Hyperbolic Topological Transitions in Photonics

• Hyperbolic topological transitions are defined by continuous changes in isofrequency contours (IFCs) through tunable refractive index modulation in GRIN lenses.
• The methodology uses spatially graded index profiles, Wick rotation, and a harmonic oscillator mapping to link material parameters with local hyperbolic states.
• This framework enables dynamic control of coexisting Type I and Type II hyperbolic domains, offering versatile pathways for on-chip photonic routing and reconfigurable optics.

Hyperbolic topological transitions describe transformations between distinct topologies of isofrequency contours (IFCs) in hyperbolic materials, typically induced by modulating system parameters such as local dielectric response or geometric structure. While traditionally explored in momentum space—where changes between elliptic, Type I hyperbolic, and Type II hyperbolic dispersions are inferred from the sign structure of the permittivity tensor—recent advances demonstrate that these transitions can also be engineered and observed directly in real space. A prime example employs spatially modulated gradient-index (GRIN) lenses, such as a hyperbolic Luneburg lens, whose refractive index profile is engineered to support continuous transitions between hyperbolic topological phases via out-of-plane permittivity control and coordinate transformations (Wick rotation). Central to this framework is the explicit mapping between model parameters (such as the “energy” parameter E in the lens profile) and the local topological character of the hyperbolic state.

1. Hyperbolic Topological Transitions through Spatially Graded Refractive Index

The GRIN (gradient-index) lens approach yields a controllable spatial modulation of the refractive index, which fundamentally determines the local IFC topology in real space. In the standard (Euclidean) Luneburg lens, the refractive index profile is

$n^2(r) = 2E - r^2$

where $r = \sqrt{x^2 + y^2}$ and E is a tunable parameter. For $n^2 > 0$, electromagnetic waves are propagating; for $n^2 < 0$, they are evanescent. By spatially engineering the out-of-plane permittivity, the local refractive index variance causes a local shift in the topology of the IFC: transitions from regions with elliptic contours (for standard positive index profiles) to hyperbolic regions (where the sign of one principal axis of the effective permittivity tensor flips).

This spatially modulated framework allows the realization and observation of multiple topological regimes (e.g., Type I and Type II hyperbolic) within a single system, as distinguished by the propagation characteristics of electromagnetic waves in different spatial locales.

2. Wick Rotation and Emergence of Real-Space Hyperbolic Dispersion

To realize hyperbolic (rather than elliptic) topology in real space, the system undergoes a Wick rotation—a formal substitution that transforms a spatial coordinate into an imaginary one: $x' = i x,\qquad y' = y$ This rotation converts the Euclidean metric $ds^2 = dx^2 + dy^2$ into a Minkowski-like metric $ds'^2 = -dx'^2 + dy'^2$. Consequently, the original refractive index profile in the $(x, y)$ plane

$n^2(x, y) = 2E - x^2 - y^2$

becomes, after Wick rotation,

$n'^2(x', y') = 2E + x'^2 - y'^2 = 2E - y^2 + x^2$

This analytically continues the spatial profile into a regime where the refractive index changes sign in both “time-like” and “space-like” directions, mimicking the underlying structure of hyperbolic materials.

The local dispersion relation likewise transforms from elliptic ($k_x^2 + k_y^2 = n^2$) to hyperbolic ($k_x^2 - k_y^2 = n'^2$), with

• $n'^2 > 0$: Type I hyperbolic regime (IFC opens along $k_x$)
• $n'^2 < 0$: Type II hyperbolic regime (IFC opens along $k_y$)

The critical line $n'^2=0$ delineates the real-space transition between these two topological phases and is tunable via spatial modulation of the refractive index.

3. Harmonic Oscillator Model: Linking Physical Parameters and Topology

The real-space topological transitions can be captured by mapping the system to a harmonic oscillator Hamiltonian, whose energy separation is determined by the parameter $E$. After appropriate coordinate transformation and variable separation, the wave equation reduces to two decoupled harmonic oscillators: $$- \frac{\hbar^2}{2m} \frac{d^2}{dx^2}\varphi(x) = A\varphi(x)$$

$$- \frac{\hbar^2}{2m} \frac{d^2}{dy^2}\chi(y) = (A + E)\chi(y)$$

Here $A$ is an integer arising from separation of variables, and E directly tunes the offset between the oscillators. The energy degeneracy and parity between the x and y oscillators determine the real-space caustic structure and, ultimately, the dominant hyperbolic topology.

• For $E = 0$: Both oscillators are degenerate, yielding four-fold symmetric caustics—a regime where Type I and Type II hyperbolic domains coexist.
• For $E > 0$: Caustics are expelled outside the “forbidden” region ($n'^2=0$), resulting in Type I topology.
• For $E < 0$: Caustics are confined within the forbidden domain, favoring Type II topology.

This mapping provides an exact correspondence between a continuous local material parameter (E) and the emergent hyperbolic topological character at any spatial point.

4. Real-Space Observation and Coexistence of Multiple Topologies

Unlike previous momentum-space studies, the graded real-space design enables observation of distinct hyperbolic (Type I, Type II, and coexistence) regimes within a single photonic device. The hyperbolic Luneburg lens, under spatially modulated permittivity and Wick rotation, contains regions separated by the curve $n'^2 = 0$, along which the topological state transitions between Type I and Type II hyperbolic dispersion. This spatial coexistence is marked by caustic patterns and focal structures corresponding to the underlying topological regime.

Such devices permit direct optical visualization and experimental paper of topological transitions, with the possibility of dynamically reconfigurable operation by varying the out-of-plane permittivity or the profile parameter E.

5. Extension to Morse Lenses and Generalization

The harmonic oscillator analogy naturally extends to a broader class of GRIN devices—generalized “Morse lenses”—where the index profile is an arbitrary quadratic (or higher-order) spatial function. The inclusion of Wick rotation and real-space gradient control generalizes the class of possible topological transitions, including both abrupt and smooth (continuous) transformations between elliptic, Type I, and Type II hyperbolic domains. The theoretical analysis for the hyperbolic Luneburg lens therefore provides a map for designing entire families of metamaterials and photonic devices with tunable, spatially dependent topological characteristics.

6. Implications for Photonics and Metamaterial Design

This continuous real-space control over IFC topology introduces a paradigm for topological photonic device engineering:

• On-chip photonic routing and focusing: Directing and steering light by sculpting local topological environments in real space.
• Reconfigurable optics: Dynamically modulating the lens parameter E or the out-of-plane dielectric to switch between hyperbolic regimes and associated functionalities.
• Edge state engineering: Facilitating formation, annihilation, or transport of topologically protected edge modes at spatially engineered topological boundaries.

The framework is generalizable to other materials (e.g., 2D van der Waals systems supporting in-plane hyperbolicity), and extends to electronic and acoustic metamaterial platforms by analogous index/properties modulation.

7. Theoretical and Computational Foundation

The formalism rests on

• Direct mapping between material/structural design parameters (spatially varying refractive index, parameter E) and the local topological state
• Transformation of the governing equation by Wick rotation to Minkowski metric, enabling Minkowski-space photonic analogs and novel dispersion topologies
• Analytical solution via harmonic oscillator eigenproblem, yielding conditions for real-space topological phase boundaries and multifold caustic patterns

The approach is conceptually robust and computationally tractable, allowing systematic exploration of device architectures with prescribed topological features.

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In summary, spatial modulation of refractive index in GRIN lenses, combined with coordinate transformations (Wick rotation) and harmonic oscillator mapping, enables continuous, real-space hyperbolic topological transitions between Type I and Type II domains. The lens parameter E acts as the tuning knob dictating local topology, and by careful design, devices can exhibit, manipulate, and probe a spectrum of coexisting hyperbolic phases within a single structure. This theoretical and experimental platform opens broad avenues for tunable topological photonic and metamaterial applications, including robust imaging, waveguiding, and real-space control of light-matter interaction (Liao et al., 2 Oct 2025).