3.1.

Measured Film Thickness Distributions

Both the optical properties and the piezoelectric response of PZT films depend explicitly on the crystal structure and microstructure, including the potential presence of second phases.^6,55 Thus, films were characterized to assess the structure and microstructure prior to ellipsometry measurements. The microstructure exhibited irregular grains with average grain sizes of $375\pm 75\mathrm{nm}$ to one standard deviation ($1\sigma$) and little to no secondary phase, as observed with SEM [Fig. 1(b)]. The structure determined by XRD was a pseudocubic perovskite, consistent with the literature for films near the morphotropic phase boundary [Fig. 1(c)].⁵⁵ AFM was used to measure the surface roughness of the films; it was confirmed that the roughness was consistent across the surface area with a root mean square (RMS) roughness of 4.6 nm, and PV surface roughness of 30 nm. This confirmation of structure, phase, and microstructure homogeneity meant that changes in the optical response of the films, measured with spectroscopic ellipsometry as a function of position on the substrate, could be unambiguously interpreted as arising from variations in the film’s thickness.

Spectroscopic ellipsometry was used to nondestructively determine the thickness of PZT films as a function of position across each substrate. Measurements were taken in an $11\times 11$ matrix (total of 121 points) across the films. The films exhibited refractive indices ($n$) values of $\sim 2.4$ (at $\lambda \sim 630\mathrm{nm}$) [Fig. 1(d)] and were in good agreement with the literature: 2.6 to 2.3 between 400 and 800 nm.^55,61–63 To determine the film thickness as a function of position, an optical model was fitted to the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ spectra collected from the center of the substrate with typical fit mean square error of $13.36\pm 0.72$ [Fig. 1(e)]. Once a good fit was achieved, optical constants and oscillator fit parameters were fixed so that the film thickness parameter could be refined to fit the $\mathrm{\Psi }$ and $\mathrm{\Delta }$ spectra from the other points on the sample.

The measured film thickness values were normalized to the center of each substrate and plotted as a function of position to create relative thickness distribution maps (Fig. 2). For the flat glass substrate, the thickness decreased radially [Fig. 2(a)]. At the corners of the substrate, where the distance from the center was $\sim 70\mathrm{mm}$, the relative film thickness was 0.88, whereas at the midpoint of each edge, the relative thickness was 0.95.

Fig. 2

Relative thickness distribution maps determined by ellipsometry for films on (a) flat, (b) convex, and (c) concave curved glass substrates.

For the curved substrate on the convex side, the film thickness distribution was elliptical with a maximum thickness at the center and a minimum thickness of 0.79 at the corners [Fig. 2(b)]. The thickness decrease was greatest in the azimuthal direction with the relative thickness of 0.94 and 0.87 at the midpoint of the edges parallel to the axial and azimuthal directions, respectively.

On the concave side, the film thickness distribution pattern was hyperbolic, with asymptotes passing diagonally through the substrate [Fig. 2(c)]. The relative thickness minimum was 0.94 at the midpoint of the edges parallel to the azimuthal direction, and a maximum of 1.02 at the midpoint of the edges parallel to the axial direction. Deviations from a perfectly symmetrical distribution pattern, seen predominantly in the lower middle region of the distribution map, were most likely related to fluctuations in the film quality caused by mechanical damage and/or contamination of the substrate prior to deposition.

3.2.

Modeled Film Thickness Distribution

3.2.1.

Geometric film thickness model for flat substrates

A mathematical model considering the geometry of the sputter tool was developed to predict the thickness distribution of films, similarly as presented in earlier literature.^{30–33,38,42,43} To determine the thickness distribution inherent to the sputter tool geometry, the flat substrate case was considered first. Sputtering parameters such as temperature, power, target chemistry, chamber pressure, and gas species were kept consistent throughout the study and were not considered in the model. Figure 3(a) provides a schematic of the sputter chamber, with labels of the geometric factors that were utilized in the model. The three critical dimensions considered were the vertical throw distance ($h$), the off-axis displacement ($l$), and the target angle ($\mathrm{\Omega }$). The magnetron sputtering tool modeled (Kurt J. Lesker CMS-18) was a commercially available system, and the key dimensions were typical of systems used for small scale or research-based sputtering. The ejected species mass flux was treated as continuous for the duration of sputtering time.

Fig. 3

A schematic of (a) the sputter chamber geometry where the circular gray arrow indicates that the substrate is rotating during deposition and (b) target erosion profile.

The ejection of the chemical species from each area element on the target ($\mathrm{d}A$) was defined using a cosine distribution, with angle ($\theta$) between the normal to the target surface and the trajectory of ejected species.^30,32 The magnetic field of the magnetron improves deposition rates by increasing the plasma density near to the target, but results in nonuniform ejection of species as a function of radial position on the target ($a$), i.e., heterogeneous target erosion.¹ This was apparent from the symmetric radial erosion profile of the target which followed the magnetic field lines [Fig. 3(b)]. The depth of the erosion profile was proportional to the target species ejection rate and was treated in the model by including an erosion function $ϵ(a)$.^30,32,38,64 The erosion function was approximated as a half sine wave [Eq. (2)], consistent with typical “race track” erosion profiles observed in the literature³⁸

Eq. (2)

$$ϵ(a)\cong \sin \left(\frac{a\pi }{R_T}\right),$$

where ($a$) is the radial position on the sputtering target, and $R_T$ is the radius of the target.

A critical consideration for the model was the distance between the target and the substrate. The ejected mass flux decreases proportionally to the inverse square of the distance it travels to the substrate (denoted by the vector $\stackrel{\rightharpoonup }{R}$). This was described via the vertical throw distance ($h$), the off-axis displacement ($l$), and the target angle ($\mathrm{\Omega }$). A Lambertian distribution term was used for the distribution of mass direction leaving the target at a given point. A cosine term to the power of one was found to achieve the best fit to the experimental thickness distributions measured for both flat and curved substrates. As a result, the mass flux reaching a point on the surface of the substrate $(\mathrm{d}{m}_s)$ is given by

Eq. (3)

$$\mathrm{d}{m}_s\propto \frac{ϵ(a)}{|\stackrel{\rightharpoonup }{R}{|}^2}\cos \theta \mathrm{d}A,$$

where $\mathrm{d}A$ is an element of area on the target from which the atomic species is being ejected, and $\theta$ is the angle between the normal of the surface of the target and the direction of species ejection.

It was assumed that mass ejected from the target travels in a straight line until it reaches the substrate surface, i.e., the effects of scatter due to atomic collisions were ignored. The rate of accumulation of mass at a point on a surface ($\mathrm{d}D$, mass per second) is proportional to the cosine of the angle between the direction of the mass trajectory ($\widehat{R}$) and the normal of the surface (${\widehat{n}}_s$). This cosine term was then expressed by the scalar product of two normalized vectors as given in

Eq. (4)

$$\frac{\mathrm{d}D(x,y)}{\mathrm{d}t}\propto (-{\widehat{n}}_s·\widehat{R})\mathrm{d}{m}_s\propto \frac{ϵ(a)}{{|\stackrel{\rightharpoonup }{R}|}^2}\cos \theta (-{\widehat{n}}_s·\widehat{R})\mathrm{d}A,$$

The cosine term for the angular emission intensity from the target can also be given by the scalar product of two vectors, the vector normal to the surface of the sputtering target (${\widehat{n}}_T$) and $\widehat{R}$. Thus changing Eq. (4) to (5):

Eq. (5)

$$\frac{\mathrm{d}D(x,y)}{\mathrm{d}t}\propto \frac{ϵ(a)}{|\stackrel{\rightharpoonup }{R}{|}^2}({\widehat{n}}_T·\widehat{R})(-{\widehat{n}}_s·\widehat{R})a\mathrm{d}a\mathrm{d}\phi .$$

The area element ($\mathrm{d}A$) could then be given in terms of polar coordinates, where $a$ and $\phi$ are the radial and angular coordinates describing a point on the target. From this, the thickness ($D$) deposited onto a point $(x,y)$ on the surface of the substrate was obtained by integrating Eq. (5) across the area of the target over a given time ($t_f$):

Eq. (6)

$$D(x,y)=C\underset{0}{\overset{t_f}{\int }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{R_T}{\int }}\frac{ϵ(a)}{|\stackrel{\rightharpoonup }{R}{|}^2}({\widehat{n}}_T·\widehat{R})(-{\widehat{n}}_s·\widehat{R})a\mathrm{d}a\mathrm{d}\phi \mathrm{d}t.$$

The constant $C$ was used to normalize the thickness to the value 1 at the center of the substrate (i.e., the $x$, $y$ coordinates 0, 0). Thus, this expression yields a relative thickness distribution across the substrate.

The vectors necessary to complete the calculations $\stackrel{\rightharpoonup }{R}$, ${\widehat{n}}_T$, and ${\widehat{n}}_s$ were defined from the geometry of the sputtering chamber as shown in Eqs. (7)–(9), respectively. We note, however, that Eq. (9) is true only in the case of flat substrates:

Eq. (7)

$$\stackrel{\rightharpoonup }{R}=\left[\begin{array}{c}\sqrt{x^2+y^2}\cos (\omega t+\delta )-l-a\cos \phi \cos \mathrm{\Omega }\\ \sqrt{x^2+y^2}\sin (\omega t+\delta )-a\sin \phi \\ a\cos \phi \cos \mathrm{\Omega }-h\end{array}\right],$$

where $\omega$ is the constant rotation rate of the substrate, and $\delta$ is the polar angle of the point $(x,y)$ at the initial time ($t=0$)

Eq. (8)

$${\widehat{n}}_T=(-\sin \mathrm{\Omega })\widehat{x}+(-\cos \mathrm{\Omega })\widehat{z},$$

Eq. (9)

$${\widehat{n}}_s=\widehat{z}.$$

To demonstrate the influence of the geometry of the sputter tool on the relative thickness distribution, thickness distributions were calculated as each parameter was introduced to the model using a $100\text{-}{\mathrm{mm}}^2$ area for the substrate (Fig. 4). For a circular target positioned directly below the substrate (i.e., $l=0$), the relative thickness distribution exhibited the expected symmetrical radially pattern, with a maximum at the center and a minimum of 0.69 at the corners [Fig. 4(a)]. When the target was moved off-axis by $\sim 1.55\mathrm{in}$. (i.e., $l=39\mathrm{mm}$), the thickness profile was likewise displaced, exhibiting a parabolic distribution along the azimuthal direction (i.e., the direction in which the target and substrate axis were still aligned) [Fig. 4(b)]. A near linear distribution was observed in the axial direction (i.e., the direction in which the target was displaced relative to the substrate) with the relative thickness decreasing from 1.56 to 0.54 at the points closest and furthest from the target [Fig. 4(b)]. The introduction of a target angle $\mathrm{\Omega }$ of 15 deg together with the off-axis position introduced almost no change to the thickness distribution along the azimuthal direction, while moderately reducing the thickness distribution in the axial direction, changing the maximum and minimum relative thicknesses to 1.45 and 0.58, respectively [Fig. 4(c)]. Finally, a continuous rotation of the substrate was introduced by making the time-dependent angular rotational frequency $\omega >0$ [Eq. (8)]; this returned the distribution to a symmetrical radial pattern with the maximum thickness at the center, a minimum relative thickness of 0.89 at the corners and a thickness at the midpoint of each edge of 0.94 [Fig. 4(d)]. The relative thickness distribution seen in Fig. 4(d) corresponds to that of a film deposited on a flat substrate in the sputter tool used for this study.

Fig. 4

Modeled relative thickness distributions of a film grown on a $100\text{-}{\mathrm{mm}}^2$ substrate for (a) a centered circular 3-in. target, (b) a circular target positioned off axis relative to the center of the substrate, (c) a circular target positioned off axis relative to the substrate and angled toward the substrate at 15 deg and, (d) a circular target positioned off-axis, angled toward the substrate with the substrate rotating continuously during the deposition. Thickness distribution is normalized to the thickness at the center of each substrate. Note different color scales are used for panels (a) and (d) versus panels (b) and (c).

3.2.2.

Geometric film thickness model for curved substrates

The thickness distribution model [Eq. (6)] was modified to account for both convex and concave substrates. The shape of the substrate was incorporated by including the substrate’s geometric features into the vector quantities for mass trajectory ($\stackrel{\rightharpoonup }{R}$) and the normal to the substrate surface (${\widehat{n}}_s$).⁶⁵ The vertical throw distance of any point on the substrate $h(x,y)$ was equal to the maximum throw distance $h$, minus the height of any point on the substrate $z(x,y)$. Thus, a new expression for $\stackrel{\rightharpoonup }{R}$ was derived. Equation (10) shows the $z$ component for $\stackrel{\rightharpoonup }{R}$, and the $x$ and $y$ components remain the same as given in Eq. (7):

Eq. (10)

$${\stackrel{\rightharpoonup }{R}}_z=a\cos \phi \cos \mathrm{\Omega }-[h-z(x,y)].$$

Figures 5(a) and 5(b) provides a cross-section schematic of convex and concave substrate geometries, respectively. The associated $z(x,y)$ equations for the convex [Eq. (11)] and concave [Eq. (12)] substrates demonstrate how the substrate ROC, $y$ position, and maximum distance from the center ($R_{\max }$) are used to determine the substrate height at a given point. Given the cylindrical geometry of the substrates (i.e., azimuthal direction curvature and no curvature in the axial direction), only one dimension ($y$) needs to be considered; however, the $z$ parameter has been kept in $(x,y)$ for application to conical mirror surfaces:

Eq. (11)

$$z(x,y)=\sqrt{{\mathrm{ROC}}^2-y^2}-\sqrt{{\mathrm{ROC}}^2-R_{\max }^2},$$

Eq. (12)

$$z(x,y)=\begin{array}{c}-\sqrt{{\mathrm{ROC}}^2-y^2}+\mathrm{ROC}\end{array}.$$

Fig. 5

(a) Convex curved substrate and (b) concave cylindrically curved substrate with equations describing the height ($z$ direction) of an arbitrary point on the substrate as a function of position in the $y$ direction.

The vector normal to the surface of the substrate ${\widehat{n}}_s$ was given by the gradient of the function defining the surface of the substrate $f(x,y,z)$:

Eq. (13)

$${\stackrel{\rightharpoonup }{n}}_s=\stackrel{\rightharpoonup }{\nabla }f(x,y,z)=\left(\frac{y}{\sqrt{{\mathrm{ROC}}^2-y^2}}\right)\widehat{y}+\widehat{z}.$$

While this definition of ${\widehat{n}}_s$ accounted for the substrate curvature, it did not account for the continuous rotation of the substrate during deposition. A rotation operator was thus applied to the vector, rotating it through the angle $\omega t$. A length preserving transformation was used for the rotation operator so that the vector remained normalized

Eq. (14)

$${\widehat{n}}_s=\left[\begin{array}{ccc}\cos \omega t& -\sin \omega t& 0\\ \sin \omega t& \cos \omega t& 0\\ 0& 0& 1\end{array}\right]·\frac{\stackrel{\rightharpoonup }{\nabla }f(x,y,z)}{|{\stackrel{\rightharpoonup }{n}}_s|}.$$

The definitions for $z(x,y)$ were used to determine different $\stackrel{\rightharpoonup }{R}$ [Eq. (10)] and ${\widehat{n}}_s$ [Eq. (14)] expressions for the convex and concave cases.

The convex and concave models were then used to estimate their respective relative thickness distributions over a $100\text{-}{\mathrm{mm}}^2$ surface area. For the convex model, the relative thickness distribution was elliptical with a maximum thickness at the center and a minimum thickness of 0.82 at the corners [Fig. 6(a)]. The thickness decrease was greatest in the azimuthal direction, with relative thicknesses of 0.94 and 0.87 at the midpoint of the edges parallel to the axial and azimuthal directions, respectively.

Fig. 6

The modeled thickness distributions for films sputter deposited onto (a) convex and (b) concave curved glass substrates with $\mathrm{ROC}=220\mathrm{mm}$. Plots show relative thickness as a fraction of the maximum thickness which was located at the center of each substrate.

For the concave model, the thickness distribution pattern was also elliptical with a maximum thickness at the center; however, the axis along which the thickness reduction was greatest rotated by 90 deg [Fig. 6(b)]. The thickness reduced to only 0.98 in the azimuthal direction, whereas the greatest thickness reduction (0.94) occurred in the axial direction. The overall thickness nonuniformity was reduced markedly on a concave substrate relative to a convex substrate.

4.1.

Comparing Measured and Modeled Thickness Distributions

The measured and the modeled relative thickness distributions were compared for each of the substrate geometries by considering the standard deviation of the relative thickness and the difference between the measured and modeled values for each case (Table 1). Relative thickness difference maps were plotted for each case (Fig. 7).

Table 1

Comparison of the modeled and measured relative thickness variation across flat, convex, and concave curved substrates.

Fig. 7

Relative thickness difference maps showing the modeled relative thickness subtracted from the measured relative thickness for (a) flat, (b) convex, and (c) concave curved substrates. Color scales are the same for all figures.

The modeled distributions were in excellent agreement with the measured results. For flat and convex substrates, the standard deviations of relative thickness over the substrate area were almost the same, with average relative thickness differences of 0.19% and $-0.10\%$ and maximum differences of 1.71% and $-1.85\%$, respectively (Table 1).

According to our model [Fig. 6(b)], concave substrates have the smallest relative thickness variation PV, and therefore, the greatest overall thickness uniformity. However, the average relative thickness difference between the measured and the modeled results was 1.45% with a standard deviation of 1.40% and a maximum of 5.74%, significantly larger than that for flat and convex substrates. The difference map clearly showed an elliptical pattern with the maximum error increase parallel with the azimuthal direction, indicating that the model mostly under estimated the film thickness [Fig. 7(c)].

The model for curved substrates is based on the principle that the substrate ROC affects the thickness distribution by changing the mass flux at the substrate (which is inversely proportional to the square of $\stackrel{\rightharpoonup }{R}$) [Eq. (7), Fig. 5], and the vector normal to the substrate surface (${\stackrel{\rightharpoonup }{n}}_s$) [Eqs. (13) and (14)]. There was some spatially correlated error visible in the maps for flat and convex substrates [Fig. 7(a) and 7(b)]. We attribute this to a small (e.g., $<1\mathrm{mm}$) substrate off-centering from the center of rotation, as indicated by the model’s off-center error considerations (see Sec. 3.2.3).

Another assumption made in the model is the erosion profile $ϵ(a)$, which is estimated using a half sine wave function.³⁰ The model does not account for the fact that over the lifetime of the target the erosion profile shifts as the magnetic field becomes stronger at the locations where the target thickness is reduced. The plasma density and deposition rates will thus vary as the target erodes, correlating with changes in the film growth rate distribution over the substrate.^32,64,66,67 Moreover, as the target erodes, the surface becomes increasingly less flat, which increases the error of the vector normal to the substrate surface ${\widehat{n}}_T$.⁶⁸ The area of the target producing the largest mass flux coincides approximately with the largest error in ${\widehat{n}}_T$, and thus may influence the measurement-model error. However, the target geometry is identical in all depositions, and hence does not explain the observed error occurring only in the concave case.

4.2.

Net Thickness Nonuniformity for Curved Mirrors

The thickness distribution differences between films deposited on the convex and concave sides of a substrate are critical for adjustable x-ray optics that employ a stress balancing approach, which aims to balance the integrated stress of films deposited on opposite sides of the mirror.^13,19 The difference in the thickness profiles of films deposited on convex and concave sides of the mirror indicates the nonuniformity of the integrated stress that will arise, assuming the stress is homogeneous throughout the films. The relative thickness distribution difference maps were analyzed by subtracting the relative thickness distributions of films on convex substrates from those of concave substrates. Both the modeled and the measured thickness distributions were considered [Figs. 8(a) and 8(b)]. In both cases, the major relative thickness difference occurred in the azimuthal direction, where the maximum difference was $-12\%$ for the modeled and $-16\%$ for the measured results. The negative sign indicates that in the azimuthal direction the film on the concave side was thicker than the film on the convex side. In the axial direction, the relative thickness difference was significantly less than in the azimuthal direction. Moreover, the measured results show a greater difference than the modeled results, with a maximum variation from the top azimuthal edge of the mirror to the bottom azimuthal edge of $<1.5\%$ down the center, and $<2.0\%$ down the axial edges.

Fig. 8

The effect of substrate curvature on (a) modeled and (b) measured relative thickness distributions, respectively, examined by subtracting the distributions of a convex substrate from a concave substrate. Note that both figures have the same color scale and PV values are given in this figure.

To maintain high x-ray reflectance, x-ray optics are operated at grazing incidence (grazing angle $\alpha <2\mathrm{deg}$). This means that the ultimate imaging quality of an x-ray optic is predominantly controlled by axial figure error and is relatively insensitive to azimuthal figure error (by a factor of the sine of the grazing angle).^10,13,14 Thus, the small relative thickness difference between convex and concave films in the axial direction ($<2.0\%$) is advantageous for adjustable x-ray optics.

4.3.

Impact of Integrated Stress Nonuniformity on Optical Performance

To quantify the impact of the thickness nonuniformity on the optical performance of the mirror, an FEA model of a prototype adjustable x-ray optic deforming under the integrated stress of thin films deposited on convex and concave sides of the mirror was constructed. The FEA model assumes a substrate of Corning Eagle-XG™ glass 101.6-mm long with an ROC of 220 mm, a half cone (graze) angle of 0.375 deg, and a thickness of 0.4 mm. Similar FEA models of adjustable x-ray optics have previously been validated with empirical measurements.^13,14

The resulting narrow aperture ($\sim 0.66\mathrm{mm}$) yields a diffracted image size of 0.3 arc sec at 1 keV (1.24 nm wavelength). The effects of aperture diffraction (i.e., the diffraction limit) are included in the calculation of the grazing incidence PSF and cannot be removed from the calculation.

For a telescope such as that proposed for Lynx, two important factors greatly mitigate the impact of diffraction on imaging. First, most of the telescope effective area at 1 keV is provided by mirror segments with much larger graze angles, and therefore, radially much larger entrance apertures ($\sim 95\%$ of the 1-keV effective area is produced at graze angles greater than $\sim 0.72\mathrm{deg}$, with $\sim 65\%$ of the effective area produced at graze angles $>1.4\mathrm{deg}$). Second, the adjustable x-ray optics mirror design utilizes 200-mm-long mirror segments, which doubles the radial span of the segment entrance aperture(s). For the full telescope design, the area-weighted diffraction contribution for 1-keV x-rays is 0.045 arc sec RMS diameter. This nonzero contribution to imaging is nearly negligible, impacting the PSF HPD and RMS diameter at the level of 1%. Thus, a 0.3-arc sec diffraction limit quoted in this work is an artifact of the shorter length and shallower graze angle for the prototype mirrors and used for the FEA and are not representative of the effective diffraction limit for Lynx optics. These prototype mirror dimensions are used in our development program in response to fabrication and test facility limitations.

To model the effects of nonuniform film thicknesses, the integrated stress at the center of the mirror on both the concave and convex sides was set to $180\mathrm{MPa}\mu \mathrm{m}$, which was the average integrated stress of the PZT layer when deposited on our adjustable x-ray optic prototypes.¹³ The integrated stress was then allowed to vary proportionally to the relative thickness distribution differences as predicted by the geometric sputtering model developed in Sec. 3.2.2 and shown in Fig. 6. With this integrated stress input, the FEA model was then solved for the radial displacement, axial slope, and azimuthal slope errors over the mirror surface. Using a Fresnel-based numerical method,⁶⁹ the PSF of the optic was calculated using these FEA outputs. A perimeter of 15 mm is excluded from the PSF calculation in relation to current fabrication development. Thus the mirror’s performance was assessed over the interior area ($71.6\mathrm{mm}\times 71.6\mathrm{mm}$), consistent with the PSF calculations previously performed for prototype adjustable x-ray optics.¹³ The PSF calculation assumes a single reflection of 1-keV x-rays from a primary Wolter-I mirror segment with a radius of 220 mm and a focal length of 8.4 m, yielding a graze angle of 0.375 deg. In order to quantify x-ray performance from the calculated PSF, both the HPD and the 68% encircled energy diameter (E68) of the calculated PSF are reported.

First, as a point of reference, the FEA model was used to predict the deformation due to films of uniform thickness applied on both the concave and convex sides of the mirror, i.e., the case in which the concave and convex sides of the mirror are subject to the same uniform integrated stress. There is a small change in this figure of the optic in this case. Figure 9(a) shows the relative change in the mirror’s figure from the nominal optical prescription, where the color bar indicates the total out-of-plane deviation from this ideal figure in nanometers. Figure 9(b) depicts the gradient of the relative figure changes taken along the optical axis of the mirror and thus represents the axial slope error of the mirror under these conditions. The axial slope error is not zero, since the finite thickness of the glass means that the ROC on the convex side is not identical to the ROC on the concave side. However, the induced slope error observed is negligible, resulting in a diffraction-limited optic (HPD of 0.3”) as would be expected in the case of equivalent stresses applied on the convex and concave surfaces.

Fig. 9

(a) Figure and (b) axial slope maps of the deformation caused by the application of uniform, stress balanced thin film coatings to the concave and convex sides of the optic. Shown here is the interior ($71.6\mathrm{mm}\times 71.6\mathrm{mm}$) area of the mirror. The integrated stress of the coating is $180\mathrm{MPa}\mu \mathrm{m}$. The single reflection performance of this optic at 1 keV is 0.3 arc sec HPD, which is consistent with the diffraction limit of an x-ray optic in this geometry.

The apparent pixelation of the slope map [Fig. 9(b)] is an artifact of the finite-element modeling process, which used both solid and shell elements in the interest of computational efficiency. Using much more computationally intense, fully solid element models on a limited number of test cases, does not alter the conclusions, changing numeric results (RMS slope, PSF size) only at the level of $<1\%$ in the axial direction and a few percent in the less significant azimuthal direction. This explanation is applicable to all the axial slope space figures.

Next, the FEA model was used to predict the deformation due to the modeled thickness variations on the concave and convex sides of the mirror, which are shown in Fig. 6. The resulting change in the figure space and axial slope space introduced is shown in Fig. 10. Despite a maximum relative thickness difference between concave and convex films in the axial direction of $<2.0\%$, the resulting figure of the mirror has a PV value of 724 nm [Fig. 10(a)], and the optical performance has degraded to 2.5 arc sec HPD. This performance is outside the requirements for Lynx if left uncompensated. However, the piezoelectric actuators of the adjustable optic can be employed to correct for the deformation induced by the nonuniform coating thickness.

Fig. 10

(a) Figure and (b) axial slope maps of the deformation caused by nonuniform coating on the concave and convex sides of the optic as calculated using the model presented in Sec. 3. The integrated stress of the coating at the central point is $180\mathrm{MPa}\mu \mathrm{m}$ on either side, and the integrated stress varies on either side according to the thickness variation presented in Fig. 6. The single-reflection performance of this mirror at 1 keV is 2.6 arc sec HPD.

Using the FEA model of the mirror, a correction to the mirror to optimize x-ray performance was simulated. This correction employs an adjuster pattern of 288 cells measuring 5 mm axially by 5 mm azimuthally. It was assumed that the strain exerted is comparable to the PZT strain exerted by cells for an existing adjustable x-ray optic. A least-squares correction algorithm, which optimizes optical performance over the interior $71.6\mathrm{mm}\times 71.6\mathrm{mm}$ area of the optic, is employed to calculate the voltages to apply to the piezoelectric cells.^13,19 The figure of merit for the optimization is a weighted RMS slope, in which the RMS axial slope is added to the RMS azimuthal slope weighted by a factor of the sine of the grazing angle, since slope errors in the axial direction dominate the optical performance of a mirror segment. The optical performance optimization is also constrained to keep the actuation voltage of any individual piezoelectric actuator cell under 10 V.¹³ The optimized mirror figure is shown in Fig. 11, and results in a figure with a PV of 418 nm, arising primarily from figure displacement which runs in the azimuthal direction are relatively constant in the axial direction [Fig. 11(a)]. In terms of axial slope, however, the mirror is free from major errors [Fig. 11(b)]. Thus, the correction results in a diffraction-limited optical performance of 0.3 arc sec HPD, similar to the uniform, stress balanced case presented in Fig. 9. Thus, thickness nonuniformity difference of the coatings deposition on convex and concave sides of the mirror results in a deformation that can be theoretically corrected by the piezoelectric actuators of the adjustable x-ray optics.

Fig. 11

(a) Figure and (b) axial slope maps of the adjustable x-ray optic following correction. The initial deformation is assumed to be that presented in Fig. 10. A least-squares routine is then employed to calculate the optimal voltage to apply to each piezoelectric cell. The discontinuities visible in the axial slope space map represent the edge of a row of cells. The single-reflection performance of this corrected mirror figure at 1 keV is 0.3 arc sec HPD, consistent with the diffraction limit for an optic in this geometry.

4.4.

Methods for Reducing Thickness Nonuniformity

A sputter tool with an angled circular off-axis planar magnetron is designed for depositions on flat substrates and is not ideal for depositing coatings of high-thickness uniformity on large ($>100{\mathrm{mm}}^2$) convex and concave substrates. However, the small difference in low-axial thickness distribution between convex and concave coatings on the curved substrate is a promising result for prototype adjustable x-ray optics.

The developed thickness distribution model indicates pathways toward reducing the thickness nonuniformity. For example, increasing the throw distance ($h$) reduces the influence that the change in substrate height $z(x,y)$ has on the thickness distribution, thus reducing the thickness nonuniformity arising due to the substrate curvature. However, sputtering rates and efficiency also reduce as the throw distance is increased [see Eq. (3)]. One method for maintaining high-sputtering rates while achieving high-thickness uniformity can be achieved in planar magnetron systems by utilizing correction masks that effectively cloak the ejected mass after leaving the target to control what reaches the substrate.^49–51,70 Another approach is to use two or more magnetron targets of the same composition and position the rotating substrate equidistant between them.⁷¹ However, such designs do not improve sputtering efficiency and require custom designs for each substrate with a different ROC.

The parameters $\widehat{R}$, ${\widehat{n}}_T$, and ${\widehat{n}}_s$ play an important role on the thickness uniformity of films deposited on curved substrates [see Eq. (6)], particularly the angular distribution of $\widehat{R}$, as discussed in Sec. 4.1. As a result, magnetron designs which restrict the angular distribution of depositing mass flux together with specific masking designs may be an effective means of improving thickness uniformity. This may be achieved using sputtering tools with collimators,^40–42 high substrate-target angles (i.e., 90 deg)⁶⁸ or hollow cathode magnetrons. Collimators placed above the sputtering target can restrict the depositing mass flux to normal incidence angles of $\pm 5\mathrm{deg}$, which might be expected to reduce the thickness nonuniformity on curved substrates. However, collimators do not necessarily overcome thickness variations resulting from target erosion profiles.⁴¹ 90-deg off-axis sputter configurations effectively remove the erosion profile’s influence from the spatial distribution of mass flux, but often at the expense of sputtering efficiency.⁶⁸ Hollow circular magnetrons utilizing unbalanced magnetic fields and a low-energy ion flux; on the other hand, have demonstrated thickness distributions with little dependence on substrate ROC.³⁹ Substrates with a 200-mm ROC recorded $<0.15\%$ variation of thickness across a 120-mm diameter area, and some studies have already indicated the feasibility of up scaling hollow magnetron sputter configurations for wider use.⁷²

For sputter deposition on large substrates or on an industrial scale, moving bar or plate planar magnetrons are often used and are likely a feasible approach for adjustable x-ray optics.⁷³ Long axial planar magnetrons were used to coat the interior (x-ray reflective side) of the optics for the Chandra x-ray Observatory.⁷⁴ For adjustable x-ray optic mirrors, which may have sizes up to $400{\mathrm{mm}}^2$, multiple mirrors could be coated simultaneously, using two or more long-plate axial magnetrons, together with specially designed collimators or shields and a method of rotating the shell past the magnetrons. However, the specific design of the sputter tool would need to vary according to ROC of the mirrors. Alternatively, a directionally controlled mass flux from a long-plate magnetron with a collimator could be coupled with a side-to-side substrate movement parallel to the azimuthal direction. To reduce the nonuniformity in the azimuthal direction, the speed of the substrate movement could be changed as a function of ROC to control the total mass flux at the substrate surface. Such a method may provide a desired higher throughput of mirrors and also allow for multiple mirrors to be coated simultaneously.

It should also be acknowledged that in addition to controlling thickness uniformity, spatially controlled stress of deposited thin films may be a useful approach for stress balancing of adjustable x-ray optics.^75,76 However, for adjustable x-ray optics, this approach is only possible for the Cr/Ir coating on the concave side of the mirrors. This is because the films deposited on the convex side of the mirror require thermal excursions to crystallize the PZT after deposition, and thus the final stress is governed predominantly by the thermal expansion coefficient mismatch between the films and the substrate, rather than the deposition parameters.