We describe a technique for experimentally determining the spatial-frequency modulation transfer function for near-field super-resolution imaging systems and present such a modulation transfer function for a 20|40|20 nm poly(vinyl alcohol)~(PVA)|Silver|PVA superlens exposed to 365 nm wavelength (i-line) radiation through a 50-nm thick tungsten mask. An extensive spectral characterization is achieved from only two exposures, with transmission coefficients determined for spatial frequencies as high as 13 µm−1, corresponding to λ / 4.75. The resulting transfer function is in good agreement with analytical models that incorporate the effects of mask-superlens interactions.
© 2012 OSA
In the year 2000 John Pendry proposed a ‘superlens’ that could focus both the propagating and evanescent components of light . Taking the form of a thin slab of silver sandwiched between dielectric layers, this superlens had the ability to reproduce images much smaller than the wavelength of light, effectively breaking the diffraction limit that constrains the performance of conventional optics. The performance of the superlens was most conveniently summarized by its modulation transfer function (MTF) in the spatial-frequency domain , which was reported by several sources according to different analytical models [3–6]. With time, questions on the accuracy of the analytical MTF arose due to the near-field nature of the superlens imaging process; in response, we proposed an alternative analytical model to take such near-field interactions into account .
In contrast to analytical characteristics, results from superlens experiments were typically reported in terms of the minimum grating half pitch (hpmin) that could be resolved [8, 9], with best performance of 30 nm half-pitch features exposed by 380 nm light reported recently . This posed a problem, as there was no way to directly compare observed results in the spatial domain with predicted performance in the spatial-frequency domain to determine which models can most adequately describe system performance. We address this issue here, presenting a method to experimentally determine the MTF of a silver-dielectric superlens and comparing this to various analytical and semi-analytical models in the case of an 80-nm thick silver superlens.
2. Prior work
Aizenberg et al.  described a method for capturing near-field intensity variations in photoresist. Using an elastomeric phase mask in contact with negative photoresist, they were able to estimate the irradiance distribution based on the depth of photoresist features after development. In a similar but independent vein, conformal masks were used in lens-less contact lithography experiments to image features with dimensions well below the diffraction limit [12, 13]. The emphasis of these latter works, however, was on the best hpmin that could be imaged, rather than on the relative intensities of different spatial frequency components incident on the photoresist. Superlens experiments have so far followed this approach, with hpmin being the most widely reported performance metric [8–10]. Although attempts have been made to translate spatial results to the spatial frequency domain using discrete Fourier transforms , the application of these techniques has so far been confined to simulated data.
In this paper we adopt and extend the approach taken by Aizenberg et al. , using both superlens-based and lens-less exposures to image the irradiance distribution below a photolithographic mask. We then convert these patterns to the spatial frequency domain, following a similar process to that described in . Finally, we compare the spectra produced by exposures taken with and without a superlens; the ratio of these two sets of data gives the superlens MTF. The significance of this result is that the MTF can be readily calculated by any one of a number of analytical or full-field simulation methods [3, 7, 14]; comparing equivalent experimental and calculated results thus opens up new possibilities for model validation and process quality measurement.
A three-step method was used to experimentally determine the MTFs of different superlenses. Firstly, superlenses were fabricated by depositing silver and dielectric layers on top of a flexible, photoresist-based substrate. Next, grating patterns were carefully transferred through the superlens and into the photoresist via contact lithography. Lastly, the patterns in the photoresist were characterized via atomic force microscopy (AFM), with the resulting data processed in MATLAB to extract the relevant MTFs. The details of these steps are outlined in the following subsections.
3.1 Sample and superlens preparation
Photoresist stacks were prepared on 30 mm wide, 3 mm thick square glass substrates. The substrates were covered by a 1.5 mm thick conformable poly(dimethyl siloxane) (PDMS) layer, which was included to absorb stresses introduced when contact was enforced between the mask, lens and photoresist stack . A 140 nm Clariant AZ BARLi II anti-reflective coating was then spun onto the PDMS layer, before the stack was completed by a 2.2 µm layer of Clariant AZ 1518 positive photoresist.
For superlens experiments ~40 nm of silver was evaporated onto the photoresist stack. The surface roughness of this metal layer was found to be between 0.5 nmRMS and 1.0 nmRMS, measured using AFM, which was a similar result to those reported in earlier superresolution experiments [8, 16]. Dual 20 nm poly(vinyl alcohol) (PVA) layers were included on either side of the silver, as shown in Fig. 1(a) . The water soluble nature of these dielectric layers allowed the superlens to be removed from the photoresist stack after exposure, so that the photoresist topography could be mapped following development. Neither the silver metal nor the PVA layers were included in control experiments, as shown in Fig. 1(b).
Photolithographic masks were fabricated by DC magnetron sputtering of 50 nm of tungsten onto 250 µm thick, 20 mm square glass slides. The tungsten was then covered by 140 nm of AZ BARLi II anti-reflective coating and 500 nm of AZ 1518 photoresist, before it was patterned with 1 µm period grating patterns via interference lithography. Subsequent plasma etching with a combination of O2 and SF6 gases transferred the patterns into the tungsten metal and removed the organic layers from the mask. For full details on this method, refer to .
3.2 Grating pattern exposure and development protocol
A tungsten mask patterned with a 1 µm period grating was brought into intimate contact with a superlens stack and was used to transfer features into the underlying photoresist. Light with a wavelength, λ0, of 365 nm was used to affect the transfer, with the exposure dose limited to 125 mJ/cm2 incident on the mask above the superlens. This controlled dose ensured that the photoresist was sensitive to variations in the near field of the superlens, which were only significant within λ0 / 10 (~35 nm) of the superlens exit plane .
After exposure, the silver and dielectric layers of the superlens were removed from the photoresist by dissolution in de-ionized water (DIW). Development of the photoresist followed for 15 s in a 4:3 solution of AZ 326 MIF developer and DIW. This weak development regime was intentionally chosen to limit the depth of the features in the photoresist, resulting in a near linear relationship between exposure dose and feature depth. After development the photoresist was imaged by atomic force microscopy (AFM).
As well as an exposure taken through a superlens, a similar impression of the mask was taken directly into a separate photoresist-covered sample, for later comparison (Fig. 1(b)). The superlens-free sample was captured by placing the mask in intimate contact with a layer of photoresist, spun over ARC and PDMS layers, as before. Since there was no superlens present to absorb light intensity between the mask and the photoresist, the exposure time was scaled down to give a 75 mJ/cm2 dose incident on the mask; this resulted in photoresist features that had a similar depth scale to those captured with a superlens. After exposure, development followed according to the same regime used for the superlens sample. The only difference between the treatments of the two samples was that the lens-free sample was not soaked in DIW after exposure, as there was no superlens structure obscuring the photoresist.
3.3 AFM measurement and data processing
After exposure and development, a Nanoscope Dimension 3100 AFM in contact mode was used to characterize the tungsten mask as well as individual photoresist samples, exposed either with or without a superlens. Height scans with dimensions of 512 × 512 pixels were captured over an area of 5 µm × 5 µm, yielding images such as those shown in Fig. 2 . Each image was then rotated about the z axis and cropped so that the feature edges were aligned and the entire image had only minimal variation along the y axis. Cropping was also used to exclude parts of the sample that had been damaged during the exposure and development steps; specifically, the mild agitation required to dissolve the superlens from the photoresist after exposure occasionally caused scratching across sections of the photoresist surface, as shown in Fig. 2(b).
Once the image was suitably cropped, the mean of the AFM height data was calculated in the y direction, yielding an average line profile for each AFM scan. A second rotation step, this time about the y axis, was then necessary to flatten the line profiles. This minimized feature elongation in the spatial frequency domain, which would otherwise cause the location of spectral features to be over-estimated. Flattened line profiles for the images shown in Fig. 2 are given in Fig. 3 . Once the two-dimensional line profiles were known, these data were converted to the spatial frequency domain via a fast Fourier transform (FFT), with examples of the resulting spectra shown in Fig. 4 .
As can be seen from Fig. 3, the average depth of the resist patterns was different for exposure completed with and without a superlens: for Fig. 3(a), where the photoresist was patterned with a superlens, the average line depth was ~7 nm, compared to ~20 nm in Fig. 3(b), which was fabricated without a superlens. This difference was present despite a higher exposure dose of 120 mJ/cm2 delivered to the superlens samples, compared to 75 mJ/cm2 for the non-superlens samples. To remove the effect of these varied average pattern depths from the subsequent spectral analysis, the calculated spectra were independently normalized to give unit depth at a nominal spatial frequency of 1 µm−1, as shown in Fig. 4. Thus the data in Fig. 4 depends only on the shape of the features in Fig. 3, rather than on the absolute depth of those features. As well as allowing the comparison of photoresist spectra exposed with and without a superlens, this normalization also allowed the spectrum of the mask profile, shown in Fig. 3(c), to be compared to both photoresist pattern spectra.
3.4 Transfer function reconstruction
Using the spatial frequency data shown in Fig. 4, an experimental superlens MTF was measured by comparing the ratio of two different spectra at various wavenumbers. Care was taken to ensure that only valid data were compared and models were used to recover the relative intensities of previously normalized spectra. The details of this process are described below.
The first step in the MTF calculation was to choose appropriate wavenumbers at which to perform comparisons, since periodic nulls in the spectra prevented the reconstruction of transmission coefficients for certain spatial frequencies. Owing to the shape of the mask grating, the patterns in the photoresist produced approximate square-wave spectra, with significant harmonics found mostly at odd multiples of the fundamental spatial frequency. For the case of a 1 µm period grating, such as the one used to produce Figs. 2 and 3, harmonics were nominally expected at 1 µm−1, 3 µm−1, 5 µm−1 etc. Large depth values were also found at some even wavenumbers, particularly when the duty cycle of the photoresist patterns deviated from the ideal value of 50%; however, these features were not uniformly present across samples and were thus not included in the analysis presented here. We note that using masks with exaggerated duty cycles of the order of 75% or so would allow the inclusion of coefficients at even multiples of the fundamental spatial frequency in this analysis and would potentially double the number of transmission coefficients that could be recovered.
Once the analysis was confined to an appropriate subset of spatial frequencies, the locations of harmonics in individual spectra had to be identified. These locations were not exact, due to rounding in the FFT algorithm and drift in the AFM, which varied between scans. For example, the first three harmonics of the photoresist pattern shown in Fig. 3(a) would be expected at 1 µm−1, 3 µm−1, and 5 µm−1; however, the corresponding peaks in Fig. 4(a) occur at 1.045 µm−1, 3.135 µm−1, and 5.224 µm−1, respectively. Similar offsets were found in spectra from other samples, with typical deviations for 5 µm × 5 µm scans amounting to between 4% and 7% of their nominal values. For this reason, harmonic coefficients were identified manually, based partly on their expected location and partly on the location of preceding harmonics in the spectrum.
The third part of the MTF measurement process involved calculating MTF coefficients from the isolated harmonic coefficients. Given an input spectrum, Ψin(n), and an output spectrum, Ψout(n), containing n coefficients each, the transfer function relating the two spectra, H(n), is given by Eq. (1):
Finally, the reconstructed MTF had to be scaled to correct for the magnitude normalization that occurred during the calculation of the depth spectra. Calibration via direct measurement of the DC transmission coefficient of the superlens was considered, with readings taken using a Süss MicroTec Model 1000 intensity meter placed above and below an isolated superlens. The rationale behind this approach was that the MTF coefficients reconstructed at 1 µm−1 were of a sufficiently low spatial frequency to approximate the DC attenuation of the superlens. This approach was abandoned, however, when analytical models identified significant variation in superlens transmission, even for fractional wavenumbers between 0 µm−1 and 1 µm−1 .
Instead, the analytical modified transfer matrix method (M-TMM) [7, 14] was used to predict the transmission coefficient of the superlens at 1 µm−1. Based on a model of a 20|40|20 nm PVA|Ag|PVA planar superlens in contact with a 50 nm thick W mask, the reconstructed MTF was scaled to have a transmission coefficient of 0.0997 × at 1 µm−1. This wavenumber was chosen as it corresponded to the fundamental spatial frequency in the mask pattern and thus had the least experimental uncertainty associated with it, compared to all higher wavenumbers. The MTFs that resulted from this scaling, as well as the particulars of the calibration model, are discussed in Sec. 4.
Initially, MTF coefficients were calculating by comparing the spectra of photoresist samples exposed with a superlens directly to the spectra of the tungsten mask, as shown in Fig. 5(a) . Unfortunately, the resulting MTF was a combination of the superlens MTF as well as the photoresist response function, which decayed exponentially with decreasing feature size. By comparing the superlens photoresist spectrum to the spectrum of a photoresist sample exposed with the same mask but without a superlens, the photoresist response function could be removed from the equation and the superlens MTF was revealed in an undiminished form, as shown in Fig. 5(b).
This experimental MTF was in good agreement with an analytical MTF based on M-TMM [7, 14], shown as a solid line in Fig. 5. Significantly, agreement was better than for the MTF calculated via the unmodified transfer matrix method (TMM) [2–6], shown in Fig. 5 as a dashed line. The difference between the two techniques is that only M-TMM accounts for mask-superlens interactions, which our experimental data show to be significant. Interestingly, the good agreement between M-TMM, which assumes a mask duty cycle of 100%, and the experimental data, which were captured with a mask duty cycle of ~50%, suggests that the mask-lens interactions that affect the MTF do not vary greatly with duty cycle. Instead, it is the corners of the mask features that have the greatest effect on the MTF, since this is where evanescent enhancement is at its strongest.
The parameters for the analytical calculations shown in Fig. 5 were a 20|40|20 nm PVA|Ag|PVA planar superlens, exposed to 365 nm wavelength light. In the case of the M-TMM method, a 50 nm slab of tungsten was also included at the entrance plane of the superlens. The electromagnetic properties for these materials are shown in Table 1 . Analytical transmission coefficients, t, were calculated from the entry plane of the superlens to the exit plane, corresponding to the start of the first PVA layer and the end of the last. Reflection coefficients, r, were calculated for the interface between the first layer of PVA and the tungsten slab and these were incorporated into the M-TMM calculation. We note here that the analytical curves shown in Fig. 5 are MTFs, based on |t|2 and |r|2, rather than amplitude transfer functions, which are functions of |t| and |r|. Further details on the implementation of this method are given in .
4.1 Experimental uncertainties
There are two significant sources of uncertainty that affect the reconstructed MTFs: firstly, mechanical drift of the AFM head during measurements causes compression or elongation in the spatial frequency domain. This effect increases with wavenumber, making the location of high wavenumber coefficients less certain than the location of coefficients at lower wavenumbers. For the AFM scans shown in Fig. 2, drift affected coefficient locations by up to 0.05 µm−1/µm−1.
The second source of uncertainty is rounding error in the location of spectral peaks, due to the finite resolution of the FFT used during data processing. This phenomenon affects all wavenumbers equally, adding an uncertainty of up to ± 0.08 µm−1, depending on the exact number of data points used in the transform. The net effect of these two types of uncertainty was that wavenumber coefficients differed from their expected locations by between 0.9% and 8%, with the greatest uncertainty of ± 0.83 µm−1 occurring for the data shown in Fig. 4(b) at a nominal wavenumber of 13 µm−1. For MTFs, which were calculated as the ratio of two spectra, the horizontal uncertainties were estimated by taking the larger of the two uncertainties from the coefficients’ constituent spectra. These data are shown as horizontal lines in Fig. 5.
As can be seen from Fig. 5, the M-TMM and experimental MTFs have good agreement at nominal wavenumbers up to 5 µm−1. In particular, the coefficients at 5 µm−1 are beyond the diffraction limit of 365 nm wavelength light, which indicates that this analysis is useful in both the propagating and evanescent parts of the spectrum. In contrast, coefficients between 5 µm−1 and 13 µm−1 are attenuated relative to the model. We suggest that this is due to the non-zero roughness of the silver and dielectric layers, which is likely to retard the plasmon resonance and subsequent evanescent enhancement . Given this attenuation due to roughness, the rate of decay of transmission for coefficients between 9 µm−1 and 13 µm−1 is consistent with that predicted by the model, once the lower peak transmission is taken into account. This suggests that the material property parameters used in the model and given in Table 1 are valid.
Finally, coefficients above 15 µm−1 do not match the model at all; this is because of the finite noise floor in the superlens exposure, which prevents the accurate measurement of spectral components below a certain threshold. In practical terms, the magnitude of the measured coefficient at 15 µm−1 in Fig. 4(a) would have to be a factor of 10 × smaller than the equivalent coefficient in Fig. 4(b) in order for the experimental data to match the M-TMM model. At 17 µm−1, a net difference of two orders of magnitude would be necessary to match the model; at 19 µm−1, the difference would need to be three orders of magnitude. Instead, the amplitudes of the measured coefficients are consistently of the same order of magnitude at similar wavenumbers, indicating that the noise floor is indeed limiting the validity of this analysis beyond 13 µm−1.
Although increasing the exposure dose and development time could lower the noise floor, this would decrease the sensitivity of the photoresist to the evanescent part of the spatial frequency spectrum and would increase distortion in the z direction. Alternatively, an improved photoresist could be used for these experiments; however, the triple requirements of i-line sensitivity, nanometer scale resolution and near linear dose response mean that such a resist is unlikely to become available in the short term.
For completeness, the photoresist response function was measured by comparing the spectrum of a photoresist sample exposed without a superlens to the spectrum of the mask itself, using the process described in Secs. 3.3 and 3.4. These data are normalized to give unit transmission at 1 µm−1 and are shown in Fig. 6 . The strong attenuation of the photoresist response function for high wavenumbers relative to a spatial frequency of 1 µm−1 indicates the extent to which the photoresist and its corresponding exposure and development conditions are limiting the efficacy of this analysis.
The method we have described comprises a novel approach to the characterization of near-field imaging structures, such as silver superlenses. We have shown that an extensive sampled magnitude transfer function can be reconstructed from only two exposures, provided photoresist sensitivity is maintained by limiting exposure and development conditions. These results are in good agreement with analytical data and validate the M-TMM model, which was designed specifically to account for the near-field interactions that occur between a lithographic mask and a superlens. Furthermore, our data provide insights into the specific limitations of practical superlenses, which are not apparent in the ideal case. Specifically, our results suggest that superlens performance is limited more by the characteristics of the photoresist used than by evanescent decay or superlens surface roughness. This improved experimental method and analysis technique pave the way for better modeling of superlenses and makes possible reconciliations with real world data that were not feasible before.
This work was supported by the MacDiarmid Institute for Advanced Materials and Nanotechnology. The authors wish to thank Helen Devereux, Gary Turner and David Healy for technical support.
References and links
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