Theoretical model of the helium zone plate microscope

Neutral helium microscopy is a new technique currently under development. Its advantages are the low energy, charge neutrality, and inertness of the helium atoms, a potential large depth of field, and the fact that at thermal energies the helium atoms do not penetrate into any solid material. This opens the possibility, among others, for the creation of an instrument that can measure surface topology on the nanoscale, even on surfaces with high aspect ratios. One of the most promising designs for helium microscopy is the zone plate microscope. It consists of a supersonic expansion helium beam collimated by an aperture (skimmer) focused by a Fresnel zone plate onto a sample. The resolution is determined by the focal spot size, which depends on the size of the skimmer, the optics of the system, and the velocity spread of the beam through the chromatic aberrations of the zone plate. An important factor for the optics of the zone plate is the width of the outermost zone, corresponding to the smallest opening in the zone plate. The width of the outermost zone is fabrication limited to around 10 nm with present-day state-of-the-art technology. Due to the high ionization potential of neutral helium atoms, it is difficult to build efficient helium detectors. Therefore, it is crucial to optimize the microscope design to maximize the intensity for a given resolution and width of the outermost zone. Here we present an optimization model for the helium zone plate microscope. Assuming constant resolution and width of the outermost zone, we are able to reduce the problem to a two-variable problem (zone plate radius and object distance) and we show that for a given beam temperature and pressure, there is always a single intensity maximum. We compare our model with the highest-resolution zone plate focusing images published and show that the intensity can be increased seven times. Reducing the width of the outermost zone to 10 nm leads to an increase in intensity of more than 8000 times. Finally, we show that with present-day state-of-the-art detector technology (ionization efficiency $1\times {10}^{-3}$), a resolution of the order of 10 nm is possible. In order to make this quantification, we have assumed a Lambertian reflecting surface and calculated the beam spot size that gives a signal 100 cts/s within a solid angle of 0.02 sr, following an existing helium microscope design.

Figure 1

Diagram of a zone plate microscope setup. The parameters kept constant are in gray boxes. $x_{\mathrm{S}}$ is the distance between the nozzle and the skimmer aperture, and $R_{\mathrm{F}}$ is the radius of the quitting surface. $b$ is the working distance, $\mathrm{\Phi }$ is the focal spot size. $r_{\mathrm{S}}$ and $r_{\mathrm{zp}}$ are the radius of the skimmer and the radius of the zone plate, respectively, and $a$ is the distance between the skimmer and the zone plate. Note that the system is rotationally symmetrical about the main axis.

Figure 2

Intensity (part/s) in a 0.9 $\mu \mathrm{m}$ focal spot of a zone plate helium microscope for a span of values of $r_{\mathrm{zp}}$ and $a$ and for a constant $\mathrm{\Delta }r=323\mathrm{nm}$. The solid line shows the subset of maximum solutions given by Eq. (21), and the dashed line shows the numerical solution using the ellipsoidal quitting-surface model. The intensity was calculated using the ellipsoidal quitting-surface model with the following parameters: $T_{\left|\right|}=0.0052$ K, $T_{\perp }=0.0035$ K, $R_{\mathrm{F}}=0.01129$ m, $x_{\mathrm{S}}=0.0113$ m, $T_0=115$ K, $P_0=101$ Bar, and $\lambda =0.089$ nm. The nozzle diameter $d_{\mathrm{n}}$ is set at $10\mu \mathrm{m}$. The maximum (black cross) lies at $(a^{\max },r_{\mathrm{zp}}^{\max },r_{\mathrm{S}}^{\max })=(0.555\mathrm{m},3.763\times {10}^{-6}\mathrm{m},1.195\times {10}^{-5}\mathrm{m})$. The yellow cross indicates the configuration corresponding to the original setup used in [33]. The optimized design would increase the intensity by seven times. An intensity increase of as much as 8000 times can be achieved by reducing $\mathrm{\Delta }r$ to 10 nm (see Fig. 3).

Figure 3

Intensity (part/s) in a $0.9\mu \mathrm{m}$ focal spot of a zone plate helium microscope for a span of values of $r_{\mathrm{zp}}$ and $a$ and for a constant $\mathrm{\Delta }r=10\mathrm{nm}$. The red line shows the subset of maximum solutions given by Eq. (21) and the dashed line shows the real line of zero derivative in the ellipsoidal quitting-surface model. The intensity was calculated using the ellipsoidal quitting-surface model with the following parameters: $T_{\left|\right|}=0.0052$ K, $T_{\perp }=0.0035$ K, $R_{\mathrm{F}}=0.01129$ m, $x_{\mathrm{S}}=0.0113$ m, $T_0=115$ K, $P_0=101$ Bar, and $\lambda =0.089$ nm. The nozzle diameter $d_{\mathrm{n}}$ is set at $10\mu \mathrm{m}$. The maximum (black cross) lies at $(a^{\max },r_{\mathrm{zp}}^{\max },r_{\mathrm{S}}^{\max })=(0.06\mathrm{m},12.22\times {10}^{-6}\mathrm{m},1.173\times {10}^{-5}\mathrm{m})$. Note how the intensity peak is at $3.5\times {10}^{11}$ (part/s), corresponding to an intensity of about 8000 times the configuration used in [33].

Figure 4

Normalized fraction of Sikora's model divided by the ellipsoidal quitting-surface model, $\zeta$, for a span of values of $r_{\mathrm{zp}}$ and $a$. The solution was calculated for a zone plate helium microscope with a resolution of 0.9 $\mu \mathrm{m}$ and a $\mathrm{\Delta }r$ of 323 nm. The red line shows the subset of maximum solutions given by Eq. (21). Both models were computed using the following parameters: $T_{\left|\right|}=0.0052$ K, $T_{\perp }=0.0035$ K, $R_{\mathrm{F}}=0.01129$ m, $x_{\mathrm{S}}=0.0113$ m, $T_0=115$ K, $P_0=101$ Bar, and $\lambda =0.089$ nm.

Figure 5

Skimmer radius $r_{\mathrm{S}}$ (m), for a zone plate helium microscope with a resolution of 0.9 $\mu \mathrm{m}$ and a $\mathrm{\Delta }r$ of 323 nm. The radius is plotted with a logarithmic scale due to the high variations in its magnitude. Note how in the areas where Sikora's approximation fails (see Fig. 4), the skimmer radius is largest. The red line indicates the subset of maximum solutions given by Eq. (21). The radius was calculated using the following parameters: $S=241.68, \lambda =8.9\times {10}^{-11}$ m.

Figure 6

Normalized optimized parameters of a zone plate helium microscope for a span of values of the speed ratio $S$. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=115$ K, $P_0=101$ Bar at a fix resolution, $\mathrm{\Phi }=0.9\mu \mathrm{m}$ and $\mathrm{\Delta }r=10$ nm. The maximum values of the parameters, used for normalization are $r_{\mathrm{zp}}=10.2\mu \mathrm{m}, r_{\mathrm{S}}=458.6\mu \mathrm{m}, a=0.05$ m, $I=2.65\times {10}^{11}$ part/s. The small fluctuations of the data are due to numerical effects.

Figure 7

Normalized optimized parameters of a zone plate helium microscope for a span of values of the focal spot size $\mathrm{\Phi }$ for a fixed resolution $\mathrm{\Delta }r=10$ nm. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=115$ K, $P_0=101$ Bar, which corresponds to $S=241.7$. The maximum values of the parameters used for normalization are $r_{\mathrm{zp}}=29.7\mu \mathrm{m}, r_{\mathrm{S}}=17.4\mu \mathrm{m}, a=0.09$ m, $I=1.68\times {10}^{12}$ part/s, and $b=0.007$ m. The small fluctuations of the data are due to numerical effects.

Figure 8

Normalized optimized parameters of a zone plate helium microscope for a span of values of the width of the outermost zone $\mathrm{\Delta }r$. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=115$ K, $P_0=101$ Bar at a fixed resolution $\mathrm{\Phi }=0.9\mu \mathrm{m}$. This corresponds to a speed ratio of $S=241.7$. In this case, $b$ has also been included to emphasize the reduction of the microscope length at high intensity setups. The maximum values of the parameters used for normalization are $r_{\mathrm{zp}}=12.5\mu \mathrm{m}, r_{\mathrm{S}}=13.9\mu \mathrm{m}, a=0.7515$ m, $I=3.81\times {10}^{11}$ part/s, and $b=0.0316$ m. The small fluctuations of the data are due to numerical effects.

Figure 9

Optimized count rate for different focal spot sizes. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=115$ K, $P_0=161$ Bar, which corresponds to $S=241.7$. The efficiency of the detector, placed at $\pi /4$ radians, is $\eta =1\times {10}^{-5}$ (lower line) or $\eta =1\times {10}^{-3}$ (upper line). The red line indicates the 100 part/s intensity limit. The constraints on the calculation are a minimum working distance of $10\mu \mathrm{m}$, a minimum skimmer radius of 100 nm, and a minimum $a$ of 1 mm. $\mathrm{\Delta }r$ is set at 10 nm.

Figure 10

Illustration of all variables used in the ellipsoidal quitting-surface model. P is a point on the quitting surface from which a particle leaves in a straight trajectory until P', a point placed on the zone plate plane. The point on the quitting surface is given by the set of Cartesian coordinates $(x_{\mathrm{R}},y_{\mathrm{R}},z_{\mathrm{R}})$, which can be related to the polar coordinates $r,\alpha ,\rho$ for further integration. $x_{\mathrm{S}}$ is the distance from the nozzle to the skimmer and $x_{\mathrm{D}}$ is the distance from the nozzle to the zone plate. Therefore, $a=x_{\mathrm{D}}-x_{\mathrm{S}}$. The angles $\beta$ and $\theta$ can also be expressed in terms of $r, \alpha$, and $\rho$.