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Abstract
The axiparabola is a novel reflective element proposed in recent years, which can generate a long focal line with high peak intensity, and has important applications in laser plasma accelerators. The off-axis design of an axiparabola has the advantage of separating the focus from incident rays. However, an off-axis axiparabola designed by the current method always produces a curved focal line. In this paper, we propose a new method to design its surface by combining geometric optics design and diffraction optics correction, which can effectively convert a curved focal line into a straight foal line. We reveal that the geometric optics design inevitably introduces an inclined wavefront, which leads to the bending of the focal line. To compensate for the tilt wavefront, we use an annealing algorithm to further correct the surface through diffraction integral operation. We also carry out numerical simulation verification based on scalar diffraction theory, which proves that the surface of this off-axis mirror designed by this method can always obtain a straight focal line. This new method has wide applicability in an axiparabola with any off-axis angle.
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1. Introduction
The long-focal-depth mirror is a reflective optical element proposed in recent years to generate quasi-Bessel beam with high peak intensity [1–4]. This reflective element essentially combines the long focal depth of an axicon and the high-energy concentration of a parabolic mirror. Therefore, in analogy to an “axilens”, K. Smartsev et al. further refers to it as an “axiparabola” [1]. The axiparabola not only solves the contradiction between long focal depth and high lateral resolution compared with conventional optical elements such as parabolic mirrors and spherical lenses, but also has the advantages of negligible dependence on wavelength and high damage threshold since it has reflective structure. When used with high-power ultra-short laser pulses, it can generate a long focal line with high peak intensity. This property makes it a promising tool for generating low-density plasma channels, which are necessary for achieving high-energy electrons in laser plasma accelerators [5,6]. Combined with a radial echelon to control the relative timing of each annulus, it can be used to produce an achromatic (or colorless) flying focus with arbitrary velocity [7,8], which has important applications in dephasingless wakefield acceleration [8–10] and diffractionless wakefield acceleration with an all-optical plasma waveguide [5,11]. Moreover, the axiparabola has many potential applications in other fields, such as THz generation [12], lightning control [13,14], ultrashort laser pulse compression [15], and extending the length of plasma filaments generated by intense laser pulses [16].
The off-axis design of a focusing mirror has the advantages of separating the focus from the incident rays. Therefore, the off-axis parabolic mirrors have been widely used to focus laser in many physics experiments [17–19]. However, the off-axis axiparabola obtained by the current design method always produces a curved focal line [1]. Moreover, the larger the off-axis angle, the larger the curvature is. Therefore, the axiparabola is always designed with on-axis case or a small value of off-axis angle at present [20,21], which limits the application of this element in some specific requirements of experimental layout. Compared with the on-axis case, the off-axis axiparabola does not have rotational symmetry, which leads it difficult to accurately describe its surface structure. The current design method is to perform numerical optimization with the local optimization tool of the Zemax software, which is according to geometrical optics. However, the off-axis axiparabola designed by this method inevitably has a curved focal line. It is generally believed that the cause of the bending of focal line is that the optimized off-axis surface did not focus all the rays precisely to their targets. If the off-axis axiparabola can be corrected to produce a straight focal line as the on-axis case, it will further expand the practical application of this novel element.
In this paper, we further obtain the exact solution of the off-axis axiparabola by solving the partial differential equation it satisfies according to geometrical optics. However, we find that even the exact solution based on geometric optics will have the curvature of the focal line when the intensity distribution of focal line is simulated by diffraction theory. Through analysis, we reveal that the real reason for this curvature is that the surface obtained by geometric optics inevitably introduces a tilted wavefront, which cannot be reflected in the partial differential equation. We prove that a straight focal line can be obtained if the incident light has the matched wavefront correction. This corrected wavefront can be produced by beam shaping, and a deformable mirror [22,23] may be a feasible technology. Furthermore, we use annealing algorithm [24] to optimize the surface of this mirror with diffraction theory to contain the wavefront correction in itself. The design method we proposed combines geometric optics design and diffraction optics correction, which has wide applicability in the design of an off-axis axiparabola. This paper is arranged as follows: Section 2 introduces the equation it satisfies, the tilted wavefront it exists, and the correction method by annealing algorithm; In Section 3, we perform numerical simulation verification based on scalar diffraction theory; Eventually, the summary is made in section 4.
2. Geometric optical design of an off-axis axiparabola
2.1 Partial differential equation satisfied by an off-axis axiparabola
To obtain the surface of an off-axis axiparabola accurately, we derive the partial differential equation that the surface needs to satisfy according to the principle of geometric optics. The geometrical parameters and schematic presentation of rays focused by an off-axis axiparabola is shown in Fig. 1. In the off-axis case, the incident light along the axis z is focused on the axis z’ by this mirror, and different annulus of incident light are focused at different focal points.
Fig. 1. The geometrical parameters and schematic presentation of rays focused by an off-axis axiparabola.
Assuming θ is the off-axis angle of an off-axis axiparabola, the relationship between the two spatial coordinates can be expressed as $x ={-} z^{\prime}\sin \theta + x^{\prime}\cos \theta$, $y = y^{\prime}$, and $z = z^{\prime}\cos \theta + x^{\prime}\sin \theta$. If the light incident at point S (xs, ys, zs) on the mirror is reflected to the focal point N (xf, yf, zf), its focal length f and the coordinate of focal point N should meet the following conditions:
Here, R is the radius of the incoming beam. Rays initially on the optical axis (r = 0) are focused at f0, while rays initially on the aperture’s edge (r = R) are focused at $z = {f_0} + \delta {z_g}$; therefore, $|{\delta {z_g}} |$ is the focal depth length. Note that the sign of δzg can be either positive or negative. In the case of δzg > 0, which is illustrated in Fig. 1, rays with small r are focused closer to the optics than rays with large r, while the opposite occurs for δzg < 0. In general, f(xs, ys) could be chosen to lead to nonconstant intensity of the focal line. Here, we are interested in a constant case, which is obtained by assuming a uniform circular illumination. It can be seen from Eq. (1) that it will lead to different correspondence between point S and point N for different values of R, f0, δz, and θ.
For any point S on the mirror, its unit inward normal vector can be expressed as:
According to the geometrical optics laws, a vector $\vec{s}$ parallel to the vector $\overrightarrow {SN}$ can be expressed as:
Then, the equation satisfied by line $\overline {SN}$ can be expressed as:
Let $A({x_s},{y_s}) = {{({{x_f} - {x_s}} )} / {({{z_f} - {z_s}} )}}$ and $B({x_s},{y_s}) = {{({{y_f} - {y_s}} )} / {({{z_f} - {z_s}} )}}$, and take Eq. (3) into Eq. (4) to get:
Take Eq. (2) into Eq. (5), and solve nx and ny in Eq. (5), we can obtain the partial derivative of the surface function to its spatial coordinates:
This partial differential equation describes the surface structure of an off-axis axiparabola with design parameters of R, f0, δzg, θ, and it satisfies the initial conditions ${z_s}(0,0) = 0$. Especially, when θ=0°, it can also describe the surface structure of an on-axis axiparabola; and when δzg =0, it can also describe the surface structure of a parabolic mirror. In Eqs. (6) and (7), the left side of this equation is the partial derivative of the surface function zs(xs, ys) to each spatial coordinate, and the right side of this equation is actually composed of the variables xs, ys and the surface function zs(xs, ys). By solving this partial differential equation accurately, we can obtain the surface function of an off-axis axiparabola under the corresponding specific design parameters.
2.2 Numerical solution based on fourth-order Runge-Kutta method
To obtain the surface function zs(xs, ys), we use the numerical methods to solve Eqs. (6) and (7). The most intuitive numerical solution method is the Euler method, which can be used to rewrite the partial differential into a forward difference. According to Euler method, Eqs. (6) and (7) can be rewritten as:
Here, hx and hy are the step lengths along the x and y directions respectively; The values of A(xs, ys), B(xs, ys), and zs(xs, ys) in step (n, m) are A(n, m), B(n, m), and zs(n, m) respectively. Based on Eq. (8), knowing the value of zs(n, m) in the previous step, we can recursively obtain the values of zs(n + 1, m) and zs(n, m + 1) in the next step. For each step, xs(n, m), ys(n, m) and zs(n, m) are known, and A(n, m) and B(n, m) can be calculated according to Eq. (1). The initial condition of recurrence is xs(0, 0) = 0, ys(0, 0) = 0, zs(0, 0) = 0. In principle, the surface function under any specific design parameters (R, f0, δzg, θ) can be calculated by using the Euler method. But due to the low accuracy of the Euler method, the surface function obtained by this method deviates greatly from its true value when applied to the solution of an off-axis axiparabola.
In order to obtain more accurate values of the surface structure, the fourth order Runge-Kutta method with higher accuracy is used to solve Eqs. (6) and (7). According to the fourth order Runge-kutta method, Eq. (6) can be re-expressed as:
This equation gives the recurrence relationship of the surface function along the x direction. Similarly, Eq. (7) can then be re-expressed as:
This equation gives the recurrence relationship of the surface function along the y direction. According to Eqs. (9) and (10), we can accurately solve the surface function of an off-axis axiparabola for different design parameters. When solving these partial differential equations in this paper, we set the number of sampling points for the mirror surface to 1999 × 1999 along the two-dimensional spatial positions, and the maximum deviation between numerical solution and the true value can be controlled below 0.01 µm, which is far less than the wavelength of incident light (λ=1µm).
For the convenience of discussion, we set the structure parameters of an off-axis axiparabola discussed in this paper as R = 27.5 mm, f0 = 185 mm, δzg = 30 mm, and these parameters are the same as Ref. [1]. Based on the fourth-order Runge-Kutta method, we obtained the surface function of the off-axis axiparabola with different off-axis angles (θ = 0°, 12°, 30°, 45°), as shown in the Fig. 2. As expected, the mirror surface is inclined more and more as the off-axis angle increases.
Fig. 2. The surface structures designed by geometric optics. (a) θ = 0°; (b) θ = 12°; (c) θ = 30°; (d) θ = 45°.
To better describe the surface of an off-axis axiparabola, we have used Zernike polynomials to fit these mirror surfaces in Fig. 2. We assume that the surface structures can be represented as: ${z_s}({x_s},{y_s}) = [{{{[{{{({x_s} + {f_0}\sin \theta )}^2} + y_s^2} ]} / {[{2(\cos \theta + 1){f_0}} ]}} - {{{f_0}(1 - \cos \theta )} / 2}} ] + \sum\limits_{m,n} {a_n^mZ_n^m({\rho _s},{\phi _s})}$. Here, the first term represents an off-axis parabolic mirror with a nominal focal length of f0 and an off-axis angle of θ, which is a known surface shape when solving the expansion coefficients of Zernike polynomials; The second term represents a linear combination of Zernike polynomials, where $Z_n^m(\rho ,\phi )$ are Zernike polynomials, ρ=r/R is the normalized radial coordinate, ϕ is the azimuthal angle, and $a_n^m$ are the expansion coefficients. By fitting with the non-linear least squares method, we obtained the expansion coefficients of the mirror surface with different off-axis angle, and the fitting results of expansion coefficients are list in Table 1.
Table 1. The expansion coefficients of mirror surface by geometric optical design.
2.3 inclined wavefront caused by geometric optical design
According to geometric optics, the surface structure of an off-axis axiparabola satisfying Eqs. (6) and (7) can perfectly converge the incident light of each annulus to the corresponding focal point N. Therefore, the focal line formed by the mirror should be straight, as described by geometric optics. However, the scalar diffraction simulation in the nest section will prove that the focal line formed by this profile is always curved, and the larger the off-axis angle, the larger the curvature is. Compared with an on-axis axiparabola with the same design parameters, we find that an off-axis axiparabola will inevitably introduce a nonlinear inclined wavefront. And it is the existence of the inclined wavefront that causes the bending of the focal line.
In this section, we show the nonlinear inclined wavefront of an off-axis axiparabola designed by geometric optics with different off-axis angle. As shown in Fig. 1, the total optical path of incident light $\overrightarrow {MS}$ made when it reaches its own focal point N is $\overline {MSN}$. We assume that $h({x_s},{y_s};\theta ) = \overline {MSN} = \overline {MS} + \overline {SN}$, which represents the total optical path traveled by the light from incident to reaching its focal point N (xf, yf, zf) for a point S (xs, ys, zs). Here, $\overline {MS}$ is the optical path taken by the incident plane wave when it reaches a point S on the mirror, which can be expressed as $\overline {MS} ={-} {z_s}({x_s},{y_s};\theta )$ when z = 0 plane is selected as the reference plane; $\overline {SN}$ is the distance from a point S on the mirror to its own focal point N, which can be calculated numerically when zs(xs, ys; θ) has been known. Then, we define that the inclined wavefront of an off-axis axiparabola with θ is ${z_w}({x_s},{y_s};\theta ) = h({x_s},{y_s};\theta ) -$ $h({x_s},{y_s};0)$, which represents the difference between the optical path traveled to reach its own focal point for a mirror with an off-axis angle θ and that for an on-axis mirror with the same parameters. It should be noted that the other design parameters (such as R, f0, δzg) are the same when calculating the inclined wavefront. Figure 3 shows the inclined wavefront introduced by geometric optical design when the off-axis angle θ is 0°, 12°, 30°, and 45°. As shown in Fig. 3, the inclined wavefront increases with the increase of the off-axis angle, and its trend of change is actually nonlinear. For an annular light with the same radius (a dashed circle) shown in Fig. 3, although it converges at the same focal point, the time (or optical path) to reach the focal point is different due to the presence of tilted wavefront. From the view of diffraction optics, the difference of optical path in an annular light will lead to a change of intensity distribution. If we correct the tilt wavefront in the incident light, the straight focal line is expected to be obtained, which will be proved in the Section 3.
Fig. 3. The inclined wavefront introduced by geometric optical design. (a) θ = 0°; (b) θ = 12°; (c) θ = 30°; (d) θ = 45°.
To better describe the nonlinear inclined wavefront of an off-axis axiparabola with different off-axis angle, we have used Zernike polynomials to fit the tilted wavefront. We assume that the tilted wavefront can be represented as ${z_w}({x_s},{y_s}) = \sum\limits_{m,n} {a_n^mZ_n^m({\rho _s},{\phi _s})}$. By fitting with the non-linear least squares method, we obtained the expansion coefficients of the tilted wavefront with different off-axis angle as shown in Fig. 3(a)-(d), and the fitting results of expansion coefficients are list in Table 2.
Table 2. The expansion coefficients of tilted wavefront by geometric optical design.
2.4 Further optimization of surface function to obtain a straight focal line
We further use the annealing algorithm to optimize the surface function of this mirror, so that this mirror can also produce a straight focal line when the plane wave is incident. We assume that the modified mirror surface is:
The intensity I (x’,z’) of an off-axis axiparabola can be obtained by the scalar diffraction simulation described by Eq. (13). To assess the design bias, an error function E(α1, α2) in the annealing algorithm is defined as the mean squared error between the diffraction intensities of the focal line and the desired intensities:
By using the annealing algorithm mentioned above, we obtained the optimized surface function of the off-axis axiparabola with different values of θ after thousands of iterations. The optimal parameters for this mirror with different θ are list in Table 3. In next section, we will prove that the surface function optimized by the annealing algorithm can obtain a straight focal line.
Table 3. The optimal parameters for the off-axis axiparabola with different θ
To better describe the surface of an off-axis axiparabola optimized by the annealing algorithm, we have also used Zernike polynomials to fit these mirror structures described by Eq. (11). As above, the surface structure optimized by the annealing algorithm can also be represented as: ${z_s}({x_s},{y_s}) = [{{{[{{{({x_s} + {f_0}\sin \theta )}^2} + y_s^2} ]} / {[{2(\cos \theta + 1){f_0}} ]}} - {{{f_0}(1 - \cos \theta )} / 2}} ] + \sum\limits_{m,n} {a_n^mZ_n^m({\rho _s},{\phi _s})}$. By fitting with the non-linear least squares method, we have obtained the expansion coefficients of the mirror surface optimized by the annealing algorithm, and the fitting results of expansion coefficients are list in Table 4.
Table 4. The expansion coefficients of mirror surface optimized by the annealing algorithm.
3. Numerical simulation verification based on scalar diffraction theory
In this section, we carry out numerical simulation verification based on scalar diffraction theory. As shown in Fig. 1, if we assume that any point near the focal line is P(xp, yp, zp), then its complex amplitude can be expressed as:
Intensity distributions of the focal line for the off-axis axiparabola with θ = 0°, 12°, 30°, 45° are shown in Fig. 4. Figure 4(a1-d1) give intensity distribution of the focal line for an off-axis axiparabola with different θ, in which the surface function of the mirror is designed by solving the partial differential equation based on geometric optics. It is obvious from these results that the off-axis axiparabola designed by geometric optics inevitably has a curved focal line, and the larger the off-axis angle, the larger the curvature is. It should be noted that the partial differential equation described by Eqs. (6) and (7) has ensured that it forms a straight focal line from the view of geometric optics. This shows the conflict between the results of geometrical optics and diffraction optics.
Fig. 4. Intensity distributions of the focal line for an off-axis axiparabola with θ = 0°, 12°, 30°, 45° from (a) to (d). (a1- d1) The surface function is designed by geometric optics; (a2-d2) The incident light has a matched wavefront correction; (a3-d3) The annealing algorithm is used to further optimization of surface function.
Figure 4(a2-d2) give intensity distribution of the focal line for an off-axis axiparabola designed by geometrical optics but the incident light contains a matched tilt wavefront correction. In the simulations, a tilt wavefront of $\exp [{ - jk({h({x_s},{y_s},\theta ) - h({x_s},{y_s},0)} )} ]$ is actually added to Eq. (13), in which this expression refers to the tilt wavefront of incident light on the mirror surface. It is obvious from these results that after considering the wavefront correction in the incident light, the curved focal line can be effectively converted into a straight focal line.
Figure 4(a3-d3) give intensity distribution of the focal line for an off-axis axiparabola designed by combining geometric optics design and diffraction optics correction. Here, the initial surface is obtained by geometrical optics as above, and then the annealing algorithm is used to further modify its surface by scalar diffraction theory. The surface obtained by this method can produce a straight focal line even when a plane wave is incident. In the simulations, we bring the final surface obtained from Eq. (11) into the diffraction integral described by Eq. (13). It is obvious from these results that the surface structure optimized by annealing algorithm can always obtain a straight focal line for an off-axis axiparabola with any off-axis angle.
4. Summary
In this paper, we propose a new method to design the surface of an off-axis axiparabola by combining geometric optics design and diffraction optics correction, which can effectively solve the problem of focal line bending caused by the current design method. We obtain the exact solution of the off-axis axiparabola by solving the partial differential equation it satisfies according to geometrical optics, and find that the geometric optics design inevitably introduces an inclined wavefront, which leads to the bending of focal line. One way to compensate the inclined wavefront is to correct it in the incident light. The results of simulation prove that this scheme is indeed feasible. Furthermore, we use annealing algorithm to optimize the surface with diffraction theory to contain the wavefront correction in the surface itself. The surface obtained by this way can produce a straight focal line even when a plane wave is incident. In a word, the new method we proposed has wide applicability in the design of an off-axis axiparabola with any off-axis angle, which can further expand the practical application of this element.
Funding
National Key Research and Development Program of China (2022YFEO3200200); National Natural Science Foundation of China (12004353, 12105268, 12175211, 12175212, 12275250, 12275253); Young Talent Fund of Laser Fusion Research Center (RCFPD6-2022-6).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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