Electrically controlled focusing of de Broglie matter waves by Fresnel zone plate

Abstract The evolution from classical optics to quantum matter wave optics has been influenced by transformative optical devices like Fresnel zone plates (FZP). This study investigates the focusing of helium atoms by an electrically biased FZP, made of chromium (Cr), using numerical solutions of the time-dependent Schrödinger equation (TDSE) with the split-step Fourier method. The n th opaque zone of an amplitude-type FZP is subject to electrostatic biasing using three methods: (i) V n = V 1 , where V 1 is the biasing voltage applied to the central zone ( n = 1), (ii) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msqrt> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> , and (iii) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mi>sin</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:msub> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , with k E being the radial modulation factor in the biasing voltage. The effects of electrostatic biasing on the transmitted wave packet distribution, transmission coefficient ( T c ), focal length ( f ), size of the focused wave packet ( σ F ), and focusing efficiency ( η ) are analyzed. Results show that electrical biasing modulates diffractive focusing via induced polarization potential. Biasing with V n = V 1 reduces the wave packet transmission and focusing efficiency. Biasing with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msqrt> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> enhances T c by ∼ 38%, f by ∼ 73.5%, and η from 6% to 20%. Biasing with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mi>sin</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:msub> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> offers better control, achieving a focusing efficiency of ∼ 10.5%. These findings suggest new methods for higher efficiency focusing of matter waves, with applications in helium microscopes, cold atom trapping, and high-precision atom lithography for quantum electronic devices.