Non-Paraxial Electromagnetic Beams

  • Rosario Martínez-HerreroEmail author
  • Pedro M. Mejías
  • Gemma Piquero
Part of the Springer Series in Optical Sciences book series (SSOS, volume 147)

In the previous chapters, we have considered electromagnetic beams whose longitudinal field component (along the propagation direction) is negligible. In other words, the electric field vector was assumed to be transverse to the z-axis and, consequently, it was represented by means of two components. This paraxial approach and the subsequent quasi-transversality assumption have provided in the above chapters a considerable simplification in both, the calculations and the characterization of this kind of beams.


Transverse Plane Longitudinal Component Evanescent Wave Initial Plane Angular Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Agrawal, G. P., Lax, M. (1983): Free-space wave propagation beyond the paraxial approximation, Phys. Rev. A 27, 1693–1695.ADSGoogle Scholar
  2. Alonso, M. A. (2004): Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space, J. Opt. Soc. Am. A 21, 2233–2243.MathSciNetADSCrossRefGoogle Scholar
  3. Alonso, M. A., Borghi, R., Santarsiero, M. (2006a): Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case, J. Opt. Soc. Am. A 23, 691–700.MathSciNetADSGoogle Scholar
  4. Alonso, M. A., Borghi, R., Santarsiero, M. (2006b): Nonparaxial fields with maximum joint spatial-directional localization. II. Vectorial case, J. Opt. Soc. Am. A 23, 701–712.MathSciNetADSGoogle Scholar
  5. April, A. (2008): Nonparaxial TM and TE beams in free space, Opt. Lett. 33, 1563–1565.ADSCrossRefGoogle Scholar
  6. Arnoldus, H. F., Foley, J. T. (2002): Traveling and evanescent fields of an electric point dipole, J. Opt. Soc. Am. A 19, 1701–1711.ADSCrossRefGoogle Scholar
  7. Belkebir, K., Chaumet, P. C., Sentetac, A. (2006): Influence of multiple scatering on three-dimensional imaging with optical diffraction tomography, J. Opt. Soc. Am. A 23, 586–595.ADSCrossRefGoogle Scholar
  8. Borghi, R., Ciattoni, A., Santarsiero, M. (2002): Exact axial electromagnetic field for vectorial Gaussian and flattened Gaussian boundary distribution, J. Opt. Soc. Am. A 19, 1207–1211.ADSCrossRefGoogle Scholar
  9. Borghi, R., Santarsiero, M. (2003): Summing lax series for nonparaxial beam propagation, Opt. Lett 28, 774–776.ADSCrossRefGoogle Scholar
  10. Borghi, R., Santarsiero, M. (2004): Nonparaxial propagation of spirally polarized optical beams, J. Opt. Soc. Am. A 21, 2029–2037.MathSciNetADSCrossRefGoogle Scholar
  11. Chaumet, P. C. (2006): Fully vectorial highly nonparaxial beam close to the waist, J. Opt. Soc. Am. A 23, 3197–3202.MathSciNetADSCrossRefGoogle Scholar
  12. Chen, C. G., Konkola, P. T., Ferrera, J., Heilman, R. K., Schattenburg, M. L. (2002): Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximation, J. Opt. Soc. Am. A 19, 404–412.ADSCrossRefGoogle Scholar
  13. Ciattoni, A., Crosignani, B., Di Porto, P. (2002): Vectorial analytical description of propagation of a highly nonparaxial beam, Opt. Commun. 202, 17–20.ADSCrossRefGoogle Scholar
  14. Deng, D. M. (2006): Nonparaxial propagation of radially polarized light beams, J. Opt. Soc. Am. B 23, 1228–1234.ADSGoogle Scholar
  15. Deng, D. M., Guo, Q. (2007): Analytical vectorial structure of radially polarized light beams, Opt. Lett. 32, 2711–2713.ADSCrossRefGoogle Scholar
  16. Diehl, D. W., Schoonover, R. W., Visser, T. D. (2006): The structure of focused radially polarized fields, Opt. Express 14, 3030–3038.ADSCrossRefGoogle Scholar
  17. Dorn, R., Quabis, S., Lenchs, G. (2003): Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91, 233901 (1–4).ADSCrossRefGoogle Scholar
  18. Duan, K., Lü, B. (2005a): Polarization properties of vectorial nonparaxial Gaussian beam in the far field, Opt. Lett. 30, 308–310.ADSCrossRefGoogle Scholar
  19. Duan, K., Lü, B. (2005b): Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams, J. Opt. Soc. Am. B 22, 1585–1593.ADSGoogle Scholar
  20. Guo, H. M., Chen, J. B., Zhuang, S. L. (2006): Vector plane wave spectrum of an arbitrary polarized electromagnetic wave, Opt. Express 14, 2095–2100.ADSCrossRefGoogle Scholar
  21. Hall, D. G. (1996): Vector-beam solutions of Maxwell’s waves equation, Opt. Lett. 21, 9–11.ADSCrossRefGoogle Scholar
  22. ISO 11146 (2005): Laser and laser related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios, Parts 1, 2 and 3 (International Organization for Standardization, Geneva, Switzerland).Google Scholar
  23. Lekner, J. (2003): Polarization of tightly focused laser beams, J. Opt. A: Pure Appl. Opt. 5, 6–14.ADSCrossRefGoogle Scholar
  24. Machavariani, G., Lumer, Y., Moshe, I., Meir, A., Jackel, S. (2008): Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams, Opt. Commun. 281, 732–738.ADSCrossRefGoogle Scholar
  25. Mandel, L., Wolf, E. (1995): Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge).Google Scholar
  26. Martínez-Herrero, R. (1979): Expansion of complex degree of coherente, Il Nuevo Cimento B, 54, 205–210.Google Scholar
  27. Martínez-Herrero, R., Mejías, P. M. (2008): Propagation of light fields with radial or azimuthal polarization distribution at a transverse plane, Opt. Express 16, 9021–9033.ADSCrossRefGoogle Scholar
  28. Martínez-Herrero, R., Mejías, P. M., Bosch, S. (2008a): On the vectorial structure of non-paraxial radially polarized light fields, Opt. Commun. 281, 3046–3050.ADSCrossRefGoogle Scholar
  29. Martínez-Herrero, R., Mejias, P. M., Carnicer, A. (2008b): Evanescent field of vectorial highly non-paraxial beams, Opt. Express 16, 2845–2858.ADSCrossRefGoogle Scholar
  30. Martínez-Herrero, R., Mejías, P. M., Bosch, S., Carnicer, A. (2001): Vectorial structure of nonparaxial electromagnetic beams, J. Opt. Soc. Am. A 18, 1678–1680.ADSCrossRefGoogle Scholar
  31. Martínez-Herrero, R., Mejías, P. M., Bosch, S., Carnicer, A. (2006): Structure of the transverse profile of Gaussian-model non-paraxial electromagnetic beams, J. Opt. A: Pure Appl. Opt. 8, 524–530.ADSCrossRefGoogle Scholar
  32. Martínez-Herrero, R., Moreu, A. F. (2006): On the polarization of non-paraxial transverse fields, Opt. Commun. 267, 20–23.ADSCrossRefGoogle Scholar
  33. Mei, Z. R., Zhao, D. M. (2008): Non-paraxial propagation of controllable dark-hollow beams, J. Opt. Soc. Am. A 25, 537–542.ADSCrossRefGoogle Scholar
  34. Mejías, P. M., Martínez-Herrero, R., Piquero, G., Movilla, J. M. (2002): Parametric characterization of the spatial structure of non-uniformly polarized laser beams, Prog. Quantum Electron. 26, 65–130.ADSCrossRefGoogle Scholar
  35. Nesterov, A. V., Niziev, V. G., Yakunin, V. P. (1999): Generation of high-power radially polarized beam, J. Phys. D 32, 2871–2875.ADSGoogle Scholar
  36. Niu, C. H., Gu, B. Y., Dong, B. Z., Zhang, Y. (2005): A new method for generating axially-symmetric and radially-polarized beams, J. Phys. D 38, 827–832.ADSGoogle Scholar
  37. Pääkkönen, P., Tervo, J., Vahimaa, P., Turunen, J., Gori, F. (2002): General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields, Opt. Express 10, 949–959.ADSGoogle Scholar
  38. Petrov, N. I. (1999): Nonparaxial focusing of wave beams in a graded-index medium, Quantum Electron. 29, 249–255.ADSCrossRefGoogle Scholar
  39. Seshadri, S. R. (1998): Electromagnetic Gaussian beam, J. Opt. Soc. Am. A 15, 2712–2719.ADSCrossRefGoogle Scholar
  40. Seshadri, S. R. (2008): Fundamental electromagnetic Gaussian beam beyond the paraxial approximation, J. Opt. Soc. Am. A 25, 2156–2164.Google Scholar
  41. Setala, T., Friberg, A. T., Kaivola, M. (1999): Decomposition of the point-dipole field into homogeneous and evanescent parts, Phys. Rev. E 59, 1200–1206.ADSGoogle Scholar
  42. Shchegrov, A. V., Carney, P. S. (1999): Far-field contribution to the electromagnetic Green’s tensor from evanescent modes, J. Opt. Soc. Am. A 16, 2583–2584.ADSCrossRefGoogle Scholar
  43. Sherman, G. C., Stamnes, J. J., Lalor, E. (1976): Asymptotic approximations to angular-spectrum representations, J. Math. Phys. 17, 760–776.MathSciNetADSCrossRefGoogle Scholar
  44. Sheppard, C. J. R. (2000): Polarization of almost-planes waves, J. Opt. Soc. Am. A 17, 335–341.MathSciNetADSCrossRefGoogle Scholar
  45. Sheppard, C. J. R., Saghafi, S. (1999): Electromagnetic Gaussian beams beyond the paraxial approximation, J. Opt. Soc. Am. A 16, 1381–1386.ADSCrossRefGoogle Scholar
  46. Siegman, A. E. (1986): Lasers (University Science Books, California).Google Scholar
  47. Simon, R., Sudarshan, E. C. G., Mukunda, N. (1987): Cross polarization in laser beams, Appl. Opt. 26, 1589–1593.ADSCrossRefGoogle Scholar
  48. Tervo, J. (2003): Azimuthal polarization and partial coherence, J. Opt. Soc. Am. A 20, 1974–1980.ADSCrossRefGoogle Scholar
  49. Tervo, J., Turunen, J. (2001): Self-imaging of electromagnetic fields, Opt. Express 9, 622–630.ADSCrossRefGoogle Scholar
  50. Varga, P., Török, P. (1996): Exact and approximate solutions of Maxwell’s equation and the validity of the scalar wave approximation, Opt. Lett. 21, 1523–1525.ADSCrossRefGoogle Scholar
  51. Varga, P., Török, P. (1998): The Gaussian wave solution of Maxwell’s equations and the validity of the scalar wave approximation, Opt. Commun. 152, 108–118.ADSCrossRefGoogle Scholar
  52. Volpe, G., Petrov, D. (2004): Optical tweezers with cylindrical vector beams produced by optical fibers, Opt. Commun. 237, 89–95.ADSCrossRefGoogle Scholar
  53. Wolf, E., Foley, J. T. (1998): Do evanescent waves contribute to the far field?, Opt. Lett. 23, 16–18.ADSCrossRefGoogle Scholar
  54. Yan, S. H., Yao, B. L. (2008): Accurate description of a radially polarized Gaussian beam, Phys. Rev. A 77, 023827(1–4).ADSGoogle Scholar
  55. Yew, E. Y. S., Sheppard, C. J. R. (2007): Tight focusing of radially polarized Gaussian and Bessel-Gauss beams, Opt. Lett. 32, 3417–3419.ADSCrossRefGoogle Scholar
  56. Zhang, Z., Pu, J., Wang, X. (2008): Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal, Appl. Opt. 47, 1963–1967.Google Scholar
  57. Zhou, G. Q. (2006): Analytical vectorial structure of Laguerre-Gaussian beam in the far field, Opt. Lett. 31, 2616–2618.ADSCrossRefGoogle Scholar
  58. Zhou, G. Q. (2008): The analytical vectorial structure of non-paraxial Gaussian beam close to the source, Opt. Express 16, 3504–3514.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Rosario Martínez-Herrero
    • 1
    Email author
  • Pedro M. Mejías
    • 1
  • Gemma Piquero
    • 1
  1. 1.Optics DepartmentUniversidad Complutense de MadridMadridSpain

Personalised recommendations