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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 23, Iss. 4 — Apr. 1, 2006
  • pp: 788–799

Complex source description of focal regions

Cesar Monzon, Donald W. Forester, and Peter Moore  »View Author Affiliations

JOSA A, Vol. 23, Issue 4, pp. 788-799 (2006)

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Closed-form solutions of the two-dimensional homogeneous wave equation are presented that provide focal-region descriptions corresponding to a converging bundle of rays. The solutions do have evanescent wave content and can be described as a source–sink pair or particle–antiparticle pair, collocated in complex space, with the complex location being critical in the determination of beam shape and focal region size. The wave solutions are not plagued by singularities, have a finite energy, and have a limitation on how small the focal size can get, with a penalty for limiting small spot sizes in the form of impractically high associated reactive energy. The electric-field-defined spot-size limiting value is 0.35 λ × 0.35 λ , which is about 38% of the Poynting-vector-defined minimum spot size ( 0.8 λ × 0.4 λ ) and corresponds to a condition related to the maximum possible beam angle. A multiple set of solutions is introduced, and the elementary solutions are used to produce new solutions via superposition, resulting in fields with chiral character or with increased depth of focus. We do not claim generality, as the size of focal regions exhibited by the closed-form solutions has a lower bound and hence is not able to account for Pendry’s “ideal lens” scenario.

© 2006 Optical Society of America

OCIS Codes
(000.6800) General : Theoretical physics
(050.1960) Diffraction and gratings : Diffraction theory
(080.2720) Geometric optics : Mathematical methods (general)
(100.6640) Image processing : Superresolution
(260.2110) Physical optics : Electromagnetic optics

Original Manuscript: May 12, 2005
Revised Manuscript: September 27, 2005
Manuscript Accepted: September 29, 2005

Cesar Monzon, Donald W. Forester, and Peter Moore, "Complex source description of focal regions," J. Opt. Soc. Am. A 23, 788-799 (2006)

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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. A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985). [CrossRef]
  3. P. A. Belanger, "Packetlike solutions of the homogeneous wave equation," J. Opt. Soc. Am. A 1, 723-724 (1984). [CrossRef]
  4. P. Hillion, "Some exotic solutions of the wave equation in unbounded isotropic media," Wave Motion 10, 143-147 (1988). [CrossRef]
  5. R. W. Ziolkowski, "Localized transmission of electromagnetic energy," Phys. Rev. A 39, 2005-2033 (1989). [CrossRef] [PubMed]
  6. P. L. Overfelt, "Helical localized wave solutions of the scalar wave equation," J. Opt. Soc. Am. A 18, 1905-1911 (2001). [CrossRef]
  7. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt. 5, 1550-1567 (1966). [CrossRef] [PubMed]
  8. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  9. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  10. J. B. Pendry and S. A. Ramakrishna, "Near field lenses in two dimensions", J. Phys.: Condens. Matter 14, 8463-8479 (2002). [CrossRef]
  11. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, "Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial," Appl. Phys. Lett. 78, 489-491 (2001). [CrossRef]
  12. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001). [CrossRef] [PubMed]
  13. F. J. Rachford, D. L. Smith, P. F. Loschialpo, and D. W. Forester, "Calculations and measurements of wire and/or split-ring negative index media," Phys. Rev. E 66, 036613 (2002). [CrossRef]
  14. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 026502 (2003). [CrossRef]
  15. P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, and C. Monzon, "Optical properties of an ideal homogeneous, causal 'left handed' material slab," Phys. Rev. E 70, 036605 (2004). [CrossRef]
  16. G. Toraldo di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-438 (1952). [CrossRef]
  17. T. Wilson, Confocal Microscopy (Academic, 1990).
  18. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).
  19. T. R. M. Sales, "Smallest focal spot," Phys. Rev. Lett. 813844-3847 (1998). [CrossRef]
  20. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987). [CrossRef] [PubMed]
  21. D. S. Jones, The Theory of Electromagnetism (MacMillan, 1964).
  22. R. F. Harrington, Time Harmonic Electromagnetic Fields, (McGraw-Hill, 1961).
  23. W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea, 1949).
  24. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).
  25. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  26. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

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