Gaussian pulsed beams with arbitrary speed

Stefano Longhi

doi:10.1364/OPEX.12.000935

Optics Express
Vol. 12,
Issue 5,
pp. 935-940
(2004)
•https://doi.org/10.1364/OPEX.12.000935

Gaussian pulsed beams with arbitrary speed

Stefano Longhi

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Stefano Longhi, "Gaussian pulsed beams with arbitrary speed," Opt. Express 12, 935-940 (2004)

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Abstract

It is shown that the homogeneous scalar wave equation under a generalized paraxial approximation admits of Gaussian beam solutions that can propagate with an arbitrary speed, either subluminal or superluminal, in free-space. In suitable moving inertial reference frames, such solutions correspond either to standard stationary Gaussian beams or to “temporal” diffracting Gaussian fields.

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Equations (19)

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\nabla_{t}^{2} ψ + \frac{\partial^{2} ψ}{\partial z^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} ψ}{\partial t^{2}} = 0,

ψ (x, y, z, t) = Φ (x, y, z, - v t) ext [i ω (t - z ⁄ c)] .

\nabla_{t}^{2} Φ + [{1 - (\frac{v}{c})}^{2}] \frac{\partial^{2} Φ}{\partial ξ^{2}} - 2 ik (1 - \frac{v}{c}) \frac{\partial Φ}{\partial ξ} = 0,

Φ (ρ, ξ) = \int d Q F (Q) J_{0} (k_{⊥} (Q) ρ) \exp (i Q ξ),

k_{⊥} (Q) = \sqrt{Q (1 - \frac{v}{c}) [2 k - Q (1 + \frac{v}{c})]},

Φ (ρ, ξ) = \int d Q F (Q) J_{0} (\sqrt{2 k Q (1 - v ⁄ c)} ρ) \exp (i Q ξ) .

\nabla_{t}^{2} Φ - 2 ik (1 - \frac{v}{c}) \frac{\partial Φ}{\partial ξ} = 0 .

Φ (ρ, ξ) = \frac{1}{{(ξ_{0} \mp i ξ)}^{n + 1}} L_{n}^{0} (\frac{k {∣ 1 - v ⁄ c ∣ ρ}^{2}}{2 (ξ_{0} \mp i ξ)}) \exp [- \frac{k ∣ 1 - v ⁄ c ∣ ρ^{2}}{2 (ξ_{0} \mp i ξ)}],

F (Q) = {\begin{matrix} \frac{1}{n!} Q^{n} \exp (- ξ_{0} Q) & Q > 0 \\ 0 & Q < 0 \end{matrix}

F (Q) = {\begin{matrix} \frac{1}{n!} {(- Q)}^{n} \exp (ξ_{0} Q) & Q < 0 \\ 0 & Q > 0 \end{matrix}

ψ (ρ, z, t) = \frac{1}{ξ_{0} \mp i (z - vt)} \int_{0}^{\infty} d ω G (ω) \exp [- ω s (ρ, z, t)],

s (ρ, z, t) = \frac{∣ 1 - v ⁄ c ∣}{2 c} \frac{ρ^{2}}{ξ_{0} \mp i (z - vt)} - i (t - z ⁄ c) .

G (ω) = {\begin{matrix} \frac{i}{Γ (α)} {(ω - ω_{0})}^{α - 1} \exp (- \frac{ω - ω_{0}}{Δ ω}) & ω > ω_{0} \\ 0 & ω < ω_{0}, \end{matrix}

ψ (ρ, z, t) = \frac{\exp [- ω_{0} s (ρ, z, t)]}{ξ_{0} \mp i (z - vt)} \times \frac{1}{{[s (ρ, z, t) + 1 ⁄ Δ ω]}^{α}} .

ψ (ρ, z, t) = \int_{0}^{\infty} d ω \int d Q F (Q, ω) J_{0} (k_{⊥} (Q, ω) ρ) \exp [it (ω - Qv) - iz (ω ⁄ c - Q)],

{\begin{matrix} x = x' \\ y = y' \\ z = γ (z' + Vt') \\ t = γ (t' + \frac{V}{c^{2}} z') \end{matrix}

ψ' (x', y', z', t') = Φ (x', y', γ (1 - vV ⁄ c^{2}) z' - γ (v - V) t') \exp [i ω' (t' - z' ⁄ c)],

ψ' (x', y', z', t') = Φ (x', y', z' ⁄ γ) \exp [i ω' (t' - z' ⁄ c)],

ψ' (x', y', z', t') = Φ (x', y', - vt' ⁄ γ) \exp [i ω' (t' - z' ⁄ c)],

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