## Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach

JOSA A, Vol. 23, Issue 6, pp. 1442-1451 (2006)

http://dx.doi.org/10.1364/JOSAA.23.001442

Enhanced HTML Acrobat PDF (156 KB)

### Abstract

We present an *ab initio* account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam’s elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

© 2006 Optical Society of America

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(080.2710) Geometric optics : Inhomogeneous optical media

(080.2720) Geometric optics : Mathematical methods (general)

**History**

Original Manuscript: June 20, 2005

Revised Manuscript: November 29, 2005

Manuscript Accepted: December 8, 2005

**Citation**

Pawel Berczynski, Konstantin Yu. Bliokh, Yuri A. Kravtsov, and Andrzej Stateczny, "Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach," J. Opt. Soc. Am. A **23**, 1442-1451 (2006)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-23-6-1442

Sort: Year | Journal | Reset

### References

- Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Medium (Springer-Verlag, 1990). [CrossRef]
- Yu. A. Kravtsov, Geometrical Optics in Engineering Physics (Alpha Science, 2004).
- Yu. A. Kravtsov, G. W. Forbes, and A. A. Asatryan, "Theory and applications of complex rays," in Progress in Optics , Vol. XXXIX, E.Wolf, ed. (Elsevier, 1999), pp. 3-62.
- S. J. Chapman, J. M. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999). [CrossRef]
- Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967). [CrossRef]
- J. B. Keller and W. Streifer, "Complex rays with application to Gaussian beams," J. Opt. Soc. Am. 61, 40-43 (1971). [CrossRef]
- G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971). [CrossRef]
- R. A. Egorchenkov and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000). [CrossRef]
- R. A. Egorchenkov and Yu. A. Kravtsov, "Complex ray-tracing algorithms with application to optical problems," J. Opt. Soc. Am. A 18, 650-656 (2001). [CrossRef]
- R. A. Egorchenkov, "Wave fields in inhomogeneous media numerically calculated on the basis of complex geometrical optics," Phys. Vib. 8, 122-127 (2000).
- V. M. Babich, "Eigenfunctions, concentrated in the vicinity of closed geodesic," in Mathematical Problems of Theory of Waves Propagation (Nauka, 1968), Vol. 9, pp. 15-63 (in Russian).
- V. M. Babich and N. J. Kirpichnikova, Boundary Layer Method in Diffraction Problems (Leningrad U. Press, 1974, in Russian; Springer-Verlag, 1980).
- V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Nauka, 1972, in Russian; Springer-Verlag, 1990).
- V. M. Babich and V. S. Buldyrev, Asymptotic Methods for Short-Wavelength Theory (Alpha Science, 2005).
- M. M. Popov and I. Psencik, "Computation of ray amplitudes in inhomogeneous media with curved interfaces," Stud. Geophys. Geod. 22, 248-258 (1978). [CrossRef]
- V. Cervený, Seismic Ray Theory (Cambridge U. Press, 2001). [CrossRef]
- M. M. Popov, Ray Theory and Gaussian Beam Method for Geophysicists (Editora da Universidade Federal da Bahia, 2002).
- M. M. Popov, "New method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-95 (1982). [CrossRef]
- V. M. Babich and M. M. Popov, "Gaussian summation method," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 32, 1447-1466 (1990) V. M. Babichc and M. M. Popov,[Radiophys. Quantum Electron. 32, 1063-1081 (1990)].
- V. Cervený, M. M. Popov, and I. Psencik, "Computation of wave fields in inhomogeneous media—Gaussian beam approach," Geophys. J. R. Astron. Soc. 70, 109-128 (1982).
- V. Cervený, "Synthetic body wave seismograms for laterally varying layered structures by the Gaussian beam method," Geophys. J. R. Astron. Soc. 73, 389-426 (1983).
- R. L. Nowack, "Calculation of synthetic seismograms with Gaussian beams," Pure Appl. Geophys. 160, 487-507 (2003). [CrossRef]
- G. V. Pereverzev, "Paraxial WKB solution of a scalar wave equation," in Reviews of Plasma Physics, B. B. Kadomtsev, ed. (Consultants Bureau, New York, 1996), Vol. 19, pp. 1-51.
- G. V. Pereverzev, "Beam tracing in inhomogeneous anisotropic plasmas," Phys. Plasmas 5, 3529-3541 (1998). [CrossRef]
- E. Poli, G. V. Pereverzev, A. G. Peeters, and M. Bornatici, "EC beam tracing in fusion plasmas," Fusion Eng. Des. 53, 9-21 (2001). [CrossRef]
- S. Choudhary and L. B. Felsen, "Asymptotic theory for inhomogeneous waves," IEEE Trans. Antennas Propag. AP-21, 827-842 (1973). [CrossRef]
- S. Choudhary and L. B. Felsen, "Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking," Proc. IEEE 62, 1530-1542 (1974). [CrossRef]
- S. Nowak and A. Orefice, "Quasi-optical treatment of electromagnetic Gaussian beams in inhomogeneous and anisotropic plasmas," Phys. Fluids B 5, 1945-1954(1993). [CrossRef]
- E. Heyman and L. B. Felsen, "Gaussian beam and pulsed-beam dynamics: complex source and complex spectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. A 18, 1588-1611 (2001). [CrossRef]
- A. B. Timofeev, "Gaussian wave beams in inhomogeneous media," Plasma Phys. Rep. 21, 610-611 (1995) A. B. Timofeev,[translated from Fiz. Plazmy 21, 646-647 (1995)].
- A. B. Timofeev, "Geometrical optics and the diffraction phenomenon," Phys. Usp. 48, 609-613 (2005) A. B. Timofeev,[translated from Usp. Fiz. Nauk 175, 637-641 (2005)]. [CrossRef]
- M. Bornatici and O. Maj, "Wave beam propagation in a weakly inhomogeneous isotropic medium: paraxial approximation and beyond," Plasma Phys. Controlled Fusion 45, 707-719 (2003). [CrossRef]
- Yu. A. Kravtsov and P. Berczynski, "Description of the 2D Gaussian beam diffraction in a free space in frame of eikonal-based complex geometrical optics," Wave Motion 40, 23-27 (2004). [CrossRef]
- P. Berczynski and Yu. A. Kravtsov, "Theory for Gaussian beam diffraction in 2D inhomogeneous medium, based on the eikonal form of complex geometrical optics," Phys. Rev. A 331, 265-268 (2004).
- Yu. A. Kravtsov and P. Berczynski, "Gaussian beam diffraction in inhomogeneous media: solution in the framework of paraxial complex geometrical optics," submitted to Stud. Geophys. Geod.
- H. Rund, The Differential Geometry of Finsler Spaces (Springer-Verlag, 1959).
- G. V. Permitin and A. I. Smirnov, "Quasioptics of smoothly inhomogeneous isotropic media," Zh. Eksp. Teor. Fiz. 109, 736-751 (1996) G. V. Permitin and A. I. Smirnov,[JETP 83, 395-402 (1996)].
- G. V. Permitin and A. I. Smirnov, "Wave packets in smoothly inhomogeneous dispersive media," Zh. Eksp. Teor. Fiz. 119, 16-30 (2001) G. V. Permitin and A. I. Smirnov,[JETP 92, 10-19 (2001)].
- S. M. Rytov, "On the transition from wave to geometrical optics," Dokl. Akad. Nauk SSSR 18, 263-267 (1938); reprinted in Topological Phase in Quantum Theory, B.L.Markovski and S.I.Vinitsky, eds. (World Scientific, 1989).
- S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160, 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)]. [CrossRef]
- J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687-1694 (1969). [CrossRef] [PubMed]
- H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966). [CrossRef]
- J. A. Arnaud, Beam and Fiber Optics (Academic, 1976).

## 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.