Cascaded-mode interferometers: spectral shape and linewidth engineering

Technical Summary: Cascaded-mode Interferometers for Spectral Shape and Linewidth Engineering

Problem Statement
Optical spectral shaping—the manipulation of the amplitude and phase of broadband light—is critical for applications ranging from pulse shaping and microwave waveform generation to biochemical sensing. Traditional approaches face significant limitations:

• Spatial Masking: Early methods using prisms and spatial light modulators offer parallel control but require large footprints and suffer from resolution limits imposed by pixel size and beam diameter, hindering miniaturization.
• Simple Interferometry: Mach-Zehnder interferometers (MZIs) are compact and widely used in integrated photonics but generate output intensities that depend sinusoidally on frequency. This limits their ability to create complex spectral shapes or narrow-linewidth filters without cascading many stages, which introduces complexity.
• Resonant Structures: Devices like Fabry-Perot interferometers, Bragg gratings, and ring resonators can achieve narrow linewidths but rely on multiple reflections. Their performance (finesse) is strictly dependent on mirror/interface reflection coefficients and is highly sensitive to optical losses. Furthermore, the amplitudes and phases in these systems are often coupled, making independent control difficult.

Methodology
The authors propose a novel framework utilizing cascaded-mode interferometers on a Silicon-on-Insulator (SOI) platform. Instead of interfering spatial modes from separate waveguides (as in an MZI), this approach exploits the interference of multiple transverse modes within a single multimode waveguide.

The core components are:

1. Transmissive Mode Converters (TMCs): These are corrugated nanogratings that act as beam splitters for optical modes. They facilitate energy exchange between orthogonal transverse modes (e.g., TE0 and TE2) based on a specific splitting ratio determined by the grating's coupling coefficient ($\kappa$) and length ($L_c$).
2. Multimode Waveguide Gap: A section of the waveguide of length $L_{gap}$ separates the TMCs. During propagation, different modes accumulate phase shifts based on their distinct effective refractive indices ($n_{eff}$).
3. Transfer Matrix Formalism: The system is modeled using a generalized transfer matrix approach. For $N$ TMCs and $M$ modes, the output amplitudes are calculated as $a_{out} = T_c (T_{wg} T_c)^{N-1} a_{in}$. Here, $T_c$ represents the mode conversion matrix, and $T_{wg}$ is a diagonal phase accumulation matrix.

The spectral response is governed by the phase difference $\Delta \phi = \Delta \beta L_{gap} = k_0 \Delta n_{eff} L_{gap}$, where $\Delta n_{eff}$ is the difference in effective indices between the interfering modes. By engineering the splitting ratios of the TMCs and the number of cascaded stages, the authors demonstrate the ability to synthesize arbitrary spectral shapes.

Key Contributions and Results
The paper presents both theoretical derivations and experimental validations on silicon photonic chips:

• Arbitrary Spectral Shaping: The authors demonstrate that by designing the transmittance coefficients ($t_{ij}$) of the TMCs, they can approximate nearly arbitrary spectral functions, overcoming the sinusoidal limitation of standard MZIs.
• Narrow-Linewidth Engineering:
  • The study shows that cascading $N$ mode converters allows for the generation of high-finesse, narrow-linewidth spectra.
  • Optimal Coupling: To achieve narrow linewidths without excessive sidebands, the coupling strength must satisfy $\kappa L_c = \pi / (2N)$.
  • Experimental Validation: Devices with $N=2, 4,$ and $8$ TMCs were fabricated. The results confirm that increasing $N$ significantly reduces the Full Width at Half Maximum (FWHM) linewidth while suppressing off-resonance transmission.
  • Finesse: The finesse ($F = \Delta \lambda_{FSR} / \Delta \lambda_{FWHM}$) is shown to be approximately equal to the number of mode converters ($N$). Crucially, unlike Fabry-Perot resonators, this finesse is loss-independent, as the device operates via transmission rather than cavity buildup.
• Independent Tuning: The Free Spectral Range (FSR) and linewidth can be tuned independently. The FSR is controlled by the gap length ($L_{gap}$), while the linewidth is controlled by the number of converters ($N$).
• Multi-Mode Parallel Processing: The framework was extended to three modes (TE0, TE1, TE2) using TMCs with dual-periodicity gratings. This enabled the simultaneous generation of distinct spectral shapes for different modes, demonstrating parallel spectral engineering capabilities.

Significance and Claims
The paper claims that cascaded-mode interferometers offer a compact, flexible, and robust alternative to traditional spectral shaping tools. Key advantages highlighted include:

• Loss Robustness: Because the device relies on transmissive interference rather than resonant cavity buildup, its spectral performance (finesse) is not degraded by optical losses, unlike Fabry-Perot interferometers. This makes it suitable for integrating switchable or active materials without compromising spectral quality.
• Compactness and Scalability: The approach eliminates the need for multiple waveguides or large spatial masks, enabling high-resolution spectral control in a single multimode waveguide.
• Distributed Sensing: The transmissive nature allows light energy to be distributed across the entire device, offering advantages for distributed sensing applications compared to local-area sensing in resonators.
• Generalized Framework: The work provides a generalized theoretical framework for spectral shaping using orthogonal modes of different orders, polarizations, and angular momenta.

The authors suggest potential applications in optical metrology, sensing, calibration, and optical computing, noting that the devices could be made programmable via thermo- or electro-optical effects for integrated computing applications.