Abstract

The detailed optics of photonic nanojets generated by normal plane-wave incidence on dielectric cylinders is discussed. These nanojets have a subwavelength beam waist and propagate with little divergence for several wavelengths. A physical explanation for this peculiar behavior is presented. Characteristic dimensions of the nanojets for a large range of physical parameters are calculated.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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  3. M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).
  4. W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  26. J. A. Stratton, “Cylindrical waves,” in Electromagnetic Theory (McGraw-Hill, 1941).
  27. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
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2004 (2)

2003 (2)

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

A. V. Itagi, D. D. Stancil, J. A. Bain, T. E. Schlesinger, “Ridge waveguide as a near-field source,” Appl. Phys. Lett. 83, 4474–4476 (2003).
[CrossRef]

2002 (1)

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

1999 (1)

E. J. Sanchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

1997 (1)

1992 (1)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

1991 (2)

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves Appl. 5, 913–926 (1991).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid drops,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
[CrossRef]

1980 (1)

1974 (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp caustic,” Philos. Mag. 37, 311–317 (1946).

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Adler, C. L.

Akiyama, Y.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Backman, V.

Bain, J. A.

A. V. Itagi, D. D. Stancil, J. A. Bain, T. E. Schlesinger, “Ridge waveguide as a near-field source,” Appl. Phys. Lett. 83, 4474–4476 (2003).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 1997).

Balistreri, M. L. M.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

Cachorro, V. E.

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves Appl. 5, 913–926 (1991).
[CrossRef]

Challener, W. A.

I. K. Sendur, W. A. Challener, C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. 96, 2743–2752 (2004).
[CrossRef]

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Chen, Z.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Debye, P.

P. Debye, “The electromagnetic field around a cylinder and the theory of the rainbow” (translation of “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 1908), in Geometrical Aspects of Scattering, P. L. Marston ed. (SPIE, 1994).

Ebbesen, T. W.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Furuki, M.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Garcia, C. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hendriks, B. H. W.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Ishimoto, T.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Itagi, A. V.

A. V. Itagi, D. D. Stancil, J. A. Bain, T. E. Schlesinger, “Ridge waveguide as a near-field source,” Appl. Phys. Lett. 83, 4474–4476 (2003).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley-IEEE, 2002).

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Kino, G.

G. Kino, “The solid immersion lens,” in Optical Engineering for Sensing and Nanotechnology (ICOSN’99), I. Yamaguchi, ed., Proc. SPIE3740, 2–6 (1999).

Kondo, T.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Lee, J. I.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Lewen, G. D.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Lezec, H. J.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Linke, R. A.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Lock, J. A.

McDaniel, T. W.

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Mihalcea, C. D.

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Mountfield, K. R.

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Nahata, A.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Nakaoki, A.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Novotny, L.

E. J. Sanchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Padiy, A. V.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp caustic,” Philos. Mag. 37, 311–317 (1946).

Pelhos, K.

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Pellerin, K. M.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Peng, C.

I. K. Sendur, W. A. Challener, C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. 96, 2743–2752 (2004).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Saito, K.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Salcedo, L. L.

V. E. Cachorro, L. L. Salcedo, “New improvements for Mie scattering calculations,” J. Electromagn. Waves Appl. 5, 913–926 (1991).
[CrossRef]

Sanchez, E. J.

E. J. Sanchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Schlesinger, T. E.

A. V. Itagi, D. D. Stancil, J. A. Bain, T. E. Schlesinger, “Ridge waveguide as a near-field source,” Appl. Phys. Lett. 83, 4474–4476 (2003).
[CrossRef]

Sendur, I. K.

I. K. Sendur, W. A. Challener, C. Peng, “Ridge waveguide as a near field aperture for high density data storage,” J. Appl. Phys. 96, 2743–2752 (2004).
[CrossRef]

W. A. Challener, T. W. McDaniel, C. D. Mihalcea, K. R. Mountfield, K. Pelhos, I. K. Sendur, “Light delivery techniques for heat-assisted magnetic recording,” Jpn. J. Appl. Phys., Part 1 42, 981–988 (2003).
[CrossRef]

Shimouma, T.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Shinoda, M.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Stancil, D. D.

A. V. Itagi, D. D. Stancil, J. A. Bain, T. E. Schlesinger, “Ridge waveguide as a near-field source,” Appl. Phys. Lett. 83, 4474–4476 (2003).
[CrossRef]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Stone, B. R.

Stratton, J. A.

J. A. Stratton, “Cylindrical waves,” in Electromagnetic Theory (McGraw-Hill, 1941).

Taflove, A.

Takeda, M.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Thio, T.

T. Thio, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, G. D. Lewen, A. Nahata, R. A. Linke, “Giant optical transmission of sub-wavelength apertures: physics and applications,” Nanotechnology 13, 429–432 (2002).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Urbach, H. P.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

van der Aa, M. A. H.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

van der Mark, M. B.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Verschuren, C. A.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Wiscombe, W. J.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

Xie, X. S.

E. J. Sanchez, L. Novotny, X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Yamamoto, M.

M. Shinoda, K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, M. Furuki, M. Takeda, Y. Akiyama, T. Shimouma, M. Yamamoto, “High-density near-field optical disc recording using phase change media and polycarbonate substrate,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 224–232 (2004).

Zijp, F.

F. Zijp, M. B. van der Mark, J. I. Lee, C. A. Verschuren, B. H. W. Hendriks, M. L. M. Balistreri, H. P. Urbach, M. A. H. van der Aa, A. V. Padiy, “Near-field read-out of a 50-GB first-surface disc with NA=1.9 and a proposal for a cover-layer-incident, dual-layer near-field system,” in Optical Data Storage 2004, B. V. K. Vijaya Kumar and Hiromichi Kobori, eds., Proc. SPIE5380, 209–223 (2004).

Appl. Opt. (1)

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Figures (14)

Fig. 1
Fig. 1

Geometry: focusing to form a caustic.

Fig. 2
Fig. 2

(a) Concentric modes. (b) Debye series: generation of the field.

Fig. 3
Fig. 3

Nanojets for ( a , n ) : (a) (100, 1.78), (b) (1000, 1.95), (c) (10, 1.35), and (d) (1, 2).

Fig. 4
Fig. 4

Characteristics of a nanojet. The peak amplitude is at point O.

Fig. 5
Fig. 5

Peak amplitude, radial shift (rs), and decay length (dl) for (a) a = 100 and (b) a = 1000 ; (c) jet width for a = 100 and a = 1000 (as a function of n).

Fig. 6
Fig. 6

Comparison of the cross section of the nanojet for ( a , n ) = ( 100 , 1.78 ) with the cross section in the focal plane for cylindrical focusing.

Fig. 7
Fig. 7

Terms in the Debye series for ( a , n ) = ( 100 , 1.78 ) : (a) incidence + R 22 term, (b) T 12 T 21 term, (c) T 21 R 11 T 12 term, and (d) T 21 R 11 2 T 12 term. The gray scale for each subfigure is different.

Fig. 8
Fig. 8

Contour in the complex spatial frequency ( s ) plane for the contour integral in the angular spectrum representation of the field.

Fig. 9
Fig. 9

(a) Normalized magnitude and (b) phase in degrees of the angular spectrum for curve I ( a , n ) = ( 100 , 1.78 ) and curve II ( a , n ) = ( 10 , 1.35 ) . The magnitude has been normalized such that the magnitude for the zero spatial frequency is unity.

Fig. 10
Fig. 10

Effect of phase distribution of the angular spectrum on the nanojet characteristics: (a) phase is a function of s, (b) phase is zero for all s.

Fig. 11
Fig. 11

Normalized far-field magnitude of the field amplitude as a function of the azimuthal angle ϕ in degrees: (a) ( a , n ) = ( 100 , 1.78 ) and (b) ( a , n ) = ( 10 , 1.35 ) .

Fig. 12
Fig. 12

Incident ray corresponding to the ray emergent along the Y axis as a function of n.

Fig. 13
Fig. 13

T 12 T 21 as a function of the mode order, for (a) ( a , n ) = ( 100 , 1.78 ) , (b) ( a , n ) = ( 10 , 1.35 ) .

Fig. 14
Fig. 14

Relation between the radial decay of the field and the width of the field used in the angular spectrum calculation.

Equations (66)

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2 h + n 2 k 2 h = 0 .
h ( r , ϕ ) = m = 0 ( A m cos m ϕ + B m sin m ϕ ) U m ( n k r ) ,
h inc ( r , ϕ ) = exp ( i n 2 k x ) = exp ( i n 2 k r cos ϕ ) = [ J 0 ( n 2 k r ) + 2 m = 1 i m J m ( n 2 k r ) cos m ϕ ] = m = 0 c m cos m ϕ J m ( n 2 k r ) ,
c m = { 1 , m = 0 2 i m , m > 0 } .
h 1 ( r , ϕ ) = m = 0 ( a m cos m ϕ + a ̃ m sin m ϕ ) J m ( n 1 k r ) ,
h 2 ( r , ϕ ) = m = 0 ( b m cos m ϕ + b ̃ m sin m ϕ ) H m ( 1 ) ( n 2 k r ) .
a ̃ m = 0 ,
b ̃ m = 0 ,
c m J m ( n 2 k a ) + b m H m ( 1 ) ( n 2 k a ) = a m J m ( n 1 k a ) ,
c m J m ( n 2 k a ) + b m H m ( 1 ) ( n 2 k a ) n 2 = a m J m ( n 1 k a ) n 1 .
a m = c m n 1 [ H m ( 1 ) ( n 2 k a ) J m ( n 2 k a ) H m ( 1 ) ( n 2 k a ) J m ( n 2 k a ) n 1 H m ( 1 ) ( n 2 k a ) J m ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) J m ( n 1 k a ) ] ,
b m = c m [ n 2 J m ( n 1 k a ) J m ( n 2 k a ) n 1 J m ( n 1 k a ) J m ( n 2 k a ) n 1 H m ( 1 ) ( n 2 k a ) J m ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) J m ( n 1 k a ) ] .
H m ( 2 ) ( n 2 k a ) + R 22 H m ( 1 ) ( n 2 k a ) = T 21 H m ( 2 ) ( n 1 k a ) ,
H m ( 2 ) ( n 2 k a ) + R 22 H m ( 1 ) ( n 2 k a ) n 2 = T 21 H m ( 2 ) ( n 1 k a ) n 1 .
H m ( 1 ) ( n 1 k a ) + R 11 H m ( 2 ) ( n 1 k a ) = T 12 H m ( 1 ) ( n 2 k a ) ,
H m ( 1 ) ( n 1 k a ) + R 11 H m ( 2 ) ( n 1 k a ) n 1 = T 12 H m ( 1 ) ( n 2 k a ) n 2 .
T 21 = n 1 [ H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 2 k a ) H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 2 k a ) n 1 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) ] ,
R 22 = n 2 H m ( 2 ) ( n 1 k a ) H m ( 2 ) ( n 2 k a ) n 1 H m ( 2 ) ( n 1 k a ) H m ( 2 ) ( n 2 k a ) n 1 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) ,
T 12 = n 2 [ H m ( 1 ) ( n 1 k a ) H m ( 2 ) ( n 1 k a ) H m ( 1 ) ( n 1 k a ) H m ( 2 ) ( n 1 k a ) n 1 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) ] ,
R 11 = n 2 H m ( 1 ) ( n 1 k a ) H m ( 1 ) ( n 2 k a ) n 1 H m ( 1 ) ( n 1 k a ) H m ( 1 ) ( n 2 k a ) n 1 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) n 2 H m ( 1 ) ( n 2 k a ) H m ( 2 ) ( n 1 k a ) .
h tot out = c m J m ( n 2 k r ) + b m H m ( 1 ) ( n 2 k r ) .
h tot out = c m 2 H m ( 2 ) ( n 2 k r ) + ( b m + c m 2 ) H m ( 1 ) ( n 2 k r ) .
h tot in = a m J m ( n 1 k r ) = a m 2 [ H m ( 1 ) ( n 1 k r ) + H m ( 2 ) ( n 1 k r ) ] .
S 1 = j = 0 T 21 ( R 11 ) j = T 21 1 R 11 .
S 2 = R 22 + j = 0 T 12 ( R 11 ) j T 21 = R 22 + T 21 T 12 1 R 11 .
S 1 = a m c m ,
S 2 = 1 + 2 b m c m .
h tot out = [ c m J m ( n 2 k r ) + c m 2 ( R 22 1 ) H m ( 1 ) ( n 2 k r ) ] + c m 2 j = 0 T 12 ( R 11 ) j T 21 H m ( 1 ) ( n 2 k r ) .
h ( x , y ) = C G ( s ) exp ( i s y ) exp ( i 1 s 2 x ) d s
= C F ( s , x ) exp ( i s y ) d s ,
F ( s , x ) = G ( s ) exp ( i 1 s 2 x ) .
F ( s , x ) = 1 2 π y = + h ( x , y ) exp ( i s y ) d s ,
h nano ( r , ϕ ) = m = 0 T 12 T 21 H m ( 1 ) ( r ) cos m ϕ m = 0 T 12 T 21 { 2 π r exp ( i r ) exp [ i ( m + 1 2 ) π 2 ] } cos m ϕ exp ( i r ) r A ( ϕ ) ,
A ( ϕ ) = [ m = 0 T 12 T 21 exp ( i m π 2 ) cos m ϕ ] .
F ( s , a ) A ( sin 1 s ) 1 s 2 .
c m J m ( n 2 k a ) + b m H m ( 1 ) ( n 2 k a ) n 1 = a m J m ( n 1 k a ) n 2 .
H = [ p H m ( 1 ) ( r ) + q H m ( 2 ) ( r ) ] cos m ϕ exp ( i ω t ) .
E = i N H r = i N [ p H m ( 1 ) ( r ) + q H m ( 2 ) ( r ) ] cos m ϕ exp ( i ω t ) .
S = 1 2 Re ( E × H * ) = 1 2 Im ( i E × H * ) .
S Im { [ p H m ( 1 ) ( r ) + q H m ( 2 ) ( r ) ] [ p * H m ( 1 ) * ( r ) + q H m ( 2 ) * ( r ) ] } cos 2 m ϕ r ̂ .
T 1 = p 2 Im [ H m ( 1 ) ( r ) H m ( 1 ) * ( r ) ] = p 2 [ J m ( r ) Y m ( r ) Y m ( r ) J m ( r ) ] = p 2 2 { [ J m ( r ) Y m 1 ( r ) Y m ( r ) J m 1 ( r ) ] + [ J m + 1 ( r ) Y m ( r ) Y m + 1 ( r ) J m ( r ) ] } = p 2 π r .
T 2 = q 2 Im [ H m ( 2 ) ( r ) H m ( 2 ) * ( r ) ] = q 2 Im { [ H m ( 1 ) ( r ) H m ( 1 ) * ( r ) ] * } = q 2 Im [ H m ( 1 ) ( r ) H m ( 1 ) * ( r ) ] = q 2 π r ,
T 3 = Im [ p q * H m ( 1 ) ( r ) H m ( 2 ) * ( r ) + q p * H m ( 2 ) ( r ) H m ( 1 ) * ( r ) ] = Im { p q * H m ( 1 ) ( r ) H m ( 2 ) * ( r ) + [ p q * H m ( 1 ) ( r ) H m ( 2 ) * ( r ) ] * } = Im { Re [ p q * H m ( 1 ) ( r ) H m ( 2 ) * ( r ) ] } = 0 .
J m ( n 1 k a ) j 1 , J m ( n 2 k a ) j 2 , H m ( 1 ) ( n 1 k a ) u 1 ,
H m ( 1 ) ( n 2 k a ) u 2 , H m ( 2 ) ( n 1 k a ) v 1 , H m ( 2 ) ( n 2 k a ) v 2 .
u i + v i = 2 j i : ( i 1 , 2 ) ,
S 1 = T 21 1 R 11 = n 1 ( u 2 v 2 u 2 v 2 ) ( n 1 u 2 v 1 n 2 u 2 v 1 ) ( n 2 u 1 u 2 n 1 u u 1 u 2 ) = n 1 [ u 2 ( 2 j 2 v 2 ) u 2 ( 2 j 2 v 2 ) ] n 1 u 2 ( u 1 + v 1 ) n 2 u 2 ( u 1 + v 1 ) = n 1 ( u 2 j 2 u 2 j 2 ) n 1 u 2 j 1 n 2 u 2 j 1 = a m c m ,
S 2 = R 22 + T 21 T 12 1 R 11 = 1 + R 22 + T 21 T 12 1 R 11 1 = 1 + 1 ( n 1 u 2 v 1 n 2 u 2 v 1 ) [ ( n 2 v 1 v 2 n 1 v 1 v 2 ) ( n 1 u 2 v 1 n 2 u 2 v 1 ) + n 1 n 2 ( u 1 v 1 u 1 v 1 ) ( u 2 j 2 u 2 j 2 ) n 1 u 2 j 1 n 2 u 2 j 1 ] = 1 + N ( n 1 u 2 v 1 u 2 v 1 ) ( n 1 u 2 j 1 n 2 u 2 j 1 ) ,
N = [ ( n 2 v 1 v 2 n 1 v 1 v 2 ) ( n 1 u 2 v 1 n 2 u 2 v 1 ) ] ( n 1 u 2 j 1 n 2 u 2 j 1 ) + n 1 n 2 ( u 1 v 1 u 1 v 1 ) ( u 2 j 2 u 2 j 2 ) .
N = 1 2 { ( n 1 u 2 v 1 n 2 u 2 v 1 ) [ n 1 ( u 2 + v 2 ) ( u 1 + v 1 ) n 2 ( u 2 + v 2 ) ( u 1 + v 1 ) ] } = 2 ( n 1 u 2 v 1 n 2 u 2 v 1 ) ( n 1 j 2 j 1 n 2 j 2 j 1 ) .
S 2 = 1 + 2 ( n 1 j 2 j 1 n 2 j 2 j 1 ) n 1 u 2 j 1 n 2 u 2 j 1 = 1 + 2 b m c m .
Q = α C m ( x ) D m ( y ) β C m ( x ) D m ( y ) ,
Q = α [ C m 1 ( x ) m x C m ( x ) ] D m ( y ) β C m ( x ) [ D m 1 ( x ) m y D m ( y ) ] = C m ( x ) D m ( y ) [ α C m 1 ( x ) C m ( x ) β D m 1 ( y ) D m ( y ) + m ( β y α x ) ] .
J m 1 ( z ) J m ( z ) m z ,
H m 1 ( j ) ( z ) H m ( j ) ( z ) z m ,
Q O [ m ] C m ( x ) D m ( y ) ,
H m ( 1 ) ( z ) H m ( 2 ) ( z ) H m ( 1 ) ( z ) H m ( 2 ) ( z ) = H m 1 ( 1 ) ( z ) H m ( 2 ) ( z ) H m ( 1 ) ( z ) H m 1 ( 2 ) ( z ) = 4 i π z .
b m j 2 u 1 ,
R 22 1 j 2 u 2 ,
1 R 11 j 1 v 1 ,
T 21 T 12 O [ m ] u 2 v 2 .
h 2 ( r , ϕ ) = m = 0 [ c m J m ( r ) + b m H m ( 1 ) ( r ) ] cos m ϕ = m = 0 { c m J m ( r ) + c m 2 [ ( R 22 1 ) + T 12 T 21 1 R 11 ] H m ( 1 ) ( r ) } cos m ϕ .
h 2 ( r , ϕ ) m = 0 { O [ J m ] + O [ j 2 u 2 H m ( 1 ) ( r ) ] + O [ m 2 v 1 H m ( 1 ) ( r ) u 2 2 v 2 2 j 1 ] } .
F ( s , X ) y = + h nano ( X , y ) exp ( i s y ) d y y = + exp ( i X 2 + y 2 ) X 2 + y 2 A ( tan 1 y X ) exp ( i s y ) d y .
F ( sin θ , X ) X ϕ = π 2 π 2 exp [ i X ( sec ϕ sin θ tan ϕ ) ] A ( ϕ ) sec 3 2 ϕ d ϕ .
F ( sin θ , X ) exp ( i X cos θ ) cos θ A ( θ ) .

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