Spatially Confined Photonic States

• Spatially confined photonic states are electromagnetic modes that are localized by disorder, defects, topology, or material effects to scales below the wavelength.
• They are quantified using techniques such as NSOM and the Inverse Participation Ratio, which measure localization lengths and capture transitions from extended to confined regimes.
• These states critically influence device engineering by modifying light transmission, enhancing quantum optical interactions, and enabling advanced photonic applications.

Spatially confined photonic states are electromagnetic modes whose spatial extent is fundamentally restricted by structural disorder, engineered defects, topology, or material effects to regions much shorter than the wavelength or propagation length characteristic of the system. These states can exist in a variety of platforms—photonic crystals, disordered waveguides, metasurfaces, topological lattices, and at material interfaces—where they can profoundly alter light–matter interactions, transmission, and device performance. Their characterization and quantitative measurement are essential in understanding localization phenomena, optical confinement limits, and the control of photonic states for applications in quantum optics, sensing, and photonic engineering.

1. Origin and Physical Basis of Spatial Confinement

Spatial confinement in photonic systems arises from a range of physical mechanisms:

• Anderson Localization: In disordered waveguides and photonic crystal structures, multiple scattering causes destructive interference, resulting in exponentially localized electromagnetic modes whose spatial extent (localization length) can be much smaller than the physically relevant dimensions of the waveguide. This mechanism gives rise to states where the optical field is effectively trapped in localized regions, sometimes not observable in standard transmission measurements (1111.5942).
• Topological Defects and Symmetry: In topological photonic lattices, specially engineered defects at symmetry-protected interfaces or boundaries (e.g., in Su–Schrieffer–Heeger-like models and higher-order topological insulators) give rise to exponentially localized, midgap, or corner states. The confinement dimensionality and protection against disorder are enforced by topological invariants such as the winding number or quantized bulk polarization (Schomerus, 2013, Li et al., 2019).
• Material and Structural Resonances: Highly confined states are also realized by coupling free-space photons to material excitations—such as surface phonon polaritons in SiC metasurfaces—where the field is squeezed into nanoscopic regions due to boundary conditions, symmetry breaking, or strong light–matter coupling. These can yield unit cell volumes orders of magnitude below the diffraction limit (Nan et al., 27 Mar 2024).
• Interface- and Defect-Induced States: Intentional alteration of local structural parameters (such as defect pore radius in 3D photonic band gap superlattices (Kozoň et al., 2023)) and engineered edge or interface conditions (e.g., staggered bianisotropy in one-dimensional arrays (Bobylev et al., 2019)) provide further routes to creating robust spatial confinement.

2. Quantitative Characterization and Measurement Techniques

Several experimental and theoretical approaches probe the degree of spatial confinement:

• Near-Field Scanning Optical Microscopy (NSOM): Use of an aluminum-coated tapered fiber probe (~200 nm diameter) in the evanescent region enables high-resolution spatial mapping of the electric field amplitude. The probe both detects and locally perturbs the photonic state, producing a resonance spectral shift whose magnitude directly encodes the spatial extent (localization length) of the mode (1111.5942).
• Perturbation Method: The fractional spectral shift, Δλ/λ, induced by the probe is given by

$$\frac{\Delta\lambda}{\lambda} \approx \frac{\alpha^{\textrm{e}}_{\textrm{eff}}}{2V_{\textrm{cav}}}\beta_E + \frac{\alpha^{\textrm{m}}_{\textrm{eff}}}{2V_{\textrm{cav}}}\beta_H$$

where $\alpha^{\textrm{e}}_{\textrm{eff}}$ and $\alpha^{\textrm{m}}_{\textrm{eff}}$ are probe polarizabilities, $V_{\textrm{cav}} = L_{\textrm{ind}} A$ is the effective cavity volume, and $\beta_E$, $\beta_H$ parameterize the field localizations at the probe versus their maxima. By inverting this relation, $L_{\textrm{ind}}$ (the localization length of an individual state) is extracted.

• Participation Ratio (Inverse Participation Ratio, IPR): The IPR, defined as

$$\textrm{IPR} = \frac{\int |E(r)|^4 dr}{\left[\int |E(r)|^2 dr\right]^2},$$

characterizes the spatial concentration of the field; higher IPR denotes greater localization. For a system of length $L_{\textrm{wav}}$, the normalized IPR relates directly to the localization length $L_{\textrm{ind}}/L_{\textrm{wav}}$ (1111.5942).

• Comparison of Methods: Both the spectral shift approach and the IPR method yield quantitatively consistent localization length measurements for individual disorder-induced states, validated by their direct comparison.

3. Regimes of Spatial Confinement and Disorder Effects

The degree of spatial confinement in photonic crystal waveguides depends strongly on wavelength (or group index):

Wavelength / Group Index	Localization Length $L_{\textrm{ind}}$	State Type
$\lambda < 1561$ nm, $n_g < 55$	$L_{\textrm{ind}} > L_{\textrm{wav}}$	Extended Bloch-like
$1561.6$–$1563.2$ nm	$L_{\textrm{ind}} \sim L_{\textrm{wav}}$	Intermediate, disorder-onset
$\lambda > 1563.2$ nm, $n_g > 80$	$L_{\textrm{ind}} \ll L_{\textrm{wav}}$	Strongly localized, hidden

As the system approaches the band edge and $n_g \to$ large, $L_{\textrm{ind}}$ sharply decreases, resulting in strongly confined states that may not contribute to transmission—demonstrating that disorder can induce photonic states that are "hidden" from standard propagation-based measurements.

4. Implications for Device Engineering and Photonic Functionality

• Waveguide Performance: The existence of highly confined localized states in slow-light photonic crystal waveguides near the band edge can compromise transmission, as $L_{\textrm{ind}}$ becomes much smaller than the device length. This is fundamentally important when designing devices intended to exploit slow group velocity for enhanced light–matter interaction.
• Quantum and Nonlinear Optics: Strong spatial confinement increases the photonic density of states locally, enhancing spontaneous emission rates (Purcell effect), enabling low-threshold lasing, and supporting cavity quantum electrodynamics regimes at the nanoscale.
• Probing and Control: The ability to measure individual localization lengths directly (via probe-induced shift or IPR) is particularly valuable for metrology in photonic devices with significant structural disorder or intentional defect engineering.

5. Generalization to Other Systems and Modalities

• Extension to Higher Dimensions and Other Fields: The perturbation and IPR methods are not limited to one-dimensional systems. They can be generalized to characterize spatial confinement in higher-dimensional disordered photonic crystals, and in other wave-based systems such as acoustics and elastic metamaterials, where localized resonant states play an analogous role (1111.5942).
• Beyond Transmission: The observation of photonic states that exist entirely within the waveguide but are not observed in transmission indicates that standard transmission and reflection spectroscopy may dramatically underestimate modal complexity in the presence of disorder.

6. Theoretical and Practical Significance

The direct, quantitative correspondence between local perturbation-induced spectral shifts and the IPR analysis demonstrates that the measurement of spatially confined photonic states can move beyond statistical averages to characterization of individual localized modes. This advances both the fundamental understanding of Anderson localization in photonic crystals and the rational design of photonic structures where disorder-induced confinement is either a limiting factor or an enabling feature. The methodologies established further enable controlled experimental access to mode confinement and light localization in nanophotonic and disordered platforms across a range of frequencies and geometries.