Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Nonlinear Abbe theory

Nature Photonics volume 7, pages 639–643 (2013)Cite this article

Subjects

Abstract

All limitations commonly associated with imaging, such as resolution, field of view and depth of field, arise from linear theory1. Nonlinear optics can break these limits by exploiting the presence and interaction of many photons at once. To date, nearly all nonlinear imaging techniques have relied on point processes, such as two-photon fluorescence2 or harmonic effects3, in which the temporal frequency is the relevant parameter. These methods ignore the spatial content of the object, typically require scanning to record a whole image, and remain restricted by linear propagation from the sample to the detector. Spatial nonlinearity can overcome these issues by mixing modes with high and low spatial frequencies. Here, we generalize Abbe's theory of diffraction1 to include nonlinear propagation and show that wave mixing can be treated as a self-induced structured illumination. We demonstrate this experimentally by nonlinearly enhancing a standard imaging system beyond its conventional limits.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Linear versus nonlinear Abbe theory.
Figure 2: Experimental set-up.
Figure 3: Field-of-view enhancement via spatial nonlinearity.
Figure 4: Bandwidth extrapolation via spatial nonlinearity.
Figure 5: Super-resolution via spatial nonlinearity.

Similar content being viewed by others

References

  1. Abbe, E. Beiträge zur Theorie des Mikroskops und der micrkoskopischen Wahrnehmung. Arch. Mikrosk. Anat. 9, 413–468 (1873).

    Article  Google Scholar 

  2. Denk, W., Strickler, J. H. & Webb, W. W. Two-photon laser scanning fluorescence microscopy. Science 248, 73–76 (1990).

    Article  ADS  Google Scholar 

  3. Campagnola, P. J., Clark, H. A., Mohler, W. A., Loew, L. M. & Lewis, A. Second-harmonic imaging microscopy of living cells. J. Biomed. Opt. 6, 277–286 (2001).

    Article  ADS  Google Scholar 

  4. Heintzmann, R. & Cremer, C. Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating. Proc. SPIE 3568, 185–196 (1999).

    Article  ADS  Google Scholar 

  5. Di Francia, T. Resolving power and information. J. Opt. Soc. Am. 45, 497–501 (1955).

    Article  ADS  Google Scholar 

  6. Gorodnitsky, I. F. & Rao, B. D. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans. Sign. Proc. 45, 600–616 (1997).

    Article  ADS  Google Scholar 

  7. Ash, E. A. & Nicholls, G. Super-resolution aperture scanning microscope. Nature 237, 510–512 (1972).

    Article  ADS  Google Scholar 

  8. Carrington, W. A. et al. Superresolution three-dimensional images of fluorescence in cells with minimal light exposure. Science 268, 1483–1487 (1995).

    Article  ADS  Google Scholar 

  9. Pavani, S. R. P. et al. Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function. Proc. Natl Acad. Sci. USA 106, 2995–2999 (2009).

    Article  ADS  Google Scholar 

  10. Gustafsson, M. G. L. Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy. J. Microsc. 198, 82–87 (2000).

    Article  Google Scholar 

  11. Klar, T. A. & Hell, S. W. Subdiffraction resolution in far-field fluorescence microscopy. Opt. Lett. 24, 954–956 (1999).

    Article  ADS  Google Scholar 

  12. Rust, M. J., Bates, M. & Zuang, X. Sub-diffraction-limit imaging by stochastic optical reconstruction microcopy (STORM). Nat. Methods 3, 793–795 (2006).

    Article  Google Scholar 

  13. Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution. Science 313, 1642–1645 (2006).

    Article  ADS  Google Scholar 

  14. Fischer, B., Cronin-Golomb, M., White, J. O. & Yariv, A. Real-time phase conjugate window for one-way optical field imaging through a distortion. Appl. Phys. Lett. 41, 141–143 (1982).

    Article  ADS  Google Scholar 

  15. Kip, D., Anastassiou, C., Eugenieva, E., Christodoulides, D. & Segev, M. Transmission of images through highly nonlinear media by gradient-index lenses formed by incoherent solitons. Opt. Lett. 26, 524–526 (2001).

    Article  ADS  Google Scholar 

  16. Yang, J., Zhang, P., Yoshihara, M., Hu, Y & Chen, Z. Image transmission using stable solitons of arbitrary shapes in photonic lattices. Opt. Lett. 36, 772–774 (2011).

    Article  ADS  Google Scholar 

  17. Hong, J. H., Chiou, A. E. & Yeh, P. Image amplification by two-wave mixing in photorefractive crystals. Appl. Opt. 29, 3026–3029 (1990).

    Article  ADS  Google Scholar 

  18. Dylov, D. V. & Fleischer, J. W. Nonlinear self-filtering of noisy images via dynamical stochastic resonance. Nature Photon. 4, 323–328 (2010).

    Article  Google Scholar 

  19. Barsi, C., Wan, W. & Fleischer, J. W. Imaging through nonlinear media using digital holography. Nature Photon. 3, 211–215 (2009).

    Article  ADS  Google Scholar 

  20. Lukosz, W. & Marchand, M. Optischen Abbildung unter Überschreitung der Beugungsbedingten Auflösungsgrenze. Opt. Acta 10, 241–245 (1963).

    Article  ADS  Google Scholar 

  21. Gustafsson, M. G. L. Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution. Proc. Natl Acad. Sci. USA 102, 13081–13086 (2005).

    Article  ADS  Google Scholar 

  22. Wan, W., Jia, S. & Fleischer, J. W. Dispersive, superfluid-like shock waves in nonlinear optics. Nature Phys. 3, 46–51 (2007).

    Article  ADS  Google Scholar 

  23. Tsang, M., Psaltis, D. & Omenetto, F. G. Reverse propagation of femtosecond pulses in optical fibers. Opt. Lett. 28, 1873–1875 (2003).

    Article  ADS  Google Scholar 

  24. Feit, M. D. & Fleck, J. A. Jr. Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams. J. Opt. Soc. Am. B 5, 633–640 (1988).

    Article  ADS  Google Scholar 

  25. Greenspan, H., Anderson, C. H. & Akber, S. Image enhancement by nonlinear extrapolation in frequency space. IEEE Trans. Image Proc. 9, 1035–1048 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  26. Renger, J., Quidant, R., van Hulst, N. & Novotny, L. Surface-enhanced nonlinear four-wave mixing. Phys. Rev. Lett. 104, 046803 (2010).

    ADS  Google Scholar 

  27. Emile, O., Galstyan, T., Le Floch, A. & Bretenaker, F. Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams. Phys. Rev. Lett. 75, 1511–1513 (1995).

    Article  ADS  Google Scholar 

  28. Sun, C. et al. Observation of the kinetic condensation of classical waves. Nature Phys. 8, 470–474 (2012).

    Article  ADS  Google Scholar 

  29. Lu, C-H. et al. Phase retrieval using nonlinear diversity. Appl. Opt. 52, D92 (2013).

    Article  ADS  Google Scholar 

  30. DelRe, E., Spinozzi, E., Agranat, A. J. & Conti, C. Scale-free optics and diffractionless waves in nanodisordered ferroelectrics. Nature Photon. 5, 39–42 (2011).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the US Air Force Office of Scientific Research. The authors thank N. Pégard for valuable discussions.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Princeton University, Princeton, 08544, New Jersey, USA

    Christopher Barsi & Jason W. Fleischer

  2. Media Lab, Massachusetts Institute of Technology, Cambridge, 02139, Massachusetts, USA

    Christopher Barsi

Authors
  1. Christopher Barsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jason W. Fleischer
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

C.B. and J.W.F. conceived and designed the experiments. C.B. performed the experiments and simulations. Both authors analysed the data and wrote the manuscript.

Corresponding author

Correspondence to Jason W. Fleischer.

Ethics declarations

Competing interests

C. Barsi and J. W. Fleischer are named inventors on US patent 8,427,650 (issued 23 April 2013), which is related to techniques described in this Letter.

Supplementary information

Supplementary information

Supplementary information (PDF 1406 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barsi, C., Fleischer, J. Nonlinear Abbe theory. Nature Photon 7, 639–643 (2013). https://doi.org/10.1038/nphoton.2013.171

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphoton.2013.171

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing