1. Introduction

Wire grid polarizers (WGP) are diattenuating gratings with a period well below the wavelength of the incident light [1,2]. The polarization component parallel to the ridges (TE-transversal electric) is strongly attenuated, whereas the polarization component orthogonal to the ridges (TM-transversal magnetic) is mostly transmitted. Compared to other polarizer concepts, such as polarizing prisms, these nano-optical elements provide advantages such as large free apertures, large acceptance angles and high integrability [2,3]. In the past decades, lots of effort was undertaken to extend the application range of WGPs from the terahertz [4–6], mid-infrared [7], infrared [8] and visible [9–12] wavelength region to the deep ultraviolet range [13–22]. On the one hand, this development is driven by the persistent trend to reduce the operating wavelengths for enhanced resolution in optical lithography [23–25] or microscopy [26]. On the other hand, the ultraviolet spectral range is the setting for many fundamental processes in atomic dynamics [27]. Therefore, it plays a major role in optical analytics, e.g. in Raman spectroscopy [28,29] and Raman scattering imaging [30] as well as in bio sensing [31].

Wire grid polarizers with a good optical performance in the ultraviolet spectral region, i.e. high extinction ratio and high transmittance, can be achieved with materials providing interband absorption due to direct band transitions [18]. However, the experimentally achieved performance of WGPs is usually much lower than the value expected from simulations. A reason for that are systematic structure deviations such as non-ideal ridge shapes [17,32,33]. Hence, improved fabrication technology [34] and in situ process control [19] have been developed. Nevertheless, mitigating solely systematic shape deviations is not sufficient for short wavelength nano-optical elements. The feature sizes of only a few tens of nanometers approach the same order of magnitude as the typical fluctuations of the line edge positions [13]. Investigating and reducing line edge roughness (LER) is therefore a crucial task to enable high-performance WGPs and other optical meta-surfaces in the far ultraviolet (FUV) spectral range or at even shorter wavelengths.

Studies on LER of nano-optical elements so far mainly base on the reconstruction of geometrical properties from scatterometry. To reduce the required computational effort Fraunhofer approximation [35,36], effective media approximation [37] or the reduction to a one dimension problem [38] have been utilized. These simplifications, however, are not applicable for measured, i.e. realistic, material properties and if the structure periods are only slightly smaller than the operating wavelength.

In this contribution, we overcome these limitations by combining a comprehensive spatial frequency dependent roughness model with finite difference time domain (FDTD) simulations to investigate the influence of line edge roughness on the optical performance of WGPs for the far ultraviolet spectral range. Thereby, the influence of LER can be separated from deterministic structure and material deviations which is virtually not feasible by experiments. Additionally, the complex problem of LER is traced back to different effects depending on the spatial frequencies, providing a vivid way to understand the particular underlying physical effects.

The article is organized as follows: First, we introduce the basic principles of WGP and important aspects of their experimental realization. Thereby, we illustrate the evolution of an initial lithographic line edge roughness to the final device. Then, we demonstrate to extract the roughness parameters standard deviation, correlation length and Hurst exponent from measurements and outline the data preparation for the FDTD simulation. Finally, we discuss the influence of the individual roughness parameters and the wavelength on the optical performance of the devices.

2. Basics on Wgps and their experimental realization

In Fig. 1 a schematic of a WGP is shown. Such an element consist of a grating structure with the period $\Lambda$, which has to obey the zero order grating condition for given application wavelength $\lambda$, angle of incidence $\phi$ and substrate refractive index $n_{\text{sub}}$:

 

Fig. 1 Schematic wire grid polarizer and coordinate system.

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The performance of a wire grid polarizer is described by the transmittance of TM polarized light $T_{\text{TM}}$ and that of TE polarized light $T_{\text{TE}}$. The extinction ratio $Er=T_{\text{TM}}$ /$T_{\text{TE}}$ denotes the suppression of TE polarized light, which is damped exponentially within the structure.

The efficient suppression of the transmittance in TE-polarization requires grating materials with a high extinction coefficient and high aspect ratio structures. Since $T_{\text{TE}}$ appears in the denominator of $Er$, the extinction ratio is very sensitive to small changes of $T_{\text{TE}}$. In the case of TM-polarized light, a high permittivity contrast of the grating and its surrounding media leads to a localization of the electric field in the gaps between the ridges. This enables a high transmittance of TM-polarized light [8].

The simulated extinction ratio and TM transmittance of a tungsten wire grid polarizer for an operation wavelength range of 230 to 400 nm is illustrated in Fig. 2. The simulations have been performed by means of the rigorous coupled wave analysis RCWA [40]. Theoretically, extinction ratios of several thousands are possible. However, the experimentally feasible optical performance of WGPs strongly depends on the fabrication process and typically differs substantially from the numerically retrieved results.

 

Fig. 2 Simulated optical performance of a tungsten WGP with a period of 100 nm, a ridge width of 23 nm and a ridge height of 150 nm. For the simulation the material data by Palik [39] were used.

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In our investigations we used a self-aligned double patterning process to fabricate the devices [41] which is schematically depicted in Fig. 3. First a layer stack consisting of polymer, chromium and electron beam resist (OEBR-CAN034A2 by Tokyo Ohka Kogyo Europe) is prepared. Afterwards the resist is patterned by electron beam lithography (Vistec SB350 OS) [42]. In the next step the subjacent chromium and the polymer are structured by ion beam etching and reactive ion beam etching (Oxford Instruments Ionfab300LC), leading to the template grating. This template grating is then coated with tungsten by ion beam deposition (Oxford Instruments IonFab300LC). Finally, the material from the horizontal surfaces and the template grating is removed by ion beam etching and reactive ion beam etching. Figure 4(a) shows top and cross-sectionional view of a tungsten WGP realized with this fabrication procedure.

 

Fig. 3 Schematic fabrication process for a WGP consisting of a) patterning of the electron beam resist on the initial layer stack, b) etching of the template grating, c) coating of the ridge material and d) removing the horizontal surfaces and the template structure by ion beam etching to achieve the final WGP structure.

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Fig. 4 a) SEM top view and cross section of a tungsten WGP. b) Measured transmittance and extinction ratio [17].

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Such a WGP shows several deviations from an ideal rectangular shape as tilted ridges, trenches in the substrate and chamfered tops originating from the fabrication process. Additionally, the actual refractive index of the deposited thin film deviates from that measured at bulk materials. These deviations substantially impair the optical properties of WGPs [16,17], which becomes clear by comparing the simulated extinction ratios in Fig. 2 with the measurement data in Fig. 4(b). Beside the systematical shape and material deviations, the top view SEM image in Fig. 4(a) shows a roughness along the ridges. It originates from several effects: The initial pattern is defined by electron beam lithography, which introduces roughness by the stochastic nature of electron shot noise as well as of the distribution, reaction and diffusion of chemical species [43]. This roughness is then transferred into the template grating by the etching processes. In Fig. 5 it is how the double patterning process translates the lithographic line edge roughness into a position roughness of the ridges. Hereafter, we assume that the tungsten layer is perfectly smooth. The edge position of the tungsten ridge may then be described by the edge position of the rough template grating that means its line edge roughness.

 

Fig. 5 a) Template grating with rough edges. b) The deposited tungsten layer resamples the roughness of the template grating. c) After removal of the template grating the position of the template edges resembles the original rough edge.

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3. Line edge roughness

Line edge roughness (LER) is defined as a random deviation of the edge position to the best-fit line. Its dependence on the spatial frequency $f$ can be described by the power spectral density (PSD) as proposed by Palasantzas [44,45]:

Here, $\sigma$ is the standard deviation, $\xi$ is the correlation length and $H$ is the Hurst exponent. $\Gamma$ represents the Gamma function. To illustrate the meaning of the individual parameters, Fig. 6 shows their influence on the PSD and on the shapes of the line edges. The red (solid) curve and drawing of a rough edge have the same PSD for comparison. According to the Parseval’s theorem the integral over the PSD is equal the variance ${\sigma }^2$. Increasing $\sigma$ approximately offsets the PSD as illustrated in Fig. 6(a). The correlation length determines the transition from frequency independent white noise at lower spatial frequencies to a decay with a power of $1/f$ characteristic at higher frequencies. Increasing the correlation length leads to a less rough appearance of the edge, although $\sigma$ is constant. The Hurst exponent determines the slope of the decay at higher spatial frequencies. Although less pronounced, a larger H results also in a slightly smoother appearance of the edge. Typical values for the roughness parameters of a patterned resist are $\sigma =\left[2,10\right]\text{nm}$, $\xi =[10,100]\text{nm}$ and H = [0.5,1] [17].

 

Fig. 6 Illustration of the roughness parameters’ influence on the power spectral density. To enable a comparison, the bold curves and right sketches (red color) of a rough edge always have the same parameters σ = 10 nm, ξ = 10 nm and H = 0.7. a) Influence of the standard deviation $\sigma$. b) Influence of the correlation length $\xi$ and c) Influence of the Hurst exponent $H$.

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The PSD of the electron beam resist profile is deduced by the evaluation of SEM images. Special care is taken not to alter the measured PSD by the image acquisition, the detection algorithm or by discretization errors [47,48]. First, the edge positions are retrieved by applying an edge detection algorithm without prior image filtering. A linear fit is applied on the edge position data to correct tilt errors. From the edge displacement $d(x)$ at the position x and the length of the detected edge L, the PSD can be achieved by a Fourier transformation:

In order to minimize statistical noise [49] this procedure is applied to 32 SEM images of 8 different samples at 4 positions for 16 edges per image, resulting in a total of 512 analyzed edges (Fig. 7). The resolution of the SEM images is 1280 px by 980 px with a pixel size $\Delta x_{px}$ of 1.24 nm. Then, the measured PSDs are averaged. The resulting statistical error is less than 26 nm³. To fit the measured PSD, we add the term ${\sigma }_n{}^2\frac{\Delta x_{px}}{2\pi }$ to Eq. (3) as proposed by Verduin et al. [49]. By this, we account for the influence of pixel noise with the standard deviation ${\sigma }_n$. From the data fit we derive the roughness parameters as shown in Table 1.

 

Fig. 7 a) SEM image of the patterned electron beam resist with a resolution of 1280 px by 980 px and a pixel size of 1.24 nm. b) PSD of the measured edge profiles and fit retrieved from 32 SEM images. The fit parameters are shown in Table 1.

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Table 1. Roughness parameters of the electron beam resist acquired by fitting of the measured PSD.

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To evaluate the impact of the measured LER on the optical performance, the PSD needs to be converted into numerical data of random edges. To this end, we utilized the algorithm by Thorsos [44]. A detailed discussion of the method is given by C. A. Mack [45]. The line edge displacement $d(x_n)$ at the position $x_n=n\Delta x$ of the $n$ ^th sampling point with spacing of $\Delta x$ according to Fig. 8 is given as:

 

Fig. 8 Magnification of a random rough edge achieved by the algorithm of Thorsos. The gray shaded area illustrates the ridge.

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Here, $L=N\Delta x$ is the domain length for N grid points and ${\eta }_1$, ${\eta }_2$ are Gaussian distributed random numbers with a mean of zero and a variance of one.

The discretization and finite domain length causes a statistical bias which is reduced by choosing an optimum spacing [45]:

In the case of an optimum spacing, the standard deviation of the data set ${\sigma }_{\text{data}}$ averaged for many realizations can be approximated as:

The average dataset standard deviation $<{\sigma }_{\text{data}}>$ is always smaller than the input standard deviation $\sigma$. This fact needs to be considered in the evaluation of the results. Since the PSD of a single, individual realization of a rough edge may significantly deviate from the input PSD, a statistical error is introduced into the simulation results. To this end, an averaging of several simulations with different realizations of rough edges is necessary. The statistical error follows a $1/\sqrt{M}$ trend, with $M$ as the number of realizations. The statistical errors $\Delta T_{\text{TE}}$ and $T_{\text{TM}}$ are calculated as standard deviations of the individual results. The resulting error of the extinction ratio $\Delta Er$ is:

For the FTDT simulation we use the FullWAVE^TM tool from Synopsys. In each simulation 11 ridges with 512 individual grid points are considered. The standard deviation of the data set is calculated as given by Eq. (7). For each individual representation of the input PSD the grid size in $x$ -direction is chosen to be ¼ of the optimal spacing $\text{Δ}{x}_{\text{opt}}$. In $y$ - and z -direction grid sizes of 0.45 nm and 10 nm are used. The grating is periodically repeated in $y$-direction. In z -direction we use 40 nm thick perfectly matched layers. These parameters are backed by careful convergence test, the model matches the prediction of 1D-RCWA of smooth gratings ($\sigma =0 nm$) and the results are self-consistent (i.e. random realization of the same LER data set achieve the same results within statistical error). For the tungsten ridge material we apply the refractive index data by Palik [39] and for the fused silica substrate the data given by Malitson [50], respectively. The simulations are conducted at a continuous wave excitation. For clarity of presentation, only data at a wavelength of 248 nm is shown, if not noted otherwise.

4. Results and discussion

4.1 Statistical error

To estimate the previously discussed statistical error, which is induced by the random realization, seven calculations are performed for an exemplary data set of $\sigma =2.5 \text{nm}$, $\xi =10 \text{nm}$ and $H=0.7$. For each simulation, individual realizations of the 11 random edges are generated. This results in relative errors of $\frac{\Delta T_{\text{TM}}}{<T_{\text{TM}}>}=0.17\text{\%}$ and $\frac{\Delta T_{\text{TE}}}{<T_{\text{TE}}>}=5.66\text{\%}$, respectively. We assume these relative errors to be approximately the same for all other simulations. For the sake of simplicity, the small error for $T_{\text{TM}}$ is omitted in the following discussions. The error for the extinction ratio is calculated by Eq. (8). For $\sigma =0 \text{nm}$ the roughness and also the statistical error in the simulations vanishes.

4.2 Standard deviation

Figure 9 shows the influence of the roughness’ standard deviation for values between 0 nm and 10 nm while the other parameters are fixed at $\xi =10 \text{nm}$ and $H=0.7$. The extinction ratio strongly degrades with increasing $\sigma$. The TM transmittance also decreases for larger standard deviations. The measured roughness with a standard deviation $\sigma =3.36 \text{nm}$ leads to a reduction of the extinction ratio by a factor of 3.6, whereas the TM transmittance decreases by about 11%.

 

Fig. 9 Dependence of the extinction ration and transmittance (TM) on the standard deviation $\sigma$ for $\xi =10 \text{nm}$ and $H=0.7$.

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To understand the influence of the standard deviation, it is instructive to separately consider three different phenomena: macroscopic effects, grating effects and effective medium effects. For a given wavelength (e.g. 248 nm), the impact of these effects on extinction ratio and transmittance depends on spatial frequency components of the roughness. As indicated in Fig. 10 for spatial frequencies much smaller than the inverse of the wavelength macroscopic effects dominate. In contrast, spatial frequencies much larger than the inverse wavelength lead to effective media effects. Grating effects are predominantly found in the intermediate spatial frequency range. In the following sections, we will discuss the influence of these three different effects on the performance of wire grid polarizers in detail.

 

Fig. 10 PSD according the measured values (see Table 1) where the inverse wavelength of incident light, the region of macroscopic disturbances and the region for effective media are indicated.

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4.3 Macroscopic effects

Roughness with small spatial frequencies can be considered as a macroscopic displacement of the grating ridge. This displacement increases the penetration depth of TE-polarized light into the grating structure as depicted in Fig. 11. In this case, the disturbed exponential decay of the TE electric field along the grating depth drastically reduces the extinction ratio. In contrast, the electric field for TM-polarized light is only slightly altered and hence the impact on the TM transmittance is small.

 

Fig. 11 a) Simulated TM transmittance and extinction ratio in dependence of the displacement. b) Normalized electrical field distribution for TE- and TM-polarized light with a ridge displaced by 5 nm.

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4.4 Grating effects

Spatial frequencies in the range of the inverse wavelength of the incident light can lead to diffraction or resonances. This applies particularly, when the zero order grating condition according to Eq. (1) is not obeyed. This results in additional loss and scattered light. To investigate the effect of scattered light, we calculate the far field distribution with FDTD simulations and deduce the angle resolved scattering. Figure 12 shows the angle resolved scattering, normalized to the zero order transmittance, for TE- and TM-polarized incident light. From these data, the amount of non-specularly transmitted light, i.e. the total integrated scattering $TIS$, is calculated. For TM-and TE-polarization we determine values of $TI{S}_{\text{TM}}=2.4 \times {10}^{-2}$ and $TI{S}_{\text{TE}}=3.9 \times {10}^{-6}$, respectively. The difference between these two results comes mainly from the much smaller overall transmittance of TE-polarized light due to the strong absorption. Consequently, also the magnitude of scattered light in transmission is much smaller for TE-polarized light. The peaks of both polarizations at angles of about $\pm$ 55° in x -direction [Fig. 12] originate from diffraction at the initial template period of 200 nm.

 

Fig. 12 Angle resolved scattering of the transmitted light normalized to the intensities of the zero order transmittances for a) TE and b) TM polarized incident light, respectively. The simulation was performed with the roughness parameters: 𝜎 = 2.5 nm, 𝜉 = 10 nm, H = 0.7.

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4.5 Effective medium effects

In this spatial frequency range, the dimension of the roughness is much smaller than the wavelength of incident light. Therefore, an effective medium approach may be used and the roughness can be described with an effective permittivity ${\epsilon }_{\text{eff}}$. In this case, the effective permittivity is calculated from the permittivity of the ridge material ${\epsilon }_{\text{W}}=3.4+i19.4$ and of air ${\epsilon }_{\text{air}}=1$ ($\lambda =248 \text{nm}$) following the Maxwell-Garnett equation [51]:

The quantity s represents the design width of the ridge and $d(x)$ denotes the edge displacement. Figure 13(a) illustrates the volume fraction $\delta w(y)$ and absolute effective permittivity $\left|{\epsilon }_{\text{eff}} \left(y\right)\right|$ of half a WGP’s unit cell. The volume fraction resembles a sigmoid function where the half width at half maximum (HWHM) matches the mean value of a rough edge depicted as gray area in Fig. 13(a). However, for the interaction of the structure with light, not the mean position of the edge (and thus the volume fraction) but the distribution of the permittivity ${\epsilon }_{\text{eff}} \left(y\right)$ is the critical parameter. The HWHM of $\left|{\epsilon }_{\text{eff}} \left(y\right)\right|$ resulting from Eqs. (9)-(11) is smaller than that of the volume fraction represented by the blue curve in Fig. 13(a). Hence, the effective optical ridge width is smaller than the value retrieved from the volume fraction. This finding corresponds to the observations for dielectric gratings reported in [52]. As shown in Fig. 13(b) a reduction of the ridge width strongly decreases the extinction ratio and thus critically affects the optical performance.

 

Fig. 13 a) Volume fraction $\delta w$ and absolute effective permittivity $\left|{\epsilon }_{\text{eff}}\right|$ for half a unit cell in dependence of the position along the ridge. The gray area resembles a rough edge viewed from top with the roughness parameters $\sigma =2.5 \text{nm}$, $\xi =10 \text{nm}$ and $H=0.7$ and b) simulated dependency of the TM transmittance and extinction ratio on the width of an ideal ridge without roughness.

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The simulation results of the ideal ridges in Fig. 13(b) suggest that the TM transmittance should increase with decreasing ridge width. In contrast, for a realistic ridge with LER, the transmittance decreases with increasing standard deviation, which is equivalent to a reduced ridge width. The reason for this discrepancy is that a large transmittance requires a sharp refractive index step such that electric field is localized within the gap [18]. Since the index profile is blurred, the localization is impaired and consequently the transmittance (TM) is not increasing with decreasing ridge width.

4.6 Correlation length

Figure 14 shows the influence of the correlation length $\xi$ on $T_{\text{TM}}$ and $Er$. The other input parameters are kept constant at $\sigma =2.5 \text{nm}$ and $H=0.7$. As given by Eq. (7), the standard deviation of the data set has a constant value of ${\sigma }_{\text{data}}=2.45 \text{nm}$ for all $\xi$. Increasing the correlation length slightly increases the transmittance of TM-polarized light, where there is seemingly a minimum for the extinction ratio (please note the statistical error). As mentioned above, the correlation length determines the amount of high and low spatial frequency roughness components. Small correlation lengths correspond to high spatial frequency roughness. As described above, effective medium effects reduce the effective ridge width. At large correlation lengths low spatial frequency roughness and thus the macroscopic displacement effects dominate. The superposition of these two effects leads to a critical range of correlation lengths where the extinction ratio becomes minimum. Consequently, an optimization of fabrication processes should pay special attention to the roughness’ correlation length.

 

Fig. 14 Variation of the correlation length $\xi$ for $\text{σ}=2.5 \text{nm}$ and $\text{H}=0.7$.

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4.7 Hurst exponent

As the third roughness parameter, the Hurst exponent and its influence on the optical performance of WGPs shall be discussed. Due to the statistical bias, a change of the Hurst exponent is always accompanied by a variation of the roughness’ standard deviation. For example, Hurst exponents of $H=0.5$ and $H=1$ lead to ${\sigma }_{\text{data }}(\text{H}=0.5)=2.43 \text{nm}$ and ${\text{σ}}_{\text{data}}(H=1)=2.47$, respectively. Although the change of ${\text{σ}}_{\text{data}}$ is small, it should be considered because the optical performance strongly depends on the roughness’ standard deviation. We increase the error of the extinction ratio by $\text{Δ}E{r}_{\text{bias}}=(Er\left(\sigma =2.47 \text{nm}\right)-Er\left(\sigma =2.43 \text{nm}\right))/2=32.6$ to account for the change of ${\text{σ}}_{\text{data}}$. The simulation results of the Hurst exponent in Fig. 15 show a constant behavior of both, the TM transmittance and the extinction ratio. This is because the Hurst exponent has only a small influence on the PSD. Consequently, the Hurst exponent’s impact on the optical performance may be neglected.

 

Fig. 15 Variation of the Hurst exponent $H$ for $\text{σ}=2.5 \text{nm}$ and $\xi =10\text{nm}$.

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4.8 Wavelength

Finally, Fig. 16 illustrates the influence of the LER in dependence of the wavelength. For comparison, the simulated data for an ideal WGP is plotted as well. Line edge roughness reduces extinction ratio and TM transmittance over the full spectrum, whereas the overall appearance is preserved. This is a consequence of the grating material’s complex refractive index which is assumed to be the same for the ideal and rough structure.

 

Fig. 16 Wavelength dependence of $T_{\text{TM}}$ and $Er$ for $\text{σ}=2.5 \text{nm}$, $\xi =10 \text{nm}$ and $\text{H}=1$ in comparison to the simulated values for an ideal WGP structure.

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Figure 17 shows the spectral extinction ratio and TM transmittance of the rough structure normalized to the respective properties of the ideal structure. As a general trend, the influence of the LER increases with decreasing wavelength. The TM transmittance is reduced by about 20% within the wavelength range from 200 nm to 450 nm. The relative extinction ratio decreases by approximately 30%. At wavelengths where the extinction ratio of the ideal structure is large (e.g. at 250… 300 nm), the influence of roughness is much more detrimental. This can be attributed to a resonant field enhancement boosting the extinction ratio of the ideal structure in this spectral range. The field enhancement is accompanied by a higher sensitivity towards structural imperfections.

 

Fig. 17 Ratio of the simulated TM transmittance and extinction ratio of rough WGP structures to ideal ridges. The calculations were performed for σ = 2.5 nm, ξ = 10 nm and H = 0.7.

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5. Conclusion

In this contribution, we investigated and quantified the detrimental effect of line edge roughness on the optical properties of far ultraviolet wire grid polarizers. For the measured roughness’ standard deviation of 3.36 nm, the transmittance of TM polarized light is reduced by 11% and the extinction ratio is reduced by a factor of 3.6 at a wavelength of 248 nm. Hence, future improvement of process technologies for the fabrication of wire grid polarizers need to focus on reducing the LER. Methods like thermal or plasma treatment of the resist structure to reduce the LER standard deviation usually also influence the correlation length and Hurst exponent [46]. Our results show that there is an unfavorable intermediate range for the correlation length, which should be avoided, whereas the Hurst exponent has no significant influence on the optical performance. The deterioration of the extinction ratio induced by the line edge roughness increases with decreasing wavelength. This imposes a challenge for the future development of wire grid polarizers operating at shorter wavelengths in the far ultraviolet. The presented method to analyze the influence of LER on the optical performance is applicable to any nano-optical meta-surfaces realized by arbitrary fabrication techniques and may provide valuable insights for these structures. Thus, the results provide an indispensable basis for a purposeful development of future fabrication strategies in the rapidly increasing field of short-wavelength optics.