Parameter selection of algorithms for supermirror design

To evaluate and compare the three types of algorithms described above, we choose a typical representative of each type: the Gukasov-Ruban-Bedrizova (GRB) algorithm [29, 30] for the continuous type, the Masalovich (Mas) algorithm [33] for the discrete type, and the ABC algorithm [27, 36] for the empirical type. These algorithms exhibit the main characteristics of their types and thus are suitable for demonstrating their performance in different design parameter spaces.

The range of the designed m value md is limited to 2-8, which covers practical values for current supermirror technology. The ABC algorithm has only one input parameter, the number of bilayers, which determines the m value of the supermirror. The GRB and Mas algorithms have two input parameters, md and the designed reflectivity Rd. Therefore, the latter two algorithms have an additional degree of freedom to adjust the designed performance. To better quantify their performance, here we use the average reflectivity for evaluation; it is defined as ${\overline{R}}_{\text{d}}=\frac{{\displaystyle \int }R\left(Q\right)\text{d}Q}{{\displaystyle \int }\text{d}Q}$. The difference in the ${\overline{R}}_{\text{d}}$ values of the three algorithms is controlled within 0.001 by adjusting the designed reflectivity in the GRB and Mas algorithms. The parameters of the three algorithms determined in this way are referred to as the initial conditions. Fig. 2 shows the average reflectivity of the three algorithms as a function of m under the initial conditions.

Fig. 2Full-screen View

(Color online) Average reflectivity of the three algorithms as a function of m under the initial conditions. The difference between the three algorithms is controlled within 0.001.

Film thickness distribution on curved surface coating

We employed the model proposed by Bishop et al. [37] to calculate the thickness distribution of sputtered films on curved substrates. This model is based on the characteristics of a magnetron sputtering system with a rotating substrate, as shown in Fig. 3. The ejected mass flux decreases in proportion to the inverse square of the distance it travels to the substrate. Therefore, it can also be used to describe coating systems with similar spatial structures, such as ion beam sputtering. In addition, the substrate rotates at a variable speed around an external center [19] (as indicated by “Revolution” in Fig. 3) so that the nonuniformity of the axial thickness distribution of the film can be ignored, and only the nonuniformity of the azimuthal thickness distribution must be considered.

Fig. 3Full-screen View

(Color online) Schematic diagram of sputtering chamber geometry. Curved gray arrow indicates substrate rotation during deposition.

Note that various methods have been proposed to reduce the thickness nonuniformity of coatings on curved surfaces, such as the use of shadow masks or differential deposition techniques [19, 38-43]. However, their implementation is complicated because they require customized designs for different sputtering configurations and substrates. Therefore, it is necessary to evaluate the effect of thickness nonuniformity on the supermirror reflectivity to determine whether these additional steps are needed.

In this model, we considered a cylindrical concave curved substrate (i.e., one with curvature in the azimuthal direction and not in the axial direction). The side length of the surface is expressed as a dimensionless number, which is the ratio of the side length to the target-substrate distance. Thus, the main parameter affecting the thickness distribution is the radius of curvature (ROC) of the substrate. Fig. 4 shows the relative thickness distribution of a layer deposited on substrates with ROCs of 50, 80, and 160 mm.

Fig. 4Full-screen View

(Color online) Relative thickness distribution of a layer deposited on substrates with ROCs of 50, 80, and 160 mm.

To simplify the calculation, we assume that each layer in the supermirror has the same relative thickness distribution. As a result, the position of a point whose thickness is normalized to the value 1 will yield different absolute thickness distributions. Therefore, the position of this normalization point is also a parameter of the reflectivity. As shown in Fig. 5, the abscissa is the azimuthal position of the normalization point. Taking m = 5 and a ROC of 80 mm as an example, we calculate the average reflectivity ${\overline{R}}_{\text{d}}$ of the supermirror designed by the three algorithms for this series of thickness distributions. The optimal value occurs when the normalization point is located at the position where the azimuthal angle is equal to 9.6. The film thickness distribution model described here will be employed to compare the three algorithms.

Fig. 5Full-screen View

(Color online) Variation of average reflectivity of supermirror (m = 5) versus position of layer thickness normalization point.

Distribution of neutron intensity with respect to Q

As noted above, the use of neutron-focusing mirrors introduces a neutron intensity distribution with respect to Q, which originates from the grazing angle distribution of incident neutrons on mirrors. Here we use a simple model to formulate this intensity distribution, as shown in Fig. 1(b). The neutron-focusing mirror has a nested structure [17] consisting of a series of concentric conical mirrors with different radii. Assuming a point neutron source, the angular distribution of neutrons grazing the surface of a mirror with radius R should be proportional to the solid angle, which is given by $p\left(\theta \right)\propto \frac{A}{L^2+R^2}$ (1) where A is the reflective area of the focusing mirror, and L is the horizontal distance between the center of the focusing mirror and the neutron source.

To simplify the calculation, we assume that the length of the focusing mirror, w, is short enough that the neutron grazing incidence angle of the mirror with the same radius can be regarded as having the same value, and mirrors with different radii will not block each other. Thus, the effective reflective area is $A=2\pi Rw\sin \theta =2\pi wL\sin \theta \tan \theta$ (2) where $L^2+R^2=\frac{L^2}{{\cos }^2\theta }$ (3) Therefore, the angular distribution is $p\left(\theta \right)\propto \frac{A}{L^2+R^2}=\frac{2\pi w}{L}{\sin }^2\theta \cos \theta$ (4) For $R\ll L$, we have $p\left(\theta \right)\propto {\theta }^2$ (5) If we denote the innermost and outermost radii of the focusing mirror as R₁ and R₂, respectively, the minimum and maximum grazing incidence angles of neutrons are ${\theta }_{\text{m}in}=\frac{R_1}{L}$ and ${\theta }_{\text{m}ax}=\frac{R_2}{L}$, respectively. Therefore, the angular distribution of neutrons can be simplified as follows: $p(\theta )=\{\begin{array}{cc}\mathrm{0,}& 0\le \theta <{\theta }_{\min }\\ C\cdot {\theta }^2\mathrm{,}& {\theta }_{\min }\le \theta \le {\theta }_{\max }\\ \mathrm{0,}& {\theta }_{\max }<\theta \le \frac{\pi }{2}\end{array}$ (6) where C is the normalization coefficient, which is set so that ${\displaystyle {\int }_0^{\frac{\pi }{2}}}p\left(\theta \right)\text{d}\theta =1$. The calculations presented in this paper are based on R₁=8 cm, R₂=12 cm, and L=4 m, which are typical values for the ongoing project at CPHS [20].

The wavelengths of incident neutrons follow the Maxwellian distribution, which peaks around the thermal wavelength at the moderator temperature T. The neutron flux ϕ(λ), with the unit of neutron counts per second per unit wavelength, is thus expressed as $\varphi \left(\lambda \right)={\varphi }_0\frac{2{\lambda }_T^4}{{\lambda }^5}\exp \left(-\frac{{\lambda }_T^2}{{\lambda }^2}\right)$ (7) where ${\lambda }_T=\frac{h}{\sqrt{\left(2m{k}_{\text{B}}T\right)}}$, h is the Planck constant, and k_B is the Boltzmann constant.

In this study, λT is set to $2\text{Å}$, which corresponds to a thermal neutron source. The neutron wavelength spectrum is weighted by the angular distribution p(θ) to obtain the distribution of neutron intensity with respect to Q, which is given by ${\displaystyle \int }I(Q)\text{d}Q={\displaystyle \int }{\displaystyle \int }\varphi \left(\lambda \right)p(\theta )\text{d}\theta \text{d}\lambda$ (8) The wavelength λ can be expressed in terms of Q as $\lambda =\frac{4\pi }{Q}\theta$. Using $\text{d}\lambda =-\frac{4\pi \theta }{Q^2}\text{d}Q$, I(Q) can be expressed as $I(Q)={\displaystyle \int }\frac{4\pi \theta }{Q^2}\varphi \left(\frac{4\pi }{Q}\theta \right)p(\theta )\text{d}\theta$ (9)

Figure 6 shows the angular distribution p(θ), neutron flux ϕ(λ), and I(Q). As shown in Fig. 6(a), the proportion of incident neutrons collected by the mirror increases with increasing θ. The neutron flux decreases rapidly with increasing wavelength, as shown in Fig. 6(b). Over the range of angles and wavelengths we consider, the angular distribution changes only by a factor of 2, whereas the neutron flux intensity changes by several orders of magnitude. Therefore, I(Q) follows the trend of ϕ(λ) and increases rapidly with increasing Q, as shown in Fig. 6(c). Note that the Q-dependent neutron intensity also depends on the size of the mirrors. Because larger mirrors can reflect more neutrons at larger grazing incidence angles, the proportion of high-Q neutrons in the neutron intensity distribution increases.

Fig. 6Full-screen View

(Color online) (a) Angular distribution, (b) neutron flux ϕ(λ), and (c) I(Q). The parameters used in the calculation are R₁=8 cm, R₂=12 cm, L=4 m, and ${\lambda }_T=2\text{Å}$.

Effect of film thickness nonuniformity on the average reflectivity of supermirrors without considering I(Q)

First, we compared the average reflectivity obtained by the three algorithms for various m values and film thickness nonuniformities. Here, the average reflectivity is defined as ${\overline{R}}_{\text{d}}=\frac{{\displaystyle \int }R(Q)\text{d}Q}{{\displaystyle \int }\text{d}Q}$ (10) The integral range is $Q\in \left[\mathrm{0,}m\cdot Q_{\text{c}}\right]$, where $Q_{\text{c}}\approx 0.022{\text{Å}}^{-1}$ is the critical Q value for total reflection by natural nickel.

The thickness deviation Δ d, which is defined as the difference between the thickest and thinnest positions, is used to quantify the nonuniformity of the film thickness. Fig. 7(a) shows the algorithm with the highest average reflectivity for each point (m,Δ d) in the parameter space.

Fig. 7Full-screen View

(Color online) (a) Algorithms with highest average reflectivity for various m and film thickness nonuniformity Δ d. (b) Differences between average reflectivity of the ABC algorithm and those of the other two algorithms.

In Fig. 7(a), an obvious dividing line appears near m = 2.8. When m < 2.8, the average reflectivity of the GRB and Mas algorithms is higher, whereas when m > 2.8, the ABC algorithm obtains the highest average reflectivity. To better show the advantages of the ABC algorithm over the other two algorithms, Fig. 7(b) shows the differences between the average reflectivity of the ABC algorithm and those of the other two algorithms at each parameter point, where $\Delta {\overline{R}}_{\text{d}}=({\overline{R}}_{\text{d}\mathrm{,}ABC}-{\overline{R}}_{\text{d}\mathrm{,}others})/{\overline{R}}_{\text{d}\mathrm{,}others}$. This result indicates that the ABC algorithm can better handle large m values and high film thickness nonuniformity. For m = 8 and a film thickness nonuniformity Δ d>25%, its average reflectivity can be 2.8% higher than those of the other two algorithms.

We further explored the origins of the observed differences between the three algorithms. Consider a supermirror with m = 5 as an example. Fig. 8 shows the reflectivity curves of the three algorithms, where the solid line is the original reflectivity curve, and the dotted line is the reflectivity curve considering the effect of film thickness nonuniformity (Δ d=8.6%). The reflectivity curve of the ABC algorithm differs from the other two. For the ABC algorithm, the reflectivity decreases continuously with increasing Q, and the degradation is more severe. By contrast, for both the GRB and Mas algorithms, the reflectivity first decreases with increasing Q and then remains stable at an almost constant value, as indicated by the high-Q plateau in the reflectivity curves. Consequently, the variation in average reflectivity with film thickness is different for the three algorithms. We selected the layers contributing to the neutron reflectivity at Q/Q_c=3 ~ 4 and calculated the corresponding reflectivities, as shown in Fig. 9. When the film thickness decreases, the reflectivity curve shifts toward higher Q. For the GRB and Mas algorithms, the slope of the fitting line is very close to 0 (approximately -0.005), regardless of whether the film thickness inhomogeneity is considered. The intercept difference is approximately 0.035 and is the source of the average reflectivity difference when the film thickness nonuniformity is considered. However, the reflectivity curve of the ABC algorithm has a noticeable slope (-0.035), and the slope also increases (to -0.055) when the film thickness nonuniformity is considered. Therefore, the decrease in reflectivity is only approximately 0.01, which is smaller than that of the GRB and Mas algorithms.

Fig. 8Full-screen View

(Color online) Reflectivity curves of the three algorithms (m=5).

Fig. 9Full-screen View

(Color online) Reflectivity curves of the three algorithms (m=5) at Q/Qc=3∼4. Straight lines are linear fits of the plateaus.

Effect of film thickness nonuniformity on weighted average reflectivity of supermirrors considering I(Q)

As demonstrated above, the reflectivity of each algorithm is distributed with respect to Q, and the intensity of incident neutrons also has a distribution with respect to Q. This result suggests decoupling between the optimal supermirror structure and the geometry of the focusing mirrors and source, which can be employed to optimize the design parameters and select the best design algorithm. Therefore, in this section, we incorporate the effect of geometric factors into the evaluations to explore the influence of film thickness nonuniformity.

Considering the neutron intensity distribution over Q, we compared the average reflectivity obtained by the three algorithms under various values of m and the film thickness nonuniformity. This comparison involves the characteristics of neutron sources, which can more accurately indicate the performance of supermirrors in practical applications. Instead of the average reflectivity written as Eq. (10) in Sec. 3.1, here the average reflectivity weighted by the neutron intensity distribution over Q is used. It is defined as ${\overline{R}}_{\text{w}}=\frac{{\displaystyle \int }I(Q)R(Q)\text{d}Q}{{\displaystyle \int }I(Q)\text{d}Q}$ (11) The integral range is also $Q\in \left[\mathrm{0,}m\cdot Q_{\text{c}}\right]$.

The same range of m and film thickness nonuniformity is used to compare the three algorithms. Fig. 10(a) shows the algorithm with the highest weighted average reflectivity for each point (m,Δ d) in the parameter space.

Fig. 10Full-screen View

(Color online) (a) Algorithms with highest weighted average reflectivity for various values of m and film thickness nonuniformity Δ d. (b) Difference between weighted average reflectivity of the Mas algorithm and ABC algorithm.

When the neutron intensity distribution over Q is considered, the selection diagram changes completely. The ABC algorithm performs well for small m and high film thickness nonuniformity. When m > 4.5, the Mas algorithm yields the highest weighted average reflectivity. Fig. 10(b) shows the difference between the weighted average reflectivities of the Mas and ABC algorithms at each parameter point, where $\Delta {\overline{R}}_{\text{w}}=({\overline{R}}_{\text{w}\mathrm{,}Mas}-{\overline{R}}_{\text{w}\mathrm{,}ABC})/{\overline{R}}_{\text{w}\mathrm{,}ABC}$. This comparison indicates that the Mas algorithm has more advantages at large m and low film thickness nonuniformity. For m = 8 and Δ d<15%, its weighted average reflectivity is 4.1% higher than that of the ABC algorithm.

This difference can still be attributed to the difference in the reflectivity curves of the three algorithms. Fig. 11 shows the reflectivity curves for nonuniform film thickness (Δ d=8.6%). Because there is a plateau in the reflectivity curves of the Mas and GRB algorithms, their reflectivity for high-Q incident neutrons is higher than that of the ABC algorithm. According to the neutron intensity versus Q curve in Fig. 6(c), the neutron intensity is higher for high Q, which results in a higher weighted average reflectivity.

Fig. 11Full-screen View

(Color online) Reflectivity curves of the three algorithms at a film thickness nonuniformity of 8.6%.

We also compared the effects of film thickness nonuniformity and neutron intensity distribution over Q on the average reflectivity of the ABC algorithm. Figure 12(a) shows the average reflectivity difference between films of nonuniform and uniform thickness. Figure 12(b) reveals difference between weighted average reflectivity with and without considering the neutron intensity distribution over Q. Even for m=8, the degradation of the reflectivity at a nonuniformity of 10% is less than 3%, which is much smaller than the effect of the Q-dependent neutron intensity distribution. The film thickness nonuniformity affects the reflectivity to the same degree as the Q distribution only at small m (m<3) and especially large film thickness nonuniformity (Δ d>25%). This result demonstrates that it is worth considering the characteristics of neutron sources in the commonly used design parameter space of supermirrors.

Fig. 12Full-screen View

(Color online) Effects of (a) film thickness nonuniformity and (b) neutron intensity distribution over Q on average reflectivity of the ABC algorithm.

Origins of differences in reflectivity

As shown in Sec. 3.1 and Sec. 3.2, the reflectivity curves of supermirrors generated by the three algorithms are different. In this section, we try to explain the reasons for these differences.

The reflectivity differences are essentially attributable to differences in the supermirror multilayer sequence, including layer thickness and layer number distribution. Fig. 13 shows the thickness distribution of each layer in the sequences designed by the three algorithms. As the ordinal number increases, the position of a layer in the multilayer sequence becomes closer to the substrate. Because the first 10 layers contribute to the reflectivity at Q/Q_c<1, where there is no difference in the output of the algorithms, we compare only the structures of other layers here. As shown in Fig. 13, the layer thicknesses in the multilayer sequences designed by the GRB and Mas algorithm are very similar, but the layer thickness of the multilayer sequence designed by the ABC algorithm is relatively large. According to the principle of Bragg diffraction, ${\lambda }_n=2d_n\sin \theta$ (12) and the definition of Q ($Q\approx \frac{4\pi }{\lambda }\theta$), we have $Q_n=\frac{2\pi }{d_n}$ (13) where dn is the thickness of the nth layer; Qn is the Q of neutrons satisfying the corresponding Bragg conditions, and the layer has high reflectivity to these neutrons. Therefore, because of the relatively thick layer, the ABC algorithm can yield higher reflectivity for low-Q neutrons.

Fig. 13Full-screen View

(Color online) Thickness distribution of each layer in supermirror multilayer sequence.

From another perspective, the curves in Fig. 13 can also be interpreted as indicating that layers with the same thickness as the other two algorithms appear later in the multilayer structure designed by the ABC algorithm. Thus, the ABC algorithm allocates more layers to the low-Q regime. Fig. 14(a) shows the distribution of the number of bilayers over Q for the three algorithms, which confirms this interpretation. For intuitive understanding, in Fig. 14(b), we divide the reflectivity curve into four Q ranges ($0<Q/Q_{\text{c}}\le 2$, $2<Q/Q_{\text{c}}\le 3$, $3<Q/Q_{\text{c}}\le 4$, $4<Q/Q_{\text{c}}\le 5$), and compare the number of bilayers that each algorithm allocates to each Q range. The ABC algorithm assigns more layers to the low-Q regime; thus, it obtains higher reflectivity for low-Q neutrons. By contrast, the GRB and Mas algorithms allocate more layers to the high-Q regime, resulting in higher reflectivity for high-Q neutrons.

Fig. 14Full-screen View

(Color online) (a) Distribution of number of bilayers over Q and (b) number of bilayers allocated to each interval in multilayer sequence of each algorithm.

In essence, these differences originate from the theoretical basis of these algorithms, which has also been discussed in [44]. The continuous and empirical algorithms were derived from the kinematic theory of neutron scattering, which considers only the Born approximation. These results are valid only for the maximum reflectivity, $R_{\text{m}ax}\ll 1$, which requires that the number of nth bilayers, Nn, is much greater than 1. By contrast, the dynamical theory of neutron diffraction considers multiple scattering events and the associated extinction effects, which are more pertinent for supermirrors. The solutions of the exact dynamical theory of reflection are used in discrete algorithms. It may be surprising that the performance of the GRB algorithm is very similar to that of the discrete algorithm. The reason is that in contrast to the Mezei algorithm [24, 25] and the original GRB algorithm [29], the improved GRB algorithm [30] and Schelten-Mika (SM) algorithm [31] introduced a refraction correction for finding the thickness of layers in a supermirror design. Thus, their thickness distribution function no longer has a power-law form, as shown in Fig. 13(a).