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Optical autocorrelation accuracy was for the first time analyzed for the silicon waveguide based autocorrelators utilizing two-photon absorption (TPA) under various short pulse conditions by numerical simulation. As for autocorrelation operation in the sub-μm silicon p-i-n rib waveguides on the 220 nm SOI (silicon on insulator) wafers, the autocorrelation error of pulse width measurement gradually increases with the increase of the peak power for both Gaussian and hyperbolic secant pulses due to the influence of free-carrier absorption (FCA). For the same pulse type, the relative error is independent of the input pulse width; however different pulse type has different peak power dependency of the accuracy. It was verified that this thin rib waveguide has a TPA responsivity >60 times higher than the thick rib waveguides and the correct pulse width can be measured with a <1% relative error for characterizing ps/sub-ps short pulses of sub-watt peak powers by utilizing the silicon wire p-i-n waveguides as the autocorrelator detector.
© 2016 Optical Society of America
1. Introduction
Integrated autocorrelators are the new emerging silicon photonic devices to realize on-chip short optical pulse measurement [1–3]. Such autocorrelators have great benefits in footprint and cost over conventional free-space optics based autocorrelators for applications in ultrafast laser systems [4]. So far, there are two mechanisms reported for silicon waveguide based autocorrelators: (1) third-harmonic generation (THG) induced by slow light in photonic crystal waveguide [1], and (2) two-photon absorption (TPA) in conventional thick rib waveguide (several μm in thickness) [2] and slow-light enhanced TPA in photonic crystal waveguides [3]. TPA responsivity in thick rib waveguides was low because the optical intensity is weak due to the large mode area. Photonic crystal waveguides could be used to enhance the responsivity by employing the slow-light enhanced light-matter interaction; but at the same time, the bandwidth becomes very narrow as the group velocity decreases. In other words, the group velocity dispersion will become more severe and broaden the correlation waveform, which will induce the measurement errors for sub-ps short pulses [3,5]. For actual autocorrelator application, both the responsivity and accuracy are important. Improving the responsivity has been one of the main purposes in these previous publications, while how accurate the short pulse width can be measured is less discussed. Especially, how the accuracy of pulse width measurement varies with pulse conditions and how it be influence by the concomitant free-carrier absorption (FCA) have not been reported yet as far as we know. For this purpose, we intend to use the 220-nm-thick sub-μm rib p-i-n waveguide (thin rib) as the autocorrelator photodetector since a strong TPA induced photocurrent has been observed in [6]. This waveguide not only offers a confinement as strong as the photonic crystal waveguide, but also shows much wider bandwidth and much lower group velocity dispersion. Thus, it is possible to characterize much shorter pulses compared to the photonic crystal waveguide using this waveguide as a TPA detector, and both high responsivity and high accuracy are promised. In addition, it is well-established for high-speed silicon photonic integrated devices using CMOS compatible process [7], which favors further integration between the autocorrelator and these integrated devices towards on-chip ultrafast pulse characterization.
In this paper, we performed autocorrelation simulation using double pulse excitation to study the autocorrelation accuracy for the TPA in the sub-μm rib p-i-n waveguide on 220 nm photonic SOI (silicon on insulator) wafers. First, we clarified the enhancement of the TPA responsivity and demonstrated correct autocorrelation operation in this thin rib waveguide. Then we examined the dependence of pulse width measurement accuracy on various pulse conditions. It is verified that both high responsivity and high accuracy autocorrelation operation can be obtained for ultrafast near-infrared pulses using this waveguide as a detector. This work will contribute to accurate characterization for weak and short pulses for on-chip silicon photonic autocorrelators.
2. Methodology
We show the studied p-i-n waveguide structure and its TE mode profile in Fig. 1(a) and 1(b), respectively. The mode profile as well as the group index was calculated by FemSIM [8]. The effective mode area (Aeff) was calculated to be 0.0875 μm2 by the equation (Eq. 1) where E is the electrical field of the mode profile. This value is much smaller than that of the traditional thick rib waveguide (Aeff ~6.2 μm2 [2]) and also smaller than the effective mode area (0.43 μm2) of the photonic crystal waveguide in [3]. For the thick rib waveguide, we considered the 1.2-μm-high and 3.2-μm-wide rib structure with a 1.8-μm-thick slab. The small area of this thin rib waveguide indicates a strong optical confinement.
Fig. 1 (a) Cross-section schematic of sub-μm silicon p-i-n waveguide. (b) Mode profile at λ = 1.55 μm, and (c) Wavelength dependent group index ng and dispersion parameter D. (d) Autocorrelation simulation schematic with a reverse-bias of 10 V and a device length of 1 mm.
The wavelength dependent group index ng is plotted in Fig. 1(c), where ng (~3.9) is almost flat within an about 100 nm spectral range. In contrast, the prominent wavelength dependence of ng was observed in the photonic crystal waveguides even with low-dispersion designs [3,5]. This spectral range is about several times wider than that (10 − 20 nm) of the photonic crystal waveguides, which means that much shorter pulses are possible to be measured using the thin rib waveguide. The dispersion parameter D ( = 1/c·dng/dλ) was also given in Fig. 1(c). With D = −3.5 × 10−4 ps/(nm·mm) at 1.55 μm, we estimated the broadening to be about 1.2 fs for the 1-ps Gaussian pulse and 1-mm device length, which is negligible in autocorrelation measurement. The schematic of autocorrelation simulation is shown in Fig. 1(d). Two correlated pulses were input into a 1-mm-long p-i-n waveguide under a reverse bias of 10 V and the correlated photocurrent signal can be obtained as a relation of the delay time between these two pulses. All calculations were done at TE polarization and λ = 1.55 μm.
The pulse propagation in the waveguide along z-axis is described by the differential Eq. (2) which takes the linear absorption, TPA, and FCA into consideration. As mentioned above, the thin rib waveguide has strong optical confinement; thus, the TPA induced photocurrent will increase. At the same time, the concomitant FCA due to the generated carriers intends to reduce the photocurrent signal. The FCA influence was not considered in [2] and assumed being fully extracted by reverse bias in [3]. The temporal carrier density response can be described by Eq. (3) by considering the carrier drift time under reverse bias. Equation (3) originates from the current continuity equation by neglecting the current flux gradient and diffusion current along the direction of electrical field [9,10]. This simplification is effective for the waveguide type diodes because the current can be assumed uniform in the small core region. Both the intensity I and carrier density N are the time and space dependent invariants and the subscript time can be interpreted to stand for the group velocity. Other symbols can be referred to in [3,9].
We solved Eqs. (2) and (3) simultaneously in time and space iterations by the central finite difference method and then the transient current response can be calculated using Eq. (4). Consequently, the photocurrent was calculated as the time average by taking the integral for Eq. (4) in which the intensity has taken the FCA into consideration by Eq. (2). Equation (4) is based on the fact that the absorbed photon number and the generated carrier number are equal. In fact, there is also an assumption for Eq. (4) that the pulse period is long enough compared to the drift time and all the generated carriers can be extracted by external circuits. In other words, the calculated responsivity stands for the intrinsic maximum efficiency. Note that the waveguide dispersion and free-carrier dispersion were not included in the simulation because of the short device length (1 mm) and the low peak power. As for the simulation parameters, the absorption coefficient α was set to the value corresponding to the propagation loss of 0.1 dB/cm for the thick rib waveguide and of 1.0 dB/cm for the thin rib one. The TPA coefficient β and FCA coefficient σ are 0.6 cm/GW and 1.47 × 10−17 cm2, respectively [3,9]. The group index ng is set to 3.9 and 4.5 for the thin and thick rib waveguides, respectively, which means that the thick rib waveguide even has a longer light-matter interaction time. As for the carrier drift time τdft in reversely biased p-i-n junctions, we determined as followed: (1) For the thin rib waveguide, a 10 V bias and a pn junction distance of 2 μm correspond to an electrical field as large as 5 × 104 V/cm at which the electron drift speed is saturated [11]. So we used an average drift speed of the saturated electrons and holes in [11], 8 × 106 cm/s, which gives a τdft of ~25 ps. (2) For the thick rib waveguide, a typical pn junction distance is usually as large as about 25 μm due to the large mode profile [12]. Thus, the τdft can be estimated to be ~625 ps using the electrical field at the voltage of 10 V and the mobility of 1000 cm2/V⋅s. Since the carrier life time in the thin rib waveguide is in nanosecond-level [6] and that in the thick rib waveguide is about several tens of nanoseconds [13], the carrier life time was not included in Eq. (3) because it is much longer than above τdft for both the thin and thick rib waveguides according to [6,13]. The repetition rate of the pulse was 40 MHz, corresponding a pulse period of 25 ns. This period is much longer than the drift time for both the thin and thick rib waveguides; thus, the assumption mentioned above is reserved.
3. Result and discussion
3.1 TPA responsivity
For the TPA responsivity calculation, a single Gaussian pulse with a 2-ps full width at half maximum (FWHM) was inputted, which can be understand as the double correlated pulses with zero delay. Therefore, the TPA responsivity means the peak power efficiency for autocorrelation operation. The peak power (Ppk) dependent photocurrent is shown in Fig. 2(a) to compare the TPA responsivity between the thin and thick rib waveguides. The photocurrents without considering FCA were also calculated for the thin rib waveguide. In the sub-ten-watts power regime, the TPA responsivity of the thin rib waveguide is about 2.4 μA/W2, about 66 times higher than that (~36 nA/W2) of the thick rib waveguide. With the increase of the peak power, the influence of FCA gradually becomes obvious for the thin rib waveguide as indicated by the photocurrent reduction at the high power side; nevertheless, its photocurrent is still about one order higher than that of the thick rib case for the peak power up to about 50 W. The deviation of photocurrent from the ideal quadratic line comes from the increase of the linear component with the increased power even without considering FCA and such a deviation is determined by the absorption coefficient and pulse intensity.
Fig. 2 (a) Photocurrent versus the peak power characteristics at the repetition rate of 40 MHz. (b) Input and output pulse intensities for the thin (top) and thick (bottom) rib waveguides at the peak power of 7.5 W.
Since the responsivity is related to three experimental factors including the pulse repetition rate, peak power, and pulse width, we should normalize the responsivity to these factors to compare the responsivities with the previously published values. For the thick rib waveguide in [2], the experimental photocurrent was about 108 nA for 1.35-ps-wide pulses with a pulse period of 56 ns and a peak power of 1.5 W. Using abovementioned simulation parameters, we normalize the calculated responsivity (36 nA/W2) of the thick rib waveguide to the same pulse condition and obtain a photocurrent of 122 nA. For the slow-light enhanced TPA in the photonic crystal waveguides in [3], the photocurrent was about 2 μA under the condition of the peak power of 1 W, 40 MHz, 2.7-ps pulse width for the highest ng. Similarly normalized to this pulse condition, the calculated responsivity (2.4 μA/W2) of the thin rib waveguide will result in a photocurrent of 3.24 μA, from which we conclude that the thin rib waveguide exhibits the responsivity as high as the photonic crystal waveguides.
In autocorrelation measurement, the detection limit of the peak power depends on the repetition rate. Taking 10 pA as the measurable current level, the peak power as low as 0.01 W can be detected at 40 MHz, as seen in Fig. 2(a); while for the repetition rates of 1 and 10 kHz, the detection limits were estimated to be about 0.3 and 0.1 W, respectively. Therefore, autocorrelation characterization for sub-watt pulses is guaranteed for > 10 kHz.
Figure 2(b) shows the pulse attenuation in both the thin and the thick rib waveguides for the peak power of 7.5 W. For the thin rib waveguide, the decrease in photocurrent caused by FCA is lower than 10% of that without FCA at the peak power less than 7.5 W. Note that the different positions of the output pulses in Fig. 2(b) correspond to the different group indices. For the thick rib waveguide, the peak absorption is only about 1%; however, for the thin rib waveguide, the peak absorption is about 40%. Such a strong attenuation in the thin rib waveguide mainly comes from the high intensity. As indicated by the effective area mentioned above, the intensity in the thin rib waveguide is about 70 times higher than that of the thick rib one, as seen in Fig. 2(b). As mentioned, the TPA responsivity is about 66 times enhanced in the thin rib waveguide, and as seen from Eq. (4), the TPA photocurrent is reversely proportional to the effective area if without considering the influences of FCA. Thus, the responsivity enhancement slightly deviates from the ideal situation. This enhanced TPA responsivity favors more effective pulse autocorrelation operation for on-chip weak pulse characterization. We will further examine how accurate the pulse width can be measured by autocorrelation operation after considering FCA in the following sections.
3.2 Autocorrelation operation
Before examining the accuracy, we intend to show the correct autocorrelation operation using the enhanced TPA responsivity in the thin rib waveguide. To demonstrate autocorrelation operation, we input two correlated pulses and adjusted the delay time. Figure 3(a) shows the autocorrelated photocurrent signal for the 2-ps Gaussian pulses with the 40 MHz repetition rate and the peak power of 12.5 W. We used this slightly higher peak power in order to clearly notice the influence of FCA on autocorrelation. The signal without considering FCA was also simulated for comparison. FCA intends to decrease the overall signal level and causes a relative current decrease on both sides of the peak during approaching or separating of two correlated pulses. The phenomenon results from that the former pulse excited carriers remain in the waveguide and cause the FCA for the following pulse. So we observed such a relative current decrease even though the two pulses are separated by a time larger than two input pulse widths. The FWHM of the autocorrelated signal in Fig. 3(a) was extracted and is shown in Fig. 3(b) in a relation of the peak power. As for autocorrelation, the autocorrelated width should be restored to the original input pulse width by being divided by the deconvolution factor of Gaussian pulse [14]. As shown in Fig. 3(b), the measured deconvoluted width almost equals the input pulse width of 2 ps in the sub-watt peak power region, demonstrating a correct autocorrelation operation. Usually sub-watt is the target peak power region for integrated autocorrelators [1,3,15]. In the whole power region, the deconvoluted width has an average value of 2.1 ps with an error of ± 0.05 ps even though the error gradually increases with the increase of the peak power.
Fig. 3 (a) Autocorrelated current signal versus the delay time for a 2-ps Gaussian pulse. (b) Autocorrelated pulse width in (a) and the deconvoluted width in a relation of the peak power.
3.3 Pulse width accuracy
Besides the responsivity, the autocorrelation accuracy, i.e., how accurate the pulse width can be measured using this waveguide autocorrelator is also important for the actual applications. To examine the dependence of accuracy on pulse conditions, in a similar way as mentioned above, the peak power dependent deconvoluted widths were also examined under various pulse conditions including both Gaussian and hyperbolic secant pulses and different widths. Note that the deconvolution factor for hyperbolic secant pulses is 1.54 [14]. The deconvoluted widths are shown in Fig. 4(a). It can be seen at the high peak power that the deviation (absolute error) of the deconvoluted width wd from the input pulse width w0 increases with the increase of w0. To easily distinguish the accuracy dependence on the pulse shape, width, and peak power, we derived the relative error (wd – w0)/w0 by normalizing the absolute error to w0 and summarized in Fig. 4(b). With the increase of the peak power, both Gaussian and hyperbolic pulses exhibit the increase in the error; however, they have different peak power dependences. For sub-watt peak powers or lower, a <1% relative error can be achieved for both types of pulses and the Gaussian pulse is more accurate than the hyperbolic secant pulse. In the sub-10 W region, both types of pulses show similar errors. For the same pulse shape, the relative error is almost independent of the input pulse width, which in other words means that the absolute error has a linear dependence on pulse width. This accuracy behavior favors correct pulse calibration for silicon waveguide autocorrelators in actual applications.
Fig. 4 (a) Deconvoluted full-width at half maximum (FWHM) versus the peak power for both Gaussian and hyperbolic secant pulses. Four input pulse widths are denoted. (b) Relative error (wd – w0)/w0 for both Gaussian and hyperbolic secant pulses under different input pulse width. wd: deconvoluted width. w0: input pulse width.
Next, we explain why the relative error increases with the increase of the peak power, which is a common feature for both Gaussian and hyperbolic secant pulses. Such an error increase results from the FCA induced pulse shape change. The shape of the output pulse becomes asymmetric because the wave front of the pulse experiences the stronger absorption than the wave tail and the carrier drifting in the p-i-n junction induces a non-instantaneous response. When such two pulses are correlated, the deconvoluted width will deviate from the original pulse width. Thus, higher peak powers result in larger errors.
Finally, the numerical error was evaluated at the 1% relative error to understand the confidence region in the accuracy analysis. We calculated the autocorrelation accuracy for the Gaussian pulse with the pulse width of 0.6 ps by decreasing the time discretization grid to the half (30 fs). The shortest pulse denotes the most severe situation. The numerical error of the accuracy is about 3% and the power dependent feature in Fig. 4(b) remains the same.
4. Summary
The TPA responsivity can be >60 times enhanced in the sub-μm silicon p-i-n waveguide on the 220 nm photonic SOI platform compared to that on thick SOI. Correct autocorrelation operation was demonstrated using this enhanced TPA and the autocorrelation accuracy was analyzed under various pulse conditions by numerical simulation. We verified that the correct autocorrelation operation with a <1% relative error can be achieved in pulse width measurement for sub-watt peak power pulses. Both high responsivity and high accuracy can be obtained in autocorrelation operation using this waveguide as the autocorrelation detector. This work promises a full integrated autocorrelator for correctly measuring ultrafast near-infrared pulses on a silicon photonic chip.
Funding
Japan Society for the Promotion of Science (JSPS) (KAKENHI 16K18097).
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