Abstract

A right-handed helical nonorthogonal coordinate system is used to determine helical localized wave solutions of the homogeneous scalar wave equation. Introducing the characteristic variables in the helical system, i.e., u=ζ-ct and v=ζ+ct, where ζ is the coordinate along the helical axis, we can use the bidirectional traveling plane wave representation and obtain sets of elementary bidirectional helical solutions to the wave equation. Not only are these sets bidirectional, i.e., based on a product of plane waves, but they may also be broken up into right-handed and left-handed solutions. The elementary helical solutions may in turn be used to create general superpositions, both Fourier and bidirectional, from which new solutions to the wave equation may be synthesized. These new solutions, based on the helical bidirectional superposition, are members of the class of localized waves. Examples of these new solutions are a helical fundamental Gaussian focus wave mode, a helical Bessel–Gauss pulse, and a helical acoustic directed energy pulse train. Some of these solutions have the interesting feature that their shape and localization properties depend not only on the wave number governing propagation along the longitudinal axis but also on the normalized helical pitch.

© 2001 Optical Society of America

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  1. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [CrossRef]
  2. P. A. Belanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [CrossRef]
  3. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [CrossRef]
  4. P. Hillion, “Solutions of Maxwell’s equations with boundary conditions on the hyperplane z-ct=0,” J. Math. Phys. 29, 2219–2222 (1988).
    [CrossRef]
  5. P. Hillion, “Some exotic solutions of the wave equation in unbounded isotropic media,” Wave Motion 10, 143–147 (1988).
    [CrossRef]
  6. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  7. I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [CrossRef]
  8. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
    [CrossRef]
  9. P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
    [CrossRef] [PubMed]
  10. R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
    [CrossRef]
  11. P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
    [CrossRef]
  12. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  13. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  14. V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen 27, 6243–6252 (1994).
    [CrossRef]
  15. P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
    [CrossRef]
  16. V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).
  17. P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory, (International Union of Radio Science, Gent, Belgium, 1998), Vol. II, pp. 802–804.
  18. P. L. Overfelt, “An approximate Bessel–Gauss pulse generated from a disk source moving more slowly than the speed of light,” J. Opt. Soc. Am. A 16, 2239–2244 (1999).
    [CrossRef]
  19. A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990).
    [CrossRef]
  20. S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
    [CrossRef]
  21. H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
    [CrossRef]
  22. W. Sichak, “Coaxial line with helical inner conductor,” Proc. IRE 42, 1315–1319 (1954).
    [CrossRef]
  23. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).
  24. B. T. Hefner, P. M. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999).
    [CrossRef]
  25. D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
    [CrossRef]
  26. R. A. Waldron, “A helical coordinate system and its applications in electromagnetic theory,” Q. J. Mech. Appl. Math. 11, 438–461 (1958).
    [CrossRef]
  27. P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992).
    [CrossRef] [PubMed]
  28. P. L. Overfelt, “Helical focus wave modes and helical nondiffracting beams,” J. Acoust. Soc. Am. 107, 2782 (2000).
    [CrossRef]
  29. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  30. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]

2000

S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
[CrossRef]

P. L. Overfelt, “Helical focus wave modes and helical nondiffracting beams,” J. Acoust. Soc. Am. 107, 2782 (2000).
[CrossRef]

1999

B. T. Hefner, P. M. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999).
[CrossRef]

P. L. Overfelt, “An approximate Bessel–Gauss pulse generated from a disk source moving more slowly than the speed of light,” J. Opt. Soc. Am. A 16, 2239–2244 (1999).
[CrossRef]

1997

D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
[CrossRef]

P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
[CrossRef]

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

1994

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen 27, 6243–6252 (1994).
[CrossRef]

1993

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

1992

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992).
[CrossRef] [PubMed]

1991

P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

1990

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

1989

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

1988

P. Hillion, “Solutions of Maxwell’s equations with boundary conditions on the hyperplane z-ct=0,” J. Math. Phys. 29, 2219–2222 (1988).
[CrossRef]

P. Hillion, “Some exotic solutions of the wave equation in unbounded isotropic media,” Wave Motion 10, 143–147 (1988).
[CrossRef]

1985

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

1984

1983

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

1958

R. A. Waldron, “A helical coordinate system and its applications in electromagnetic theory,” Q. J. Mech. Appl. Math. 11, 438–461 (1958).
[CrossRef]

1954

W. Sichak, “Coaxial line with helical inner conductor,” Proc. IRE 42, 1315–1319 (1954).
[CrossRef]

Agius, A. A.

S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
[CrossRef]

Altintas, A.

A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

Belanger, P. A.

Besieris, I. M.

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

Borisov, V. V.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen 27, 6243–6252 (1994).
[CrossRef]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Donnelly, R.

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Flatters, D.

D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Hefner, B. T.

B. T. Hefner, P. M. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999).
[CrossRef]

Hillion, P.

P. Hillion, “Solutions of Maxwell’s equations with boundary conditions on the hyperplane z-ct=0,” J. Math. Phys. 29, 2219–2222 (1988).
[CrossRef]

P. Hillion, “Some exotic solutions of the wave equation in unbounded isotropic media,” Wave Motion 10, 143–147 (1988).
[CrossRef]

Kenney, C. S.

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory, (International Union of Radio Science, Gent, Belgium, 1998), Vol. II, pp. 802–804.

Kitamura, Y.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).

Lavery, R.

D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
[CrossRef]

Leach, S. M.

S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
[CrossRef]

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Love, J. D.

A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

Marston, P. M.

B. T. Hefner, P. M. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999).
[CrossRef]

Mimaki, H.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

Nakano, H.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

Overfelt, P. L.

P. L. Overfelt, “Helical focus wave modes and helical nondiffracting beams,” J. Acoust. Soc. Am. 107, 2782 (2000).
[CrossRef]

P. L. Overfelt, “An approximate Bessel–Gauss pulse generated from a disk source moving more slowly than the speed of light,” J. Opt. Soc. Am. A 16, 2239–2244 (1999).
[CrossRef]

P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
[CrossRef]

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992).
[CrossRef] [PubMed]

P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef] [PubMed]

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory, (International Union of Radio Science, Gent, Belgium, 1998), Vol. II, pp. 802–804.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Saunders, S. R.

S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
[CrossRef]

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

Shaarawi, A. M.

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

Sichak, W.

W. Sichak, “Coaxial line with helical inner conductor,” Proc. IRE 42, 1315–1319 (1954).
[CrossRef]

Simonenko, I. I.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen 27, 6243–6252 (1994).
[CrossRef]

Takeda, T.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

Waldron, R. A.

R. A. Waldron, “A helical coordinate system and its applications in electromagnetic theory,” Q. J. Mech. Appl. Math. 11, 438–461 (1958).
[CrossRef]

Yamauchi, J.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

Zakrzewska, K.

D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[CrossRef]

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Can. J. Phys.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

IEE Proc. H Microw. Antennas Propag.

S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000).
[CrossRef]

IEEE Trans. Antennas Propag.

H. Nakano, T. Takeda, Y. Kitamura, H. Mimaki, J. Yamauchi, “Low profile helical array antenna fed from a radial waveguide,” IEEE Trans. Antennas Propag. 40, 279–284 (1992).
[CrossRef]

J. Acoust. Soc. Am.

B. T. Hefner, P. M. Marston, “An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems,” J. Acoust. Soc. Am. 106, 3313–3316 (1999).
[CrossRef]

P. L. Overfelt, “Helical focus wave modes and helical nondiffracting beams,” J. Acoust. Soc. Am. 107, 2782 (2000).
[CrossRef]

J. Appl. Phys.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[CrossRef]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[CrossRef]

J. Comp. Chem.

D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997).
[CrossRef]

J. Math. Phys.

P. Hillion, “Solutions of Maxwell’s equations with boundary conditions on the hyperplane z-ct=0,” J. Math. Phys. 29, 2219–2222 (1988).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A Math. Gen

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen 27, 6243–6252 (1994).
[CrossRef]

Opt. Quantum Electron.

A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990).
[CrossRef]

Phys. Rev. A

P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

Phys. Rev. Lett.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[CrossRef] [PubMed]

Proc. IRE

W. Sichak, “Coaxial line with helical inner conductor,” Proc. IRE 42, 1315–1319 (1954).
[CrossRef]

Proc. R. Soc. London Ser. A

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Q. J. Mech. Appl. Math.

R. A. Waldron, “A helical coordinate system and its applications in electromagnetic theory,” Q. J. Mech. Appl. Math. 11, 438–461 (1958).
[CrossRef]

Wave Motion

P. Hillion, “Some exotic solutions of the wave equation in unbounded isotropic media,” Wave Motion 10, 143–147 (1988).
[CrossRef]

Other

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory, (International Union of Radio Science, Gent, Belgium, 1998), Vol. II, pp. 802–804.

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (2)

Fig. 1
Fig. 1

Nonorthogonal helical coordinate system (see Ref. 26).

Fig. 2
Fig. 2

Helical fundamental Gaussian focus-wave mode (t=0,β=1,a1=1,ϕ=0).

Equations (46)

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

ρ=(x2+y2)1/2,
ϕ=tan-1(y/x),
ζ=z-p¯ tan-1(y/x),
2=2ρ2+1ρρ+1ρ22ϕ2+1+p¯2ρ22ζ2-2p¯ρ22ζϕ.
2-2(ct)2ψ(ρ, t)=0,
ψ1(ρ, t)=[A1J(μ-pk¯ζ)(κρ)+B1J-(μ-pk¯ζ)(κρ)]×exp[i(-μϕ-kζζ+ωt)]
κ2=(ω/c)2-kζ2.
ψ2(ρ, t)=[A2J(μ+pk¯ζ)(κρ)+B2J-(μ+pk¯ζ)(κρ)]×exp[i(μϕ-kζζ+ωt)],
u=ζ-ct;v=ζ+ct
Φ1(1)(ρ, t)=[C1Jq(κρ)+D1J-q(κρ)]×exp[i(-μϕ-αu+βv)],
q2=[μ-p¯(α-β)]2,
κ2=4αβ.
Φ2(1)(ρ, t)=[C2Jq(κρ)+D2J-q(κρ)]×exp[i(μϕ-αu+βv)],
q2=[μ+p¯(α-β)]2
Φ1(2)(ρ, t)=Jμ+p¯β(κρ)exp{i[-ϕ(μ+p¯α)-αu+βv]},
Φ2(2)(ρ, t)=Jμ-p¯β(κρ)exp{i[ϕ(μ-p¯α)-αu+βv]},
Φ1(3)(ρ, t)=Jμ-p¯α(κρ)exp{i[-ϕ(μ-p¯β)-αu+βv]},
Φ2(3)(ρ, t)=Jμ+p¯α(κρ)exp{i[ϕ(μ+p¯β)-αu+βv]},
Φ1(4)(ρ, t)=Jμ(κρ)exp(i{ϕ[μ-p¯(α-β)]-αu+βv}),
Φ2(4)(ρ, t)=Jμ(κρ)exp(i{ϕ[μ+p¯(α-β)]-αu+βv}),
Φ(1)(ρ, t)=Jp¯(α-β)(κρ)exp[i(-αu+βv)],
Φ(2)(ρ, t)=Jp¯β(κρ)exp[i(-p¯αϕ-αu+βv)],
Φ(3)(ρ, t)=Jp¯α(κρ)exp[i(p¯βϕ-αu+βv)],
Φ(4)(ρ, t)=J0(κρ)exp{i[p¯(α-β)ϕ-αu+βv]},
Ψ1(ρ, t)=1(2π)20dμ0dκ-dω-dkζ×A(μ, ω, kζ, κ)κJμ-pk¯ζ(κρ)×exp[i(-μϕ-kζζ+ωt)]×δ(ω2/c2-κ2-κζ2)
Ψ1(ρ, ϕ, u, v)=1(2π)20dμ0dκ0dα0dβ×B(μ, α, β, κ)κJμ-p¯(α-β)(κρ)×exp[i(-μϕ-αu+βv)]×δ(αβ-κ2/4),
Ψ2(ρ, ϕ, u, v)=1(2π)20dμ0dκ0dα0dβ×B(μ, α, β, κ)κJμ+p¯β(κρ)×exp{i[-ϕ(μ+p¯α)-αu+βv]}δ(αβ-κ2/4).
B1(α, β, κ)=π2 δ(β-β)κσexp(-a1α),
f1(σ)(ρ, ϕ, u, v)=18π0dκ0dβ κσ+1β Jp¯β(κρ)×exp-κ24β [a1+i(u+pϕ¯)]×exp(iβv)δ(β-β).
f1(σ)(ρ, ϕ, u, v)=18πβ ρp¯βΓp¯β+σ+22×1F1[(p¯β+σ+2)/2; p¯β+1; -ρ2/4ξ]2p¯β+1ξ(σ+p¯β+2)/2Γ(p¯β+1),
Re[ξ]>0,Re[p¯β+σ+2] > -1.
ξ=a1+i(u+pϕ¯)4β.
σ=p¯β.
f1(p¯β)(ρ, ϕ, u, v)=(2βρ)p¯β4πexp[-β(ρ2/Vϕ-iv)]Vϕp¯β+1,
Vϕ=a1+i(u+pϕ¯).
limp¯0f1(p¯β)(ρ, ϕ, u, v)=14πV exp[-β(ρ2/V-iv)],
VϕV=a1+iu,ζ  z.
B2(α, β, κ)=π2 δ(β-β) exp(-a1α)Iν(κa2),
π(ρ, ϕ, u, v)=18π0dκ0dβκβ Jp¯β(κρ)Iν(κa2)×exp(-ξκ2+iβv)δ(β-β),
π(ρ, ϕ, u, v)=1(4π)2βξ Jp¯βa2ρ2ξ×expa22-ρ24ξexp(iβv).
a2=κa12β,
π(ρ, ϕ, u, v)=14πVϕ Jp¯βκa1ρVϕ×exp-βρ2Vϕ-ivexp(κa1)24βVϕ,
F(β)=4πiγ(γβ-b)a-1βp¯βΓ(a)exp[-a(γβ-b)]Hβ-bγ.
g(ρ, t)=14πiVϕ0dβF(β) (2βρ)ρβ¯exp[-β(ρ2/Vϕ-iv)]Vϕp¯β,
g(ρ, t)=1Vϕexp{-b/γ[s(ρ)+p¯ ln(Vϕ/2ρ)]}s(ρ)/γ+p¯γ ln(Vϕ/2ρ)+aa,
s(ρ)=ρ2/Vϕ-i(ζ+ct).

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