Theoretical model of the helium pinhole microscope

In recent years, the development of neutral helium microscopes has gained increasing interest. The low energy, charge neutrality, and inertness of the helium atoms makes helium microscopy an attractive candidate for the imaging of a range of samples. The simplest neutral helium microscope is the so-called pinhole microscope. It consists of a supersonic expansion helium beam collimated by two consecutive apertures (skimmer and pinhole), which together determine the beam spot size and hence the resolution at a given working distance to the sample. 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 working distance. Here we present an optimization model for the helium pinhole microscope system. We show that for a given resolution and working distance, there is a single intensity maximum. Further we show that with present-day state-of-the-art detector technology (ionization efficiency $1\times {10}^{-3}$), a resolution of the order of 600 nm at a working distance of 3 mm 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 $0.02\pi$ sr, following an existing design. Reducing the working distance to the micron range leads to an improved resolution of around 40 nm.

Figure 1

Simplified illustration of a pinhole microscope setup. The constants of the problem are marked in gray boxes. $W_{\mathrm{D}}$ is the working distance, $\delta$ is the geometrical spread of the beam, and $\mathrm{\Phi }$ is the focal spot size. Due to diffraction effects, $2\delta$ is not always equal to $\mathrm{\Phi }$. $r_{\mathrm{S}}$ and $r_{\mathrm{ph}}$ are the radius of the skimmer and the radius of the pinhole, respectively, and $a$ is the distance between the skimmer and the pinhole. Note that the system is cylindrically symmetric about the main axis.

Figure 2

Intensity (part/s) crossing a focal spot $(\mathrm{\Phi }=5\mu \mathrm{m})$ of a pinhole helium microscope for a span of values of $r_{\mathrm{S}}$ and $a$. The dashed line shows the subset of maximal solutions given by Eq. (16). The intensity was calculated using the ellipsoidal quitting surface model with the following parameters: $T_{\left|\right|}=0.00802$ K, $T_{\perp }=0.00209$ K, $R_{\mathrm{F}}=0.0112$ m, $x_{\mathrm{S}}=0.0113$ m, $T_0=309$ K, and $P_0=161$ Bar. The working distance of the microscope is set at 0.003 m, and the nozzle diameter $d_{\mathrm{n}}$ at $10\mu \mathrm{m}$. The maximum lies at $(a^{\max },r_{\mathrm{ph}}^{\max },r_{\mathrm{S}}^{\max })=(0.035\mathrm{m},1.7886\times {10}^{-6}\mathrm{m},2.0250\times {10}^{-5}\mathrm{m})$. The black cross indicates the configuration corresponding to the microscope designed by Barr et al. [22]. The optimized design would increase the intensity by 75%. However, a distance of only 35 mm between the two pinholes would not be so easy to realize experimentally with a good pumping speed.

Figure 3

Fresnel number for a span of values of $r_{\mathrm{S}}$ and $a$. The Fresnel number was calculated assuming an average wavelength of $\overline{\lambda }=8.2535\times {10}^{-11}$ m. Note that the line of maximal intensity given by Eq. (16) spans a region where $F\gtrsim 10$. The working distance of the microscope is set to 3 mm, and the resolution to 5 $\mu \mathrm{m}$.

Figure 4

Sikora's approximation intensity divided by the ellipsoidal model, $\zeta$ for a span of values of $r_{\mathrm{S}}$ and $a$ [Eq. (20)]. Note how both models diverge at high apertures due to the omission of the perpendicular temperature effect in Sikora's approximation. At the intensity maximum, both models behave similarly with $\zeta =0.9722$.

Figure 5

Intensity (part/s) crossing the focal spot of a pinhole helium microscope for a span of values of $r_{\mathrm{S}}$ and $a$. The dashed line shows the subset of maximal solutions given by Eq. (16). The intensity was calculated using the ellipsoidal quitting surface model with the following parameters: $T_{\left|\right|}=0.0054$ K, $T_{\perp }=0.0039$ K, $R_{\mathrm{F}}=0.0112$ m, $x_{\mathrm{s}}=0.0113$ m, $T_0=131.3$ K, and $P_0=161$ Bar. The working distance of the microscope is set to 3 mm, and the resolution to 5 $\mu \mathrm{m}$. The maximum lies at $(a^{\max },r_{\mathrm{ph}}^{\max },r_{\mathrm{S}}^{\max })=(0.0260,1.7714\times {10}^{-6},1.4750\times {10}^{-5}$ m). The black cross indicates the configuration corresponding to the microscope designed by Barr et al. [22].

Figure 6

Normalized optimized parameters of a pinhole helium microscope for a span of values of the speed ratio $S$. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=131.3$ K and $P_0=161$ Bar. The working distance of the microscope is set at 0.003 m.

Figure 7

Normalized optimized parameters of a pinhole helium microscope for a span of values of the working distance $W_{\mathrm{D}}$. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=131.3$ K and $P_0=161$ Bar. The working distance of the microscope is set at 0.003 m.

Figure 8

Normalized optimized parameters of a pinhole helium microscope for a span of values of the focal spot $\mathrm{\Phi }$. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=131.3$ K and $P_0=161$ Bar. The working distance of the microscope is set at 0.003 m.

Figure 9

Optimized count rate for different focal spot sizes. The intensity was calculated using Sikora's approximation with the following parameters: $T_0=131.3$ K and $P_0=161$ Bar. The working distance of the microscope is set at 0.003 m. 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).

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 ${\text{P}}^{\prime }$, a point placed on the pinhole 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 pinhole. Therefore, $a=x_{\mathrm{D}}-x_{\mathrm{S}}$. The angles $\beta$ and $\theta$ can also be expressed in terms of $r,\alpha$, and $\rho$.