On the theory of spectral compression-assisted optical temporal differentiation

Bandwidth limitation represents a significant factor that degrades the performance of optical devices. The dimensions, composition and configuration of optical devices impose intrinsic constraints on processing broadband optical pulse signals. The enhancement of the response bandwidth of optical devices represents a significant challenge. In this study, we put forward the theory of self-similar spectral compression (SSSC), which involves solving the nonlinear Schrödinger equation with variable coefficients by using the Taylor expansion and residual theorem. The spectral waveform can be precisely preserved in the process of SSSC, leading to a predictable compression factor without pedestals. To demonstrate the effectiveness of the proposed SSSC, we present a case study by designing an on-chip optical time-domain differentiator (OTD) system including a silicon-based tapered spiral waveguide. A 200-fs chirped pulse is well differentiated at multiple orders in the OTD system. Although the linear loss of spiral waveguide has a detrimental impact on SSSC, the broadband spectrum can still be self-similarly compressed, leading to a reduction of differentiation deviation of 22.5 times. The proposed SSSC theory offers valuable guidance for designing all-optical signal processing systems with high spectral resolution and low signal error.