2 Theoretical model

As typical in the field of ultrashort laser beam focusing, we consider in our model an OAP mirror whose boundary, projected onto a plane orthogonal to the (parent) paraboloid axis, is a circle. In other words, the mirror boundary results from the intersection of a revolution paraboloid surface and a cylinder with axis parallel to the paraboloid axis (we denote the distance between the two axes as $d_{OAD}$ ). The axis of the cylinder intersects the OAP surface at a point which from now on we will refer to as the ‘OAP centre’. Figure 1 provides some conventions used in the following. In particular, the system of Cartesian orthogonal coordinates $Oxyz$ is centred on the parent paraboloid vertex and is oriented in such a way that the parabola focus and the OAP centre ( $O^{\prime }$ in Figure 1) are located at the points $\mathbf{x}_{f}=(0,0,z_{f})$ , with $z_{f}=f>0$ , and $\mathbf{x}_{c}=(d_{OAD},0,d_{OAD}^{2}/4f)$ , respectively. The meridional (sagittal) plane is thus the plane $x$ – $z$ ( $y$ – $z$ ). The OAP surface ( $S_{\text{OAP}}$ ) is therefore identified by the equation

(1) $$\begin{eqnarray}z=\frac{1}{4f}(x^{2}+y^{2})=a(x^{2}+y^{2})=s(x,y)\cdot f\end{eqnarray}$$

together with the condition $(x-d_{OAD})^{2}+y^{2}\leqslant (d/2)^{2}$ , where $d$ is the OAP diameter (we have defined $a=1/4f$ and $s(x,y)=(x^{2}+y^{2})/4f^{2}$ , which will be useful in the following). We can also introduce the so-called off-axis angle $\unicode[STIX]{x1D717}_{OA}$ , defined by $\tan \unicode[STIX]{x1D717}_{OA}=d_{OAD}/(f-ad_{OAD}^{2})$ . In the following, two further systems of coordinates will be used, both having the origin at the OAP centre (see the inset of Figure 1): the system $O^{\prime }x^{\prime }y^{\prime }z^{\prime }$ , which is obtained from $Oxyz$ with just a translation, and the system $O^{\prime }XYZ$ , which encompasses a further rotation of an angle $-\unicode[STIX]{x1D717}_{OA}$ around the $y$ -axis ( $Z$ lies thus along the direction of the ray reflected from the OAP centre, which will be occasionally called the ‘central ray’ in the following).

Figure 1. Sketch showing the systems of coordinates used throughout the text.

We now consider a monochromatic beam, with a super-Gaussian transverse profile, incident along the $-z^{\prime }$ direction; its electric and magnetic fields can thus be written, in the $O^{\prime }x^{\prime }y^{\prime }z^{\prime }$ system, as

(2) $$\begin{eqnarray}\displaystyle \mathbf{E}(\mathbf{x}^{\prime },t) & = & \displaystyle A^{\prime }(x^{\prime },y^{\prime })(\cos \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{x^{\prime }}+\sin \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{y^{\prime }})\nonumber\\ \displaystyle & & \displaystyle \times \,e^{-ik(z^{\prime }-z_{0}^{\prime })}e^{-i\unicode[STIX]{x1D714}t}\equiv \mathbf{E}_{\text{inc}}(\mathbf{x}^{\prime })e^{-i\unicode[STIX]{x1D714}t},\end{eqnarray}$$

(3) $$\begin{eqnarray}\displaystyle \mathbf{B}(\mathbf{x}^{\prime },t) & = & \displaystyle A^{\prime }(x^{\prime },y^{\prime })(\sin \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{x^{\prime }}-\cos \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{y^{\prime }})\nonumber\\ \displaystyle & & \displaystyle \times \,e^{-ik(z^{\prime }-z_{0}^{\prime })}e^{-i\unicode[STIX]{x1D714}t}\equiv \mathbf{B}_{\text{inc}}(\mathbf{x}^{\prime })e^{-i\unicode[STIX]{x1D714}t},\end{eqnarray}$$

with

(4) $$\begin{eqnarray}\displaystyle A^{\prime }(x^{\prime },y^{\prime })=A_{0}\exp \left\{-\frac{1}{2}\left[\left(\frac{x^{\prime }}{\unicode[STIX]{x1D70E}_{x}}\right)^{2}+\left(\frac{y^{\prime }}{\unicode[STIX]{x1D70E}_{y}}\right)^{2}\right]^{n}\right\}, & & \displaystyle\end{eqnarray}$$

$n$ being the super-Gaussian order of the spatial profile of the beam. In equations (2) and (3) we have assumed that the phase of the incoming beam is equal to zero at $t=0$ on a reference plane $z^{\prime }=z_{0}^{\prime }$ , with $z_{0}^{\prime }>0$ (also, we have implicitly ruled out any deviation from a perfect planar wavefront). The angle $\unicode[STIX]{x1D6FF}$ was introduced in order to account for different polarization directions; in particular, $\unicode[STIX]{x1D6FF}=0$ ( $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D70B}/2$ ) corresponds to a polarization in the meridional (sagittal) plane.

We are now interested in the time-dependent behaviour of the electromagnetic fields in the focal region upon reflection off the OAP surface. As it is well known, the problem can be formally factorized into the time and space domains, and a suitable diffraction approach can be used to deal with this latter domain. As a consequence, we can write the field at the point $\mathbf{x}_{P}$ at time $t$ as $\mathbf{E}(\mathbf{x}_{P},t)=\mathbf{E}_{SC}(\mathbf{x}_{P})e^{-i\unicode[STIX]{x1D714}t}$ (a similar equation holds for $\mathbf{B}$ ), where the spatial part has to be calculated using a suitable diffraction formulation; in our case, we use a full vector diffraction approach based on the Stratton–Chu theory (hence the subscript SC). As recently discussed in Ref. [Reference Peatross, Berrondo, Smith and Ware32], the Stratton–Chu approach allows, generally speaking, beams with sharper transverse profiles to be dealt with, with respect to a more direct approach based on Green’s theorem applied to each field. It can be shown (see for instance our recent paper^{[Reference Labate, Ferrara, Fulgentini and Gizzi24]}) that, assuming a perfect (100%) reflection, the boundary fields (on the OAP surface) appearing in the Stratton–Chu theory can be related to the incident fields, and the resulting integrals read

(5) $$\begin{eqnarray}\displaystyle \mathbf{E}_{SC}(\mathbf{x}_{P}) & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D70B}}\int _{\text{OAP}}[ik(\hat{\mathbf{n}}\times \mathbf{B}_{\text{inc}})G\nonumber\\ \displaystyle & & \displaystyle +\,(\hat{\mathbf{n}}\cdot \mathbf{E}_{\text{inc}})\unicode[STIX]{x1D6FB}G]\,\text{d}A,\end{eqnarray}$$

(6) $$\begin{eqnarray}\displaystyle \mathbf{B}_{SC}(\mathbf{x}_{P}) & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D70B}}\int _{\text{OAP}}[(\hat{\mathbf{n}}\cdot \unicode[STIX]{x1D6FB}G)\mathbf{B}_{\text{inc}}\nonumber\\ \displaystyle & & \displaystyle -\,(\mathbf{B}_{\text{inc}}\cdot \unicode[STIX]{x1D6FB}G)\hat{\mathbf{n}}]\,\text{d}A.\end{eqnarray}$$

Here $G$ is the Green function for the Helmholtz equation, $G=\exp (ik|\mathbf{x}-\mathbf{x}_{P}|)/|\mathbf{x}-\mathbf{x}_{P}|\equiv \exp (ik|\mathbf{u}|)/|\mathbf{u}|$ , with $\mathbf{u}=\mathbf{x}-\mathbf{x}_{P}$ . The integrals are of course carried out over the OAP surface.

Using as parameters of the OAP surface just the $x,y$ coordinates of each point, we can write the (inward) normal to the surface as $\hat{\mathbf{n}}=(-x/2f,-y/2f,1)/\sqrt{1+s(x,y)}$ and the area element as $\text{d}A=\sqrt{1+s(x,y)}\,\text{d}x\,\text{d}y$ . The incident fields in the above integrals must be expressed in the system $Oxyz$ and can be easily retrieved from equations (2) and (3); the electric field, for instance, can be written as $\mathbf{E}_{\text{inc}}(\mathbf{x})=A(x,y)(\cos \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{x}+\sin \unicode[STIX]{x1D6FF}\hat{\mathbf{e}}_{y})e^{ikp(\mathbf{x})}$ . Here $p(\mathbf{x})$ is the optical path from the point $(x,y,z_{0}=ad_{OAD}^{2}+z_{0}^{\prime })$ on the reference plane to the point $(x,y,a(x^{2}+y^{2}))$ on the OAP surface (notice that we are assuming $z_{0}>a(x^{2}+y^{2}),\forall x,y\in S_{\text{OAP}}$ ). The transverse field amplitude is obtained as $A(x,y)=A^{\prime }(x-d_{OAD},y)$ .

On substituting all these expressions into equations (5) and (6) one thus gets, after some cumbersome algebra, the following equations for the (time-dependent) electric and magnetic field components focused by the OAP:

(7) $$\begin{eqnarray}\displaystyle E_{j}(\mathbf{x}_{P},t) & = & \displaystyle e^{-i\unicode[STIX]{x1D714}t}\frac{i}{\unicode[STIX]{x1D706}}\int _{\text{OAP}}A(x,y)e^{ik(p(\mathbf{x})+u(\mathbf{x},\mathbf{x}_{P}))}\nonumber\\ \displaystyle & & \displaystyle \times \,g^{(E_{j})}(\mathbf{x},\mathbf{x}_{P})\,\text{d}x\,\text{d}y,\end{eqnarray}$$

(8) $$\begin{eqnarray}\displaystyle B_{j}(\mathbf{x}_{P},t) & = & \displaystyle e^{-i\unicode[STIX]{x1D714}t}\frac{i}{\unicode[STIX]{x1D706}}\int _{\text{OAP}}A(x,y)e^{ik(p(\mathbf{x})+u(\mathbf{x},\mathbf{x}_{P}))}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{1}{u^{2}}\left(1-\frac{1}{iku}\right)g^{(B_{j})}(\mathbf{x},\mathbf{x}_{P})\,\text{d}x\,\text{d}y,\end{eqnarray}$$

with $j=x,y,z$ . In these expressions $u=u(\mathbf{x},\mathbf{x}_{P})=|\mathbf{u}|=|\mathbf{x}-\mathbf{x}_{P}|$ and the functions $g^{(E_{j})}(u(\mathbf{x},\mathbf{x}_{P}))$ , $g^{(B_{j})}(u(\mathbf{x},\mathbf{x}_{P}))$ can be written as

(9) $$\begin{eqnarray}\displaystyle g^{(E_{x})} & = & \displaystyle \frac{1}{u}\cos \unicode[STIX]{x1D6FF}-\left(1-\frac{1}{iku}\right)\frac{1}{u^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}+\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)(x-x_{P}),\end{eqnarray}$$

(10) $$\begin{eqnarray}\displaystyle g^{(E_{y})} & = & \displaystyle \frac{1}{u}\sin \unicode[STIX]{x1D6FF}-\left(1-\frac{1}{iku}\right)\frac{1}{u^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}+\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)(y-y_{P}),\end{eqnarray}$$

(11) $$\begin{eqnarray}\displaystyle g^{(E_{z})} & = & \displaystyle \frac{1}{u}\left(\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}+\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)-\left(1-\frac{1}{iku}\right)\frac{1}{u^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}+\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)(z-z_{P}),\end{eqnarray}$$

and

(12) $$\begin{eqnarray}\displaystyle g^{(B_{x})} & = & \displaystyle \left(-\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}-\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)(y-y_{P})\nonumber\\ \displaystyle & & \displaystyle +\,(\sin \unicode[STIX]{x1D6FF})(z-z_{P}),\end{eqnarray}$$

(13) $$\begin{eqnarray}\displaystyle g^{(B_{y})} & = & \displaystyle \left(\frac{x}{2f}\cos \unicode[STIX]{x1D6FF}+\frac{y}{2f}\sin \unicode[STIX]{x1D6FF}\right)(x-x_{P})\nonumber\\ \displaystyle & & \displaystyle -\,(\cos \unicode[STIX]{x1D6FF})(z-z_{P}),\end{eqnarray}$$

(14) $$\begin{eqnarray}\displaystyle g^{(B_{z})} & = & \displaystyle -(\sin \unicode[STIX]{x1D6FF})(x-x_{P})+(\cos \unicode[STIX]{x1D6FF})(y-y_{P}).\end{eqnarray}$$

The real part of the fields, which we are going to use in the following, can be easily calculated from the above equations. In particular, it can be readily verified that

(15) $$\begin{eqnarray}\displaystyle E_{j,\text{r}}(\mathbf{x}_{P},t) & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D706}}\int _{\text{OAP}}\left[g_{\text{r}}^{(E_{j})}\sin (kv-\unicode[STIX]{x1D714}t)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\,g_{\text{i}}^{(E_{j})}\cos (kv-\unicode[STIX]{x1D714}t)\right]A(\mathbf{x})\,\text{d}x\,\text{d}y,\end{eqnarray}$$

(16) $$\begin{eqnarray}\displaystyle B_{j,\text{r}}(\mathbf{x}_{P},t) & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D706}}\int _{\text{OAP}}g^{(B_{j})}\frac{1}{u^{2}}\left[\!\vphantom{\frac{1}{ku}}\sin (kv-\unicode[STIX]{x1D714}t)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{1}{ku}\cos (kv-\unicode[STIX]{x1D714}t)\right]A(\mathbf{x})\,\text{d}x\,\text{d}y,\end{eqnarray}$$

where we have defined $v\equiv v(\mathbf{x},\mathbf{x}_{P})\equiv p(\mathbf{x})+u(\mathbf{x},\mathbf{x}_{P})$ , $g_{\text{r}}^{(E_{j})}(\mathbf{x},\mathbf{x}_{P})=\text{Re}(g^{(E_{j})}(\mathbf{x},\mathbf{x}_{P}))$ and $g_{\text{i}}^{(E_{j})}(\mathbf{x},\mathbf{x}_{P})=\text{Im}(g^{(E_{j})}(\mathbf{x},\mathbf{x}_{P}))$ (notice that the functions $g^{(B_{j})}$ are real valued). Equations (15) and (16), along with the expressions for the $g^{(E_{j})}$ and $g^{(B_{j})}$ functions (9)–(11) and (12)–(14), allow the time-dependent electric and magnetic field values at any space–time point to be calculated. It is worth to point out that, although not providing a closed expression, and thus requiring a numerical approach to be solved, no approximation or Taylor expansion has been used in our treatment, so that they retain the validity of the original Stratton–Chu formulation as detailed in Ref. [Reference Varga and Török14].

In the field of high-intensity laser–matter interaction, one is in general interested in the study of the field components along longitudinal and transverse directions with respect to the focused beam propagation direction, that is the direction along $O^{\prime }O^{\prime \prime }$ in Figure 1. With our conventions, this basically requires the knowledge of the field components in the system of coordinates $O^{\prime }XYZ$ ; it is readily verified that these components can be retrieved from the components in the $Oxyz$ system, provided by the integrals (15) and (16), using the obvious transformation $\mathbf{F}^{(O^{\prime }XYZ)}={\mathcal{R}}(\unicode[STIX]{x1D717}_{OA})\mathbf{F}^{(Oxyz)}$ , where ${\mathcal{R}}(\unicode[STIX]{x1D717}_{OA})$ is the matrix accounting for the rotation of an angle $\unicode[STIX]{x1D717}_{OA}$ around the $y$ -axis.

In the following discussion, we use the coordinates $u$ and $v$ shown in Figure 1 to label the directions on a plane orthogonal to the central ray.

Figure 2. Left and middle columns: pattern of the $E$ (top – plots A and B) and $B$ (bottom – plots D and E) vector fields at $t_{0}$  and $t_{0}+T/2$ for an $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ OAP. Right column: intensity (top – plot C) and $E$ field pattern (bottom – plot F) at $t_{max}$ . The beam incident on the OAP is supposed to be polarized along $x$ (or $u$ , corresponding to $\unicode[STIX]{x1D6FF}=0^{\circ }$ ).

3 Intra-cycle behaviour of the electromagnetic fields

3.1 General discussion

In this section, we discuss some general features of the electric and magnetic fields in the focal plane of the OAP, starting from a numerical integration of equations (15) and (16) at different times during the optical cycle. The numerical integration was performed using a multi-dimensional adaptive integration scheme based on the algorithm described in Ref. [Reference Bernsten, Espelid and Genz33]. For the sake of the following discussion, we used a beam with $\unicode[STIX]{x1D706}=800~\text{nm}$ and with a transverse amplitude profile given by formula (4) with $n=4$ and $A_{0}=1$ ; we also consider a rotationally symmetric beam ( $\unicode[STIX]{x1D70E}_{x}=\unicode[STIX]{x1D70E}_{y}=\unicode[STIX]{x1D70E}$ ) and set the value of $\unicode[STIX]{x1D70E}$ so as to have a beam with an intensity full width at half-maximum (FWHM) of $40~\text{mm}$ . The integration algorithm was implemented in a C $++$ code; with the above parameters, the integration required for each field component (at a given point and time) took typically a few hundreds of milliseconds to complete on a (Linux based) desktop PC equipped with a pretty standard CPU.

As said above, we are interested here in the field behaviour on the focal plane at different times of the optical cycle. For the sake of conciseness, from now on we will refer to the time at which the electric and magnetic fields at the centre of the observation plane (that is, at the point which the central ray is supposed to pass through) take on their maximum amplitude as $t_{max}$ ; conversely, we refer to the time at which both fields are supposed to vanish as $t_{0}$ .

The bottom right plot of Figure 2 (plot $F$ ) shows the direction and amplitude (normalized to 1) of the electric field (in the focal plane) at $t_{max}$ , for a beam polarized along $x$ ( $\unicode[STIX]{x1D6FF}=0$ in equations (2) and (3)) and focused by an OAP with $f/\#=2$ and $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ . As expected, the electric field is directed along the $u$ direction. The top right plot (plot $C$ ) shows, for the sake of a visual aid in considering the importance of the effects we are going to discuss, the intensity of the focused beam.

The first (left) column of Figure 2 shows the direction and amplitude of the electric (plot $A$ ) and magnetic (plot $D$ ) fields at $t_{0}$ . Here, a not obvious effect can be observed. Indeed, both the electric and magnetic fields actually only vanish in the surroundings of the central point, while a complex pattern is observed out of this point. Looking at the colour scale of these plots, one can see that the nonzero field components reach typical values of a few percent of those at $t_{max}$ ; we will discuss later how the importance of this effect depends upon the OAP parameters such as the $f$ number and the off-axis angle.

Figure 3. Pattern of the $E$ vector field at the times $t_{0}-T/200$ (top) and $t_{0}+T/200$ (bottom), for an $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ OAP.

The middle column of Figure 2 shows the $E$ and $B$ field patterns at the other minimum within the optical cycle, that is at $t_{0}+T/2$ , being $T$ the radiation period. As it is rather predictable, the field patterns are similar to those encountered at $t_{0}$ , with the sign of the fields reversed.

It is interesting to look, at this point, at the behaviour of the fields for times very close to $t_{0}$ . Figure 3 shows the $E$ -field pattern for the times $t_{0}-T/200$ (top) and $t_{0}+T/200$ (bottom) (this time span corresponds to a few tens of attoseconds for a typical infrared laser beam). As it can be easily realized from this Figure, the region where the field actually vanishes does describe a sort of sweep along the meridional ( $x$ – $z$ ) plane. In other words, for a small neighbour of the points in the meridional plane, a time instant exists, close to $t_{0}$ , at which the field is zero; this time instant corresponds to $t_{0}$ only for the focal point.

Figure 4. $E$ (top) and $B$ (bottom) field patterns at $t_{0}$  for an $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ OAP, with an incident beam polarized along $y$ (or $v$ , corresponding to $\unicode[STIX]{x1D6FF}=90^{\circ }$ ).

The pattern of $E$ and $B$ is interchanged when an incident beam polarized along $y$ is considered; this can be seen in Figure 4, which shows the electric and magnetic field patterns at $t_{0}$  for the same OAP and beam parameters as the ones considered in Figure 2 but with $\unicode[STIX]{x1D6FF}=90^{\circ }$ . Finally, as shown in Figure 5, the situation is somewhat intermediate for a beam incident on the OAP with a polarization at $45^{\circ }$ with respect to the $x$ – $z$ plane; in this case, the typical convergent/divergent pattern seen for the $E$ field (in the case $\unicode[STIX]{x1D6FF}=0^{\circ }$ ) or for the $B$ field (in the case $\unicode[STIX]{x1D6FF}=90^{\circ }$ ) is not encountered any more. However, it should be observed that the typical maximum amplitude of the fields at $t_{0}$ , which is of the order of a few percent of that at $t_{max}$ , does not depend on the beam polarization.

Figure 5. $E$ (top) and $B$ (bottom) field patterns at $t_{0}$  for an $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ OAP, with an incident beam polarized at $45^{\circ }$ with respect to $x$ (or $u$ , corresponding to $\unicode[STIX]{x1D6FF}=45^{\circ }$ ).

Figure 6. Map of the ratio (calculated at $t_{0}$ ) of the $E$ field longitudinal component $E_{Z}$ to the transverse component $|E_{tr}|$ , for an $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ OAP and a beam polarized along $x$ .

It is worth to observe that a longitudinal electric field component is also appearing at $t_{0}$ . Figure 6 shows a density map of the ratio of the longitudinal component $E_{Z}$ to the transverse component $|E_{tr}|=\sqrt{|E_{u}|^{2}+|E_{v}|^{2}}$ , calculated at $t_{0}$ . As it can be realized by comparing with the top left plot of Figure 2, a longitudinal field component appears in the regions where the transverse field component $|E_{tr}|$ is smaller (except for the neighbour of the central point).

Notice that in Figure 6 we have restricted our attention to a region of interest (ROI) over which the focused beam intensity keeps at a level greater than 0.1 of its maximum; in other words, we have only considered points $(u,v)$ such that $I(u,v)\geqslant 0.1I_{max}$ and forced to a zero value all the points outside this ROI. Beside enabling a better readability of the plots, this procedure allows us to only consider a spatial region where the field observed at $t_{0}$  has enough magnitude to potentially lead to nonnegligible physical effects in real laser–plasma interaction experiments. Unless otherwise specified, this procedure will be adopted in the following discussion.

3.2 Depolarization dependence upon the OAP parameters

We are now interested in investigating how the anomalous field patterns observed at the time $t_{0}$  depend upon the OAP parameters, namely the $f$ number and the off-axis angle. To this purpose, we first look at the ratio of the square modulus of the electric field transverse component ( $|E_{tr}|^{2}=|E_{u}|^{2}+|E_{v}|^{2}$ ) at $t_{0}$  to the corresponding quantity at $t_{max}$ . Notice that for this discussion, we restrict our attention to the $E$ field, since similar results obviously hold for the $B$ field.

Figure 7. Maps of the ratio of $|E_{tr}|^{2}$ at $t_{0}$  to the corresponding value at $t_{max}$ , calculated for a beam polarized along $x$ ( $\unicode[STIX]{x1D6FF}=0^{\circ }$ ) and focused with the following OAPs: $f/2$ , $\unicode[STIX]{x1D717}_{OA}=10^{\circ }$ (plot A), $f/2$ , $\unicode[STIX]{x1D717}_{OA}=90^{\circ }$ (plot B), $f/2$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ (plot C), $f/10$ , $\unicode[STIX]{x1D717}_{OA}=40^{\circ }$ (plot D).

Figure 8. Plots of the $\unicode[STIX]{x1D705}$ parameter (defined in the text) vs. the off-axis angle $\unicode[STIX]{x1D717}_{OA}$ (top) and the $f/\#$ (bottom). In the first plot, data for both the $x$ ( $\unicode[STIX]{x1D6FF}=0^{\circ }$ ) and $y$ ( $\unicode[STIX]{x1D6FF}=90^{\circ }$ ) polarizations are shown, while only the data for the $x$ polarizations are shown in the second plot. The results of fits with functions of the form given in equation (17) are also shown for the case $\unicode[STIX]{x1D6FF}=0^{\circ }$ .

Figure 7 shows the maps (restricted to the ROI defined above) of this ratio for an increasing off-axis angle $\unicode[STIX]{x1D717}_{OA}$ (top row) and for an increasing $f$ number (bottom row). In particular, the top row shows the maps of $|E_{tr}|^{2}|_{t_{0}}/|E_{tr}|^{2}|_{t_{max}}$ for an $f/2$ OAP with $\unicode[STIX]{x1D717}_{OA}=10^{\circ }$ (plot A) and $\unicode[STIX]{x1D717}_{OA}=90^{\circ }$ (plot B). It can be seen that the amplitude of the field obtained at $t_{0}$  is strongly dependent on the off-axis angle. In fact, by integrating the equations (15) and (16) for an on-axis parabola we find that the fields at $t_{0}$  almost vanish across all the plane; the effects observed at $t_{0}$  are thus a consequence of the off-axis focusing scheme. Analogously, the maps in the bottom row of Figure 7 show that the $f/\#$ plays a role as well: tighter focusing causes larger field components to appear at $t_{0}$ .

As it is clear from Figure 7, the (relative) magnitude of the anomalous fields at $t_{0}$  is not uniform across the ROI. For a quantitative assessment of the dependence upon the $f/\#$ and $\unicode[STIX]{x1D717}_{OA}$ of the observed phenomena, we thus need a spatially averaged quantity; we can consider, for instance, the integral of the square modulus of the transverse $E$ field averaged over the ROI using the local intensity as a weight:

$$\begin{eqnarray}\displaystyle \langle |E_{tr}|^{2}\rangle :=\frac{\int _{ROI}|E_{tr}|^{2}I(u,v)\,\text{d}u\,\text{d}v}{\int _{ROI}I(u,v)\,\text{d}u\,\text{d}v}. & & \displaystyle \nonumber\end{eqnarray}$$

In particular, we define the parameter $\unicode[STIX]{x1D705}$ as the ratio of this quantity at $t_{0}$  to the corresponding value at $t_{max}$ : $\unicode[STIX]{x1D705}:=\langle |E_{tr}|^{2}\rangle |_{t_{0}}/\langle |E_{tr}|^{2}\rangle |_{t_{max}}$ . In Figure 8 we plot the $\unicode[STIX]{x1D705}$ parameters as a function of the off-axis angle (top) and of the $f$ number (bottom). Fitting the data, the following scaling laws can be obtained for the $x$ polarization:

(17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}\propto (\unicode[STIX]{x1D717}_{OA})^{\unicode[STIX]{x1D6FC}},\quad \unicode[STIX]{x1D705}\propto \frac{1}{(f/\#)^{\unicode[STIX]{x1D6FD}}}, & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}\simeq 2.75$ and $\unicode[STIX]{x1D6FD}\simeq 1.66$ . Finally, we notice that a weak difference between the two orthogonal polarizations of the incoming beam ( $\unicode[STIX]{x1D6FF}=0^{\circ }$ and $\unicode[STIX]{x1D6FF}=90^{\circ }$ ) can be observed; the corresponding $\unicode[STIX]{x1D6FC}$ value for the $y$ polarization is $\unicode[STIX]{x1D6FC}\simeq 2.77$ .

4 Conclusions and open issues

Starting from an exact time-dependent, vector diffraction based model developed on purpose, we have studied the electromagnetic field behaviour, at different times within the optical cycle, of a beam focused by an OAP mirror. In particular, we have investigated the electric and magnetic field patterns across planes orthogonal to the beam propagation direction.

A behaviour far from trivial was found, in the focal region, at the time ( $t_{0}$ ) at which the electric and magnetic fields are supposed to vanish; actually, this zero field value only occurs in a small neighbour of the focus, while a complex electromagnetic field pattern exists at farther points. Such a complex pattern basically results in the appearance of field components orthogonal to the original polarization (or magnetic field) direction; furthermore, longitudinal field components (that is, directed along the original propagation direction) can also appear.

What seems to be relevant for laser–matter interaction experiments at relativistic intensities is the fact that the amplitude of these ‘anomalous’ electric and magnetic fields can reach, depending on the focusing conditions, values of a few percent of the maximum values expected during the optical cycle. Beside the boundaries of the beam, where the intensity (and thus the field amplitude) drops down to negligible values, this may occur, under some circumstances, even within a transverse spatial region where the field values are supposed to be high enough so as to potentially lead to nonnegligible effects on the laser–matter interaction dynamics.

As mentioned in the Introduction, such effects are to be possibly expected for laser–plasma interaction processes dependent on the laser polarization, in particular when tight focusing is employed, such as proton acceleration via either target normal sheath acceleration or radiation pressure acceleration. On the other hand, according to our results, the phenomena discussed in this paper are expected to be negligible at high $f$ numbers, so that, for instance, no departure from an ‘ideal’ laser beam is expected to occur in the context of Laser WakeField Acceleration experiments, where long focal length OAPs are commonly employed in order to sustain a long laser beam propagation.

As a final remark we observe that, in order to theoretically investigate possible effects in the laser–matter interaction at ultrahigh intensity, a full knowledge of the temporal dynamics of the field patterns discussed here would be needed. The discussion of a theory allowing such a study to be carried out is beyond the scope of the current paper and will be reported elsewhere. According to preliminary investigations carried out by numerically calculating the field integrals given above at different times close to the time $t_{0}$  and studying the resulting patterns, we can estimate that the features observed around this time have typical timescales of the order of $10^{-2}$ of the pulse cycle.