Performance Evaluation of Elliptical and Multi-part Mirrors for Concentrating Diffused Light onto Collector Surface from Half Celestial Sphere

An elliptical mirror has a mirror surface at the first focal point on a plane perpendicular to its central axis, which, in turn, passes through the first and second focal points. It can collect diffused light from a half celestial sphere, which extends 360◦ in the axial rotation direction and 180◦ in the axial direction at the second focal point. An elliptical mirror allows only the light incident on the first focal point to be concentrated on only the second focal point. A single elliptical mirror consists of only one elliptical mirror. A multi-part mirror is composed of multiple elliptical mirrors. In a multi-part elliptical mirror, the first focal point lies on its surface, and the second focal point is so positioned that it is aligned with the top of its surface. A multi-part mirror can concentrate diffused light from a wider area than a single elliptical mirror because its first focal points are dispersed. Furthermore, in a single elliptical mirror, only the light incident on the first focal point is concentrated on only the second focal point; therefore, the light-concentrating efficiency is low. In this study, we installed a collector surface on the second focal point of an elliptical mirror to increase its light-concentrating efficiency. Subsequently, we evaluated single elliptical and multi-part mirrors that capture the diffused light from a half celestial sphere not incident on the first focal point and concentrate it on the collector surface as well as the second focal point. We configured a coordinate system for the design of single elliptical and multi-part mirrors and subsequently analyzed the mechanism by which the light will strike these designed mirrors and concentrate on their collector surfaces. Based on the design values, we manufactured actual single elliptical and multi-part mirrors, which were illuminated with diffused light, and measured the illuminance of the light concentrated on their collector surfaces. Finally, we evaluated the light-concentrating performance of these mirrors based on the analysis results and measured illumination.


Introduction
Light-concentrating collectors, which employ a concentrator to concentrate direct sunlight and convert this concentrated light energy into heat energy by emitting it onto a heat collector at the collector surface are widely used [1]. Plane mirrors, concave mirrors, composite parabolic mirrors, and Fresnel lenses are used as concentrators [1]. However, a light-concentrating collector only concentrates direct light from the sun and cannot concentrate diffused light from the entire sky, which is a half celestial sphere [2]. The proportion of diffused light is high in most parts of the world, except in desert areas where the weather is sunny all year round [2]. In Japan, approximately half of the total annual solar radiation is diffused light [2]. In particular, on the Sea of Japan side from San'in to Hokkaido, there is an average cloud cover of 8.5 or more for more than 180 days during which only diffused light irradiates [3]. Therefore, locations such as the Sea of Japan side of Japan require a concentrator that can concentrate the diffused light from a half celestial sphere on the collector surface.
A variety of research has been conducted on concentrators for concentrating diffused light. Winston [5]. Yamada et al. proposed Fresnel lenses, cylindrical lenses, water cylinder lenses, and parabolic mirrors [17]. However, all these studies suggested two-dimensional concentrators, which are  [18]. However, it emphasized the injection of direct light into a solar cell. Matsumoto presented a multi-part mirror arranged with an array of elliptical mirrors [19]. An elliptical mirror has a mirror surface at the first focal point on a plane perpendicular to its central axis, which, in turn, passes through the first and second focal points. It can collect diffused light from a half celestial sphere that extends 360 • in the axial rotation direction and 180 • in the axial direction at the second focal point. An elliptical mirror allows only the light incident on the first focal point to be concentrated on only the second focal point. A multi-part mirror is composed of multiple elliptical mirrors. In all the elliptical mirrors of a multi-part mirror, the first focal point is located on the surface of the multi-part mirror, and the second focal point is positioned so that it is aligned to its top. A multi-part mirror can concentrate diffused light from a wider area than a single elliptical mirror because the first focal points of the former are dispersed on its surface. However, in a multipart mirror, only the light incident on the first focal point is concentrated on only the second focal point; therefore, the light-concentrating efficiency is low.
In this study, we installed a collector surface on the second focal point of an elliptical mirror, to increase its lightconcentrating efficiency. Subsequently, we evaluated single elliptical and multi-part mirrors that capture the diffused light from a half celestial sphere not incident on the first focal point and concentrate it on the collector surface as well as the second focal point. We configured a coordinate system for the design of single elliptical mirrors and multipart mirrors and subsequently analyzed the mechanism by which the light is incident on our designed single elliptical and multi-part mirrors and concentrates on their collector surfaces. Based on the design values, we manufactured actual single elliptical and multi-part mirrors. Using our experimental set-up, we illuminated the manufactured mirrors with diffused light and measured the illuminance of the light concentrated on their collector surfaces. Finally, we evaluated the light-concentrating performance of these single elliptical and multi-part mirrors based on the analysis results and the measured illumination. Table 1 lists the variables in the equations used in this study. We set up the coordinate system of an elliptical mirror to design single elliptical mirrors and elliptical mirrors for multi-part mirrors. Figure 1 presents the elliptical mirror coordinate system and design method. Figures 1(a) and (b) show the front and longitudinal views, respectively. In the coordinate system, F jk , and the origin, O jk , are the first and second focal points of the elliptical mirror, respectively. The central axis of the elliptical mirror passes through the first focal point, F jk , and the origin, O jk . The coordinate system of the elliptical mirror is R jk -Θ jk -Z jk , where the R jk axis is the radial direction, Θ jk axis is the direction of rotation around the center axis, and Z jk axis is the center axis direction. The elliptical mirrors are indexed as j and k. Coordinates in the R jk -Θ jk -Z jk co-  ordinate system are denoted as (r jk , θ jk , z jk ).

Elliptical Mirror
An elliptical mirror has a three-dimensional shape obtained by rotating an ellipse around the Z jk axis on the R jk -Z jk plane in the R jk -Θ jk -Z jk coordinate system, which allows axisymmetry to be achieved with respect to the Z jk axis. The focal length, length of the major axis, and length of the minor axis of the ellipse are denoted as f , 2l, and 2s, respectively. The mirror surface is at the first focal point, F jk , on a plane perpendicular to the Z jk axis and parallel to the R jk -Θ jk plane in the R jk -Θ jk -Z jk coordinate system passing through the first focal point, F jk . The radius and height of the elliptical mirror are denoted as r E and h, respectively.
The range of the light incident on the elliptical mirror is a half celestial sphere from the first focal point, F jk , in the direction of the origin, O jk . In the design of the elliptical mirror, focal length f , radius r E , and height h are determined using the equations below. The focal length, f , can be expressed as Using Eq. (1), the equation of the ellipse can be expressed as Using Eq. (2), the radius, r E , can be expressed as Using Eq. (1), the height, h, can be expressed as

Single Elliptical Mirror and Multi-part Mirrors
We set up a coordinate system for designing single elliptical and multi-part mirrors. Figure 2 shows the coordinate system and design method for these single elliptical and multipart mirrors. Figures 2(a) and (b) present the front and longitudinal views, respectively. A single elliptical mirror has a first focal point F 00 . A multi-part mirror is composed of multiple elliptical mirrors with multiple first focal points, F jk . For each elliptical mirror of a multi-part mirror, the first focal point, F jk , lies on the surface of the multi-part mirror. The elliptical mirrors are positioned on the surface of the multi-part mirror. Adjacent elliptical mirrors are arranged with a certain distance between them. The origin, O jk , which is the second focal point of all the elliptical mirrors, is aligned with the top of the multi-part mirror surface. In the coordinate system, the origin, O, denotes the second focal point of all the elliptical mirrors, and the central axis of a single elliptical or multi-part mirror passes through this point. The coordinate system is defined as R-Θ-Z, where R axis is the radial direction, Θ axis is the direction of rotation around the center axis, and Z axis is the center axis direction. Coordinates in the R-Θ-Z coordinate system are denoted as (r, θ, z). For an elliptical mirror with a first focal point F 00 and origin O 00 , the R 00 , Θ 00 , and Z 00 axes in the R 00 -Θ 00 -Z 00 coordinate system coincide with the origin, O, and the R, Θ, and Z axes in the R-Θ-Z coordinate system. Coordinates in the R 00 -Θ 00 -Z 00 coordinate system are denoted as (r F00 , θ F00 , z F00 ).
For an elliptical mirror having a first focal point F j0 , the R j0 -Θ j0 -Z j0 coordinate system is the coordinate system obtained by rotating the R 00 -Θ 00 -Z 00 coordinate system at an angle of ξ F j about the origin, O 00 , on the R 00 -Z 00 plane. The index, j, denotes the number of elliptical mirrors rotated accordingly, where j is a natural number including 0. In this rotation, g denotes the edge-to-edge distance on the mirror surface from the first focal point, F 00 , to first the focal point of an adjacent elliptical mirror, F 10 , and ξ F denotes the corresponding angle. Angle ξ F is found using trigonometry based on the geometry of angle ξ F , focal length f , radius r E , and distance g, which are shown in Fig. 2(b). For this rotation, the number of elliptical mirrors is denoted as n R .
For an elliptical mirror with a first focal point F jk , the R jk -Θ jk -Z jk coordinate system is defined as a coordinate system rotated at an angle θ F jk about the Z 00 axis centered in the R j0 -Θ j0 -Z j0 coordinate system. Index k denotes the number of elliptical mirrors rotated accordingly, where k is a natural number including 0. In this rotation, θ F j denotes the angle from the first focal point, F j0 , to the first focal point of an adjacent elliptical mirror, F j1 . Angle θ F j is found using trigonometry based on the geometry of angle θ F j , radius r E , and radius r F j0 , which are presented in Fig. 2(a). For this rotation, the number of elliptical mirrors is denoted as n Θ j . The number, n Θ j , is a natural number rounded off to the nearest whole number. Coordinates in the R jk -Θ jk -Z jk coordinate system are denoted as (r F jk , θ F jk , z F jk ).
The total number of elliptical mirrors is denoted as n M . Note that when j and k are 0, it is a single elliptical mirror. A multi-part mirror disperses the first focal points over its surface more widely than a single elliptical mirror. Using a multi-part mirror, it is possible to concentrate diffused light from a wider area than using a single elliptical mirror. In the designs of the single elliptical and multi-part mirrors, the number of mirrors n M , angle ξ F j , number n Θ j , angle θ F j , coordinate r F jk , and coordinate z F jk are determined using the equations below. Using Eq. (1), coordinates (r F00 , θ F00 , z F00 ) can be expressed as Using Eqs. (1) and (3), the relationship between angle ξ F , focal length f , radius r E , and distance g can be expressed as Using Eq. (6), angle ξ F j can be expressed as Using Eqs. (5) and (7), coordinates (r F j0 , θ F j0 , z F j0 ) can be expressed as Using Eqs. (3) and (8), the relationship between angle θ F j , radius r E , and radius r F j0 can be expressed as Using Eq. (9), number n Θ j can be expressed as Using Eq. (10), angle θ F j can be expressed as Using Eqs. (10) and (11), angle θ F jk can be expressed as Using Eqs. (8) and (12), Using Eq. (10), the number of mirrors, n M , can be expressed as To evaluate the light-concentrating performance of a single elliptical mirror, we designed three types of it with different focal lengths f and radii r E . Table 2 lists the design values for these single elliptical mirrors. The design values of the single elliptical mirror with a focal length f of 199.929 mm and radius r E of 1 mm are based on the values provided by Matsumoto [19]. To evaluate the lightconcentrating performance of the multi-part mirrors, we designed two types with different focal lengths f . The design values of the multi-part mirrors are summarized in Tables  3 and 4. Table 3 and 4 list the data for multi-part mirrors with focal lengths f of 17.503 mm and 199.929 mm, respectively. The design values in Table 4 are based on the values obtained by Matsumoto [19]. In Tables 2-4, the design values are determined using Eqs. (1), (3), (4), (14), (7), (10), (11), and (13) for focal length f , radius r E , height h, number of mirrors n M , angle ξ F j , number n Θ j , angle θ F j , and coordinates r F jk and z F jk , respectively.

Analysis Method
In single elliptical and multi-part mirrors, the diffused light incident on the elliptical mirrors is reflected by the mirrored surface and concentrated on the collector surface at the second focal point. To evaluate the light-concentrating performance of the designed single elliptical and multi-part mirrors, we analyzed how much of the light incident on them was concentrated on their collector surface. Figure 3 shows the analysis method for these single elliptical and multi-part Table 2: Design values of length of major axis, length of minor axis, focal length, radius of elliptical mirror, height, number of elliptical mirrors in R-axis, number of all elliptical mirrors, angle from R 00 -Θ 00 -Z 00 to R 00 -Θ 00 -Z 00 , number of elliptical mirrors in Θ-axis, angle from F 00 to F 01 , and coordinate of F 00 in single elliptical mirrors.  Table 3: Design values of length of major axis, length of minor axis, focal length, radius of elliptical mirror, height, distance from one edge to another edge of mirror surface, number of elliptical mirrors in R-axis, number of all elliptical mirrors, number indicating elliptical mirror in R-axis, angle from R 00 -Θ 00 -Z 00 to R j0 -Θ j0 -Z j0 , number of elliptical mirrors in Θ-axis, angle from F j0 to F j1 , and coordinates of F jk in multi-part mirror when f is 17.503 mm.  mirrors. An elliptical mirror is three-dimensional in shape, which is obtained by rotating an ellipse around the Z jk axis on the R jk -Z jk plane of the R jk -Θ jk -Z jk coordinate system. Therefore, to achieve axisymmetry with respect to the Z jk axis, each elliptical mirror is analyzed in the R jk -Z jk plane in the R jk -Θ jk -Z jk coordinate system. We analyzed the designed single elliptical and multi-part mirrors in the R-Z plane in the R-Θ-Z coordinate system. The number of reflections of the light in the mirror surface is denoted as m, where m is a natural number including 0. When m is 0, it is assumed that the light enters the elliptical mirror. This occurs from a line perpendicular to the Z jk axis that passes through the first focal point, F jk , and is parallel to the R jk axis. The point of incidence R 0 is defined as the point where the parallel line intersects with the incident light. The coordinates of the point of incidence, R 0 , are (r R0 , -2 f ). Note that the absolute value of the coordinate, r R0 , is less than or equal to the radius, r E . The angle between the point of incidence of the light, R 0 , and the R jk axis is denoted as ϕ R0 . The reflection points on the mirror surface are denoted as R m−1 and R m . The coordinates of reflection point R m−1 are (r Rm−1 , z Rm−1 ). The angle between the normal of the ellipse and the R jk axis at the point of reflection, R m−1 , is denoted as ϕ Nm−1 . The angle between the light reflected at the point of reflection, R m−1 , and the R jk axis is denoted as ϕ Rm−1 . The coordinates of the reflection point R m are (r Rm , z Rm ). The angle between the normal of the ellipse and the R jk axis at the point of reflection, R m , is denoted as ϕ Nm . The angle between the light reflected at the point of reflection, R m , and the R jk axis is denoted as ϕ Rm .
The collector surface is a line on the R axis, perpendicular to the Z axis, and passing through the origin, O, which is the second focal point. The R-Z plane is a plane obtained by rotating the R 00 -Z 00 plane about the origin O 00 by an angle ξ F j . The point of concentration, C, is defined as the point of intersection of the light rays on the collector surface at coordinates (r C , 0).
In the analysis of the designed single elliptical and multipart mirrors, the relationship between the angle, ϕ R0 , of the coordinate, r R0 , of the light incident on the elliptical mirrors and the coordinate, r C , of the surface where the incoming light collects is found using the equations below. The equation describing the light passing through the point of reflection R m−1 in the R jk -Z jk plane can be expressed as follows, where a and b are variables.
Variable a can be expressed as a = tan ϕ Rm−1 (16) Variable b can be expressed as Using Eqs. (2) and (15), coordinate r Rm can be expressed as IIAE Journal, Vol.9, No.4, 2021   Using Eqs. (15) and (18), coordinate z Rm can be expressed as Using Eqs. (18) and (19), angle ϕ Nm can be expressed as Using Eq. (20), angle ϕ Rm can be expressed as , the coordinates (r, z) can be expressed as Using Eqs. (7), (22), and (23), the equation describing the light passing through the point of reflection, R m , in the R-Z plane can be expressed as Using Eq. (25), the coordinate, r C , can be expressed as

Experimental Method 4.1 Manufacturing Mirrors
Using the design values listed in Table 2, we manufactured actual single elliptical mirrors with a focal length f of 17.503 mm and a radius r E of 0.98 mm and with a focal length f of 17.504 mm and a radius r E of 32.697 mm. Fig. 4 shows images of the single elliptical mirror. Fig. 4(a) shows a single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm, and Fig. 4(b) shows a single elliptical mirror with a focal length f of 17.504 mm and a radius r E of 32.697 mm. For the single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm, the substrate is black polycarbonate. Aluminum was vapor-deposited on   the mirror surface, after which silicon dioxide was vapordeposited as a protective film. The arithmetic mean roughness of the mirror surface is 1.17 µm. For the single elliptical mirror with a focal length f of 17.504 mm and a radius r E of 32.697 mm, the substrate is heat-resistant glass. Aluminum was vapor-deposited on the mirror surface, after which silicon dioxide was vapor-deposited as a protective film. The reflectivity of the mirror surface is more than 88% at wavelengths from 400 nm to 700 nm. The parts other than the mirror surface of this single elliptical mirror were covered with a light-shielding sheet.
Using the design values listed in Table 3, we manufactured an actual multi-part mirror with a focal length f of 17.503 mm, whose images are displayed in Fig. 5. The multi-part mirror is composed of multiple elliptical mirrors and a base. The elliptical mirrors of the multi-part mirror are identical to the manufactured single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm. The base of the multi-part mirror is made of an aluminum alloy. The surface of the base was anodized in matte black to prevent light from reflecting off it and subsequently covered with a light-shielding sheet. Figure 6 shows the structure of the experimental set-up. Figures 6(a)   the measurements. An LED light source and a diffuser are installed inside the dome light. The diffuser is in the shape of a half celestial sphere created by rotating a semicircle on the R-Z plane in the R-Θ-Z coordinate system around the Z axis. The equation of the semicircle can be expressed as

Experimental Set-up
The diffuser can generate diffused light inside the dome light. The measured object and the light receiver are positioned with the Z axis as the central axis. Depending on what is to be measured, the measured object and the light receiver are fixed at positions along the Z axis. Table 5 lists the specifications of the measured objects in the experimental set-up. Figure 6 shows the positions of the light-shielding sheet, measured object, and light receiver in the measured object in the experimental set-up. The light receivers are installed at the focal point, F, and the origin, O, and oriented in the positive and negative direction of the Z axis, respectively. The light-shielding sheet is installed at the focal point, F, for the light-shielded single elliptical mirror and at point S for the light-shielded multi-part mirror. For the light-shielded single elliptical and multi-part mirrors and the single elliptical and multi-part mirrors, the first focal point is set at the focal point, F, and the second focal point is set at the origin, O. Figure 7 shows the images of the experimental set-up, with Figs. 7(a)-(c) displaying the plan, side, and front views, respectively. In Fig. 7, the measured object is the single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm. The dome light is connected to a power source. The light receiver is connected to an illuminance meter. The dome light is model HPD2-400SW-CR14, and the power source is model PD2-5024, both manufactured by CCS Inc. HPD2-400SW-CR14 can generate diffused light by illuminating the diffuser inside the dome light from the white LED installed inside the dome light. PD2-5024 allows a dimmer control of the dome light. A total of 16 levels of coarse dimming and 16 levels of fine dimming can be combined to produce 256 levels of dimming. In the experimental set-up, we used coarse dimming to adjust the dimming level and fixed the fine dimming to one level. The light receiver and illuminance meter are model T-10MA from Konica Minolta Inc. T-10MA can measure the illuminance at the light receiver and display the measured illuminance on the illuminance meter. The light receiver has a radius of 7 mm, and the light-receiving direction is hemispherical. The range of illuminance that can be measured is from 0.01 lx to 299,900 lx. Figure 8 shows an image of the diffused light on the inside of the dome light. The dimming level of PD2-5024 was set as 2. The image was captured with Kodak SP360. SP360 can shoot at a Θ-axis direction of 360 • and a Z-axis direction of 214 • . The lens of SP360 was placed at the focal point, F, and oriented in the positive Z-axis direction. The figure shows the diffused light generated from the diffuser at 360 • in the Θ direction and 180 • in the Z direction, except at the light receiver.

Measurement
The illuminance of the dome light, light-shielded measured object, measured object, and measured object only and the illuminance ratio indicating the light-concentrating efficiency are denoted as i D , i S , i M , i T , and e, respectively. Illuminance i T is the value obtained by subtracting illuminance i S from illuminance i M . The illuminance ratio, e, is the ratio of illuminance i T to illuminance i D . Illuminance i T can be expressed as Using Eq. (28), the illuminance ratio, e, can be expressed as We experimentally determined the relationship between the dimming level and illuminance i D of the dome light. We measured illuminance i D with the dimming level of the dome light set as 2. We compared the measured illuminance i S and i M of the measured object to each other and to illuminance i T of the measured object only. Subsequently, we determined illuminance i T from illuminance i S and i M . We compared the illuminance ratio, e, in the measured object and then determined it from illuminance i T .

Performance Evaluation 5.1 Analysis Results
We analyzed the relationship between angle ϕ R0 and coordinate r C of the single elliptical and multi-part mirrors. We used Eqs. (15)-(26) for this by calculating the value of coordinate r C at 1 • increments of angle ϕ R0 . The range of angle ϕ R0 was set from -90 • to 90 • to cover a half celestial sphere of 180 • in the Z-axis direction. Coordinate r C ranged from -7 mm to 7 mm to cover the radius of the light receiver. We compared the single elliptical mirrors with a focal length f of 17.503 mm and a radius r E of 0.98 mm and with a focal length f of 199.929 mm and a radius r E of 1 mm. We also compared the multipart mirrors with focal lengths f of 17.503 mm and 199.929 mm. The ratio of coordinate r R0 to radius r E is denoted as r R0 /r E . For the single elliptical and multi-part mirrors, we compared the relationship between angle ϕ R0 and coordinate r C symmetrically about the Z axis at the following values of r R0 /r E : 0, 0.0979, 0.5, and 1. For the multi-part mirrors, we compared the relationship between angle ϕ R0 and coordinate r C symmetrically about the Z axis at the following values of r R0 /r E : -1, -0.5, -0.0979, 0, 0.0979, 0.5 and 1. In each case, we took measurements at angles ξ F0 , ξ F8 , and ξ F16 or alternatively at angles ξ F0 and ξ F11 . We plotted all the relationships between angle ϕ R0 and coordinate r C at r R0 /r E values of -0.0979 and 0.0979 for the single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm and the multi-part mirror with a focal length f of 17.503 mm at angle ξ F0 . In these figures when the ratio of r R0 /r E is greater than the range from 0 to 1 or smaller than the range from 0 to -1, less of the light incident from -90 • to 90 • , which is the range of angle ϕ R0 , is concentrated from -7 mm to 7 mm, which is the range of coordinate r C . When the focal length, f , and the radius, r E , of the single elliptical mirrors are increased, less light is concentrated at each Figure 9: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is 0 in single elliptical mirrors.  ratio r R0 /r E . In the case of the multi-part mirrors, less light is concentrated as the angle, ξ F j , and the focal length, f , increase. Figure 21 shows the appearance of the measured objects inside the experimental set-up during the experiment. Figures 21(a)-(c) present the dome light, light-shielded single elliptical mirrors with a focal length f of 17.503 mm and a radius r E of 0.98 mm and with a focal length f of 17.504 mm and a radius r E of 32.697 mm, and (c) light-shielded multi-part mirror with a focal Figure 12: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is 1 in single elliptical mirrors. Figure 13: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is 0 at ξ F0 , ξ F8 , and ξ F16 in multi-part mirror when f is 17.503 mm. Figure 14: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is -0.0979 and 0.0979 at ξ F0 , ξ F8 , and ξ F16 in multi-part mirror when f is 17.503 mm.

Experiment Results
length f of 17.503 mm, respectively. Figures 21(d)-(f) show the single elliptical mirror with a focal length f of 17.503 mm and radius r E of 0.98 mm, single elliptical mirror with a focal length f of 17.504 mm and radius r E of 32.697 mm, and multi-part mirror with a focal length f of 17.503 mm, respectively. This is the view from the negative Z-axis direction with the dimming level set as 2. The images of the dome light and the light-shielded single elliptical and multipart mirrors demonstrate the light-shielding sheet reflecting the diffused light. Therefore, it is necessary to determine illuminance i T , which is the difference between illuminance i M and i S . The images of the single elliptical and multipart mirrors show that the mirrors can brightly reflect the diffused light. Table 6 lists the dome light dimming level, illuminance i D , i S , i M , and i T values, and illuminance ratio e for the sin- Figure 15: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is -0.5 and 0.5 at ξ F0 , ξ F8 , and ξ F16 in multi-part mirror when f is 17.503 mm. Figure 16: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is -1 and 1 at ξ F0 , ξ F8 , and ξ F16 in multi-part mirror when f is 17.503 mm. Figure 17: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is 0 at ξ F0 and ξ F11 in multi-part mirrors. gle elliptical mirrors. Illuminance i M is greater than illuminance i S . Therefore, it is necessary to obtain illuminance i T , which is the difference between illuminance i M and illuminance i S . The illuminance ratio, e, for the single elliptical mirror with a focal length f of 199.929 mm and radius r E of 1 mm was estimated based on previous results of Matsumoto [19]. The relationship between the number of mirrors, n M , and the illuminance ratio, e, can be expressed as IIAE Journal, Vol.9, No.4, 2021 Figure 18: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is -0.0979 and 0.0979 at ξ F0 and ξ F11 in multi-part mirrors. Figure 19: Comparison of relationship of angle between incidentlight at R 0 and R jk and coordinate of C when r R0 /r E is -0.5 and 0.5 at ξ F0 and ξ F11 in multi-part mirrors.
Figure 20: Comparison of relationship of angle between incident light at R 0 and R jk and coordinate of C when r R0 /r E is -1 and 1 at ξ F0 and ξ F11 in multi-part mirrors.
Using Eq. (30), the illuminance ratio, e, when n M is 1 is estimated as 0.0000479. Because illuminance i D is different for the three single elliptical mirrors, we compared their values of the illuminance ratio, e. First, we compared the single elliptical mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm to that with a focal length f of 17.504 mm and a radius r E of 32.697 mm. As the radius, r E , increases, the illuminance ratio, e, also increases. When the radius, r E , is 33.3 times as large, the illuminance ratio, e, is 986 times larger. Next, we compared the single elliptical  mirror with a focal length f of 17.503 mm and a radius r E of 0.98 mm to that with a focal length f of 199.929 mm and a radius r E of 1 mm. As the focal length f decreases, the illuminance ratio, e, increases. When the focal length f is 0.0875 times as large, the illuminance ratio, e, is 6.07 times larger. This shows that to increase the illuminance ratio, e, it is more effective to change the radius, r E , than modify the focal length, f . Figure 22 compares the relationship between the number of mirrors, n M , and the illuminance ratio, e, for the multipart mirrors. We also compared the multi-part mirrors with focal lengths f of 17.503 mm and 199.929 mm. For the multi-part mirror with a focal length f of 199.929 mm, the relationship between the number of mirrors, n M , and the illuminance ratio, e, is the result obtained by Matsumoto [19]. As the number of mirrors, n M , increases, the illuminance ratio, e, increases. The rate of increase in the illuminance ratio, e, is greater for the multi-part mirror with a focal length f of 17.503 mm than that for the multi-part mirror with a focal length f of 199.929 mm. Furthermore, the increase in the illuminance ratio, e, is nonlinear. As the number of mirrors, n M , increases, the rate of increase in the illuminance ratio, e, decreases. The analysis results in Figs. [13][14][15][16][17][18][19][20] show that as the angle, ξ F j , and the focal length, f , increase, less light is concentrated. To solve this problem, it is necessary for the multi-part mirror to possess a three-dimensional collector instead of a flat collector surface to accommodate a large angle ξ F j .

Conclusion
In this study, we installed a collector surface on the second focal point of an elliptical mirror to increase its lightconcentrating efficiency. Subsequently we evaluated single elliptical and multi-part mirrors that capture diffused light from a half celestial sphere not incident on the first focal point and concentrate it on the collector surface as well as the second focal point. The results obtained in this study are as follows: