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]

3.1 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]