Perfectly rectangular grating profiles, as assumed in the derivation of Eq. (7.3), are an idealization. In reality, the grating profiles will deviate from a rectangular form, where the most obvious deviation is obtained by the rounding of the edges of the bars. Since a rectangular shape carries infinite spatial frequencies, rounded edges can be introduced by a suitable low-pass filter in frequency domain. In the following, we will model rounded edges by the convolution of transmission function of a grating by a disturbance Tσ(x) of Gaussian shape with a standard deviation ofσ, as shown in Fig. 54 (a). The convolution with a Gaussian function introduces an additional Gaussian factor in the visibility calculation of Eq. (7.2)
νn= 2· ∞
∑
n=1 sinc3 (n 2 ) · exp(−2π2·σ2· n2) , (7.13)where we have assumed that all gratings have a duty cycle ofτ = 1/2. Using Eq. (7.13), the dependence of the visibility on the width of the disturbing Gaussian functionν(σ) can thus be calculated analytically. It is shown in Fig. 54 (b).
Fig. 54: (a) Rounded deviations from a rectangular profile, modeled with the convolution of a Gaussian
with widthσ, (b) dependence of the visibility (solid black line) and its first order approximation (solid red line) on the widthσ of the Gaussian modeling the rounded bars.
Fig. 55: (a) Trapezoidal deviations from a rectangular profile, modeled with the convolution of a
rectangular function with a duty cycle τσ, (b) dependence of the visibility (solid black line) and its first order approximation (solid red line) on the duty cycleτσ of the rectangular function modeling the trapezoidal bars.
Fig. 56: (a) Caused by inhomogeneities in current density during the galvanization procedure, the metal
structures are higher near the resist (schematic courtesy of J. Kenntner). (b) Grating height as measured with an x-ray fluorescence spectrometer. The average bar height decreases towards the middle of the grating.
Imaging with imperfect gratings 101
Trapezoidal bars:
Similar to the rounded grating profiles, trapezoidal profiles, as shown in Fig. 55 (a), can be modeled by the convolution of the transmission function of the gratings with an additional rectangle function of width p·τσ and a duty cycle τσ. The additional rectangle function introduces another sinc-function in Eq. (7.2)
νn= 2· ∞
∑
n=1 sinc (n 2 )3 · sinc(n ·τσ) , (7.14)where we have again assumed that the three interferometer gratings have a duty cycle of τ= 1/2. The visibility as a function of the duty cycleτσ of the rectangular disturbance as given by Eq. (7.14) can be seen in Fig. 55 (b). From Figs. 54 and 55, it can be seen that rounded and trapezoidal deviations from the ideal rectangular grating profile introduce only a marginally decreasing visibility and can thus be tolerated for a wide range of deviations. Additionally, the difference between the visibilityν and the usually applied first order approximationν1 is
reduced with growing grating errors. The shape of the oscillation in each pixel thus approaches a harmonic oscillation with increasing grating errors.
Bathtub effect:
Another deviation from the ideal grating shape is introduced by the so-called bathtub effect, as shown schematically in Fig. 56 (a): Due to the structure of the grating, the current density during galvanization is higher near the resist structures than in the middle of the grating bar. This causes the galvanized metal ions Me+to be inhomogeneously deposited within one grating bar. Instead of having a constant height, the bars are overgrown near the resist structures, creating a lateral profile that resembles a ’bathtub’. This effect is observed within every bar, as well as over the entire area of the grating, where the structures at the grating’s perimeter are higher than in the middle of the grating.
The effect is negligible for attenuation gratings, since overgrown structures in each bar will be equivalent to a mean increase of bar height and thus a higher attenuation, if the pixels are big enough to average over multiple grating bars. The bathtub effect for attenuation gratings therefore exhibits a negligible influence on imaging. Considered over the entire surface of the grating, the bathtub effect will create higher grating structures at the outer ranges of the grating, thus potentially increasing the visibility there in a measurement.
The situation becomes more complicated for phase gratings. The bathtub-effect increases with the height of the galvanized structures. As the bars of the phase grating are comparatively low, the bathtub effect within a single bar can be assumed to be negligible and leads only to a mean increase of bar height. If the bathtub effect is considered over the entire surface of the phase grating and the nominal grating height is achieved in the center of the grating, the outer reaches of the grating will exhibit a phase shift that is higher than the design phase shift at the design energy. This will lead to an overall visibility decrease when close to the edges of the grating, which can be calculated using the visibility spectrum of the grating, as explained in section 7.2. Exemplarily, the bathtub effect for a nickel (Ni) phase grating with a nominal height of h0≈ 8
µm was measured, exploiting the Ni Kα fluorescence radiation of the grating material. For the measurement, a FischerScope X-Ray XDV SD fluorescence spectrometer was used. From the fluorescence yield of the sample, the material thickness under the assumption of homogeneous material layer was calculated. As the grating surface is not homogeneous but structured into bar material and resist structure, the grating heights that are calculated from the measured data are reduced by a factor of approximately 0.5. The grating was scanned at 30× 15 points horiz×vert, each for 60 s integration time. The fluorescence signal was averaged over the area
of the beam focus at the sample, which had a diameter of d= 1 mm. No filtration was used. The resulting measured heights for all points were rendered numerically into the surface shown in Fig. 56 (b). It can be seen that the bar height changes by approximately 20 % from the nominal height in the middle of the grating towards the outside. Additionally, it can be seen that the bathtub effect is higher along the long side of the grating. The reason for this is the higher inhomogeneity in current density in this case, as shown in Fig. 56 (a).