Perturbative optical nonlinearities induced by static electric fields1 have proven useful in visualizing dynamical function in systems including operating circuits2,3, electric and magnetic domain walls4, and biological matter5, and in controlling light for applications in silicon photonics6. Here, we extend field-induced second-harmonic generation to the non-perturbative regime. We demonstrate that static or transient fields up to terahertz (THz) frequencies applied to silicon and ZnO crystals generate even-order high harmonics. Images of the even harmonics confirm that static fields delivered with conventional electronics control the spatial properties of the high-harmonic emission. Extending our methodology to higher-harmonic photon energies7,8 paves the way for realizing active optics in the extreme ultraviolet and will allow imaging of operating electronic circuits9, of Si-photonic devices10 and of other functional materials11,12, with higher spatio-temporal resolution than perturbative methods. For THz spectroscopy, our method has the bandwidth to allow measurement of attosecond transients imprinted on THz waveforms.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Terhune, R. W., Maker, P. D. & Savage, C. M. Optical harmonic generation in calcite. Phys. Rev. Lett. 8, 404–406 (1962).

  2. 2.

    Ruzicka, B. A. et al. Second-harmonic generation induced by electric currents in GaAs. Phys. Rev. Lett. 108, 077403 (2012).

  3. 3.

    Manaka, T., Lim, E., Tamura, R. & Iwamoto, M. Direct imaging of carrier motion in organic transistors by optical second-harmonic generation. Nat. Photon. 1, 581–584 (2007).

  4. 4.

    Fiebig, M., Lottermoser, T., Fröhlich, D., Goltsev, A. V. & Pisarev, R. V. Observation of coupled magnetic and electric domains. Nature 419, 818–820 (2002).

  5. 5.

    Peterka, D. S., Takahashi, H. & Yuste, R. Imaging voltage in neurons. Neuron 69, 9–21 (2011).

  6. 6.

    Timurdogan, E. et al. Electric field-induced second-order nonlinear optical effects in silicon waveguides. Nat. Photon. 11, 200–206 (2017).

  7. 7.

    Luu, T. T. et al. Extreme ultraviolet high-harmonic spectroscopy of solids. Nature 521, 498–502 (2015).

  8. 8.

    You, Y. S., Reis, D. A. & Ghimire, S. Anisotropic high-harmonic generation in bulk crystals. Nat. Phys. 13, 345–349 (2016).

  9. 9.

    Holler, M. et al. High-resolution non-destructive three-dimensional imaging of integrated circuits. Nature 543, 402–406 (2017).

  10. 10.

    Englund, D. et al. Controlling cavity reflectivity with a single quantum dot. Nature 450, 857–861 (2007).

  11. 11.

    Raja, A. et al. Coulomb engineering of the bandgap and excitons in two-dimensional materials. Nat. Commun. 8, 15251 (2017).

  12. 12.

    Tetienne, J.-P. et al. Quantum imaging of current flow in graphene. Sci. Adv. 3, e1602429 (2017).

  13. 13.

    Boyd, R. W. Nonlinear Optics (Academic Press, London, 2003).

  14. 14.

    Bloembergen, N. & Pershan, P. S. Light waves at the boundary of nonlinear media. Phys. Rev. 128, 606–622 (1962).

  15. 15.

    Mourou, G. A. & Meyer, K. E. Subpicosecond electro-optic sampling using coplanar strip transmission lines. Appl. Phys. Lett. 45, 492–494 (1984).

  16. 16.

    Valdmanis, J. A. 1 THz bandwidth prober for high-speed devices and integrated circuits. Electron. Lett. 23, 1308–1310 (1987).

  17. 17.

    Ryabov, A. & Baum, P. Electron microscopy of electromagnetic waveforms. Science 353, 374–377 (2016).

  18. 18.

    Feist, A. et al. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature 521, 200–203 (2015).

  19. 19.

    Hassan, M. Th. et al. High-temporal-resolution electron microscopy for imaging ultrafast electron dynamics. Nat. Photon. 11, 425–430 (2017).

  20. 20.

    Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2011).

  21. 21.

    Vampa, G. et al. Linking high harmonics from gases and solids. Nature 522, 462–464 (2015).

  22. 22.

    Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations. Nat. Photon. 8, 119–123 (2014).

  23. 23.

    Chapman, H. N. et al. Femtosecond diffractive imaging with a soft-X-ray free-electron laser. Nat. Phys. 2, 839–843 (2006).

  24. 24.

    Tung, R. T. Recent advances in Schottky barrier concepts. Mater. Sci. Eng. E 35, 1–138 (2001).

  25. 25.

    Niikura, H., Dudovich, N., Villeneuve, D. M. & Corkum, P. B. Mapping molecular orbital symmetry on high-order harmonic generation spectrum using two-color laser fields. Phys. Rev. Lett. 105, 053003 (2010).

  26. 26.

    Dudovich, N. et al. Measuring and controlling the birth of attosecond XUV pulses. Nat. Phys. 2, 781–786 (2006).

  27. 27.

    Kim, K. K. et al. Petahertz optical oscilloscope. Nat. Photon. 7, 958–962 (2013).

  28. 28.

    Wilke, I. & Sengupta, S. in Terahertz Spectroscopy: Principles and Applications (ed. Dexheimer, S. L.) 48–49 (CRC, Boca Raton, 2007).

  29. 29.

    Lu, X. & Zhang, X.-C. Balanced terahertz wave air-biased-coherent-detection. Appl. Phys. Lett. 98, 151111 (2011).

  30. 30.

    Sivis, M. et al. Tailored semiconductors for high-harmonic optoelectronics. Science 357, 303–306 (2017).

Download references


We thank A. Laramée and C. A. Couture (Advanced Laser Light Source), D. Crane and B. Avery (NRC Canada) for technical support, and G. Lopinski (NRC Canada) for discussions and for lending electrical equipment. This material is based on work supported by the US Air Force Office of Scientific Research under award numbers FA9550-16-1-0109 and FA9550-15-1-0037; and by Canada’s National Research Council (NRC), the Natural Sciences and Engineering Research Council (NSERC), Canada Foundation for Innovation (CFI) and the Ontario Research Fund (ORF).

Author information


  1. Department of Physics, University of Ottawa, Ottawa, Ontario, Canada

    • G. Vampa
    • , T. J. Hammond
    • , M. Taucer
    • , Xiaoyan Ding
    •  & P. B. Corkum
  2. Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

    • G. Vampa
  3. INRS-EMT, Varennes, Quebec, Canada

    • X. Ropagnol
    • , T. Ozaki
    • , S. Delprat
    • , M. Chaker
    • , N. Thiré
    •  & F. Légaré
  4. few-cycle Inc., Montreal, Quebec, Canada

    • B. E. Schmidt
  5. National Research Council of Canada, Ottawa, Ontario, Canada

    • D. D. Klug
    • , A. Yu. Naumov
    • , D. M. Villeneuve
    • , A. Staudte
    •  & P. B. Corkum


  1. Search for G. Vampa in:

  2. Search for T. J. Hammond in:

  3. Search for M. Taucer in:

  4. Search for Xiaoyan Ding in:

  5. Search for X. Ropagnol in:

  6. Search for T. Ozaki in:

  7. Search for S. Delprat in:

  8. Search for M. Chaker in:

  9. Search for N. Thiré in:

  10. Search for B. E. Schmidt in:

  11. Search for F. Légaré in:

  12. Search for D. D. Klug in:

  13. Search for A. Yu. Naumov in:

  14. Search for D. M. Villeneuve in:

  15. Search for A. Staudte in:

  16. Search for P. B. Corkum in:


G.V. and P.B.C. conceived the experiment. G.V., T.J.H., M.T., X.D. and N.T. performed the measurements with static fields. G.V. and X.R. performed the THz experiment. S.D. fabricated the electrodes. N.T., B.E.S. and A.S. maintained the laser sources. D.D.K. provided theoretical support. T.O., M.C., A.Yu.N., D.M.V., A.S., F.L. and P.B.C. supervised the experiments. All authors contributed to the manuscript.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to G. Vampa.

Supplementary information

  1. Supplementary Information

    Supplementary notes and figures.

  2. Supplementary Video 1

    Electric field build up.

About this article

Publication history




Issue Date



Further reading