Abstract
Photonic time crystals (PTCs) exhibit momentum band gaps and modulation-induced amplification. When uniform periodicity is relaxed, their spectral and dynamical response becomes considerably richer. In this work, we investigate modified-Fibonacci PTCs in which the ratio of two elementary optical-time durations serves as a continuous tuning parameter between periodic and deterministic quasiperiodic temporal order. We derive closed-form analytical predictions for the positions and relative strengths of the momentum bandgaps, and use them to characterize how gap widths, amplification rates, and band-edge mode localization evolve as the modulation is tuned away from the periodic limit. This analytical control provides explicit design rules for activating, suppressing, or rebalancing selected momentum gaps. These findings establish modified-Fibonacci modulation as a versatile route for engineering quasiperiodic order in PTCs and tailoring their spectral and dynamical properties.