Flat-Top Multi-Pole Filters: Principles & Applications

Updated 20 October 2025

Flat-top multi-pole filters are engineered structures that achieve a maximally flat Butterworth response with minimal in-band ripple and steep out-of-band attenuation.
They leverage classical circuit analogies and advanced electromagnetic as well as photonic synthesis methods to finely tune bandwidth, rejection levels, and device compactness across various frequencies.
Implementations using metatronic design, resonator cascades, and coupled Fabry–Pérot configurations offer practical benefits for high-sensitivity detection, signal processing, and integrated photonic systems.

Flat-top multi-pole filters are engineered structures designed to provide maximally flat transmission or reflection spectra across a defined passband, while steeply attenuating signals outside that band. Their defining feature is the “flat-top” response—where the transmission (or reflection) within the passband closely approximates an ideal rectangular profile—and minimal ripple. Implementation approaches leverage both classical circuit analogies and more advanced electromagnetic and photonic architectures, permitting precise control over bandwidth, rejection, and device compactness from microwave to optical frequencies.

1. Maximally Flat Butterworth Response and Spectral Lineshapes

The canonical flat-top filter response is described mathematically by the Butterworth transfer function. For a filter of order $N$ , the squared modulus of the transmission or reflection coefficient at angular frequency $\omega$ (centered on $\omega_0$ and with bandwidth parameter $\Delta$ ) is:

$T_{\mathrm{BW},N}(\omega) = \frac{1}{1 + \left( \frac{\omega - \omega_0}{\Delta} \right)^{2N}}$

This polynomial response ensures zero ripple in the passband and progressively steeper spectral edges as $N$ increases. In real implementations, the Butterworth response arises from constructive and destructive interference among multiple resonances, engineered either via lumped-element analogies (capacitance/inductance in electronics, dielectric/plasmonic meta-atoms in photonics) or by direct coupling of physical resonator modes.

The maximally flat property is essential for applications in high-sensitivity detection, signal processing, and background suppression, as detailed extensively for both microwave and optical filter systems (Li et al., 2016, Li et al., 16 Oct 2025, Doskolovich et al., 2019, Doskolovich et al., 2019).

2. Metatronic and Photonic Synthesis of Multi-Pole Flat-Top Filters

A leading approach for realizing flat-top responses at optical frequencies is the metatronic design framework, which adapts electronic filter synthesis procedures to nanoscale photonic systems (Li et al., 2016). In this method:

Dielectric metasurfaces act as shunt capacitors: their effective capacitance $C_{\text{slab}} = a\epsilon_d$ is set by thickness $a$ and permittivity $\epsilon_d$ .
Plasmonic metasurfaces serve as shunt inductors: with inductance $L_{\text{slab}} \approx a/(-\epsilon_m)$ for permittivity $\epsilon_m$ negative and weakly dispersive.
Filters are synthesized by stacking these metasurfaces, arranging them according to normalized Butterworth coefficients $\{g_n\}$ : for a low-pass prototype, $C_n = g_n/(Z_0\omega_{3\mathrm{dB}})$ and $L_n = g_n Z_0/\omega_{3\mathrm{dB}}$ .

Transformations (e.g., frequency and impedance mapping) generalize structures to high-pass, band-pass, and band-stop responses while preserving the flat-top lineshape. Theoretical designs are validated via full-wave electromagnetic simulations, exhibiting accurate bandwidth and steep attenuation outside the passband (Li et al., 2016). This modular, circuit-inspired approach enables compact and integrated devices with customizable dispersion.

3. Multi-Resonator Photonic Implementations: Scattering Matrix Composition

Beyond lumped-element analogies, flat-top transmission and reflection responses can be realized by combining several identical resonant structures with precisely engineered phase relationships (Doskolovich et al., 2019, Doskolovich et al., 2019). For example:

“W-structures” (core/cladding multilayers) or resonant dielectric ridges are arranged in series, with phase-shift layers separating each resonator.
The composite response is derived using the scattering matrix formalism and the Redheffer star product:
- The resonance parameters of each building block are captured by complex poles ( $\omega_p$ , $\lambda_p$ ).
- The thickness and refractive index of phase-shift layers set the accumulated phase ( $\psi$ ), enabling Butterworth-like pole-zero cancellation in the global transfer function.

Careful phase tuning (e.g., $\psi + \phi = \pi(m-\frac12)$ for second-order filters) guarantees exact correspondence to Butterworth filter responses. Increasing the number of constituent resonators enhances passband flatness and bandwidth control; for instance, two or four resonators yield second- or third-order Butterworth profiles, respectively. The design is robustly validated via rigorous numerical simulations, showing rectangular passbands with subnanometer widths and sharp spectral edges (Doskolovich et al., 2019). When implemented in waveguides, near-bound-state-in-the-continuum (BIC) regimes permit arbitrarily narrow flat-top bands without device size expansion (Doskolovich et al., 2019).

4. Strongly Coupled Microwave and Coaxial Filter Designs

In the microwave domain, flat-top multi-pole filters are conventionally achieved via macroscopic resonator assemblies (posts, cavities) with tunable coupling (Amari et al., 21 May 2025). Notably:

Strong coupling between posts necessitates describing the system via transverse EM eigenmodes rather than localized lumped resonances.
Dual-post and triple-post units exhibit “even” and “odd” modes; transmission zeros (TZs) critical for spectral steepness and rejection are set by the interplay between these modes, governed by formulae such as $\omega_z = \omega_{\text{od}} + p(\omega_{\text{od}}-\omega_{\text{sp}})$ .
The location of TZs is manipulated through physical port placement and coupling symmetry, directly controlling passband equiripple and skirt sharpness.

While similarity transformations allow mathematical equivalence between circuit topologies, only EM boundary-condition-respecting models yield physically predictive filter designs. Full-wave simulation and empirical tuning ensure accurate realization of the desired spectral features. The versatility of this approach is demonstrated in 2-order in-line filters, which can create transmission zeros for enhanced flat-top characteristics (Amari et al., 21 May 2025).

5. Multi-Pole Flat-Top Filters via Multipolar Resonance Engineering

All-dielectric metasurfaces utilizing core-shell geometries support multiple Mie resonances (electric dipole, magnetic dipole, quadrupole) (Monti et al., 2021). By balancing these resonances:

A broad reflection band is established, with a narrow transmission window (“passband”) embedded within, set by the quadrupole resonance.
The surface impedance homogenization formalism models the metasurface as a thin sheet supporting electric and magnetic currents; transmission and reflection coefficients are derived from symmetric ( $Z_\mathrm{symm}$ ) and asymmetric ( $Z_\mathrm{asymm}$ ) impedance components.

Analytical and full-wave simulations confirm that properly tuned core-shell parameters (permittivity, geometry) offer Q-factors as high as 80 (for $\sim$ 2% fractional bandwidth), with materials ranging from high-permittivity ceramics (microwave) to silicon or GaP (optical). These filters find utility in self-filtering antennas (“filtennas”), where the metasurface enhances out-of-band rejection without compromising main-beam directivity (Monti et al., 2021).

6. Fabry-Pérot, BIC, and Polarization Mode Coupling

A recent development, notably in optical filtering, is the use of coupled polarization modes within a single Fabry–Pérot cavity to realize flat-top (second-order Butterworth) filters (Li et al., 16 Oct 2025). Key elements include:

Two orthogonal polarization modes are coupled via intracavity birefringence (e.g., tilted wave plate or stressed glass), forming “virtual” resonators.
Transmission is described by $T(\delta) = 1/(1 + (2\delta/\kappa)^4)$ , where $\delta$ is detuning and $\kappa$ sets FWHM.
The architecture achieves passband widths (e.g., 2.68 GHz), stopband suppression (up to 43 dB), and low insertion loss (2.2 dB), outperforming traditional multilayer dielectric stacks with passbands typically exceeding 100 GHz.

Furthermore, in composite photonic structures, Fabry–Pérot BIC conditions ( $\psi^{(1)} + \phi = \pi m$ ) give rise to sharp electromagnetically induced transparency (EIT)-like peaks within a broad rejection dip. These effects, stemming from controlled modal interference, provide additional flexibility for high-resolution spectral engineering (Doskolovich et al., 2019).

7. Wideband Flat-Top Filters via Multipole Resonance Merging

In microwave and THz domains, spoof localized surface plasmon (SLSP) structures are used for wideband flat-top filtering (Zhang et al., 2018). SLSP disks with dense groove arrays support numerous multipole standing-wave modes, whose frequencies are compressed by nonlinear dispersion. Carefully tapered interlayer microstrip couplings blend these discrete resonances, yielding a continuous, nearly rectangular passband with up to 73% fractional bandwidth and excellent spurious rejection. Geometric parameters such as disk radius allow continuous tuning of passband width and location, while group delay is maintained within 1 ns except at band edges (Zhang et al., 2018).

Summary Table: Core Flat-Top Multi-Pole Filter Architectures

Architecture	Implementation Domain	Key Principle
Metatronic Butterworth Filters	Optical; Metasurface stacks	Lumped circuit analog; impedance transforms
Resonator Cascade (Scattering)	Photonic (thin-films, W-structure)	Scattering matrix, phase-shift engineered
Strongly Coupled Post Arrays	Microwave/coaxial	EM eigenmodes; TZ via mode symmetry
Multipolar Dielectric Metasurface	Microwave/Optical	Mie resonance balancing; surface impedance
Coupled Polarization Fabry-Pérot	Optical	Birefringent mode coupling; destructive interference
SLSP Multi-Resonance	Microwave/THz	Multipole merging; tapered microstrip coupling

Flat-top multi-pole filters provide the spectral selectivity, background rejection, and tunability essential for advanced signal processing, communication, sensing, and photonic integration. Diverse implementation paradigms—ranging from nanocircuit metasurfaces and cascaded resonator matrices to multipolar metasurfaces and polarization-coupled cavities—permit finely controlled passband topologies and performance, accessible across all frequency domains.