## Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space

JOSA A, Vol. 24, Issue 10, pp. 3317-3325 (2007)

http://dx.doi.org/10.1364/JOSAA.24.003317

Enhanced HTML Acrobat PDF (197 KB)

### Abstract

Starting from the vector Maxwell’s equations and applying the multiscale singular perturbation method, the nonparaxial beam propagation is studied in free space. Two new equations have been derived for transverse and longitudinal electric fields of an arbitrary polarized electromagnetic wave even in the case of tightly focused nonparaxial laser beams. By using the analogy of the optical beam in the space domain and the optical pulse in the time domain, the higher-order diffraction term is introduced. For strongly nonparaxial beams that are characterized by large values of the perturbative parameter, our correction solutions yield an accurate description of the field in the near-field region and are consistent with all other correction results obtained by others in the far-field region. For weakly nonparaxial beams, our correction solutions can be expressed in a very simple form that is proved to be exactly consistent with the solutions obtained by Cao and Deng [J. Opt. Soc. Am. A 15, 1144 (1998) ]. In addition, the lowest-order correction to the paraxial approximation can be found to be in good agreement with the results of Lax et al. [Phys. Rev. A 11, 1365 (1975) ] and Seshadri [J. Opt. Soc. Am. A 19, 2134 (2002) ].

© 2007 Optical Society of America

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 22, 2007

Revised Manuscript: June 26, 2007

Manuscript Accepted: June 26, 2007

Published: September 24, 2007

**Citation**

Dongmei Deng, Qi Guo, Sheng Lan, and Xiangbo Yang, "Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space," J. Opt. Soc. Am. A **24**, 3317-3325 (2007)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-24-10-3317

Sort: Year | Journal | Reset

### References

- L. Cicchitelli, H. Hora, and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990). [CrossRef] [PubMed]
- B. Quesnel and P. Mora, "Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum," Phys. Rev. E 58, 3719-3732 (1998). [CrossRef]
- S. Banerjee, S. Sepke, R. Shah, A. Valenzuela, A. Maksimchuk, and D. Umstadter, "Optical deflection and temporal characterization of ultrafast laser-produced electron beams," Phys. Rev. Lett. 95, 035004 (2005). [CrossRef] [PubMed]
- S. Weber, G. Riazuelo, P. Michel, R. Loubere, F. Walraet, V. T. Tikhonchuk, V. Malka, J. Ovadia, and G. Bonnaud, "Modeling of laser-plasma interaction on hydrodynamic scales: physics development and comparison with experiments," Laser Part. Beams 22, 189-195 (2004). [CrossRef]
- S. Sepke and D. Umstadter, "Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes," Opt. Lett. 31, 1447-1449 (2006). [CrossRef] [PubMed]
- S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of tightly focused laser beams of arbitrary pulse length," Opt. Lett. 31, 2589-2591 (2006). [CrossRef] [PubMed]
- S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. I. Small F-number laser focusing," J. Opt. Soc. Am. B 23, 2157-2165 (2006). [CrossRef]
- S. Sepke and D. Umstadter, "Analytical solutions for the electromagnetic fields of flattened and annular Gaussian laser modes. II. Large F-number laser focusing," J. Opt. Soc. Am. B 23, 2166-2173 (2006). [CrossRef]
- M. Lax, W. H. Louisell, W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975). [CrossRef]
- G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979). [CrossRef]
- L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979). [CrossRef]
- M. Couture and P. A. Belanger, "From Gaussian beam to complex-source-point spherical wave," Phys. Rev. A 24, 355-359 (1981). [CrossRef]
- G. P. Agrawal and M. Lax, "Free-space wave propagation beyond the paraxial approximation," Phys. Rev. A 27, 1693-1695 (1983). [CrossRef]
- T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826-829 (1985). [CrossRef]
- E. Zauderer, "Complex argument Hermite-Gaussian and Laguerre-Gaussian beams," J. Opt. Soc. Am. A 3, 465-469 (1986). [CrossRef]
- A. Wünsche, "Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765-774 (1992). [CrossRef]
- Q. Cao and X. M. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998). [CrossRef]
- S. R. Seshadri, "Nonparaxial corrections for the fundamental Gaussian beam," J. Opt. Soc. Am. A 19, 2134-2141 (2002). [CrossRef]
- R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28, 774-776 (2003). [CrossRef] [PubMed]
- R. Borghi and M. Santarsiero, "Nonparaxial propagation of spirally polarized optical beams," J. Opt. Soc. Am. A 21, 2029-2037 (2004). [CrossRef]
- D. M. Deng, "Nonparaxial propagation of radially polarized light beams," J. Opt. Soc. Am. B 23, 1228-1234 (2006). [CrossRef]
- D. M. Deng, Q. Guo, L. J. Wu, X. B. Yang, "Propagation of radially polarized elegant light beams," J. Opt. Soc. Am. B 24, 636-643 (2007). [CrossRef]
- P. Varga and P. Török, "The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation," Opt. Commun. 152, 108-118 (1998). [CrossRef]
- J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, "Fifth-order corrected field descriptions of the Hermite-Gaussian (0,0) and (0,1) mode laser beam," Phys. Rev. E 64, 066612 (2001). [CrossRef]
- Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5, 101301 (2002). [CrossRef]
- J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989). [CrossRef]
- A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, "Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation," J. Opt. Soc. Am. B 17, 809-819 (2000). [CrossRef]
- A. Ciattoni, B. Crosignani, and P. Di Porto, "Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections," Opt. Commun. 177, 9-13 (2000). [CrossRef]
- E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989). [CrossRef]
- A. H. Nayfeh, Perturbation Methods (Wiley, 1973), Chap. 6 pp. 231-243.
- Q. Guo, G. S. Zhou, W. G. Lin, S. H. Liu, C. J. Liao, and H. C. Jin, "Evolution equation of optical solitons in femtosecond regime," Sci. China A34, 1365-1377 (1991).
- Q. Guo, C. J. Liao, and S. H. Liu, "Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions," Fiber Integr. Opt. 12, 71-81 (1993). [CrossRef]
- S. Chi, and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1989), Chap. 3.
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1989), Chap. 4, 5.
- S. Mookherjea and A. Yariv, "Analysis of optical pulse propagation with two-by-two ABCD matrices," Phys. Rev. E 64, 016611 (2001). [CrossRef]
- Q. Guo, "Optical beams in media with spatial dispersion," Chin. Phys. Lasers 20, 64-67 (2003).
- D. M. Deng, H. Guo, D. A. Han, C. F. Li, "Analysis of beam propagation and pulse propagation with variance matrix treatment," Opt. Commun. 238, 205-213 (2004). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.