Flux and Energy Density of a Collapsing Electromagnetic Pulse in the Free Space

Abstract

On the basis of the exact solution of the Maxwell equations, we obtained a relationship between the energy and spectrum of a collapsing electromagnetic wave and the energy density in the focusing point. The presented theoretical approach can be useful for a study of ultra short laser pulses without approximation of slowly varying amplitudes.

Annex: Formulas Useful for the Frequency and Bandwidth Analysis (See Sect. 28.5)

Let $g\left( t \right)$ be a real-bounded signal with the power equal (or proportional) to $\left[ {g\left( t \right)} \right]^{2}$ and the total energy G equal to:

$$ G = \mathop \int \limits_{ - \infty }^{\infty } \text{d}t\left[ {g\left( t \right)} \right]^{2} . $$

(28.35)

Then we can write the following expressions for the Fourier transform $g\left( \omega \right)$:

$$ g\left( t \right) = \mathop \int \limits_{ - \infty }^{\infty } e^{ - i\omega t} g\left( \omega \right)\text{d}\omega ,\quad g\left( \omega \right) = \frac{1}{2\pi }\mathop \int \limits_{ - \infty }^{\infty } e^{i\omega t} g\left( t \right)\text{d}t, $$

(28.36)

$$ g\left( { - \omega } \right) = g^{*} \left( \omega \right),\left| {g\left( { - \omega } \right)} \right|^{2} = \left| {g\left( \omega \right)} \right|^{2} . $$

(28.37)

$$ G = 2\pi \mathop \int \limits_{ - \infty }^{\infty } \text{d}\omega \left| {g\left( \omega \right)} \right|^{2} = 4\pi \mathop \int \limits_{0}^{\infty } \text{d}\omega \left| {g\left( \omega \right)} \right|^{2 } = \mathop \int \limits_{0}^{\infty } \varepsilon \left( \omega \right)\text{d}\omega , $$

(28.38)

where $\varepsilon \left( \omega \right)$ is the spectral density of the energy.

$$ \varepsilon \left( \omega \right) = 4\pi \left| {g\left( \omega \right)} \right|^{2} . $$

(28.39)

The frequency $\omega_{0}$ and the bandwidth $\delta \omega$ can be expressed in terms of the first and second moments of the spectral function $\varepsilon \left( \omega \right).$

$$ \omega_{0} = G^{ - 1} \mathop \int \limits_{0}^{\infty } \omega \varepsilon \left( \omega \right)\text{d}\omega = G^{ - 1} 4\pi \mathop \int \limits_{0}^{\infty } \omega \text{d}\omega \left| {g\left( \omega \right)} \right|^{2} , $$

(28.40)

$$ \left( {\delta \omega } \right)^{2} = G^{ - 1} \mathop \int \limits_{0}^{\infty } \left( {\omega - \omega_{0} } \right)^{2} \varepsilon \left( \omega \right)\text{d}\omega = G^{ - 1} 4\pi . $$

(28.41)