Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces

Mikhail A. Moiseev; Egor V. Byzov; Sergey V. Kravchenko; Leonid L. Doskolovich

doi:10.1364/OE.23.0A1140

1. Introduction

Light-emitting diodes (LEDs) have become one of the most popular light sources on the growing market of illumination devices. LED-based lighting devices usually include so-called secondary optics – reflective or refractive optical elements redirecting the emitted flux from the LED into illuminated area. Using such optical elements improves the optical performance of the lighting device and, particularly, increases its illumination efficacy.

Design of secondary optics is an inverse problem of geometrical optics. Even in the case of a single refractive surface and a point light source, this problem can be reduced to solving a nonlinear differential equation in partial derivatives of Monge–Ampère type [1, 2]. This equation has analytical solutions for trivial problems with axial or cylindrical symmetry [3, 4], while in other cases different numerical methods are usually used [5–8]. The refractive secondary optical element has inner and outer surfaces, and each of them can be free-form to obtain the best optical performance. The universal techniques for designing such complex optical solutions are underdeveloped. Nowadays, only few papers concerning this problem are published [9–17]. Michaelis, Schreiber, and Bräuer [9] discover a method for computation of a single free-form refractive surface focusing an arbitrary incident wavefront into a set of points. Sending a number of points to infinity can lead the discrete problem to the continuous one, which enables generation of any complex illuminance distributions by a single surface. The Simultaneous Multiple Surface design method [10] makes it possible to compute two free-form optical surfaces that transform two prescribed incident wavefronts into two required outgoing wavefronts. Bruneton, Bäuerle, and colleagues [11, 12] propose a numerical approach for the design of the optical elements with two refractive surfaces and present several examples proving the reliability of the presented technique. Nevertheless, none of the published papers discuss how to build the most compact or the most efficient optical element with two free-form surfaces. Indeed, if Oliker [13] shows us that in the case of a single optical surface only one exact solution exists, in the case of double free-form surfaces the number of solutions is infinite because the work on ray deflection can be divided between surfaces in an infinite number of ways.

In this paper, we present a novel method for designing refractive optical elements with two surfaces that makes it possible to control the balance of work between the inner and outer surfaces of the optical element. After presentation of the method we consider a standard illumination problem and obtain several solutions, demonstrating how the ratio of work between surfaces influences the optical performance.

2. Refractive collimator with two working surfaces

Before considering the problem of generating a continuous intensity distribution, let us derive the shape of the refractive optical element with two surfaces collimating the light flux from a point light source in the direction of the axis $O z$ . For efficient collimation, as discovered in Appendix A, the inner surface $R (ψ)$ should perform ray transformation $γ = γ (ψ) = ψ / 2$ and the outer surface should provide transformation $β = β (γ) \equiv 0$ (Fig. 1).

Fig. 1 Working principle of collimator with two refractive surfaces.

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The explicit expression for the inner profile implementing the prescribed ray-correspondence function can be obtained by solving the following explicit differential equation [18]:

\frac{d R (ψ)}{d ψ} = - R (ψ) \frac{\sin (ψ - γ (ψ))}{1 / n - \cos (ψ - γ (ψ))} .

The substitution function

γ (ψ) = k ψ

(we use

k

instead of

1 / 2

for the sake of generality) and integration of Eq. (1) provide the following analytical solution:

R = R_{0} {(\frac{n - 1}{n \cos ((1 - k) ψ) - 1})}^{\frac{1}{1 - k}},

where

R_{0} = R (0)

is a parameter defining the size of the inner surface.

To obtain the explicit expression for the outer surface let us proceed from the fact that the collimator should generate a planar wavefront after the second refractive surface. This means that the length of the optical path between point $O$ and any arbitrary point $(x, z_{0})$ in the plane $z = z_{0}$ is constant (Fig. 1):

R_{0} + n l_{0} = R (ψ) + n l (ψ) + (R_{0} + l_{0} - R (ψ) \cos ψ - l (ψ) \cos γ (ψ)),

where

l_{0} = l (0)

is a parameter defining the size of the outer surface, and

l (ψ)

is the distance between the inner and outer surfaces along the ray refracted on the inner surface (Fig. 1). Rearranging Eq. (3), we obtain the explicit definition for the function

l (ψ)

:

l (ψ) = \frac{l_{0} (n - 1) - R (ψ) (1 - \cos ψ)}{n - \cos k ψ} .

Expressions (2) and (4) with the geometry shown in Fig. 1 enable the following parametric definition for the collimator’s surfaces:

\begin{matrix} R (ψ) = R (ψ) s_{0}, \\ r (ψ) = R (ψ) + l (ψ) s_{1}, \end{matrix}

where

s_{0} (ψ) = {\sin ψ, \cos ψ}

is an arbitrary unit vector of the ray emitted by the source,

s_{1} = {\sin k ψ, \cos k ψ}

is a unit vector of the corresponding ray refracted by the inner surface,

R (φ, ψ)

is a radius-vector of the inner surface, and

r (φ, ψ)

is a radius-vector of the outer surface.

The results of Eqs. (2)–(5) can be easily generalized for an arbitrary direction of collimation $p$ by substitution of the dot product $(s_{0}, p)$ instead of $\cos ψ$ and the dot product $(s_{1} (s_{0}), p)$ instead of $\cos k ψ$ :

\begin{matrix} R (s_{0}) = R (s_{0}) s_{0}, \\ r (s_{0}) = R (s_{0}) + l (s_{0}) s_{1} (s_{0}), \\ R (s_{0}) = R_{0} {(\frac{n - 1}{n \cos ((1 - k) arc \cos (s_{0}, p)) - 1})}^{\frac{1}{1 - k}}, \\ l (s_{0}) = \frac{l_{0} (n - 1) - R (s_{0}) (1 - (s_{0}, p))}{n - (s_{1} (s_{0}), p)} . \end{matrix}

The unit vector

s_{1}

of the ray refracted by the first surface can be computed analytically using Fresnel’s law [19].

Let us analyze the dependence of the integrated efficiency of the collimator Eq. (6) on the coefficient $k$ which balances the deflection angles (Fig. 2). The value $k = 0$ corresponds to the case when all collimation is performed by the inner surface, with further growth in coefficient $k$ the outer surface performs more work on the ray rotation, and, finally, in the case of $k = 1$ the inner surface has a spherical shape and all collimation is performed by the outer surface. As was proved in Appendix A, the most efficient case corresponds to the value of $k = 1 / 2$ . As we can see in Fig. 2, a single working surface has very small collimation efficiency due to the high Fresnel losses and the total internal reflection (TIR) effect occurring for most of the rays.

Fig. 2 The dependence of collimator integrated efficiency on the coefficient k.

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3. Design of the free-form optical element

Let us consider the computation of smooth free-form surfaces of the optical element generating a continuous intensity distribution $I (p)$ . The inner surface should perform a fraction of the work equal to $1 - k$ on ray deflection, as in the case of collimator Eq. (6), and the outer surface should perform the remaining $k$ of the work. For this purpose, we propose to use the following algorithm:

1. The continuous intensity distribution $I (p)$ is approximated by the discrete light distribution $(p_{i}, Φ_{i})$ , $i = 1, ..., N$ that corresponds to the set of collimated light beams with fluxes $Φ_{i}$ propagating in the directions $p_{i}$ . The range of definition of function $I (p)$ should be divided into small areas $δ Ω_{i}$ , and after that the values $Φ_{i}$ and $p_{i}$ are computed as the integrated flux and the weighted direction in the area $δ Ω_{i}$ , respectively: $\begin{matrix} Φ_{i} = \int_{δ Ω_{i}} I (p) d Ω, \\ p_{i} = \frac{\int_{δ Ω_{i}} p I (p) d Ω}{Φ_{i}} . \end{matrix}$
2. The piecewise smooth inner surface $R_{p} (s_{0})$ of the optical element is defined as following using the representation of the inner surface of the collimator Eq. (6) as a base primitive [13]: $\begin{matrix} R_{p} (s_{0}) = R_{i} (s_{0}) s_{0}, \\ i = argmin R_{j} (s_{0}), \\ R_{j} (s_{0}) = R_{0 j} {(\frac{n - 1}{n \cos ((1 - k) arc \cos (s_{0}, p_{j})) - 1})}^{\frac{1}{1 - k}} . \end{matrix}$
All rays which are incident on the segment $i$ of the inner surface are partially deflected to the corresponding direction $p_{i}$ . The set of parameters $R_{0 j}$ determining the shape of the inner surface and fluxes in each direction $p_{i}$ can be computed using the well-known Oliker’s method [13]. After that, we can state that the inner surface performs a fraction of the work of ray deflection equal to $1 - k$ to generate the required discrete intensity distribution $(p_{i}, Φ_{i})$ and the rest of the work should be performed by the outer surface.
3. The smooth inner surface $R (s_{0})$ is obtained by fitting a NURBS spline on the resulting piecewise smooth surface $R_{p} (s_{0})$ . Let us note that the smooth inner surface can be obtained by tending the number of points $N$ to the infinity, but this way cannot be used due to the computational limitations. Fitting a spline does not guarantee that the inner surface will perform a fraction of the work equal to exactly $1 - k$ , but in the case of large values of $N$ (at least one to two thousand) it provides an acceptable result.
4. The piecewise smooth outer surface is defined similarly to the expressions (8), but as a base primitive we use the outer surface of the collimator Eq. (6): $\begin{matrix} r (s_{0}) = R (s_{0}) + l_{i} (s_{0}) s_{1} (s_{0}), \\ i = \arg \min l_{j} (s_{0}), \\ l_{j} (s_{0}) = \frac{l_{0 j} (n - 1) - R (s_{0}) (1 - (s_{0}, p_{j}))}{n - (s_{1} (s_{0}), p_{j})} . \end{matrix}$
As in the case of the inner surface, the set of parameters $l_{0 j}$ completely defines the shape of the outer surface and should be computed using the same algorithm [13]. The smooth inner surface $R (s_{0})$ and piecewise-smooth outer surface $r (s_{0})$ together accurately generate the required discrete intensity distribution $(p_{i}, Φ_{i})$ .
5. The smooth outer surface is obtained by fitting a NURBS spline on the resulted piecewise solution.

4. Simulation analysis

The method proposed above was implemented in MATLAB and several examples were computed for demonstration of its workability and versatility.

Let us discuss how the balance of Fresnel losses between the inner and outer surfaces influences the shape of the optical element. The ratio of work on ray deflection is defined by the coefficient $k$ in the Eq. (8). The case of $k = 1$ corresponds to the spherical inner surface, which does not change the direction of the rays after refraction. Further, a decrease of $k$ changes the balance of work between surfaces and, finally, in the case of $k = 0$ the outer surface does not deflect rays, and its shape just repeats the shape of the wavefront outgoing from the inner refractive surface. For the illustration of this fact, we have solved the standard problem of generating a uniformly illuminated square area (with sides of 5500 mm at a distance of z = 1000 mm from the source, an angular size of 140°) with different values of coefficient $k$ and combined the cross-sections of the obtained optical elements in a single plot (Fig. 3). Let us note that the intensity function in the case of the prescribed illuminance distribution $E (x, y)$ in a distant plane $z = f$ is given by

I (p) = E (x, y) \frac{x^{2} + y^{2} + f^{2}}{p_{z}},

where

(x, y)

are the Cartesian coordinates in the plane

z = f

.

Fig. 3 The cross-sections of the designed optical elements.

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The height of the outer surface is fixed. Figure 3 shows that increasing the value of coefficient $k$ leads to growth in the width of the optical element, which can be important for applications with limited overall dimensions of the optical element. In addition, high values of coefficient $k$ also provide greater variations in thickness which leads to low tolerance of the injection molding manufacturing process.

Let us analyze the influence of the coefficient $k$ on the optical performance. For every obtained solution, we have simulated the illuminance distribution in the target plane for an extended light source of 1 × 1 mm (the side is 20% of the height of the optical element) and for an extended source of 2 × 2 mm (the side is 40% of the height of the optical element). Simulation data are presented in Fig. 4 and demonstrate that the obtained point-source design remains valid also for extended light source at a proper choise of the the coefficient $k$ . In particular, Fig. 4 shows that the illuminance distributions obtained for the extended light source have much better uniformity for higher values of coefficient $k$ . This fact is particularly noticeable for the huge extended source of 2 × 2 mm: the shape of the illuminance distribution is not squared for $k$ values of 0.5 or less and the uniformity is quite low. Greater values of $k$ provide solutions which are more tolerant to the size of the light source, which can be explained by the movement of the forming surface away from the source. For high values of $k$ , the inner surface barely influences the rays’ trajectory, while the outer surface carries out most of the deflection.

Fig. 4 Optical elements with different values of k and simulation results.

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The example considered above corresponds to the case of generating wide angular illuminance distribution. Analysis of the obtained results gives us the following recommendation for the choice of the $k$ value: the best optical performance is provided by the range of $0.6 - 0.9$ . If the problem has dimensional limitations, the lower $k$ values are preferable due to the compactness of the solution. In the cases of generating narrow angular distributions, the optical performance depends on the $k$ value similarly, but the size of the final solution diminishes with growth in the coefficient $k$ . Let us note that, while for wide angular light distribution the efficiency of the optical element changes slightly with alteration of $k$ (in the presented example, the efficiency is more than 90% for all solutions), in the case of narrow angular light distributions the dependence of the optical efficiency on the coefficient $k$ is similar to that in Fig. 2.

5. Conclusion

We have presented a novel design method for computation of LED secondary optics with two free-form surfaces. The method makes it possible to control the balance of work on ray deflection between the inner and outer surfaces. For investigation of the developed computation technique we have obtained several solutions to the problem of generating a uniformly illuminated square with an angular size of 140°. Despite the fact that the computation approach assumes a point light source, all solutions were simulated with two extended light sources: 1 × 1 mm (20% of the height of the optical element) and 2 × 2 mm (40% of the height). Analysis of different solutions has shown that the best optical performance is provided by the optical elements where the inner surface performs 60–80% of the work on ray deflection and the outer surface performs the remaining 20–40%. Otherwise, if the problem has dimensional limitations, the balance of work should be shifted to the inner surface for more compactness.

Appendix A

Let us discover the most efficient way to deflect a ray by an arbitrary angle $δ$ using two refractive surfaces. Hereinafter, under a deflection of ray at the angle $δ$ we understand the change of the ray direction such that $(s_{0}, s_{1}) = \cos δ$ , where $s_{0}$ and $s_{1}$ are unit vectors of the ray before and after refraction, correspondingly. Infinite number of solutions can perform this operation: the first surface can rotate the ray at any arbitrary angle $α$ and then the second surface should rotate the ray at an angle $δ - α$ to complete the prescribed transformation. To understand which way provides minimal Fresnel losses, the total transmittance function $T_{T} (α) = T (α) T (δ - α)$ should be analyzed (here, the function $T (α)$ describes the dependence of Fresnel losses at the single refractive surface on the ray rotation angle $α$ ). The only existing extremal of this function can be found by setting its first derivative to zero:

{T^{'}}_{T} (α) = T^{'} (α) T (δ - α) - T (α) T^{'} (δ - α) = 0.

The solution of the Eq. (11) is

α = δ / 2

. This point is a global maximum that can be easily proved by computing the second derivative

{T^{″}}_{T} (α)

in it:

{{T^{″}}_{T} (α) |}_{δ / 2} = 2 T^{″} (δ / 2) T (δ / 2) - 2 T^{'} {(δ / 2)}^{2} .

The second summand in Eq. (12) is always negative. The first one is also always negative because the multiplier

T (δ / 2)

is positive and the multiplier

T^{″} (δ / 2)

is negative due to the concaveness of the function

T (α)

(the plot of the concave function

T (α)

is presented in Fig. 5 for the refractive index of 1.49). Therefore, as the first derivative of

{T^{'}}_{T} (α)

is equal to zero at the point

α = δ / 2

and the second derivative

T^{″} (δ / 2)

is negative, this point is a maximum of the function

T_{T} (α)

. This result means that the most efficient way to deflect a ray at the angle

α

using two refractive surfaces is to deflect it at the angle

α / 2

by each surface.

Fig. 5 Dependence of Fresnel transmission coefficient at the single refractive surface on the ray rotation angle.

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Acknowledgments

The work was funded by the Russian Science Foundation under project # 14-19-00969.

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Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces

Abstract

1. Introduction

2. Refractive collimator with two working surfaces

3. Design of the free-form optical element

4. Simulation analysis

5. Conclusion

Appendix A

Acknowledgments

References and links

Cited By

Figures (5)

Equations (12)

Optics Express