## 2 Oscillating Dipolar Source and Integral Expressions for the Radiation Field
### Steps Needed to Write the Integral Expression for the Electric Field
where \(E_{i}^{\phi}(t,\mathbf{x})\) is the contribution of the scalar potential:  
\[E_{i}^{\phi}(t,\mathbf{x})=-qK^{2}\int d^{3}z\int d^{3}y\frac{f(\mathbf{z})(x_{i }-y_{i})}{|\mathbf{y}-\mathbf{z}||\mathbf{x}-\mathbf{y}|^{3}}\;(K|\mathbf{x}- \mathbf{y}|\sin\Omega-\cos\Omega) \tag{19}\]  
and the contribution of the vector potential is  
\[E_{i}^{A}(t,\mathbf{x})=-qK^{2}\int d^{3}z\int d^{3}y\frac{f(\mathbf{z})(y_{i}- z_{i})}{|\mathbf{y}-\mathbf{z}|^{3}|\mathbf{x}-\mathbf{y}|}\;(K|\mathbf{y}- \mathbf{z}|\sin\Omega-\cos\Omega) \tag{20}\]  
Here \(K\) is the wavenumber: \(K=k\omega=c^{-1}\omega=2\pi\lambda^{-1}\); the phase \(\Omega\) is given by  
\[\Omega=\omega(t-k|\mathbf{x}-\mathbf{y}|-k|\mathbf{y}-\mathbf{z}|) \tag{21}\]  
and \(f(\mathbf{z})\) is a regularized representation of the double \(\delta\)-function of the dipolar source in Equation (15) (because, as discussed in [17], extra-sources originating from a non-local wavefunction are smooth):  
\[f(\mathbf{z})=\frac{1}{\sqrt{(2\pi)^{3}}\varepsilon d^{2}}\left[e^{-\frac{1 }{2}\left(\frac{\pi^{2}}{d^{2}}+\frac{\pi^{2}}{d^{2}}+\frac{(5\alpha-\mu)^{2} }{\varepsilon^{2}}\right)}-e^{-\frac{1}{2}\left(\frac{\pi^{2}}{d^{2}}+\frac{ \pi^{2}}{d^{2}}+\frac{(5\alpha+\mu)^{2}}{\varepsilon^{2}}\right)}\right] \tag{22}\]  
The parameter \(a\) represents the length of the dipole and is taken to equal \(2.5\times 10^{-7}\) cm (in the following, \(a_{1}=a_{2}=0\); \(a_{3}=a\)). The parameter \(\varepsilon\) represents the size in the 3-direction of the dipole charges, and \(d\) their size in the 1- and 2-directions. At the beginning, the oscillation frequency is set to \(\omega=2\pi\times 10^{10}\) Hz and \(d=\varepsilon=10^{-7}\) cm. The field is computed at the point \(\mathbf{x}=\left(\frac{r}{\sqrt{2}},0,\frac{r}{\sqrt{2}}\right)\); at the beginning we set \(r=10\) cm (approximately equal to three wavelengths), then \(r\) is increased to 40 cm. The transverse and longitudinal components of the electric field at this position are defined by the expressions  
\[E_{T}=\frac{1}{\sqrt{2}}(-E_{1}+E_{3});\;\;\;E_{L}=\frac{1}{\sqrt{2}}(E_{1}+E _{3}) \tag{23}\]  
In addition to the integral in Equation (19) there is also another contribution to \(E^{\phi}\), due to the normal density \(\rho\) (not \(\rho^{non-loc}\)) of the source (15). The corresponding \(\phi\) is given in Section 4 (Equation (60)), in the limit when the function \(f\) becomes a double delta-function. It turns out to be of the order of \(E_{T}^{rms}\) and will therefore be disregarded here in the computation of \(E_{L}\). The presence of the Gaussian function \(f(z)\) restricts the effective range of the integration in \(d^{3}z\) approximately to \(\varepsilon\) in directions 1, 2, and to \((a+\varepsilon)\) in direction 3. Therefore, in the Monte Carlo integration procedure we just set the range of \(\mathbf{z}\) accordingly, and the corresponding integration volume is small. Setting the range of \(\mathbf{y}\) is much more difficult, because there is no exponential cutoff in \(\mathbf{y}\) in the integrand, but only a decrease according to a power law. So we can only proceed empirically by integrating over an increasing range \(R_{y}\) until the result stabilizes. All our trials give a stabilization value of \(R_{y}\) (with the parameters employed) between approx. 100 and 200 cm.