Abstract

High-aperture focusing through a spherical interface has been employed in optical data storage, photolithography, and especially microscopy. This paper first forms an approximate model, based on geometrical optics and Fourier optics, for evaluating focal fields of the focusing systems. This approximate model helps to clarify some doubts existing in literature. We then propose a rigorous model that is applicable to more general systems. Our model is based on multipole theory, which expands the electromagnetic fields into spherical harmonics.

© 2013 Optical Society of America

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  1. S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
    [CrossRef]
  2. S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
    [CrossRef]
  3. Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE. 88, 1491–1498 (2000).
  4. E. Ramsey, N. Pleynet, D. Xiao, R. J. Wadburton, and D. T. Reid, “Two-photon optical-beam-induced current solid-immersion imaging of a silicon flip chip with a resolution of 325 nm,” Opt. Lett. 30, 26–28 (2005).
    [CrossRef]
  5. E. Ramsey, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizaheh, R. J. Warburton, and D. T. Reid, “Three-dimentional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90, 131101 (2007).
    [CrossRef]
  6. K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
    [CrossRef]
  7. F. H. Koklu, B. B. Goldberg, and M. S. Ünlü, “Dielectric interface effects in subsurface microscopy of integrated circuits,” Opt. Commun. 285, 1675–1679 (2012).
    [CrossRef]
  8. S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97, 053105 (2005).
    [CrossRef]
  9. Y. Zhang, C. Zheng, and Y. Zou, “Focal-field distribution of the solid immersion lens system with an annular filter,” Optik 115, 277–280 (2004).
    [CrossRef]
  10. I. Ichimura, S. Hayashi, and G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
    [CrossRef]
  11. B. D. Terris, H. J. Mamin, and D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
    [CrossRef]
  12. Y. Zhang, X. Xu, and Y. Okuno, “Theoretical study of optical recording with a solid immersion lens illuminated by focused double-ring-shaped radially-polarized beam,” Opt. Commun. 282, 4481–4485 (2009).
    [CrossRef]
  13. L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74, 501–503 (1999).
    [CrossRef]
  14. E. Pavel, S. Jinga, E. Andronescu, B. S. Vasile, G. Kada, A. Sasahara, N. Tosa, A. Matei, M. Dinescu, A. Dinescu, and O. R. Vasile, “2 nm Quantum optical lithography,” Opt. Commun. 291, 259–263 (2013).
    [CrossRef]
  15. S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
    [CrossRef]
  16. K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “The effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011).
    [CrossRef]
  17. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
    [CrossRef]
  18. A. S. van de Nes, L. Billy, S. F. Pereira, and J. J. M. Braat, “Calculation of the vectorial field distribution in a stratified focal region of a high numerical aperture imaging system,” Opt. Express 12, 1281–1293 (2004).
    [CrossRef]
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    [CrossRef]
  21. R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Resolution of aplanatic solid immersion lens based microscopy,” J. Opt. Soc. Am. A 29, 1059–1070 (2012).
    [CrossRef]
  22. R. Chen, K. Agarwal, Y. Zhong, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
    [CrossRef]
  23. B. Richards and 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]
  24. C. J. R. Sheppard and S. H. Goh, “Comment on ‘Theoretical analysis of numerical aperture increasing lens microscopy’ [J. Appl. Phys. 97, 053105 (2005)],” J. Appl. Phys. 100, 086108 (2006).
    [CrossRef]
  25. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  26. M. S. Ünlü, S. B. Ippolito, A. N. Vamivakas, and B. B. Goldberg, “Response to ‘Comment on “Theoretical analysis of numerical aperture increasing lens microscopy’” [J. Appl. Phys. 97, 053105 (2005)],” J. Appl. Phys. 100, 086109 (2006).
    [CrossRef]
  27. A. N. Vamivakas, R. D. Younger, B. B. Golderg, A. K. Swan, M. S. Ünlü, E. R. Behringer, and S. B. Ippolito, “A case study for optics: the solid immersion microscope,” Am. J. Phys. 76, 758–768 (2008).
    [CrossRef]
  28. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  29. S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express 16, 3397–3407 (2008).
    [CrossRef]
  30. M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye–Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
    [CrossRef]
  31. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
    [CrossRef]
  32. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. 86, 033817 (2012).
    [CrossRef]
  33. W. C. Chew and Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
    [CrossRef]
  34. D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with wavelength-scale solid immersion lenses,” Opt. Lett. 35, 2007–2009 (2010).
    [CrossRef]
  35. M. S. Kim, T. Scharf, M. T. Haq, W. Nakagawa, and H. P. Herzig, “Subwavelength-size solid immersion lens,” Opt. Lett. 36, 3930–3932 (2011).
    [CrossRef]
  36. A. Vlad, I. Huynen, and S. Melinte, “Wavelength-scale lens microscopy via thermal reshaping of colloidal particles,” Nanotechnology 23, 285708 (2012).
    [CrossRef]
  37. T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 79, 1005–1017 (2003).
  38. C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
    [CrossRef]
  39. L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” Acta Numer. 6, 229–269 (1997).
    [CrossRef]
  40. W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).
  41. H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “A wideband fast multipole method for the Helmholtz equation in three dimensions,” J. Comput. Phys. 216, 300–325 (2006).
    [CrossRef]
  42. J. M. Song and W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
    [CrossRef]
  43. B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
  44. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  45. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  46. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I—multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. 19, 378–390 (1971).
    [CrossRef]
  47. F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
    [CrossRef]
  48. B. U. Felderhof and R. B. Jones, “Addition theorems for spherical wave solutions of the vector Helmholtz equation,” J. Math. Phys. 28, 836–839 (1987).
    [CrossRef]
  49. R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
    [CrossRef]
  50. W. C. Chew, Waves and Fields in Inhomogeneuos Media (IEEE, 1995).
  51. W. C. Chew, “Recurrence relations for three-dimensional scalar addition theorem,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
    [CrossRef]
  52. K. T. Kim, “Symmetry relations of the translation coefficients of the scalar and vector spherical multipole fields,” Prog. Electromagn. Res. 48, 45–66 (2004).
    [CrossRef]
  53. J. M. Song and W. C. Chew, “Error analysis for the truncation of multipole expansion of vector green’s functions,” IEEE Microw. Wirel. Compon. Lett. 11, 311–313 (2001).
  54. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef]
  55. K. Agarwal, R. Chen, L. S. Koh, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Experimental validation of the computational model of aplanatic solid immersion lens scanning microscope,” presented at Focus On Microscopy 2013, Maastricht, The Netherlands, March 24–27, 2013.

2013 (1)

E. Pavel, S. Jinga, E. Andronescu, B. S. Vasile, G. Kada, A. Sasahara, N. Tosa, A. Matei, M. Dinescu, A. Dinescu, and O. R. Vasile, “2 nm Quantum optical lithography,” Opt. Commun. 291, 259–263 (2013).
[CrossRef]

2012 (6)

F. H. Koklu, B. B. Goldberg, and M. S. Ünlü, “Dielectric interface effects in subsurface microscopy of integrated circuits,” Opt. Commun. 285, 1675–1679 (2012).
[CrossRef]

R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Resolution of aplanatic solid immersion lens based microscopy,” J. Opt. Soc. Am. A 29, 1059–1070 (2012).
[CrossRef]

R. Chen, K. Agarwal, Y. Zhong, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
[CrossRef]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
[CrossRef]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. 86, 033817 (2012).
[CrossRef]

A. Vlad, I. Huynen, and S. Melinte, “Wavelength-scale lens microscopy via thermal reshaping of colloidal particles,” Nanotechnology 23, 285708 (2012).
[CrossRef]

2011 (4)

2010 (1)

2009 (3)

S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
[CrossRef]

Y. Zhang, X. Xu, and Y. Okuno, “Theoretical study of optical recording with a solid immersion lens illuminated by focused double-ring-shaped radially-polarized beam,” Opt. Commun. 282, 4481–4485 (2009).
[CrossRef]

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80, 013703 (2009).
[CrossRef]

2008 (4)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2, 311–314 (2008).
[CrossRef]

A. N. Vamivakas, R. D. Younger, B. B. Golderg, A. K. Swan, M. S. Ünlü, E. R. Behringer, and S. B. Ippolito, “A case study for optics: the solid immersion microscope,” Am. J. Phys. 76, 758–768 (2008).
[CrossRef]

S. S. Sherif, M. R. Foreman, and P. Török, “Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system,” Opt. Express 16, 3397–3407 (2008).
[CrossRef]

M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye–Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[CrossRef]

2007 (1)

E. Ramsey, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizaheh, R. J. Warburton, and D. T. Reid, “Three-dimentional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90, 131101 (2007).
[CrossRef]

2006 (3)

C. J. R. Sheppard and S. H. Goh, “Comment on ‘Theoretical analysis of numerical aperture increasing lens microscopy’ [J. Appl. Phys. 97, 053105 (2005)],” J. Appl. Phys. 100, 086108 (2006).
[CrossRef]

M. S. Ünlü, S. B. Ippolito, A. N. Vamivakas, and B. B. Goldberg, “Response to ‘Comment on “Theoretical analysis of numerical aperture increasing lens microscopy’” [J. Appl. Phys. 97, 053105 (2005)],” J. Appl. Phys. 100, 086109 (2006).
[CrossRef]

H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “A wideband fast multipole method for the Helmholtz equation in three dimensions,” J. Comput. Phys. 216, 300–325 (2006).
[CrossRef]

2005 (2)

E. Ramsey, N. Pleynet, D. Xiao, R. J. Wadburton, and D. T. Reid, “Two-photon optical-beam-induced current solid-immersion imaging of a silicon flip chip with a resolution of 325 nm,” Opt. Lett. 30, 26–28 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97, 053105 (2005).
[CrossRef]

2004 (3)

Y. Zhang, C. Zheng, and Y. Zou, “Focal-field distribution of the solid immersion lens system with an annular filter,” Optik 115, 277–280 (2004).
[CrossRef]

A. S. van de Nes, L. Billy, S. F. Pereira, and J. J. M. Braat, “Calculation of the vectorial field distribution in a stratified focal region of a high numerical aperture imaging system,” Opt. Express 12, 1281–1293 (2004).
[CrossRef]

K. T. Kim, “Symmetry relations of the translation coefficients of the scalar and vector spherical multipole fields,” Prog. Electromagn. Res. 48, 45–66 (2004).
[CrossRef]

2003 (1)

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 79, 1005–1017 (2003).

2001 (2)

J. M. Song and W. C. Chew, “Error analysis for the truncation of multipole expansion of vector green’s functions,” IEEE Microw. Wirel. Compon. Lett. 11, 311–313 (2001).

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

2000 (1)

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE. 88, 1491–1498 (2000).

1999 (1)

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74, 501–503 (1999).
[CrossRef]

1997 (2)

I. Ichimura, S. Hayashi, and G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
[CrossRef]

L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” Acta Numer. 6, 229–269 (1997).
[CrossRef]

1996 (1)

B. D. Terris, H. J. Mamin, and D. Rugar, “Near-field optical data storage,” Appl. Phys. Lett. 68, 141–143 (1996).
[CrossRef]

1995 (2)

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

J. M. Song and W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

1994 (1)

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

1993 (1)

W. C. Chew and Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

1992 (1)

W. C. Chew, “Recurrence relations for three-dimensional scalar addition theorem,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
[CrossRef]

1990 (1)

S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

1988 (1)

R. C. Wittmann, “Spherical wave operators and the translation formulas,” IEEE Trans. Antennas Propag. 36, 1078–1087 (1988).
[CrossRef]

1987 (1)

B. U. Felderhof and R. B. Jones, “Addition theorems for spherical wave solutions of the vector Helmholtz equation,” J. Math. Phys. 28, 836–839 (1987).
[CrossRef]

1980 (2)

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef]

1971 (1)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I—multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. 19, 378–390 (1971).
[CrossRef]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards and 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]

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).

Agarwal, K.

Andronescu, E.

E. Pavel, S. Jinga, E. Andronescu, B. S. Vasile, G. Kada, A. Sasahara, N. Tosa, A. Matei, M. Dinescu, A. Dinescu, and O. R. Vasile, “2 nm Quantum optical lithography,” Opt. Commun. 291, 259–263 (2013).
[CrossRef]

Behringer, E. R.

A. N. Vamivakas, R. D. Younger, B. B. Golderg, A. K. Swan, M. S. Ünlü, E. R. Behringer, and S. B. Ippolito, “A case study for optics: the solid immersion microscope,” Am. J. Phys. 76, 758–768 (2008).
[CrossRef]

Billy, L.

Booker, G. R.

Borghese, F.

F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Braat, J. J. M.

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres, part I—multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. 19, 378–390 (1971).
[CrossRef]

Chen, R.

Chen, X.

R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Resolution of aplanatic solid immersion lens based microscopy,” J. Opt. Soc. Am. A 29, 1059–1070 (2012).
[CrossRef]

R. Chen, K. Agarwal, Y. Zhong, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Complete modeling of subsurface microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).
[CrossRef]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).
[CrossRef]

T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a focused beam,” Phys. Rev. 86, 033817 (2012).
[CrossRef]

L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011).
[CrossRef]

K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “The effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011).
[CrossRef]

K. Agarwal, R. Chen, L. S. Koh, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Experimental validation of the computational model of aplanatic solid immersion lens scanning microscope,” presented at Focus On Microscopy 2013, Maastricht, The Netherlands, March 24–27, 2013.

Cheng, H.

H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “A wideband fast multipole method for the Helmholtz equation in three dimensions,” J. Comput. Phys. 216, 300–325 (2006).
[CrossRef]

Chew, W. C.

J. M. Song and W. C. Chew, “Error analysis for the truncation of multipole expansion of vector green’s functions,” IEEE Microw. Wirel. Compon. Lett. 11, 311–313 (2001).

J. M. Song and W. C. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

W. C. Chew and Y. M. Wang, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Waves Appl. 7, 651–665 (1993).
[CrossRef]

W. C. Chew, “Recurrence relations for three-dimensional scalar addition theorem,” J. Electromagn. Waves Appl. 6, 133–142 (1992).
[CrossRef]

W. C. Chew, Waves and Fields in Inhomogeneuos Media (IEEE, 1995).

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

Chua, C. M.

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T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focused laser beam,” J. Quant. Spectrosc. Radiat. Transfer 79, 1005–1017 (2003).

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Figures (8)

Fig. 1.
Fig. 1.

SIL configurations: (a) hemispherical SIL configuration, (b) general SIL configuration.

Fig. 2.
Fig. 2.

Individual plane wave incident on the spherical surface.

Fig. 3.
Fig. 3.

Transversal electric intensity distributions with different radii of HSILs: R 1 = 10 , R 2 = 50 , R 3 = 100 , R 4 = 500 .

Fig. 4.
Fig. 4.

Longitudinal electric intensity distributions with different radii of HSILs: R 1 = 10 , R 2 = 50 , R 3 = 100 , R 4 = 500 .

Fig. 5.
Fig. 5.

Transversal electric intensity distributions with different radii of ASILs: R 1 = 10 , R 2 = 50 , R 3 = 100 , R 4 = 500 .

Fig. 6.
Fig. 6.

Longitudinal electric intensity distributions with different radii of ASILs: R 1 = 10 , R 2 = 50 , R 3 = 100 , R 4 = 500 .

Fig. 7.
Fig. 7.

Transversal electric intensity distributions with an arbitrary thicknesses: R = 500 , d 1 = 0 , d 2 = R / 2 , d 3 = 2 R , d 4 = 3.5 R .

Fig. 8.
Fig. 8.

Longitudinal electric intensity distributions with arbitrary thicknesses: R = 500 , d 1 = 0 , d 2 = R / 2 , d 3 = 2 R , d 4 = 3.5 R .

Equations (96)

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E ( x , y , z ) = i 2 π Ω a ( k x , k y ) k z e i ( k x x + k y y + k z z ) d k x d k y ,
a ( k x , k y ) = r e i k r E + ( k x , k y ) .
E ( x , y , z ) = i 2 π Ω r e i k r E + ( k x , k y ) k z e i ( k x x + k y y + k z z ) d k x d k y .
k ^ = r ^
a ( k x , k y ) = f e i k f [ E α α ^ + E β β ^ ] .
E + out ( k x , k y ) = f e i k f [ E α α ^ + E β β ^ ] e i k r R = f R e i k ( f R ) [ E α α ^ + E β β ^ ] .
E + in ( k s x , k s y ) = f R e i k ( f R ) [ t p E α α ^ + t s E β β ^ ] .
a in ( k x , k y ) = R e i k s R E + in ( k s x , k s y ) = f e i k f e i ( k s k ) R [ t p E α α ^ + t s E β β ^ ] .
E s ( x , y , z ) = i k s f e i k f 2 π 0 α m sin α d α 0 2 π d β [ t p E α ( cos β cos α sin β cos α sin α ) + t s E β ( sin β cos β 0 ) ] e i ( k s k ) R e i k s [ ρ sin α cos ( β ϕ ) + z cos α ] .
a in ( s x , s y ) = f e i k f [ t p E α α ^ + t s E β β ^ ] .
E ( x , y , z ) = i f e i k f 2 π Ω E + GRS ( k x , k y ) e i ( k x x + k y y + k z z ) 1 k z d k x d k y .
E ( k x , k y ) = i f e i k f d k x d k y 2 π k z E + GRS ( k x , k y ) e i ( k x x + k y y + k z z ) = i f e i k f d k x d k y 2 π k z [ E α α ^ + E β β ^ ] e i k ¯ · r ¯ .
E in ( k x , k y ) = i f e i k f d k x d k y 2 π k z t [ E α α ^ + E β β ^ ] e i k ¯ s · r ¯ .
E ( r ¯ ) = l = 1 m = l l [ p E l m N l m ( r ¯ ) + p M l m M l m ( r ¯ ) ] ,
E r = i ω ε ( 2 r 2 r Π e + k 2 r Π e ) , E θ = i ω ε 1 r 2 r θ r Π e + 1 sin θ ϕ Π m , E ϕ = i ω ε 1 r sin θ 2 r ϕ r Π e θ Π m .
Π e = i ω ε l = 1 m = l l p E l m h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) , Π m = i k l = 1 m = l l p M l m h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) ,
p E l m = i l f e i k f l ( l + 1 ) c l m 0 2 π 0 π ( d P l m ( cos α ) d α E α i m P l m ( cos α ) sin α E β ) e i m β sin α d α d β , p M l m = i l f e i k f l ( l + 1 ) c l m 0 2 π 0 π ( d P l m ( cos α ) d α E β + i m P l m ( cos α ) sin α E α ) e i m β sin α d α d β .
E ( r ¯ ) = l = 1 m = l l [ p E l m N l m ( r ¯ ) + p M l m M l m ( r ¯ ) ] ,
p E l m = l = 1 m = l l [ A l m l m p E l m + i B l m l m p M l m ] , p M l m = l = 1 m = l l [ A l m l m p M l m i B l m l m p E l m ] .
Π e = i ω ε l = 1 m = l l p E l m h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) , Π m = i k l = 1 m = l l p M l m h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) .
Π e = i ω ε l = 1 m = l l p E l m j l ( k r ) Y l m ( θ , ϕ ) , Π m = i k l = 1 m = l l p M l m j l ( k r ) Y l m ( θ , ϕ ) .
E ( r ¯ ) = a ( α ) exp ( i n β ) α ^ .
E ( r ¯ ) = a ( α ) exp ( i n β ) β ^ .
p E l m = p E l n δ m n and p M l m = p M l n δ m n .
E ( r ¯ ) = a ( α ) cos 2 α 2 { ( 1 S ( α ) ) cos β α ^ ( 1 + S ( α ) ) sin β β ^ } ,
p E l m = p E l 1 ( δ m 1 δ m 1 ) and p M l m = p M l 1 ( δ m 1 + δ m 1 ) .
E ( r ¯ ) = l = | n | [ ε ε s c l 2 p E l n N l n ( r ¯ ) + k k s d l 2 p M l n M l n ( r ¯ ) ] .
E ( r ¯ ) = l = 1 m = ± 1 [ ε ε s c l 2 p E l m N l m ( r ¯ ) + k k s d l 2 p M l m M l m ( r ¯ ) ] .
p E l m = δ m n l = | n | [ A l n l n p E l n + i B l n l n p M l n ] , p M l m = δ m n l = | n | [ A l n l n p M l n i B l n l n p E l n ] .
E ( r ¯ ) = l = | n | [ ε ε s c l 2 p E l n N l n ( r ¯ ) + k k s d l 2 p M l n M l n ( r ¯ ) ] .
E ( r ¯ ) = l = | n | [ ε ε s c l p E l n N l n ( r ¯ ) + k k s d l p M l n M l n ( r ¯ ) ] .
p E l m = p E l 1 ( δ m 1 δ m 1 ) , p M l m = p M l 1 ( δ m 1 + δ m 1 ) ,
p E l 1 = l = 1 [ A l 1 l 1 p E l 1 + i B l 1 l 1 p M l 1 ] , p M l 1 = l = 1 [ A l 1 l 1 p M l 1 i B l 1 l 1 p E l 1 ] .
E ( r ¯ ) = l = 1 m = ± 1 [ ε ε s c l 2 p E l m N l m ( r ¯ ) + k k s d l 2 p M l m M l m ( r ¯ ) ] .
E ( r ¯ ) = l = 1 m = ± 1 [ ε ε s c l p E l m N l m ( r ¯ ) + k k s d l p M l m M l m ( r ¯ ) ] .
λ = 1.34 , μ s = μ , ε s = 3.5 2 ε .
E r = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( p E l m [ h l 1 ( 2 ) ( k r ) + h l + 1 ( 2 ) ( k r ) ] P l m ( cos θ ) ) exp ( i m ϕ ) E θ = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( p E l m [ h l 1 ( 2 ) ( k r ) l h l + 1 ( 2 ) ( k r ) l + 1 ] d P l m ( cos θ ) d θ m 2 l + 1 l ( l + 1 ) p M l m h l ( 2 ) ( k r ) P l m ( cos θ ) sin θ ) exp ( i m ϕ ) , E ϕ = i k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( m p E l m [ h l 1 ( 2 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] P l m ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) p M l m h l ( 2 ) ( k r ) d P l m ( cos θ ) d θ ) exp ( i m ϕ ) .
h l ( 2 ) ( k r ) i l + 1 e i k r k r .
E r ( r , α , β ) 0 , E α ( r , α , β ) e i k r r l = 1 m = l l ( i ) l c l m [ p E l m d P l m ( cos α ) d α + i m p M l m P l m ( cos α ) sin α ] e i m β , E β ( r , α , β ) e i k r r l = 1 m = l l ( i ) l c l m [ i m p E l m P l m ( cos α ) sin α p M l m d P l m ( cos α ) d α ] e i m β .
E ( R ¯ ) f R e i k ( f R ) [ E α α ^ + E β β ^ ] .
E r r = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( a l 2 p E l m [ h l 1 ( 1 ) ( k r ) + h l + 1 ( 1 ) ( k r ) ] P l m ( cos θ ) ) exp ( i m ϕ ) , E θ r = k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( a l 2 p E l m [ h l 1 ( 1 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] d P l m ( cos θ ) d θ m 2 l + 1 l ( l + 1 ) b l 2 p M l m h l ( 1 ) ( k r ) P l m ( cos θ ) sin θ ) exp ( i m ϕ ) , E ϕ r = i k l = 1 m = l l l ( l + 1 ) 2 l + 1 c l m ( m a l 2 p E l m [ h l 1 ( 1 ) ( k r ) l h l + 1 ( 1 ) ( k r ) l + 1 ] P l m ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) b l 2 p M l m h l ( 1 ) ( k r ) d P l m ( cos θ ) d θ ) exp ( i m ϕ ) .
a l 2 ( 1 ) l e 2 i k R ε ε s k s k 1 ε ε s k s k + 1 , b l 2 ( 1 ) l e 2 i k R μ μ s k s k 1 μ μ s k s k + 1 , h l ( 1 ) ( k R ) ( i ) l + 1 e i k r k R ,
E r r 0 , E α r ε ε s k s k 1 ε ε s k s k + 1 e i k R R l = 1 m = l l ( i ) l c l m [ p E l m d P l m ( cos α ) d α + i m p M l m P l m ( cos α ) sin α ] e i m β , E β r ε ε s k s k 1 ε ε s k s k + 1 e i k R R l = 1 m = l l ( i ) l c l m [ i m p E l m P l m ( cos α ) sin α p M l m d P l m ( cos α ) d α ] e i m β .
E r ( R ¯ ) ε ε s k s k 1 ε ε s k s k + 1 f R e i k ( f R ) [ E α α ^ + E β β ^ ] .
c l 2 2 k s k + ε ε s k s e i ( k s k ) R , d l 2 2 k s k + μ μ s k s e i ( k s k ) R , h l ( 2 ) ( k R ) i l + 1 e i k R k R ,
E t ( R ¯ ) 2 ε ε s k s k ε ε s k s k + 1 f R e i k ( f R ) [ E α α ^ + E β β ^ ] .
E ( R ¯ ) + E r ( R ¯ ) = E t ( R ¯ ) .
r p = r s = ε ε s k s k 1 ε ε s k s k + 1 , t p = t s = 2 ε ε s k s k ε ε s k s k + 1 .
E ( r ¯ ) = a ( α ) α ^ ,
a ( α ) = sin α for α α m ; and a ( α ) = 0 for α > α m
p E l m = i l f e i k f [ π ( 2 l + 1 ) ] 1 2 l ( l + 1 ) δ m 0 0 α m a ( α ) d P l ( cos α ) d α sin α d α , p M l m = 0 .
E r = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l 2 p E l 0 [ j l 1 ( k s r ) + j l + 1 ( k s r ) ] P l ( cos θ ) , E θ = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l 2 p E l 0 [ j l 1 ( k s r ) l j l + 1 ( k s r ) l + 1 ] d P l ( cos θ ) d θ , E ϕ = 0 .
E ( x , y , z ) = k s f e i k f e i ( k s k ) R ( I 1 cos ϕ I 1 sin ϕ i I 0 ) ,
I 0 = 0 α m t p sin α J 0 ( k s ρ sin α ) e i k s z cos α sin 2 α d α , I 1 = 0 α m t p cos α J 1 ( k s ρ sin α ) e i k s z cos α sin 2 α d α ,
E ( x , y , z ) = k s f e i k f ( I 1 cos ϕ I 1 sin ϕ i I 0 ) .
E ( x , y , z ) = k f e i k f ( I 1 cos ϕ I 1 sin ϕ i I 0 ) .
E ( r ¯ A ) = f r A e i k ( f r A ) [ E α α ^ + E β β ^ ] .
E t ( r ¯ A ) = f r A e i k ( f r A ) [ t p E α α ^ + t s E β β ^ ] .
p E l m = δ m 0 l = 1 A l 0 l 0 p E l 0 , p M l m = 0 .
E r = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l 2 p E l 0 [ j l 1 ( k s r ) + j l + 1 ( k s r ) ] P l ( cos θ ) , E θ = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l 2 p E l 0 [ j l 1 ( k s r ) l j l + 1 ( k s r ) l + 1 ] d P l ( cos θ ) d θ , E ϕ = 0 .
E r = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l p E l 0 [ j l 1 ( k s r ) + j l + 1 ( k s r ) ] P l ( cos θ ) , E θ = k s π l = 1 l ( l + 1 ) 2 l + 1 ε ε s c l p E l 0 [ j l 1 ( k s r ) l j l + 1 ( k s r ) l + 1 ] d P l ( cos θ ) d θ , E ϕ = 0 .
E ( x 1 , y 1 , z 1 ) = k s f e i k f ( I 1 cos ϕ 1 I 1 sin ϕ 1 i I 0 ) ,
I 0 = 0 α m t p E α sin α J 0 ( k s ρ 1 sin α ) e i k s z 1 cos α tan α cos α d α , I 1 = 0 α m t p E α cos α J 1 ( k s ρ 1 sin α ) e i k s z 1 cos α tan α cos α d α .
Ψ l m ( r ¯ ) = l = 0 m = l l Ψ l m ( r ¯ ) α l m l m ,
Ψ l m ( r ¯ ) = { h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) , r < r j l ( k r ) Y l m ( θ , ϕ ) , r > r ,
α l m 00 = { ( 1 ) l + m 4 π j l ( k r ) Y l m ( θ , ϕ ) , r < r ( 1 ) l + m 4 π h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) , r > r
a l m + α l m l + 1 , m = a l m α l m l 1 , m + a l 1 , m + α l 1 , m l m + a l + 1 , m α l + 1 , m l m ,
b l m + α l m l + 1 , m + 1 = b l m α l m l 1 , m + 1 + b l 1 , m 1 + α l 1 m 1 l m + b l + 1 , m 1 α l + 1 , m 1 l m ,
b l l + α l m l + 1 , l + 1 = b l 1 , m 1 + α l 1 , m 1 l l + b l + 1 , m 1 α l + 1 , m 1 l l .
α l m 00 = { 2 l + 1 j l ( k d ) δ m 0 , d < r 2 l + 1 h l ( 2 ) ( k d ) δ m 0 , d > r .
α l m l l = ( 2 l + 1 ) ! ! ( 2 l ) ! ! ( l + l ) ! ( l l ) ! α l 0 00 ( k d ) l δ m l .
α l m m m = ( 2 m + 1 ) ! ! ( 2 m ) ! ! ( l + m ) ! ( l m ) ! α l 0 00 ( k d ) m δ m m , α l m m + 1 , m = ( 2 m + 3 ) ! ! ( 2 m ) ! ! ( l + m ) ! ( l m ) ! α l 0 00 ( k d ) m ( j l ( k d ) j l ( k d ) m k d ) δ m m .
α l m l m = α l m l m δ m m .
α l , m l , m = α l m l m .
M l m ( r ¯ ) = × [ r