An inverse method for color uniformity in white LED spotlights
© Prins et al.; licensee Springer. 2014
Received: 13 December 2012
Accepted: 24 March 2014
Published: 3 June 2014
Color over Angle (CoA) variation in the light output of white phosphor-converted LEDs is a common problem in LED lighting technology. In this article we propose an inverse method to design an optical element that eliminates the color variation for a point light source. The method in this article is an improved version of an earlier method by the same authors, and provides more design freedom than the original method. We derive a mathematical model for color mixing in a collimator and present a numerical algorithm to solve it. We verify the results using Monte-Carlo ray tracing.
LED is a rising technology in the field of lighting. In the past, LEDs were only suitable as indicator lights, but the enormous improvements in energy efficiency, cost and light output now allow the use of LEDs for lighting applications . Additionally, LED lighting benefits from low maintenance cost and long lifetime.
Because LED is a rising technology, companies and researchers are constantly searching for methods to reduce the production cost and increase the efficiency, light output and light quality of LED-based lamps. An important issue for white LED lamps is color variation of the emitted light. This is caused by color variation in the light output of the most common type of white LED, the phosphor-converted LED. This type of LED consists of a blue LED with on top a so-called phosphor layer which converts part of the blue light into yellow and red. The resulting output is white light. The distance that a light ray travels through the phosphor depends on the angle of emission. As a result, the light emitted normal to the LED surface is more bluish, while the light emitted nearly parallel to the surface is more yellowish [, pp.353-357]. This phenomenon is called Color over Angle (CoA) variation.
A lot of research has been done to reduce this color variation. Introduction of bubbles in the phosphor layer causes scattering of light, reducing the color variation . Another common method is the application of a dichroic coating on the LED . However, these methods reduce the efficiency of the LED and increase the production costs. Wang et al.  proposed a modification of the optics on the LED to improve the color uniformity. In the case of a spot light, the LED is combined with a collimator. A collimator is an optical component that reduces the angular width of the light emitted by the LED. A common technique is to add a microstructure on top of the collimator. However, this microstructure introduces extra costs in the production process of the collimator, makes the collimator look unattractive and broadens the light beam.
None of the methods mentioned above rigorously solve the problem of color variation, and all methods reduce the efficiency of the optical system. In earlier work , we introduced an inverse method to design a specific type of collimator, the so-called TIR (total internal reflection) collimator. The TIR collimator designed with this method mixes light from a point source such that the color variation is completely eliminated. The collimator requires no microstructures nor scattering techniques. However, the inverse method left very little design freedom for optical designers. An optical designer wants to influence the height and width of the collimator, for example, to fit it into the available space in a lamp design. Also, optical designers want a color mixing collimator which resembles a standard collimator as closely as possible. The inverse method introduced in this paper is an improvement of the method introduced in . The collimator has three free surfaces instead of two. As a result, the improved method offers more design freedom, and it is nearly impossible to distinguish the resulting collimator with the naked eye from a collimator without color correction.
The contents of this paper is the following. First we give a thorough introduction to inverse methods for optical systems and the theory of color mixing in Section 2. In Section 3 we explain the improved inverse method. Section 4 describes three examples where the new method is used. Finally, we end with concluding remarks in Section 5.
2 Design of a TIR collimator using inverse methods
2.1 Source and target intensities
The effective intensity has unit [lm/rad]. For an LED, the effective intensity is typically positive for .
where for monotonically decreasing transfer functions and for monotonically increasing transfer functions.
2.2 Color mixing
The second requirement on transfer functions is related to the color of the resulting beam from the collimator. First we give a short introduction to the theory of color perception, then we derive an ordinary differential equation describing the color of the beam.
The resulting chromaticity coordinates are weighted averages of the chromaticity coordinates of the original beam with weights and . Note that a point is on the straight line segment between and .
2.3 Free surface calculations
Light propagates through the collimator by two type of routes. In the ‘TIR route’, light is refracted by surface S, reflected by surface B or C by total internal reflection and finally refracted by surface T. In the ‘lens route’, light is refracted by surface A and subsequently refracted by surface T.
3 A TIR collimator with three transfer functions
The system (22a)-(22c) with boundary conditions (19a) and (19b) has three unknown functions , and . The functions and are known from measurements on the LED. The function can be chosen by the optical designer as a finite function on . The constants c, and r cannot be chosen freely, we will show that their values follow from conservation of luminous flux, the law of color mixing and the choice of and , respectively. Also the values of and cannot be chosen freely, we will derive an inequality that guarantees monotonicity of the transfer functions.
We choose the positive sign in front of the square root since should be positive. Here and are the right derivatives of at and of at , respectively. These right derivatives are positive because and are positive at and . We have by definition of chromaticity coordinates, and we assume based on measurements that and thus . From this we see that we need to choose such that , so the right hand side of (23) is positive and real.
3.1 The values of c, and r
The system (21) with boundary conditions (19a) and (19b) appears to be overdetermined. However, the system contains three unknown parameters which still need to be chosen. We derive values for three constants c, and r given the boundary conditions and assuming monotonicity of the transfer functions. Later we show that our choice of the constants c, and r imply that three of the boundary conditions are superfluous.
The function is known from measurements on the LED, the function is chosen by the optical designer, so from this relation we derive the value of the constant c. This relation corresponds to conservation of luminous flux (equation (2)).
This relation shows that is the weighted harmonic average of the y-chromaticity coordinate of the light source. Like , the function is known from measurements on the LED, thus we can derive the value of .
This relations corresponds to conservation of luminous flux for the second transfer function.
3.2 Monotonicity of the transfer functions
The transfer functions calculated from (22a)-(22c) should be monotonic, otherwise they have no physical meaning. From (22b) we can easily see that because , and , thus, is monotonically decreasing. The monotonicity of and is more complicated to show and we need some additional assumptions to derive a sufficient condition for monotonicity.
then and are monotonically increasing.
The function it is monotonically increasing because is monotonically decreasing. Therefore, if , we find using (27) and (28) that for all θ. Similarly, if , then for all θ. The inequalities and are equivalent to the second and third inequality in (29). □
3.3 The initial value problem
The ODE-system (21) with the boundary conditions (19a) and (19b) can be solved as an initial value problem. We remark that solving the system as an end value problem has no advantages or disadvantages. We discard the end conditions and solve the initial value problem using a Runge-Kutta method. The end conditions are satisfied as a result of our choices of c, and r.
Theorem 2 Assume monotonicity of the transfer functions. The solution of the initial value problem defined by the ODE system (21) and the initial conditions , satisfies the end conditions , .
Because for all t except for two points at the boundary, we can conclude . Note that this implies .
4 Numerical procedure and results
We solved the mathematical model described in the previous section to design three different TIR collimators. The collimators were designed for two different LEDs, which we refer to as LED16 and LED02. Both of them are Luxeon Rebel IES white LEDs without a dichroic coating, and have a larger than usual CoA variation. The intensity and chromaticity-coordinates of the LEDs were measured, and the measured data were interpolated. The interpolation polynomials have been used to approximate and in (22a)-(22c). The first two collimators were designed for LED16 and have a Gaussian-shaped target intensity profile. The two collimators differ in their values for and . The third collimator was designed for LED02 and has a block-shaped target intensity profile. The collimators were evaluated using the LightTools software package .
4.1 Modelling of the LEDs
Coefficients from the linear least squares fits
4.2 Computation of the transfer functions
with , . The collimator was designed for LED16. The choice of and is restricted by (29). This relation is highly nonlinear. A scatter plot of values of and that satisfy (29) for LED16 is shown in Figure 4.
Parameter values and characteristics for the three different collimators
Gaussian, small 2nd segment
Gaussian, large 2nd segment
4.3 Performance of the TIR collimators
Average chromaticity coordinates of the LEDs and the maximum difference with the chromaticity coordinates in the simulations
Gaussian, small 2nd segment
Gaussian, large 2nd segment
We introduced an inverse method to design a TIR collimator that eliminates CoA variation for a point light source. This method improves the method introduced earlier in  by producing collimators that closely resemble standard collimators and at the same time have more parameters for optical design. In Section 3 we discussed which choices for these design parameters give meaningful results. In Section 4 we tested the method and verified the resulting collimators with Monte-Carlo raytracing using the software package LightTools. The simulations show color variations that are not visible with the human eye.
Unfortunately, LEDs are too large to be treated as a point light source. In future research, we would like to extend this method to take the finite size of the light source into account using iterative methods such as described in for example [18, 19]. This point source method will be an important building block in such an iterative method.
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