## Abstract

Absolute optical instruments with rotation symmetry and corresponding absolute geodesic lenses have drawn considerable attention for their property of perfect imaging of light rays. In this paper, we systematically explore a series of absolute geodesic lenses which is mapped from generalized Maxwell’s fish-eye lenses with a rational number index {*p*}. Moreover, we construct new types of duplex absolute geodesic lenses by splicing two different half absolute geodesic lenses, which is inspired by the work [Huiyan Peng, et al Phys. Rev. Applied **13**, 034050 (2020)]. Also, we fabricate some samples of absolute geodesic lenses based on the 3D printing technique and observe light rays on them. Our findings enlarge the family of absolute geodesic lenses and might find an application on classical imaging systems.

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

## 1. Introduction

Conventional lenses like our glasses are suffering from optical aberration. However, there exist optical devices that can make stigmatical images in geometrical optics, which are called absolute optical instruments (AOIs) [1–6]. A common AOI is a mirror, which is based on light reflection. Another AOI is the perfect lens [7] made of negative refractive index [8]. Widely-studied AOIs are lenses with gradient refractive index, such as Maxwell’s fish-eye lens [9], Luneburg lens [2], Eaton lens [10] and Miñano lens [3]. Those lenses have spherical symmetry in three-dimensional (3D) space or rotation symmetry in two-dimensional (2D) space, whose gradient refractive index only depends on the distance from the origin point [4]. By exploring the close analogy between classical mechanics and geometrical optics, AOIs without continuous symmetry are investigated based on classical superintegrability and separability of the Hamilton-Jacobi equation [11], which flourish the family of AOIs both in 3D space and 2D space. More recently, two special kinds of AOIs in 2D space, duplex Mikalien lenses and duplex Maxwell’s fish-eye lenses [6], are proposed based on the well-known Mikalien lenses [12] and generalized Maxwell’s fish-eye lenses [13]. However, the gradient refractive index of some new AOIs is ranging from 0 to infinity, which is hard to fabricate for broadband frequency. Hence their experiments and applications are limited.

In addition, transformation optics [14–16] has established the relation between electromagnetic property and the geometry property from geometrical perspective based on coordinate transformations. A lot of unconventional devices are designed by transformation optics and fabricated by metamaterials, such as invisibility cloaks [15,17], carpet cloaks [18,19], rotators [20,21] and concentrators [22,23]. For the special case, conformal transformation optics [14,24] connects the gradient refractive index and conformal structure of 2D space, hence offering us an alternative way to manipulate light propagation with either gradient refractive index or curved surface (without gradient refractive index) [25]. There exists a geodesic conformal mapping (see more details in Sec. 2) between rotation gradient refractive index of a 2D plane and a surface of revolution (also called a geodesic lens) [26–28]. Based on this mapping, conformal singularities with the gradient refractive index ranging from 1 to infinity achieved by cone surfaces are investigated both theoretically and experimentally [29].

From the above discussion, we understand that AOIs with rotation gradient refractive index is hard to fabricate. However, they can be equivalently replaced by absolute geodesic lenses (AGLs). In this paper, we systematically explore AGLs from generalized Maxwell’s fish-eye lenses, which can make perfect images for light rays. Moreover, we construct different types of duplex AGLs by splicing two different half AGLs, which can also perform perfectly imaging. Intuitive illustrations of these AGLs and their duplex AGLs are presented. Compared to the AOIs, these AGLs are much easier to fabricate in current 3D print technology, which can work in broadband of frequencies. This paper is organized as follows. In Sec 2, we introduce Maxwell’s fish-eye lenses with rational number indexes and their corresponding AGLs based on geodesic conformal mapping. In Sec. 3, we construct the duplex AGLs and investigate their caustic effects. In Sec. 4, we fabricate AGLs and observe light rays on them. In Sec. 5. we give a conclusion.

## 2. Maxwell’s fish-eye lenses and their corresponding AGLs

The famous Maxwell’s fish-eye lens was originally studied in 1884 by Maxwell J. C. [9]. Its perfect imaging property for waves with drain is revisited by Leonhardt U. in 2009 [30]. The generalization of Maxwell’s fish-eye lenses is further discussed in several works of literature [4,13,31]. Generalized Maxwell’s fish-eye lenses in 2D space are AOIs marked with a rational number index $\{p \}$ [6], whose gradient refractive index is written as,

where ${r_0}$ is a reference length scale. In the following discussion, ${r_0}$ is set as 1mm. We will explain why*p*should be a rational number in Sec. 4, more details can be found in Ref. [4,13,27] and a heuristic discussion from exponential conformal mapping [6]. When $p = p1 = 1$, Eq. (1) denotes a conventional Maxwell’s fish-eye lens, whose gradient refractive index is plotted in the complex plane (denoted by $w = u + vi$) in Fig. 1(a). The blue and green lines represent Maxwell’s fish-eye lenses with index $p\textrm{2} = 1/2$ and $p3 = 2$, respectively. When

*p*becomes smaller, the line of gradient refractive is approached to the magenta line, which corresponds to $p\textrm{4} = 0$ with the form of $n(r) = 1/r$. It has been reported that an isotropic material with this gradient refractive index has radially polarized modes with the same perfect focusing properties as the Maxwell fish-eye lens with $\{1 \}$ [32]. When

*p*approaches to $\infty $, the gradient refractive index is close to a delta function $n(r) = \delta (r - r0)$, where we use a black point to denote in Fig. 1(a). As we can see, the gradient refractive is the monotonic function with

*r*for $p \le \textrm{1}$, while there is the extreme value at $0 < r < 1$ for $p > 1$ (see the green line).

Maxwell’s fish-eye lens with index $\{p \}$ can be cast into a AGL based on geodesic conformal mapping, which is written as [33,34],

where $\rho $ is the radial coordinate and*h*is the length along the meridian from the original point as shown in Fig. 1(d). The shape of an AGL is determined by a function $Z = Z(\rho )$, where

*Z*is the third coordinate obtained from $\rho = \rho (h )$. The angle coordinate is unchanged under Eq. (2) because of rotation symmetry. In Fig. 1(b), the value of $\rho = n(r )r$ from Eq. (2) is plotted in colored lines corresponding to different index $\{p \}$. As we can see from the straight horizontal line (in magenta) with $p = p4 = 0$ in Fig. 1(b), the $\rho $ is unchanged with

*r*. Hence its corresponding geodesic lens is a cylinder, as shown its magenta meridian in Fig. 1(d). For $p = \infty $, the AGL of a point can be treated as a circle, see the black point in Fig. 1(d). According to Eq. (2), we can obtain the following expression,

For better understanding, we plot light rays in Maxwell’s fish-eye lenses with different index $\{p \}$ and their corresponding AGLs in Fig. 2. In Fig. 2(a), it is a conventional Maxwell’s fish-eye lens with $\{1 \}$, and its AGL is a sphere. Light rays (in orange and purple) are circles on both of them. The dashed meridian in Fig. 2(e) comes from the half red circle in Fig. 1(d). Similarly, Fig. 2(b) and Fig. 2(f) are with an index of $\{2 \}$. As shown in Fig. 2(c) and Fig. 2(g) with an index of $\{{1/2} \}$, the geodesic lens is not a compact manifold, but a ring-like surface. However, the light rays confined in the ring-like surface can still form closed trajectories, hence make perfect images for rays. Maxwell’s fish-eye lens and its AGLs with $\{0 \}$ are circles, where light rays can wind on it clockwise or anti-clockwise. Maxwell’s fish-eye lens with $\{\infty \}$ is finally shown in Fig. 2(d) and its corresponding AGL is a cylinder as shown in Fig. 2(h).

In Fig. 2, we have demonstrated Maxwell’s fish-eye lenses and corresponding AGLs with some simple index $\{p \}$ which is either an integer or the inverse of an integer. Here, we want to give a heuristic explanation about Maxwell’s fish-eye lenses and corresponding AGLs with an arbitrary rational number $\{{p = {p_u}/{p_d}} \}$ (${p_u}$ and ${p_d}$ are coprime) based on exponential conformal mapping, which has already been discussed by Ref. [6]. The exponential conformal transformation $w = \exp (z )$ can make a one-to-one correspondence between virtual space (denoted with $w = u + vi$ in Fig. 3(a)) and a ribbon region between two dashed lines with a period of $L\textrm{ = }2\pi $ along *y* axis in physical space (denoted with $z = x + yi$ in Fig. 3(d)). By this mapping, Maxwell’s fish-eye lens with $\{{{p_u}/{p_d}} \}$ can be cast to Mikaelian lens with $\{{m = {p_u}/{p_d}} \}$, whose gradient refractive index is ${n_z} = 1/\cosh ({x{p_u}/{p_d}} )$. The period of this Mikaelian lens is ${L_D} = 2\pi {p_d}/{p_u}$. Hence, we can always find a large period ${L_g} = {p_d}L = {p_u}{L_D}$, which has ${p_d}$ ribbon regions and ${p_u}$ periods for this Mikaelian lens. Therefore, in the region of a large period ${L_g}$ in virtual $w$-space, light rays from one point source have ${p_d}$ periods, which corresponds to wind around the original point of $z$-space ${p_u}$ times and then recover to the original propagation condition. In this way, Maxwell’s fish-eye lenses with any rational number $\{p \}$ can make perfect images [6]. Once *p* is not a rational number [35], we cannot have a large period. Hence gradient refractive index of Eq. (1) with irrational number $\{p \}$ is not an AOI, which is excluded from generalized Maxwell’s fish-eye lenses. In this paper, we can further make AGLs from Maxwell’s fish-eye lenses with rational number $\{p \}$ based on geodesic conformal mapping in Eq. (2). Although the results of Maxwell’s fish-eye Lenses with $p \le 1$ and their corresponding AGLs have been discussed [4,28], the main purpose here is to give a systematical description of generalized Maxwell’s fish-eye lenses as well as their corresponding AGLs with the rational number $\{p \}$, and to construct more AGLs.

## 3. Duplex AGLs and their caustic effects

Following the strategy of duplex Mikealian lenses and their exponential conformal mapping [6], we can also construct a duplex AGL by splicing two half of AGL with different index $\{{{p_u}/{p_d},{q_u}/{q_d}} \}$, where ${p_u}/{p_d}$ and ${q_u}/{q_d}$ are the rational index of the lower half and upper half AGL, respectively. It is obvious that a duplex AGL with indexes of $\{{{p_u}/{p_d},{q_u}/{q_d}} \}$ has its corresponding duplex Maxwell’s fish-eye Lens and duplex Mikealian lens based on geodesic and exponential conformal mapping. As shown by quantities in Fig. 1, their value and the first derivative are both continuous, hence the duplex geodesic lens with $\{{{p_u}/{p_d},{q_u}/{q_d}} \}$ are also good enough for light rays to propagate along geodesics smoothly based on geodesic equations [28]. Let us look at a concrete example. In Fig. 3(a), the light rays starting from the blue point source in the duplex Maxwell’s fish-eye Lens with $\{{2/3,3/4} \}$are plotted with purple and orange lines, which form closed trajectories. The corresponding AGL can also make perfect imaging as shown in Fig. 3(b). Under the mapping of Eq. (4), we can find that the period of a duplex Mikealian lens is ${L_D} = \pi {p_u}/{p_d} + \pi {q_u}/{q_d}$. Hence the large period is ${L_{Dg}} = ({p_u}{q_d} + {p_d}{q_u})L = 2{p_u}{q_u}{L_D}$. As shown in Fig. 3(d), there are ${p_u}{q_d} + {p_d}{q_u} = 17$ periods of *L* and $2{p_u}{q_u} = 12$ periods of ${L_D}$ during the region of a large period. Moreover, there are $12$ blue focusing points and $12$ regions marked with red circles where the caustic effect takes place. The caustic effect is defined as the envelope of a system of orthotomic rays that cannot focus perfectly [36–38]. Figure 3(c) shows an example of the caustic effect, where the inner surface of cylinder can reflect the exposure light the desk to form a cardiac-shape caustic curve. The caustic effect is clearer in Fig. 3(e) with more light rays, and its enlarged region of the first period of *L* is shown in Fig. 3(c). The caustic effect in this duplex Mikealian lens can be treated as reminiscences of imaging points on the other side of *y* axis due to the asymmetric gradient refractive index. More details about this caustic effect could be founded in Ref. [6]. However, the caustic effect is not obvious in duplex AGLs with complex indexes $\{{2/3,3/4} \}$ due to many periods of *L* and ${L_D}$ in the large period. After exponential conformal mapping, all light rays in Fig. 3(e) will heavily overlap and diminish the caustic effect in the corresponding AGL.

The possible situation for obvious caustic effect in duplex AGLs is that its index has a simple rational number. Taking the value 1 for example, the large period ${L_{Dg}}$ of these duplex AGLs cannot be too many times of the period *L*, which could eliminate the caustic effect in the duplex AGLs. In Fig. 4(a), we can see two focusing points (in red and blue) and two cardioid caustic regions marked with red circles in the duplex AGL with $\{{\textrm{1},\textrm{1/2}} \}$. The upside-down view of Fig. 4(d) shows two cardioid caustics. As a comparison, one focusing point (in blue) and one cardioid caustic region marked with a red circle in duplex AGL with $\{{\textrm{1},1/\textrm{3}} \}$ is plotted in Fig. 4(b). Its upside-down view in Fig. 4(e) only shows one cardioid caustic region. It is worth mentioning that the duplex AGL with an index larger than 1 cannot have a caustic effect, which results from its uncompact geometry. As an example, for the duplex AGL with $\{{\textrm{1/2},\textrm{2}} \}$ shown in Fig. 4(c), there is no caustic effect. The light rays in its corresponding duplex Maxwell’s fish-eye lens are shown in Fig. 4(f), which are bounded by a dashed red circle.

## 4. Fabrication of AGLs and observation of closed light rays

There are a lot of AGLs and duplex AGLs which can serve as AOIs. These duplex AGLs provide more platforms to investigate the optical phenomenon. In this section, we fabricate some AGLs and made of photosensitive resin by the Stereo Lithography Apparatus (SLA) method, which is a kind of 3D printing technique. The photograph of one sample of an AGL with $\{2 \}$ is shown in Fig. 5(a). Its diameter of the largest equator is $D = 30mm$, while its thickness is $1mm$. The schematic of experimental measurement set-up is shown in Fig. 5(d). The sample is placed on the adjust platform, whose position could be tuned in three directions. A beam from He-Ne laser with a wavelength of $\textrm{632}\textrm{.8}$ nm is coupled to the sample at grazing incidence position (marked with a star). Light trajectories on the sample are caught by the camera. We catch one photograph when a laser beam is impinging into this sample, which is shown in Fig. 5(b). In this photograph, there is a screw on the up-left side, which serves as a reference during the following result fitting. We use a magenta geodesic curve of an AGL with $\{2 \}$, see the inserted upside figure of Fig. 5(e), to fit the light trajectories in commercial software Mathematica. As we can see the fitting results in the downside of Fig. 5(e), part of the magenta geodesic curve fits the light trajectory very well. We cannot observe the whole light trajectory in our measurement, which is mainly due to the loss and scattering during the propagation of the laser beam on the sample. To be more convincing about our fitting, the opposite view of Fig. 5(b) and Fig. 5(e) is shown in Fig. 5(c) and Fig. 5(f), respectively. The light trajectory is almost covered by the magenta geodesic curve, as shown in the down figure of Fig. 5(f).

We also fabricate another sample of an AGL with $\{2 \}$, whose size is the same as the previous one. By adjusting the position of the sample as shown in Fig. 6, we fit the trajectory of the AGL with $\{2 \}$ at different positions, which demonstrates the evolution of one light trajectory on this sample.

Also, we fabricate some other AGLs and duplex AGLs as shown in Fig. 7. The length of the largest equator is $\textrm{20}mm$, while its thickness is $\textrm{0}\textrm{.8}mm$. The photograph of Fig. 7(a) is the AGL with $\{1 \}$, which is cut off at both sides for the convenience of fabrication. Its light trajectory is shown in Fig. 7(e). The photograph of Fig. 7(b) shows the AGL with $\{{1/2} \}$, which is also cut off at both sides. Its light trajectory is shown in Fig. 7(f). The photograph of a cylinder of Fig. 7(c) is corresponding to the AGL with $\{0 \}$. We can see a typical cardioid caustic on the desk because of the global flection of light rays at the surface of the cylinder under the exposure light. Such cardioid caustic is similar to that of Fig. 4(d) and (e) in the previous section. We can also observe one part of a circular light trajectory in Fig. 7(g). Figure 7(d) shows the photograph of a duplex AGL with $\{{1,2} \}$. One light trajectory on such duplex AGL is also caught by the camera as shown in Fig. 7(h). Although light trajectories in Fig. 7(e-h) are all eliminated during a certain length, it still shows the ability of AGLs. They can serve as AOIs, which makes perfect imaging for light rays.

## 5. Conclusion

In this paper, we systematically explore a series of AGLs with a rational number index $\{p \}$ and duplex AGLs based on generalized Maxwell’s fish-eye lenses. Besides, we fabricate some samples of absolute geodesic lenses based on the 3D printing technique and observe closed light rays on them. Although we haven’t observed closed light trajectories on it, it might encourage more effect on the experimental work to the perfect imaging on AGLs and duplex AGLs, which is hard to investigate in their corresponding Maxwell’s fish-eye lenses with gradient refractive index ranging from 0 to infinity. Our findings enlarge the family of AOIs, which might find an application on nano-photonics and surface plasmonics.

## Funding

National Natural Science Foundation of China (No. 11690033, No. 11904006, No. 41875158, No. 61675005, No. 61705002, No.11874311); Natural Science Foundation of Anhui Province (No. 1908085QA20); the Recruitment Program for Leading Talent Team of Anhui Province (No. 2019-16).

## Disclosures

The authors declare no conflicts of interest.

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