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Abstract
Laser beam splitting by freeform optics is promising but less studied. Instead of directly forming a target spot array, we propose to first convert the input beam into a closely connected Gaussian sub-beam array. All the Gaussian sub-beams have the same optical field distributions which thus can produce identical discrete spots on the target plane. Such a design concept is very beneficial to ensure the consistency for laser processing. Importantly, the introduction of an intermediate Gaussian sub-beam array can reduce diffraction effects when the size of each Gaussian sub-beam is sufficiently larger than that of the corresponding sub-area within the input beam. The desired transformation can be achieved by two typical systems. The first system consists of two plano-freeform lenses. The second system is composed of a plano-freeform lens and a lens with an entrance freeform surface and an exit surface of freeform lens array. The two freeform beam splitting systems can be determined based on appropriate ray mappings among the input, intermediate and target irradiance distributions and a subsequent double-surface construction. Geometrical and physical simulations verify the effectivenesses of the two beam splitting systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser beam splitter, which is also called spot array generator, plays an important role in the fields of laser parallel processing, medical cosmetic, laser detection, laser projection, optical communication and structured light illumination [1–4]. By dividing one laser beam into a one-dimensional or two-dimensional beam array, a beam splitter avoids the alignment challenge of multiple lasers in related application scenarios, and conforms to the development trend of system integration and miniaturization.
Diffractive optical elements (DOEs), mainly diffraction gratings, are commonly used to realize beam splitting [5,6]. The periodic phase structure of a beam splitting DOE designed based on the scalar diffraction theory can help the element inject the beam energy into specified diffraction orders, so as to obtain laser array beams propagating in different directions. Diffraction efficiency and manufacturing difficulty are the two main factors to be balanced. Binary phase gratings are easier to fabricate, but the beam energy can not be efficiently injected into the required orders because of the limitation of the quantization structures [7,8]. Continuous phase gratings are the opposite i.e. they are high in efficiency but difficult in manufacturing [9,10]. Only a few companies can produce high-quality continuous phase gratings [11–13]. Multi-level phase grating can better balance efficiency and manufacturability [14]. Other optical elements, including polarization gratings [15,16] and metasurfaces [17], designed by the vectorial diffraction theory, can theoretically achieve much better performances (e.g., higher efficiency and larger field of view) than conventional DOEs designed using the scalar diffraction theory. However, mass processing and production of these elements are currently difficult.
Geometrical optical elements can also be used for generating laser spot arrays. They have advantages including relatively high efficiency, high damage threshold, insensitive to wavelength and low cost. Spherical lens arrays or crossed cylindrical lens arrays are conventional geometrical optical elements for beam splitting, which generate spot arrays based on geometrical aperture splitting or pupil division [18]. However, the generated spots on the target will vary in irradiance distribution depending on the incident beam profile, and it is necessary to employ a collimated beam homogenizer to be ahead of the lens array for achieving a uniform energy distribution over the entire target spot array [18].
Freeform surfaces have been applied successfully to geometrical optical elements. Freeform surface refers to a surface without translational or rotational symmetry, which can provide sufficient freedom degrees to achieve a precise beam control. The design of freeform optical elements for illumination and beam shaping applications have been extensively studied in recent years. However, only a few studies have been focused on beam splitting applications. Jarczynski et al. [19] designed a monolithic multi-facet beam splitter generating one dimensional spot array. The entrance surface is described by an acylinder and each facet of the exit surface is described by a $XY$ polynomial freeform surface with the coefficients obtained by optimization. Maksimovic et al. [20] designed a compact freeform beam splitter where the entrance surface is functioned as a collimator and the exit surface converts the collimated beam into three (or five) converging sub-beams. Each sub-beam corresponds to an independent freeform surface which is also specified based on parametric optimization. In a previous paper, we designed a plano-freeform beam splitter that can convert a collimated incident beam into two-dimensional spot array with a given energy ratio based on a modified ray mapping method [21]. The input beam domain is divided into a series of sub-domains, each of which contains the same energy as the corresponding target spot. Each sub-surface is constructed following a variable-separable ray mapping between the corresponding pair of the input sub-beam and the target spot. The whole exit surface of the beam splitter is generated by combining all the sub-surfaces where the height differences between adjacent sub-surfaces are set to be very small. It is necessary to note that the well-known supporting quadric method (SQM), which can design a piecewise smooth freeform reflective or refractive surface for generating a given discrete set of points [22–25], can also be applied for designing beam splitters.
The array beams obtained by the above freeform beam splitting design methods are not necessarily identical. In addition, diffraction effects should be considered in the design process since the laser beam is typically coherent. For the problem of reshaping a Gaussian beam into a flat-top one, a parameter $\beta$ can be used to predict the impact of diffraction [26]:
where $\lambda$ is the wavelength, $r_0$ is the $1/r^2$ beam radius or waist, $Y_0$ denotes the half width of output beam, and $d$ is the distance between the optical element and the target. Generally, geometrical optics can be accurate enough when $\beta > 32$; diffraction effects become significant when $4< \beta < 32$; and when $\beta < 4$, beam shaping will not produce acceptable results due to the uncertainty principle. In laser splitting applications, $\beta$ should be evaluated for each sub-aperture. For designs generating square or rectangular spots, aperture radii of $2r_0$ could effectively avoid truncation effects [26]. A larger $\beta$ value can be beneficial to reduce the influence of truncation effects when the aperture radius is less than $2r_0$.To demonstrate the influence of diffraction effects, we design a plano-freeform beam splitter as shown in Fig. 1(a) for converting a Gaussian beam irradiance distribution produced by a point source into a $5 \times 5$ flat-top spot array using the method similar as [21], and provide the geometrical and physical optics simulation results in Figs. 1(b) and 1(c) respectively. The reason why we are interested in generating multiple flat-top spots is that they can be desired for a fast and parallel processing in many laser processing applications e.g., thin film scribing in photovoltaic [12]. In this design, the input beam, the target spot array and their locations are the same as those defined in Section 3, which is convenient for comparison. As can be seen from Fig. 1(a), the freeform exit surface can be considered as the splicing of 25 sub-surfaces with different sizes. Each sub-surface corresponds to a single spot on the target. We perform Monte-Carlo ray tracing in LightTools to generate the geometrical optical simulation results, where we can observe very uniform target spots (see Fig. 1(b)). However, since the $\beta$ values for all the sub-apertures are sufficiently small, diffraction brings devastating effects on the shapes and energy distributions of the target spots as shown in Fig. 1(c). The physical optics simulations are implemented in VirtualLab Fusion using the the second generation field tracing engine (see [27] for an overview). There are heavy truncations of the input beam by the sub-apertures, but we still employ Eq. (1) to calculate the $\beta$ values, where the $r_0$ values are chosen as the half widths of the sub-apertures. The $\beta$ value for the central aperture is $\sim 3.7$, where $\lambda$ = 532 nm, r0 ≈ 0.274 mm is the half width of the central sub-aperture, $Y_0$ = 0.1 mm is the half width of the central spot on the target and $d$ = 70 mm is the distance between the input aperture and the target. We can clearly see from Fig. 1(c) that the central spot on the target is far from uniform and also has a serious truncation effect. For the lager sub-aperture at the lower left corner, we can observe a nearly square spot from Fig. 1(c), but diffraction effect is still very significant. There exist apparent ripples for both of the spots, which are resulted from the truncations of the input Gaussian beam by the two sub-apertures.
Fig. 1. A single freeform surface beam splitter directly generate a spot array on the target, where diffraction effects are very significant: (a) lens model and ray tracing diagram and (b) geometric optics simulation results using LightTools; (c) physical optics simulation results using VirtualLab Fusion.
The main purpose of this paper is to design freeform optical systems for generating identical sub-beams from a given input beam while reducing the influences of diffractions through increasing the $\beta$ values. We propose two layouts of freeform beam splitting systems for achieving such a design purpose. The design of the two freeform beam splitting systems is detailedly described in Section 2. In Section 3, we provide two design examples respectively for the two beam splitting systems, where geometrical and physical optics simulations are implemented to verify the effectivenesses. Finally, the conclusion and a brief discussion is given in Section 4.
2. Freeform beam splitting system design
2.1 Intermediate Gaussian sub-beam array
Suppose we have an input beam with the irradiance distribution of $I_0(u,v)$ at $z=d_0$, where $(u,v) \in \Omega _0$. The irradiance distribution of the target spot array at $z=d_t$ is denoted as $I_t(x,y)$, where $(x,y) \in \Omega _t$. $I_t(x,y)$ is the sum of all the spots’ irradiance distributions:
where $I^{(n,m)}_t$ refers to the irradiance distribution of the $(n,m)$ spot.Our strategy is illustrated in Fig. 2. The input beam is first converted into a Gaussian sub-beam array on an intermediate plane at $z=d_c$. This intermediate irradiance distribution is denoted as $I_c(\xi ,\eta )$, where $(\xi ,\eta )\in \Omega _c$. If the number of the target spots is $N \times M$, there will also be $N \times M$ Gaussian sub-beams on the intermediate plane. $I_c(\xi ,\eta )$ can also be expressed as a sum of all the Gaussian sub-beams:
where $I^{(n,m)}_c (\xi ,\eta )$ refers to the irradiance distribution of the $(n,m)$ Gaussian sub-beam. All the Gaussian sub-beams have the same irradiance and wavefront distributions, which thus form an identical spot array on the target plane. The identity of all the sub-beams can be very beneficial to ensure the consistency of parallel laser processing.Fig. 2. (a) The input beam is directly converted into a spot array on the target plane; (b) the input beam is first converted into an intermediate Gaussian sub-beam array, which then forms an identical spot array on the target plane.
For the direct beam splitting, $\beta$ could be very small (even < 4) which makes geometrical optical method far from being accurate (see Fig. 1). For our beam splitting strategy, let $\beta _1$ indicates the diffraction effects between the input plane and the intermediate plane and $\beta _2$ indicates the diffraction effects between the intermediate plane and the target plane. The size of each Gaussian sub-beam is set to be enough larger than that of the corresponding sub-area of the input beam and both the the distances $d_c-d_0$ and $d_t-d_c$ are smaller than $d_t-d_0$. Thus, according to Eq. (1), we can easily make $\beta _1$ and $\beta _2$ to be much larger than $\beta$ so that the influence of the diffraction effects can be greatly reduced. Such a $\beta$ increase strategy is also valid in reducing diffraction influences for those cases that either the input beams or the intermediate sub-beams are not of Gaussian distributions.
2.2 Two freeform beam splitting systems for achieving the beam transformation
The design can be thought as an integration of two tasks: 1) mapping the input beam irradiance into a closely-arranged sub-beam array with equal Gaussian irradiance distributions; 2) generating an identical wavefront (or phase) distribution for each Gaussian sub-beam so that the target spots are identical. At least two freeform surfaces are required to realize the desired transformation.
We propose two typical freeform beam splitting systems as demonstrated in Fig. 3. Freeform beam splitting system 1 (FBSS1) shown in Fig. 3(a) is a plano-freeform lens pair. Such a system can simultaneously generate a Gaussian sub-beam array on the intermediate plane and a undulating wavefront distribution, which together form an identical array spots at the target plane. As can be seen Fig. 3(a), both the exit surface of the first lens and the the entrance surface of the second lens are freeform, which are responsible for the required beam reshaping. Notice that we can realize the same goal using a single lens with double freeform surfaces or a double-freeform-mirror system.
Fig. 3. (a) Freeform beam splitting system 1 (FBSS1): a plano-freeform lens pair; (b) freeform beam splitting system 2 (FBSS2): a combination of a plano-freeform lens and a lens with an entrance surface and an exit surface of freeform lens array.
Unlike FBSS1, freeform beam splitting system 2 (FBSS2) shown in Fig. 3(b) first generates a planer wavefront inside the second freeform lens. The exit surface of the second lens of FBSS2 is a freeform lens array which can focus the planar beam into a spot array on the target plane. Compared with FBSS1, the existence of the freeform lens array in FBSS2 can undertake part of the ray bending power to facilitate the fabrication of the entrance freeform surface of the second lens.
Take FBSS1 for example, we introduce the notation of the system. Suppose a light source is located at the original point and let $\mathbf {W}$ denote the input wavefront. The refractive indices of the air environment and freeform lenses are $n_1$ and $n_2$, respectively. An arbitrary ray emitted from the source first reaches point $\mathbf {S}$ on the incident plane at $z=d_0$, which is also the entrance surface of first lens, and is refracted to point $\mathbf {P}$ on the first freeform surface. Redirected by the first freeform surface, the ray hits on the second freeform surface at point $\mathbf {P}'$, and then refracted to point $\mathbf {S}'$ on the planar surface at $z=d_e$. After that, the output ray passes through point $\mathbf {Q}$ on the intermediate irradiance plane at $z=d_c$ and arrived at point $\mathbf {T}$ on the target plane at $z=d_t$ forming the output wavefront $\mathbf {W}'$. As illustrated in Fig. 3, let $\hat {\mathbf {I}}$ and $\hat {\mathbf {O}}$ denote the unit vectors of the input and output ray sequences, respectively, and let $\hat {\mathbf {R}}_1$, $\hat {\mathbf {R}}_2$ and $\hat {\mathbf {R}}_3$ describe the unit ray vectors inside the first lens, between the two lenses and inside the second lens, respectively. In the following, we will introduce the construction of FBSS1 detailedly and then supplement the differences in the construction procedure of FBSS2.
2.3 Determination of the two beam splitting optical systems
A direct determination of the double freeform surfaces in FBSS1 is very difficult. There are very few studies which can deal with the beam shaping problem of not only generating a prescribed irradiance distribution but also forming a special wavefront distribution. Feng et al. provided a fast and effective ray mapping method in designing a single lens with double freeform surfaces for producing two different irradiance distributions on two successive target planes [28]. This method and its modified version in designing a pair of plano-freeform lenses for laser diode beam shaping [29] are adapted here to tackle the beam splitting system design problem.
In fact, we can simplify the design further by restricting that both the input beam and the target beam are variable separable. The intermediate irradiance distribution $I_c(\xi ,\eta )$ and each sub-beam irradiance distribution $I_c^{(n,m)}(\xi ,\eta )$ can also be factorized in $\xi$ and $\eta$ coordinates (variable separable). Thus, we can simply apply variable separation to acquire the coordinates relationships among the three planes.
Energy conservation among the three planes can be written as:
Our ray mapping strategy among the input, intermediate and target planes is demonstrated in Fig. 4. Based on such a ray mapping strategy, Fig. 5 shows the design diagram for determining the double freeform surfaces in FBSS1. A detailed description for the computation steps is provided in the following.
Step 1. The ray mapping calculation begins with discretizing $\Omega _c$ on the intermediate plane into a uniform rectangular grid with the number of the grid points of $I\times J$, where the grid points coordinates are $(\xi _i,\eta _j)$, $i=1,2,\ldots ,I, j=1,2,\ldots ,J$. According to the variable separability of the irradiance distribution on the two planes, we can compute the inverse ray map $\psi : \Omega _{c} \to \Omega _{0}$, and acquire the corresponding grid points $(u_i,v_j)$ on the input plane as:
Step 2. After calculating the ray mapping from the intermediate plane to the incident plane, we then calculate the ray mapping from the intermediate plane to the target plane. Since the target plane has a discrete irradiance distribution $I_t(x,y)$, a direct calculation of the overall ray mapping from the intermediate plane to the target plane is difficult. We thus adopt a one-to-one ray mapping $\phi ^{(n,m)}:\Omega _{c}^{(n,m)}\to \Omega _{t}^{(n,m)}$ from a single Gaussian sub-beam to its corresponding spot on the target. Since the size of each spot is identical, they occupy equivalent number of uniform grid points, which is $(I/N)\times (J/M)$. We can calculate the grid division of the $(n,m)$ target spot based on:
Step 3. After the computation of the grid points $(u_i,v_j)$, $(\xi _i,\eta _j)$ and $(x_i,y_j)$, we will specify the input and output ray sequences. For a divergent input beam with spherical wavefront $\mathbf {W}$ emitted from a point source, the incident ray sequence $\mathbf {\hat I}_{i,j}$ can be computed as:
According to Snell’s law, we can compute the unit ray vector $\mathbf {\hat R}_{1,i,j}$ inside the first freeform lens as:
Because the light propagation is reversible, we can compute $-\mathbf {\hat R}_{3,i,j}$ by substituting $\mathbf {\hat I}_{i,j}$ with $-\mathbf {\hat O}_{i,j}$ and replace $\mathbf {\hat N}_{s}$ with $-\mathbf {\hat N}_{s}$ in Eq. (8). The output wavefront $\mathbf {W}'_{i,j}$ perpendicular to the output ray sequence $\mathbf {\hat O}_{i,j}$ can be reconstructed by a least squares method requiring that the chord joining two adjacent output wavefront points is perpendicular to the average of the two unit vectors of the output rays passing through the two points [28]:
Step 4. After specifying the input and output ray sequences and wavefronts, we now turn to the iterative construction procedure of the double freeform surfaces. We first give an estimate of the first freeform surface, which can be simply set as a planar surface.
Step 5. The corresponding second freeform surface $\mathbf {P}_{i,j}'$ can be acquired based on the constancy of optical path length (OPL):
Step 6. The refracted ray sequence $\mathbf {\hat R}_{2,i,j}$ can be simply computed as:
Then, the normal vectors $\mathbf {\hat N}_{i,j}$ of the first freeform surface can be recalculated according to the Snell’s law in vector form:
Step 7. After specifying a normal field $\mathbf {\hat N}_{i,j}$, the least squares method mentioned earlier is employed again to acquire the first freeform surface point $\mathbf {P}_{i,j}$, where the basic relationships are [28]:
We can recalculate $\mathbf {P}_{i,j}'$ by inserting the obtained $\mathbf {P}_{i,j}$ into Step 5, update the required normals to the first freeform surface according to Step 6 and obtain a new data of $\mathbf {P}_{i,j}$ in Step 7. This process is repeated multiple times until satisfying a stop criterion.
The construction of FBSS2 folowing the ray mapping is very similar with that of FBSS1. We first establish a plano-freeform lens pair which can form an intermediate irradiance distribution and a planar wavefront. We then replace the plano surface at $z=d_{e}$ with a freeform lens array, where each freeform lens surface can be specified by the ray mapping $\phi ^{(n,m)}:\Omega _{c}^{(n,m)}\to \Omega _{t}^{(n,m)}$ and a least squares surface construction method.
3. Design examples
Here, we demonstrate the optical performances of FBSS1 and FBSS2 with specific parameters. Suppose we have a 532 nm divergent laser beam with the full width divergence angle ($1/e^2$ intensity) of 12.47$^{\circ }$. The material for all the lenses is set as PMMA, whose refractive index is set as 1.4947. The input plane is located at $d_0=20$ mm, and the input beam irradiance distribution is shown in Fig. 6(a), where the $1/e^2$ radius is 2.1850 mm and the aperture width is 5.6811 mm. Figure 6(b) shows the desired $5 \times 5$ Gaussian sub-beam array on the intermediate plane within a square domain of $12 \times 12 \ \textrm {mm}^2$. For FBSS1, the intermediate plane coincides with the exit plano surface at $d_c=d_e=45$ mm. The waist radius and aperture width for each Gaussian sub-beam are set as 1 mm and 2.4 mm, respectively. The prescribed target flat-top spot array at $d_t=$ 90 mm is shown in Fig. 6(c), where the side length of each spot is set as 0.2 mm.
Fig. 6. Irradiance distributions of the (a) given input beam, (b) intermediate Gaussian sub-beam array and (c) target flat-top spot array. The color limits for (c) and all the the following figures of the target spots are set as [0,2].
The calculations of the double freeform surfaces in FBSS1 are implemented in Maltab. The algorithm is started by discretizing the domain $\Omega _c$ for the intermediate irradiance distribution into a uniformly sampled grid with $2005 \times 2005$ grid points. The corresponding non-uniformly sampled grid points on the input plane can be acquired based on solving Eq. (5). After that, we compute the ray mapping $\phi ^{(3,3)}: \Omega _c^{(3,3)} \rightarrow \Omega _t^{(3,3)}$ between the central Gaussian sub-beam and the central target spot with the number of grid points of $401 \times 401$. The coordinates relationships between the other Gaussian sub-beams and their target spots can be determined by just shifting the ray mapping $\phi ^{(3,3)}$, and thus the overall ray mapping $\phi : \Omega _c \rightarrow \Omega _t$ can be specified. After determining the input and output ray sequences and wavefronts following the ray mappings, we then specify the double freeform surfaces following the iterative surface constructions procedure provided in Section 2-3.
Figure 7(a) illustrates the required output wavefront that can transfer the Gaussian sub-beam array into the corresponding top-hat spot array. The final constructed double freeform surfaces are shown in Figs. 7(b) and 7(c), respectively. Figures 7(d) and 7(e) show the 3D models of the two designed lenses. The central thicknesses of the first and second freeform lenses are 1 mm and 3.346 mm, respectively. The first freeform surface, whose major function is to redistribute the input Gaussian beam into the closely connected Gaussian sub-beam array, is at least G1 smooth. However, you may observe from Figs. 7(b) and 7(d) that the first freeform surface acts like a combination of 25 sub-surfaces which is mainly resulted from the undulating intermediate irradiance distribution of Gaussian sub-beam array. A problem appears on the second freeform surface that adjacent sub-surfaces can intersect at their boundaries, and we have to remove all the intersecting segments between adjacent sub-surfaces to make a reasonable integrated surface. This can be simply realized by first removing the intersecting-segment points columns by columns and then removing the remaining intersecting-segment points rows by rows. The intersecting segments, which account for small area proportions and correspond to the edges of the Gaussian sub-beams, have little influences on the target spots.
Fig. 7. Design results of FBSS1: (a) the required output wavefront, the designed (b) first and (c) second freeform surfaces; 3D models of the designed (d) first and (e) second freeform lenses.
We implement Monte-Carlo ray tracing in LightTools with $1 \times 10^7$ rays. Figure 8(a) illustrates the ray tracing of FBSS1 in LightTools, and Fig. 8(b) provides the simulated target spots (with resolutions of 2001 $\times$ 2001), which are close to the prescribed ones shown in Fig. 6(c).
Fig. 8. Simulation results of FBSS1: (a) geometrical ray tracing illustration and (b) simulation results using LightTools; (c) physical optics simulation results using VirtualLab Fusion.
We can observe a very uniform central spot from Fig. 8(b). We can also see that there is a deformation for the target spot on the lower left corner (and the other three corners). This is caused by the fact that the ray mapping method we employed here is only accurate for the paraxial or small-angle approximation which is not fulfilled well by rays through the four sub-apertures near the corners.
The calculated scattered grid points of the double freeform surfaces are interpolated and extrapolated into equidistant and square grid points with the number of $2501 \times 2501$, which are then inserted into VirtualLab Fusion to implement physical optics simulation. As stated before, one of the roles of the intermediate plane is to increase the parameter $\beta$. The central sub-aperture on the input beam irradiance distribution has the minimum half width of $\sim 0.274$ mm. The $\beta _1$ value between the central sub-aperture and the central Gaussian sub-beam is $\sim 103$. The $\beta _2$ value between the central Gaussian sub-beam and the central target spot is $\sim 21$, which is less than 32 but still much higher than that for the single lens design shown in Fig. 1. Thus, diffraction effects can be efficiently reduced, which is demonstrated by physical optics simulations shown in Fig. 8(c). We can see that the physical optics simulation result shown in Fig. 8(c) is accordant with the geometrical optics result shown in Fig. 8(b). In the simulation settings for the second generation field tracing engine in VirtualLab Fusion, both the "Sampling Accuracy" and "Fourier Transformation Accuracy" are set as 16, and the number of the sampling points for the "Camera Detector" is set as $8192 \times 8192$.
For FBSS2, the intermediate irradiance plane is set to be located at $d_c$=48 mm. Figure 9 shows the three surface profiles of FBSS2 to form an equal energy $5 \times 5$ top-hat spot array. The first freeform surface shown in Fig. 9(a) is similar to that of FBSS1. Compared with the second freeform surface of FBSS1 (see Fig. 7(c)), the second freeform surface of FBSS2 becomes smoother, as observed in Fig. 9(b). Figure 9(c) displays the calculated freeform lens array, which realizes the irradiance conversion from the intermediate plane to the target plane. The 3D model of the first freeform lens is illustrated in Fig. 9(d). The 3D model of the second freeform lens is illustrated in Figs. 9(e) and 9(f). The central thicknesses of the first and second freeform lenses are 1 mm and 3.344 mm, respectively.
Fig. 9. Design results of FBSS1: the designed (a) first and (b) second freeform surfaces, and the exit surface of the second freeform lens; (d) 3D model of the designed first freeform lens; 3D model of the designed second freeform lens, which is (e) viewed from the entrance surface and (f) viewed from the exit surface.
Figure 10(a) is the ray tracing illustration of FBSS2 in LightTools. The geometrical optics simulated target spots shown in Fig. 10(b) is very similar with that of FBSS1 shown in Fig. 8(b). Physical optics simulation result is shown in Fig. 10(c), where both the "Sampling Accuracy" and "Fourier Transformation Accuracy" are also set as 16. Compared with the single lens design shown in Fig. 1, the similarity between the geometrical and physical optics simulations demonstrates that FBSS2 can also resist diffraction effects.
Fig. 10. Simulation results of FBSS2: (a) geometrical ray tracing illustration and the (b) simulation results using LightTools; (c) physical optical simulation results using VirtualLab Fusion.
For each beam splitting system, the ratio of the optical power on the target region of $12 \times 12$ mm$^2$ to the input optical power within the defined aperture is around $85\%$ considering Fresnel losses, which could be improved by surface coating.
Since each beam splitting system is composed of two lenses, alignment errors of the two lenses can influence the beam splitting performances. Generally, there are three possibilities: incorrect lens separation ($\Delta z$), centration offset ($\Delta x$ and $\Delta y$), and tilt (rotation angles $\Delta \alpha$, $\Delta \beta$ and $\Delta \gamma$ around $x$, $y$ and $z$ axes, respectively). For the centration offset, we just analyse the influences of $\Delta x$ because of the system symmetry. The influences of $\Delta \alpha$ and $\Delta \beta$ are also equivalent. Figure 11 illustrates the change of the performance of FBSS1 when we change $\Delta x$, $\Delta z$, $\Delta \alpha$ and $\Delta \gamma$ for the two freeform lenses respectively. Figure 11(a) shows the maximum irradiance value of the target spot array increases as $\Delta x$ changes from 0 mm to 0.01 mm for the first freeform lens. As can be observed from Fig. 11(a), the maximum irradiance value for $\Delta x$=0.01 mm is increased to be higher than 1.5 times of the maximum irradiance value for $\Delta x$ = 0 mm. The corresponding target spot array for $\Delta x$=0.01 mm is shown in Fig. 11(e), where we can see a change of the spot sizes along $x$-axis. The spots of the leftmost column are shrinked while the spots of the rightmost column are enlarged. Figure 11(b) shows the variation of the maximum irradiance value as $\Delta z$ changes from −0.1 mm to 0.006 mm for the first lens. It seems that the increase of $\Delta z$ has a bigger influence on the maximum irradiance value than the decrease of $\Delta z$. Figure 11(f) provides the target spot array for $\Delta z$=−0.1 mm, where we can observe a radial variation of the spot sizes. The central spot has the smallest size while the spots at the four corners have the largest size. Figure 11(c) demonstrates that the target performance is gradually decreased when $\Delta \alpha$ is increased from $0^\circ$ to $0.5^\circ$ for the first lens. The target spot array for $\Delta \alpha =0.5^\circ$ is provided in Fig. 11(g), which has a similar variation as Fig. 11(e). Figure 11(d) shows the change of the maximum irradiance value when we rotate the first lens along z-axis ($\Delta \gamma$). The corresponding target spot array for $\Delta \gamma =0.2^\circ$ is shown in Fig. 11(h), where we can see a rotation of the spots at the four corners. Figures 11(i)-(l) show the variation of the system performances when we move and rotate the second lens. The corresponding target spot arrays for the maximum position and angle variations are shown in Figs. 11(m)-11(p). Compared with Figs. 11(e)-11(h), it seems that there are contrary variations in Figs. 11(m)-11(p), because of the relativity of two lens positions. Figure 12 demonstrates the change of the performances of FBSS2 when we change $\Delta x$, $\Delta z$, $\Delta \alpha$ and $\Delta \gamma$ for the first and second freeform lenses respectively, which have very similar variations compared to those of FBSS1 shown in Fig. 11.
Fig. 11. Evolution of the maximum irradiance values of FBSS1 as we change (a) $\Delta x$, (b) $\Delta z$, (c) $\Delta \alpha$, (d) $\Delta \gamma$ for the first freeform lens, and the physical simulated target spot arrays for (a) $\Delta x$=0.01 mm, (b) $\Delta z$=−0.1 mm, (c) $\Delta \alpha =0.5^\circ$ and (d) $\Delta \gamma =0.2^\circ$, respectively; Evolution of the maximum irradiance values of FBSS1 as we change (i) $\Delta x$, (j) $\Delta z$, (k) $\Delta \alpha$, (l) $\Delta \gamma$ for the second freeform lens, and the physical simulated target spot arrays for (m) $\Delta x$=0.02 mm, (n) $\Delta z$=−0.08 mm, (o) $\Delta \alpha =0.6^\circ$ and (p) $\Delta \gamma =0.2^\circ$, respectively. P= 1.717 W/mm$^2$ denotes the peak irradiance value of the target spots for FBSS1 without any changes.
Fig. 12. Evolution of the maximum irradiance values of FBSS2 as we change (a) $\Delta x$, (b) $\Delta z$, (c) $\Delta \alpha$, (d) $\Delta \gamma$ for the first freeform lens, and the physical simulated target spot arrays for (a) $\Delta x$=0.01 mm, (b) $\Delta z$=0.06 mm, (c) $\Delta \alpha =0.5^\circ$ and (d) $\Delta \gamma =0.2^\circ$, respectively; Evolution of the maximum irradiance values of FBSS2 as we change (i) $\Delta x$, (j) $\Delta z$, (k) $\Delta \alpha$, (l) $\Delta \gamma$ for the second freeform lens, and the physical simulated target spot arrays for (m) $\Delta x$=0.02 mm, (n) $\Delta z$=0.16 mm, (o) $\Delta \alpha =0.6^\circ$ and (p) $\Delta \gamma =0.2^\circ$, respectively. P=1.738 W/mm$^2$ denotes the peak irradiance value of the target spots for FBBS2 without any changes.
4. Conclusion and discussion
We have demonstrated how to generate identical beam splitting using freeform optics while resisting the influence of diffraction effects. Instead of directly generating the target spot array, the input beam was first transformed into a closely connected Gaussian sub-beam array on the intermediate plane. All the Gaussian sub-beams have the same irradiance and wavefront distributions which can thus produce identical spots on the target plane. To realize such a beam splitting purpose, we proposed two beam splitting systems (FBSS1 and FBSS2) each of which is composed of two freeform lenses. FBSS1 is a plano-freeform lens pair, which can simultaneously generate a Gaussian sub-beam array and a special wavefront together forming identical array spots on the target. FBSS2 first generates a Gaussian sub-beam array with a planar wavefront via the first lens and the entrance freeform surface of the second lens, and then focus the parallel beam into a target spot array through the exit surface of the second lens. The two beam splitting systems can be designed using an efficient ray mapping method: ray mappings among the input beam, intermediate Gaussian sub-beam array and the target spot array are first computed based on variable-separation, and then the double freeform surfaces can be constructed based on a least squares-based iterative procedure. Simulation results show that, compared with the single lens design, the two proposed beam splitting systems can effectively reduce the diffraction effects.
The two beam splitting systems could be fabricated using the methods demonstrated in [19]. Alignment errors can reduce the beam splitting performances. A suitable lens barrel structure, which is similar with those used in beam shaping and beam expanding optical systems, can be employed to overcome alignment errors.
Our design method is also applicable to other distributions (e.g, uniform distribution) for the input and the intermediate sub-beams but not suitable to deal with the design requirements that can not obtain two large $\beta$ values. Both FBSS1 and FBSS2 have their own pros and cons. FBSS1 only needs two freeform surfaces and the other optical surfaces are planar, but the second freeform surface can be discontinuous. FBSS2 has a smoother second freeform surface but requiring a freeform lens array as the exit surface of the second lens to form the target spot array.
Note that the ray mapping method we used is only applicable for separable irradiance distributions and accurate for paraxial or small-angle approximations. However, such restrictions can greatly simplify the design and are fulfilled by many practical laser applications. We can increase $\beta$ further by employing a larger intermediate irradiance distribution and enlarge the distance between the two lenses to not increase the surface errors. The freeform beam splitting system can deviate much from small angle-deflections when system compactness is an important consideration. This require developing a more accurate design method.
We need to mention that it is not strictly true to calculate $\beta$ following Eq. (1) for each sub-aperture which only accounts for a small part of the Gaussian beam. A more appropriate expression of $\beta$ considering truncation effects and more general beam irradiance distributions is worth further research.
Funding
National Key Research and Development Program of China (2017YFA0701200); National Natural Science Foundation of China (11704030).
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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