An inverse method for color uniformity in white LED spotlights

Abstract

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.

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R E S E A R C H Open Access

An inverse method for color uniformity in white LED spotlights

Corien Prins1*, Jan ten Thije Boonkkamp1, Teus Tukker2 and Wilbert IJzerman3

*Correspondence: mailto:[email protected]

Web End [email protected]

1Centre for Analysis, Scientic computing and Applications (CASA), Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The NetherlandsFull list of author information is available at the end of the article

Abstract

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.

1 Introduction

LED is a rising technology in the eld of lighting. In the past, LEDs were only suitable as indicator lights, but the enormous improvements in energy eciency, cost and light output now allow the use of LEDs for lighting applications []. Additionally, LED lighting benets 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 eciency, 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.-]. 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 eciency of the LED and increase the production costs. Wang et al. [] proposed a modication 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.

2014 Prins et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu

tion License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

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None of the methods mentioned above rigorously solve the problem of color variation, and all methods reduce the eciency of the optical system. In earlier work [], we introduced an inverse method to design a specic type of collimator, the so-called TIR (total internal reection) 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 inuence the height and width of the collimator, for example, to t 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 oers 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 . In Section we explain the improved inverse method. Section describes three examples where the new method is used. Finally, we end with concluding remarks in Section .

2 Design of a TIR collimator using inverse methods

A TIR collimator is a rotationally symmetric lens, usually made of a transparent plastic like polycarbonate (PC) or polymethyl methacrylate (PMMA), that is used to collimate the light of an LED into a compact beam. A prole of a TIR collimator can be seen in Figure . The design procedure using inverse methods consists of two steps: rst we choose a relation between the angles t of rays leaving the LED and the angles of rays leaving the collimator, the so-called transfer functions. Subsequently we use these transfer functions to calculate the free surfaces of the TIR collimator such that the light is redirected according to the relation dened by the transfer functions. In Figure these free surfaces are denoted by A, B and C.

2.1 Source and target intensities

The rst requirement that determines the choice of the transfer functions is the intensity pattern of the light emitted from the TIR collimator. Let I(t, u) [lm/sr] be the intensity

Figure 1 Prole of a TIR collimator. A full TIR collimator can be obtained by rotating the prole around the z-axis. Surface B and C are separated by the ray with angle = 0.

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distribution of the light source. The unit lm stands for lumen, and is the unit to denote energy ux corrected for the sensitivity of the eye at dierent wavelengths, and sr stands for steradian, the unit of solid angle. The angle t [, /] is the angle with respect to the z-axis (inclination), and u [, ) is the angle that rotates around the z-axis (azimuth).

Because of the symmetry of the system, the intensity I(t, u) is independent of u and denoted by I(t). We introduce an eective intensity I(t), which is the ux per rad through the

circular strip [t, t + dt] on the unit sphere divided by . We calculate I(t) by integrating

I(t) over the angle u:

I(t) =

max

G(

segments [i, i], i = , , . . . , N. For each segment we dene a transfer function i : [, max] [i, i] [, /]. For a certain , i() gives the emission angle t of the LED in [i, i]. We choose each transfer function to be strictly monotonic and thus invertible. The luminous ux emitted from the collimator in the interval [, + d] must be equal to the sum over i of the luminous uxes emitted from the source in each interval [i(), i( + d)]. This leads to the following relation:

N

i=iI

i()

I(t)

sin(t) du = I(t) sin(t). ()

The eective intensity has unit [lm/rad]. For an LED, the eective intensity is typically positive for t (, /).

The light emitted from the TIR collimator has a desired pattern in the far eld, meaning that the TIR collimator itself can be considered a point source. The desired intensity prole is denoted by G(, ) [lm/sr], where [, max] is the inclination for some maximum inclination angle < max /, and [, ) is the azimuth. We only consider intensity proles that are rotationally symmetric and thus independent of . Integration over the angle results in an eective intensity G() = sin()G() [lm/rad]. A more in-depth

discussion of eective intensity distributions can be found in Maes []. The target intensity is multiplied by a constant c > such that we have conservation of luminous ux for the optical system:

/

) d. ()

The angular space [, /] of the light emitted by the LED is partitioned into N

N

I(t) dt = c

i() = cG(), ()

where i = for monotonically decreasing transfer functions and i = 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 dierential equation describing the color of the beam.

Color perception is described extensively in [, ]. The human perception of a beam of light can be fully described by its luminous ux (in lm) and the two so-called chromaticity coordinates < x, y < . There is a simple rule to calculate the chromaticity coordinates

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Figure 2 Scatter plot of the measured x and y chromaticity coordinates of the LED used in the numerical experiments in this article. The size of the circles corresponds to the eective intensity. The measured data in the lower left corner correspond small angles of t. The measured data for values of t close to /2 around (0.423, 0.4) are unreliable because of the low light intensities, causing the irregularity.

(x, y) of the beam resulting from mixing two beams of light with luminous uxes L and L and chromaticity coordinates (x, y) and (x, y), respectively:

x = xL/y + xL/yL/y + L/y , (a)

y = L + LL/y + L/y . (b)

The resulting chromaticity coordinates are weighted averages of the chromaticity coordinates of the original beam with weights L/y and L/y. Note that a point (x, y) is on the straight line segment between (x, y) and (x, y).

The chromaticity coordinates of the light emitted from an LED are not constant, but depend on the angle of emission t and are described by functions x(t) and y(t). From measured data we have observed an approximate linear relationship between x(t) and y(t), see Figure . From the color mixing rule, we conclude that if we mix light from dierent angles of the LED into a single beam with color coordinates (xT, yT), these coordinates must be on the straight line segment relating x(t) and y(t). Therefore, given yT, the chromaticity coordinate xT is fully determined and we only need that the y-coordinate of the mixed light equals a certain constant target value yT. The light in the interval [, + d] emitted from the TIR collimator is the sum of beams with intensity iI(i()) di(). The y-coordinate

of this light is therefore

yT =

N

. ()

Using () we nd the following dierential equation:

Ni= iI(i()) di()

Ni= iI(i())/y(i()) di()

i=i I(i())y(i()) i() = cG()/yT. ()

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Figure 3 Geometry of the free surfaces B and C.

The grey arrow shows the TIR route of the light.

2.3 Free surface calculations

The light paths in the TIR collimator shown in Figure correspond to three transfer functions, one transfer function for each of the free surfaces A, B or C. These free surfaces can be calculated from the transfer functions using the generalized functional method developed by Bortz and Shatz [, ]. They derived a dierential equation that describes the location of a free surface, given a surface S from which the rays depart with a given angle t:

dfds =

dt ds

. ()

Here f is the distance a light ray travels from the surface S to the free surface, s is the arc-length along S, t is the angle of the ray leaving S with respect to the z-axis, and is the ray-emission angle measured counterclockwise with respect to the normal of S. The angle is the angle of incidence on the free surface with respect to the surface normal. For the reective surfaces B and C, the variables are illustrated in Figure . We like to formulate this dierential equation in terms of t instead of s. Multiplication by ds/dt gives

dfdt =

dt dt

tan()f +

tan() cos() + sin()

tan()f + ds

dt

tan() cos() sin()

. ()

The parameters and depend on t and are derived below.

Light propagates through the collimator by two type of routes. In the TIR route, light is refracted by surface S, reected by surface B or C by total internal reection and nally refracted by surface T. In the lens route, light is refracted by surface A and subsequently refracted by surface T.

First consider the surfaces B and C. These surfaces are on the TIR route, which is shown in Figure . Surface B is bounded at one side by the rays that leave the source at angle t = . The boundary between surface B and C is marked by the rays that leave the collimator at angle = , and we dene the angle of this ray when leaving the light source to be t = . The angles and are illustrated in Figure . First the light is refracted at surface S. Let d be the distance from the left of surface S to the LED and the clockwise angle of this surface with respect to the symmetry-axis. A ray that leaves the LED at angle t, will hit

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surface S at (xS(t), yS(t)) and leave surface S with angle t(t) given by

xS(t) = d

tan(t) + tan(), (a)

yS(t) = d

tan(t)tan(t) + tan(), (b)

t(t) = arccos

cos( + t) n

. (c)

Relation (c) was derived using Snells law of refraction. The refractive index of the material of the collimator is denoted by n. From (a) and (b) we nd s, which is dened to be at t = /. Also we calculate :

s(t) = d

cos() tan(t) + sin(), (a)

(t) = t(t) +

. (b)

Subsequently, the rays are reected at surface B or C. For a reective surface we have []

(t) =

t(t) (t)

. ()

is the angle of the rays with respect to the z-axis after reection. Before the rays leave the TIR collimator, they are once more refracted by surface T. Rays that leave the collimator at angle = (t) must enter surface T at angle

(t) = arcsin

Here

, ()

where the sign is negative if the rays cross the z-axis, and positive otherwise. Equation () is derived using Snells law. Now we can calculate f (t) by numerically integrating the ODE () backwards, starting at t = /. The parameters in () are given by (c), (a), (b), () and (). For surface C, a plus sign is chosen in () and for surface B a minus sign. The integration for surface C starts with f (/) = b , which is usually chosen larger than

to prevent a sharp edge of the collimator for manufacturing purposes. At t = , the nal value f () of the calculation of surface C is chosen as starting value for the calculation of surface B. The coordinates of the surfaces B and C can be calculated as

xB/C(t) = xS(t) + f (t) cos t(t)

, (a)

sin

(t)

/n

. (b)

Now consider surface A, the lens route of the collimator. The light incident on surface A comes from a single point, therefore the arc-length along the source surface is , so we take s(t) = . Furthermore, we have t(t) = t. For a refractive surface we need the following expression for []:

tan() =

sin(

t) /n cos(

t)

. ()

yB/C(t) = yS(t) + f (t) sin

t(t)

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Using (), the dierential equation () is now

dfdt =

. ()

The initial and end conditions for the transfer functions follow from the signs i and the boundaries and between the segments:

() = , () = , () = , (a)

(max) = , (max) = , (max) = /. (b)

The system () is underdetermined, therefore we add an extra equation. We choose an equation which is as simple as possible, has an obvious physical interpretation and yields a regular coecient matrix for the ODE system. The equation we choose corresponds to the requirement that the intensity resulting from one of the transfer functions contributes a factor r (, ) to the total target intensity. Let j be the index of this transfer function, then we impose

jIj() j() = rcG(). ()

For j = , the coecient matrix of the system is singular for = , and for j = , the coecient matrix is singular for = max. Such a singular coecient matrix does not occur for

t) /n cos(

t)

sin(

f , ()

which we solve by numerically integrating backwards subject to the end condition

f () = xS() + yS(). ()

Surface A can be calculated according to

xA(t) = f (t) cos(t), (a)

yA(t) = f (t) sin(t). (b)

3 A TIR collimator with three transfer functions

Our goal is to design a TIR collimator that has a beam with a specied intensity output

G() and uniform chromaticity coordinates (xT, yT). To achieve this, the transfer functions must satisfy () and (). The layout of the TIR collimator as shown in Figure corresponds to three transfer functions, so N = . For the lens part and surface C, t increases with , so = = . For surface B, t decreases for increasing values of , and thus = . We use the following convention: Ii() = I(i()) and yi() = y(i()). We now have the following system of dierential equations:

I() I() I()

I()/y() I()/y() I()/y()

() ()

()

= cG()

/yT

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j = , so this will be our choice. The ODE system is now

I() I() I() I()/y() I()/y() I()/y() I()

() ()

()

= cG()

/yT r

. ()

The system can be inverted, yielding the following explicit system

() = cG(

)

I()

y()

y() y()

r y()yT + r y() y()

, (a)

() = r cG(

)

I() , (b)

() = cG(

)

I()

y()

y() y()

r y()yT + r y() y()

. (c)

The system (a)-(c) with boundary conditions (a) and (b) has three unknown functions (), () and (). The functions I(t) and y(t) are known from measurements on

the LED. The function G() can be chosen by the optical designer as a nite function on

[, max]. The constants c, yT and r cannot be chosen freely, we will show that their values follow from conservation of luminous ux, 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.

Equation (a) has a removable singularity at = , because G() = and the initial

values of the transfer functions imply () = and thus I() = I() = . We calculate

() using lHpitals rule:

() =

cG +()

I +()

y() y() y()

y() yT

. ()

We choose the positive sign in front of the square root since () should be positive. Here

G +() and I +() are the right derivatives of G() at = and of I(t) at t = , respectively.

These right derivatives are positive because I(t) and G() are positive at t > and > .

We have y(t) > by denition of chromaticity coordinates, and we assume based on measurements that y() < y() and thus y() < y(). From this we see that we need to choose such that y() > yT, so the right hand side of () is positive and real.

3.1 The values of c, yT and r

The system () with boundary conditions (a) and (b) appears to be overdetermined. However, the system contains three unknown parameters which still need to be chosen. We derive values for three constants c, yT and r given the boundary conditions and assuming monotonicity of the transfer functions. Later we show that our choice of the constants c, yT and r imply that three of the boundary conditions are superuous.

The rst unknown value is the constant c. Integration of the rst row of (), using the given boundary conditions and Ii() = I(i()), yields

c

max

G(

) d =

i=iIi()i() d = / I(t) dt. ()

max

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The function I(t) is known from measurements on the LED, the function G() 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 ux (equation ()).

The second unknown is the target chromaticity value yT. Integration of the second row of () using the given boundary conditions and substitution of () yields

/yT

/

I(t) dt =

/

I(t)/y(t) dt. ()

This relation shows that yT is the weighted harmonic average of the y-chromaticity coordinate of the light source. Like I(t), the function y(t) is known from measurements on the

LED, thus we can derive the value of yT.

The third unknown is r. Integration of the third row of () with the given boundary conditions and substitution of () yields

r

/

I(t) dt =

I(t) dt. ()

This relations corresponds to conservation of luminous ux for the second transfer function.

3.2 Monotonicity of the transfer functions

The transfer functions calculated from (a)-(c) should be monotonic, otherwise they have no physical meaning. From (b) we can easily see that () because r > ,

G() and I(t) , thus, () is monotonically decreasing. The monotonicity of

and is more complicated to show and we need some additional assumptions to derive a sucient condition for monotonicity.

Theorem Assume that the chromaticity coordinate function y(t) satises the inequalities

< y(t) < y() < y(t) < y() < y(t)

t, t, t s.t. < t < < t < < t < /, ()

and the transfer functions satisfy the bounds

() , () , () /. ()

If

I(t) dt

/

I(t) dt

= r < min

y()yT ,y()/yT y()/y() , y()/yT y()/y()

, ()

then () and () are monotonically increasing.

Proof We need to prove that the derivatives of and are positive. From (a), using assumptions () and (), we nd that is monotonically increasing if

r y(

)yT + r

y()

y() ,

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and likewise is monotonically increasing if

r y(

)yT + r

y() y()

Using () and () we nd from the assumption r < y()/yT that

r y()

Dene

M() = r

/yT r/y().

Using r

y()

yT , the inequality for the monotonicity of the rst transfer function can be rewritten as M() y() and the inequality for the third transfer function as y() M().

We combine these two results to obtain

y() M() y().

The function M() it is monotonically increasing because () is monotonically decreasing. Therefore, if y() M(), we nd using () and () that y() M() for all .

Similarly, if M(max) y(), then M() y() for all . The inequalities y() M() and M(max) y() are equivalent to the second and third inequality in ().

Figure shows a scatter plot of values of and for which () is satised for an LED which was also used in the numerical experiments. The acceptable values of and are bounded by the lines = tav and = tav, where tav is the such that y(tav) = yT. In this case, the value of tav is unique. From () we see that > tav results in y() > yT, and thus r < . Therefore we cannot guarantee the monotonicity of the transfer functions. Using (c) we can verify that indeed the third transfer function is not monotonic at = max. Similarly, < tav results in r < , and we can verify using () that the rst transfer function is not monotonic at = .

3.3 The initial value problem

The ODE-system () with the boundary conditions (a) and (b) 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 satised as a result of our choices of c, yT and r.

Theorem Assume monotonicity of the transfer functions. The solution of the initial value problem dened by the ODE system () and the initial conditions () = , () = () =

satises the end conditions (max) = (max) = , (max) = /.

y()

y() .

Subtracting the second inequality from the rst we obtain

t

yT

+ r

y()

y() y()

.

yT .

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Figure 4 Scatter plot of the values of 1 and 2 for LED16 that guarantee monotonic transfer functions. tav is such that y(tav) = yT .

Proof First we show that (max) = is satised. Integration of the last row of () from = to = max using () gives

max

I(

) () d =

I(t) dt = rc

max

G(

) d =

(max)

I(t) dt.

So we nd

(max)

I(t) dt +

I(t) dt = .

Because I(t) > for all t except for two points at the boundary, we can conclude () = .

Note that this implies () .

Using the monotonicity of the transfer functions, we can integrate the left hand side of the rst row of ():

max

I() () I() () + I() ()

d

= I(t) dt

I(t) dt +

(max)

(max)

I(t) dt.

For the right hand side we have due to ()

c

max

G(

) d =

/

I(t) dt.

By subtracting the last two relations we nd

(max)

I(t) dt +

(max)

/ I(t) dt = . (*)

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Similarly, we can derive from the second row of () using equation ()

(max)

I(t)

y(t) dt +

(max)

/

I(t)

y(t) dt = .

The functions I(t) and /y(t) are continuous, and I(t) does not change sign in the interval (, /). Using the expanded rst mean-value theorem for integrals [, p.], we nd that for some t ((max), ) and some t ((max), /) we have

y(t)

(max)

I(t) dt +

y(t)

(max)

/ I(t) dt = . (**)

The equations (*) and (**) form a linear system for the integrals of I(t) over [, (max)]

and [/, (max)]. If t < , we nd from the assumption y(t) < y(t) that

det

/y(t) /y(t)

= ,

and thus

(max)

I(t) dt = ,

(max)

/ I(t) dt = .

Because I(t) > for all t except at the boundary points, we conclude

(max) = ,

(max) = /.

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 dierent LEDs, which we refer to as LED and LED. 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 I(t) and y(t)

in (a)-(c). The rst two collimators were designed for LED and have a Gaussian-shaped target intensity prole. The two collimators dier in their values for and . The third collimator was designed for LED and has a block-shaped target intensity prole. The collimators were evaluated using the LightTools software package [].

4.1 Modelling of the LEDs

The LEDs were measured using a goniophotometer []. A goniophotometer is a device that measures intensity, chromaticity coordinates and many other characteristics of light at dierent solid angles. Our LEDs were measured at dierent angles t and dierent angles u. For each LED, the chromaticity values were averaged and the intensities were

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Table 1 Coefcients from the linear least squares ts

LED i 0 2 3 4 5 6 7

16 Ci 0 181.8279 76.0797 221.9411 624.0461 499.8253 127.0787 Dxi 0.4013 0.0578 0.0367 0.0271 0.0564 0.0444 0.0116

Dyi 0.3546 0.0988 0.0557 0.0255 0.0371 0.0196 0.0034

02 Ci 0 148.8533 60.3797 520.5934 913.6760 636.8623 153.5863 Dxi 0.4591 0.0452 0.0590 0.0916 0.1399 0.1030 0.0271

Dyi 0.3792 0.0892 0.1790 0.3596 0.4585 0.2823 0.0658

summed over the angle u. These data have been interpolated with a least squares t using

the following polynomials:

I(t) =

), for k = , . . . , we have ((k) , (k) ) and nally for k = we have ( , ). The intensity and chromaticity coordinates of the LED models in these subintervals correspond to the measured data of the real LEDs at the angles t = (k) . The size of the LED model was reduced to . mm by . mm to simulate a point light source. A comparison of the measured data, the least squares t and the raytracing results of the LightTools model of LED without collimator can be seen in Figures and . A scatter plot of the measured x and y chromaticity coordinates for this LED was shown earlier in Figure . The plot shows the near-linear relationship between x and y, indeed.

4.2 Computation of the transfer functions

Three example collimators have been calculated. The rst collimator was designed for a Gaussian target intensity [] with full width at half maximum (FWHM) [] at /. This yields the following eective target intensity:

G() = sin() exp

ln()

ti (/)i

,

x(t) = Dx +

i= Dxiti,

y(t) = Dy +

i= Dyiti.

The polynomial for the intensity was chosen because it equals at t = / and has zero derivative at t = , both properties are characteristic for the intensity distribution of an LED. The eective intensity equals I(t) = sin(t)I(t). The polynomials for the chromaticity

coordinates were chosen because their derivative equals at t = . The coecients for the two LEDs can be found in Table .

In the LightTools software package, two three-dimensional models were built to simulate the LEDs. The range (, ) of the angle t was discretized into dierent subintervals, labeled k = , , . . . , . For k = we have the interval (,

FWHM

, ()

with .FWHM = max, FWHM = /. The collimator was designed for LED. The choice of and is restricted by (). This relation is highly nonlinear. A scatter plot of values of and that satisfy () for LED is shown in Figure .

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Figure 5 Comparison of the measured effective intensity I, the least squares t and the LightTools

model of LED16. The graph of the LightTools model is not visible because it is hidden behind the least squares t.

Figure 6 Comparison of measured chromaticity coordinates x and y, the least squares t and the LightTools Model of LED16. The graph of the LightTools model is not visible because it is hidden behind the least squares t.

We chose = . and = .. The second collimator was designed for the same LED and target intensity, but this time we chose = . and = ., which gives a larger second segment. The third collimator was designed for LED. The target intensity was chosen to be a block function, yielding the eective intensity G() = sin(), with

max = /. We chose = . and = ., which satises (). An overview of the values chosen and calculated for the three collimators is shown in Table .

The ODE system (a)-(c) with initial conditions (a) was solved using the ODE-solver ode45 in Matlab. The calculation times were a few seconds on a laptop computer

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Table 2 Parameter values and characteristics for the three different collimators

Collimator LED 1 2 r yT c max

Gaussian, small 2nd segment LED16 0.20 0.25 0.1630 0.3862 5,585.9 11.25 Gaussian, large 2nd segment LED16 0.16 0.30 0.4417 0.3862 5,585.9 11.25 Block prole LED02 0.20 0.25 0.1612 0.3986 4,843.3 9

Figure 7 Transfer functions for the collimator with Gaussian prole and a small second segment.

Figure 8 Transfer functions for the collimator with Gaussian prole and a large second segment.

with a . GHz processor and GB RAM. The calculated transfer functions are shown in Figure , and . The transfer functions are indeed monotonic, as expected. Also, (max) = (max) = and (max) = /, as anticipated.

4.3 Performance of the TIR collimators

Subsequently, a TIR collimator was designed for each set of transfer functions, and evaluated using LightTools. We chose for all the collimators d = mm, b = . mm and

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Figure 9 Transfer functions for the collimator with block prole.

Figure 10 LightTools model of the rst collimator.

= /. For every collimator, each free surface was discretized using points and converted into a LightTools model. A screenshot of the LightTools model of the rst collimator can be seen in Figure . Results of the simulations can be seen in Figures , and . In these gures, we see the expected proles of the eective intensity and chromaticity. Figures and look very similar, because the rst and second collimator were designed with the same output specications. In these gures, an irregularity is visible in the chromaticity coordinates near = max. This can be explained as follows. Every bar in the graph corresponds to a range of one degree (/ rad). We chose max = /,

and thus the ux at this angle should be zero. Due to small errors in the free surfaces, a small number of rays exits the collimator at angles larger than max. This happens at surface C, and therefore the chromaticity coordinates at > max are larger than the target values. Because the luminous ux of this light is very small, the irregularity is not visible. A similar irregularity is visible for the collimator with the block prole, only with a smaller chromaticity dierence.

Apart from this small irregularity, the variation in chromaticity is very small. The maximum dierence between the average chromaticity of the LEDs and chromaticity coordi-

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Figure 11 LightTools simulation results for the collimator with Gaussian prole and a small second segment.

Figure 12 LightTools simulation results for the collimator with Gaussian prole and a large second segment.

nates in the simulations are shown in Table . A color dierence of . is considered very good by optical designers and is invisible for the human eye [, ]. The measured color dierences in the simulations are comfortably below this value, thus the color variation in the beam is eliminated.

5 Conclusions

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

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Figure 13 LightTools simulation results for the collimator with Block prole for LED02.

Table 3 Average chromaticity coordinates of the LEDs and the maximum difference with the chromaticity coordinates in the simulations

Collimator LED Target x max(| x|) Target y max(| y|)

Gaussian, small 2nd segment LED16 0.4181 4 104 0.3862 7 104

Gaussian, large 2nd segment LED16 0.4181 4 104 0.3862 1 103

Block prole LED02 0.4691 4 104 0.3985 5 104

have more parameters for optical design. In Section we discussed which choices for these design parameters give meaningful results. In Section we tested the method and veried 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 nite size of the light source into account using iterative methods such as described in for example [, ]. This point source method will be an important building block in such an iterative method.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

The main idea in this paper was proposed by CP, who also prepared the manuscript initially. The proofs were elaborated by CP and JtTB. Technical assistance was provided by TT. All authors provided feedback on the manuscript, and read and approved the nal manuscript.

Author details

1Centre for Analysis, Scientic computing and Applications (CASA), Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. 2Philips Research, High Tech Campus 44, 5656 AE Eindhoven, The Netherlands.

3Philips Lighting, High Tech Campus 44, 5656 AE Eindhoven, The Netherlands.

Received: 13 December 2012 Accepted: 24 March 2014 Published: 3 June 2014

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doi:10.1186/2190-5983-4-5Cite this article as: Prins et al.: An inverse method for color uniformity in white LED spotlights. Journal of Mathematics in Industry 2014 4:5.