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1.INTRODUCTIONNowadays, many optical applications employ diffraction gratings, which have turned into one of the most challenging components. Especially in high-performance spectrometry, the gratings, which are used as dispersive elements, are required to fulfill strong demands on the diffraction efficiency, the bandwidth, and the spectral dispersion. It has been shown that these requirements can be addressed by different grating types, e.g., binary phase gratings, echelle gratings, or blazed gratings. Nevertheless, light scattering of such gratings becomes increasingly critical as it constrains the radiometric accuracy of spectroscopic measurements and, thus, limits the accuracy of a spectrometer. In case of spectrometer gratings, there are plenty of stray light sources due to fabrication imperfections and shape deviations of the microscopic grating structure. These deviations strongly depend on the spectrometer concept (Offner type spectrometer, echelle spectrometer, etc.) and the required grating structural feature type (e.g. binary transmission gratings, echelle type reflection gratings, blazed low-resolution gratings, etc.), respectively, and the fabrication technology (electron beam lithography, interferometry, ultra-precision diamond turning, etc.). Depending on the particular type of grating imperfection, either stochastic or more systematic and periodic light scattering effects can occur. In this sense, it is basically distinguished between a homogeneous scattering background and scattering singularities known as grating ghosts.1 An analysis of these fundamentally different phenomena and their dependence on grating type, grating design, and grating fabrication will be presented. In particular, our contribution will be three-fold. It will be shown that…
Throughout the presented analysis, different grating structural feature types will be investigated. These grating structures are chosen according to the most commonly used spectrometer types:
2.ANALYSIS OF STRAYLIGHT MEASUREMENTSThe angle resolved scattering ARS is usually measured by focussing the illuminating beam onto an aperture (slit or pinhole) in front of the detector.6 The light source and the detector are mounted onto a goniometer, which allows to scan the full scattering half space defined by the azimutal angle θ and conical angle φ. The ARS is than calculated by with PS being the signal power measured by the detector, P0 the power of the incident beam and Ω the aperture angle, which is in particular determined by the pinhole size. In order to illustrate the impact of different set-up configurations, several ARS measurements with different aperture angles Ω were performed. The measurements were done on a SFT2-grating according to the configuration as described in Fig. 1b, i.e., AOI = 55°, λ = 633 nm, Λ = 2070 nm. The measurement results are shown in Figure 2. In the applied configuration, the grating shows 5 propagating diffraction orders in reflection half space, which can be clearly identified. In Figure 2a three main properties can be identified: First, a homogeneous straylight background especially rising around the –5th DO (useful order) is detected. Second, the dispersion plane (defined by φ = 0° and θ = −90° … 90°) shows an increased straylight level. Third, along dispersion plane and in between the main diffraction orders, several grating ghosts are slightly visible. Figure 2b shows the ARS-measurements along dispersion plane around the −5th DO. The measurements have been performed with an oversampling, i.e., with a step width of only 0.02°, in order to ensure a full detection of the ghosts. The grating ghosts can now be clearly identified as they are rising out of a continous scattering background. It is further clearly visible that the scattering background is always detected on the same level and, thus, is independent of the measurement set-up. Though, the strength of the grating ghosts strongly depends on the aperture angle; they are the stronger, the smaller the aperture angle. This becomes clear when considering the ghosts as spurious diffraction orders that arise from a super-period in the diffraction grating [RefHeusinger]: In this sense, the measured signal power of the ghost and the ghost efficiency ηS, respectively, should be independent of Ω as long as the ghost is fully detected (meaning that the aperture must be big enough to allow measuring the full ghost). The corresponding measurement will be referred to as “angle resolved efficiency” (ARE) measurement, which simply can be derived from the ARS measurements by ARE = ARS · Ω. Though, oversampling and a suited aperture has to be ensured. The corresponding ARE of the SFT2 test grating is shown in Figure 2c. The Figure contains several comments, which explain the features visible in the measurement. Regarding the ghost characterization, we find that the strength of the ghosts is in the range ARE < 10 −4 with the strongest ghost in close vicinity to the −5th DO. 3.THEORETICAL ANALYSIS OF SCATTERED LIGHT DISTRIBUTION3.1Basic model for straylight simulationFor 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 = Np · 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 Np = 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 Np = 1. However, a disturbance of the ideal grating geometry results in a certain amount of energy in every SO. The diffraction efficiency ηm(θm) is calculated by the RCWA algorithm and the angle resolved scattering6 (ARS) can be estimated by The corresponding scattering angle θm is calculated using the grating equation with m = mmin,…, mmax enumerating the propagating straylight and diffraction orders. More details of this model and the extend to 2D-disturbances can be found in.8 3.2Simulating ghostsIn 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 Pseg ≫ 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) σseg of the sub-segments and, second, a deterministic alignment error ΔPseg resulting in a gap (ΔPseg > 0) or overlay (ΔPseg < 0) between adjacent sub-segments. The effect of this error is calculated for a sub-segment size of Pseg ≈ 35 μm as it occurs in the used e-beam writer. Figure 4 shows the results of the simulation. The main findings from the simulation are the following:
3.3Simulating backgroundThe 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 σp describing the Shot positioning error and σb describing the stochastic error of the groove width. In the following, the effect of these errors onto the straylight spectrum is presented. 3.3.1Binary high resolution gratingIt 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 –1st 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, θi = 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 σp = 5nm, and σb = 5nm are evaluated. 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 –1st and 0th 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 0th 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. 3.3.2Echelle gratingThe 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 σp and the error of the line width σb. Such an investigation is shown in Figure 6. 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 −5th 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.GRATING FABRICATION4.1BINARY HIGH RESOLUTION GRATINGOne 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 ΔPseg 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 –1st 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 Pseg = 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. 4.2ECHELLE GRATINGThe 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. 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. 5.CONCLUSIONThe conclusion is three-fold as is the total paper: First, it was shown that homogeneous straylight background and grating ghosts of significant difference as the background is a continuous effect while the ghosts are singularities. As such, the background must be evaluated in terms of ARS-measurements while the ghosts need to be evaluated in terms of efficiency measurements. Within the paper we used the term ”angle resolved efficiency” for such a measurement as it is closely related to ARS. Second, a simulation method is presented that allows to calculate straylight in diffraction gratings. The method is applied for ghost and background analysis and it is shown that not only the particular type of disturbance but also the grating geometry itself affects the straylight level and distribution. A comparison of ARS-measurement and ARS-simulation verifies the model. Third, investigations on straylight optimiation in binary transmission gratings and echelle reflection gratings are presented. By means of calibrating the deflection system of the applied e-beam writer a reduction of the grating ghosts by more than 2 orders of magnitude was achieved. ACKNOWLEDGMENTSThis work was funded by the European Space Agency (ESA) and the European Space Research and Technology Centre (ESTEC). REFERENCESPalmer, C. A., Loewen, E. G., and Thermo, R., Diffraction grating handbook, Newport Corporation Springfield, Ohio, USA
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