## Abstract

We propose a simple technique of cutting short pulses out of a sharp edge input signal. The technique is based on the Fano antiresonance buildup dynamics. The output pulse duration is inverse proportional to the coupling strength to the resonator. We show that this coupling can be effectively increased by using more than one resonator and exploiting the antiresonance coalescence phenomenon. Analytical calculations for a model of standing-wave resonators and whispering gallery mode (WGM) resonators are performed within the coupled mode theory. We show that the latter can provide better pulse compression. Analytical results for WGM resonators are verified numerically by finite difference time domain method. Ability to generate pulses as short as a few hundreds of femtoseconds at 1.55 *μ*m wavelength has been demonstrated for a potentially CMOS compatible silicon waveguide, which does not require optical nonlinearities to operate.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The development of photonic integrated circuits (PhIC) is required to fulfill the growing demand for information processing and transmission performance. For instance, PhICs can provide optical time division multiplexing [1], which is crucial for high rate data transmission. This technique essentially requires efficient pulse compression devices. Today many different principles of operation and configurations of picosecond and subpicosecond pulse compression devices are being developed [2–6]. Among them compact solid-state devices compatible with silicon CMOS technology are of prior interest. Pulse compression has been demonstrated in solid-state nonlinear optical systems: dispersive silicon waveguides (WG) with phase self-modulation due to the third-order Kerr nonlinearity [5] and photonic crystals with asymmetric Fano resonance, which experienced blue shift as a result of two-photon absorption [7]. However, pulse compression mechanisms based on nonlinear effects are power dependant and hence impose bounds on large-scale integration because of energy consumption and dissipation restrictions.

In this paper, we present a new concept of subpicosecond pulse formation based on the buildup process of Fano antiresonance that is of interference nature and linear. Fano resonances represent a hot topic of the modern optics [8,9] with a variety of applications including all-optical switches [10], refractive index sensors [11], efficient third-harmonic generation [12], etc. A lot of different schemes of Fano waveguides were proposed in the last decades, such as plasmonic WGs [13], ring WGs [14], photonic crystal WGs [15], and silicon slot WGs [6]. Here we consider a WG with evanescently side-coupled cavity with either a standing wave (SW) or a whispering gallery mode (WGM) resonator. Destructive interference in such systems takes place between the passing-by wave and the field outgoing from the resonator. However, the peculiarities of transmission characteristics of SW and WGM resonators, as well as the role of loss and dephasing in these systems are quite different. The proposed mechanism is qualitatively independent of the Fano resonance spectrum profile (either symmetric or asymmetric), and the only requirement is the presence of an antiresonance. In this paper, we show that during the initial buildup of a Fano antiresonance (field accumulation in the resonator), the transmission of the WG will be high until the resonator acquires enough field amplitude. Thus, a Fano WG with a side-coupled spherical resonator can be considered as a simple device, which can cut pulses as short as a few hundreds of femtoseconds at $1.55$ $\mu$m wavelength as our numerical modeling shows.

## 2. Theoretical model: standing-wave resonators

#### 2.1 Stationary transmission of a single resonator waveguide

Consider a single symmetric SW resonator evanescently coupled to a lossless WG (Fig. 1(a)). Dynamics of an electromagnetic pulse propagation can be described by the coupled mode theory (CMT) [16,17]. The resonator mode field amplitude $a(t)$ evolves according to the following equation:

Considering a resonator coupled to a waveguide, we should distinguish its unloaded and external $Q$-factors (see, e.g., Refs. [18,19]), which characterize losses due to intrinsic absorption/radiation (proportional to $\tau _{0}^{-1}$) and coupling to waveguide (proportional to $\tau _{c}^{-1}$) correspondingly. The total (loaded) $Q$-factor is calculated as an inverse sum of inverse unloaded and external $Q$-factors. Therefore, the loaded $Q$-factor can be measured from the transmission spectrum peak/dip as the inverse of its FWHM [20,21].

The amplitude of the outgoing wave $s_{out}(t)$ can be calculated as

#### 2.2 Single resonator coupled to a waveguide

As an example of a realistic input pulse signal waveform, which provides an analytic solution to the CMT equations, we consider a finite duration pulse with linear switching on and off edges and carrier frequency $\omega$ (Fig. 1(c)). Formally, in this case the input signal can be written as

Let us begin with a simple but illustrative case of an input signal with linear rising edge of duration $\Delta _{up}$ (Fig. 1(d)). For $t\rightarrow \infty$ fraction of the electromagnetic energy transmitted through WG will tend to $T^{(1)}_{SW}(\omega )$, which is zero under the perfect conditions ($\omega =\omega _{r}$ and $\tau _{0}\rightarrow \infty$). However, initially, the pulse will be transmitted until the resonator accumulates enough field amplitude $a\sim \sqrt {\tau _{c}}$. Thus, the output signal will be nonzero only during the antiresonance buildup process, and hence its duration is determined by the antiresonance buildup time. That is the key idea of the short pulse formation method proposed in this paper.

Equations (1–2) for this input waveform can be solved with the natural initial condition $a(0)=0$ and the time resolved transmitted energy can be calculated. For $\omega =\omega _{r}$ and $\tau _{0}\rightarrow \infty$ we have

Above, we have shown a clear physical picture of short pulse formation from sharp edges of the input signal. Now we focus on implementing this result for compression of incident short pulses of the form (8) having linear rise and fall edges with duration $\Delta _{up}$ and $\Delta _{down}$ correspondingly (Fig. 1(e)). Each incident pulse edge leads to a separate output pulse. The mechanism of pulse formation from the rising edge is described above, and it is related to the Fano antiresonance buildup. The pulse from the falling edge has the opposite nature – it arises from the field escaping from the resonator after the passing-by wave has been turned off. Surely, for a smooth input pulse with $\Delta _{up,down}\gg \tau _{c}$ the evolution is quasi-stationary and the output pulse amplitude is close to $T^{(1)}_{SW,min}$ and hence is negligible. We can estimate that the output pulse from the raising edge is dominant (by amplitude) as $\Delta _{up}<\Delta _{down}$. Moreover, if $\Delta _{up}<\tau _{c}[2+\mathcal {W}_{0}(-2e^{-2})]\approx 1.59362\tau _{c}$, where $\mathcal {W}_{0}$ is the principal branch of the Lambert $W$-function, then the rising edge provides a dominant pulse regardless of the value of $\Delta _{down}$. The pulse from the falling edge, in this case, plays a parasitic role.

One can estimate the duration (at half maximum of intensity) of the transmitted pulse and calculate the compression ratio $\eta ^{(1)}_{SW}$ compared to the input signal duration. Superscript $(1)$ and subscript $SW$ again indicate a WG with a single SW resonator. Figure 3 shows a plot of the compression ratio $\eta ^{(1)}_{SW}$ vs. uprising front duration. In the limiting case $\Delta _{up}/\tau _{c}\rightarrow 0$ compression ratio becomes

It is illustrative to discuss the pulse shortening from the frequency domain point of view. We consider a linear system, and hence components of signal with different frequencies propagate independently through the system. Thus, the output signal spectrum $s_{out}(\omega )$ is just the spectrum of the input signal $s_{in}(\omega )$ multiplied by the stationary transmission amplitude $t_{in-out}(\omega )$:

Fig. 2(b) shows an example of the input and output signal waveforms for the waveguide with a single standing wave (SW) resonator. In Fig. 4(a), we show calculated spectra of these signal waveforms. The stationary transmission coefficient of the waveguide with a single SW resonator is depicted in Fig. 4(b). It is clearly seen that Eq. (11) is fulfilled. In the optimal case of a SW resonator with high unloaded $Q$-factor ($\tau _{0}\gg \tau _{c}$), we have $|t_{in-out}(\omega _{r})|^{2}=|T_{SW}^{(1)}(\omega _{r})|^{2}\approx 0$ at the carrier frequency of the signal $\omega _{r}$ (Fig. 4(c)) and the output signal spectrum splits into two parts. The partial contribution of the high-frequency components becomes relatively more significant, and hence the most smooth input signal edge becomes sharper in the output. In Fig. 2(b), one can see that the smoothest falling edge of the input signal becomes sharper in the output.#### 2.3 Two resonators coupled to a waveguide

Coupling time $\tau _{c}$ between the resonator and the WG cannot be arbitrary small due to technological difficulties of producing a small gap between them. Nevertheless, the spectral width of the antiresonance, which is inversely proportional to $\tau _{c}$, can be increased by adding more resonators. Thus, one can use more than one resonator to effectively reduce $\tau _{c}$ below technological limits. Consider now a fully transparent waveguide ($t_{b}=1$) with two equal evanescently coupled SW resonators (Fig. 1(b)). Solving the CMT equations, one can get the stationary transmission coefficient of this system as

whereAnalyzing Eq. (13), one can get that, zero-valued transmission minimums are possible in this system only in the following conditions:

- 1. for $\tau _{0}\rightarrow \infty$ and $\kappa \geq 2\tau _{c}^{-1}\cos {\phi }$ at $\omega =\omega _{r}\pm \sqrt {\kappa (\kappa -2\tau _{c}^{-1}\cos {\phi })}$;
- 2. for $\cos {\phi }\geq \tau _{c}\tau _{0}^{-1}$ and $\kappa =\tau _{c}^{-1}\cos {\phi }-\sqrt {\tau _{c}^{-2}\cos {\phi }-\tau _{0}^{-2}}$ at $\omega =\omega _{r}$.

On the other hand, the increase of a transmission dip spectral width in the case of two resonators can be regarded as an effective decrease of $Q$-factor in a single resonator configuration. It is known that the lower $Q$-factor is the lower field amplitude the resonator can acquire. For pulse propagation problem, this means a faster process of field accumulation in the resonator, which leads to the faster relaxation to a stationary regime of transmission. Hence, a smaller amplitude of the output pulse can be achieved. For instance, according to Eq. (12) in the case of two almost lossless ($\tau _{0}\gg \tau _{c}$) SW resonators in the regime of antiresonance coalescence ($\kappa =2\tau _{c}^{-1}\cos {\phi }$), we have a transmission dip $\sqrt {2}$ times wider than in the case of a single SW resonator. This can be interpreted as an effective reduction of $\tau _{0,c}$ (and $Q$-factor correspondingly) of a single resonator by $\sqrt {2}$ times. Using Eq. (9), one can conclude that the amplitude of the output pulse will be reduced by a factor about $1-(\sqrt {2}-1)\Delta _{up}/\tau _{c}$ if the condition $\Delta _{up}/\tau _{c}\ll 1$ is fulfilled. Moreover, the effective decrease of $\tau _{0}$ will lead to an increase of losses and degradation of the output pulse waveform. This fact makes the usage of more than several resonators and coalescence of corresponding antiresonances to be impractical.

Consider an input signal described by Eq. (8) with $\Delta _{peak}\rightarrow \infty$. In regime of coalescence of the transmission zeroes of the first type ($\kappa =2\tau _{c}^{-1}\cos {\phi }$), one can derive the output intensity evolution as (in the high-$Q$ resonator limit, i.e. $\tau _{0}\rightarrow \infty$):

In the case of an asymmetric input pulse with $\Delta _{up}<\Delta _{down}$ (Fig. 1(e)) one can evaluate the half-maximum duration of the output pulse (for $\kappa =2\tau _{c}^{-1}\cos {\phi }$ and $\tau _{0}\rightarrow \infty$) and calculate the compression ratio $\eta ^{(2)}_{SW}$, which is shown in Fig. 3. In the limit $\Delta _{up}/\tau _{c}\ll 1$ on can see that $\eta ^{(2)}_{SW}<\eta ^{(1)}_{SW}$ (see Fig. 3), which highlights the advantage of using multiple resonators and the antiresonance coalescence phenomenon. Example of a CMT calculated time-resolved output signal plotted by the blue dot-dashed line in Fig. 2(b). The output pulse peak value is again about $0.907$ times the peak value in a single resonator case.

Thus, we have shown that usage of two resonators can provide an antiresonance coalescence regime, which enhances the pulse compression effect. However, the coalescence of antiresonances of the first type requires precise adjustment of parameters (phase shift $\phi$, coupling $\kappa$, and characteristic time $\tau _{c}$) and very high $Q$-factor of resonators (to provide $\tau _{0}\gg \tau _{c}$). On the other hand, the transmission zero of the second type demonstrates a similar output pulse duration (see Fig. 2(c),d) and does not require high-$Q$ resonators. Precise adjustment of the parameters is still needed to fulfill the condition $\kappa =\tau _{c}^{-1}\cos {\phi }-\sqrt {\tau _{c}^{-2}\cos {\phi }-\tau _{0}^{-2}}$. Fortunately, this condition can be easily fulfilled in a whispering gallery mode (WGM) resonator, which has two degenerate eigenmodes propagating clockwise and counter-clockwise. Therefore, one can expect that a single WGM resonator can be as effective in pulse compression as a pair of SW resonators.

## 3. Theoretical model: whispering gallery mode resonators

#### 3.1 Single resonator

Consider a single dielectric spherical WGM resonator evanescently coupled to a lossless WG (Fig. 5(a)). Using the CMT equations adapted for a WGM resonator, one can get the following equation for the resonator field amplitude $a(t)$:

In the stationary regime Eqs. (16–17) give the transmission:

In case of a linearly rising input signal with rising time $\Delta _{up}$, Eqs. (16–17) can be solved, and the time-resolved transmitted energy can be calculated:

For $\Delta _{up}\ll \tau _{0,c}$ duration of the output pulse at half maximum $\Delta t^{(1)}_{WGM}$ is governed by the stationary antiresonance spectrum width $W$ as $\Delta t^{(1)}_{WGM}\sim W^{-1}$. Using this dependence, one can obtain a short pulse with a wide transmission dip by taking small $\tau _{c}$. However, $\tau _{c}$ cannot be arbitrarily small as $\tau _{c}$ and $\tau _{0}$ should be roughly equal ($\tau _{c}\approx \tau _{0}$) to make $T_{WGM,min}^{(1)}\approx 0$ and provide high enough pulse contrast. In the optimal case of $\tau _{0}=\tau _{c}$ and $\Delta _{up}\ll \tau _{0,c}$ the pulse duration becomes $\Delta t^{(1)}_{WGM}\approx \tau _{c}\frac {\log {2}}{4}\approx 0.17329\tau _{c}$. Assuming the unloaded $Q$-factor of the resonator to be $Q\sim 10^{3}$ one can estimate $\tau _{0}$ for a standard telecommunication wavelength $\lambda =1.55$ $\mu$m as $\tau _{0}\sim 1$ ps, which leads to $\Delta t^{(1)}_{WGM}\sim 200$ fs in the optimal configuration of the structure ($\tau _{0}=\tau _{c}$). Red dashed line in Fig. 5(c) shows an example of the output waveform in this case for $\tau _{0}=\tau _{c}$ and $\Delta _{up}=0.1\tau _{c}$. One can easily check, that output signal waveform is the same as in the case of two SW resonators for $\phi =0$, $\kappa =\tau _{c}^{-1}$, and $\tau _{0}=\tau _{c}$ (compare red dashed lines in Fig. 5(c) with blue dot-dashed lines in Fig. 2(c)).

Pulse shaped input signal again provides an output of two pulses (one for each signal edge). Output pulse from the rising edge is dominant (by amplitude) when $\Delta _{up}<\Delta _{down}$ and if $\Delta _{up}<\tau _{c}[1+\frac 12\mathcal {W}_{0}(-2e^{-2})]\approx 0.79681\tau _{c}$, then the uprising front is dominant regardless of the value of $\Delta _{down}$. The pulse from the falling edge, in this case, is a side effect. One can estimate the duration (at half maximum of intensity) of the transmitted pulse and calculate the compression ratio $\eta ^{(1)}_{WGM}$ compared to the input signal duration. Figure 3 shows the plots of the compression ratio $\eta ^{(1)}_{WGM}$ vs. uprising front duration. In the limiting case $\Delta _{up}/\tau _{c}\rightarrow 0$ compression ratio becomes

Similar to SW resonators, here compression becomes effective for highly asymmetric input pulses ($\Delta _{up}\ll \tau _{c}\ll \Delta _{down}$). From Eqs. (10) and (20) one can see that in this case WGM resonator provides a twice more effective pulse compression than a SW resonator does (see Fig. 3). Red dashed line in Fig. 5(d) shows an example of the output intensity for $\tau _{0}=\tau _{c}$, $\Delta _{up}=0.1\tau _{c}$ and $\Delta _{down}=3\tau _{c}$. One can easily check, that output signal waveform is the same as in the case of two SW resonators for $\phi =0$, $\kappa =\tau _{c}^{-1}$, and $\tau _{0}=\tau _{c}$ (compare red dashed lines in Fig. 5(d) with blue dot-dashed lines in Fig. 2(d)) For extremely short pulses with $\Delta _{up,down}\ll \tau _{c}$ we again get that $\eta ^{(1)}_{WGM}\lesssim 1$ and $\eta ^{(1)}_{WGM}\rightarrow 1$ as $\Delta _{up,down}/\tau _{c}\rightarrow 0$.

#### 3.2 Two resonators

Similar to SW resonators, one can use several WGM resonators and the antiresonance coalescence phenomenon to increase the spectral width of a transmission dip. In the single resonator interpretation, this also leads to the effective decrease of the $Q$-factor, and hence decreases the duration of the output pulses. Consider the dynamics of the field amplitudes for two side-coupled resonators (see Fig. 5(b)). In this case, if the evanescent coupling between the resonators is negligible, then the stationary transmission is $T^{(2)}_{WGM}(\omega )=[T^{(1)}_{WGM}(\omega )]^{2}$, where $T^{(1)}_{WGM}(\omega )$ is described in Eq. (3). One can estimate that in the optimal configuration with $\tau _{0}=\tau _{c}$, the width of the transmission dip of two WGM resonators will be $\sqrt {1+\sqrt {2}}$ times bigger than for a single resonator. Thus, according to Eq. (19), the output pulse in the specific case of two WGM resonators is suppressed by a factor of $1-2(\sqrt {\sqrt {2}+1}-1)\Delta _{up}/\tau _{c}$ for $\Delta _{up}/\tau _{c}\ll 1$ compared to the single WGM resonator case. One can see that the peak suppression factor is more significant for WGM resonators rather than for SW resonators, and hence a smaller number of WGM resonators can be efficiently used compared to SW resonators. However, WGM resonators provide almost a twice shorter duration of output pulses and, therefore, in each case, the optimum configuration and number of resonators should be determined.

Consider a linearly rising input signal waveform with rising time $\Delta _{up}$. Time-resolved transmitted energy of the output signal derived from the CMT equations is the following:

In the case $\Delta _{up}\ll \tau _{0,c}$ the output signal has the dominant pulse from the rising edge and a pulse from the fall edge with a much lower amplitude plays a parasitic role. The dominant pulse has a slightly lower peak than in the case of one resonator configuration, however, it can have smaller duration $\Delta t^{(2)}_{WGM}$. An example of the output waveform for $\tau _{0}=\tau _{c}$ and $\Delta _{up}=0.1\tau _{c}$ is shown by the blue dot-dashed line in Fig. 5(c). In the optimal configuration ($\tau _{0}=\tau _{c}$ and $\Delta _{up}\ll \tau _{0,c}$) it tends to $\Delta t^{(2)}_{WGM}\approx 0.082784\tau _{c}$, which is approximately twice less than $\Delta t^{(1)}_{WGM}$. The output pulse peak value in the case of two resonators is about $0.815$ times the peak value in a single resonator case. This ratio is pretty close to the estimation by $Q$-factor reduction, which in this case gives $1-2(\sqrt {\sqrt {2}+1}-1)\Delta _{up}/\tau _{c}\approx 0.889$.

In the case of an asymmetric input pulse with $\Delta _{up}<\Delta _{down}$ (Fig. 1(e)) one can estimate compression ratio $\eta ^{(2)}_{WGM}$ (in optimal configuration $\tau _{0}=\tau _{c}$), which is shown in Fig. 3. In the limiting case $\Delta _{up}/\tau _{c}\rightarrow 0$ it becomes

## 4. Numerical analysis

#### 4.1 Transmission spectra

As shown above, WGM resonators can provide more effective pulse compression compared to SW resonators. Thus, to validate our theoretical results, we performed a numerical analysis of the WGM resonator Fano WG using finite difference time domain (FDTD) software MEEP [27] and Harminv mode analyzer [28]. As the WGM resonator we took a silicon ball with radius $487$ nm and $\varepsilon =12$. The lossless silicon WG has a square cross-section with side length $278.3$ nm. The WGM resonator has an eigenmode near wavelength $\lambda _{0}=1.55$ $\mu$m, which is indicated by an antiresonance at $\lambda _{0}$ in the stationary transmission spectra (Fig. 6(a)). The transmission loss factor of silicon square cross-section waveguides at $1.55$ $\mu$m wavelength is typically less than $10$ dB/cm [29,30], which becomes negligible for a system about several $\mu$m long (as in our case), and hence assumption of a lossless WG is valid.

As it is shown in Fig. 6(a), the antiresonance becomes wider with decreasing distance between the waveguide and the resonator. If the resonator is too close to the WG, the transmission dip minimum value naturally grows up. In our calculations, we have achieved the minimal transmission dip value around $0.02$ for $278.3$ nm distance between the resonator and the waveguide, and hence we choose this configuration as optimal.

#### 4.2 Fano antiresonance buildup

To establish the correspondence between the numerical model developed in the previous subsection and the CMT equations, one should fit the parameters $\tau _{0}$ and $\tau _{c}$. Therefore, we study the Fano antiresonance buildup process and the electromagnetic energy transmission evolution of a zero rise time monochromatic light source with $\lambda =\lambda _{0}=1.55$ $\mu$m.

Due to dispersion in the waveguide, perfectly sharp rise edge of the initial input signal is smeared by approximately $102$ fs even with no resonators as it can be seen in Fig. 7(a), where black dots show the FDTD simulated evolution of the transmitted electromagnetic energy through the bare WG (without resonators). Dispersion in the waveguide plays, surely, a negative role in the proposed method. For a more realistic device, optimization can be performed, and the dispersion can be reduced sufficiently in such waveguide [31]. To take into account dispersion in our analytic model, given by Eq. (19), we choose $\Delta _{up}=102$ fs, which corresponds to the smeared output of the bare WG, instead of the perfect input signal parameter $\Delta _{up}=0$. Now, the analytic model perfectly fits the simulation with $\tau _{0}=982.3$ fs and $\tau _{c}=885.8$ fs (blue line and blue squares in Fig. 7(a)). Time origin corresponds to the moment when the light from the source comes to the end of the bare WG, where the energy transmittance is measured. From these values of $\tau _{0}$ and $\tau _{c}$, one can estimate the loaded and unloaded $Q$-factors of the resonator as $Q\approx 283$ and $Q\approx 597$ correspondingly.

Red triangles in Fig. 7(a) correspond to the FDTD simulation of the system with two WGM resonators separated by $1.809$ $\mu$m. Two resonators provide a wider transmission dip (Fig. 6(b)), and hence shorter pulse times are expected. Evanescent coupling between the resonators results in little splitting of their eigenmodes, which can be seen as a little split in the transmission dip in Fig. 6(b). This splitting naturally vanishes as resonators are moved far from each other, but even particular spacing we take ($1.809\mu$m) is enough to make this splitting have a negligible influence on the pulse propagation process. CMT model (21) for the same values of $\tau _{0}$, $\tau _{c}$ and $\Delta _{up}$ as in the single resonator case also shows good consistency with the simulation results (red dashed line in Fig. 7(a)). One can estimate duration (at half maximum) of the transmitted pulses as $\Delta t^{(1)}_{WGM}\approx 183.4$ fs and $\Delta t^{(2)}_{WGM}\approx 103.4$ fs in the case of one and two WGM resonators correspondingly. Due to the effective decrease of $Q$-factor, the peak value of the output pulse in the case of two resonators is lower than in the case of a single resonator, as expected. From FDTD simulations we have peak value reduction by a factor of $0.831$, which is close to the simple theoretical estimation for a given values of $\Delta _{up}$ and $\tau _{c}$: $1-2(\sqrt {\sqrt {2}+1}-1)\Delta _{up}/\tau _{c}\approx 0.873$.

#### 4.3 Compression of short pulse

Now consider a $\lambda _{0}=1.55$ $\mu$m input signal, which is modulated asymmetrically: sharp (initially abrupt) rise edge and linear fall during $465$ fs. The simulated evolution of the electromagnetic energy transmission through the bare WG and the Fano WG with one and two resonators is shown in Fig. 7(b) by black dots, blue squares, and red triangles correspondingly. It can be clearly seen that pulse is compressed in both of the cases. Here again, due to the dispersion, the initially sharp rising edge of the pulse smears as it goes through the WG. Figure 7(b) also depicts time evolution of the transmitted pulse energy calculated from the CMT model with the same fitting parameters ($\tau _{0}=982.3$ fs, $\tau _{c}=885.8$ fs, and $\Delta _{up}=102$ fs). The compression ratios of the input pulse are $\eta ^{(1)}_{WGM}\approx 0.601$ and $\eta ^{(2)}_{WGM}\approx 0.467$ for structures with one and two resonators correspondingly. Again the output pulse peak value in the case of two resonators is lower than in the single resonator case. The FDTD calculated factor in this case is $0.842$ is even closer to the mentioned above estimation $0.873$.

## 5. Discussion and summary

In this paper, we have proposed a simple and potentially CMOS compatible technique for generation and compression of subpicosecond light pulses using the Fano antiresonance buildup process and the phenomenon of antiresonance coalescence. From a frequency domain point of view, this corresponds to a transmission spectral intensity redistribution within the given bandwidth, which enhances the relative partial contribution of the high-frequency components. FDTD simulation of a silicon waveguide with side-coupled WGM resonators showed that this technique is capable of providing output pulses about $100$ fs long for input with the abrupt rising front. Also, simulation showed an approximately $2$ times pulse compression for an asymmetric input pulse of about $150$ fs duration (at half maximum). This result seems to be superior to recent achievements in the field of short pulse formation even compared with non-CMOS compatible techniques [2,32].

Recently another CMOS compatible method for short pulse carving has been published, which uses WGs with a side-coupled resonator. It exploits the transmission spectrum shift due to two-photon absorption process in a nonlinear dielectric standing wave resonator [7]. Thus, spectral separation $\Delta \omega$ between the transmission maximum and minimum defines the maximum bandwidth and hence the minimum duration of the input signal, which can be effectively carved by this method [33]. It turns out, that $\Delta \omega \sim \tau _{c}^{-1}$ and therefore strong coupling to resonator (small $\tau _{c}$) can lead to shorter pulse curving. On the other hand, the field amplitude in the resonator $a$ is $a\sim \sqrt {\tau _{c}}$ for small $\tau _{c}$ (strong coupling) and in order to make the nonlinear spectrum shift sufficient $\tau _{c}$ cannot be arbitrary small. This implies a fundamental lower bound for the shortest pulse duration, which is possible to produce by this technique (e.g., $5$ ps in Ref. [7]).

Our method is of interference nature, and hence there is no fundamental limit on the field amplitude in the resonator. Therefore, we can use strongly coupled resonators with small $\tau _{c}$. Moreover, WGM resonators with lower unloaded $Q$-factor provide shorter pulses, if the agreement between $\tau _{0}$ and $\tau _{c}$ is fulfilled. The proposed technique utilizes basically interference phenomena, and hence can be adapted to another frequency range if the dielectric permittivity does not show significant frequency dispersion in this range and can be approximated as a constant. However, it is essential to note that the lowering of the resonators $Q$-factor results in the spectral broadening of the transmission coefficient dip. Consequently, the input signal with a wider bandwidth must be used to have efficient short pulse formation. In time domain formulation, we can conclude that our approach requires incident pulses with a sharp signal edge (uprising preferably) and asymmetric waveform to be effective. This effect leads to the suppression of the output pulse peak value when two or more resonators are used, due to the antiresonance coalescence. Therefore, there is an optimal number of resonators, which can be used for short pulse formation. This number should be determined individually in each particular configuration.

From this point of view, we believe that the most efficient way to use the proposed system is attaching it to the output of a primary pulse compression device, which utilizes nonlinear optical effects and provides asymmetric output pulses. It can be used to form short pulses from those, which have passed through the primary device, but were too short for it, and the compression failed, but their waveform become asymmetric. For instance, $5$ ps pulse passed through the device in Ref. [7] result in an asymmetric output, which is not compressed (see Fig. 6(b) in [7]). We also propose to exploit the antiresonance coalescence effect to enhance the spectral width of the transmission dip and thus to make shorter output pulses. In particular, we have shown that usage of two resonators instead of one can almost double the compression ratio of the WG. Different configurations with more than two resonators will be considered elsewhere.

## Funding

Russian Foundation for Basic Research (18-32-00453); Russian Academy of Sciences.

## Disclosures

The authors declare no conflicts of interest.

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