfectly periodic AR structures [18], all of the following calculations in this section have been carried out at a grating period of 80 nm. Figure 3 depicts the different unit cells with the different profile geometries ranging from conical to Gaussian, as well as 2nd order and 3rd order Super-Gaussian geometry from upper left to lower right. As profile determining factors the half-width of the Super-Gaussian profiles and the basis radius of the conical profile was adjusted to guarantee a minimum reflection lower than 0.05 % for one set of parameters. In detail we chose: w = 0.6 for Ω = 1, w = 0.7 for Ω = 2, w = 0.75 for Ω = 3 and a basis radius of 0.7 grating periods for the conical structure. In the following, these parameters are used as optimized profiles.
Fig. 3 Different profile shapes used as an input for the rigorous simulation. From upper left to lower right: conical structure, Gaussian profile (w = 0.6), 2nd order Super-Gaussian profile (w = 0.7) and 3rd order Super-Gaussian profile (w = 0.75)
To compare the performance of the different optimized profiles, the reflectance was calculated with a variation of the structure heights between 0 nm and 470 nm (Fig. 4). The 3rd order Super-Gaussian profile shows the strongest decay of reflectance with increasing structure height allowing for the fabrication of AR structures with a low structure height of approximately 90 nm. However, this effective medium appears similar to a single homogeneous layer and therefore small changes in the structure height lead to dramatic changes in the antireflective performance. In contrast, the reflectance of conical structures remains nearly unaffected with increasing structure height but shows a slower decay with increasing wavelength. Also the Gaussian profile shows a similar behavior, the required structure height is even higher.
Fig. 4 Calculated reflectance for the selected profile geometries in dependency of structure height. Incidence wavelength was assumed to be 325 nm (perpendicular incidence).
Figure 5 shows the reflectance as a function of the wavelength for an optimized conical and 3rd order Super-Gaussian profile. For the two selected heights of the conical sub-wavelength structures a reflectance below 0.1 % was calculated in the wavelength region below 400 nm. At longer wavelengths a transition is observable for both curves in which the reflectance is continuously increasing with increasing wavelength. Obviously, for the higher structures the transition point of the rising reflectance is shifted to the longer wavelength range. A structure height of 450 nm guarantees a low reflectivity up to the far red visible region, whereas a structure height of 235 nm shows low reflectivity only up to the blue visible region. The 3rd order Super-Gaussian structures show a significantly different behavior. The AR structures exhibit a local minimum of the reflectance at 325 nm but not a broadband low reflectance - independent of the structure height. The 95 nm height structure shows a simple concave reflectance curve over the investigated spectral range. With the original at 325 nm the reflectance is increasing for shorter and longer wavelengths. Additional minima emerge with increasing height of the Super-Gaussian profile so that a modulated dependency is observable. The number of oscillations increases with increasing structure height while the reflectance of the local maxima and the width of the local minima are reduced.
Fig. 5 Calculated reflectance as a function of wavelength for different conical and 3rd order Super-Gaussian profiles. The structure heights are optimized for minimum reflectance at 325 nm (perpendicular incidence angle).
To summarize: Higher order Super-Gaussian structures offer the advantage of reduced structure height required for optimum reflectance reduction compared to conically shaped profiles. In order to achieve a very low reflectance (< 0.1 %) at 325 nm, a structure height of 95 nm is already sufficient in the Super-Gaussian case, whereas a structure height of 235 nm is needed for conical shaped sub-wavelength profiles. However, conical structures possess broadband antireflective properties, which cannot be achieved with high-order Super-Gaussian profiles.
A comparison of the calculated and measured reflectance spectra of AR structures are displayed in Fig. 6. The AR structures have been prepared by a combination of BCML and reactive ion etching using a plane fused silica substrate which was structured on both sides, with a resulting profile that is comparable to the profile shown in Fig. 1(b). The profile height of the simulated and the measured profile is about 235 nm. The spectral transmittance measurement was carried out at normal incidence and shows two maxima at 325 nm and 750 nm and a local minimum at 450 nm. Furthermore a major reduction in transmittance is observable in the wavelength range below 300 nm.
Fig. 6 Comparison of measured and calculated transmittance spectra. The fused silica substrate was structured on both sides with BCML and reactive ion etching. The half-width of the optimized 2nd order Super-Gaussian profile was reduced from w = 0.7 to w = 0.5 to fit the profile of the manufactured structure.
Following the previous discussion we simulated the transmittance spectrum of an AR structure with an optimized 2nd Super-Gaussian profile (w = 0.7), which enables the prediction of the appearance and the wavelength position of the measured extrema. To describe the accurate spectral transmittance we tried to reduce the half-width of the profile form while keeping the height of the structure constant at the measured value of 235 nm. This resulted in a shift of transmittance, but hardly altered the spectral position of the characteristic extrema. The result for a half-width of w = 0.5 is in accordance with the experimental determined transmittance in the wavelength range above 300 nm. The decay of the transmittance in the deep-UV was not correctly reflected by this simulation.
3. Transmittance reduction in the deep-UV caused by inhomogeneous local profile heights
An ideal sub-wavelength grating is explicitly described by the periodicity of a basic unit cell. In contrast, the homogeneity in period and structure height of the samples depends on the manufacturing process. Thus individual protuberances of the sub-wavelength structures may deviate slightly from each other. The appearance of such imperfectness is related to additional spatial frequencies in the frequency spectrum of the grating. If these additional frequencies are related to periods for which the zero-order condition is not fulfilled any more, higher propagating diffraction orders are introduced and the 0th-order efficiency is decreased. Especially, periods which are slightly larger than the limiting period of the zero-order condition will induce a deflection in large spatial angles. On the other hand very large periods are also associated with diffraction orders in small angels. In general the introduced irregularities will be described statistically and the correlated diffraction orders show a dense distribution. From this follows a continuous spatial intensity distribution which is equivalent to a scattering effect.
To simulate the influence of a statistical height variation on the transmittance of an AR structure by the rigorous method we had to change the simulation approach. Instead of a single unit cell containing only one protuberance, we artificially extended the period of the unit cell extremely, so that a large number N of individual protuberances with the lateral extension of Λ0 are accommodated in a single unit cell. The origin lateral dimensions of the protuberances Λ0 remain unaltered. Therefore, the grating period Λ is artificially increased without changing the physical properties of the grating (Λ = N Λ0). The first two drawings in Fig. 7 are showing the approach with N = 4 schematically. The situation on the left illustrates a single unit cell with a single protuberance. In the center of Fig. 7, the size of the unit cell was increased by a factor of 4 without altering the lateral size of the protuberances. The introduced lateral extension of the grating period is accompanied by the theoretical appearance of higher diffraction orders. But there is no energy attended with the apparent higher diffraction orders, due to the equivalence of the sub-structures.
Fig. 7 Schematic transition from the perfect periodic structure to a height varied approach. Here 4 single protuberances are displayed. For the calculation 128 units were used.
In a second step [Fig. 7(right)] the height of the protuberances was varied in the extended unit cell. The height variation f (h) was assumed to follow a statistical Gaussian distribution:
Here h is the height of the individual protuberance and σ the standard deviation. With the introduction of the Gaussian height distribution additional spatial frequencies occur which will not fulfill the zero-order condition. As a consequence the energy is redistributed in the related higher diffraction orders. The fraction of light which is not transmitted or reflected into the 0th order is considered as scattered light. Regarding the convergence criteria of the rigorous simulation the extension of the unit cell requires the consideration of an increased number of Rayleigh orders which is accompanied with an enormous rise of the necessary computing time. Hence, a fully 2-dimensional approach was not applicable. As a reasonable compromise between computing effort and simulated significance we applied a 1-dimensional approach which allows at least a qualitative characterization of the optical performance of the AR structures. Three different spacings of the protuberances were simulated (80 nm, 139 nm, and 150 nm) varying the wavelength between 200 nm and 1 μm.A consecutive increase of the number of protuberances per unit cell leads to a confluence of the calculated efficiencies for the relevant diffraction orders, as the high spatial frequencies which are equivalent to small periodic structures are already introduced with small numbers of N. These high spatial frequencies give rise to large diffraction angles and are responsible for scattering effects. With increasing N the contribution of the high spatial frequencies remains unaffected, but long range periodic structures are added. The correlated low spatial frequencies have only a minor effect on the light propagation, thus a convergence of the calculated data can be expected with an increasing count of sub-units N. To test the convergence we calculated the transmittance of an 1-dim grating with a Gaussian profile in dependence of the mean structure height and the number of sub-units (Fig. 8(left)). The wavelength was chosen to be 325 nm and the spacing of the protuberances was set to 80 nm. The angle of incidence was perpendicular to the surface. Each calculated data point represents an average of 10 single simulation steps with a new selected height variation but an unchanged Gaussian distribution. Under these conditions, a number of N = 128 offers a sufficient convergence of the simulation. In Fig. 8(right) the calculated data is plotted with and without averaging over 10 single simulation steps for the highest number of sub-units (N = 128). This comparison shows that the averaging procedure is resulting in a smoothening of the calculated data without altering the overall trend of the calculated values. Of course this smoothening could be achieved by increasing the number of sub-units, but this would far more increase the calculation time without changing the results concerning the scattering losses. If the ratio between wavelength and dimension of the single protuberance is increased, it is also necessary to increase the number of sub-units.
Fig. 8 Calculated transmittance of a 1-dim grating with a Gaussian profile in dependence of the mean structure height (wavelength: 325 nm, grating period: 80 nm, direction of incidence: perpendicular to the surface). Left: Increasing number of sub-units and averaging of each data point over 10 single simulation steps. Right: With and without averaging over 10 simulation steps (N = 128).
As mentioned above, the amount of scattered light will be defined as the total integral scattering (TIS) which is given by the difference of the incoming light and the sum of transmitted and reflected zero order light (T + R):
In Fig. 9 the simulated dependency of scattering loss as a function of the structure height is shown for different periods of the individual protuberances (80 nm, 139 nm and 150 nm). The indicated height is related to the mean value of the Gaussian distribution and has been varied between 0 nm and 430 nm. Simulations were carried out for transmittance at normal incidence using a wavelength of 325 nm. A Gaussian profile, which shows a good antireflective performance, was assumed for each single protuberance with a height distribution of σ = 5 % (standard deviation). For comparison the dotted line in Fig. 9 shows the 100 % value for R+T which results without statistical height variations (Λ0 = 80 nm, σ = 0 %).
Fig. 9 Calculated sum of reflectance and transmittance of a 1-dim AR grating with local height variations as function of the mean structure height. The data was calculated for different grating periods and smoothed before drawing. The dotted line represents an antireflection grating without height variations.
According to Eq. (4) Fig. 9 shows that the scattering losses intensify with increasing mean structure height and with larger structure periods. The standard deviation is a measure for the relative height variation and therefore the absolute height difference will increase with increasing mean structure height. This effect can also be found in manufactured structures e.g. when a mask transfer process is included in the fabrication method. By increasing the grating period Λ0, the whole frequency spectrum is shifted towards lower spatial frequencies. This leads to increased scattering effects, because more spatial frequencies have to be considered, which will not fulfill the zero order condition anymore.
In order to gain a better understanding of the spectral dependency of the scattering effects we simulated a wavelength variation between 200 nm and 1000 nm for a fixed mean structure height of 235 nm, which correlates to a reflectance minimum at 325 nm. The Gaussian shaped protuberances had a distance of 80 nm for these simulations. The calculations were carried out for three values of the standard deviation (0 %, 5 %, and 15 %).
On the left of Fig. 10 the calculated reflectance spectra are displayed which show the weak influence of the standard deviation on the positions of the extreme values. However, with increasing standard deviation the averaging effects become more prominent and the amplitudes of the modulated curves are damped. The right side of Fig. 10 shows the sum of zero order reflected and transmitted light demonstrating a strong increase of the scattering effects with increasing standard deviation and decreasing wavelength.
Fig. 10 Left: Reflectance spectrum for 1-dim AR structure with local height variations. A Gaussian profile with a structure height of 235 nm was assumed. Right: Sum of reflectance and transmittance in dependency on wavelength. The different curves are related to different standard deviations.
4. Local statistical position deviations of single protuberances
Depending on the manufacturing process, another apparent imperfectness of AR structures is a statistical variation of the lateral spacing between single protuberances. Similar to the height variations, these imperfectness leads to spatial frequencies in the frequency domain violating the zero-order condition and therefore leading to scattering effects. To simulate this effect and confirm the identical effects compared to statistical height variations an analogous approach with an extended unit cell was used for the calculation. Figure 11 depicts the procedure for N = 4. Instead of altering the height with a Gaussian distribution [Eq. (5)], the spacing between the protuberances was altered by a similar Gaussian distribution:
Fig. 11 Schematic transition from the perfect periodic structure to a statistical variation of the grating period. Here 4 single protuberances are displayed. For the calculation 128 units were used.
In this expression Δx is specifying the deviation of a protuberance to the position of a protuberances in a perfect periodic grating.
To avoid an overlapping of the protuberances and ensure a defined length of the unit cell (Λ = NΛ0) additional conditions had to be considered. Thus, the minimal spacing between the protuberances was limited to 0.3 Λ0 and the position of the outmost protuberances were leaved unaltered.
As in Section 3 the according simulation was carried out by a Gaussian profile (height: 200 nm, mean period Λ0: 80 nm). Because the simulation is very similar to the previous simulation the parameters regarding the convergence of this method have been left unaltered (N=128 and averaging each data point over 10 simulation steps). The wavelength of the light was varied from 180 nm to 950 nm. The standard deviation σ was chosen to be 5 %, 15 % and 30 %. Fig. 12 displays the calculated reflectance (left) and the sum of reflectance and transmittance (right). The results are in conformance with the results originating from the height variations. The reflectance curve shows an averaging effect with increasing standard deviation and the scattering losses are similar to the calculated losses before. The equivalence of variations in height and displacement, related to scattering effects, is therefore confirmed by the simulations.
Fig. 12 Left: Reflectance spectrum for 1-dim AR structure with local displacement variations. A Gaussian profile with a structure height of 200 nm was assumed. Right: Sum of reflectance and transmittance in dependency on wavelength. The different curves are related to different standard deviations.
5. Comparison of simulations and experimental determined transmittance of AR structures
In the preliminary sections we found that a 2nd order Super-Gaussian profile with a particular half-width describes the measured reflectance very well in the long wavelength range, especially it verifies the position of the transmittance maximum. In the short wavelength range a significant deviation was observed. In addition, the calculations of the scattering effects induced by the height and displacement variations of the individual protuberances correlate to the observed decay of the reflectance in the short wavelength range of manufactured samples. To qualitatively include the scattering losses of imperfect AR Structures in the simulation of the 2nd order Super-Gaussian profile, the results concerning the shape dependence (Fig. 6) and the curve describing the scattering losses caused by height variations (Fig. 10) were combined [Eq. (6)].
The multiplication of the calculated transmittance for a perfectly periodic 2nd order Super-Gaussian profile T2D (λ) (w = 0.7, h = 235 nm, Λ0 = 80 nm) with the scattering losses of a 1-dim Gaussian profile TIS1D(λ) (σ = 15 %, h = 235 nm, Λ0 = 80 nm, σ = 15 %) leads to a transmittance spectrum Tscatterd(λ) which is in accordance with the experimental determined transmittance of AR structures fabricated by BCML and RIE (Fig. 13). This combination cannot give a quantitative description of fabricated AR structures because the scattering losses were derived from a 1-dim model. But it is obvious that the simulation describes the modulated transmittance in the long wavelength range and verifies the decay with shorter wavelengths as well.
Fig. 13 Comparison of measured and calculated transmittance spectra. The fused silica substrate was structured on both sides with BCML and RIE (solid line). The transmittance was calculated for a 2nd order Super-Gaussian profile (w = 0.7, h = 235 nm, Λ0 = 80 nm) with perfect periodicity (dashed line) and was multiplicatively combined with the scattering (TIS) of a 1-dim Gaussian profile (σ = 15 %, h = 235 nm, Λ0 = 80 nm, σ = 15 %) (dash-dotted line).
6. Conclusion
To summarize, the optical performance of manufactured AR structures was successfully described by simulations using RCWA. It was found that the antireflective behavior of sub-wavelength AR structures in the long wavelength region is dominated by the profile shape of the protuberances. Scattering effects observed in the short wavelength region are associated with additional spatial frequencies based on local irregularities such as height and displacement variations of the protuberances which are found in AR structures fabricated by a combination of BCML and RIE. Both considered irregularities have the same influence on the AR structures optical performance. Hence, to guarantee a strong antireflective behavior in the UV, or even deep-UV, irregularities of each kind, with spatial frequencies violating the zero order condition, have to be avoided.
Acknowledgments
The authors thank the BMBF and the Max Planck Society for financial support (contract No. 13N9713, “EFFET”).
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