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We theoretically investigate the operation bandwidth limitation of the photonic differentiator including the upper limitation, which is restrained by the device operation bandwidth and the lower limitation, which is restrained by the energy efficiency (EE) and detecting noise level. Taking the silicon photonic crystal L3 nano-cavity (PCN) as an example, for the first time, we experimentally demonstrate that the lower limitation of the operation bandwidth does exist and differentiators with different bandwidths have significantly different acceptable pulse width range of input signals, which are consistent to the theoretical prediction. Furthermore, we put forward a novel photonic differentiator scheme employing cascaded PCNs with different Q factors, which is likely to expand the operation bandwidth range of photonic differentiator dramatically.
© 2015 Optical Society of America
1. Introduction
In addition to conventional advantages of reduced system size and dramatically lower cost, employing photons as information carriers also benefits from higher processing speed and lower power consumption [1, 2 ]. Great efforts have been paid to replace the traditional electronic computing circuit with integrated photonic circuit due to its tremendous potential in the future interconnection systems [3]. As an indispensable computing block in all-optical computing, photonic field differentiator has long been a focus since it was first proposed [4]. It has numerous application potential in ultra-short pulse generation [5, 6 ], odd-symmetry Hermite-Gaussian waveform generation [4] and pulse edge recognition [7]. Until now, the photonic differentiation was implemented by lots of optoelectronic devices, including fiber Bragg gratings (FBGs) [8, 9 ], long-period Bragg gratings (LPBGs) [4, 10 ], semiconductor optical amplifiers [6], Mach-Zehnder interferometers [11] and micro-ring resonators (MRRs) [12, 13 ]. Moreover, higher order photonic differentiators were realized by employing specially-designed FBGs [9], phase-shifted LPBGs [14], titled FBGs [15], programmable pulse shaper [16], cascaded silicon Bragg gratings [17], and so forth. For all these differentiators, operation bandwidth and processing accuracy are the most important parameters to evaluate the performance of the differentiator [18]. To date, most reported photonic differentiators are chasing to boost the upper bandwidth limitation, beyond which the processing accuracy will decrease greatly. For example, M. Li et al demonstrated a photonic differentiator with ultra-wide operation bandwidth of 25 THz [19]. Fortunately, Azaña et al analyzed the minimum operation bandwidth caused by the resonance depth of the differentiator [18]. It should be noted that, only the upper limitation of the operation bandwidth exists if the transmittance at the carrier frequency is null. However, in the practical application of photonic differentiator, the minimum operation bandwidth should always exist even if the notch transmittance is null because the energy efficiency should be considered [20]. If the bandwidth of input signal is much narrower than the bandwidth of the differentiator, the photonic differentiator with a notch response will attenuate the energy strongly, which may cause the output power to be lower than the noise level. Thus, the output signal cannot be recovered even if an optical amplifier is used. Hence, a high-speed differentiator is inappropriate to process very low speed signal. It means every photonic differentiator has an optimal input pulse width (OIPW), taking both device operation bandwidth (DOB) and energy efficiency into account. Unfortunately, the OIPW of the photonic differentiator was not analyzed and concerned yet.
In this paper, we theoretically analyze operation bandwidth range because of both the energy efficiency and the DOB of the photonic differentiator. By introducing the noise model of photodetectors into our calculations, we infer that the photonic differentiator has the lower limitation of operation bandwidth, which is restrained by the energy efficiency. Associated with the upper bandwidth limitation restrained by the DOB, the operation bandwidth of the photonic differentiator should be a band-pass response. In addition, we fabricated three samples of silicon photonic crystal nano-cavities (PCNs) with different Q factors to verify our theoretical prediction. Furthermore, we design a cascaded PCN structure to expand the operation bandwidth of the differentiator.
2. Theoretical model
An ideal first order differentiator (DIFF) should have a transfer function in the frequency domain as follows:
where is the optical frequency variable, is the carrier frequency and. In our model, to avoid the inaccuracy produced by finite resonance depth and emphasize the effect of the energy efficiency, we assume that the transmittance at the carrier frequency is null. Meanwhile, a finite bandwidth is set on the transfer function to meet the practical case. Now the transfer function in our model can be rewritten as follows:The processing accuracy of the photonic differentiator is evaluated by calculating the similarity of the measured output waveform and ideal output waveform, which is defined as the cross-correlation coefficient () [21]:
where and are the output intensity profiles and the ideal differentiation intensity profiles, respectively. Meanwhile, the energy efficiency (EE) is defined as the ratio between the power of output signal and input signal [20],In the modeling, the input signal profile is set as the Gaussian function, whose full width half-maximum (FWHM) varies from 1 ps to 35 ps. Meanwhile, the carrier wavelength is set as 1550 nm. According to Eqs. (1)-(4) , and EE of three DIFFs with different BWs (BW = 50 GHz, 100 GHz and 250 GHz) can be obtained respectively.
Figure 1(a) describes the as a function of the pulse width, which indicates that the processing accuracy will significantly decrease if the signal bandwidth is larger than the DOB [18]. Nevertheless, is not affected no matter how the signal bandwidth is reduced within the range of DOB. In this case, we ignore the impact of photodector noise. Figure 1(b) shows the EE as a function of input pulse width. We can see that the EE monotonically decreases with the pulse width increasing, which means input signal of lower speed has lower energy efficiency. From Fig. 1(b), we find that EE of 250 GHz DIFF is quite low when the pulse width is 30 ps, which can be as low as 0.01%. Thus, we can infer that when the EE is extremely low, the differentiator will filter out most energy of the input signal and the following devices are unable to distinguish the differentiation signals from the noise, which indicates that every DIFF should have a minimum operation bandwidth due to the EE restraint.
Fig. 1 Theoretical calculation of photonic differentiators. (a) The cross-correlation coefficient. (b) The energy efficiency.
To evaluate practical impact of EE on , a noise model is introduced into our simulation. Without loss of generality, we assume that the noise here mainly comes from thermal noise and shot noise, which obeys Gaussian distribution and is considered as white noise [22]. Meanwhile, the noise equivalence power (NEP) is set as 1.5 [23] and the input signal power is set as 10dBm. Figures 2(a)-2(c) calculate the of DIFF with BWs of 50 GHz, 100 GHz and 250 GHz, respectively. Different from Fig. 1(a), here the will decrease if the pulse width keeps increasing. This is caused by the extremely low EE and additional noise contribution. Therefore, this calculation exactly verifies our prediction that the minimum operation bandwidth exists due to the restraint of EE. In our calculation results in Fig. 2, every DIFF will have an OIPW, which is defined as the acceptable pulse width range of the input signal when the is larger than 80%. In Fig. 2, we can see that the minimum of OIPW is limited by the conventional DOB and the maximum of OIPW is determined by the restraint of EE. Table 1 summarizes the OIPW for three typical DIFFs of different BW where the input power is set as 10 dBm.
Fig. 2 Cross-correlation Coefficient calculation results with noise under different BW of the DIFF. (a) BW = 50 GHz. (b) BW = 100 GHz. (c) BW = 250 GHz.
Table 1. OIPW for DIFFs with different BWs
According to Table 1, DIFFs with different BWs have significantly different OIPW. The wider the BW is, the narrower the OIPW will be. The existence of OIPW is vitally important in the practical application of DIFF. For instance, if we aim to process a Gaussian signal with the FWHM of 20 ps, we should choose the DIFF with BW of 50 GHz rather than 250 GHz. Despite the fact that the wider BW of 250 GHz enables it to process signal with higher speed, processing relatively lower speed signal is out of the OIPW due to the extremely low EE, which will greatly degrade the operation performance.
Moreover, we investigate the relationship between the maximum acceptable pulse width and NEP in these three DIFFs with different BWs, as shown in Fig. 3 . The maximum acceptable pulse width is defined as the maximum pulse width of input signal when decreases to 80%. The signal peak powers were set as 10 dBm, 13 dBm and 16 dBm respectively in the calculations. One can see that, the maximum pulse width decreases with the NEP increasing. Meanwhile, for the same noise power level, the maximum acceptable pulse width will increase if we increase the input signal power. The maximum acceptable pulse width represents the lower limitation of operation bandwidth, which is restrained by the noise level of DIFF system.
Fig. 3 Maximum acceptable pulse width as a function of NEP under different BWs of DIFFs. (a) BW = 50 GHz. (b) BW = 100 GHz. (c) BW = 250 GHz.
3. Device design and fabrication
To experimentally validate our theoretical prediction, silicon PCNs with different Q factors were employed. According to previous research, the PCN has various applications in Hilbert transformation [24], optical temporal differentiation [25, 26 ], wavelength division signal demultiplexing [27] and dynamic optical nonlinear process [28]. We choose silicon L3 PCN as an experimental example due to its unique advantage in controlling the resonant bandwidth. In our experimental demonstration, DIFFs with various distinctly different bandwidths are required to verify the theoretical results. Fortunately, the Q factor of the PCN can be easily manipulated by adjusting the position of adjacent holes and the position of the cavity with respect to the waveguide according to previous study [29]. Therefore, PCNs with different Q factors were designed and fabricated to validate our theoretical prediction.
The transfer function of the two-dimensional photonic crystal slab including a L3 cavity and a waveguide can be described as follows [29, 30 ]:
where is a constant phase delay, which can be ignored for simplicity. The decay rates from the cavity into the waveguide and into the free space are denoted by and , respectively. The decay rates are related to the in-plane quality factor () and vertical quality factor () by and . The propagation constant is denoted by. Considering the similarities of the transfer function with MRR [12], the transfer function of photonic Nano-crystal (PCN) can be also approximated as the transfer function of the first-order differentiation, implying that such PCN is capable of differentiating the optical signal.
The designed PCNs were fabricated on three commercial silicon-on-isolator (SOI) wafers with 220 nm thick silicon on 3μm thick silica. We used E-beam lithography (Vistec EBPG 5000 Plus) to define the photonic crystal structures on a ZEP520A resist. Then the pattern was transferred to the top silicon layer by inductively coupled plasma (ICP) etching using SF6 and C4F8 gases. The upper silicon layer was etched downward for 220 nm to form a silicon ridge waveguide and etched downward for 70 nm to form input/output grating couplers. The couplers have a period of 630 nm, and the duty cycle is 50%. The coupling loss for one coupler is about 5.1 dB, thus the maximum transmission loss includes two couplers is about 10.2 dB [31]. In order to strengthen optical confinement in normal direction and increase the symmetry of the structure, the buried silicon oxide (BOX) layer was removed by dilute hydrofluoric acid solution. Finally, we fabricated a L3 cavity with a lattice constant of a = 420 nm and a hole radius of r/a = 0.3. In order to obtain PCNs with distinctly different Q factors, we optimized the position of three holes adjacent to the cavity and the position of the cavity with respect to the waveguide in the design [29]. Figure 4(a) shows the scanning electron microscope (SEM) image of the fabricated PCN, Figs. 4(b)-4(d) are SEM images of cavity region of different PCN samples and Figs. 4(e)-4(g) are measured transmission spectra of these three PCNs, respectively. The resonance wavelengths of three PCNs are 1562.9 nm, 1556.1 nm and 1561.1 nm respectively, which are slightly different due to the fabrication error. Meanwhile, the Q factors of three PCNs are 10000, 4500 and 1000, respectively. Thus the 3dB-bandwidth of three PCNs is 19.4GHz, 43.1GHz and 193.8GHz, respectively.
Fig. 4 (a) Scanning electron microscope of the PCN. (b)-(d) Zoom in region of three different nano-cavities with different Q factors. (e)-(g) Measured transmission spectra of three PCNs, respectively.
4. Experimental results and discussion
4.1 Experimental results
In order to verify the theoretical analysis results, we carried out experiments with the configurations depicted in Fig. 5 . A Mach-Zehnder modulator (MZM) and a phase modulator (PM) externally modulated a continuous wave (CW) beam generated by the tunable laser source (TLS) with a precisely tuning resolution of 100 kHz, successively. The frequency of the CW beam was aligned to the resonance frequency of the PCN. Two polarization controllers (PCs) were placed before the MZM to optimize the polarization states of incident light. The MZM and PM have a bandwidth of 20 GHz and 40 GHz respectively and were both driven by a tunable radio frequency (RF) signal (frequency range: 5 GHz ~20 GHz). A single mode fiber (SMF, 5km) was used to compensate the chirps of incident signal to generate a Gaussian pulse with narrow pulse width. The pulse width of input Gaussian pulse can be tuned by changing the frequency of the RF signal [32]. The first erbium doped fiber amplifier (EDFA) was used to boost the input optical power. Afterwards, the input signal was divided into two portions by a 50:50 optical coupler (OC), with one part directly injected to the chip and the other monitored by an optical power meter. To observe the impact of EE, the input signal power was fixed at 10 dBm by adjusting a variable optical attenuator (VOA1) during the whole experiment. To calculate the EE, the power of output signal was measured by another optical power meter before the second EDFA amplified the output signal. Finally, a high-speed oscilloscope (OSC) with a bandwidth of 500 GHz (EYE-1000C) recorded the temporal waveforms.
In the experiment, we first adjusted the frequency of the RF signal to generate Gaussian waveforms with different FWHMs. Frequencies of 5 GHz, 7 GHz, 10 GHz and 20 GHz were chosen to generate Gaussian pulses with the pulse width of 75 ps, 40 ps, 25 ps and 12 ps, respectively. Then we finely tuned the wavelength of the TLS to align to the resonance notch of the PCN. The VOA was adjusted to keep the optical power injected to the chip constant as 10dBm in every measurement. Meanwhile, the output signal waveform and power were recorded by the OSC and optical power meter 2, respectively. We measured and analyzed the performance of PCNs with Q of 10000, 4000 and 1000 successively. Higher Q factor implies narrower operation bandwidth and lower operation speed, thus we defined the PCN with Q of 10000 as low speed differentiator (LS DIFF), PCN with Q of 4000 as moderate speed differentiator (MS DIFF) and PCN with Q of 1000 as high-speed differentiator (HS DIFF).
We first measured the performance of LS DIFF under the input signal with different pulse width. The wavelength of the TLS was set as 1562.9 nm. The experimental results are shown in Fig. 6 , where the measured input and output temporal waveforms are depicted in Figs. 6(a)-6(f) and the measured spectra are shown in Figs. 6(g)-6(i). In Fig. 6, the measured results are depicted solid curves and the fitted or ideal waveforms are depicted in dotted curves. We can see that the processing error of differentiators increases along with the input pulse width decreasing. When the input FWHM reduces to 25 ps, the output waveform begins to diverge significantly from the ideal output. The calculated for every input is 96.74%, 98.15% and 89.75%, respectively. Meanwhile, the calculated EE is 19.54%, 26.67% and 51.17%, respectively.
Fig. 6 Time domain and spectrum measurement results of LS DIFF. (a)-(c). Input temporal waveforms. (d)-(f). Output temporal waveforms. (g)-(i): Measured spectra.
Next, we tested the performance of the MS DIFF. The wavelength of TLS was tuned to 1556.1nm while the input power was fixed at 10 dBm. Figure 7 shows the input waveforms Figs. 7(a)-7(c), the output waveforms Figs. 7(d)-7(f), and the spectra Figs. 7(g)-7(i) when varying the input pulse width. Meanwhile, the calculated is 97.26%, 99.12% and 92.93% when the FWHM of input signal is 75 ps, 40 ps and 25 ps, respectively. We can see the processing accuracy of MS DIFF is higher compared to the LS DIFF according to both waveforms and , especially when the FWHM of input signal is 25 ps. The EE is 7.31%, 14.96% and 15.07% when the FWHM varies from 75 ps to 25 ps, respectively.
Fig. 7 Time domain and spectrum measurement results of MS DIFF. (a)-(c). Input temporal waveforms. (d)-(f). Output temporal waveforms. (g)-(i). Measured spectra.
The HS DIFF was measured at last after the wavelength of TLS was set as 1561.1 nm. The measured results of the input waveforms Figs. 8(a)-8(c) , output waveforms Figs. 8(d)-8(f) and the spectra Figs. 8(g)-8(i) are illustrated in Fig. 8, respectively. According to the results, the HS DIFF is able to process the high-speed input signal with the FWHM of 12 ps. However, when the FWHM of the input signal is 75 ps, the differentiated signal cannot be identified because the optical power of output signal is too low compared to the noise level. The is calculated to be 0, 99.82%, 93.29% and 91.38% when the input signal FWHM varies from 75 ps to 12 ps. Meanwhile, the EE is 0.1954%, 0.2667%, 0.5117% and 0.8730%.
Fig. 8 Time domain and spectrum measurement results of HS DIFF. (a)-(d). Input temporal waveforms. (e)-(h): Output temporal waveforms. (i)-(l). Measured spectra.
4.2 Discussion
Figures 9(a) and 9(b) show the as a function of input pulse width under different Q factors. From Fig. 9(a), we can see that when the input pulse width is 75 ps, the of HS DIFF suddenly drops to zero. The reason is that the EE under this condition is extremely low, which causes the signal to be buried with noise and thus photodetector is unable to recognize the output signal. In Fig. 9(b), we can see that for the same input pulse width, increases with the bandwidth of the DIFF, which is consistent to the prediction of Fig. 1(a). Meanwhile, the of MS and LS DIFFs also start to decrease slightly when the FWHM of the input signal is larger than 40ps, which is consistent to our theoretical analysis of Fig. 2. Figure 9(c) depicts the relationship between EE and input pulse width under different Q factors. In Fig. 9(c), it can be seen that HS DIFF has very low EE due to its large bandwidth, which coincides well with our numerical analysis in Fig. 1(b). Moreover, the narrow bandwidth of low speed input signal makes more energy filtered out by the differentiators. Thus, the HS DIFF has a limitation of maximum pulse width that was observed at 75 ps. Unfortunately, since the maximum pulse width was limited by the minimum driving RF signal frequency (5 GHz) in our experiments, the lower limitation of the operation bandwidth of MS and LS DIFFs was not obtained precisely in our experiment. However, according to the existing trends of the and EE of LS and MS DIFFs, we can conclude that there must be a pass band range of the operation bandwidth. Particularly, it is obvious that HS DIFF is unable to differentiate low speed signal in the practical cases, which we have rarely been aware of before.
Fig. 9 (a). for different DIFFs with different BW and (b) its zoom in. (c). EE for different DIFFs with different BW.
Despite the fact that three PCNs we fabricated have different resonant depth, which may play an important role in the differentiation process [18], it may not have a significant effect on our conclusion. On the one hand, in our theoretical analysis, the resonant depth has been set as ideal to ensure the calculation results free from the impact of the resonant depth. Therefore, we can conclude that the energy efficiency is the only factor that can have influence on the processing accuracy when the input signal bandwidth is narrow in the theoretical analysis. On the other hand, in the experiments, when the input pulse width increases, the output signal will become completely overlapped by noise due to the low energy efficiency, which is shown in Fig. 8(e). Therefore, the cross-correlation coefficient decreases dramatically, which indicates the lower limitation of the operation bandwidth. However, the processing error caused by the resonant depth mainly introduces a deviation from the ideal temporal waveform rather than being overlapped by the noise. By discriminating the two different forms of processing error caused by energy efficiency and resonant depth, we can validate that the resonant depth does not have a significant influence on our experimental results.
To break the limited operation bandwidth caused by EE, a feasible solution is to enhance the input signal power when EE is low. However, the optical signal with high power is very likely to damage the integrated devices. More importantly, if resonant devices such as MRRs and PCNs are employed as DIFFs, significant red-shift of the resonance wavelength can easily happen due to the optical-thermal effect [33], which brings troubles in aligning to the resonance notch. Meanwhile, active medium may be deployed in the integrated devices to offer additional gain [34]. However, this may increase the complexity of fabrication process and another pump source is required.
Considering the ultra-compact size and the diversity of Q factor of the PCN, we believe the cascaded PCN structure is an ideal solution. As Fig. 10 illustrates, we can integrate two PCN units with different BWs to form a DIFF with a broader bandwidth. By adjusting the position of the adjacent holes of the cavity as well as the position of the cavity, we can obtain two cavities (defined as PCN1 and PCN2) with distinctly different Q factors, such as the typical Q of 10000 and 1000, respectively. Therefore, PCN1 is expected to act as LS DIFF and PCN2 is expected to act as HS DIFF. Meanwhile, the resonant wavelengths of two PCNs are modified to be different by altering the lattice constant of the photonic crystal. On the one hand, if the input signal pulse width is about 75 ps (defined as low-speed signal), we should tune the signal wavelength aligned to the resonant wavelength of PCN1. On the other hand, if the input signal pulse width is about 10 ps (defined as high-speed signal), the signal wavelength will be aligned to the resonant wavelength of PCN2. In this way, the operation bandwidth of DIFF can be extended employing a single all-passive chip. Moreover, the Q factors and the number of PCNs can also be optimized to provide even wider operation bandwidth.
5. Conclusion
In this article, we theoretically investigate the operation bandwidth range because of both energy efficiency and processing accuracy of the photonic differentiator. Meanwhile, by employing silicon photonic crystal Nano-cavities of different Q factors as differentiators, we experimentally validate our prediction that the operation bandwidth of photonic DIFF is a band-pass response rather than merely concerning maximum DOB. Therefore, according to our verifications, we propose that the bandwidth of DIFF should be carefully chosen with different signal speed. Furthermore, we put forward a cascaded PCN scheme to break the bandwidth range limitation of DIFF introduced by the energy efficiency.
Acknowledgment
This work is partially supported by the National Basic Research Program of China (Grant No. 2011CB301704), the Program for New Century Excellent Talents in Ministry of Education of China (Grant No. NCET-11-0168), a Foundation for Author of National Excellent Doctoral Dissertation of China (Grant No. 201139), the National Natural Science Foundation of China (Grant No. 11174096 and 61475052), the Major State Basic Research Development Program of China (Grant No. 2013CB632104 and 2013CB933303) and National Natural Science Foundation of China (Grant No. 61335002).
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