Designing freeform TIR optical elements using supporting quadric method

Ksenya V. Andreeva; Sergey V. Kravchenko; Mikhail A. Moiseev; Leonid L. Doskolovich

doi:10.1364/OE.25.023465

1. Introduction

Usually, LEDs cannot be directly applied to general lighting because of their Lambertian emission pattern. To obtain high optical performance, modern LED-based lighting devices usually contain a secondary refractive or reflective optical element redirecting the emitted light flux to a desired target. The computation of such an element is an inverse problem of nonimaging optics, which consists in the computation of the optical element surfaces from the condition of generating a prescribed light distribution. Even in the case of a point light source and a single optical surface, this problem is reduced to solving complex nonlinear partial differential equation of Monge-Ampère type [1,2]. In addition to sophisticated numerical techniques for directly solving this equation, various iterative and heuristic methods are widely used to compute the desired optical surfaces [3–14]. One of the general and commonly used iterative methods is the supporting quadric method (SQM) [8–16]. The most known SQM modification is intended for the design of mirrors or refractive optical elements generating a discrete intensity distribution corresponding to a set of points. Depending on the problem solved, paraboloids, ellipsoids or hyperboloids are used as quadrics. In the SQM framework, a continuous freeform optical surface is represented as a set of quadric segments with certain parameters, the number of quadrics being equal to the number of points in the target distribution. The quadric parameters are determined by an efficient iterative algorithm, the convergence of which has been rigorously proven [8–11].

When the angular size of the generated light distribution is less than 40–50°, the use of “conventional” refractive optics in LED illumination becomes less efficient due to increasing Fresnel reflection losses. For light distributions with narrow angular size, the most efficient solution is provided by total internal reflection (TIR) optical elements [17–24]. Nowadays, many approaches for designing TIR optical elements generating axisymmetrical light distributions or collimated light beams are proposed [17–22]. However, various application areas such as the automotive or architectural lighting require generation of asymmetrical light distributions. Only a few methods for designing TIR optical elements generating nonrotational illumination patterns have been proposed so far. In [2,23], optical elements corresponding to axisymmetrical TIR-collimator with free-form outer surface are considered. In this case, the nonrotational light pattern is formed by the outer surface. This approach has some potential issues, such as relatively large dimensions of the optical element and hot spots in the illumination pattern. The paper [24] presents a ray mapping method for the computation of an optical element with a freeform TIR-surface. However, the ray mapping in [24] is proposed for rectangular target distribution only, and the generalization of this mapping for arbitrary light distributions is a challenging problem.

In this paper, we consider for the first time the design of TIR optical elements using the supporting quadric method. In the framework of the method, both inner and lateral TIR surfaces of the optical element are computed within the SQM methodology. Versatility and high performance of the proposed method are confirmed by two design examples of optical elements generating uniformly illuminated square and a bat-shaped pattern with small angular dimensions.

2. Problem statement

Let us consider the problem of designing an optical element with a free-form TIR-surface from the condition of generating a required continuous intensity distribution $I (q)$ , where $q$ is a unit direction vector. The $q$ vector can be considered as a radius vector of a point from the area $Ω$ on a unit sphere corresponding to the support of the function $I (q)$ . The configuration of the optical element is presented in Fig. 1; the light source is located at the origin.

Fig. 1 Geometry of the optical element.

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The inner free-form refractive surface includes two parts: surface (a) generating the specified light distribution $I (q)$ and surface (b) redirecting the source rays on the lateral surface (c). Surface (c) is a free-form surface working on the total internal reflection principle and generating the entire light distribution, similar to the surface (a). The outer surface (d) is flat. Thus, the design of optical element is reduced to the sequential computation of surfaces (a), (b), and (c). Each of the surfaces (a) and (c) independently generates the entire light distribution.

3. Inner surface design

For the computation of the inner surface (a), we propose to use the supporting quadric method [8–15]. Typically, SQM is utilized for the design of refractive or reflective surfaces generating discrete intensity distributions corresponding to a set of points. For this case, an efficient algorithm for the design of optical surface was proposed and the convergence of this algorithm was rigorously proven [8–11]. To obtain a continuous intensity distribution, we propose to use the following approach [12,14–16]. The target continuous distribution is approximated by a discrete one consisting of $N$ points. Then the piecewise smooth surface generating the discrete distribution is designed with the use of SQM. The designed surface consists of $N$ quadric segments, each of which generates one of the points of the discrete light distribution. The obtained segmented surface is then approximated by a smooth surface [14–16]. When the number of points $N$ is sufficiently large, the intensity distribution generated by the obtained smoothed surface approximates well the required continuous one.

Thus, before we start computing piecewise smooth (segmented) optical surfaces we have to approximate the required continuous intensity distribution by a discrete light distribution $(q_{i}, W_{i})$ . Such a distribution corresponds to a set of collimated light beams with fluxes $W_{i}$ propagating in the directions $q_{i}, i = 1, \dots, N$ [15,16]:

\begin{array}{l} W_{i} = \int_{Δ Ω_{i}} I (q) ​ ​ d Ω, \\ q_{i} = \frac{1}{W_{i}} \int_{Δ Ω_{i}} q I (q) ​ ​ d Ω, \end{array}

where

Δ Ω_{i}

is the solid angle element,

\cup_{i = 1}^{N} Δ Ω_{i} = Ω

.

For the computation of surface (a), it is necessary to obtain the discrete light distribution inside the optical element $(p_{i}, W_{i})$ that, after refraction on the plane surface (d), is transformed into the required “outer” distribution $(q_{i}, W_{i})$ of Eq. (1). In this case, each light beam propagating inside the optical element in the $p_{i}$ direction refracts on the flat surface (d) to the collimated beam propagating in the $q_{i}$ direction. The expression for the unit vector $p_{i}$ can be derived using the vector form of Snell’s law [25]:

p_{i} = \frac{1}{n} [N \times (- N \times q_{i})] - N \cdot \sqrt{1 - \frac{1}{n^{2}} (N \times q_{i}) (N \times q_{i})},

where

N = (0, 0, - 1)

is the normal vector to the outer surface (d), and

n

is the refractive index of the optical element.

Let us consider the computation of the piecewise smooth surface (a) generating a discrete light distribution defined by Eq. (1). The surface to be designed consists of $N$ segments, each of which should generate a collimated beam propagating inside the optical element in the specified direction $p_{i}$ , Eq. (2). The shape of such a surface segment can be easily obtained from Fermat’s principle. The planar wave front with the direction $p_{i}$ is written as follows:

(x, p_{i}) = C,

where

C

is a constant. According to the Fermat’s principle, optical path length

Ψ_{a, i}

of the rays from the source to the plane of Eq. (3) should be constant:

r_{a, i} (s_{0}) + n h_{i} (s_{0}) = Ψ_{a,}_{i},

where

r_{a, i} (s_{0})

is the radius-vector length of surface (a) segment collimating the light from the source in the direction

p_{i}

,

s_{0}

is the unit vector of the emitted ray, and

h_{i} (s_{0})

is the distance between the surface (a) and the considered plane Fig. 2 along the direction

p_{i}

. Let us write the parametric equation of the ray with the direction

p_{i}

originated from the point

r_{a, i} (s_{0}) \cdot s_{0}

of the surface (a):

Fig. 2 Geometry of the problem of designing the segment of surface (a)

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x (h) = r_{a, i} (s_{0}) \cdot s_{0} + p_{i} h .

Substituting Eq. (5) into Eq. (3), we obtain the distance $h_{i} (s_{0}) = C - r_{a, i} (s_{0}) \cdot (s_{0}, p_{i})$ . Then to derive the equation for the radius-vector of the surface segment, we plug the expression for $h_{i} (s_{0})$ into Eq. (4):

r_{a, i} (s_{0}) = \frac{C_{a,}_{i}}{1 - n (s_{0}, p_{i})},

where

C_{a,}_{i}

is a constant that defines the size of a segment. It can be seen that Eq. (6) represents a hyperboloid of revolution [8].

Defining vector $s_{0}$ in spherical coordinates, we can derive the equation for the surface segment in the following form:

r_{a, i} (φ, ψ; C_{a,}_{i}) = s_{0} (φ, ψ) r_{a, i} (φ, ψ) = s_{0} (φ, ψ) \frac{C_{a,}_{i}}{1 - n (s_{0} (φ, ψ), p_{i})},

where

s_{0} (φ, ψ) = (\cos φ \sin ψ, \sin φ \sin ψ, \cos ψ)

and the constant

C_{a,}_{i}

is considered as a parameter.

In the context of SQM, the piecewise smooth surface (a) consisting of hyperboloid segments, Eq. (7), can be represented in the following way [8,15, 16]:

r_{a} (φ, ψ) = \min_{1 \leq i \leq N} r_{a, i} (φ, ψ; C_{a,}_{i})

The parameters $C_{a,}_{i}, i = 1, ..., N$ in Eqs. (7), (8) define the intensity distribution generated by the surface (a). Indeed, according to Eqs. (7), (8), decreasing the $C_{a,}_{i}$ value enlarges the $i$ -th segment and, therefore, increases the current light flux propagating in the direction $q_{i}$ . Calculation of the $C_{a,}_{i}$ parameters can be performed using the iterative algorithm [15, 16] from the condition of forming the prescribed light distribution $W_{i}, i = 1, ..., N$ . It is important to note that the limiting case of infinite number of points $N$ corresponds to the generation of a continuous intensity distribution.

When the number of points $N$ is sufficiently large, the required continuous intensity distribution $I (q) $ can be obtained with the use of approximation of the segmented surface of Eq. (8) by a smooth surface [15, 16]. In the present paper, the smooth inner surface is constructed by fitting the NURBS spline to the piecewise smooth surface $r_{a} (φ, ψ)$ [26]. Practical experience has shown that 1 000 – 2 000 points (depending on the complexity of the problem) with subsequent spline approximation usually provides a nearly perfect continuous distribution.

The surface (b) redirecting the source rays on the lateral TIR surface can be defined in different ways. Here we propose to represent it by a set of straight line segments with small apex angle $α$ Fig. 3. This representation of the surface meets the requirements of the injection molding technology and allows to obtain a compact optical element.

Fig. 3 Geometry of the problem of designing the segment of surface (c).

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4. TIR-surface design

The surface (c), working on the total internal reflection principle, can be computed similarly to the surface (a): first, using the supporting quadric method, a piecewise smooth solution is obtained; then the smooth surface is constructed by fitting a NURBS spline to it.

Let us consider the computation of the $i - th$ segment of the TIR surface (c) generating a collimated beam propagating in the direction $p_{i}$ . The radius vector of the $i - th$ segment can be represented by the following equation Fig. 3:

r_{c, i} (φ, ψ) = r_{b} (φ, ψ) + l_{i} (φ, ψ) \cdot s_{1} (φ, ψ),

where

r_{b} (φ, ψ)

is the radius-vector of surface (b),

s_{1} (φ, ψ)

is the unit vector of the ray refracted on the surface (b), and

l_{i} (φ, ψ)

is a scalar function defining the distance from surface (b) to the

i - th

segment of surface (c) along the ray

s_{1} (φ, ψ)

. The ray-vector

s_{1} (φ, ψ)

can be easily calculated using the Snell’s law. The

l_{i} (φ, ψ)

function in Eq. (9) can be derived from Fermat’s principle. According to this principle, for any ray reflected on the i-th segment of the TIR surface, the optical path length

Ψ_{c, i}

from the source to the planar wave front of Eq. (3) should be constant:

r_{b} (φ, ψ) + n l_{i} (φ, ψ) + n h_{i} (φ, ψ) = Ψ_{c, i},

where

h_{i} (φ, ψ)

is the distance between the corresponding point

r_{с, i} (φ, ψ)

on the surface (c) and the planar wave front Fig. 3. Let us write the parametric equation of the ray with the direction

p_{i}

originated from the point

r_{с, i} (φ, ψ)

:

x (h) = r_{b} (φ, ψ) + l_{i} (φ, ψ) \cdot s_{1} (φ, ψ) + p_{i} h .

Substituting Eq. (11) into Eq. (3), we obtain the distance

h_{i} (φ, ψ) = C - r_{b} (φ, ψ) \cdot (s_{0} (φ, ψ), p_{i}) - l_{i} (φ, ψ) \cdot (s_{1} (φ, ψ), p_{i})

. Plugging this expression into Eq. (10), we find the function

l_{i} (φ, ψ)

in the following form:

l_{i} (φ, ψ; C_{c, i}) = \frac{C_{c, i} - r_{b} (φ, ψ) (1 - n (p_{i}, s_{0} (φ, ψ)))}{n (1 - (p_{i}, s_{1} (φ, ψ)))},

where

C_{c, i}

is a constant parameter defining the size of a surface segment.

In the framework of SQM, the piecewise smooth surface consisting of segments defined by Eqs. (9), (12) can be represented in the following way [8,15,16]:

r_{c} (φ, ψ) = \min_{1 \leq i \leq N} [r_{b} (φ, ψ) + l_{i} (φ, ψ; C_{c, i}) \cdot s_{1} (φ, ψ)] .

The segment parameters $C_{c, i}, i = 1, ..., N$ completely defining the piecewise smooth solution can be calculated by the iterative algorithm [15, 16] from the condition of generating the prescribed light distribution $W_{i}, i = 1, ..., N$ . An approximation of the piecewise surface (computed for a sufficiently large number of points N) by NURBS provides generation of a continuous light distribution that is close to the required distribution $I (q)$ .

5. Designed examples

The presented method allows generating a prescribed illuminance distribution in the far field. In such approximation, the required intensity distribution $I (q)$ can be expressed in terms of the illuminance distribution $E (x, y)$ in the following way:

I (q) = E (x, y) \frac{x^{2} + y^{2} + H^{2}}{q_{z}},

where

(x, y)

are the Cartesian coordinates in the target plane

z = H

and

q = (x, y, H) / \sqrt{x^{2} + y^{2} + H^{2}}

is the unit direction vector.

To demonstrate the efficiency of the proposed method for generating narrow-angle light distributions, let us consider two design examples. As the first example, the optical element generating a uniform illuminance distribution in a square area with a side of 3 000 mm located in a distant plane z = 10 000 mm (the transverse angular size is 17°) was computed. The element refractive index is $n = 1.493$ (PMMA).

The specified intensity distribution was approximated by 1 600 points (Eq. (1)). Then the optical surfaces (a) and (c) were computed by the SQM. Brightness of discrete points in the simulated distribution slightly differs due to the tolerances of CAD system and limited computational resources of the raytracing system. We have traced 500 000 rays in the simulation, that averagely corresponds just to 312 rays per discrete spot in the distribution.

Further the piecewise smooth surfaces (a) and (c) were fitted by a NURBS-splines to obtain the optical element with smooth surfaces. The inner lateral surface (b) was represented as a conical surface with the apex angle $α = 2 ° .$ The designed optical element is presented in Fig. 4(d), its overall dimensions are 21.84 x 21.84 x 11.93 mm³. The illuminance distribution generated by the piecewise smooth TIR surface and simulated with point light source is shown in Fig. 5. The illuminance distribution generated by the element in Fig. 4(d) is shown in Fig. 6 and fully confirms the efficiency of the proposed design method. The illuminance distribution was simulated in the commercial software TracePro by tracing 1 000 000 rays from a compact Lambertian source.

Fig. 4 Optical surfaces of the optical element generating a uniformly illuminated square: (a) inner free-form refractive surface; (b) inner lateral refractive surface; (с) lateral free-form TIR surface; (d) full optical element with colored surfaces.

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Fig. 5 Grayscale illuminance distribution generated by a segmented TIR surface of optical element for focusing into a set of points in a square domain.

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Fig. 6 Simulated illuminance distribution generated by the optical element in Fig. 4: (a) grayscale illuminance distribution; (b) illuminance cross sections along the coordinate axes.

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The lighting efficiency of the optical element (the fraction of emitted light flux reaching the target plane) is equal to 91.0%, and the relative root-mean-square deviation (RRMSD) of the generated light distribution from the required one is less than 5.2%. The uniformity of the illuminance distribution (ratio of minimal illuminance to the average illuminance) is 93.2%.

It is instructive to compare the performance and dimensions of the designed element with “conventional” TIR optics containing axisymmetrical TIR-surface. Two other TIR optical elements were computed to solve the same illumination problem. The first one consists of a TIR collimator and a lens array Fig. 7(a) [23]. The array contains identical square-shaped lenses generating a uniformly illuminated square region for collimated incident beam [27]. The second one includes the same TIR collimator and a free-form upper (output) surface Fig. 7(b). The upper surface was calculated by the SQM within the considered methodology. The illuminance distributions for these two optical elements simulated with point light source are presented in Figs. 8(a) and 8(b), respectively.

Fig. 7 Optical elements with axisymmetrical TIR surface and different output surfaces: a) lens array; b) single free-form surface.

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Fig. 8 Simulated illuminance distributions generated by the optical elements in Fig. 7 with the outer lens array (a) and the outer free-form surface (b).

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The performance and dimensions of all three designed optical elements are summarized in Table 1. For all the considered elements, the values of lighting efficiency are close to 91%, while the relative root-mean-square deviations (RRMSD) of the generated illuminance distributions from the required one are not more than 8%.

Table 1. Dimensions and optical performance of the computed optical elements.

View Table

For the modern lighting systems, the size of the optical elements plays an important role [28]. Compactization of the optical element reduces its cost, improves tolerances appearing in the injection molding process [29], allows to increase density of LEDs on the PCB that leads to the minimization and the cost decrease of the entire luminaire. To compare the compactness of the considered designs (Figs. 4 and 7), the height of the inner surface was set to 4.5 mm in all three cases. According to the Table 1, the most compact solution is the optical element with lateral free-form TIR surface (column 1), which also provides slightly better optical performance (in terms of RRMSD) in comparison with the other two designs. Volume (that means weight) of new optical element with TIR free-form surface is less than the volume of two conventional designs by 38% and 45% correspondingly.

Let us consider the second, more intriguing example of optical element intended for generating complex bat-shaped uniform illuminance distribution with the angular dimensions of 43.6° x 22.6°.

For the purpose of the optical element design, the light distribution was approximated by 684 points (Eq. (1)) and a piecewise smooth optical element was computed by the SQM. The illuminance distribution generated by the piecewise smooth TIR surface is shown in Fig. 9. After that, a NURBS-spline was fitted to the obtained solution Fig. 10. The designed optical element has the dimensions of 24.08 × 21.61 × 13.00 mm³. The illuminance distribution simulated with a point Lambertian source at the distance of 10 000 mm is presented in Fig. 11 and demonstrates excellent performance of the proposed method for the case of a complex illuminance distribution. The lighting efficiency of the designed optical element is 91.4% and the RRMSD of the generated illuminance distribution from the required one is less than 4.6%. The uniformity value is 96.5%.

Fig. 9 Grayscale illuminance distribution generated by a segmented TIR surface of optical element for focusing into a set of points in a bat-shaped domain

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Fig. 10 Optical surfaces of the optical element generating a uniformly bat-shaped region: (a) inner free-form refractive surface; (b) inner lateral refractive surface; (с) lateral free-form TIR surface; (d) full optical element with colored surfaces.

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Fig. 11 Simulated illuminance distribution generated by the optical element in Fig. 10(d).

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The solutions obtained by using the presented modification of SQM method are very accurate for the case of point light source, and the accuracy level depends on the number of points in the discrete intensity distribution mostly. Let us note, that the most of existing computation methods (e.g. ray-mapping methods) are also oriented at the point source approximation. To consider the size of the light source during the optical design, standard optimization procedures are usually used. Performance of such optimization methods depends very heavily on the initial approximation. Given that the presented SQM modification provides an accurate solution for the case of the point light source, it is highly recommended to use it as an initial approximation for further optimization when required.

6. Conclusions

A new SQM-based method for the computation of TIR optical elements generating arbitrary narrow-angle light distributions is proposed. High performance of the presented method is confirmed by two designed optical elements generating uniformly illuminated square and bat-shaped areas with narrow angular dimensions. The lighting efficiency of the elements exceeds 90%, while the RRMSD of the generated light distributions is less than 6.0%.

The comparison of TIR optical elements of various types demonstrates that the proposed design with free-form TIR surface provides the most compact overall dimensions preserving the high optical performance. The weight of new optical element with TIR free-form surface is substantially smaller than the weight of TIR lenses with traditional design, that is a great advantage for further manufacturing of optical element by injection molding technology.

Funding

Ministry of Education and Science of Russian Federation; Russian Foundation for Basic Research (RFBR) grant 17-47-630164.

References and links

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	Optical element with TIR free-form surface	Optical element with axisymmetrical TIR surface and outer free-form surface	Optical element with axisymmetrical TIR surface and outer lens array
RRMSD	5.2	7.7	7.3
Lighting efficiency	91.0	91.1	90.7
Maximal width × Height, mm	21.84 × 11.93	25.11 × 16.88	25.67 × 14.59
Volume of optical element, mm³	2401	3911	4404

Designing freeform TIR optical elements using supporting quadric method

Abstract

1. Introduction

2. Problem statement

3. Inner surface design

4. TIR-surface design

5. Designed examples

6. Conclusions

Funding

References and links

Cited By

Figures (11)

Tables (1)

Equations (14)

Optics Express