## Abstract

A novel photonic fractional-order temporal differentiator is proposed based on the inverse Raman scattering (IRS) in the side-coupled silicon microring resonator. By controlling the power of the pump light-wave, the intracavity loss is adjusted and the coupling state of the microring resonator can be changed, so the continuously tunable differentiation order is achieved. The influences of input pulse width on the differentiation order and the output deviation are discussed. Due to the narrow bandwidth of IRS in silicon, the intracavity loss can be adjusted on a specific resonance while keeping the adjacent resonances undisturbed. It can be expected that the proposed scheme has the potential to realize different differentiation orders simultaneously at different resonant wavelengths.

© 2015 Optical Society of America

## 1. Introduction

With the rapid development of high-speed optical communication networks, all-optical signal processing technology is an effect approach to overcome the inherent speed limitations of conventional electronics. The temporal photonic differentiator is a basic element which can implement the time derivative of the complex envelope of the input optical pulse. It can be used in a wide range of applications, such as ultrafast all-optical computing [1], pulse shaping and coding [2–4], dark-soliton detection [5], and Hermite-Gaussian waveforms generation [6]. Different schemes have been proposed to realize the temporal differentiation based on the interferometers, fiber gratings, programmable pulse shapers, photonic crystal nanocavities, and microring resonators [7–13]. In particular, the silicon based microring resonator has been considered as a promising scheme with the advantages of compactness, mature fabrication, and photonic-electronic integration [12–15].

In addition to the regular integer-order temporal differentiator, the fractional-order differentiator has also attracted considerable research interest [16–21]. It has been recently demonstrated that the fractional-order differentiation can be implemented by using the under-coupled and over-coupled microring resonators [19,20]. However, the differentiation order can only be tuned electrically in the scheme reported in [20], which limits the response speed. In [19], the differentiation order is controlled by the polarization state of the input light-wave. As the polarization convertor and the multimode interference (MMI) coupler are required in this scheme, the device complexity is increased.

In the inverse Raman scattering (IRS) process, light at the anti-Stokes wavelength is strongly attenuated in the presence of an intense pump [22]. In silicon, IRS has already been used for wavelength-selective all-optical switching, where the intracavity loss of an optical resonator is precisely controlled in real time [23–25]. In this paper, a novel fractional-order temporal differentiator is proposed in a microring resonator on the silicon-on-insulator (SOI) platform. The coupling coefficient of the resonator is kept constant, while the intracavity loss is adjusted by the pump intensity using the IRS effect. Thus, the continually tunable differentiation order can be realized. By increasing the pump power continuously from 60 mW to 95 mW, the Gaussian pulse with a full width at half-maximum (FWHM) of 50 ps can be differentiated with the order from 1.6 to 0.3, and the output deviation from an ideal fractional-order differentiator is maintained less than 5%. Due to the narrow bandwidth of IRS in silicon, the adjustment of intracavity loss can be implemented on a specific resonance while leaving the adjacent resonances undisturbed. As a result, the proposed scheme has the potential to achieve different differentiation orders simultaneously at different resonant wavelengths. Furthermore, the effects of input pulse width on the differentiation order and the average deviation are discussed in detail, which have not been analyzed in previous literatures on the microring-based fractional-order differentiators.

## 2. Operational principle

For an input signal $x(t)$, the $n$th-order differentiator with an output $d{x}^{n}(t)/d{t}^{n}$ can be considered as an optical filter. The transfer function ${H}_{n}(\nu )$ is given by [16,18]:

In the IRS process, anti-Stokes photons are transferred to the pump, which leads to an attenuation of the anti-Stokes light-wave. In silicon, the anti-Stokes wavelength is blue-shifted by 15.6 THz with respect to the pump wavelength, and the bandwidth of spectral loss is 105 GHz [22,25]. The IRS process is governed by the generalized nonlinear Schrödinger equation (NLSE) in [22]. However, the loss coefficient ${\alpha}_{R}$ induced by IRS near the anti-Stokes frequency ${\omega}_{a}$ can be simplified and described as [25]:

In the absence of the pump light-wave, as the signal power is rather small, the power loss in the microring resonator is mainly caused by the linear loss of the bended waveguide, i.e. $\alpha ={\alpha}_{L}$. However, when the pump and the signal light-waves are launched into the waveguide simultaneously, the intense pump light-wave will result in the nonlinear processes, such as the Kerr effect, IRS, two-photon absorption (TPA), free carrier absorption (FCA), and free carrier dispersion (FCD). Beside IRS, TPA and FCA lead to an additional attenuation of the signal light-wave, while the Kerr effect and FCD cause a change of the refractive index [31]. Therefore, the power loss coefficient at the signal wavelength in the microring resonator can be written as $\alpha ={\alpha}_{L}+{\alpha}_{R}+{\alpha}_{T}+{\alpha}_{F}$, where ${\alpha}_{L}$ is the linear loss coefficient, ${\alpha}_{T}=2{\beta}_{TPA}I$ and ${\alpha}_{F}$ are caused by TPA and FCA respectively, and ${\beta}_{TPA}=0.45$ cm/GW is the TPA coefficient. In addition, the FCA loss coefficient can be calculated by ${\alpha}_{F}=1.45\times {10}^{-21}{({\lambda}_{s}/{\lambda}_{ref})}^{2}N$, where ${\lambda}_{ref}=1550$ nm, ${\lambda}_{s}$ is the signal wavelength, the free carrier density $N$ obeys the rate equation $\partial N/\partial t={\beta}_{TPA}{I}^{2}/(2\hslash {\omega}_{p})-N/{\tau}_{0}$, and ${\tau}_{0}$ is the effective carrier lifetime [31,32]. By adopting the reverse-biased p-i-n structures, the free carrier lifetime can be decreased significantly, so that FCA and FCD are both reduced [33]. The refractive index change induced by FCD can be cancelled out by the Kerr effect, so the resonant wavelength is kept constant [23–25].

## 3. Results and discussion

In this work, the silicon waveguides have a cross section of 450 nm × 250 nm. The TE polarized pump and the TM polarized signal are launched at 1556.1 nm and 1439.5 nm, respectively. The TE-TM pump-signal combination can reduce the difference of coupling coefficients at the two wavelengths, while increase the Raman induced loss by a factor of 1.5 [25]. A microring with the radius $R=50$ μm is used, so both the pump and the signal are on resonance. In the absence of the pump, the microring resonator is over-coupled at the signal wavelength with ${\alpha}_{L}=10.77$ dB/cm and $r=0.83$. As the coupling strength and the linear loss coefficient are related to the wavelength and the polarization state, the resonator is operated near the critical coupling condition at the pump wavelength with ${\alpha}_{L}=5$ dB/cm and $r=0.97$ [25,34]. The free carrier lifetime is ${\tau}_{0}=50$ ps.

Figure 2 shows the power transmission and the phase response of the microring resonator. When the difference between $r$ and $\tau $ is increased, the slope of phase response across the resonance and the resonant attenuation are both decreased. Moreover, the 3-dB bandwidth of the transmission spectrum is increased with the decrease of $\tau $.

As mentioned above, for the continuous-wave pump, ${\alpha}_{R}$ and ${\alpha}_{F}$ are proportional to $I$ and ${I}^{2}$ respectively, and ${\alpha}_{T}$ is much smaller than ${\alpha}_{R}$. When the pump light-wave is injected into the resonator, as shown in Fig. 3(a), the intracavity loss at the signal wavelength is nearly linear with respect to the pump power, which indicates the signal attenuation is predominated by the IRS effect. With the increase of the pump power, the round-trip amplitude transmission coefficient $\tau $ is decreased, and the coupling state at the signal wavelength will be altered from the over-coupled regime to the critical-coupled regime, and then to the under-coupled regime.

The effect of pump light-wave on the intracavity loss at 1442.3 nm is shown in Fig. 3(b), where 1442.3 nm is the resonance adjacent to the anti-Stokes wavelength. The bandwidth of IRS is much smaller than the free spectral range (FSR) of the microring resonator. Therefore, the optical loss induced by IRS is decreased significantly compared with that at the anti-Stokes wavelength. The optical loss induced by TPA and FCA is kept almost constant within the FSR. Thus, for the resonances which do not coincide with the anti-Stokes wavelength, it can be seen that TPA and FCA plays an important role in the intracavity loss.

When a Gaussian pulse with a temporal FWHM of 50 ps is launched into the resonator as the signal, by properly adjusting the pump power, differentiated pulses with the order *n* = 0.4, 0.7, 1, 1.2, and 1.5 can be generated as shown in Fig. 4. Compared with the outputs of the time-reversed ideal differentiators, it can be seen that the waveform deviation is relevant to the differentiation order $n$. A good agreement can be observed when the differentiation order is close to 1. In addition, the waveform deviation with $n>1$ is greater than that with $n<1$. The dependence of waveform deviation on the differentiation order will be further discussed in the later part of this section.

The effect of pump power on the differentiation order is described in Fig. 5. When the pump power is varied from 60 mW to 95 mW, the input pulse with FWHM = 50 ps can be differentiated with the order from 1.6 to 0.3. There is approximately a linear correlation between the pump power and the differentiation order with $n>1$. When $n<1$, the differentiation order is decreased more and more slowly with the increase of the pump power. Besides, under the critical coupling condition where $r=\tau =0.83$, the differentiation order is equal to $n=1$ for all the input signal pulses with FWHM ≥ 50 ps. However, a small discrepancy from the differentiation order $n=1$ can be observed for the input pulse with FWHM = 20 ps.

The differentiation order is also affected by the pulse width of the input signal light-wave. The input with larger pulse width has less bandwidth in its spectrum. If the input spectrum is concentrated more densely to one of the resonances, the output will be more sensitive to the change of resonator response which is induced by the variation of $\tau $. Therefore, when the input pulse width is increased, the curve in Fig. 5 becomes steeper indicating the differentiation order is more sensitive to the pump power. In practical applications, we can increase the input pulse width to reduce the required variation range of the pump power. On the contrary, decreasing the input pulse width is beneficial to improve the tuning accuracy of the differentiation order.

As seen from Fig. 5, one pump power can correspond to more than one differentiation orders due to the variation of the input pulse width. This phenomenon can be attributed to two factors. One is the insufficient bandwidth of the microring resonator. Figure 6(a) shows the output under the critical coupling condition for various input pulse widths. A discrepancy from the first-order differentiator can be observed for the input pulse with FWHM = 20 ps, where the two peaks of the output have different maximums. In fact, the output can be fitted using an ideal fractional-order differentiator with $n=0.94$. It is noteworthy that this discrepancy will not emerge when the input pulse width exceeds FWHM = 50 ps. On the resonance, the critical-coupled microring resonator is quite ideal for the first-order differentiator, as the transmission is completely repressed and the phase is shifted by exact $\pi $ radians with almost infinite slope. Hence, it can be deduced that the discrepancy in Fig. 6(a) is derived from the insufficient bandwidth of the resonator. The bandwidth can be increased by decreasing $R$ and $r$, while greater pump power will be required to achieve the same differentiation order. The second factor is the response of the over-coupled and under-coupled microring resonator does not perfectly coincide with that of an ideal fractional-order differentiator. The transmission of the resonator cannot be attenuated to 0 on the resonance. Furthermore, the phase response is more crucial where the slope of the phase shift across the resonance declines with the increase of $|r-\tau |$. In Fig. 6(b), the differentiation orders with $\tau =0.834$ are $n=1.11$ and $n=1.2$ for the input pulses with FWHM = 50 ps and FWHM = 80 ps, respectively. When $\tau =0.816$, the differentiation orders are $n=0.64$ and $n=0.5$ for these pulse widths as shown in Fig. 6(c). The differentiation orders for the input pulse with FWHM = 20 ps are $n=1$ and $n=0.77$ in Fig. 6(b) and Fig. 6(c) respectively, which are affected by both of the two factors simultaneously.

To quantitatively evaluate the performance of the proposed fractional-order differentiator, the average deviation ${D}_{ave}$ can be defined as the mean absolute deviation between the outputs of the ideal and the microring-based differentiators [8]:

where ${P}_{MR}(t)$ and ${P}_{i}(t)$ are both normalized output powers. In this work, to guarantee the successive output pulses do not overlap with each other, the time repetition period of the input waveform is $T=6\times FWHM$, where FWHM is the input pulse width.The average deviation of the proposed differentiator for various input pulse widths is presented in Fig. 7. To demonstrate the effect of differentiation order on the average deviation, we focus on the curve for the input FWHM = 80 ps, where the bandwidth of the microring resonator is sufficient for the input pulse. As discussed before, the critical-coupled microring resonator has the most similar spectral response compared to the ideal differentiator, so the average deviation has a minimum at $n=1$. With the differentiation order departing from 1, the spectral response of the resonator deteriorates more and more seriously from the ideal fractional-order differentiator, so the average deviation is increased. Quite interestingly, the average deviation has a local maximum near $n=0.6$. As seen from Fig. 4(a) and Fig. 4(b), when $n<1$, the deviation is mainly derived from the minor peak of the output, while the minor peak of output pulse for the ideal differentiator declines with the decrease of $n$. The discrepancy of spectral response promotes the growth of the deviation, and on the contrary, the declining of the minor peak of output diminishes the average deviation. Thus, the two effects counteract with each other, and form the local maximum of the average deviation. However, when $n>1$, the average deviation is monotonous with respect to the differentiation order $n$.

Figure 7 also depicts the effect of input pulse width on the average deviation. With the decrease of the input FWHM, the minimum of the deviation curve is increased, while the differentiation order corresponding to the minimum deviation is reduced. In addition, although less deviation can be obtained for small differentiation orders, the deviation with $n>1$ will be increased. Hence, there is a trade-off for the deviations with small and large differentiation orders. For the input FWHM ≥ 50 ps where sufficient bandwidth can be provided by the microring resonator, the curve shift caused by the input pulse width is inconspicuous. It can be seen that the deviation curves for input FWHM = 80 ps and FWHM = 120 ps are almost coincident with each other. In this case, compared to Fig. 5, although the required pump power is varied with the input pulse width to achieve a particular differentiation order $n$, the average deviation ${D}_{ave}$ can be held relatively constant.

As discussed before, in the microring-based differentiator, the signal wavelength should coincide with one of the resonant wavelengths. However, the resonant wavelengths can be shifted by many factors, such as the light intensity, the free carrier density, and the thermal effect. To analyze the effect of resonance shift on the output waveform, without loss of generality, we assume the resonant wavelength is kept at 1439.5 nm, and the anti-Stokes signal pulses are launched at 1439.51 nm. The differentiated pulses with $n=0.5$, $n=1$, and $n=1.5$ are depicted in Fig. 8. When the signal is detuned from the resonance, it can be seen that the optical power cannot drop to zero in the notch part of the output waveform. Therefore, to reduce the output deviation of the microring-based differentiator, it is crucial to decrease the mismatch between the signal and the resonant wavelength.

As shown in Fig. 3(b), due to the narrow bandwidth of IRS, the intracavity loss induced by IRS is rather small except for the anti-Stokes resonance. Thus, different differentiation orders can be realized simultaneously at different resonant wavelengths by launching multiple pump light-waves. For instance, two pump light-waves at ${\lambda}_{P1}=1556.1$ nm and ${\lambda}_{P2}=1559.2$ nm are injected into the resonator with the power ${P}_{P1}=50$ mW and ${P}_{P2}=70$ mW. As shown in Fig. 9, when the Gaussian pulses with FWHM = 50 ps are launched as the anti-Stokes input signals, the 1.23-order differentiated pulse can be obtained at 1439.5 nm with the average deviation ${D}_{ave}=2.7\%$, and at the same time, the 0.49-order differentiated pulse can be obtained at 1442.3 nm with ${D}_{ave}=2.0\%$. The TPA process occurs between the signal and each pump light-wave, and the free carrier is generated by both of the pump light-waves. Therefore, compared with the signal-pump configuration with ${P}_{P1}=50$ mW or ${P}_{P2}=70$ mW, the intracavity loss is increased and the differentiation order is decreased. Besides, the four-wave mixing (FWM) effect is negligible, as the signal wavelengths are far from its gain-bandwidth.

## 4. Conclusion

In summary, using the IRS effect in silicon waveguide, a novel photonic fractional-order temporal differentiator is proposed based on the side-coupled microng resonator. By controlling the power of the pump light-wave, the intracavity loss is adjusted, and the coupling state of the microring resonator can be changed. Thus, the differentiation order can be tuned continuously. As the microring resonator has limited bandwidth and non-ideal spectral response as a fractional-order differentiator, the required pump power is varied with the input pulse width to obtain a particular differentiation order. In addition, with the increase of the input pulse width, the differentiation order becomes more sensitive to the variation of pump power. Compared with the output of an ideal fractional-order differentiator, the average deviation is minimal when the differentiation order is close to 1. The average deviation is monotonous with respect to the differentiation order when $n>1$, while it has a local maximum when $n<1$. While altering the input pulse width, the correspondence between the average deviation and the differentiation order is shifted, and there is a trade-off for the deviations with small and large differentiation orders. However, the shift is inconspicuous when sufficient bandwidth can be provided by the microring resonator. The proposed scheme offers a new approach to realize the optically tunable fractional-order differentiation in the microring resonator. The narrow attenuation bandwidth of the IRS process in silicon provides the potential to achieve different differentiation orders simultaneously at different resonant wavelengths. Furthermore, this work has discussed the influencing factors on the differentiation order and the average deviation, which can be applied to all the fractional-order differentiators based on microring resonators.

## Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (61307109 and 61475023), the Natural Science Foundation of Beijing (4152037), the National High-Technology Research and Development Program of China (2013AA031501), the Specialized Research Fund for the Doctoral Program of Higher Education (20120005120021), the Fundamental Research Funds for the Central Universities (2013RC1202), the China Postdoctoral Science Foundations (2012M511826 and 2014M552213), the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) P. R. China, the Hong Kong Scholars Program 2013 (PolyU G-YZ45), the Research Grant Council of the Hong Kong Special Administrative Region China (PolyU5272/12E), and the Science Foundation Ireland (SFI) under the International Strategic Cooperation Award Grant Number SFI/13/ISCA/2845.

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