Objective  Optical frequency comb (OFC) is widely used in optical communication system and spectroscopy. OFC can be generated by using mode-locked laser and electro-optic modulators (EOMs). Although the EOM-based OFC has good performance of flexibility, its flatness performance can be improved. The flatness of OFC is determined mainly by the modulation index and phase of driving microwave signal in phase modulation, which is the fundamental process of electro-optic modulation. Therefore, the optimized modulation index and phase of driving signal as well as other parameters are critical to the generation of flat OFC. In addition, due to the periodicity of driving signals' phases, one of the phases can be left without adjustment in the experiment to achieve flat OFC. However, to the best of the authors' knowledge, this phenomenon has seldom been investigated.

  Methods  An approach to the generation of flat OFC is proposed, where cascaded phase modulator (PM) and intensity modulator (IM) are driven by combined harmonics (Fig.1). The parameters of driving harmonics and IM are optimized, where the optimization problem is formulated to minimize the variance of power for the OFC with certain number of comb lines. Differential evolution (DE) algorithm is applied to solve the optimization problem. Feasible solutions for combined harmonics are investigated in simulation. Experiment is also carried out to verify the feasibility of proposed approach (Fig.13). Although the phases of all the combined driving harmonics can be optimized, it is found that one of the driving harmonics' phases can be left without optimization while the performance of flatness is not affected.

  Results and Discussions   When the fundamental tone and third harmonic are combined to drive the PM and the fourth harmonic drives the IM, a 13-line OFC is generated, where the flatness is 0.27 dB and 0.83 dB under simulation and experiment respectively (Fig.3, Fig.14). When the fundamental tone, third harmonic, and fifth harmonic are combined to drive both the PM and IM, a 19-line OFC with 0.56 dB flatness is achieved in simulation (Fig.4). The rest feasible solutions to generate flat OFC are listed (Tab.1). The relationship between number of comb lines and flatness is also investigated (Fig.12). When the number of comb lines is no larger than 7, the flatness is 0 dB; when the number of comb lines is larger than 7, the flatness increases as the number of comb lines grows. If the phase of one of the driving harmonics is fixed when the optimization problem is being solved, the flatness performances for the cases listed (Tab.1) are not affected (Fig.5, Fig.9-11). Regarding the case in Tab.1(b), the modulation indices for all the three driving harmonics are also investigated, when one of three harmonics' phases is fixed in the simulation (Fig.6-8).

  Conclusions  This work investigates the approach to generate flat OFC by using cascaded PM and IM. These EOMs are driven by combined harmonics, where the parameters are optimized, such as the modulation index and phase of each driving harmonic as well as the bias phase caused by the bias voltage of IM. The optimized parameters are obtained by using DE algorithm, which solves the optimization problem to minimize the variance of power for the OFC. Both simulation and experiment have been carried out to achieve flat OFC. When the combined fundamental tone as well as third harmonic drive the PM and the fourth harmonic drives the IM, a 13-line OFC is generated with 0.27 dB and 0.83 dB flatness under simulation and experiment respectively. It is found that when one of the driving harmonics' phases is fixed, flat OFC can also be achieved by solving the same optimization problem. This phenomenon makes it possible to generate flat OFC in the experiment without adjusting that phase. Therefore, both the feasibility and robustness of the proposed approach are guaranteed.