A Concept of Two-Point Propagation Field of a Single Photon: A Way to Picometer X-ray Displacement Sensing and Nanometer Resolution 3D X-ray Micro-Tomography

Abstract

We introduce the two-point propagation field (TPPF), a real-valued, phase-sensitive quantity defined as the functional derivative of the single-photon detection probability with respect to an infinitesimal opaque perturbation placed between source and detection slits. The TPPF is analytically derived and shown to exhibit a stable, high-frequency sinusoidal structure (6.7 nm period) near the detection slit. This structure enables shot-noise-limited displacement sensing at ~15 pm precision using routinely available synchrotron fluxes and practical nanofabricated slit/comb geometries, requiring mechanical stability only over the final 0.5 mm. The same principle provides a foundation for future nanometer-resolution 3D X-ray microtomography of bulk samples, potentially resulting in a reduced radiation dose. Two conceptual strategies, a central blocker and off-axis multi-slit arrays, are estimated to lower the required incident fluence by more than one order of magnitude each, yielding combined reductions of two to three orders of magnitude with near-term detector development. The TPPF concept, originally developed in a perturbative study of single-particle propagation, thus bridges fundamental quantum measurement questions with practical high-resolution X-ray metrology and imaging.

Figures (8)

• Figure 1: (a) Experimental geometry $\textbf{(not to scale)}$. The wave functions at slits 1,2 at $z_{1},z_{2}$ with apertures $\sigma_{1},\sigma_{2}$ and corresponding transverse displacements $s_{1},s_{2}$ in the x-direction, and a pin $\chi(x)$ at $z,x_{p}$ of width $\sigma_{\chi}$. The distances between the slits and the pin are $z,L-z,$ and $L.$ The longitudinal is the $z$-axis, the x-axis is vertical in this figure. The slits are perpendicular to the plane of the figure, parallel to the $y$-axis. We use $\psi_{1}(x_{1},z_{1})$, $\psi(x,z),$$\psi_{2a}(x_{2},z_{2})$ and $\psi_{2b}(x_{2},z_{2})$ to represent the wave function at the entrance, the pin and the exit, respectively. The subscripts $a$ and $b$ represent before and after slit 2. We use $f_{1}(x_{1})$ and $f_{2}(x_{2})$ to represent the effect of the slits such that $\psi_{1}(x_{1},z_{1})\equiv f_{1}(x_{1}),\psi_{2b}(x_{2},z_{2})=f_{2}(x_{2})\psi_{2a}(x_{2},z_{2})$. If we choose the slit with the hard-edged opening, $f_{1}(x_{1})$ and $f_{2}(x_{2})$ would be zero outside the slits and equal to 1 within the slits. To simplify the calculation, we assume they are Gaussian with peak value 1, except that we choose $f_{1}(x_{1})$ to normalize $\psi_{1}$ as $P_{1}=\int dx_{1}|\psi_{1}(x_{1},z_{1}=0)|^{2}=1$. The pin profile is $\chi(x)=1$ when it is removed. When inserted, $\chi(x)=1-\exp(-\frac{1}{2\sigma_{\chi}^{2}}\left(x-x_{p}\right)^{2})$; effective width (equivalent hard-edged slit width) is $\Delta x=\sqrt{2\pi}\sigma_{\chi}$. (b)The wave function of a single particle spreads over a wide region after emission and collapses instantaneously upon detection
• Figure 2: For a setup in Fig.\ref{['fig:(b)-The-pond-wave']}(a), take $\lambda=0.541nm,\sigma_{2}=0.8nm,\sigma_{1}=0.5\mu m,L=0.5m,L-z=0.5mm$, Fig.2(a): $\frac{\delta P_{2b}}{\delta\chi(x,z)}$ vs. $x,z$ in color scale for $s_{2}=50\text{$\mu$}m$, $P_{2b}=9.47\times10^{-6}$. Some elliptical patterns are artifact due to the limited number of points of the plot and the nearly periodic structure of the function $\frac{\delta P_{2b}}{\delta\chi(x,z)}$. The patterns change with the number of points of the plot, but it is hard to avoid even with pixels increased to $4\times10^{6}$ in the plot. Fig.2(b): A narrow region within 2mm from the slit 2 in (a) showing the details not visible in (a). The detailed fringe structure is not visible in this plot because it is visible only when magnified, as given in the following (c) and (d) plots. The region between the two white lines (45$\mu m<x<55\text{$\mu m$})$ is given in (c) with details. The hardly visible white dot, which is too small to be recognized as a box, indicates the region (pointed to by the arrow in Fig.2(b)) shown in (d) with fringe details. Fig.2(c): The region (45$\mu m<x<55\text{$\mu m$})$ indicated by the two white lines in (b). Fig.2(d): The region indicated in (b) by an arrow as a white dot in a box size of $10\mu m\times1\mu m$ (60$\mu m<x<61\text{$\mu m,-500<z-L<-490\text{$\mu m$})$}$ shows the fringe structure. The most pronounced feature is that the amplitude ($\pm30m^{-1})$indicated by the color scale is comparable to the peak amplitude in Fig.2(c).
• Figure 3: The contours of $\frac{\delta P}{\delta\chi(x,z)}$. The colored lines are the centroids $x_{c0}$ for $s_{2}=0,25,$$50,75\mu$m, respectively. The cyan colored contours $x_{c0}\pm\frac{1}{2}x_{\pi}$ represent the contours of the main peaks of TPPF for various $s_{2}$. $x_{\pi}$ here is the distance from the the centroid $x_{c0}$ to the point with phase shift from the centroid $x_{c0}$ by $\pi$. For $s_{2}=50\mu$m, this contour corresponds to the red colored region in Fig.\ref{['fig:dPdx_color']}(a). As a comparison, the RMS of the wave function $|\psi(x,t)|^{2}$ is the thick dashed cyan line showing its width continues to spread till the end.
• Figure 4: $\lambda=0.541nm,\sigma_{2}=0.8nm,\sigma_{1}=0.5\mu,L=0.5m,z_{1}-z=0.5mm,$$\Delta\chi\Delta x=3$nm (a) around peak at $|x|<60\mu m$. (b) around peak at $|x|<3\mu m$. (c) in reginon around $x=x_{max}+\sigma_{w}$, $x_{\pi}=0.52\mu m$, $\sqrt{6012}x_{\pi}-\sqrt{6011}x_{\pi}=3.35$nm , $\sigma_{w}\approx40\mu$m. $P_{2b}=1.86\times10^{-5}$ for this configuration.
• Figure 5: Width $x_{\pi},\sigma_{w}$ for the case of $\lambda=0.541nm$, $\sigma_{1}=0.5\mu m,\sigma_{2}=0.8nm,$$s_{1}=0,$$z_{2}=L=0.5m$. The maximum width $x_{\pi}$ of phase shift $\pi$ is in the middle point $z=0.25m$. Fig. \ref{['fig:Width--for']}(a): within the valid region of $x_{\pi}=\sqrt{\frac{\pi}{\alpha_{\chi i}}}$ for $0.01m<z<0.5m-10\mu m$. When $z$ close to $0,$ paraxial approximation is invalid. When $z$ is too close to $0.5m$, $\alpha_{\chi i}=0,$$x_{\pi}$ does not exist. Fig. \ref{['fig:Width--for']}(b) : $\sigma_{w}=\sqrt{-\frac{1}{2\alpha_{\chi r}}}$ as the function of $z$ (the green curve) is continuous as it converges to the exit slit near $z=L=0.5m$ as shown with details near slit 2 for $50mm<L-z<10\mu m$, as compared with (a). Fig. \ref{['fig:Width--for']}(c): same plot as Fig. \ref{['fig:Width--for']}(b) except $z$-axis is replaced by a log scale of $L-z$, to see how fast the TPPF converges into the exit slit.