No single "best" design exists, as the optimal approach will depend on the specific needs of the vertical farm, including the crops being grown, the available space, and the budget. However, by focusing on the key principles outlined above, you can design an LED grow light system that delivers efficient and effective plant growth in a vertical farming environment.

## Introduction

With population growth and increasing urbanization, the world is facing increasing food scarcity. This calls for finding innovative solutions to increase crop yields and improve their nutritional value. In this regard, the development of agricultural lighting technologies has significant potential. Agricultural lighting technology utilizes artificial light sources (e.g., LED lights) to provide the light needed for plant growth.

Compared to natural light in conventional agriculture, agricultural lighting technology can provide more precise and controllable light intensity and light cycles. This provides ideal conditions for plant growth and photosynthesis. By optimizing light conditions, photosynthetic efficiency and nutrient uptake can be improved, resulting in faster plant growth and increased yields. Therefore, the development of agricultural lighting technology is of great significance in increasing the yield and nutritional value of crops. Agricultural supplemental lighting is a lighting device used during plant growth and is designed to provide sufficient light to promote plant growth and development.

In this environment, uniformity of illumination is very important. Since different plants require different light intensities and durations, and if the light is not evenly distributed, it may result in some plants being adversely affected in terms of growth and development. For example, if plants require the same light intensity, but some of them receive more light than others, they will grow faster than others, resulting in uneven growth and harvest.

In addition, if the light is not evenly distributed, it may result in some plant parts being over-exposed to problems such as heat damage and photodamage, which can affect plant health and yield. Therefore, uniformity of illumination must be taken into account in agricultural breeding supplemental lighting to ensure that each plant receives the full amount of light it needs to ensure uniform growth and ultimately high quality yields.

In order to realize the illumination uniformity of the planting surface, the current common method is to lay multiple lamps horizontally on the culture frame. Although the use of multiple lamps can make the planting surface's PPFD increase dramatically, this will also bring some problems, such as the number of lamps and lanterns, the project's initial one-time investment cost is high; the edge of the light leakage is serious, resulting in an increase in the cost of electricity; the number of lamps and lanterns, the cost of operation and maintenance increases, and so on. Therefore, if the optical system can be designed so that a single fill light can form a uniform PPFD on the culture frame, and the photosynthetic photons are only distributed on the planting surface and do not overflow, then the above three problems can be solved under the premise of retaining all the advantages of plant growth lights. In this paper, we will discuss the design methodology of the optical system, and make samples of the lamps for testing according to the design results.

## Free-form lens design based on digital light field

In order to spatially redistribute the light emitted from the filler lamps so that it is efficiently and uniformly distributed on the growing surface, this paper uses a low-cost single distribution lens approach. If it is desired to be able to achieve uniform distribution of light energy through a single lens (2 refractive surfaces), the lens cannot use a conventional spherical mirror, but must be designed with a free-form surface that has greater control over light.

There are several commonly used methods for the design of free-form lenses. The iterative method utilizes numerical simulation methods such as the finite element method or ray tracing method for the design, where the lens surface is continuously adjusted to meet preset requirements until the lens is able to focus or scatter the incident light in a predetermined manner.The Zernike polynomial or higher order aspheric method utilizes the Zernike polynomials or higher order aspheric expressions to describe the shape of the lens surface, and by adjusting the polynomial coefficients, the adjustment of the shape of the lens surface can be realized to control the incident light. The optimization design method based on genetic algorithms transforms the design problem into a parameter optimization problem by considering the lens surface shape as a parameter space, and searches through optimization algorithms, such as genetic algorithms, to finally obtain the optimal design result. The above method, due to the use of optimization methods in the design process, in order to speed up the calculation speed, usually can only simplify the light source into a point light source first, and then carry out the surface design of the lens, and then bring the designed lens into the simulation software, and set the light source in the simulation software as an extended surface light source and then carry out the simulation, and then compare the difference between the simulation results and the design goal, and then carry out corrections.

When the size of the optical system is 5 to 10 times larger than the size of the light source, at this time the light source will be simplified as a point light source for design, the difference between the results obtained and the design goal will not be very large, but if the size of the optical system compared to the size of the light source is less than 3 times, the point light source design results will be very different from the results of the actual use of the extended light source. In the case of the light source has been determined, the larger the lens size, the use of point light source to simplify the design results will be more accurate, but the lens size is too large will bring some effects: longer process time and the use of injection molding materials will significantly increase the cost of the lens; at the same time, too large a lens will lead to an increase in the size of the final product.

In this paper, a 3535 dome packaged LED light source is used as the plant growth light source. In order to ensure that the lens has a small volume and high precision, this paper will use the digital light field based free-form surface design method developed by Hangzhou Hoshino Optics for the design of the lens, which is an all-digital design method for the extended light source, and is especially suitable for the design of small volume and high precision lens design.

### Digital Lighting Light Field Concepts and Connotations

Photometry is the foundation of lighting optics, in which the basic quantities of lighting optics are defined. Although light intensity is the basic quantity value of the international unit, from the practical logic, the practical application is to take the luminous flux as the de facto basic quantity, and its commonly used quantitative values such as illuminance, light intensity and luminance are used to describe the luminous flux distribution density of the given luminous surface in the target surface or the target angular space, and these quantitative values are focused on the description and evaluation of the distribution of the light field and the quality of illumination, and it is more difficult to be directly applied in the free-form surface of the optical The design of free-form surface optics is more difficult to apply directly.

The digital illumination light field theory is a theory that abstracts the basic unit of light field distribution in an optical system, and establishes the non-imaging digital light field function NDLFF to describe the optical system in a digitized and discretized way. The basic concept of digital illumination light field covers the following aspects: first, the overall illumination light field space needs to be discretized, especially for the surface or space we are interested in. Second, each microelementary subfacet corresponds to a light field distribution, and the overall light field is described using a light field function, whose core features include the light cone passing through that subfacet and the face normal of the subfacet. Then, with the use of a digital light field description, the goal of illumination optical design shifts to the tuning of the light field function on a specific surface, which can be achieved by combining single or multiple free-form surfaces. Hoshino Optics has developed a free-form surface algorithm based on the digital lighting light field, which can accurately control the light field function on the target surface by combining multiple free-form surfaces, and thus accurately control the illuminance and light intensity distribution on the target surface, providing a more flexible, accurate, and customized lighting solution for indoor lighting design, automotive lighting design, stage lighting design, and other fields.

### Calculation of Light Sources and Layout and Optimal Illumination Distribution of Single Lens

Vertical farms are an innovative model of agricultural production that improves crop productivity by vertically stacking multiple layers of growing areas in a limited space. Such farms are usually built inside buildings in or near cities and utilize indoor lighting and automated systems to provide the light, water, and nutrients needed for cultivation. In order to achieve both high efficiency and uniformity, a plant grow light lens scheme is proposed in this paper. In this scheme, the light source is a 3535 encapsulated high power LED with particles spaced 48mm apart and arranged in a single row in the direction of the long axis of the lamp body. Each of these LEDs is equipped with a lens above it. The lamp has a total of 25 LEDs with a total length of 1.2 m. The target surface is the culture shelf at a distance of 30 cm below the lamp body, and the width is set to be slightly larger than 60 cm. The design goal is to form a uniform PPFD distribution in the target surface. For the combination of a single LED plus a lens, if it simply forms a uniform PPFD spot on the target surface, the spot formed by 25 LEDs together will show a PPFD distribution with a bright center and dark surroundings. Therefore, before proceeding to the free-form surface design of the lens, it is necessary to first obtain the optimal illuminance distribution of a single lens, under which 25 LEDs will jointly form a uniform spot.

If the analytical method is cumbersome to get the optimal illumination distribution, the optimization method can be simply used to get the result quickly. The specific process is as follows: First, we assume that the optical system is rotationally symmetric structure, which means that the light intensity distribution of the spot is also rotationally symmetric. Then, we set the normalized PPFD distribution function of the light spot formed by a single LED as P (r), r ∈ [0, 300], P ∈ [0, 1], which describes the distribution law of the LED light spot in space. Then, we discretize the whole illuminance region r at equal intervals into a set of sequences ｛r1, r2, ..., rn｝, and each discretized point r corresponds to a value of PPFD P (r). In this way, we can transform the illumination distribution problem into a problem solved at discretized points. The PPFD distribution formed by a single LED is P(r), and next, we follow the different positions of the 25 LEDs in Matlab to calculate the total PPFD distribution formed by their superposition. By considering the positions and spot distributions of multiple LEDs, we can get a more accurate illumination distribution. Then, we set P(r) as the optimization variable with the minimum statistical mean square deviation of the total PPFD distribution as the optimization objective. By using the optimization function in Matlab for optimization, we can get the best P (r) and thus the best illuminance distribution.

In summary, through the above steps, we can use the optimization method to quickly obtain the optimal illuminance distribution without using complex analytical methods. This method can effectively simplify the problem and get the illuminance distribution results that meet the design requirements. Finally get the optimized PPFD distribution of a single LED, a single LED needs to form a PPFD distribution with a dark center and bright surroundings in order to form a uniform PPFD distribution on the planting surface under the lamp.

### Single Lens Design

In order to allow a single lens to achieve the ideal spot distribution optimized in section 1.2, this paper adopts Hoshino Optical's "secondary light source surface method" for the design, and the designed model is imported into the optical simulation software, the lens cross-section and the main light line. It can be seen that the main ray angle gradient is smooth, and the light density is densely distributed at large angles to form the external highlight circle corresponding to the ideal PPFD distribution. The ray tracing simulation based on Monte Carlo method is carried out for the above lens to obtain the PPFD on the target surface, and the result is highly consistent with the ideal distribution calculated in section 1.2.

### Analysis of the whole lamp design results

In LightTools software, the combination of LEDs and lenses in the previous section is arrayed into 25 combinations with equal intervals of 48 mm, and the irradiance distribution of the whole lamp on the target surface is simulated. The width of the spot is slightly larger than the width of the 60 cm planting frame, so that the spot can cover the entire planting frame well, which is consistent with the design. Moreover, the spot has a good edge cutoff, and the theoretical energy utilization (PPF on the grow frame/PPF of the light source) is more than 92%, which means that 92% of the photosynthetic photons emitted by the LEDs will fall into the plant canopy, which greatly improves the energy utilization compared to the traditional plant growth lamps, thus reducing the electricity bill and lowering the cost of planting. There are PPFD gradient areas at both ends of the spot corresponding to the head and tail of the fixture. In practical application, usually more than one planting frame longitudinal head-to-tail arrangement placement, at this time more than one fill light will also form a head-to-tail connection of the lamp mode, more than one spot head-to-tail superposition, the longitudinal uniformity will reach a very high value, two lamps head-to-tail connection of the planting surface of the PPFD distribution, the two lamps overlap the part of the light spot of the PPFD mutual enhancement, the formation of the overall uniformity of distribution. The distribution of PPFD on the planting surface when two lamps are connected at the head and the tail.

### Test analysis

The above design to get the lens to open the mold trial production, and at the same time designed the aluminum extrusion profile of the heat dissipation bracket and lamps and lanterns at both ends of the plug, and finally get the lamps and lanterns and their light spot. The measured results show that the plant grow light has excellent performance in terms of efficiency, which can reach a high efficiency level of more than 92%, and the photosynthetic photons on the planting surface account for more than 86% of the photosynthesized photons emitted by the light source, indicating that the light source can provide a large amount of light that can be used for plant photosynthesis. At the same time, the light spot uniformity in the planting surface is also very good, the ratio between the minimum light intensity and the average light intensity is more than 82%, which ensures that each plant can receive light uniformly. The plant grow light has been designed to perform well on a number of metrics, and can be used to meet the needs and perform well in vertical farms.

## Results and Conclusions

Lamp efficiency and PPFD uniformity are important metrics for evaluating plant grow lights. In this paper, a new type of plant grow light lens for culture racks was obtained by a free-form lens design method based on digital light field theory developed by Hoshino Optics. The design of this plant grow light lens is aimed at high efficiency and energy saving, and the free-form surface lens design based on digital light field reduces energy loss and waste, improves the light efficiency of the lamp, and converts more light energy into usable energy required by the plants to provide a better growing environment. The design of the lens also focuses on the uniformity of the light spot. By accurately controlling the propagation path and scattering angle of the light, the light can be evenly distributed on the culture frame, avoiding the uneven phenomenon of overpowering or overweakening of the light intensity. This ensures that the plants in different locations in the cultivation and planting area can receive sufficient light to ensure the consistency of the growth rate of the crops in the cultivation area, which can reduce the increase in operation and maintenance costs brought about by the inconsistent growth rate of the crops with different quality and the need for sorting, grading, and other treatments. In addition, the lamp in the planting frame only the center of a lamp can meet the width of 60 cm of illumination needs, easy to clean and maintain, reducing the installation and maintenance costs.

Overall, the free-form lens design method based on digital light field theory makes the plant grow light energy efficient, highly uniform, and easy to operate and maintain. It can provide a high-quality light environment and is an ideal supplemental lighting device for vertical farms.