Abstract

A delay line fabricated of a chain of SNAP (Surface Nanoscale Axial Photonics) coupled microresonators is demonstrated. In contrast to resonant delay lines demonstrated to date, the slow light in this structure is enhanced by the 2R (Rotation + Reflection) effect realized due to the 3D propagation of light along the surface of a SNAP fiber. Here, the delay line coupled to a single input/output waveguide (i.e., operating in the reflection mode) is considered. Depending on the coupling parameters and loss, the delay time in this device is either proportional to the density of resonances averaged over the pulse spectrum or tends to zero. The delay line is fabricated of 20 coupled microresonators with the total length of 1.2 mm and footprint area of 0.05 mm2. It exhibits the record low insertion loss (< 3 dB), small speed of light (<c/250), and large (>1 ns) delay time along the 0.1 nm bandwidth achieved for the miniature microresonator delay lines. The feasibility of significant improvement of the SNAP delay line characteristics (larger delay time and bandwidth, smaller losses and dimensions, and anti-reflecting apodization) is discussed.

© 2013 OSA

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References

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    [Crossref]
  2. F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
    [Crossref]
  3. M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011).
    [Crossref]
  4. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
    [Crossref]
  5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
    [Crossref]
  6. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
    [Crossref]
  7. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, Y. A. Vlasov, and S. Mookherjea, “Statistics of light transport in 235-ring silicon coupled-resonator optical waveguides,” Opt. Express 18(25), 26505–26516 (2010).
    [Crossref]
  8. M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.
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    [Crossref]
  10. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. 36(24), 4824–4826 (2011).
    [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]
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    [Crossref]
  22. T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
    [Crossref]
  23. M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010).
    [Crossref]
  24. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003).
    [Crossref]
  25. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
    [Crossref]
  26. A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
    [Crossref]
  27. N. K. Fontaine, J. Yang, Z. Pan, S. Chu, W. Chen, B. E. Little, and S. J. B. Yoo, “Continuously Tunable Optical Buffering at 40 Gb/s for Optical Packet Switching Networks,” J. Lightwave Technol. 26(23), 3776–3783 (2008).
    [Crossref]
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2012 (5)

2011 (3)

2010 (4)

2009 (1)

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

2008 (3)

2007 (1)

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[Crossref]

2006 (1)

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[Crossref]

2005 (2)

2003 (1)

2002 (1)

2001 (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

2000 (1)

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[Crossref]

1999 (1)

Abedin, K.

Assefa, S.

Birks, T. A.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[Crossref]

Boyd, R. W.

Canciamilla, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

Chen, W.

Chu, S.

Cooper, M. L.

De La Rue, R. M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

DiGiovanni, D. J.

Dimmick, T. E.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[Crossref]

Dulashko, Y.

Eggleton, B.

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

Ferrari, C.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[Crossref]

Fini, J. M.

Fontaine, N. K.

Green, W. M. J.

Gupta, G.

Heebner, J. E.

Ilchenko, V. S.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[Crossref]

Khurgin, J. B.

Knight, J. C.

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[Crossref]

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[Crossref]

Lee, R. K.

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

Little, B. E.

Liu, X.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

Maleki, L.

Martinelli, M.

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[Crossref]

A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Vertically coupled whispering-gallery-mode resonator waveguide,” Opt. Lett. 30(22), 3066–3068 (2005).
[Crossref]

Melloni, A.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[Crossref]

Monberg, E. M.

Mookherjea, S.

Morichetti, F.

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
[Crossref]

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett. 33(20), 2389–2391 (2008).
[Crossref]

Notomi, M.

M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011).
[Crossref]

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[Crossref]

O’Shea, D.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

Pan, Z.

Park, Q.-H.

Pöllinger, M.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

Rauschenbeutel, A.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

Samarelli, A.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

Savchenkov, A. A.

Scherer, A.

Schneider, M. A.

Sekaric, L.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[Crossref]

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

Sorel, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

Sumetsky, M.

M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, X. Liu, E. M. Monberg, and T. F. Taunay, “Photo-induced SNAP: fabrication, trimming, and tuning of microresonator chains,” Opt. Express 20(10), 10684–10691 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-10-10684 .
[Crossref]

M. Sumetsky, K. Abedin, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, and E. M. Monberg, “Coupled high Q-factor surface nanoscale axial photonics (SNAP) microresonators,” Opt. Lett. 37(6), 990–992 (2012).
[Crossref]

M. Sumetsky, “Theory of SNAP devices: basic equations and comparison with the experiment,” Opt. Express 20(20), 22537–22554 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-20-20-22537 .
[Crossref]

M. Sumetsky and Y. Dulashko, “SNAP: Fabrication of long coupled microresonator chains with sub-angstrom precision,” Opt. Express 20(25), 27896–27901 (2012).
[Crossref]

M. Sumetsky and J. M. Fini, “Surface nanoscale axial photonics,” Opt. Express 19(27), 26470–26485 (2011).
[Crossref]

M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, “Surface nanoscale axial photonics: robust fabrication of high-quality-factor microresonators,” Opt. Lett. 36(24), 4824–4826 (2011).
[Crossref]

M. Sumetsky and Y. Dulashko, “Radius variation of optical fibers with angstrom accuracy,” Opt. Lett. 35(23), 4006–4008 (2010).
[Crossref]

M. Sumetsky, “Vertically-stacked multi-ring resonator,” Opt. Express 13(17), 6354–6375 (2005).
[Crossref]

M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express 11(4), 381–391 (2003).
[Crossref]

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[Crossref]

Taunay, T. F.

Torregiani, M.

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

Vlasov, Y.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[Crossref]

Vlasov, Y. A.

Warken, F.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

Xia, F.

Xia, F. N.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[Crossref]

Xu, Y.

Yang, J.

Yariv, A.

Yoo, S. J. B.

Adv. Opt. Photon. (1)

IEEE J. Quantum Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37(4), 525–532 (2001).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes - Part II: Applications,” IEEE J. Sel. Top. Quantum Electron. 12(1), 15–32 (2006).
[Crossref]

IEEE Photon. Technol. Lett. (1)

T. A. Birks, J. C. Knight, and T. E. Dimmick, “High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment,” IEEE Photon. Technol. Lett. 12(2), 182–183 (2000).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (1)

A. Canciamilla, M. Torregiani, C. Ferrari, F. Morichetti, R. M. De La Rue, A. Samarelli, M. Sorel, and A. Melloni, “Silicon coupled-ring resonator structures for slow light applications: potential, impairments and ultimate limits,” J. Opt. 12(10), 104008 (2010).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

F. Morichetti, C. Ferrari, A. Canciamilla, and A. Melloni, “The first decade of coupled resonator optical waveguides: bringing slow light to applications,” Laser Photonics Rev. 6(1), 74–96 (2012).
[Crossref]

Nat. Photonics (2)

M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics 2(12), 741–747 (2008).
[Crossref]

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[Crossref]

Opt. Express (7)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103(5), 053901 (2009).
[Crossref]

Proc. IEEE (1)

M. Notomi, “Strong light confinement with periodicity,” Proc. IEEE 99(10), 1768–1779 (2011).
[Crossref]

Other (3)

M. Sumetsky, “Localization of light in an optical fiber with nanoscale radius variation,” in CLEO/Europe and EQEC 2011 Conference Digest, postdeadline paper PDA_8.

J. U. Nöckel, “2-d Microcavities: Theory and Experiments,” in Cavity-Enhanced Spectroscopies, R. D. van Zee and J. P. Looney, eds. (Academic Press, San Diego, 2002).

M. Sumetsky, “Dispersionless impedance-matched low-loss optical bottle resonator slow light delay line,” arXiv:1305.6591, http://arxiv.org/abs/1305.6591 .

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Figures (8)

Fig. 1
Fig. 1

Illustration of a SNAP device.

Fig. 2
Fig. 2

Group delay as a function of frequency in the vicinity of an all-pass resonance calculated from Eq. (3) for σc = 2σa corresponding to DBP = 1 (upper blue curve) and for σc = 0.5σa corresponding to DBP = 0 (lower red curve).

Fig. 3
Fig. 3

Experimental characterization (a) and theoretical modeling (b) of the resonant transmission amplitude of the fabricated 20 coupled MR chain. The surface plots of experimental data are obtained with 2 μm resolution along the fiber axis and 1.24 pm wavelength resolution. The theoretical modeling is performed with the same spatial resolution and 0.62 pm spectral resolution. Black line is the calculated ERV.

Fig. 4
Fig. 4

(a) – Experimentally measured surface plot of the group delay of the device considered in Section 3. (b) Comparison of the experimental and theoretical group delay spectra along the blue dashed line in Fig. 4(a). (c) – Spectra shown in Fig. 4(b) averaged over 6 pm. (d) – The transmission amplitude spectra corresponding to the group delay spectra in Fig. 4(b) found from data depicted in Fig. 3.

Fig. 5
Fig. 5

Experimental characterization (a) and theoretical modeling (b) of the resonant transmission amplitude of the fabricated MR chain with coupling parameters defined by Eq. (10). As in Fig. 3, the experimental data is obtained with 2 μm resolution along the fiber axis and 1.24 pm wavelength resolution, while the theoretical modeling is performed with the same spatial resolution and 0.62 pm spectral resolution.

Fig. 6
Fig. 6

(a) – Experimentally measured surface plot of the group delay of the MR chain with modified coupling parameters described in Section 4. (b) – Comparison of the experimental and theoretical group delay spectra along the blue dashed line in Fig. 6(a). Green bold curve – the spectrum of the pulse considered in Section 5. (c) – Spectra shown in Fig. 6(b) averaged over 6 pm. (d) – Spectra shown in Fig. 6(b) averaged over 0.03 nm. (e) – The transmission amplitude spectra corresponding to the group delay spectra in Fig. 6(b) found from data depicted in Fig. 5.

Fig. 7
Fig. 7

(a) – Pulse propagation modeling using the experimental transmission amplitude spectrum depicted in Figs. 6(b) and 6(e). Solid curve – the input pulse, dashed curve – the output pulses. The directly transmitted pulse and the pulse, which was transmitted after one reflection from the coupling region back into the MR chain, are circled. (b) – Modeling of the pulse propagation for the theoretical transmission amplitude shown in Figs. 6(b) and 6(e) (green dashed curve) and for the model of the transmission amplitude defined by Eq. (12).

Fig. 8
Fig. 8

The cartoon illustrating combination of SNAP delay lines with planar photonic circuits.

Tables (1)

Tables Icon

Table 1 Comparison of characteristics of the state of the art resonance delay lines having micron-scale [5,6,26] and millimeter-scale [25,27] elements with the SNAP delay line demonstrated in this paper.

Equations (12)

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τ = 1 2 π Im ( 1 S S ν ) = λ 2 2 π c Im ( 1 S S λ )
S ( ν ) = ν ν 0 i ( σ a σ c ) ν ν 0 i ( σ a + σ c ) .
τ ( ν ) = 2 σ c [ ( ν ν 0 ) 2 σ a 2 + σ c 2 ] [ ( ν ν 0 ) 2 + σ a 2 + σ c 2 ] 2 4 σ a 2 σ c 2 .
τ ( ν ) d ν = { 1 if σ a < σ c , 0 if σ a > σ c .
S ( λ , z 1 ) = S 0 i | C | 2 G 0 ( λ , z 1 , z 1 ) 1 + D G 0 ( λ , z 1 , z 1 )
Ψ z z + ( E ( λ ) V ( z ) ) Ψ = 0 , E ( λ ) = κ λ λ r e s i γ λ r e s , V ( z ) = κ Δ r e f f ( z ) r 0 , κ = 2 ( 2 π n λ r e s ) 2 ,
ν 1 < ν < ν N + 1 τ ( ν ) d ν = n = 1 N ν n ν n + 1 d ν G ν [ 1 [ G + S 0 ( S 0 D i | C | 2 ) 1 ] 1 ( G + D 1 ) ] = N d G [ 1 [ G + S 0 ( S 0 D i | C | 2 ) 1 ] 1 ( G + D 1 ) ] = { N if Λ > 0 0 if Λ < 0
Λ = | C | 2 | S 0 | 2 Re ( S 0 ) Im D
S 0 = 0.95 0.19 i , | C | 2 = 0.01 μ m 1 , D = 0.03 + 0.03 i μ m 1
γ = 0.6 pm, S 0 = 0.85 0.1 i , | C | 2 = 0.042 μ m 1 , D = 0.021 + 0.024 i μ m 1 .
E ¯ i n ( t ) = E i n ( ν ) exp ( 2 i π ν t ) d ν E ¯ o u t ( t ) = E o u t ( ν ) exp ( 2 i π ν t ) d ν = S ( ν ) E i n ( ν ) exp ( 2 i π ν t ) d ν
S ( ν ) = exp [ 2 i π τ 0 ( ν ν 0 ) + i ζ cos ( 2 π τ 0 ( ν ν 0 ) ) ] ,

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