Open Access
Add to BibSonomy
Abstract
A comparative study of three different tight-focusing schemes for high-power lasers is performed numerically. Using the Stratton-Chu formulation, the electromagnetic field in the vicinity of the focus is evaluated for a short-pulse laser beam incident upon an on-axis high numerical aperture parabola (HNAP), an off-axis parabola (OAP), and a transmission parabola (TP). Linearly- and radially-polarized incident beams are considered. It is demonstrated that while all the focusing configurations yield intensities above 1023 W/cm2 for a 1 PW incident beam, the nature of the focused field can be drastically modified. In particular, it is shown that the TP, with its focal point behind the parabola, actually converts an incoming linearly-polarized beam into an m = 2 vector beam. The strengths and weaknesses of each configuration are discussed in the context of future laser-matter interaction experiments. Finally, a generalization of NA calculations up to 4π-illumination is proposed through the solid angle formulation, providing a universal way to compare light cones from any kind of optics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The advent of Petawatt (PW) laser infrastructures, among them are the Texas Petawatt [1], BELLA [2], SULF [3] and CoReLS [4] facilities, allow for unprecedented intensity levels above $10^{22}$ W/cm$^{2}$. Several ultra-high-power laser facilities are as well under construction for delivering 10 PW and more at APOLLON [5], ELI-NP [6], EP-OPAL [7] and SEL [8]. This opens the possibility of an experimental detection of quantum electrodynamics (QED) effects including vacuum polarization, Breit-Wheeler pair production and nonlinear inverse Compton scattering [9]. To reach such ultra-high intensities, high Numerical Aperture (NA) parabolic reflectors are used to focus ultrashort (fs) incident laser pulses close to the diffraction limit, in the so-called $\lambda ^3$-regime [10]. For example, the record-breaking laser intensity of $1.1\times 10^{23}$ W/cm$^2$, recently shown at CoReLS using 4 PW of peak laser power, focused the beam down to a spot of 1.1 $\mu$m at Full-Width-Half-Maximum (FWHM) using an $f/1.1$ Off-Axis Parabola (OAP) [4].
Reaching such levels of focusing where the focal volume radius is approximately one wavelength is a feat that requires extensive technical skills and the deployment of numerous diagnostic tools. Compounded with the high cost of large optics, the optimization of the focusing configuration becomes a costly process in terms of human and financial resources. In general, the OAP satisfies the requirements in PW infrastructures because it can have a relatively high NA while giving an access to the focal region, which is essential for placing diagnostics in laser-matter experiments. However, other parabolic reflector configurations do exist and can be viable alternatives for certain applications. In this work, we perform an investigation of three tight-focusing parabolic reflector schemes for high power lasers: the on-axis High Numerical Aperture Parabolic (HNAP) mirror, the OAP and the Transmission Parabola (TP). Using numerical simulations of the focused fields, the peak intensity and field distribution at focus are compared to highlight the strengths and weaknesses of each configuration.
To obtain a reliable description of the non-Gaussian beams generated by these high NA focusing schemes, numerical simulations based on the Stratton-Chu formulation [11] are used. When an electromagnetic (EM) wave is strongly focused by a reflector, the paraxial approximation is not valid and therefore, a special theoretical treatment is required to evaluate the field in the vicinity of the focus. Explicit solutions [12–18] and analytical techniques based on perturbative developments such as Lax series [19–25] have been investigated to solve this problem. While these analytic solutions can be evaluated efficiently, they cannot accommodate for arbitrary incident fields or reflector geometries. This flexibility can be provided by numerical approaches derived from the finite-difference time-dependent (FDTD) method [26], the Richards-Wolf (RW) vector diffraction theory [14,27–34] or the Stratton-Chu (SC) formulation [11]. The FDTD method can be very costly for the multiscale focusing problem considered in our work because a fine mesh is required to resolve the field at the focus. The RW has been used successfully in many investigations, however is an approximation of the SC formulation that can be recovered in the Debye limit [35]. In light of this, our choice has fallen on the SC formulation in order to provide the highest reliability of the calculated EM fields. In addition, the SC technique can be applied to any type of scatterer and thus, has been specialized to study the field structure generated by parabolic mirrors like the ones considered in our work [36–38].
This article is separated as follows. In Section 2, the numerical method and a description of the three focusing schemes is given. Since the TP has an additional parameter from its inner opening, its shape optimization is shown in order to better compare with the two other reflectors. This necessitates a proper evaluation of the NA and re-visiting its definition. The results of the comparative study for linear and radial incident polarizations are given in Section 3. Then, the focal spot intensity distribution of a linearly-polarized beam incident on the TP leads to the analysis of vector beams incident on the reflectors. Section 4 is devoted to the comparison of the three optics implemented in an experimental context. Finally, a conclusion is provided in Section 5.
2. Methodology
2.1 Stratton-Chu diffraction for tightly focused beams
The SC integrals express the EM radiation emitted by an arbitrarily shaped conducting surface in terms of currents and charge densities. After applying boundary conditions for a perfect conductor and the physical optics approximation (where the radius of curvature of the reflecting surface $\mathcal {R}$ is much larger than the wavelength, i.e., $\mathcal {R} \gg \lambda$ [39]), the SC integral equations reduce to [37]:
The time dependence of the electromagnetic field can be recovered from spectral components. For a short-pulse broadband laser beam with $n_\textrm {tot}$ spectral components, the EM fields can be written as:
We have implemented the SC field formulation for broadband laser pulses in a high-performance code, called the Strattocalculator [40]. The flexible code architecture allows for many incident field models with various spatio-temporal profiles and polarization modes. The incident field is projected on the reflector surface mesh and the SC integrals in Eqs. (1) and (2) are evaluated numerically using a standard Gauss-Legendre quadrature. The convergence of numerical results is then verified by varying the mesh size.
2.2 Incident field and parabolic reflector geometries
All the focusing configurations studied in this work consist of a simple collimated incident laser beam reflected by a parabolic mirror (see Fig. 1). While many polarization and spatial profiles will be considered, other properties of the laser beam are fixed to perform a comparative study of different reflector geometries. In particular, we consider a 20 J pulse with a spectral energy density modeled as a super-Gaussian $\frac {d\mathcal {E}}{d\omega } = Ae^{-\left \vert \frac {\omega -\omega _0}{\Delta \omega }\right \vert ^j}$ of order $j=7$ centered at $\lambda _0= 2\pi c/\omega _0 = 800$ nm with $\Delta \lambda = 2\pi c \Delta \omega /\omega _0^2= 90$ nm of bandwidth at FWHM and $A$ being an energy normalization constant. This leads to a Fourier-limited pulse duration of 20 fs at FWHM in intensity, making for a 1 PW laser pulse. The polarization and spatial profiles are described in the next section. The laser beams are incident on three different focusing optics (depicted in Fig. 1):
- (a) High Numerical Aperture Parabola (HNAP)
- (b) 90$^\circ$ Off-Axis Parabola (OAP90)
- (c) Transmission Parabola (TP)
All three geometries have an outer diameter of $D = 220$ mm to ensure that external beam clipping is the same in all configurations. The first parabolic reflector studied is the HNAP with a focal length of $f_{0}=58$ mm and a focus located $\Delta =5$ mm in front of the parabola on the propagation axis ($z$-direction), as shown in Fig. 1.A. In practice, this geometry is inconvenient for experiments since the focal spot is located within the incident beam path. Moreover, the last turning mirror blocks the view of detectors looking at the focus, implying that a through-hole in the mirror is often required. Nevertheless, it provides the tightest focusing geometry (NA $\approx 1$) and therefore, it is expected to yield the highest laser intensity reachable with parabolic mirrors. In our numerical study, we exclude HNAPs with NA $>1$ (i.e., with focusing solid angle $\Omega >2\pi$) because they would reflect some of the EM energy back into the incident path, which could damage the laser system at high power. The second geometry studied is a 90$^\circ$ OAP (OAP90) with an effective focal length of $f_\textrm {eff} = 115$ mm, and a focus located $\Delta =5$ mm outside the outer edge of the optic, as depicted in Fig. 1.B. This geometry is investigated because the 90$^\circ$ OAP with $f_{0} \approx D/2$ is widely used in experimental schemes as the focus is not hindered by the incident beam. At the same time, its NA is relatively large which makes for a tight focusing geometry and leads to high intensities. Figure 1.C displays the third geometry called the Transmission Parabola (TP), which has an inner opening $D_\textrm {in}$ and a focus located at $\Delta =5$ mm behind the reflector on the longitudinal axis. This geometry is convenient for detectors owing to its colinearity with the incident laser beam direction and because the focus can be accessed. Moreover, it provides tight-focusing (near $90^\circ$ reflections) from the inner part of the beam and also has a rotational symmetry, in contrast to the OAP. Its most obvious drawback however is the energy loss from its central opening due to $D_\textrm {in}$.
Fig. 1. Three considered parabolic reflector geometries with $D = 220$ mm to focus the incident $w_{\textrm {FWHM}} = 200$ mm, 1 PW linearly-polarized laser beam. (A) On-axis parabola with NA $\approx 1$. (B) 90$^\circ$ OAP with NA $= 0.87$ and (C) through-hole TP with NA $= 0.94$. Each geometry includes a space margin of $\Delta = 5$ mm.
For all reflectors, the fields are evaluated in the coordinate system centered at the geometrical focus. Then, the parabolic equation describing the mirror surface $\mathcal {S}$ is expressed as $z = \frac {r^2}{4f_0} - f_0$, where $r=\sqrt {x^{2} + y^{2}}$ is the radial distance and $f_{0}$ is the focal length. Since the TP is characterized by an additional parameter, the inner opening diameter $D_\textrm {in}=2r_\textrm {min}$, the geometry can be optimized to maximize the focused intensity for a reliable comparison with the HNAP and OAP90. For this purpose, we define the energy loss from the central hole of the TP as the following obstruction ratio:
Hence, varying the obstruction ratio changes the reflector’s focal length $f_0$, while parabola length varies inversely to $\alpha$.
2.3 Numerical aperture calculation
In order to properly compare the focusing geometries, a precise calculation of the NA is required for all reflector types. Specific calculations for all three geometries can be found in Supplement 1. For the case of a reflector with a hole such as the TP geometry, we calculate the NA through its effective conical solid angle as:
and using:Since the NA is a metric that evaluates the range of angles from a light cone in an optical system, its definition naturally refers to the solid angle of the light cone, i.e., $\Omega = 2\pi (1-\cos \theta )$. The solid angle has an additive property as it comes from an integral definition, whereas the usual expression:
containing $\sin \theta$ prevents adding and subtracting light cones, i.e., $\sin (\theta _1) + \sin (\theta _2) = \textrm {NA}_1 + \textrm {NA}_2 \not = \textrm {NA}_{1+2}$. Inserting the solid angle of the light cone into the NA definition (8) enables a fully universal way to compare the NA of different optical systems. One can express the fill factor of the sphere illumination around the focus, up to $4\pi$-illumination (NA = 2). Also, this definition reduces to the usual NA = $\sin \theta _\textrm {max}$ when using a single unobstructed light cone with $\theta _\textrm {max} \leq 90^\circ$, i.e., the HNAP case. In the general case, one can calculate the effective solid angle of the system $\Omega _\textrm {eff} = \iint \sin \theta \, \mathrm {d} \theta \mathrm {d} \phi$, and then insert it in Eq. (8). When the effective light cone half-angle $\theta _\textrm {eff} = \arccos \left (1- \frac {\Omega }{2\pi }\right )>90^\circ$, one must apply NA = $2-\sin \theta _\textrm {eff}$ in order to properly evaluate the metric (see Supplement 1 for NA calculation of the three reflectors and further discussion).We emphasize that Eq. (10) should be used with caution, for example when calculating the NA of the widely used OAP. In the paraxial case and for small off-axis angles, this expression yields similar results to Eq. (8), but this is not the case outside of this regime. Another incorrect use of NA$_{\mathrm {standard}}$ is to apply the largest focusing angle definition, NA = $\sin \theta _\textrm {max}$, for annular pupils having a physical obstruction in the center. When an optical system has $\theta _\textrm {max} > 90^\circ$ or $\theta _\textrm {min} > 0^\circ$, it is essential to rely on the unsimplified version of the NA through the solid angle.
3. Results
In this section, numerical results are presented for different incident fields and reflector types. A comparative study is performed to highlight the advantages of each focusing configuration. In all cases, the intensity is evaluated from the focused field envelope as obtained from the complex norm of the analytic signal (twice the sum over positive frequency components). The latter is denoted as $\tilde {\boldsymbol {E}}(\boldsymbol {r},t),\tilde {\boldsymbol {B}}(\boldsymbol {r},t)$, and the electric field intensity is given by the vectorial norm squared as:
As many configurations are possible with the TP due to its extra parameter (i.e., its inner opening), we first optimize its obstruction ratio $\alpha$ by maximizing the focused intensity for both linear and radial field models in Sections 3.1.1 and 3.2.1, respectively. This permits a better comparison with the two other reflectors considered in this work.
3.1 Linearly-polarized super-Gaussian beam
3.1.1 TP shape optimization for linear polarization
The first configuration studied is a $p$-polarized ($x$-direction) collimated laser beam with a super-Gaussian transverse profile. It has a beamsize of $w^\textrm {inc}_{\textrm {FWHM}} = 200$ mm and EM fields given by:
The field strengths $E_{0,n}$ for all spectral components are set such that the total energy in the incident beam is $\mathcal {E}_\textrm {L}=20$ J with the super-Gaussian spectral density described above. In other words, the normalization of each spectral component is chosen to satisfy:
Figure 2 shows the optimization of the transmission parabola geometry as a function of the obstruction ratio $\alpha$. We can first note the presence of a peak intensity optimum $I_0 = \max _{\boldsymbol {r},t}\left [I(\boldsymbol {r},t)\right ]$ for $\alpha \approx \alpha _{\mathrm {I}} = 20\%$ in Fig. 2.A, emerging from a balance between energy loss and focusing tightness. This value corresponds to an inner opening of $D_\textrm {in} = \sqrt {\alpha }w^\textrm {inc}_{\textrm {FWHM}} \approx 89$ mm. For large obstruction ratios at which the TP has a wide inner opening but is short in length $\Delta z$, the energy loss is high. However the focusing is tight since light rays are all reflected at nearly 90$^\circ$. For $\alpha < 5\%$, more energy is focused because the TP has a smaller opening. However, its longitudinal size and effective focal length become large and therefore, the focusing is not optimal because most of the focused rays have $\theta \gg 90^\circ$ reflection angles. This effect is confirmed by looking at geometrical parameters of the parabola in Fig. 2.B. This figure displays the NA (blue circles), calculated from Eq. (8), and the parabola length $\Delta z$ (red triangles) as a function of $\alpha$. The value of NA has an optimum at $\alpha \approx \alpha _{\mathrm {NA}} = 10$%, approximately 10% lower than $\alpha _{\mathrm {I}}$. Therefore, NA is not the only critical parameter to reach the maximum intensity. Below about $\alpha \approx 15$%, the parabola length also plays a predominant role. In this range, $\Delta z$ increases significantly with decreasing $\alpha$, going up to 80 cm-long for an obstruction ratio of 1.25%, and this results in lower intensities due to the larger spot size, even if the NA has a relatively high value above 0.9. For the focused beam parameters shown in Fig. 2.C, an inverted but highly correlated trend is observed for the Rayleigh length $z_\textrm {R}$ when compared to the NA in Fig. 2.B, more precisely $z_\textrm {R} \propto 1/\textrm {NA}$. The focused spot size $w_{\textrm {FWHM}}$ follows the same trend as the parabola length $\Delta z$, namely decreasing for increasing $\alpha$ values, and does not follow the Rayleigh length’s behavior as is expected using conventional Gaussian optics. In this latter case, the tightness of focus typically increases with NA because a higher NA leads to wider focusing angles generating a smaller focal spot. In the case of the TP, the trend is reversed for $\alpha > 10\%$ as the light rays are focused more tightly with a wider inner opening (i.e., shorter parabola length), also leading to a long Rayleigh length due to the smaller focusing solid angle. The most tightly focused rays are located in the vicinity of the inner opening rather than on the outer edges of the beam as with conventional focusing optics. Hence for the TP case with $\alpha > 10\%$, the trend is reversed between the Rayleigh length and the spot size, i.e., $z_\textrm {R} \not \propto w_{\textrm {FWHM}}$. It is observed in Fig. 2.A that the inverse of the focal volume $V_\textrm {focus}^{-1} = z^{-1}_\textrm {R}w^{-2}_{\textrm {FWHM}}$ also has its maximum at $\alpha =20\%$ since $I_0 \propto \mathcal {E}_\textrm {L}c/V_\textrm {focus}$. This explains why the peak intensity does not share the same optimum point as the numerical aperture around $\alpha _{\mathrm {NA}} = 10\%$, and shows that the highest NA does not necessarily yield the highest peak intensity for this type of parabolic reflector.
Fig. 2. Optimization of the Transmission Parabola as a function of the obstruction ratio $\alpha$. (A) Peak intensity $I_0$ (black squares) and inverse focal volume $V_\textrm {focus}^{-1}$ (green diamonds) variation, both showing an optimum at $\alpha = 20\%$ (i.e., $D_\textrm {in} \approx 89$ mm). (B) TP geometrical parameters: Numerical Aperture (blue circles) and parabola length $\Delta z$ (red triangles). (C) Focused beam parameters: Rayleigh length $z_\textrm {R}$ (blue circles) and spot size $w_{\textrm {FWHM}}$ (red triangles). Note that $z_\textrm {R} \propto 1/\textrm {NA}$ and $w_{\textrm {FWHM}} \propto \Delta z$.
As observed, the Rayleigh length depends directly on the total focusing solid angle, i.e., $z_\textrm {R} \propto 1/\textrm {NA}(\Omega )$, whereas the spot size is mostly dependent on the maximum sine of the angular light ray distribution, i.e., $w_{\textrm {FWHM}} \propto 1/\max \left [\sin \theta \right ]$, which is not necessarily equal to $1/\sin \theta _\textrm {max}$ since $\theta _\textrm {max}$ can be greater than $90^\circ$ and go up to $180^\circ$. The generalization of focused beam properties with focusing angles is as follows: for a single unobstructed light cone ($\theta _\textrm {min} = 0^\circ$), $\textrm {NA} = \sin \left [\arccos \left (1- \frac {\Omega }{2\pi }\right )\right ] = \sin \theta _\textrm {max}$ for $\theta _\textrm {max}\leq 90^\circ$, and $\textrm {NA} = 2 - \sin \theta _\textrm {max}$ for $\theta _\textrm {max}>90^\circ$. For annular reflectors ($\theta _\textrm {min} > 0^\circ$, like an HNAP with a hole or the TP for instance), $\textrm {NA}(\Omega ) \not = \sin \theta _\textrm {max}$ and must be calculated through the solid angle of Eq. (8). In all cases, $\textrm {NA}(\Omega )$ determines the spatial extent of the Rayleigh length $z_\textrm {R}$, whereas the spot size $w_{\textrm {FWHM}}$ is dictated by $\max \left [\sin \theta \right ]$ (i.e., the rays focusing the closest to 90$^\circ$). For a focusing geometry similar to HNAP with $\theta _\textrm {max}\leq 90^\circ$ (as for Gaussian optics in the paraxial regime), $\max \left [\sin \theta \right ] = \sin \theta _\textrm {max}$ and therefore $z_\textrm {R} \propto w^2_{\textrm {FWHM}}$, whereas for cases with $\theta _\textrm {max} > 90^\circ$, $z_\textrm {R}$ and $w_{\textrm {FWHM}}$ can go opposite directions with $\textrm {NA}(\Omega )$. For example with the TP, $\theta _\textrm {max}$ has its highest value of 172$^\circ$ (i.e., smallest $\sin \theta _\textrm {max}$) for $\alpha =1.25\%$ with the largest focal spot, whereas $\max \left [\sin \theta \right ]$ is maximal for $\alpha =80\%$ with the smallest focal spot.
The use of the solid angle definition of NA has enlightened a direct correlation with the Rayleigh length, whereas it is not necessarily the case for the spot size, as opposed to conventional observations in Gaussian optics. As a matter of fact, it is observed in Fig. 2 that a smaller NA leads to a smaller focal spot, which is counter-intuitive. To the best of our knowledge, this effect was never put forth in previous literature. In the quest for reaching the highest laser intensities, it is beneficial to achieve a smaller focal spot owing to its square relationship with peak intensity $I_0 \propto w_\textrm {FWHM}^{-2}$. Since smaller focal spots often implies high NA, this requires the use of the more general definition of the metric, as suggested in this work.
3.1.2 Comparison of reflectors for linear polarization
Now choosing the optimal TP geometry with $\alpha = 20\%$, Fig. 3 shows the intensity distribution in the focal plane at $z=0$ for the three parabolic reflector geometries with the same incident fields. The intensity distribution of the HNAP displayed in Fig. 3.A and its related field components are similar to previous calculations [28,41–48]. The elliptical shape is typical of a non-paraxial linearly-polarized beam focusing, which induces a strong dipolar-shaped $E_z$ component with lobes positioned along the polarization axis (see Supplement 1). A slight ellipticity is also observed for the field component in the polarization direction $E_x$, consistent with other work [40,46]. In Fig. 3.B, a larger elliptical spot is observed for the $90^\circ$ OAP. Again, the field distribution is in agreement with previous work on OAPs [44,48].
Fig. 3. Focused electric field intensity distribution in the focal plane ($z=0$) and $t=0$ (time of maximum intensity) for (A) HNAP (B) OAP90 and (C) TP. All three parabolic reflectors produce a peak intensity $I_0 > 10^{23}$ W/cm$^2$ using a single PW of peak power. The TP exhibits an annular shape with a local minimum at the center of its focal spot due to destructive interference.
In the case of the TP, two intensity lobes and an annular shape appear in Fig. 3.C, with a local minimum in the center of the spot. This peculiar shape has been noted in the work of Sheppard [42] with a thin annular parabolic reflector and in the thesis of Person [49] using an ellipsoidal reflector. This effect is specific to an on-axis parabola section with $\theta >90^\circ$, and therefore it only applies to the TP reflector in this paper. More generally, the effect described in the following will be observed for any circularly symmetric reflector with obtuse focusing angles. Before reflection, an incident linearly-polarized beam has the same polarization direction over the entire transverse plane. Upon reflection, the field component along the polarization direction, $E_x$ here, cancels out when combining at the focus. This results in a destructive interference at $r=0$, as shown by the schematic diagram of Fig. 4. In Fig. 4.A-B, the reflection of two light rays in the horizontal plane ($xz$-plane) for the HNAP case is shown. We notice that the vector components from different light rays add up constructively in the focus along the polarization axis, yielding a vector sum in the same direction as the incident polarization. For two light rays in the vertical plane ($yz$-plane), the reflections are $s$-type and therefore the field component along the polarization axis remains unchanged after reflection. In the TP case, as observed in Fig. 4.C-D, the polarization direction in the horizontal plane undergoes a $\pi$ phase shift after reflection, whereas in the vertical plane it remains unchanged, which leads to vectorial cancellation. More generally, reflections from any two orthogonal directions in the transverse plane result in destructive interference at the focus, explaining the local intensity minimum at $r=0$. Note that the width of the annular shape is on the order of $\lambda _0/4$. Finally, the two bright intensity regions oriented along the polarization axis $x$, as observed in Fig. 3.C, emerge from the dominant dipolar $E_z$ component due to tight-focusing.
Fig. 4. Top (A-C) and transverse (B-D) views of polarization orientations on HNAP (A-B) and TP reflectors (C-D). The fields add up constructively in the same direction as the incident polarization for HNAP, whereas vectorial cancellation occurs for the TP. The blue dot is the focal point and the red dots are the two red light rays in the $xz$-plane (#1 and #2) represented in A-C. The two other light rays in the $yz$-plane (#3 and #4) undergo an $s$-type reflection and therefore remain unchanged after reflection.
A summary of the focused beam characteristics is shown in Table 1. All three geometries produce field intensities with $I_0 > 10^{23}$ W/cm$^2$ using a single PW of peak power. OAP90 and TP geometries yield peak intensities $1.9\times$ and $3.4\times$ lower than HNAP, respectively. Again, note that the peak intensity trend does not follow the NA trend directly, as opposed to expectations from Gaussian optics in the paraxial regime. A direct correlation is observed between NA and the Rayleigh length however, as previously remarked for the TP optimization in Fig. 3.B-C. For the spot size, the main difference between HNAP and OAP90 resides in the looser focus in the $y$-direction. The TP has a slightly larger spot size which reduces its peak intensity, due to the annular shape of the spot. Nevertheless, its Rayleigh length is shorter than OAP90 resulting from its larger NA.
Table 1. Focal volume characteristics (peak intensity $I_0$, spot size $w_\textrm {FWHM}$ along $x-y$ directions, and Rayleigh length $z_\textrm {R}$) evaluated for the three parabolic reflector geometries for a 1 PW laser beam.a
3.2 Radial polarization
3.2.1 TP shape optimization for radial polarization
In our second investigation, we consider a radially-polarized incident beam (TM$_{01}$ mode) for which the EM fields are defined as [25]:
This field model has a doughnut-shaped radial component $E_r$ emerging from the $re^{-(r/w_0)^2}$ factor, with a maximum field value located at $r(E_r^\textrm {max})=w_0/\sqrt {2}$. We used the same incident waist as for the linearly-polarized field model, namely $w_0 = w^\textrm {inc}_{\textrm {FWHM}}/2\left [\ln (2)/2\right ]^{1/p} \approx$ 107 mm.
As a first step, we optimize the TP again with the obstruction ratio $\alpha$ to ensure a fair comparison between the three reflectors. The results of this optimization are shown in Fig. 5. The optimization of the geometrical parameters, NA and parabola length $\Delta z$, are not displayed in Fig. 5 since they are independent of the incident field model, thus we refer the reader to Fig. 2.B. We observe in Fig. 5.A that the maximum peak intensity has an optimum at $\alpha \approx 20$%, the same as for linearly-polarized beam, and now reaches a higher value of $2.5\times 10^{23}$ W/cm$^2$. This occurs because the doughnut-shaped incident beam has less energy loss in the inner opening of the TP. The focused beam parameters $z_\textrm {R}$ and $w_{\textrm {FWHM}}$ show a similar trend as for a linearly-polarized beam, namely that the Rayleigh increases and the spot decreases with increasing obstruction ratio above 10%, in agreement with other work in the literature [50,51]. In the limit where $\alpha \to 100\%$, we obtain the needle of light case with pure $E_z$ described in the work of Dehez et al. [52]. Compared to the linearly-polarized beam, we can see a steeper decrease of the spot size down to smaller values for the TM$_{01}$ polarization. The optimal case with $\alpha =20\%$ has an inner diameter of $D_\textrm {in}\approx 89$ mm, which corresponds to a central radius of $r_\textrm {center} =(D + D_\textrm {in})/4 \approx$ 77 mm. Interestingly, this is very close to the radius of maximum radial field of $r(E_r^\textrm {max})=w_0/\sqrt {2} \approx 76$ mm, showing that the optimization of $\alpha$ maximizes the focusing of the incident EM energy.
Fig. 5. Optimization of the Transmission Parabola as a function of the obstruction ratio $\alpha$ using a TM$_{01}$ incident field model. (A) Peak intensity $I_0$ variation, showing again an optimum at $\alpha = 20\%$ (i.e., $D_\textrm {in} \approx 89$ mm). (B) Focused beam parameters: Rayleigh length $z_\textrm {R}$ (blue circles) and spot size $w_{\textrm {FWHM}}$ (red triangles).
3.2.2 Comparison of reflectors for radial polarization
Figure 6 shows the intensity distributions from the three parabolic reflectors using a TM$_{01}$ incident field. Figure 6.A has a well-known shape observed several times in the literature [28,30,45,53–55]. More precisely, it has been demonstrated that this focusing geometry (HNAP) with an incident radially-polarized beam provides the smallest focal spot possible with $w_{\textrm {FWHM}} = 0.36\lambda _0$ [30], because it is the closest to a dipole-like emission. For the OAP90 in Fig. 6.B, the largest spot size is observed. It is also slightly elliptical because the reflector has an asymmetrical shape. It is clear that this focusing geometry used with a TM$_{01}$ mode does not combine the fields optimally at the focus. A non-negligible $E_\phi$ component emerges from the asymmetry (see Supplement 1). As shown in Fig. 6.C, the TP now produces a very small and circular spot, without a local minimum at $r=0$. This is due to the same rotational symmetry between the incident field and the reflector: it prevents the destructive interference. The fields combine to produce a strong longitudinal $E_z$ component following a typical Bessel-like distribution (see Supplement 1). The main difference in the field components between HNAP and TP is the $\pi$ phase shift (i.e., reversed polarity) of their $E_z$ component.
Fig. 6. Focused electric field intensity distributions in the focal plane ($z=0$) and $t=0$ (time of maximum intensity) with TM$_{01}$ mode as incident field model for (A) HNAP (B) OAP90 and (C) TP. All three reflectors yield a very circular spot mainly composed of the longitudinal field component $E_z$, except OAP90 that exhibits a slight ellipticity due its asymmetric shape with respect to the incident beam.
Table 2 summarizes the focused beam characteristics for the three cases. Again, all three geometries produce peak intensities $I_0 > 10^{23}$ W/cm$^2$ for a single PW of peak power. The peak intensity of the HNAP is reduced by 18% with respect to the linearly-polarized beam, whereas OAP90 lost 60% and the TP gained 64%. We observe very small spot sizes for HNAP and TP, smaller than $\lambda _0/2=0.4$ $\mu$m, which are in very good agreement with values reported in the work of Quabis et al. [30]. Both HNAP and TP geometries exhibit similar focused beam characteristics, making the TP a very good alternative to the HNAP with a radially-polarized beam. The Rayleigh length of OAP90 is shorter than the TP, and this is most probably due to the emerging $E_\phi$ component that differentiates this geometry from the others. Finally, it is important to note that the doughnut-shaped nature of the radially-polarized beam model used in this Section is impractical to produce for ultrashort pulses. A more realistic approximation of this field model produced with a sectored assembly of achromatic half-wave plates [56,57].
Table 2. Focal volume characteristics (peak intensity $I_0$, spot size $w_\textrm {FWHM}$ and Rayleigh length $z_\textrm {R}$) evaluated for the three parabolic reflector geometries using an incident TM$_{01}$ mode.a
3.3 Vector beam property of the Transmission Parabola
In this section, we investigate further the field structure generated by the Transmission Parabola when using a linearly-polarized incident beam. As shown in Fig. 4, the polarization is shifted upon reflection and this shift depends on the polar angle $\phi$: the resulting polarization is displayed in Fig. 7.A. The polarization direction undergoes two full rotations per $2\pi$ radians in the transverse plane. Solutions of Maxwell’s equation satisfying this property where the polarization state changes locally with the polar angle are called cylindrical vector beams of order $m$ [58]. The radially-polarized beam given in Section 3.2 with $m=1$ is the quintessential member of this class of solutions [59]. From the analysis of the polarization direction, we can conclude that the TP converts the collimated incident linearly-polarized beam to a tightly-focused vector beam of order $m=2$. The tight focusing of the latter gives the field distribution given in Fig. 3.B-C.
Fig. 7. Effect of a vector beam of order $m=2$ incident on the Transmission Parabola. (A) Transverse polarization directions upon reflection on TP from an incident linearly-polarized beam. (B) Intensity distribution at the focal plane for a vector beam of order $m=2$ incident on the Transmission Parabola. No more destructive interference is observed and the distribution is equivalent to a linearly-polarized beam incident on the HNAP geometry. (C) Focal spot transverse profiles in $x$ (blue) and $y$ (red) directions. The focused beam characteristics are of $I_0 = 2.52 \times 10^{23}$ W/cm$^2$, $w_{\textrm {FWHM}}$ = 0.63 (0.31) $\mu$m ($y$-direction) and $z_\textrm {R}$ = 0.59 $\mu$m.
To verify this numerically, we simulated the tight focusing of a vector beam of order $m=2$ from a linearly-polarized input converted by a Vortex Half-Wave Plate (VHWP), as is experimentally investigated in the works of Qi et al. [60,61]. A VHWP consists of a half-wave plate for which the fast axis does not have a constant orientation over the transverse plane but rather varies with the polar angle $\phi$. A vector field of order $m$ can be generated from a waveplate with $m/2$ full rotations of the fast-axis over $2\pi$. Thus, the orientation of the fast axis $\psi$ for a VHWP is defined as:
where $\psi _0$ is the fast axis offset angle at $\phi =0$. The transformation from the linear polarization to a vector beam of order $m$ is performed using the Jones matrix:We then apply the transformation $\boldsymbol {E^\prime _\perp }(\boldsymbol {r}) = \boldsymbol {J}_{m,\psi _0}(\boldsymbol {r}) \boldsymbol {E_\perp }(\boldsymbol {r})$ with $\boldsymbol {E_\perp } = \left [E_x \, E_y\right ]^\intercal$ over the transverse $xy$-plane. This transformation is applied achromatically with the same phase shift for all wavelengths. As previously noticed when using a radially-polarized beam incident on the TP, the destructive interference vanishes when the symmetry of the reflector is similar to that of the incident field.
Using an incident vector beam of order $m=2$ on the TP gives the intensity distribution shown in Fig. 7.B-C. Note that the destructive interference is now completely inverted and the distribution is equivalent to the HNAP geometry in Fig. 3.A. The peak intensity now reaches $I_0 = 2.52 \times 10^{23}$ W/cm$^2$, similar to the TM$_{01}$ beam incident on the TP. Even when considering the 20% energy loss in the inner part of the TP, the peak intensity is still about half of the HNAP case with a linearly-polarized beam because a substantial proportion of the EM energy is in the outer region of the HNAP where it generates a tighter focus. Nevertheless, the TP geometry exhibits much better focused beam characteristics when a vector beam of the same order as the reflector is incident on its surface. More precisely its Rayleigh length, $z_\textrm {R}$ = 0.59 $\mu$m, and spot size, $w_{\textrm {FWHM}}$ = 0.63 (0.31) $\mu$m ($y$-direction), are now very similar to the HNAP case with an incident linearly-polarized beam. We have also separately verified (the numerical results are not displayed for simplicity) that a vector beam of order $m=2$ incident on the HNAP geometry generates an intensity distribution equivalent to the TP with a linearly-polarized beam, as shown in Fig. 3.C. These results confirm that the TP generates a tightly-focused vector beam of order 2 and that a tightly-focused linearly-polarized beam ($m=0$) can be obtained from the combination of the VHWP and the TP. Finally, note that the liquid crystal technology does not allow to produce achromatic Vortex Half-Wave Plates, as is possible with standard achromatic half-wave plates. However, the use of the TP geometry opens the possibility to produce a reflective achromatic $m=2$ vector beam generator that is suitable for high-intensity ultrashort pulses.
4. Discussion: Comparing focusing geometries
In light of the simulation results presented previously, we now position the three different parabolic reflector geometries into an experimental context to better evaluate their potential for future use. Table 3 summarizes the strengths and weaknesses of each configuration. The HNAP geometry is attractive because it provides the highest intensity. Moreover, it is endowed with rotational symmetry, making it suitable for any kind of cylindrical vector beam. However, the focal spot is within the incident beam path, which is detrimental for many experimental investigations. This is a major challenge in laser-matter interactions where targets need to be placed at the focus and detection systems need to probe the interaction region. A common solution to this problem is to drill a hole in the last turning mirror to enable the positioning of particle detectors, which results in energy loss and lower intensities. Moreover, not only focal spot measurements are impractical to perform, but also wavefront measurements. As this focusing geometry inherently has a strong curvature, this may lead to surface fabrication errors while wavefront corrections by a deformable mirror are very difficult to achieve.
Table 3. Strengths and weaknesses of HNAP, OAP90 and TP reflectors.
For the vast majority of high-intensity laser-matter interaction experiments, the OAP geometry is chosen because its focus lies outside the incident beam path. Among all the OAPs, the 90$^\circ$ OAP considered in this article is the one that has the shortest path out of the incident beam, hence also the smallest focal length and one of the highest NA. Based on the results of Section 3, we conclude that this geometry is well-suited for linearly-polarized beams. In addition, the free access to the focal spot allows for direct measurements of the beam intensity, as opposed to the HNAP where this is not possible without interfering with the incident beam. Even though it provides effective focusing and high-intensity, the OAP90 is rather complex to fabricate. It often requires the construction of an entire mother-parabola from which cylindrical sections are extracted and this process can become very costly for large OAPs. Finally, its lack of rotational symmetry induces a skewed wavefront, which has to be fixed using substantial corrections from a deformable mirror for optimal performance, as shown in the works of Bahk et al. [38,44]. This strong wavefront distortion can result in lower effective intensities because the spot fluence becomes very asymmetric in out-of-focus planes. Then, target tilt positioning becomes a much more challenging task. Aside from a reduction of peak intensity, the intensity distribution exhibits a strong gradient on the target surface which can be detrimental for the laser-matter interaction. Hence, this case requires more stringent tilt positioning capabilities.
In contrast to the OAP90, one of the main advantages of the TP is its rotational symmetry. In principle, this avoids the slant wavefront of the OAP90, while offering good focusing properties with vector beams ($m>0$). Moreover, having its focus located on the other side of the reflector on the longitudinal axis, colinear to the incident beam, makes this geometry practical for numerous experimental schemes. It confers a direct access to the focal spot and facilitates the positioning of particle detectors. Even though its intensity is slightly lower than HNAP and OAP90 with a linear polarization, partly due to its central energy loss and annular focal spot, it nonetheless provides intensities $>10^{23}$ W/cm$^2$ per PW of peak power. If used with incident vector beams, this geometry is efficient because it provides tight-focusing without the drawbacks of the HNAP. So far, the TP is under-used in experimental schemes of high-intensity laser-matter interactions, and hence requires further investigation. Outside of the high intensity context, the TP may also find applications for a variety of pump-probe experiments at intermediate intensities or for non-linear microscopy at lower intensities. This geometry is unexploited even for CW lasers, where a series of VHWP can be used to switch from one polarization mode to another with the same focusing optic. Optical trapping or optical tweezing can also be advantageous schemes for the TP with CW lasers due the practical focal spot position after the optic. Moreover, it is expected that using a phase-delay in conjunction with the vector beam properties will results in the production of vortex beams with angular momentum.
5. Conclusion
In this study, we have simulated the focused field of three tight-focusing parabolic reflector geometries (HNAP, OAP90 and TP) with an incident beam of either linear or radial polarization. All three reflectors generated peak intensities above $10^{23}$ W/cm$^2$ for a single PW of peak power. The results showed a direct correlation between the Rayleigh and the Numerical Aperture of the focusing optic, which is related to the focusing solid angle of the light cone, while this correlation is not observed for the spot size. More precisely, it is shown that in some cases a smaller NA can generate a tighter focal spot, which is counter-intuitive and is put forth for the first time in literature with this work. A generalization of NA calculations up to $4\pi$-illumination has been proposed through the solid angle formulation, providing a universal way to compare light cones from any kind of optics. The shape optimization of the TP yielded an intensity optimum at an obstruction ratio of 20%, resulting in the smallest focal volume. A local minimum was found at the center of the TP’s focal spot (i.e., annular spot) with a linearly-polarized beam, due to production of a vector beam of order 2 which induces destructive interference. This effect was inverted when sending either a TM$_{01}$ mode or a vector beam of order 2 on the TP. With its focusing axis colinear to its incident beam path, the TP is a promising reflector for numerous applications ranging from laser-matter interactions, pump-probe experiments, non-linear microscopy and optical trapping. It is widely unexplored in experimental implementations at the present time, and its many advantages make it a viable option to be considered for tight-focusing in future ultra-high intensity experiments on multi-PW laser facilities.
Funding
Alliance de recherche numérique du Canada; Fonds de recherche du Québec – Nature et technologies; Natural Sciences and Engineering Research Council of Canada (NSERC).
Acknowledgments
The authors would like to thank Joey Dumont for the help with the code development.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data that support the findings of this study are available from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. G. Tiwari, E. Gaul, M. Martinez, G. Dyer, J. Gordon, M. Spinks, T. Toncian, B. Bowers, X. Jiao, R. Kupfer, L. Lisi, E. McCary, R. Roycroft, A. Yandow, G. Glenn, M. Donovan, T. Ditmire, and B. Hegelich, “Beam distortion effects upon focusing an ultrashort petawatt laser pulse to greater than 1022 W/cm2,” Opt. Lett. 44(11), 2764–2767 (2019). [CrossRef]
2. S. Hakimi, L. Obst-Huebl, A. Huebl, K. Nakamura, S. S. Bulanov, S. Steinke, W. P. Leemans, Z. Kober, T. M. Ostermayr, T. Schenkel, A. J. Gonsalves, J.-L. Vay, J. van Tilborg, C. Toth, C. B. Schroeder, E. Esarey, and C. G. Geddes, “Laser–solid interaction studies enabled by the new capabilities of the iP2 BELLA PW beamline,” Phys. Plasmas 29(8), 083102 (2022). [CrossRef]
3. Z. Guo, L. Yu, J. Wang, C. Wang, Y. Liu, Z. Gan, W. Li, Y. Leng, X. Liang, and R. Li, “Improvement of the focusing ability by double deformable mirrors for 10-PW-level Ti:sapphire chirped pulse amplification laser system,” Opt. Express 26(20), 26776–26786 (2018). [CrossRef]
4. J. W. Yoon, Y. G. Kim, I. W. Choi, J. H. Sung, H. W. Lee, S. K. Lee, and C. H. Nam, “Realization of laser intensity over 1023 W/cm2,” Optica 8(5), 630–635 (2021). [CrossRef]
5. K. Burdonov, A. Fazzini, V. Lelasseux, et al., “Characterization and performance of the Apollon Short-Focal-Area facility following its commissioning at 1 PW level,” Matter Radiat. Extremes 6(6), 064402 (2021). [CrossRef]
6. C. Radier, O. Chalus, M. Charbonneau, et al., “10 PW peak power femtosecond laser pulses at ELI-NP,” High Power Laser Sci. Eng. 10, e21 (2022). [CrossRef]
7. J. Bromage, S.-W. Bahk, I. Begishev, C. Dorrer, M. Guardalben, B. Hoffman, J. Oliver, R. Roides, E. Schiesser, M. Shoup III, M. Spilatro, B. Webb, D. Weiner, and J. Zuegel, “Technology development for ultraintense all-OPCPA systems,” High Power Laser Sci. Eng. 7, e4 (2019). [CrossRef]
8. E. Cartlidge, “The light fantastic,” Science 359(6374), 382–385 (2018). [CrossRef]
9. A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84(3), 1177–1228 (2012). [CrossRef]
10. N. M. Naumova, J. A. Nees, I. V. Sokolov, B. Hou, and G. A. Mourou, “Relativistic Generation of Isolated Attosecond Pulses in a λ3 Focal Volume,” Phys. Rev. Lett. 92(6), 063902 (2004). [CrossRef]
11. J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56(1), 99–107 (1939). [CrossRef]
12. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16(6), 1381–1386 (1999). [CrossRef]
13. A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008). [CrossRef]
14. A. April and M. Piché, “4π Focusing of TM01 beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010). [CrossRef]
15. A. April, “Bessel–Gauss beams as rigorous solutions of the Helmholtz equation,” J. Opt. Soc. Am. A 28(10), 2100–2107 (2011). [CrossRef]
16. A. April, “Ultrashort, Strongly Focused Laser Pulses in Free Space,” Coherence and Ultrashort Pulse Laser Emission, Duarte, F. J., Ed. (InTech, 2010), pp. 355–382.
17. J.-X. Li, Y. I. Salamin, K. Z. Hatsagortsyan, and C. H. Keitel, “Fields of an ultrashort tightly focused laser pulse,” J. Opt. Soc. Am. B 33(3), 405–411 (2016). [CrossRef]
18. J. Lekner, “Tight focusing of light beams: a set of exact solutions,” Proc. R. Soc. A 472(2195), 20160538 (2016). [CrossRef]
19. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]
20. Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86(2), 319–326 (2007). [CrossRef]
21. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31(17), 2619–2621 (2006). [CrossRef]
22. Y. I. Salamin, “Fields and propagation characteristics in vacuum of an ultrashort tightly focused radially polarized laser pulse,” Phys. Rev. A 92(5), 053836 (2015). [CrossRef]
23. J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004). [CrossRef]
24. U. Levy and Y. Silberberg, “Weakly diverging to tightly focused gaussian beams: a single set of analytic expressions,” J. Opt. Soc. Am. A 33(10), 1999–2009 (2016). [CrossRef]
25. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys. 8(8), 133 (2006). [CrossRef]
26. İ. R. Çapoğlu, A. Taflove, and V. Backman, “Computation of tightly-focused laser beams in the FDTD method,” Opt. Express 21(1), 87–101 (2013). [CrossRef]
27. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959). [CrossRef]
28. M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001). [CrossRef]
29. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]
30. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000). [CrossRef]
31. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72(1), 109–113 (2001). [CrossRef]
32. N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008). [CrossRef]
33. Z. Chen, L. Hua, and J. Pu, “Tight Focusing of Light Beams: Effect of Polarization, Phase, and Coherence,” in Progress in Optics, vol. 57E. Wolf, ed. (Elsevier, 2012), Chapter 4 pp. 219–260.
34. T. M. Jeong, S. Weber, B. L. Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015). [CrossRef]
35. J. Peatross, M. Berrondo, D. Smith, and M. Ware, “Vector fields in a tight laser focus: comparison of models,” Opt. Express 25(13), 13990–14007 (2017). [CrossRef]
36. W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11(2), 623–629 (1994). [CrossRef]
37. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17(11), 2081–2089 (2000). [CrossRef]
38. S.-W. Bahk, P. Rousseau, T. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022 W/cm2),” Appl. Phys. B 80(7), 823–832 (2005). [CrossRef]
39. S. Silver, Microwave Antenna Theory and Design, 19 (McGraw-Hill, 1949).
40. J. Dumont, F. Fillion-Gourdeau, C. Lefebvre, D. Gagnon, and S. MacLean, “Efficiently parallelized modeling of tightly focused, large bandwidth laser pulses,” J. Opt. 19(2), 025604 (2017). [CrossRef]
41. C. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. UK 1(4), 129–132 (1977). [CrossRef]
42. C. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. Microw. Opt. Acoust. UK 2(5), 163–166 (1978). [CrossRef]
43. P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. ii. numerical results,” J. Opt. Soc. Am. A 17(11), 2090–2095 (2000). [CrossRef]
44. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (1022 W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004). [CrossRef]
45. C. Stanciu, M. Sackrow, and A. Meixner, “High NA particle-and tip-enhanced nanoscale Raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229(2), 247–253 (2008). [CrossRef]
46. S. G. Bochkarev and V. Y. Bychenkov, “Acceleration of electrons by tightly focused femtosecond laser pulses,” Quantum Electron. 37(3), 273–284 (2007). [CrossRef]
47. K. Popov, V. Y. Bychenkov, W. Rozmus, and R. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008). [CrossRef]
48. O. Vais and V. Y. Bychenkov, “Direct electron acceleration for diagnostics of a laser pulse focused by an off-axis parabolic mirror,” Appl. Phys. B 124(11), 211 (2018). [CrossRef]
49. S. M. Person, “Optical methods for nanoparticle detection and imaging,” Ph.D. thesis, University of Rochester (2014).
50. P. Suresh, K. Rajesh, T. S. Pillai, and Z. Jaroszewicz, “Effect of annular obstruction and numerical aperture in the focal region of high NA objective lens,” Opt. Commun. 318, 137–141 (2014). [CrossRef]
51. C. Mariyal, P. Suresh, K. Rajesh, T. S. Pillai, and M. Lavanya, “Effect of annular obstruction on tight focusing of partially coherent radially polarized Vortex beam,” Optik 125(12), 2934–2937 (2014). [CrossRef]
52. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20(14), 14891–14905 (2012). [CrossRef]
53. J. Stadler, C. Stanciu, C. Stupperich, and A. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008). [CrossRef]
54. X. Gao, Q. Wang, M. Yun, J. Yu, H. Guo, and S. Zhuang, “Conditions for tighter focusing and higher focal depth of radially polarized vector beam,” Opt. Appl.42 (2012).
55. N. Lindlein, M. Sondermann, R. Maiwald, H. Konermann, U. Peschel, and G. Leuchs, “Focusing light with a deep parabolic mirror,” arXiv, arXiv:1905.05997 (2019). [CrossRef]
56. S. Payeur, S. Fourmaux, B. Schmidt, J. MacLean, C. Tchervenkov, F. Légaré, M. Piché, and J. Kieffer, “Generation of a beam of fast electrons by tightly focusing a radially polarized ultrashort laser pulse,” Appl. Phys. Lett. 101(4), 041105 (2012). [CrossRef]
57. J. Powell, S. Payeur, S. Fourmaux, H. Ibrahim, J. C. Kieffer, S. MacLean, and F. Légaré, “100 kev electron beam generation by direct laser acceleration using longitudinal electric fields,” in Conference on Lasers and Electro-Optics (2021), pp. 1–2.
58. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18(7), 1612–1617 (2001). [CrossRef]
59. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
60. J. Qi, W. Wang, B. Shi, H. Zhang, Y. Shen, H. Deng, W. Pu, X. Liu, H. Shan, X. Ma, L. Zhang, W. Lu, M. Fu, and X. Li, “Concise and efficient direct-view generation of arbitrary cylindrical vector beams by a vortex half-wave plate,” Photonics Res. 9(5), 803–813 (2021). [CrossRef]
61. J. Qi, W. Yi, M. Fu, M. Zhu, J. Liu, G. Huang, J. Pan, S. Zhu, X. Chen, W. Tabg, H. Zhang, B. Shi, H. Deng, W. Wang, and X. Li, “Practical generation of arbitrary high-order cylindrical vector beams by cascading vortex half-wave plates,” Opt. Express 29(16), 25365–25376 (2021). [CrossRef]
