## Abstract

In this paper, we present a systematic investigation of the characterization of tightly focused vector fields formed by an off-axis parabolic mirror. Based on the Stratton-Chu integral of Green’s theorem, the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror were derived in detail. In addition, explicit analytical expressions for the far-field vector diffraction electric and magnetic fields suitable for an off-axis parabolic mirror were also obtained. It is shown that there are significant differences in the vector diffraction characterizations between on- and off-axis parabolic mirrors. When the off-axis rate is greater than 4, the longitudinal field is predominant, and the maximum peak intensity ratio between the longitudinal field and the transverse field at the focus is approximately 10^{3}. This property is valuable for all applications in which a strong longitudinal field component is desirable. The effective focal length increases with increasing off-axis rate, and the depth of focus and the convergence angle of the focused beam are strongly dependent on${f}^{\prime}/2\omega $: smaller values of ${f}^{\prime}/2\omega $ lead to shorter focal depths and larger beam convergence (or tighter focusing).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a focusing device, a parabolic mirror can perfectly collect collimated incident light—the axis of propagation of which is parallel to the axis of revolution of the mirror—into the focal point of the mirror without any chromatic aberration within the accuracy of the Gaussian optics, which is even capable of focusing light nearly within the 4π solid angle [1,2]. Owing to these distinctive features, parabolic mirrors have attracted the attention of increasing numbers of researchers over the past several decades and are widely used in astronomical telescope design [3], telecommunications [4], optical tweezers [5], confocal microscopy [6,7], and high-intensity laser systems [8–10]. When a beam is tightly focused by a parabolic mirror, the vector character of the light becomes crucial to correctly describe such a non-paraxial beam. Furthermore, the expressions of its electromagnetic field must satisfy Maxwell’s four equations beyond the paraxial regime. This analysis of the vector field focusing properties is of particular relevance to the high-intensity laser community, which expends significant effort to achieve the highest possible laser intensities. Due to the evolution of high-intensity laser systems, laser intensities in the focal spot of 10^{23}–10^{24} W/cm^{2} have been achieved [10–15]. This highly focused laser intensity can be achieved by tightly focusing the beam in a vacuum using an off-axis parabolic mirror (OAP), which creates many exciting opportunities for particle laser acceleration [16,17], materials processing, and the study of fundamental forces in nature. To model such phenomena, an accurate accounting of the electric and magnetic vector field distributions throughout the region of tight focus is often critical [18].

A rigorous vector diffraction theory of tightly focused light from parabolic mirrors [9,19–28] has been developed over the course of nearly a century, beginning with a treatment in 1920 by Ignatovsky [19]. Ignatovsky first developed vector diffraction for a uniformly polarized collimated beam reflected from a parabolic mirror. Owing to the limitations of the contemporary computing power, he computed the intensity only in the focal plane. Another theoretical approach was provided by Richards and Wolf [29,30], by which strongly focused beams were accurately described, given the field distribution of the collimated input beam at the entrance pupil of the focusing system. In addition, another alternate method for analyzing tightly focused beams is based on the Stratton-Chu formulation of Green’s theorem [31], which involves both the electric and magnetic fields after substitutions from Maxwell’s equations.

To date, most investigations of vector diffraction field properties in the focal region of a parabolic reflector have mainly considered the case of an on-axis parabolic mirror [19–28]. The vector diffraction properties of an OAP have rarely been explored. However, Bahk and colleagues [9] derived the far-field vector diffraction formulae of the focal electromagnetic field formed by an OAP for a circular incident beam based on the Stratton-Chu integral formula [31]. However, they did not provide a detailed investigation of the vectorial diffraction properties of an OAP, likely because they were mainly concerned with the generation of an extreme power density by correcting the characterized aberrations. To the best of our knowledge, a detailed theoretical study of the vectorial field focusing properties of an off-axis parabolic mirror is not available in the literature, whether the incident beam is circular or square.

Seeking solutions to the abovementioned problems was the aim of this study. In Sections 2 and 3, we first outline the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror based on the Stratton-Chu integral. In Section 4, we derive the expressions of the far-field vector diffraction electromagnetic fields suitable for an OAP. In Section 5, we use the expression derived in Section 4 to investigate the characterization of tightly focused vector fields formed by an OAP, including tightly focused vector field component profiles in the focal plane, the three-dimensional (3D) intensity distribution in the vicinity of the focus, the focal depth and convergence angle of a highly confined optical field, and the effective focal length (EFL) of the OAP, among other characteristics.

## 2. General formulas of on-axis and off-axis parabolic mirror and total field

We first consider an ideal on-axis rotational parabolic mirror with its axis of revolution symmetry coinciding with the $z$ axis and its focus coinciding with the origin of a Cartesian coordinate system$S(x,y,z)$, such that its surface intersects the $z$ axis a distance $f$ to the left of the origin, as shown in Fig. 1. The equation of the paraboloidal surface is then given by

where ${x}_{\text{o}}$, ${y}_{\text{o}}$, and ${z}_{\text{o}}$ are the transverse and longitudinal coordinates of any point $O$ on the paraboloidal surface in the Cartesian system, respectively. $f$is the parent focal length of the parabolic mirror. In this paper, we assume that the incident wave is reflected only once, i.e., the field scattered off the surface leaves the paraboloid, since a parabolic mirror as a focusing device encounters this circumstance most commonly. The unit inward normal vector to this paraboloidal surface is expressed asNext, we consider the total field and the field reflected by the parabolic mirror. Based on the electromagnetic boundary conditions, as discussed in the work of Varga and Tцrцk [25], upon reflection the normal components of the electric field and the tangential components of the magnetic field remain unchanged; that is

where ${E}_{i}$ and ${H}_{i}$ denote the incident electric and magnetic fields, respectively. The normal components of the incident and reflected electric fields are denoted ${E}_{i,n}$ and ${E}_{r,n}$. ${H}_{i,t}$ and ${H}_{r,t}$ represent the tangential components of the incident and reflected magnetic fields, respectively. However, the tangential component ${E}_{r,t}$ of the electric field and the normal component ${H}_{r,n}$of the magnetic field change sign. Consequently, the tangential component of the reflected electric field may be written asand the normal component of the reflected magnetic field may be written asUsing Eqs. (5)–(7), one obtains the total reflected field is expressed as Therefore, the total electric field is given as the sum of the incident and reflected fields:A similar procedure yields the following for the total magnetic field:## 3. Vector diffraction formulas for a parabolic mirror

In this section, we derive an expression for the vector diffraction field from a parabolic mirror, for which we use the Stratton-Chu integral formula [31]:

Substituting Eqs. (9)–(11) and (15)–(19) into (13) and (14), we obtain the vector diffraction field from a parabolic mirror:

## 4. Vector diffraction formulas near the focus of an off-axis parabolic mirror

High-intensity laser systems can currently achieve a focused peak intensity of approximately 10^{22} W/cm^{2} [8,9] and will reach 10^{23}–10^{24} W/cm^{2} in the near future [10–15]. The intensity range can be reached by tightly focusing the beam in a vacuum using an OAP [8–10]. This high focused laser intensity creates many exciting opportunities for laser-matter interactions in the relativistic and even ultra-relativistic regimes [15–18]. To model such interactions, an accurate and rapid accounting of the electric and magnetic vector field distribution throughout a tight focus is often critical.

Therefore, the aim in this section is to determine the vector diffraction formulas near the focus of an OAP. For this purpose, we simplify Eqs. (20) and (21) and make approximations. Considering a point $P\left({x}_{P},{y}_{P},{z}_{P}\right)$ on the observation plane lying in the vicinity of focus, due to this focus coinciding with the origin of a Cartesian coordinate system, we make the following approximation: ${x}_{P}\to 0$, ${y}_{P}\to 0$, and ${z}_{P}\to 0$, which results in $\Delta x\to {x}_{\text{o}}$, $\Delta y\to {y}_{\text{o}}$, and $\Delta z\to {z}_{\text{o}}$ in Eqs. (16), (20), and (21). The reason for these approximate treatments is that the incident wave on a paraboloid is a plane wave propagating in a direction parallel to the optical axis and the reflected wave will converge toward the focus, which is readily verified by calculating the Poynting vector of the reflected wave:

For the second approximation, it is assumed that $(ik-1/{r}_{OP})\to ik$. This approximation is justified if the field is at a distance many wavelengths from the paraboloidal surface.

The third approximation comprises the well-known Debye approximations, which state the following: a) The exponent of the Green’s function $G$ can be approximately expressed as $ik{r}_{OP}=ik\left|{r}_{P}-{r}_{O}\right|\approx ik\left({r}_{\text{o}}-{r}_{\text{O}}\xb7{r}_{\text{P}}/{r}_{\text{o}}\right)$. Based on this approximation, spherical wavelets originating from the diffraction paraboloidal surface are replaced by plane wavelets. b) The denominator ${r}_{OP}$ of $G$can be replaced approximately by ${r}_{\text{o}}$. On the paraboloidal surface, when ${z}_{\text{o}}$ is replaced by the Eq. (1), we have ${r}_{\text{o}}=f+\left({x}_{\text{o}}^{2}+{y}_{\text{o}}^{2}\right)/4f$.

Under the Debye and other two above-mentioned approximations, we can reduce Eqs. (20) and (21) to

Using Eqs. (27)–(29), we can obtain the inversion formulas from ($\alpha ,\beta ,\gamma $) to (${x}_{\text{o}},{y}_{\text{o}},{z}_{\text{o}}$):

## 5. Numerical vector diffraction field calculations

The vector diffraction electromagnetic field distributions of linearly polarized collimated light focused by a parabolic mirror have been computed for different values of the dimensionless parameters that characterize the optical system. The first parameter is defined by $\alpha =h/R$, which quantifies the amount of the offset from the $z$ axis to the center of the incoming beam. We call this parameter the off-axis rate. When $h/R=0$, the parabolic mirror is fully on-axis, i.e., the surface of the parabolic mirror under this circumstance is rotationally symmetric about the $z$ axis. When $0<\left|h/R\right|<1$, the parabolic mirror is partially off-axis. However, for $\left|h/R\right|\ge 1$, the parabolic mirror is entirely off-axis. It should be noted that the rotation angle $\phi $ of the coordinate system is closely related to this parameter, and is determined by ${\scriptscriptstyle \frac{1}{2}}[\mathrm{arctan}((h+R)/{z}_{0|h+R})+\mathrm{arctan}((h-R)/{z}_{0|h-R})]$. The second dimensionless parameter is defined by ${f}^{\prime}/2\omega $, where ${f}^{\prime}$ is the effective focal length (EFL) of the OAP system and $2\omega $ is the waist diameter of the incident beam. For larger values of ${f}^{\prime}/2\omega $, the ratio is similar to the f-number of the optical system. Note that the EFL ${f}^{\prime}$ is different from the parent focal length $f$ of the parabolic mirror, as shown in Fig. 1. This parameter is related to the depth of the focus and the convergence angle of the focused beam: smaller values of ${f}^{\prime}/2\omega $ lead to a shorter focal depth and larger beam convergence (or tighter focusing). The EFL of the OAP system is different from that of the lens or on-axis parabolic mirror system, which increases with increasing off-axis rate $\alpha $, as will be shown below.

We present profiles of the electromagnetic field intensities below, which are defined by ${I}_{{E}^{\prime}}\equiv {\left|{E}^{\prime}\right|}^{2}={\left|{E}_{{x}^{\prime}}\right|}^{2}+{\left|{E}_{{y}^{\prime}}\right|}^{2}+{\left|{E}_{{z}^{\prime}}\right|}^{2}$ and ${I}_{{H}^{\prime}}\equiv {\left|{H}^{\prime}\right|}^{2}={\left|{H}_{{x}^{\prime}}\right|}^{2}+{\left|{H}_{{y}^{\prime}}\right|}^{2}+{\left|{H}_{{z}^{\prime}}\right|}^{2}$. Since high-power laser beams generally have square cross-sections [32], we assume that the illuminating beam has a super-Gaussian top-hat beam profile of ${E}_{0}\mathrm{exp}\{-[((x-{x}_{\text{center}})/{\omega}_{x0}{)}^{2n}+(y-{y}_{\text{center}})/{\omega}_{y0}{)}^{2n}]\}$with polarization in the $+x$direction, where ${E}_{0}$ is constant amplitude and $2{\omega}_{x0}$ and $2{\omega}_{y0}$ are the beam waists in the $x$ and $y$ directions, respectively. For a square beam profile, ${\omega}_{x0}={\omega}_{y0}$. ${x}_{\text{center}}$ and ${y}_{\text{center}}$ are the transverse coordinates of the center of the incoming beam. The wavelength $\lambda $ of the incident beam is 1.053 µm. Unless otherwise stated, the size of the square incident beam is 320 mm Ч 320 mm, and the parent focal length of the parabolic mirror is 800 mm. It is beyond the scope of this paper to discuss the optimum values of the beam size and parent focal length of the parabolic mirror.

An animation showing the contour plots of the electromagnetic field intensity distribution of an aberration-free square focused super-Gaussian top-hat beam in the focal plane (the *x*'-*y*' plane with ${{z}^{\prime}}_{P}=0$) by an OAP with different off-axis rates is presented in Fig. 2. The top four panels show ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, ${\left|{E}_{{z}^{\prime}}\right|}^{2}$, and ${\left|{E}^{\prime}\right|}^{2}$, and the four lower panels show ${\left|{H}_{{x}^{\prime}}\right|}^{2}$, ${\left|{H}_{{y}^{\prime}}\right|}^{2}$, ${\left|{H}_{{z}^{\prime}}\right|}^{2}$, and ${\left|{H}^{\prime}\right|}^{2}$ in the *x*'-*y*' plane with ${{z}^{\prime}}_{P}=0$. The intensity is indicated by the “heat” of the color, which goes from black (characterizing the minimum intensity) to red, orange, yellow, and finally, white (characterizing the maximum intensity). As the animation progresses, the off-axis rate $\alpha $ is varied from 0 to 22.5, or the offset $h$ is varied from 0 to 3600 mm. This animation reveals the following: 1) for the intensity distribution of the field component${\left|{E}_{{x}^{\prime}}\right|}^{2}$, which possesses lobes of relatively weak amplitude surrounding a central high amplitude peak, the width of the central peak of the intensity distribution and the side-lobe distributions shown in the focal plane along the *x'* and *y'* directions widen with increasing off-axis rate $\alpha $or offset $h$, except for the intensity distribution in the vicinity of $\alpha =2f/R$(the rotation angle $\phi =\pi /2$). 2) For the intensity distribution of the field component${\left|{E}_{{z}^{\prime}}\right|}^{2}$, which is very similar to that of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, the width of the central peak and that of the side-lobe distributions shown in the focal plane along the *x'* and *y'* directions also widen as the off-axis rate $\alpha $or offset $h$ increases, except in the range $0\le \alpha <0.4$. 3) For the intensity distribution of the field component ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, which exhibits a linear dark center along the *x'*-axis direction, i.e., a local minimum of intensity will occur along the *x'* axis with *y'* = 0, the intensity distribution exhibits bilateral symmetry with respect to the *x'* and *y'* axes. Except for the $0\le \alpha <0.4$ region, two peak intensities are symmetrically distributed along the *y'* axis in the first ring of main lobes around the dark core, and the width of the linear dark center and the width of the side-lobe distributions shown in the focal plane along the *y'* direction widen with increasing off-axis rate $\alpha $ or offset $h$. These results can be seen more clearly in Fig. 3. In addition, we can also see the influence of the intensity distribution of magnetic field components on the off-axis rate $\alpha $. However, at optical frequencies, the magnetic field intensity is negligible compared to the electric field intensity, and the electric field is primarily responsibility for interacting with matter. Hence, in the following section, we mainly focus on the electric field.

Figure 3 shows the dependence of the transverse intensity distribution in the focal plane on the offset $h$. The intensity distribution of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ along the dashed lines shown in Fig. 2 are plotted as functions of the offset $h$ in Figs. 3(a), 3(b), and 3(c), respectively. It is noted that the intensity along the dashed lines shown in Fig. 2 for each offset $h$ is continuously rescaled and normalized to the corresponding maximum amplitude. This scaling allows the relatively weak intensity distribution to be plotted on the same scale as the high-intensity distribution for clarity. As shown in Fig. 3, the size of the focal spot and the width of the side-lobe distributions increase with increasing off-axis rate $\alpha $ or offset $h$, excluding the special intensity distribution characteristics shown in the above three cases. For the intensity distribution of ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ in the region $0\le h<64\text{mm}$ (or $0\le \alpha <0.4$), although it has four peak intensities in the focal plane, the intensity has a very small value along the *y'* axis (*x'* = 0). To show the extraordinary features in the other two cases more clearly, we enlarged the intensity distributions for ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ in the region $1400\text{mm}\le h\le 1800\text{mm}$ (in the vicinity of $\alpha =2f/R$) shown in Fig. 3(a) and for ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ in the region $0\le h\le 100\text{mm}$ (or $0\le \alpha \le 0.6$) shown in Fig. 3(b), which are shown in Figs. 3(d) and 3(e), respectively. From the result shown in Fig. 3(d), it can be seen that the electric field intensity ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ in the center decreases and the dark center gradually appears with increasing *h*, until *h* = 1608 mm (the rotation angle$\phi =\pi /2$). As the value of *h* increases, the electric field intensity is enhanced, and the dark center gradually disappears (also see Fig. 4). However, from the result shown in Fig. 3(e), it can be seen that the intensity distribution of ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ in the center is small compared to its maximum value, providing an intensity distribution with a dark center; however, as the value of *h* increases, the electric field intensity is enhanced (also see Fig. 4) and the dark center gradually disappears.

Figure 4 shows the peak intensity of the electric field components${\left|{E}_{{x}^{\prime}}\right|}^{2}$, ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$for the focused square super-Gaussian top-hat beam polarized along the + *x* direction by the OAP as a function of the off-axis rate $\alpha $ [see Fig. 4(a)] and the rotation angle $\phi $ [see Fig. 4(b)]. The dependence of the EFL ${f}^{\prime}$ of the OAP system on the rotation angle is also presented in the Fig. 4(b). It is noted that a logarithmic scale for the intensity of field components (${\left|{E}_{{x}^{\prime}}\right|}^{2}$, ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, and${\left|{E}_{{z}^{\prime}}\right|}^{2}$) is used in Fig. 4(a), but an absolute value scale for the intensity of field components is used in Fig. 4(b). From the results shown in Fig. 4, it is easily seen that, based on the overall variation trend, the peak intensity of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ decreases with increasing off-axis rate $\alpha $ or rotation angle $\phi $, of course, excluding the region in the vicinity of $\alpha =2f/R$(in this circumstance, the rotation angle $\phi =\pi /2$). For a rotation angle of approximately$\pi /2$, there is a process in which the peak intensity of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ first decreases and then increases as $\alpha $ or $\phi $ increases. This observation can be understood by the fact that a central main lobe is split into two lobes first, and then the two main lobes are combined into one with increasing off-axis rate $\alpha $ or rotation angle $\phi $, which is verified by Figs. 2, 3(a), and 3(d). These figures also show that when $\alpha >6$ (or$\phi >1.07\text{rad}$), the peak intensities of ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$decrease with increasing $\alpha $ or $\phi $. However, for $\alpha <6$ (or $\phi <1.07\text{rad}$), their peak intensities increase. Furthermore, the OAP with $\alpha \approx 4$(in this circumstance, the rotation angle $\phi \approx \pi /4$) produces ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ with comparable peak intensities. For $\alpha <4$ (or $\phi <\pi /4$), the field component ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ is predominant. However, when $\alpha >4$ (or $\phi >\pi /4$), the longitudinal field ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ is predominant, and the maximum peak intensity ratio between the longitudinal field and transverse fields at the focus is approximately 10^{3}. This property is valuable for all applications in which a strong longitudinal field component is desirable, e.g., (1) for generation of plasma channels [33], or (2) for particle acceleration [16,17]. In addition, we can see that the EFL ${f}^{\prime}$ increases with increasing rotation angle $\phi $(or the off-axis rate $\alpha $).

To comprehensively understand the vectorial diffraction properties of the electric field near the focus, the 3D intensity distribution of a focused square super-Gaussian top-hat beam by an OAP for $\alpha =1.844$ (in this circumstance the rotation angle $\phi =0.36\text{rad}$) is shown in Fig. 5. To clearly show the focused field 3D intensity distributions along the propagating direction, we plot the intensity distributions in the ${y}^{\prime}-{z}^{\prime}$plane with ${x}^{\prime}=0$ and cross-sections (${x}^{\prime}-{y}^{\prime}$planes) with different values of ${z}^{\prime}$ before and after the focal plane (${z}^{\prime}=0$). Figure 5(a) presents the intensity profile of the total diffraction electric field ${E}^{\prime}$on the *y'*-*z'* plane (${x}^{\prime}=0$) and the transverse intensity cross-sections at *z'* = $-40\lambda $,$-20\lambda $,$0\lambda $,$20\lambda $, and $40\lambda $. The intensity distribution exhibits a central square main spot and bilateral symmetric off-axis rectangular side lobes. Clearly, the central main spot is formed by constructive interference on the optical axis ${z}^{\prime}$ and the appearance of off-axis side lobes is caused by the diffraction effect. This figure clearly reveals the diffraction characteristics of the electric field in the vicinity of the focus. In addition, the focused electromagnetic fields indeed propagate along the ${z}^{\prime}$ axis and not the $z$ axis [also see Fig. 5(b)]. In the focal plane, the intensity distribution is very sharp, but it becomes increasingly more blurred far from the focal plane.

To clearly reveal the differences in the vectorial diffraction characterizations of the electric field in the vicinity of the focus in the coordinate system $S(x,y,z)$ and the new coordinate system ${S}^{\prime}\left({x}^{\prime},{y}^{\prime},{z}^{\prime}\right)$, the 3D intensity distributions in the coordinate system $S(x,y,z)$for the focused square super-Gaussian top-hat beam by an OAP with $\alpha =1.844$ are shown in Fig. 5(b). This numerical computation is based on Eqs. (23), (24), and (4), and was obtained using the same parameters used in Fig. 5(a), apart from the coordinate system. Here, the electric field intensities are defined by ${I}_{E}\equiv {\left|E\right|}^{2}={\left|{E}_{x}\right|}^{2}+{\left|{E}_{y}\right|}^{2}+{\left|{E}_{z}\right|}^{2}$. Discrepancies in the electric field intensity distributions between the coordinate systems $S(x,y,z)$ and ${S}^{\prime}\left({x}^{\prime},{y}^{\prime},{z}^{\prime}\right)$ are apparent. The first difference is that the intensity distribution in the coordinate system $S$ has lost bilateral symmetry. An extended tail develops along the $+x$axis before the focal plane. However, after the focal plane, this behavior is the opposite, i.e., the extended tail develops along the $-x$axis. This result can be explained by the fact that the electric energy converges to the focus on the $z$ axis from the $+x$ direction, while after the focal plane it diverges from the focus to the $-x$ direction deviating from the $z$ axis. The second difference is that the focused electromagnetic fields propagate along the yellow short dashed line shown in Fig. 5(b), not the $z$axis. Thus, the propagation of the electromagnetic field in the coordinate system $S$ is non-paraxial. It should be noted that the angle between the dashed line and the $z$ axis is the rotation angle $\phi $ of the coordinate previously defined, such that analytical expressions (34) and (35) are more suitable for characterizing the vectorial diffraction fields focused by an OAP than Eqs. (23) and (24) are.

In the following section, we consider the main features in the behavior of the depth of focus and the size of the central focal spot of the focused beam as a function of ${f}^{\prime}/2\omega $. Figure 6 presents the contour plots of the electric field intensity distribution along the ${z}^{\prime}$ axis (${x}^{\prime}=0$,${y}^{\prime}=0$,$-100\lambda \le {z}^{\prime}\le 100\lambda $) of a focused square super-Gaussian top-hat beam with a fixed $\omega $ by an OAP as function of off-set *h*. The contour of the electric field intensity distribution ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ along the ${z}^{\prime}$axis is plotted in Fig. 6(a) and for ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ in Fig. 6(b). Each profile is on the ${z}^{\prime}$ axis (${y}^{\prime}=0$), such that ${\left|{E}_{{y}^{\prime}}\right|}^{2}=0$(see Figs. 2, 3, and 5). Note that the intensity along the ${z}^{\prime}$ axis for each offset $h$ is continuously rescaled and normalized to the corresponding maximum amplitudes. Although Fig. 6 shows only the dependence of the intensity distribution along the ${z}^{\prime}$ axis on offset *h*, it indirectly manifests the behavior of the depth of focus as a function of ${f}^{\prime}/2\omega $. This is because the intensity distribution along the ${z}^{\prime}$ axis characterizes the depth of focus, and the EFL ${f}^{\prime}$ is closely related to offset *h*. As shown in Fig. 4, a larger value of $h$ leads to a longer EFL${f}^{\prime}$. Thus, from these figures, we readily find that the depth of focus increases with increasing ${f}^{\prime}/2\omega $ ($\omega $is fixed), i.e., smaller values of ${f}^{\prime}/2\omega $ lead to a shorter focal depth. It is necessary to note that a dark center appears in the intensity distribution of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ for *h* = 1608 mm (the rotation angle $\phi =\pi /2$), as shown in Fig. 6(a), which means that focal depth does not exist due to the appearance of the dark center.

To further demonstrate the dependences of the depth of focus and the size of the central focal spot of the focused beam on ${f}^{\prime}/2\omega $, Fig. 7 presents contour plots of the electric field intensity distributions ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ along the ${x}^{\prime}$ and ${z}^{\prime}$ axes of a focused square super-Gaussian top-hat beam by an OAP with fixed EFL ${f}^{\prime}=1088\text{mm}$(corroding$h=960\text{mm}$) as a function of ${f}^{\prime}/2\omega $. For clarity, the intensity is continuously rescaled and normalized to the corresponding maximum amplitudes, which is indicated by the “heat” of the color. Figures 7(a) and 7(b) show contour plots of the electric field intensity distributions ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ along the ${x}^{\prime}$ axis (${y}^{\prime}=0$) of the focused beam as function of ${f}^{\prime}/2\omega $, respectively. These electric field intensity distributions are computed in the focal plane ${z}^{\prime}=0$. These figures clearly reveal that the size of the central focal spot of the focused beam decreases with decreasing ${f}^{\prime}/2\omega $, until it reaches the diffraction limit; that is, smaller values of ${f}^{\prime}/2\omega $ lead to tighter focus. Figures 7(c) and 7(d) show contour plots of the electric field intensity distributions ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ and ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ along the ${z}^{\prime}$ axis (${x}^{\prime}=0,{y}^{\prime}=0$) of the focused beam as a function of ${f}^{\prime}/2\omega $, respectively. Therefore, the depth of focus increases with increasing ${f}^{\prime}/2\omega $.

Finally, to provide a comparison of the vectorial diffraction field properties formed by an OAP for square and circular incident beams, a movie showing the contour plots of the electromagnetic field intensity distributions near the focus of an aberration-free circularly focused super-Gaussian top-hat beam in the focal plane (the *x*'-*y*' plane with ${{z}^{\prime}}_{P}=0$) by an OAP with different off-axis rates is presented as Fig. 8. The top four cells plot ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, ${\left|{E}_{{z}^{\prime}}\right|}^{2}$, and ${\left|{E}^{\prime}\right|}^{2}$, and the four lower cells plot ${\left|{H}_{{x}^{\prime}}\right|}^{2}$, ${\left|{H}_{{y}^{\prime}}\right|}^{2}$, ${\left|{H}_{{z}^{\prime}}\right|}^{2}$, and ${\left|{H}^{\prime}\right|}^{2}$ in the *x*'-*y*' plane with ${{z}^{\prime}}_{P}=0$. The intensity is indicated by the “heat” of the color. Numerical computations are based on Eqs. (34), (35), and (3). The vector diffraction field properties formed by an OAP for a circular incident beam have been investigated and discussed by many researchers [19–28]. However, only in the work of Bahk and associates was the focal vector field formed by an OAP for a circular incident beam considered. Nonetheless, these researchers did not provide a detailed investigation of the intensity distribution of a focused circular beam by an OAP for different off-axis rates [8]. We present this analysis here for direct comparison with the intensity distributions in the focal region for a square incident beam. From the results shown in Fig. 8, one can see the following: 1) For the intensity distribution of the field component ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, which possesses a circular main spot and concentric ring-shaped side-lobes, the size of the central circular main spot and width of the side-lobe distributions progressively increase with increasing $\alpha $or $h$, except for the intensity distribution in the vicinity of $\alpha =2f/R$. 2) For the intensity distribution of the field component ${\left|{E}_{{z}^{\prime}}\right|}^{2}$, which is similar to that of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, the width of the central circular main spot and width of the side-lobe distributions also widen as $\alpha $or $h$ increases, except in the range $0\le \alpha <0.6$. 3) For the intensity distribution of the field component ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, which exhibits a linear dark center along the *x'*-axis direction, the intensity distribution exhibits bilateral symmetry with respect to the *x'* and *y'* axes. Except for the $0\le \alpha <0.4$range, two peak intensities are symmetrically distributed along the *y'* axis in the first ring side-lobes around the dark core, and the width of the linear dark center and that of the side-lobe distributions widen with increasing $\alpha $or $h$. 4) The intensity distribution changes in the focused vector field components ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ before and after $\alpha =2f/R$(or $\phi =\pi /2$), ${\left|{E}_{{z}^{\prime}}\right|}^{2}$for $0\le \alpha <0.6$, and ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ for $0\le \alpha <0.4$ exhibit behaviors similar to those of the vector-focused field components for a square incident beam. Moreover, we can also see the influence of the intensity distribution of the magnetic field components on $\alpha $. By comparing the vectorial electromagnetic field intensity distribution properties for circular and square incident beams, it can be determined that the influence of $\alpha $ on the circular field intensity distribution properties is the same as in the case with a square incident beam, other than a different focal spot pattern. In addition, we observed that the numerical intensity distribution of the vector field components formed by an on-axis parabolic mirror based on Eqs. (34), (35), and (3) is identical to that based on other rigorous vector diffraction theories [19–28].

## 6. Conclusions

In this paper, we presented an exact analytical approach for the description of the vector electromagnetic fields formed by an on- and off-axis paraboloidal mirror for circular and square incident beams based on the Stratton-Chu integral. General analytical expressions for the spatial distributions of electromagnetic field components anywhere within and outside of the focus were derived in detail. Using several approximations, including the well-known Debye approximations, the far-field vector diffraction formulas suitable for an OAP were obtained from the general analytical expressions derived herein.

According to our rigorous vector diffraction theory, the characterizations of the tightly focused vector fields formed by off- and on-axis parabolic mirrors for square and circular incident beams with a super-Gaussian top-hat profile polarized along the positive *x* direction were analyzed and discussed in detail. The results show that there are significant differences in vector diffraction characterizations between on- and off-axis parabolic mirrors. For the intensity distribution of the field component ${\left|{E}_{{x}^{\prime}}\right|}^{2}$, which possesses lobes of relatively weak amplitude surrounding a central high-amplitude peak, the width of the central peak of the intensity distribution and that of the side-lobe distributions in the focal plane along the *x'* and *y'* directions widen with an increasing off-axis rate $\alpha $. However, before and after $\alpha =2f/R$ (or$\phi =\pi /2$), the intensity of ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ in the center decreases, and the dark center gradually appears with increasing $\alpha $, until $\alpha =2f/R$. As the value of $\alpha $further increases, the field intensity is enhanced, and the dark center gradually disappears. For the intensity distribution of ${\left|{E}_{{z}^{\prime}}\right|}^{2}$, which also possesses lobes of relatively weak amplitude surrounding a central high-amplitude peak, the width of the central peak and that of the side-lobe distributions shown in the focal plane also widen as $\alpha $ increases overall. However, for $0\le \alpha <0.6$, the intensity distribution of ${\left|{E}_{{z}^{\prime}}\right|}^{2}$has a dark center, and as the value of $\alpha $ increases, the center intensity is enhanced and the dark center gradually disappears. For the intensity distribution of ${\left|{E}_{{y}^{\prime}}\right|}^{2}$, which exhibits a linear dark center along the *x'*-axis direction, the intensity distribution exhibits bilateral symmetry with respect to the *x'* and *y'* axes. Apart from $0\le \alpha <0.4$, two peak intensities are symmetrically distributed along the *y'* axis in the first-ring main lobes around the dark core, and the width of the linear dark center and that of the side-lobe distributions shown in the focal plane along the *y'* direction widen with increasing $\alpha $. Although the intensity distribution of ${\left|{E}_{{y}^{\prime}}\right|}^{2}$ has four peak intensities in the focal plane for $0\le \alpha <0.4$, the intensity has a very small value along the *y'* axis (*x'* = 0). In addition, the influence of the intensity distribution of the magnetic field components on $\alpha $was clearly shown.

We found from our calculations that, for $\alpha <4$ (or $\phi <\pi /4$), the field component ${\left|{E}_{{x}^{\prime}}\right|}^{2}$ is predominant. However, when $\alpha >4$ (or $\phi >\pi /4$), the longitudinal field ${\left|{E}_{{z}^{\prime}}\right|}^{2}$ is predominant, and the maximum peak intensity ratio between the longitudinal field and transverse fields at the focus is approximately 10^{3}. This property is valuable for all applications in which a strong longitudinal field component is desirable.

In addition, we explored the focal depth and convergence angle of a highly confined optical field as well as the EFL of the OAP. We found that the EFL increases with increasing off-axis rate $\alpha $, and the depth of focus and convergence angle of the focused beam are strongly dependent on ${f}^{\prime}/2\omega $: smaller values of ${f}^{\prime}/2\omega $ lead to shorter focal depths and larger beams convergence (or tighter focusing).

## Funding

Fujian Provincial Natural Science Foundation of China (2013J05095, 2017J01560); Science Foundation for the Youth Scholars of Minjiang University (Mj9n201602); National Science and Technology Major Project of the Ministry of Science and Technology of China.

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