Abstract

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

1. Introduction

All-optical fully integrated on-chip computing components will increase the speed of information processing by several orders of magnitude [1]. 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 [2], ring resonator [3], a time-spectrum convolution system [4] have been proposed. Such integrators can be used in both digital and analog signal processing. Among the digital signal processing applications are pulse-shaping [5], optical dark soliton detection [6], pulse counting, photonic analog-to-digital conversion [7] and ultrafast optical memory [8]. Analog signal processing applications include all-optical solution of differential equations of various orders [9].

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; κi1 and κi, 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 pin, prf, and ptr are the amplitude of the input, reflected, and transmitted fields, respectively.

 figure: Fig. 1

Fig. 1 Scheme of coupled-resonator optical waveguide.

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Consider an array in which the resonators have identical resonant frequencies. Then, according to the temporal coupled-mode theory [12], the amplitudes of the resonant mode in each resonator are connected by an equation that is written as follows in matrix form:

Ma=pin,
whereM=(s1+κ0iκ1000iκ1s2iκ2000iκ2s300iκN20000sN1iκN1000iκN1sN+κN).

Further, sj=i(ωω0)+rj=s+rj 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, a=[a1a2aN]T, pin=[i2k0pin00]T, and i is the imaginary unit.

The transmission function of the system can be written as [13]

TN(s)ptrpin=2(i)N1κ0κNκ1κ2κN1det(M),
where ptr=i2κNaN=2κ0κN[M-1]N,1pin, and det(M) is the determinant of the matrix M.

For N = 1, Eq. (3) is reduced to the form

T1(s)=2κ0s+2κ0

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 Pin(t) can be written as

E(x,t)=Pin(tx/vg)exp(im0xiω0t)==R(ωω0)exp(im(ω)xiωt)dω,,
where R(ω) is the envelope spectrum signal, m(ω) is the wave number [m0=m(ω0)], and vg is the group velocity.

A linear system described by the complex transfer function (TF) H(ω) converts the envelope of the input pulse [Eq. (5)] to

Ptr(t)=R(ω)H(ω)exp(iωt)dω==Pin(t)h(t),,
where the symbol * denotes the convolution operation, and h(t) is the spectrum of the TF H(ω).

The impulse response of a linear system with the TF T1(s) is

h1(t)=κ0exp(κ0t)u(t),
where u(t) is the Heaviside step function.

Substituting Eq. (7) into Eq. (6), we obtain an expression for the envelope of the output pulse:

Ptr(t)=κ0tPin(T)exp(iκ0(tT))dT.

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 κ0 by the ratio Q=ω0/(4κ0). 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.

 figure: Fig. 2

Fig. 2 Result of first-order integration of optical pulse with duration of 1 ps by resonators with Q-factors of 3 × 104 and 5 × 104.

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For N = 2, Eq. (3) can be written as

T2(s)=i2κ0κ1(s+κ0)2+κ12.

For an array of three resonators, TF is equal to

T3(s)=2κ0κ1κ2s(s+κ0)2+(s+κ0)(κ12+κ22).

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

h2(t)=κ0(κ0+iκ1)t(2iκ1t1)u(t).

You can obtain the relation

h2(t)2iκ0κ1texp(κ0t)u(t).

Substituting Eq. (12) into Eq. (6), we obtain an expression for the output of a second-order integrator:

Ptr(t)~iκ0κ1T2=tT1=T2Pin(T1)iκ0(tT1)dT1dT2.

The general formula for high-order integrators is derived in [14].

The TF of an ideal integrator for a signal with a carrier frequency of ω0 is written as

Hint(s)=1/s.

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.

 figure: Fig. 3

Fig. 3 Result of integration by resonators with Q-factors of 104,105 and 106. Input optical signal is first derivative of Gaussian pulse with duration of 100 ps.

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 figure: Fig. 4

Fig. 4 Gaussian pulse with duration of 1 ps and its first three derivatives.

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The use of a PC resonator as an optical differentiator is described in [15]. The function of the complex reflection of such a resonator can be written as

Hdf(s)=ss+κ0.

The value of ω0 is zero for the ideal differentiator function [Hidf(s)=s]. 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.

 figure: Fig. 5

Fig. 5 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.

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For an array of two isolated resonators, TF can be approximated as

T12(s)=2κ0s+2κ02κ0s+2κ0.

The denominator in Eq. (9) is s2+2sκ0+2κ02. Whereas, the denominator in Eq. (16) is s2+4sκ0+4κ02. 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 [10], PC nanobeam cavities [11] 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 ntap is the number of holes in the tapering region. The coupling value between resonators in the array is determined by nreg, the number of holes with the maximum radius between defects. It can be shown [16] that for two adjacent resonators with quality factors Q1 and Q2, the coupling coefficient is

κ=ω04Q1Q2=ω04Q0anreg,
where Q0 is the Q-factor of the resonator containing only the hole defect zone (nreg=0), and ω0 is the resonant frequency corresponding to the Bragg wavelength. The value of a can be approximated from the calculation of the Q-factor of a single resonator with different values of nreg.

 figure: Fig. 6

Fig. 6 Schemes of (a) PC nanobeam cavity and (b) array of two such cavities.

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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 [17] 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 nreg, 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 [16]. 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 (ntap=12). 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 nreg=6. 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.

Tables Icon

Table 1. Geometric Parameters of PC Resonator Shown in Fig. 6(a)

 figure: Fig. 7

Fig. 7 (a) The amplitude and (b) phase of the frequency response of the integrator.

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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) [3]. 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 [20] and ring resonator [3]. 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 nreg. 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 [21]. The considered PC cavity structure allows for the development of an integrated on-chip active cavity with electrical pumping [22].

If the number of left and right holes is equal to nreg/2, as shown in Fig. 6(b), then the following condition holds:

κ0=κN=κj,j=1,N1.

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.

 figure: Fig. 8

Fig. 8 Results of integration of corresponding derivatives of Gaussian pulse with duration of 150 fs by (a) one PC resonator, (b) an array of two PC resonators, and (c) an array of three PC resonators and (d) the result of integration of the third derivative of a Gaussian pulse with a duration of 200 fs by an array of three PC resonators. Q-factor of resonators is 3.6 × 104.

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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 κj is, the closer the corresponding TFs are to that of the ideal integrator. This is also described in [23] for integrators based on Bragg gratings. The values of κj,j=1,N1 may be adjusted by changing the distance between cavities in the array. For example, by reducing the value of nreg, 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 κj. 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 6.31κ0=6.31κN=κj,j=1,N1. Here, the value 6.31 corresponds to a3=1.853. 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: Fig. 9

Fig. 9 TF of three-resonator array for two variants of the coupling coefficients: (a) amplitude of the TF, (b) phase of the TF.

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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 κj degrades the accuracy of integration.

 figure: Fig. 10

Fig. 10 Results of third-order integration for two values of the coupling coefficients.

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4. Conclusion

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.

Acknowledgments

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.

References and links

1. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010). [CrossRef]  

2. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007). [CrossRef]   [PubMed]  

3. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010). [CrossRef]   [PubMed]  

4. A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, and J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. 37(8), 1355–1357 (2012). [CrossRef]   [PubMed]  

5. Y. Park, T. J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008). [CrossRef]   [PubMed]  

6. N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006). [CrossRef]   [PubMed]  

7. Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010). [CrossRef]  

8. Y. Ding, X. Zhang, X. Zhang, and D. Huang, “Active microring optical integrator associated with electroabsorption modulators for high speed low light power loadable and erasable optical memory unit,” Opt. Express 17(15), 12835–12848 (2009). [CrossRef]   [PubMed]  

9. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008). [CrossRef]   [PubMed]  

10. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005). [CrossRef]   [PubMed]  

11. P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006). [CrossRef]  

12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

13. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef]   [PubMed]  

14. M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009). [CrossRef]  

15. N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013). [CrossRef]   [PubMed]  

16. H. C. Liu and A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express 20(8), 9249–9263 (2012). [CrossRef]   [PubMed]  

17. Q. Quan and M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(5), 18529–18542 (2011). [CrossRef]   [PubMed]  

18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

19. D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009). [CrossRef]  

20. M. H. Asghari, C. Wang, J. Yao, and J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010). [CrossRef]   [PubMed]  

21. N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, and N. Zhu, “Active Fabry-Perot cavity for photonic temporal integrator with ultra-long operation time window,” Opt. Express 22(3), 3105–3116 (2014). [CrossRef]   [PubMed]  

22. G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011). [CrossRef]  

23. M. H. Asghari and J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008). [CrossRef]   [PubMed]  

References

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  1. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
    [Crossref]
  2. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
    [Crossref] [PubMed]
  3. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
    [Crossref] [PubMed]
  4. A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, and J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. 37(8), 1355–1357 (2012).
    [Crossref] [PubMed]
  5. Y. Park, T. J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008).
    [Crossref] [PubMed]
  6. N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006).
    [Crossref] [PubMed]
  7. Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
    [Crossref]
  8. Y. Ding, X. Zhang, X. Zhang, and D. Huang, “Active microring optical integrator associated with electroabsorption modulators for high speed low light power loadable and erasable optical memory unit,” Opt. Express 17(15), 12835–12848 (2009).
    [Crossref] [PubMed]
  9. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
    [Crossref] [PubMed]
  10. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005).
    [Crossref] [PubMed]
  11. P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
    [Crossref]
  12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  13. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
    [Crossref] [PubMed]
  14. M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
    [Crossref]
  15. N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
    [Crossref] [PubMed]
  16. H. C. Liu and A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express 20(8), 9249–9263 (2012).
    [Crossref] [PubMed]
  17. Q. Quan and M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(5), 18529–18542 (2011).
    [Crossref] [PubMed]
  18. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  19. D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
    [Crossref]
  20. M. H. Asghari, C. Wang, J. Yao, and J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
    [Crossref] [PubMed]
  21. N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, and N. Zhu, “Active Fabry-Perot cavity for photonic temporal integrator with ultra-long operation time window,” Opt. Express 22(3), 3105–3116 (2014).
    [Crossref] [PubMed]
  22. G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
    [Crossref]
  23. M. H. Asghari and J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
    [Crossref] [PubMed]

2014 (1)

2013 (1)

2012 (2)

2011 (3)

2010 (4)

M. H. Asghari, C. Wang, J. Yao, and J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
[Crossref] [PubMed]

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[Crossref]

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

2009 (3)

Y. Ding, X. Zhang, X. Zhang, and D. Huang, “Active microring optical integrator associated with electroabsorption modulators for high speed low light power loadable and erasable optical memory unit,” Opt. Express 17(15), 12835–12848 (2009).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[Crossref]

D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[Crossref]

2008 (3)

2007 (1)

2006 (2)

N. Q. Ngo, “Optical integrator for optical dark-soliton detection and pulse shaping,” Appl. Opt. 45(26), 6785–6791 (2006).
[Crossref] [PubMed]

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

2005 (1)

Ahn, T. J.

Akahane, Y.

Asano, T.

Asghari, M. H.

Ashrafi, R.

Ayotte, N.

Azaña, J.

N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, and N. Zhu, “Active Fabry-Perot cavity for photonic temporal integrator with ultra-long operation time window,” Opt. Express 22(3), 3105–3116 (2014).
[Crossref] [PubMed]

A. Malacarne, R. Ashrafi, M. Li, S. LaRochelle, J. Yao, and J. Azaña, “Single-shot photonic time-intensity integration based on a time-spectrum convolution system,” Opt. Lett. 37(8), 1355–1357 (2012).
[Crossref] [PubMed]

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

M. H. Asghari, C. Wang, J. Yao, and J. Azaña, “High-order passive photonic temporal integrators,” Opt. Lett. 35(8), 1191–1193 (2010).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[Crossref]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[Crossref] [PubMed]

Y. Park, T. J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
[Crossref] [PubMed]

Caulfield, H. J.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[Crossref]

Charvolin, T.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Chu, S. T.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Costanzo-Caso, P.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

Dai, Y.

Ding, Y.

Dolev, S.

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[Crossref]

Doucet, S.

Ellis, B.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Ferrera, M.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Golovashkin, D. L.

D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[Crossref]

Granieri, S.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

Hadji, E.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Haller, E. E.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Harris, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Huang, D.

Huang, N.

Hugonin, J. P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Jin, Y.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

Kazanskiy, N. L.

N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
[Crossref] [PubMed]

D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[Crossref]

Khonina, S. N.

Lalanne, P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

LaRochelle, S.

Li, M.

Little, B. E.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Liu, H. C.

Loncar, M.

Malacarne, A.

Mayer, M.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Morandotti, R.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Moss, D. J.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Ngo, N. Q.

Noda, S.

Park, Y.

Petykiewicz, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Peyrade, D.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Picard, E.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Quan, Q.

Quoc Ngo, N.

Razzari, L.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Rodier, J. C.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Sarmiento, T.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Serafimovich, P. G.

Shambat, G.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Siahmakoun, A.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

Slavík, R.

Song, B.-S.

Velha, P.

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Vuckovic, J.

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

Wang, C.

Wang, L.

Wang, X.

Yao, J.

Yariv, A.

Zhang, X.

Zhu, N.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. Shambat, B. Ellis, J. Petykiewicz, M. Mayer, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Nanobeam photonic crystal cavity light-emitting diodes,” Appl. Phys. Lett. 99(7), 071105 (2011).
[Crossref]

J. Lightwave Technol. J. (1)

M. H. Asghari and J. Azaña, “On the design of efficient and accurate arbitrary-order temporal optical integrators using fiber bragg gratings,” J. Lightwave Technol. J. 27(17), 3888–3895 (2009).
[Crossref]

Nat. Commun. (1)

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaña, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun. 1(3), 29 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010).
[Crossref]

New J. Phys. (1)

P. Velha, J. C. Rodier, P. Lalanne, J. P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, and E. Hadji, “Ultra-high-reflectivity photonic-bandgap mirrors in a ridge SOI waveguide,” New J. Phys. 8(9), 204 (2006).
[Crossref]

Opt. Express (9)

Y. Park, T. J. Ahn, Y. Dai, J. Yao, and J. Azaña, “All-optical temporal integration of ultrafast pulse waveforms,” Opt. Express 16(22), 17817–17825 (2008).
[Crossref] [PubMed]

H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011).
[Crossref] [PubMed]

H. C. Liu and A. Yariv, “Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators,” Opt. Express 20(8), 9249–9263 (2012).
[Crossref] [PubMed]

Q. Quan and M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(5), 18529–18542 (2011).
[Crossref] [PubMed]

Y. Ding, X. Zhang, X. Zhang, and D. Huang, “Active microring optical integrator associated with electroabsorption modulators for high speed low light power loadable and erasable optical memory unit,” Opt. Express 17(15), 12835–12848 (2009).
[Crossref] [PubMed]

R. Slavík, Y. Park, N. Ayotte, S. Doucet, T. J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008).
[Crossref] [PubMed]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13(4), 1202–1214 (2005).
[Crossref] [PubMed]

M. H. Asghari and J. Azaña, “Design of all-optical high-order temporal integrators based on multiple-phase-shifted Bragg gratings,” Opt. Express 16(15), 11459–11469 (2008).
[Crossref] [PubMed]

N. Huang, M. Li, R. Ashrafi, L. Wang, X. Wang, J. Azaña, and N. Zhu, “Active Fabry-Perot cavity for photonic temporal integrator with ultra-long operation time window,” Opt. Express 22(3), 3105–3116 (2014).
[Crossref] [PubMed]

Opt. Lett. (4)

Opt. Mem. Neural Networks (1)

D. L. Golovashkin and N. L. Kazanskiy, “Mesh domain decomposition in the finite-difference solution of Maxwell’s equations,” Opt. Mem. Neural Networks 18(3), 203–211 (2009).
[Crossref]

Proc. SPIE (1)

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 77970J (2010).
[Crossref]

Other (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

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Figures (10)

Fig. 1
Fig. 1 Scheme of coupled-resonator optical waveguide.
Fig. 2
Fig. 2 Result of first-order integration of optical pulse with duration of 1 ps by resonators with Q-factors of 3 × 104 and 5 × 104.
Fig. 3
Fig. 3 Result of integration by resonators with Q-factors of 104,105 and 106. Input optical signal is first derivative of Gaussian pulse with duration of 100 ps.
Fig. 4
Fig. 4 Gaussian pulse with duration of 1 ps and its first three derivatives.
Fig. 5
Fig. 5 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.
Fig. 6
Fig. 6 Schemes of (a) PC nanobeam cavity and (b) array of two such cavities.
Fig. 7
Fig. 7 (a) The amplitude and (b) phase of the frequency response of the integrator.
Fig. 8
Fig. 8 Results of integration of corresponding derivatives of Gaussian pulse with duration of 150 fs by (a) one PC resonator, (b) an array of two PC resonators, and (c) an array of three PC resonators and (d) the result of integration of the third derivative of a Gaussian pulse with a duration of 200 fs by an array of three PC resonators. Q-factor of resonators is 3.6 × 104.
Fig. 9
Fig. 9 TF of three-resonator array for two variants of the coupling coefficients: (a) amplitude of the TF, (b) phase of the TF.
Fig. 10
Fig. 10 Results of third-order integration for two values of the coupling coefficients.

Tables (1)

Tables Icon

Table 1 Geometric Parameters of PC Resonator Shown in Fig. 6(a)

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Ma= p in ,
where M=( s 1 + κ 0 i κ 1 0 0 0 i κ 1 s 2 i κ 2 0 0 0 i κ 2 s 3 0 0 i κ N2 0 0 0 0 s N1 i κ N1 0 0 0 i κ N1 s N + κ N ).
T N ( s ) p tr p in = 2 ( i ) N1 κ 0 κ N κ 1 κ 2 κ N1 det(M) ,
T 1 ( s )= 2 κ 0 s+2 κ 0
E( x,t )= P in ( tx/ v g )exp( i m 0 xi ω 0 t )= = R( ω ω 0 )exp( im( ω )xiωt )dω , ,
P tr ( t )= R( ω )H( ω )exp( iωt )dω = = P in ( t )h( t ), ,
h 1 ( t )= κ 0 exp( κ 0 t )u(t),
P tr ( t )= κ 0 t P in ( T )exp( i κ 0 ( tT ) )dT .
T 2 ( s )= i2 κ 0 κ 1 ( s+ κ 0 ) 2 + κ 1 2 .
T 3 ( s )= 2 κ 0 κ 1 κ 2 s ( s+ κ 0 ) 2 +( s+ κ 0 )( κ 1 2 + κ 2 2 ) .
h 2 ( t )= κ 0 ( κ 0 +i κ 1 )t ( 2i κ 1 t 1 )u(t).
h 2 ( t )2i κ 0 κ 1 texp( κ 0 t )u(t).
P tr ( t )~i κ 0 κ 1 T 2 = t T 1 = T 2 P in ( T 1 ) i κ 0 ( t T 1 ) d T 1 d T 2 .
H int ( s )=1/s .
H df ( s )= s s+ κ 0 .
T 1 2 ( s )= 2 κ 0 s+2 κ 0 2 κ 0 s+2 κ 0 .
κ= ω 0 4 Q 1 Q 2 = ω 0 4 Q 0 a n reg ,
κ 0 = κ N = κ j ,j=1,N1.

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