Acoustical Physics

, Volume 64, Issue 6, pp 684–691 | Cite as

An Analytical Inverse Approach to Design GRIN Lenses

  • Kang ZhangEmail author


An analytical inverse method to design lenses of isotropic inhomogeneous refractive index (RI) distribution is presented, where the wave ray propagation is described by the eikonal equation. We show that some particular RI distributions can be obtained by the angles of incidence and emergence when the rays pass through the surfaces of the lenses. This method is applied to design lenses that perfectly focus rays or bend them to arbitrary angles. In addition, gradient refractive index (GRIN) devices are proposed, able to generate self-bending acoustic beams and obtain illusion shadows of arbitrary objects. The ray tracing and finite elements method simulation results indicate the validity of the method. The method may have potential applications in designing acoustic and optic GRIN devices for controlling energy flux, such as medical imaging, therapeutic ultrasound, acoustic levitation, energy isolation, acoustic and optic camouflaging, etc.


GRIN lenses inverse design ray tracing eikonal equation and Snell’s law perfect focus ray bending self-bending beams acoustic and optic illusions 


  1. 1.
    D. T. Moore, Appl. Opt. 19, 1035 (1980).ADSCrossRefGoogle Scholar
  2. 2.
    D. Schurig and D. R. Smith, Phys. Rev. E 70, 065601 (2004).ADSCrossRefGoogle Scholar
  3. 3.
    A. C. Urness, K. Anderson, C. Ye, W. L. Wilson and R. R. McLeod, Opt. Express 23, 264 (2015).ADSCrossRefGoogle Scholar
  4. 4.
    A. A. Abramovich, Acoust. Phys. 55 (3), 353 (2009).ADSCrossRefGoogle Scholar
  5. 5.
    E. V. Glushkov, N. V. Glushkova, S. I. Fomenko, and C. Zhang, Acoust. Phys. 58 (3), 339 (2012).ADSCrossRefGoogle Scholar
  6. 6.
    J. M. Gordon, Appl. Opt. 39, 3825 (2000).ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Merlin, J. Opt. 13, 024017 (2011).ADSCrossRefGoogle Scholar
  8. 8.
    G. Savini, P. A. Ade, and J. Zhang, Opt. Express 20 (23), 25766 (2012).ADSCrossRefGoogle Scholar
  9. 9.
    T. M. Chang, G. Dupont, S. Enoch, and S. Guenneau, New J. Phys. 14, 035011 (2012).ADSCrossRefGoogle Scholar
  10. 10.
    R. Q. Li, B. Liang, Y. Li, W. W. Kan, X. Y. Zou, and J. C. Cheng, Appl. Phys. Lett. 101, 263502 (2012).ADSCrossRefGoogle Scholar
  11. 11.
    R. Q. Li, X. F. Zhu, B. Liang, Y. Li, X. Y. Zou, and J. C. Cheng, Appl. Phys. Lett. 99, 193507 (2011).ADSCrossRefGoogle Scholar
  12. 12.
    Y. Li, B. Liang, X. Tao, X. F. Zhu, X. Y. Zou, and J. C. Cheng, Appl. Phys. Lett. 101, 233508 (2012).ADSCrossRefGoogle Scholar
  13. 13.
    J. R. Hensler, PhD Thesis (1972).Google Scholar
  14. 14.
    R. K. Mohr, P. K. Gupta, M. G. Drexhage, H. Hojaji, J. H. Simmons, and P. B. Macedo, Strengthening of Optical Fibers by Molecular Stuffing. Fiber Optics (Springer, 1979).Google Scholar
  15. 15.
    F. Gaufillet and É. Akmansoy, J. Appl. Phys. 114 (8), 083105 (2013).ADSCrossRefGoogle Scholar
  16. 16.
    H. Liu, C. Sheng, S. Zhu, and D. Genov, Nat. Photonics 7 (11), 902 (2013).ADSCrossRefGoogle Scholar
  17. 17.
    S. Chong, R. Bekenstein, L. Hui, S. Zhu, and M. Segev, Nat. Commun. 7, 10747 (2016).CrossRefGoogle Scholar
  18. 18.
    A. Žukauskas, I. Matulaitienė, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, Laser Photonics Rev. 9 (6), 706 (2016).CrossRefGoogle Scholar
  19. 19.
    S. C. S. Lin, T. J. Huang, J. H. Sun, and T. T. Wu, Phys. Rev. B 79, 094302 (2009).ADSCrossRefGoogle Scholar
  20. 20.
    Z. Liang and J. Li, Phys. Rev. Lett. 108 (11), 114301 (2012).ADSCrossRefGoogle Scholar
  21. 21.
    Y. Nishidate, J. Opt. Soc. Am. A 30 (7), 1373 (2013).ADSCrossRefGoogle Scholar
  22. 22.
    P. J. Sands, J. Opt. Soc. Am. 60, 1436 (1970).ADSCrossRefGoogle Scholar
  23. 23.
    A. Gupta, K. Thyagarajan, I. C. Goval, and A. K. Ghatak, J. Opt. Soc. Am. 66, 1320 (1976).ADSCrossRefGoogle Scholar
  24. 24.
    D. T. Moore, J. Opt. Soc. Am. 65, 451 (1975).ADSCrossRefGoogle Scholar
  25. 25.
    T. Sakamoto, J. Mod. Opt. 40, 503 (1993).ADSCrossRefGoogle Scholar
  26. 26.
    R. Ilinsky, J. Opt. A: Pure Appl. Opt. 2, 449 (2000).ADSCrossRefGoogle Scholar
  27. 27.
    A. Fletcher, T. Murphy, and A. Young, Proc. R. Soc. London, Ser. A 223, 216 (1954).ADSCrossRefGoogle Scholar
  28. 28.
    P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1970).ADSCrossRefGoogle Scholar
  29. 29.
    F. Borghero and G. Bozis, J. Phys. A: Math. Gen. 38, 175 (2005).ADSCrossRefGoogle Scholar
  30. 30.
    P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin and X. Zhang, Nat. Commun. 5, 4316 (2014).ADSCrossRefGoogle Scholar
  31. 31.
    J. Lu and J. F. Greenleaf, IEEE Trans. Ultrason., Ferroelectr. Freq. Control 37, 438 (1990).CrossRefGoogle Scholar
  32. 32.
    C. A. Speed, Rheumatology 40, 1331 (2001).CrossRefGoogle Scholar
  33. 33.
    A. A. Anosov, O. Y. Nemchenko, Y. A. Less, A. S. Kazanskii, and A. D. Mansfel’d, Acoust. Phys. 61 (4), 488 (2015).ADSCrossRefGoogle Scholar
  34. 34.
    J. Li, X. Chen, Y. Wang, Y. Shi, and D. Yu, Acoust. Phys. 63 (2), 229 (2017).ADSCrossRefGoogle Scholar
  35. 35.
    A. P. Sarvazyan, S. N. Tsyuryupa, M. Calhoun, and A. Utter, Acoust. Phys. 62 (4), 514 (2016).ADSCrossRefGoogle Scholar
  36. 36.
    E. H. Brandt, Science 243, 349 (1989).ADSCrossRefGoogle Scholar
  37. 37.
    Y. Lai, Jack Ng, H. Chen, D. Han, J. Xiao, Z. Zhang, and C. T. Chan, Phys. Rev. Lett. 102 (25), 253902 (2009).ADSCrossRefGoogle Scholar
  38. 38.
    H. Chen, B. Hou, S. Chen, X. Ao, W. Wen, and C. T. Chan, Phys. Rev. Lett. 102 (18), 183903 (2009).ADSCrossRefGoogle Scholar
  39. 39.
    W. Jiang, F. Hui, C. Qiang, and T. J. Cui, Appl. Phys. Lett. 96 (12), 1780. (2010).Google Scholar
  40. 40.
    W. Xiang and T. J. Cui, Opt. Express 18 (5), 5161 (2010).ADSCrossRefGoogle Scholar
  41. 41.
    W. Kan, B. Liang, X. Zhu, R. Li, X. Zou, H. Wu, J. Yang, and J. C. Cheng, Sci. Rep. 3 (3), 1427 (2013).ADSCrossRefGoogle Scholar
  42. 42.
    S. Zhang, S. Gan, J. Xiong, X. Zhang, and K. Wang, Phys. Rev. A 82 (2), 362 (2010).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Modern Acoustics, MOE, Department of Physics, Collaborative Innovation Center for Advanced Microstructures, Nanjing UniversityNanjingChina

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