Bohm approach to the Gouy phase shift

doi:10.1016/j.ijleo.2021.168468

Optik

Volume 252, February 2022, 168468

https://doi.org/10.1016/j.ijleo.2021.168468 Get rights and content

Highlights

•
A new explanation of the Gouy phase shift is presented
•
We adapt the Madelung-Bohm formalism to the paraxial wave propagation approximation
•
The effective refraction generates a medium that produces the focusing needed
•
We expect that these ideas give hints on Gouy phases for different kinds of beams
•
It is expected that these ideas inspire more optical simulations of quantum systems

Abstract

By adapting the Madelung–Bohm formalism to paraxial wave propagation we show, by using Ermakov–Lewis techniques, that the Gouy phase is related to the form of the phase chosen in order to produce a Gaussian function as a propagated field. For this, we introduce a quantum mechanical invariant, that it is explicitly time dependent. We finally show that the effective Bohm index of refraction generates a GRIN medium that produces the focusing needed for the Gouy phase.

Introduction

The Gouy phase [1], [2], [3], [4], [5] is a phase gradually acquired by a beam around the focal region and it results in an increase in the apparent wavelength near the waist ( $z \approx 0$ ) deriving in the fact that the phase velocity in such region formally exceeds the speed of light. It has been explicitly shown that the Gouy phase shift of any focused beam originates from transverse spatial confinement, that introduces a spread in the transverse momentum and therefore a shift in the expectation value of the axial propagation constant. The simultaneous effect of confinement in space combined with spread in momentum is, of course, reminiscent of Heisenberg’s uncertainty principle. [3]. The Gouy phase has been used to generate arbitrary cylindrical vector beams [6], to shape vectorially structured light with custom propagation–evolution properties [7] and to give a new Bateman–Hillion solution to the Dirac equation for a relativistic Gaussian electron beam [8], among other applications. Moreover, its origin has been investigated over the years [3], [9], [10].

On the other hand, it is well known that during free propagation light bends when an initial ( $z = 0$ ) Airy field is considered [11], [12], [13], [14], [15], [16], which may be explained by using Madelung–Bohm theory [17], [18]. This formalism has been applied to solve the Schrödinger equation for different systems by taking advantage of their non-vanishing Bohm potentials [19], [20], [21], [22]. We may therefore ask whether Madelung–Bohm theory [23], [24] could provide us a way to get a better understanding of the origin of Gouy’s phase. We show that indeed it may be explained and to this end we use Ermakov–Lewis techniques in order to use the invariant introduced by Lewis for a time dependent harmonic oscillator [25], [26], and thus use it for a free particle.

The article is organized as follows: In the next Section, we show that for a free particle an invariant of the Ermakov–Lewis form may be written which, remarkably, is explicitly time dependent; we then give a solution to the Ermakov equation for this particular case. In Section 3, by taking advantage of the similarity between the Schrödinger equation and the paraxial wave equation, we translate the Madelung–Bohm theory to classical optics. In Section 4, we give an operator solution to the (optical) Bohm equations and propose a Gaussian initial field; by using the Ermakov solution obtained in Section 1, we calculate the Gouy phase. Finally, Section 5 is left for conclusions.

Section snippets

Ermakov–Lewis invariant

In the sixties, Lewis [25], [26] introduced an invariant quantity that has the form (we have set $ħ = 1$ and in this Section dot means derivative with respect to time) $I = \frac{1}{2} [\frac{x^{2}}{ρ^{2}} + {(ρ p - \dot{ρ} x)}^{2}],$ with $ρ$ an auxiliary function that obeys the Ermakov equation [27], [28], [29], [30], [31] $\ddot{ρ} + Ω^{2} (t) ρ = \frac{1}{ρ^{3}};$ consequently (1) takes the name Ermakov–Lewis invariant. The operator given in (1) is invariant in the sense that $\frac{d I}{d t} = \frac{\partial I}{\partial t} + i [H, I] = 0 .$ It is interesting that even though the time-dependent frequency is zero, i.e. a

Madelung–Bohm approach to paraxial wave propagation

The optical paraxial propagation equation in one dimension is given by [36] $i \frac{\partial E (x, z)}{\partial z} = - \frac{1}{2 k} \frac{\partial^{2} E (x, z)}{\partial x^{2}} - \frac{1}{2} n^{2} (x, z) E (x, z),$ where $E (x, z)$ is the propagated field, $k$ is the wave number and $n (x, z)$ is the index of refraction. This equation is formally equivalent to the Schrödinger equation by doing $z \to t$ , $k \to m$ and $- \frac{1}{2} n^{2} \to V (x, z)$ , with $m$ the particle mass, $V (x, z)$ the quantum potential and $n$ the refractive index. We may paraphrase Feynman [37], stating that there is always the hope that this point of view will

Operator solution of the continuity equation

It is possible to rewrite Eq. (9) as a Schrodinger-like equation; for this, we do $\frac{\partial A}{\partial z} = - \frac{1}{2 k} (2 S^{'} \frac{\partial}{\partial x} + S^{''}) A,$ that, by using the operator $\hat{p} = - i \frac{\partial}{\partial x}$ , may be taken to the form $\frac{\partial A}{\partial z} = - \frac{1}{2 k} (i 2 S^{'} \hat{p} + S^{''}) A .$ By choosing $S (x, z) = Q (x) \dot{ν} (z) + μ (z),$ where $Q, ν$ and $μ$ are arbitrary well behaved real functions, such that $S^{'} = Q^{'} \dot{ν}$ , and using the property $[f (x), \hat{p}] = i f^{'} (x)$ to rearrange terms, we have $2 S^{'} \hat{p} = \dot{ν} (Q^{'} \hat{p} + Q^{'} \hat{p}) = \dot{ν} [Q^{'} \hat{p} + \hat{p} Q^{'} + i Q^{''}];$ so that, we may write Eq. (12) as $\frac{\partial A}{\partial z} = - i \frac{\dot{ν}}{2 k} (Q^{'} \hat{p} + \hat{p} Q^{'}) A .$ The function $μ (z)$ in Eq. (13) will

Conclusions

We have used Ermakov–Lewis techniques to show that if we choose a (Bohm) phase that produces a Gaussian field, the Gouy phase arises naturally in the Madelung–Bohm approach to paraxial wave propagation. This is done with the help of the solution of the Ermakov equation coming from the explicitly time dependent invariant introduced by Lewis [25], [26], in our case for the Hamiltonian of the free particle (paraxial free propagation in classical optics). Although the Gouy phase (27) has been known

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

Original Research ArticleBohm approach to the Gouy phase shift

Highlights

Abstract

Introduction

Section snippets

Ermakov–Lewis invariant

Madelung–Bohm approach to paraxial wave propagation

Operator solution of the continuity equation

Conclusions

Declaration of Competing Interest

Opt. Commun.

Optik

Phys. Lett. A

Phys. Lett. A

Prog. Quantum Electron.

Sur une propriete nouvelle des ondes lumineuses

C. R. Academie Des Sci.

Sur la propagation anomale des ondes

Ann. Chim. Phys. Ser.

Physical origin of the gouy phase shift

Opt. Lett.

Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses: errata

Opt. Lett.

Lasers

Arbitrary cylindrical vector beam generation enabled by polarization-selective gouy phase shifter

Photonics Res.

Gouy-phase-mediated propagation variations and revivals of transverse structure in vectorially structured light

Phys. Rev. A

Fractional angular momenta,gouy and berry phases in relativistic bateman-hillion-gaussian beams of electrons

Phys. Rev. Lett.

Clarifications on the gouy phase of radially polarized laser beams

J. Opt. Soc. Amer. A

Origin of gouy phase shift identified by laser-generated focused ultrasound

ACS Photonics

Observation of accelerating airy beams

Phys. Rev. Lett.

A geometric optics description of airy beams

Front. Optic.

Airy beams and accelerating waves: an overview of recent advances

Optica

Generation of airy solitary-like wave beams by acceleration control in inhomogeneous media

Opt. Express

On the general properties of symmetric incomplete airy beams

J. Opt. Soc. Amer. A

Propagating airy wavelet-related patterns

J. Opt.

Quantum particles that behave as free classical particles

Phys. Rev. A

Original Research Article
Bohm approach to the Gouy phase shift