Abstract

A vectorial diffraction theory that considers light polarization is essential to predict the performance of optical systems that have a high numerical aperture or use engineered polarization or phase. Vectorial diffraction integrals to describe light diffraction typically require boundary fields on aperture surfaces. Estimating such boundary fields can be challenging in complex systems that induce multiple depolarizations, unless vectorial ray tracing using 3×3 Jones matrices is employed. The tracing method, however, has not been sufficiently detailed to cover complex systems and, more importantly, seems influenced by system geometry (transmission versus reflection). Here, we provide a full tutorial on vectorial diffraction calculation in optical systems. We revisit vectorial diffraction integrals and present our approach of consistent vectorial ray tracing irrespective of the system geometry, where both electromagnetic field vectors and ray vectors are traced. Our method is demonstrated in simple optical systems to better deliver our idea, and then in a complex system where point spread function broadening by a conjugate reflector is studied.

© 2018 Optical Society of America

Full Article  |  PDF Article

Corrections

12 March 2018: A typographical correction was made to the final paragraph of page 527.


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References

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2016 (2)

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

2012 (1)

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

2011 (3)

2008 (1)

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

2007 (1)

2005 (3)

2004 (2)

1999 (1)

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

1998 (4)

P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[Crossref]

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

1997 (6)

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5, 205–218 (1997).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[Crossref]

C. J. R. Sheppard and P. Torok, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[Crossref]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[Crossref]

P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36, 2305–2312 (1997).
[Crossref]

1996 (1)

1995 (2)

1994 (1)

1993 (2)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–347 (1993).
[Crossref]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2310 (1993).
[Crossref]

1991 (1)

1987 (1)

1984 (1)

1982 (1)

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[Crossref]

1981 (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
[Crossref]

1966 (1)

1959 (2)

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

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

1952 (1)

S. Inoué, “Studies on depolarization of light at microscope lens surfaces,” Exp. Cell Res. 3, 199–208 (1952).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Acklin, B.

Aieta, F.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Arbabi, A.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Arbabi, E.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Backlund, M. P.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 3rd ed. (AMS Chelsea Publishing, 1987).

Billy, L.

Blanchard, R.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Booker, G. R.

Booth, M. J.

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Botcherby, E. J.

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

Braat, J. J. M.

Capasso, F.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Chen, W. T.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Chipman, R. A.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
[Crossref]

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 3rd ed. (AMS Chelsea Publishing, 1987).

Crabtree, K.

Devlin, R. C.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Dupertuis, M. A.

Faraon, A.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Foreman, M. R.

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58, 339–364 (2011).
[Crossref]

Gaburro, Z.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
[Crossref]

Genevet, P.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Gu, M.

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

Hecht, B.

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

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

Higdon, P.

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[Crossref]

Higdon, P. D.

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

Inoué, S.

S. Inoué, “Studies on depolarization of light at microscope lens surfaces,” Exp. Cell Res. 3, 199–208 (1952).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Juskaitis, R.

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

Juškaitis, R.

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[Crossref]

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2310 (1993).
[Crossref]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–347 (1993).
[Crossref]

Karczewski, B.

Kats, M. A.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Khorasaninejad, M.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Kim, J.

J. Kim, Y. Wang, and X. Zhang, “Comment on ‘Comparison of different theories for focusing through a plane interface’,” J. Opt. Soc. Am. A (in press).

Laczik, Z.

Li, Y.

Matthews, H. J.

McClain, S. C.

Moerner, W. E.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Munro, P. R. T.

Novotny, L.

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

Oh, J.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Peatross, J.

J. Peatross and M. Ware, Physics of Light and Optics (2015), available at http://optics.byu.edu/ .

Pereira, S. F.

Petrov, P. N.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Proctor, M.

Richards, B.

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

Saurabh, S.

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Torok, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5, 205–218 (1997).
[Crossref]

C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[Crossref]

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[Crossref]

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
[Crossref]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Torok, P.

C. J. R. Sheppard and P. Torok, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

Török, P.

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58, 339–364 (2011).
[Crossref]

P. R. T. Munro and P. Török, “Calculation of the image of an arbitrary vectorial electromagnetic field,” Opt. Express 15, 9293–9307 (2007).
[Crossref]

P. R. T. Munro and P. Török, “Vectorial, high-numerical-aperture study of phase-contrast microscopes,” J. Opt. Soc. Am. A 21, 1714–1723 (2004).
[Crossref]

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
[Crossref]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[Crossref]

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5, 205–218 (1997).
[Crossref]

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[Crossref]

P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. 36, 2305–2312 (1997).
[Crossref]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[Crossref]

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]

van de Nes, A. S.

Varga, P.

Visser, T. D.

Wang, L.

Wang, Y.

J. Kim, Y. Wang, and X. Zhang, “Comment on ‘Comparison of different theories for focusing through a plane interface’,” J. Opt. Soc. Am. A (in press).

Ware, M.

J. Peatross and M. Ware, Physics of Light and Optics (2015), available at http://optics.byu.edu/ .

Wiersma, S. H.

Wilson, T.

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[Crossref]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[Crossref]

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[Crossref]

Wolf, E.

Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

B. Karczewski and E. Wolf, “Comparison of three theories of electromagnetic diffraction at an aperture. Part I: coherence matrices,” J. Opt. Soc. Am. 56, 1207–1214 (1966).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London 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 A 253, 358–379 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yu, N.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Yun, G.

Zhang, X.

J. Kim, Y. Wang, and X. Zhang, “Comment on ‘Comparison of different theories for focusing through a plane interface’,” J. Opt. Soc. Am. A (in press).

Zhang, Y.

Zheng, C.

Zhu, A. Y.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Appl. Opt. (3)

Bioimaging (1)

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5, 205–218 (1997).
[Crossref]

Exp. Cell Res. (1)

S. Inoué, “Studies on depolarization of light at microscope lens surfaces,” Exp. Cell Res. 3, 199–208 (1952).
[Crossref]

IEE J. Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
[Crossref]

J. Microsc. (2)

C. J. R. Sheppard and P. Torok, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

P. D. Higdon, P. Török, and T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning optical microscopes,” J. Microsc. 193, 127–141 (1999).
[Crossref]

J. Mod. Opt. (4)

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[Crossref]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2310 (1993).
[Crossref]

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58, 339–364 (2011).
[Crossref]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–347 (1993).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (12)

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[Crossref]

Y. Zhang, L. Wang, and C. Zheng, “Vector propagation of radially polarized Gaussian beams diffracted by an axicon,” J. Opt. Soc. Am. A 22, 2542–2546 (2005).
[Crossref]

P. R. T. Munro and P. Török, “Vectorial, high-numerical-aperture study of phase-contrast microscopes,” J. Opt. Soc. Am. A 21, 1714–1723 (2004).
[Crossref]

Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[Crossref]

P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
[Crossref]

Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22, 68–76 (2005).
[Crossref]

Y. Li, “Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory,” J. Opt. Soc. Am. A 22, 77–83 (2005).
[Crossref]

C. J. R. Sheppard and H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[Crossref]

T. D. Visser and S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
[Crossref]

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]

S. H. Wiersma and T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[Crossref]

M. A. Dupertuis, B. Acklin, and M. Proctor, “Generalization of complex Snell–Descartes and Fresnel laws,” J. Opt. Soc. Am. A 11, 1159–1166 (1994).
[Crossref]

Nano Lett. (1)

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Nat. Photonics (1)

M. P. Backlund, A. Arbabi, P. N. Petrov, E. Arbabi, S. Saurabh, A. Faraon, and W. E. Moerner, “Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask,” Nat. Photonics 10, 459–462 (2016).
[Crossref]

Opt. Commun. (6)

E. J. Botcherby, R. Juskaitis, M. J. Booth, and T. Wilson, “An optical technique for remote focusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[Crossref]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[Crossref]

P. Török, P. Higdon, R. Juškaitis, and T. Wilson, “Optimising the image contrast of conventional and confocal optical microscopes imaging finite sized spherical gold scatterers,” Opt. Commun. 155, 335–341 (1998).
[Crossref]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[Crossref]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[Crossref]

T. Wilson, R. Juškaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997).
[Crossref]

Opt. Eng. (1)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Proc. R. Soc. London A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London 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 A 253, 358–379 (1959).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

C. J. R. Sheppard and T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[Crossref]

Science (1)

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352, 1190–1194 (2016).
[Crossref]

Other (9)

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 3rd ed. (AMS Chelsea Publishing, 1987).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. Kim, Y. Wang, and X. Zhang, “Comment on ‘Comparison of different theories for focusing through a plane interface’,” J. Opt. Soc. Am. A (in press).

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

J. Peatross and M. Ware, Physics of Light and Optics (2015), available at http://optics.byu.edu/ .

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic for diffraction integrals where boundary electromagnetic fields on x determine a field at x . The enclosed surface Σ can be practically reduced to a diffraction aperture in optical systems [2,18]. (b) Light diffraction at an aperture stop in general optical systems could be assumed to occur equivalently at the exit pupil. The field in the image space is calculated by diffraction integrals with E Σ and k ^ Σ on Σ (typically the Gaussian reference sphere surface) traced from the source. ( x , y , z ) is the reference Cartesian coordinate and W ( θ , ϕ ) is the wavefront error in spherical pupil coordinate ( f , θ , ϕ ).
Fig. 2.
Fig. 2. (a) Schematic of aplanatic focusing through a dielectric interface on a meridional plane ( NA = n 1 sin α 1 = 1.4 , λ 0 = 488    nm (vacuum), n 1 = 1.522 , n 2 = 1.337 , and f = 1.8    mm ). Meridional/sagittal fields are marked by red arrows and blue concentric circles, respectively. (b) Comparison of axial intensity when z 1 = 20    μm for x -polarized, uniformly incident light.
Fig. 3.
Fig. 3. Schematic of microscopic imaging of an electric dipole p emitting an object field of E o = ( k ^ o × p ) × k ^ o . All the lenses are assumed aplanatic. Red arrows and blue concentric circles indicate meridional and sagittal fields, respectively.
Fig. 4.
Fig. 4. Schematic of microscopic imaging of an electric dipole object through a linear polarizer.
Fig. 5.
Fig. 5. Theoretical versus experimental PSF in (a) microscopic imaging (theoretical FWHM: 275.4 nm) and (b) imaging with a vertical polarizer [theoretical FWHM: 234.9 nm ( x ), and 329.3 nm ( y )]. Two insets are images of an identical fluorescent bead at 1.4 NA. The anisotropic PSF caused by the polarizer is smaller (larger) along x ( y ) than the isotropic PSF in (a). The paraxial PSF, 2 J 1 ( r ) / r , is for comparison.
Fig. 6.
Fig. 6. Microscopic imaging with a reflector placed at an intermediate focus. In ray tracing, reflection at the PBS’s hypotenuse to image space was neglected and instead a backward propagation model in the green box was considered.
Fig. 7.
Fig. 7. Theoretical PSF with a metallic reflector at intermediate focus. (a) Amplitude and phase of ( r s r p ) / 2 in a silver mirror ( n M = 0.0409 + 2.676 i , oil n 4 = 1.525 , λ 0 = 450    nm ) included in the PSF model. (b) Each intensity term of PSF in Eq. (27) at 1.4 NA. The top two distributions mainly influence the anisotropic PSF shown in the inset in (c). (c) PSF cross-sections for perfect and silver mirrors. Scale bars are 200 nm.

Tables (1)

Tables Icon

Table 1. 3 × 3 Jones Matrices for Vectorial Ray Tracing a

Equations (27)

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

E ( x ) = 1 4 π Σ [ i ω ( N ^ × B Σ ( x ) ) G + ( N ^ × E Σ ( x ) ) × G + ( N ^ · E Σ ( x ) ) G ] d 2 x .
E ( x ) = i k 4 π Σ G [ ( N ^ · k ^ Σ ) E Σ ( N ^ · E Σ ) k ^ Σ ( E Σ · R ^ ) N ^ + ( N ^ · R ^ ) E Σ + ( N ^ · E Σ ) R ^ ] d 2 x .
E ( x ) = i k 2 π Ω E Σ e i k Σ · x d Ω ,
2 π [ cos ( m ϕ ) sin ( m ϕ ) ] e i ρ cos ( ϕ ϕ ) d ϕ = 2 π i m J m ( ρ ) [ cos ( m ϕ ) sin ( m ϕ ) ] ,
E x , y ( x ) = 1 2 π Σ E x , y ( x ) G z d 2 x , E z ( x ) = 1 2 π Σ ( E x ( x ) G x + E y ( x ) G y ) d 2 x ,
E ( x ) = 1 2 π × G ( N ^ × E Σ ) d 2 x .
r p = n 2 cos θ i n 1 cos θ t n 1 cos θ t + n 2 cos θ i , r s = n 1 cos θ i n 2 cos θ t n 1 cos θ i + n 2 cos θ t ,
t p = 2 n 1 cos θ i n 1 cos θ t + n 2 cos θ i , t s = 2 n 1 cos θ i n 1 cos θ i + n 2 cos θ t .
E 2 = R z 1 ( ϕ 0 ) R y s ( θ 2 ) F T R y s ( θ 1 ) C p L ( θ 1 ) R z ( ϕ 0 ) E i = ( n 1 1 cos θ 1 ) 1 2 C p × [ t p cos θ 2 cos 2 ϕ 0 + t s sin 2 ϕ 0 ( t p cos θ 2 t s ) cos ϕ 0 sin ϕ 0 ( t p cos θ 2 t s ) cos ϕ 0 sin ϕ 0 t p cos θ 2 sin 2 ϕ 0 + t s cos 2 ϕ 0 t p sin θ 2 cos ϕ 0 t p sin θ 2 sin ϕ 0 ] [ cos ϕ p sin ϕ p ] , k ^ 2 R z 1 ( ϕ 0 ) R y s ( θ 2 ) F T R y s ( θ 1 ) L ( θ 1 ) R z ( ϕ 0 ) k ^ i [ sin θ 2 cos ϕ 0 ; sin θ 2 sin ϕ 0 ; cos θ 2 ] ,
E x ( z ) = i k 2 4 π o 2 π 0 a e i k 2 R R [ ( R z R + cos θ 2 ) E 2 , x + ( R x R + sin θ 2 cos ϕ 0 ) E 2 , z ] ρ 1 d ρ 1 d ϕ 1 ,
E ( z ) = i k 2 4 0 a C p cos θ 1 n 1 e i k 2 R R [ ( z z 1 R + cos θ 2 ) ( t p cos θ 2 + t s ) + ( ρ 1 R + sin θ 2 ) t p sin θ 2 ] ρ 1 d ρ 1 ,
E 2 = R z 1 ( ϕ o ) L 2 ( θ 2 ) L 1 ( θ 1 ) R z ( ϕ o ) E o = n 1 cos θ 2 cos 1 θ 1 × [ p x ( cos θ 1 cos θ 2 cos 2 ϕ o + sin 2 ϕ o ) + p y ( cos θ 1 × cos θ 2 1 ) cos ϕ o sin ϕ o p z sin θ 1 cos θ 2 cos ϕ o p x ( cos θ 1 cos θ 2 1 ) cos ϕ o sin ϕ o + p y ( cos θ 1 × cos θ 2 sin 2 ϕ o + cos 2 ϕ o ) p z sin θ 1 cos θ 2 sin ϕ o p x cos θ 1 sin θ 2 cos ϕ o + p y cos θ 1 sin θ 2 sin ϕ o p z sin θ 1 sin θ 2 ] , k ^ 2 R z 1 ( ϕ o ) L 2 ( θ 2 ) L 1 ( θ 1 ) R z ( ϕ o ) k ^ o ,
E = π [ U 0 ( 1 ) + U 2 ( 1 ) cos ( 2 ϕ ) U 2 ( 1 ) sin ( 2 ϕ ) 2 i U 1 ( 1 ) cos ϕ U 2 ( 1 ) sin ( 2 ϕ ) U 0 ( 1 ) U 2 ( 1 ) cos ( 2 ϕ ) 2 i U 1 ( 1 ) sin ϕ 2 i U 1 ( 2 ) cos ϕ 2 i U 1 ( 2 ) sin ϕ 2 U 0 ( 2 ) ] p ,
U p ( q ) = i k 2 2 π 0 α 2 n 1 cos θ 2 cos θ 1 F p ( q ) J p ( k 2 ρ sin θ 2 ) × exp ( i k 2 z cos θ 2 ) sin θ 2 d θ 2
F 0 ( 1 ) = 1 + cos θ 1 cos θ 2 , F 0 ( 2 ) = sin θ 1 sin θ 2 , F 1 ( 1 ) = sin θ 1 cos θ 2 , F 1 ( 2 ) = cos θ 1 sin θ 2 , F 2 ( 1 ) = 1 cos θ 1 cos θ 2 .
E c = R z 1 ( π + ϕ o ) L 3 ( θ 2 ) R z ( π + ϕ o ) × R z 1 ( ϕ o ) L 2 ( θ 2 ) L 1 ( θ 1 ) R z ( ϕ o ) E o ,
E 4 = R z 1 ( π + ϕ o ) L 4 ( θ 4 ) R z ( π + ϕ o ) P ( ψ ) E c .
E = π 4 [ p x ( U 0 ( 1 ) U 4 ( 1 ) cos ( 4 ϕ ) ) + p y ( U 2 ( 1 ) sin ( 2 ϕ ) U 4 ( 1 ) sin ( 4 ϕ ) ) + p z ( i U 1 ( 1 ) cos ϕ + i U 3 ( 1 ) cos ( 3 ϕ ) ) p x ( U 2 ( 2 ) sin ( 2 ϕ ) U 4 ( 1 ) sin ( 4 ϕ ) ) + p y ( U 0 ( 2 ) U 2 ( 3 ) cos ( 2 ϕ ) + U 4 ( 1 ) cos ( 4 ϕ ) ) + p z ( i U 1 ( 2 ) sin ϕ + i U 3 ( 1 ) sin ( 3 ϕ ) ) p x ( i U 1 ( 3 ) cos ϕ + i U 3 ( 2 ) cos ( 3 ϕ ) ) + p y ( i U 1 ( 4 ) sin ϕ + i U 3 ( 2 ) sin ( 3 ϕ ) ) + p z ( U 0 ( 3 ) + U 2 ( 4 ) cos ( 2 ϕ ) ) ] ,
U p ( q ) = i k 4 2 π 0 α 4 n 1 cos θ 4 cos θ 1 F p ( q ) J p ( k 4 ρ sin θ 4 ) × exp ( i k 4 z cos θ 4 ) sin θ 4 d θ 4
F 0 ( 1 ) = F 4 ( 1 ) = ( 1 cos θ 1 ) ( 1 cos θ 4 ) , F 0 ( 2 ) = 3 + cos θ 1 + cos θ 4 + 3 cos θ 1 cos θ 4 , F 0 ( 3 ) = F 2 ( 4 ) = 4 sin θ 1 sin θ 4 , F 1 ( 1 ) = F 3 ( 1 ) = 2 sin θ 1 ( 1 cos θ 4 ) , F 1 ( 2 ) = 2 sin θ 1 ( 1 + 3 cos θ 4 ) , F 1 ( 3 ) = F 3 ( 2 ) = 2 ( 1 cos θ 1 ) sin θ 4 , F 1 ( 4 ) = 2 ( 1 + 3 cos θ 1 ) sin θ 4 , F 2 ( 1 ) = 2 ( 1 + cos θ 1 ) ( 1 cos θ 4 ) , F 2 ( 2 ) = 2 ( 1 cos θ 1 ) ( 1 + cos θ 4 ) , F 2 ( 3 ) = 4 ( 1 cos θ 1 cos θ 4 ) .
I = | U 0 ( 2 ) | 2 + | U 1 ( 2 ) | 2 sin 2 ϕ 2 R { U 0 ( 2 ) U 2 ( 3 ) * } cos ( 2 ϕ ) + | U 2 ( 3 ) | 2 ,
E c 2 = P ( 90 ° ) W ( 90 ° , 45 ° ) R z 1 ( ϕ o ) L 4 ( θ 4 ) × R y s 1 ( π θ 4 ) F R R y s ( θ 4 ) R z ( ϕ o ) × R z 1 ( π + ϕ o ) L 4 ( θ 4 ) R z ( π + ϕ o ) W ( 90 ° , 45 ° ) P ( 0 ° ) E c = [ 0 0 0 i ( r p r s ) 2 0 0 0 0 1 ] E c ,
r p = n M 2 cos θ 4 n 4 n M 2 n 4 2 sin 2 θ 4 n M 2 cos θ 4 + n 4 n M 2 n 4 2 sin 2 θ 4 , r s = n 4 cos θ 4 n M 2 n 4 2 sin 2 θ 4 n 4 cos θ 4 + n M 2 n 4 2 sin 2 θ 4 .
E = π 4 [ p x ( U 2 ( 3 ) sin ( 2 ϕ ) + U 4 ( 1 ) sin ( 4 ϕ ) ) + p y ( U 0 ( 1 ) U 4 ( 1 ) cos ( 4 ϕ ) ) + p z ( i U 1 ( 1 ) sin ϕ + i U 3 ( 1 ) sin ( 3 ϕ ) ) p x ( U 0 ( 2 ) + U 2 ( 1 ) cos ( 2 ϕ ) U 4 ( 1 ) cos ( 4 ϕ ) ) + p y ( U 2 ( 2 ) sin ( 2 ϕ ) U 4 ( 1 ) sin ( 4 ϕ ) ) + p z ( i U 1 ( 2 ) cos ϕ i U 3 ( 1 ) cos ( 3 ϕ ) ) p x ( i U 1 ( 4 ) sin ϕ + i U 3 ( 2 ) sin ( 3 ϕ ) ) + p y ( i U 1 ( 3 ) cos ϕ i U 3 ( 2 ) cos ( 3 ϕ ) ) + p z ( U 2 ( 4 ) sin ( 2 ϕ ) ) ] ,
U p ( q ) = i k 5 2 π 0 α 5 n 1 cos θ 5 cos θ 1 i ( r s r p ) 2 F p ( q ) J p ( k 5 ρ sin θ 5 ) × exp ( i k 5 z cos θ 5 ) sin θ 5 d θ 5
F 0 ( 1 ) = F 4 ( 1 ) = ( 1 cos θ 1 ) ( 1 cos θ 5 ) , F 0 ( 2 ) = 1 + 3 cos θ 1 + 3 cos θ 5 + cos θ 1 cos θ 5 , F 1 ( 1 ) = F 3 ( 1 ) = 2 sin θ 1 ( 1 cos θ 5 ) , F 1 ( 2 ) = 2 sin θ 1 ( 3 + cos θ 5 ) , F 1 ( 3 ) = F 3 ( 2 ) = 2 ( 1 cos θ 1 ) sin θ 5 , F 1 ( 4 ) = 2 ( 3 + cos θ 1 ) sin θ 5 , F 2 ( 1 ) = 4 ( cos θ 5 cos θ 1 ) , F 2 ( 2 ) = 2 ( 1 cos θ 1 ) ( 1 + cos θ 5 ) , F 2 ( 3 ) = 2 ( 1 + cos θ 1 ) ( 1 cos θ 5 ) , F 2 ( 4 ) = 4 sin θ 1 sin θ 5 ,
I = | U 0 ( 2 ) | 2 + | U 1 ( 2 ) | 2 cos 2 ϕ + 2 R { U 0 ( 2 ) U 2 ( 1 ) * } cos ( 2 ϕ ) + | U 2 ( 1 ) | 2 ,

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