Abstract

A superluminal space-to-time mapping process is reported and numerically validated in grating-assisted (GA) co-directional couplers, e.g. fiber/waveguide long-period gratings (LPGs). We demonstrate that under weak-coupling conditions, the amplitude and phase of the grating complex apodization profile of a GA co-directional coupling device can be directly mapped into the device’s temporal impulse response. In contrast to GA counter-directional couplers, this mapping occurs with a space-to-time scaling factor that is much higher than the propagation speed of light in vacuum. This phenomenon opens up a promising new avenue to overcome the fundamental time-resolution limitations of present in-fiber and on-chip optical waveform generation (shaping) and processing devices, which are intrinsically limited by the achievable spatial resolution of fabrication technologies. We numerically demonstrate the straightforward application of the phenomenon for synthesizing customized femtosecond-regime complex optical waveforms using readily feasible fiber LPG designs, e.g. with sub-centimeter resolutions.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Gillooly, “Photosensitive fibres: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
    [CrossRef]
  2. J. Azaña, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J.2(3), 359–386 (2010).
    [CrossRef]
  3. A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron.4(2), 317–331 (1998).
    [CrossRef]
  4. S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J.33, 661–719 (1954).
  5. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J.55(1), 109–126 (1976).
  6. K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26(11), 1918–1929 (1990).
    [CrossRef]
  7. J. Azaña and L. R. Chen, “Synthesis of temporal optical waveforms by fiber Bragg gratings: a new approach based on space-to-frequency-to-time mapping,” J. Opt. Soc. Am. B19(11), 2758–2769 (2002).
    [CrossRef]
  8. E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
    [CrossRef]
  9. J. Skaar and K. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol.16(10), 1928–1932 (1998).
    [CrossRef]
  10. J. K. Brenne and J. Skaar, “Design of grating-assisted codirectional couplers with discrete inverse-scattering algorithms,” J. Lightwave Technol.21(1), 254–263 (2003).
    [CrossRef]
  11. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9(9), 919–933 (1973).
    [CrossRef]
  12. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997).
    [CrossRef]
  13. J. Jiang, C. L. Callender, J. P. Noad, and J. Ding, “Hybrid silica/polymer long period gratings for wavelength filtering and power distribution,” Appl. Opt.48(26), 4866–4873 (2009).
    [CrossRef] [PubMed]
  14. R. Slavík, M. Kulishov, Y. Park, and J. Azaña, “Long-period-fiber-grating-based filter configuration enabling arbitrary linear filtering characteristics,” Opt. Lett.34(7), 1045–1047 (2009).
    [CrossRef] [PubMed]
  15. M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
    [CrossRef]

2011 (2)

A. Gillooly, “Photosensitive fibres: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

2010 (1)

J. Azaña, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J.2(3), 359–386 (2010).
[CrossRef]

2009 (2)

2003 (1)

2002 (1)

1998 (2)

J. Skaar and K. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol.16(10), 1928–1932 (1998).
[CrossRef]

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron.4(2), 317–331 (1998).
[CrossRef]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997).
[CrossRef]

1996 (1)

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
[CrossRef]

1990 (1)

K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26(11), 1918–1929 (1990).
[CrossRef]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J.55(1), 109–126 (1976).

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9(9), 919–933 (1973).
[CrossRef]

1954 (1)

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J.33, 661–719 (1954).

Azaña, J.

Bock, W. J.

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

Brenne, J. K.

Callender, C. L.

Capmany, J.

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
[CrossRef]

Chen, J.

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

Chen, L. R.

Ding, J.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997).
[CrossRef]

Gillooly, A.

A. Gillooly, “Photosensitive fibres: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

Jiang, J.

Kan’an, A. M.

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron.4(2), 317–331 (1998).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J.55(1), 109–126 (1976).

Kulishov, M.

Marti, J.

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
[CrossRef]

Mikulic, P.

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

Miller, S. E.

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J.33, 661–719 (1954).

Noad, J. P.

Park, Y.

Peral, E.

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
[CrossRef]

Risvik, K.

Roman, J. E.

K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26(11), 1918–1929 (1990).
[CrossRef]

Skaar, J.

Slavík, R.

Smietana, M.

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

Weiner, A. M.

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron.4(2), 317–331 (1998).
[CrossRef]

Winick, K. A.

K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26(11), 1918–1929 (1990).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9(9), 919–933 (1973).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

S. E. Miller, “Coupled wave theory and waveguide applications,” Bell Syst. Tech. J.33, 661–719 (1954).

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J.55(1), 109–126 (1976).

IEEE J. Quantum Electron. (3)

K. A. Winick and J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26(11), 1918–1929 (1990).
[CrossRef]

E. Peral, J. Capmany, and J. Marti, “Iterative solution to the Gel'Fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron.32(12), 2078–2084 (1996).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9(9), 919–933 (1973).
[CrossRef]

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

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron.4(2), 317–331 (1998).
[CrossRef]

IEEE Photon. J. (1)

J. Azaña, “Ultrafast analog all-optical signal processors based on fiber-grating devices,” IEEE Photon. J.2(3), 359–386 (2010).
[CrossRef]

J. Lightwave Technol. (3)

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

Meas. Sci. Technol. (1)

M. Smietana, W. J. Bock, P. Mikulic, and J. Chen, “Increasing sensitivity of arc-induced long-period gratings—pushing the fabrication technique toward its limits,” Meas. Sci. Technol.22(1), 015201 (2011).
[CrossRef]

Nat. Photonics (1)

A. Gillooly, “Photosensitive fibres: Growing gratings,” Nat. Photonics5(8), 468–469 (2011).
[CrossRef]

Opt. Lett. (1)

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Illustration of the space-to-time mapping phenomena in GA counter- and co-directional couplers (fiber BG and LPG cases, respectively).

Fig. 2
Fig. 2

Illustration of the temporal resolutions (dt) that are associated to a prescribed spatial resolution (dz) in the grating apodization profile for the GA co-directional and counter-directional cases. In each case, the temporal resolution is defined by the difference between the arrival times of the impulses coupled at the input and output ends of the corresponding spatial-resolution element.

Fig. 3
Fig. 3

An illustration of the speed difference of the two pulse shaping approaches based on space-to-time mapping in realistic fiber BGs and LPGs for a target optical OOK bit stream pattern generation.

Fig. 4
Fig. 4

Proof-of-concept numerical simulation for (a) three LPGs with the same length of 5cm and different amounts of kmax. The corresponding spectral power responses (b) and temporal impulse response amplitudes (c) of the LPGs.

Fig. 5
Fig. 5

Simulation results of a designed fiber LPG (a) for generation of a 4-symbol optical 8-QAM (b) data stream pattern, i.e. 4137, with a speed of 4Tsymbol/s from an input ultra-short optical Gaussian pulse with a (full width at 10% of the peak amplitude) duration of 100fs (@1550nm). (c) The spectral power response of the designed LPG. (d) The output temporal amplitude and phase response to the mentioned ultra-short optical input pulse in the linear regime.

Tables (1)

Tables Icon

Table 1 The estimated space-to-time mapping speeds for the considered BG and LPG made in SMF28 fiber.

Equations (14)

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

{ dR(z) / dz +jδR(z)=j|k(z)| e jφ(z) S(z) dS(z) / dz jδS(z)=j|k(z)| e jφ(z) R(z) .
δ(ω)=(1/2)[ β R (ω) β S (ω) 2π /Λ ],
ω 0 = 2πc / (ΛΔN) .
δ(ω)= ωΔN / ( 2c ) .
H local (ω,z)=S(ω,z)/ R 0 (ω).
R(ω,z) R 0 (ω) e jωz n eff1 /c 1 | H local (ω,z) | 2 R(ω,z) R 0 (ω) e jωz n eff1 /c
H local (ω,z)[ S(ω,z)/R(ω,z) ] e jωz n eff1 /c .
H local (ω,z)[ S(ω,z)/R(ω,z) ] e jωzΔN/c .
d H local (ω,z) / dz j| k(z) |[ e j( ωzΔN/cφ ) H local 2 (ω,z) e j( ωzΔN/cφ ) ].
d H local (ω,z)j| k(z) | e jφ(z) e jωzΔN/c dz.
H local (ω, z 0 )j 0 z 0 | k(z) | e jφ(z) e jωzΔN/c dz .
H(ω)j 0 Δz | k(z) | e jφ(z) e jωzΔN/c dz .
H(ω)= 0 Δt h(t) e jωt dt ,
h(t) { | k(z) | e jφ(z) } z=tc/ΔN ,

Metrics