Abstract

We propose and experimentally demonstrate an all-optical temporal differential-equation solver that can be used to solve ordinary differential equations (ODEs) characterizing general linear time-invariant (LTI) systems. The photonic device implemented by an add-drop microring resonator (MRR) with two tunable interferometric couplers is monolithically integrated on a silicon-on-insulator (SOI) wafer with a compact footprint of ~60 μm × 120 μm. By thermally tuning the phase shifts along the bus arms of the two interferometric couplers, the proposed device is capable of solving first-order ODEs with two variable coefficients. The operation principle is theoretically analyzed, and system testing of solving ODE with tunable coefficients is carried out for 10-Gb/s optical Gaussian-like pulses. The experimental results verify the effectiveness of the fabricated device as a tunable photonic ODE solver.

© 2014 Optical Society of America

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References

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  1. L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003).
    [Crossref]
  2. F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
    [Crossref] [PubMed]
  3. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007).
    [Crossref] [PubMed]
  4. 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(29), 29 (2010).
    [PubMed]
  5. 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]
  6. J. Azaña, “Proposal of a uniform fiber Bragg grating as an ultrafast all-optical integrator,” Opt. Lett. 33(1), 4–6 (2008).
    [Crossref] [PubMed]
  7. 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]
  8. R. Ashrafi and J. Azaña, “Terahertz bandwidth all-optical Hilbert transformers based on long-period gratings,” Opt. Lett. 37(13), 2604–2606 (2012).
    [Crossref] [PubMed]
  9. L. Zhuang, M. R. Khan, W. Beeker, A. Leinse, R. Heideman, and C. Roeloffzen, “Novel microwave photonic fractional Hilbert transformer using a ring resonator-based optical all-pass filter,” Opt. Express 20(24), 26499–26510 (2012).
    [Crossref] [PubMed]
  10. G. F. Simmons, Differential Equations With Applications and Historical Notes (McGraw-Hill, 1991), Chap. 1.
  11. A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems (Prentice-Hall, 1996), Chap. 2.
  12. K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
    [Crossref]
  13. S. Tan, Z. Wu, L. Lei, S. Hu, J. Dong, and X. Zhang, “All-optical computing system for solving differential equations based on optical intensity differentiator,” Opt. Express 21(6), 7008–7013 (2013).
    [Crossref]
  14. S. Tan, L. Xiang, J. Zou, Q. Zhang, Z. Wu, Y. Yu, J. Dong, and X. Zhang, “High-order all-optical differential equation solver based on microring resonators,” Opt. Lett. 38(19), 3735–3738 (2013).
    [Crossref] [PubMed]
  15. T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
    [PubMed]
  16. A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems (Prentice-Hall, 1996), Chap. 4.
  17. C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
    [Crossref]
  18. L. Chen, N. Sherwood-Droz, and M. Lipson, “Compact bandwidth-tunable microring resonators,” Opt. Lett. 32(22), 3361–3363 (2007).
    [Crossref] [PubMed]
  19. P. Koonath, T. Indukuri, and B. Jalali, “Monolithic 3-D silicon photonics,” J. Lightwave Technol. 24(4), 1796–1804 (2006).
    [Crossref]
  20. P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
    [Crossref]
  21. J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
    [Crossref]

2014 (1)

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

2013 (3)

2012 (2)

2010 (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(29), 29 (2010).
[PubMed]

2008 (3)

2007 (3)

2006 (2)

P. Koonath, T. Indukuri, and B. Jalali, “Monolithic 3-D silicon photonics,” J. Lightwave Technol. 24(4), 1796–1804 (2006).
[Crossref]

P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
[Crossref]

2003 (1)

L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003).
[Crossref]

1999 (1)

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

1998 (1)

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Ahn, T. J.

Arceo, J.

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Ashrafi, R.

Ayotte, N.

Azaña, J.

Beeker, W.

Beerel, P. A.

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Berger, N. K.

Cao, P.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Chen, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Chen, L.

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(29), 29 (2010).
[PubMed]

Dong, J.

Dooply, A. E.

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Doucet, S.

Essiambre, R. J.

P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
[Crossref]

Fan, S.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[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(29), 29 (2010).
[PubMed]

Fischer, B.

Haus, H.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Heideman, R.

Hu, S.

Hu, X.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Indukuri, T.

Jalali, B.

Jiang, X.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Joannopoulos, J.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Khan, M.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Khan, M. R.

Koonath, P.

Kulishov, M.

LaRochelle, S.

Lei, L.

Leinse, A.

Levit, B.

Li, F.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Lipson, 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(29), 29 (2010).
[PubMed]

Liu, F.

Lu, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Manolatou, C.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[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(29), 29 (2010).
[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(29), 29 (2010).
[PubMed]

Park, Y.

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(29), 29 (2010).
[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]

Plant, D. V.

Qiang, L.

Qiu, M.

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(29), 29 (2010).
[PubMed]

Roeloffzen, C.

Sherwood-Droz, N.

Slavík, R.

Su, Y.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref] [PubMed]

Tan, S.

Vakilotojar, V.

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Venema, L.

L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003).
[Crossref]

Villeneuve, P.

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

Wang, T.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

F. Liu, T. Wang, L. Qiang, T. Ye, Z. Zhang, M. Qiu, and Y. Su, “Compact optical temporal differentiator based on silicon microring resonator,” Opt. Express 16(20), 15880–15886 (2008).
[Crossref] [PubMed]

Winzer, P. J.

P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
[Crossref]

Wu, J.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Wu, Z.

Xiang, L.

Xu, M.

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Yang, T.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Ye, T.

Yu, Y.

Yun, K. Y.

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

Zhang, Q.

Zhang, X.

Zhang, Z.

Zheng, A.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zhou, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

Zhuang, L.

Zou, J.

IEEE J. Quantum Electron. (1)

C. Manolatou, M. Khan, S. Fan, P. Villeneuve, H. Haus, and J. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Wu, P. Cao, X. Hu, T. Wang, M. Xu, X. Jiang, F. Li, L. Zhou, and Y. Su, “Nested configuration of silicon microring resonator with multiple coupling regimes,” IEEE Photon. Technol. Lett. 25(6), 580–583 (2013).
[Crossref]

IEEE Trans. VLIS Syst. (1)

K. Y. Yun, P. A. Beerel, V. Vakilotojar, A. E. Dooply, and J. Arceo, “The design and verification of a high-performance low-control-overhead asynchronous differential equation solver,” IEEE Trans. VLIS Syst. 6(4), 643–655 (1998).
[Crossref]

J. Lightwave Technol. (1)

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(29), 29 (2010).
[PubMed]

Nature (1)

L. Venema, “Photonic technologies,” Nature 424(6950), 809 (2003).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Proc. IEEE (1)

P. J. Winzer and R. J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE 94(5), 952–985 (2006).
[Crossref]

Sci. Rep. (1)

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Other (3)

A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems (Prentice-Hall, 1996), Chap. 4.

G. F. Simmons, Differential Equations With Applications and Historical Notes (McGraw-Hill, 1991), Chap. 1.

A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems (Prentice-Hall, 1996), Chap. 2.

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic illustration of the proposed tunable first-order ODE solver based on an add-drop MRR with two interferometric couplers. (b) Add-drop MRR equivalent to (a) with two directional couplers. (c) Schematic illustration of the second-order ODE solver implemented by two cascaded add-drop MRRs with interferometric couplers.
Fig. 2
Fig. 2 Calculated coefficients (a) a0 and (b) b0 of the ODE in Eq. (7) for various nb1 and nb2.
Fig. 3
Fig. 3 (a) Micrograph of the fabricated device. (b) Experimentally measured through port transmission spectrum of the fabricated device. (c) Simulated effective power coupling coefficients κ1,2 in the 1540 nm ~1560 nm wavelength range.
Fig. 4
Fig. 4 Experimental setup for system testing of solving ODE with tunable coefficients. PC: polarization controller. MZM: Mach-Zehnder modulator. PPG: pulse pattern generator. RFS: radio frequency synthesizer. EDFA: erbium-doped fiber amplifier. BPF: band pass filter. DUT: device under test. VOA: variable optical attenuator. OSA: optical spectrum analyzer. The experimentally observed 10-Gb/s optical Gaussian-like input pulse and the fitted one are shown in the inset by the black solid and red dashed curves, respectively.
Fig. 5
Fig. 5 Experimental results of the first set of system testing with varied P1 and unchanged P2. (a-I)–(a-III) Measured (solid curves) and fitted (dashed curves) transmission spectra of testings I, II, and III, respectively. (b-I)–(b-III) Experimentally observed temporal output pulse waveforms (solid curves) and numerical solutions of the ODE in Eq. (7) with fitted coefficients a0 and b0 in (a-I)–(a-III) (dashed curves), respectively. (c)–(d) Fitted transmission spectra and numerical solutions of the ODE in Eq. (7) for the three testings, respectively.
Fig. 6
Fig. 6 Experimental results of the second set of system testing with unchanged P1 and varied P2. (a-I)–(a-III) Measured (solid curves) and fitted (dashed curves) transmission spectra of testings I, II, and III, respectively. (b-I)–(b-III) Experimentally observed temporal output pulse waveforms (solid curves) and numerical solutions of the ODE in Eq. (7) with fitted coefficients a0 and b0 in (a-I)–(a-III) (dashed curves), respectively. (c)–(d) Fitted transmission spectra and numerical solutions of the ODE in Eq. (7) for the three testings, respectively.
Fig. 7
Fig. 7 Experimental results of the third set of system testing with constant a0 and varied b0. (a-I)–(a-III) Measured (solid curves) and fitted (dashed curves) transmission spectra of testings I, II, and III, respectively. (b-I)–(b-III) Experimentally observed temporal output pulse waveforms (solid curves) and numerical solutions of the ODE in Eq. (7) with fitted coefficients a0 and b0 in (a-I)–(a-III) (dashed curves), respectively. (c)–(d) Fitted transmission spectra and numerical solutions of the ODE in Eq. (7) for the three testings, respectively.
Fig. 8
Fig. 8 Experimental results of the fourth set of system testing with varied a0 and constant b0. (a-I)–(a-III) Measured (solid curves) and fitted (dashed curves) transmission spectra of testings I, II, and III, respectively. (b-I)–(b-III) Experimentally observed temporal output pulse waveforms (solid curves) and numerical solutions of the ODE in Eq. (7) with fitted coefficients a0 and b0 in (a-I)–(a-III) (dashed curves), respectively. (c)–(d) Fitted transmission spectra and numerical solutions of the ODE in Eq. (7) for the three testings, respectively.

Equations (10)

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

d n y(t) d t n + i=0 n1 a i d i y(t) d t i = k=0 m b k d k x(t) d t k ,
dy(t) dt + a 0 y(t)= b 1 dx(t) dt + b 0 x(t).
H(ω)= b 1 jω+ b 0 jω+ a 0 jω+ b 0 / b 1 jω+ a 0 ,
T(ω)= j(ω ω 0 )+( ω 0 2 Q i + ω 0 2 Q e2 ω 0 2 Q e1 ) j(ω ω 0 )+( ω 0 2 Q i + ω 0 2 Q e1 + ω 0 2 Q e2 ) ,
Q i = ω 0 n g L/[ cln( 1η ) ] Q e1 = ω 0 n g L/[ cln( 1 κ 1 ) ], Q e2 = ω 0 n g L/[ cln( 1 κ 2 ) ]
dy(t) dt e j ω 0 t + a 0 [y(t) e j ω 0 t ]= dx(t) dt e j ω 0 t + b 0 [x(t) e j ω 0 t ],
dy(t) dt + a 0 y(t)= dx(t) dt + b 0 x(t),
κ 1,2 = κ 0 (1 κ 0 )[ T b1,2 + T r1,2 +2 T b1,2 T r1,2 cos( φ b1,2 φ r1,2 )],
d 2 y(t) d t 2 + a 1 dy(t) dt + a 0 y(t)= b 2 d 2 x(t) d t 2 + b 1 dx(t) dt + b 0 x(t),
a 0 = a 10 a 20 , a 1 = a 10 + a 20 , b 0 = b 10 b 20 , b 1 = b 10 + b 20 , b 2 =1,

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