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Imaging through nonlinear media using digital holography

Nature Photonics volume 3, pages 211–215 (2009)Cite this article

Abstract

It is well known that one cannot image directly through a nonlinear medium, as intensity-dependent phase changes distort signals as they propagate. Indirect methods can be used1,2,3,4,5,6, but none has allowed for the measurement of internal wave mixing and dynamics. Recently, the reconstruction of nonlinear pulse propagation in fibres was demonstrated by generalizing the techniques of digital holography7,8 to the nonlinear domain9. The method involves two steps: (1) recording the total field (both amplitude and phase) exiting a nonlinear medium and (2) numerically back-propagating the wavefunction. Here, we extend this process to two-dimensional spatial beams and experimentally demonstrate it in a self-defocusing photorefractive crystal, giving examples in soliton formation, dispersive radiation and imaging. For known nonlinearity, the technique enables reconstruction of wave dynamics within the medium and suggests new methods of super-resolved imaging, including subwavelength microscopy and lithography. For unknown nonlinearity, the method facilitates modelling and characterization of the optical response.

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Figure 1: Experimental set-up.
Figure 2: Nonlinear digital reconstruction of a self-defocused Gaussian beam.
Figure 3: Nonlinear reconstruction of evolution of a dark stripe.
Figure 4: Nonlinear digital reconstruction of a self-defocused USAF 1951 resolution chart.
Figure 5: Schematic for nonlinear holographic microscopy.

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References

  1. Yariv, A. Three-dimensional pictorial transmission in optical fibers. Appl. Phys. Lett. 28, 88–89 (1976).

    Article  ADS  Google Scholar 

  2. Yariv, A. Compensation for atmospheric degradation of optical beam transmission by nonlinear optical mixing. Opt. Commun. 21, 49–50 (1977).

    Article  ADS  Google Scholar 

  3. 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 

  4. Khyzniak, A. et al. Phase conjugation by degenerate forward four-wave mixing. J. Opt. Soc. Am. A 1, 169–175 (1984).

    Article  ADS  Google Scholar 

  5. Jones, D. C., Lyuksyutov, S. F. & Solymar, L. Three-wave and four-wave forward phase-conjugate imaging in photorefractive bismuth silicon oxide. Opt. Lett. 15, 935–937 (1990).

    Article  ADS  Google Scholar 

  6. 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 

  7. Schnars, U. & Jüptner, W. Direct recording of holograms by a CCD target and numerical reconstruction. Appl. Opt. 33, 179–181 (1994).

    Article  ADS  Google Scholar 

  8. Schnars, U. & Jüptner, W. P. O. Digital recording and numerical reconstruction of holograms. J. Meas. Sci. Technol. 13, R85–R101 (2002).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. Yamaguchi, I. & Zhang, T. Phase-shifting digital holography. Opt. Lett. 22, 1268–1270 (1997).

    Article  ADS  Google Scholar 

  11. Goldfarb, G. & Li, G. Demonstration of fibre impairment compensation using split-step infinite-impulse-response filtering method. Electron. Lett. 44, 814–815 (2008).

    Article  Google Scholar 

  12. Mateo, E., Zhu, L. & Li, G. Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation. Opt. Express 16, 16124–16137 (2008).

    Article  ADS  Google Scholar 

  13. Segev, M., Valley, G. C., Crosignani, B., Diporto, P. & Yariv, A. Steady-state spatial screening solitons in photorefractive materials with external applied field. Phys. Rev. Lett. 73, 3211–3214 (1994).

    Article  ADS  Google Scholar 

  14. Christodoulides, D. N. & Carvalho, M. I. Bright, dark and gray spatial soliton states in photorefractive media. J. Opt. Soc. Am. B 12, 1628–1633 (1995).

    Article  ADS  Google Scholar 

  15. 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 

  16. Ghofraniha, N., Conti, C., Ruocco, G. & Trillo, S. Shocks in nonlocal media. Phys. Rev. Lett. 99, 043903 (2007).

    Article  ADS  Google Scholar 

  17. Shih, M. et al. Two-dimensional steady-state photorefractive screening solitons. Opt. Lett. 21, 324–326 (1996).

    Article  ADS  Google Scholar 

  18. Linzon, Y. et al. Near-field imaging of nonlinear pulse propagation in planar silica waveguides. Phys. Rev. E 72, 066607 (2005).

    Article  ADS  Google Scholar 

  19. Krökel, D., Halas, N. J., Giuliani, G. & Grischkowsky, D. Dark-pulse propagation in optical fibers. Phys. Rev. Lett. 60, 29–32 (1988).

    Article  ADS  Google Scholar 

  20. Oppenheim, A. V. & Lim, J. S. The importance of phase in signals. Proc. IEEE 69, 529–541 (1981).

    Article  ADS  Google Scholar 

  21. Agrawal, G. P. Modulation instability induced by cross-phase modulation. Phys. Rev. Lett. 59, 880–883 (1987).

    Article  ADS  Google Scholar 

  22. Stentz, A. J., Kauranen, M., Maki, J. J., Agrawal, G. P. & Boyd, R. W. Induced focusing and spatial wave breaking from cross-phase modulation in a self-defocusing medium. Opt. Lett. 17, 19–21 (1992).

    Article  ADS  Google Scholar 

  23. Hickmann, J. M., Gomes, A. S. L. & de Araújo, C. B. Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium. Phys. Rev. Lett. 68, 3547–3550 (1992).

    Article  ADS  Google Scholar 

  24. Jia, S., Wan, W. & Fleischer, J. W. Forward four-wave mixing with defocusing nonlinearity. Opt. Lett. 32, 1668–1670 (2007).

    Article  ADS  Google Scholar 

  25. Goos, F. & Hänchen, H. Ein neuer und fundamentaler versuch zur totalreflexion. Ann. Phys. 1, 333–346 (1947).

    Article  Google Scholar 

  26. Artmann, K. Berechnung der seitenversetzung des totalreflektierten strahles. Ann. Phys. 2, 87–102 (1947).

    MATH  Google Scholar 

  27. Tomlinson, W. J., Gordon, J. P., Smith, P. W. & Kaplan, A. E. Reflection of a Gaussian beam at a nonlinear interface. Appl. Opt. 21, 2041–2051 (1982).

    Article  ADS  Google Scholar 

  28. 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 

  29. Jost, B. M., Al-Rashed, A.-A. R. & Saleh, B. E. A. Observation of the Goos–Hänchen effect in a phase-conjugate mirror. Phys. Rev. Lett. 81, 2233–2235 (1998).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Science Foundation, the Department of Energy and the Air Force Office of Scientific Research. C.B. would like to thank the Army Research Office for support through a National Defense Science and Engineering Graduate Fellowship.

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Authors and Affiliations

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

    Christopher Barsi, Wenjie Wan & Jason W. Fleischer

  2. Program in Applied and Computational Mathematics, Princeton University, Princeton, 08544, New Jersey, USA

    Jason W. Fleischer

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  2. Wenjie Wan
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  3. Jason W. Fleischer
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Correspondence to Jason W. Fleischer.

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Barsi, C., Wan, W. & Fleischer, J. Imaging through nonlinear media using digital holography. Nature Photon 3, 211–215 (2009). https://doi.org/10.1038/nphoton.2009.29

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