## Abstract

Incoherent collimated beam has a wide application, and reshaping the collimated beam with freeform optics has become a popular and challenging topic of noniamging design. In this paper, we address this issue, embedded in three-dimensional space without any symmetry, with a freeform surface from a new perspective. A mathematical model is established for achieving the one-freeform surface design based on the problem of optimal mass transport. A numerical technique for solving this design model is disclosed for the first time, and boundary conditions for balancing light are presented. Besides, some key issues in achieving complex illuminations are addressed, and the influence of caustic surface on this design model is also discussed. Design examples are given to verify these theories. The results show elegance of the design model in tackling complex illumination tasks. The conclusions obtained in this paper can be generalized to achieve LED illumination and tackle multiple freeform surfaces illumination design.

© 2013 OSA

## 1. Introduction

Collimated beam shaping requires that a collimated beam should be directed to produce a target illumination. This technique has a wide range of applications. For example, as an important application, this technique is usually employed in an exposure system to produce an off-axis illumination to double the resolution and improve the depth of focus and image contrast [1–3]. Compared with the conventional technologies, such as the pupil filtering and the diffractive optical elements, the freeform surface becomes more attractive in reshaping the collimated beam due to its high degrees of design freedom that can be used to achieve a compact design with an excellent optical performance. Deriving freeform surfaces to direct light from a given source to a target is an inverse design problem. If it is only desired to control the irradiance distribution at the target, one freeform surface is sufficient. Generally, there are two main kinds of methods for solving this problem: optimization design and “Partial Differential Equation (PDE)” method. Optimization design is an iteration of successively finding some appropriate variable values to reduce the merit function with certain optimization algorithms to achieve a better design. Thousands of rays are usually needed to reduce statistic noise during the Monte-Carlo raytracing optimization process, and the optimization results are strongly determined by the initial design and robustness of the optimization algorithm [4–6]. Besides, it is not possible for an optimization method to generate a complex illumination with hundreds of optimization variables. So, the optimization method is not the best choice in some freeform surface design problems. For PDE method, the main idea is to establish a set of first-order partial differential equations or a second-order partial differential equation, by which the freeform surface is governed, to represent this inverse problem [7–17]. Compared with the optimization design method, PDE method is more efficient and can be used to tackle complex illumination tasks. When the inverse problem is converted into a set of first-order partial differential equations, a key step is to find an energy mapping between the light source and the target illumination [7–11]. Continuity of the freeform surface is strongly determined by integrability of the mapping [7–11]. A smooth freeform surface can only be obtained with an integrable mapping, however, it is very difficult to find such an integrable mapping [9]. This inverse problem can also be represented by an elliptic Monge-Ampére equation which is a nonlinear second-order partial differential equation [12–17]. With this method, the energy mapping is not required. Unfortunately, numerical techniques for solving this equation have not been detailedly disclosed so far [12,13], or only weak solutions can be obtained by iteratively determining lots of pieces of paraboloids (ellipsoids or hyperboloids) [14–17]. This kind of method is still covered with a veil of mystery.

In this paper, we focus on the one-freeform surface design for the collimated beam shaping, and will establish an effective and complete mathematical model for solving this problem from a new perspective based on a novel design idea reported in [18]. We will disclose for the first time a numerical technique for solving this mathematical problem. And, boundary conditions for balancing light in such a design will be detailed. Also, some key issues in achieving high resolution designs will be addressed and the influence of caustic surface on this design model will be detailedly discussed. The mathematical model, the numerical technique and these conclusions obtained in this paper can be easily generalized to tackle the problem of freeform illumination design in other applications (for example, the LED freeform optics). This paper is organized as follows; Section 2 briefly introduce the relationship between the problem of freeform surface illumination design and the problem of optimal mass transport, and detailedly presents how to establish a mathematical model for the one-freeform surface design problem of the collimated beam shaping from the Snell’s law and the conservation law of energy. A numerical technique for solving this mathematical problem is introduced in Section 3. Also in this section, boundary conditions for balancing light in such a design are discussed. In Section 4, design examples are given to verify this design model, and elaborate analyses of the optical performance of this design are given. Then, some key issues in achieving high resolution designs and the influence of caustic surface on this design model are discussed in Section 5, before concluding the paper.

## 2. Establish the mathematical model

With a single freeform surface, the intensity I(*x*,*y*) of the collimated beam is redistributed to produce a target E(*t _{x}*,

*t*) on a given illumination plane and the energy is conserved during the beam-shaping process, as shown in Fig. 1.

_{y}*S*

_{1}and

*S*

_{2}denote the cross section of the collimated beam and the illumination area, respectively. ∂

*S*

_{1}and ∂

*S*

_{2}are, respectively, the boundaries of

*S*

_{1}and

*S*

_{2}. Figure 1 indicates that not only the target irradiance distribution should be produced, but also the boundary of the illumination pattern should be ensured by the freeform surface during the beam-shaping process. Such a beam-shaping process is similar to the problem of optimal mass transport [19]. In this section, we show how to establish a mathematical model for this beam-shaping process from the Snell’s law and the conservation law of energy.

Assume that the entrance surface of the freeform lens is planar, and the exit surface of the lens is a freeform surface. A Cartesian coordinate system with *z*-axis along the optical axis is established, as shown in Fig. 2. An arbitrary ray of the collimated beam intersects the freeform surface at point P, and then is refracted to point T on the target plane. Since the incident beam propagates along the *+ z*-axis, the unit vector of the incident ray **I** = (0,0,1). The coordinates of point P are assumed to be (*x*,*y*,*z*(*x*,*y*)). Then, the unit normal vector **N** of the freeform surface at point P is given by

*z*and

_{x}*z*are the first-order partial derivatives of the coordinate

_{y}*z*with respect to

*x*and

*y*, respectively. According to the Snell’s law, the relationship between the unit vector

**I**of the incident ray, the unit vector

**O**of the emergent ray and the unit normal vector

**N**at point P could be expressed aswhere

*n*is the refractive index of the medium surrounding the freeform lens, and

_{o}*n*is the refractive index of the freeform lens. Then, for a freeform reflector in air

_{i}*n*= −1 and

_{o}*n*= 1 hold. The parameter

_{i}*P*

_{1}is given by

Obviously, *P*_{1} is negative for both freeform reflector design and freeform lens design. According to Eqs. (1)-(3), we obtain the unit vector **O**

**T**)| is the Jacobian matrix of the position vector

**T**. The quantity 1/|J(

**T**)| represents the expansion (or contraction) of an infinitesimal tube of rays due to the refraction of the freeform surface. Without loss of energy, then we can obtain a local energy conservation which is given byIntegrating this equation over the domain

*S*

_{1}yields

Equation (8) is a second-order partial differential equation. Simplify this equation, and we obtain

## 3. Boundary conditions and numerical technique for solving the mathematical model

As highlighted above, the nonlinear boundary condition specifies that the incident rays on $\partial {S}_{1}$ is refracted by the freeform surface to $\partial {S}_{2}$. Assume that the boundary of the illumination pattern has an analytical formula *f*(*t _{x}*,

*t*). According to the boundary condition, the intercept points of the boundary incident rays on the target plane should satisfy

_{y}*f*(

*t*,

_{x}*t*). Take an elliptical illumination pattern for example. Since the boundary of the pattern is elliptical, the boundary condition can be expressed as

_{y}*a*and

*b*are one-half of the ellipse's major and minor axes respectively; the point (

*C*,

_{x}*C*) represents the center of the ellipse, and the point (

_{y}*t*) is the intercept point defined by Eq. (6). This equation shows clearly that the specific position of each boundary ray on $\partial {S}_{2}$ is not predefined.

_{x},t_{y}After establishing an appropriate boundary condition, we use a numerical technique, which has not been disclosed before, to solve this design model. For such a design model, only the numerical solution can be obtained. First, we have to discretize the elliptic Monge-Ampére equation and the nonlinear boundary condition. Take the discretization of a rectangular domain *S*_{1} as an example. Assume the domain *S*_{1} = {(*x*,*y*)|*x*_{min}≤*x*≤*x*_{max}, *y*_{min}≤*y*≤*y*_{max}}. Discretizing this rectangular domain yields a net of grid points *S*_{1} = {(*x _{i}*,

*y*)|

_{j}*x*=

*x*

_{min}+

*ih*

_{1},

*y*=

*y*

_{min}+

*jh*

_{2},

*i*= 0,1,…,

*m*;

*j*= 0,1,…,

*n*}.

*h*

_{1}= (

*x*

_{max}-

*x*

_{min})/

*m*and

*h*

_{2}= (

*y*

_{max}-

*y*

_{min})/

*n*are, respectively, the spacing in the

*x*-axis and the

*y*-axis, and gridding will align parallel to the

*x*and

*y*coordinate system, as depicted in Fig. 3. The gridd points on $\partial {S}_{1}$ are called the boundary points, and the grid points inside

*S*

_{1}are called the interior points. So, each interior point should satisfy the elliptic Monge-Ampére equation, and each boundary point should satisfy the boundary condition. Then, we use difference formula for derivatives to discretize the elliptic Monge-Ampére equation and the nonlinear boundary condition.

For the interior points, the 9-point finite difference scheme with second-order error is used for derivatives. As depicted in Fig. 4(a), this scheme is defined as

*x*=

*x*

_{max}shown in Fig. 4(b) for example, a backward difference approximation with second-order error is used at the boundary point

*z*

_{m}_{,}

*for*

_{j}*z*, and a centered difference approximation with second-order error is used for

_{x}*z*, which are given by

_{y}Of course, one can use a higher-order finite difference scheme for derivatives. With these finite difference schemes, we convert the mathematical model shown in Eq. (13) into a set of nonlinear equations. Write these nonlinear equations in the form

where**X**represents the variables of the nonlinear equations, which are the

*z*-coordinates of all the discrete data points. We use the Newton’s method to solve these nonlinear equations, and an approximate solution of this nonlinear problem can be obtained. Then, the smooth freeform surface is constructed to pass these (m + 1) × (n + 1) discrete data points with a B-spline surface.

## 4. Verify the design model and the numerical technique

A challenging illumination where a freeform lens is required for casting the lowercase letters “mao” on an elliptical background with the irradiance ratio of 4 (the letters) to 1 (background) to zero (outside), is given here to verify this mathematical model. Assume that the source is a parallel beam with uniform intensity distribution. The target pattern is shown in Fig. 5, and the other design parameters are listed in Table 1. Since the Newton’s method is used here to solve the nonlinear equations, we employ an initial design which produces a uniform rectangular illumination with the size of 100mm × 50mm for the iterative scheme. This initial design may have two optical configurations both designed by an approach which is similar to the one disclosed in Ref [11]. with two different mappings, as shown in Fig. 6. In this section, we want to explore the characteristics of this design model by employing the first configuration shown in Fig. 6(a). And, the optical performance of the second configuration shown in Fig. 6(b) will be discussed in the next section.

With this numerical technique, we can obtain an approximate solution of this design problem shown in Fig. 7(a). Six million rays are traced, and the obtained illumination pattern is shown in Fig. 7(b). According to this pattern, we depict the irradiance distribution along the line *y* = −3.4 mm, as shown in Fig. 8. The red solid curve shows clearly that the irradiance ratio is almost 4 (the letters) to 1 (background) to zero (outside), even though there are a few differences between the actual ratio and the target one. But we also find that the difference will become more obvious for the design with a larger irradiance ratio. We will address this issue in Section 5. As stated above, the specific position of each boundary ray on $\partial {S}_{2}$ is not predefined by the nonlinear boundary condition. So what will happen to the boundary rays during the iterative process? Figure 9 shows the change of the position of the boundary ray (*x* = 2, *y* = 2) on the target plane. It is amazing that the boundary ray can automatically adjust its position on $\partial {S}_{2}$ to meet the design requirements during the iterative process.

To further explore the elegance of this freeform lens, we change the lighting distance between the lens and the target plane and get the results shown in Fig. 10. It shows clearly that although the size of the elliptical pattern varies with the lighting distance, the irradiance ratio almost keeps unchanged. Obviously, we obtain an excellent design with the mathematical model presented above.

In this design, undoubtedly such a complex illumination task will pose huge challenges for the existing design methods of freeform surface. The results of this example show clearly that the target illumination is achieved. Besides, the more complex an illumination is, the more discrete data points one needs to accurately construct the freeform surface. For example, 90 × 90 data points are used here to construct the freeform surface. It is hard to imagine that an optimization design method based on a few parameters for such a complex task would be feasible. In practical applications, a collimated beam usually has a certain spread angle. So, it is necessary to further explore the characteristics of the freeform lens designed by this mathematical model with an actual collimated beam. The influence of spread angle of the collimated beam on the irradiance distribution is shown in Fig. 8. The red solid line, the black dashed line and the blue dot line represent a spread angle of 0 mrad, 3 mrad and 5 mrad, respectively. The corresponding illumination patterns are shown in Figs. 7(b), 11(a) and 11(b). These figures clearly show that there are few differences between these three irradiance curves. When the spread angle increases to 10 mrad, the actual ratio is a little smaller that the target one shown in Fig. 11(c). And, the illumination pattern becomes a little blurred shown in 11(d), when the spread angle increases to 15 mrad. On the whole, this design has a large tolerance to spread angle and can satisfy the requirements of practical applications. These analyses show the elegance of this mathematical model in solving the three-dimensional design problem of collimated beam shaping.

## 5. Discussions

#### 5.1 How to achieve a high resolution design

In this subsection, some key issues in achieving high resolution designs are discussed. In the first case, let *m* = *n* = 39, and the spacing *h*_{1} = *h*_{2} = 0.1026 mm. We also assume that *m* = *n* = 59 and obtain the spacing *h*_{1} = *h*_{2} = 0.0678 mm in the second case. The illumination patterns of these two cases are shown in Fig. 12. Compare Fig. 12 to Fig. 7, and we can find that a smaller spacing can ensure a better design. As mentioned in Section 3, some finite difference schemes are used here for the derivatives. The approximation error of a finite difference scheme is strongly determined by the spacing. A smaller spacing means a smaller approximation error. That is just the reason why the designs with *m* = *n* = 89 and *m* = *n* = 59 are better than the design with *m* = *n* = 39.

Keep this in mind, and we could speculate that a more complex illumination would require a smaller spacing to accurately represent the curvature of the freeform surface. We change the irradiance ratio of the design with *m* = *n* = 89, and obtain four other designs with the target ratio of 2:1, 3:1, 5:1 and 6:1 respectively, as shown in Fig. 13. Since the spacing is same in the four designs, one can find that the differences between the actual ratio and the target one are a little more obvious for a larger target ratio, as shown in Figs. 13(c) and 13(d). But a smaller spacing usually means more computation time. Thus, it is necessary to determine anappropriate spacing to make a trade-off between the optical performance of a design and the design efficiency. One can also find that there could still be a few differences for a smaller ratio, as shown in Figs. 13(a) and 13(b). The main reason is that the numerical technique presented in Section 3 employs the Newton’s method which can only converge towards a local minimum here. So, the result of this numerical technique might be determined by the initial value to some extent. As stated in Section 4, we use a design which produces a uniform rectangular illumination with the size of 100mm × 50mm as the initial value. Actually, it is hard to imagine that these high resolution designs shown in Figs. 7 and 13 are derived from such an initial design. These designs strongly demonstrate the elegance of the design model and the numerical technique presented above in tackling complex illuminations. Of course, a better initial design can ensure a better result for the numerical technique introduced above. And, we will further address these issues in our future work.

#### 5.2 Caustic surface: the second optical configuration

There is a caustic surface between the freeform lens and the target plane in the second configuration shown in Fig. 6(b). In this subsection, the optical performance of this configuration is explored. The model and the illumination pattern obtained from simulation are shown in Fig. 14. Obviously, this is a desirable design. Changing the lighting distance between the lens and the target plane, we obtain the change of the illumination pattern, as shown in Fig. 15. We can also find that the size of the elliptical pattern varies with the lighting distance, and the irradiance ratio almost keeps unchanged though. What will happen to this design if we change the spread angle of the collimated beam? Figure 16 can give us the answer. This figure clearly shows the design of the second configuration also has a large tolerance to spread angle. According to these analyses, we find that the second configuration has the same optical characteristics of the first configuration and the two optical configurations both have excellent performance. And, the caustic surface has little influence on the illumination in this design.

## 6. Conclusion

Collimated beam shaping which requires a collimated beam should be directed to produce a target illumination, is a challenging topic of nonimaging optics. In this paper, we aim to solve this problem with a freeform surface from a new perspective. Based on a concept that the beam-shaping process of a collimated beam is similar to the problem of optimal mass transport, we establish a mathematical model for such a beam-shaping process without any symmetry for the beam-shaping system. With this model, the beam shaping problem can be converted into a nonlinear boundary problem for the elliptic Monge-Ampére equation. A numerical technique for solving this mathematical model is disclosed for the first time, and boundary conditions for balancing light are discussed. Some key issues in achieving high resolution designs are addressed and the influence of caustic surface on the illumination is also presented. The work reported in this paper is a further research on the method presented in our previous publication [18]. This paper pierces the veil of mystery surrounding this kind of design method of freeform surface based on the elliptic Monge-Ampére equation. Being different from the first-order PDE methods, an energy mapping is not required for this design model. The position of intercept point of each incident ray on the target plane is automatically adjusted to produce the target illumination, and smooth freeform surfaces can be obtained. In the design examples, complex illumination tasks are easily tackled. This mathematical model shows its elegance in solving the three-dimensional design problem of collimated beam shaping with one freeform surface. In future work, we will focus on establishing a mathematical model for solving the two-freeform surface design problem and explore this technique in laser beam shaping.

## Appendix: Ellipticity of the equation

Let D=|J(**T**)|, then D could be expressed as

*A*

_{6}=

*A*

_{5}+

*I*(

*x*,

*y*)/

*E*(

*t*,

_{x}*t*). Thus, Eq. (8) can be rewritten asThe definition of ellipticity [20] of the second order partial differential equation requires that the expression

_{y}According to Eq. (18), we can obtain

*D*>0, whether

*Q*is positive or negative is determined by

*A*

_{1}.

- 1). For a freeform reflector design,
*n*= −1,_{o}*n*= 1 and ${O}_{z}={n}_{o}b+{n}_{i}({z}_{x}^{2}+{z}_{y}^{2})<0$. Then, A_{i}_{1}>0. - 2). For a freeform lens design,
*n*_{o}= 1 and ${O}_{z}={n}_{o}b+{n}_{i}({z}_{x}^{2}+{z}_{y}^{2})>0$. Then, A_{1}>0.

Thus, we could conclude that Eq. (11) is an elliptic Monge-Ampére equation.

## Acknowledgment

We thank K.W. Liang from Department of Mathematics of Zhejiang University for discussions and acknowledge support from the National High Technology Research and Development Program of China (863 Program No. 2012AA10A503), the National Natural Science Foundation of China (No. 61177015) and the Fundamental Research Funds for the Central Universities of China (2012XZZX013).

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