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Self-organized nonlinear gratings for ultrafast nanophotonics


As devices utilizing femtosecond-duration laser pulses become more commonplace, there is a need for next-generation nonlinear-photonics technologies that enable low-energy femtosecond pulses to be converted from one wavelength to another with high efficiency. However, designing nonlinear materials to operate with femtosecond pulses is challenging, because it is necessary to match both the phase velocities and group velocities of the light. Here, we show that femtosecond laser pulses can generate self-organized nonlinear gratings in nanophotonic waveguides, thereby providing a nonlinear optical device with both quasi-phase-matching and group-velocity matching for second-harmonic generation. We use nonlinear microscopy to uniquely characterize the self-organized nonlinear gratings and demonstrate that these waveguides enable simultaneous χ(2) and χ(3) nonlinear processes for laser-frequency-comb stabilization. Finally, we derive the equations that govern self-organized grating formation for femtosecond pulses and uncover the crucial role of group-velocity matching. In the future, nanophotonics with self-organized gratings could enable scalable, reconfigurable nonlinear photonics.

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The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

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Competing interests

D.D.H. is currently employed by KMLabs, Inc., a company that manufactures femtosecond lasers. D.R.C. is an owner of Octave Photonics, a company specializing in nonlinear nanophotonics.

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  1. 1.

    New, G. Introduction to Nonlinear Optics (Cambridge University Press, 2011).

  2. 2.

    Schneider, T. Nonlinear Optics in Telecommunications (Springer, 2004).

  3. 3.

    Weston, M. M. et al. Efficient and pure femtosecond-pulse-length source of polarization-entangled photons. Opt. Express 24, 10869–10879 (2016).

  4. 4.

    Garmire, E. Nonlinear optics in daily life. Opt. Express 21, 30532–30544 (2013).

  5. 5.

    Dianov, E. & Starodubov, D. Photoinduced second-harmonic generation in glasses and glass optical fibers. Opt. Fiber Technol. 1, 3 (1994).

  6. 6.

    Balakirev, M. K., Vostrikova, L. I. & Smirnov, V. A. Photoelectric instability in oxide glass. JETP Lett. 66, 809–815 (1997).

  7. 7.

    Anderson, D. Z., Mizrahi, V. & Sipe, J. E. Model for second-harmonic generation in glass optical fibers based on asymmetric photoelectron emission from defect sites. Opt. Lett. 16, 796 (1991).

  8. 8.

    Stolen, R. H. & Tom, H. W. K. Self-organized phase-matched harmonic generation in optical fibers. Opt. Lett. 12, 585 (1987).

  9. 9.

    Billat, A. et al. Large second harmonic generation enhancement in Si3N4 waveguides by all-optically induced quasi-phase-matching. Nat. Commun. 8, 1016 (2017).

  10. 10.

    Porcel, M. A. G. et al. Photo-induced second-order nonlinearity in stoichiometric silicon nitride waveguides. Opt. Express 25, 33143 (2017).

  11. 11.

    Carlson, D. R. et al. Photonic-chip supercontinuum with tailored spectra for counting optical frequencies. Phys. Rev. Appl. 8, 014027 (2017).

  12. 12.

    Carlson, D. R. et al. Self-referenced frequency combs using high-efficiency silicon-nitride waveguides. Opt. Lett. 42, 2314–2317 (2017).

  13. 13.

    Ji, X. et al. Ultra-low-loss on-chip resonators with sub-milliwatt parametric oscillation threshold. Optica 4, 619 (2017).

  14. 14.

    Mayer, A. S. et al. Frequency comb offset detection using supercontinuum generation in silicon nitride waveguides. Opt. Express 23, 15440–15451 (2015).

  15. 15.

    Porcel, M. A. G. et al. Two-octave spanning supercontinuum generation in stoichiometric silicon nitride waveguides pumped at telecom wavelengths. Opt. Express 25, 1542–1554 (2017).

  16. 16.

    Briles, T. C. et al. Interlocking Kerr-microresonator frequency combs for microwave to optical synthesis. Opt. Lett. 43, 2933 (2018).

  17. 17.

    Dudley, J. M., Genty, G. & Coen, S. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135 (2006).

  18. 18.

    Margulis, W., Laurell, F. & Lesche, B. Imaging the nonlinear grating in frequency-doubling fibres. Nature 378, 699–701 (1995).

  19. 19.

    Zhu, M. et al. Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides. Opt. Express 20, 15899–15907 (2012).

  20. 20.

    Zhang, L., Yue, Y., Beausoleil, R. G. & Willner, A. E. Flattened dispersion in silicon slot waveguides. Opt. Express 18, 20529–20534 (2010).

  21. 21.

    Carlson, D. R. et al. Ultrafast electro-optic light with subcycle control. Science 361, 1358–1363 (2018).

  22. 22.

    Metcalf, A. J. et al. Stellar spectroscopy in the near-infrared with a laser frequency comb. Optica 6, 233–239 (2019).

  23. 23.

    Lamb, E. S. et al. Optical-frequency measurements with a Kerr microcomb and photonic-chip supercontinuum. Phys. Rev. Appl. 9, 024030 (2018).

  24. 24.

    Timurdogan, E., Poulton, C. V., Byrd, M. J. & Watts, M. R. Electric field-induced second-order nonlinear optical effects in silicon waveguides. Nat. Photon. 11, 200–206 (2017).

  25. 25.

    Zel’dovich, B. Y. & Chudinov, A. N. Interference of fields with frequencies ω and 2ω in external photoelectric effect. JETP Lett. 50, 439–441 (1989).

  26. 26.

    Baranova, N., Chudinov, A. & Zel’dovich, B. Polar asymmetry of photoionization by a field with <E 3> ≠ 0. Theory and experiment. Opt. Commun. 79, 116 (1990).

  27. 27.

    Hickstein, D. D. et al. Ultrabroadband supercontinuum generation and frequency-comb stabilization using on-chip waveguides with both cubic and quadratic nonlinearities. Phys. Rev. Appl. 8, 014025 (2017).

  28. 28.

    Tzeng, S.-D. & Gwo, S. Charge trapping properties at silicon nitride/silicon oxide interface studied by variable-temperature electrostatic force microscopy. J. Appl. Phys. 100, 023711 (2006).

  29. 29.

    Fujita, S. & Sasaki, A. Dangling bonds in memory-quality silicon nitride films. J. Electrochem. Soc. 132, 398–402 (1985).

  30. 30.

    Krick, D. T., Lenahan, P. M. & Kanicki, J. Nature of the dominant deep trap in amorphous silicon nitride. Phys. Rev. B 38, 8226–8229 (1988).

  31. 31.

    Warren, W. L. & Lenahan, P. M. Electron-nuclear double-resonance and electron-spin-resonance study of silicon dangling-bond centers in silicon nitride. Phys. Rev. B 42, 1773–1780 (1990).

  32. 32.

    Lenahan, P. M. & Curry, S. E. First observation of the 29Si hyperfine spectra of silicon dangling bond centers in silicon nitride. Appl. Phys. Lett. 56, 157–159 (1990).

  33. 33.

    Warren, W. L., Lenahan, P. M. & Curry, S. E. First observation of paramagnetic nitrogen dangling-bond centers in silicon nitride. Phys. Rev. Lett. 65, 207–210 (1990).

  34. 34.

    Warren, W. L., Rong, F. C., Poindexter, E. H., Gerardi, G. J. & Kanicki, J. Structural identification of the silicon and nitrogen dangling‐bond centers in amorphous silicon nitride. J. Appl. Phys. 70, 346–354 (1991).

  35. 35.

    Wang, M., Li, D., Yuan, Z., Yang, D. & Que, D. Photoluminescence of Si-rich silicon nitride: defect-related states and silicon nanoclusters. Appl. Phys. Lett. 90, 131903 (2007).

  36. 36.

    Seol, K. S., Futami, T., Watanabe, T., Ohki, Y. & Takiyama, M. Effects of ion implantation and thermal annealing on the photoluminescence in amorphous silicon nitride. J. Appl. Phys. 85, 6746–6750 (1999).

  37. 37.

    Hafsi, N., Bouridah, H., Beghoul, M. R. & Haoues, H. Photoluminescence from silicon nanocrystals embedded in silicon nitride fabricated by low-pressure chemical vapor deposition followed by high-temperature annealing. J. Appl. Phys. 117, 063105 (2015).

  38. 38.

    Park, Y. C., Jackson, W. B., Johnson, N. M. & Hagstrom, S. B. Spatial profiling of electron traps in silicon nitride thin films. J. Appl. Phys. 68, 5212 (1990).

  39. 39.

    Gritsenko, V. et al. Silicon dots/clusters in silicon nitride: photoluminescence and electron spin resonance. Thin Solid Films 353, 20–24 (1999).

  40. 40.

    Fallahkhair, A. B., Li, K. S. & Murphy, T. E. Vector finite difference modesolver for anisotropic dielectric waveguides. J. Lightwave Technol. 26, 1423–1431 (2008).

  41. 41.

    Bolla, L. Empy: Electromagnetic Python (2017).

  42. 42.

    Luke, K., Okawachi, Y., Lamont, M. R. E., Gaeta, A. L. & Lipson, M. Broadband mid-infrared frequency comb generation in a Si3N4 microresonator. Opt. Lett. 40, 4823–4826 (2015).

  43. 43.

    Malitson, I. H. Interspecimen comparison of the refractive index of fused silica. J. Opt. Soc. Am. 55, 1205–1209 (1965).

  44. 44.

    Sinclair, L. C. et al. Invited article: a compact optically coherent fiber frequency comb. Rev. Sci. Instrum. 86, 081301 (2015).

  45. 45.

    Tourigny-Plante, A. et al. An open and flexible digital phase-locked loop for optical metrology. Rev. Sci. Instrum. 89, 093103 (2018).

  46. 46.

    Weiner, A. Ultrafast Optics (Wiley, 2009).

  47. 47.

    Suhara, T. & Fujimura, M. Waveguide Nonlinear-Optic Devices (Wiley, 2003).

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The authors thank G. Moille, N. Sanford and the National Institute of Standards and Technology (NIST) Boulder Editorial Review Board for providing helpful feedback on this manuscript, and K. Dorney, J. Ellis, H. Kapteyn and M. Murnane for the timely loan of a polarizer. This work is supported by AFOSR under award no. FA9550-16-1-0016, DARPA (DODOS and ACES programmes), NIST and NRC. Nonlinear optical imaging instrumentation at CU-Boulder was supported by the National Science Foundation Grant DMR-1420736. Certain commercial equipment, instruments or materials are identified here in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by NIST, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

Author information

D.D.H. and D.R.C. conducted the SHG experiments. D.D.H., H.M. and I.I.S. conceived and conducted the microscopy experiment. K.S. and D.W. designed, fabricated and characterized the SiN waveguides. D.D.H., D.R.C., S.B.P., S.A.D. and A.K. analysed and interpreted the data. J.B.K. developed the theoretical models.

Competing interests

D.D.H. is currently employed by KMLabs, Inc., a company that manufactures femtosecond lasers. D.R.C. is an owner of Octave Photonics, a company specializing in nonlinear nanophotonics.

Correspondence to Daniel D. Hickstein or Scott B. Papp.

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This file contains more information about the work and Supplementary Figs. 1–10.

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Fig. 1: SONG formation in nanophotonic waveguide.
Fig. 2: SHG in amorphous SiN waveguides using femtosecond pulses.
Fig. 3: Theoretical estimate of SHG with SiN waveguides.
Fig. 4: Frequency comb stabilization through one-step f–2f self-referencing.