Multi-objective time-domain adjoint via temporal convolution for band-selective electromagnetic topology optimization

Adjoint-based topology optimization has emerged as a powerful approach for electromagnetic device design, enabling gradient evaluation with respect to all design variables from only two simulations, independent of problem size. Conventional frequency-domain adjoint methods are well-suited to single-frequency objectives, but for broadband goals, they incur computational costs that scale linearly with spectral resolution. Time-domain adjoint methods efficiently capture broadband responses; however, their native gradients integrate over the entire excitation bandwidth, preventing independent control of multiple spectral bands. As a result, existing multi-band formulations require separate forward--adjoint simulation pairs per band, forfeiting the principal efficiency advantage of time-domain approaches.We present a multi-objective time-domain adjoint method that computes band-selective gradients via temporal convolution of stored electromagnetic fields. By the convolution theorem, post-processing with band-pass filters isolates designated spectral content without altering the simulation setup. We validate the method on three nanophotonic systems. First, a wavelength-division demultiplexer achieves port efficiencies exceeding 90% with a per-iteration time of 8.74s, outperforming conventional time-domain (14.32s) and frequency-domain (120s) methods. Second, a metalens demonstrates independently prescribed numerical apertures across four spectral bands with high focusing efficiency. Third, a spectral router achieves 70-75% routing efficiency, approximately (2x) the theoretical upper limit of absorption-based designs. Collectively, these results establish that convolution-enabled, band-selective gradients restore the computational advantage of time-domain adjoint optimization for multi-band electromagnetic design.