Abstract

We demonstrate all-optical temporal integration of arbitrary optical waveforms with temporal features as short as ~1.9ps. By using a four-port micro-ring resonator based on CMOS compatible doped glass technology we perform the 1st- and 2nd-order cumulative time integral of optical signals over a bandwidth that exceeds 400GHz. This device has applications for a wide range of ultra-fast data processing and pulse shaping functions as well as in the field of optical computing for the real-time analysis of differential equations.

© 2011 OSA

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References

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    [CrossRef]
  6. 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. 27(17), 3888–3895 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2011 (1)

M. H. Asghari and J. Azaña, “Photonic Integrator-Based Optical Memory Unit,” IEEE Photon. Technol. Lett. 23(4), 209–211 (2011).
[CrossRef]

2010 (2)

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

2009 (2)

2008 (5)

2007 (1)

2006 (2)

N. Q. Ngo and L. N. Binh, “Optical realization of Newton-Cotes-Based Integrators for Dark Soliton Generation,” J. Lightwave Technol. 24(1), 563–572 (2006).
[CrossRef]

Y. Park, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. 18(17), 1798–1800 (2006).
[CrossRef]

2004 (2)

C. W. Hsue, L. C. Tsai, and K.-L. Chen, “Implementation of First-Order and Second-Order Microwave Differentiator,” IEEE Trans. Microw. Theory Tech. 52(5), 1443–1448 (2004).
[CrossRef]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

2000 (1)

1995 (2)

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75(13), 2598–2601 (1995).
[CrossRef] [PubMed]

L. Lepetit, G. Chériaux, and M. Joffre, “Linear technique of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12), 2467–2474 (1995).
[CrossRef]

Ahn, T. J.

Ahn, T.-J.

Asghari, M. H.

Ayotte, N.

Azaña, J.

Baumberg, J. J.

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75(13), 2598–2601 (1995).
[CrossRef] [PubMed]

Belabas, N.

Binh, L. N.

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Chen, K.-L.

C. W. Hsue, L. C. Tsai, and K.-L. Chen, “Implementation of First-Order and Second-Order Microwave Differentiator,” IEEE Trans. Microw. Theory Tech. 52(5), 1443–1448 (2004).
[CrossRef]

Chériaux, G.

Chu, S. T.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Costanzo-Caso, P.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

Dai, Y.

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Ding, Y.

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Dorrer, C.

Doucet, S.

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

Ferrera, M.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Granieri, S.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

Heberle, A. P.

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75(13), 2598–2601 (1995).
[CrossRef] [PubMed]

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Hsue, C. W.

C. W. Hsue, L. C. Tsai, and K.-L. Chen, “Implementation of First-Order and Second-Order Microwave Differentiator,” IEEE Trans. Microw. Theory Tech. 52(5), 1443–1448 (2004).
[CrossRef]

Huang, D.

Jin, Y.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

Joffre, M.

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Köhler, K.

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75(13), 2598–2601 (1995).
[CrossRef] [PubMed]

LaRochelle, S.

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Lepetit, L.

Li, F.

Y. Park, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. 18(17), 1798–1800 (2006).
[CrossRef]

Likforman, J. P.

Liscidini, M.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

Little, B. E.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Morandotti, R.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Moss, D. J.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Muriel, M. A.

Ngo, N. Q.

Oei, Y.-S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Park, Y.

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]

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, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. 18(17), 1798–1800 (2006).
[CrossRef]

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 (2010)
[CrossRef]

Preciado, M. A.

Quoc Ngo, N.

Razzari, L.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

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 (2010)
[CrossRef]

Siahmakoun, A.

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

Sipe, J. E.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

Slavík, R.

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Smit, M. K.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Tsai, L. C.

C. W. Hsue, L. C. Tsai, and K.-L. Chen, “Implementation of First-Order and Second-Order Microwave Differentiator,” IEEE Trans. Microw. Theory Tech. 52(5), 1443–1448 (2004).
[CrossRef]

Yang, Z.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

Yao, J.

Zhang, X.

IEEE Photon. Technol. Lett. (2)

M. H. Asghari and J. Azaña, “Photonic Integrator-Based Optical Memory Unit,” IEEE Photon. Technol. Lett. 23(4), 209–211 (2011).
[CrossRef]

Y. Park, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. 18(17), 1798–1800 (2006).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

C. W. Hsue, L. C. Tsai, and K.-L. Chen, “Implementation of First-Order and Second-Order Microwave Differentiator,” IEEE Trans. Microw. Theory Tech. 52(5), 1443–1448 (2004).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. T. Chu, B. E. Little, and D. J. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photonics 2(12), 737–740 (2008).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. T. Chu, B. E. Little, and D. J. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4(1), 41–45 (2010).
[CrossRef]

Nature (1)

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oei, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75(13), 2598–2601 (1995).
[CrossRef] [PubMed]

Proc. SPIE (1)

Y. Jin, P. Costanzo-Caso, S. Granieri, and A. Siahmakoun, “Photonic integrator for A/D conversion,” Proc. SPIE 7797, 1–8 (2010).

Other (6)

G. P. Agrawal, “Fiber-optic Communication Systems,” in Microwave and Optical Engineering, 3rd ed. (John Wiley & Sons, Inc. New York, 2002).

P. Kinget and M. Steyaert, “Analog VLSI Integration of Massive Parallel Processing Systems,” in The Springer International Series in Engineering and Computer Science, ed. (Kluwer Academic Publishers, Boston, Dordrecht, London, 2010).

M. Tooley, “Electronic Circuits - Fundamentals & Applications,” in Advanced Technological and Higher National Certificates Kingston University, ed. (Elsevier Ltd., Oxford UK, 2006).

A. Mehrotra and A. L. Sangiovanni-Vincentelli, Noise Analysis of Radio Frequency Circuits, ed. (Kluwer Academic Publishers, Massachusetts, 2010).

G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd ed. (McGraw-Hill, New York, 1991).

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 (2010)
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Integrator transfer function, showing a comparison between the spectral transfer function of an ideal integrator (black curve) with that of a Fabry-Perot cavity (red curve) in which one resonance matches the integrator operative frequency ω0. The figure shows the main discrepancies between these two curves around the operative frequency and at the lobes of the characteristic resonance. It also shows (blue arrows) how largely the operation (processing) bandwidth of the resonator can exceed its linewidth when it is used as integrator.

Fig. 2
Fig. 2

Schematic of the all-optical integrator. The image on the left shows a SEM image of the micro-ring resonator cross section (taken prior to the deposition of the SiO2 cladding). The figure on the right depicts the ring resonator device and illustrates the overall I/O scheme of the cavity when it operates as an integrator.

Fig. 3
Fig. 3

Experimental set-up. Block (a) and block (b) in the blue box represent the 1st- and the 2nd-order integration modules, respectively. During the experiment we used a mode-locked fiber laser (Pritel) emitted nearly Gaussian pulses with its central wavelength in resonance with the ring @ 1553.37nm. The optical pulses had a time duration of ~1.9ps and repetition rate of 16.9MHz. The laser output was split into two by means of a 50/50 fiber splitter, one arm of the splitter directed the beam towards a Michelson interferometer that shaped the input waveforms to be processed. The second arm of the splitter was a reference line used to obtain a phase sensitive spectrum of the device output in order to reconstruct the temporal output [21]. In the reference arm, the signal was first delayed and then the polarization was set to TM to align the signal with the selected TM resonance @ 1553.37nm. The reference signal was then interfered with the device output by means of a 50/50 fiber coupler and the result recorded on an optical spectrum analyzer (OSA). The resulting spectra were then used to retrieve the complex-field (amplitude and phase) temporal information of the cavity (integrator) output by using Fourier transform spectral interferometry (FTSI) [22,23].

Fig. 4
Fig. 4

(a) Optical spectra of the laser pulses (input of the integrator, dashed line) and output of the resonator (solid line) recorded at the drop port. (b) The experimentally measured temporal profile of the output of the device (black solid curve), representing the 1st-order integral of the laser pulses, as well as the theoretically calculated integral (blue dashed curve), and the theoretical response of an equivalent cavity (magenta solid curve) with the same photon-life time of our resonator (~12.5ps). The inset in (b) is the input laser pulse temporal profile (red curve). All experimental measurements were obtained using the FTSI based approach mentioned in the text.

Fig. 5
Fig. 5

Experimental results. (a) the 1st- and (b) 2nd-order integrals of the in-phase signal; (c) the 1st- and (d) the 2nd-order integrals for the π-shifted pulses. For all these plots, the solid black curves represent the experimentally measured temporal profiles obtained using the FTSI based method described in the text. The dashed blue lines correspond to the theoretical cumulative time integrals. Finally, the solid green lines are temporal convolution products. For the case of the 1st order integral, the convolution was performed between the corresponding ideal input and the impulse response of Fig. 4b. For the 2nd order integral, the convolution is performed between the FTSI-recovered signal of the 1st integral and the same impulse response. All the input waveforms represented by the red curves of the insets in Fig. 4b and Fig. 5a,c are ideal Gaussian approximations of the experimentally measured input signals, and they were used to evaluate the correspondent theoretical cumulative time integral.

Equations (3)

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H(ω)= 1 j(ω ω 0 )
u(t)={ 0 for t<0; 1 for t0;
h cavity (t)=exp(Kt)u(t)

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