###### Abstract  
The significance of this work lies in the development of a novel code-based, detailed, and deterministic geometrical approach that couples the optimization of the Fresnel lens primary optical element (POE) and the dome-shaped secondary optical element (SOE). The objective was to maximize the concentration acceptance product (CAP), while using the minimum SOE and receiver geometry at a given f-number and incidence angle (also referred to as the tracking error angle). The laws of polychromatic light refraction along with trigonometry and spherical geometry were utilized to optimize the POE grooves, SOE radius, receiver size, and SOE-receiver spacing. Two literature case studies were analyzed to verify this work's optimization, both with a spot Fresnel lens POE and a spherical dome SOE. Case 1 had a 625 cm\({}^{2}\) POE at an f-number of 1.5, and Case 2 had a 314.2 cm\({}^{2}\) POE at an f-number of 1.34. The equivalent POE designed by this work, with optimized SOE radiuses of 13.6 and 11.4 mm, respectively, enhanced the CAP value of Case 1 by 52% to 0.426 and that of Case 2 by 32.4% to 0.45. The SOE's analytical optimization of Case 1 was checked by a simulated comparative analysis to ensure the validity of the results. Fine-tuning this design for thermal applications and concentrated photovoltaics is also discussed in this paper. The algorithm can be further improved for more optimization parameters and other SOE shapes. solar concentrator; solar energy; renewable energy; optics; Fresnel lens; secondary optics; non-imaging optics +
Footnote †: journal: Applied Soft  
concentrator was designed by the edge-ray principle in a four-part symmetry relating the POE and the SOE [5; 6; 7]. On the same note, the edge-ray principle has been used in the SOE designs of a ball-lens [8; 9], a dome [10; 11], a kaleidoscope with a flat top surface [12; 13; 14; 15; 16; 17; 18] and a domed top surface [12; 13; 14; 18; 19], a half-egg [12; 13], a refractive truncated pyramid (RTP) [15; 16; 17], a single-lens optical element (SILO) pyramid [15; 16; 17; 19], and a refractive dielectric-crossed compound parabolic concentrator (DCCPC) [15; 16; 17]. While a few prior studies have used iterative simulations to optimize the SOE geometry [20], the analytical design in much of the literature has not been typically explained in detailed equations, which makes it hard to replicate, enhance the SOE optimization, or include it in a simulative computer code. Refractive SOEs have advantages over reflective ones, because they are less material-intensive and are easier to manufacture--not only from plastic but also from glass [11]. However, the receiver (which is usually a solar cell) is normally attached with an optical adhesive at the bottom of the refractive SOE [6; 13; 17; 19]. This might jeopardize the optical performance and limit the long-term system reliability. Instead, introducing an air gap between the receiver and the SOE can reduce those risks [8]. It can also be noted that some solar literature does not account for a solar incidence angle in the design phases of the POE and/or SOE [8; 9], while other work does not optimize the POE along with the SOE to further fine-tune the SOE geometry [8; 10], and thus this does not enhance the optical performance. Another issue has been the lack of the polychromatic representation of light in the POE and/or SOE design phases [9; 14], which can deviate the simulated analysis from the experimental results. This work aims to enhance and detail the use of geometrical optics in an algorithmic approach to collectively design a solar concentrator with a POE and a single hemispherical SOE. The goal was to achieve a high concentration acceptance product (CAP) at a predetermined incidence (tracking error) angle and f-number, while minimizing the geometry of the SOE and the receiver, and introducing an air gap in-between them. The optimization employed polychromatic ray tracing, starting with a light incident at the upper POE surface until refracted through the bottom SOE surface.