1. Introduction

Optical elements with long focal depth and high lateral resolution are required for many applications [1–3]. For conventional optical elements such as parabolic mirrors and spherical lenses, there is a contradiction between long focal depth and high lateral resolution, since a high lateral resolution requires a high numerical aperture and a long focal depth requires a low numerical aperture. To overcome this limitation, various methods such as diffractive axicons [4,5], holographic elements [6] and diffractive optical elements [7,8] have been proposed, which could generate quasi-Bessel beam with both long focal depth and high lateral resolution. However, these refractive or diffractive elements are wavelength dependent and have a low damage threshold, which are not suitable to focus ultra-short laser pulses with broadband spectra and high intensity. To solve this problem, a long-focal-depth mirror with a special profile has been proposed in recent years [9,10]. Since it is a reflective element, it has negligible dependence on wavelength and high damage threshold. Due to these advantages, the long-focal-depth mirror can be applied in ultra-short laser pulse compression [11], laser plasma accelerators [12] and so on.

At present, all experimental and theoretical studies on the long-focal-depth mirrors are focusing on the case of low numerical aperture with NA < 0.45. In this case the paraxial approximation is satisfied, and the scalar diffraction theory such as Fresnel-Kirchoff diffraction integrals [13] and angular spectrum theory [14] are suitable to describe the focusing properties, in which the change of polarization direction can be ignored. Tight focusing with high numerical aperture is an important way to further improve the peak power intensity of laser pulses [15]. It can also be expanded to the long-focal-depth mirrors in principle, but we do not know whether the tight focusing will adversely affect the focusing properties of this mirror. Moreover, with the increase of numerical aperture, the rigorous vector diffraction theory must be used to describe the vector field in focusing region correctly, in which the change of polarization direction must be considered [16,17]. So far, several vector diffraction methods have been proposed, including Unidirectional Hertz vector Propagation Equation (UHPE) [18,19], the Bluestein method [20], the Rayleigh-Sommerfeld integrals [21,22] and the Richards-Wolf integrals [23,24]. Based on the above vector diffraction theories, a large number of studies have been carried out internationally on the focusing properties of different kinds of incident beams passing through parabolic or ellipsoidal mirrors with high numerical aperture [25–28]. But to the best of our knowledge, we haven't found any literature to study the focusing properties of a long-focal-depth mirror with high numerical aperture. Carrying out relevant study can fill this blank area of research and expand the fields of application for the long-focal-depth mirrors.

In this paper, the tight focusing properties of a long-focal-depth mirror are studied by the Richards-Wolf integrals, and it is proved that an accelerating superluminal laser focus can be always generated by a long-focal-depth mirror with high numerical aperture. For different kinds of incident light, we have carried out numerical simulations to obtain the intensity distribution of each polarization component, which reveals that accelerating superluminal laser focus with different intensity distribution can be generated by different kinds of incident light. Compared with those of a parabolic mirror, the axial intensity distributions of each component are obviously different since its properties of long focal depth and tight focusing, but the lateral intensity distributions of each component are always similar. This paper is arranged as follows: Section 2 gives the working principle of the Richard and Wolf integrals and the apodization factor of a long-focal-depth mirror; In Section 3, we prove that a long-focal-depth mirror with high numerical aperture always generates an accelerating superluminal laser focus; Section 4 gives the numerical results of intensity distribution for different kinds of incident light; Eventually the summary is made in section 5.

2. Theory of simulations

2.1 Richards-Wolf integrals for a long-focal-depth mirror

Figure 1 gives the basic principle of the tight focusing of a long-focal-depth mirror with high numerical aperture. In the case of tight focusing, the change of polarization direction before and after the light is reflected by the mirror must be considered. As shown in Fig. 1, after the light $\overrightarrow {MS}$ is perpendicularly incident on a point S on the mirror, the light is reflected by the mirror and intersects with the optical axis z at point N. Then, $\overline {SN}$ is used to stand for the distance between point S and point N, and $\overrightarrow s = \overrightarrow {SN} /\overline {SN}$ is used to stand for the unit vector of the propagation direction of reflected light. Since a long-focal-depth mirror focuses the incident light at different positions at different focal points, different position S also corresponds to different focal length $\overline {ON}$. In Fig. 1, the vectors of the incident light and the reflected light are also marked, which need to be used in the vector field simulations. For example, ${\overrightarrow e _0}$ and ${\overrightarrow e _1}$ represent the polarization directions of incident light and reflected light respectively;${\overrightarrow g _0}$ is unit vector both perpendicular to the light $\overrightarrow {MS}$ and ${\overrightarrow e _0}$, and ${\overrightarrow g _1}$ is unit vector both perpendicular to the light $\overrightarrow {SN}$ and ${\overrightarrow e _1}$; θ is the convergence angle for point S; r is the radial component of point S in cylindrical coordinates; R is the radius of the mirror; and P (r_p, φ_p, z_p) is a point in focal range, whose vector field is need to be calculated.

 

Fig. 1. The basic principle of the tight focusing of a long-focal-depth mirror with high numerical aperture.

Download Full Size | PDF

In this paper, the Richards-Wolf vector integrals are used for numerical simulations, which can not only describe the electromagnetic field for the optical system with low numerical aperture, but also accurately solve the three-dimensional vector field for the optical system with large numerical aperture. In the Richards-Wolf vector integrals, the electric field of point P in the focal region can be written as:

Here, $k = 2\pi /\lambda$ is wave vector; $\lambda$ is the wavelength of incident light; $\vec{A}(S)$ represents the electric field vector of the incident light at point S after reflection; $\overrightarrow s =$ (s_x, s_y, s_z) is the unit vector of the propagation direction of reflected light, which has been mentioned above; $\overrightarrow r = \overrightarrow {SP}$ is the vector from point S to point P; $d\Omega = d{s_x}d{s_y}/{s_z}$ represents the solid angle for integration. The physical meaning of this integration formula is that the electric field vector at point P near the focal region is equal to the vector superposition of the electric field vectors at each point S on the mirror.

In the Eq. (1), the electric field vector of the light at point S after reflection $\vec{A}(S)$ can be expressed as:

Here, $\Phi ({s_x},{s_y}) = \overline {MS} \cdot 2\pi /\lambda$ represents the initial phase of $\vec{A}(S)$ at the point S; $\overline {\textrm{O}N}$ is the focal length of the reflected light which focuses on the position of point N; ${\overrightarrow e _1}$ indicates the polarization direction of the reflected light, which is closely related to the polarization direction of incident light ${\overrightarrow e _0}$; ${L_1}$ is the amplitude of the reflected light at the point S; ${L_0}$ is the amplitude of the incident light at the point S; $L(\theta )$ is the apodization factor, which describes the change of light energy in the process of incident light to reflected light.

As shown in Eq. (1) and Eq. (2), many parameters mentioned above are related to the mirror structure, such as θ, $\overrightarrow s$, $\Phi ({s_x},{s_y})$, $\overline {\textrm{O}N}$, $L(\theta )$ and so on. In this paper, the structure of the mirror can be expressed as z = f(r, φ)=f(x, y). Here (r, φ) is the value in cylindrical coordinate system, and (x, y) is the value in Cartesian coordinate system. The relationship of two coordinate systems is x = rcosφ, and y = rsinφ. Figure 2 shows the geometric relationship required to obtain the value of above parameters, in which $\vec{n}$ represents the normal vector of the infinitesimal reflection element on point S. These relationships are $\beta (r) = \arctan (1/z^{\prime}(r))$, $\alpha (r) = \pi /2 - \beta (r)$, $\theta (r) = \pi - 2\beta (r)$, ${k_1}(r) = \tan (\theta (r))$ and ${z_N}(r) ={-} [{r - {k_1}\textrm{(}r\textrm{)} \cdot \textrm{z(}r\textrm{)}} ]/{k_1}\textrm{(}r\textrm{)}$, in which all these parameters can be regarded as functions of radius r. Here, z’(r) represents the derivative of z(r); z(r) is the curve structure of the long-focal-depth mirror; β is the grazing incidence angle of incident light; α is the incident (or reflected) angle of incident (or reflected) light; θ is the convergence angle as mentioned above; k₁ is the slope of $\overline {SN}$; r is the radial component of point S; and z_N is the position of point N. Besides, it also has $\overline {MS} = \overline {AB} - z(r)$, $\overline {SN} = \sqrt {({z_S} - {z_N}) - ({r_S} - {r_N})}$, $\overline {AB} = z(R) = z(D/2)$, and $\overline {\textrm{O}N} = {z_N}(r)$.

 

Fig. 2. The geometric relationship required for numerical calculation.

Download Full Size | PDF

If the values of ${L_0}$, $L(\theta )$ and ${\overrightarrow e _1}$ are known, and the above parameters calculated with mirror structure z = f(x, y) are brought into Eq. (1) and Eq. (2), we can obtain the electric field vector of a point P in the focal region by numerical calculations of the Richards-Wolf integrals. In this paper, it is assumed that the incident light is uniformly distributed, which means ${L_0}\textrm{ = }1$. The apodization factor $L(\theta )$ will be discussed in detail in the following section. Since the expression of the polarization direction for the reflected light ${\overrightarrow e _1}$ is quite different for the incident light with different polarization ${\overrightarrow e _0}$, the polarization directions of linearly polarized (LP), radially polarized (RP) and azimuthal polarized (AP) light in the Cartesian coordinate system (x, y, z) after reflection are calculated and given below. The focusing characteristics of different polarized light after passing through the long-focal-depth mirror will be discussed in detail later in the numerical simulation section.

For the light linearly polarized (LP) along the x-axis, the polarization components are:

If the mirror is illuminated by radially polarized (RP) light, the polarization components are rewritten as:

In the case of azimuthally polarized (AP) incident light, the polarization components are expressed as:

Here, θ is the convergence angle corresponding to point S; φ is the angular component of point S in the cylindrical coordinate system; and ${\vec{e}_x}$, ${\vec{e}_y}$, ${\vec{e}_z}$ are the unit vector of x, y, z axis respectively.

2.2 Surface structure of a long-focal-depth mirror

In this paper, a long-focal-depth mirror with high numerical aperture is designed and the tight focusing properties of this mirror are studied for different kinds of incident light. The beam aperture of this mirror is D = 400 mm, the focal length is f = 200 mm, and the focal depth is Δf = 20 mm. Therefore, the numerical aperture of this mirror is NA = 0.707. Here we choose NA = 0.707 just as an example, which meets the condition of high numerical aperture. It is also possible to choose other values for NA. With the increase of NA, the proportion of other components different from the incident light will increase, which is just like the behavior of a parabolic mirror [29]. In order to obtain the mirror with the above parameters, we use the method in the literature [9] to design the surface structure, and the results (red line) are shown in Fig. 3(a). This method is based on the principle of energy conservation and equal optical path, and will not be described in detail here. Figure 3(a) also gives the curve of a parabolic mirror with f = 200 mm (green line), it can be seen that on the whole, the difference between the two structures is not large. Figure 3(b) shows the subtraction of the two mirrors, and it can be seen that the maximum difference between the two mirrors is about 0.7 mm.

 

Fig. 3. (a) The curves of a long-focal-depth mirror and a parabolic mirror with similar parameters; (b) The difference of the surface structure between the two mirrors.

Download Full Size | PDF

Since the mirror structure cannot be expressed analytically, we use the higher even order aspherical equation to fit its surface structure. The higher even order aspherical equation can be expressed as:

The fitting parameters of a long-focal-depth mirror with different parameters (such as D, f, Δf) can be obtained through formula (6). To verify the tight focusing characteristics of a long-focal-depth mirror with high numerical aperture, we just take a long-focal-depth with f = 200 mm and Δf = 20 mm as an example to carry out our numerical simulations in Section 4. In order to obtain the influence of different structural parameters on the apodization factor and the velocity of laser focus, we have also compared the results with those of other structures in the corresponding sections. In Table 1, we have completely given the fitting parameters of the long-focal-depth mirrors with different parameters, and all of these structures are used for comparisons and simulations in this paper.

Table 1. The fitting parameters of the long-focal-depth mirror with different parameters

View Table

2.3 Apodization factor of a long-focal-depth mirror

Apodization factor is an important parameter for the tight focusing, which describes the relationship between the amplitude before reflection ${L_0}$ and the amplitude after reflection ${L_1}$. Below, we have calculated the apodization factor of the long-focal-depth mirror and compared it with the apodization factor of a parabolic mirror. The calculation method of the apodization factor is given below. For the tight focusing of a long-focal-depth mirror, the incident light is assumed to be a plane wave when apodization factor is discussed. According to the law of energy conservation, the relationship between ${L_0}$ and ${L_1}$ is ${L_1}^2\Delta {S_1} = {L_0}^2\Delta {S_0}$, where $\Delta {S_0}$ is the infinitesimal plane element of the incident beam in front of the mirror, and accordingly we define $\Delta {S_1}$ as the infinitesimal plane element of the beam in front of the converging wave after reflect by the mirror. Based on the geometric relationships, we can get the expressions of $\Delta {S_0} = 2\pi r\Delta r$ and $\Delta {S_1} = 2\pi {\overline {\textrm{O}N} ^2}\sin \theta \Delta \theta$, respectively. After calculations with the above expressions and simplification using the geometric relationship in Fig. 2, the expression of apodization factor is as follows:

Here, z(r) is the curve structure of the long-focal-depth mirror, which can be expressed by Eq. (6); θ(r) is the convergence angle corresponding to point S, and according to the geometric relationship in Fig. 2 there is $\theta (r) = \pi - 2\arctan ({{1 / {z^{\prime}(r)}}} )$; z’(r) represents the derivative of z(r); and θ’(r) represents the derivative of θ(r). Since the expressions of z(r) and θ(r) are complex, it is difficult to calculate the derivative of θ(r) analytically. Therefore, we use the numerical solution of above formula to obtain the apodization factor of a long-focal-depth mirror. It is worth noting that the apodization factor of a parabolic mirror can also be calculated by formula (7). For a parabolic mirror with $z(r) = {{{r^2}} / {({4{f_0}} )}}$, there are $\theta (r) = \pi - 2\arctan ({{{2{f_0}} / r}} )$, and ${{[{r - \tan (\theta (r)) \cdot z(r)} ]} / {\tan (\theta (r))}} = {f_0}$. Transforming the former formula of θ(r), we can get $r(\theta ) = {{2{f_0}} / {\tan({{{({\pi - \theta } )} / 2}} )}} = {{2{f_0}\sin \theta } / {({1 + \cos \theta } )}}$. Then it has $\theta ^{\prime}(r)\textrm{ = 1/}r^{\prime}(\theta ) = {{({1 + \cos \theta } )} / {(2{f_0})}}$. Therefore, the apodization factor of a parabolic mirror can be expressed as $L(\theta )\textrm{ = }\sqrt {\textrm{1/}f_0^2 \cdot {{2{f_0}\sin \theta } / {({1 + \cos \theta } )\cdot {{2{f_0}} / {({\sin \theta ({1 + \cos \theta } )} )}}}}} = {2 / {({1 + \cos \theta } )}}$.

As shown in Fig. 4(a), the apodization factor of three mirror structures are calculated by formula (7). These mirrors are a long-focal-depth mirror with f = 200 mm and Δf = 20 mm, a long-focal-depth mirror with f = 200 mm and Δf = 10 mm, and a parabolic mirror with f = 200 mm, and all of these mirrors have the same beam aperture D = 400 mm. In Fig. 4(a), L(θ) will be close to 1 if θ is small enough to 0, which means the energy density remains constant before and after reflection. But if L(θ) is greater than 1, the incident wave in an infinite small area will be compressed after reflection, that is, the energy density will increase. By comparisons of the three mirrors with different structures in Fig. 4(a), we can conclude that the apodization factor of a long-focal-depth mirror is always larger than a parabolic mirror with similar parameters, and it gradually approaches that of the parabolic mirror with the decreasing of its focal depth Δf. To show this feature more clearly, Fig. 4(b) gives the apodization factor of a long-focal-depth mirror with similar parameters with the variation of focal depth Δf for a fixed convergence angle θ=0.7. In Fig. 4(b), it should be noted that the circle point with Δf = 0 represents the apodization factor of a parabolic mirror. It is obvious from Fig. 4(b) that the apodization factor of a long-focal-depth mirror decreases nearly linearly with the decrease of its focal depth Δf.

 

Fig. 4. (a) The variation of apodization factor with θ for different mirrors; (b) The variation of apodization factor with Δf for a fixed angle θ=0.7.

Download Full Size | PDF

3. Velocity of laser focus

In this section, we prove that a long-focal-depth mirror with high numerical aperture always generates an accelerating superluminal laser focus, and the degree of acceleration increases with the increase of numerical aperture. According to the geometric relationship we obtained in Fig. 2, the time to reach any point N in focal region can be expressed as $T = {( \overline {MS} + \overline {SN} )} \left/ {c}\right.$, and the focal length for any point N is known as $\overline {ON}$. Therefore, the velocity of laser focus along the z direction can be expressed as:

Figure 5 gives the numerical results of the velocity of laser focus for long-focal-depth mirrors with different parameters. It can be seen that the velocity of laser focus v_f always increases approximately linearly with the increase of distance z from the speed of light, which indicates that the long-focal-depth mirror always generates an accelerating beam. Figure 5(a) shows the variation of v_f for different Δf and a fixed f, and it can be seen that the increase of v_f in focal region is basically the same for a fixed f (or a fixed NA). This implies that the value of NA (namely D and f) mainly determines the increase of v_f in focal region. Figure 5(b) shows the variation of v_f for different f and a fixed Δf, and it can be seen that the value of v_f increases significantly with the increase of NA. This implies that the degree of acceleration becomes more and more obvious with the increase of NA. For low numerical aperture, the increase of v_f can be almost ignored. But for high numerical aperture with NA = 0.707, the maximum value of v_f is about 1.6 times the speed of light c, which is obviously different from that under low numerical aperture (${v_f} \approx c$).

 

Fig. 5. The propagation velocity of focal spot along the z direction. (a) For different values of focal depth (Δf) at f = 200 mm; (b) For different values of focal length (f) at Δf = 20 mm.

Download Full Size | PDF

The superluminal laser focus obtained by a long-focal-depth mirror has different characteristics from that obtained by the dispersive focusing optic with a chirped laser pulse [30,31]. For the latter, the dispersive focusing optic mainly determines the focal depth (Δf), and the velocity of laser focus (v_f) can be adjusted by controlling the group delay dispersion (GDD) of a chirped laser pulse, and the value of v_f can reach hundreds of times of c when GDD is appropriate. In addition, the velocity of laser focus is generally constant in the whole range of focal depth, and there is an internal relationship between the pulse duration and the velocity of laser focus, as shown in Fig. 4 of Ref. [30] and in Fig. 3 of Ref. [31]. But for the former, the superluminal laser focus is only determined by the structure of the mirror, and it is always approximately linearly accelerated. In addition, the pulse duration of laser focus is basically the same as that of the incident laser. To further control the velocity of laser focus obtained by a long-focal-depth mirror, it is necessary to add a stepped mirror (or a stepped echelon) with appropriate design in front of the long-focal-depth mirror, which has been discussed in Refs. [32,12]. The accelerating superluminal laser focus generated by the long-focal-depth mirror under the tight focusing condition has potential applications in the strong field physics.

4. Numerical results for different kinds of incident light

For different kinds of incident light, accelerating superluminal laser focus with different intensity distributions can be generated by the long-focal-depth mirror with high numerical aperture. In this section, the intensity distributions of laser focus for different kinds of incident lasers are simulated and discussed, and the Richards-Wolf vector diffraction theory has been used to carry out relevant studies, in which the theory of simulations has been introduced in Section 2. In all the following simulations, the parameters of mirror with D = 400 mm, f = 200 mm and Δf = 20 mm (NA = 0.707) in Table 1 are selected, and the wavelength of the incident light is set as λ=1µm. Five kinds of incident light have been considered and discussed, and these are linearly polarized light, radially polarized light, azimuthally polarized light, linearly polarized light with spiral phase, and linearly polarized light with ultrashort pulses. Through comparisons of numerical results, we can see that the focusing properties of different kinds of incident light are obviously different. It should be noted that, since the structure of the mirror has been determined as fixed values, it should always be an accelerating superluminal laser focus in the following simulations, and its velocity at the tail of the focal region is about 1.6c for every kind of incident light. For a long pulse with pulse duration much greater than the focal depth of this mirror, this acceleration feature is not easy to see. However, it can be seen clearly in the case of ultrashort pulse.

4.1 Linearly polarized light along the x-axis

If the incident light is the linearly polarized light along the x-axis, the vector field of the reflected light ${\overrightarrow e _1}$ can be expressed in formula (3). The intensity distribution of each component in the focal region can be obtained by the Richards-Wolf vector diffraction theory, and the numerical results of each component |E_x|², |E_y|², |E_z|² and the total intensity distribution |E|² have been shown in Fig. 6.

 

Fig. 6. The intensity distribution and focal spot size of the linearly polarized incident light (along the x-axis) in the focal range. (a) The intensity distributions of |E_x|², |E_y|², |E_z|², |E|² in the X-Y plane at z = 200 mm, and the intensity distributions of |E_x|², |E_y|², |E_z|², |E|² in the Z-X plane. (b) The 3D distribution of the total intensity |E|² focused by the long-focal-depth mirror; (c) The normalized focal spot for every component with the variation of position z, in which the second-order intensity moments ($\langle$ω_x²(z)$\rangle$ and $\langle$ω_y²(z)$\rangle$) are used to define the size of focal spot and their values at each position are normalized by $\langle$ω_x²(z = 200 mm)$\rangle$ and $\langle$ω_y²(z = 200 mm)$\rangle$.

Download Full Size | PDF

Due to tight focusing, the long-focal-depth mirror changes part of the incident light to other polarization directions. For the linearly polarized light along x-axis (E_x), the reflected light polarized in y direction and z direction will be generated, where it marked as E_y and E_z. Figure 6(a) shows the intensity distributions of |E_x|², |E_y|², |E_z|² and |E|² in the X-Y plane at z = 200 mm (left) and in the Z-X plane at y = 0 (right), and it has |E|²=|E_x|²+|E_y|²+|E_z|². For linearly polarized light along the x-axis, the simulation results of the intensity distribution of the E_x component can also be approximately regarded as the simulation results of the total light intensity by using the scalar diffraction theory, because the scalar diffraction theory does not consider the change of the polarization direction of the incident light. Through comparisons, the simulation results of two theories are obviously different. And as we know, vector diffraction theory is more accurate. In the X-Y plane at z = 200 mm, it has I_{z_max}= 0.18I_{x_max} and I_{y_max}= 4 × 10⁻³I_{x_max}, where I_{x_max}, I_{y_max} and I_{z_max} represent the maximum value of |E_x|², |E_y|² and E_z|² in this plane, respectively. From the left of Fig. 6(a), it can be seen that a long-focal-depth mirror and a parabolic mirror with the same numerical aperture have the similar lateral distribution for each component, and the tight focusing properties of a parabolic mirror with different numerical aperture (or f-number) can be found in Fig. 4 of the Ref. [29]. From the right of Fig. 6(a), we can see that the focal depth obtained by numerical calculation is 20 mm, which is consistent with the design value. This shows that the long-focal-depth mirror can also work well in the case of tight focusing. The yellow solid line gives the intensity distribution along the white dashed line of |E|², and there is high frequency oscillation in the intensity distribution along z axis. As discussed in Ref. [33], the oscillation is mainly caused by interference of incident light, and it can be eliminated by the use of the apodization or incoherent light.

Figure 6(b) gives the three-dimensional intensity profile of the total vector field |E|², in which the lateral intensity of the focal spot at different positions are shown clearly. From Fig. 5(b), it can be seen that the light field always maintains a small focal spot in the whole focal depth (Δf = 20 mm), and the lateral resolution becomes better as the distance z increases. To further quantitatively display this feature described above, Fig. 6(c) gives the normalized size of focal spot for every component (|E_x|², |E_y|², |E_z|²) with the variation of position z. Here, the second-order intensity moments ($\langle$ω_x²(z)$\rangle$ and $\langle$ω_y²(z)$\rangle$) are used to define the size of focal spot in the horizontal and the vertical directions respectively, and their definitions can be found in the Ref. [34]; And the focal spot sizes at each position are normalized by the focal spot size at z = 200 mm for every component. For |E_x|², the second-order intensity moments are $\langle$ω_x²(z)$\rangle$= 0.22µm² and $\langle$ω_y²(z)$\rangle$= 0.20µm² at z = 200 mm. For |E_y|², the size of focal spot are $\langle$ω_x²(z)$\rangle$= 1.41µm² and $\langle$ω_y²(z)$\rangle$= 1.41µm² at z = 200 mm. For |E_z|², the second-order intensity moments are $\langle$ω_x²(z)$\rangle$= 0.99µm² and $\langle$ω_y²(z)$\rangle$= 0.41µm² at z = 200 mm. From Fig. 6(c), we can clearly see that the focal spot size decreases very quickly in the first half of the focal depth; However, the focal spot size decreases very slowly in the second half of the focal depth. Moreover, the change trend of focal spot size of the three components is basically consistent, especially in the second half of the focal depth.

4.2 Radially polarized light and azimuthally polarized light

If the incident light is the radially polarized light and the azimuthally polarized light, the vector field of the reflected light ${\overrightarrow e _1}$ can be expressed in formula (4) and (5) respectively. As mentioned above, the intensity distribution of each component can also be obtained by the Richards-Wolf vector diffraction theory. Since it is easier to represent the radially polarized light and the azimuthally polarized light in the cylindrical coordinate system, we convert the three-dimensional vector field calculated by simulations from the cartesian coordinate system to the cylindrical coordinate system. The relationship between two coordinate systems is shown in Fig. 7, which can be expressed in Eq. (9).

 

Fig. 7. The relationship between cartesian coordinate system and cylindrical coordinate system.

Download Full Size | PDF

Here, the value of every point P (E_x, E_y, E_z) in cartesian coordinate system can be calculated by the numerical simulations. Then the corresponding P (E_ρ, E_φ, E_z) in cylindrical coordinate system can be calculated by combining Eq. (9) and Eq. (10).

For the radially polarized light and the azimuthally polarized light, the numerical results of each component |E_ρ|², |E_φ|², |E_z|² and the total intensity distribution |E|² have been shown in Fig. 8(a) and Fig. 8(b), respectively. And the intensity distribution of the E_ρ component and the E_φ component in Fig. 8(a) and Fig. 8(b) can also be approximately regarded as the simulation results of the total light intensity of radially polarized light and azimuthally polarized light by using the scalar diffraction theory, respectively. Compared with the results of Fig. 6, we can see that the intensity distribution of the vector field in focal range changes greatly by changing the polarization of incident light. As shown in Fig. 8(a), after the radially polarized beam is tightly focused by the mirror, the light field in focal range has only radial component |E_ρ|² and axial component |E_z|², and there is no angular component |E_φ|². For the radial component |E_ρ|², it is a hollow beam, whose light intensity is basically evenly distributed along the z-axis, with the focal depth of 20 mm consistent with the design, and the spot size decreases with the increase of distance. For the axial component |E_z|², it is a solid beam, whose light intensity increases gradually along the z-axis, and it has I_{z_max} / I_{ρ_max} ≈ 5, which means most of the energy of the incident radially polarized light is transferred to the z component |E_z|². Here, I_{z_max} is the maximum of |E_z|², and I_{ρ_max} is the maximum of |E_ρ|². Since it has |E|²=|E_ρ|²+|E_φ|²+|E_z|², the total intensity distribution |E|² is a hollow conical beam that changes from hollow to solid as the z-axis increases, which is very different from the focusing properties of a parabolic mirror for the radially polarized light [35–37]. As shown in Fig. 8(b), after the angular polarized incident light is tightly focused by the mirror, there is only the angular component |E_φ|², and the hollow beam with basically evenly distribution is formed. The hollow beam with azimuthal polarization also has a focal depth of 20 mm, and the spot size also decreases with the increase of distance. The simulation results show that, for the angular polarized incident light, there is no transfer of beam energy to other polarization directions during the tight focusing of a long-focal-depth mirror.

 

Fig. 8. (a) The intensity distribution of radially polarized light in cylindrical coordinate system. The intensity distributions of |Eρ|², |Eφ|², |Ez|², |E|² in the X-Y plane at z = 200 mm. The intensity distributions of |E_ρ|², |E_φ|², |E_z|², |E|² in the Z-X plane. (b) The intensity distribution of angularly polarized light in cylindrical coordinate system. The intensity distributions of |E_ρ|², |E_φ|², |E_z|², |E|² in the X-Y plane at z = 200 mm. The intensity distributions of |E_ρ|², |E_φ|², |E_z|², |E|² in the Z-X plane.

Download Full Size | PDF

4.3 Linearly polarized light with spiral phase

If the incident light is the linearly polarized light with spiral phase, the vector field of the reflected light ${\overrightarrow e _1}$ can be expressed in formula (3) by multiplying the spiral phase exp(ipφ), where p represents topological charge and φ represents azimuth. As mentioned above, the intensity distribution of each component |E_x|², |E_y|², |E_z|² and the total intensity of the vector field |E|² can also be obtained by the Richards-Wolf vector diffraction theory.

Figure 9 gives the numerical results with different topological charges (p = 1, 2) for the intensity distribution of |E_x|², |E_y|², |E_z|² in the X-Y plane at z = 200 mm and |E|² in the Z-X plane. And the intensity distribution of the E_x component in Fig. 9(a) and Fig. 9(b) can also be approximately regarded as the simulation results of the total light intensity of linearly polarized light with spiral phase by using the scalar diffraction theory. Compared with the E_x component of Fig. 6(a), the effect of spiral phase is to convert a solid beam with x polarization into a hollow beam with x polarization. For |E_x|², the intensity is basically evenly distributed along the z-axis, the spot size decreases with the increase of distance, and the focal depth in simulation is 20 mm as well, which is consistent with the design. For |E_y|² and |E_z|², the intensity of each increase gradually along the z-axis, and we can find that the value of |E_z|² is much larger than that of |E_y|². For the linearly polarized light with p = 1 in Fig. 9(a), the |E_y|² component is a hollow beam and the |E_z|² component is a solid beam. But for the linearly polarized light with p = 2 in Fig. 9(b), the |E_y|² component is a solid beam and the |E_z|² component is a hollow beam. Since the intensity of |E_z|² is much larger than that of |E_y|² in both case of p = 1 and 2, it makes the total intensity of the vector field |E|² a hollow conical beam for p = 1, but still a hollow beam for p = 2. Compared with X-ray vortices generated by spiral photon sieves and spiral zone plates [38,39], optical vortices produced by the long-focal-depth mirror with high numerical aperture have strong other polarization components, such as |E_y|² and |E_z|². However, spiral photon sieves and spiral zone plates work under the condition that the scalar diffraction theory is applicable, in which they do not change the polarization direction of the incident light.

 

Fig. 9. (a) and (b) Topological charge p = 1 and 2. The intensity distributions of |E_x|², |E_y|², |E_z|² and |E|² in the X-Y plane at z = 200 mm and intensity distributions of |E_x|², |E_y|², |E_z|² and |E|² in the Z-X plane.

Download Full Size | PDF

4.4 Linearly polarized light with ultrashort pulse

If the incident light is the linearly polarized light with ultrashort pulse, the vector field of the reflected light ${\overrightarrow e _1}$ can be expressed in formula (3) by further multiplying the time-dependent correction of its amplitude for each point S, which can be expressed as ${E_T}(t) = \sqrt {{I_T}(t - L/c)} \exp ( - i\omega t)$. Here, t is the moment for simulation; $L = \overline {MS} + \overline {SP}$ is the delay length of a point P for a point S on the mirror; and ${I_T}(t - L/c) = \exp [{{{ - {{({t - L/c} )}^2}} / {2{\sigma^2}}}} ]$ is the distribution of pulse duration, which is Gaussian distribution. As mentioned above, the intensity distribution of focal spot at every moment can also be obtained by the Richards-Wolf vector diffraction theory. In the following simulations, we assume that the pulse duration (half height and full width) of incident light is 471fs (σ=200fs), and the moment when the focal spot reaches z = 200 mm is set to t = 0ps.

Figure 10 shows the evolution of ultrashort pulse with time in the focal region, and we can see that the focal spot of each component propagates along the z axis with the increase of time, which always keeps a small focal spot that does not diverge. Five moments are considered, and these are t = -20ps, -10ps, 0ps, 10ps, and 20ps. As shown in Fig. 10(a)-(c), the peak intensity of |E_x|² and |E|² remains basically unchanged with the increase of distance z, the peak intensity of |E_z|² increases linearly with the increase of distance z, and these two components have exactly the same propagation velocity. Compared with Fig. 6(a), we can find that these characteristics are consistent for both long pulses and ultrashort pulses. It implies that the intensity distribution in focal region is mainly determined by the structure of mirror and the effect of ultrashort pulse is only to shorten the pulse duration of focal spot. Figure 10(d) gives the spatial position of peak intensity of focal spot at different times, and the agreement between analytical formula and numerical simulation implies that this mirror does produce accelerating superluminal laser focus, which can be more clearly demonstrated in the case of ultrashort pulses. Figure 10(e) gives the pulse duration of focal spot at different times, and we can see that the pulse duration of focal spot will gradually widen with the increase of evolution time, in which the pulse duration of focal spot at t = 20ps becomes 1.4 times of the initial value.

 

Fig. 10. (a)-(c) The intensity distributions in the Z-X plane at different times for |E_x|², |E_z|² and |E|², respectively; (d) The spatial position of peak intensity of focal spot at different times, the red square is numerical results from simulation by Richards-Wolf integrals, and the dashed line is analytical results from formula (8); (e) The pulse duration of focal spot at different times, and ‘Initial’ stands for the pulse duration of incident light.

Download Full Size | PDF

5. Summary

In this paper, the tight focusing properties of a long-focal-depth mirror with NA = 0.707 are studied by the Richards-Wolf integrals, and it is proved that an accelerating superluminal laser focus can be always generated by a long-focal-depth mirror with high numerical aperture, in which the degree of acceleration increases with the increase of its numerical aperture. And for NA = 0.707, the velocity of laser focus (v_f) increases approximately linearly from c to 1.6c in focal region. The apodization factor of a long-focal-depth mirror is also obtained numerically, which is always larger than that of a parabolic mirror for the same value of θ, and gradually approaches the value of a parabolic mirror with the decreasing of its focal depth Δf. In all the simulations, we used a long-focal-depth mirror with the same parameters such as D = 400 mm, f = 200 mm, and Δf = 20 mm. And five kinds of incident light have been considered in the simulations. These are linearly polarized light, radially polarized light, azimuthally polarized light, linearly polarized light with spiral phase and linearly polarized light with ultrashort pulse. The intensity distribution of the total intensity distribution and its each component for every kind of incident light are given in Fig. 6, Fig. 8, Fig. 9 and Fig. 10. And as shown in Fig. 10, the acceleration feature of this mirror under tight focusing condition can be clearly seen in the case of ultrashort pulse.

From comparisons of these results for different kinds of incident light, we can conclude that the lateral intensity distribution of each component is basically determined by incident light, and they are always similar to those of a parabolic mirror; Besides, the intensity distribution along z-axis for each component is closely determined by the structure of a long-focal-depth mirror, which lead to the generation of a long-focal-depth field with acceleration feature. However, the axial distribution along z-axis for different polarization component is different. For the component with the same polarization as the incident light, the axial intensity distribution of a long-focal-depth mirror is always similar to that obtained in the case of low numerical aperture, in which they are basically evenly distributed along the z axis. But for other component that different from incident light, the axial intensity distribution is always gradually increasing with the increase of the distance z, which can be mainly attributed to the influence of tight focusing. Under the combined influence of incident light and the tight focusing, it may produce an accelerating beam with special structure, such as the hollow conical beam produced in Fig. 8(a) and Fig. 9(a). And these accelerating beams with special structure may have potential applications in the intense field physics. We believe that our studies can provide a valuable reference for the application of the long-focal-depth mirror in the case of tight focusing.