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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 20974–20989
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A mathematical model of the single freeform surface design for collimated beam shaping

Rengmao Wu, Peng Liu, Yaqin Zhang, Zhenrong Zheng, Haifeng Li, and Xu Liu  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 20974-20989 (2013)
http://dx.doi.org/10.1364/OE.21.020974


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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

1. T. Uemura, US Patent No. 2008/006, 254, 1 A1 (2008).

3

3. F. M. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004). [CrossRef]

]. 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

4. F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. 47(7), 957–966 (2008). [CrossRef] [PubMed]

6

6. A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011). [CrossRef]

]. 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

7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

17

17. L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008). [CrossRef]

]. 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

7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

11

11. R. M. Wu, H. F. Li, Z. R. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. 50(5), 725–732 (2011). [CrossRef] [PubMed]

]. Continuity of the freeform surface is strongly determined by integrability of the mapping [7

7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

11

11. R. M. Wu, H. F. Li, Z. R. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. 50(5), 725–732 (2011). [CrossRef] [PubMed]

]. A smooth freeform surface can only be obtained with an integrable mapping, however, it is very difficult to find such an integrable mapping [9

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

]. This inverse problem can also be represented by an elliptic Monge-Ampére equation which is a nonlinear second-order partial differential equation [12

12. J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. 62(12), 1498–1501 (1972). [CrossRef]

17

17. L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008). [CrossRef]

]. With this method, the energy mapping is not required. Unfortunately, numerical techniques for solving this equation have not been detailedly disclosed so far [12

12. J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. 62(12), 1498–1501 (1972). [CrossRef]

,13

13. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

], or only weak solutions can be obtained by iteratively determining lots of pieces of paraboloids (ellipsoids or hyperboloids) [14

14. X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996). [CrossRef]

17

17. L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008). [CrossRef]

]. 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

18. R. M. Wu, L. Xu, P. Liu, Y. Q. Zhang, Z. R. Zheng, H. F. 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] [PubMed]

]. 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(tx,ty) on a given illumination plane and the energy is conserved during the beam-shaping process, as shown in Fig. 1
Fig. 1 Schematic illustration of the collimated beam shaping process.
. S1 and S2 denote the cross section of the collimated beam and the illumination area, respectively. ∂S1 and ∂S2 are, respectively, the boundaries of S1 and S2. 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

19. T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003). [CrossRef]

]. 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
Fig. 2 The geometrical design layout of the freeform surface.
. 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
N=1zx2+zy2+1(zx,zy,1)
(1)
where zx and zy are the first-order partial derivatives of the coordinate 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 as
noO=niI+P1N
(2)
where no is the refractive index of the medium surrounding the freeform lens, and ni is the refractive index of the freeform lens. Then, for a freeform reflector in air no = −1 and ni = 1 hold. The parameter P1 is given by

P1=no(1ni2no2)(zx2+zy2)+1nizx2+zy2+1
(3)

Equation (8) is a second-order partial differential equation. Simplify this equation, and we obtain
A1(zxxzyyzxy2)+A2zxx+A3zyy+A4zxy+A5=0where,b=a(zx2+zy2)+1,A1=(ztz)2nob[1+(zx2+zy2)](nobni)2[nob+ni(zx2+zy2)]3,A2=(ztz)(nobni)[nob(zy2+1)nizx2]+noniabzx2(zx2+zy2+1)[nob+ni(zx2+zy2)]2,A3=(ztz)(nobni)[nob(zx2+1)nizy2]+noniabzy2(zx2+zy2+1)[nob+ni(zx2+zy2)]2,A4=2(ztz)zxzy[noniab(zx2+zy2+1)(no2b2ni2)][nob+ni(zx2+zy2)]2,A5=nob(zx2+zy2+1)nob+ni(zx2+zy2)I(x,y)E(tx,ty)
(11)
This nonlinear second-order partial differential equation is an elliptic Monge-Ampére equation, and ellipticity of this equation will be proved in the appendix. This elliptic Monge-Ampére equation describes the conservation and redistribution of energy during the beam-shaping process. Since the shape of the target illumination pattern should be ensured, the beam-shaping process should also satisfy a boundary condition which is defined by
{tx=tx(x,y,z,zx,zy)ty=ty(x,y,z,zx,zy):S1S2
(12)
This is a nonlinear boundary condition which specifies that the incident rays on S1 is refracted by the freeform surface to S2, however, the specific position of each ray on S2 is not predefined. How does this boundary condition work will be discussed in following sections. Then, we obtain a mathematical model of the single freeform surface design problem of collimated beam shaping, which is given by

{A1(zxxzyyzxy2)+A2zxx+A3zyy+A4zxy+A5=0BC:{tx=tx(x,y,z,zx,zy)ty=ty(x,y,z,zx,zy):S1S2
(13)

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

As highlighted above, the nonlinear boundary condition specifies that the incident rays on S1 is refracted by the freeform surface to S2. Assume that the boundary of the illumination pattern has an analytical formula f(tx,ty). According to the boundary condition, the intercept points of the boundary incident rays on the target plane should satisfy f(tx,ty). Take an elliptical illumination pattern for example. Since the boundary of the pattern is elliptical, the boundary condition can be expressed as
(txCx)2a2+(tyCy)2b2=1
(14)
where, a and b are one-half of the ellipse's major and minor axes respectively; the point (Cx,Cy) represents the center of the ellipse, and the point (tx,ty) is the intercept point defined by Eq. (6). This equation shows clearly that the specific position of each boundary ray on S2 is not predefined.

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 S1 as an example. Assume the domain S1 = {(x,y)|xminxxmax, yminyymax}. Discretizing this rectangular domain yields a net of grid points S1 = {(xi,yj)|x = xmin + ih1, y = ymin + jh2, i = 0,1,…,m; j = 0,1,…,n}. h1 = (xmax-xmin)/m and h2 = (ymax-ymin)/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
Fig. 3 The discretization of the domain S1.
. The gridd points on S1 are called the boundary points, and the grid points inside S1 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)
Fig. 4 (a) The 9-point finite difference scheme for the interior points and (b) the forward (or backward) difference approximation with second-order error for the boundary points.
, this scheme is defined as
zx=zi+1,jzi1,j2h1,zy=zi,j+1zi,j12h2,zxx=zi+1,j2zi,j+zi1,jh12zyy=zi,j+12zi,j+zi,j1h22,zxy=zi+1,j+1zi+1,j1zi1,j+1+zi1,j14h1h2
(15)
To ensure that the order of approximation error is same at each grid point, a forward (or backward) difference approximation with second-order error is used at each boundary point. Take the boundary line x = xmax shown in Fig. 4(b) for example, a backward difference approximation with second-order error is used at the boundary point zm,j for zx, and a centered difference approximation with second-order error is used for zy, which are given by

zx=3zm,j4zm1,j+zm2,j2h1zy=zm,j+1zm,j12h2
(16)

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
F(X)=0
(17)
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

With this numerical technique, we can obtain an approximate solution of this design problem shown in Fig. 7(a)
Fig. 7 (a) The model of the freeform lens and (b) the pseudocolor plot of the irradiance distribution obtained from the simulation for the first optical configuration.
. 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
Fig. 8 Irradiance distribution along the line y = −3.4 mm. This figure shows the influence of the spread angle of the collimated beam on the irradiance distribution. The red solid line, the black dashed line and the blue dot line represent that the spread angle of the collimated beam is 0 mrad, 3 mrad and 5 mrad, respectively. The irradiance ratio is almost 4 (the letters) to 1 (background) to zero (outside). This design has a large tolerance to the spread angle.
. 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 S2 is not predefined by the nonlinear boundary condition. So what will happen to the boundary rays during the iterative process? Figure 9
Fig. 9 The change of the position of the boundary ray (x = 2,y = 2) on the target plane. In the initial design, the position is (49.871,25.1194). Its position is automatically adjusted to meet the design requirements during the iterative process by the numerical technique presented above.
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 S2 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
Fig. 10 The influence of lighting distance on the illumination. The lighting distance (a) tz = 200mm and (b) tz = 400mm.
. 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)
Fig. 11 The influence of the spread angle on the illumination pattern. The spread angle is (a) 3 mrad; (b) 5 mrad; (c) 10 mrad; (d) 15 mrad. This design has a large tolerance to spread angle.
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 h1 = h2 = 0.1026 mm. We also assume that m = n = 59 and obtain the spacing h1 = h2 = 0.0678 mm in the second case. The illumination patterns of these two cases are shown in Fig. 12
Fig. 12 (a) m = n = 39, and h1 = h2 = 0.1025mm; (b) m = n = 59, and h1 = h2 = 0.0678 mm. The irradiance curve represents the irradiance distribution along the line y = −3.4 mm. The optical performance of a design is strongly determined by the spacing.
. 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
Fig. 13 The irradiance ratio is (a) 2:1, (b) 3:1, (c) 5:1 and (d) 6:1, respectively. The irradiance curve represents the irradiance distribution along the line y = −3.4 mm.
. 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
Fig. 14 (a) The model of the freeform lens and (b) the pseudocolor plot of the irradiance distribution obtained from simulation for the second optical configuration.
. 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
Fig. 15 The influence of the lighting distance on the illumination for the second configuration. The lighting distance (a) tz = 200mm and (b) tz = 400mm.
. 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
Fig. 16 The influence of the spread angle on the illumination pattern for the second configuration. The spread angle is (a) 3 mrad; (b) 5 mrad; (c) 10 mrad; (d) 15 mrad. There are few differences in the results shown in Figs. 14(b), 16(a) and 16(b). The illumination pattern becomes a little blurred, when the spread angle increases to 15 mrad.
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

Appendix: Ellipticity of the equation

Let D=|J(T)|, then D could be expressed as
D=A1(zxxzyyzxy2)+A2zxx+A3zyy+A4zxy+A6
(18)
Where A6 = A5 + I(x,y)/E(tx,ty). Thus, Eq. (8) can be rewritten as
DE(tx,ty)=I(x,y)
(19)
The definition of ellipticity [20

20. L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math. 6(1), 103–156 (1953). [CrossRef]

] of the second order partial differential equation requires that the expression

Q=4DzxxDzyyDzxy2>0where,Dzxx=D/zxx,Dzyy=D/zyyandDzxy=D/zxy.
(20)

According to Eq. (18), we can obtain
Dzxx=A1zyy+A2,Dzyy=A1zxx+A3,Dzxy=2A1zxy+A4
(21)
Then, we get
Q=4A1D
(22)
Since D>0, whether Q is positive or negative is determined by A1.

  • 1). For a freeform reflector design, no = −1, ni = 1 and Oz=nob+ni(zx2+zy2)<0. Then, A1>0.
  • 2). For a freeform lens design, no = 1 and Oz=nob+ni(zx2+zy2)>0. Then, A1>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).

References and links

1.

T. Uemura, US Patent No. 2008/006, 254, 1 A1 (2008).

2.

H. Tanaka, US Patent No. 2009/000, 267, 1 A1 (2009).

3.

F. M. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004). [CrossRef]

4.

F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. 47(7), 957–966 (2008). [CrossRef] [PubMed]

5.

R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimization design of irradiance array for LED uniform rectangular illumination,” Appl. Opt. 51(13), 2257–2263 (2012). [CrossRef] [PubMed]

6.

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011). [CrossRef]

7.

L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

8.

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

9.

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

10.

R. M. Wu, H. Li, Z. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. 50(5), 725–732 (2011). [CrossRef] [PubMed]

11.

R. M. Wu, H. F. Li, Z. R. Zheng, and X. Liu, “Freeform lens arrays for off-axis illumination in an optical lithography system,” Appl. Opt. 50(5), 725–732 (2011). [CrossRef] [PubMed]

12.

J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. 62(12), 1498–1501 (1972). [CrossRef]

13.

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

14.

X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996). [CrossRef]

15.

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998). [CrossRef]

16.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999). [CrossRef]

17.

L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008). [CrossRef]

18.

R. M. Wu, L. Xu, P. Liu, Y. Q. Zhang, Z. R. Zheng, H. F. 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] [PubMed]

19.

T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003). [CrossRef]

20.

L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math. 6(1), 103–156 (1953). [CrossRef]

OCIS Codes
(220.1250) Optical design and fabrication : Aspherics
(080.1753) Geometric optics : Computation methods
(220.2945) Optical design and fabrication : Illumination design
(220.4298) Optical design and fabrication : Nonimaging optics

ToC Category:
Geometric Optics

History
Original Manuscript: June 14, 2013
Revised Manuscript: August 22, 2013
Manuscript Accepted: August 22, 2013
Published: August 30, 2013

Citation
Rengmao Wu, Peng Liu, Yaqin Zhang, Zhenrong Zheng, Haifeng Li, and Xu Liu, "A mathematical model of the single freeform surface design for collimated beam shaping," Opt. Express 21, 20974-20989 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20974


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References

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  17. L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci.154(1), 39–49 (2008). [CrossRef]
  18. R. M. Wu, L. Xu, P. Liu, Y. Q. Zhang, Z. R. Zheng, H. F. 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] [PubMed]
  19. T. Glimm and V. I. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci.117(3), 4096–4108 (2003). [CrossRef]
  20. L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math.6(1), 103–156 (1953). [CrossRef]

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