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Fourier series and an energy mapping method were used in this study to design a lens that produces a light pattern of multiple concentric circles (LPMCC) for a light-emitting diode (LED) fishing lamp. Fourier series were used to represent the light intensity distribution curve (LIDC) of the LPMCC light pattern. Energy mapping involves performing angular energy mapping based on the LIDCs of an LED light source and LPMCC to design a freeform lens. Type I and Type II LPMCC lenses were designed according to the phototaxis behavior of fish to create a LPMCC light pattern of interleaving light–dark zones that attracts fish shoals to stay in an area for a long period. The experimental results indicated that, in comparing the LIDCs of the Type I and II lenses with the respective simulation values, the normalized cross-correlation (NCC) value reached 96%. According to a 24-hour observation of the phototaxis of Poecilia reticulata to evaluate the effectiveness of the proposed light pattern to attract fish, when a fish shoal was habituated to a light source that emitted constant illumination light, it gradually moved away from the intense light zone and hovered around the junction of the light and dark zones. In the future, the design used in this study can be applied to LED fishing lamps to replace traditional fishing lamps.
© 2014 Optical Society of America
1. Introduction
Light emitted into the sea or ocean affects the feeding habits, growth, reproduction, survival, phototaxis, and aggregation behavior of fish; thus, fish react differently toward light stimuli in terms of feeding and aggregation [1]. The phototaxis of fish comprises two phases. In the first phase, when stimulated by a light source, fish first approach the light source and then hover around the light source. In the second phase, after a period of time, the fish shoal becomes habituated to the light source and experiences fatigue [2]. Subsequently, the fish shoal gradually leaves the light source and remains in a dark area, waiting to catch plankton. Phototactic fish demonstrating such behavior include squid and Poecilia reticulata. By using the directivity of a light-emitting-diode (LED) and an optical lens to design an underwater light pattern that consists of multiple interleaved light and dark zones, fish shoals can be attracted to remain in a specific zone for a long period and, therefore, the feasibility of replacing traditional fishing lamps with LED fishing lamps can be enhanced. Regarding the use of a fishing lamp to attract fish, Arakawa et al. indicated that squid react to a light stimulus by clustering in a less-bright area [3]. Masuda et al. applied LEDs to stationary fishing nets, using the phototaxis of fish to increase fish catch [4]. Shikata et al. reported that underwater creatures (e.g., squid) prefer to stay in the junction of the light and dark zones in a light field [5]. Based on the phototaxis of fish, Shen used an LED array configuration combined with a light intensity function to produce a shadowed area underwater for attracting fish shoals [6].
In recent years, LED light sources have been applied in undersea areas. However, a substantial difference exists between LED and traditional light sources. The optical design and lens structure of traditional lighting cannot provide all the advantages of LED light. Therefore, a specific lens must be designed according to the characteristics of LED light to enhance the energy utilization efficiency of LED lighting. This study focused on the optical design and the strong directivity of LEDs to design a freeform lens that produces the desired light pattern and distribution. Current studies have adopted various methods to create freeform lens based on the predicted light pattern of a target plane; these methods involved trial and error, tailoring, the edge ray principle, and virtual reflecting/refracting surfaces. Schruben designed a freeform lens by using a 3D coordinate system to convert the relationship between the incident vector and reflection vector of a freeform surface into a differential Eq [7]. In addition, Schruben numerically solved the differential Eq. for a freeform surface to create an axisymmetric light pattern and a freeform lens [8]. Thereafter, studies on the design of optical lenses have focused on employing new mathematical theories to represent the freeform surface of a lens. Over the past decade, a ray tracing method has often been used to design optical lenses. For example, Maris applied the Maxwell Eq. to derive an Eq. that can trace uniaxial birefringent optical elements, and subsequently applied this Eq. to Wollaston prisms [9]. Shirley et al. used Monte Carlo ray tracing to simulate the change of geometrical optical paths and obtain uniform light illumination distribution [10]. Timinger et al. employed a tailoring method to segment a curved surface into several small surfaces according to the light pattern of streetlights to design a freeform lens for LED streetlights [11]. Ding et al. developed an optical freeform lens for an LED miniprojector by using a tailoring method and the laws of coordinate transformations and the conservation of energy [12]. Feng et al. described a numerical double freeform lens surface design method using energy mapping concept for separable irradiance distributions applications [13]. Domhardt et al. designed optical lenses for LED car headlights according to virtual reflecting/refracting surfaces and the light pattern required for car headlights [14,15]. Shatz used a trial-and-error method combined with total internal reflection, designing optical lenses that can produce uniform illumination light pattern [16]. After 2009, by using a single lens module, Sun and Lee adopted nonaxisymmetric freeform surfaces to create a rectangular light pattern specifically for LED streetlights [17]. Guttsait created an illumination pattern using a numerical method, in which functions of light intensity distribution curves (LIDCs) and light intensity for LED arrays were devised and the attenuation of light transmission in space was considered to estimate illumination distribution and uniformity over a long distance [18,19].
The design goals of the aforementioned studies were to create a light pattern and achieve illumination uniformity. In these studies, complex mathematical functions and repeated calculations rendered the processes of designing an LED freeform lens time consuming. In addition, no studies have demonstrated that a single lens can be used to achieve regular light illumination distribution within a light pattern. In other words, the current methods for designing lenses focus only on the illumination uniformity provided by a light pattern and cannot be used to establish equations for controlling the light illumination distribution of a light pattern. The light pattern and light illumination distribution required by a target plane were used in this study to design an LED freeform lens. Specifically, the light illumination distribution of interleaving light–dark zones was regarded as a periodic function, and Fourier series were used to represent the LIDC and modify the light illumination distribution value of a light pattern. Subsequently, angular energy mapping between the LIDCs of the LED light source and the light pattern of interleaving light–dark zones was conducted. Optical lenses with a light pattern of multiple concentric circles (LPMCC) were created, the light illumination distribution of which in a particular light pattern changed periodically. In addition, this study applied the optical lenses to LED fish lighting attractor to replace traditional energy-consuming lighting, and thereby creating an underwater phenomenon that would effectively attract fish shoals.
2. Design of the LPMCC lens
2.1 Design and analysis
The LIDC is the most crucial characteristic of a light source (i.e., the luminous intensity distribution of a light source or the light emitted through a lens in all directions in space). The corresponding LIDC for the light pattern of a target plane (i.e., the LIDC produced by the light passing through a lens or the LIDC of an LED) was required for the design of an LED freeform lens. By conducting energy mapping, in which complex mathematical functions were simplified to angular energy mapping equations, and using the Snell’s Law, the coordinate of each point on the freeform lens and the corresponding normal vector were calculated, and the results were used to construct a complete freeform lens. In the design process, designing and representing the LIDC of a target plane was highly crucial. Therefore, a regular light pattern of interleaving light–dark zones was regarded as a periodic energy function, and then Fourier series were used to represent the respective LIDC (i.e., the illumination distribution of a light pattern), Finally, the research using the Fourier series can to improve the current LED-lens design method that focuses on illumination uniformity and is incapable of controlling the light illumination distribution of a light pattern.
As shown in Fig. 1, the freeform lens was developed based on the directivity of LEDs and the required light illumination distribution of the target plane. Given 1) the LIDC function I(θLED) for an LED light source, 2) the LIDC function I(θLens) for the taget plane, and 3) the distance between the highest point of the lens and the light source L, where θ and ϕ are the elevation angle (i.e., the angle against the x axis) and the azimuth (i.e., the angle against the z axis), respectively, the freeform lens design can be formulated using the law of conservation of energy, energy mapping, and the Snell’s Law. Figure 1 shows the schematic design for using the LPMCC as the target plane. In this design, the light pattern shows axisymmetric distribution and, therefore, only the lens curve on the x-z plane was considered. Subsequently, a 3D diagram for the entire lens was obtained by rotating the curve about the z axis.
Lambertian LEDs are typically used as light sources. According to the LIDC function (1), the luminous intensity distribution varies according to various angles:
where is the luminous intensity for each angle and is the axial luminous intensity.The LED used in this study was a XM-L. The light emission angle was approximately 125°. According to Eq. (1), the LIDC of the light source can be represented as follows:
In the Fig. 2, the LPMCC design concept, including the Type I and Type II light patterns. Figure 2(a) shows the light illumination distribution for the Type I light pattern. The center of the circle is a dark zone that is encircled by a light ring, a dark ring, and finally an outermost light ring. Regarding the LIDC design, when the light emission angle is ± 20° and ± 50°, high luminous intensity is outputted. Figure 2(b) shows the light illumination distribution for the Type II light pattern. The center of the circle is a light zone that is encircled by dark, light, dark, light, and dark rings in that order. When the light emission angle is 0°, ± 30°, and ± 50°, high luminous intensity outputs occur. Figures 3(a) and 3(b) shows the LIDC diagram for Type I and II light patterns.
This study considered the LIDCs for the Type I and II light patterns to be periodic functions. The LIDCs are symmetrical across the central axes of the curves and are represented by the sine and cosine functions in Fourier series. The LIDC of the LPMCC is an even function with a period of π. The Eq. for the fit between the LIDC of the LPMCC and the ideal LIDC is expressed in Eq. (3).
where The Eq. (3) can be expanded as follows:The various parameters – for the Type I and II light patterns are presented in Table 1.
Table 1. Parameters of the Equations for the LIDCs of the Type I and II Light Patterns
After the LIDCs for the Type I and II light patterns are obtained, angular energy mapping is conducted according to the law of conservation of energy. Ideally, the total energy of the LED light passing through the lens is constant. Therefore, the relationships between the LIDCs of the light source and the light that passes through a lens on the vertical and horizontal planes in the first quadrant are expressed as follows:
Equation (7) and Eq. (8) are the mapping relationship between the LIDCs on the vertical plane and horizontal plane, respectively. In these equations, θn and Φn denote the energy mapping relationships for specific angles θ and Φ.
To design a freeform lens, the LIDCs of the light source in the θ and Φ directions of the first quadrant (0 ≦ θ ≦ π/2 and 0 ≦ Φ ≦ π/2) can be divided into equal parts, m and n, respectively. Then, Eq. (7) and Eq. (8) can be used to obtain the energy mapping relationships between (Φ) and (Φ) and between (θ) and (θ), revealing the mapping relationships for various angles of the light passing through a lens. Subsequently, the Snell’s Law is employed to obtain each point on the lens and the corresponding normal vector and thereby construct an entire freeform lens. The vector Eq. of the Snell’s Law is written as follows:
where denotes the unit refraction vector; I denotes the unit incident vector; n denotes the refractive index of the lens; and denotes the normal vector corresponding to the incident and refraction vectors. In Eq. (9), , , and n are known and, therefore, Eq. (9) can be used to solve for the vector .
Using the normal vectors that correspond to each point on the lens, the main curve of the freeform lens can be constructed, as shown in Fig. 4. First, an initial value L must be determined. This value represents the height of the lens at θ = 0 and Φ = 0 or the distance between the highest point of the freeform lens and the surface of the LED. Accordingly, an initial point P11 can be obtained, based on which the tangent plane T11 that passes through P11 perpendicular to the first incident vector I11 is identified. The second point P12 of the main curve is derived by identifying the intersection point of T11 and the second incident vector I12. Using this method, the subsequent points on the main curve (i.e., P13 to P1n) can be determined. Once the main curve of the freeform lens is constructed, other subcurves can be generated using the same method. Therefore, each point on the curves on the x-z plane of the lens and the corresponding normal vector are determined by using angular energy mapping between the LIDC Eq. (2) for the light source and the LIDC Eq. (3) for the lens and then substituting the mapping relationship into the Snell’s Law Eq. (9). Figures 5(a) and 5(b) show the 3D diagrams for the Type I and II light patterns.
2.1 Simulation of the LPMCC lens
The LED lamps of the Type I and II LPMCCs were constructed using a Cree® XM-L as the light source combined with Type I and Type II lenses. Regarding the LPMCC simulation analysis (Fig. 5), the 3D models of the Type I and II lenses were imported into TracePro to simulate the light illumination distribution. The lens material was optical polymethyl methacrylate; the total flux of the light source was 75 lm. In addition, a 1 × 1m2 target plane was installed 45 cm away from the light source to observe the light illumination distributions of the Type I and II LPMCCs. As shown in Fig. 6(a), the total luminous flux of the Type I lens on the target plane was 71.53 lm, the light energy efficiency was 95.37%, the most light illumination occurred in the inner ring of light at 766 lux, the light illumination of the outer ring of light was 200 lux, the width of the outer ring of light was 100 mm, the diameter of the inner dark zone was 100 mm, and the distance between the first and second ring of light was 170 mm. As shown in Fig. 6(b), the total luminous flux of the Type II lens on the target plane was 71.90 lm; the light energy efficiency was 95.86%; the LPMCC contained a light zone and two rings of light; the light illumination from the inner part to the outer part was 606, 400, and 150 lux; the diameter of the light zone was 110 mm; and the widths of the two rings of light were 110 and 100 mm. Regarding the dark rings, the widths of the inner and outer rings were 80 mm and 150 mm, respectively.
Fig. 6 Simulation diagrams for the light illumination distributions of the (a) Type I and (b) Type II LPMCC lenses.
Figure 7 shows the simulated LIDCs of the Type I and II LPMCCs. Compared with the ideal LIDCs shown in Fig. 3, weak light illumination occurred in the dark ring and the relative light intensity for the Type I and II lenses was approximately below 7% and 20%, respectively. The reason for this was that in the freeform lens design, LEDs were viewed as a point source. However, the LEDs were a surface light source; therefore, trace of light energy was observed in the dark zone of the light pattern. In other words, random light was emitted by the point and surface light sources in the simulation. In contrast with the geometric shape of the freeform lens, each point of the surface light source was considered to be an approximate origin of light. By regarding a surface light source as a collection of point source, using this point source to calculate the parameters for designing freeform lens and stacking the light patterns reveals why light patterns were distorted. The difference in relative light intensity between the Type I and II lenses resulted from the positional residual errors of coordinate values in the vector computation.
3. Results and discussion
Figure 8 shows the diagrams for the Type I and II lenses that had undergone computer numeric control precision processing. The diameter and height of the lenses were 10.7 mm and 9.7 mm, respectively. The bases of the lenses allowed the lenses to be combined with the LED light source, thereby reducing optical loss. The LIDCs of the Type I and II LPMCC lenses were first measured and the light intensity of each light emission angle was compared with the simulation results. Next, in an underwater environment, light illumination was measured using a light intensity meter and LPMCC was observed using a charge-coupled device (CCD). Finally, a fish-attracting assessment was conducted.
3.1 Measurement of the LPMCC LIDC
The LIDCs of the Type I and II lenses were measured using the IGM-500 optical system (Isuzu Optics Corporation), which comprises an illumination sensor that is placed 2.25 meters apart from a rotation platform. The vertical and horizontal rotation angles of the rotation platform are ± 150° and 0° to 360°, respectively. The rotation precision is ± 0.2° and the resolution is 0.05°. The rotation platform allows the light emitted from every angle to be received by the illumination sensor. After signal processing, the LIDCs of the Type I and II lenses can be obtained.
The Lambertian LED light source was used to measure LIDC. As shown in Fig. 9(a), when LED light passed through the Type I lens, the LIDC exhibited multiple concentric circles. When the light intensity peaks were at ± 20°, the relative light intensity was the highest at 100%. When the light intensity peaks were at ± 50°, the relative light intensity was approximately 85%. Figure 9(b) shows the LIDC of the Type II lens. When the relative light intensity peak was at 0°, the relative light intensity was the highest at 100%. The relative light intensity for the peaks at ± 30° and ± 50° was approximately 80% and 45%, respectively. Normalized cross-correlation (NCC) was used to investigate the relationship between the simulated and the measured LIDCs. Equation (10) shows how the NCC was calculated.
where Axy and Bxy are the relative light intensity of the simulation value (A) and experimental value (B). α and β are the mean of A and B.Fig. 9 Comparison between the measured and the simulated LIDCs of the (a) Type I and (b) Type II lenses.
The correlation coefficient between two vectors was first calculated and then normalized to eliminate the effect of the vector mean; thus, the similarity between the two vectors was determined [20]. By comparing Figs. 9(a) and 9(b), the results indicated that, the simulated and measured LIDCs of the Type I and II lenses were highly similar. The NCC values for these two types of lenses were 96.2% and 96.7%, respectively, indicating that the LIDCs of the Type I and II lenses met the design requirement. Therefore, integrating Fourier series and the energy mapping method not only facilitated the design of an arbitrary light pattern but also enabled the control of the light illumination distribution to achieve a light pattern of interleaving light–dark zones.
3.2 Measurement of the LPMCC
The Type I or Type II LED lamp was placed directly above the center of a 1.2 × 0.6 × 0.45 m water tank. Light-absorbing black fabric was placed in the water tank to reduce light reflection from the tank wall. Figure 10 shows images of the light transmission in the water after LED light passed through the Type I and II lenses. An underwater light intensity meter (Konica Minolta) was used to measure the light patterns and light illumination distributions (cd/m2) of the Type I and II lenses. This intensity meter must be placed directly below and normal to the LED lamp. Figure 11(a) shows the Type I LPMCC on the water surface, which produced a light illumination distribution of interleaving light–dark zones; the bottom of the water tank exhibited an LPMCC, in which the center of the circle was a dark zone that was encircled by (in order) light, dark, light, and dark rings. Figure 11(b) shows the Type II LPMCC on the water surface and the bottom of the tank. The center of the circle was a light zone encircled by dark, light, dark, and light rings in that order. As shown in Fig. 11, for both the Type I and II light patterns, the illumination uniformity in the light zone of the outermost ring deteriorated. The main reason was that microparticulates were suspended in the water, causing light to scatter. The light pattern considerably changed according to the transmission distance of the light and the frequency of light scattering. In other words, a large light emission angle increases the transmission distance of light, which increases the frequency of light scattering. Therefore, the outer ring of the LPMCC became uneven. In addition, an underwater CCD was used in this experiment to capture photographs. As shown in Fig. 12, an LPMCC was observed on the target plane as light passed through a lens. The measurement results were consistent with the measurement results shown in Fig. 11.
Fig. 11 Energy images for the LPMCCs of the (a) Type I and (b) Type II lenses measured using an underwater light intensity meter.
Fig. 12 Photographs of the LPMCCs of the (a) Type I and (b) Type II lenses taken using an underwater CCD camera.
3.3 Experiment on the LPMCC used for attracting fish
The ability of the Type I lens to attract fish, which was Poecilia reticulata in this study, was evaluated. Twenty-five fish were used and the experiment lasted for 24 hours. Figures 13(a) and 13(b) show the experimental results. The fish shoal was first attracted to the light source and hovered in a light zone, as shown in Fig. 13(a). After 3 hours, the fish shoal was habituated to the light stimulus and became fatigued; thereafter, the fish shoal swam toward the dark zone close to the edge of the light source (Figs. 13(b)–13(e)). After 24 hours, the fish shoal remained clustered around the junction of the light and dark zones, waiting to catch plankton (Fig. 13(f)). Finally, the ability of the Type II lens to attract fish was evaluated and similar results were obtained, as illustrated in Fig. 14. Overall, a single LED light source was used to control the light illumination distribution within a light-pattern zone, and a fish-attracting light pattern of interleaving light–dark zones was designed to replace traditional energy-consuming fishing lamps.
4. Conclusions
Replacing traditional fishing lamps with LED is a global trend. In this study, Fourier series were successfully used to represent the LIDC of the fish-attracting light pattern such as LPMCC light pattern. In the men time, a LED was used to design a freeform lens that produces a LPMCC light pattern. Fourier series and the energy mapping method were integrated and the LPMCC lens design was simplified to reflect the angular energy mapping relationship between the LIDCs of a light source and the light pattern of LPMCC. Based on the phototaxis of fish, Type I and II LPMCC lenses that complement each other were designed to create a fish-attracting light pattern of interleaving light–dark zones, which successfully prompted fish shoals to hover near the junction of these zones for a long period. Comparing the LIDCs of the Type I and II lenses with the simulation values showed a NCC value of 96% and light energy efficiency of 95%. According to the 24-hour experiment conducted to evaluate the fish-attracting ability of the lenses, once the fish shoal was habituated to a light source of identical intensity, the fish shoals moved to the junction of the light and dark zones, and waited to catch plankton. Overall, the proposed method can be used to not only design arbitrary light patterns but also control the light illumination distribution of a light pattern, therefore facilitating the replacement of traditional fishing lamps with LED fishing lamps.
Acknowledgments
The authors would like to thank National Science Council (NSC) and Research Center for Energy Technology and Strategy, National Cheng Kung University for their financial supports to the project (granted number: NSC 100-2628-E-006-019-MY3).
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