In this paper, we propose and numerically investigate an all-optical temporal integrator based on a photonic crystal cavity. We show that an array of photonic crystal cavities enables high-order temporal integration. The effect of the value of the cavity’s free spectral range on the accuracy of the integration is considered. The influence of the coupling coefficients in the resonator array on the integration accuracy is demonstrated. A compact integrator based on a photonic crystal nanobeam cavity is designed, which allows high-precision integration of optical pulses of subpicosecond duration.
© 2014 Optical Society of America
All-optical fully integrated on-chip computing components will increase the speed of information processing by several orders of magnitude . Moreover, such components enable the processing of not only real, but also complex values. In this regard, it is important to implement the basic computing operations optically. In recent years, all-optical integrators based on Bragg grating , ring resonator , a time-spectrum convolution system  have been proposed. Such integrators can be used in both digital and analog signal processing. Among the digital signal processing applications are pulse-shaping , optical dark soliton detection , pulse counting, photonic analog-to-digital conversion  and ultrafast optical memory . Analog signal processing applications include all-optical solution of differential equations of various orders .
Integrators based on Bragg gratings are a few millimeters in size. Integrators based on ring resonators are more compact. Their size is on the order of tens of micrometers on the chip plane. In this paper we propose and study numerically the most compact optical integrators based on photonic crystal (PC) cavities [10,11].
2. Problem statement
Figure 1 shows a scheme of the coupled-resonator optical waveguide (CROW). The variable ai, i = [1, N] is the complex amplitude of the resonant mode in the i-th resonator; and , i = [1, N], are the left and right coupling coefficients of the i-th resonator, respectively; ri, i = [1, N] is the energy loss of the i-th resonator to the exterior space; and , , and are the amplitude of the input, reflected, and transmitted fields, respectively.
Consider an array in which the resonators have identical resonant frequencies. Then, according to the temporal coupled-mode theory , the amplitudes of the resonant mode in each resonator are connected by an equation that is written as follows in matrix form:
Further, is a variable that takes into account the mismatch relative to the frequency of the resonant mode and the energy losses to the exterior space, , , and is the imaginary unit.
The transmission function of the system can be written as 
For N = 1, Eq. (3) is reduced to the form
Hereafter, for simplicity, we neglect the losses to the exterior space.
Let us consider how accurately Eq. (4) approximates the integrator of the first order. The polarized electric field with envelope can be written as
A linear system described by the complex transfer function (TF) converts the envelope of the input pulse [Eq. (5)] to
The impulse response of a linear system with the TF is
The right side of this equation expresses the integral of the input pulse envelope with exponential weight.
Figure 2 shows the result of integration of the envelope of an optical pulse with a duration of 100 ps by resonators with Q-factors of 3 × 104 and 5 × 104. The Q-factor is related to by the ratio . The figure shows that the higher Q-factor of the resonator is, the more slowly the integrated signal envelope decays. For resonators with Q-factors of 3 × 104 and 5 × 104 we estimate an integration time window (defined as the decay time required to reach 80% of the maximum intensity) of 12.5 ps and 19.5 ps, respectively.
For N = 2, Eq. (3) can be written as
For an array of three resonators, TF is equal to
As an example, we show that the system with the TF in Eq. (9) allows second-order signal integration. The impulse response of a linear system with this TF is
You can obtain the relation
The general formula for high-order integrators is derived in .
The TF of an ideal integrator for a signal with a carrier frequency of ω0 is written as
The greatest differences between Eqs. (4) and (14) are observed near ω0, which is the pole of Eq. (14). This causes a large error when pulses with a narrow spectral range are integrated. The difference between Eqs. (4) and (14) near ω0, and thus the integration error, increase with decreasing Q-factor of the resonator
Figure 3 shows the result of integration of the first derivative of a Gaussian pulse with a duration of 100 ps by resonators with Q-factors of 104, 105 and 106. Figure 4 shows examples of the input test pulses. This is the original Gaussian pulse with a duration of 1 ps and its first three derivatives. Thus, the first-order integrator restores a Gaussian pulse from its first derivative. The second- and third-order integrators must restore the Gaussian pulse from the second and third derivatives, respectively.
The use of a PC resonator as an optical differentiator is described in . The function of the complex reflection of such a resonator can be written as
The value of ω0 is zero for the ideal differentiator function . Thus, Eq. (15) differs from the ideal differentiator function near ω0 quantitatively but not qualitatively. Therefore, differentiation of signals with a narrow spectral range by a high-Q resonator does not lead to errors like those shown in Fig. 3.
The reasons for the differences between Eqs. (4) and (14) outside of the neighborhood of ω0 vary with the type of resonator. For example, for a ring resonator, the error of integration at high frequencies is determined by the free spectral range (FSR) and depends in particular on the radius of the resonator. For PC resonators, the error of integration at high frequencies is influenced by the PC bandgap and neighboring resonant modes.
Figure 5 shows the root-mean-square error (RMSE) of integration of the first derivative of a Gaussian pulses with a durations of 1 ps and 150 fs for different FSR values. The Q-factor of the resonator is 5 × 104. The resonators with FSR values of 12 nm (RMSE = 14%) and 15 nm (RMSE = 6%) integrate the 1 ps impulse with large distortions. High-quality integration was achieved at an FSR of 25 nm. Similarly, a good quality of integration for 150 fs impulse is clearly achieved for FSR values as large as 120 nm (RMSE = 4%). The development of ring resonators with such a large FSR is associated with certain difficulties. PC resonators are a suitable candidate for the integration of subpicosecond optical pulses.
For an array of two isolated resonators, TF can be approximated as
The denominator in Eq. (9) is . Whereas, the denominator in Eq. (16) is . Thus, the TF of an array of coupled resonators approximates the TF of the ideal integrator (1/s2) more accurately than an array of isolated resonators.
3. Example of integrator based on PC nanobeam cavity
Let us calculate the parameters of the particular PC nanobeam cavity that integrates the optical signal. Compared with the resonators in the two-dimensional PC layer , PC nanobeam cavities  have a smaller area and are naturally integrated into the waveguide geometry of the chip.
Figure 6(a) illustrates a possible embodiment of a resonator based on a PC nanobeam. The decreasing radius of holes in the tapering region forms a defect in which the resonant mode is excited. An array of two PC resonators is shown in Fig. 6(b), where is the number of holes in the tapering region. The coupling value between resonators in the array is determined by , the number of holes with the maximum radius between defects. It can be shown  that for two adjacent resonators with quality factors Q1 and Q2, the coupling coefficient is
To create a high-Q PC nanocavity, it is necessary to reduce the radiation of the resonant mode in space. This is achieved by optimizing the shape of the envelope of the resonant mode. The spectrum of the electromagnetic field distribution directly above the waveguide determines the energy distribution in the far zone. This spectrum consists of two peaks. Energy is dissipated from the cavity through the light cone of the waveguide, which is located between the spectral peaks. Therefore, the width of the spectral peaks determines the cavity losses by scattering. In [16,17], it is proposed that a resonant mode can be formed with an envelope matching the Gaussian function. One of the same papers  showed that a resonant mode with a Gaussian envelope can be realized by changing the PC nanobeam material fill factor quadratically.
The resonance cavity characteristics were computed using the parallel 3D finite-difference time-domain method [18,19]. The waveguide in our simulations has a width of 490 nm and a height of 220 nm. It is composed of silicon and deposited on silica substrate. Air-filled holes in the regular part of the waveguide have a radius of 100 nm and are spaced 330 nm apart. The Bragg wavelength of such nanobeam grating is 1.57 µm. To ensure that the resonant wavelength will not change with , we need to design each cavity of the array with the resonant frequency at the Bragg wavelength of the nanobeam grating. Otherwise an additional waveguide section between the two resonators is required for appropriate coupling . Thus, both the radii and the periods of the tapering region are varied to ensure that the resonant wavelength is equal to the Bragg wavelength. Furthermore, the nanobeam grating strength is maximal at the Bragg wavelength, which enables the shortest possible device length.
Table 1 lists the lattice parameters and the radii of the holes near the defect (). These parameters demonstrate the existence of an energy bandgap for transverse electric polarization in the waveguide. The left and right borders of the bandgap are 1.46 µm and 1.67 µm, respectively. The size of the bandgap is about 210 nm. The resonant mode wavelength (1.57 µm) is placed in the center of this region. The FSR of the cavity can be estimated to 100 nm (~12.5 THz). The Q-factor of the resonator is 3.6 × 104, and . The integration time window for the first-order integrator based on such cavity is about 14 ps. Figure 7 shows the frequency response of the integrator.
The time-bandwidth product (TBP) of the integrator can be defined as the multiplication of the integration time window (in ns units) and the FSR of the cavity (in GHz units) . Thus, the TBP of the calculated integrator is 0.014 × 12500 = 175. That value is comparable to the TBP of the passive integrators based on Bragg grating  and ring resonator . The suggested integrator based on PC cavity has a large potential of increasing of the achieved TBP value. At least two methods can be proposed. Both of those are related to increasing of Q-factor of the resonator while the size of the bandgap remains the same. The first one is based on the fine-tuning of the tapering region and increasing of . By this way, Q-factor value achieved in this work can be enhanced by several orders of magnitude [11,16]. Another way is to use an active cavity . The considered PC cavity structure allows for the development of an integrated on-chip active cavity with electrical pumping .
If the number of left and right holes is equal to , as shown in Fig. 6(b), then the following condition holds:
Figure 8(a) shows the result of integration of the first derivative of a Gaussian pulse with a duration of 150 fs. The results of integration of the second and third derivatives of a pulse with the same duration by second- and third-order integrators are shown in Figs. 8(b) and 8(c), respectively. The second-order integrator consists of two resonators, as shown in Fig. 6(b), and the third-order integrator consists of three such resonators. The coupling coefficients between the resonators in each integrator are given by Eq. (18). Figures 8(b) and 8(c) show that the integration quality deteriorates with increasing order of integration. Figure 8(d) shows the result of third-order integration of a pulse with a duration of 200 fs. Thus, the suggested PC cavity is capable of qualitatively integrating subpicosecond pulses.
The dimensions of the first-order integrator based on the PC resonator in Fig. 6(a) are about 6.0 × 0.5 × 0.2 µm3 for a wavelength of 1.57 µm. The second- and third-order integrators have dimensions of 12.0 × 0.5 × 0.2 µm3 and 18.0 × 0.5 × 0.2 µm3, respectively. Thus, these integrators are at least 10 times more compact than any of those previously suggested.
To this point in this paper, we have assumed that the values of the coupling coefficients for the resonator array are given by Eq. (18). According to Eqs. (4), (9), and (10), the smaller the value of is, the closer the corresponding TFs are to that of the ideal integrator. This is also described in  for integrators based on Bragg gratings. The values of may be adjusted by changing the distance between cavities in the array. For example, by reducing the value of , we can increase the coupling coefficients. The coupling coefficients also affect the amount of energy at the output of the integrator. The energy efficiency of the integrator can be enhanced by increasing . However, this causes additional distortions of the corresponding TF. Figure 9(a) shows the amplitude of the TF of the three-resonator array for two variants of the coupling coefficients: those for the embodiment described by Eq. (18) and for the condition . Here, the value 6.31 corresponds to . Thus, we increase the coupling coefficients, leaving only holes with a variable radius between cavities in the array and removing six holes with a constant radius. Figure 9(b) shows the corresponding values of the TF phase.
Figure 9(b) shows that increasing the coupling coefficients extends the zone of an additional phase jump in the center of the TF. Figure 10 shows the results of integration of the third derivative of the Gaussian pulse with a duration of 1 ps by the resonator arrays with the TFs shown in Fig. 9. Thus, increasing degrades the accuracy of integration.
This paper presents a compact integrator of the complex envelope of an optical signal based on a PC waveguide. This integrator is more compact than any of those previously suggested. Its dimensions depend linearly on the order of integration. The proposed integrator may have a bandgap of more than 200 nm. This allows high-precision integration of optical pulses of subpicosecond duration.
This work was supported by the RFBR grants 13-07-97002, 13-07-13166, 14-07-97008, 14-07-97009, Ministry of Education and Science of the Russian Federation, and Program ONIT RAS No. 5.
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