1. Introduction

Many fields such as plasmonic [1], photonic crystals [2], and metamaterials [3] require the creation of periodic organisations of features such that coupling phenomena occurring between nanostructures can be controlled by changing the period of the array. This level of control is usually achieved through a top-down lithography process.

Deep-UV immersion lithography systems have been commonly used to pattern feature sizes down to 40 nm [4] with the future plan that extreme-UV sources will reduce the feature size further [5]. However, the high cost and maintenance of these techniques make them unavailable for research. Electron beam [6] and Nanoimprint Lithography (NIL) [7] are more affordable alternative techniques often used in research for the fabrication of nanostructures. The first technique allows feature sizes down to $\sim$ 10-100 nm, but the processing time for large samples is problematic at the wafer scale. NIL achieves similar feature sizes, but the technique is limited by the mould’s lifetime and the difficulty of achieving certain patterns such as rings and high or low pattern densities [8]. For feature sizes $\sim$ 100-300 nm and large-area patterning, no practical, affordable solution is widespread.

Displacement Talbot Lithography (DTL) is a lower-cost technique to pattern wafer-scale samples with sub-micron periodic structures [9]. 125-300 nm features and 75 nm features have been achieved with near and deep-UV sources, respectively [9,10]. The optimal mask configurations to reduce the feature size have been found computationally [11]. DTL has demonstrated its versatility through the creation of metamaterials [3], III-V semiconductor photonic materials in the form of core-shell structures [12,13], faceted AlN nanostructures [14] and nanotube cavities [15], the formation of neuronal networks [16] and creation of nanoimprint masters [17].

DTL is based on the Talbot effect, whereby a coherent light source illuminates a periodic mask such that the periodic diffracted beams interfere and create a periodic 3D interference pattern. A repeated self-imaging of the mask is obtained with a repetition distance called the Talbot length. During Talbot lithography (TL), a sample is placed at a specific interference plane to either replicate the mask or even obtain more complex structures between the self-images [18,19]. With TL, the cost of the alignment system becomes an issue for wafer-scale patterning. Instead, with DTL, by introducing a displacement of the sample equal to a multiple of the Talbot length during the illumination, the light distribution becomes independent of the gap. Therefore, the process is insensitive to the parallelism of the sample with the mask and the process repeatability and robustness are improved. Achromatic Talbot Lithography, a technique closely related to DTL, recreates this infinite depth-of-field by using a broad laser source and a large gap between the mask and sample [20].

Compared to conventional photo-lithography, DTL is depth-of-field free and tolerant to sample roughness [9]. However, for masks with pitches larger than two times the laser wavelength, the uniformity of the 3D interference pattern deteriorates since the introduction of higher diffraction orders makes the interference much more complex [21]. More specifically, spatial periodicities that are smaller than the main Talbot length appear. Non-paraxial modelling of the Talbot effect is then needed to understand the impact of this occurrence fully [22]. In practise, these spatial periodicities impact the contrast, feature size, and final aerial image of the DTL process.

More generally, in lithography, angled illuminations are commonly used to remove some diffraction orders [23]. In the case of Talbot lithography, a recent paper demonstrated the possibility of doing this to create line illumination, and thus, high-aspect-ratio structures [24]. However, the angled illumination leads to angled patterns, limiting the possibilities for pattern transfer. An alternative, more pragmatic, solution is to increase the integration distance [25] so that by diminishing the impact of the spatial periodicities the patterning quality is improved.

In this paper, the impact of the spatial periodicities created by the higher diffraction orders is studied more fundamentally as a function of the mask topology to control the Talbot effect. The mask conditions leading to enhanced feature size, contrast and uniformity for lithography purposes are reported.

2. Modelling method and framework

2.1 Modelling framework

The simulation of the Talbot carpet has been performed with MATLAB and assumes that a homogeneous, coherent, incident light source illuminates the mask. The laser wavelength is chosen to be 375 nm, corresponding to the system at the University of Bath (PhableR 100, EULITHA). Still, the results and application presented in this paper are transferable to other wavelengths.

This study is focused on one dimensional masks: gratings. With such periodic configuration, a shifted replication of the mask appears at half the Talbot length. Consequently, the periodicity obtained with DTL [9] as well as with Achromatic Talbot Lithography [26] is divided by two. Such a reduction is not obtained for TL.

Experimentally, the gap used during the DTL process is typically larger than 50 $\mu$m. At this distance, light propagates in the Fourier regime. For this reason, the Talbot effect can be modelled with a Fourier propagation technique [27]. It has been demonstrated that the Talbot effects obtained for the TE and TM modes propagating after the mask are different when the pitch is smaller than three times the wavelength [28]. Indeed, plasmonic and wall interactions at the mask level lead to modifications of the light distribution [29,30] and the final transmission. Structuring of the metal layer can even be performed to further enhance the light transmission through the mask beyond the actual opening size limitation [31]. Still, the possibility of modulating the Talbot carpet shape has been demonstrated by controlling the interference between the TE and TM polarisations [32].

Nevertheless, a polarised light source parallel to the aperture plane is usually applied to reduce the impact of the polarisation. It has been demonstrated that spatially uniform incident light inside a circular aperture propagates in the near-field, far-field and Fraunhofer region the same way as if it is polarised. It then can be modelled as a scalar field [33]. This scalar assumption has been applied even if the aperture type is different and plasmonic effects could impact the light uniformity within the opening. Therefore, this study presents a first step for demonstrating the control of the Talbot carpet shape through mask topology design.

Under the scalar approximation, the light distribution over any plane parallel to the mask can be calculated. Mask openings can be coded as 1 or -1 as a function of the light phase, with the absorbent part of the mask coded as 0 [22,27,34]. This type of modelling agrees with many other modelling studies of the Talbot effect [9,18] and has been validated experimentally for 2D masks [11,15,35].

The spectra of the spatial periodicities appearing inside the Talbot carpet have been calculated with a Fourier spectrum analysis on 1000 Talbot lengths to ensure all spatial periodicities are integrated. The Fourier transform is performed on the light intensity distributed along the line perpendicular to the middle of the 0-phase opening. The resolution per Talbot length is 100 pixels. The DTL modelling presented in the discussion section has been performed on 60 Talbot lengths with a starting gap of 50 $\mu$m. The resolution of the width of each mask is 400 pixels. After the FFT, the x-axis unit is in 1/$\mu$m. By inverting the values of the array generated, the actual spatial spectrum in $\mu$m is generated. The reading of the spectra is then eased since the largest spatial periodicity corresponds to the Talbot length.

2.2 Talbot length formula of the spatial periodicity

A periodic mask illuminated by a monochromatic coherent light source generates a periodic 3D interference pattern known as the Talbot effect. The number of diffraction orders existing inside the interference carpet increases when the pitch distance increases. The impact of the higher diffraction orders on the spatial periodicity inside the interference distribution is often neglected for pitches exceeding ten times the wavelength; the paraxial approximation. However, this assumption cannot be kept for pitches closer to the wavelength [34]. For this reason, the origin of the interference needs to be explored to understand the impact of the diffraction orders on the Talbot effect.

The largest spatial periodicity inside the carpet is the Talbot length. This distance corresponds to the first phase matching between the zeroth and projected first diffraction order along the mask’s perpendicular axis. It is formulated by [36]:

A generalised phase cycling between any two diffraction orders can be expressed through the same method [21,37]. For example, between diffraction orders $m$ and $n$, Eq. (1) becomes:

This leads to the generalised Talbot length formula:

To avoid any negative spatial periods with no physical meaning, the condition $n>m$ is applied. The number of diffraction orders existing inside the Talbot carpet is dependent on the pitch of the mask, $d$, and the wavelength of the system, $\lambda$. Increasing the pitch on the mask leads to new spatial periodicities inside the Talbot carpet since new diffraction orders are introduced. This phenomenon is illustrated in Fig. 1.

 

Fig. 1. Value of the Talbot length spatial periodicities and appearance as a function of the ratio of pitch to wavelength. A wavelength of 375 nm is used for the calculation of the spatial periodicities.

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Below a pitch of two wavelengths on the mask, only the first and zeroth diffraction orders are present. As a result, only $Z_{0,1}$ exists. For pitches between two and three wavelengths, for which the second diffraction order appears, the spatial periodicities $Z_{0,2}$ and $Z_{1,2}$ are added. In the same way, the appearance of the third diffraction-order for pitches larger than three-wavelengths leads to three new spatial periodicities: $Z_{0,3}$, $Z_{1,3}$ and $Z_{2,3}$.

The plane wave’s intensity of any diffraction order evolves as a function of the mask topology. By changing the size of the mask’s opening, it can even be cancelled [38]. The intensity of a diffraction order $n$ is expressed in the Fourier-regime by:

3. Result

Three periodicities have been studied: 600 nm, 800 nm, and 1200 nm, such that a new diffraction order is introduced for each increase in pitch. The impacts of the opening width and the type of mask, amplitude or phase, are studied. For the larger periodicity listed, an additional study for a mask combining phase and amplitude properties is presented.

The light distribution has been normalised to the more intense part of the carpet to help the figure’s reading. This normalisation often occurs at the mask plane, and thus, some carpets can look faint as in Fig. 2(a).

 

Fig. 2. Simulation of Talbot effects of 600-nm-period masks, with a) 200, b) 300, and c) 400 nm size width of the main opening. The bottom left corresponds to amplitude masks, and the top right to phase masks. The carpets have been plotted for four Talbot lengths $Z_{0,1}$. The opening and 0-phase are represented by the white rectangle, and the filled part the absorbent or $\pi$-phase section.

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3.1 600 nm pitch mask

600 nm is smaller than twice the wavelength so that only the zeroth and first diffraction orders are present. Figure 2 shows the Talbot effect generated for different mask designs at this pitch. The bottom left of each sub-figure corresponds to the case of an amplitude mask, and the top-right part to a phase mask. 200, 300 and 400 nm openings with an amplitude of 1 have been used for the 0-degree phase on the mask. Amplitude masks are opaque on the remaining part of the mask, and phase masks have a -1 amplitude to represent the phase inversion. The schematics above each carpet represent the transmission distribution.

Figure 2 shows that for amplitude masks, the expansion of the opening width leads to a wider self-replication shape along the x-axis. If $I_0$ and $I_1$, the intensity of both diffraction orders, are not equal, then the interference between them is not perfect. Equation (5) reveals that the larger is the opening, the smaller $I_1$ becomes. Hence, the widening of the features. On the other hand, thinner self-images and so potentially smaller transferred features are obtained if $I_0 = I_1$. This condition is obtained for $\beta$ tending to 0; i.e. small opening widths. This opening criterion for an enhanced feature size was also reported for 2-D mask organisations [11].

The phase mask response to the change of opening width is, however, different. The Talbot carpets obtained in Fig. 2(a) and Fig. 2(c) are equivalent since the phase masks are inversions of each other. In contrast, Fig. 2(b) has no Talbot effect visible inside the carpet. Since each phase of the mask has the same width, the zeroth-order generated by each area has the same intensity. Thus, a cancellation of the zeroth diffraction order is obtained since both planar waves are $\pi$-phase shifted. This type of illumination has the potential for creating high-aspect-ratio structures [39–41].

Similarly, the same cancellation phenomenon is obtainable for masks with pitches larger than two wavelengths. A specific diffraction order can be cancelled through Eq. (5) by using a specific opening width. In particular, by using an opening equal to half of the pitch, all even diffraction orders are removed to leave only odd diffraction orders.

To show the control of the diffraction order presence and so, the modification of the Talbot carpet shape, masks with larger pitches need to be studied. The following section presents an investigation of masks where the second diffraction order is added.

3.2 800 nm pitch mask

When the periodicity of the mask is between two and three times the wavelength, $Z_{0,1}$, $Z_{0,2}$ and $Z_{1,2}$ can exist inside the Talbot carpet. Figure 3 presents the Talbot carpet obtained for amplitude and phase masks with openings of 200, 300, 400, and 500 nm in the same way as Fig. 2.

 

Fig. 3. Simulation of Talbot effects of 800-nm-period masks, with a) 200, b) 300, c) 400, and d) 500 nm size width of the main opening. The bottom left corresponds to an amplitude mask, and the top right to a phase mask. The carpets have been plotted on four Talbot lengths, $Z_{0,1}$. The opening and 0-phase are represented by the white rectangle, and the filled part the absorbent or $\pi$-phase section. The spatial Fourier analysis of the Talbot effect for the c) 400 nm amplitude and d) 500 nm phase masks are represented alongside the corresponding Talbot carpets.

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The opening width impacts, even more, the Talbot carpet shapes. For instance, Fig. 3(a) shows that the repeatability of the fundamental self-replication pattern is affected across the Talbot carpet. Indeed, for both phase and amplitude masks, the replica shapes along the z-axis are not identical but are modulated by a finer structuring.

When the opening width is equal to 400 nm, or half of the pitch, the second diffraction order is cancelled so that only $Z_{0,1}$ remains: with the amplitude mask, a simpler Talbot effect is obtained, similar to that from a 600 nm mask; with the phase mask, a line illumination with an infinite depth of field is obtained since the zeroth diffraction order has been cancelled.

The Talbot carpets for the different opening widths and mask types vary due to the different intensities of the different diffraction orders. For example, the Talbot carpets from phase masks with 300 and 500 nm openings (equivalent masks) are dominated by the smaller spatial periodicity $Z_{0,2}$ and lead to a narrower feature size for lithography compared with carpets without this periodicity.

3.3 1200 nm pitch mask

The study has been extended to masks where the third diffraction order is added: 1200 nm such that the spatial periodicities $Z_{0,1}$, $Z_{0,2}$, $Z_{0,3}$, $Z_{1,2}$, $Z_{1,3}$ and $Z_{2,3}$ can exist. Figure 4 presents the Talbot carpet obtained for 200, 300, 400, 500 and 600 nm openings.

 

Fig. 4. Simulation of Talbot effects of 1200-nm-period masks, with a) 200, b) 300, c) 400, d) 500 nm and e) 600 nm size width of the main opening. The bottom left corresponds to amplitude masks, and the top right to phase masks. The carpets have been plotted on four Talbot lengths $Z_{0,1}$. The opening and 0-phase are represented by the white rectangle, and the filled part the absorbent or $\pi$-phase section. f) Spatial Fourier analysis of the Talbot effect for the 200 nm opening amplitude mask.

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Figure 4(f) presents the Fourier spectrum obtained for the 200 nm opening where all the spatial periodicities are present. The spatial periodicities involving the third diffraction order are the shorter ones in the spectrum.

The Talbot carpets obtained for an opening of 200 nm are different for the amplitude and phase mask. Figure 4(a) shows the width of the nodes obtained with the amplitude mask is smaller than with the phase mask. With a 300 nm opening width, no clear distinction between both mask types is visible. Then, for the 400 nm opening mask, a diffraction order cancellation occurs, such that the third diffraction order is removed and only $Z_{0,1}$, $Z_{0,2}$ and $Z_{1,2}$ are present in the carpets. Thus, a simplified Talbot effect, similar to the 800 nm case, is obtained.

Increasing the opening width to 500 nm openings (Fig. 4(d)) leads to the presence of even stronger tiny wings around the self-replications that are strongly linked to the addition of the third diffraction order. Equation (5) shows that the intensities of the spatial periodicities of the third diffraction order have a local maximum around openings of 570 nm. Therefore, the presence of this secondary interference is stronger for 500-600 nm openings.

For the 600 nm opening design in Fig. 4(e), the second diffraction order is cancelled so that only $Z_{0,3}$, $Z_{0,3}$ and $Z_{1,3}$ spatial periodicities are present. For the amplitude mask, it leads to a Talbot carpet with wide self-replications that are thinly subdivided, resulting from both $Z_{0,3}$ and $Z_{1,3}$. In contrast, only $Z_{1,3}$ remains for the phase mask since the zeroth-diffraction order has also been cancelled. Therefore, a new, simplified, periodic Talbot effect is obtained. The spatial periodicity appearing, $Z_{1,3}$, is smaller than $Z_{0,1}$ so that a new Talbot length should be considered in TL and DTL. Since the spatial periodicity value is now smaller, a shorter integration can be applied for DTL.

3.4 Combination of amplitude and phase mask

We have highlighted that, on the one hand, the feature size is enhanced when the opening on an amplitude mask is small, whilst on the other, larger openings lead to simpler Talbot carpets since some diffraction orders can be cancelled. Therefore, using solely amplitude or phase masks alone, no combination of both enhanced feature size and homogenised Talbot carpet can be obtained for wide pitches. Instead, this section presents a new mask configuration combining amplitude and phase characteristics to achieve such outcomes.

The schematics above the carpets in Fig. 5 presents the new mask’s designs. Alternating phases of equal width are placed at a half-pitch distance, $d/2$. If the openings’ width is smaller than half the pitch of the mask, then an absorbing opaque layer, represented by the black layer on the schematic, is implemented between the holes.

 

Fig. 5. Simulation of Talbot effects of combined amplitude and phase masks. The Talbot effects for 1200-nm-period masks with a) 200, b) 400 and c) 600 nm openings are presented. The carpets have been plotted on four Talbot lengths $Z_{0,1}$. d) represents the integrated intensity profile for a DTL process performed for each mask over 60 Talbot lengths.

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With such a design, the zeroth-diffraction order planar waves generated by each phase destructively interfere for any opening width. For this same reason, any even diffraction order is also cancelled. Consequently, for a mask with a pitch of 1200 nm, only $Z_{1,3}$ can exist. Figure 5 presents the Talbot carpets and integrated DTL intensity profiles obtained for openings of 200, 400 and 600 nm, where the 600 nm results correspond to the same phase mask design presented in Fig. 4(e), i.e. with no absorbing component. $Z_{1,3}$ is visible in Fig. 5(a) and Fig. 5(c) for 200 and 600 nm openings, respectively, since no cancellation of the third diffraction order occurs. In contrast, with 400 nm openings in Fig. 5(b), a cancellation occurs, creating a line-illumination with infinite depth of field.

Figure 5(d) presents the integrated intensity profile after DTL for each design. Thanks to the mask’s phase configuration, the normalised light distribution evolves from 1 to 0, assuring a high selectivity for the transfer into photo-resist, and so, an enhanced process contrast. As a function of the opening width, the intensity profile shape changes. The smallest feature size, or the sharper curve around the maxima, is obtained when the opening width is the smallest. Such occurrence is visible in the Talbot carpet, Fig. 5(a), since sharper self-replicas are obtained. Furthermore, the light distribution is symmetrical, meaning that the patterning of small and large openings within the same photo-resist should be possible.

Inside the Talbot carpet for the 200 nm mask configuration, local bright secondary interference occurs, leading to shoulders on the red curve in Fig. 5(d). These would lead to an imprecise process if an exposure dose corresponding to this region were applied.

The combination of phase and amplitude masks has allowed the combination of an improved process contrast in addition to small feature size. In the next section, we discuss the results obtained and how one might choose a mask configuration for a lithography process depending on the application targeted.

4. Discussion

Whilst the feature size achievable is often the main focus, flexibility in the feature shape and size achievable can also be targeted in a lithography process. One of the advantages of DTL and TL is their capability, with a single mask, to create a range of openings by changing the exposure dose applied or the gap distance used. By improving the contrast of the light distribution, the feature range achievable can then be increased without any complex mask design. Phase masks with equal phase width should be favoured to achieve this goal with DTL. Indeed, our results showed an enhanced contrast in addition to spatially wider self-replications. Due to these two elements, the range of features achievable reaches its maximum. In contrast, this phase width ratio should be avoided for TL since the complexity of the Talbot effect reduces. This simplification results in a reduction of the fractional plane diversity between self-images; thus, pattern transferable.

In contrast, if the feature size is the most important aspect, the best choice for the mask design is not as simple. For both TL and DTL, having small openings leads to smaller feature size. However, TL becomes strongly dependent on the gap when additional diffraction orders are introduced; a study of the gap choice for the patterning is necessary to enhance the feature size further. DTL, however, removes this dependence by integrating over the Talbot carpet. Figure 6 presents the intensity profiles obtained after the DTL process corresponding to the smallest feature size that can be achieved for the 600, 800, and 1200 nm mask designs. For 600 nm pitch, Fig. 6(a) proves that the smallest feature size is obtained with the phase mask once contrast is also considered. When the pitch is increased to 800 nm, Fig. 6(b) shows that the light distribution profiles for 200 nm openings for amplitude and phase masks overlay. The feature size can be further improved by using small openings, but the background intensity (minimum on the profile) remains constant for amplitude masks and increases with phase masks. This is due to the zeroth diffraction order of the largest phase saturating the illumination. Therefore, amplitude masks should be favoured for this pitch range. Finally, Fig. 6(c) presents the best curve for amplitude, phase, and combination masks for the 1200 nm pitch. An opening width of 200 nm leads, for any mask, to the smallest feature size, but the combination mask shows a better result due to its greater contrast. Nevertheless, amplitude mask would be favoured since they are easier to manufacture.

 

Fig. 6. DTL intensity profiles chosen for the smallest feature size. a) 200 nm opening amplitude and 300 nm opening phase 600 nm pitch mask, b) 200 nm opening amplitude and phase 800 nm pitch mask, and c) 200 nm opening amplitude, phase and combined 1200 nm pitch mask. The DTL calculations have been performed on 60 Talbot lengths.

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5. Conclusion

The complex periodicity inside the Talbot effect is a phenomenon that has been reported experimentally and computationally. It is caused by the smallest spatial periodicities created by the interference of the highest diffraction order that disturb the largest periodicity inside the Talbot effect and, in turn, reduce the uniformity of the interference carpet. In this paper, we show, via computer simulations, how changing the mask configuration can modify the contributions of different diffraction orders in order to balance the Talbot effect uniformity with the feature size and contrast that is achievable through a lithography process. The best mask designs are reported for enhancing the smallest feature size and the versatility of features patternable with both Talbot Lithography and Displacement Talbot Lithography; valuable information for designing masks within research and industry.