Abstract

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 0.35λ×0.35λ, which is about 38% of the Poynting-vector-defined minimum spot size (0.8λ×0.4λ) 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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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).

2004 (1)

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]

2003 (1)

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]

2002 (2)

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]

J. B. Pendry and S. A. Ramakrishna, "Near field lenses in two dimensions", J. Phys.: Condens. Matter 14, 8463-8479 (2002).
[CrossRef]

2001 (3)

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]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001).
[CrossRef] [PubMed]

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

2000 (1)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998 (1)

T. R. M. Sales, "Smallest focal spot," Phys. Rev. Lett. 813844-3847 (1998).
[CrossRef]

1989 (1)

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

1988 (1)

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

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987).
[CrossRef] [PubMed]

1985 (1)

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

1984 (1)

1983 (1)

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

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

1966 (1)

1952 (1)

G. Toraldo di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Belanger, P. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

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]

di Francia, G. Toraldo

G. Toraldo di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987).
[CrossRef] [PubMed]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Forester, D. W.

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]

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]

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]

Gradshteyn, I. S.

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

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields, (McGraw-Hill, 1961).

Hillion, P.

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

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (MacMillan, 1964).

Kogelnik, H.

Li, T.

Loschialpo, P. F.

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]

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]

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]

Magnus, W.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea, 1949).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987).
[CrossRef] [PubMed]

Monzon, C.

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]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Nemat-Nasser, S. C.

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]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea, 1949).

Overfelt, P. L.

Pendry, J. B.

J. B. Pendry and S. A. Ramakrishna, "Near field lenses in two dimensions", J. Phys.: Condens. Matter 14, 8463-8479 (2002).
[CrossRef]

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Rachford, F. J.

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]

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]

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]

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, "Near field lenses in two dimensions", J. Phys.: Condens. Matter 14, 8463-8479 (2002).
[CrossRef]

Ryzhik, I. M.

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

Sales, T. R.

T. R. M. Sales, "Smallest focal spot," Phys. Rev. Lett. 813844-3847 (1998).
[CrossRef]

Schelleng, J.

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]

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]

Schultz, S.

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]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001).
[CrossRef] [PubMed]

Sezginer, A.

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

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001).
[CrossRef] [PubMed]

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]

Smith, D. L.

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]

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]

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]

Smith, D. R.

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]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Wilson, T.

T. Wilson, Confocal Microscopy (Academic, 1990).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Ziolkowski, R. W.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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]

J. Appl. Phys. (2)

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. Opt. Soc. Am. A (2)

J. Phys.: Condens. Matter (1)

J. B. Pendry and S. A. Ramakrishna, "Near field lenses in two dimensions", J. Phys.: Condens. Matter 14, 8463-8479 (2002).
[CrossRef]

Nuovo Cimento, Suppl. (1)

G. Toraldo di Francia, "Super-gain antennas and optical resolving power," Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E (3)

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]

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]

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]

Phys. Rev. Lett. (3)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

T. R. M. Sales, "Smallest focal spot," Phys. Rev. Lett. 813844-3847 (1998).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 581499-1501 (1987).
[CrossRef] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-99 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsi and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Wave Motion (1)

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

Other (8)

T. Wilson, Confocal Microscopy (Academic, 1990).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

D. S. Jones, The Theory of Electromagnetism (MacMillan, 1964).

R. F. Harrington, Time Harmonic Electromagnetic Fields, (McGraw-Hill, 1961).

W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea, 1949).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Electric field magnitude of a complex source with ξ ¯ = 0.4 λ y ̂ and x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , for λ = 10 mm . The x and y coordinates are in millimeters. Radiation is clearly in the ξ ̂ direction. The discontinuous character of the field and the singular points at ( x ¯ x ¯ 0 ) = ± ξ v ̂ are clearly visible.

Fig. 2
Fig. 2

Electric field magnitude of a sink for ξ ¯ = 0.4 λ y ̂ and x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , where λ = 10 mm . The x and y coordinates are in millimeters.

Fig. 3
Fig. 3

Electric field magnitude of a source–sink pair for ξ ¯ = 0.4 λ y ̂ and x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , where λ = 10 mm . The x and y coordinates are in millimeters.

Fig. 4
Fig. 4

Magnitude of the Poynting vector for a source–sink pair with ξ ¯ = 0.4 λ y ̂ , x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , and λ = 10 mm .

Fig. 5
Fig. 5

Poynting vector and its magnitude in the neighborhood of the source-sink pair. Here ξ ¯ = 0.4 λ y ̂ , x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , and λ = 10 mm . The power flow corresponds to that of a focal region.

Fig. 6
Fig. 6

Amplitudes of electric field and Poynting vector for a source-sink pair with k ξ = 2 π , x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , and λ = 10 mm . The electric field is normalized per Eq. (7).

Fig. 7
Fig. 7

Amplitudes of electric field and Poynting vector for a source-sink pair with k ξ = π , x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , and λ = 10 mm . The electric field is normalized per Eq. (7).

Fig. 8
Fig. 8

Amplitudes of electric field and Poynting vector for a source-sink pair with k ξ = π 2 , x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , and λ = 10 mm . The electric field is normalized per Eq. (7).

Fig. 9
Fig. 9

Normalized amplitude of the Poynting vector in the principal planes and for different values of ξ λ . Coordinates are with respect to x ¯ 0 and in wavelengths. (a) Y-plane cut (y fixed, variable x), (b) X-plane cut (x fixed, variable y). The different curves correspond to ξ λ = 10 3 , 10 2 , 10 1 , 0.25, 0.5 and 1.

Fig. 10
Fig. 10

Amplitudes of normalized electric field and Poynting vector for a source–sink pair characterized by α = (a) 60°, (b) 90°. It is shown that as critical conditions are approached (b), the electric fields become sharper and rotationally invariant, while the Poynting vector assumes the almost limiting 0.8 λ × 0.4 λ spot size. Here x ¯ 0 = 3 λ ( x ̂ + y ̂ ) , λ = 10 mm .

Fig. 11
Fig. 11

Amplitude of normalized electric field and Poynting vector field for ξ = 0.01 λ . The Poynting vector field is presented over a small, 2 λ × 2 λ area around the focal point x ¯ 0 = 3 λ ( x ̂ + y ̂ ) . λ = 10 mm .

Fig. 12
Fig. 12

Normalized square amplitude of the electric field in an X-plane cut (x fixed, variable y) centered at x ¯ 0 . Coordinates are with respect to x ¯ 0 and in wavelengths. The different curves correspond to ξ λ = 10 4 , 10 3 , 10 2 , and 10 1 . Past the critical point the electric field focal area (defined as the distance between points that are 50% of the peak) converges quickly to Δ Y 0.35 λ .

Fig. 13
Fig. 13

Magnitude of normalized reactive power flow field Q ¯ for ξ λ = 1 π , 1 4 π , 1 16 π and 1 64 π (clockwise from top left). The figure shows that values of ξ below the critical point λ 4 π result in large reactive field energy concentrated about the focal point x ¯ 0 = 3 λ ( x ̂ + y ̂ ) . λ = 10 mm

Fig. 14
Fig. 14

Reactive power vector field Q ¯ for ξ = 0.01 λ . The field is presented over a small, 2 λ × 2 λ area around the focal point x ¯ 0 = 3 λ ( x ̂ + y ̂ ) . λ = 10 mm

Fig. 15
Fig. 15

Higher-order n = 1 electric field modes. The magnitudes of the even and odd modes are presented corresponding to (a) ξ λ = 2 π , (b) ξ λ = 3 4 π , (c) ξ λ = 1 4 π . The coordinates are referred to the focal point, and are expressed in wavelengths. The figure demonstrates that multiple foci are possible in the focal region. The size of each focus is, however, bounded and cannot become infinitesimally small.

Fig. 16
Fig. 16

Magnitude of electric field modes obtained by superposition of the n = 0 and n = 1 modes. The E 0 mode can be combined with E 1 ODD to produce an electric field focal region ξ λ = 0.1 with chiral character (left), and can be combined with E 1 EVEN to increase the depth of focus significantly ξ λ = 0.01 . The equation at the top of each figure presents the mixing ratios. This is merely an illustration of what can be obtained via superposition.

Equations (16)

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

E = i H 0 ( 2 ) ( k s ) , s = s ¯ = x ¯ x ¯ s ,
s = ( x ¯ x ¯ 0 ) 2 ξ ¯ 2 2 i ( x ¯ x ¯ 0 ) ξ ¯ .
E ( x ¯ 0 ) = { i H 0 ( 2 ) ( i k ξ ) = 2 π K 0 ( k ξ ) on the + ξ ̂ side i H 0 ( 2 ) ( i k ξ ) = 2 π K 0 ( k ξ ) on the ξ ̂ side } .
E i 2 i π k x ¯ x ¯ 0 exp [ i k x ¯ x ¯ 0 k ξ cos ( ϑ ) ] ,
α = arccos ( 1 1 2 k ξ ) .
E = i H 0 ( 1 ) ( k s ) .
E = J 0 ( k s ) I 0 ( k ξ ) ,
E ξ x ¯ x ¯ 0 exp { i ( k x ¯ x ¯ 0 π 4 ) sgn [ cos ( ϑ ) ] } exp [ 2 k ξ sin 2 ( ϑ 2 ) ] .
E ( x ¯ x ¯ 0 ) ξ ¯ = 1 2 π I 0 ( k ξ ) exp [ i k ( x ¯ x ¯ 0 ) 2 ξ ¯ 2 cos ( ψ ) ] d ψ .
P ¯ = 1 2 η Re { i J 0 ( k s ) J 1 * ( k s ) I 0 2 ( k ξ ) [ ( s ̂ * x ̂ ) x ̂ + ( s ̂ * y ̂ ) y ̂ ] } ,
S ¯ 2 i k ξ J 0 ( z ) J 1 ( z ) r ̂ 2 r ̂ sin ( φ ) { [ J 1 ( z ) ] 2 + J 0 ( z ) J 1 ( z ) } 2 φ ̂ J 0 ( z ) J 1 ( z ) z cos ( φ ) ,
Q ¯ 2 k ξ J 0 ( z ) J 1 ( z ) r ̂ .
Q ¯ m a x P ¯ m a x 0.678 k ξ .
E n = J n ( k s ) { sin ( n β ) cos ( n β ) } ,
E = ∫∫∫ C o m p l e x S p a c e x ¯ s 3 J 0 ( k s ) A ( x ¯ s ) d x ¯ s 3 .
ψ n , m 3 D = j n ( k s ) P n m ( cos ϴ ) { sin ( m β ) cos ( m β ) } ,

Metrics