SPIE Proceedings [SPIE Optical Systems Design - Glasgow, Scotland, United Kingdom (Monday 1 September 2008)] Illumination Optics - Secondary optical design for LED illumination using

Secondary optical design for LED illumination using freeform lens

Yi Ding*, Xu Liu, Zhen-rong Zheng, Pei-fu Gu

State Key Laboratory of Modern Optical Instrumentation, 38 Zheda Road, Hang Zhou, Zhejiang, China 310027

ABSTRACT

The purpose of the secondary optical design for light emitting diodes (LED) illumination is to rearrange the direct output of LED and then achieve the desired illumination. Usually a lens is applied to realize this function in an LED illumination system, and a freeform lens can do better than the traditional spherical ones. The method in this paper can be used to design a freeform lens in short time, just less than 10 seconds. And this freeform lens is constructed using the numerical solutions of a set of first-order partial differential equations, which are deduced from the Snell’s law according to the conservation of energy. Using an LED as the source while immersing it in the lens, a uniform rectangle can be gotten through single refraction, and with uniformity near to 90%. The rectangular illumination also has a relatively clear cut-off line with little blur at the edge. This method can shorten the designing time and improve the performance of LED illumination system. Furthermore, not only rectangle, but other illumination figures can be achieved by freeform lens designed by this method, which can broaden the scope of freeform lens’s usage.

Keywords: secondary optical design, LED illumination, freeform lens, uniform illumination, first-order partial differential equation sets

1. INTRODUCTION LED will become the main lighting source in future for its continuous increase of luminous flux and many other merits. But there’s still some shortage in today’s LED illumination system: its light distribution is quite different from other sources, so the traditional illumination system may cause deterioration on illumination plane and energy loss. Besides, the direct output light flux of an LED is still a circle spot while the desired illumination is always a rectangle, for example, in projector illumination system. So the output of LED needs to be rearranged through secondary optical design in order to achieve the desired illumination. In a projector system, fly eyes or light tubes are applied to change the circle illumination into a uniform rectangle [1, 2]. But multiple reflections or refractions of light by these elements also would cause loss. Freeform optical components may successfully solve these problems, and there are several reports discussed freeform lens designing method. Essentially, there are two ways to design them. One is trial and error method, and generally speaking it always requires a lot of time [3, 4]. The other is to express the lens by a set of partial differential equations [5, 6]. These previous methods have lots of limitations on forming the desired illumination or saving the calculation time. Especially on the occasion of a uniform rectangular illumination with high energy efficiency is needed, it’s hard to use the above methods to meet the demands.

In this paper we have proposed a new set of first-order differential equations for the freeform lens design. These equations can be deduced from Snell’s law according to the conservation of energy, on condition that the characteristics of the source and the desired illumination are fixed ahead [6-9]. By numerically solving these equations using C programming language, we can get the freeform lens in short time, just less than 10 seconds. By adopting an LED as the source and immersing it in the lens, a uniform rectangle can be gotten through single refraction with uniformity near to 90%. The rectangular illumination also has a relatively clear cut-off line with little blur at the edge, which can help to reduce the energy loss when illuminating a rectangle. Besides, in some particular situations where a clear cut-off line is very necessary, such as the illumination of a dipped headlight, this merit could be quite helpful. Furthermore, not only rectangle, but other illumination figures such as an octagon, can be achieved by freeform lens designed by this method. Examples will be given in the paper to show how to design such a lens in detail and ASAP will be adopted as the software to simulate the illumination results.

* [email protected]; phone 86-571-87951758; fax 86-571-87951758

Illumination Optics, edited by Tina E. Kidger, Stuart R. DavidProc. of SPIE Vol. 7103, 71030K · © 2008 SPIE · CCC code: 0277-786X/08/$18 · doi: 10.1117/12.797810

Proc. of SPIE Vol. 7103 71030K-1

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2. PARTIAL DIFFERENTIAL EQUATION SETS DEDUCING Before deducing the partial differential equation sets, the luminous characteristics and the illumination on the target plane should be fixed. We assume that the luminous distribution of the source S is rotational symmetric and the intensity versus polar angles is I(φ). The center of LED source is located at the origin of an orthogonal coordinate system. So points on the target plane T can be expressed as t(x, y, z), and the illumination on this plane is E(t(x, y, z)). The freeform reflector P is located in a spherical coordinate system, which shares the same origin with the orthogonal system mentioned above [10]. Thus, the points on P can be expressed as p(θ, φ, ρ(θ, φ)), and the normal vector at point p of the lens is N. we assume that I is the vector of the incident light from source to point p, while O is the vector of refractive light by the freeform lens from point p to point t. Fig.1 shows the relationships between these vectors and points.

Fig. 1. Vectors in refraction.

2.1 First-order partial differential equations

With the definition of vectors N, I and O, it’s easy to know that the direction of I is decided by the position of the source and point p, while the direction of O is decided by the position of point p and t. Here N stands for the slope of the lens at point p [11]. These three vectors are connected to each other via the Snell’s law:

( )1

2 2 2n n n n 2 n n .O I O I O I N⎡ ⎤× − × = + − × × × • ×⎣ ⎦O I O I (1)

Here, nI and nO indicate the refractive index of the incidence and emergence medium respectively. Once source and point t are known, the position of point p can be calculated. Assuming that these three vectors are unit vectors, they can be expressed as follows in the orthogonal coordinate system:

( ), .x y zN ,N N=N i j k (2)

( ), .x y zI ,I I=I i j k (3)

.−=

−t pOt p

(4)

After substituting Eq.2, Eq.3 and Eq.4 for the corresponding items in Eq.1, and using x, y and z to express point t, the coordinates of point t can be expressed by that of point p and the slope of the freeform lens, just as Eq.5 and Eq.6 show:



( )n n n

.n n

x O z I z Ix x

O z O

N z p Ix p I

N⎡ − − − ⎤⎣ ⎦= + + −

t pt p (5)

( )n n n

.n n

y O z I z Iy y

O z O

N z p Iy p I

N⎡ − − − ⎤⎣ ⎦= + + −

t pt p (6)

Eq.5 and Eq.6 are the partial differential equations used for the freeform lens design. And the subscript of N include the first-order partial derivative of ρ(θ, φ) on the directions of θ and φ. It could be calculated from:

.θ ϕ

θ ϕ

ρ ρ

ρ ρ

⊗=

⊗N (7)

And the subscript of p is the vector component of p on x, y and z directions. In orthogonal coordinates they can be expressed as: ( ) ( ) ( )( )( , , ) , sin cos , , sin sin , , cos .x y z= p p p ρ θ ϕ ϕ θ ρ θ ϕ ϕ θ ρ θ ϕ ϕ=p i j k i j k (8)

2.2 Conservation of energy

In order to solve Eq.5 and Eq.6, the relationship between the position of source and point t should be known. Without regard to the energy loss in refraction, the process of light transmission should enable the observation of the energy conservation, which means the output of the source is equal to the flux inside the target plane. Assuming that the half viewing angle of the source is φMAX and considering its luminous distribution is rotational symmetric, then:

( ) ( )2

0 0d sin d d .MAX I E A

π ϕθ ϕ ϕ ϕ =∫ ∫ ∫ t (9)

Here, I(φ) is the intensity in the direction of φ and it’s tested at the position far away from the source that the area of the source could be ignored. A is the area of the illuminated target plane. Eq.9 indicates the relationship between θ, φ and x, y, z, and its exact form depends on the topological mapping from the source to the target plane.

2.3 Solving method for equations

When the source S and the target plane T are fixed, which means I(φ) and the range of θ, φ and x, y, z are known, then x, y, z can be expressed by θ and φ though Eq.9. After replacing the corresponding items in Eq.5 and Eq.6 by θ and φ, they become the first-order partial differential equation sets of ρ(θ, φ) on the directions of θ and φ. It is difficult to obtain the analytic solution from these partial differential equations, so numerical methods are employed [12]. And the vertex of the freeform lens, whose polar angle is zero, would be applied as the initial condition for solving the equations:

( ) 0,0 .ρ θ ρ= (10)

Boundary condition also should be confirmed before solving the equations sets. And this is decided according to the shape of the target plane and the light distribution of the source. A pure surface-emitting LED is applied as the source in this paper, whose light distribution is rotational symmetric. In order to collect the light from the source efficiently, the projection of the boundary on the x-y plane should also be a circle, which means the boundary of freeform lens is on a conical surface.

3. FREEFORM LENS DESIGN We will discuss the freeform lens design and the equations solution in detail in this section. A pure surface-emitting LED is adopted as the source and would be immersed in the freeform lens, whose refractive index is 1.5. In order to simplify the calculation, we assume that the light emitting surface of LED is in the x–y plane with its center located at the origin. The LED has a Lambertian radiation distribution. Its luminous surface is 1×1 mm2 and the viewing angle is 120°. The illuminated target plane T is a rectangle with the ratio of 4:3, and its diagonal is 100 mm long. The target plane is perpendicular to the z-axis, and the coordinate of its center is (0, 0, 20). The vertex of the freeform lens is 10 mm above the LED, just as Fig.2 shows.



I ILL) !-'-1iir iiiiiiiciocn iii ri icIGU2

Lt4ULW I tiUc

j9LEPG

Fig. 2. The LED refractive illumination system.

3.1 Equation simplifying

Considering the height of the freeform lens and the size of the LED, the emitting area of the LED is sufficiently small compared with the freeform lens. So the source can be treated as a point source, then the direction of the incident light on the freeform lens is exclusively decided by the coordinates of point p itself, which means:

( )sin cos sin sin ,cos .,ϕ θ ϕ θ ϕ=I i j k (11)

And then:

( )( ) ( ), , , ,θ ϕ ρ θ ϕ ρ θ ϕ= = ×p I. (12)

When a uniform illumination on the target plane is desired, E(t) in Eq.9 becomes a constant. Considering the Lambertian characteristics of the LED, then I(φ) in Eq.9 becomes:

( ) cos .I Iϕ ϕ= × (13)

Here, I is the intensity value in direction of (0, 0, 1). Considering the boundary conditions of the freeform lens and the rectangular shape of the illumination, the topological mapping as Fig.3 shows is applied. In order to reach the maximum efficiency, lights of equal φ would be refracted to the same edge of the rectangle [13]. Considering the viewing angle of LED is 120° and the target plane is perpendicular to the z-axis, after substituting Eq.13 into Eq.9, Eq.9 can be changed to:

( )( )1 cos 28 .

43

IX Y

EXY

ϕ⎧ − × × ×π× =⎪⎪ ×⎨

⎪ =⎪⎩

(14)

Here X and Y are the coordinates of the corner of the rectangle, and both of them are >0. Once the corner has been confirmed, points on target plane in the first quadrant can be expressed as:

'

, 0 .'

x XYy

θ

θ

⎧ =π⎪ ≤ ≤⎨ 4 4= ×⎪ π⎩

(15)

' 2

, .'

x X

y Y

θθ

⎧ 4⎛ ⎞= − × × π π⎪ ⎜ ⎟ ≤ ≤π⎝ ⎠⎨ 4 2⎪ =⎩

(16)



( Y — (°"))r 1

TV/ \NN

Y,2O)'V I I

Here x’ and y’ are coordinates of the borders of the rectangle, and their values in the other quadrants can be easily expressed in the same way by θ and φ.

Fig. 3. The topological mapping from source to target plane.

In this paper we assume that the luminous distribution of LED is Lambertian, but in most cases the distribution of LED is just an assembly of measuring data, which are luminous intensity values in different φ directions. If these measuring points are dense enough, the luminous intensity can be considered as constant in the range of φn±dφ/2. Assume the intensity in direction of φn as I(φn), and the number of measuring points is N, then the luminous flux of LED between φ0 and φN can be expressed as:

( ) ( ) ( )0

0

122

01 22

2 sin d sin d sin d .n N

Nn

dd N

ddLED n Nn

I I Iϕϕ ϕϕ ϕ

ϕϕϕ ϕϕπ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

− ++

−−=

⎛ ⎞Φ = × + +⎜ ⎟

⎝ ⎠∑∫ ∫ ∫ (17)

Here dφ=(φN-φ0)/N, φn=φ0+n*dφ. Some software, such as ASAP, treat the intensity distribution between φn and φn+1 as linear. Then I(φ) between φn and φn+1 becomes:

( ) ( ) ( ) ( ) ( )( )1 1 11

1 .n n n n n nLINEARn n

I I I I Iϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕϕ ϕ + + +

+

⎡ ⎤= × × − × + × −⎣ ⎦− (18)

In this situation the luminous flux of LED between φ0 and φN can be expressed as:

( )11

_0

2 sin d .n

n

N

LED LINEAR LINEARn

Iϕ

ϕπ ϕ ϕ ϕ+

−

=

Φ = ×∑∫ (19)

By substituting Eq.17 or Eq.19 into Eq.9, the coordinates on the target plane can also be expressed by θ and φ. After using the methods mentioned above, the equations for non-Lambertian LEDs can be deduced.

3.2 Numerical resolving

After substituting Eq.15 and Eq.16 into Eq.5 and Eq.6, these two equations become the first-order partial differential equation sets of ρ(θ, φ) in the direction of θ, φ. Because the source is known, the range of θ and φ is fixed. Then by using the difference scheme of ρ(θ, φ) to replace its first derivatives, while discretizing the equations with the grids depicted in Fig.3, the partial differential equation sets turn into a set of nonlinear equations whose unknown is ρ(θ, φ).

It only takes less than 10 seconds to solve the set of nonlinear equations mentioned above by using C language and a computer with a 2.40 GHz Celeron CPU, and the numerical results stand for the contour of the freeform lens

3.3 Surface modeling

We adopted non-uniform rational B-splines (NURBS) method to model the freeform surface for its merits in optical design [14, 15]. First the calculated data points are used to construct lines, and then these lines construct the freeform surface through loft. Theoretically the freeform surface constructed from these data should be smooth. But in practical, if the freeform surface was modeled as a whole, the simulated result was not good because of modeling errors.



t

This problem could be avoided by modeling the surface with some discrete sub-surfaces, as Fig.4 shows. In this case, the freeform lens consists of 450 pieces. And each piece is lofted from 3 lines while lines are constructed from points. Points on the central line are calculated in the direction of φ, while points on the two fringe lines are calculated in the direction of θ. All 3 lines cross at the vertex point of the freeform lens, which means 450 sub-surfaces share the same vertex. And the more sub-surfaces, the better simulated results. But too many sub-surfaces would cost much time to model them, so 450 are just fine. The projective length of freeform lens on the x-axis is about 34 mm, on the y axis it is about 26 mm, with vertex point 10 mm from LED.

Fig. 4. 3D model of the freeform lens and LED.

3.4 Simulated results

A lambertian emitting surface was used as the LED source, and 1 million light rays are traced by ASAP software to verify this freeform lens illumination performance. The Fig.5 shows the simulated results.

Fig.5(a) shows the rectangular illumination on the target plane, which is a 4:3 rectangle whose diagonal length is about 100 mm. And Fig.5(b) and 5(c) show the energy distribution cross the center of the rectangle in x and y directions. The vertical axis in graphs stands for the normalized illumination, while the horizontal axis stands for the spatial distribution of the illumination on its center line.

From Fig.5(b) and 5(c), we can see that the uniformity is quite well, near 90%. The energy inside the desired rectangle is about 98.17% of the output of the source, if there is no energy loss when refracting. But there’s still some insufficiency: the illumination on the diagonal of the rectangle is a little lower compared with their surroundings according to Fig.5(a), which was caused by the absorptions of the gaps between those sub-surfaces of the freeform lens.

(a)



t. I .qn for flO -

1.DD

D.8D

D.6D

D.DD5D —4D —3D 2D 1D D 1D 2D 3D 4D SD

X ml ilimet ers

t. I .q-n for flO-

D.8D

D. 2D

4D —3D 2D 1D DV ml ilimet ers

1D 2D 3D 4D

(b) (c)

Fig. 5. The simulated results.

Not only limit to rectangle, this method can also be used to design freeform lens illuminating different figures, such as square or octagon. Fig.6 shows that the same LED forms a square and octagon using different freeform lens. Here a uniform illumination with very good clear boundary can also be gotten.

Fig. 6. Simulated results of a square and an octagon illumination.

3.5 Machining discussion

There are great differences between freeform and traditional optical elements in the aspect of design and manufacture. And the conventional machining equipments and technologies are hard to meet the demand for a complicated freeform surface which needs high machining precision [16]. Especially the freeform surface in this paper in non-rotational symmetric; besides is consisted of multi sub-surfaces. The ultra precision multi-axes diamond machining systems by Precitech Inc. can handle non-axes symmetric freeform optics, with a sub-micron-grade figure precision and nano-grade surface roughness [17].

Before the Numerical Control (NC) machine process the freeform surface, first of all the cutter contact points (CC points) should be calculated, then the cutting tool path was calculated by NURBS surface interpolator using those CC points. And the CC data were converted to the cutter location (CL) data, and then are fed to the computer numerical control (CNC) machine tools [18]. The surface data calculated from the first-order partial differential equations above can be considered as CC points and used for NC machining directly. And because all the CC points lie on the machined surface, the proposed method would not involve the cumulated errors.



The NC machine uses NURBS surface for interpolation, which also is applied by the 3D modeling software. So the cutting tool path would be similar with the 3D CAD/CAM model, and the machining errors can be minimized. Moreover, the angle interval of the discrete CC points is uniform, and the size of the lens is small, which means the distance of any two adjacent CC points is nearly equivalent. So the CC velocity is basically kept constant and consecutive, which would be helpful to improve machining efficiency and quality.

4. CONCLUSION Using freeform lens in the secondary optical design for LED can improve the performance of the LED illumination system, and we proposed a new designing method in this paper. The first-order partial differential equation sets presented can be applied to construct a freeform lens for uniform illumination of various figures, on condition that the characters of the source and the desired illumination are know. The numerical results of the partial differential equations stand for the figure of the freeform surface. Without professional optical designing software, it only takes less than 10 seconds to obtain the solution by C programming language and the simulated uniformity by ASAP is quite good. Its clear cut-off line can not only help to reduce the energy loss when illuminating a specific figure, but also can be quite useful in some particular situations where a clear cut-off line is desired, such as the dipped headlight. Moreover this method can be used to design freeform lens illuminating various uniform figures, broadening the scope of freeform lens’s usage.

REFERENCES

[1] Zhou Pin, Lu Wei, Lin Yuxiang, Zheng zhenrong, Li Haifeng and Gu Peifu, “Fly eye lens array used in liquid crystal projection display with high light efficiency,” Acta Optica Sinica 24(5), 587-591 (2004).

[2] Shen Mo, Li Haifeng, Lu Wei and Liu Xu, “Method of reflective fly eye lens design for LED illuminating projection system,” Acta Photonica Sinica 35(1), 93-95 (2006).

[3] J. Bortz, N. Shatz and D. Pitou, "Optimal design of a nonimaging projection lens for use with an LED source and a rectangular target," Proc. SPIE 4092, 130-138 (2000).

[4] B. A. Jacobson and R. D. Gengelbach, “Lens for uniform LED illumination: an example of automated optimization using Monte Carlo ray-tracing of an LED source,” Proc. SPIE 4446, 130-138 (2002).

[5] B. Parkyn and D. Pelka, “Free-form illumination lens designed by a pseudo-rectangular lawnmower algorithm,” Proc. SPIE 6338, 633808 (2005).

[6] H. Ries and J. Muschaweck, “Tailoring freeform lenses for illumination,” Proc. SPIE 6338, 633808 (2001). [7] H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590-595 (2002). [8] Ding Yi and Gu Peifu, “The Freeform Reflector for Uniform Illumination,” Acta Optica Sinica 27(3), 540-544

(2007). [9] Ding Yi, Liu Xu, Li Haifeng and Gu Peifu, “The design of the freeform reflector for uniform illumination,” ASID

2007, 735-738 (2007). [10] J. Schruben, “Formulation of a reflector design problem for a lighting fixture,” J. Opt. Soc. Am 62(12), 1498-1501

(1972). [11] Chen Weiheng, Introduction of Differential Geometry (Beijing University, 1990), Chap. 4. [12] Su Yucheng, Numerical Solutions of Partial Differential Equations (Weather, 1989), Chap. 1. [13] W. A. Parkyn, "Segmented illumination lenses for steplighting and wall-washing," Proc. SPIE 3779, 363-370 (1999). [14] H. Chase, "Optical Design with Rotationally Symmetric NURBS," Proc. SPIE 4832, 10-24 (2002). [15] T. L. R. Davenport, "3D NURBS representation of surface for illumination," Proc. SPIE 4832, 293-301 (2002). [16] T. L. R. Davenport, " Generation of NC Tool Path for Subdivision Surface," in Proceedings of CAD/Graphics 2001,

Q. Peng, ed. (International Academic, Kunming, China, 2001), pp. 1-7. [17] Precitech product features, “Freeform 700G,” http://www.precitech.com/Precitech_ff700G_features.html. [18] Y. Z. Wang and L. J. Chen, “A real-time NURBS surface interpolator for 5-axis surface machining,” Chinese

Journal of Aeronautics 18, 263-272 (2005).



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SPIE Proceedings [SPIE Optical Systems Design - Glasgow, Scotland, United Kingdom (Monday 1 September 2008)] Illumination Optics - Secondary optical design for LED illumination using