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Certificate #1
Efficiency Certificate for a Blazed Holographic Quartz Grating in the Soft X-Ray-EUV Range
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Certificate #2
Efficiency Certificate for a Replica of a Blazed Holographic Grating in the Soft X-ray-EUV Range
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Certificate #3
Efficiency Certificate for a Replica of a Red-Blazed Ruled Grating in the NUVľNIR Range
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Certificate #4
Efficiency Certificate for a Replica of a Layered Holographic Grating in the VUVľNUV Range
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Certificate #5
Efficiency Certtificate for a Multilayer MoRu-Be Blazed Holographic Grating in the Soft X-rayľEUV Range
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Efficiency Certificate #6
Efficiency Certificate for a Flight Multilayer Mo/Si Lamellar Grating in the EUV Range and TM polarization
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Efficiency Certificate for a Blazed Holographic Quartz Grating in the Soft X-Ray-EUV Range  
 
 
General Parameters of the Grating  
Here we present an example of the soft X-ray-EUV master grating efficiency29 based on the average atomic force microscopy (AFM) groove profile measurements, exact PCGrate-S(X) v.6.1 Series modeling, and efficiency synchrotron measurements. The grating was fabricated by Spectrogon UK Limited (formerly Tayside Optical Technology). The groove pattern was recorded using a holographic technique onto a fused silica (Spectrosil B) blank with a concave radius of curvature of 2.0 m. The pattern frequency was 2400 grooves/mm (g/mm), and the pattern covered an area of 45 mm by 35 mm. The sinusoidal groove profile was then ion-etched into the fused silica to produce a blazed groove profile with a desired blaze angle of 2.5░. However, ion etching results in a groove profile closer to a triangular form than the ideal blazed (sawtooth) profile with the other facet inclined at an angle of 5.5░ to the surface.29  
 
Groove Profile Measurements  
The groove profile was identified using a Topometrix Explorer scanning probe microscope, i.e., a type of AFM. The probe was a supertip which was tapered to a size much less 0.1 mkm. The AFM images were 500 pixels square with pixel sizes between 10 ┼ and 20 ┼. Using standard Topometrix software, the images were leveled with a second-order polynomial fitting technique.  
 
A typical groove profile derived from the AFM image, recorded using 16-┼ pixels with 210 points of the master grating, is shown in Fig. # 11. The groove profile is approximately triangular in shape with rounded corners and troughs and with facet angles of 2.5░ and 5.5░. To provide a groove profile for the efficiency calculation, a representative AFM scan perpendicular to the grooves was chosen at random and scaled to the average groove height (70-80 ┼). The resulting groove profile is shown in Fig. # 14. This groove profile has 120 points.  
 
Layer Thicknesses Investigation  
The thickness of the SiO2 layer was assumed to be semi-infinite for modeling purposes.  
 
Investigation of Random Roughness and Interdiffusion  
The microroughness, determined by integrating the power spectral density function (PSD)71 over 4-40 mkm-1 spatial frequency range, was 3.2 ┼.29 Most of the microroughness is concentrated at low spatial frequencies, as is apparent from measurements.  
 
Efficiency Measurements  
The grating efficiency was measured with the Naval Research Laboratory beamline X24C39 at the National Synchrotron Light Source (Brookhaven National Laboratory). Measurements were performed at 15 wavelengths in the 125-146 ┼ range using a silicon filter and at 11 wavelengths in the 170-225 ┼ range using an aluminum filter.29 The grating was mounted in a reflectometer which admitted precision sample and detector rotational motions.39,40 For purposes of measurement, the grating was rotated so that the radiation was incident at an angle of 10░ to normal to the grating surface. The grating was oriented so that the 5.5░. facets faced the incident radiation. Owing to the small angle of the opposite facets (2.5░), the radiation also illuminated the facets that faced away from the incident radiation. In this orientation, the outside orders of the 2.5░ facet were close to the blaze condition. An area at the center of the grating approximately 1 mm in size was illuminated. The radiation was approximately 80-90% polarized with the electric field vector in the plane of incidence. In this orientation, the electric field vector was almost perpendicular to the grating grooves.  
 
Scattering Light Measurements  
The scattering light has a very low level and was not measured for this grating.  
 
Sources of the Refractive Indices  
Initially,29 refractive indices (RIs) were taken from the Henke et al. tables.10 For this Efficiency Certificate, the updated RIs were derived from the CXRO compilations. To compare the calculated results with the measured ones, we also used RI data from the handbook of Palik.11  
 
Efficiency Modeling  
The preliminary modeling was performed for a plane grating model, because ratios of the grating sizes to the radius of curvature are small. The calculation accounted for the finite conductivity of the grating surface, and the optical constants of fused silica (SiO2) were derived from the compilation of CXRO. The calculations were performed using the average AFM groove shape model of 70 ┼ depth shown in Fig. # 14 with 120 points on it. Results were calculated for radiation that was incident at an angle of -10░ to normal to the grating surface. The grating facets were oriented identically to those during the measurements, with the steeper facet (at an angle of 5.5░) facing the incident radiation. The polarization angle was taken equal to 63.43░. In this orientation, the electric field vector was nearly perpendicular to the grooves (80% of TM and 20% of TE). Topography of the random residual grating roughness was included in the efficiency model by multiplying orders intensities by the Debye-Waller factor with the rms roughness of 0.32 nm.5 It turns out to play a minor role, even for the shortest wavelengths from the working range because of small values of the ratio of microroughness rms to the wavelength used. However, the correlated groove component from the average AFM shape was included (automatically) by accounting for the real groove profile with a high level of accuracy.  
 
Convergence of the efficiency results was investigated in the type of the calculation mode ("normal" or "resonance"), in the type of lower border conductivity ("perfect" or "finite"), in the main accuracy parameter N (number of collocation points), in including (or non-including) options for accelerating convergence, in the type of integration step (equal along X axis (x) or along groove profile (s)), and in the source of RI data with the type of their interpolation (constant, linear, or cubic splines).9 A high rate of convergence was observed for the efficiency results in the entire wavelength range of interest (Fig. # 24). Discrepancies between the results for all calculating models in all the points considered are several times less than the corresponding distinctions between the measured and calculated data (Figs. # 25, 26). For further investigation and calculation we choose the model based on the "normal" calculation mode for the "finite" type of lower border conductivity with N = 121 and without any convergence acceleration. In the covered range, this model yields accurate results at a high calculation speed. The linear extra- interpolation type was chosen for approximation of RI data between and well apart from the tabulated points.  
 
The total error for all points derived from the energy balance was about 1.E-5. The time elapsed to calculate one point was less than 6 s using IBM« Think Pad with an Intel« Pentium« M 1700 MHz processor, 1 MB !ache, 400 MHz Bus Clock, 512 MB RAM, and controlled by OS Windows« XP Pro.  
 
Comparison Between Calculated and Measured Efficiencies  
The effects of the assumed groove depth, its shape and refractive indices on the calculated efficiency in the (-1)st order are presented in Figs. # 25, 26. The efficiencies are very sensitive to the groove shape and depend on the refractive indices as well. For better fit, calculations using the same AFM groove profile but scaled to different assumed groove depths were performed with different RI libraries. The calculated efficiencies in the (-1)st order tend to decrease with increasing groove depth in the short and medium working wavelength regions. A correlation between the measured data and accurate calculations using two "ideal" groove profiles (a sawtooth profile with 2.5 deg. and a triangle profile with 2.5 and 5.5 deg.) is weak (Fig. # 25). Using RIs from the Palik book11 together with the average AFM groove profile also does not result in a good agreement with the experiment (Fig. # 26). Calculated efficiencies are in good overall correlation with the measured ones for the RI data of Henke10 and for the AFM groove profile of depth 75 ┼.  
 
The best overall agreement between the calculated and measured efficiencies in the (-1)st order was achieved with a peak-to-valley AFM-measured groove depth of 70 ┼ and the CXRO RI data. The overall (-1)st order efficiency is well predicted by the PCGrate-S(X) v.6.1 based on the average AFM groove profile and the CXRO RI data without any other assumptions adduced (Fig. # 26). The average relative error for the (-1)st order efficiencies in the whole working wavelength range was 6.5% and did not exceed 14% at the worst point. The validity of the model was supported by comparing the experimental and predicted efficiency data in -2, 0, +1, and +2 orders. In Fig. # 27 we can see the measured data designated by markers and the calculated data indicated by curves for all evaluated orders. A good agreement was achieved therewith for all orders and wavelengths.  
 
Efficiency Certificate  
For issuance of the Efficiency Certificate, calculations based on the best grating model were performed in the wavelength range under investigation with the step of 0.1 nm. There were no free parameters in the calculation. Fig. # 28 demonstrates the final efficiency results for ▒2, ▒1, and 0 orders of a 2400 g/mm concave quartz master grating working in the 12.5-22.5 nm wavelength range.  
 
Summary  
  • A good quantitative agreement between the experiment and exact modeling based on a real groove profile (AFM-measured) and appropriate RI libraries can be obtained for such types of gratings.  
  • Calculations are very sensitive to a groove profile, with the best result obtained for the average AFM model.  
  • The impact of small changes of the real and imaginary parts of the RIs upon the efficiency is comparatively large.  
  • Low values of the random roughness (rms ~ 3.2 ┼ ) are of little importance for the grating efficiency in the wavelength range covered and can be ignored in calculations.  
  • The electromagnetic code can be used effectively for precise modeling of real groove profile bulk gratings taking into account the finite conductivity of the surface in soft X-ray and EUV ranges despite small wavelength-to-period ratios.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation process can be obtained from the downloading page (Cert_2400_holo-master_SiO2_soft-X-ray--EUV.zip file). Various files both in *.xls (MS Excel«) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.1) formats relating to this grating efficiency certification are included in the package. You can view the later formats by the PCGrate-S v.6.1 Complete Demo (updated on the 3rd May, 2005).  
 
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Efficiency Certificate for a Replica of a Blazed Holographic Grating in the Soft X-ray-EUV Range  
 
 
General Parameters of the Grating  
Here we present an example of the soft X-ray-EUV replica grating efficiency13 standardized using synchrotron radiation, atomic force microscopy (AFM) and PCGrate-S(X) v.6.1. The master grating was fabricated by Spectrogon UK Limited (formerly Tayside Optical Technology). The groove pattern was fabricated in fused silica using a holographic technique. The groove pattern was ion-beam etched to produce an approximately triangular, blazed groove profile. The grating has 2400 grooves/mm (g/mm), a concave radius of curvature of 2.0 m, and a patterned area that is 45 mm by 35 mm. The master grating was uncoated. The replica of the master grating was produced by Hyperfine, Inc. As a result of the replication process, the grating had an oxidized aluminum surface. A thin SiO2 overcoating was applied to the replica grating for the purpose of reducing the microroughness of the groove facets.  
 
Border Profiles Measurements
The surface of the replica grating was characterized using a Topometrix Explorer scanning probe microscope, a type of AFM. The scan was performed across the grooves over a range of 1 mkm (20-┼ pixels) - see Fig. # 15. The silicon probe had a pyramid shape. The base of the pyramid was 3 to 6 mkm in size, the height of the pyramid was 10 to 20 mkm, and the height to base ratio was approximately 3. The tip of the pyramid had a radius of curvature less than 200 ┼. The AFM scans were performed using the non-contact resonating mode, where the change in the oscillation amplitude of the probe is sensed by the instrument. The grating topography was measured merely for the upper interface.  
 
A typical groove profile derived from the AFM image (1 mkm in size) of the replica grating is shown in Fig. # 17. The groove profile is approximately triangular in shape with rounded corners and troughs and with facet angles of 3.4░ and 6.2░ . The average groove depths derived from the AFM images are in the range 85 to 95 ┼. These values of the facet angles and the groove depth are larger than the corresponding values for the master grating, 2.5░ and 5.5░ facet angles and 75 ┼ average groove depth.29 Thus the grooves of the replica grating are deeper and the facet angles are steeper compared to those of the master grating.  
 
Layer Thicknesses Investigation  
The efficiencies in the 0, ▒1, and ▒2 orders, measured using synchrotron radiation, have an interesting oscillatory behavior that results from the presence of the thin SiO2 layer on the highly reflecting oxidized aluminum surface (Fig. # 29). The thicknesses of the SiO2 and Al2O3 layers were inferred from the frequency and amplitude of the zero-order efficiency in the 100 ┼ to 350 ┼ wavelength range.13 The inferred thicknesses of the SiO2 layer and the Al2O3 layer were 743 ┼ and 30 ┼ respectively. The thickness of the Al layer was assumed to be semi-infinite for modeling purposes.  
 
Investigation of Random Roughness and Interdiffusion
The average microroughness was derived from a 2 mkm image spanning nearly five grooves.13 The rms microroughness of 7 ┼ was determined by integrating the power spectral density function (PSD)71 over the 4-40 mkm-1 spatial frequency range. Most of the microroughness is concentrated at low spatial frequencies, as is apparent from Fig. # 16.  
 
No information is available about layers interdiffusion.  
 
Efficiency Measurements
The efficiency of the replica grating was measured using the Naval Research Laboratory beamline X24C39 at the National Synchrotron Light Source at the Brookhaven National Laboratory. The synchrotron radiation was dispersed by a monochromator that had a resolving power of 600.40 The replica grating was mounted in the reflectometer so that the dispersed radiation from the monochromator was incident on the grating at an angle of 15.2░ measured from the normal to the surface of the grating. The grating was oriented so that the groove facet with the larger facet angle (measured from the surface of the grating substrate) should face the incident beam. In this orientation, the inside orders were closest to satisfying the on-blaze condition for these facets, where the radiation is specularly reflected from the facets. At fixed wavelengths, the detector was scanned in angle about the grating. Measurements were performed at 23 wavelengths in the 125-182 ┼ range using a silicon filter and at 17 wavelengths in the 172-225 ┼ range using an aluminum filter.13 The incident radiation was approximately 80-90% polarized with the electric field vector in the plane of incidence (p polarization). In this orientation, the electric field vector was perpendicular to the grating grooves (i.e. TM(S) polarization).
 
The measured efficiencies were determined by fitting a background curve and five Gaussian profiles to the 0, ▒1, and ▒2 orders (Fig. # 30). The background and the five Gaussian profiles were fitted to the data points using the least-squares technique. The heights of the Gaussian profiles (above the background curve) were determined for each of the 40 detector scans that were performed at fixed wavelengths in the 125-225 ┼ wavelength range.13  
 
Scattering Light Measurements  
Some scattering light was observed during the efficiency measurements. Its level between the orders is low for diffraction angles in the off-blaze direction (> 15.2░) and higher in the on-blaze direction (< 15.2░).13 The scattering light level was low and was not measured for this grating.  
 
Sources of the Refractive Indices  
Initially13 the refractive indices (RIs) required for calculations were taken from the Henke et al. tables.10 For this Efficiency Certificate, the updated RIs were derived from the CXRO compilations. For better agreement between the calculated results and the measured ones in the long wavelength range (> 15 nm), we used RI data from the handbook of Palik.11  
 
Efficiency Modeling
The preliminary theoretical investigation was performed for different calculation plane grating models (due to small ratios of the grating sizes to the radius of curvature) using the same AFM replica groove shape (Fig. # 17) of 85 ┼ depth13 for all 3 borders with 120 points per each. The conformal layer approximation was used for the calculation purposes.9 The random roughness topography of the grating was taken into account by applying the amplitude Debye-Waller factor with the rms roughness of 0.7 nm for all interfaces.5 The periodical lateral-correlated component of the border roughness from the average AFM groove shape was included automatically by accounting the real groove profile with a high degree of accuracy. An assumption about absence of the vertical correlation between the border roughnesses was applied. The RI data throughout the entire wavelength range under investigation were derived from the Palik handbook.11 Results were calculated for radiation that was incident at an angle of -15.2░ to the normal to the grating surface. Then the grating facets were oriented identically to those during the measurements, with the steeper facet (at an angle of 6.2░) facing the incident radiation. The polarization angle was taken equal to 63.43░. In this orientation, the electric field vector was nearly perpendicular to the grooves (80% of TM and 20% of TE). Thirty five wavelengths that were used in the experiment were included in the theoretical investigation and for comparison.  
 
Convergence of the efficiency results was investigated in the type of the calculation mode ("normal" or "resonance"), in the type of lower border conductivity ("perfect" or "finite"), in the main accuracy parameter N (number of collocation points), in including (or non-including) options for accelerating convergence, in the type of integration step (equal along X axis (x) or along groove profile (s)), and in the source of RI data with the type of their interpolation (constant, linear, or cubic splines).9 A high rate of convergence was observed for the efficiency results in the entire wavelength range of interest (Fig. # 31). Discrepancies between the results for all calculating models in all the points considered are several times less than the corresponding distinctions between the measured and calculated data (Figs. # 32, 33). For further investigation and calculation we choose the model based on the "normal" calculation mode for the "perfect" type of lower border conductivity with N = 241 and without any convergence acceleration. In the covered range, this model yields accurate results at a very high calculation speed with respect to the "resonance" mode (a few orders faster). The linear extra- interpolation type was chosen for approximation of RI data between and well apart from the tabulated points. The total error for all points derived from the energy balance was an order of 1.E-5.  
 
The time required to calculate one point for a model grating was about 12 s using an IBM« Think Pad with an Intel« Pentium« M 1700 MHz processor, 1 MB Cache, 400 MHz Bus Clock, 512 MB RAM, and controlled by OS Windows« XP Pro.  
 
Comparison Between Calculated and Measured Efficiencies  
A comparison between calculated and measured (-1)st order efficiency data was made using the least-squares technique for 35 wavelengths. The fitting parameters were as follows: the RIs, the groove depth, and the groove shape. The best agreement (Fig. # 32) was obtained for the model having an average AFM replica groove shape with a depth of 85 ┼ and RI for all the layer materials taken from the Palik tables.11 This model was improved by applying the RI data to all the materials from the CXRO compilation for wavelengths less than 15 nm (Fig. # 33). Palik's RI data for Al were extrapolated in the range from 15 to 16.5 nm.  
 
The rms efficiency for the best model over the entire wavelength range involved was about 3.1E-5 in the (-1)st order. This value is approximately 20% of the average (-1)st order efficiency in the range covered. The validity of the model was supported by comparing the experimental and predicted efficiency data in -2, 0, +1, and +2 orders. A good agreement was achieved therewith for all orders and wavelengths (Fig. #34).  
 
Efficiency Certificate  
For issuance of the Efficiency Certificate, calculations based on the best grating model were performed in the wavelength range under investigation with the step of 0.1 nm. There were no free parameters in the calculation. The calculation was performed for an average AFM replica groove shape with a depth of 85 ┼, random roughness, and the optical constants) for all the materials taken from the compilation of CXRO (< 15 nm) and from the Palik handbook11 (≥ 15 nm). The Efficiency Certificate in ▒2, ▒1, and 0 orders of a replica of the 2400 g/mm concave holographic blazed grating for the wavelength range of 12.5-22.5 nm is presented in Fig. # 35.  
 
Summary  
  • A good quantitative overall agreement between the experiment and the predicted modeling efficiency based on both a real groove profile (AFM-measured) and appropriate RI libraries can be obtained for these types of gratings and wavelengths.  
  • Calculations are very sensitive to the groove profile, with the best results obtained for the average AFM model.  
  • The impact of small changes of the real and imaginary parts of the RIs upon the efficiency is comparatively large. Using RIs from the Palik handbook11 gives more accurate efficiency results (in respect to using CXRO data) in the wavelength range larger than 15 nm, all of which agrees with the last EUV RI investigations in design and fabrication of multilayer-coated gratings.1  
  • Intermediate values of the random roughness (rms ~ 7 ┼ ) are of some importance for the grating efficiency in the wavelength range covered and can not be ignored in calculations.  
  • The electromagnetic code can be used effectively for a precise modeling of real groove profile layered gratings in the soft X-ray and EUV ranges despite small wavelength-to-period ratios.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation process can be obtained from the downloading page (Cert_2400_holo-replica_Al-Al2O3-SiO2_soft-X-ray--EUV.zip file). Various files both *.xls (MS Excel«) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.1) formats relating to this grating efficiency certification are included in the package. You can view the later by the PCGrate-S v.6.1 Complete Demo (updated on the 3rd May, 2005).  
 
 
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Efficiency Certificate for a Replica of a Red-Blazed Ruled Grating in the NUVľNIR Range  
 
 
General Parameters of the Grating  
The aforementioned methods are applied to simulate the efficiency of an individual grating from a specific grating set which is mounted in the Space Telescope Imaging Spectrograph (STIS) flown aboard the Hubble Space Telescope (HST).72 A first order reflection grating with 67.556 grooves/mm (g/mm) blazed for 750 nm (1.44░ nominal blaze angle) working in the range from 500-1000 nm at 8░ incident angle was chosen for certification.6 The pattern size was 1.5 inches by 1.5 inches, and the pattern ruled an area of 30 mm by 30 mm. A sister replica to this grating, designated "Ng41M" or by its manufacturers' (Richardson Gratings of Newport Corp.) serial number, 1528, is in use on HST/STIS as a red survey grating (blazed for red visible and near infrared wavelengths). In its flight application, this grating had a reflective overcoating of 100 nm Al plus 25 nm MgF2. However, in this wavelength range the effect of the MgF2 layer is minimal and simulations showed no significant difference (to a small fraction of the accuracy in the efficiency measurements) with or without it. This simulation included the MgF2 layer.  

Border Profile Measurements
The groove profile was characterized using atomic force microscopy (AFM) measurements. The tips used here were 20 nm in radius. An example of the typical groove profile of a 1528 grating is presented in Fig. # 9. The figure shows that the groove minima are clearly resolved in the AFM image. If the conventional choice of the groove boundary as the minimum value is made, there are two complete grooves in each scan line. The resulting average groove profile (with averaging performed both across the multiple grooves and along the grooves) is shown in Fig. # 36. The solid line is based on the AFM data, and the dotted line is based on the stylus profilometer data (the groove tops are aligned in the comparison; the relatively sharp groove bottom is not as well resolved by the stylus profilometer). The periodicity of this profile is shown by comparing a simulation of the averaged scan line based on the average groove shape to the average scan line. This is demonstrated by dotted lines overplotted against the raw data in Fig. # 6 (microinterferometer) and Fig. # 7 (stylus profilometer). Once the average profile has been determined, the fitting routine finds the sawtooth and two-angle shape fits by least squares. In this case the blaze angle is 1.45░ and the antiblaze angle is found to be 30░ (Fig. # 37). The efficiency in general is fairly insensitive to the antiblaze angle, and the fitting routine does not fit it as consistently as it does the blaze angle. The final groove profile has 100 points.  
 
Layer Thicknesses Investigation  
This grating had a protective overcoating of 25 nm MgF2. The thickness of the Al layer was assumed to be semi-infinite for modeling purposes.  

Investigation of Random Roughness and Interdiffusion
The AFM average groove profile was used to simulate the overall grating roughness.71 The rms roughness for the residual image is around 20 nm from the frequency range 100-2000 mm-1.
 
No information is available about layers interdiffusion.  

Efficiency Measurements
Measurements of the diffraction efficiency were made at Richardson Gratings of Newport Corp. using a highly automated spectrograph (AEC - Fig. # 18) and detector on a movable arm. Measurements were made in spectrograph mode, in which the detector is moved to follow the spectrum vs. wavelength for a fixed incidence angle. Polarizers were used over the extended wavelength range of 300-1500 nm to allow independent tests of the transverse electric (TE) and transverse magnetic (TM) polarizations.  
 
The grating was measured using the efficiency apparatus described above, in both TE and TM polarization, using a fixed incidence angle of 8░, similar to its mount on HST/STIS. The step between the measured points was 10 nm. In order to prevent mechanical interference between the illuminating beam and the measurement arm, the illumination is slightly (a few degrees or less) out of plane. Studies of this effect with codes that can model conical diffraction show that the effect of this small out of plane angle on the efficiencies is in general very low, with a cosine-type behavior. Reproducibility in the measured efficiency was within 1%.  
 
Scattering Light Measurements  
The scattering light was not measured directly for this grating. However, the roughness extrapolated from the fit to the AFM residual agrees with the roughness in the residual image; this provides a basic consistency check. This analysis attempted to discover the "other" or "side" components of the power spectral density function (PSD)71 in an attempt to predict scatter away from the principal diffracted order.  
 
Sources of the Refractive Indices  
For this investigation, the refractive indices (RIs) of Al and MgF2 were taken from the handbooks of Palik11 and AIP.12  
 
Efficiency Modeling
The calculation was performed using the average AFM groove shape model shown in Fig. # 36. The efficiency models for the nominal profile and for the nominal profile plus or minus one standard error in the height profile (by combining repeatability and calibration uncertainties) were computed using PCGrate-S(X) v.6.1. The results were calculated for two basic states of polarization, with the radiation being incident at an angle of 8░ to the normal to the grating surface. The optical constants) were derived from the Palik handbook11 and the linear interpolation was chosen for approximation of RI data between the tabulated points. The grating facets were oriented with the blazed facet (at a nominal angle of 1.44░) facing the incident radiation. The calculation was run over the extended range of 300-1500 nm. In the process of calculation, the finite conductivity of the grating surface was taken into account using the (RIs for aluminum and magnesium fluoride. The random roughness topography of the grating was not included in the efficiency model because of the small value of rms roughness compared to the working wavelengths. The correlated groove component from the AFM average groove profile was included automatically in the computation by working out the real groove profile with a high degree of accuracy.  
 
Convergence of the efficiency results was investigated in the type of the calculation mode ("normal" or "resonance"), in the type of lower border conductivity ("perfect" or "finite"), in the main accuracy parameter N (number of collocation points), in including (or non-including) options for accelerating convergence, in the type of integration step (equal along X axis (x) or along groove profile (s)), and in the source of RI data with the type of their interpolation (constant, linear, or cubic splines).9 The convergence and accuracy were checked for the efficiency results with N = 601 & N = 801 in the "resonance finite" calculation mode and with N = 201 & N = 301 in the "normal finite" calculation mode. Discrepancies between the results in both polarizations for appropriate calculation mode throughout the entire wavelength region (Figs. # 38, # 39) are much less than corresponding deviations between the measured and calculated data. For the final results we chose the model based on the "resonance finite" calculation mode with all the above-listed options for convergence acceleration and with the s type of integration step. In the range covered the model yields accurate results (with the energy balance within 0.1%) at a low calculation speed with respect to the "normal finite" mode. However, the resulting TM-efficiencies obtained using the "resonance finite" mode are closer to the measured data in the vicinity anomalies (Fig. # 43).  
 
The time required to calculate one point for the Ng41M grating efficiency was about 20 s (N = 201) for the "normal finite" calculation mode and about 17 min (N = 601) for the "resonance finite" mode using an IBM« Think Pad with Intel« Pentium« M 1700 MHz processor, 1 MB Cache, 400 MHz Bus Clock, 512 MB RAM, and controlled by OS Windows« XP Pro.  
 
Comparison Between Calculated and Measured Efficiencies  
The grating efficiency is sensitive to the shape of the groove profile model and less sensitive to the RIs. The data from the sawtooth and two angle blaze models are far from the best average AFM model prediction - see Figs. # 40, # 41. The handfit AFM model (Table 1) yields better prediction than the sawtooth and two angle models do. Nonetheless it deserves more attention when compared with the average one.  
 
Table 1. Coordinates for the handfit AFM model  
Points  
1  
2  
3  
4  
5  
6  
7  
8  
X  
0  
0.04  
0.16  
0.2  
0.4  
0.8  
0.955  
1  
Y  
0  
0.003  
0.007  
0.012  
0.0155  
0.025  
0.025  
0  
 
The best agreements (Figs. # 42, # 43) were obtained for the models using the "resonance finite" calculation mode, the average AFM shape with the depth of 402 nm for both borders, and RIs for the layer materials taken from the Palik or AIP tables.11,12 For short wavelengths, the difference may be explained mostly by a stronger impact of deviations from the exact groove profile shape on the efficiency, while for long wavelengths - by unsuitable values of the RIs used in modeling. Weak anomalies in the TM polarization case for wavelengths above ~ 900 nm were measured and observed in the simulation, but not exactly at the same points or with the same amplitudes (this can easily be seen from the polarization efficiency curves in Fig. # 44). The efficiency in the TM polarization calculated using the Palik's RI data are closer to the measured data in the vicinity anomalies.  
 
Statistics of the residual efficiency for different polarizations (residual value = simulated value - measured value) over the extended wavelength range (300-1500 nm) are listed in Table 2. The agreement is within a few percent over the entire range covered, but departs slightly from the measurements at the extreme wavelengths not included in the working range.  
 
Table 2. Statistics for efficiency discrepancies between the PCGrate model and measurements  
Difference in Efficiencies  
Average  
Standard deviation  
PCGrate - measurement, TE  
-0.00389  
0.02634  
PCGrate - measurement, TM  
0.00429  
0.02226  
PCGrate - measurement, NP  
0.0002  
0.0243  

Efficiency Certificate  
An example of the first order red blazed grating efficiency standardized using AFM and PCGrate-S(X) v.6.1 is shown in Fig. # 45 for the unpolarized light. The Efficiency Certificate is based on the best TE/TM grating model and performed in the extended wavelength range under investigation with the step of 2.5 nm. Error bars from average groove profile and RI uncertainties are displayed in Fig. # 45. There were no free parameters in the calculation.  
 
Summary  
  • An overall disagreement between the experiment and the accurate modeling based on the real groove profile (AFM-measured) can be obtained for this type of gratings within about 1%-2% of the absolute efficiency throughout the working wavelength range.  
  • Calculations are sensitive to the groove profile shape, with the best results achieved for the average (AFM model.  
  • The impact of the refractive indices taken from well-known sources11,12on the efficiency is small in the wavelength range covered.  
  • The modeling requires the electromagnetic theory to be used for taking into account real groove profiles, finite conductivity of materials, and different polarization states, even with small wavelength-to-period and depth-to-period ratios for such NUV-NIR gratings.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation process can be obtained from the downloading page (Cert_67_ruled-replica_Al-MgF2_NUV--NIR.zip file). Various files both *.xls (MS Excel«) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.1) formats relating to this grating efficiency certification are included in the package. You can view the later by the PCGrate-S v.6.1 Complete Demo (updated on the 3rd May, 2005).  
 
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Efficiency Certificate for a Replica of a Layered Holographic Grating in the VUV-NUV Range  
 
 
General Parameters of the Grating  
The aforementioned methods are applied to simulate the efficiency of a 5870 grooves/mm (g/mm) G185M grating intended for operation at vacuum-ultraviolet (VUV) wavelengths below 200 nm.18 This grating has the highest groove density and the shortest operational wavelength range of all Cosmic Origins Spectrograph (COS)16 gratings planned for the last servicing mission to the Hubble Space Telescope (HST). The G185M master grating was recorded holographically on 40 mm by 15 mm rectangular fused silica blank and the Pt coated at HORIBA Jobin Yvon Inc. An adhesive Cr coating, a working Al coating, and a protective (from oxidation) MgF2 coating were deposited on Au-coated replica gratings at NASA/GSFC. Resonance efficiency anomalies associated with waveguide funneling modes inside the MgF2 dielectric layer degrading the G185M COS NUV grating performance were measured and qualitatively described at NASA/GSFC.3 We used PCGrate-S(X) v.6.1 to model the efficiency of the G185M subwavelength grating with real boundary profiles [measured by atomic force microscopy (AFM)] and RIs taken from different sources, including best fits of the calculated efficiency data to experimental ones.18  
 
Border Profile Measurements  
The border profiles were characterized using AFM measurements.18 The profile of the G185M grating (replica C) intended for operation in the 170-200-nm range was AFM-measured before and after deposition of the Cr/Al/MgF2 coating (Fig. # 4). As seen from the figure, after the deposition the profile depth decreased by about a factor of 2.05 (46.4 nm against 22.6 nm), and the profile shape changed noticeably too, thus evidencing the case of nonconformal layering of the grating. For the reason that all G185M gratings were manufactured from the same master and by the same technology, one may suggest that all of them share before- and after-coating profiles. The average before-coating groove profile had 165 points and the average after-coating profile had 163 points.  
 
Layer Thicknesses Determination  
The nominal layers deposited on the SiO2 substrate are 160 nm Au, 5.4 nm Cr, 71.2 nm Al, and 40.1 nm MgF2.  

Investigation of Random Roughness and Interdiffusion  
The AFM average topography indicates a very low level of the residual grating roughness.71 The same conclusion results from the direct scattering light measurements.  
No information is available about layers interdiffusion.  
 
Efficiency Measurements  
The efficiency measurements were performed using a Fully Automated Ultraviolet Scatter Tester38 (FAUST) setup configured for a vacuum environment (Fig. # 20). Light from a Pt-Ne hollow cathode lamp reduced by the Cherny-Turner monochromator to 1 nm FWHM quasi-monochromatic irradiation was collimated to a 2'' beam, then reflected from the sample and re-focused as a monochromator exit slit image onto a Multi-channel Multi-Anode (MAMA) detector (a CsTe cathode with 1024 x 1024 pixels) by the camera mirror. Proper light source baffling and electronic noise reduction techniques (cable shielding, separate high- and low-voltage circuit grounding) were applied, thus keeping the average detector background noise to less then 10-2 counts/pixel/sec level. Each recorded data set consisted of at least five rounds of counts acquisition (more if necessary): dark counts measurements, diffracted beam measurements, reflected beam measurements and again diffracted beam and dark counts measurements, with all acquisitions performed in the fastest possible succession. All sets (except a very few ones made in a weak Pt-Ne lamp spectral region around 210-220 nm) acquired at least a few 10K diffracted image counts to achieve 0.1% accuracy of the statistics. A measurement was considered successful if it satisfied the criterion of the maximum allowable 0.5% drift between the first and second diffracted and dark counts acquisitions, otherwise the measurement was repeated.  
The measurements were done by comparing the light power of the grating order of interest (minus one for all tests at 34.7 deg. incidence) to the light power reflected onto the very same detector area by reference plane mirror (grating and mirror mounted on computer-controlled mechanical stages allowing them for easy interchanging). In this way we obtain a "relative" efficiency value. Grating relative efficiencies were calculated by dividing the light intensity diffracted to the grating order of interest (noise background subtracted) by the light intensity reflected from a flat witness mirror coated exactly the same way as the grating (noise background also subtracted). Absolute efficiencies were then derived by multiplying the relative efficiency by the mirror reflectance at the given wavelength. Later on, absolute efficiency measurements were performed as a ''sanity check'' to verify the witness sample reflectance: in these tests, the intensity of the diffracted beam was compared directly with the incident beam intensity. Both direct and indirect measurements produced absolute efficiency values identical within 0.5% accuracy. Witness samples and reference mirrors reflectivity measurements were verified independently at outside facilities.3 The light source used in the measurements was unpolarized within what is measurable at this waveband accuracy of ▒2%. The experimental efficiency data for the G185M gratings (replicas A and B) are presented in Fig. # 46. The measurements were performed in the extended VUV-NUV wavelength range from 120 nm to 255 nm.  
 
Scattering Light Measurements  
The scattering light was measured directly for this grating using the FAUST (Fig. # 20). For scatter tests the grating was kept tip/tilted such a way that the dispersion plane (e.g. series of images at a grating order) never falls outside the very center of the detector imaging area. The long detector/camera mirror rotational arm provides high angular resolution for the measurements, but at the same time demands 10-20 μrad grating tilt adjustment accuracy. The gratings scatter is lower than 10-5 (at GDF) 1 nm away from the specular maximum and 10-6 at 10 nm away from the specular maximum at any test wavelengths.3 This G185M grating meets the required scatter specification of < 2Î10-5/┼ for all COS NUV flight gratings.  
 
Sources of the Refractive Indices  
For this investigation, the refractive indices (RIs) of Al and MgF2 were taken from the handbooks of Palik11 and AIP.12 RIs for bulk MgF2 taken from well-known references11, 12 were found to be not suitable for thin optical layers at wavelengths between 115 and 170 nm.17,18 The method referred to below and based on scale fitting of the calculated and measured grating efficiencies was outlined for derivation of the thin-film optical constants at hard to measure wavelengths. The values of the real and imaginary parts of the RI for MgF2 obtained that way are listed in the Table.  
Table. MgF2 RI for evaporated thin-films with layer thicknesses ~40 nm, derived from efficiency modeling  
λ, nm  
Re(RI)  
Im(RI)  
120  
1.759  
0.12  
130  
1.653  
0.1  
140  
1.603  
0.06  
150  
1.554  
0.04  
160  
1.482  
0.001  
170  
1.468  
0  
180  
1.451  
0  
190  
1.442  
0  
200  
1.439  
0  
212.5  
1.437  
0  
225  
1.434  
0  
237.5  
1.432  
0  
250  
1.43  
0  
262.5  
1.4275  
0  
 
Efficiency Modeling  
The investigated efficiency models for different nominal border profiles and vertical shifts (layer thicknesses) as well as for scaled border profiles and vertical shifts were computed by PCGrate-S(X) v.6.1 using various RI libraries.18 As the study has shown, the discrepancy between the calculated efficiencies in the two-border grating model with a semi-infinite Al layer and in the complete five-border model does not exceed a few hundredths of a percent throughout the wave band because of a small depth of light penetration into the metal. The two-border model with protective MgF2 and semi-infinite Al layers was assumed for further modeling purposes (Fig. # 47). The calculation was performed using the average AFM border profile shapes shown in Fig. # 48. The lower Al-MgF2 border was scaled by factor of 1.04 from the nominal value. The thickness of the MgF2 layer used in calculations was 40.1 nm and the vertical shift between the border reference levels (a vertical displacement of one boundary with respect to the other) was 68.5 nm. The “resonance finite” mode taken into account the finite conductivity of the both grating layers without any approximation was used in calculations.9 RIs for Al were taken from the handbooks of Palik.11 RIs for MgF2 were derived from the numerical procedure based on comparing experimental efficiencies data with calculated values from MIM-based modeling using precise AFM-measured groove profile for the particular example of the G185M grating.18 The random roughness topography of the grating borders was not included in the efficiency model because of the very small value of rms roughness compared to the working wavelengths. The correlated components from the average AFM border profiles were included automatically in the computation by working out the real border profiles with a high degree of accuracy. The results were calculated for the nonpolarized light incident at an angle of 34.7 to the normal to the grating surface as shown in Fig. # 47. The calculation was run in the measured points over the extended range of 120-255 nm.  
Convergence of the efficiency results was investigated in the main accuracy parameter N (number of collocation points), in including (or non-including) options for accelerating convergence, in the type of integration step (equal along X axis (x) or along border profile (s)), in the type of RI data interpolation (constant, linear, or cubic splines), and in smoothing border profiles by trigonometric harmonics.9 The convergence and accuracy were checked for the efficiency results with N being taken from 164 (upper border) and 166 (lower border) to N = 653 and 661 respectively. The convergence of the efficiencies is fast enough and the discrepancies between the results for appropriate calculation models throughout the entire wavelength region (Fig. # 49) are much less than corresponding deviations between the measured and calculated data (Fig. # 52). For comparison with the measured data, we chose the model with N = 164&166 for polygonal border profiles, with all the checked options for convergence acceleration, with the s type of integration step, and with the RI linear interpolation. In the range covered, the model yields accurate results (with the energy balance within a few tenths of a percent at a high calculation speed).  
The time required to calculate one point for the G185M grating efficiency was about 17 s for N = 164&166 using an IBM« Think Pad with Intel« Pentium« M 1700 MHz processor, 1 MB Cache, 400 MHz Bus Clock, 512 MB RAM, and controlled by OS Windows« XP Pro.  
 
Comparison Between Calculated and Measured Efficiencies  
A dielectric coating applied over a metallic grating brings about, other conditions being equal, the appearance of resonance anomalies associated with energy transport by leaky waveguide modes forming inside the dielectric.18 The position and strength of these anomalies is intimately connected with trajectories of the scattering matrix poles and zeros of diffraction amplitudes in the complex plane, which are different for different polarizations (e.g., Ref. 42, Sections 5.3.2 and 5.4.1).  
 
Influence of Layer Shapes on Efficiency  
To determine which of the two AFM-measured boundary profiles, MgF2 (border profile 1 measured after Cr/Al/MgF2 coating) or (Cr)-Au (border profile 2 measured before Cr/Al/MgF2 coating.), is closer to the MgF2-Al boundary, we started with modeling the NP efficiency of a two-boundary grating. We assume a conformal MgF2 layer (the lower MgF2-Al boundary is identical in shape to the MgF2 one) with the 40.1 nm thickness. The calculated efficiencies (Fig. # 50, pink curve) differ from the measured values in time throughout the whole wavelength range, thus implying3 invalidity of a model with a conformal layer. All calculated efficiency data presented in Fig. # 50 were obtained with the RIs of Al and MgF2 taken from the handbook of Palik.11 Although hereinafter the experimental efficiency data of two grating replicas (A and B) are displayed, we will focus primarily on discussing the grating A data (solid dark blue squares in Figs. # 50, # 51, and # 52), because replica A is the grating on which more measurements were performed.  
The next step is to use two models with nonconformal layers, one with the lower boundary being the same as border 2 (Fig. # 50, yellow curve) and the other with the boundary scaled from border 2 at all points by a factor of 0.488 to the profile depth of border 1 (Fig. # 50, bright green curve). In both cases, a vertical displacement of one boundary with respect to the other (shift of the boundary reference levels) was 40.1 nm, as in the conformal model. As evident from Fig. # 50, the nonconformal model with unscaled lower boundary yields a noticeably superior qualitative agreement with experimental data. This suggests that the MgF2-Al boundary more closely resembles border profile 2 than border profile 1. The model takes into account the fact that the thickness difference of 23.8 nm between the lower and upper boundaries should be added to the conformal vertical displacement (40.1 nm) to obtain an adequate vertical displacement for the nonconformal MgF2 layer. In this way the period-averaged thickness of the nonconformal MgF2 layer is kept approximately equal to 40.1 nm within the boundary shape distortion.  
To determine the effect of profile shape, we set up models with equal depths and vertical shifts. The first one has border 1 scaled to the depth of border 2 (making it grater by a factor of 2.05) and a vertical displacement between the zero boundary levels equal to 63.9 nm. As seen from Fig. # 50, the efficiency of this model (orange curve) is close to that of another model with unscaled border 2 and a vertical shift of 63.9 nm (sky blue curve), while it is inferior by 40% or more as far as matching the experimental efficiencies. The latter suggests that, to set up an exact model, one has not only to determine the depth of the MgF2-Al boundary but also to take into account the shape of its profile.  
Having determined the type of the MgF2-Al boundary profile, we have to refine it by scaling the shape in depth and then comparing the efficiencies obtained for each model with experimental data. Another fitting parameter is the vertical displacement of the boundaries. By automatic modeling of the efficiency over a small-meshed grid of these two parameters and wavelength, one can determine the average thickness of the MgF2 layer from the best fit between the calculated and the experimental efficiencies. Even slight changes (with a few nanometers) in profile depth and vertical displacement give a noticeable rise to the efficiency at fixed wavelengths, particularly in resonance regions. Fig. # 50 presents an efficiency curve (heavy dark blue) for the model with a lower-boundary scaling factor of 1.04 and a vertical displacement of 68.5 nm. The model with these parameters of the layer geometry provides the better least-squares fit (not worse than 20%) of calculated efficiency to experimental data, both in the medium and in the long-wavelength ranges. As to the short-wavelength part, no variations in the lower boundary profile chosen within our approach yield theoretical values of the efficiency close enough to the measured ones.  
 
Influence of RIs on Efficiency  
The efficiency curves calculated for the best model in Fig. # 50 (heavy dark blue curve) for various combinations of the RIs of the materials taken from Refs. 11, 12, and 17 are presented in Fig. # 51. A twofold better fit to measured efficiencies in the short-wavelength range is reached for the MgF2 RI taken from Ref. 17 rather then from Refs. 11, 12. The radical improvement is due to the noticeable absorption evident at wavelengths beyond 112 nm for the MgF2 RI taken from Ref. 17. Figure # 1 illustrates the difference trend of imaginary parts of the MgF2 RI taken from different sources. Still, despite the considerably better agreement between theoretical and experimental data at the short-wavelength region, the absorption coefficients taken Ref. 17 are smaller near the absorption edge for the model to produce a real quantitative fit. The curve at Fig. # 51 (heavy dark blue) with the Al RI taken from Palik's handbook demonstrates a good quantitative agreement with experiment at all points in the medium- and long-wavelength parts of the range, with the exclusion of points near 163 nm. The behavior of this theoretical curve over the 160-180-nm interval and the comparison of the calculated and measured data suggest an overestimation of absorption in this region from use of the MgF2 RI taken from Ref. 17. On the whole, however, the model in Fig. # 51 with the RI of MgF2 from Ref. 17 and of Al from Ref. 11 provide a qualitatively correct fit to the behavior of efficiency throughout the wavelength range from 120 to 255 nm.  
 
Deriving Factual MgF2 Refractive Indices from Efficiency Modeling  
The result described above stimulates further refinement of the model now aimed at obtaining a correct RI of MgF2. A comparison of the exactly calculated grating efficiency  
with measured values offers a possibility to not only find discrepancies between the tabulated and factual values of optical constants, but also to solve the inverse problem, i.e., to derive the RI of a layer material from the grating efficiency data.15,75 The idea that underlies the proposed method is based on the nonscalar properties of the grating efficiency inherent in certain modes of its operation. Given other fixed parameters, the efficiency behavior of a grating cannot be described by the scalar theory of diffraction unless the ratio between relative and absolute grating efficiencies is proportional to the coating agent reflectance. We will illustrate the point by the following example explaining the grating efficiency modeling as an instrument used to extract the RI of MgF2 from the measured grating efficiencies data. In setting up the final grating model, we shall start from the layer's geometry of the model presented in Fig. # 50 (heavy dark blue curve). As was done with the best models until now, we use the Al RI from Ref. 11. In view of the fact that the RI of MgF2 from Ref. 17 provided a better fit to the measured efficiencies in the previous section throughout the wavelength range covered, we kept the real part of the MgF2 RI from Ref. 17 in the new grating model unchanged. We shall be searching for the unknown imaginary part of the RI in the form of a piecewise linear function with nodes through every 10 nm, starting from 120 nm. Considering that the absorption at the MgF2 layer beyond 170 nm (as is evident from a comparison of calculations with experiment) is small, we will set the imaginary part of the MgF2 RI, starting from 170 nm and further into the long wavelength region, to zero. We now have to determine the slopes of the piecewise linear function over the 120-170 nm interval. Since only three experimental points were measured for grating A in this range, we shall improve the accuracy by adding an other data point, measured on grating B at 134.8 nm. Next we perform least squares fitting of the calculated efficiency curve for those four points with a step of 0.01 for the imaginary part of the RI. The values of the imaginary part of the RI for MgF2 obtained this way are listed in Table. To smooth out the function we then replaced the derived modeling zero value of the imaginary part of the RI of MgF2 at 160 nm with the 0.001 value; such a small fit practically does not change the efficiency value at that wavelength. Figure # 52 plots the efficiency curve obtained for the final model RIs (heavy dark blue curve).  
 
Influence of Fine Layer Parameters on Final Efficiency  
What only remains is to check whether the average-thickness parameters of the MgF2 nonconformal layer used in the final model provide a better fit between the calculated and experimental values of efficiency throughout the wavelength range with a new RI library. To do this, we scale the vertical displacement and boundary parameters for the final model. Graphical results of this three-parameter optimization (scale, shift, and wavelength) are displayed in Fig. # 52. An analysis of these results shows that the parameters of the final model do indeed provide the best agreement between the measured and calculated values of efficiency throughout the wavelength range. The relative deviation of experiment from theory for all wavelengths at which grating A was studied does not exceed 10% throughout the wavelength range. Finally, we will demonstrate the outcome of the imaginary part of the MgF2 RI changes on the G185M grating efficiency. To do that we compile a library similar to the one from Table but with the imaginary parts of the RIs decreasing linearly from 0.1 to 0 at the 120-170 nm wavelength region. This function is in fact a result of our averaging the Table function and, as can be readily verified, differs from it at all points by no more than 0.02. Figure # 52 presents an efficiency curve (sky blue curve) calculated by use of these approximate values of the MgF2 absorption index; all other parameters of the final model remain intact. A comparison of the curve efficiencies (Fig. # 52) based on scaled (sky blue curve) and exactly calculated (heavy dark blue curve) values of absorption shows that the efficiency changes at the wavelengths where the RI imaginary values scale only slightly are indeed appreciable.  
 
Efficiency Certificate  
An example of the -1st order subwavelength grating efficiency standardized in the VUV-NUV range using AFM, RIs derived in a variety of ways, and PCGrate-S(X) v.6.1 is shown in Fig. # 53 for different polarizations. The Efficiency Certificate of a resonant G185M Al+MgF2 grating is based on the best NP model and performed in the extended wavelength range under investigation with a step of 1 nm. There were no free parameters in the calculation.  
 
Summary  
  • An overall disagreement between the experiment and the accurate modeling based on AFM profile measurements and RIs derived using best fits between the calculated efficiency data and the experimental results can be obtained for this type of gratings within less than 10% of the absolute efficiency throughout the wide wavelength range in VUV-NUV.  
  • The calculations are sensitive to the MgF2-Al border profile depth and shape, with the best results achieved for the nonconformal model using a scaled AFM border and a respective vertical shift between borders.  
  • The impact of the refractive indices taken from well-known sources11,12,17 on the efficiency is strong in the wavelength range covered.  
  • In some sensitive cases there is a need for detailed numerical efficiency modeling to obtain accurate values of the unknown optical constants of thin layers; linear extrapolation or standard interpolations of known RIs may induce significant errors into the calculated efficiency values.9  
  • Anywhere away from scalar domain, the finite conductivity of VUV-NUV grating substrates and coatings leads to very high polarization sensitivity.  
  • The modeling requires the electromagnetic theory to be used for taking into account AFM-measured border profiles, finite conductivity of grating materials, and different polarization states. Resonance efficiency anomalies associated with waveguide funneling modes inside dielectric layers with real shapes and optical properties can be exactly modeled using the boundary integral equation code PCGrate-S(X), which proved to be a reliable tool to study complex problems of diffraction on multilayer gratings.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation process can be obtained from the downloading page (Cert_5870_holo-replica_Al-MgF2_VUV--NUV.zip file). Various files both *.xls (MS Excel«) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.1) formats relating to this grating efficiency certification are included in the package. You can view the later by the PCGrate-S(X) v.6.1 Complete Demo (updated on the 3rd May, 2005).  
 
 
 
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Efficiency Certificate for a Multilayer Mo4Ru6/Be Blazed Holographic Grating in the Soft X-ray-EUV Range  
 
General Parameters of the Grating  
Here we present an example of the soft x-ray-EUV multilayer grating efficiency5,75,76 standardized using synchrotron radiation, atomic force microscopy (AFM) and PCGrate-S(X) v.6.1 software. The master grating was fabricated by Spectrogon UK Limited (formerly Tayside Optical Technology). The groove pattern was fabricated in fused silica using a holographic technique. The groove pattern was ion-beam etched to produce an approximately triangular, blazed groove profile. The grating has 2400 grooves/mm (/mm), a concave radius of curvature of 2.0 m, and a patterned area that is 45 mm by 35 mm. The master grating was uncoated. The surfaces of the replica gratings were oxidized aluminum. A thin overcoating of SiO2 was applied to the final concave replicas for the purpose of reducing the nanoroughness of the oxidized aluminum surface. Multilayer gratings were produced by application of Mo4Ru6/Be multilayer coatings5 to two replicas of the holographic master grating. Beryllium-based multilayer coatings can provide substantial reflectance at wavelengths near 11 nm. Such a Mo4Ru6/Be multilayer coating with 50 bi-layers was applied to the grating substrate. The coating was deposited by the magnetron-sputtering technique. Here we investigate one of the multilayer gratings - Grating # 1.5  
 
Border Profiles Measurements
The surface of the multilayer grating was characterized using a Topometrix Explorer scanning probe microscope, a type of AFM. The grating topography was measured merely for the master, replica, and multilayer gratings. The scan was performed across the grooves over a range of 1 mcm (2-nm pixels) - see Figs. 11, 15, 55. The silicon probe had a pyramid shape. The base of the pyramid was 3 to 6 mcm in size, the height of the pyramid was 10 to 20 mcm, and the height to base ratio was approximately 3. The tip of the pyramid had a radius of curvature less than 20 nm. The AFM scans were performed using the non-contact resonating mode, where the change in the oscillation amplitude of the probe is sensed by the instrument.  
 
A typical groove profile derived from the AFM image (1 mcm in size) of the master, replica, and multilayer gratings are shown in Figs. 14, 17, 56. These groove profiles have from 210 to 120 points. The groove profiles are approximately triangular in shape with rounded corners and troughs and with facet angles of 2.5░ & 5.5░, 3.4░ & 6.2░, and 3.0░ & 4.1░ respectively.13,29,76 The average groove depths derived from the AFM images are in the range 7 to 8 nm, 8.5 to 9.5 nm, and 8 to 9 nm respectively. Within the AFM groove-to-groove variation of the facet angles, the border shapes did not significantly change after multilayer coating.76 As determined below the average surface of the multilayer grating was characterized using a scaled replica AFM profile (Fig. 57).  
 
Multilayer Coating Investigation  
A newly developed multilayer coating, Mo4Ru6/Be, has several beneficial properties.35 The absorber layer consists of an alloy material with a composition of Mo4Ru6. A normal-incidence reflectance of 69.3% at a wavelength of 11.4 nm was achieved. The multilayer's stress, which is composition dependent, varies between -30 and +30 MPA and can be designed to be nearly zero. Owing to the microstructural properties of the layers (the alloy layers are amorphous and the Be layers are polycrystalline), the coating smoothes the high-frequency roughness of the substrate. Since the efficiency of the multilayer-coated grating in the short-wavelength region strongly depends on roughness, a multilayer coating that can smooth the grating surface is beneficial. From the maximum reflectance of a multilayer Mo4Ru6/Be coating with 50 layers, a bilayer thickness of 6.03 nm and the ratio of the beryllium layer thickness to the bilayer thickness equal to 0.58 were determined.76 The total thickness of the coating on the grating was 301.5 nm. This coating thickness was much larger than the calculated grating average groove depth of 6.5 nm.  
 
Investigation of Random Roughness and Interdiffusion
The nanoroughness was determined by integrating the power spectral density (PSD) function over the 4-40 /mcm spatial frequency range for the AFM data in the center of a grating before and after application of the Mo4Ru6/Be multilayer coating. The rms value of the nanoroughness derived from the PSD function before coating was 1.35 nm and included spikes that contributed significantly to the nanoroughness value.76 The rms nanoroughness derived from the AFM image after coating (Fig. 55), which included bump-type features, was 0.93 nm in the integrated spatial frequency range.  
 
No information is available about layers interdiffusion.  
 
Efficiency Measurements
The efficiencies of the multilayer replica gratings were measured with the Naval Research Laboratory beamline X24C at the National Synchrotron Light Source at the Brookhaven National Laboratory. The synchrotron radiation was dispersed by a monochromator that had a resolving power of 600. Thin filters suppressed the radiation from the monochromator in the higher harmonics. The wavelength scale was established by the geometry of the monochromator and the absorption edges of the filters. The gratings were oriented so that the groove facets with the larger facet angle (measured from the surface of the grating substrate) faced the incident beam. The incident radiation was ~80% polarized with the electric-field vector in the plane of incidence. In this orientation the electric-field vector was almost perpendicular to the grating grooves. The 13.9░ angle of incidence, used when measuring the multilayer grating, permitted observation of a wide angular range without obscuration of the higher diffraction orders by the detector.76 At fixed incident wavelengths the detector was scanned in angle about the grating. The incident beam was ~1 mm in size. Measurements were performed at a number of fixed wavelengths in the short (10.5-13.0-nm) and in the long (25-50-nm) wavelength ranges. From the efficiency measurements performed at a number of fixed incident wavelengths, it was found that the peak -2nd efficiency of the multilayer grating occurred at a wavelength of 11.4 nm. Low efficiencies could be measured because of the relatively high multilayer reflectance and groove efficiency of the grating and the high sensitivity and low background current (3pA) of the silicon photo-diode detector (type AXUV-100G provided by International Radiation Detectors, Inc.).5  
 
As an example the efficiencies of the multilayer grating measured in different orders at an incident wavelength of 11.37 nm are shown in Fig. 58. The 0.5░ widths (at the half-peak efficiency levels) of the orders shown in Fig. 58 result from the 1-mm slit over the detector.5  
 
Scattering Light Measurements  
Some scattering light was observed during the efficiency measurements. Its level between the orders is low for diffraction angles in the off-blaze direction (>13.9*) and higher in the on-blaze direction (<13.9*).13 The scattering light level was low and was not measured for this grating.  
 
Sources of the Refractive Indices  
The choice of the refractive indices of the layer materials is very important in development of the efficiency calculation model for multilayer gratings in the Extreme Ultraviolet wavelength range.1 Efficiency investigations for the multilayer grating were made in two widely separated wavelength regions. For wavelengths as long as 39 nm, optical constants derived from Ref. 10 were used for efficiency calculations. For wavelengths longer than 39 nm the compilation of experimental values from Ref. 11 was used.  
 
Efficiency Model
The preliminary theoretical efficiency investigation was performed for different calculation models of plane gratings due to small ratios of the grating sizes to the radius of curvature. It was found5,75,76 that the calculated short wavelength efficiencies of the Mo4Ru6/Be multilayer grating were very sensitive to variations in the groove profile. Small changes in the assumed groove shape, groove height, and facet angles resulted in significant changes in the calculated efficiencies. In this case, the average depth of modulation of the layers was scaled to achieve the best agreement with the experimental data near 11.4 nm where the efficiency was maximal. The depth of the profile of the replica and multilayer grating surfaces was ~8.5 nm, as derived from the AFM measurements in Figs. 17, 56. During the process of coating the grating substrate with multiple layers to produce a multilayer grating, a smoothing of the groove profile takes place.76 As a result the depth of modulation of the upper layers decreases appreciably. Because the influence of diffraction by the upper layers on an absolute grating efficiency is higher than that by the lower layers, and because there was no information on a change in the profile modulation depth (and the profile deformation) from one layer to another, a model with average with respect to all layers profile modulation depth was used. By scaling the initial groove profile of the replica grating, a groove profile was determined that resulted in the smallest least-squares difference between the calculated and measured efficiencies at a wavelength of 11.375 nm. The resulting depth was ~6.5 nm, which means that the profile becomes appreciably smoother in the upper layers. A scaled averaged groove profile used in the efficiency calculations of the multilayer grating is shown in Fig. 57. The polarization angle was taken equal to 63.43░. In this orientation, the electric field vector was nearly perpendicular to the grooves (80% of TM and 20% of TE).  
 
It was also determined that the calculated efficiency in the long-wavelength range depended significantly on the degree of the upper-beryllium-layer oxidation.5 For example, the efficiency in 0 and ▒1 orders for the unoxidized upper beryllium layer and the completely oxidized upper beryllium layer differ by a factor of ~3 for a wavelength of 40 nm. Since the effect of the beryllium oxide (BeO) film thickness for short wavelengths is insignificant, we determined the oxide film thickness by the best agreement of the efficiency values measured in the 25-40-nm range with those calculated with the least-quares method. The calculated value of the BeO film thickness thus obtained is equal with a small uncertainty to one third of the Be layer thickness. The model with the upper one-third oxidized beryllium layer has been used in all calculations. Because of the lack of refractive indices data for BeO in the 40-50-nm range, for example, in Ref. 11, the uppermost BeO layer was not taking into account in this wavelength range. Note that a model with a completely oxidized upper layer and another type of the average groove profile was used in efficiency calculations of the very similar grating in Ref. 76.  
 
Random Roughness Accounting  
The Strehl factor77 calculated for the same plane multilayer stack, the ratio of the reflectance maxima reduced by rough interfaces and the reflectance maxima obtained with perfect interfaces, was used to account approximately for the layer interdiffusion and random nanoroughness of the Mo4Ru6/Be multilayer grating. The inferred nanoroughness, resulting in the best agreement between the peak values of the calculated and measured efficiency in the -2nd order near 11.375 nm, was 1.03 nm. This is an intermediate value between the measured AFM data before and after coating. The random roughness topography of the grating was taken into account in the fit by applying the amplitude Debye-Waller factor5 with the rms roughness of 1.03 nm for all interfaces. The periodical lateral-correlated component of the border roughness from the average AFM groove shape was included automatically by accounting for the average groove profile with a high degree of accuracy. An assumption about the absence of the vertical correlation between the border random roughness components was applied in this model.  
 
Efficiency Modeling
Convergence and accuracy of the efficiency results of a 103-boundary grating was investigated in the different type of the calculation mode ("normal" or "resonance" in v.6.1), in the type of lower border conductivity ("perfect" or "finite"), in the main accuracy parameter N (number of collocation points), in including or non-including options for accelerating convergence, in the type of integration step (equal along X axis (x) or along groove profile (s) and with or without a half-segment shift), and in the source of RI data with the type of their interpolation (constant, linear, or cubic splines).9 The thicknesses of SiO2 and Al2O3 layers were chosen in accordance with the Replica Certificate (74.3 nm and 3 nm respectively). Figures 59, 60, 61 show the measured and different calculated efficiencies in the diffraction orders of the multilayer grating with 50 Mo4Ru6/Be periods in the short wavelength range. The measured inside and outside orders (Fig. 59) are separated in wavelength while the efficiencies calculated5,76 using the asymptotic formula for multilayer gratings61 (“normal” mode) are not (Fig. 60). In contrast, the efficiencies calculated using the fully rigorous approach (“resonance” mode) used further in this Efficiency Certificate in the short wavelength range (Fig. 61) are in good agreement with the measured data and with the phenomenological approach.75 As seen from comparison of curves in Figs. 59 and 61, the calculated efficiency curves of some orders tend to be larger and wider than the respective measured efficiency curves. This may result substantially from scaling and averaging of the border profiles as well as from degree of uniformity. The measured efficiencies of basic orders are significantly different in spectral widths and their maxima shifts for a similar grating investigated, as seen from another comparison.76 When the asymptotic approach for efficiencies of multilayer gratings is used, the computations execute much faster (by a few orders), owing to the implementation of approximate algorithms, but only the heights of order maxima are predicted well. To determine the spectral separation of the inside and outside orders1,75 and the exact shape and position of efficiency curves, rigorous electromagnetic theory for high frequency multilayer-coated gratings must be applied (“resonance” mode). However, the very fast “normal” mode with non-including options for accelerating convergence, "perfect" type of lower border conductivity, and x-type integration step with a half-segment shift was used for calculations in the long wavelength range (13-50 nm) with the higher rate of convergence and the same accuracy as compared with the “resonance” mode.  
 
A high rate of convergence of the efficiency modeling was observed for the “resonance” mode, non-including options for accelerating convergence, "finite" type of lower border conductivity, and x-type integration step with a half-segment shift (the “best” model in Fig. 62). The linear extrapolation and interpolation types were chosen for approximation of RI data between and well apart from the tabulated points. Even the minimal number of collocation points of N = 121 is enough to obtain accurate efficiency results in all orders of interest. The difference between data obtained with high accuracy (N = 481) and low one (N = 121) is ~1% in the -2nd order. The similar difference is observed for other diffraction orders, between efficiency results obtained for x-type or s-type integration step, and including (N = 481) or non-including (N = 121) acceleration convergence options.  
 
The total error for all points and ranges derived from the energy balance was an order of 1.E-4. The time taken up by one rigorous computation (one scanning point in the short wavelength range) in the “resonance” calculation mode for N = 121 on a workstation with two Quad-Core Intel« Xeon« 2.66 GHz processors, 8 MB L2 Cache, 1333 MHz FSB, and 16 GB RAM, is ~55 sec when operating on Windows Vista« Ultimate 64-bit and employing eightfold paralleling. The time taken up by one computation in the long wavelength range (“normal” mode) for N = 121 is ~0.1 sec using the same workstation.  
 
Comparison Between Calculated and Measured Efficiencies  
For further investigations and calculations we choose the best multilayer grating model described above. In the covered range, this model yields accurate results at a high calculation speed. The validity of the model was supported by comparing the experimental and predicted efficiency data in ▒5, ▒4, ▒3 ▒2, ▒1, and 0 orders in the short (11-12 nm) and ▒1 and 0 orders in the long (25-50 nm) wavelength ranges (Figs. 59, 61, 63). The measured and modeling efficiency maxima are close for principal orders with uncertainties of ~20% in the short wavelength range covered. The maxima positions coincide for the principal orders over the entire wavelength range. The rms (measured to calculated) efficiencies for +1, -1, and 0 orders in the long wavelength range involved are 23.67%, 28.7%, and 7.36% respectively.  
 
Efficiency Certificate  
For issuance of the present Efficiency Certificate, calculations were performed in the extended wavelength range from 10 nm to 50 nm. There were no free parameters in the calculation. The calculation was performed for the scaled average AFM replica groove shape with a depth of 6.5 nm, random roughness, derived layers' thicknesses and the optical constants for all the materials taken from the compilation of CXRO10 (<40 nm) and from the Palik's handbook11 (e40 nm). The Efficiency Certificate in the principal orders of the 2400 /mm multilayer Mo4Ru6/Be blazed holographic grating for the extended wavelength range is presented in Figs. 64 and 65. In addition, relative efficiencies are calculated in the approximation of perfectly-conductive grooves of the considered multilayer grating for -6, ▒5, ▒4, ▒3 ▒2, ▒1, and 0 diffraction orders (Fig. 66) in the interesting wavelength range of 4.5-30 nm.  
 
Summary  
  • A good quantitative overall agreement between the experiment and the predicted modeling efficiency based on both a real groove profile (AFM-measured) and appropriate RI libraries can be obtained for these types of gratings and wavelengths.  
  • Calculations are very sensitive to boundary profiles, with the best results obtained for the scaled average AFM-measured grooves of a replica grating.  
  • It was determined that the calculated efficiency in the long-wavelength range depended significantly on the degree of the upper-beryllium-layer oxidation.  
  • Intermediate values of the random roughness (rms ~1 nm) are of some importance for the multilayer grating efficiency in the short wavelength range covered and cannot be ignored in calculations.  
  • The impact of small changes of the real and imaginary parts of the RIs upon the efficiency is not large. Using RIs from CXRO data gives accurate efficiency results in the wavelength range smaller than 40 nm. RIs from the Palik's handbook11 gives more accurate efficiency results (in respect to using other available data1) in the wavelength range starting from 40 nm.  
  • The rigorous electromagnetic code can be used effectively for a precise modeling of layered gratings in the soft x-ray and EUV ranges despite small wavelength-to-period ratios and a big number of boundary profiles with real shapes.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation and measurement processes can be obtained from the downloading page (Cert_2400_holo-blaz_MoRu-Be_soft-X-ray--EUV.zip file). Various files both *.xlsx (MS Excel« 2010) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.1) formats relating to this grating efficiency certification are included in the package. You can view the later by the PCGrate Demo v.6.1 Complete (updated on the 16th April, 2010).  
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Efficiency Certificate for a Flight Multilayer Mo/Si Lamellar Grating in the EUV Range and TM polarization  
 
General Parameters of the Grating  
Here we present examples of the Mo/Si lamellar grating efficiency standardized for the Extreme-Ultraviolet Imaging Spectrometer (EIS) on the Hinode (former Solar-B) mission, the first implementation of a multilayer grating on a satellite instrument.78 We describe the performance of the flight grating that has an optimized groove profile and matching multilayer coatings developed for the EIS instrument. The Efficiency Certificate was created using data of synchrotron EUV radiation and atomic force microscopy (AFM) measurements and of PCGrate-S(X) v.6.5 software calculations. The grating blank, fabricated by Zeiss Laser Optics GmbH, is of fused silica and has toroidal radii of 1182.9 mm in the dispersion direction and 1178.3 mm in the perpendicular direction. The slope error in the figure is 0.49 arc sec rms, and the nanoroughness of the blank is ~ 0.25 nm rms. Mo/Si multilayers with different periods cover each half of the grating. The multilayer coatings have high normal-incidence reflectance in two wavelength regions, 17-21 and 25-29 nm, which include many intense solar spectral lines that are emitted over a wide range of plasma temperature. The toroidal grating is used at a magnification of 1.44 and forms stigmatic images of the slit plane on two CCD detectors for the two bandpasses centered at 19.5- and 27.0-nm wavelengths. The diameter of the grating blank is 100 mm, and the usable patterned area is specified to have a 90-mm diameter. Here we investigate one of the 4200 /mm multilayer gratings - FL1. The theoretical and experimental efficiencies of the backup flight grating FL7 are presented in Refs. 1 and 78.  
 
Border Profile Measurements
The grating substrates were fabricated by Zeiss with a holographic technique to form a sinusoidal 4200 groove/mm pattern and ion-beam etching to shape the laminar grooves and to achieve the specified groove depth ~ 6.0 nm. The trapezoidal groove profile based on sampling AFM measurements of the FL1 master grating (before coating) has a depth of ~ 5.8 nm with side slopes and equal top and groove widths (Fig. 67). The groove profile was assumed to have sloping sides with an angle of 35░ as indicated by AFM studies of a Zeiss test grating.78  
 
Multilayer Coating Investigation  
Multilayer coatings were applied to the two halves of the pair of flight-quality optics, the FL1 grating and the M2 parabolic mirror, at the Columbia University multilayer deposition facility by using magnetron sputtering.79 The two multilayers with 20 bilayers of different periods, optimized to reflect the EIS short and long wave bands, were deposited onto the FL1 flight grating and the M2 flight mirror on sequential deposition runs with a mask used to cover half of the optic. The deposition parameters were optimized on the basis of the measured reflectances of small test flat mirrors at grazing (in the hard x-ray) and normal incidence (in the EUV). The reflectance of the M2 mirror was measured at the Naval Research Laboratory X24C beamline (see below) at 41 points on each half of the mirror total of 82 points. The points were on a square grid with 15-mm separation and extended to the edge of the usable area (diameter of 150 mm). The angle of incidence at the center of the mirror was 2.15░, the same as the angle used in the EIS flight instrument. It was found that the reflectances measured at several points near the edges of the usable areas of M2 were low. Excluding these points, the average reflectances on the short-band and long-band sides of M2 are shown in Fig. 68 together with the theoretical fits obtained by PCGrate-SX v.6.5. The peak values are 35.2% at 19.4 nm and 21.8% at 26.7 nm.  
 
Investigation of Random Roughness and Interdiffusion
The structural parameters of the layers were derived from the reflectance fits in the EUV. They depend crucially on the accuracy of the optical constants used in the calculation, on detailed knowledge of the layer and interface morphology, and on the condition of the surface (especially in the EUV where thin surface contamination layers can strongly affect the normal-incidence reflectance). The best fit obtained by using the optical constants for Mo from Ref. 80, along with the CXRO data for Si, are shown as the bulk curve in Fig. 68. The M2 short-band half extracted parameters, which are close to the design parameters are: 20 Mo/Si layer pairs with the bilayer period D = 10.3 nm, Mo thickness to D ratio Γ = 0.37, Si-Mo interface rms roughness σ = 0.2 nm, and Mo-Si rms roughness σ = 0.85 nm. The Si protective capping layer is 2 nm thick. The M2 long-band half extracted parameters, which are also close to the design parameters are: 20 Mo/Si layer pairs with the bilayer period D = 14.35 nm, Mo thickness to D ratio Γ = 0.34, Si-Mo interface rms roughness σ = 0.2 nm, and Mo-Si rms roughness σ = 0.85 nm. The Si protective capping layer of 2 nm was modeled by using 1.5-nm-thick amorphous SiO2 on 1.5-nm Si in order to account for the oxidation of the Si capping layer.  
No information is available about layers interdiffusion.  
 
Efficiency Measurements
The efficiency of the multilayer FL1 grating was measured with the Naval Research Laboratory beamline X24C at the National Synchrotron Light Source at the Brookhaven National Laboratory. The synchrotron radiation was dispersed by a monochromator that had a resolving power of 600-1000. Thin filters suppressed the radiation from the monochromator in the higher harmonics. The wavelength scale was established by the geometry of the monochromator and the absorption edges of the filters. The incident radiation was ~ 80% polarized with the electric-field vector in the plane of incidence. In this orientation the electric-field vector was almost perpendicular to the grating grooves (the TM polarization).  
The efficiency of the multilayer-coated FL1 grating was measured at 30 points (10 mm apart) on each side of the grating(total of 60 points). The points extended to the edge of the usable area at a diameter of 90 mm. The efficiencies were measured at nine wavelengths on the short-band side (from 17.1 to 22.0 nm) and eight wavelengths on the long-band side (from 25.1 to 29.3 nm). The efficiencies measured at the central points on the short-band and long-band sides of FL1 are shown in Fig. 69, Fig. 70. The angle of incidence at the center of the grating was 6.5░, whereas the corresponding central angle of incidence on the grating in the EIS spectrometer is 4.48░. The measurement angle was set to 6.5░ to place the obscuration of the direct beam by the detector between the measured orders. Because the grating efficiency is a weak function of angle of incidence near normal incidence, the small difference between the measurement angle 6.5░ and the operating angle 4.48░ is not significant. The FL1 grating efficiencies measured at two extreme points near the edge of the usable area on each side of the grating were found to be low. As expected on the basis of the FL1 calculated efficiencies shown in Fig. 78 and Fig. 79, the zero-order efficiency is relatively low in both wave bands owing to the near optimal value of the FL1 groove depth, ~ 6.0 nm. The measured peak efficiencies in the short and long bands are 12.4% at 19.25 nm and 8.3% at 26.15 nm, respectively.  
 
Scattering Light Measurements  
Some scattering light was observed during the efficiency measurements. The scattered light intensity level between the orders is more than a factor of 1000 lower than the -1 order efficiency level.  
 
Sources of the Refractive Indices  
The choice of the refractive indices (RIs) of the layer materials is very important in development of the efficiency calculation model for multilayer gratings in the Extreme Ultraviolet wavelength range.1 Efficiency investigations for the multilayer grating were made in two separated wavelength regions. For the both wavelength ranges, optical constants of Si derived from the updated data of Ref. 10 were used for efficiency calculations. While the CXRO and NIST80 RIs for Si are in good agreement (the Palik data11 are also close), except near the Si L3 absorption edge at 12.4 nm wavelength, the CXRO and NIST optical constants for Mo significantly differ, particularly at the longer wavelengths.1 This illustrates how discrepancies among published RIs can occur even for the widely utilized Mo/Si multilayer coating. RIs of Mo derived from the NIST data, RIs of Si derived from the CXRO data, and RIs of SiO2 derived from the Palik data were used for efficiency calculations in the working wavelength ranges.  
 
Efficiency Model
To determine the exact shape and position of efficiency curves including the spectral separation of the inside and outside orders1,75 rigorous electromagnetic theory for multilayer-coated gratings with realistic data of boundary profiles and RIs must be applied. The preliminary theoretical efficiency investigation based on the boundary integral equation method was performed for different calculation models of plane gratings due to small ratios of the grating sizes to the radius of curvature.75,78,81 It was found78 that the calculated efficiencies of the Mo/Si multilayer grating are sensitive to variations in the groove profile depth. Also some changes in the assumed groove shape (ridge-to-period ratio) and slope angle resulted in valuable changes in the calculated efficiencies. During the process of coating the grating substrate with multiple layers to produce a multilayer grating, a smoothing of the groove profile is not takes place, as derived from the comparing between efficiency measurements and calculations of various EIS gratings and coatings.78 Thus the depth of all the boundary profiles of the multilayer grating was 6.0 nm with side slopes of 35░ and equal top and groove widths, as derived from the AFM and efficiency measurements. Because polarization effects are small near normal incidence, the efficiencies are presented for the case of TM-polarized radiation.  
To determine the absolute values of order efficiencies, a model of the two-period-randomized-trapezium grating describing the realistic boundary shape and roughness can be applied. For the rigorous accounting of the random roughness impact on the efficiency, the model with 41 randomly rough borders of the period of ~ 476.19 nm having 400 random sampling points on two trapezoidal grooves with the same Gaussian surface roughness height statistics and Gaussian autocorrelation function (see a boundary shape example in Fig. 71) was applied. For both the short-band and long-band multilayers the rough boundary parameters are: Si-Mo interface rms roughness σ = 0.2 nm and Mo-Si rms roughness σ = 0.85 nm. The lateral correlation length of ζ = 5 nm was chosen from the detailed microscopic analysis82 and the growth model83 of typical Mo/Si layers obtained by using magnetron sputtering. An assumption about the absence of a vertical correlation between the border random roughness components was applied in this model. There were generated seven sets of 41 rough border profiles to compute exact efficiencies of the FL1 multilayer grating.  
The Si protective capping layer of 2 nm was modeled by using 1.5-nm-thick amorphous SiO2 on 1.5-nm Si in order to account for the oxidation of the Si capping layer.79 The FL1 short-band half multilayer parameters extracted from the M2 mirror investigation are: 20 Mo/Si layer pairs with the bilayer period D = 10.3 nm, Mo thickness to D ratio Γ = 0.37. The FL1 long-band half multilayer parameters are: 20 Mo/Si layer pairs with the bilayer period D = 14.35 nm, Mo thickness to D ratio Γ = 0.34.  
 
Rigorous Accounting of Random Roughness  
To determine the absolute values of scattering light intensities between orders,84,85 a model of ten-period-randomized-trapezium (low-frequency) grating allowing a fine-diffraction-angle discretization and describing the realistic boundary shape and roughness can be applied. For the rigorous accounting of random roughnesses, the model with 41 randomly rough borders of the period of ~ 2381 nm having 800 random sampling points on 10 trapezoidal grooves with the same Gaussian surface height statistics and Gaussian autocorrelation function was applied (for an example, see Fig. 72).  
For short-band and long-band multilayers the rough boundary parameters are: Si-Mo interface rms roughness σ = 0.2 nm, Mo-Si rms roughness σ = 0.85 nm, and lateral correlation length ζ = 5 nm. There were generated 105 sets of 41 non-correlated vertically border profiles to compute exact scattering light intensities between orders of the FL1 multilayer grating. The same layer parameters as for the Efficiency Model (see above) are used.  
 
Efficiency and Scattering Intensity Calculations
Convergence and accuracy of the efficiency results of the randomly-rough 41-boundary grating were investigated using the Penetrating solver, Gauss computation algorithm, Finite Type of low border conductivity, and Accelerating convergence On with Equal S-interval in Accuracy optimization options of PCGrate-S(X) v.6.5. The From library type of RI sources and the Linear Type of data inter/extrapolation were chosen. A high rate of convergence of the efficiency results was observed for the developed grating model (Fig. 73). Only several sets of 41 rough border profiles and the medium number of discretization points per boundary are enough to compute exact efficiencies in all orders of interest. The differences between short-band principal order efficiencies obtained with seven boundary sets with low (N = 600), medium (N = 800), and high (N = 1000) accuracy are about a few relative %% for all orders under study (Fig. 74). The differences between short-band efficiencies obtained with three, five, and seven statistical boundary sets (N = 800) are also about a few relative %% for all diffraction orders under study (Fig. 75). For long-band data the convergence of the efficiency results (not presented) are even better. For the further efficiency modelling, N = 800 and seven (for the short wavelength range) or five (for the long wavelength range) random boundary sets are used. The total error for all points and ranges derived from the energy balance was an order of 1.E-3. The time taken up by one rigorous computation (one scanning point in the short wavelength range) for N = 800 on a workstation with two Quad-Core Intel« Xeon« 2.66 GHz processors, 8 MB L2 Cache, 1333 MHz FSB, and 16 GB RAM, is ~ 45 min when operating on Windows Vista« Ultimate 64-bit and employing eightfold paralleling.  
Convergence and accuracy of the scattering intensity results of a randomly-rough 41-boundary grating were investigated using the Penetrating solver, Gauss computation algorithm, Finite Type of low border conductivity, and Accelerating convergence Off with Equal S-interval in Accuracy optimization options of PCGrate-S(X) v.6.5. A medium rate of convergence of the light intensity results was observed for a wavelength of 19.25 nm and the above computation model. More than 100 sets of 41 rough border profiles and the medium number of discretization points are enough to compute exact values of scattering light intensities between orders. The difference between scattered light intensities data obtained with seven boundary sets using medium (N = 1000) and high (N = 1200) accuracy is about 1.E-5 for almost all diffraction orders (Fig. 76). The differences between scattered light intensities obtained with different numbers of statistical boundary sets (35, 70, 98, 105) and N = 1000 for all diffraction (scattering) angles are shown in Fig. 77. For the final scattering intensity modelling, N = 1000 and 105 random boundary sets were chosen. The total error for all points and ranges derived from the energy balance was an order of 1.E-4. The time taken up by one rigorous computation (one scanning point in the short wavelength range) for N = 1000 on the mentioned above workstation is about two hours.  
 
Comparison between Calculated and Measured Results  
For further efficiency computations we choose the complex multilayer grating model described above. In the covered wavelength ranges, this model yields accurate results at a medium calculation speed. The validity of the model was supported by comparing the experimental and predicted efficiency data in ▒1 and 0 orders in the short (17-22.25 nm) and -1, -3, and 0 orders in the long (25-29.5 nm) wavelength ranges (Fig. 78 and Fig. 79). The measured and modelling efficiency maxima are close for the principal orders with uncertainties of several %% in both wavelength ranges covered. The maxima positions coincide for all the orders investigated over the entire wavelength range. Some non-significant rms (measured to calculated) efficiencies for principal orders are observed in the long wavelength range, basically, due to a lack of realistic data of the material refractive indices.  
The calculated scattered light intensities between the orders (see Fig. 80) is about ~ 1000 times lower than the -1 order minimal efficiency that is in a quantitative agreement with the measured level.  
 
Efficiency Certificate  
For issuance of the present Efficiency Certificate, the -1 order efficiency results were obtained using the above model in the extended wavelength range from 17 nm to 30 nm covered both the sort-wave band and the long-wave band of the two multilayers (see Fig. 81). There were no free parameters in the calculation. The scattered light intensity level calculated between the principal orders is ~ 1.E-5 for a wavelength of 19.25 nm.  
 
Summary  
  • A good quantitative agreement between the experiment and the theoretical efficiency and scattering light intensity between orders based on the real groove profile shape (AFM-measured), randomization, and appropriate RI libraries can be obtained for these types of gratings and wavelengths.  
  • Calculations are very sensitive to boundary profiles, i.e. to the trapezium depth, side slopes, and ridge-to-land ratio.  
  • It was determined that the calculated efficiency in the short wavelength range depended significantly on the degree of the upper-silicon-layer oxidation.  
  • One of the interface values of the random roughness (rms = 0.85 nm) is of some importance for the multilayer grating efficiency and scattering light intensity that cannot be ignored in calculations.  
  • The effect of random roughness on the grating efficiency and scattering light can be exactly taken into account with the model in which an uneven surface is represented by a grating with a large period, which includes a sufficient number of random asperities required for Monte Carlo calculus. Rigorous calculations revealed that diffuse x-ray-EUV scattering intensities obtained for randomly-rough surfaces with boundary profiles having different statistics (autocorrelation functions, correlation lengths, etc.) differ noticeably.83,84,85  
  • The impact of some diversity of the real and imaginary parts of the RIs upon the efficiency is not very large. Using RIs for Si from CXRO data gives accurate efficiency results in the wavelength range under study. RIs of Mo derived from the NIST data80 give more accurate efficiency results (in respect to the CXRO data), particularly at the longer wavelengths.  
  • The rigorous electromagnetic code can be used effectively for a precise modeling of layered gratings in the EUV range despite small wavelength-to-period ratios and a large number of boundary profiles with real randomized shapes.  
 
IMPORTANT NOTE  
The complete results of the grating efficiency investigation and all relevant information about the computation and measurement processes can be obtained from the downloading page (Cert_4200_holo-lam_Mo-Si_EUV.zip file). Various files both *.xlsx (MS Excel« 2010) and *.grt, *.ggp, *.ri, *.ari, & *.pcg (PCGrate«-S(X)Ö v.6.5) formats relating to this grating efficiency certification are included in the package. You can view the later by the PCGrate Demo v.6.5 Complete (updated on the 25th July, 2012).  
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