Energy feedback freeform lenses for uniform illumination of extended light source LEDs

ZONGTAO LI, ¹ SHUDONG YU,¹ LIWEI LIN , ² YONG TANG,¹ XINRUI DING,^1,* WEI

1

YUAN, ¹ BINHAI YU

¹ Key Laboratory of Surface Functional Structure Manufacturing of Guangdong High Education Institutes, South China University of Technology, Guangzhou 510640, China ² Department of Mechanical Engineering, University of California, Berkeley, California 94720, United States

*Corresponding author: [email protected]

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

Using freeform lenses to construct uniform illumination systems is important in light-emitting diode (LED) devices. In this paper, the energy feedback design is used for freeform lens (EFFL) constructions by solving a set of partial differential equations that describe the mapping relationships between the source and the illumination pattern. The simulation results show that the method can overcome the illumination deviation caused by the extended light source (ELS) problem. Furthermore, a uniformity of 95.6% is obtained for chip-on-board (COB) compact LED devices. As such, prototype LEDs manufactured with the proposed freeform lenses demonstrate significant improvements in luminous efficiency and emission uniformity. © 2016 Optical Society of America

OCIS codes: (220.4298) Nonimaging optics; (220.2945) Illumination design; (230.3670) Light-emitting diodes.

sources (PLSs). For LED applications, the brightness of these LEDs is not sufficient. Although a module with multiple sources makes it

1. Introduction

possible to obtain sufficient luminous flux, Kuo found that such a
Solid-state lighting devices, especially light-emitting diodes (LEDs), solution would lead to the “multi-shadow” phenomenon, which causes have gained attention recently as a result of their outstanding features the eyes to feel tired [9].
[1], which include low energy consumption, ultra-fast response,
For high-quality illumination, chip-on-board (COB) LEDs achieve a environmental friendliness, and tunable emitting spectra. These high-power output from one source. In such components, the large size features have made LEDs penetrate deeply into our daily life, in forms of the luminous area has greatly worsened the performance of the such as large-scale backlights, high-performance projectors, freeform lens system [4, 10], which causes the so-called “extended light automobile headlamps, and especially general lighting systems. At source (ELS) problem”. The simultaneous multiple surfaces (SMS) present, the angular intensity patterns of most LEDs are circularly method [11], in which two freeform surfaces are simultaneously symmetric, with the highest energy output lying in the normal designed to refract or reflect a set of input/output ray bundles, direction (i.e., Lambertian distributions), which cause glare upon provides a viable solution. Nevertheless, for compact packages in human eyes. On the other hand, LED lighting systems are always which only single freeform lenses are available, the SMS method is not installed at a certain height, and only the downward-emitting light suitable. On the other hand, optimizations based on tailoring method fulfills the illuminating function, whereas the light emitted at large also demonstrated potentials for solving the COB ELS problems [5], but viewing angles leads to energy losses. Therefore, specially designed calculation errors would be introduced due to the pre-discrete optical lenses are required in order to collect and guide the light into feedback, lowering the design accuracy and efficiency. uniform illumination patterns.
In this study, we further develop the PDE freeform lens design method,
The freeform lens is one of the most popular optical devices for present the concept of “effective rectification function” for COB performing this job owing to its distinct advantages, such as compact compact sources based on a set of energy feedback functions. size and precise light controlling [2–5]. Compared with the traditional
Additionally, an engineering example is discussed, and the trial and error method [6] or the tailoring method [7], solving a set of performance of the lens is experimentally investigated. The results partial differential equations (PDE) [2,8] that describe the mapping show that the improved design method is capable of handling the ELS relationships between the source and the illumination pattern is of problems, and that it exhibits excellent system efficiency. The findings higher accuracy and extensibility for different illumination patterns of this study will offer a valuable reference to LED optics researchers.
(such as rectangle or octagon). More importantly, this method is high

efficient, and fast [2]. However, existing studies are limited to LEDs with small emitting areas (single chip component) – e.g., point light

2. Modeling

Figure 1 shows the lighting model of a freeform lens designed for uniform illumination. The light source O is located at the origin of the coordinate system. The target plane T is located at a distance z₀ from the origin, and is perpendicular to the z-axis. The unit normal vector of flux of the specific area on the target plane to the total luminous flux on the target plane equals that of the luminous flux of the specific solid angle to the total luminous flux:

θ

² 2πI₀ cosθ sinθdθ

ꢂꢂꢂꢂꢃ

E₀π(y₂² − y²)

the target plane T is ꢀ_ꢁ =(0, 0, −1), and the points on plane T can be



₀

θ₁

1

=

,

(5)

expressed as ꢄ(ꢅ, ꢆ, ꢇ_ꢈ). The points on the freeform lens P can be expressed in spherical coordinates as ꢉ(ꢊ, ꢋ, ꢌ(ꢊ, ꢋ)), where γ is the

θ

E₀πR^{2 max} 2πI₀ cosθ sinθdθ

ꢂꢂꢂꢂꢂꢃ angle between the x-axis and the projection of ꢍꢉ on the x-y plane, and

where ꢋ_ꢙꢚꢛ is the angle corresponding to the maximum light emission, and R is the radius of the illumination pattern. Equation (5) is the socalled “energy mapping function.” This mapping function is very important in the PDE freeform lens design method, which contains information regarding the illumination distribution on the target plane. Altering the energy mapping function can alter the designed illumination distribution. Therefore, Eq. (3) can be transformed into:
ꢂꢂꢂꢂꢂꢃ θ is the angle between the z-axis and ꢍꢉ. The unit normal vector at

• ꢂꢂꢂꢂꢂꢃ
• ꢂꢂꢂꢂꢂꢃ
• ꢃ

point P is ꢀ_ꢎ, and ρ(γ , θ) is the module of ꢍꢉ. Let ꢏ be the unit vector

• ꢂꢂꢂꢂꢂꢃ ꢂꢃ
• ꢂꢂꢂꢂꢂꢃ

of ꢍꢉ, and ꢍ be the direction vector of ꢉꢄ. According to Snell’s law:

•    
• 
• 

n² +n² −2n n   O⋅I N = n O−n I

.

(1)

• (
• )

• o
• I

0

• I
• p

0

I

z₀ − ρ cosθ sin θ (n₀

− n_I cosθ ) −

z

( f (θ ) − ρ sin θ )² + (z₀ − ρ cosθ )²

（x，y，z₀）

T

f (θ ) − ρ sin θ
( f (θ ) − ρ sin θ )² + (z₀ − ρ cosθ )²

, (6)

cosθ (n₀

− n_I sin θ )

∂ρ ∂θ

uuur

N_p

= ρ

z₀ − ρ cosθ
( f (θ ) − ρ sin θ )² + (z₀ − ρ cosθ )² cosθ (n₀

− n_I cosθ ) +

(γ，θ，ρ(γ，θ))

P

f (θ ) − ρ sin θ
( f (θ ) − ρ sin θ )² + (z₀ − ρ cosθ )²

O

sin θ (n₀

− n_I sin θ )

y

By solving Eq. (6), the profile of the freeform lens can be obtained, as shown in Fig. 2.

x

Fig. 1. The lighting model of the freeform lens for uniform illumination. The relationship between the incident light and the refracted light can be derived:

∂ρ ∂γ
∂ρ ∂θ ∂ρ ∂θ



−

• sin γ +
• sinθ cosθ cosγ − ρ sin² θ cosγ

n₀O_x − n_I I_x n₀O_z − n_I I_z

=

• −
• sin² θ − ρ sinθ cosθ

, (2)

∂ρ ∂γ
∂ρ cosγ +

sinθ cosθ sin γ − ρ sin² θ sin γ

Fig. 2. Molding of freeform light-emitting diodes (LEDs).

n₀O_y − n_I I_y

∂θ ∂ρ ∂θ
=

n₀O_z − n_I I_z

• −
• sin² θ − ρ sinθ cosθ

where ꢐ_ꢑ and ꢐ_ꢒ are the refractive indices of the lens material (which is set as 1.54) and air, respectively. In order to simply the discussion, we focus on a uniform circular illumination pattern, and the specific value of ꢊ = 90° ꢓs selected. We therefore obtain:

sinθ(n₀O_z − n_I I_z ) − cosθ(n₀O_x − n_I I_x ) cosθ(n₀O_z − n_I I_z ) + sinθ(n₀O_x − n_I I_x )

∂ρ ∂θ
= ρ

,

(3)

which can be solved as ordinary differential equations in this specific situation. In order to solve Eq. (3), an additional condition ꢆ = ꢔ(ꢋ) should be introduced. The illuminance E₀ on the target plane is constant since uniform illumination is expected. Therefore, the total luminous flux on the target plane can be written as:
Fig. 3. The simulation results of the illumination distribution for the case of a point light source (PLS).

For the convenience of discussion, the ratio of the lens central-height to the source diameter is defined as the L-S factor. Figure 3 shows the simulated (Monte Carlo ray tracing (MCRT),typical wavelength of 450 nm is utilized) results for a PLS, from which it can be seen that the illumination on the target plane is quite uniform; this is rather suitable for LEDs, especially concerning indoor applications. However, as the L- S factor decreases (i.e., as the PLS gradually transforms into an ELS),
,

(4)

φ_total

I (γ ,θ )dΩ

^{= E S  = φ =} 

0

• l
• v

where ꢕ_ꢖ is the illumination area, ꢏ(ꢊ, ꢋ) is the luminous intensity at the viewing angle (ꢊ, ꢋ) from the light source, and ꢗꢘ is the unit solid angle relative to the light source. Generally, angular intensity patterns of LEDs are Lambertian distributions.Considering the energy loss during light traveling, it can be derived that the ratio of the luminous the light spots become bright in the center and dark along the edge; this is referred to as PLS-ELS deviation, and is illustrated in Fig. 4. Obviously, the freeform lens designed by the PLS method is not suitable for the case of an ELS. Further modifications are required in order to obtain uniform illumination.
Let ꢜ = ꢔ(ꢆ) be the input illumination distribution function (IIDF) of the circular light spot on a given target plane. The energy mapping function is therefore:

_y^y

_θ^θ

• ² 2π f (y)ydy
• ² 2π I(θ )sinθdθ

• 1
• 1

(8)

=

₀^R

₀^θ

• 2π f (y)ydy
• ^max 2π I(θ)sinθdθ

To solve Eq. (6), the energy mapping function is reconstructed as follows:.

_y^y

_θ^θ

• ² 2π (k₁E₀ + k₂ f₂ (y))ydy
• ² 2π I(θ )sinθ dθ

• 1
• 1

,

(9)

=

₀^R

₀^θ

• 2π (k₁E₀ + k₂ f₂ (y))ydy
• ^max 2π I(θ )sinθ dθ

where k₁ and k₂ are the weighting coefficients, such that k₁+ k₂ = 1 (k₁=k₂=0.5 is selected in this study), but in practice k₁<k₂ is preferred when the PLS-ELS deviation is large, and a k₁>k₂ combination is recommended when the optimization object is close. For Eq. (9) the

( )

IIDF is ꢜ = ꢔ ꢆ = ꢝ_ꢞꢜ_ꢈ + ꢝ_ꢟꢔ (ꢆ). There must be a special form of

ꢟ

ꢔ (ꢆ) = ꢔ (ꢆ) that will make the calculated freeform lens meet the
Fig. 4. The effect of the L-S factor on illumination distribution along the

light spot diameter on the target plane.

• ꢟ
• ꢠ

design requirements. The equation ꢔ (ꢆ) is therefore called the

ꢠ

“effective rectification function” for the process of transforming a PLS into an ELS. In this work, the rectification function is calculated from the PLS-ELS deviation, so it is also called the “energy feedback function.” Consequently, the task of designing the freeform lens for an ELS is to find the appropriate energy feedback function. Considering a more general condition, the uniform light spot is realized after n generations of rectification, and we obtain:

3. Design of energy feedback freeform lens (EFFL)

Based on these considerations, we develop an energy feedback freeform lens (EFFL) method to further optimize the freeform lens, as shown in Fig. 5. The PLS-ELS deviation is first quantitatively calculated, and then applied as an additional input (i.e., an energy feedback function) for the process of optimization. Simply put, the primary purpose of the EFFL method is to darken the bright area and brighten the dark area. Since the energy appears to be focused at the center of the light spot, we can expect that, if the input of the energy mapping function contains an illumination distribution with a dark center and bright edges (i.e., a non-uniform distribution), then the PLS-ELS deviation should be corrected to a certain degree. The key components of the EFFL method are described in what follows of this section.

n

_y^y

² 2π k f (y)ydy

θ

² 2π I(θ)sinθdθ



• i
• i

_θ

1

i=1

1

=

(10)

n

θ

^max 2π I(θ)sinθdθ

₀^R

2π k f (y)ydy

₀



• i
• i

i=1

Equation (10) is called the “effective energy mapping function” for the process of transforming a PLS into an ELS.

B. Modeling of the ELS illumination system
A. Calculating the freeform lens

The model of the ELS illumination system consists of the lens, the light sources, and the target plane. The freeform lens is established by solving Eq. (6). Next, it is imported into MCRT. The ELS is then simplified to a circular or square light-emitting plane with certain dimensions with the Lambertian distribution. For the purpose of calculating the PLS-ELS deviation in the next step, a far-field target plane is required. The axis of the lens, the light source, and the target plane lie along the same line.
The method of calculating the freeform lens for an ELS is similar to the case of a PLS. The general form of the energy mapping function is:

_θ^θ

² 2π I(θ)sinθdθ
E₀π (y₂² − y₁² )

1

=

(7)

E₀π R²