Reflective surface design for uniform LED lighting

As a typical representative of the fourth generation of new energy-saving light sources, LEDs have advantages that other light sources cannot match, such as long life, energy saving, environmental protection, and good color rendering. Moreover, with the rapid development of high-power LEDs, they are increasingly widely used in various indications, displays, backlight sources, general lighting, urban night scenes, and many other fields. However, the light emitted by LED light sources is uneven. Therefore, in special lighting occasions with strict requirements on lighting, a secondary light distribution structure must be designed based on the luminous characteristics of the LED itself and the expected light intensity distribution to redistribute the light intensity of the LED.

There are generally two types of light distribution structures: reflective surface and lens. Currently, most of the secondary light distribution structures of LEDs use lens structures. Considering that the lens has two light-transmitting surfaces and a certain thickness, for any curved surface, if there is a slight deviation in the design or production process of the curved surface, or there is a little impurity in the lens, it will have a great impact on the refraction and energy distribution of light. In practical applications, the lens has a high absorption rate for light and causes energy loss. Using a reflective surface as a secondary light distribution structure, theoretically only one reflective surface is needed, which is easy to process and has less energy loss, and better solves the problem of uniform lighting of LEDs. This article aims to design a reflective surface for secondary light distribution of LEDs.

The traditional reflective surface light distribution structure relies on a quadratic surface or a combination of quadratic surfaces with analytical solutions. However, for light sources with uneven light emission, simple quadratic surfaces or combinations of quadratic surfaces cannot achieve a specific light intensity distribution. This paper adopts the method of free-form surface design, establishes a differential equation based on the light emission characteristics of the light source and the expected light intensity distribution, solves the equation using a numerical solution method, fits the free-form reflective surface based on the solution results, and then uses optical software to simulate the results.

2 Reflective surface design

In this paper, part of the light emitted by the light source directly irradiates the illumination surface, and the other part is reflected by the designed reflective surface to the illumination surface, and finally a rectangular uniform illumination area is realized on the illumination surface. The law of reflection of light is used to obtain the vector relationship between the outgoing light and the incident light, and then the energy carried by the reflected light is calculated using the energy conservation and illumination formula. Finally, the differential equation is listed, and a series of points for determining the free-form surface are obtained by solving the differential equation, and finally the required free-form surface is determined.

As shown in Figure 1, o' is the position of the light source, and the coordinate system x'o'y' is established. 1 is the illumination surface. In order to make the calculation more convenient, the distance from the light source o' to the illumination surface 1 is set as unit length 1. With o' as the center, a unit sphere is made, and the unit sphere is tangent to plane 1 at point o. Plane 2 is a tangent plane of the unit sphere parallel to plane 1. In planes 1 and 2, draw x, y axes and u, v axes parallel to x' and y' respectively. The origin of the z' axis is at o'.

Any ray intersects the unit sphere at point p, then op is a unit vector, denoted as i. Connect points o and p and extend the intersection plane 2 to point (u, v, 1). Let the coordinates of point p be (x′, y′, z′), then i = (x′, y′, z′).

As shown in the figure, we can see that x′ / u = y′ / v = (z′ + 1) /2, and x′2 + y′2 + z′2 = 1, so we can obtain: i = (1 + ω2 /4) (u, v, 1 - ω2 /4), where ω2 = u2 + v2.

Figure 1 Established coordinate system

2.1 Law of reflection of light

As shown in Figure 2, I is the light emitted by the LED, incident on point q on the reflection surface, R is the reflected light, reflected to point t on the illumination surface, T is the vector from o′ to t, and N is the normal at point q on the reflection surface.

Figure 2 Optical path of reflected light irradiating point t

According to the law of reflection in non-imaging optics theory:

Where I = ρi, the expression of the vector T can be obtained as T = T (ρ, u, v).

From the geometric relationship, we can see that the normal vector of the reflecting surface is the product of two vectors in the tangent plane, that is, N = Au × Av = (ρi)u × (ρi)v.

From the i vector we can find:

Substituting the above three equations into the law of reflection and defining p = ρu, q = ρv, we can obtain T = T (u, v, ρ, p, q), then the coordinates of the point where the reflected light is reflected on the illuminated surface are:

in:

2.2 Energy Conservation

The LED lighting surface is perpendicular to the z-axis and faces downward. This paper uses a Lambertian LED light source. Part of the light emitted directly hits the lighting surface, and the other part is reflected and then hits the lighting surface, generating a rectangular uniform lighting area with a length and width of a and b respectively. According to the law of energy conservation, the radiant flux of the light source should be equal to the radiant flux on the lighting surface.

The radiant flux radiated by the light source to the illumination surface is Φ1 =∫I0 cosΦdΩ1 + ∫μI0 cosΦdΩ2, where the first term is the radiant flux directly irradiated to the illumination surface, Ω1 is the solid angle corresponding to the incident light, and the second term is the radiant flux reflected to the illumination surface by the reflection surface, Ω2 is the solid angle corresponding to the incident light, and μ is the reflection coefficient of the reflection surface. The radiant flux received on the illumination surface is Φ2 = ∫ EdS, E is the average illuminance on the illumination surface, and according to the energy conservation law, Φ1 = Φ2. Since the reflectivity of the reflection surface can be as high as 95% or more, for the sake of simplicity, the reflection energy loss is not considered here, and the luminous angle of the Lambertian light source is 120 degrees, then , Φ1 = Φ2, so E = πI0 /4ab.

There are two beams of light shining on point t, one is the T light shown in the figure, and the other is the reflected light of the I light shown in the figure. Then E = E1 +E2, where E1 is the illuminance generated by the T light beam at point t, and E2 is the illuminance generated by the B light beam.

As shown in Figure 3, let the luminous intensity of the point light source be I, and the solid angle of the illuminated area element dS′ to it be dΩ, then the luminous flux irradiated on dS′ is dΦ′ = IdΩ = IdS′cosθ′, so the illuminance is E=. From this, the illuminance of the light beam T directly irradiating the illuminated surface at point t can be obtained as:

Figure 3 Illuminance of light source

The illumination of the light beam R reflected on the illuminated surface can be calculated as:

2.3 Partial differential equations

In this reflection process, the intensity I of the incident light I and the illumination E2 of the reflected light R at point t satisfy the partial differential equation:

in:

x and y are given by the results in 2.1. The intensity of the ray I incident on the reflecting surface is:

E2 is given by the result in 2.2. Substitute the above results into the partial differential equation and list the specific form of the partial differential equation.

The boundary conditions of the partial differential equation are determined by the luminous characteristics of the Lambertian light source and the rectangular uniform illumination area. First, the lower edge of the reflective surface is set to be a rectangle with a length-width ratio of a: b, so that the light directly irradiated on the illuminated surface is a rectangular spot. The luminous angle of the Lambertian LED light source is 120°, so the upper edge of the reflective surface is on a conical surface with a vertex angle of 120°. The above formula is discretized using the numerical solution method to calculate the partial differential equation. The coordinates of the initial point of the lower edge of the reflective surface are set as the initial point conditions, combined with the boundary conditions, and brought into the computer program to iteratively calculate the equation to obtain the coordinates of a series of points on the reflective surface.

3 Simulation and Emulation

The series of points of the solved reflection surface are input into the modeling software ProE, and fitted into a solid reflection surface, as shown in Figure 4. Then the reflection surface is imported into the optical simulation software Tracepro, and the final effect is simulated and verified to obtain the illumination distribution diagram on the illumination surface, as shown in Figure 5. In the middle illumination area, the uniformity of the illumination can reach more than 85%, which can achieve a relatively ideal lighting effect. Because in the modeling process, points are first input to form a spline curve, and then the final reflection surface is formed by the spline curve. In this process, a smoothing algorithm is used, which introduces errors and makes the final result a little distorted, but the degree of distortion is not large.

Figure 4 Reflection surface structure diagram

Figure 5 Illuminance distribution

4 Conclusion

This paper obtains the surface shape of free reflection surface by numerically solving differential equations. Different boundary conditions and specific equations are obtained according to different light sources and lighting requirements, and the reflection surface can be freely designed.

The reflective surface type light distribution structure can avoid the shortcomings of the lens such as high processing cost and high energy absorption rate.