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We compare various techniques for determining (numerically) the profile of a surface relief grating. Several instruments applicable for this purpose are compared. The advantage of using direct groove metrology to predict grating efficiency characteristics lies in the ability to more quickly reject unusable fabrication attempts. This is in contrast to the typical approach, wherein a master grating is fabricated (whether by mechanical burnishing with a ruling engine, holographic recording, or various newer writing techniques), perhaps replicated, and then tested for efficiency. For a mechanically ruled grating, a `test' ruling can quickly be evaluated using this new approach, whereas a complete ruling can require up to several weeks of continuous use of the (expensive) ruling machine. Even for holographic gratings, the considerable expense of coating the grating with specialized coatings can be deferred until metrology assures that the coated part will suffice in the applications. In our experience, many gratings are currently rejected for poor performance after completion, at the loss of a considerable investment.  
After a discussion of the three proposed direct metrology techniques (microinterferometry, stylus profilometry, and atomic force microscopy (AFM)), this text describes the sequence by which raw images are analyzed for grating profile, profile consistency, and profile roughness. Examples of groove metrology results are presented.  
Any method for measuring the profile of a surface relief grating entails calibration requirements. These instruments are often more commonly used for surface roughness determination. Trying to measure, for example, the actual groove depth, adds to the requirements for this application. An analogy can be drawn to the use of figure measuring phase shift interferometers, which are commonly used to measure surface irregularity, but, with more work and calibrated stages, can be used to measure the lowest order parameters such as absolute tilt or radius of curvature.  
Of course, measuring the depth of a grating, which is the first parameter that helps to determine the wavelength for peak efficiency (in a given mounting geometry) depends on accurate vertical calibration. What is less apparent is that the use of grating efficiency simulation codes also requires accurate lateral calibration. This is because the groove vertical geometry is often expressed in dimensionless units, relative to the grating period. Any lateral error becomes vertical error in the required transformation. Typically, however, lateral errors are fairly well known. And in most grating work, the grating period is well known beforehand and thus the grating data itself gives a calibration factor to correct the lateral scale. But typical errors, of the order 1-4%, in the lateral calibration, can affect the calculation of the blaze wavelength, which can be toleranced closely. Table 1 summarizes the capabilities and limits of the three metrology devices we characterized for grating metrology. The AFM has the finest lateral resolution. The stylus profilometer and the microinterferometer have comparable vertical ranges. The stylus profilometer has significantly larger lateral range for probing to millimeter spatial scales.  
Table 1. Measurement characteristics of 3 surface profiling instruments  
Microinterferometer 50x  
Stylus Profilometer  
Atomic Force Microscope  
vertical resolution  
vertical range  
lateral resolution  
lateral range  
163 (more w/stitching)  
limiting factor(s) for lateral resolution  
MTF, sampling, need for retroreflection over the whole profile  
tip radius & angle  
tip radius  
upper slope limit  
The atomic force microscope data gives an example of non-linearity. We obtained a series of step height calibration standards (some NIST-traceable, some not)19,20. The vertical axis was calibrated using one of the smallest steps (10 nm). Then the rest of the step height series was measured. Table 2 summarizes the results: Slight errors, up to 8%, were observed for heights much higher than that used to calibrate the AFM. We use a fit to this nonlinear fit to correct for linearity when we believe it is a significant contribution.  
Notes to step heights (in nm) as measured on the 3 devices (Table 2):  
1. Nonlinear at ~8% at highest step when calibrated to a 10 nm step.  
2. Using 0.1 micron tip, could not resolve depth of 3.3 micron period, AFM step height standard.  
3. Used at 50x magnification.  
4. At 100x (highest magnification), did not have lateral resolution to see the 3 micron period samples tested on the AFM.  
Table 2. Step height data  
Nominal height (nm)  
Stylus profilometer  
See Note  
3, 4  
The microinterferometer has two lateral resolution-limiting factors that do not arise in the other methods - the optical resolution (including diffraction) and pixel sampling. To some extent, using different magnifications can separate these; the sampling can differ but all magnifications are subject to common optical elements and diffraction limits. To some extent, this discussion is similar to explorations of these instruments for measurement of the power spectrum for smooth mirrors as applied to short wavelength imaging systems21. The mathematical limit on lateral resolution in these instruments is the wavelength used, or, typically, ~0.2-0.3 m. Tip convolution, which limits the resolution of both the AFM and the stylus profilometer, has also been much studied and an algorithm to account for this effect has been developed [see, for example,22]. We have not attempted the use of this algorithm in this work; rather, the approach is to ensure that the tip radius is known, and restrict measurements for a given tip to periods that are much larger than the tip radius. For example, the radius of a fresh AFM tip radius is 10-15 nm; we do not believe the profiles for any grating with period < 20 tip radii (or 300 nm, a density of 3300/mm) should be trusted without at least adding the tip deconvolution step. However, the depth is probably correctly recorded for periods down to ~10 tip radii, or , for an AFM tip of 15 nm, to periods of 6600/mm.  
Anyone who has worked with these instruments is aware that a vibration free, quiet environment is essential. Noise effects can also be (to some extent) averaged out by repeated measurements. Discussion of these effects is deferred to the special references.  
This instrument, also called an optical profilometer, is essentially an interferometric head on a microscope, where the reference arm of the interferometer views a small, highly polished reference flat. Typically, this reference can be removed from the results of measurements on highly polished surfaces. For grating measurements, this is not necessary, as gratings are inherently much rougher than the reference. A good discussion of the basics of this instrument can be found in23; however, newer models take advantage of vastly increasing computing power to expand the vertical range. We have used a Phase Shift Instruments model MicroXAM for this work. It has variable magnification from 2x to 100x; values of range and resolution for the 50x magnification is listed in Table 1. This instrument does not use phase stitching reconstruction algorithms the way some of the older implementations of this measurement architecture did e.g. Wyko TOPO-3D); rather, the zero path difference is calculated (independently for each pixel from a series of images acquired during a vertical sweep. This increases the vertical range and slope range accessible significantly. Portion of a trace of grating 1528 taken with the microinterferometer is shown in Fig.6. Both the depth and the profile shape are severely distorted. Compare with the profiles in Figs.7 and 9. However, an essential limitation is that light must be returned to the interferometer from each location to be measured. Gratings are of course designed to diffract light away from the incident beam. This prevented us from obtaining good measurements over the whole profile for several steeply blazed gratings, including the deep echelle shown in Fig.8.  
This instrument can be thought of as a precision phonograph. A diamond tip is brought into direct contact with the surface, with calibrated contact force. As the tip moves across the surface, the motion of the tip is amplified, filtered, and detected. A good discussion of the basic limits inherent to and results obtainable with this class of device is given in23. Care must be exercised to prevent indentations of the surface by the tip, depending on materials and forces used. The model used in this work is a Tencor P-10. Table 1 presents the basic lateral and vertical ranges and resolutions typical for the instrument. Typical measurement parameters are: tip radius 0.1 micron (in grating dispersion direction), 5 microns/sec tip speed, 2kHz digital sampling, 0.25 milligram tip force, and profile lengths of at least 100 microns (depending on groove period). Fig.7 shows an example, again the red reflection grating 1528-1-2-3. Again, the overall groove depth and profile are evident. Also evident now are that the profile noise is higher on the upper sloped portion than on the steep edges. Difficulties in holding the sample steady during `flyback' prevented reproducibility measurements for this paper. Portion of the stylus profilometer trace for a deep echelle grating for IR spectroscopy is shown in Fig.8. This grating, including deep (~57) sloped facets, a flat top, and sharp bottom, is well resolved due to its extremely long period of ~140 microns.  
AFM metrology has been around for some time. Several AFM profile measurements on surface-relief diffraction gratings have been presented8,24-28. However, a careful analysis of the use of the AFM method for surface relief grating groove metrology has not really been undertaken. Nor do we claim to do so here; that would include tip. deconvolution, noise exploration, and other discussions we have not completed. The model used here is a Digital Instruments Nanoscope III. Fig.9 shows an example of AFM data for a portion of the surface of a ruled grating; the grating is a 67.556/mm, ~1.4 blazed grating (blazed for red and near infrared wavelengths). The basic groove shape (profile) is evident, along with portions of the profile that are rougher than others, and some roughness along the grooves is indicated as well. This grating was chosen because (1) high quality efficiency data exists for it, and (2) it should be measurable by all three methods for a direct comparison.  
Instead, we have measured the tip radius of the tips used before and after measuring profiles of gratings, and found the radius to be in the range of 10-20 nm. One measurement found a fresh tip to be ~10 nm radius and a used one to be ~20 nm. Therefore, we have restricted our AFM profile work to gratings of period much longer than 10 nm as discussed above. The rms difference of two consecutive measurements as shown in Fig.9 is 37 nm; clearly this places a lower limit on the smoothness of the groove facets that can be verified. By examination of the power spectrum of direct and difference measurements, the effective frequency limits for shape metrology in a given measurement can be ascertained [see, for example,21].  
Contour plot of AFM data on a 3600/mm ruled grating for EUV normal incidence spectroscopy is shown in Fig.10. Units are microns for the lateral scale and nm in the vertical scale bar. Rough groove edges are apparent.  
The surface of the master29 grating and the replica13 grating, as an example, were characterized using a Topometrix Explorer Scanning Probe Microscope, a type of atomic force microscope (AFM). 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. Ion etching results in a groove profile closer to triangular than the ideal blazed (sawtooth) profile. The grating has 2400 g/mm, a concave radius of curvature of 2.0 m, and a patterned area of size 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 replica grating had an aluminum surface. A thin SiO2) coating was applied to the aluminum surface for the purpose of reducing the microroughness.  
The AFM images typically had 500x500 pixels and a scan range of 1 to 20 mkm (pixel size 20 to 400 ). 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 100 to 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. A surface topology reference sample was used to optimize the AFM scanning parameters, to calibrate the height scaling of the instrument, and to evaluate the performance of the AFM. This was essential for the accurate characterization of the gratings. The surface topology reference sample consisted of an array of approximately square holes fabricated on the silicon dioxide surface of a silicon die by VLSI Standards Inc.19 The top surface of the die was coated with a thin layer of platinum. The hole array had a pitch of 3 mkm and a hole depth of 180 .  
A typical AFM image of the master grating recorded using 16 pixels is shown in Fig.11, where the vertical scale has been exaggerated to reveal the texture of the groove surface. The microroughness, determined by integrating the power spectral density function over the 2 to 40 1/mkm spatial frequency range, was 3.2 rms. Most of the microroughness is concentrated at low spatial frequencies as is apparent in Fig.11. The central portion of the AFM image shown in Fig.11 that covers one period of the grating pattern was selected for further study. An analysis program was written in the Interactive Display Language (IDL) for this purpose and will be discussed in detail in the publication13. The histogram of the pixel heights, for one period of the grating pattern, is shown in Fig.12. The maxima at 10 and 85 in Fig.12 are caused by rounding of the groove profile at the peaks and the troughs which is a result of the pattern fabrication process. An ideal groove profile, either sawtooth or triangular, would have a flat height histogram. The separation between the peaks in Fig.12 represents the average groove height, approximately 75 . The local blaze angle at each pixel was determined by using a least squares algorithm to fit a linear curve to the data points in a sliding window. The window was 25 pixels (400 ) long in the direction perpendicular to the grooves and one pixel wide parallel to the grooves. The blaze angle is the arctangent of the fitted slope. The histogram of the blaze angles, for all rows of data in one period of the grating, is shown in Fig.13. The peak at 2.5 deg. represents the classical blaze angle, and the peak at 5.5 deg. represents the steep facet of the ideal sawtooth profile as modified by the ion etching process. For a density of 2400 grooves/mm and for facet angles of 2.5 deg. and 5.5 deg., an ideal grating would have an groove height of 125 . However, the measured value of 75 deg. (Fig.12) indicates a significant degree of rounding at the peaks and troughs of the groove profile. In addition, the measured ratio of the heights of the 2.5 deg. and 5.5 deg. features in the angle histogram (Fig.13) is approximately 3, greater than the ratio of approximately 2 that is expected based on the average facet angles. The interpretation of the widths of the features in Fig.13 is difficult because they are complicated functions of the surface roughness, the width of the sliding window, and the probe geometry. This is addressed in the publications13,30,31. The feature at -2 deg. in Fig.13 results from the fits to the peaks and troughs of the groove profile, where the local slope is changing rapidly but has an average value near zero. Simulations show that the -2 deg. offset of this feature from zero is a consequence of the unequal average blaze angles of the two facets. 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 (75 ). The resulting groove profile is shown in Fig.14. This groove profile has 210 points.  
An AFM image of two grooves of the replica grating is shown in Fig.15. The scan was performed across the grooves over a range of 1 mkm (20 pixels). The vertical scale in Fig.15 has been expanded to reveal the texture of the grating surface. The PSD derived from a 2 m size image spanning nearly 5 grooves is shown in Fig.16. The peak in the 2 to 3 m-1 frequency range results from the 0.4167 m groove period. The rms microroughness is 7 in the 4-40 1/mkm frequency range. By comparison, the microroughness of the master grating measured by the same type of AFM instrument was 3.2 , and this implies that the replica grating is significantly rougher than the master grating. This may result from the replication process, which for a concave grating is at least a two step process. Furthermore, the master grating was fabricated on a fused silica surface by a holographic technique and was ion-beam polished, while the aluminum surface of the replica grating may contribute to its larger microroughness. The replica grating without the SiO2) coating was not characterized by AFM. 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 deg. and 6.2 deg. 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 deg and 5.5 deg. facet angles and 75 average groove depth. Thus the grooves of the replica grating are deeper and the facet angles are steeper compared to those of the master grating.  
Other methods have been used for determining groove angle and profile. Examples of these include (1) `shadow casting' (see, for example,32,33), where fibers on the surface are viewed at a glancing angle, (2) Transmission electron microscope analysis of broken gratings, and (3) indirect methods, whereby a profile is fit to a set of parameters, and a forward calculation of the efficiency for each combination of the parameters is done, and a least-squares determination of the best fit of actual efficiency data to that set of parameters is performed (see, e.g.34-36). The first two of these are detrimental to the grating (or at least to a replica), and are not very quantitative; the last requires substantial foreknowledge of the profile and significant computer time. However, it does allow very rapid determination of profile parameters and has been shown useful as a diagnostic for low-linewidth lithography tools.  
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