## Abstract

Freeform optics constitutes a new technology that is currently driving substantial changes in beam shaping. Most of the current beam shaping systems are elaborately tailored for fixed optical properties, which means the output light distribution of a beam shaping system usually cannot be changed. What we present here is a class of beam shaping systems, the optical properties of which can be changed to meet the requirements for different applications. The proposed beam shaping system is composed of a freeform lens and a non-classical zoom system which is designed by ray aiming and the conservation of energy instead of aberration control. The freeform lens includes two elaborately designed freeform optical surfaces, by which both the intensity distribution and wave-front of an incident light beam are manipulated in a desired manner. The light beam after propagating through the non-classical zoom system produces an illumination pattern on a fixed observation plane with a variable pattern size and an unchanged irradiance distribution at different zoom positions. Two design examples are presented to demonstrate the effectiveness of the proposed beam shaping systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Beam shaping, the art of controlling the intensity/phase distributions of a light beam, is an enabling technology used in various applications, such as automotive lighting [1–5], street lighting [6–10], laser beam shaping [11–13], super-resolution imaging [14–16], visible light communication [17,18], *et al*. In these applications, it may require different intensity distributions or phase profiles of light beam. The purpose of beam shaping is to produce a prescribed irradiance/intensity distribution with a phase profile by a means of some elaborately designed optical surfaces. Freeform surfaces are optical surfaces without linear or rotational symmetry [19] that can be used to avoid restrictions on surface geometry and create compact, yet efficient designs with better performance. More importantly, the use of freeform surfaces can produce new designs that cannot be achieved by the use of spherical or aspherical surfaces, satisfying the ever-growing demand for advanced beam shaping systems. Due to the unique merits of freeform surfaces, freeform optics constitutes a new technology that is currently driving substantial changes in beam shaping. The most current designs of freeform beam shapers can be broken into two groups: (1) only the intensity distribution of incident light beam is redistributed to produce a prescribed irradiance on a target plane or a prescribed intensity distribution in far field [20–33]; (2) both the intensity and phase distributions of incident light beam are redistributed to produce a prescribed irradiance distribution with a predefined output phase distribution [34–40]. The two different types of designs have one thing in common: the output light distribution produced by the freeform beam shapers is fixed for a given incident beam. That means, for a fixed output light distribution, the illumination pattern produced by the output light beam on a target plane cannot be changed once the working distance between the freeform beam shaper and the target plane is fixed. Consequently, the working distance has to be adjusted to change the illumination pattern size in order to match the different region of interest (RoI) on the target plane (see Fig. 1). However, significant changes in irradiance distribution usually cannot be avoided due to the change of lighting distance, and a remarkable change of lighting distance is usually not allowed in industrial applications due to cost control and ease of use. Obviously, those beam shaping systems with variable illumination properties, whose output light distribution can be easily adjusted without changing the working distance, are preferred. However, it is not a simple task to design this class of beam shaping systems. A variable-diameter refractive beam shaping system was presented in Ref. [41]. The Gaussian irradiance of an incident laser beam was converted into a circular uniform irradiance distribution with the capability to vary the spot diameter by laterally shifting two plano-freeform lenses. A zoom XY-beam expander based on movable freeform lenses was introduced in Ref. [42], which allows one to simultaneously vary the magnification in *x*- and *y*- direction, respectively. Li employed three movable aspherical lenses to convert a Gaussian beam into flat-top beams with different magnifications in Ref. [43]. A non-classical zoom system with a constant focal length but various image locations was presented in Ref. [44].

In this paper, we propose a class of beam shaping systems which produce variable illumination properties on a fixed target plane. The proposed beam shaping system consists of a freeform lens which includes two elaborately designed freeform optical surfaces and a non-classical zoom system which is designed by ray aiming and the conservation of energy instead of aberration control. The freeform lens is employed here to do freeform irradiance tailoring, and the illumination pattern produced by the freeform lens is further magnified in a desired manner by the non-classical zoom system. The illumination pattern produced by the beam shaping system on the target plane can be scaled up and down by moving the lens elements included in the zoom system to match the different regions of interest on the fixed target plane without changing the working distance. The rest of this paper is organized as follows. The proposed beam shaping system is introduced in Section 2, and the design methodologies of the freeform lens and the non-classical zoom system are presented in this section. Then, two design examples are presented in Section 3 to verify the effectiveness of the proposed beam shaping system and the design methodologies, and elaborate analyses of the two designs are also made in this section before we conclude our work in Section 4.

## 2. Beam shaping system and design methodology

The geometric layout of the proposed beam shaping system is given in Fig. 2. The proposed beam shaping system consists of two components: a freeform lens and a non-classical zoom system. The freeform lens includes two elaborately designed freeform optical surfaces, and the zoom system contains several lens elements. Due to the nature of a zoom system, at least three lens elements should be included in the zoom system. A divergent beam of light emanating from a point-like source becomes a parallel beam which is parallel to the optical axis (the *z*-axis in Fig. 2) with a predefined irradiance distribution after refraction by the entrance and exit surfaces of the freeform lens. Then, the parallel beam propagates through the zoom system and produce a prescribed illumination pattern on the target plane. Here, the position of the target plane is fixed, meaning the working distance between the target plane and the zoom system remains unchanged. For a different RoI on the target plane, the illumination pattern produced by the beam shaping system on the target plane can be scaled up and down to match the RoI by moving the lens elements included in the zoom system. For a conventional zoom system, it is known that an incident parallel beam emanating from an object point at infinity converges to a point on the image plane. Obviously, the zoom system included in the proposed beam shaping system is different from the conventional zoom systems. In the rest of this section, we will give more physical insights into the designs of the freeform lens and the non-classical zoom system.

#### 2.1 Design of the freeform lens

As mentioned above, the entrance and exit surfaces of the freeform lens are freeform surfaces and the divergent beam emanating from the point-like source becomes a parallel beam with a prescribed irradiance distribution after refraction by the two freeform surfaces. The design of the freeform lens is shown in Fig. 3. The light source is placed at the origin O of the coordinate system. A light ray emanating from the light source strikes the entrance surface of the freeform lens at point P and passes through the exit surface at point Q. After refraction by the freeform lens, the outgoing ray is incident normally on the virtual plane at point T. The virtual plane is placed somewhere between the freeform lens and the zoom system and is perpendicular to the *z*-axis. B_{1} and B_{2} are, respectively, the vertexes of the entrance and exit surfaces of the freeform lens. Obviously, both the intensity distribution and phase profile of the incident beam are controlled in a desired manner by the two freeform surfaces. In our previous publications, we demonstrated that the design of freeform illumination optics for controlling the intensity/phase distributions of a light beam can be converted into an Monge–Ampère equation with a nonlinear boundary condition by use of the conservation law of energy and Snell’s law [28–31,35,38]. Similarly, the conservation law of energy and Snell’s law are also employed to design the proposed freeform lens shown in Fig. 2. A brief introduction to the design process of the proposed freeform lens is presented here.

Since the divergent beam from the light source is converted into a parallel beam, the optical path length (OPL) between the light source and the virtual plane should be a constant, which can be written as

where*n*is the refractive index of the freeform lens and OP=

*ρ*. Here,

*ρ*is the radial distance between the origin O and P, which is a function of the azimuthal angle

*θ*and the polar angle

*φ*. We further assume that the illumination system is lossless, meaning neither Fresnel nor absorption losses are considered here. It means that the energy of an infinitesimal light beam is conserved when this infinitesimal light beam passes through the freeform lens. Then, the local energy conservation of the infinitesimal light beam can be written as where,

*t*and

_{x}*t*are

_{y}*x*- and

*y*- coordinates of T,

*I*(

*θ*,

*φ*) is the intensity of the incident light beam,

*E*(

*t*,

_{x}*t*) is the irradiance produced by the output light beam at T on the virtual plane; |J(

_{y}**T**)| is the determinant of the Jacobian matrix of the position vector

**T**[28], which is given by

*ρ*and

_{θ}*ρ*are the first-order derivatives of

_{φ}*ρ*with respect to

*θ*and

*φ*. The coefficients A

_{i}(i=1,…,5) are functions of

*ρ*

_{φ},

*ρ*

_{θ,}

*ρ*,

*θ*and

*φ*. Equation (4) is an elliptic Monge–Ampère equation and tells us that those rays inside the domain of incident beam should satisfy this equation. For those boundary rays, an additional condition should be defined to make the boundary rays strike the boundary of the illumination pattern after refraction by the freeform lens, which is given by

*t*is the

_{z}*z*-coordinate of T which is also the intersection point of the optical axis and the virtual plane; $\partial \Omega_{1}$ and $\partial \Omega_{2}$ are the boundaries of Ω

_{1}and Ω

_{2}which are, respectively, the domains on which

*I*(

*θ*,

*φ*) and

*E*(

*t*,

_{x}*t*) are defined. The boundary condition says that the incident boundary rays will be mapped to the boundary of the illumination pattern on the target plane. With this additional boundary condition, all the light rays from the source are fully well controlled. From the derivation presented above we can see the design of two freeform surfaces can still be formulated into an Monge–Ampère equation with a nonlinear boundary condition.

_{y}It should be noted that it is not a simple task to find an analytic solution due to the high nonlinearity of the equation. We have developed a numerical method to calculate a numerical solution to a Monge–Ampère equation in our previous publication [28,29]. Since this numerical method is very effective and easy to implement, we employ this method to find a numerical solution to Eqs. (4) and (5). After the entrance surface of the freeform lens is obtained, the exit surface can be numerically calculated by Eq. (1).

#### 2.2 Design of the non-classical zoom system

The zoom system introduced in Fig. 2 is not a typical optical imaging zoom system which is optimized for aberration control. The parallel beam emerging from the freeform lens produces a prescribed illumination pattern on the target plane after passing through the non-classical zoom system, and the pattern size can be scaled up and down by moving the lens elements in the zoom system. The ray aiming and conservation of energy are used here to design the proposed non-classical zoom system. The design process of the proposed zoom system is shown in Fig. 4. There are two design phases in this method. The purpose of the first phase is to find an initial point based on the first-order optics. Application of the first-order optics yields a set of nonlinear equations, and solving the nonlinear equations gives us a spherical lens system which is a starting point for subsequent optimization. In the second phase, a ray mapping which represents the relationship between the ray height of an incident ray on the virtual plane and that of an outgoing ray on the observation plane is established by use of the conservation of energy, the spherical surfaces are replaced by aspherical surfaces, and then a ray aiming based optimization is employed to improve the performance of the beam shaping system. More details about the proposed method are introduced below.

Due to the nature of a zoom system, a zoom system includes at least two lens elements. Undoubtedly, more lens elements included in the zoom system can provide more degrees of design freedom. In this paper, three lens elements are employed to build the non-classical zoom system. In the first design phase, the ideal lenses are used to represent the lens elements, as shown in Fig. 5. The distance *d*_{4} between the lens 3 and the target plane is fixed, and the distance between the virtual plane and the target plane also remains unchanged. The lens 1 and lens 2 can be moved along the optical axis to change *d*_{2} and *d*_{3}. As mentioned above, the light beam is incident normally on the virtual plane after refraction by the freeform lens. We trace a marginal ray which is parallel to the *z*-axis, as shown in Fig. 5. The ray is to be refracted by the three ideal lenses and strikes the target plane at a prescribed ray height *h*_{4}.

In the territory of paraxial optics [45], we know that the trajectory of this marginal ray between the lens 1 and the target plane can be described by the two recursion formulas

where*u*

_{i}is the slop of the marginal ray,

*h*

_{i}is the ray height at the

*i*-th lens element,

*d*

_{i}is the

*i*-th air space, and

*ϕ*

_{i}is the optical power of the

*i*-th lens element. Since the marginal ray is parallel to the

*z*-axis, we have that

*u*

_{0}=0. Thus, Eq. (6) can be written as

*u*

_{1},

*u*

_{2}and

*u*

_{3}, Eq. (7) becomes

*d*

_{1},

*d*

_{2}and

*d*

_{3}should satisfy the condition where

*d*

_{total}is a constant. It should be noted that the ray height

*h*

_{1}at the lens 1 remains unchanged; however, the ray height

*h*

_{4}are different at different zoom positions. Since the distance

*d*

_{4}between the lens 3 and the target plane is fixed, the extent of the illumination pattern can be represented by the angle

*γ*shown in Fig. 5. For ease of calculation, we could sample two zoom positions at

*γ*=

*r*and

_{min}*γ*=

*r*, respectively. It is apparent that Eqs. (8) and (9) should be hold at the two sampled zoom positions. That means the two zoom positions give us four equations with nine unknowns (

_{max}*ϕ*

_{1},

*ϕ*

_{2},

*ϕ*

_{3}, and two groups of

*d*

_{1},

*d*

_{2},

*d*

_{3}). Obviously, this system of equations is underdetermined and has infinitely many solutions, because there are more unknowns than equations. In this paper, the downhill simplex method is used to find a solution [46]. The downhill simplex method allows us to set a reasonable interval for the unknowns in order to avoid unreasonable parameters. When the optical powers and the air spaces are obtained, the three ideal lenses are replaced by three spherical lenses, and an initial zoom system is built. After that, we can proceed to the second design phase.

In the second design phase, we do a ray aiming based optimization instead of aberration control to optimize the zoom system. It is worth mentioning that the lens elements included in the proposed non-classical zoom system could be aspherical cylindrical lenses with linear symmetry or aspherical lenses with rotational symmetry. The use of aspherical cylindrical lenses in the zoom system allows us to elongate the illumination pattern only in one direction (e.g., the *x*-direction in the first example in Section 3). We assume that an incident parallel beam propagates in the + *z-*axis direction, and the incident beam has a uniform intensity within its circular cross section. Due to the nature of the proposed zoom system, we know that the irradiance distribution of the magnified illumination pattern is still uniform. When a magnified illumination pattern is predefined at a zoom position, it is a simple task to establish a ray mapping which represents the relationship between the ray height of an incident ray on the virtual plane and that of its outgoing ray on the observation plane by conservation law of energy. In the early optimization of the zoom system, several meridian rays which are parallel to the optical axis are uniformly sampled on the interval [0,*h*_{1}]. The target ray heights of these meridian rays at the target plane are calculated by use of the ray mapping. Then, the zoom system is optimized to minimize the merit function which represents the differences between the actual ray heights and the target heights

*Num*

_{1}is the number of zoom positions employed in the optimization of the zoom system,

*Num*

_{2}is the number of sampled meridian rays,

*x*is the ray height of the

_{k}*k*-th meridian ray on the target plane, and

*x*

_{ok}is the target ray height of the

*k*-th meridian ray. After the early optimization, more meridian rays are sampled and the spherical surfaces are changed to aspherical surfaces in the refinement optimization. The two design phases allow one to obtain a high-performance zoom system.

## 3. Design examples and discussions

In this section, two examples are given here to verify the effectiveness of the proposed beam shaping system and the proposed methods. In the first example, our goal is to design a beam shaping system which can produce a lath-shaped illumination pattern with uniform irradiance distribution along the length direction of the pattern. The length of the pattern can be scaled up and down by moving the lens elements included in the zoom system to meet different requirements; however, the width of the pattern is assumed to be unchanged. This beam shaping system may have great potential application in machine vision and flexible-display production [47]. Due to the large aspect ratio of the lath-shaped illumination pattern, for ease of design, we assume that the light beam after refraction by the freeform lens produces a square illumination pattern on the virtual plane with size of 10*mm*×10 *mm*. The irradiance distributions along the length and width directions of the pattern are, respectively, controlled in a desired manner. After that, the parallel beam is to be spread out by a non-classical zoom system which consists of three aspherical cylindrical lenses, to produce the predefined lath-shaped illumination pattern on a fixed target plane. The light source is a Lambertain emitter. The other parameters are listed in Table 1.

Due to the C_{0} continuity of the piecewise boundary of a square pattern, the boundary of the square illumination pattern is approximated by a superellipse in order to avoid unreasonable restrictions on the propagation of boundary rays. The profiles of the freeform lens are calculated by numerically solving Eqs. (4) and (5). Figure 6 gives the entrance and exit surfaces of the freeform lens. Ten million rays are traced to reduce statistic noise. The illumination pattern produced on the virtual plane and the irradiance distributions along the lines *x*=0 *mm* and *y*=0 *mm* are depicted in Fig. 7(a). From Fig. 7(a), we can see a good agreement between the actual irradiance distribution and the target.

After the freeform lens is obtained, we then proceed to the design of the non-classical zoom system. We assume that the distance *d*_{4} between the lens 3 and the target plane equals 500 *mm*, *d*_{total}=50 *mm* and *h*_{1}=5 *mm*. In the second phase of the proposed design method presented in Fig. 4, we trace 20 meridian rays which are parallel to the optical axis and uniformly sampled on the interval [0 *mm*,5 *mm*]. The zoom system is optimized at three different zoom positions with γ=20°, 30°, and 40°, respectively. Figure 8(a) shows the optimized zoom system, and the zoom curves of the non-classical zoom system are depicted in Fig. 8(b). The optimized parameters of the zoom system at the zoom position 1 are given in Table 2, and the values of *d*_{1}, *d*_{2} and *d*_{3} at three zoom positions are listed in Table 3. The overall length of the zoom system is less than 40 *mm*.

After both the freeform lens and the zoom system are ready, the two subsystems are integrated to build the beam shaping system. Ten million rays are traced to evaluate the performance of the beam shaping system. Figures 7(b)–7(d) give the illumination patterns and the irradiance distributions along the lines *x*=0 *mm* and *y*=0 *mm* at Zoom 1, 2 and 3. The fractional RMS is employed here to quantify the uniformity of the irradiance distribution along the line *y*=0 *mm*, which is defined by

*num*is the number of sample points,

*E*is the irradiance value at the

_{q}*q-*th sampling point, and

*E*is the mean value of irradiance. A smaller value of RMS represents a better uniformity. Here, 128 points are sampled on the interval –

_{a}*h*

_{4}≤

*x*≤

*h*

_{4}to compute RMS. From Figs. 7(b)-7(d), we know that RMS=0.0062 at Zoom 1, RMS=0.0081 at Zoom 2, and RMS=0.0071 at Zoom 3, indicating a high uniformity at each zoom position. We also evaluate the performance of the beam shaping system at the other two zoom positions which are not used in the optimization of the zoom system, as shown in Figs. 7(e) and 7(f). γ, respectively, equals 24.58° and 34.63° at the two zoom positions. Figures 7(e) and 7(f) tell us that RMS=0.0120 at Zoom 4 and RMS=0.0168 at Zoom 5. From Fig. 7 we can clearly see that the lath-shaped illumination pattern produced by the beam shaping system on the target plane can be scaled up and down by moving the lens 1 and lens 2 to match the different regions of interest on the fixed target plane without changing the working distance between the beam shaping system and the target plane. It is of great interest to mention that the uniformity of the irradiance distribution along the line

*y*=0

*mm*almost remains unchanged at different zoom positions.

A common way to generate a lath-shaped illumination pattern is to extend a circular illumination pattern produced by a collimated beam along the length direction of the lath-shaped pattern. The collimated beam is usually generated by placing a light source at the front focal plane of a spherical/aspherical lens. This conventional design may have two main limitations: (1) the uniformity of the irradiance distribution along the length direction of the lath-shaped pattern is very limited due to the limited uniformity of the circular pattern produced by a collimation lens; (2) high power consumption cannot be avoided. Figure 9 shows the comparison between the proposed beam system presented above and the conventional design. For ease of calculation, we assume a uniform circular illumination is produced by the collimated beam. It should be noted that a uniform circular pattern requires relatively low power consumption as compared to a non-uniform circular pattern. We define a rectangle with size of *L*_{1}×*dy* on the circular and square patterns shown in Fig. 9(a) and 9(c). The center of the rectangle coincides with the centers of the circular and square patterns, respectively. Then, the two different light beams propagate through an identical non-classical zoom system, and the elongated rectangles shown in Figs. 9(b) and 9(d) have an identical size of *L*_{2}×*dy*. We assume the width of the rectangle, *dy*, is small so that the irradiance distribution on the elongated rectangular region can be represented by the irradiance distribution along the line *y*=0 *mm*. If identical mean irradiance values of the two elongated rectangular region in Figs. 9(b) and 9 (d) are required, we know that the ratio of radiant power of the proposed beam system to that of the conventional design equals 0.46. It means that the elaborately designed freeform lens included in the proposed beam shaping system leads to a lower power consumption. This is of great significance in practical applications.

In the second design, a uniform rectangular illumination with aspect ratio of 2:1 is to be produced on a fixed target plane. The illumination pattern can be scaled up and down at different zoom positions and the uniform irradiance distribution remains unchanged. The non-classical zoom system designed in the first example is also employed here. The three aspherical cylindrical lenses included in the zoom system are replaced by three aspherical lenses which are constructed by rotating the profiles of the cylindrical lenses around the *z*-axis. It means that the zoom system is rotationally symmetric about the *z*-axis, and therefore the boundary shape of the magnified illumination pattern is fully governed by the freeform lens rather than the zoom system. The other parameters are kept unchanged. The diagonal of the uniform rectangular pattern produced by the freeform lens on the virtual plane equals 10 *mm*, which is equal to the entrance pupil diameter of the zoom system. Figure 10 gives the entrance and exit surfaces of the freeform lens. From this figure we can clearly see plane symmetry of the freeform lens. Ten million rays are traced, and the irradiance distributions produced by the freeform lens on the virtual plane are given in Fig. 11(a). This figure tells us that the actual irradiance distribution agrees well with the target.

Figures 11(b)–11(d) give the illumination patterns and irradiance distributions along the line *x*=0 *mm* and *y*=0 *mm* at Zoom 1, 2 and 3. 128 × 128 sample points are used to calculate the RMS which represents the uniformity of the illumination pattern. From these figures we know that RMS=0.0154 at Zoom 1, RMS=0.0160 at Zoom 2, and RMS=0.0090 at Zoom 3, indicating a high uniformity at each zoom position. Similarly, we also evaluate the performance of the beam shaping system at Zoom 4 and 5 which are not used in the optimization of the zoom system, as mentioned above. Figures 11(e) and 11(f) tell us that RMS=0.0149 at Zoom 4 and RMS=0.0206 at Zoom 5. Figure 11 tells us that the beam shaping system produces a uniform rectangular illumination pattern on a fixed observation plane with a variable pattern size and an unchanged irradiance distribution at different zoom positions. It is of great interest to mention that the proposed method allows us to create illumination patterns with more complex boundary shapes, because the boundary shape of a magnified illumination pattern is fully governed by the freeform lens rather than the zoom system, and the Monge–Ampère equation method introduced in subsection 2.1 is very effective in freeform irradiance tailoring [30,31].

## 4. Conclusion

In this paper, we present the optical design of a class of beam shaping systems which can produce variable illumination properties on a fixed target plane. The beam shaping system is composed of a freeform lens and a non-classical zoom system. The freeform lens consists of two elaborately designed freeform optical surfaces which allow us to control both the intensity and phase distributions of an incident light beam. The non-classical zoom system, which is designed by the conservation of energy and ray aiming, is not a typical imaging system. The illumination pattern produced by the beam shaping system can be scaled up and down by moving the lens elements included in the zoom system to match the different regions of interest on the fixed target plane without changing the working distance. The designs of the freeform lens and the non-classical zoom system are introduced in detail. Two high-performance beam shaping systems are presented to show the effectiveness of the proposed method. This class of beam shaping systems may have great potential in industrial applications.

## Funding

National Natural Science Foundation of China (11804299, 12074338, 62022071); “the Fundamental Research Funds for the Zhejiang Provincial Universities” (2021XZZX020).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

No data were generated or analyzed in the presented research.

## References

**1. **F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, and S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express **18**(20), 20926–20938 (2010). [CrossRef]

**2. **C. Hung, Y. Fang, M. Huang, B. Hsueh, S. Wang, B. Wu, W. Lai, and Y. Chen, “Optical design of automotive headlight system incorporating digital micromirror device,” Appl. Opt. **49**(22), 4182–4187 (2010). [CrossRef]

**3. **C. Hsieh, Y. Li, and C. Hung, “Modular design of the LED vehicle projector headlamp system,” Appl. Opt. **52**(21), 5221–5229 (2013). [CrossRef]

**4. **P. Ge, Y. Li, Z. Chen, and H. Wang, “LED high-beam headlamp based on free-form microlenses,” Appl. Opt. **53**(24), 5570–5575 (2014). [CrossRef]

**5. **W. Sun, C. Tien, W. Lo, and P. Chu, “Optical design of an LED motorcycle headlamp with compound reflectors and a toric lens,” Appl. Opt. **54**(28), E102–E108 (2015). [CrossRef]

**6. **W. B. Elmer, “A Study in Street Lighting Reflector Design,” Appl. Opt. **5**(2), 343–349 (1966). [CrossRef]

**7. **X. Lee, I. Moreno, and C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express **21**(9), 10612–10621 (2013). [CrossRef]

**8. **I. Moreno, M. Avendano-Alejo, T. Saucedo-A, and A. Bugarin, “Modeling LED street lighting,” Appl. Opt. **53**(20), 4420–4430 (2014). [CrossRef]

**9. **Z. Feng, Y. Luo, and Y. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express **18**(21), 22020–22031 (2010). [CrossRef]

**10. **S. Wang, K. Wang, F. Chen, and S. Liu, “Design of primary optics for LED chip array in road lighting application,” Opt. Express **19**(S4), A716–A724 (2011). [CrossRef]

**11. **F. M. Dickey and S Holswade, *Laser Beam Shaping Applications* (CRC Press, Boca Raton, FL2006), Chapter 8.

**12. **S. Zhang, G. Neil, and M. Shinn, “Single-element laser beam shaper for uniform flat-top profiles,” Opt. Express **11**(16), 1942–1948 (2003). [CrossRef]

**13. **J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. **27**(16), 1463–1465 (2002). [CrossRef]

**14. **P. W. Winter, A. G. York, D. D. Nogare, M. Ingaramo, R. Christensen, A. Chitnis, G. H. Patterson, and H. Shroff, “Two-photon instant structured illumination microscopy improves the depth penetration of super-resolution imaging in thick scattering samples,” Optica **1**(3), 181–191 (2014). [CrossRef]

**15. **H. Luo, H. Yu, Y. Wen, T. Zhang, P. Li, F. Wang, and L. Liu, “Enhanced high-quality super-resolution imaging in air using microsphere lens groups,” Opt. Lett. **45**(11), 2981–2984 (2020). [CrossRef]

**16. **Y. Xue and R. T. C. So, “Three-dimensional super-resolution high-throughput imaging by structured illumination STED microscopy,” Opt. Express **26**(16), 20920–20928 (2018). [CrossRef]

**17. **K. Wang, A. Nirmalathas, C. Lim, and E. Skafidas, “Impact of background light induced shot noise in high-speed full-duplex indoor optical wireless communication systems,” Opt. Express **19**(22), 21321–21332 (2011). [CrossRef]

**18. **S. Kim, “Visible Light Communication Employing Optical Beamforming: A Review,” Curr. Opt. Photon. **2**(4), 308–314 (2018). [CrossRef]

**19. **J. Reimers, A. Bauer, K. P. Thompson, and J. P. Rolland, “Freeform spectrometer enabling increased compactness,” Light Sci Appl **6**(7), e17026 (2017). [CrossRef]

**20. **Z. Feng, D. Cheng, and Y. Wang, “Iterative wavefront tailoring to simplify freeform optical design for prescribed irradiance,” Opt. Lett. **44**(9), 2274–2277 (2019). [CrossRef]

**21. **K. Desnijder, P. Hanselaer, and Y. Meuret, “Ray mapping method for off-axis and non-paraxial freeform illumination lens design,” Opt. Lett. **44**(4), 771–774 (2019). [CrossRef]

**22. **L. B. Romijn, J. H. M. T. Boonkkamp, and W. L. Ijzerman, “Inverse reflector design for a point source and far-field target,” J. Comput. Phys. **408**, 109283 (2020). [CrossRef]

**23. **D. H. Lippman and G. R. Schmidt, “Prescribed irradiance distributions with freeform gradient-index optics,” Opt. Express **28**(20), 29132–29147 (2020). [CrossRef]

**24. **Y. Ding, X. Liu, Z. Zheng, and P. Gu, “Freeform LED lens for uniform illumination,” Opt. Express **16**(17), 12958–12966 (2008). [CrossRef]

**25. **Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express **18**(9), 9055–9063 (2010). [CrossRef]

**26. **F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express **18**(5), 5295–5304 (2010). [CrossRef]

**27. **M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of high-efficient freeform LED lens for illumination of elongated rectangular regions,” Opt. Express **19**(S3), A225–233 (2011). [CrossRef]

**28. **R. Wu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “A mathematical model of the single freeform surface design for collimated beam shaping,” Opt. Express **21**(18), 20974–20989 (2013). [CrossRef]

**29. **R. Wu, K. Li, P. Liu, Z. Zheng, H. Li, and X. Liu, “Conceptual design of dedicated road lighting for city park and housing estate,” Appl. Opt. **52**(21), 5272–5278 (2013). [CrossRef]

**30. **R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. **38**(2), 229–231 (2013). [CrossRef]

**31. **R. Wu, L. Yang, Z. Ding, L. Zhao, D. Wang, K. Li, F. Wu, Y. Li, Z. Zheng, and X. Liu, “Precise light control in highly tilted geometry by freeform illumination optics,” Opt. Lett. **44**(11), 2887–2890 (2019). [CrossRef]

**32. **A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express **21**(9), 10563–10571 (2013). [CrossRef]

**33. **R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express **22**(S2), A552–A560 (2014). [CrossRef]

**34. **K. Desnijder, P. Hanselaer, and Y. Meuret, “Flexible design method for freeform lenses with an arbitrary lens contour,” Opt. Lett. **42**(24), 5238–5241 (2017). [CrossRef]

**35. **L. Yang, Y. Liu, Z. Ding, J. Zhang, X. Tao, Z. Zheng, and R. Wu, “Design of freeform lenses for illuminating hard-to-reach areas through a light-guiding system,” Opt. Express **28**(25), 38155–38168 (2020). [CrossRef]

**36. **Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express **21**(23), 28693–28701 (2013). [CrossRef]

**37. **S. Wei, Z. Zhu, Z. Fan, Y. Yan, and D. Ma, “Double freeform surfaces design for beam shaping with non-planar wavefront using an integrable ray mapping method,” Opt. Express **27**(19), 26757–26771 (2019). [CrossRef]

**38. **S. Chang, R. Wu, L. An, and Z. Zheng, “Design beam shapers with double freeform surfaces to form a desiredwavefront with prescribed illumination pattern by solving a Monge-Ampère type equation,” J. Opt. **18**(12), 125602 (2016). [CrossRef]

**39. **C. Bösel and H. Gross, “Double freeform illumination design for prescribed wavefronts and irradiances,” J. Opt. Soc. Am. A **35**(2), 236–243 (2018). [CrossRef]

**40. **V. I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mech. Anal. **201**(3), 1013–1045 (2011). [CrossRef]

**41. **P. J. Smilie and T. J. Suleski, “Variable-diameter refractive beam shaping with freeform optical surfaces,” Opt. Lett. **36**(21), 4170–4172 (2011). [CrossRef]

**42. **F. Duerr and H. Thienpont, “Analytic design of a zoom XY-beam expander with freeform optical surfaces,” Opt. Express **23**(23), 30438–30447 (2015). [CrossRef]

**43. **W. Li, S. Wei, Z. Fan, Z. Zhu, and D. Ma, “Variable-diameter beam-shaping system design with high zoom ratio containing aspheric optical components,” Appl. Opt. **60**(3), 705–713 (2021). [CrossRef]

**44. **S. Chang, X. Tao, R. Wu, Y. Sun, and Z. Zheng, “Generation of propagation-invariant and intensity-controlled dark hollow beams by a refractive beam shaping system,” Opt. Express **26**(20), 26385–26399 (2018). [CrossRef]

**45. **J. M. Geary, *Introduction to lens design: with practical ZEMAX examples* (Willmann-Bell: Richmond, 2002), Chap. 4.

**46. **R. J. Koshel, “Enhancement of the downhill simplex method of optimization,” in International Optical Design Conference, 2002 OSA Technical Digest Series (Optical Society of America, 2002), paper ITuC2.

**47. **B. Cheon, J. Kim, and M. Oh, “Plastic optical touch panels for large-scale flexible display,” Opt. Express **21**(4), 4734–4739 (2013). [CrossRef]