## 2 Radially Polarized Laser Beams  
In the paraxial limit, the intensity profile of an RPLB of order \(p\)--also referred to as a \(\mathrm{TM}_{p1}\) beam--is characterized by \(p+1\) rotationally symmetric concentric rings [24,25]. There exists several ways to produce RPLBs (see Appendix A). In particular, the lowest-order RPLB (\(\mathrm{TM}_{01}\)) is often represented as a superposition of two Hermite-Gaussian modes with orders \((0,1)\) and \((1,0)\)[26]. As the name says, the electric field oscillations of RPLBs are radially polarized (in the weak focusing limit). This feature confers to this beam family very particular properties. The most striking characteristic is the central dark intensity region that turns into a bright spot of sub-wavelength diameter under tight focusing conditions [27, 28, 29]. This behaviour is explained by the particular beam symmetry that favors a strong axial longitudinal electric field component [30]. In complex notation (\(E=\mathrm{Re}[\tilde{E}\exp(j\omega_{0}t+j\phi_{0})]\), where \(\omega_{0}\) is the frequency of maximum spectral amplitude and \(\phi_{0}\) is a constant phase delay), the field components of the lowest-order RPLB can be written in the very compact form [25]:  
\[\tilde{E}_{r} = A_{0}j_{2}(k_{0}\tilde{R})\sin(2\tilde{\theta})\exp(-k_{0}z_{0}) \tag{1a}\] \[\tilde{E}_{z} = \frac{4}{3}A_{0}\left[j_{0}(k_{0}\tilde{R})+j_{2}(k_{0}\tilde{R} )P_{2}(\cos\tilde{\theta})\right]\exp(-k_{0}z_{0})\] (1b) \[\tilde{B}_{\theta} = \frac{2}{c}jA_{0}j_{1}(k_{0}\tilde{R})\sin\tilde{\theta}\exp(-k_ {0}z_{0}) \tag{1c}\]  
where \(A_{0}\) is a normalization constant, \(k_{0}=\omega_{0}/c\) is the wave number, \(j_{n}(k_{0}\tilde{R})\) is the order-\(n\) spherical Bessel function of the first kind, and \(P_{2}(\cos\tilde{\theta})=\frac{1}{4}[1+3\cos(2\tilde{\theta})]\) is the Legendre polynomial of degree 2. The complex coordinates \((\tilde{R},\tilde{\theta})\) are defined as \(\tilde{R}=[x^{2}+y^{2}+(z+jz_{0})^{2}]^{1/2}\) and \(\cos\tilde{\theta}=(z+jz_{0})/\tilde{R}\), respectively. The confocal parameter \(z_{0}\) is related to the Gaussian beam waist size \(w_{0}\) by \(z_{0}=w_{0}[1+(\frac{1}{2}k_{0}w_{0})^{2}]^{1/2}\). All other field components (\(\tilde{E}_{\theta}\), \(\tilde{B}_{r}\), and \(\tilde{B}_{z}\)) are zero. The beam waist corresponds to \(z=0\). The set of Equations (1) is a rigorous closed form solution to Maxwell equations and is valid for arbitrary beam waist sizes [25,31] (see Appendix B). In the paraxial limit (\(k_{0}z_{0}\gg 1\)), the electromagnetic field components can be expanded into infinite series [30,32,33] whose leading terms are:  
\[\tilde{E}_{r}^{(0)} = -j\frac{A_{0}}{k_{0}}\;\frac{r}{\tilde{q}^{2}}\exp(-\frac{jk_{0}r ^{2}}{2\tilde{q}})\exp(-jk_{0}z) \tag{2a}\] \[\tilde{E}_{z}^{(0)} = -2\;\frac{A_{0}}{k_{0}^{2}}\;\frac{1}{\tilde{q}^{2}}\left(1-\frac {jk_{0}r^{2}}{2\tilde{q}}\right)\exp(-\frac{jk_{0}r^{2}}{2\tilde{q}})\exp(-jk_ {0}z)\] (2b) \[\tilde{B}_{\theta}^{(0)} = \tilde{E}_{r}^{(0)}/c \tag{2c}\]  
where \(\tilde{q}=z+jz_{0}\) is the Gaussian beam complex parameter and \(c\) the speed of light in free space. It should be noted that in the paraxial limit \(z_{0}\simeq z_{R}=k_{0}w_{0}^{2}/2\), where \(z_{R}\) is the Rayleigh distance [34]. In the paraxial limit, the parameter \(A_{0}\) is related to the peak value \(E_{0}\) (in V/m) of the \(\tilde{E}_{r}^{(0)}\) field component by \(A_{0}=-j(\sqrt{2}/4)k_{0}^{3}w_{0}^{3}\exp(1/2)E_{0}\). The power carried by an RPLB can be obtained through the integral of the projection of the average Poynting vector over a surface perpendicular to the beam propagation [35].