Floquet-Engineered Directional Couplers

• The paper demonstrates that periodic spatial modulation in silicon waveguides enables selective suppression of TE modes while maintaining efficient TM coupling.
• Using coupled-mode theory and Floquet analysis, the approach achieves an ultra-compact (~20 μm) footprint with sub-dB insertion losses and >20 dB polarization extinction ratios over >130 nm bandwidth.
• The technique is highly CMOS-compatible, offering robust manufacturability and practical integration for advanced photonic circuits in applications like wavelength multiplexing and quantum photonics.

Floquet-engineered directional couplers are silicon photonic devices that leverage periodic spatial modulation to control optical power exchange between adjacent waveguides, enabling precise, broadband, and polarization-selective coupling. Recently, Floquet engineering—a technique generally associated with temporal modulation in quantum and photonic systems—has been adapted to the spatial domain in nanophotonics, yielding directional couplers with engineered coupling profiles for transverse-electric (TE) and transverse-magnetic (TM) modes. These structures offer ultra-compact footprints ($\sim$20 μm coupling length), sub-dB insertion losses, >20 dB polarization extinction ratios, and broad operational bandwidths (>$130$ nm), positioning them as high-performance polarization beam splitters within integrated photonic circuits (Ma et al., 17 Jan 2026).

1. Fundamental Principles of Floquet-Engineered Coupling

Floquet engineering in silicon directional couplers exploits periodic modulation of waveguide geometry (typically the core width) along the propagation axis. The mutual coupling coefficient $\kappa(z)$ between parallel waveguides is modulated as: $$\kappa(z) = \kappa_0 + \Delta \kappa \cos(\Omega z + \varphi)$$ with $\Omega = 2\pi/\Lambda$ (modulation spatial frequency), $\Lambda$ (modulation period), $\Delta \kappa$ (coupling amplitude), and $\varphi$ (relative phase offset between core modulations).

Coupled-mode theory describes the evolution of modal amplitudes $(A, B)$ in each waveguide for both TE and TM polarizations:

• TE mode:

$$\frac{dA_\text{TE}}{dz} = -i\kappa_\text{TE}(z) B_\text{TE}, \;\; \frac{dB_\text{TE}}{dz} = -i\kappa_\text{TE}(z) A_\text{TE}$$

• TM mode (analogous with $\kappa_\text{TM}(z)$).

Floquet theory yields an effective coupling: $$\kappa_\text{eff} = \frac{1}{\Lambda} \int_0^\Lambda \kappa(z) e^{i \Delta \beta z} dz$$ where $\Delta \beta$ is the instantaneous difference in propagation constants, itself modulated by the periodic geometry.

With Jacobi–Anger expansion, the effective coupling for each polarization reduces to: $\kappa_\text{eff} = \kappa_0 J_0(\xi)$ where $J_0$ is the zeroth-order Bessel function and $\xi = 2\Delta\beta/\Omega$. By selecting $\xi_\text{TE}$ at a zero of $J_0$ (e.g., $\xi \approx 2.405$), one can fully suppress TE-mode power transfer, while choosing $\xi_\text{TM}$ far from the root so that TM coupling remains efficient (Ma et al., 17 Jan 2026).

2. Device Architecture and Key Parameters

The prototypical Floquet-engineered PBS comprises two parallel strip waveguides on a 220 nm SOI device layer, each subjected to periodic width modulation out-of-phase ($\varphi_2-\varphi_1=\pi$). Design variables include:

Parameter	Typical Value / Role	Reference
Device layer	220 nm Si on 3 μm BOX, air cladding	(Ma et al., 17 Jan 2026)
Avg. waveguide width	$W_0\approx 450$ nm	(Ma et al., 17 Jan 2026)
Modulation amplitude	$\Delta w = 50$ nm	(Ma et al., 17 Jan 2026)
Modulation period	$\Lambda = 7$ μm	(Ma et al., 17 Jan 2026)
Core separation	$215$ nm gap	(Ma et al., 17 Jan 2026)
Coupler length	$L_{\text{tot}}=20$ μm (2 × 10 μm sections)	(Ma et al., 17 Jan 2026)
Output grating params	TE: 630 nm, TM: 980 nm period	(Ma et al., 17 Jan 2026)

Two sequential coupler sections serve to enhance TM cross-port extinction and clean up residual signal. The periodic modulation is implemented as $W(z) = W_0 \pm \Delta w \sin(2\pi z/\Lambda)$ in the two waveguide cores.

3. Performance Characteristics and Experimental Metrics

Floquet-engineered PBSs demonstrate sub-dB insertion loss and high polarization extinction ratio across broad telecom bands:

Wavelength Range (nm)	PER$_\text{TE}$ (dB)	PER$_{\text{TM}}$ (dB)	IL$_\text{TE}$ (dB)	IL$_\text{TM}$ (dB)
1483–1620	$>$20	$>$20	0.15	1.2

PER is defined as $10\log_{10}(P_\text{desired}/P_\text{undesired})$ at a given port, while insertion loss is $-10\log_{10}(P_\text{out}/P_\text{in})$. Both figures remain robust across the $>130$ nm bandwidth, with device response at $1550$ nm showing near-ideal mode separation (Ma et al., 17 Jan 2026).

Bandwidth is tunable via:

• Modulation amplitude $\Delta w$: Higher amplitudes sharpen TE suppression but can narrow the effective bandwidth for the zero-coupling condition.
• Period $\Lambda$: Longer periods increase bandwidth, but require longer couplers or may dilute extinction strength.
• Gap: Smaller gap increases TM coupling rate but can introduce dispersion affecting bandwidth.

4. Comparison with Alternative Broadband PBS Architectures

Floquet-engineered couplers represent a distinct operational regime when compared to:

• Slot-waveguide splitters: Rely on single-mode slot supermodes and symmetric adiabatic tapers for dual-polarization, ultra-broadband 50:50 splitting, but do not exploit dynamic coupling modulation (González-Andrade et al., 2018).
• Topographically anisotropic photonics (TAP): Utilize embedded form-birefringent multilayer stacks (MLS) to establish polarization-selective modal indices within compact adiabatic splitter configurations. TAP PBSs define split states (B, ACH, ASH) and employ anisotropic effective-medium theory for high fractional bandwidth, but their bandwidth and device length trade-offs differ (Chiles et al., 2017).
• Inverse-designed splitters: Employ adjoint-based FDTD optimization to yield sub-$3\,\mu$m² footprint PBSs with >16 dB ER and 200 nm bandwidth, agnostic to theoretical mechanism but robust to fabrication errors (Goudarzi et al., 2021).
• Metasurface PBSs: Use arrays of Si Mie resonators to realize polarization-mode separation via magnetic resonant reflection/transmission; effective in planar geometry for far-field applications (Slovick et al., 2016).

A plausible implication is that Floquet PBSs offer a compact, robust, and lithography-tolerant solution in the regime where background index and device footprint constraints preclude high-order anisotropic or inverse-designed realizations.

5. Fabrication Considerations and CMOS Compatibility

Fabrication proceeds via electron-beam lithography and ICP dry etching of SOI wafers. Modulated waveguide widths (sub-60 nm features) and etch depths are well within modern CMOS foundry tolerances. Shallow-etched grating couplers are used to separately address TE and TM launch conditions.

No multi-level processing or non-binary etch steps are required, contrasting with the MLS refill steps in TAP (Chiles et al., 2017) or quasi-continuous permittivity patterns in inverse-designed PBSs (Goudarzi et al., 2021). This suggests high process yield and compatibility with volume manufacturing for photonic integrated circuits.

6. Practical Applications and Trade-Offs

Floquet-engineered directional couplers define a paradigm for broadband polarization beam splitting in photonic integrated circuits, suitable for wavelength-division multiplexing, quantum photonics, and remote polarimetric sensing. Device miniaturization ($<25$ μm), broad operational range (C/L-band), and high extinction support cascaded circuit architectures.

Trade-offs inherent in Floquet PBS design include:

• Wavelength vs. extinction optimization: The zero-coupling band for TE is finite; balancing $\Delta w$ and $\Lambda$ is critical.
• TM insertion loss: Although TM coupling remains strong, insertion loss approaches 1.2 dB due to residual mismatch or modal overlap.
• Fabrication tolerance: Modulation must be accurately patterned, but sub-10 nm deviations do not dominate performance (Ma et al., 17 Jan 2026).

Floquet PBSs co-exist in literature alongside slot-based, anisotropic-material, and inverse-designed architectures, offering a unique combination of theoretical basis (Floquet suppression/enhancement), manufacturability, and integration potential for next-generation photonic circuits.