Broadband Achromatic Resonance

• Broadband achromatic resonance is a phenomenon where optical systems maintain resonant enhancement over a wide spectral range with minimal chromatic aberration.
• It employs techniques such as angular dispersion engineering, metasurface design, and inverse optimization to achieve uniform resonance and high field efficiency across broad bandwidths.
• This approach enables advanced applications in imaging, quantum photonics, and flat optics by overcoming conventional dispersion limitations with practical design strategies.

Broadband achromatic resonance refers to optical (or, by extension, acoustic or quantum) systems that exhibit resonant enhancement or tailored wavefront control across a broad spectral bandwidth with minimal or no chromatic aberration. Critically, such systems retain resonance properties—such as constructive interference, deep nulls, or focusing efficiency—across a large range of input frequencies, surpassing the bandwidths allowed by conventional material or structural dispersion. Achieving broadband achromatic resonance underpins advances in ultrathin flat optics, high-contrast astronomical instrumentation, quantum photonics, metasurfaces, and nonlinear optics.

1. Foundational Concepts and Motivation

In standard resonant structures—such as Fabry–Pérot (FP) cavities, diffractive lenses, or multimode beam combiners—resonance conditions are governed by the phase accumulated over specific paths or the interplay of structural and material dispersion. For instance, an FP microcavity transmits only at discrete optical frequencies where the roundtrip phase matches an integer multiple of $2\pi$, yielding resonant linewidths determined by the cavity finesse. This property severely limits the bandwidth for enhanced field buildup, frequency conversion, or imaging, especially under broadband or white-light illumination.

Broadband achromatic resonance aims to circumvent these limits, enabling simultaneous multiwavelength resonance, deep nulling, or diffraction-limited focusing over bands vastly exceeding the structure’s natural spectral selectivity. Approaches span engineered optical dispersion in meta-atoms (Chen et al., 2017, Hu et al., 2022, Lin et al., 26 Apr 2024, Hou et al., 27 Jul 2024), structured substrates (Lin et al., 26 Apr 2024), angular dispersion pre-conditioning (Shabahang et al., 2016, Hall et al., 2022, Turo et al., 2 Oct 2025), mode-selective coupling (Hsiao et al., 2010, Klinner-Teo et al., 2022), inverse design (Chung et al., 2019, Pan et al., 2023, Dong et al., 2020), geometric phase control (Bos et al., 2018, Chen et al., 2017), and hybrid metasurface-multilayer strategies (Pan et al., 2023).

2. Angular Dispersion Engineering and Omni-Resonance

An effective means to achieve achromatic resonance in planar FP cavities is the introduction of angular dispersion (AD) into the incident field (Shabahang et al., 2016, Hall et al., 2022, Turo et al., 2 Oct 2025). This approach "remaps" each wavelength $\lambda$ to strike the cavity at a unique angle $\varphi(\lambda)$ so that the resonance condition,

$4\pi n d / \lambda = 2\pi m$

(for integer $m$, cavity length $d$, and refractive index $n$) is fulfilled over the entire desired spectrum. The roundtrip phase for arbitrary incidence is

$$\chi(\lambda, \varphi) = (4\pi d/\lambda) \sqrt{n^2 - \sin^2\varphi}.$$

By programmatically pre-conditioning the incident field such that

$$\varphi(\lambda) = \psi + \beta_m (\lambda - \lambda_c)$$

for a chosen $\beta_m$ (AD coefficient) and central wavelength $\lambda_c$, every spectral component simultaneously satisfies the resonance for a single longitudinal mode, vastly exceeding the linewidth set by finesse $\mathcal{F}$. The resultant "omni-resonant" state enables, for example, $>$100 nm bandwidth resonance in the visible while maintaining the high field enhancement typical of narrowband cavities (Hall et al., 2022).

Such AD schemes enable broadband color imaging through a cavity, white-light nonlinear optics, or photon-starved quantum applications, even inserting nonclassical broadband entangled-photon states into a single FP mode without disturbing their spectral correlations (Turo et al., 2 Oct 2025). Experimental implementations typically employ dispersive gratings followed by afocal lens systems to achieve the required $\beta_m$ (e.g., $0.23^\circ/$nm at 810 nm), with the cavity axis ($\psi$) set to the degenerate wavelength for SPDC-generated biphoton states.

3. Dispersion-Engineered Metalenses and Flat Optics

Broadband achromatic focusing with metasurfaces relies on engineering both phase and dispersion at the level of individual meta-atoms. In standard designs, focusing phase profiles such as

$$\phi(r, \omega) = -\frac{\omega}{c} \left( \sqrt{r^2 + f^2} - f \right)$$

(where $r$ is radial distance and $f$ the focal length) must be imparted at all $\omega$. To accomplish achromatism, each meta-atom's geometry is chosen to independently control both phase (via geometric rotation or propagation phase) and group delay (dispersion). For instance, nanofins of varying geometry provide a set of available phase delays and dispersions, with the overall response optimized such that group delay ($\partial\phi/\partial\omega$) and higher derivatives (group delay dispersion) match target lens profiles across wavelengths (Chen et al., 2017).

Optimization strategies—such as global minimization of root-mean-square phase error across wavelengths, multi-wavelength particle swarm optimization for phase assignment (Hou et al., 27 Jul 2024), or neural network-based (GAN) searches (Lio et al., 2020)—facilitate high efficiency, focal length invariance, and arbitrary polarization performance over bandwidths from ultraviolet to near-infrared. Multilayer or zone-multiplexed substrates augment the phase compensation provided by meta-atoms alone, enabling metalenses with NA approaching unity and bandwidths exceeding 350 nm (Lin et al., 26 Apr 2024, Pan et al., 2023, Chung et al., 2019).

4. Broadband Achromatic Resonance in Quantum and Nonlinear Photonics

Angular dispersion strategies are particularly potent in quantum optics where entangled photon states, such as frequency-anticorrelated photon pairs from SPDC, are otherwise poorly admitted to narrowband resonant cavities. AD can precondition both members of an entangled pair so that the entire biphoton spectrum is coupled to a single FP longitudinal mode, preserving spectral and temporal correlations (Turo et al., 2 Oct 2025). Critical equations include:

• Cavity resonance at normal incidence: $k_m = m\pi/(nd)$, $\lambda_m=2nd/m$.
• At angle $\varphi$: $$\lambda_m(\varphi) = (2nd/m)\sqrt{1-\sin^2\varphi/n^2}$$.
• With AD: $$\varphi(\lambda) \approx \gamma (\lambda - \lambda_0)$$ for degenerate wavelength $\lambda_0$ and AD coefficient $\gamma$.

Consequences include the enabling of omnidirectional resonant enhancement for quantum nonlinear effects, photon-number-resolved detection, and coherent perfect absorption in broadband, few-photon scenarios.

5. Device Architectures and Applications

Integrated Optics and Beam Combiners

Symmetric multi-waveguide structures (e.g., three-waveguide combiners (Hsiao et al., 2010)) or photonic tricouplers (Klinner-Teo et al., 2022) are designed to achieve broadband, achromatic, and polarization-insensitive interference. Careful adiabatic variation in coupling coefficients (achieved via gradual tapering and tailored spacing) ensures minimal nonadiabatic mixing, preserving deep broadband nulls required for applications such as mid-infrared nulling interferometry for exoplanet detection.

Metasurface Mirrors and Polarization Devices

Anomalous mirrors utilizing metal–insulator–metal resonators or broadband achromatic half-wave plates (Nemilentsau et al., 2016, Zhang et al., 2011) achieve frequency-invariant phase gradients or phase delays by exploiting geometric or propagation phase, multilayer architectures, and careful design of inductive and capacitive responses for phase-control over large spectral regions. For instance, anomalous mirrors achieve $>$98% steering efficiency for reflection angles up to $40^\circ$, with Joule losses $<$10% (Nemilentsau et al., 2016), while CMB-polarimetry half-wave plates offer achromatic $\sim180^\circ$ phase-shift over $\sim$125–250 GHz (Zhang et al., 2011).

Advancements in Flat Optical and Quantum Devices

Multiplexed substrates (zone-division multiplexing) (Lin et al., 26 Apr 2024) and multilayer 3D-printed meta-optics (Pan et al., 2023) decouple the required phase dispersion from meta-atom limitations, enabling high-NA and broadband operation suitable for miniaturized microscopy and imaging. In the quantum regime, omnidirectional cavity resonance enabled by angular dispersion (Turo et al., 2 Oct 2025) preserves the entangled structure of single- or biphoton states.

Applications include:

• Omnidirectional, broadband imaging in solar window architectures (Hall et al., 2022).
• High-contrast astronomical nulling interferometers (Hsiao et al., 2010, Klinner-Teo et al., 2022).
• Achromatic, polarization-insensitive and high-efficiency metasurfaces for AR/VR, telecommunications, and spectral imaging (Chen et al., 2017, Hou et al., 27 Jul 2024).

6. Limitations, Trade-offs, and Future Directions

Key trade-offs remain between achievable NA, focusing efficiency, spectral bandwidth, and device complexity. For instance, in diffractive architectures, the maximum Fresnel number for achromatic operation is tied to the phase step height and refractive index contrast, typically expressed as $FN_\text{max} = p\Delta n + 1$ with $p = h/\lambda_0$ (Engelberg et al., 2020). Metalens performance (empirically, $\Delta\omega \lesssim C_1/\text{NA}$) has been improved via inverse design and multilayer strategies, although efficiency at very high NA often falls below bulk optics benchmarks unless leveraging freeform 3D structures or additional phase management (Chung et al., 2019, Pan et al., 2023).

Integration of metasurfaces with programmable dispersion functions, ultrafast dynamic tuning, and inherent nonlinearities is an open direction. Further reduction in device thickness, expansion of bandwidth to cover the entire visible or mid-IR, deepening nulls for interferometry, and realization of spectrally multiplexed omnidirectional cavities for quantum applications are active research areas.

7. Summary Table: Representative Broadband Achromatic Resonance Approaches

Approach	Structural Principle	Achieved Bandwidth
Angular dispersion (FP cavity)	AD pre-conditioning, resonance remapping	$>$100 nm (visible), $>$20 nm (entangled photons) (Hall et al., 2022, Turo et al., 2 Oct 2025)
Geometric/propagation phase metasurfaces	Phase and group delay engineering	200 nm (470–670 nm), efficiency $>$80% (Chen et al., 2017, Hou et al., 27 Jul 2024)
Multilayer zone multiplexing	Stepwise substrate phase, meta-atom hybrid	NA = 0.9, 650–1000 nm (Lin et al., 26 Apr 2024)
Inverse design metalens	Topology optimization, multilayer/freeform	NA up to 0.99, visible to NIR (Chung et al., 2019, Pan et al., 2023)
Symmetric waveguide combiner	Adiabatic mode evolution, coupled-waveguides	Mid-IR, nulling over broad band (Hsiao et al., 2010)
Tricoupler for nulling	Transfer symmetry, near-achromatic phase shifter	$1.4$–$1.7~\mu$m, $0.6^\circ$ phase error (Klinner-Teo et al., 2022)

8. Concluding Perspective

Broadband achromatic resonance combines rigorous dispersion engineering, structural symmetry, material selection, and often advanced computational/inverse design to overcome fundamental spectral limits in resonant photonic systems. The realized architectures accelerate ultrathin flat-optics, quantum-light interfaces, robust astronomical instrumentation, and compact imaging, often achieving simultaneous high efficiency, polarization insensitivity, and spectral invariance in device responses. Ongoing challenges include managing fabrication tolerances, further improving efficiency at ultra-high NA, integrating programmable dispersion, and broadening applicability to new functional regimes and quantum states.