Freeform and precise irradiance tailoring in arbitrarily oriented planes

Fanqi Shen; Lin Yang; Guangyin Hu; Zhanghao Ding; Jun She; Yu Zhang; Rengmao Wu

doi:10.1364/OE.445593

1. Introduction

The purpose of beam shaping is to redirect the light rays emitted from a given source to produce a prescribed irradiance/intensity distribution by a means of some elaborately designed optical surfaces [1]. Freeform surfaces are optical surfaces without linear or rotational symmetry [2]. Their flexible surface geometry offers high degrees of freedom, which can be used to avoid restrictions on surface geometry and create compact yet efficient designs with better performance [3–7]. Irradiance tailoring using freeform optics is a process of calculating a freeform surface (or multiple freeform surfaces) to produce a prescribed irradiance distribution in a target plane. The typical freeform lens designs for irradiance tailoring can be divided into two groups: non-tilted designs and tilted designs (see Fig. 1). In a non-tilted design, the optical axis of a beam shaping system is perpendicular to the target plane [8–28]. Due to the fact that the orthogonality of the target plane with respect to the optical axis could simplify irradiance tailoring, the non-tilted designs have been extensively investigated. However, the application of non-tilted designs may be relatively limited due to the perpendicular arrangement of a non-tilted design. Transcending the orthogonality of the target plane with respect to the optical axis leads to tilted designs [29]. Compared to the non-tilted designs, the tilted designs allow a more flexible geometrical arrangement of a beam shaping system and therefore have greater potential in practical applications. However, the tailoring of freeform surfaces in tilted designs is not a simple task due to the difficulties raised by the tilted arrangement. In our previous work [29], we imposed a planar symmetry restriction on the geometrical arrangement of the beam shaping system, where the tilted target plane is obtained by rotating a non-tilted plane around an axis of the coordinate system established at the light source [see Fig. 1(b)], and developed an algorithm for irradiance tailoring in the tilted target plane using one freeform surface [29]. Although the tilted designs presented in Ref. [29] significantly outperform the non-tilted designs in highly tilted geometry, the application of those tilted designs is still limited due to the planar symmetry restriction imposed on the geometrical arrangement of the beam shaping system and the limitation of one freeform refractive surface in those designs where the angles of deflection between the incident rays and their refracted ray are large. Breaking through this constraint will undoubtedly lead to a more flexible geometrical arrangement of the beam shaping system [see Fig. 1(c)]; however, freeform and precise irradiance tailoring in those tilted target planes without any symmetric constraints is more challenging and still faces many unresolved challenges.

Fig. 1. Typical geometrical arrangements of a beam shaping system. (a) The optical axis must be perpendicular to the target plane in non-tilted designs. (b) Although the optical axis is not perpendicular to the target plane, the geometrical arrangement of the system is still symmetric about a coordinate plane in the tilted case 1. For example, the geometrical arrangement of the system is symmetric about the global coordinate plane yz which coincides with the local coordinate plane y’z’ shown in this figure. (c) Breaking through the orthogonality and plane symmetry leads to the tilted case 2 which has a more flexible geometrical arrangement without any geometrical constraints. For better illustration, a global coordinate system xyz is placed at the light source with the z-axis coinciding with the optical axis, and a local coordinate system x’y’z’ is placed in the target plane.

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In this paper, we present a method for freeform and precise irradiance tailoring in arbitrarily oriented target planes in 3D space using multiple freeform surfaces. The main contributions of this work are trifold: (1) a universal algorithm for freeform irradiance tailor in tilted target planes without any geometrical constraints is developed; (2) the proposed method is applicable to the design of both one freeform surface and multiple freeform surfaces, and is valid in both near field and far field; (3) Fresnel losses occurring on the freeform refractive surfaces are considered and reasonable ray deflection on each freeform surface has been achieved, yielding high-performance beam shaping systems. The robustness and effectiveness of the proposed method are verified by both simulation and experimental results. The rest of this paper is organized as follows. We develop a general formulation for freeform and precise irradiance tailoring in three-dimensional (3D) space using freeform lenses in Section 2, and also introduce some characteristics of the proposed method in this section. Then, two interesting but challenging designs are presented in Section 3 to verify the effectiveness of the proposed method. After that, we conclude our work in Section 4.

2. Design methodology

The design of the freeform lens is shown schematically in Fig. 2. A point-like light source is placed at the origin of the global coordinate system xyz, and the optical axis of the beam shaping system coincides with the z-axis. A target plane is placed at B₃ which is the intersection point between the optical axis and the target plane. We assume that no constraints are imposed on the relative position of the target plane. That means the target plane could be an arbitrarily oriented plane in 3D space. A local coordinate system x’y’z’ is placed at B₃, and the x’y’ coordinate plane is located in the target plane. α denotes the angle between the x-axis and the projection of the z’-axis in the xy coordinate plane. β is the angle between the z’-axis and z-axis. Obviously, the tilt (the relative position) of the target plane in 3D space is fully determined by α and β. An incident ray emitted from the light source strikes the entrance surface of the freeform lens at P and hits the exit surface of the lens at Q(x,y,z). After refraction by the freeform lens, the outgoing ray intersects the arbitrarily oriented target plane at T, as shown in Fig. 2.

Fig. 2. Schematic diagram of the design of the freeform lens.

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We assume that r₁ is the distance between S and P, which is a function of the azimuthal angle θ and the polar angle φ. The position vector P of point P can be written as P = r₁×I₁. Here, I₁ is the unit vector of the incident ray. Then, the unit normal to the entrance surface at P is given by

(1)$${{\textbf N}_1} = ({{\textbf P}_\theta }\times {{\textbf P}_\varphi })/|{{\textbf P}_\theta }\times {{\textbf P}_\varphi }|$$

where P_θ and P_φ are the first-order derivatives of P with respect to θ and φ. Then, the unit vector O₁ of the outgoing ray at P can be calculated by use of Snell's law. We let I₂ denote the unit vector of the incident ray at Q, and r₂ denote the distance between P and Q. It is apparent that O₁=I₂, and r₂ is also a function of θ and φ. Similarly, the position vector Q of point Q can be written as Q = P + r₂×I₂, and the unit vector O₂ of the outgoing ray at Q can be calculated also by use of Snell's law. Obviously, the exit surface is fully determined by the entrance surface. After we obtain O₂, the coordinates of T in the xyz coordinate system can be written as

(2)$${t_x} = x + ({{t_z} - z} ){O_{2x}}/{O_{2z}} ,\,\, {t_y} = y + ({{t_z} - z} ){O_{2y}}/{O_{2z}}$$

where t_x, t_y, and t_z are the x-, y-, and z-coordinates of T; O_2x, O_2y and O_2z are the three components of O₂. We assume the position of the arbitrarily oriented target plane in 3D space is governed by the equation that Ax + By + Cz + D=0. Here, A = sinβcosα, B = sinβsinα, C = cosβ, D=-Lcosβ, and L is the distance between S and B₃. Obviously, A and B equals zero in a non-tilted design, and both A and B are non-zero in a tilted design without any symmetric constraints, which is the case shown in Fig. 1(c). Since T is located in the target plane, the coordinates of T should satisfy the equation of the target plane. Substituting Eq. (2) into the equation of the target plane gives us

(3)$$\left\{ \begin{array}{l} {t_x} = x - \frac{{({Ax + By + Cz + D} ){O_{2x}}}}{{A{O_{2x}} + B{O_{2y}} + C{O_{2z}}}} ,\\ {t_y} = y - \frac{{({Ax + By + Cz + D} ){O_{2y}}}}{{A{O_{2x}} + B{O_{2y}} + C{O_{2z}}}}, \\ {t_z} = \frac{{({A{O_{2x}} + B{O_{2y}}} )z - ({Ax + By + D} ){O_{2z}}}}{{A{O_{2x}} + B{O_{2y}} + C{O_{2z}}}} \end{array} \right.\textrm{ }$$

It is obvious that the coordinates of T in the global coordinate system xyz is fully governed by Eq. (3). Further, we transform the global coordinates of T into local coordinates with respect to the local coordinate system x’y’z’

(4)$$\left\{ \begin{array}{l} t{^{\prime}_x} = \frac{{Ex + Fy}}{C} - \frac{{({Ax + By + Cz + D} )({E{O_{2x}} + F{O_{2y}}} )}}{{C({A{O_{2x}} + B{O_{2y}} + C{O_{2z}}} )}}\\ t{^{\prime}_y} = Ey - Fx + \frac{{({Ax + By + Cz + D} )({F{O_{2x}} - E{O_{2y}}} )}}{{A{O_{2x}} + B{O_{2y}} + C{O_{2z}}}} \\ t{^{\prime}_z} = 0 \end{array} \right.\textrm{ }$$

where E = cosα and F = sinα. Equations (2)–(4) tell us that the location of T on the target plane is fully determined by P and Q, which indicates that Eq. (4) holds in both near field and far field. Therefore, it allows the proposed method to be valid in both near field and far field. We assume that the freeform lens is lossless. 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

(5)$$E({t{^{\prime}_x},t{^{\prime}_y}} )|{J({\textbf T} )} |- {I_1}({\theta ,\varphi } )\sin \varphi = 0$$

where E(t^’_x, t^’_y) is the irradiance produced by the output light beam at T, I₁ (θ, φ) is the intensity of the incident beam, and J(T) is the Jacobian matrix of the position vector T, which is given by

(6)$$J({\textbf T} )\textrm{ = }\left|\begin{array}{l} {{\partial t_x^{\prime}} / {\partial \varphi }} {{\partial t_x^{\prime}} / {\partial \theta }}\\ {{\partial t_y^{\prime}} / {\partial \varphi }} {{\partial t_y^{\prime}} / {\partial \theta }} \end{array} \right|$$

It should be noted that there are an infinite number of combinations of r₁ and r₂ which satisfy this equation, due to the nature of a prescribed irradiance design. Thus, how to determine r₁(θ, φ) and r₂(θ, φ) is a challenging problem facing us. To solve this problem, we need to predefine the entrance surface which could be an aspherical surface or a freeform surface. However, it does not mean r₁ can be arbitrarily selected. We will introduce a reasonable selection of the entrance surface later in this paper. With a predefined entrance surface, reorganizing and simplifying Eq. (5) yields an elliptical Monge–Ampère equation

(7)$${A_1}({{r_2}_{\theta \theta }{r_2}_{\varphi \varphi } - {r_2}_{^{\theta \varphi }}^2} )+ {A_2}{r_2}_{\varphi \varphi } + {A_3}{r_2}_{\theta \theta } + {A_4}{r_2}_{\theta \varphi } + {A_5} = 0$$

where the coefficients A_i (i=1,…,5) are functions of r₂_φ, r₂_θ, r₂, θ and φ. This equation tells us that those rays inside the domain of incident beam should satisfy Eq. (7). 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

(8)$$\left\{ \begin{array}{l} t{^{\prime}_x} = t{^{\prime}_x}({{r_2}_\theta ,{r_2}_\varphi ,{r_2},\theta ,\varphi } )\\ t{^{\prime}_y} = t{^{\prime}_y}({{r_2}_\theta ,{r_2}_\varphi ,{r_2},\theta ,\varphi } )\end{array} \right.:\textrm{ }\partial {\varOmega _{\textrm{ }1}} \to \partial {\varOmega _{\textrm{ }2}}\textrm{ }$$

where $ \partial \Omega_1$ and $ \partial\Omega_2 $ are the boundaries of the domains Ω₁ and Ω₂, on which I₁(θ,φ) and E(t’_x,t’_y) are defined. The derivation presented above shows that the freeform irradiance tailoring in arbitrarily oriented planes using two freeform surfaces can be formulated into an elliptical Monge–Ampère equation with a nonlinear boundary condition based on point source assumption, and the profiles of the freeform surfaces are fully determined by I₁(θ,φ), E(t’_x,t’_y) and the relative position of the target plane in 3D space. It is not a simple task to find an analytic solution due to the high nonlinearity of Eqs. (7) and (8). We have developed a numerical method to calculate a numerical solution to a Monge–Ampère equation in our previous publication [16], as shown in Fig. 3. This numerical method is very effective and easy to implement. Firstly, Ω₁ is discretized along the θ and φ directions, yielding a net of grid points. Since we have a partial differential equation (PDE) at each grid point, the discretization operation gives us a set of PDEs. Then, the PDEs are converted into a set of nonlinear equations after we use difference formula for derivatives in the PDEs. After that, the Newton’s method is employed to numerically solve the nonlinear equations, and yields a set of discrete data points of the freeform surface.

Fig. 3. Numerical calculation of the freeform surface.

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As mentioned above, we should obtain the entrance surface before calculating the exit surface. Although Eqs. (7) and (8) are established for the exit surface, these equations can still be used to design the entrance surface. When designing the entrance surface, we predefine an intermediate irradiance on a virtual plane, and place a virtual spherical surface between the source and the entrance surface with the center of curvature of the spherical surface coinciding with the source. Then, the spherical surface becomes a virtual entrance surface, and r₂ denotes the distance between point P and the striking point of the light ray on the virtual entrance surface. Calculating the entrance surface of the lens using Eqs. (7) and (8) becomes very straightforward. It is worth mentioning that this method is applicable to the design of multiple (more than two) freeform surfaces. When multiple freeform surfaces are employed, we predefine an intermediate irradiance for each freeform surface, and then the freeform surfaces are calculated in a one-by-one manner by repeating the use of Eqs. (7) and (8).

For irradiance tailoring using two freeform surfaces, there are an infinite number of combinations of r₁(θ,φ) and r₂(θ,φ) which could produce the prescribed irradiance distribution. Since the profile of the exit surface is fully governed by the entrance surface, the performance of the freeform lens also strongly depends upon the entrance surface. That means the entrance surface cannot be arbitrarily selected. Here, we propose a method to make a reasonable selection for the entrance surface. It is well known that Fresnel reflection occurs at refractive surfaces, and Fresnel loss caused by Fresnel reflection depends upon the index difference and the angle of incidence, as shown in Fig. 4(a). Thus, we can reduce the Fresnel loss of the freeform lens by reducing the angle of incidence of the light rays on the two freeform surfaces. Since the input intensity I₁(θ,φ) and the target irradiance E(t^’_x,t^’_y) are both predefined, the ray deflection caused by each freeform surface is strongly determined by the pattern size of the intermediate irradiance distribution. This allows us to make a reasonable selection for the entrance surface by choosing an appropriate size of the intermediate illumination pattern. As mentioned above, the profiles of the freeform surfaces are numerically calculated. This numerical calculation yields a set of discrete data points of the freeform surfaces, which allow us to calculate the angle of incidence at each data point and therefore the total Fresnel loss of the freeform lens. Further, we calculate the ratio of the total Fresnel loss of the freeform lens to the total power collected by the freeform lens, which can be written as

(9)$$R = \frac{{\sum\limits_{i = 1}^{{m_1}} {\sum\limits_{j = 1}^{{n_1}} {{I_1}({\theta _i},{\varphi _j})\{{{R_{en}}({\theta_i},{\varphi_j}) + {R_{ex}}({\theta_i},{\varphi_j})[{1 - {R_{en}}({\theta_i},{\varphi_j})} ]} \}} } }}{{\sum\limits_{i = 1}^{{m_1}} {\sum\limits_{j = 1}^{{n_1}} {{I_1}({\theta _i},{\varphi _j})} } }} \times 100\%$$

where m₁ and n₁ are the number of the sample points along the θ and φ directions, respectively; R_en(θ_ij,φ_j) denotes the Fresnel reflectivity at point P_i,j on the entrance surface and R_ex denotes the Fresnel reflectivity at point Q_i,j on the exit surface. As shown in Fig. 4(a), a smaller angle of incidence leads to a smaller Fresnel reflectivity. In order to reduce R to an acceptable level, we change the pattern size of the intermediate irradiance distribution and repeat the calculation of the total Fresnel loss, as shown in Fig. 4(b). Since the Newton iteration gives us the data points of the freeform surfaces and converges very fast [16], it is a simple task to calculate the total Fresnel loss. This algorithm shown in Fig. 4(b) can yield high-performance beam shaping systems and can be straightforwardly generalized to the design of multiple freeform surfaces.

Fig. 4. Control of Fresnel losses. (a) Fresnel reflectivity as a function of angle of incidence on the optical surface. The refractive index of the lens material is equal to 1.584. R_en denotes the Fresnel reflectivity at point P on the entrance surface and R_ex denotes the Fresnel reflectivity at point Q on the exit surface. The subscripts (s and p) denote the Fresnel reflectivity of s- and p-polarized light, respectively. The purple dashed line and the green solid line denote the average Fresnel reflectivity of s- and p-polarized light. (b) Flow chart of the algorithm for controlling Fresnel losses.

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3. Simulation and experimental results

Two interesting but challenging designs are presented in this section to show the effectiveness of the proposed method. In the first example, a freeform lens which includes two freeform surfaces is elaborately tailored to produce a prescribed uniform rectangular irradiance distribution on a tilted target plane. The geometrical arrangement of this design is depicted in Fig. 5(a). No symmetric restrictions are imposed on the geometrical arrangement of the beam shaping system. The optical axis of the freeform lens which is also the z-axis of the global coordinate system placed at the light source. The projection of the light source onto the target plane, which is point S’ in Fig. 5(a), lies on the boundary of the pattern. The pattern size equals 12m×10 m, the distance between S’ and the line CD which is also the boundary of the pattern equals 1.5 m, and SS’=5 m. The distance between the light source and the intersection point of optical axis and target plane equals 7.89 m, α=−35° and β=−50.7°. This again indicates that the target plane is a highly tilted plane in 3D space without any symmetric constraints.

Fig. 5. Simulation verification of the first design: (a) the geometrical arrangement of the beam shaping system; (b) the surface models; (c) the entrance surface profile, and (d) the exit surface profile; (e) the irradiance distribution on the target plane; (f) the normalized irradiance distributions along the line x₁=0 m and y₁=0 m; (g) illustration of the great potential application of the proposed method in tunnel and underpass lighting.

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The lens material is PMMA, and the light source is a Lambertian point-like source. The maximum collection angle of the freeform lens equals 120° which means φ_max=60°. The z-coordinates of the vertexes of the entrance and exit surfaces of the freeform lens (B₁ and B₂) equal 10 mm and 25 mm, respectively. Given these input design parameters, the entrance and exit surfaces of the freeform lens can be calculated by numerically solving Eqs. (7) and (8). The surface models are shown in Fig. 5(b), the entrance and exit surface profiles are depicted in Figs. 5(c) and 5(d). From these figures we can observe that the two freeform surfaces are smooth and therefore are easy to fabricate. It is worth mentioning that the Fresnel loss of the freeform lens has been reduced to an acceptable level with Ratio₁ of 10.13%. The green solid line and the purple dashed line depicted in Fig. 4(a) tell us that the minimum Fresnel reflectivity occurs when the angle of incidence equals zero. It means that the minimum R occurs when the angles of incidence of all light rays on the entrance and exit surfaces equal zeros. This is a case where both the entrance and exit surfaces are spherical surfaces with the two centers of curvature coinciding with the light source. Equation (9) can be used to calculate the minimum R in this special case, which is equal to 9.96% for a Lambertian light source with a half spreading angle of 60°. Obviously, R is always greater than 9.96% in freeform irradiance tailoring designs, and a R of 10.13% indicates that the Fresnel loss is controlled very well in the first design. Ten million rays are traced to reduce statistic noise caused by Monte Carlo ray tracing. The irradiance distribution produced by the freeform lens on the tilted target plane is shown in Fig. 5(e), and the normalized irradiance distributions along the lines x₁=0 m and y₁=0 m are given in Fig. 5(f). From Figs. 5(e) and 5(f) we can see a very good agreement between the actual irradiance distribution and the prescribed one. This design shows clearly that the proposed method allows us to do freeform and precise irradiance tailoring in a highly tilted plane with a flexible geometrical arrangement of the beam shaping system. It is of great interest to mention that the proposed method may have great potential application in tunnel and underpass lighting, as shown in Fig. 5(g). The flexible geometrical arrangement of the beam shaping system allows us to cast all the light beams emitted from the light source down to the ground in the direction of vehicle travel with a predefined light distribution. Therefore, the glare caused by the light beams coming from the luminaires and falling directly on our eyes, which is a big problem in the current tunnel and underpass lighting, can be dramatically reduced.

From the first design presented above we can see that freeform irradiance tailoring in a highly tilted plane in 3D space can be achieved by use of the proposed method. In the second example, a freeform lens which also includes two freeform surfaces is elaborately designed to produce a prescribed irradiance distribution on a compounded screen which consists of several highly tilted planes in 3D space. For better demonstrating the effectiveness of the proposed method, we assume that the compounded screen includes three tilted planes which are perpendicular to each other, as shown in Fig. 6(a). The light beams emitted from a Lambertian source after refraction by the lens produces a compounded irradiance which includes a complex pattern at the center of a square background with size of 300mm×300 mm in the three tilted planes, respectively. The ratio of irradiance of the complex patterns to that of the background equals 2.5:1. The three square illumination patterns are connected to avoid singularities on the freeform surfaces. The light source lies on the diagonal of the cube defined by the three illumination patterns. The optical axis of the freeform lens passes through the intersection of the three planes, and the distance between this intersection point and the light source is equal to 600 mm. β=54.7° for the three tilted planes, α equals 60°, 180° and 300°, respectively. The z-coordinates of B₁ and B₂, respectively, equal 10 mm and 22 mm. The other parameters remain unchanged.

Fig. 6. The second design: (a) the geometrical arrangement of the beam shaping system; (b) the prescribed irradiance distributions in the three target planes; and (c) the surface models of the freeform lens.

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Numerically solving Eqs. (7) and (8) yields a set of discrete data points. Here, we use 67951 data points to construct the entrance and exit surfaces of the freeform lens, respectively. Figure 6(c) gives the surface models, and the entrance and exit surface profiles are depicted in Figs. 7(a) and 7(b). From these figures we can clearly see the entrance and exit surfaces are geometrically smooth at every point on the surface, and all light rays pass through the freeform lens without any blockage. That means the freeform lens designed by the proposed method is highly efficient. In order to provide more physical insights into the irradiance tailoring in this challenging design, we calculate the Gaussian curvature of the exit surface, as shown in Fig. 7(c). Figure 7(c) and the definition of the Gaussian curvature [30] tell us that the power density transported by the radiation along the direction of propagation of outgoing rays is fully governed by the Gaussian curvature, which is also indicated by the Monge–Ampère equation in Eq. (7). Again, the algorithm given in Fig. 4(b) is employed here to reduce Fresnel loss. R=10.18%, indicating that the Fresnel loss is also controlled very well in the second design. We do Monte Carlo ray tracing to evaluate the performance of the freeform lens. One hundred million rays are traced to reduce statistic noise. The irradiance distributions in the three target planes are given in Fig. 7(d), and the irradiance distributions along the lines y_i=150 mm (i=1,2,3) are plotted in Fig. 7(e). From Figs. 7(d) and 7(e) we see a very good agreement between the simulated irradiance distribution and the prescribed one. Again, this design shows clearly the robustness and effectiveness of the proposed method in freeform irradiance tailoring in 3D space.

Fig. 7. Simulation verification of the second design: (a) the entrance surface profile, and (b) the exit surface profile; (c) the Gaussian curvature of the exit surface; (d) the irradiance distributions in the three target planes; and (e) the irradiance distributions along the lines y_i=150 mm (i=1,2,3).

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The freeform lens presented in the second design is fabricated by injection molding. The prototype of the freeform lens is shown in Fig. 8(a). A white LED light source is used here [31], and the experimental setup is shown in Fig. 8(b). The lighting effect produced by the prototype is given in Fig. 8(c). From Fig. 8(c) we clearly see the three complex patterns on the compounded screen.

Fig. 8. Experimental verification of the second design: (a) the freeform lens, (b) the experimental setup, and (c) the lighting effect produced by the prototype.

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The irradiance distribution on each target plane is recorded by use of a camera (Baumer VCXU-53C). Figure 9(a) gives the recorded illumination patterns. The normalized irradiance distributions of the illumination patterns are shown in Fig. 9(b), and the measured irradiance distributions along the lines y_i=150 mm (i=1,2,3) are plotted in Fig. 9(c). From Fig. 9 we can see that a good agreement between the experimental and the simulated results has been achieved, showing the effectiveness of the proposed method. The chromatic aberration, fabrication errors of the two freeform surfaces, and alignment errors are the main error sources that contribute to the differences between the actual illumination and the target.

Fig. 9. Experimental results of the second design: (a) the recorded patterns on the three target planes; (b) the normalized irradiance distributions of the recorded patterns; (c) the measured irradiance distributions along the lines y_i=150 mm (i=1,2,3). The red dashed line denotes the simulated irradiance distribution and the blue solid line denotes the tested one.

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4. Conclusion

In summary, we developed a general formulation for arbitrary and precise irradiance tailoring in 3D space using freeform lenses. Both the position of the arbitrarily oriented target plane in 3D space and Fresnel losses occurring on the freeform surfaces are considered. This method allows one to do freeform irradiance tailoring in arbitrarily oriented target planes without any symmetric restrictions on the geometrical arrangement of the system, and can yield high-performance beam shaping systems with new functions and flexible geometrical arrangements in 3D space. The robustness and effectiveness of the proposed method are verified by both simulation and experimental results. The proposed method is valid in both near field and far field, and can be straightforwardly generalized to the design of multiple freeform surfaces. This method is important from both mathematical and practical standpoints, and paves a way for the broad application of freeform optics.

Funding

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

Disclosures

The authors declare no competing financial interests.

Data availability

No data were generated or analyzed in the presented research.

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Freeform and precise irradiance tailoring in arbitrarily oriented planes

Abstract

1. Introduction

2. Design methodology

3. Simulation and experimental results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (9)

Optics Express