The compound eye of insects consists of a large number of sub-eyes, each of which can form a independent photosensitivity^[1]. Due to the special structure of compound eye, insect compound eye has the advantages of small size^{[2, 3]}, large field of view and sensitivity to moving objects^{[4, 5]}. At present, there are many researches on compound eye system on these advantages^{[6, 7]}. However, most of these researches ignore another advantage of insect compound eye-broad spectrum^[8]. Studies have shown that the compound eyes of many insects have good imaging in the ultraviolet, infrared^{[9, 10]}. Wide-spectrum compound eyes are of great significance for insects to adapt to natural environment. Especially for insects that live at night or in dense forests, or other low light environments, the wide-spectrum compound eye can capture more effective information^[11]. Besides, wide-spectrum also play an important role in insect foraging^{[12, 13]}, flight navigation and other behaviors^{[14, 15]}. Therefore, in order to improve the complex environmental adaptability of compound eye optical system, and receive target information more effectively and comprehensively, it is necessary to design wide-spectrum compound eye optical system.

This paper proposes a wide-spectrum compound eye structure capable of receiving both visible and long-wave infrared bands, and specifically designs the ommatidia and receiving optical system. Using the focal power matching method, the imaging positions of the two bands of light passing through the ommatidia system are made identical, and the matching conditions are used to ensure that the light of the ommatidia system can enter the receiving system. The compound eye system has a wide imaging range and compact structure, which improves the capabilities of system image acquisition.

In order to receive the image formed by the ommatidia, the image planes of the visible light and the long-wave infrared band are required to be at the same position. However, the refractive indices of the same material in the different wavelength bands are dissimilar, as showed in Fig 1. According to the focal power Eq. (1) of the single lens and focal power Eq.(2) of the single lens, for the same lens, the degree of deflection of each band of light is different, and the final image planes of the two bands will not be at the same position.

1 $ \left[ {$$\begin{array}{ll}l& 0\\ 0& l^{\prime }\end{array}$$} \right] = \left[ {$$\begin{array}{cc}l& l^{\prime }\\ f^{\prime }& f^{\prime }\end{array}$$} \right] $

where l′ is the image distance, l is the object distance, and f′ is the focal length of the lens.

2 $ \left[ {$$\begin{array}{cc}{f}^{\prime }n& d\\ f^{\prime }\left(1-n\right)& r_1-r_2\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\frac{n}{n-1}& 0\\ 0& r_1{r}_2\end{array}$$} \right] $

where n is the refractive index, r₁, r₂ are the curvature of the two faces of the lens, and d is the lens thickness.

According to the above formula, we obtain

3 $ \left[ {$$\begin{array}{cc}{l}^{\prime }n(n-1)& -r_1{r}_2\\ n& l\left(r_1-r_2\right)\end{array}$$} \right] + \left[ {$$\begin{array}{cc}{l}^{\prime }n(n-1)& dl\\ l^{\prime }(n-1)& dl\end{array}$$} \right] = \left[ {$$\begin{array}{cc}{r}_1{r}_2& 0\\ 0& nl\end{array}$$} \right] $

To image at the common positionat different bands, the imaging planes to coincide

4 $ \left\{ $$\begin{array}{l}\left[\begin{array}{cc}\frac{1}{{l^{\prime }}_{\lambda 1}}& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\left(r_1-r_2\right)n_{\lambda 1}& \left(r_1-r_2\right)\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}\right]+\left[\begin{array}{cc}d{n}_{\lambda 1}& 2d\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}\right]+\left[\begin{array}{cc}d& 1\\ \frac{1}{l}& \frac{1}{r_1{r}_2{n}_{\lambda 1}}\end{array}\right]\\ \left[\begin{array}{cc}\frac{1}{{l^{\prime }}_{\lambda 2}}& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\left(r_1-r_2\right)n_{\lambda 2}& \left(r_1-r_2\right)\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}\right]+\left[\begin{array}{cc}d{n}_{\lambda 2}& 2d\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}\right]+\left[\begin{array}{cc}d& 1\\ \frac{1}{l}& \frac{1}{r_1{r}_2{n}_{\lambda 2}}\end{array}\right]\end{array}$$ \right. $

where l′_λ1, l′_λ2 are the image distances of the two bands, n′_λ1, n′_λ2 are the refractive indices of the two bands.

Subtracting the two parts of Eq. (4), we obtained

5 $ \left[ {$$\begin{array}{cc}{l}_{\lambda 2}^{\prime }& l_{\lambda 1}^{\prime }\\ \frac{1}{l_{\lambda 1}^{\prime }{l}_{\lambda 2}^{\prime }}& \frac{1}{l_{\lambda 1}^{\prime }{l}_{\lambda 2}^{\prime }}\end{array}$$} \right] = \left[ {$$\begin{array}{cc}{r}_1\left(n_{\lambda 1}-n_{\lambda 2}\right)& r_2\left(n_{\lambda 1}-n_{\lambda 2}\right)\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}$$} \right] + \left[ {$$\begin{array}{cc}d\left(n_{\lambda 1}-n_{\lambda 2}\right)& d\frac{\left(n_{\lambda 1}-n_{\lambda 2}\right)}{n_{\lambda 1}{n}_{\lambda 2}}\\ \frac{1}{r_1{r}_2}& \frac{1}{r_1{r}_2}\end{array}$$} \right] $

when l′_λ2=l′_λ1-Δl′, we obtained

6 $ \left[ {$$\begin{array}{cc}\mathrm{\Delta }{l}^{\prime }& 0\\ 1& \frac{1}{l_{\lambda 1}^{\prime }\left(l_{\lambda 1}^{\prime }-\mathrm{\Delta }{l}^{\prime }\right)}\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\mathrm{\Delta }n& \mathrm{\Delta }n\\ \frac{1}{r_1}& \frac{1}{r_2}\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\mathrm{\Delta }n& \mathrm{\Delta }n\\ \frac{d}{r_1{r}_2{n}_{\lambda 1}{n}_{\lambda 2}}& \frac{1}{r_1{r}_2}\end{array}$$} \right] $

Since Δl′ is far less than l′_λ1, l′_λ2, when l′_λ1l′_λ2≈l′_m², n_λ1n_λ2≈n_λm², l′_m, n_λm are the image distance and refractive index of intermediate wavelength $\left( {{\lambda _m} = \frac{{{\lambda _1} + {\lambda _2}}}{2}} \right)$

7 $ \left[ {$$\begin{array}{cc}\mathrm{\Delta }{l}^{\prime }& 0\\ 0& 1\end{array}$$} \right] \approx \left[ {$$\begin{array}{cc}\mathrm{\Delta }n& \mathrm{\Delta }n\\ \frac{l_m^{\prime 2}}{r_1}& \frac{l_m^{\prime 2}}{r_1}\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\mathrm{\Delta }n& \mathrm{\Delta }n\\ \frac{d{l}_m^{\prime 2}}{r_1{r}_2}& \frac{d{l}_m^{\prime 2}}{r_1{r}_2{n}_{\lambda m}^2}\end{array}$$} \right] $

It can be seen from the Eq. (7) that the imaging position inconsistency of the system in different bands is determined by the ideal imaging position, lens curvature, thickness and focal length difference of the two bands. According to Eq. (7), in order to make the single lens have the same imaging position in both bands, which should be satisfied r₁－r₂+d=0 and d=0 at the same time. At this time, the lens focal power is zero, which has no practical significance.

Considering that the actual optical system is generally composed of several lenses, and each lens produces an imaging position deviation which is different in magnitude and sign, the final image plane position can be achieved by canceling each other by different lens image plane differences. Combined with the optical system superposition theory, the total image plane position difference between the two bands is the accumulation of the image plane position difference produced by each lens, which can be expressed as

8 $ \left[ {$$\begin{array}{cc}\mathrm{\Delta }{S}^{\prime }& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\mathrm{\Delta }{l}_1^{\prime }& 0\\ 0& {\alpha }_2{\alpha }_3\cdots {\alpha }_n\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\mathrm{\Delta }{l}_2^{\prime }& 0\\ 0& {\alpha }_3{\alpha }_4\cdots {\alpha }_n\end{array}$$} \right] \ldots \left[ {$$\begin{array}{cc}\mathrm{\Delta }{l}_i^{\prime }& 0\\ 0& {\alpha }_{i+1}{\alpha }_{i+2}\cdots {\alpha }_n\end{array}$$} \right] $

where ΔS′ is the total imaging position difference of the system in the two bands, α_i is the magnification of the difference in imaging position of the i-th lens. Considering Δl′ is the distance in the axial direction, α_i can be approximated by the intermediate wavelength axial magnification $\left( {{\alpha _i} \approx {\alpha _{im}} = \frac{{l_m^\prime }}{{{l_m}}}} \right)$ in practical applications.

In order to ensure that the ommatidia system and the receiving system are compatible, the object surface of the receiving system should be equal to or greater than the image plane of the ommatidia combination system. Additionally, the curvatures of the two systems need to be identical. This ensures that the images of each sub-eye system can enter the receiving system. At the same time, the aperture of the receiving system should be greater than the aperture of the edge sub-eye light, so that the light of the edge sub-eye system can enter the receiving system. Known ommatidia system aperture is D_s, the focal length is f_s, the curvature of the distribution of the ommatidia system is R, aperture is D₀, the distance from the sub-eye to the receiving system along the axis is L, the receiving system has a diameter D, and the field of view is ω, the projection aperture of the edge of the ommatidia in the vertical direction of the receiving system is ΔD, the angular magnification of the first lens of the receiving system is γ.

Considering a single lens for the sub-eye, the distance to the image plane is approximately equal to the focal length, and the imaging surface of the sub-eye combination system should be a spherical surface. The geometric imaging principle can be used to obtain the matching formula of the sub-eye and the receiving system which is based on the aperture division structure.

9 $ \left[ {$$\begin{array}{ll}\omega & 0\\ 0& r\end{array}$$} \right] = \left[ {$$\begin{array}{cc}{D}_0& 0\\ 0& \frac{1}{2}\end{array}$$} \right] $

10 $ \left[ {$$\begin{array}{cc}D& -2h\\ 1& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}{D}_0& -\mathrm{\Delta }D\\ 1& \frac{1}{2}\end{array}$$} \right] $

where h is the difference between the aperture of the receiving system and the curved image surface is satisfied h=tanω·γ·s.

According to the geometric considerations, the following relationship is satisfied between the known quantity and the intermediate quantity in the formula

11 $ \left\{ {$$\begin{array}{c}\left[\begin{array}{ll}s& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}L-f_s& r_{\mathrm{c}}\\ \left(1-\cos {\omega }_0\right)& 1\end{array}\right]\\ \left[\begin{array}{cc}\cos {\omega }_0& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\sqrt{4R^2-D_0^2}& 0\\ 0& \frac{1}{2R}\end{array}\right]\\ \left[\begin{array}{cc}{r}_{\mathrm{c}}& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}R& f_{\mathrm{s}}\\ 1& 1\end{array}\right]\\ \left[\begin{array}{cc}{D}_2& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}{r}_{\mathrm{c}}& 0\\ 0& \sin {\omega }_0\end{array}\right]\\ \left[\begin{array}{cc}\mathrm{\Delta }D& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}{D}_{\mathrm{s}}& D_{\mathrm{s}}\\ 1& \frac{l}{f_s}\end{array}\right]\end{array}$$} \right. $

Substituting the Eq. (11) into the conditional Eqs. (9) and (10), the matching requirements for the compound-eye optical system based on the aperture division are satisfied

12 $ \left[ {$$\begin{array}{ll}\omega & 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}{D}_0& 0\\ 0& \frac{1}{2R}\end{array}$$} \right] $

13 $ \left[ {$$\begin{array}{cc}D& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}2\tan \omega \cdot \gamma & 2\tan \omega \cdot \gamma \\ R& L\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\sqrt{4R^2-D_0^2}& \sqrt{4R^2-D_0^2}\\ \frac{1}{2}& f_s\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\left(R-f_{\mathrm{s}}\right)D_0& \left(L-f_s\right)D_{\mathrm{s}}\\ \frac{1}{f_s}& \frac{1}{2R}\end{array}$$} \right] $

In order to verify the correctness of the above method, acommon imaging position visible and long wave infrared band wide spectrum sub-eye optical system is designed, as shown in Table 1, the focal length is f′=5 mm, the relative aperture is D/f′=1/3, and the FOV is 10°.

The optical system consist of three-pieces lens is used as the initial structure of the dual-band sub-eye system. The wavelength of the visible light center is 0.587 μm, whereas the wavelength of the long-wave infrared is 8.2 μm. After several iterations, a two-band sub-eye initial structure with the same image plane position is obtained. The image positions of the two bands are at 3.0 mm. The parameters are shown in Table 2.

The initial structure only considers the on-axis point (the FOV is 0°). The imaging process of the apposition compound eye is to mosaic a plurality of images obtained by the ommatidia to obtain a complete image with a larger FOV. The angle between the ommatidia axis and the FOV of the sub-eye is one of the important factors influencing the overall structure of the compound eye optical system. Therefore, analyzing the relationship between the FOV of the sub-eye system and the total FOV of the compound eye is the key to determining the parameters and the distribution type of sub-eye. The sub-eye lenses are equally spaced in a honeycomb arrangement. Known sub-lens lens distribution surface caliber D_LA=88 mm, sub-eye lens aperture D=2 mm. Considering the size of the mechanical structure and the spacing between the two sub-eye lenses d_v=4 mm, it can be seen that the number of sub-eye lenses is

14 $ n = \left[ {$$\begin{array}{ll}1& 0\\ 0& 1\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\pi & 0\\ 0& \frac{1}{\tan 1/2}\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\pi & 0\\ 0& \frac{1}{\tan 1/4}\end{array}$$} \right] + \cdots \left[ {$$\begin{array}{cc}\pi & 0\\ 0& \frac{1}{\tan 1/2i}\end{array}$$} \right] $

where i=1, 2, 3… and i≤$\left[ {\frac{{2{D_{{\rm{LA}}}} - {D_m}}}{{2{D_m}}}} \right]$, the number of sub-eye lenses calculated by the Eq. (14) is 650.

Since the distribution is equally spaced, the angle between two neighboring ommatidia is not equal, and its size is

15 $ \left[ {$$\begin{array}{cc}\mathrm{\Delta }{\theta }_i& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\arctan \frac{2\left(i+1\right)D_m}{R}& 0\\ 0& \arctan \frac{2i{D}_m}{R}\end{array}$$} \right] $

Considering the material properties and design requirements, the system uses BaF2/PBF2/CSBR materials that can transmit both visible and medium-wave infrared light. To improve image quality, a diffractive surface is introduced on the fourth surface of the system. The designed sub-eye optical system can now receive both visible light and infrared light. Through the common image surface design, the imaging positions of visible light and long-wave infrared are both at 2.92 mm. System volume is 2 mm×5 mm, which is comparable to the single-band ommatidia system. The use of the dual-band sub-eye system can obtain more comprehensive target information, which is beneficial towards the improvement of the detection and recognition ability of the compound eye system. The final sub-eye lens structure is shown in Fig. 3.

The designed dual-band sub-eye system parameters and diffraction surface coefficient are shown in Table 3 and Table 4.

Fig. 4 shows the Modulation Transfer Function (MTF) curve of the optical system under different fields of view. It can be seen that the system MTF value is greater than 0.5 at the Nyquist frequency 50 lp/mm of the visible light band, and the system MTF value can reach 0.4 at Nyquist frequency 17 lp/mm of the infrared band and the imaging quality can approach the diffraction limit.

Fig. 5 shows a spot diagram of the overall system. It can be seen that the RMS at the visible light band is less than 9.38 μm, and the RMS value is no more than 13.27 μm in the mid-infrared band, which are all less than one pixel size (25 μm), thereby satisfying the matching requirements of the system and the detector.

The receiving system has a surface curvature of 50 mm and a diameter of 80 mm. According to Eq. (13), the FOV of the receiving system is determined to be greater than 80°, and the aperture is not less than 16 mm. The specific parameters of the acceptance system are shown in Table 5.

In order to reduce the system volume while ensuring the consistency of the received images in the visible and infrared bands, the receiving optical system adopts a partial common aperture structure. The final design of the receiving system has a focal length of 4 mm, a relative aperture of 1:3 and an FOV of 80°. Synthetic FOV is 2ω_c=2ω_r+2ω=90°. The receiving system consists of five lenses, three for the common part, one for the visible and one for the infrared band, with a total length of 31 mm. In order to enhance the image quality, a diffractive surface is used on the third lens. The structure of the receiving system is shown in Fig. 6. The structural parameters of the receiving optical system is listed in Table 6.

The parameters of the dual band receiving system are shown in Table 6.The diffraction surface coefficient is shown in Table 7.

Fig. 7 shows the MTF curve of the optical system in different fields of view. It can be seen that the system MTF value is greater than 0.4 at the Nyquist frequency 150 lp/mm at the visible light band and the system MTF can reach 0.4 at the Nyquist frequency 17 lp/mm at the mid-infrared band. The imaging quality here also is close to the diffraction limit.

Fig. 8 shows a spot diagram of the overall system, where the RMS value of the visible light band is less than 2.47 μm, and the RMS value of the mid-infrared band is at most 13.82 μm, which are all in a pixel (25 μm). It satisfies the matching requirements of the system and the detector.

The field of view of X-direction and Y-direction of the compound eye system are 110° and 90° respectively. Systematic sub-eyes are distributed in hexagonal shape, with a total of 650 sub-eyes. The overall picture of compound eye system and sub-eye distribution are shown in Fig. 9.

Since the system contains both visible and infrared bands, it is challenging to design the system without heating effects. It is proposed to adopt the optical passive athermalization method, in order tosuppress heating by mutual compensation between materials. Fig. 10 and Fig. 13 show the MTF diagram of the chronograph system in the visible and long-wave infrared in the temperature range of -40 ℃ and +60 ℃. The value of the MTF is greater than 0.2 at the both visible and long-wave infrared band, which still meets the requirements of general infrared system imaging.

In order to reduce the difficulty of image fusion, the images of different sub-eye lenses should be separated. That is, the image height of the single sub-eye lens at the receiving system 2y′_s should be less than the spacing between the two sub-eye images d_s. Analysis of the designed compound eye system, known sub-lens field of view 2ω=10°, the focal length f_s=5 mm, sub-eye synthetic surface caliber D_c=80 mm, Curvature r_c=50 mm, receiving system image height 2y′=6 mm. From geometric optics, the image height of a single sub-eye in the receiving system is given by

16 $ \left[ {$$\begin{array}{cc}{y}_{\mathrm{s}}^{\prime }& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\tan \omega \cdot f_{\mathrm{s}}& 0\\ 0& \frac{2y^{\prime }}{D_{\mathrm{c}}}\end{array}$$} \right] $

According to the Eq.(16), it can be calculated that y′_s=0.078 mm.

The spacing of the images of the two sub-eyes on the receiver of the receiving system is

17 $ \left[ {$$\begin{array}{cc}{d}_{\mathrm{s}\mathrm{v}}& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}\mathrm{\Delta }\theta \cdot r_{\mathrm{c}}& 0\\ 0& \frac{2y^{\prime }}{D_{\mathrm{s}}}\end{array}$$} \right] $

It can be calculated according to the Eq. (17) that d_sv=0.272 mm, satisfy the requirement that the images of sub-eyes do not overlap.

The diffractive surface is a phase-type optical surface, and its rotationally symmetric aspherical surface diffractive equationcan be expressed as

18 $ \left[ {$$\begin{array}{cc}Z(r)& 0\\ 0& 1\end{array}$$} \right] = \left[ {$$\begin{array}{cc}c{r}^2& -\left(A_4{Y}_4+\cdots +A_N{Y}_N\right)\\ 1& \frac{1}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}\end{array}$$} \right] + \left[ {$$\begin{array}{cc}\frac{{\lambda }_0}{n\left({\lambda }_0\right)-1}& \mathrm{i}\mathrm{n}\mathrm{t}\left[\frac{\phi (r{)}^{\prime }}{2\mathsf{\pi }}\right]\\ \frac{{\lambda }_0}{n\left({\lambda }_0\right)-1}& \frac{D_n{Y}^2}{2\mathsf{\pi }}\end{array}$$} \right] $

where r is the radius of surface; Y is the ordinate; k is the conic coefficient; c is the reciprocal of the curvature; φ(r)′ is the basic formula of the diffractive surface element; the phase function φ(r)′ represents the amplitude of the surface error; A_n is the aspherical coefficient; λ₀ is the wavelength; n(λ₀) is the refractive index; D_n is the phase coefficient. The width of the three diffractive faces of the design are 0.266 mm, 0. 219 mm and 0. 293 mm, and the depth of the diffractive surface is 16.706 μm. The aspect ratio satisfies the requirement of less than 1/10^[11]. The phase fitting model of the diffractive surface is obtained as shown in Fig. 14, in which blue region represents the diffraction phase morphology and red region represents the complete diffractive surface morphology after superposition.

In this work, a visible light and long wave infrared wide spectrum compound-eye optical system is designed by theoretical derivation of the dual-band common imaging position. The system can simultaneously receive target information in the visible and long-wave infrared bands, thereby expanding the spectral receiving range of the compound eye system. In the designed sub-eye lens, the visible light band transfer function value is higher than 0.4 at the Nyquist frequency50 lp/mm, and the infrared band transfer function value reaches 0.4 at the Nyquist frequency17 lp/mm. In the designed receiving system, the value of the visible band transfer function is higher than 0.4 at the Nyquist frequency 150 lp/mm, and the value of the infrared band transfer function reaches 0.4 at the Nyquist frequency17 lp/mm. Athermalization is achieved over the temperature range －40°~+60 ℃. The results of the diffractive surface analysis prove that the system can be processed.