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Observing angular deviations in the specular reflection of a light beam

Abstract

The Law of Reflection of a light ray incident upon a mirror (θin = θout) was first formulated by Euclid around 300 bc in his book Catoptrics1; it has been a tenet of geometrical optics ever since. However, more recently, a small angular deviation of the Law of Reflection has been predicted for a physical light beam when this is regarded as the implementation of a ray2,3,4,5. The deviation is a diffractive consequence of the angular dependence of the reflectivity and should occur for any mirror with less than 100% reflectivity. We report here experimental proof of this angular deviation by determining the direction of an optical beam after reflection from an air–glass interface, using a position detector with nanometre resolution. Our results are relevant for angular metrology in general and cantilever-based surface microscopies in particular. Analogous angular deviations are expected for reflection of acoustic waves and quantum matter waves.

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Figure 1: Schematic representation of non-specular angular reflection.
Figure 2: Experimental set-up.
Figure 3: Angular deviation far from the Brewster angle θB.
Figure 4: Angular deviation near the Brewster angle θB
Figure 5: Verification of the angular nature of the deviation.

References

  1. Hecht, E. Optics 4th edn, 1 (Addison-Wesley, 2002).

    Google Scholar 

  2. Ra, J. W., Bertoni, H. L. & Felsen, L. B. Reflection and transmission of beams at a dielectric interface. SIAM J. Appl. Math. 24, 396–413 (1973).

    Article  Google Scholar 

  3. Antar, Y. M. & Boerner, W. M. Gaussian beam interaction with a planar dielectric interface. Can. J. Phys. 52, 962–972 (1974).

    ADS  Article  Google Scholar 

  4. Chan, C. C. & Tamir, C. Angular shift of a Gaussian beam reflected near the Brewster angle. Opt. Lett. 10, 378–380 (1985).

    ADS  Article  Google Scholar 

  5. Aiello, A. & Woerdman, J. P. Role of beam propagation in Goos–Hänchen and Imbert–Fedorov shifts. Opt. Lett. 33, 1437–1439 (2008).

    ADS  Article  Google Scholar 

  6. Goos, F. & Hänchen, H. Ein neuer und fundamentaler Versuch zur Totalreflexion. Ann. Phys. (Leipzig) 1, 333–346 (1947).

    Article  Google Scholar 

  7. Artmann, K. Berechnung der Seitenversetzung des totalreflektierten Strahles. Ann. Phys. 2, 87–102 (1948).

    Article  Google Scholar 

  8. Bonnet, C., Chauvat, D., Emile, O., Bretenaker, F. & Le Floch, A. Measurement of positive and negative Goos–Hänchen effects for metallic gratings near Wood anomalies. Opt. Lett. 26, 666–668 (2001).

    ADS  Article  Google Scholar 

  9. Berman, P. R. Goos–Hänchen shifts in negatively refracting media. Phys. Rev. E 66, 067603 (2002).

    ADS  Article  Google Scholar 

  10. Bliokh, K. Y., Shadrivov, I. V. & Kivshar, Y. S. Goos–Hänchen and Imbert–Fedorov shifts of polarized vortex beams. Opt. Lett. 34, 389–391 (2009).

    ADS  Article  Google Scholar 

  11. 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).

    ADS  Article  Google Scholar 

  12. Huang, J., Duan, Z., Ling, H. & Zhang, W. Goos–Hänchen-like shifts in atom optics. Phys. Rev. A 77, 063608 (2008).

    ADS  Article  Google Scholar 

  13. Felbacq, D., Moreau, A. & Smaâli, R. Goos–Hänchen effect in the gaps of photonic crystals. Opt. Lett. 28, 1633–1635 (2003).

    ADS  Article  Google Scholar 

  14. Emile, O., Galstyan, T., Le Floch, A. & Bretenaker, F. Measurement of the nonlinear Goos–Hänchen effect for Gaussian optical beams. Phys. Rev. Lett. 75, 1511–1514 (1995).

    ADS  Article  Google Scholar 

  15. Yin, X. & Hesselink, L. Large positive and negative lateral optical beam displacements due to surface plasmon resonance. Appl. Phys. Lett. 85, 372–374 (2004).

    ADS  Article  Google Scholar 

  16. Gragg, R. F. The total reflection of a compact wave group: Long-range transmission in a waveguide. Am. J. Phys. 56, 1092–1094 (1988).

    ADS  MathSciNet  Article  Google Scholar 

  17. Merano, M. et al. Observation of Goos–Hänchen shifts in metallic reflection. Opt. Express 15, 15928–15934 (2007).

    ADS  Article  Google Scholar 

  18. Aiello, A. & Woerdman, J. P. Theory of angular Goos–Hänchen shift near Brewster incidence. Preprint at <http://arxiv.org/pdf/0903.3730>.

  19. Müller, D., Tharanga D., Stahlhofen, A. A. & Nimtz, G. Nonspecular shifts of microwaves in partial reflection. Europhys. Lett. 73, 526–532 (2006).

    ADS  Article  Google Scholar 

  20. Mandel, L. & Wolf, E. Optical Coherence and Quantum Optics Vol. 143, 271 (Cambridge Univ. Press, 1995).

    Book  Google Scholar 

  21. Fainman, Y. & Shamir, J. Polarization of nonplanar wavefronts. Appl. Opt. 23, 3188–3195 (1984).

    ADS  Article  Google Scholar 

  22. Li, Q. & Vernon, R. J. Theoretical and experimental investigation of Gaussian beam transmission and reflection by a dielectric slab at 110 GHz. IEEE Trans. Antennas Propag. 54, 3449–3457 (2006).

    ADS  Article  Google Scholar 

  23. Mueller, F., Heugel, S. & Wang, L. J. Femto-newton light force measurement at the thermal noise limit. Opt. Lett. 33, 539–541 (2008).

    ADS  Article  Google Scholar 

  24. Putman, C. A. J., De Grooth, B. G., Van Hulst, N. F. & Greve, J. A theoretical comparison between interferometric and optical beam deflection technique for the measurement of cantilever displacement in AFM. Ultramicroscopy 42, 1509–1513 (1992).

    Article  Google Scholar 

  25. Centrella, J. M. Resource letter: GrW-1: Gravitational waves. Am. J. Phys. 71, 520–524 (2003).

    ADS  Article  Google Scholar 

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Acknowledgements

This work was supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) and by the Seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement HIDEAS, no. FP7-ICT-221906. We acknowledge E.R. Eliel and G.'t Hooft for useful discussions.

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Correspondence to A. Aiello or J. P. Woerdman.

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Merano, M., Aiello, A., van Exter, M. et al. Observing angular deviations in the specular reflection of a light beam. Nature Photon 3, 337–340 (2009). https://doi.org/10.1038/nphoton.2009.75

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