1. Introduction

Freeform optical elements play a key role in modern lighting systems. The problem of designing a freeform reflecting or refracting surface generating a required irradiance or intensity distribution belongs to the inverse problems of nonimaging optics. This problem is extremely challenging and can be reduced to finding a solution to a nonlinear partial differential equation of second order [1,2]. Analytical solutions to this equation have been obtained only for the problems possessing axial symmetry, as well as for some problems of generation of one-parameter irradiance distributions [3–8]. In the general case, various numerical and iterative methods are used for the solution of this inverse problem [1,2,9–15]. One of the widely used methods consists in the construction of an equi-flux mapping between the coordinates of the rays emitted by the source and the points of the illuminated domain [12–17]. Usually, the rays emitted by the source are represented in spherical coordinates $\left(\phi ,\theta \right)\in \Omega$, where $\Omega$ is a domain on the unit sphere which corresponds to the emission solid angle of the source, while the points of the illuminated domain $D$ are described in polar coordinates $\left(\rho ,\psi \right)\in D$. For the construction of a mapping $\Omega \to D$, the light flux conservation law is used. The mapping has to be constructed in a form allowing the separation of variables. In the most frequently used mapping, the “meridians” (straight lines $\phi ={\phi }_0$ on the unit sphere in the domain $\Omega$) correspond to the straight lines $\psi ={\psi }_0$ in the illuminated domain $D$, see Fig. 1(b) in [13]. This mapping can be considered as a heuristic generalization of the mappings for the problems possessing axial symmetry. In the absence of axial symmetry, such a mapping does not meet the “integrability condition” (the possibility of the construction of an optical surface implementing the mapping). Despite this, the considered mapping allows one to calculate refracting optical surfaces generating illuminated domains with a given shape (rectangle, ellipse, rhombus) with acceptable accuracy. This method is commonly used for the domains with relatively low aspect ratio (the ratio between the longitudinal and transverse dimensions) not exceeding 1–2. At the aspect ratio of 2, typical error of the method (relative standard deviation (RSD) of the generated irradiance distribution from the required distribution) amounts to 20–25% [13]. To improve the distribution quality, so-called iterative feedback modification methods are used [13–15]. In some cases, these heuristic techniques can reduce the relative error down to 10%; however, they require a fine tuning of the optimization parameters.

 

Fig. 1 Geometry of a typical illuminated domain.

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The use of more sophisticated coordinate systems can lead to more accurate source-target mappings. For example, the so-called double-pole coordinate system was used for the design of freeform refractive elements [17]. In the considered case of rectangular irradiance distribution, the RSD did not exceed 10%.

The existing papers do not describe the performance of these methods in the problem of calculation of the mirrors generating prescribed irradiance distributions [11–17]. In [12,18], it was mentioned that the error in this case is large, but the corresponding detailed results were not presented.

In the present work, a novel source-target mapping method for the design of freeform mirrors generating required 2D intensity distributions is presented. This mapping is a generalization of a mapping corresponding to the mirrors generating one-parameter intensity distributions (i.e. the distributions which depend on one angular coordinate only) [5,6]. Since such distributions are defined on a line on the unit sphere, they will be referred to as the line-shaped intensity distributions. In [5,6], it was shown that the rays directed by the mirror to each point of the line are located on a cone with the vertex at the light source. In this work, we propose to use this source-target mapping for the generation of 2D intensity distributions with high aspect ratio. The illuminated 2D domain is obtained from the line segment by direct replacement of each point of the segment by a line segment perpendicular to the original one [Fig. 1]. The simulation results presented below demonstrate high performance of the proposed method, including the domains with relatively low aspect ratio. For rectangular and elliptical domains with angular dimensions from ${80}^{\text{o}}\times {20}^{\text{o}}$ to ${40}^{\text{o}}\times {20}^{\text{o}}$, RSD does not exceed 8.5%. It is important to note that the mirror surface is expressed in an analytical form.

2. Calculation of the mirror generating a line-shaped intensity distribution

Analytical solution to the problem of the calculation of a mirror generating one-parameter line-shaped intensity distribution was presented in [6]. In the present section, we briefly review this solution for the presentation consistency. The problem of generating a prescribed intensity distribution consists in the calculation of the mirror surface $S\left(u,\sigma \right)=\left(x\left(u,\sigma \right),y\left(u,\sigma \right),z\left(u,\sigma \right)\right)$, where $\left(u,\sigma \right)$ are certain curvilinear coordinates, from the condition that the directions of the rays reflected by the mirror have the form [Fig. 2(a)]:

 

Fig. 2 Mirror for generating one-parameter intensity distribution (a) and its central profile (b).

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The mirror $S(u,\sigma )$ is the envelope of a one-parameter family of paraboloids with the focus at the origin (at the location of the point light source O) and the axes direction $p\left(\beta \right)$. In [5,6], it was shown that the envelope surface can be regarded as a family of parabolas $L\left(u;\beta \right)$ with respect to parameter $\beta$. The rays from the point source incident on the parabola $L\left(u;\beta \right)$, which are shown with blue lines in Fig. 2, lie on a cone $K\left(u,\sigma ;\beta \right)$ with the vertex at the light source. On each parabola, the reflected rays have the same direction $p\left(\beta \right)$, lie in the same plane and are shown with green lines in Fig. 2. To better illustrate the cone geometry, let us consider the central profile of the mirror $r\left(\sigma \right)$ which corresponds to the cross-section of the mirror $S\left(u,\sigma \right)$ by the xOz plane, where $\sigma$ is the angle that determines the direction of the incident ray [Fig. 2(b)]. Let us denote by $\beta \left(\sigma \right)$ the angle between the reflected ray and the $z$ axis. The cross-section of the ray cone by the xOz plane is shown in Fig. 2(b) with blue lines: one of the generators of the cone coincides with the incident ray, while the other one is parallel to the reflected ray. To describe the position of a ray on the cone, it is convenient to use the u angle which defines the position of a point on the circle at the cone base and is measured from the xOz plane. In the $\left(u,\sigma \right)$ coordinates, the envelope equation takes the form [6]

For a given function $\beta \left(\sigma \right)$, the functions $r\left(\sigma \right)$ and $f\left(\sigma \right)$ are defined by the following relations [6]:

The function $\beta \left(\sigma \right)$ is calculated from the condition of generation of a prescribed intensity distribution $I\left(\beta \right)$ using the light flux conservation law. Calculation of $\beta \left(\sigma \right)$ is reduced to the solution of an ordinary differential equation [6]:

3. Calculation of the mirror generating a 2D intensity distribution

Let us now turn to the main part of the paper and consider the problem of generation of a prescribed 2D intensity distribution $I\left(\beta ,\gamma \right)$, $\left(\beta ,\gamma \right)\in D$, where $D=\left\{\left(\beta ,\gamma \right)|\beta \in \left[-{\beta }_d,{\beta }_d\right],\gamma \in \left[{\gamma }_1\left(\beta \right),{\gamma }_2\left(\beta \right)\right]\right\}$, $\gamma$ is the angle between the ray and the xOz plane, and the functions ${\gamma }_1\left(\beta \right)$, ${\gamma }_2\left(\beta \right)$ define the boundaries of the domain $D$ along the $\gamma$ coordinate [Fig. 1]. In contrast to the problem of generating a line-shaped intensity distribution considered above, the angle $\beta$ is the angle between the ray projection onto the plane xOz and the z axis. In the limiting case ${\gamma }_1\left(\beta \right)={\gamma }_2\left(\beta \right)=0$, the domain $D$ corresponds to a line segment (1), and the definition of the angle $\beta$ coincides with the one in Section 2. For the generation of the intensity distribution $I\left(\beta ,\gamma \right)$, the directions of the reflected rays must have the form

Let us define a mapping between the coordinates of the reflected rays $\left(\beta ,\gamma \right)$ and the coordinates of the incident rays $\left(\sigma ,u\right)$. We propose to find the functions $\beta =\beta \left(\sigma ,u\right)$ and $\gamma =\gamma \left(\sigma ,u\right)$ from the condition that the ray cones (3) are transformed into the segments $\gamma \in \left[{\gamma }_1\left(\beta \right),{\gamma }_2\left(\beta \right)\right]$. A similar type of mapping was previously used by some of the present authors for the design of diffractive optical elements [19]. The element of the solid angle corresponding to the vector function (9) is easily written as $d\Omega =\cos \gamma d\beta d\gamma$. We set $\beta \left(\sigma ,u\right)=\beta \left(\sigma \right)$, where $\beta \left(\sigma \right)$ is found from Eqs. (6), (7) for the following line-shaped intensity distribution:

Let us write the element of the solid angle corresponding to the ray cones (3):

In the problem of generation of the intensity distribution $I\left(\beta \right)$ defined by Eq. (10), the light flux conservation law takes the form [6]

Differential Eq. (6) defining the function $\beta \left(\sigma \right)$ is obtained precisely from this law. Taking into account Eq. (11), we can express the light flux conservation law in the problem of generation of the 2D intensity distribution $I\left(\beta ,\gamma \right)$ as

Substituting the relations $\tau \left(\sigma \right)=\left[\sigma +\beta \left(\sigma \right)\right]/2$ and $\alpha \left(\sigma \right)=\pi /2-\left[\sigma -\beta \left(\sigma \right)\right]/2$ into Eq. (12) and taking into account Eq. (6), we obtain:

It can be shown that in the limiting case ${\gamma }_1\left(\beta \right)={\gamma }_2\left(\beta \right)=0$ the obtained functions $\beta \left(\sigma ,u\right)=\beta \left(\sigma \right)$, $\gamma \left(\sigma ,u\right)\equiv 0$ correspond to the case of generation of a line-shaped intensity distribution. This suggests that for a 2D domain with a small transverse size $\Delta ={\gamma }_2-{\gamma }_1$ (and with a high aspect ratio ${\beta }_d/\Delta$) the proposed source-target mapping will provide the solution to the problem of generation of a prescribed 2D intensity distribution $I\left(\beta ,\gamma \right)$ with acceptable accuracy.

Let us now consider the reconstruction of the mirror surface from the functions $\beta \left(\sigma \right)$ and $\gamma \left(\sigma ,u\right)$. In the general case, the mirror generating a prescribed 2D intensity distribution $I\left(\beta ,\gamma \right),\left(\beta ,\gamma \right)\in D$ corresponds to an envelope of a two-parameter family of paraboloids with the parameters $\left(\beta ,\gamma \right)\in D$. The envelope is defined by the equations [11]

Equation (15a) defines a paraboloid with the focal length $f\left(\beta ,\gamma \right)$ and the axis $p\left(\beta ,\gamma \right)$. Equations (15b) can be rewritten in the following form:

To restore the mirror surface, we write the derivatives of the function $G\left(\sigma ,u\right)=\ln f\left(\beta \left(\sigma \right),\gamma \left(\sigma ,\gamma \right)\right)$ with respect to $\left(\sigma ,u\right)$:

According to Eq. (16), the calculation of the function $G\left(\sigma ,u\right)$ is reduced to the restoration of the function from its exact (total) differential. This problem has an exact solution only when the condition ${\partial }^{2}G\left(\sigma ,u\right)/\partial \sigma \partial u\equiv {\partial }^{2}G\left(\sigma ,u\right)/\partial u\partial \sigma$ holds. The last identity corresponds to the integrability condition (the condition of the existence of a surface implementing the mapping $\beta =\beta \left(\sigma \right),\gamma =\gamma \left(\sigma ,u\right)$). For the obtained functions $\beta \left(\sigma \right)$ and $\gamma \left(\sigma ,u\right)$, the integrability condition strictly holds only in the limiting case $\gamma \left(\sigma ,u\right)=0$. Nevertheless, for a domain with high aspect ratio $\left({\beta }_d/\Delta >>1\right)$ one can expect that the error will be small. Therefore, we formally restore the function $G\left(\sigma ,u\right)$ from Eqs. (16). Let us note that $G\left(\sigma ,0\right)=\ln f\left(\sigma \right)$, where the function $f\left(\sigma \right)$ can be expressed through $\beta \left(\sigma \right)$ using Eqs. (4), (5). Thus, integrating the last equation in (16), we obtain:

Substituting $\beta =\beta \left(\sigma \right)$, $\gamma =\gamma \left(\sigma ,u\right)$ and $G\left(\sigma ,u\right)$ into Eq. (15a), we finally arrive to the following expression for the surface generating a 2D intensity distribution:

It is worth mentioning that in the existing papers, no analytical expressions for the mirror surfaces similar to Eq. (18) are presented. There, a numerical technique based on the sequential construction of a segmented surface from multiple planes is used for the reconstruction of the optical surface from the given source-target mapping [13]. In contrast to these works, we provide an analytical approach to the surface reconstruction.

4. Examples of the mirror calculation

To study the performance of the proposed method, we applied it to the calculation of several mirrors generating uniform intensity distributions in rectangular domains with different angular dimensions from ${80}^{\text{o}}\times {20}^{\text{o}}$ to ${20}^{\text{o}}\times {20}^{\text{o}}$. The mirrors were calculated using Eqs. (18) and (3) for an isotropic light source $\left(I_0\left(u,\sigma \right)\equiv I_0\right)$.

Before discussing the intensity distributions generated by the designed mirrors, let us illustrate the proposed source-target mapping for the case of the following illuminated domain: $D=\left[-30°,30°\right]\times \left[-10°,10°\right]$. Figures 3(a) and 3(b) depict one half of the mirror surface (at $x>0$) and the corresponding half of the illuminated domain D (at $\beta >0$). The angular dimensions of the mirror in Fig. 3(a) in the two central cross-sections corresponding to the xOz and yOz planes amount to $\sigma \in \left[-105°,105°\right]$ and $u\in \left[-110°,110°\right]$, respectively. The light source position is indicated by a small black sphere. The area $\Omega$ on a sphere surrounding the source indicates its emission solid angle. Similarly to the mirror surface, a half of this area is shown in Fig. 3(a). Thin solid curves in the area $\Omega$ depict the intersections between the ray cones $e\left(\sigma ,u\right)$ defined by Eq. (3) and the sphere at a set of equally spaced $\sigma$ values from the range $\sigma \in \left[0,105°\right]$. The curves $S\left(\sigma =\mathrm{const},u\right)$ on the mirror surface are also shown. In the proposed mapping, these curves have to be mapped to the corresponding segments $\gamma \in \left[-10°,10°\right]$ shown by solid horizontal lines in Fig. 3(b). It follows from Eq. (11) that equally spaced $\sigma$ values correspond to non-uniformly spaced solid angles $\Delta {\Omega }_c$ in the area $\Omega$. This leads to different sizes of “stripes” in the area $\Omega$ and mirror surface S and of the corresponding rectangles in the illuminated domain D.

 

Fig. 3 Theoretical ray mapping for a mirror generating uniform rectangular intensity distribution with angular dimensions of ${60}^{\text{o}}\times {20}^{\text{o}}$: one half of the mirror surface and the light source emission solid angle $\Omega$ (a), one half of the illuminated domain D (b). The light source is depicted with a small black sphere in Fig. (a). The mapping is indicated by color: the “stripes” in area $\Omega$ (and the corresponding “stripes” on the mirror surface) in Fig. (a) are mapped to the corresponding rectangles in Fig. (b). One particular pair of “stripes” in the area $\Omega$ and on the mirror surface as well as the corresponding rectangle in the illuminated domain are marked with red and connected by black arrows.

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The mapping is additionally indicated by color in Figs. 3(a) and 3(b): the “stripes” in area $\Omega$ (and the corresponding “stripes” on the mirror surface) in Fig. 3(a) are mapped to the corresponding rectangles in Fig. 3(b) having the same color. One particular pair of “stripes” in the area $\Omega$ and on the mirror surface as well as the corresponding rectangle in the illuminated domain are marked with red and connected by black arrows.

Let us now consider the performance of the designed mirrors generating uniform intensity distributions in rectangular domains with the following angular dimensions: ${80}^{\text{o}}\times {20}^{\text{o}}$, ${60}^{\text{o}}\times {20}^{\text{o}}$, ${40}^{\text{o}}\times {20}^{\text{o}}$, and ${20}^{\text{o}}\times {20}^{\text{o}}$ [Figs. 4(a)–4(d)]. Note that the mirror in Fig. 4(b) is the same as in Fig. 3(a). The intensity distributions generated by these mirrors in the case of a compact light source with the diameter of 0.06 mm are shown in Figs. 4(e)–4(h). Source positions in Figs. 4(a)–4(d) are depicted with black spheres; the distance from the source to the mirror vertex amounts to 1 mm for all examples. The intensity distributions in Figs. 4(e)–4(h) were calculated using the commercial software TracePro implementing a ray tracing technique [20]. The cross-sections of the obtained distributions demonstrate their high uniformity.

 

Fig. 4 (a)–(d) Mirrors generating uniform intensity distributions in rectangular domains with the following angular dimensions: ${80}^{\text{o}}\times {20}^{\text{o}}$ (a), ${60}^{\text{o}}\times {20}^{\text{o}}$ (b), ${40}^{\text{o}}\times {20}^{\text{o}}$ (c), ${20}^{\text{o}}\times {20}^{\text{o}}$ (d). Mirror dimensions along the coordinate axes: 12.90 x 4.94 x 3.42 mm (a), 9.42 x 4.92 x 2.77 mm (b), 7.41 x 4.91 x 2.39 mm (c), 6.12 x 4.89 x 2.15 mm (d). (e)–(h) Intensity distributions generated by the mirrors in Figs. (a)–(d) in the case of a compact light source with the diameter of 0.06 mm (depicted with black spheres in Figs. (a)–(d)). The distance from the source to the mirror vertex equals to 1 mm for all examples.

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RSD amounts to 6.2%, 7.1%, 8.5% and 13.5% for the distributions in Figs. 4(e)–4(h), respectively. Let us note that the presented technique is intended for the generation of illuminated domains with small transverse size (and with high aspect ratio). Despite this, the RSD in the case of generation of uniform intensity distribution becomes less than 10% even for the rectangular domain with the angular dimensions of ${40}^{\text{o}}\times {20}^{\text{o}}$ [Fig. 4(g)]. For a square domain of ${20}^{\text{o}}\times {20}^{\text{o}}$ (unity aspect ratio), the proposed method can still be used, even though high-intensity peaks (roughly 120% of the mean intensity value) are present at the corners of the illuminated domain.

It is interesting to compare the performance of the mirror in Figs. 4(d) and 4(h) with the mirror calculated using the well-known source-target mapping shown in Fig. 1(b) in [13]. In that case, the mirror is described in polar coordinates $\left(\phi ,\theta \right)\in \Omega$, and the mapping is constructed from the condition that the “meridians” (lines $\phi ={\phi }_0$ on a unit sphere in the domain $\Omega$) are transformed to radial lines $\gamma =\alpha \beta$ in the illuminated domain $D$. The mirror calculated on the basis of this mapping and the corresponding intensity distribution are shown in Fig. 5. The intensity distribution in Fig. 5(b) is significantly less uniform than the intensity distribution in Fig. 4(h). Let us note that the low-quality distribution in Fig. 5(b) is consistent with the results presented in [12,18] (in particular, Fig. 2 in [18]) demonstrating very limited applicability of this approach to the problem of mirror design.

 

Fig. 5 (а) Mirror calculated using the conventional source-target mapping (see Fig. 1(b) in [13]) generating uniform intensity distribution in a rectangular domain with the angular dimensions of ${20}^{\text{o}}\times {20}^{\text{o}}$. Mirror dimensions along the coordinate axes: 6.77 x 6.77 x 2.33 mm. (b) Intensity distribution generated by the mirror in the case of a compact light source with the diameter of 0.06 mm (depicted with a black sphere). The distance from the source to the mirror vertex equals to 1 mm.

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Thus, the mapping proposed in the present work demonstrates a much better performance even in the “worst” considered case when the aspect ratio is unity.

The developed method enables the calculation of the mirrors for the illumination of domains having different shapes. As an example, the mirrors generating uniform intensity distributions in an elliptical domain with angular dimensions of ${60}^{\text{o}}\times {20}^{\text{o}}$ and in a domain in the form of a half ellipse with angular dimensions of ${60}^{\text{o}}\times {10}^{\text{o}}$ were also calculated [Fig. 6]. The generated intensity distributions are shown in Figs. 6(c) and 6(d) and are highly uniform. For these distributions, RSD amounts to 5.4% and 6.5%, respectively.

 

Fig. 6 (a), (b) Mirrors generating uniform intensity distributions in an elliptical domain with angular dimensions of ${60}^{\text{o}}\times {20}^{\text{o}}$ (a) and in a domain in the form of a half ellipse with angular dimensions of ${60}^{\text{o}}\times {10}^{\text{o}}$ (b). Mirror dimensions along the coordinate axes: 9.51 x 4.92 x 2.77 mm (a) and 9.51 x 5.85 x 2.80 mm (b). (c), (d) Intensity distributions generated by the mirrors in Figs. (a), (b) in the case of a compact light source with the diameter of 0.06 mm (depicted with black spheres in Figs. (a), (b)). The distance from the source to the mirror vertex equals to 1 mm for both examples.

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5. Conclusion

A novel source-target mapping for the solution of the problem of generating prescribed 2D intensity distributions was proposed. This mapping is a generalization of the source-target mapping obtained from the solution of the problem of generating one-parameter intensity distributions. The surface of the mirror implementing the proposed mapping was expressed in an analytical form. The presented simulation results demonstrate high performance of the proposed method. In the case of generation of uniform rectangular and elliptical intensity distributions with angular dimensions from ${80}^{\text{o}}\times {20}^{\text{o}}$ to ${40}^{\text{o}}\times {20}^{\text{o}}$, the RSD of the obtained distribution did not exceed 8.5%. The proposed method can be extended to the calculation of refractive optical elements. In this case, the corresponding source-target mapping will be based on the analytical solution to the problem of calculation of refracting surfaces generating one-parameter intensity distributions [7,8].