Hyperbolic absolute instruments

Abstract

As a lens capable of sending images of deep sub-wavelength objects to the far field, the hyperlens has garnered significant attention for its super-resolution and magnification capabilities. However, traditional hyperlenses require extreme permittivity ratios and fail to achieve geometrically perfect imaging, significantly constraining their practical applications. In this paper, we introduce the generalized versions of hyperbolic absolute instruments from the perspective of dispersion and fundamental optical principles. These instruments support the formation of closed orbits in geometric optics, thereby enabling hyperlenses to realize aberration-free perfect imaging. This development not only provides a flexible and practical tool for enhancing the performance of traditional hyperlens, but also opens the possibilities for new optoelectronics applications based on hyperbolic ray dynamics

Graphical abstract

Keywords

absolute instrument / hyperbolic dispersion / ray dynamics

1 Introduction

The resolution of a conventional optical system is limited to approximately half of the working wavelength [1, 2], which stems from that evanescent waves carrying the detailed information decays rapidly and cannot reach the far field. In recent years, various advanced imaging lenses have been proposed to overcome this resolution barrier by effectively manipulating evanescent waves. These imaging lenses are generally classified into two categories: flat lenses [3−11] and cylindrical lenses [12−18]. While flat lenses are more convenient for microscopy applications [19], they are unable to transmit information to the far field without external assistance as the waves outside the lens remain evanescent. In contrast, cylindrical lenses can magnify images and address this limitation [20] by compressing the waves with high wavevectors to propagate to the far field as they move towards the outer edge of the cylindrical lens.

Here we primarily focus on two significant types of cylindrical imaging lens: absolute optical instrument (AI) [21] and hyperlens. AIs can provide aberration-free imaging of all points within a certain spatial region, but they are inherently incapable of supporting the propagation of evanescent waves. Hyperlens [22, 23], on the other hand, has been proposed to convert the evanescent waves into propagating waves, thereby enabling super-resolution imaging. However, hyperlens requires extreme permittivity ratio and cannot achieve geometrically perfect imaging, often resulting in severe caustics. The recently proposed perfect hyperlens [24] underscores the potential of combining the advantages of hyperlens and perfect lens [3], yet it is still a flat lens. Thus, developing a method for designing AIs in hyperbolic systems is highly desirable for optical imaging systems but remains unexplored.

In this paper, we propose the concept and designing methodology of hyperbolic absolute instrument (HAI). By analyzing the hyperbolic dispersion, we demonstrate that HAI can circumvent the topological singularity of traditional hyperlens and support closed orbits in geometric optics. To further elucidate hyperbolic ray dynamics, we derive the expressions for the angular momentum and the turning parameter in radially hyperbolic anisotropic media (RHAM). Finally, we present the general forms of electromagnetic parameters for HAIs and demonstrate several specific profiles that exhibit these desirable properties. Compared to traditional AIs based on isotropic dispersion, HAIs support beamlike radiation patterns and offer additional controlling degrees of freedom, enabling superior field regulation and enhanced imaging ability in wave optics.

2 Results and discussion

Firstly, we present the metric of perfect hyperlens in our recent work [24]

(1)\mathrm{d}{s}^2=n^2( \mathrm{d}{x}^2-\mathrm{d}{y}^2)= {\displaystyle \frac{n_0^2}{{\cos }^{2}y}}(\mathrm{d}{x}^2-\mathrm{d}{y}^2).

Notably, this lens, inspired from the de Sitter spacetime in cosmology, inherits the self-focusing properties of Mikaelian lens [25] and achieves enhanced resolution through hyperbolic dispersion. However, it remains a flat lens and cannot send information to the far field [see Fig.1(a)].

Furthermore, we apply an exponential conformal mapping w = e^z [26], transforming the flat metric into a cylindrical one \mathrm{d}{s}^2=n^2(-\mathrm{d}{r}^2+r\mathrm{d}{\phi }^2)={\left(\frac{1}{r\cos (\mathrm{l}\mathrm{n}r)}\right)}^2(-\mathrm{d}{r}^2+r\mathrm{d}{\phi }^2). In the following discussions, we demonstrate that it can support closed orbits in geometric optics as the same as Maxwell’s fish-eye (MFE) [27], a well-known AI with constant curvature. Beyond this specific example, more general absolute instruments do not necessarily exhibit constant curvature but can still support closed trajectories. This raises a fundamental question: How can we develop a systematic approach for designing such AIs with radially hyperbolic dispersions, i.e., HAIs?

2.1 Hyperbolic dispersion analysis

To begin, we focus on the RHAM where permeabilities exhibit opposite signs in the tangential and radial directions (µ_r < 0, µ_φ > 0 or µ_r > 0, µ_φ < 0). The corresponding dispersion relations [22] are

(2)\begin{array}{rl}& \frac{k_r^2}{{\mu }_{\phi }}-\frac{k_{\phi }^2}{\left|{\mu }_r\right|}=n^2\frac{{\omega }^2}{c^2},\\ & -\frac{k_r^2}{\left|{\mu }_{\phi }\right|}+\frac{k_{\phi }^2}{{\mu }_r}=n^2{k}_0^2,\end{array}

where k_r and k_φ = l/r represent the radial and tangential components of the wave vector k, and l is the angular momentum mode number. For convenience, we discretize the real space [see Fig.2(a)] and plot the dispersion between k_r and k_φr (l) at different radii in Fig.2(b)−(d). Here we consider two incident beams with wave vectors k₁ (k_r < 0) and k₂ (k_r > 0). According to phase-matching conditions (l_in = l_out = l₀), the propagation of wave in real space corresponds to the transitions between different isofrequency contours. Notably, the propagation directions in anisotropic systems are determined by the Poynting vector rather than the wave vector.

Turning point [28] is a key physical quantity for analyzing the ray motions in AIs. In general, the centrifugal/centripetal rays will transition into centripetal/centrifugal rays at the turning points. Hence, the rays satisfy dr/dt = 0 or k_r = 0 at these points, i.e., the intersections of isofrequency contours with the l-axis. Previous works told that there are two distinct turning points (r₊ and r₋) or infinite turning points (r₊ = r₋) during the propagation of rays in the AIs. For simplicity, we first consider the simplest case of n = 1. In the traditional hyperlens (µ_r < 0, µ_φ > 0), hyperbolic dispersion causes wave to either travel toward the center or diverge, as they cannot bypass the turning points for any l₀ [see Fig.1(b) and Fig.2(b)]. Consequently, traditional hyperlenses fail to achieve perfect imaging. For the other case (µ_r > 0, µ_φ < 0) in Fig.2(c), the beam with wave vector k₁ encounters the only turning point (r₊) once before traveling toward the center (path 1), while the beam with wave vector k₂ is eventually trapped at the center (path 2).

Obviously, this latter case aligns more closely with the design requirements of HAIs. Then we introduce an ideal refractive index profile n(r) ≠ 1 for the case of (µ_r > 0, µ_φ < 0), generating two isofrequency contours (r = r_±) intersecting on the l-axis. Under these conditions, the beam with wave vector k₁ reaches the outer turning point r₊ before turning into the center (path 1), while the beam with wave vector k₂ reaches the inner turning point r₋ and turns away from the center (path 2). Eventually, the beams following the path 1 and path 2 reconverge at the incident point r = r₀, forming closed orbits [see Fig.2(d)]. Therefore, the above hypothetical profile meets our preview of HAIs [see Fig.1(c)].

2.2 Methods for designing HAIs

To derive the exact solution of the above refractive index, we start from a special hyperbolic line element with an assumed index n(r)

(3)\mathrm{d}{s}^2=n^2( -{\gamma }^2\mathrm{d}{r}^2+r^2\mathrm{d}{\phi }^2).

Here we define a physical space (r, φ) and a virtual space (r, φ'), where the coordinate transformation between them satisfies φ' = φ/(iγ), with γ being a real and positive constant. Under this transformation, the line element becomes

(4)\mathrm{d}{s}^2=-n^2{\gamma }^2(\mathrm{d}{r}^2+r^2\mathrm{d}\phi {}^{\prime 2}) =-n^2{\gamma }^2\mathrm{d}{l}^2,

where dl can be interpreted as the particle trajectory in virtual space. In physical space, the angular momentum of anisotropic system [29] can be written as

(5)\begin{array}{rl}L=& n^2{r}^2\frac{\mathrm{d}\phi }{\mathrm{d}s}\\ =& nr\frac{r\mathrm{d}{\phi }^{\prime }}{\mathrm{d}l}\\ =& nr\sin {\alpha }^{\prime },\end{array}

where α' represents the angle between the tangent to the particle trajectory and the radius vector in the virtual space. For a cylindrically symmetric medium, we introduce a turning parameter [28] defined as N(r) = nr. Considering a light ray propagating with angular momentum L in the physical space, it follows from Eq. (5) that L = Nsinα' and the radius r will reach the maximum or minimum at the turning points r_±, where N(r = r_±) = L. At a general point along a ray trajectory, the derivative of the polar angle φ satisfies

(6)\tan \alpha ={\displaystyle \frac{r\mathrm{d}\phi }{\mathrm{d}r}}= \mathrm{i}\gamma {\displaystyle \frac{r\mathrm{d}{\phi }^{\prime }}{\mathrm{d}r}}=\mathrm{i}\gamma \tan {\alpha }^{\prime }= \pm {\displaystyle \frac{\gamma L}{\sqrt{L^2-{(nr)}^2}}} ,

where α represents the angle between the tangent to the particle trajectory and the radius vector in the physical space. Using Eq. (6), we express α' as α' = arctan(tanα/iγ). It is worth mentioning that for any free trajectory (i.e., without external force, collision), the quantity L remains invariant once the initial position r₀ and incident angle α₀ are given. Consequently, the angular momentum of RHAM can also be written as

(7)L=N( r_0)\sin {\alpha }_0^{\prime }= N(r_0)\sin (\arctan (\tan {\alpha }_0/(\mathrm{i}\gamma ) )).

In Fig.3, we illustrate the relationship between angular momentum L of RHAM and incident angle α₀ for γ = 1, with the isotropic cases provided for comparison. We find that the angular momentum of radially hyperbolic system takes the real value only when α₀ ∈ {π/4, 3π/4}; otherwise, it takes the imaginary value. This indicates that light can propagate stably only for the incidence angle α₀ within this range {π/4, 3π/4} which closely corresponds to the opening angle of the hyperbolic dispersion discussed above. Furthermore, the angular momentums in RHAM and isotropic media (IM) always satisfy the relationship L_RHAM ≥ N(r₀) ≥ L_IM with their curves intersecting at the special point α₀ = π/2. In other words, the incident angle in virtual space needs to satisfy a special condition |\sin {\alpha }_0^{\prime }| ≥ 1 to ensure ray propagation in physical space. Therefore, different from the isotropic case L ≤ N(r) discussed in previous studies [30], all the points along the propagation path need to satisfy the condition L ≥ N(r).

Considering a light ray originating from the origin (r₀, φ₀), we obtain from Eq. (6) that

(8)\phi -{\phi }_0= \gamma L{\displaystyle {\int }_{r_0}^r{\displaystyle \frac{\mathrm{d}r}{r\sqrt{L^2-N{(r)}^2}}},}

which serves as a fundamental equation of light ray in the RHAM with cylindrical symmetry. In AI, the ray follows a closed trajectories, oscillating between two circles of radii r₋ and r₊, touching them at equal angular intervals [see Fig.4(a)]. To ensure such a trajectory, we assume that the function N(r) decreases for r ≤ r_m and increases for r ≥ r_m with the radius r_m > 0 [see Fig.4(b)]. Consequently, N(r) attains a global minimum L₀ at the point r = r_m, i.e., N(r_m) = L₀. In order to transform Eq. (8) into a known integral equation, let us introduce the variable τ = lnr. During the motion between two turning points r₋(L) and r₊(L), the increment of the polar angle in physical space can be expressed as

(9)\mathrm{\Delta }\phi (L) =\gamma L{\displaystyle {\int }_{{\tau }_{-}}^{{\tau }_{+}}{\displaystyle \frac{\mathrm{d}\tau }{\sqrt{L^2-N{(\tau )}^2}}}.}

Here we set Δφ = π/m' = γπ/m, where m' is a rational number ensuring the closed rays in HAI. In Note S1 of Supplemental materials, we solve this integral equation of Abel’s type and obtain the general index profile as

(10)n(r)={\displaystyle \frac{L_0}{r\cos \left({\displaystyle \frac{m}{2}}\mathrm{l}\mathrm{n}(f(r)/r)\right)}}.

Here f(f(r)) = r and graph of f(r) is symmetric with respect to the axis f(r) = r. By substituting Eq. (10) into Eqs. (3) and (5), we obtain the complete expressions for angular momentum and line element

(11)\begin{array}{rl}& L=nr\sin {\alpha }^{\prime }=\frac{L_0\sin {\alpha }^{\prime }}{\cos \left({\displaystyle \frac{m}{2}}\mathrm{l}\mathrm{n}(f(r)/r)\right)},\\ & \mathrm{d}{s}^2={\left(\frac{L_0}{r\cos \left({\displaystyle \frac{m}{2}}\mathrm{l}\mathrm{n}(f(r)/r)\right)}\right)}^2(-{\gamma }^2\mathrm{d}{r}^2+r^2\mathrm{d}{\phi }^2).\end{array}

When γ and m are set as i, Eq. (11) will degenerate into the case of traditional AIs [28]. Based on transformation optics, the permittivity and permeability of HAI can be expressed as

(12){\epsilon }^{ij}={\mu }^{ij}=\pm \sqrt{\left|g\right|} g^{ij}=\pm \left(\begin{array}{ccc}-1/\gamma & & \\ & \gamma & \\ & & \gamma n^2\end{array}\right).

Considering the 2D TE polarization and the dispersion requirement (µ_r > 0, µ_φ < 0), we select the dominant parameters {μ_r, μ_φ, ε_z} = {1, −γ², n²} to screen out the self-focusing waves in wave optics. To simplify the complexity of the permittivity, an alternative selection {μ_r, μ_φ, ε_z} = {n², −γ²n², 1} achieves the same effect. Similarly, the dominant parameters in 2D TM polarization can be written by replacing the position of permittivity and permeability. With these formulations, we establish a framework for HAI design. Furthermore, our approaches extend naturally to corresponding flat hyperlens by a logarithmic conformal mapping z = ln w, finding exciting applications in cosmology [31] and polariton regulation [24]. Their metric and the corresponding dominant parameters of 2D TE polarization can be written as

(13)\begin{array}{rl}\mathrm{d}{s}^2=& n_f^2(\mathrm{d}{x}^2-{\gamma }^2\mathrm{d}{y}^2)\\ =& {\left({\displaystyle \frac{L_0}{\cos ({\displaystyle \frac{m}{2}}(\mathrm{l}\mathrm{n}(f(\exp (y)))-y))}}\right)}^2(\mathrm{d}{x}^2-{\gamma }^2\mathrm{d}{y}^2),\end{array}

and {μ_x, μ_y, ε_z} = {−1, γ², n_f²} or {μ_x, μ_y, ε_z} = {−n_f², γ²n_f², 1}.

Notably, there are beamlike radiation patterns and well-defined boundaries in HAI systems, strictly constraining the incident direction and propagation regions. In geometrical optics, the incident directions need to be satisfied , which can be derived by Eq. (6) and the divergence angle of hyperbolic dispersion θ_c = arctan(√|μ_r/μ_φ|) [32]. Moreover, in order to avoid singularity n = ∞, the radius r must satisfy the conditions r ≠ 0 and f(r)/r ≠ exp(2(pπ + π/2)/m) (p is an arbitrary integer), leading to periodic limiting boundaries in the ray propagation. Interestingly, the allowed range of the incident direction only depends on the value of γ, regardless of m and f(r), while the positions of the boundaries only depends on the value of f(r) and m, independent of γ. In the following section, we will demonstrate the designing methodology and some unique characteristics of HAIs by discussing several famous examples. For convenience, we set L₀ = 1 in the subsequent discussions.

2.3 Examples of HAIs

Above all, we consider the case of traditional MFE, where the refractive index profile is given by n = 1/(r² + 1). In this case, the object position r_A and the image position r_B are also turning points, satisfying the relation r_Ar_B = 1 [see Fig.5(a)]. To model this in the hyperbolic framework, we set f(r) = 1/r and m = 1 for the case of γ = 1. This leads to the gradient refractive index of hyperbolic Maxwell’s fish-eye (HMFE) n=\frac{1}{r\cos (\ln r)}. Interestingly, it is also the cylindrical analog of the perfect hyperlens [24], confirming the universality of our method. With the increasing of radius r, the index function of HMFE follows a parabolic variation, in contrast to the monotonically decreasing profile of the traditional MFE lens [see Fig.1(d)]. For demonstration, we take 2D TE mode for example, where the dominant parameter of HMFE are given by {μ_r, μ_φ, ε_z} = {1, −γ², (\frac{1}{r\cos (\ln r)}{)}^2}. To verify its geometric effect, we set the incident point (r₀, φ₀) = (0.8, 0) to launch rays along different directions. Among them, The red and blue curves represent the rays propagating along the incident directions {\left|\frac{\mathrm{d}r}{r\mathrm{d}\phi }\right|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}>1 and , respectively. In wave simulation, a line source along the z-axis is placed at the same position (r₀, φ₀) = (0.8, 0) to stimulate the field. Due to the beamlike radiation patterns and boundaries in hyperbolic systems, some rays from the object (r, φ) focus on the images (r⁻¹, φ+π) and return back to the origin, while others diverge towards the boundaries [see Fig.5(c)]. Nevertheless, we can observe perfect hyperbolic focusing wavefront in Fig.5(d).

In the above analysis, HAI provides an additional degree of freedom γ to control light compared with traditional AI. To further understand the effects of γ and m, we show two types of hyperbolic generalized Maxwell’s fish eye (HGMFE) (I[m = 2, γ = 1] and II[m = 1, γ = 0.5]) with the same Δφ in Fig.5(e)−(h). Notably, there are three images at the same positions in HGMFE as in the traditional generalized Maxwell’s fish eye (GMFE) in Fig.5(b). However, the type II support the propagating wave with a broader beam range and wider boundary compared to type I influenced by the value of m and γ. This demonstrates that the flexible control of m and γ can greatly enhance the design and application potential of HAIs in future.

In Note S2 of Supplemental materials, we also explore other HAIs that focuses all the rays from the object r_A to the image r_B with the same radius or back to the object, which corresponding to the Luneburg lens profile and Eaton/Miñano lens profile.

3 Conclusion

In this paper, we extend the concept of traditional absolute optical instrument into the hyperbolic domain by introducing the general forms of HAIs and the corresponding flat hyperlenses. We demonstrate that HAIs retain the geometrically perfect imaging property of traditional AI while uniquely enabling the propagation of evanescent wave with high wave vector. This capability paves the ways for numerous applications, such as super-resolution real-time imaging, sensing, and so on. Compared with traditional hyperlenses, HAIs do not require extreme permittivity ratio |ε_r/ε_θ|, offering greater flexibility in the selection of operating frequencies and mitigating the impact of weak material losses on imaging resolution. Although the general electromagnetic forms of HAIs are complex, their realization may be feasible in the future through carefully manipulating the thickness of two-dimensional materials [33], the filling factor of concentric multilayer metamaterials [34, 35] or the geometry of split-ring resonators [36].

Previous studies suggest that incident beams in the RHAM are attracted towards the origin along spirallike trajectories due to topological singularity [32, 35, 37]. Notably, our work circumvents the problem by employing special refractive indexes and derives the propagation processes of light rays in RHAM with multiple degrees of freedom, which are expected to stir up great interest in exploring more intriguing hyperbolic devices with novel functionalities, such as revolution, magnification, invisibility, and so on. Looking ahead, we anticipate the discovery of new classes of AIs with complex electromagnetic parameters, potentially capable of self-focusing on complex manifold, thereby opening exciting avenues for imaging in non-Hermitian systems.

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