With the recent development of ultrahigh-speed optical communication systems, techniques for generating and processing high repetition rate optical pulse trains are attracting considerable interest ^[1]. Since conventional mode-locking or direct-modulating methods are hard to operate at such ultrahigh speed, optical pulse repetition rate multiplication (PRRM) becomes necessary ^[2]. Most PRRM schemes are based on periodically changing the amplitude and/or phase of the input pulse spectrum. For example, by spectral amplitude filtering, a double-passing Fabry–Perot filter ^[3], an array waveguide grating ^[4], and a lattice-form Mach–Zehnder interferometer ^[5] have been used for generating high repetition rate optical pulse trains. For phase-filtering-based PRRM, chirped-fiber-Bragg gratings ^[6] have been employed to implement the fractional temporal Talbot effect. Due to the merits of small size and compatibility with the integration process, ring-resonator-based optical PRRM exhibits better applicable prospects. In Ref. ^[7], optical pulse trains are generated by using a multiple-microring-based time interleaved architecture. In Refs. ^[8,9], doubling and quadrupling PRRM are obtained by microring-induced beating resonance. In Ref. ^[10], PRRM and shaping are achieved by ring resonator arrays. However, device fabrication and stability become more difficult as the number of ring resonators increases. Therefore, in this paper, by simulating the spectral characteristic of the series-coupled double-ring resonator, we propose a much more compact optical PRRM with better power uniformity.

Figure 1(a) is a typical configuration of a series-coupled double-ring resonator. The transfer function can be obtained by dividing the series-coupled double-ring resonator into three directional couplers (coupling coefficients are K0, K1, and K2) and two uncoupled half-rings (ring radii are R1 and R2, respectively), as shown in Figs. 1(b) and 1(c).

Based on the transfer matrix method, the transfer function of a series-coupled double-ring resonator can be obtained as ^[11] where β=2π·neff/λ is the propagating constant, neff is the effective refractive index, λ is the wavelength, τk=exp(−α·2πRk) is the waveguide normalized loss, and Tk and Kk are the normalized transferring coefficient and coupling coefficient, which have the relationship Tk2+Kk2=1.

By assuming no input signal at port 3 (E3=0) and defining one-round phase within the ring as ϕ1,2=(2π·neff/λ)·2πR1,2, the spectral responses at drop port D(λ) can be obtained as

Due to the resonant characteristic, D(λ) is periodic with respect to the inverse of wavelength. Defining FSR=λm+1−λm as the periodic resonant spacing and using the inverse Fourier transform (F−1), the temporal domain impulse response of D(t) is discrete-time, as shown by ^[5]

When a periodic input pulse train ain(t)=∑q=−∞∞a0(t−qT) (where a0(t) represents the complex envelope of an individual pulse and T is the temporal period) is injected into port 1, the output is the convolution (⊗) between the input ain(t) and the impulse response of D(t), as shown by

When the periodic resonant spacing (FSR) of the series-coupled double-ring resonator and the temporal period T of the input pulse train meet the condition T·FSR=N, the repetition rate of the output pulse train can be multiplied by a factor of N if the input pulsewidth is sufficiently narrow ^[12].

In the following numerical analyses, we examine optical PRRM using the aforementioned series-coupled double-ring resonator. Due to the high refractive index contrast of silicon-on-insulator (SOI) technology, a wavelength-scale microring resonator can be fabricated with very small bend loss. By setting the width and height of the top silicon waveguides as 500 and 230 nm, respectively, and assuming polymer SU8 as the top cladding layer, the group index is calculated approximately to be 3.901 at 1550 nm wavelength ^[13].

First, the spectral responses of the series-coupled double-ring resonator are studied. When both the rings have the same radius R, similarly to the single-ring resonator, the series-coupled double-ring exhibits a periodic resonant spacing FSR, as shown in Fig. 2(a). However, different from the Lorentzian spectral response of the single-ring resonator, the resonant passband within each period of the series-coupled double-ring resonator is divided into two narrower subresonant passbands, whose spacing and shaping depend on the coupling coefficients. As demonstrated in Figs. 2(b) and 2(c), when the internal coupling coefficients of the adjacent rings K1 increase, the subresonant spacing expands, whereas, when the external coupling coefficients between the rings and the input/output waveguides (K0 and K2) increase, the subresonant passbands will recombine and evolve into one broader flat-top passband if the following two conditions are satisfied ^[14]: K0=K2 and K1=K02/(2−K02)≈K02/2.

Next, PRRM is investigated by employing a series-coupled double-ring resonator with a flat-top resonant passband. We assume that a 10 GHz Gaussian-pulse train is input and the full width at half-maximum (FWHM) of each pulse is 1 ps, which is narrow enough to avoid interfering with each other. When the radii of the series-coupled double-ring resonator are set to be 612 μm, its periodic resonant spacing FSR equals twice that of the spectral comb of the input pulse train, so that every other spectral line will be filtered out, which results in the output spectral line spacing being doubled and repetition rate multiplication with a factor of 2 being achieved, as shown in Fig. 3. According to the law of conservation of energy, only half of the incident power remains for ×2 repetition rate multiplication. In addition, due to the fabrication difficulty, it is hard to set the resonator radius accurately. Figure 3(e) shows the output pulse trains for resonator radii with ±100 and ±200nm fabrication deviation. As the resonator radii deviating from the ideal values increase, the output pulse trains decrease and system power loss deteriorates. Although the power loss can be compensated by adding an optical amplifier, we suggest that the fabrication deviation should be kept as small as possible in order to obtain better PRRM performance. Compared with other reported systems ^[3,4], series-coupled double-ring resonators have the advantage that the shape of each individual output pulse remains unchanged from the original input pulses.

Since the periodic resonant spacing FSR is approximately inversely proportional to the ring radii ^[15], larger FSR and higher PRRM can be achieved by using a resonator with smaller size. As shown in Fig. 4, output pulse trains with multiplication factors of 3, 4, and 5 are obtained by employing series-coupled double-ring resonators with radii of 408, 306, and 245 μm, respectively. To judge the quality of multiplied pulses, the power nonuniformity of the output pulse train is calculated as ^[16] Δ=var(O(t))/mean(O(t))×100%. It is obvious that the power uniformity is much better for the series-coupled double-ring resonator compared with the single-ring resonator as the multiplication factor increases, as demonstrated in Fig. 5. This is because of the broader and flatter passband characteristic of the series-coupled double-ring resonator, which allows a higher tolerance to external noise and fabrication errors.

In this paper, based on the transfer matrix method, the spectral characteristic of the series-coupled double-ring resonator is simulated. The influence of the internal and external coupling coefficients on the subresonant passband is analyzed, and the optimum coupling coefficients to achieve the flat-top passband filtering response are obtained. Then, high-quality PRRM is realized by a series-coupled double-ring resonator to periodically filter out the spectral lines of the input pulse train. In addition, by adjusting ring radii, output pulse trains with different multiplication factors are achieved, which exhibit much better power uniformity compared with the single-ring resonator.

Acknowledgment. This project is supported by the National Natural Science Foundation of China (61007007), the talents of North China University of Technology (CCXZ201307), and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (CIT&TCD201304001).