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Abstract
A method for modeling the irradiance spatial distribution by light-emitting diodes (LEDs) on near distance targets has been developed. The model can easily handle the complex simulation of non-homogenous emitting LEDs, multichip LEDs, LED arrays, and phosphor coated LEDs. The LED irradiation profile is obtained by image processing one photograph of the emitting LED, taken with a smartphone. The method uses image convolution or image correlation between the LED image and a special kernel. The model provides the irradiation spatial pattern in function of the irradiation distance. And the model is tested both with theory and with experimental measurements.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Novel and emerging LED applications require short-distance irradiation when the space between the LED source and the target is small [1–9]. A recent and important application is the rapid inactivation of coronavirus SARS-CoV-2 with LED irradiation [1]. Other applications include the design and optimization of dental light curing devices, photodynamic inactivation of bacteria, photodynamic therapy, micro-devices for neural stimulation, and irradiation with metalenses [2–9]. In these applications the short distance is required to produce intense irradiance profiles or because the small size of the systems. Then modeling the irradiance spatial profile produced by a LED source at near distances may be useful for analyzing, designing or optimizing the light irradiation in these applications.
Several mathematical models for describing the spatial radiation profile of LEDs have been reported [10–14], but are not valid for short distances. Some give the radiant intensity as function of the emitting angle [10–12], and some give the irradiance (or illuminance) as function of Cartesian coordinates [13,14]. There are other models valid for short distances, which are based on Monte Carlo ray-tracing numerical simulations, and are fundamental for secondary optics design [15–16]. But until recently, fast and easy modeling tools for short distances were lacking.
Recently, we derived an exact formula of the irradiation profile produced by Lambertian LEDs at short distances [17]. Also, we proposed a mathematical model of the irradiance pattern, valid for Lambertian and directional LEDs at short distances. However, these results apply to single-chip LEDs with rectangular shape [17], but are difficult to apply to: multichip LEDs, phosphor coated LEDs, and LED arrays. In such cases, the irradiance spatial distribution is quite more complex to calculate or to analytically model. One possible solution could be a sum of single-chip LED models of irradiance Ei [17], i.e. ΣwiEi, with wi being weight factors for each LED chip. But this approach could become not practical in real LEDs, where: chips may be rotated or arranged in not periodic positions, chips may have emission in-homogeneities, and other imperfections of LEDs. Here we propose an alternative approach that overcomes these problems using image convolution or cross correlation of one photo.
2. Model formulation
The irradiance produced by a light source is the power per unit area that is incident on a surface illuminated by the source. Its usual symbol is E [W/m2], and mathematically E = dΦ/dA. The irradiation pattern is given by the spatial distribution of irradiance, i.e. E(x, y, z), where (x, y, z) are the coordinates of any point in the space illuminated by the light source. Ideally, the irradiance pattern produced by an LED source may be obtained by solving the integral equation of the theory of radiation transfer [18], i.e. by integrating the LED radiance Ls [W/m2sr] over the LED emitting area As, by:
The irradiance pattern over a flat irradiated surface in front of the LED, at the target plane (Figure 1(a)), can be calculated by making cosθs=cosθ. And then, the irradiance pattern on each point (x, y) over a surface at distance z =h, which is irradiated by an LED with a radiance Ls is:
Fig. 1. (a) Shows the geometry between an LED and a target (flat surface under irradiation) at an irradiation distance h. Here the irradiance spatial distribution is function of Cartesian coordinates (x, y). (b) Summary of LED irradiance modeling based on image convolution (or correlation) of an image captured by a smartphone camera. Here IMAGE-0 represents the recorded image, IMAGE is the linearized and normalized image, and K is the convolution (or correlation) kernel that is function of (x, y, h).
The radiance Ls is in general a function of both emitting points (xs, ys), and emitting angles (θs, φs). However, most LEDs have a constant radiance in the azimuthal angle φs, and it is only function of θs. Therefore, the radiance can be approached by Ls =Ms(xs, ys)Ds(θs), where Ms is the LED emittance [W/m2], and Ds is a function of directionality of the light emission. And since θs is a function of (xs - x, ys -y), the radiance can be written as Ls =Ms(xs, ys)Ds(xs -x, ys -y). Therefore, Eq. (2) can be written as
We can observe that Eq. (3) is the integral of the product Ms(xs, ys) K(xs - x, ys -y), where K(xs - x, ys - y)=Ds(xs - x, ys - y)[(xs - x)2+(ys - y)2+h2]-2. By observing that the mathematical operation of convolution is defined as f * g = ∫f(xs)g(xs-x)dxs, the irradiance spatial distribution may be obtained from a two-dimensional convolution E(x, y, h) =h2Ms =K. The interesting thing is that Ms can be approached by a photograph of the emitting LED, i.e. Ms(xs, ys)=M0 IMAGE(xs, ys). Here IMAGE(xs, ys) is a normalized and linearized image of the photo, and M0=Φs/As, where Φs [W] is the LED light flux, and As [m2] is the LED emitting area. In non-homogenous emitting multichip LEDs, or in phosphor coated LEDs M0=Φs[∫IMAGE(xs, ys)dAs]-1.
Therefore, the LED irradiation pattern can be determined by the 2D convolution of IMAGE and the convolution kernel K:
If the image is recorded with a smartphone, it must be first corrected for gamma to make the light distribution linear [19]. Also the image must be normalized to its maximum gray level. This means that IMAGE = [IMAGE-0 / max(IMAGE-0)]γ, where IMAGE-0 is the image recorded with a smartphone, and γ is the gamma value. In smartphones, γ=2.2 is a standard value. The calculating process is graphically depicted in Figure 1(b). Image convolution is a 2D convolution, where the kernel or mask is a 2D array or matrix. Its xy-dimensions determine the number of neighbors to be included in the weighted sum calculation. We used a square N×N kernel mask for simplicity in calculations.
For practical purposes, the kernel K may be evaluated with a directional radiance that depends only on the view angle θs, given by Ls=L0cosm−1θs, where L0=Ms(m+1)/2π. In this case, the kernel K is given by (see Appendix A):
The light emission directionality is given by m = −ln2/ln(cosΘ0.5), where Θ0.5 is the half-intensity angle [10]. If Θ0.5=60° (m=1), the emitting points are Lambertian emitters. Because this kernel has azimuthal symmetry (does not depend on φs), Eq. (4) may be performed by a convolution or by a cross-correlation, i.e. E(x, y, h) =M0h2[IMAGE✶K]. Although irradiance is typically simulated by the convolution of radiance in the fields of computer vision and computer graphics [20], as far as we know it has not been applied in the radiometric modeling of LEDs.
Convolution and cross correlation functions are available in most mathematical software, making these image processing tools available to most of users. The cross-correlation is similar in nature to the convolution. The basic difference between convolution and correlation is that the convolution process rotates the kernel by 180 degrees. If the kernel has rotational symmetry, then convolution and correlation achieve the same result, because folding the kernel would give the same function, and then correlating the image will give the same result. In what follows we use the symmetrical kernel given by Eq. (5), and then we use cross-correlation in all calculations.
3. Modeling LED irradiation patterns
First, let us compare the irradiance pattern obtained by the model and the exact radiometric equation for a squared-shape light source with directional emitting radiance (Ls=L0cosm−1θs). Figure 2 plots Equations (2) and (4), for a source with emittance M0=1 [Arbitrary Units], and with emitting area As=S×S. The figure plots the irradiance pattern for sources with different half-intensity angles Θ0.5 at irradiation distance h=0.5S. Figure 2(a) shows the 2D irradiation pattern for half-intensity angles Θ0.5=20° and 40°. Figure 2(b) shows the irradiance in x-axis for several Θ0.5 half angles. Here, kernel size N was determined as the distance at which the kernel reduces at 99.9% of its maximum. And the image size of the simulated source was 200 × 200 pixels. In this example the model and exact equation give the same result. But even more, the computation is significantly faster with the model than with the exact solution (∼10 times faster), and achieves high accuracy.
Fig. 2. Irradiance spatial distribution of a square-shaped light source with directional radiance. Graphs display the exact equation Eq. (2) and model Eq. (4). (a) Shows two-dimensional irradiance pattern E(x,y) at distance h=0.5S, for two beam directionalities Θ0.5. (b) Shows E(x) in y=0 for several directionalities.
Next, we model real LEDs. Figure 3 displays the LED photos, captured with a smartphone camera, which are labeled as IMAGE-0. The smartphone used was a Xiaomi Redmi Note 8 Pro, with its camera in manual mode (also called Pro mode). Manual mode was adjusted (ISO value and shutter speed S) to correctly expose the captured image for the LED brightness. The position of the camera was parallel to the LED surface to avoid the effect of image distortion.
Fig. 3. Modeled LED irradiance pattern E(x, y) on a target for several short distances. A photo of the modeled LED is captured by a smartphone, which is IMAGE-0. (a) Shows the irradiance of a blue multichip LED with 4 chips, at four irradiation distances h=0.1, 0.5, 1.0, and 1.5 mm. (b) Shows the irradiance of a white multichip LED with 56 chips at h=0.1, 1.0, 5.0, and 10 mm.
Figure 3 shows the modeled irradiance spatial distribution of two LEDs at four short distances. Fig. 3(a) shows the irradiance of a blue multichip LED with 4 chips, and Fig. 3(b) shows the irradiance of a phosphor-coated multichip LED with 56 blue chips. Here, kernel size (N) was determined as the distance at which the kernel reduces at 99% of its maximum. The size of the emitting LED images in Fig. 3 were: 520 × 420 pixels (blue LED), and 900 × 900 pixels (white LED). The LED modeled in Fig. 3(a) is a blue light multichip LED (LE-B-P2W OSRAM OSTAR), with an emitting area of 3.2 × 2.6 mm2. And Fig. 3(b) models a white light multichip LED (Cree XLamp CMT1930), which has an emitting source with a 14.5mm diameter. Because all LEDs are encapsulated in a transparent package, there is a chip-to-window distance, which for the blue LED is ∼0.6mm according to the datasheet provided by the supplier. And then, the effective h distance for calculating Eq. (4) was h+0.6/n, where n=1.45 is the refractive index of encapsulating plate. In the case of the white LED, the source-to-window distance is ∼0.55mm (according to the datasheet), and then the effective h distance was h+0.55/n, where n=1.45. The emitting angle used in kernel was Θ0.5=57° (according to the datasheets).
After several calculations, the model was found well behaved, becoming the IMAGE at h=0, and converging to a Gaussian function at large irradiation distances [10]. The image processing takes longer as image resolution increases, however it improves the sharpness of the irradiance pattern, which is important at very short irradiation distances (at less than one tenth of the emitting size of LED). In the other hand, at very short distances the size of the convolution (correlation) kernel becomes smaller, which reduces the image processing time. An example of the calculation is included in Appendix B; it is a computation code in Mathcad software.
4. Comparison of modeling with experimental data
Let’s now to validate the model's consistency and ability in reproducing experimental measurements of the LED irradiance pattern at short distances for LEDs modeled in Fig. 3.
Several measurements of the irradiance E(x, 0, h) were realized with a scanning detector. Figure 4(a) shows the geometry of the measuring set up with the emitting LED and light detector. This figure depicts x-coordinate of a measuring point, and h denotes the distance between the LED and the measuring plane. The detector is moved across x direction to measure the irradiance profile, and the origin (x=0, y=0) is at the center of LED (Fig. 4(b)). The used detector was a NIST calibrated radiometer (ILT 1400) with high dynamic range and high linearity, 7 decades and 0.2%, respectively. A stop aperture with 0.4 mm diameter (circular) was used for near distance measurements of irradiance. The measured field of view of the detector with the small stop aperture was ±15 degrees, and its angular variation is shown in Fig. 4(c). The detector was not calibrated with the tiny aperture for absolute measurements, obtaining relative irradiance measurements, and therefore we used arbitrary units.
Fig. 4. Measurement of the irradiation pattern. (a) Shows a schematic diagram. (b) Shows xy-axes across measured LEDs. (c) Shows the measured angular response of the detector with the small stop aperture.
Figures 5 and 6 show the measured and modeled irradiance patterns for several h distances. The modeled irradiance includes the narrow field of view of the detector by multiplying the kernel K by the function of Fig. 4(c). The measurements are normalized with respect to the central point at h=0.5mm, i.e. E(0, 0, 0.5) = 1 [A.U.]. It can be seen that there is a very good agreement between the proposed model and measurements, showing a small RMSE of 0.04 for Fig. 5, and of 0.03 for Fig. 6, thus demonstrating that this simple and well-behaved model reproduces the irradiance patterns of LEDs at short distances.
Fig. 5. Modeled and experimentally measured LED irradiance pattern of a blue LED with 4 chips. The irradiance E(x) in y=0, is shown for different h irradiation distances: (a) 0.5mm, (b) 1.0mm, (c) 1.5mm, and (d) 2.5mm.
Fig. 6. Modeled and experimentally measured LED irradiance pattern of a white multichip LED with 56 chips. The irradiance E(x) in y=0, is shown for different h irradiation distances: (a) 0.5mm, (b) 1.0mm, (c) 1.5mm, and (d) 2.0mm.
5. Summary
Emerging LED applications require short-distance irradiation, and one that stands out is the inactivation of coronavirus COVID-19 by LED irradiation. To accurately analyze the irradiation in such applications, it is useful to model the irradiance spatial profile produced by LEDs. Here we developed a new method for modeling the LED irradiance on near distance targets. The model used one photography of the emitting LED, taken with a smartphone. And the LED irradiance pattern was obtained by the convolution (or cross-correlation) operation between the photo of the emitting LED and a simple kernel. Because the symmetry of the angular emission pattern of LEDs, the irradiation pattern can be obtained from the convolution or from the cross correlation. The accuracy of the irradiance pattern model was tested both with radiometric theory and with experimental measurements. Because the model used an LED image, it eliminated the complexity of modeling non-homogenous emitting LEDs, multichip LEDs, and phosphor coated LEDs. The simplicity of the model makes it easy to understand, improve, and apply in new LED applications. Further work may include the modeling of LEDs encapsulated in a spherical dome, the modeling of 3D irradiance maps, etc.
Appendix A: derivation of kernel K
The generalized Lambertian radiance Ls=L0cosm−1θs, comes from considering the generalized Lambertian radiant intensity Is=I0cosmθs, which has been widely used as the simplest radiometric model of LEDs [21]. The value of L0 can be obtained by calculating the radiometric integral of the radiant power Φs emitted by a light source with radiance Ls and uniform emittance Ms,=Φs/As, which gives L0=Ms(m+1)/2π. By using Ls=MsDs, we obtain Ds=[(m+1)/2π]cosm−1θs, where cosθs= h [(xs - x)2+(ys - y)2+h2]-0.5. Finally, because the kernel is given by K = Ds [(xs - x)2+(ys - y)2+h2]-2, we obtain
Appendix B: example of the calculation in Mathcad software
Disclosures
The author declares no conflicts of interest.
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