Rectangular Glass Optical Fiber for Transmitting Sunlight in a Hybrid Concentrator Photovoltaic and Daylighting System

In this paper, we propose to use glass optical fibers with a rectangular cross-section for the application in a concentrator photovoltaic and daylighting system (CPVD) due to the unique characteristics of rectangular fibers with the capability to provide a uniform rectangular beam shape and a top-hat profile at the output. A mathematical model of rectangular optical fibers has been formulated in this study for different incident angles, and the results are compared with those of round optical fibers. Furthermore, the performance of the bundle of RGOFs is compared with that of the bundle of round optical fibers via simulation by using the ray-tracing method. The mathematical modelling and numerical simulation have demonstrated that the RGOF has advantages in terms of the improvement in relative transmission and reduction in energy leakage for the transmission through the optical fiber. The simulation result also shows that a higher flux of sunlight can be transmitted via the bundle of RGOFs as compared to the bundle of round optical fibers due to the higher coupling efficiency. The experiment results on the relative transmission in different incident angles for both round optical fibers and RGOFs have validated both the simulation and the mathematical modelling. The beam profile of our fabricated RGOF has also been measured via our laboratory facility. The flexibility test on the fabricated RGOF has been carried out to bend at a radius of 150 mm and twist at 90° at a fiber length of 2.2m.


Introduction
The initial idea of the optical fiber used for transmitting sunlight can be dated back to forty years ago [1]. Recently, significant advances in the technology of optical fibers have extended the application to transmit the sunlight with a wide band of electromagnetic waves. The application of transmitting sunlight by a round optical fiber and the bundle of round optical fibers have been studied [2][3][4][5][6]. The optical fibers can transmit sunlight with reasonable low loss for a distance fewer than 10 meters, but the optical loss in the optical fibers becomes noticeable at longer distances [7]. Considering the said limitation, there are still many applications especially transmitting daylight via fiber optics for residential and commercial buildings, which are still regarded as a promising method [7][8][9].
In transmitting sunlight through a dish and a round optical fiber, many components such as mirror accuracy and tracker precision, as well as the material, diameter, length, and type of optical fibers, can affect the efficiency of the whole system [10]. The material of the core and cladding of the optical fiber are the major factors to determine the acceptance angle of the optical fiber [11]. A parabolic dish with a high precision sun tracker can provide a solar concentration ratio of more than 8000 [12]. For transmitting the high flux of concentrated sunlight, a larger core diameter is usually preferable. The one-millimeter core diameter of fused silica with a low refractive index of hard polymer resin as cladding can transmit a reasonable flux of sunlight with lower loss than that of polymer optical fibers [13,14].
Transmitting sunlight with a wide spectrum of wavelengths via optical fibers encounters more challenges as compared to the application of optical fibers in telecommunications. The large core diameter of the optical fiber transmits light in multimode. The optical loss of the multimode fiber is higher in comparison to that of single-mode fibers made of the same material and transmitting the same wavelength of light [15]. Attenuation of power transmission in optical fibers with fused silica as the core and a hard polymer as the cladding is independent of the shape of the optical fiber [16]. Dugas et al. found that the leakage of power in sunlight transmission via a round optical fiber was attributed to optical loss due to too many reflections between the core and the clad with angles near the numerical aperture of the optical fiber [17]. To minimize the light leakage, the incident angle of sunlight relative to the numerical aperture of the optical fiber should be significantly less than the acceptance angle of the optical fiber. Feuermann et al. studied the dependence of light leakage within the nominal numerical aperture of the solar fiber optic on several parameters including incidence angle, optical properties of the core and cladding, and fiber length [13].
Moreover, the optical loss can occur when coupling the sunlight to a bundle of optical fibers. The method of bundling optical fibers will determine the amount of coupling loss. If a bundle of round optical fibers is fabricated by fusing all the optical fibers, the loss is lesser than that of the bundle of optical fibers being joined by epoxy adhesive. In the fusing process, the gaps among the optical fibers are diminished by forming one whole solid block, which has improved the coupling efficiency of light to the bundle of optical fibers [18]. However, the method of fusing optical fibers is much more expensive than by simply joining with epoxy adhesive.
A multijunction solar cell with a square dimension can convert sunlight into electricity in a concentrate photovoltaic (CPV) system. The dimensions of the multijunction cells that are available in the market are 3 mm × 3 mm, 5 mm × 5 mm, and 10 mm × 10 mm. Nevertheless, the matching between the concentrated light profile and the geometry of solar cells is essential to optimize the performance of the CPV system [19]. Many works have been carried to study the design of solar concentrators for mapping the focused sunlight with the square shape of multijunction solar cells. Baig et al. and Yu et al. studied the compound parabolic concentrator (CPC) as a nonimaging concentrator to harness solar energy and mapping sunlight to square solar cells [20,21]. A compound truncated pyramid and cone may be used for concentrating and coupling sunlight with the optical fiber. A bundle of optical fibers can be used as a coupler of concentrated sunlight to the multijunction solar cells in the CPVD system [12]. The optical fiber must be made of glass to tolerate the high temperature of concentrated light. A bundle of RGOFs with a square shape can map well the focused sunlight onto square solar cells.
Rectangular fibers were first studied for transmitting Xrays and light-wave circuit. Marcatili studied the rectangular waveguide for integrated optics in 1969 [22]. In the meantime, a square core optical fiber was developed for matching with the laser diode output beam. Cherny et al. (1979) studied waveguide characterization of a rectangular core and round clad for the cases of multi-mode and singlemode waves including the effect of single and double clads on dispersion of a pulse [23]. Blomster and Blomqvist studied square fiber for high-power laser applications in 2007 [24]. Konishi et al. developed a rectangular core optical fiber for a high-power laser in 2010 [25]. Ambran et al. demonstrated a physical micromachining technique to fabricate a flat fiber substrate for a light circuit [26]. The width of the flat fiber was 1612 μm, and the thickness of the core was about 10 μm.
Rectangular core fibers in the market have round clads. The core is made of fused silica for its broadband UV to NIR transmittance while the clad may be made of glass or a low refractive polymer. Figure 1 shows the attenuation of a square core in the market at different wavelengths. The core of the fiber is fused silica, and the clad is a hard polymer. These rectangular fibers are developed for applications in transmitting laser. The published data same as the published data of round optical fibers are extracted from propagating a beam of laser with a specific bandgap. Figure 1 reveals that attenuation is as low as 20 dB/km at 600 nm while the loss of fiber is more than the mentioned loss in transmitting sunlight.
In this paper, we would like to introduce RGOF that is specially designed for a CPVD system to transmit sunlight through the optical fiber. The core of RGOF can be fabricated by drawing a fused silica preform at the temperature of 2250°C. The preform must have a rectangular cross-section before drawing. The clad is a polymer with refractive index of 1.37 that is coated on the fused silica core. The length of    International Journal of Photoenergy the rectangular core is between 5 mm and 10 mm with the width ranging from 0.2 mm to 0.5 mm. The thickness of the clad is about 0.1 mm.
The CPVD system has two functions including generating electricity and daylighting. The bundles of RGOF transmit sunlight from the focal point of the solar concentrator to the multijunction cells and the remote target. Furthermore, RGOF can also be used in other applications, i.e. sensors and delivery of a high-power laser beam.
A comprehensive manner composed of three major aspects embracing mathematical modelling, numerical simulation, and experimental verification is used for investigating the mechanism of leakage in transmitting sunlight in the RGOF. Based on previous studies, relative transmission of round optical fibers and RGOF in transmitting sunlight and the loss due to gaps between the fibers in a bundle are also studied. In Section 2, the mathematical model of the relative transmission is presented to compare between the round optical fiber and RGOF for different angles of incidence. In Section 3, the performance of RGOF is simulated by using the ray-tracing method. In Section 4, the effect of gaps is simulated for a bundle of RGOFs and the result is compared with that of a bundle of round optical fibers. In Section 5, the relative transmission, beam profile, and bending of the RGOF are investigated via experiments.
In this work, transmitting sunlight through an optical fiber with a rectangular glass core and a rectangular polymer clad has been investigated. Other researches on rectangular glass core fibers were only focused on the characterizations of the rectangular fiber for transmitting a high-power laser and pulse dispersion that involves a narrow band of wavelengths in the application of optical communication. For the originality of our study, the relative transmission of sunlight with a wide spectrum of wavelengths in the rectangular optical fiber is studied in a comprehensive manner in three major aspects embracing mathematical modelling, numerical simulation, and experimental verification.

Analytical Approach on Leakage of the Optical Fiber during Transmission of Sunlight
The efficiency of an optical fiber under the propagation of a full spectrum is different from the efficiency of the optical fiber under a collimated laser beam with a narrow bandgap. Manufacturers of optical fibers only provide information about the attenuation of light power at specific wavelengths but not the efficiency of transmitting a broad spectrum of light. The optical loss of an optical fiber with a specific attenuation varies with the length of the fiber. The light is coupled with the fiber with an incident angle of 0°. We may consider a source of light beam larger than the diameter of the optical fiber. If the angle of propagation increases up to the acceptance angle, the loss is changed with the cosine of incident angle. The loss of the fiber under the propagation of a wide spectrum for a specific incident angle is less than the expected loss. Feuermann measured the leakage of optical fibers under the propagation of a wide spectrum with different angles of propagation and different ratios of the length of the fiber to the diameter of the core.

International Journal of Photoenergy
For a fiber with a specific numerical aperture and specific angle of propagation, the ratio of the length of fiber to the diameter of the core directly affects the amount of leakage.
The mechanism of leakage in transmitting sunlight by optical fibers was studied, and it was found that the reflection of light between the core and the clad is not perfect and that the number of reflections of light determines the amount of leakage in the optical fiber. Concentrating sunlight leads to an increase in the incident angle between the rays of sunlight and the aperture of optical fibers. The number of reflections increases in such a condition, and the effect of the very small loss due to reflection is noticeable. The number of reflections increases with addition of the angle of incidence and the ratio of the length to the diameter of the fiber.
Although ray-tracing is a popular method, a mathematical analysis of loss is useful for understanding the behavior of fibers in various conditions and for assessing the accuracy of numerical methods. For the mathematical analysis of loss due to the number of reflections between the core and the clad, the optical fibers have been modelled to be exposed to the collimated light of a tilted source, which is likely to happen for the bundle of fibers used as the receiver of a CPVD system. Figure 2 shows the setup of a collimated light with an angle of incidence relative to the axis of the optical fiber. The average number of reflections for all rays is a variation that affects the relative transmission and determines the loss of the fiber due to the angle of incidence. Figure 3(a) shows that the azimuth angle of the rays is 90°a nd the rays skew along the length of the fiber. Figure 3(b) shows the rays in the rectangular fiber. The rays in the round optical fiber skew around the axis of the fiber, but the rays move in a zigzag form in the rectangular fiber. Figure 3(c) shows the relation between 2x, dL, θ, and θ d in a round optical fiber.
2x is the distance between two sequential reflections in the plane normal to the round optical fiber's axis. R is the radius of the optical fiber, and dL is the distance between two sequential reflections in the plane including a line parallel to the axis of the optical fiber and 2x. θ is the angle between dL and the trajectory onto the plane through the dL including to 2x.

International Journal of Photoenergy
where N i is the number of reflections for a ray in the length of the fiber, L is the length of the optical fiber, and d is the diameter of the fiber. The total number of reflections of all rays is where r and ω represent the polar coordinates of the interface of the fiber Considering the uniform distribution of rays on the surface of the interface of the fiber, we have where N av is the average number of reflections for all rays. The angle of α min corresponds to the rays with the highest reflection and the lowest acceptable energy. We may consider that rays with a lower angle than α min are ignored. If the diameter of the propagated light is larger than the diameter of the optical fiber, then the power of incident light is where R av is the arithmetic average of reflectivity and θ d is the angle of incidence relative to the axis of the optical fiber (see Figure 3). τ 0 is the transmission of the optical fiber at the angle of zero degree.
In the flat fiber, we do not have skew rays. The direction of the rectangular fiber to the propagated light determines the transmission of light in the zigzag direction. Figure 4(a) shows the effect of the zigzag direction on the number of reflections. Figure 4(b) shows the relation between the length of rectangle b, dL, and θ in a rectangular optical fiber.
where a and b are the length and the width of the rectangular fiber, respectively.    International Journal of Photoenergy where dx and dy represent the Cartesian coordinate of the interface of the rectangular fiber For another side, we have If the diameter of the propagated light is larger than the width of the rectangle in the fiber, then the power of incident light is proportional to the cosine of the incident angle. For one side, the relative transmission is For another side, the relative transmission is The direction of the rectangular fiber to the propagated light determines the zigzag transmission of light direction. Figure 4 shows that the direction of zigzag affects the number of reflections.
The designed RGOF is a glass optical fiber with a rectangular shape of the core and polymer clad; the ratio of the length to width of the rectangular fiber is more than 5 where the width must have a limit to prevent breakage during bending. As an example, the fabricated RGOF has a width or thickness of 0.4 mm and the minimum bending radius of 150 mm. If the diameter of the propagated light is large enough to cover the aperture of the optical fibers and the length of the fiber is to be short, the cosine of the incident angle plays an important role. Figure 5 shows the loss of a 100 mm length of a round fiber and a rectangular fiber where the light covers the aperture of the fibers. The diameter of the light is large enough and the cosine of angle of incidence should be considered. The ratio of the length of the rectangle to the diameter of the round fiber affects the relative transmission of the rectangular fiber. The effect of loss due to the number of reflections between the core and the clad is very low and the trends of optical fibers are near to each other and near to the cosine of the incident angle. Figure 6 shows the relative transmission of fibers with a length of 10,000 mm, and light covers the surface of the fibers. The rectangular fiber with the length of the rectangle 10 times the diameter of the round fiber has a lower loss due to the number of reflections between the core and the clad and shows higher relative transmission than other fibers. The zigzag movement is in the longer trajectory ( Figure 4).
If the diameter of the light is to be equal to the diameter of the round fiber, a rectangular fiber with the length of the rectangle 10 times the diameter of the round fiber shows higher relative transmission as compared to the round fiber. Figure 7 shows the relative transmission of a round and a rectangular fiber. The diameter of the round fiber is one millimeter, and the length of the rectangle of the rectangular fiber is 10 mm. the length of both fibers is 10,000 mm, and the diameter of the light is smaller than the diameter of the round fiber. The difference between the relative transmission of two fibers increases when the source is more tilted relative to the axis of the fiber.

Simulation of RGOF and Round Optical Fiber
The ray-tracing method has been applied to simulate the responses of the flat fiber (RGOF) and the conventional round fiber to a tilted source of light. For both cases of the flat and round fibers that consist of a fused silica core and a low refractive index resin clad, the parameters including the flux, relative transmission, and optical loss of the light have been investigated for different incident angles relative to the axis of the fiber, i.e., from 0°to 30°.    10 International Journal of Photoenergy contains a significant part of solar spectral irradiance that is required in energy harnessing for a high concentrator photovoltaic receiver and daylighting system, where the glass optical fibers with the core material made of fused silica has a high transmission coefficient in this range. Figure 9(a) shows the schematic diagram of the round optical fiber, and Figure 9(b) shows the schematic diagram of RGOF in the simulation. Table 1 shows the input data of the simulation for both the round optical fiber and the RGOF. The schematic diagram of simulation shows that the flat fiber does not produce skew rays. The result of the simulation shows that the leakage of the round fiber for incident angles ranging from 0°to 20°is more than that of RGOF. Figure 10 shows that the relative transmission in the round fiber is 3% less than that in RGOF for the incident angles ranging from 0°to 10°. For the incident angle between 10°a nd 20°, the difference of the relative transmission between the round optical fiber and the RGOF can increase up to 17% at the angle of 20°. Considering NA = 0:39 and an incident angle less than half of the acceptance angle, we conclude that the round optical fiber shows low angular loss similar to the RGOF.
To investigate the angular response and distribution of light at the endpoint of both types of fibers, Figures 11 and  12 show the simulated results at the incident angles 0°, 15°, and 25°for the round optical fiber and RGOF, respectively. The diameter of the source is 2 mm, and the maximum angle of the source with reasonable intensity and uniform output of the round optical fiber occurred at 10°which is less than half of the critical angle. Based on the results of the simulation, collimating the light before the light is transmitted via the optical fiber can increase the relative transmissions significantly for the incident angles 0°to 20°.
For the incident angles 0°-20°, the intensity of light in RGOF was higher than that in the round optical fiber. For an incident angle of 25°, the intensity of light dropped dramatically to nearly zero while the skew rays caused a maximum intensity of 4 watts per steradian in the round optical fiber. However, the amount of intensity is not considerable.

Investigation of the Effects of Gaps for Bundle of RGOFs and Bundle of Round Optical Fibers
In this section, we will assess the performances of the bundle of RGOFs and the bundle of round optical fibers under the same conditions. For the application of optical fibers in transmitting concentrated sunlight, the concentrator photovoltaic and daylighting (CPVD) system encounters several possible optical losses when using the bundle of optical fibers. First, the optical loss can occur during the coupling of sunlight to the aperture of the bundle of optical fibers. Second, the gap spacing between the fibers tied in a bundle can cause the energy loss when the light falls into the gaps. Third is the optical losses during the process of transmitting sunlight through optical fibers. Lastly, at the end of the fibers in a bundle, the geometrical matching between the aperture of fibers in the bundle and the dimension of the solar cell may also cause the energy loss.   The optical characteristics of round optical fibers and RGOFs in the bundle have been investigated using the raytracing method via Zemax software. To compare the performance of round and rectangular optical fibers, same core and cladding materials have been used, which are fused silica with a refractive index of 1.458464 at D-line and resin XPC-373 AP clad with a low refractive index of 1.387939 at 852nm [27]. For both cases of fibers, the propagation of light rays has been investigated at the incident angle of 0°relative to the central line of the optical fiber. A 1-watt light source based on blackbody radiation at the temperature of 5780 K (wavelengths ranging from 400 nm to 1600 nm) is applied to illuminate the bundles of optical fibers. The maximum divergence of the light source is 0.27°.  12 International Journal of Photoenergy The active area of the photodetector is square (12 mm × 12 mm) and the thickness of the cladding is 25 μm. The area of the bundle of RGOFs was 101.6064 mm 2 while the area of the bundle of round optical fibers was 101.9172 mm 2 , where the difference between the areas of both bundles is negligible, at about 0.3%. The performance including power transmission and distribution of light intensity and flux of light are evaluated in both cases. Figure 13 shows a row of fifteen units of RGOF with a cross-sectional area of 10:08 mm × 0:67 mm for each fiber to form a total cross-sectional area of 10:08 mm × 10:08 mm. The light is well mapped to the square dimension of the photodetector in both directions of X and Y. For both cases, we have applied the same testing conditions, i.e., light source, photodetector, length of fibers, material, and cross-sectional area of bothbundles. Figure 14(a) shows images of the flux on the active surface of the photodetector for both cases, and Figure 14(b) shows the intensity distribution in the X and Y directions. A comparison between the flux distributions in both cases reveals that the transmitted light rays via flat fibers are more uniform than those via round optical fibers. Figure 15 shows the total powers for both bundles of RGOF and round optical fibers for the lengths between 50 mm and 300 mm. The result shows that the bundle of RGOF fibers transmits more power as compared to the bundle of round optical fibers because of lower optical loss at the entrance aperture of the optical fibers. The effective area of the cores in the bundle of RGOFs is 93.279 mm 2 , and the effective area of the cores in the bundle of roundoptical fibers is 78.539 mm 2 .

Experimental Investigation on a Round Optical Fiber and a RGOF
With regard to the profile of the beam, relative transmission of RGOF for incident angles ranging from 0°to 25°has been investigated experimentally by using collimated white light, focused white light, and the laser beam. A supercontinuum white-light laser of NKT Photonics Company and a beam profiler by Thorlabs Company have been employed as a light source and plotter for the beam profile, respectively. Two samples of RGOF have been fabricated by using two different materials: the first sample has fused silica as the core material and the XPC 373 low refractive index polymer as the clad material; the second sample has borosilicate 3.3 as the core material and Teflon as the clad material. Table 2 lists down the characteristics of the supercontinuum laser. Figure 16 shows the experimental setup of the beam profile measuring system, where the light source illuminates the light from one end side of the RGOF, and the beam profiler is positioned at the other end of the RGOF fiber. Figure 17(a) shows the results of the measured beam profile along the Y -axis. The intensity of light was distributed across the surface of RGOF and the maximum intensity occurred at the center of the fiber. The intensity was distributed from zero to 1000 μm along the Y-axis. The amplitude of the peak point   Figure 17(b) shows the measurement of the beam profiler along the X-axis.
Relative transmission of RGOF for incident angles between 0°and 20°has been measured under the propagation of a white light produced by a xenon lamp. Figure 18(a) shows the experiment setup of the test using a xenon lamp, and Figure 18(b) shows the spectral irradiance of the xenon lamp in the test that is extracted by an Ocean 2002 spectrometer. A total length of 740 mm RGOF with a cross-sectional area of 10 mm × 0:4 mm and a total length of 740 mm round optical fiber with 1 mm diameter have been fabricated for measuring the relative transmission of RGOF at different angular propagations of a focused light. Table 3 shows the conditions of the test. Figure 19 shows that round optical fibers follow the cosine of incident light while RGOF has a higher relative transmission. The length of rectangle in RGOF is 10 mm that makes it suitable for a focused light with a diameter of less than 2 mm. The result may be used in harnessing the light around the solar cells in the receiver of a CPVD. The light at the middle of the focal plane is uniform enough for multijunction solar cells, and the light around the cells may be transmitted by bundles of RGOF.
The fabricated RGOF has just a thin layer of polymer as the clad where it has been examined with bending and twisting tests. Figure 20 shows the demonstration on the bending and twisting tests of RGOF. The fabricated fiber is bent on a cylinder with 350 mm diameter and twisted 90°along its longitude axis with a length of 2.2 m. The minimum radius of bending for the fabricated RGOF was 150 mm with a crosssectional area of 10 mm × 0:4 mm. A lower radius may be achieved by an additional coating process.

Conclusion
In this study, the concept of introducing RGOF to transmit sunlight in a CPVD system is proposed and analyzed in detail. Relative transmission of RGOF for incident angles ranging from 0°to 25°has been studied via a mathematical model, numerical simulation, and experiment. The simulation result reveals that a RGOF with a cross-sectional area of 10 mm × 0:5 mm has a higher relative transmission as compared to that of a glass round optical fiber with a core diameter of 1 mm while it is exposed to the focused light with a diameter of spot size less than 2 mm. The results of experiments and simulations are limited to the collimated and focused white-light source with the spectral irradiance close to the standard solar spectrum. Moreover, the second limitation is that the relative transmission of RGOF is measured for the incident angles of light beam less than the acceptance angle of the optical fiber. A bundle of RGOFs can achieve higher efficiency in both coupling and transmitting light power as compared to a bundle of round fibers. The bundle of RGOFs has less gap spacing between fibers. The simulated result shows a good matching between of output flux distribution and the dimension of the solar cell for the case of RGOF, which is consistent with the higher coupling efficiency of RGOF. The beam profiles of both the RGOF and the round glass optical fiber under propagation of a focused light have been validated via analytical formula and numerical simulation. The profile of the output light of RGOF has   14 International Journal of Photoenergy been investigated experimentally. A top-hat profile matches with the result of the propagating laser beam to the rectangular glass optical fiber. In the last section of our study, tests on the flexibility of our designed and fabricated RGOF in both bending and twisting have been carried out. In this test, RGOF can be bent with a radius of 150 mm and twisted 90°a long the axis of the fiber at a fiber length of 2.2 m.

Data Availability
The data will be available upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.