Fig. 1. Schematic of micro-ring differentiator structures. (a) Single micro-ring; (b) cascaded double micro-rings

Fig. 2. Specific process of the optimization method

Fig. 3. 1.5 order differential waveforms for single micro-ring and cascaded dual micro-ring structures. (a) Input signal; (b) output signal

Fig. 4. Transmission responses of a single micro-ring and the cascaded dual micro-ring. (a) Amplitude spectra; (b) phase spectra

Fig. 5. Amplitude and phase responses of cascaded dual micro-ring structure with different orders. (a) Amplitude spectra; (b) phase spectra

Fig. 6. Differential waveforms of cascaded dual micro-ring structures with different differentiation orders

Fig. 7. Mean deviation characteristics of different differentiation orders

Fig. 8. Cross-correlation coefficients properties of different differentiation orders

Fig. 9. Energy efficiency characteristics and D_ave/E_E index under different differentiation orders. (a) Energy efficiency characteristics differentiation; (b) D_ave/E_E index differentiation

Fig. 10. Effects of the input pulse width on the differentiation result. (a) Differential waveforms under different input pulse widths; (b) effect of input signal pulse width on differentiation order

Fig. 11. Comparison of the differentiations based on a single micro-ring and the cascaded dual micro-ring. (a) Cross-correlation coefficient as a function of the input pulse width; (b) energy efficiency as a function of the input pulse width