Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape

Xianglong Mao; Hongtao Li; Yanjun Han; Yi Luo

doi:10.1364/OE.23.004313

1. Introduction

Freeform-surface optical system plays a key role in various illumination applications [1–16], wherein the freeform surface is inherently governed by a set of Monge-Ampére type partial differential equations [2–4]. Up to now, no analytic solution of these complex Monge-Ampére equations has been found. Instead, several numerical techniques are utilized to predigest the solving process [3–16]. Among which, a substantial proportion of the methods are based on the mapping of equal-flux grids between the source and the target [10–16], as shown in Fig. 1.

Fig. 1 Two types of source-target ray mappings.

Download Full Size | PDF

The source-target mapping is utilized to prescribe the flux transportation route from the source to the target, which inherently obeys the energy conservation. However, as there are many ways to transport the flux from the source to the target, the mapping is somewhat arbitrary [10–13]. When constructing the grids of the source space and the target field, one rule to obey is that the two grids must have the same topology [13] for the convenience of establishing the mapping relations. Before calculating the grid of the source, a specified two-parameter coordinate system is usually adopted to describe the directions of the rays emitted from the source. Then, the emitting grid of the source is divided respectively along two orthogonal directions related to the two parameters. Finally, the corresponding grid of the target is derived by keeping its topology as well as the cell-flux the same as the source grid. The source-target mapping to some degree depends on the choice of the coordinate system. Two types of coordinate systems have been reported to obtain the emitting grid of the source, as illustrated in Fig. 2. One commonly used is the (u, v) coordinate system as shown in Fig. 2(a), where u is the angle between y-axis and the ray while v is formed by z-axis and the plane containing the ray and y-axis [10]. Another is the conventional (θ, φ) spherical coordinate system as shown in Fig. 2(b), where θ is the zenith angle and φ is the azimuth angle [13–16].

Fig. 2 (a) (u, v) coordinate system and (b) (θ, φ) coordinate system.

Download Full Size | PDF

As for the (u, v) coordinate system, the corresponding target grid is usually desired to be rectangular to maintain a similar topology as the source grid, as shown in Fig. 1(a). Our group has previously proposed a general variable separation method to construct the rectangular target grid and obtain the source-target mapping from (u, v) to (x, y), where (x, y) are Cartesian coordinates used to specify the position on the target plane [10]. The method has shown its capability in forming complex-boundary target distributions by casting a letter “E” on a far-field target plane.

In fact, the source usually has a rotational symmetric intensity distribution and the circular edge of the intensity distribution is usually corresponding to a constant zenith angle θ rather than a constant u or v. In this case, the (θ, φ) grid will be easier to establish than the (u, v) grid to cover the emitting space of the source. On the other hand, the spherical coordinate system (θ, φ) will be a better choice to describe the emitting features of the source. References [13–16] have used the coordinate system (θ, φ) to specify the emitting space of the source, and they use the Cartesian coordinate system (x, y) to describe the position on the target plane. Both the grids are radial to maintain a similar topology, as shown in Fig. 1(b). Once the grids are established, the source-target mapping is obtained by matching the two grids one-to-one. Specifying an initial point, the optical surface is then generated according to the mapping and the Snell’s law [10, 15].

However, only several simple cases have been investigated where the target boundary is usually a circle or a rectangle, when the mapping is quite easy to be established [13–16]. In fact, the target domain to be illuminated usually has various boundary shapes in the actual lighting applications, such as an irregularly shaped theme park or a curved road. To our best knowledge, no general method has been specially demonstrated to establish the polar grids mapping for arbitrarily shaped target. Moreover, due to the somewhat arbitrary choice of the mapping, it usually does not satisfy the integrability condition for a smooth continuous surface, and as a result, step discontinuities must be introduced [13–16]. The discontinuous patches will bring a great trouble for the low cost injection molding manufacture of the freeform-surface optical system. In the meantime, they will also bring about considerable manufacturing defects, which will generate obvious dark stripes on the target illumination pattern [17]. If we artificially force the surface to be smooth, errors of the normal vectors will be produced, and a dramatic performance degradation of the system will be the result. Besides, the extension of the source will affect the system performance.

This paper focuses on handling these problems. Firstly, a general method is proposed to establish the polar grids based flux transportation mapping from the source to an arbitrarily-shaped target, which is the main contribution of this paper. The method is realized in mainly two steps. In the first step, the source grid is divided along φ and θ respectively using the spherical coordinate system (θ, φ). In the second step, the target grid is accordingly divided by separating the two variables (polar angle and polar radius) of the flux integral equations of the target in polar coordinate system. When dividing the target grid along the polar radius, a strategy based on uniformly scaling down the external contour of the target is introduced. Secondly, an iterative feedback strategy [18, 19] is utilized to compensate for the impact caused by the normal errors of the smooth surface as well as the source extension. And this is realized by iteratively modifying the divided polar grids based on the difference between the simulated and the desired illumination distribution without introducing the surface-discontinuities. As examples, smooth freeform lenses are respectively designed for a 1 × 1 mm² LED source to form prescribed illumination patterns within the target fields with a variety of boundaries, such as superellipse, cardioid-shaped and multiple rings. High performances of the optical systems with the light utilization efficiency η (Fresnel loss ignored) over 0.8 and the relative standard deviation (RSD) of the measured illumination distribution less than 0.1 are obtained simultaneously for all the cases, where η is defined as the ratio of the flux within the desired field to the total flux emitted by the source and RSD is defined as:

R S D = \sqrt{\frac{1}{N_{P}} \sum_{i = 1}^{N_{P}} {[\frac{P_{S} (i) - P_{0} (i)}{P_{0} (i)}]}^{2}},

where N_P is the number of the sample points, P_S(i) is the simulated illuminance at the i-th sample point and P₀(i) is the desired illuminance at the i-th sample point.

2. Design method

The design process of the proposed method is illustrated in Fig. 3. It mainly contains four steps: source-target mapping establishment, freeform surface construction, ray-tracing simulation, and iterative feedback modification [18, 19]. Among the four steps, deriving the mapping is the first and key procedure, which is detailed in Section 2.1. Once the mapping is established, a smooth freeform surface is then generated with a geometric method to realize the mapping, as described in Section 2.2. After that, the performance of the optical system is simulated and the iterative feedback technique is introduced to compensate the deviation of the target distribution from the expected, as expounded in Section 2.3.

Fig. 3 A flow diagram of the design process.

Download Full Size | PDF

2.1 Polar grids based source-target mapping establishment

In our method, the polar grids are used to establish the flux mapping. The grid of the source emitting space is divided based on the spherical polar coordinates (θ, φ), where θ is the zenith angle and φ is the azimuth angle. Generally speaking, the luminous intensity distribution of the source is rotationally symmetric, i.e., it is independent of the azimuth angle φ. Therefore, the luminous intensity of the source is assumed to be I(θ) here. To keep the grid topology of the target in accordance with that of the source, the polar coordinate system (ρ, γ) is adopted to calculate a corresponding radial grid on the target plane, where ρ is the radius and γ is the polar angle. Assuming the target illumination distribution is P(ρ, γ), for a lossless optical system, the flux conservation between the source and the target can be described as:

\iint_{Ω_{S}} I (θ) \sin θ d θ d φ = \iint_{Ω_{T}} P (ρ, γ) ρ d ρ d γ,

where Ω_S denotes the solid angle of the source ray directions and Ω_T is the corresponding target region to be illuminated. Then, the flux transportation routes from the source to the target can be obtained by establishing the mapping relationship between a set of equal-flux source and target grids respectively divided according to each side of Eq. (2).

By separating variables of the left side of Eq. (2), the source grid is divided into M × N cells along θ and φ respectively, and the flux contained in each cell is prescribed as E(i, j) (i = 1, 2, …, M, j = 1, 2, …, N). The stripe-like φ-grid can be firstly calculated according to Eq. (3):

\frac{\int_{φ_{\min}}^{φ_{k + 1}} d φ \int_{θ_{\min}}^{θ_{\max}} I (θ) \sin θ d θ}{\int_{φ_{\min}}^{φ_{\max}} d φ \int_{θ_{\min}}^{θ_{\max}} I (θ) \sin θ d θ} = \frac{\int_{φ_{\min}}^{φ_{k + 1}} d φ}{\int_{φ_{\min}}^{φ_{\max}} d φ} = \frac{\sum_{j = 1}^{k} \sum_{i = 1}^{M} E (i, j)}{\sum_{j = 1}^{N} \sum_{i = 1}^{M} E (i, j)},

where φ_min and φ_max are the minimum and maximum azimuth angles of the source, θ_min and θ_max are the minimum and maximum zenith angles of the source, and φ_k₊₁ is the (k + 1)^th azimuth angle (see Fig. 4). Then, each φ-grid is subdivided along θ according to Eq. (4), and the θ-grid for each φ is derived:

\frac{\int_{φ_{k}}^{φ_{k + 1}} d φ \int_{θ_{\min}}^{θ_{_{s + 1}}^{k + 1}} I (θ) \sin θ d θ}{\int_{φ_{k}}^{φ_{k + 1}} d φ \int_{θ_{\min}}^{θ_{\max}} I (θ) \sin θ d θ} = \frac{\int_{θ_{\min}}^{θ_{_{s + 1}}^{k + 1}} I (θ) \sin θ d θ}{\int_{θ_{\min}}^{θ_{\max}} I (θ) \sin θ d θ} = \frac{\sum_{i = 1}^{s} E (i, k)}{\sum_{i = 1}^{M} E (i, k)},

where θ_s₊₁^k⁺¹ is the (s + 1)^th zenith angle corresponding to φ_k₊₁, as shown in Fig. 4. As a result, each cell-grid of the source is formed by two adjacent equal-φ arcs (the black curves in Fig. 4) and two adjacent equal-θ arcs (the green curves in Fig. 4). Moreover, from Eq. (4), it can be derived that the end point of each equal-φ profile is located on the circular edge (the red curve in Fig. 4) of the source intensity distribution, i.e., θ_M₊₁^k⁺¹ = θ_max where k = 1, 2, …, N.

Fig. 4 Grid division of the source based on spherical coordinates (θ, φ). The black curves (equal-φ arcs) are the grid boundaries of the azimuth angles. The green curves (equal-θ arcs) are the grid boundaries of the zenith angles. The red curve is the circular edge of the source intensity distribution.

Download Full Size | PDF

Once the source grid is generated, the corresponding equal-flux polar grid of the target can then be established by separating variables of the right side of Eq. (2). Target fields with two general types of boundaries are investigated here: simply connected and doubly connected.

1). Simply connected target field

In the (ρ, γ) polar coordinate system, the boundary of a simply connected target field can be described as:

ρ_{S C} = f (γ),

where ρ_SC is the radius of the boundary, depending on the polar angle γ (see Fig. 5).

Fig. 5 Grid division of a simply connected target field based on polar coordinates (ρ, γ). The black lines are the grid boundaries of the polar angles. The green curves are the grid boundaries of the polar radii. The red curve is the boundary of the target.

Download Full Size | PDF

Considering a target illumination distribution P(ρ, γ), the fan-shaped γ-grid of the target is firstly divided by solving the double flux integral equation of the target according to Eq. (6), in accordance with the φ-grid of the source:

\frac{\int_{0}^{γ_{k + 1}} d γ \int_{0}^{f (γ)} P (ρ, γ) ρ d ρ}{\int_{0}^{2 π} d γ \int_{0}^{f (γ)} P (ρ, γ) ρ d ρ} = \frac{\sum_{j = 1}^{k} \sum_{i = 1}^{M} E (i, j)}{\sum_{j = 1}^{N} \sum_{i = 1}^{M} E (i, j)},

where γ_k₊₁ is the (k + 1)^th polar angle, as shown in Fig. 5. Then, each γ-grid is subdivided along ρ according to Eq. (7) by scaling down the external contour of the γ-grid, and the ρ-grid for each γ is obtained:

\begin{array}{l} \frac{\int_{γ_{k}}^{γ_{k + 1}} d γ \int_{0}^{ω_{s + 1}^{k + 1} f (γ)} P (ρ, γ) ρ d ρ}{\int_{γ_{k}}^{γ_{k + 1}} d γ \int_{0}^{f (γ)} P (ρ, γ) ρ d ρ} = \frac{\sum_{i = 1}^{s} E (i, k)}{\sum_{i = 1}^{M} E (i, k)} \\ ρ_{s + 1}^{k + 1} = ω_{s + 1}^{k + 1} f (γ_{k + 1}) \end{array}},

where ω_s₊₁^k⁺¹ is the (s + 1)^th scaling factor of ρ_SC along γ_k₊₁, and ρ_s₊₁^k⁺¹ is the corresponding radius, as shown in Fig. 5.

2). Doubly connected target field

The inner and outer boundaries of a doubly connected target field can be described as Eq. (8):

{\begin{cases} ρ_{D C 1} = f_{1} (γ) \\ ρ_{D C 2} = f_{2} (γ) \end{cases},

where ρ_DC₁ and ρ_DC₂ are the radii of the inner and outer boundaries respectively (see Fig. 6).

Fig. 6 Grid division of a doubly connected target field based on polar coordinates (ρ, γ). The black lines are the grid boundaries of the polar angles. The green curves are the grid boundaries of the polar radii. The red curves are respectively the inner and outer boundaries of the target.

Download Full Size | PDF

For a target illumination distribution P(ρ, γ), the divisions of γ and ρ for a doubly connected target can also be achieved from the prescribed matrix E by Eqs. (9) and (10) respectively, similar to the case of a simply connected target:

\frac{\int_{0}^{γ_{k + 1}} d γ \int_{f_{1} (γ)}^{f_{2} (γ)} P (ρ, γ) ρ d ρ}{\int_{0}^{2 π} d γ \int_{f_{1} (γ)}^{f_{2} (γ)} P (ρ, γ) ρ d ρ} = \frac{\sum_{j = 1}^{k} \sum_{i = 1}^{M} E (i, j)}{\sum_{j = 1}^{N} \sum_{i = 1}^{M} E (i, j)},

\begin{array}{l} \frac{\int_{γ_{k}}^{γ_{k + 1}} d γ \int_{f_{1} (γ)}^{ω_{s + 1}^{k + 1} f_{2} (γ)} P (ρ, γ) ρ d ρ}{\int_{γ_{k}}^{γ_{k + 1}} d γ \int_{f_{1} (γ)}^{f_{2} (γ)} P (ρ, γ) ρ d ρ} = \frac{\sum_{i = 1}^{s} E (k, j)}{\sum_{i = 1}^{M} E (k, j)} \\ ρ_{s + 1}^{k + 1} = ω_{s + 1}^{k + 1} f_{2} (γ_{k + 1}) \end{array}},

where γ_k₊₁ is the (k + 1)^th polar angle, ω_s₊₁^k⁺¹ is the (s + 1)^th scaling factor of ρ_DC₂ along γ_k₊₁, and ρ_s₊₁^k⁺¹ is the corresponding radius, as shown in Fig. 6.

Once the polar grid of the target is obtained, we can easily convert it to the Cartesian coordinates (x, y):

(x_{s}^{k}, y_{s}^{k}) = (ρ_{s}^{k} \cos γ_{k}, ρ_{s}^{k} \sin γ_{k}), s = 1, 2, \dots, M + 1, k = 1, 2, \dots, N + 1.

Then, the source-target mapping (θ, φ)-(x, y) is established.

In summary, by separating the variables of the flux integral equation in polar coordinate system, a general and simple formulated method is proposed in this section to achieve the polar grids based flux transportation mapping between the source and an arbitrarily-shaped target. The cell-flux matrix E is used in our method to construct the source-target mapping, which makes the feedback process much easier and clearer [20], as described later in Section 2.3. And it is worth noting that, the elements of the matrix E can be all equal (in general) or properly adjusted according to the luminous property of the source, making the design more flexible. Besides, if the boundary shape of the target cannot be given in an explicit expression, we can firstly sample the boundary in a discrete form, and then either fit the discrete boundary points with an explicit polar equation or use numerical integral combined with interpolation to obtain the target grid. Therefore, the method is quite general and flexible.

2.2 Optical system construction

Herein, the optical system is constructed by a geometric method [10, 15] to realize the source-target mapping (θ, φ)-(x, y), as shown in Fig. 7.

Fig. 7 Geometric configuration of the freeform surface. The black curves are the lens profiles along the zenith angle. The green curves are the lens profiles along the azimuth angle.

Download Full Size | PDF

Once the source grid is established, a corresponding set of rays emitted from the source are then obtained. Since all these rays will fall on the freeform surface, the calculation of the surface points is then converted to determining the end point of each of these rays. There are mainly three steps. Firstly, a seed curve C¹ is constructed along θ. It is started by specifying an initial point P₁¹ as the vertex, which is on the trajectory of the incident ray I₁¹. Assuming the emergent ray at P₁¹ is O₁¹, the normal vector N₁¹ can be calculated by the Snell’s law expressed as Eq. (12):

N = \frac{n_{o} O - n_{i} I}{| n_{o} O - n_{i} I |},

where n_i and n_o are the refractive indices of the incidence and emergence media, I and O are the unit vectors of the incident and emergent rays, and N is the unit normal vector. The next point P₂¹ on C¹ can be calculated as the intersection of the incident ray I₂¹ and the tangent plane at the previous point P₁¹. Repeating the above procedure, all the other points and the corresponding normal vectors on the curve C¹ are obtained. Secondly, taking all the points on the seed curve C¹ as initial points, all the other curves can be generated along φ based on the same algorithm. Finally, a smooth optical surface is obtained by lofting the curves. However, because the mapping does not meet the integrability condition, the forced smooth surface will cause normal errors [10, 11].

2.3 Iterative feedback modification with fixed target grid

After 2.2, a smooth freeform optical system is constructed for a hypothetically perfect point source. Then we simulate its performance by Monte Carlo ray tracing with the real source which has a finite size. The simulated illumination distribution P_S(x, y) usually has a large deviation from the desired distribution P₀(x, y), mainly caused by the normal errors and the non-ignorable dimension of the source. An iterative feedback process [18, 19] is employed here to alleviate this deviation by iteratively modifying the source grid with a feedback function. The feedback function β(x, y) used here is defined as Eq. (13):

β_{k} (i, j) = {P_{0} (i, j) / [λ_{1} \cdot P_{S k} (i, j) + (1 - λ_{1}) \cdot P_{0} (i, j)]}^{λ_{2}}, i = 1, 2, \dots, M, j = 1, 2, \dots, N,

where 0 ≤ λ₁ ≤ 1, λ₂ ≥ 0 are the weighting parameters, P₀ and P_Sk are respectively the prescribed and simulated illumination distribution matrixes of the k^th iteration, interpolated with the fixed target grid, and β_k is the corresponding feedback matrix. After k times of iterations, the modified cell-flux matrix of the source grid can be described as Eq. (14):

E_{k} (i, j) = Π_{l = 1}^{k} β_{l} (i, j) \cdot E (i, j), i = 1, 2, \dots, M, j = 1, 2, \dots, N .

Then the source grid is redivided according to Eqs. (3) and (4) using the modified cell-flux matrix in Eq. (14), and a new source-target mapping is obtained. After that, the smooth optical system is reconstructed according to the new mapping. Usually less than 10 times of iterations are needed until the final result becomes acceptable.

3. Design examples

To show the effectiveness of the proposed method, several lenses are designed to generate prescribed illumination distributions within the target regions with different boundaries. Design parameters are specified in Fig. 8. The source is a 1 × 1 mm² square-shaped Lambertian LED source with a divergence half-angle of 90°. The central height h of the lens is 5 mm with a refractive index n_l of 1.59. The target plane is placed H = 10 m away from the source. And the shape of the target will be given respectively in the following examples. The other parameters used in the design are: M = 50, N = 200, and λ₁ = λ₂ = 0.5. Moreover, the Fresnel loss is ignored in the ray-tracing simulation.

Fig. 8 Design parameters.

Download Full Size | PDF

According to Eqs. (3) and (4), an equal-flux grid is firstly divided for the source as illustrated in Fig. 9, i.e., the cell-flux matrix E is a matrix with each element equals to a constant E₀. And, each cell-grid is bounded by equal-φ and equal-θ arcs. Moreover, the edge of the source grid is rightly located on the edge of the source intensity distribution (the red curve in Fig. 9). Unless otherwise specified, the initially divided source grid is kept the same for all the following design examples.

Fig. 9 (a) The equal-flux source grid divided over the unit emitting hemisphere of the source; (b) The source grid projected onto the x-y plane. The black curves are equal-φ curves. The green curves are equal-θ curves. The red curve is the edge of the source intensity distribution.

Download Full Size | PDF

3.1 Simply connected target

Firstly, the case, where the boundary of the target is a superellipse, is investigated. The superellipse is a curve with the Cartesian equation:

{| \frac{x}{a} |}^{n} + {| \frac{y}{b} |}^{n} = 1,

where n, a and b are positive numbers. It is given in polar coordinates by Eq. (16):

ρ_{S C} = {({| \frac{\cos γ}{a} |}^{n} + {| \frac{\sin γ}{b} |}^{n})}^{- 1 / n} .

The curve is an ordinary ellipse when n is 2, and it becomes a rectangle as n tends to infinity. For a = 20 m, b = 10 m, and n = 1, 2, 4, and infinity respectively, four smooth-surface lenses are designed with the proposed method. The radial target grids are shown in Figs. 10(a)-10(d). In accordance with the source grid, each cell of the target grids contains the same flux. Namely, the area of each cell is equal. Two basic lighting parameters, light utilization efficiency η and relative standard deviation (RSD) of the illumination distribution, are calculated from the simulation results.

Fig. 10 (a)-(d) Equal-flux target grids; (e)-(h) Simulated illuminations without feedback modification; (i)-(l) Simulated illuminations with feedback modification; (m)-(p) Lens models.

Download Full Size | PDF

The simulated illumination distributions without feedback modification are shown in Figs. 10(e)-10(h). We can see that η is high which approximates 0.9, however RSD is also very large which is about 0.2. The simulated illumination distributions after feedback modification are shown in Figs. 10(i)-10(l). Uniform illumination distributions with precisely controlled boundaries are obtained for all the four cases, where η is over 0.8 and RSD is below 0.1. The corresponding lens models with smooth freeform surfaces are shown in Figs. 10(m)-10(p).

To show the capability of the proposed method in handling target field with more complex boundary, a smooth lens is also designed to form a uniform pattern within a cardioid-shaped target field. A cardioid is a curve with the polar equation:

ρ_{S C} = R_{1} (1 - \sin γ),

where R₁ is a positive number. To avoid zero-radius at the origin, we translate the cardioid along y-axis by a distance R₂. Taking R₁ = 10 m and R₂ = 10 m, a smooth lens is designed based on the proposed method, as shown in Fig. 11. The polar grid of the heart-shaped target is shown in Fig. 11(a), and each grid cell contains the same flux. The freeform lens model is shown in Fig. 11(b). The dimensions (length, width, and central height) of the model are 9.3 mm × 8.2 mm × 5 mm. The simulated illumination distribution is shown in Figs. 11(c) and 11(d). A uniform illumination distribution is achieved within the specified domain, while the boundary of the target is slightly blurred caused by the extension of the source. High lighting performance with η = 0.81 and RSD = 0.057 is obtained.

Fig. 11 (a) Equal-flux target grid; (b) Lens model; (c) Simulated illumination; (d) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).

Download Full Size | PDF

To show the universality of the proposed method, a smooth freeform lens is also designed to form a nonuniform pattern on a far-field target plane. Here, an elliptical Gaussian illumination distribution is considered:

P (ρ, γ) = P_{0} \exp {- 2 [\frac{{(ρ \cos γ)}^{2}}{a_{0}^{2}} + \frac{{(ρ \sin γ)}^{2}}{b_{0}^{2}}]},

where P₀ is a constant, a₀ = 20 m, b₀ = 10m, and the definition domain is an ellipse in the form of Eq. (16) with a = 20 m, b = 15 m, and n = 2. The normalized desired illumination distribution and the divided polar grid of the target is shown in Fig. 12(a). Based on the target grid, a smooth freeform lens is constructed using the proposed method, as shown in Fig. 12(b). The dimensions of the lens are 8.1 mm × 6.9 mm × 5 mm. The normalized simulated illumination distribution [see Fig. 12(c)] and the corresponding cross-section curves [see Fig. 12(d)] show that a good match of the simulated distribution to the desired distribution is achieved. For this design, η is 0.91 and RSD is 0.042.

Fig. 12 (a) The desired illumination distribution (normalized) under the divided grid; (b) Lens model; (c) Simulated illumination distribution (normalized); (d) The cross-sections of the simulated distribution (normalized) at x = 0 (red line) and y = 0 (black line), and the corresponding cross-sections of the desired distribution (dashed blue line).

Download Full Size | PDF

To verify the effectiveness of the method for discrete target boundary, a freeform lens is also designed to form a uniform illumination pattern within the target domain bounded by a series of discrete points. The optionally selected points are shown in Fig. 13(a) (the red points) using polar coordinates (ρ, γ). Then, a cubic spline interpolation is adopted to fitting the discrete points as illustrated in Fig. 13(a) (the black curve), i.e., a spline function SF(γ) is used to define the boundary radius ρ_SC as a replacement of the discrete form. The discrete points and the fitted spline curve under Cartesian coordinate system (x, y) are shown in Fig. 13(b). Finally, the target mesh is generated according to Eqs. (6) and (7) by replacing f(γ) with SF(γ).

Fig. 13 (a) Fit the discrete boundary points (the red points) using spline curve (the black curve) under polar coordinate system (ρ, γ); (b) Transform (a) to Cartesian coordinate system (x, y).

Download Full Size | PDF

The calculated equal-flux target mesh is shown in Fig. 14(a). The top and front views of the freeform lens are respective shown in Figs. 14(c) and 14(d), and the dimensions of the lens are 10.0 mm × 8.9 mm × 5 mm. The simulated illumination pattern is shown in Fig. 14(b) and a high lighting performance with η = 0.89 and RSD = 0.055 is achieved.

Fig. 14 (a) Equal-flux target grid; (b) Simulated illumination; (c) Top view of the lens; (d) Front view of the lens.

Download Full Size | PDF

3.2 Doubly connected target

In this section, the doubly connected target field is investigated. Both the inner and the outer boundaries are described by superellipses, as expressed in Eq. (19):

{\begin{cases} ρ_{D C 1} = {({| \frac{\cos γ}{a_{1}} |}^{n_{1}} + {| \frac{\sin γ}{b_{1}} |}^{n_{1}})}^{- 1 / n_{1}} \\ ρ_{D C 2} = {({| \frac{\cos γ}{a_{2}} |}^{n_{2}} + {| \frac{\sin γ}{b_{2}} |}^{n_{2}})}^{- 1 / n_{2}} \end{cases},

where a₁, b₁, a₂, b₂, n₁ and n₂ are positive numbers. Taking a₁ = b₁ = 5 m, a₂ = b₂ = 15m, n₁ = 2, and n₂ = 4, a smooth lens is designed based on the proposed method. The equal-flux polar grid of the target is shown in Fig. 15(a). The dimensions of the lens are 12.5 mm × 12.5 mm × 5 mm, as shown in Fig. 15(b). A uniform illumination distribution is also generated within the specified region with η = 0.88 and RSD = 0.10, as shown in Figs. 15(c) and 15(d).

Fig. 15 (a) Equal-flux target grid; (b) Lens model; (c) Simulated illumination; (d) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).

Download Full Size | PDF

Sometimes, a light pattern of multiple alternately dark and bright zones is desired for fishing task to attract the fish shoals [21]. Here, a multiple-rings target pattern is considered, as shown in Fig. 16(a). The target consists of three concentric quartic superellipses [see Eq. (15)] with a₁ = b₁ = 4 m, a₂ = b₂ = 8 m and a₃ = b₃ = 12 m respectively. The three superellipses form three regions (I, II, and III) where the illuminance ratio is 1:0.5:1. Corresponding to each one of the three regions, the proposed method is used to divide the source and the target grids, as shown in Figs. 16(b) and 16(c). The source grid is divided using Eqs. (3) and (4), where the upper and lower bounds of the zenith angle θ for each region are firstly calculated based on the energy conservation between the source and the target region. As for the target regions, region I is a simply connected field, therefore the target grid of this part can be achieved by solving Eqs. (6) and (7). Regions II and III are doubly connected fields whose grids can be obtained by solving Eqs. (9) and (10).

Fig. 16 (a) The target pattern; (b) The source grid and (c) the target grid are respectively divided corresponding to each ring of the target using the proposed method.

Download Full Size | PDF

Using the mapping between the source and the target grids shown in Figs. 16(b) and 16(c), a smooth freeform lens is constructed by the proposed method, as shown in Fig. 17(a). The dimensions of the lens are 8.7 mm × 8.7 mm × 5 mm. The simulated illumination pattern and the illumination curves across the target center are shown in Figs. 17(b) and 17(c) respectively. The average illumination ratio of the three regions is 1.08:0.50:0.92 which approaches the prescribed ratio. For this design, η is 0.89 and RSD is 0.061.

Fig. 17 (a) Lens model; (b) Simulated illumination; (c) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).

Download Full Size | PDF

4. Conclusions

In summary, we proposed a general and easy-handling flux mapping construction method to design freeform illumination optical system for extended LED light source. The mapping was based on equal-flux polar grids, which were achieved by separating variables of the flux integral equations in polar coordinates. Moreover, instead of utilizing step discontinuities, an iterative feedback method was employed to minimize the impact caused by the normal errors as well as the source extension, and as a result a smooth-surface optical system was obtained. As examples, several freeform lenses were designed for a finite-sized LED source to form prescribed illumination patterns within the target regions with various boundaries. High lighting performance is obtained for all these cases. As shown from the examples, the method is quite flexible and effective, and it can be very useful in illumination design.

Acknowledgment

This work was supported by the National Basic Research Program of China (Grant Nos. 2015CB351900, 2011CB301902, and 2011CB301903), the National Natural Science Foundation of China (Grant Nos. 61307024, 61176015, and 61176059), the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant Nos. 2011BAE01B07, and 2012BAE01B03), the Opened Fund of the State Key Laboratory on Integrated Optoelectronics (Grant No. IOSKL2014KF06), the High Technology Research and Development Program of China (Grant Nos. 2011AA03A106, 2011AA03A112, and 2011AA03A105).

References and links

1. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.

2. J. S. Schruben, “Formulation a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. A 62(12), 1498–1501 (1972). [CrossRef]

3. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

4. R. M. Wu, L. Xu, P. Liu, Y. Q. Zhang, Z. R. Zheng, H. F. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013). [CrossRef] [PubMed]

5. V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE 5942, 594207 (2005). [CrossRef]

6. D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011). [CrossRef] [PubMed]

7. C. Canavesi, W. J. Cassarly, and J. P. Rolland, “Target flux estimation by calculating intersections between neighboring conic reflector patches,” Opt. Lett. 38(23), 5012–5015 (2013). [CrossRef] [PubMed]

8. H. R. Ries and R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11(4), 1260–1264 (1994). [CrossRef]

9. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004). [CrossRef]

10. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]

11. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010). [CrossRef] [PubMed]

12. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012). [CrossRef] [PubMed]

13. W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE 3779, 363–370 (1999). [CrossRef]

14. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]

15. K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009). [CrossRef]

16. L. Hongtao, C. Shichao, H. Yanjun, and L. Yi, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013). [CrossRef] [PubMed]

17. K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009). [CrossRef] [PubMed]

18. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010). [CrossRef] [PubMed]

19. X. L. Mao, H. T. Li, Y. J. Han, and Y. Luo, “Two-step design method for highly compact three-dimensional freeform optical system for LED surface light source,” Opt. Express 22(Suppl 6), A1491–A1506 (2014). [PubMed]

20. W. C. Situ, Y. J. Han, H. T. Li, and Y. Luo, “Combined feedback method for designing a free-form optical system with complicated illumination patterns for an extended LED source,” Opt. Express 19(S5), A1022–A1030 (2011). [CrossRef] [PubMed]

21. S. C. Shen, J. S. Li, and M. C. Huang, “Design a light pattern of multiple concentric circles for LED fishing lamps using Fourier series and an energy mapping method,” Opt. Express 22(11), 13460–13471 (2014). [PubMed]

Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape

Abstract

1. Introduction

2. Design method

2.1 Polar grids based source-target mapping establishment

1). Simply connected target field

2). Doubly connected target field

2.2 Optical system construction

2.3 Iterative feedback modification with fixed target grid

3. Design examples

3.1 Simply connected target

3.2 Doubly connected target

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (17)

Equations (19)

Optics Express