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

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  7. G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
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  8. R. A. Egorchenkov and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000).
    [CrossRef]
  9. 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).
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  10. R. A. Egorchenkov, "Wave fields in inhomogeneous media numerically calculated on the basis of complex geometrical optics," Phys. Vib. 8, 122-127 (2000).
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  12. V. M. Babich and N. J. Kirpichnikova, Boundary Layer Method in Diffraction Problems (Leningrad U. Press, 1974, in Russian; Springer-Verlag, 1980).
  13. V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Nauka, 1972, in Russian; Springer-Verlag, 1990).
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  19. 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)].
  20. 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).
  21. 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).
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  28. 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]
  29. 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).
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  32. 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]
  33. 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]
  34. 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).
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  38. 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)].
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  40. 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)].
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2005 (1)

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]

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]

2004 (2)

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).

2003 (2)

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]

R. L. Nowack, "Calculation of synthetic seismograms with Gaussian beams," Pure Appl. Geophys. 160, 487-507 (2003).
[CrossRef]

2001 (4)

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]

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]

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]

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)].

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)].

2000 (2)

R. A. Egorchenkov, "Wave fields in inhomogeneous media numerically calculated on the basis of complex geometrical optics," Phys. Vib. 8, 122-127 (2000).

R. A. Egorchenkov and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000).
[CrossRef]

1999 (1)

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]

1998 (1)

G. V. Pereverzev, "Beam tracing in inhomogeneous anisotropic plasmas," Phys. Plasmas 5, 3529-3541 (1998).
[CrossRef]

1996 (1)

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, "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)].

1995 (1)

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, "Gaussian wave beams in inhomogeneous media," Plasma Phys. Rep. 21, 610-611 (1995) A. B. Timofeev,[translated from Fiz. Plazmy 21, 646-647 (1995)].

1993 (1)

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]

1990 (2)

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]

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]

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. 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)].

1983 (1)

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).

1982 (2)

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).

M. M. Popov, "New method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-95 (1982).
[CrossRef]

1978 (1)

M. M. Popov and I. Psencik, "Computation of ray amplitudes in inhomogeneous media with curved interfaces," Stud. Geophys. Geod. 22, 248-258 (1978).
[CrossRef]

1974 (1)

S. Choudhary and L. B. Felsen, "Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking," Proc. IEEE 62, 1530-1542 (1974).
[CrossRef]

1973 (1)

S. Choudhary and L. B. Felsen, "Asymptotic theory for inhomogeneous waves," IEEE Trans. Antennas Propag. AP-21, 827-842 (1973).
[CrossRef]

1971 (2)

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]

1969 (1)

1967 (1)

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

1966 (1)

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

1938 (1)

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).

Arnaud, J. A.

Asatryan, A. A.

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.

Babich, V. M.

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. 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, Asymptotic Methods for Short-Wavelength Theory (Alpha Science, 2005).

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Nauka, 1972, in Russian; Springer-Verlag, 1990).

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).

Babichc, V. M.

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)].

Berczynski, P.

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.

Bornatici, M.

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]

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]

Buldyrev, V. S.

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).

Cervený, V.

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).

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, Seismic Ray Theory (Cambridge U. Press, 2001).
[CrossRef]

Chapman, S. J.

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]

Choudhary, S.

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. Choudhary and L. B. Felsen, "Asymptotic theory for inhomogeneous waves," IEEE Trans. Antennas Propag. AP-21, 827-842 (1973).
[CrossRef]

Debrov, V. L.

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]

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]

Deschamps, G. A.

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Dubovik, V. M.

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]

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]

Egorchenkov, R. A.

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).

R. A. Egorchenkov and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000).
[CrossRef]

Felsen, L. B.

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]

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. Choudhary and L. B. Felsen, "Asymptotic theory for inhomogeneous waves," IEEE Trans. Antennas Propag. AP-21, 827-842 (1973).
[CrossRef]

Forbes, G. W.

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.

Heyman, E.

Keller, J. B.

Kirpichnikova, N. J.

V. M. Babich and N. J. Kirpichnikova, Boundary Layer Method in Diffraction Problems (Leningrad U. Press, 1974, in Russian; Springer-Verlag, 1980).

Kogelnik, H.

Kravtsov, Yu. A.

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).

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 and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000).
[CrossRef]

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

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.

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.

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).

Lawry, J. M.

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]

Li, T.

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Maj, O.

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]

Markovski, B. L.

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]

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]

Nowack, R. L.

R. L. Nowack, "Calculation of synthetic seismograms with Gaussian beams," Pure Appl. Geophys. 160, 487-507 (2003).
[CrossRef]

Nowak, S.

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]

Ockendon, J. R.

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]

Orefice, A.

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]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Medium (Springer-Verlag, 1990).
[CrossRef]

Peeters, A. G.

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]

Pereverzev, G. V.

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]

G. V. Pereverzev, "Beam tracing in inhomogeneous anisotropic plasmas," Phys. Plasmas 5, 3529-3541 (1998).
[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.

Permitin, G. V.

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)].

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)].

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, "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)].

Poli, E.

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]

Popov, M. M.

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. 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)].

M. M. Popov, "New method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-95 (1982).
[CrossRef]

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).

M. M. Popov and I. Psencik, "Computation of ray amplitudes in inhomogeneous media with curved interfaces," Stud. Geophys. Geod. 22, 248-258 (1978).
[CrossRef]

M. M. Popov, Ray Theory and Gaussian Beam Method for Geophysicists (Editora da Universidade Federal da Bahia, 2002).

Psencik, I.

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).

M. M. Popov and I. Psencik, "Computation of ray amplitudes in inhomogeneous media with curved interfaces," Stud. Geophys. Geod. 22, 248-258 (1978).
[CrossRef]

Rund, H.

H. Rund, The Differential Geometry of Finsler Spaces (Springer-Verlag, 1959).

Rytov, S. M.

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).

Smirnov, A. I.

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)].

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)].

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, "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)].

Stepanovskii, Yu. P.

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]

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]

Streifer, W.

Tew, R. H.

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]

Timofeev, A. B.

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]

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]

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, "Gaussian wave beams in inhomogeneous media," Plasma Phys. Rep. 21, 610-611 (1995) A. B. Timofeev,[translated from Fiz. Plazmy 21, 646-647 (1995)].

Vinitsky, S. I.

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]

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]

Appl. Opt. (1)

Dokl. Akad. Nauk SSSR (1)

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).

Electron. Lett. (1)

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Fusion Eng. Des. (1)

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]

Geophys. J. R. Astron. Soc. (2)

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).

IEEE Trans. Antennas Propag. (1)

S. Choudhary and L. B. Felsen, "Asymptotic theory for inhomogeneous waves," IEEE Trans. Antennas Propag. AP-21, 827-842 (1973).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

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)].

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Phys. Fluids B (1)

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]

Phys. Plasmas (1)

G. V. Pereverzev, "Beam tracing in inhomogeneous anisotropic plasmas," Phys. Plasmas 5, 3529-3541 (1998).
[CrossRef]

Phys. Rev. A (1)

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).

Phys. Usp. (1)

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]

Phys. Vib. (1)

R. A. Egorchenkov, "Wave fields in inhomogeneous media numerically calculated on the basis of complex geometrical optics," Phys. Vib. 8, 122-127 (2000).

Plasma Phys. Controlled Fusion (1)

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]

Plasma Phys. Rep. (1)

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)].

Proc. IEEE (2)

S. Choudhary and L. B. Felsen, "Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking," Proc. IEEE 62, 1530-1542 (1974).
[CrossRef]

H. Kogelnik and T. Li, "Laser beams and resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Pure Appl. Geophys. (1)

R. L. Nowack, "Calculation of synthetic seismograms with Gaussian beams," Pure Appl. Geophys. 160, 487-507 (2003).
[CrossRef]

Radiophys. Quantum Electron. (2)

R. A. Egorchenkov and Yu. A. Kravtsov, "Numerical realization of complex geometrical optics method," Radiophys. Quantum Electron. 43, 512-517 (2000).
[CrossRef]

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

SIAM Rev. (1)

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]

Stud. Geophys. Geod. (1)

M. M. Popov and I. Psencik, "Computation of ray amplitudes in inhomogeneous media with curved interfaces," Stud. Geophys. Geod. 22, 248-258 (1978).
[CrossRef]

Usp. Fiz. Nauk (1)

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]

Wave Motion (2)

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]

M. M. Popov, "New method of computation of wave fields using Gaussian beams," Wave Motion 4, 85-95 (1982).
[CrossRef]

Zh. Eksp. Teor. Fiz. (2)

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)].

Other (13)

J. A. Arnaud, Beam and Fiber Optics (Academic, 1976).

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).

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.

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).

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).

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.

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Figures (4)

Fig. 1
Fig. 1

Numerical results for widths w 1 , 2 versus distance τ (from 0 to 3 L R 1 , where L R 1 = π w 10 2 λ is the Rayleigh distance for smaller width) for the Gaussian beam propagating along the helical ray described in Section 6. The parameters are as follows: ϵ 0 = 1 , L = 200 λ , l φ = 1 2 , w 10 = 10 λ , w 20 = 15 λ ; the initial phase front is flat: ρ i 0 = .

Fig. 2
Fig. 2

GB phase-front curvatures ρ i 1 versus distance τ. The interval and parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Evolutions of the intensity and the phase-front cross sections (solid and dashed curves, respectively) for the GB propagating along the helical ray in the Levi–Civita (parallel-transport) coordinates. The parameters are the same as in Fig. 1; (a)–(d) correspond to the values of τ, for which the angle θ, Eqs. (48), equals 0, π 3 , 2 π 3 , π , respectively. For the given parameters’ values, this range of τ equals that in Figs. 1, 2.

Fig. 4
Fig. 4

Same as in Fig. 3 but in the coordinates attached to the normal and binormal, Eq. (47).

Equations (70)

Equations on this page are rendered with MathJax. Learn more.

μ GO λ L 1 .
μ DIF λ w 1 ,
μ REF w L 1 ,
μ GO μ DIF , μ REF .
u ( τ , ξ 1 , ξ 2 ) = A ( τ ) exp [ i k 0 ψ ( τ , ξ 1 , ξ 2 ) ] ,
ψ ( τ , ξ 1 , ξ 2 ) = ψ c ( τ ) + δ ψ ( τ , ξ 1 , ξ 2 ) ,
δ ψ ( τ , ξ 1 , ξ 2 ) = 1 2 B i j ( τ ) ξ i ξ j .
( ψ ) 2 = ϵ ,
div [ A 2 ψ ] = 0 ,
e 1 = n cos θ + b sin θ , e 2 = b cos θ n sin θ ,
h τ h = ϵ [ 1 ( ξ ) ϵ 2 ϵ ] r = r c , h ξ 1 = h ξ 2 = 1 ,
1 h 2 { ( ψ ̇ c ) 2 + ψ ̇ c B ̇ i j ξ i ξ j + ( 1 2 B ̇ i j ξ i ξ j ) 2 } + ( B 1 i ξ i ) 2 + ( B 2 i , ξ i ) 2 = ϵ .
ϵ ( r ) = ϵ c + ( ξ ) ϵ r = r c + 1 2 ( ξ ) 2 ϵ r = r c + O ( μ REF 3 ) .
ψ ̇ c = ϵ c .
d ( l ϵ ) d τ = 1 2 ϵ , d r d τ = 1 ϵ .
B ̇ i j ξ i ξ j + ( B 1 i ξ i ) 2 + ( B 2 i ξ i ) 2 = [ ( ξ ) 2 ϵ 2 3 ( ξ ϵ ) 2 4 ϵ ] r = r c .
B ̇ i j + B i k B k j = α i j ,
B ̇ 11 + ( B 11 2 + B 12 2 ) = α 11 ,
B ̇ 12 + B 12 ( B 11 + B 22 ) = α 12 ,
B ̇ 22 + ( B 22 2 + B 12 2 ) = α 22 .
α i j ( τ ) = ( 1 2 2 ϵ ξ i ξ j 3 4 ϵ ϵ ξ i ϵ ξ j ) r = r c .
R ̇ i j + R i k R k j I i k I k j = α i j ,
I ̇ i j + R i k I k j + I i k R k j = 0 ,
R ̇ 11 + R 11 2 + R 12 2 I 11 2 I 12 2 = α 11 ,
R ̇ 12 + R 12 ( R 11 + R 22 ) I 12 ( I 11 + I 22 ) = α 12 ,
R ̇ 22 + R 22 2 + R 12 2 I 22 2 I 12 2 = α 22 ,
I ̇ 11 + 2 R 11 I 11 + 2 R 12 I 12 = 0 ,
I ̇ 12 + R 12 ( I 11 + I 22 ) + I 12 ( R 11 + R 22 ) = 0 ,
I ̇ 22 + 2 R 22 I 22 + 2 R 12 I 12 = 0 .
R i = ϵ c ρ i , I i = 1 k 0 w i 2 .
w 1 , 2 2 = 2 k 0 [ I 11 + I 22 ± ( I 11 I 22 ) 2 + 4 I 12 2 ] ,
ρ 1 , 2 = 2 ϵ c R 11 + R 22 ± ( R 11 R 22 ) 2 + 4 R 12 2 .
I 11 + I 22 > ( I 11 I 22 ) 2 + 4 I 12 2 ,
2 h 2 ψ τ d A d τ + [ 1 h τ ( 1 h ψ τ ) + 2 ψ ξ 1 2 + 2 ψ ξ 2 2 ] A = 0 .
A ̃ ̇ + 1 2 Tr B i j A ̃ = 0 ,
A ̃ ( τ ) = A ̃ 0 exp [ 1 2 Tr B i j d τ ] ,
A ̃ = A ̃ 0 exp [ 1 2 Tr B i j d τ ] .
D ̇ + 2 Tr R i j D = 0 .
Tr R i j d τ = ln D D 0 ,
A ̃ 2 = A ̃ 0 2 D D 0 .
A ̃ 2 = A ̃ 0 2 w 10 w 20 w 1 w 2 .
Π = ϵ u 2 d ξ 1 d ξ 2 A ̃ 2 D A ̃ 2 w 1 w 2 = const. ,
B ̂ ( τ ) = [ B ̂ 1 ( 0 ) + I ̂ τ ] 1 ,
B i = B i 0 1 + B i 0 τ ,
A ̃ ̇ + 1 2 ( B 10 1 + B 10 τ + B 20 1 + B 20 τ ) A ̃ = 0 ,
A ̃ = A ̃ 0 ( 1 + B 10 τ ) ( 1 + B 20 τ ) = A ̃ 0 B 1 B 2 B 10 B 20 .
ρ i = τ [ 1 + ( L Ri τ ) 2 ] , w i = w i 0 1 + ( τ L Ri i ) 2 ,
w i w i 0 = const. , ρ i L Ri 2 τ .
w i w i 0 L Ri τ , ρ i τ .
A A 0 = const. ,
A k 0 w 10 w 20 τ A 0 = w 10 w 20 w 1 w 2 A 0 .
B ̇ i + B i 2 = α i ,
B i = α i C i e 2 α i τ 1 C i e 2 α i τ + 1 , C i = α i + B i 0 α i B i 0 ,
w i w i 0 2 1 + ( L i L Ri ) 2 e τ L i .
τ L i min ( ln μ REF 1 , ln μ DIF 1 ) .
B i = 1 L i i ( L i L Ri ) tan ( τ L i ) i ( L i L Ri ) tan ( τ L i ) + 1 .
w i = w i 0 1 + [ ( L i L Ri ) 2 1 ] sin 2 ( τ L i ) ,
ρ i = L i 1 + ( L i L Ri ) 2 tan 2 ( τ L i ) [ ( L i L Ri ) 2 1 ] tan ( τ L i ) .
ϵ ( r ) = ϵ 0 r 2 L 2 > 0 .
α r ϵ 2 3 ϵ 2 4 ϵ = 1 L 2 3 r 2 ϵ L 4 < 0 ,
r c = l φ 2 2 ϵ ϵ r = r c .
r c = l φ ϵ 0 1 + l φ 2 L .
χ = l z l φ r c = 1 l φ 4 ϵ 0 L = const .
n e r , b = ( l z e ϕ l φ e z ) .
e 1 = e r cos θ ( l z e φ l φ e z ) sin θ ,
e 2 = ( l z e φ l φ e z ) cos θ + e r sin θ .
θ = 1 l φ 2 L τ
α i j = ( e i e r ) ( e j e r ) [ ϵ 2 3 ( ϵ ) 2 4 ϵ ] .
α 11 = 1 + 3 l φ 2 L 2 cos 2 θ , α 22 = 1 + 3 l φ 2 L 2 sin 2 θ ,
α 12 = α 21 = 1 + 3 l φ 2 L 2 sin θ cos θ .

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