Spectrally smooth broadband response via autocorrelation-constrained inverse design

Technical Summary: Spectrally Smooth Broadband Response via Autocorrelation-Constrained Inverse Design

Problem Statement
Time-domain inverse design in photonics is widely recognized for its efficiency in maximizing optical device performance across broad frequency ranges. Standard approaches utilize time-integrated field quantities (e.g., energy, flux, or dissipated power) derived from Poynting's theorem. Through Parseval's theorem, these time-domain objectives correspond to frequency-integrated responses. However, the authors identify a critical limitation: while computationally efficient, these integrated objectives measure only the total response over a bandwidth. They fail to capture or penalize undesired in-band spectral ripples, narrow spectral features (such as dips), or sidelobes. Consequently, an optimizer may converge to a local maximum of total energy while producing a spectrally "rough" response with sharp defects that are effectively invisible to the objective function. Existing solutions, such as adaptive spectral weighting, require explicit identification of underperforming regions and dynamic modification of the excitation pulse during optimization, which adds complexity.

Methodology
The authors propose a time-domain metric designed to quantify spectral smoothness without requiring pulse adaptation. The core of the methodology relies on the Wiener–Khinchin theorem, which establishes that the autocorrelation function of a signal and its energy spectral density form a Fourier-transform pair.

• The Metric ($T_{AC}$): The authors define a metric based on the weighted long-lag autocorrelation energy. A broad, smooth spectrum corresponds to a rapidly decaying autocorrelation function. Conversely, narrow spectral features (like dips or ripples) manifest as slowly decaying, oscillatory tails in the autocorrelation function.
• Formulation: The metric $T_{AC}[s; s_0]$ is defined as the ratio of the weighted long-lag autocorrelation energy of the response signal $s(t)$ to that of a broadband reference signal $s_0(t)$. The weighting function increases quadratically with lag ($\tau^2$) to emphasize late-time "ringing" tails.
  • $T_{AC} = 1$ implies a response similar to the smooth reference.
  • $T_{AC} > 1$ indicates the presence of sharp spectral features.
• Optimization Framework: This metric is incorporated into a density-based topology optimization (TopOpt) framework using the Finite-Difference Time-Domain (FDTD) method. The metric is applied in two ways:
  1. As a Constraint: Maximizing reflectance efficiency ($\eta_r$) subject to $T_{AC} \leq 1$.
  2. As a Penalization Term: Maximizing a weighted objective $J = (1-\alpha)\eta_r - \alpha T_{AC}$, where $\alpha$ balances efficiency against spectral smoothness.

Key Contributions

1. Novel Time-Domain Metric: Introduction of a metric that translates the detection of narrow spectral features into a time-domain quantity (long-lag autocorrelation energy), making them accessible to standard time-domain gradient-based optimizers.
2. Demonstration on Bragg Gratings: Application of the metric to the inverse design of one-dimensional dielectric Bragg mirrors. The study demonstrates that maximizing total energy alone is insufficient to prevent spectral defects, whereas the autocorrelation constraint effectively suppresses them.
3. Physical Insight: The methodology is shown to recover known physical design strategies (apodized and chirped gratings) purely through optimization, validating the metric's ability to guide the solver toward physically meaningful, smooth broadband solutions.

Results
The authors evaluated the approach using 1D dielectric gratings with a weak refractive index contrast ($\Delta n \approx 0.22$) over a bandwidth of 105 THz centered at 300 THz.

• Unconstrained vs. Constrained: In unconstrained optimizations, the resulting reflectance spectra often exhibited prominent narrow spectral features (resembling $\pi$-shifted grating resonances) with high $T_{AC}$ values (e.g., 8.6). Enforcing the $T_{AC} \leq 1$ constraint consistently removed these features, producing smooth, flat-top reflectance spectra.
• Statistical Robustness: A statistical evaluation over 100 random initial densities showed that the constraint did not significantly reduce the median reflectance efficiency ($\eta_r$). The median $\eta_r$ remained nearly identical (0.8096 unconstrained vs. 0.8093 constrained), indicating the constraint acts as a regularizer rather than a performance limiter.
• Pareto Trade-offs: By varying the weighting parameter $\alpha$ in the penalized objective, the authors mapped the trade-off between efficiency and smoothness. A weak penalty ($\alpha = 10^{-2}$) actually improved the local optimum for efficiency while achieving near-unity $T_{AC}$. Stronger penalties led to apodized profiles, trading some peak reflectance for extreme smoothness.
• Discovery of Chirped Structures: In a longer design domain ($L = 17.2 \mu m$), the optimization naturally converged to a chirped grating structure. Parametric analysis of the resulting binary design revealed a linear variation in the local grating period, matching the Bragg vectors of the target spectral bounds. A reconstructed chirped grating based on this pattern achieved comparable performance to the raw TopOpt design, confirming the physical validity of the solution.

Significance and Claims
The paper claims that the proposed metric addresses a central limitation of integrated time-domain objectives by providing direct sensitivity to localized spectral defects. The authors assert that:

• The metric serves as an efficient time-domain measure for assessing and suppressing broadband spectral fluctuations without the need to adapt the excitation pulse.
• It can be utilized as a weighted objective, a constraint, or a post-evaluation tool.
• Unlike methods that prescribe a specific temporal envelope, penalizing $T_{AC}$ imposes a weaker constraint, allowing the optimizer freedom to determine the detailed temporal shape while ensuring spectral uniformity.
• The approach is applicable beyond simple gratings to other challenging inverse-design problems, such as wide-band near-field coupling, achromatic focusing, and broadband absorption.

The work is supported by VILLUM FONDEN, and the underlying code is made publicly available.