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Ideal shape of Fresnel lens for visible solar light concentration

Xinglong Ma, Rihui Jin, Shen Liang, and Hongfei Zheng

Open Access Open Access

Abstract

This paper presents theoretical research based on the optimal transmittance condition of a prism to find an ideal shape for Fresnel lenses to concentrate visible solar light. First, the ideal-shape equation was derived out through a simplified method that uses one refraction on the midline of a prism to replace the two refractions, respectively, on its upper and lower interfaces. It has been assumed that the Fresnel lens is thin enough to consider each prism as a point, then all the simplified points form a curve. The differential equation of this curve was built up, which has been solved and expressed by a parametric formula. The parametric formula is defined as the ideal-shape equation of Fresnel lens. Second, the optimal combination of the total refracted angle θ and refractive index n has been analyzed to determine the maximal transmittance. The quantitative analysis has indicated that only one group of (θ, n) can achieve the optimal Fresnel lens’ transmittance. Finally, the maximal geometrical concentration ratio Cg of ideal Fresnel lens has been discussed. When material is defined, there is a unique θ that makes the geometric concentration maximal for visible solar light. Generally, materials with low refractive index can be used to design a Fresnel lens with larger Cg.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Fresnel lenses can be used as solar concentrators to effectively converge lights onto a designated area for high-quality energy generation. It has several advantages such as lightweight, low material consumption, large permissive tracking error etc [1]. Chromatic aberration and transmittance declination from its center to two edges of the Fresnel lens are the two unavoidable problems those influence the concentration. Solar antireflection coating and doublet lenses are effective ways to reduce the chromatic aberration and increase transmittance. Guido presented an achromatic lens which makes of plastic and elastomer. It gets high geometric concentration by reducing a little on optical efficiency [2]. Fabian obtained super-high concentration by combining Polycarbonate (PC) and Polymethyl Methacrylate (PMMA). Concomitantly, the energy distribution on the focus is more uniform [3]. Optimization on the shape of Fresnel lens is another popular way to address the two problems because chromatic aberration and transmittance are strongly associated with the structure of Fresnel lens [4]. For example, Khalil designed a flat plate line-focus Fresnel lens. The experimental results illustrate that the middle part of the lens has high optical efficiency. However, optical efficiency drops rapidly when it gets close to the edge of the Fresnel lens because of the sharp increase in reflection loss. The average optical efficiency is barely 58% [5]. By contrast, Fresnel lenses with curve shape have better transmittance. O’ Neil early designed the curve Fresnel lens that was used for concentrating photovoltaic in space. The newly designed curve Fresnel lens has lighter weight and higher optical efficiency [6]. Naichia developed a mechanism of a curve-based, point-focused Fresnel lens concentrating system and used it to examine each spectral segment’s distribution patterns on the lens’ focal area. He pointed out that a Fresnel lens would have good thermal performance when it is designed with red light (average wavelength of 675 nm) [7,8]. Hongfei presented a method to design a Fresnel lens based on the cylindrical surface. Three circles are used to limit the thickness of the Fresnel lens, by which the maximum thickness would not lead to too much energy adsorption and the minimum thickness is not easily broken [9]. Because of the differences in designing and processing, it is difficult to compare the advantages and disadvantages among the convex-shaped Fresnel lenses. But for the locally optimal structure on a Fresnel lens, there are some references providing positive information. Born firstly demonstrated the condition of minimum chromatic dispersion when a light going through a prism, which was called “minimum deviation” [10]. The minimum deviation illustrates that only the incident angle on the upper surface of a prism equals the emergent angle on its lower surface will the chromatic dispersion be minimum. Later, Leutz designed a convex-shaped non-imaging Fresnel lens following the edge ray principle [11]. The “minimum deviation” was used to make its transmittance “optimal”. However, it is not very persuasive because the minimum deviation was proposed without mentioning energy transfer. Xinglong furtherly analyzed the energy transfer process of the prism, then derived out the optimal transmittance condition of prism mathematically [4]. Coincidently, the optimal transmittance condition of prism is the same with the condition of minimum deviation. This paper will work on the ideal shape of Fresnel lens based on the optimal transmittance condition. A detailed analysis on transmittance of interfaces of prism will be expatiated.

2. Ideal-shape equation of Fresnel lens

The optimal transmittance condition of prism leads to the minimal reflection loss on surfaces when lights go through the prism, which also points out the relationship between incident angle and emergent angle. As shown in Fig. 1(a), incident angle is the angle between incident light and upper surface normal, denoted as i1. Emergent angle is the angle between refracted light and lower surface normal, denoted as i2’. When the incident angle equals to the emergent angle, namely i1=i2’, the transmittance of two surfaces will be equal and the total transmittance reaches a maximal value.

 figure: Fig. 1.

Fig. 1. (a) The optimum transmittance condition of prism; (b) Profile of the ideal shape of Fresnel lens. Drawing the normal line pass through the point A, it can be found that the incident light and emergent light are distributing in the same side of the normal line, which means the simplified condition conforms to the law of reflection of light. In this case, both the incident angle and the emergent angle are equivalent to θ/2.

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Transmittance calculation is based on Fresnel Eq. (1) [4], which is given by separating natural lights into two orthogonal directions: s-polarized vibration and p-polarized vibration, as shown in Fig. 2. The reflectance of s-polarized vibration on the interface of two media is greater than p-polarized vibration and it keeps monotonously increasing. Reference [4] also proved that the total reflectance is monotonously changing with the increase of incident angle, which means it has the same tendency with s-polarized vibration.

$$\left\{ \begin{array}{l} {R_p} = {\left( {\frac{{{n_{21}}\cos {i_1} - \cos (\arcsin (\sin {i_1}/{n_{21}}))}}{{{n_{21}}\cos {i_1} + \cos (\arcsin (\sin {i_1}/{n_{21}}))}}} \right)^2}\\ {R_s} = {\left( {\frac{{\cos {i_1} - {n_{21}}\cos (\arcsin (\sin {i_1}/{n_{21}}))}}{{\cos {i_1} + {n_{21}}\cos (\arcsin (\sin {i_1}/{n_{21}}))}}} \right)^2} \end{array} \right.$$

It indicates that the transmittance difference between only considering s-polarized vibration and both considering s-polarized vibration and p-polarized vibration is no greater than 0.5%, when refractive index ranges in 1∼2 [4,12]. Thus, to simplify the calculation, s-polarized vibration-based transmittance will be used in the following analysis. Equation (2) is given to calculate the transmittance of s-polarized vibration τs [12].

$$\begin{array}{l} {\tau _\textrm{s}} = \left( {\frac{{n\cos ({\arcsin ({\sin {i_1}/n} )} )}}{{\cos {i_1}}}} \right.\\ {\left. { \times {{\left( {\frac{{2\cos {i_1}}}{{\cos {i_1} + n\cos ({\arcsin ({\sin {i_1}/n} )} )}}} \right)}^2}} \right)^\textrm{2}} \end{array}$$

 figure: Fig. 2.

Fig. 2. (a) Graph of s-polarized and p-polarized vibrations

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Then, referring to Fig. 1(a), the incident angle equals to the emergent angle, that is i1=i2’. The first refractive angle in the upper interface also equals to the incident angle in the lower interface, that is i2=i1’. Then the total refracted angle of emergent light θ, which is the angle of a light went through the prism, and its original direction, can be expressed as Eq. (3) upon the geometrical relationship.

$$\theta = 2({i_1} - {i_2})$$

According to the Snell’s Law, the refractive index n is shown as Eq. (4).

$$n = \sin i{}_1/\sin {i_2}$$

Combining Eqs. (3) and (4), the relationship of i1-θ can be derived out as follow:

$${i_1} = \arctan \left( {\frac{{\sin (\theta /2)}}{{\cos (\theta /2) - 1/n}}} \right)$$

The included angle γ between upper and lower surfaces of the prism is:

$$\gamma = 2\arcsin ({\sin {i_1}/n} )$$

If both of the incident light path and emergent light path are elongated simultaneously, two elongated lines will intercross at the point A as shown in Fig. 1(a), which is on the midline BC of the isosceles prism. Then the two refractions are simplified respectively into one refraction at the point A. Referring to Fig. 1(b), it is assumed that point A is on the profile curve of a Fresnel lens, the midline of the isosceles triangle would be the tangent line of the profile curve at point A. For the coordinate of point A, one equation can be presented as follow:

$$\tan \theta = {x \mathord{\left/ {\vphantom {x y}} \right.} y}$$

Setting the equation of the profile curve as y = f(x), then the slope of the curve at point A is:

$$\frac{{dy}}{{dx}} = y^{\prime} ={-} \tan {\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2}$$

There are three variables (θ, x, y) in Eqs. (7) and (8), the intermediate variable θ could be eliminated to make (x, y) solvable. Equation (9) is giving the trigonometric relationship between θ and θ/2.

$$\tan \theta = \frac{{2\tan {\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2}}}{{1 - {{\tan }^2}{\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2}}}$$

Considering θ∈[0, π), that means tan(θ/2) ≥0, Eq. (9) can be converted as follow:

$$\tan {\theta \mathord{\left/ {\vphantom {\theta 2}} \right.} 2} = \frac{{\sqrt {1 + {{\tan }^2}\theta } - 1}}{{\tan \theta }}$$

Combining the Eqs. (7), (8) and (10) into one, a differential Eq. (11) could be achieved.

$$\frac{x}{y}y^{\prime} = 1 - \sqrt {1 + {{\left( {\frac{x}{y}} \right)}^2}}$$

By solving the differential Eq. (11), a parametric equation could be gained depicted in Eq. (12). The total refracted angle θ is a parametric variable and C is the integral constant number. Therefore, the Eq. (12) is for determining the ideal-shape of Fresnel lens. When C is changed, the curve equation will illustrate a series of confocal Fresnel lens. The intercept in y-axis equals to C/2, which can represent the focal length.

$$\left\{ {\begin{array}{c} {x = C\sqrt {\frac{{1 - \cos \theta }}{{1 + \cos \theta }}} }\\ {y = C\sqrt {\frac{{1 - \cos \theta }}{{1 + \cos \theta }}} \cdot \frac{1}{{\tan \theta }}} \end{array}} \right.\left( \begin{array}{l} \theta \in [0,\pi ),\\ C \in \textrm{Positive number} \end{array} \right)$$

Figure 3 shows a group of curves (Some particular solutions from Eq. (12)). It can be found that the theoretical ideal curves can run through the whole four quadrants. However, the present Fresnel lenses can only perform in the first and second quadrants, and within the third and fourth quadrants, the transmittance is quite low. The detailed reasons will be explained in the section 3.

 figure: Fig. 3.

Fig. 3. Curves of particular solutions of Eq. (12).

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3. Analysis and discussion on ideal Fresnel lens

The equation for ideal-shape of Fresnel lens shows a reference shape to design Fresnel lens. However, even the sequence of prisms in a Fresnel lens underlines the ideal-shape equation and the structure is designed under the optimal transmittance condition, the transmittance would not be optimal unless the proper material is selected. In other words, the refractive index of material also has a significant influence on the transmittance.

As shown in Fig. 4(a), the included angle γ presents the angle between the upper and lower interfaces of the prism. n is assumed as a continuous variable, if a higher refractive index n is selected, the light would go through the prism along B1C1. If a lower refractive index n is selected, the light would go through the prism along B2C2. Both prisms can refract incident light onto the focus. The question is which prism has a bigger transmittance. Because of that the thickness of the Fresnel lens can be thin enough and the materials are highly transparent so that the absorption can be ignored. Thus, reflection loss will be the only factor impact the transmittance.

 figure: Fig. 4.

Fig. 4. (a) Prisms on one point of the ideal curve; (b) Variations of transmittance with θ and n.

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Considering both of the Eq. (2) and Eq. (5), the variations of transmittance can be calculated out as presented in Fig. 4(b). The transmittance will be zero when n=∞ or θ=90°. If the refractive index increases to infinity, the Fresnel lens would turn into a sun shield. Another significant change is that 3D transmittance surface has a ridge, on which the transmittance reaches its maximal. It illustrates that at different part of the Fresnel lens, there is a unique refractive index n to make its transmittance locally optimal. The most appropriate values of refracted angle θ and refractive index n should be chosen from the ridge line to achieve the optimal transmittance everywhere on the Fresnel lens. For instance, for the location of θ=30°, the appropriate material should be chosen with n = 1.3 and the corresponding incident angle i1 and included angle γ are 54° and 77°, respectively, to obtain the locally optimal transmittance 0.89 . While if θ=50°, the parameters should be n = 1.5, i1=62° and γ=72°, and the locally optimal transmittance is 0.72.

In addition, the transmittance is much lower when θ>90°, so Fig. 4(b) doesn’t present the results when θ>90°.

The transmittance of the Fresnel lens would be optimal only when each of the prism has the optimal transmittance. In order to presents the explicit parameters of the whole ideal Fresnel lens, Fig. 5(a) is generated by combining the Eq. (5) and Eq. (6) with the ridge line data in Fig. 4(b). It can be seen that the incident angle is not starting from 0, which is beyond our normal thoughts. Throughout the total refracted angle from 0° to 90°, the incident angle is changing linearly from 60° to 69°, the increment is less than 10°. The included angle also linearly changes due to the increase of the refractive index. The red curve indicates the maximum transmittance of each corresponding total refracted angle. Figure 5(b) is an explicit structure to clearly show the structure of the ideal Fresnel lens.

 figure: Fig. 5.

Fig. 5. (a) Parameters of ideal Fresnel lens; (b) Structure of ideal Fresnel lens.

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4. Geometric concentration

Geometric concentration Cg is the ratio of the aperture width w to the focal width d (linear-focusing), which is:

$${C_\textrm{g}} = \frac{{\textrm{Aperture}\; \textrm{width}}}{{\textrm{Focal}\; \textrm{width}}}$$

The extreme Cg of the solar concentrator is 1/sinδ for linear-focusing solar concentrator, in which δ is the parallactic angle from the earth to the sun. It is (1/sinδ)2 for the point-focusing solar concentrator. As for the reflection type solar concentrator, this extremity can be reached unless the design and manufacture are perfectly processed. However, because of the existence of chromatic dispersion, the focal width of the Fresnel lens would become wider, the Cg of the Fresnel lens would never reach the extremity. The prism at the edge has the largest chromatic dispersion due to its largest refraction. The light with a shorter wavelength will have a larger refractive index and bend more than the light with longer wavelength when it goes through a prism. θe,sw and θe,lw are used to indicate the total refracted angles of visible light with shortest and longest wavelength, respectively, which can pass through the material of Fresnel lens. As shown in Fig. 6, for the first Fresnel lens, two lights go through s along through the point A (xe, ye) and reach to the x-axis. Considering the δ is a small angle, by referring to formula (3), θe,sw and θe,lw can be expressed as follow:

$$\left\{ \begin{array}{l} {\theta_{\textrm{e,lw}}} = 2({i_{1,\textrm{e,lw}}} - \arcsin (\sin{i_{1,\textrm{e,lw}}}/{n_{\textrm{lw}}})) - \delta \\ {\theta_{\textrm{e,sw}}} = 2({i_{1,\textrm{e,sw}}} - \arcsin (\sin{i_{1,\textrm{e,sw}}}/{n_{\textrm{sw}}})) + \delta \end{array} \right.$$

 figure: Fig. 6.

Fig. 6. Refractions at the edge of ideal Fresnel lenses.

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In which i1,e,lw is the light incident angle with longest wavelength at the edge of prism and i1,e,sw is incident angle for the light with shortest wavelength.

According to the geometrical relationship in Fig. 6, both of the incident angles can be calculated as follow:

$$\left\{ \begin{array}{l} {i_{1,\textrm{e,lw}}} = {i_{1,\textrm{e}}} + \delta \\ {i_{1,\textrm{e,sw}}} = {i_{1,\textrm{e}}} - \delta \end{array} \right.$$

Then the intercepts of two lights on the x-axis is follow:

$$\left\{ {\begin{array}{c} {{x_{\textrm{e,lw}}}\textrm{ = }{y_\textrm{e}} \cdot \tan{\theta_{\textrm{e,lw}}}}\\ {{x_{\textrm{e,sw}}}\textrm{ = }{y_\textrm{e}} \cdot \tan{\theta_{\textrm{e,sw}}}} \end{array}} \right.$$

Because of the Fresnel lens is symmetric, Cg can be calculated as the quotient of half-width xe and the maximum intercept at the x-axis, that is:

$${C_g} = \frac{{{x_\textrm{e}}}}{{\max \{{|{{x_{\textrm{e,lw}}}\textrm{ - }{x_\textrm{e}}} |, |{{x_{\textrm{e,sw}}}\textrm{ - }{x_\textrm{e}}} |} \}}}$$

The Fresnel lens 2 is considered confocal with Fresnel lens 1. The incident angle would not change and the refraction keeps the same with Fresnel 1, which means the triangle ABC is similar to the triangle A'B'C’. Therefore, the two Fresnel lens has the same Cg. In other words, all of the confocal Fresnel lenses should have the same Cg. In this case, considering the incident light beam as the visible light with within the wavelength range from 400nm to 700nm, the Cg of the ideal Fresnel lens can be calculated out. Two frequently used materials PMMA and PC have been chosen as the calculating example. Figure 7(a) gives the refractive index changing with the increase of light wavelength. The center wavelength of visible light (λ=550nm) has been selected as the reference point). The intercepts difference between the refractive indexes of λ=400nm and λ=550nm is much larger than that of λ=700nm and λ=550nm. It could be expressed as |xe,lw-xe|<|xe,sw-xe|. Thus, the Eq. (17) can be simplified as following:

$${C_g} = \frac{{{x_\textrm{e}}}}{{|{{x_{\textrm{e,sw}}}\textrm{ - }{x_\textrm{e}}} |}}$$

 figure: Fig. 7.

Fig. 7. (a) Variations of n-λ; (b) Variations of Cg-θ.

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Figure 7(b) shows the variations of Cg with the total refracted angle for linear-focusing Fresnel lens. Both of the two kinds of Fresnel lens have a maximum Cg. The material with low refractive index has a larger Cg. In general, it can be designed with a larger total refracted angle with a winder aperture width once the focal length is fixed. For example, if the PMMA is selected to manufacture a Fresnel lens, the maximum Cg can be 41.8 when the total refracted angle is around 28°. For the PC, the maximum Cg can be 29.2 when the total refracted angle is around 20°. All of these results are calculated by only considering the visible light. If a wider wavelength range of sunlight is considered, the Cg value will be lower than the present calculated results.

5. Conclusion

In conclusion, the ideal shape of the Fresnel lens has been theoretically based on the optimal transmittance condition of prism. A differential equation has been obtained and solved to express the series of confocal curves. To make sure the ideal Fresnel lens having a maximal transmittance for a certain θ, the refractive index must be chosen properly. Besides, it is found that the geometric concentration Cg is not growing with the increase of the aperture width of the Fresnel lens for visible light. For linear-focusing lenses, the maximum Cgs for PMMA and PC are 41.8 and 29.2, corresponding to the total refracted angle around 28° and 20°, respectively. Generally, materials with low refractive index can be used to design a Fresnel lens with larger Cg.

However, the ideal Fresnel lenses are difficult to be manufactured, this work still provides a referable standard for Fresnel lens’ design. Fresnel lens with optimal transmittance should -meet two conditions: the shape conforms to the ideal shape equation and the material (refractive index, n) is chosen properly. In practical, a Fresnel lens is usually made of one material, so if its shape is close to the ideal shape, it will have the optimal transmittance.

Funding

National Natural Science Foundation of China (51976013).

Acknowledgments

Xinglong Ma is very thankful his post-doctoral contract under the Hongfei Zheng’s project.

Disclosures

The authors declare no conflicts of interest.

References

1. W. T. Xie, Y. J. Dai, and R. Z. Wan, “Concentrated solar energy applications using Fresnel lenses: A review,” Renewable Sustainable Energy Rev. 15(6), 2588–2606 (2011). [CrossRef]  

2. G. Vallerotto, M. Victoria, and S. Askins, “Design and modeling of a cost-effective achromatic Fresnel lens for concentrating photovoltaics,” Opt. Express 24(18), A1245–1256 (2016). [CrossRef]  

3. F. Languy and S. Habraken, “Nonimaging achromatic shaped Fresnel lenses for ultrahigh solar concentration,” Opt. Lett. 38(10), 1730–1732 (2013). [CrossRef]  

4. X. Ma, H. Zheng, and M. Tian, “Optimize the shape of curved-Fresnel lens to maximize its transmittance,” Sol. Energy 127, 285–293 (2016). [CrossRef]  

5. E. J. Khalil, K. A. Munadhila, and A. Kaysi, “The study of the performance and efficiency of flat linear Fresnel lens collector with sun tracking system in Iraq,” Renewable Energy 14(1-4), 41–48 (1998). [CrossRef]  

6. M. J. O’Neil, M. F. Piszczor, and M. I. Eskenazi, “Ultra-light stretched Fresnel lens solar concentrator for space power applications,” Proc. SPIE 5179, 116–126 (2003). [CrossRef]  

7. Y. Naichia, “Optical geometry approach for elliptical Fresnel lens design and chromatic aberration,” Sol. Energy Mater. Sol. Cells 93(8), 1309–1317 (2009). [CrossRef]  

8. Y. Naichia, “Illumination uniformity issue explored via two-stage solar concentrator system based on Fresnel lens and compound flat concentrator,” Energy 95, 542–549 (2016). [CrossRef]  

9. H. Zheng, C. Feng, and Y. Su, “Design and experimental analysis of a cylindrical compound Fresnel solar concentrator,” Sol. Energy 107, 26–37 (2014). [CrossRef]  

10. M. Born and E. Wolf, “Principles of Optics,” in Principles of Optics, 6th edition, (Oxford, UK, 1989). Chap. 3.

11. R. Leutz, A. Suzuki, and A. Akisawa, “Design of a non-imaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999). [CrossRef]  

12. X. Ma, H. Zheng, and S. Liu, “Optimization on a cylindrical Fresnel lens and its validation in a medium temperature solar steam generation system,” Renewable Energy 134, 1332–1343 (2019). [CrossRef]  

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Figures (7)
Fig. 1.
Fig. 1. (a) The optimum transmittance condition of prism; (b) Profile of the ideal shape of Fresnel lens. Drawing the normal line pass through the point A, it can be found that the incident light and emergent light are distributing in the same side of the normal line, which means the simplified condition conforms to the law of reflection of light. In this case, both the incident angle and the emergent angle are equivalent to θ/2.

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Fig. 2.
Fig. 2. (a) Graph of s-polarized and p-polarized vibrations

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Fig. 3.
Fig. 3. Curves of particular solutions of Eq. (12).

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Fig. 4.
Fig. 4. (a) Prisms on one point of the ideal curve; (b) Variations of transmittance with θ and n.

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Fig. 5.
Fig. 5. (a) Parameters of ideal Fresnel lens; (b) Structure of ideal Fresnel lens.

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Fig. 6.
Fig. 6. Refractions at the edge of ideal Fresnel lenses.

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Fig. 7.
Fig. 7. (a) Variations of n-λ; (b) Variations of Cg-θ.

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Equations (18)

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{ R p = ( n 21 cos ⁡ i 1 − cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n 21 ) ) n 21 cos ⁡ i 1 + cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n 21 ) ) ) 2 R s = ( cos ⁡ i 1 − n 21 cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n 21 ) ) cos ⁡ i 1 + n 21 cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n 21 ) ) ) 2
τ s = ( n cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n ) ) cos ⁡ i 1 × ( 2 cos ⁡ i 1 cos ⁡ i 1 + n cos ⁡ ( arcsin ⁡ ( sin ⁡ i 1 / n ) ) ) 2 ) 2
θ = 2 ( i 1 − i 2 )
n = sin ⁡ i 1 / sin ⁡ i 2
i 1 = arctan ⁡ ( sin ⁡ ( θ / 2 ) cos ⁡ ( θ / 2 ) − 1 / n )
γ = 2 arcsin ⁡ ( sin ⁡ i 1 / n )
tan ⁡ θ = x / x y y
d y d x = y ′ = − tan ⁡ θ / θ 2 2
tan ⁡ θ = 2 tan ⁡ θ / θ 2 2 1 − tan 2 θ / θ 2 2
tan ⁡ θ / θ 2 2 = 1 + tan 2 θ − 1 tan ⁡ θ
x y y ′ = 1 − 1 + ( x y ) 2
{ x = C 1 − cos ⁡ θ 1 + cos ⁡ θ y = C 1 − cos ⁡ θ 1 + cos ⁡ θ ⋅ 1 tan ⁡ θ ( θ ∈ [ 0 , π ) , C ∈ Positive number )
C g = Aperture width Focal width
{ θ e,lw = 2 ( i 1 , e,lw − arcsin ⁡ ( sin ⁡ i 1 , e,lw / n lw ) ) − δ θ e,sw = 2 ( i 1 , e,sw − arcsin ⁡ ( sin ⁡ i 1 , e,sw / n sw ) ) + δ
{ i 1 , e,lw = i 1 , e + δ i 1 , e,sw = i 1 , e − δ
{ x e,lw  =  y e ⋅ tan ⁡ θ e,lw x e,sw  =  y e ⋅ tan ⁡ θ e,sw
C g = x e max { | x e,lw  -  x e | , | x e,sw  -  x e | }
C g = x e | x e,sw  -  x e |
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