Abstract

High intensity sub-wavelength spots and low divergence nanojets are observed in a system of Si3N4 microdisks illuminated from the side with laser light of wavelengths 488 nm, 532 nm and 633 nm. The disks are of height 400 nm with diameters ranging from 1μm to 10μm. Light scattered from the disk and substrate is observed by imaging from above. In free space light is focused inside the disks and a sub wavelength spot is observed, whereas in water the refractive index contrast is such that photonic nanojets are formed. The angular distribution of the intensity compares well to the analytical solution for the case of an infinite cylinder. Two distinct cases of scattering pattern are observed with even and odd numbers of lobes. Finally when the disks are illuminated with a focused Gaussian beam perpendicular to the substrate an extremely low divergence beam is observed. This beam has a divergence angle over 10 times smaller than a focused Gaussian in free space with the same waist.

©2011 Optical Society of America

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References

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  1. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Annalen der Physik 25(3), 377–445 (1908).
    [Crossref]
  2. Mie Theory 1908–2008 Present developments and interdisciplinary aspects of light scattering, W. Hergert and T. Wriedt eds.(Univ. Bremen, 2008).
  3. J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
    [Crossref]
  4. P. Chýlek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24(23), 3940–3942 (1985).
    [Crossref] [PubMed]
  5. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W.-F. Hsieh, and R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26(7), 1348–1356 (1987).
    [Crossref] [PubMed]
  6. Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004).
    [Crossref] [PubMed]
  7. X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005).
    [Crossref] [PubMed]
  8. A. V. Itagi and W. A. Challener, “Optics of photonic nanojets,” J. Opt. Soc. Am. A 22(12), 2847–2858 (2005).
    [Crossref] [PubMed]
  9. D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,” Opt. Lett. 26(7), 399–401 (2001).
    [Crossref] [PubMed]
  10. A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008).
    [Crossref] [PubMed]
  11. Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. 31(2), 196–198 (2006).
    [Crossref] [PubMed]
  12. S. Lecler, Y. Takakura, and P. Meyrueis, “Properties of a three-dimensional photonic jet,” Opt. Lett. 30(19), 2641–2643 (2005).
    [Crossref] [PubMed]
  13. A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
    [Crossref]
  14. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008).
    [Crossref] [PubMed]
  15. M. Gerlach, Y. P. Rakovich, and J. F. Donegan, “Nanojets and directional emission in symmetric photonic molecules,” Opt. Express 15(25), 17343–17350 (2007).
    [Crossref] [PubMed]
  16. A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
    [Crossref] [PubMed]
  17. V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Photonic Microresonator Research and Applications, L. Chremmos, O. Schwelb, and N.Uzunoglu, eds. (Springer Series in Optical Sciences 156, 2010), pp.423–457.
  18. S. Yang, A. Taflove, and V. Backman, “Experimental confirmation at visible light wavelengths of the backscattering enhancement phenomenon of the photonic nanojet,” Opt. Express 19(8), 7084–7093 (2011).
    [Crossref] [PubMed]
  19. E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
    [Crossref] [PubMed]
  20. S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
    [Crossref] [PubMed]
  21. D. Gérard, J. Wenger, A. Devilez, D. Gachet, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Strong electromagnetic confinement near dielectric microspheres to enhance single-molecule fluorescence,” Opt. Express 16(19), 15297–15303 (2008).
    [Crossref] [PubMed]
  22. Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
    [Crossref]
  23. A. M. Kapitonov and V. N. Astratov, “Observation of nanojet-induced modes with small propagation losses in chains of coupled spherical cavities,” Opt. Lett. 32(4), 409–411 (2007).
    [Crossref] [PubMed]
  24. A. Darafsheh, A. Fardad, N. M. Fried, A. N. Antoszyk, H. S. Ying, and V. N. Astratov, “Contact focusing multimodal microprobes for ultraprecise laser tissue surgery,” Opt. Express 19(4), 3440–3448 (2011).
    [Crossref] [PubMed]
  25. P. W. Barber and S. C. Hill, “Light Scattering by Particles: Computational Methods,” in Advanced Series in Applied Physics (World Scientific, 1990)
  26. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998)
  27. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980).
    [Crossref] [PubMed]
  28. M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011).
    [Crossref] [PubMed]
  29. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2006)

2011 (3)

2009 (1)

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

2008 (5)

2007 (2)

2006 (3)

Z. Chen, A. Taflove, X. Li, and V. Backman, “Superenhanced backscattering of light by nanoparticles,” Opt. Lett. 31(2), 196–198 (2006).
[Crossref] [PubMed]

J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
[Crossref]

A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
[Crossref]

2005 (3)

2004 (1)

2001 (1)

1987 (1)

1985 (1)

1980 (1)

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Annalen der Physik 25(3), 377–445 (1908).
[Crossref]

Antoszyk, A. N.

Arnold, C. B.

E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

Arnold, N.

J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
[Crossref]

Astratov, V. N.

Backman, V.

Barber, P. W.

Benincasa, D. S.

Bonod, N.

Challener, W. A.

Chang, R. K.

Chen, Z.

Chýlek, P.

Darafsheh, A.

Devilez, A.

Donegan, J. F.

Fardad, A.

Ferrand, P.

Fletcher, D. A.

Fried, N. M.

Gachet, D.

Gérard, D.

Gerlach, M.

Goodson, K. E.

Guo, W.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Heifetz, A.

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
[Crossref]

Herzig, H. P.

Hong, M.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Hsieh, W.-F.

Huang, K.

A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
[Crossref]

Itagi, A. V.

Kapitonov, A. M.

Khan, A.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Kim, M.-S.

Kino, G. S.

Kofler, J.

J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
[Crossref]

Kong, S. C.

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

Kong, S.-C.

Lecler, S.

Li, L.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Li, X.

Liu, Z.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Luk'yanchuk, B.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

McLeod, E.

E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

Meyrueis, P.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Annalen der Physik 25(3), 377–445 (1908).
[Crossref]

Mühlig, S.

Pendleton, J. D.

Pianta, M.

Pinnick, R. G.

Popov, E.

Rakovich, Y. P.

Rigneault, H.

Rockstuhl, C.

Sahakian, A.

Sahakian, A. V.

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
[Crossref]

Scharf, T.

Stout, B.

Taflove, A.

Takakura, Y.

Wang, Z.

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

Wenger, J.

Wiscombe, W. J.

Yang, S.

Ying, H. S.

Zhang, J.-Z.

Annalen der Physik (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Annalen der Physik 25(3), 377–445 (1908).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

A. Heifetz, K. Huang, A. V. Sahakian, X. Li, A. Taflove, and V. Backman, “Experimental confirmation of backscattering enhancement induced by a photonic jet,” Appl. Phys. Lett. 89(22), 221118 (2006).
[Crossref]

J Comput Theor Nanosci (1)

A. Heifetz, S. C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J Comput Theor Nanosci 6(9), 1979–1992 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Nat. Nanotechnol. (1)

E. McLeod and C. B. Arnold, “Subwavelength direct-write nanopatterning using optically trapped microspheres,” Nat. Nanotechnol. 3(7), 413–417 (2008).
[Crossref] [PubMed]

Opt. Express (10)

Z. Chen, A. Taflove, and V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Opt. Express 12(7), 1214–1220 (2004).
[Crossref] [PubMed]

X. Li, Z. Chen, A. Taflove, and V. Backman, “Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets,” Opt. Express 13(2), 526–533 (2005).
[Crossref] [PubMed]

M. Gerlach, Y. P. Rakovich, and J. F. Donegan, “Nanojets and directional emission in symmetric photonic molecules,” Opt. Express 15(25), 17343–17350 (2007).
[Crossref] [PubMed]

P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16(10), 6930–6940 (2008).
[Crossref] [PubMed]

S.-C. Kong, A. Sahakian, A. Taflove, and V. Backman, “Photonic nanojet-enabled optical data storage,” Opt. Express 16(18), 13713–13719 (2008).
[Crossref] [PubMed]

A. Devilez, B. Stout, N. Bonod, and E. Popov, “Spectral analysis of three-dimensional photonic jets,” Opt. Express 16(18), 14200–14212 (2008).
[Crossref] [PubMed]

D. Gérard, J. Wenger, A. Devilez, D. Gachet, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Strong electromagnetic confinement near dielectric microspheres to enhance single-molecule fluorescence,” Opt. Express 16(19), 15297–15303 (2008).
[Crossref] [PubMed]

A. Darafsheh, A. Fardad, N. M. Fried, A. N. Antoszyk, H. S. Ying, and V. N. Astratov, “Contact focusing multimodal microprobes for ultraprecise laser tissue surgery,” Opt. Express 19(4), 3440–3448 (2011).
[Crossref] [PubMed]

S. Yang, A. Taflove, and V. Backman, “Experimental confirmation at visible light wavelengths of the backscattering enhancement phenomenon of the photonic nanojet,” Opt. Express 19(8), 7084–7093 (2011).
[Crossref] [PubMed]

M.-S. Kim, T. Scharf, S. Mühlig, C. Rockstuhl, and H. P. Herzig, “Engineering photonic nanojets,” Opt. Express 19(11), 10206–10220 (2011).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Rev. B (1)

J. Kofler and N. Arnold, “Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie,” Phys. Rev. B 73(23), 235401 (2006).
[Crossref]

Other (6)

Z. Wang, W. Guo, L. Li, B. Luk'yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, (2011).
[Crossref]

P. W. Barber and S. C. Hill, “Light Scattering by Particles: Computational Methods,” in Advanced Series in Applied Physics (World Scientific, 1990)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge, 2006)

V. N. Astratov, “Fundamentals and Applications of Microsphere Resonator Circuits,” in Photonic Microresonator Research and Applications, L. Chremmos, O. Schwelb, and N.Uzunoglu, eds. (Springer Series in Optical Sciences 156, 2010), pp.423–457.

Mie Theory 1908–2008 Present developments and interdisciplinary aspects of light scattering, W. Hergert and T. Wriedt eds.(Univ. Bremen, 2008).

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Figures (12)

Fig. 1
Fig. 1 (a) SEM image of a single microdisk of diameter 8μm. (b) FIB milling was used to cut through the disk. The disk was covered in a thin layer of Pt in order to protect the dimensions of the structure. The side profile shows the disks are of height 400 nm with a sidewall angle of 56°.

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Fig. 2
Fig. 2 (a): Laser light is introduced with an angle of incidence of 80°. Most of the light is reflected specularly, but a small percentage is scattered at large angles by the disk. The scattered light from the structure is collected using the microscope objective (MO) and imaged onto a CCD as indicated by the marginal ray bundle in green. The input beam is a Gaussian with a large beam diameter of 2 mm, making the excitation effectively a plane wave on the scale of the microdisk. Images of the disk can be obtained by introducing white light through the beam splitter. (b) Raw image from a microdisk of diameter 6.5μm, with 532 nm incident wavelength. Light is incident from the top.

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Fig. 3
Fig. 3 Co-ordinate systems used in specific form of the analytical solution. Incident light is linearly polarised along the z-axis in the TM case, and propagates along the x-axis.

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Fig. 4
Fig. 4 Experimental images and analytical results of the scattered intensity for individual disks.All images are normalised to the maximum intensity . Light is incident from the top of the images at a wavelength of 532nm. Top row from left: Experimental linear image, and log10 of 1.5μm diameter disk, 4.5μm diameter disk and 8.5μm diameter disk. Bottom row from left: Analytical results for infinite cylinder of same dimensions as above experimental results, plotted in linear and log scales. The scale is the same for all images with a scale bar given in the top left experimental image.

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Fig. 5
Fig. 5 Norm of expansion co-efficient of the scattered field Es for disk of diameters (a) 1.5μm, (b) 8.5μm.

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Fig. 6
Fig. 6 Raw experimental images. Incident light of wavelength 532nm from bottom left corner of image. (a) Disk of diameter 9μm (b) Disk of Diameter 9.5μm. (c) Axial and transverse profile of the central peak in (b) fitted with Gaussians. This gives a diffraction limited spot of FWHM 460nm±60nm by 508±65nm.

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Fig. 7
Fig. 7 Calculated total intensity for an infinite cylinder of index 2.1 in air for diameters of 1.5μm, 4.5μm and 8.5μm plotted on a linear scale.

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Fig. 8
Fig. 8 Experimental setup used to measure the scattering properties of the microdisks in water. Some spherical aberration is introduced in this setup, but is accounted for in the scaling of the image pixel size. The rest of the optical system from the back of the MO is identical to that of Fig. 2(a). The laser is input at 80° angle of incidence through the cover-slip. The shaded red region indicates a marginal ray bundle.

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Fig. 9
Fig. 9 Scattering distribution from microdisks in water. The incident wavelength is 633nm which is effectively reduced to 475nm in water. The refractive index contrast is now 1.5. In this case the maximum of the scattered light is outside of the disk and we see the photonic nanojet effect. The normalised intensity plots are shown for 3 different disk diameters (a) 1.5μm (b) 3μm (c) 6μm on a normalised linear scale.

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Fig. 10
Fig. 10 (a) Experimental setup. Laser light of wavelength 488nm is focused onto the rim of the disk (b)The white light image of the 9μm diameter microdisk before illumination. (c) Image with 488nm CW laser illumination. The Nanojet forms in air on the opposite side of the disk to the illumination.

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Fig. 11
Fig. 11 (a) Standard fitting of the nanojet to extract FWHM. The FWHM is this case is 510nm corresponding to a w0 value of 205nm. (b) The profile of the intensity along the θ = 0 axis. The data was fitted with a Lorentzian function.

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Fig. 12
Fig. 12 The transverse FWHM is plotted as a function of distance, with 0 corresponding to the point of highest intensity (smallest FWHM). We can see the extremely low divergence angle. (b) Comparison between the divergence angle of the experimentally measured Jet and that of a Gaussian with the same waist. The dashed line represents the Gaussian divergence with an angle of 47°. This is over 10 times the divergence angle of the nanojet.

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Equations (6)

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

E z i ( kr )= E 0 ∑ n=−∞ ∞ i n J n (kr)exp(inθ)
E z s ( r, θ )= − E 0 ∑ n=−∞ ∞ i n b n cyl H n (1) (kr)exp(inθ)
E z int ( r, θ )=  E 0 ∑ n=−∞ ∞ i n d n cyl J n (mkr)exp(inθ)
b n cyl = m J n ' ( m x s )− J n ( x s ) J n ' ( x s ) m J n ' ( m x s ) H n ( 1 ) ( x s   )–  J n ( m x s ) H n '( 1 ) ( x s )
d n cyl = 1 J n (m x s ) [ J n ( x s )− b n cyl H n (1) ( x s )]
E z s ( r, θ )= − E 0 b 0 H 0 (1) (kr)−2 E 0 ∑ n=1 n c i n b n cyl H n (1) (kr)cos(nθ)

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