(Jae-Wook Ahn)
1iD
(Byeong-Seok Yu)
†iD
(Uh-Chan Ryu)
††iD
-
(Ph.D. Student, Department of Electrical Engineering, Pukyong National University,
Korea)
Copyright © The Korean Institute of Illuminating and Electrical Engineers(KIIEE)
Key words
3-dimensional space vector, Inverse reflector design, Lambertian distribution, Simulation
1. Introduction
Fabricating an optical system is an inherently time-consuming and costly process,
posing significant limitations on conducting experiments under diverse conditions.
Consequently, computer-supported simulations have become increasingly critical in
optical design[1]. Simulations enable performance verification and design optimization without the
need for physical prototyping. However, optimization-focused simulations rely on iterative
trial-and-error across extensive experimental ranges, resulting in substantial computational
demands[2].
To address this challenge, various methods have been developed to improve simulation
efficiency, including statistical approaches such as Design of Experiments (DOE)[3]. More recently, machine learning-based techniques have also been explored[4]. While these iterative simulations are frequently automated using simulator-provided
application programming interfaces (APIs), the computational burden can become prohibitive
especially when employing machine learning algorithms or handling complex systems
with multiple components. In such cases, standard laboratory PCs are often inadequate,
necessitating high-performance computational systems.
Providing appropriate initial values for optimization can significantly reduce computational
costs and streamline the simulation process, particularly in resource-constrained
environments. In this study, the basic shape of the optical system was derived using
a straightforward approach based on three- dimensional spatial vectors. The initial
design was then used as the starting point for optimization within the simulation.
The efficiency of this method was validated by comparing the total number of simulations
required when using the proposed approach against conventional trial-and-error and
DOE methods.
A Study on Combining Simulation and 3D Space Vector for Efficient UV-C LED Reflector
Design
2. Theoretical Background
2.1 Lambertian Distribution
In this study, a reflector was designed to redirect light emitted from an LED with
a Lambertian distribution onto a rectangular target area(35㎝×35㎝) was examined, as
shown in Fig. 1.
Fig. 1. Basic configuration of a reflector and LED source
A UV-C LED with a Lambertian distribution follows Lambert’s the cosine law[5], as given in Equation (1).
where $I$ is light intensity, $D$ is distance between
light source and the object, and $\phi$ is radiant flux.
As shown in Equation (1), the intensity of emitted light varies with the angle, and this variation is graphically
illustrated in Fig. 2, where approximately 95% of the total light is confined within a range of ±60° (i.e.,
120°). Therefore, under the assumption of a Lambertian distribution, it can be reasonably
assumed that the LED emits light only within this ±60° range without significantly
impacting the overall simulation results[6-8].
Fig. 2. Light distribution as a function of angle $\theta$
2.2 3D Vector Calculation
When considering a ray from the LED as a single line, the light reflection process
between the source, reflector, and target can be represented using two straight vectors[9, 10], as illustrated in Fig. 3.
Fig. 3. Reflection process between source-reflector-target
Assume $P_{1}$ is a position of the light source, $P_{2}$ is the reflection point,
and $P_{3}$ is a point in the target area. The reflection process can be expressed
using two vectors going from $P_{1}$ → $P_{2}$ and $P_{3}$ → $P_{2}$ , respectively.
Since the distance between a reflector and a target area (≥200mm) is significantly
larger than the light source size (4mm), the light source was approximated as a point.
If the former is vector $\vec{V_{1}}$ and the latter is vector $\vec{V_{2}}$ , using
the vector expression method using parameters[11], $\vec{V_{1}}$ and $\vec{V_{2}}$ can be expressed as Equation (2) and (3).
$\vec{r_{1}}$ and $\vec{r_{2}}$ denote the vectors extending from the origin to point
$P_{1}$ and $P_{2}$, respectively. Similarly, $\vec{v_{1}}$ and $\vec{v_{2}}$ represent
the unit vectors corresponding to $\vec{V_{1}}$ and $\vec{V_{2}}$. The parameters
$s$ and $t$ can be determined using the cross product of the vectors, as shown in
Equation (4) and (5)[11].
Additionally, the vector that bisects the angle between two intersecting vectors can
be obtained by summing their respective unit vectors, as presented in Equation (6)[11].
A unit vector has a magnitude of 1, and the sum of two unit vectors aligns with the
diagonal of the rhombus formed by these vectors. Since the diagonal of a rhombus inherently
bisects its angles, the sum of these unit vectors also acts as the angle bisector[12]. Consequently, the angle bisector can be computed as shown in Equation (6).
2.3 Material for the UV-C Band
The design method presented in Section 2.2 is mathematically derived from the general
characteristics of visible LEDs. Since the reflector designed in this study is intended
for use with LEDs operating in the UV-C band, the material choice for the reflector
is critical.
The UV-C band, characterized by wavelengths ranging from 200 to 280nm[13], is known for the low reflectance with most materials[14]. As illustrated in Fig. 4, aluminum is the only material that exhibits high reflectance within the UV-C band.
Therefore, aluminum is recommended as the optimal material for reflectors operating
in this wavelength range. However, prior research on aluminum surfaces has shown that
they lack smoothness, leading to diffuse reflection. So at points where the surface
is uneven, the angle of incidence and the angle of reflection are not equal[15]. Thus, Snell’s Law of Reflection cannot be directly applied in such cases.
To design an accurate reflector, it is essential to account for light scattering on
the aluminum surface. Mathematical models used to describe light scattering on rough
surfaces include Kirchhoff Diffraction Theory and the Rayleigh Approach[16]. However, these methods are often difficult to intuitively understand and have significant
limitations when applied to practical design scenarios. Alternatively, the Bidirectional
Reflectance Distribution Function (BRDF) can be used to represent surface scattering.
The BRDF characterizes the distribution of reflected light relative to incident light
and is mathematically defined as a function of parameters such as the incident angle
vector, the reflection point, and the reflection angle vector[17, 18]. In recent industrial applications, the BRDF is commonly measured directly, using
specialized equipment.
The scattering characteristics of the reflector surface were modeled based on the
BRDF properties of aluminum, as determined from previous research [19]. The detailed scattering characteristics are shown in Fig. 5.
Fig. 4. Comparison of reflectance by wavelength among each material[14]
Fig. 5. Measured BRDF of Al-on-steel[19]
3. Experiments and Results
3.1 Designing a Reflector
In Fig. 3, $P_{1}$ and $P_{3}$ are predetermined values, while $P_{2}$ must be determined through
calculation. The coordinates of $P_{2}$ can be obtained using Equation (2) to (6), but multiple possible solutions for $P_{2}$ may exist. Hence, to calculate the value,
several reference points were set, as shown in Fig. 6. Therefore, among the multiple possible values of $P_{2}$, it is necessary to determine
the appropriate coordinates that enable the continuous construction of the reflector’s
curve. Assuming a radiation angle of 120° for the LED, the light originating from
$P_{1}$ reaches x-Start and x-End respectively, as illustrated in Fig. 6.
Points labeled $P_{2}'$ and $P_{2}''$ in Fig. 6. represent some of the possible solutions for $P_{2}$ near x-Start. Similar to Fig. 3, the bisectors $n_{1}$ and $n_{2}$ as well as their corresponding tangential lines
can be drawn at these points. Since the tangential line at $P_{2}''$ intersects x-Start,
$P_{2}''$ can be selected as the value for $P_{2}$.
By repeating this process to x-End, an entire line of points for constructing the
reflector can be obtained. Subsequently, by iterating this procedure along the y-axis,
the full surface coordinates of the reflector can be determined. These sequential
steps were automated using Python, and the resulting reflector shape was visualized
using the 3D modeling software Blender, as shown in Fig. 7.
Fig. 6. A schematic representation of the method for determining the value of $P_{2}$
Fig. 7. The shape of the reflector implemented through 3D modeling
3.2 Simulation Results
A simulation was conducted using LightTools software to evaluate the performance of
the reflector designed in Section 3.1. The 3D-designed geometry was implemented in
the simulation using the Freeform Entity function in LightTools. The simulation results
are shown in Fig. 8.
The reflector material was specified as aluminum, and the BRDF characteristics shown
in Fig. 5. were applied to account for surface scattering. Under these conditions, a simulation
was conducted, and the resulting irradiance [W/m²] distribution in the target area
is shown in Fig. 9(a).
The irradiance chart in Fig. 9(a) demonstrates that, while some light is concentrated at the center, the overall irradiance
distribution is highly uniform. The target area was divided into a 10 × 10 grid, and
the number of incident rays in each section was measured in the simulations. The uniformity
$U$, as calculated using Equation (7), was found to be approximately 70%. Fig. 9(b) presents the measured results from a previous study[20], showing a uniformity of approximately 43%.
This study confirms that the designed reflector effectively fulfills its intended
purpose. Without prior design considerations, directly fabricating the reflector’s
shape would require numerous iterative experiments. For instance, the "Trough Parabola,"
a basic reflector included in LightTools, requires the specification of 14 design
parameters. Table 1 summarizes these parameters, categorizing them by data type, range, and level.
For categorical data, the number of levels was set equal to the number of selectable
items available in the simulation, while for numerical data, the levels were fixed
at three. Under these conditions, performing a full factorial design of DOE would
require a total of 8,748 runs[21]. Assuming each simulation takes 10seconds, the total simulation time would be approximately
24hours, with additional time potentially required, depending on the specifications
of the computing resources. In contrast, the proposed method, which includes automated
reflector design followed by minor adjustments in the simulation, required approximately
30minutes. This demonstrates that reflector design can be achieved more efficiently
compared to traditional trial-and-error methods or statistical DOE approaches.
Fig. 8. The shape of the reflector implemented through simulation
Fig. 9. (a) Measurement results from this study (b) Measurement results from previous
study [20]
Table 1. Summary of experimental factors for the trough parabola
Factor
|
Data type
|
Range
|
Level
|
Half width
|
Numerical
data
|
0-50mm
|
3
|
Half height
|
0-50mm
|
x Eccentricity
|
0-1
|
y Eccentricity
|
0-1
|
Rotate
|
0-90˚
|
Radius
|
0-50mm
|
Offset
|
0-10mm
|
Shape
|
Categorical
data
|
Circle, Oval,
Square,
Rectangle
|
4
|
4. Conclusion
In this study, a reflector was designed using an inverse reflector design approach
based on three-dimensional vector calculations to achieve uniform illumination of
a target area from a UV-C LED with a Lambertian distribution. The performance of the
designed reflector was evaluated through simulations, and the resulting irradiance
measurements confirmed an overall uniform distribution, with minor deviations at the
center.
The proposed method enables the derivation of a reflector’s basic shape through relatively
simple calculations, and the design process can be efficiently automated using tools
such as Python. Compared to traditional trial-and-error methods or full-factorial
DOE designs without initial values, this approach offers significant time savings
and improved efficiency.
Future research will focus on experimentally validating a reflecter fabricatedusing
aluminum. Additionally, precise BRDF measurements tailored to the material properties
will be conducted and incorporated into both the design and fabrication processes
to enhance accuracy and effectiveness.
Acknowledgement
This work was supported by a Research Grant of Pukyong National University(2024).
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Biography
He received a B.S. degree in electronic engineering from Korea Aviation University,
Korea, in 2004, and his M.S. degree in electrical engineering from Pukyong National
University, Busan, Korea, in 2021. He is now a Ph.D. student in the Department of
Electrical Engineering at Pukyong National University, Busan, Korea, and a government
worker of ROKAF (Republic of Korea, Air Force).
He received B.S., M.S., and Ph.D. degrees in Naval Architecture and Ocean Engineering
from Seoul National University, Seoul, Korea, in 1990, 1993, and 1998, respectively.
Until 2019, he worked for Hanjin Heavy Industries and Construction. Since Sep. 2019,
he has been a professor in the Major of Smart Mechanics and Mobility Engineering at
Pukyong National University, Busan, Korea.
He received B.S. and M.S. degrees in electrical engineering from Seoul National
University, Seoul, Korea, in 1997 and 1999, respectively, and his Ph.D. degree in
the School of Information and Communications from Gwangju Institute of Science and
Technology, Gwangju, Korea. Since Sep. 2013, he has been a professor in the Major
of Electrical, Electronic, and Software Engineering at Pukyong National University,
Busan, Korea.