## Helical localized wave solutions of the scalar wave equation

JOSA A, Vol. 18, Issue 8, pp. 1905-1911 (2001)

http://dx.doi.org/10.1364/JOSAA.18.001905

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### 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., *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

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(260.2110) Physical optics : Electromagnetic optics

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: September 22, 2000

Manuscript Accepted: January 24, 2001

Published: August 1, 2001

**Citation**

P. L. Overfelt, "Helical localized wave solutions of the scalar wave equation," J. Opt. Soc. Am. A **18**, 1905-1911 (2001)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-8-1905

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### References

- J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983). [CrossRef]
- P. A. Belanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984). [CrossRef]
- A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985). [CrossRef]
- 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]
- 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]
- 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]
- P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991). [CrossRef] [PubMed]
- 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]
- P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993). [CrossRef]
- R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991). [CrossRef] [PubMed]
- 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]
- 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]
- 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).
- 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.
- 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]
- A. Altintas, J. D. Love, “Effective cutoffs for modes on helical fibers,” Opt. Quantum Electron. 22, 213–226 (1990). [CrossRef]
- S. M. Leach, A. A. Agius, S. R. Saunders, “Intelligent quadrifilar helix antenna,” IEE Proc. H Microw. Antennas Propag. 147, 219–223 (2000). [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]
- W. Sichak, “Coaxial line with helical inner conductor,” Proc. IRE 42, 1315–1319 (1954). [CrossRef]
- A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).
- 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]
- D. Flatters, K. Zakrzewska, R. Lavery, “Internal coordinate modeling of DNA: force field comparisons,” J. Comp. Chem. 18, 1043–1055 (1997). [CrossRef]
- R. A. Waldron, “A helical coordinate system and its applications in electromagnetic theory,” Q. J. Mech. Appl. Math. 11, 438–461 (1958). [CrossRef]
- P. L. Overfelt, “Scalar optical beams with helical symmetry,” Phys. Rev. A 46, 3516–3522 (1992). [CrossRef] [PubMed]
- P. L. Overfelt, “Helical focus wave modes and helical nondiffracting beams,” J. Acoust. Soc. Am. 107, 2782 (2000). [CrossRef]
- I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
- R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989). [CrossRef] [PubMed]

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