3.1

Basic model for straylight simulation

For simulating the light propagation and eventually the intensity distribution in the half-space of a transmission grating we used the rigorous coupled wave analysis (RCWA).⁷ This algorithm numerically calculates the light propagation of an electromagnetic plane wave incident upon a periodically structured surface with period Λ and in particular the diffraction efficiency of the propagating diffraction orders. E.g., the ideal SFT1-grating (period p = 667 nm, incidence angle AOI = 23.8° in air and wavelength λ = 633 nm) possesses only 2 diffraction orders in transmission half space.

Scattered light is now caused by large scale variations of the ideal grating structure with typical length scales P ≫ Λ. In order to simulate the intensity distribution of the diffusely scattered light we still use RCWA by introducing a super-lattice with period P = N_p · Λ. Such a compound of many single periods Λ possesses additional diffraction orders. For example, in case of the SFT1-grating there are 8 propagating orders for N_p = 4.

The effect of increasing P is depicted in Fig. 3a. More and more diffraction orders appear and in the limit of N → ∞ we end up with a quasi-continuous scattering background. Here, we will denote the additional diffraction orders as straylight orders (SO). In the case of an ideal undisturbed grating, the SOs have an intensity of zero and thus give the same result as N_p = 1. However, a disturbance of the ideal grating geometry results in a certain amount of energy in every SO. The diffraction efficiency η_m(θ_m) is calculated by the RCWA algorithm and the angle resolved scattering⁶ (ARS) can be estimated by

Figure 3.

RCWA simulation approach for simulating straylight of disturbed gratings. A super lattice with extent P = N_p · Λ is introduced, which allows to consider gratings with local geometrical distrurbances of their mikrostructure (a) and illustration of the stochastic (b) and deterministic (c) segment alignment error (“stitching error”) occuring in sequential writing processes.

The corresponding scattering angle θ_m is calculated using the grating equation

with m = m_min,…, m_max enumerating the propagating straylight and diffraction orders. More details of this model and the extend to 2D-disturbances can be found in.⁸

3.2

Simulating ghosts

In case of SFT1 and also SFT2, we apply electron beam lithography and in particular the ”variable-shaped-beam” approach (VSB) for defining the microstructure. Within this method, the broadened electron beam is shaped by means of two rectangular apertures (this shaped beam is called a ”Shot” with adjustable dimensions of maximum 2.5 μm × 2.5 μm). The precise position of the Shot on the sample is controlled by different electromagnetic deflection systems in the electron column. Additionally, the substrate to be exposed is mounted on a substrate stage moving along x- and y-direction. The different positioning systems generate grating sub-segments with size P_seg ≫ Λ, which are stitched together to the final full size grating. An imperfect alignment of these segments causes the super-periods and eventually the grating ghosts.

Within this work, we simulate the effects of, first, a stochastic alignment error (AE) σ_seg of the sub-segments and, second, a deterministic alignment error ΔP_seg resulting in a gap (ΔP_seg > 0) or overlay (ΔP_seg < 0) between adjacent sub-segments. The effect of this error is calculated for a sub-segment size of P_seg ≈ 35 μm as it occurs in the used e-beam writer. Figure 4 shows the results of the simulation.

Figure 4.

Effect of a deterministic and systematic positioning/alignment error occuring in sequential fabrication technologies (such as EBL) for the SFT1 and SFT2 grating type. The sub-segment size is P_seg = 35 μm ≈ 52 · Λ_SFT1 ≈ 17 · Λ_SFT2 and the alignment errors have the strength σ = 5 nm and ΔP_seg = 5 nm.

The main findings from the simulation are the following:

3.3

Simulating background

The fabrication of the SFT1- and SFT2-grating by EBL also results in slight local errors of the Shot positioning and Shot format, which results in a locally varying grating period and groove width. The straylight simulation can account for these fabrication error by introducing the parameter σ_p describing the Shot positioning error and σ_b describing the stochastic error of the groove width. In the following, the effect of these errors onto the straylight spectrum is presented.

3.3.1

Binary high resolution grating

It is commonly known that binary gratings (with periods in the range of the wavelength) illuminated in Littrow mount show very high diffraction efficiencies in the –1^st diffraction order for various combinations of grating depth d and dutycycle .^{9, 10} E.g., For TE-polarized light the dependency η₋₁(d, FF) of the investigated SFT1-grating (p = 667nm, Λ = 633nm, θ_i = 28.3°) is shown in Fig. 5a. The figure reveals that there are several possible grating designs that allow diffraction efficiencies of more than 90%. The presented straylight simulation method allows for investigating the straylight performance depending on the grating geometry and, hence, to consider straylight specifications already in the grating design process. For this purpose, the scattering spectra for two different possible grating geometries with maximum η_–1 and specified Shot inacurracies defined by σ_p = 5nm, and σ_b = 5nm are evaluated.

Figure 5.

(a) Efficiency of the –1^st DO of the SFTl-grating dependent on the grating depth d and the dutycycle FF (for λ = 633nm, AOI = 23.8° (in air) and TE-polarization). (b) Straylight simulations for very different FF and constant d with maximum η₋₁ (marks in (a)).

Figure 5b shows the scattered light distribution of gratings with different dutycycle FF = 0.2 and FF = 0.53, but constant depth d = 1220 nm (marks in Fig. 5a). The corresponding straylight spectra (Fig. 5c) show a very different angular distribution along dispersion plane. Whereas the curve for FF = 0.2 possesses a very high straylight level in particular between the –1^st and 0^th DO (θ = [−23.8°,…, 23.8°]) and decreases by two orders of magnitude for higher scattering angles, the grating with FF = 0.53 shows basically the reverse behaviour. There, we find a weak straylight level in the angular range between the main diffraction orders and especially around the 0^th DO.

This investigation shows, that the grating geometry and the grating design, respectively, have an influence onto the straylight distribution. With the simple presented simulation method it is possible to evaluate the straylight generated by a certain grating structure already during design process.

3.3.2

Echelle grating

The same simulation principle can be applied for the presented SFT2-grating. The grating geometry of this grating type is predetermined by the grating period Λ and the crystallographic structure of the silicon substrate. Therefore, there are no degrees of freedom in the grating geometry as there are for the SFT1 grating (depth and dutycycle as investigated in Sec. 3.3.1). However, a straylight simulation for the fixed geometry can be compared to a straylight measurement in order to evaluate the error of the line positioning σ_p and the error of the line width σ_b. Such an investigation is shown in Figure 6.

Figure 6.

Straylight simulation along the dispersion plane (i.e. φ = 0°) of the SFT2-grating (echelle type grating in reflexion with Λ = 2070 nm, AOI = 55°, λ = 633 nm according to Fig. 1b) disturbed by a stochastic line position error σ_p = 0.4 nm and line width error σ_b = 0.4 nm (red curve) compared with a straylight measurement of the very same test grating as presented in Fig. 2 (black curve).

Measurement and simulation fit very well: even the local minima within the continuous straylight background are met by the simulation. Though, the absolute straylight level around the −5^th DO differs slightly. Further, the simulation curve of course does not possess ghosts as no deterministic errors were considered within the applied simulation. The measured ghosts show an unexpected distribution as they are the strongest in middle of adjacent DOs. As explained in Sec. 3.2, an increase close to the main diffraction orders would be expected. The reason is probably a different formation of super periods during holography, which was applied for the investigated test grating. The mechanisms responsible for super period formation that finally leads to ghosts as observed in Fig. 6 is still unknown and under investigation.

4.1

BINARY HIGH RESOLUTION GRATING

One approach to reduce the straylight level and especially to lower the Rowland ghosts is the direct improvement of the positioning accuracy in the EBL writing process, i.e. the improvement of the stitching of the single subsegments. In this investigation we aim for controlling the gap betwee adjacent segments as illustrated in Fig. 3c. The e-beam-writer offers several calibration parameters that control the segment alignment. The calibration of the micro deflection system and in this way the gap ΔP_seg is controlled by the parameter ΔMDS. An experiment was performed, in which several grating with different ΔMDS were fabricated and the corresponding straylight performance around the –1^st DO was measured. The measurement result of the best and worst calibration state is shown in Fig. 7a. The blue arrows in this graph mark the expected angular positions of the ghosts that correspond to the applied segment sie of P_seg = 35 μm. In the graph, there occur a lot of weaker ghosts that correspond to second deflection system (”macro deflection system”), which is not investigated here. However, as we see in Fig. 7a, the calibration of the micro deflection system strongly affects the strength of the corresponding peaks, but also reduces the strength of the ghosts originating form the macro deflection system. Further, we applied a so-called multi-pass exposure¹¹ that allows to reduce the ghosts further as shown in Fig. 7b.

Figure 7.

Straylight measurement of the grating ghosts around the –1^st DO (at Δθ = 0°) of the SFTl-grating with the blue arrows marking the calculated angular positions of the ghosts corresponing to the ”micro deflection system” (segment size of P_seg = 35 μm): (a) Straylight curves of the best and worst calibration state. (b) Additionally applied Mulipass-exposure for further improvement of the best calibration state.

4.2

ECHELLE GRATING

The fabrication of echelle gratings in a silicon crystal substrate applies two crucial steps. First, the realization of the lateral grating pattern into a grating hard mask (usually chromium or silicon nitride) by means of a suited lithography process, e.g., holography or electron beam lithography. Within this work, we use EBL for lateral structuring. Second, the transfer of the lateral structure into the silicon crystal by wet anisotropic etching. The second step is done by KOH-etching of the [100]-cSi-substrate, which inherently leads to the formation of the desired echelle profile according to the crystallographic planes of the Si-crystal. During the second step, a correct alignment of the grating lines with respect to the crystal orientation is mandatory. The alignment can either be realized by a rotation of the grating pattern during EBL-data praperation or by rotation of the substrate during lithography.

Within this work, the effect of different rotation methods onto the straylight pattern was tested:

The straylight measurements of the full transmission hemisphere according to Fig. 1b and an SEM-image of the grating profile inspection is shown in Fig. 8.

Figure 8.

Top: Full ARS measurments of the complete reflection hemisphere of the SFT2 grating (echelle type grating in reflexion with Λ = 2070 nm, AOI = 55°, λ = 633 nm according to Fig. 1b). The dark blue region obscuration of the light source. Bottom: Corresponding SEM-images of the grating structure.

It is clearly found that the third method produces the lowest straylight level. The first two methods produce grating ghosts within the whole half space. The ghosts of grating 1 (standard tilting) are more pronounced than the ghosts of grating 2, which are rather blurred. However, the total straylight level of grating 2 and espacially the straylight background is considerably higher. This can also be confirmed by SEM-inspection of the grating structure. Grating 2 shows a significantly increased facet roughness. The facet roughness of grating 3 is almost not visible within the SEM image.