Introduction

A compound parabolic concentrator (CPC) is a non-imaging optical device that can improve the efficiency of light energy utilization and reduce light energy loss [1,2,3]. As an optical energy harvesting front-end, CPC is widely used in solar concentrating collectors, solar photovoltaic systems, and optical antennas in visible light communication (VLC) systems [4,5,6,7]. Winston proposed a CPC for circular absorbers in 1974, which was designed based on the principle of edge rays, and the concentrating energy collection characteristics were verified [8]. Chen et al. [9] proposed a Multi-section Compound Parabolic Concentrator (M-CPC) with optimized structure, and the experimental results showed that the effective time of gathering solar beam radiation was increased and the adaptability to weather changes was improved. Chang et al. [10] proposed a novel compound parabolic concentration photothermal and photoelectricity device CTPV, which adopts solar energy central heating technology to solve the agricultural soil heating problem, which provides a reference for the application of the compound parabolic concentrator in agricultural soil heating system. Wan Roshdan Wan Nur Adilah introduced an asymmetric compound parabolic concentrator (PV/T-ACPC) for building solar applications to enhance the performance of dual-pass photovoltaic/thermal air solar collectors. So that the solar incidence angle of the symmetric CPC must be within the full acceptance angle of the CPC, which improves the performance of the dual-pass PV/T-ACPC under different climate conditions [11]. Ma et al. [12] proposed a mirror field design method without shading to improve the geometric optical efficiency of CPC by using the half-receiving angle of CPC to limit its aspect ratio. Peng et al. proposed an advanced optical receiver antenna based on a composite paraboloidal concentrator (CPC). Taguchi method and ANOVA technique are used to explore and optimize the receiver lens structure parameters to improve the optical efficiency of VLC system and match the channel performance requirements [13]. Wang et al. proposed a design of a gradient lens that can extend the field of view and receive higher optical power. The compact structure and small size are suitable for indoor visible light wireless communication [14]. Wang et al. [15] proposed a new optical receiving system capable of reducing the received energy spot size by integrating a composite paraboloidal concentrator with a hemispherical lens.

CPC is a non-imaging concentrator designed according to the principle of edge ray, whose structure is made of two symmetric parabolas rotating around the symmetry axis. Its maximum field-of-view range for collecting light radiation is limited by the principle of edge ray and can only collect light energy from a narrow field of view, thus limiting the efficiency and performance of some applications where CPC is used as an optical antenna, such as the power generation efficiency of solar cells and the performance of LED wireless optical communication systems.

Fisheye lens is a large field-of-view angle of light radiation receiving device, which the largest role is to extend the field of view of light radiation receiving range [16,17,18,19]. In order to enhance the performance of a system that applies a conventional CPC as an optical antenna, this paper proposes a novel wide-field fisheye lens combined with a conventional CPC concentrator to increase the field of view of the light energy received by the concentrator. According to the authors’ research, there is no international literature on the use of this method to increase the field of view. The new wide field-of-view concentrator is suitable for solar concentrating collectors, photovoltaic panels and visible light communications.

Geometric structure model of a new wide-field concentrator

The novel wide-field concentrator proposed in this paper is composed of a negative meniscus fisheye lens and a CPC concentrator, and the whole structure is composed of three major parts, where I is a meniscus fisheye lens (FGHCDE), II is CPC (CBAD), and III is the photosensitive surface (AB) of the photoelectric device placed at the CPC concentrator, whose geometric structure is shown in Fig. 1. The main function of the negative meniscus fisheye lens is to expand the field angle range of the received optical radiation. The optical radiation incident port of the fisheye lens receives the optical radiation energy within the field Angle range and then refracts through the fisheye lens and enters the traditional CPC concentrator. The ray outlet of the fisheye lens is closely coordinated with that of the CPC concentrator. The CPC concentrates the optical radiation energy entering from the fisheye lens onto the photosensitive surface of the photoelectric device in the application system.

Fig. 1
figure 1

Schematic diagram of new wider-field concentrators

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Figure 1 shows a beam of chief rays impinges the first surface at an angle of \(\omega_{0}\) with the axis. \(r_{1}\) and \(r_{2}\) are the radius of curvature of the front and rear optical surfaces of the fisheye lens. \(\omega_{1}\), \(\omega_{2}\) are the image field angles of the front and rear optical planes. \(h_{1}\), \(h_{2}\) are the distance from the intersection of the main ray with the front and back optical planes to the optical axis. \(\alpha_{1}\), \(\beta_{1}\) are the incidence Angle and refraction Angle of the beam on the front optical plane.\(\alpha_{2}\), \(\beta_{2}\) are the incidence Angle and refraction Angle of the beam on the rear optical plane. \(\theta_{\max }\) is the maximum receiving Angle of the CPC concentrator. \(h_{2}\), \(h_{3}\) are the radii of the incoming and outgoing ports of CPC. \(l\) is the length of CPC.

Figure 1 shows a ray with an initial field angle of view \(\omega_{0}\) is refracted twice through the front and rear optical surfaces of the fisheye lens and then enters the CPC concentrator. If the field Angle \(\omega_{2}\) of the image side of the rear optical plane is less than or equal to \(\theta_{\max }\), the light can reach the lower opening of CPC directly or through reflection and be received and utilized by the receiver. Therefore, the maximum received field-of-view (FOV) angle \(\theta_{\max }\) is an important parameter of CPC, which directly affects the light concentration performance of CPC.

The maximum field-of-view (FOV) Angle of the new concentrator was set as \(\omega_{0}\) and the maximum received FOV Angle was set as \(\theta_{\max }\), and the concentrator model of the new structure was constructed by applying the photosensitive surface size \(h_{3}\) (CPC’s ejector radius) of the system, and the model size calculation formula was derived.

The focal length of the CPC parabola is:

$$ f = h_{3} (1 + \sin \theta_{\max } ) $$
(1)

As shown in Fig. 1, according to trigonometric functions, CPC \(l_{1}\), \(l_{2}\) and CPC total height \(l\) are deduced as follows:

$$ \left\{ {\begin{array}{*{20}l} {l_{1} = \frac{{h_{2} \cos \theta_{\max } }}{{\sin \theta_{\max } }}} \hfill \\ {l_{2} = \frac{{h_{3} \cos \theta_{\max } }}{{\sin \theta_{\max } }}} \hfill \\ {l = l_{1} + l_{2} = \frac{{h_{2} \cos \theta_{\max } }}{{\sin \theta_{\max } }} + \frac{{h_{3} \cos \theta_{\max } }}{{\sin \theta_{\max } }} = \frac{{h_{2} \cos \theta_{\max } + h_{3} \cos \theta_{\max } }}{{\sin \theta_{\max } }} = \frac{{h_{3} \frac{1}{{\sin \theta_{\max } }}\cos \theta_{\max } + h_{3} \cos \theta_{\max } }}{{\sin \theta_{\max } }} = \frac{{h_{3} (1 + \sin \theta_{\max } )\cos \theta_{\max } }}{{\sin^{2} \theta_{\max } }} = \frac{{f\cos \theta_{\max } }}{{\sin^{2} \theta_{\max } }}} \hfill \\ \end{array} } \right. $$
(2)

The CPC whose length satisfies Eq. (2) is called standard CPC. In practical applications, CPC is often truncated, that is, its left part is truncated to reduce its height. The ratio \(k_{0}\) of CPC height \(l^{\prime}\) to \(l\) after truncation is the interception ratio. Proper truncation (\(k_{0}\) is generally 0.5–0.8) has little effect on the performance of CPC. Although the concentration ratio will be slightly reduced, the concentration Angle can be increased to receive more light, and at the same time, materials can be saved and the volume and processing difficulty can be reduced. Therefore, practical CPC is generally truncated.

As shown in Fig. 1, the equation of CPC parabola AD in the rectangular coordinate system of XOY plane is as follows:

$$ \left( {2\sqrt {2h_{3} f - f^{2} } x + 2(f - h_{3} )y} \right)^{2} + 8h_{3} fx - 8h_{3} (f + h_{3} )\sqrt {2h_{3} f - f^{2} } y = 4fh_{3}^{2} (f + 2h_{3} ) $$
(3)

The focal length \(f\) and outlet radius \(h_{3}\) are known to be determined parameters. When the length of the CPC concentrator is \(l^{\prime} = k_{0} l\), \(h_{2}\) is the radius of the incident port. Through derivation of Eq. (3) and optimization, the following formula can be obtained:

$$ h_{2} = \frac{{k_{0} l(h_{3} - f)\sqrt {2fh_{3} - f^{2} } - h_{3} f^{2} + \sqrt {fh_{3}^{3} (4k_{0} l\sqrt {2h_{3} f - f^{2} } ) + 4h_{3} f} }}{{f(2h_{3} - f)}} $$
(4)

Let θ be the Angle between the normal of the primary ray at the intersection of the second optical plane and the optical axis:

$$ \sin \theta = \frac{{h_{2} }}{{r_{2} }} \equiv k_{1} $$
(5)

According to the design requirements of the initial FOV and compression ratio, the system structural parameters need to meet the conditions. Under the conditions of the initial FOV Angle \(\omega_{0}\), FOV compression ratio \(\omega_{0} /\omega_{2}\), \(k_{1}\) defined by Eq. (5) (\(k_{1}\) = 0.75–0.95 [20] for general lens) and the refractive index \(n\) of the material, it can be deduced according to the geometric relation and trigonometric function of light propagation.

$$ \left\{ {\begin{array}{*{20}l} {\sin \beta_{2} = \sin \left( {\theta - \omega_{2} } \right)} \hfill \\ {\sin \alpha_{2} = \sin \beta_{2} /n} \hfill \\ {\omega_{1} = \omega_{2} + \beta_{2} - \alpha_{2} } \hfill \\ {\beta_{1} - \alpha_{1} = \omega_{0} - \omega_{1} } \hfill \\ {\sin \alpha_{1} = n\sin \beta_{1} } \hfill \\ \end{array} } \right. $$
(6)

According to Eq. (6), it can be obtained.

$$ \left\{ {\begin{array}{*{20}l} {\beta_{2} = \arcsin \left( {\theta - \omega_{2} } \right)} \hfill \\ {\alpha_{2} = \arcsin \left( {\sin \beta_{2} /n} \right)} \hfill \\ {\omega_{1} = \omega_{2} + \beta_{2} - \alpha_{2} = \theta_{\max } + \beta_{2} - \alpha_{2} } \hfill \\ {\beta_{1} = \arctan \left( {\sin \left( {\omega_{0} - \omega_{1} } \right)/\left( {n - \cos \left( {\omega_{0} - \omega_{1} } \right)} \right)} \right)} \hfill \\ {\alpha_{1} = \beta_{1} + \omega_{0} - \omega_{1} } \hfill \\ \end{array} } \right. $$
(7)

According to Eq. (6), the ratio of the curvature radius of the front and rear optical surfaces of the negative crescent lens can be obtained.

$$ \frac{{r_{2} }}{{r_{1} }} = \frac{{\sin \omega_{1} - \sin \beta_{2} }}{{\sin \alpha_{2} + \sin \omega_{1} }} $$
(8)

According to Eqs. (5), (6) and (7), the curvature radius of the front and rear optical surfaces of the negative crescent lens can be deduced.

$$ r_{2} = h_{2} /k_{1} $$
(9)
$$ r1 = r_{2} (\sin \alpha_{2} + \sin \omega_{1} )/(\sin \omega_{1} - \sin \beta_{2} ) $$
(10)

According to Fig. 1, the transverse dimension of the first negative crescent lens of the fisheye lens is

$$ h_{1} = r_{1} \sin (\omega_{0} - \alpha_{1} ) $$
(11)

According to Eqs. (1–11), the concentrator’s concentrator model can be determined according to the concentrator’s receiving radius \(h_{3}\), maximum receiving Angle \(\theta_{\max }\), field Angle \(\omega_{0}\) parameter, \(k_{1}\), truncation coefficient \(k_{0}\), material refractive index \(n\) and other parameters.

Modeling of a novel wide-field concentrator

The coordinate system is established with the center of the second optical surface of the fisheye lens as the center point, as shown in Fig. 2.

Fig. 2
figure 2

Coordinate system of new wider-field concentrators

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Set the entrance angle \(\omega_{0}\) of the new wide-field concentrator as 160°, the photosensitive surface size \(h_{3}\) (the radius of the outlet) of the application system as 5 mm, the maximum receiving Angle \(\theta_{\max }\) as 30°, the value of \(k_{1}\) as 0.95, the thickness of the fisheye lens \(d_{1}\) as 3 cm, and the length truncation coefficient \(k_{0}\) of the CPC concentrator as 0.8.

The structural parameters of the new wide-field concentrator are calculated according to Eqs. (1 –10), as shown in Table 1.

Table 1 Design parameters of new wide field concentrator with a field of view of 80°
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In this paper, the fisheye lens model is constructed by inserting different basic geometries for Boolean operations such as intersection, difference and concatenation. According to the parameters in Table 1, the center point and radius of the outer and inner circles of the fisheye are calculated, and the inner and outer circles as well as for the rectangle are constructed. The fisheye lens model is obtained by making the difference set of the outer circle with the rectangular body and then with the inner circle. On the basis of the fisheye lens model, the CPC concentrator model is constructed. Through the above derivation and calculation of the parameters of the new wide-field concentrator, the focal length of the non-canonical CPC is obtained as f = 7.5 mm, the maximum reception angle of the CPC is 30°, and the length is 20.7846 mm. The parameters to be set in TracePro are as follows: the Front length is 20.7846, the Back length is 0, the Lateral focal shift is 5 mm, Thickness is 1 mm, Axis tilt is 30, and Focal length is 7.5 mm. The contour, 3D frame, and coloring diagrams of the new concentrator model are shown in Fig. 3.

Fig. 3
figure 3

Model of new wider-field concentrators

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Performance analysis of a new wide-field concentrator

Analysis of concentrating performance

The case discussed in this paper is the addition of a fisheye lens to a conventional CPC concentrator with a maximum receiving angle of 30°, making the field of view reach 80°. Horizontal boundary extremes are used to verify the concentrating performance, and the intersection of the incident light and the Z-axis is the center point, plus or minus 80° (the field of view is 160°) to establish a grid point light source, to trace the light, and to compare the field of view of the traditional CPC concentrator and the new wide field of view. The distribution of light sources is shown in Fig. 4.

Fig. 4
figure 4

Distribution of grid light source

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The ray tracing results of the traditional CPC concentrator and the new wide-field concentrator are shown in Figs. 5 and 6. From Figs. 5, 6, it can be seen that in the traditional CPC concentrator in the light incident angle exceeds the maximum receiving angle, the light enters the CPC, after multi-angle reflection into and out of the injection port; the new wide field-of-view concentrator in the light incident angle is less than or equal to 80°, after the fisheye lens refraction, and finally from the incident port.

Fig. 5
figure 5

Ray tracing diagram of CPC

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Fig. 6
figure 6

Ray tracing diagram of new wider-field concentrators

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Taking the fisheye lens of the new wide field-of-view concentrator as the optical front-end receiving surface, and the photosensitive surface at the outlet of the CPC as the optical back-end receiving surface, the light energy distribution diagrams obtained at an incident angle of 80°, for example, are shown in Figs. 7 and 8. From the figure, we can get that the number of incident light rays on the receiving surface of the front-end of the fisheye lens is 50, and the number of light rays received on the photosensitive surface is also 50, which indicates that the light rays enter into the photosensitive surface comprehensively at the incidence angle of 80°, and that the new wide-field concentrator enlarges the field-of-view angle of the concentrator.

Fig. 7
figure 7

Power distribution of optical front

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Fig. 8
figure 8

Power distribution of optical back-end

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Analysis of light loss

The intersection of the incident light and the Z-axis is used as the center point, and the grid point light source is rotated by 80° (the field of view is 160°) to simulate the sun’s east rise and west fall. The grid point light source changes from [− 80 ~ 80], 10° each time, and the energy of the grid point light source is 50 W/m2. The fisheye lens of the new wide-field concentrator is used as the receiving surface of the front-end of the optics, and the light-sensitive surface at the outlet of the CPC is the receiving surface of the back-end of the optics, and the ratio of the energy of the receiving surface of the light-sensitive surface to the receiving surface of the front-end of the optics is obtained by the irradiance analysis of Tracepro, and the receiving irradiance of the light-sensitive surface of the front-end is shown in Fig. 1, and the partially light-sensitive surface is shown in Fig. 3, which is a part of the receiving surface. Surface receiving irradiance is shown in Fig. 9, and the ratio is shown in Fig. 10.

Fig. 9
figure 9

Irradiance maps of photosensitive surfaces

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Fig. 10
figure 10

Distribution of light energy loss

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Case study-wireless receiver channel performance analysis

In indoor VLC system, without considering wall reflection, the direct line-of-sight (LOS) channel DC gain defined by Eq. (12).

$$ H_{{{\text{LOS}}}} (0) = \left\{ {\begin{array}{*{20}l} {\frac{{(m + 1)A_{d} }}{{2\pi D_{d} }}\cos^{m} (\phi )g(\theta )T_{s} (\theta )\cos (\theta )} \hfill & {(0 \le \theta \le \theta_{C} )} \hfill \\ 0 \hfill & {(\theta > \theta_{C} )} \hfill \\ \end{array} } \right. $$
(12)

where \(A_{d}\) is the effective receiving area of the photodetector, \(D_{d}\) is the distance from the light source to the detector, \(\phi\) is the radiation Angle of the light source, \(\theta\) is the Angle of incidence of the ray, \(T_{s} (\theta )\) is the gain of the optical filter, \(\theta_{C}\) is the field-of-view Angle at the optical receiver. \(g(\theta )\) is the gain of the concentrator defined by Eq. (13).

$$ g(\theta ) = \left\{ {\begin{array}{*{20}l} {\frac{{n^{2} }}{{\sin \theta_{C} }}} \hfill & {(0 \le \theta \le \theta_{C} )} \hfill \\ 0 \hfill & {(\theta > \theta_{C} )} \hfill \\ \end{array} } \right. $$
(13)

where \(n\) is the refractive index of the optical receiver. Therefore, the optical received power of the system in the direct line-of-sight is defined by Eq. (14).

$$ P_{{r({\text{LOS}})}} = \sum\limits_{i = 1}^{N} {P_{{{\text{ti}}}} H_{{({\text{LOS}}i)}} (0)} $$
(14)

where \(i\) indicates the ith LED, \(N\) is the total number of LED lights, \(P_{{{\text{ti}}}}\) is the transmitted power of the ith LED.

According to the study of Toshihiko and Masao [20], for indirect line-of-sight links (NLOS), the optical power occupied by LOS propagation is higher than 95%, the primary wall reflection accounts for about 3.37% of the total, and the secondary wall reflection light power only accounts for 1.27%. In this paper, the case of one or more wall reflections is negligible.

According to Eq. (14), a new wide-field concentrator is used as the optical receiving front-end to establish the indoor VLC system model, and Matlab simulation software is used to conduct channel modeling and simulation analysis for the VLC system. The indoor visible light communication model is shown in Fig. 11, \(5m \times 5m \times 3m\) empty room, four groups of white LED arrays are used to distribute evenly and symmetrically in the room. Each group of arrays has an LED unit. The optical power of a single LED lamp is 20 mW, the transmitting half-power Angle is 70°, and the central light intensity is 0.73 cd.

Fig. 11
figure 11

Indoor visible light communication model

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The optical received power distribution under the four conditions of no optical antenna, using a CPC with a field of view of 30°, using a CPC with a field of view of 60° and using new wider-field concentrators was obtained by calculation, as shown in Fig. 12. The maximum, minimum and average values of the received power distribution are shown in Table 2.

Fig. 12
figure 12

Power distribution in the room

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Table 2 Receiving power comparison for different optical antennas
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It is seen from Fig. 12 and Table 2 that the maximum received power of the installed CPC is increased by 8.2333 dBm compared with that of the direct received CPC, but there is a large blind area in the received CPC. In the case of the installed CPC, the maximum received power is increased by 4.7221 dBm compared with the case of the direct received power. Although the maximum received power is decreased by 3.5112 dBm compared with the case of the installed CPC, the reception blind area is reduced. When the new wide-field concentrator is used as the optical receiving antenna, the maximum received power is 6.9933 dBm, the minimum is 3.8601 dBm, and the average is 6.1690 dBm. Compared with the direct receiving situation, the maximum received power is increased by 9.5424 dBm, and the installed CPC is increased by 4.8203 dBm. The new wide field-of-view concentrator has a large field-of-view Angle, and the spot energy distribution is uniform, so the power distribution in the room is uniform, and the communication blind area is greatly reduced.

Conclusion

The new wide-field concentrator is simulated by TracePro simulation software and Matlab software to confirm that the design method of the new wide-field concentrator is effective.

  1. (1)

    The key parameters of the new wide-field concentrator are analyzed and derived to obtain the structural design and theoretical formula of the new wide-field concentrator.

  2. (2)

    On the basis of analyzing the structure and optical characteristics of the new concentrator, designing the new concentrator with a field-of-view angle of 80° as an example on the basis of a CPC of 30° incidence angle, the structure of the new concentrator is designed with each geometric dimension by calculating the theoretical formula. Optical simulation software TracePro was used to analyze the performance of the new concentrator in terms of light concentration and light loss.

  3. (3)

    Channel modeling of the indoor VLC system is carried out by Matlab software to obtain the received power distribution of different optical antenna systems. The simulation results show that the maximum power of the new wide-field-of-view concentrator is increased by 9.5424 dBm compared with the case of direct reception, and the power distribution uniformity in the room is good, which greatly reduces the blind zone of the communication, and can satisfy the high-speed and stable communication requirements of the indoor VLC system.