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Natural Sciences

Free‐Space Diffraction and Interference in a Transformed Frame

J S Briggs

Published Jan 7, 2025• 1 citation
Publisher

Abstract

ABSTRACT For free propagation from a focus, the Hermite–Gauss wave functions of optics spread in space. In quantum mechanics, the Hermite–Gauss functions are referred to as the harmonic oscillator eigenfunctions. These functions are used here to describe the interference of wave packets. It has been shown that when transformed to a frame moving with the normal to the wave front trajectories, the Hermite–Gauss functions are constant up to a phase factor which is the Gouy phase. The Gouy phase itself assumes the role of proper space or time coordinate. Along the whole of such a trajectory, the space wave function is proportional to the wave number or momentum function. An arbitrary normalizable wave packet can be expanded using the Hermite–Gauss functions as a basis. As example, it is shown that in the transformed frame, a displaced Gaussian does not spread but rather becomes a coherent state. This allows a particularly simple representation of the Young's interference pattern from two or more slits.

DOI
10.1002/ntls.20240028

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ABSTRACT For free propagation from a focus, the Hermite–Gauss wave functions of optics spread in space. In quantum mechanics, the Hermite–Gauss functions are referred to as the harmonic oscillator eigenfunctions. These functions are used here to describe the interference of wave packets. It has been shown that when transformed to a frame moving with the normal to the wave front trajectories, the Hermite–Gauss functions are constant up to a phase factor which is the Gouy phase. The Gouy phase itself assumes the role of proper space or time coordinate. Along the whole of such a trajectory, the space wave function is proportional to the wave number or momentum function. An arbitrary normalizable wave packet can be expanded using the Hermite–Gauss functions as a basis. As example, it is shown that in the transformed frame, a displaced Gaussian does not spread but rather becomes a coherent state. This allows a particularly simple representation of the Young's interference pattern from two or more slits.
For free propagation from a focus the Hermite-Gauss wave functions of optics spread in space. In quantum mechanics the Hermite-Gauss functions are referred to as the harmonic oscillator eigen- functions. These functions are used here to describe the interference of wave packets. It has been shown that when transformed to a frame moving with the normal to the wave front trajectories, the Hermite-Gauss functions are constant up to a phase factor which is the Gouy phase. The Gouy phase itself assumes the role of proper space or time coordinate. Along the whole of such a trajectory, the space wave function is proportional to the wave number function. An arbitrary normalisable wave packet can be expanded using the Hermite-Gauss functions as a basis. As example, it is shown that in the co-moving frame, a displaced Gaussian does not spread but rather becomes a coherent state. This allows a particularly simple representation of the Young's interference pattern from two or more slits.
Abstract The two‐dimensional paraxial equation of optics and the two‐dimensional time‐dependent Schrödinger equation, derived as approximations of the three‐dimensional Helmholtz equation and the three‐dimensional time‐independent Schrödinger equation, respectively, are identical. Here the free propagation in space and time of Hermite–Gauss wave packets (optics) or harmonic oscillator eigenfunctions (quantum mechanics) is examined in detail. The Gouy phase is shown to be a dynamic phase, appearing as the integral of the adiabatic eigenfrequency or eigenenergy. The wave packets propagate adiabatically in that at each space or time point they are solutions of the instantaneous harmonic problem. In both cases, it is shown that the form of the wave function is unchanged along the loci of the normals to wave fronts. This invariance along such trajectories is connected to the propagation of the invariant amplitude of the corresponding free wave number (optics) or momentum (quantum mechanics) wave packets. It is shown that the van Vleck classical density of trajectories function appears in the wave function amplitude over the complete trajectory. A transformation to the co‐moving frame along a trajectory gives a constant wave function multiplied by a simple energy or frequency phase factor. The Gouy phase becomes the proper time in this frame. Key Points This paper builds a bridge between quantum mechanics (QM) and classical optics in that the identity of the paraxial equation of optics and the Schrödinger equation of QM is shown, the Bohmian trajectories of QM are defined in optics, the Gouy phase of optics is defined in QM and given a new interpretation and The space and momentum wave functions are equivalent along a trajectory.
ABSTRACT We revisit multi‐slit diffraction with a scaling of space and time to exploit an equivalence of wave packets describing quantum free and harmonic oscillator (HO) motion. We introduce a co‐moving frame of space and time coordinates to define an effective, time‐independent, HO potential that confines and directs initially displaced wave packets along the classical phase space of the oscillator. ‐slit diffraction in the lab frame from the slits to a distant detector and the familiar spreading of the propagating wave front is then described as the propagation of HO wave packets over just a quarter cycle of the oscillator and confined to the HO potential well. We show that diffraction features are preserved in the co‐moving frame and well characterized in both the near and far fields by the classical trajectories of the HO phase space.
We revisit multi-slit diffraction with a scaling of space and time to exploit an equivalence of wave packets describing quantum free and harmonic oscillator (HO) motion. We introduce a co-moving frame of space and time coordinates to define an effective, time-independent, HO potential that confines and directs initially displaced wave packets along the classical phase space of the oscillator. NNN-slit diffraction in the lab frame from the slits to a distant detector and the familiar spreading of the propagating wave front is then described as the propagation of HO wave packets over just a quarter cycle of the oscillator and confined to the HO potential well. This connection with the co-moving frame suggests a simplified time of flight experiment wherein the arrival time distribution of diffracted particles hitting a \emph{single point detector} in the lab frame images in the co-moving frame as the \emph{full} diffraction pattern from grating to detector.
Different families of states, which are solutions of the time-dependent free Schrödinger equation, are imported from the harmonic oscillator using the quantum Arnold transformation introduced in Aldaya et al (2011 J. Phys. A: Math. Theor. 44 065302). Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite–Gauss and Laguerre–Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.
Different families of states, which are solutions of the time-dependent free Schr\"odinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.