Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves

Michel Zamboni-Rached

doi:10.1364/OPEX.12.004001

Optics Express
Vol. 12,
Issue 17,
pp. 4001-4006
(2004)
•https://doi.org/10.1364/OPEX.12.004001

Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves

Michel Zamboni-Rached

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Michel Zamboni-Rached, "Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves," Opt. Express 12, 4001-4006 (2004)

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Abstract

In this paper it is shown how one can use Bessel beams to obtain a stationary localized wave field with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0≤z≤L of the propagation axis. This intensity envelope remains static, i.e., with velocity υ=0; and because of this we call “Frozen Waves” the new solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc.

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Fig. 1. (a) Comparison between the intensity of the desired longitudinal function F(z) and that of our Frozen Wave (FW), Ψ(ρ=0, z, t), obtained from Eq. (9). The solid line represents the function F(z), and the dotted one our FW. (b) 3D-plot of the field intensity of the FW chosen in this case by us.

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Fig. 2. (a) The Frozen Wave with Q=0.99996ω ₀/c and N=30, approximately reproducing the chosen longitudinal pattern represented by Eq. (14). (b) A different Frozen wave, now with Q=0.99980ω0/c (but still with N=30) forwarding the same longitudinal pattern. We can observe that in this case (with a lower value for Q) a higher transverse localization is obtained.

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Equations (16)

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ψ (ρ, z, t) = J_{0} (k_{ρ} ρ) e^{i β z} e^{- i ω t}

k_{ρ}^{2} = \frac{ω^{2}}{c^{2}} - β^{2},

ω ⁄ β > 0 and k_{ρ}^{2} \geq 0

Ψ (ρ, z, t) = e^{- i ω_{0} t} \sum_{n = - N}^{N} A_{n} J_{0} (K_{ρ n} ρ) e^{i β_{n} z},

0 \leq β_{n} \leq \frac{ω_{0}}{c}

\sum_{n = - N}^{N} A_{n} e^{i β_{n} z} \approx F (z) with 0 \leq z \leq L

β_{n} = Q + \frac{2 π}{L} n,

0 \leq Q \pm \frac{2 π}{L} N \leq \frac{ω_{0}}{c}

Ψ (ρ = 0, z, t) = e^{- i ω_{0} t} e^{i Q z} \sum_{n = - N}^{N} A_{n} e^{i \frac{2 π}{L} n z},

A_{n} = \frac{1}{L} \int_{0}^{L} F (z) e^{- i \frac{2 π}{L} n z} d z

Ψ (ρ, z, t) = e^{- i ω_{0} t} e^{i Q z} \sum_{n = - N}^{N} A_{n} J_{0} (K_{ρ n} ρ) e^{i \frac{2 π}{L} n z},

k_{ρ n}^{2} = ω_{0}^{2} - {(Q + \frac{2 π n}{L})}^{2}

F (z) = {\begin{matrix} - 4 \frac{(z - l_{1}) (z - l_{2})}{{(l_{2} - l_{1})}^{2}} & for l_{1} \leq z \leq l_{2} \\ 1 & for l_{3} \leq z \leq l_{4} \\ - 4 \frac{(z - l_{5}) (z - l_{6})}{{(l_{6} - l_{5})}^{2}} & for l_{5} \leq z \leq l_{6} \\ 0 & elsewhere, \end{matrix}

F (z) = {\begin{matrix} - 4 \frac{(z - l_{1}) (z - l_{2})}{{(l_{2} - l_{1})}^{2}} & for l_{1} \leq z \leq l_{2} \\ 0 & elsewhere \end{matrix},

Z = \frac{R}{\tan θ},

R \geq L \sqrt{\frac{ω_{0}^{2}}{c^{2} β_{n = - N}^{2}} - 1}

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Equations (16)

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