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 Å and 20 Å. 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-Å 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 Å). 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 Å.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 Å range using a silicon filter and at 11 wavelengths in the 170-225 Å 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 Å 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 Å.  

 

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 Å 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  

 

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)™ 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 3^rd May, 2005).  

 

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-Å 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 Å. 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 Å. 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 Å 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 Å to 350 Å wavelength range.¹³ The inferred thicknesses of the SiO₂ layer and the Al₂O₃ layer were 743 Å and 30 Å 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 Å 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 Å range using a silicon filter and at 17 wavelengths in the 172-225 Å 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 Å 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 Å 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 Å 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 Å, 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  

 

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)™ 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 3^rd May, 2005).  

 

 

Efficiency Certificate for a Replica of a Red-Blazed Ruled Grating in the NUV–NIR 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  

 

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  

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  

 

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)™ 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 3^rd May, 2005).  

 

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 < 2×10^-5/Å 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 references^{11, 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  

 

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 -1^st 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. T