Abstract

The details of a model used to predict the scattering of a plane polarized wave by a spherical particle as observed with a microscope are presented. The model accounts for the effect of a refractive interface on the outgoing scattered field and determines the image produced by a lens with high numerical aperture. The predictions of the model are verified by direct comparison with the experimentally observed scattering from polystyrene spheres in a fluid.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. R. Sheppard, 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]
  2. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  3. T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopes and Related Imaging Systems (Academic, New York, 1996).
  4. P. Török, T. Wilson, “Rigorous theory of axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
    [CrossRef]
  5. T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
    [CrossRef]
  6. J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
    [CrossRef]
  7. T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
    [CrossRef]
  8. M. Hammer, D. Schweitzer, B. Michel, E. Thamm, A. Kolb, “Single scattering by red blood cells,” Appl. Opt. 37, 7410–7418 (1998).
    [CrossRef]
  9. S. A. Schaub, D. R. Alexander, J. P. Barton, “Theoretical model of the laser imaging of small aerosols: applications to aerosol sizing,” Appl. Opt. 30, 4777–4784 (1991).
    [CrossRef] [PubMed]
  10. M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
    [CrossRef]
  11. W. Weise, P. Zinin, T. Wilson, A. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
    [CrossRef] [PubMed]
  12. Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
    [CrossRef] [PubMed]
  13. 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]
  14. R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–347 (1993).
    [CrossRef]
  15. P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
    [CrossRef]
  16. B. Ovryn, J. D. Khaydarov, “Forward scattering particle image velocimetry (FSPIV): application of Mie and imaging theory to measure 3D velocities in microscopic flows using partially coherent illumination and high aperture optics,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 243–254 (1997).
    [CrossRef]
  17. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  18. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  19. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  20. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  21. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  22. D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
    [CrossRef]
  23. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [CrossRef]
  24. H. Osterberg, J. E. Wilkins, “The resolving power of a coated objective,” J. Opt. Soc. Am. 39, 553–557 (1949).
    [CrossRef]
  25. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  26. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  27. S. H. Izen, B. Ovryn, “Imaging spheres with general incident wavefronts using a dipole decomposition,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. Conchello, T. Wilson, T. T. Lu, J. M. Lerner, eds., Proc. SPIE3261, 7–16 (1998).
    [CrossRef]
  28. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  29. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

1998 (1)

1997 (5)

T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
[CrossRef]

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

T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[CrossRef]

1996 (3)

1993 (1)

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

1991 (1)

1990 (1)

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

1989 (1)

1987 (1)

1986 (1)

1982 (1)

C. J. R. Sheppard, 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]

1980 (1)

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]

1949 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Agard, D. A.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Alexander, D. R.

Barnes, M. D.

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[CrossRef]

Barton, J. P.

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Boseck, S.

Briggs, A.

Corle, T. R.

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopes and Related Imaging Systems (Academic, New York, 1996).

Crocker, J. C.

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

Flagello, D. G.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grier, D. G.

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

Hammer, M.

Hewlett, S. J.

P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
[CrossRef]

Higdon, P.

T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Hiraoka, Y.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Huffman, D. R.

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

Izen, S. H.

S. H. Izen, B. Ovryn, “Imaging spheres with general incident wavefronts using a dipole decomposition,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. Conchello, T. Wilson, T. T. Lu, J. M. Lerner, eds., Proc. SPIE3261, 7–16 (1998).
[CrossRef]

Juskaitis, R.

T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Kant, R.

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

Khaydarov, J. D.

B. Ovryn, J. D. Khaydarov, “Forward scattering particle image velocimetry (FSPIV): application of Mie and imaging theory to measure 3D velocities in microscopic flows using partially coherent illumination and high aperture optics,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 243–254 (1997).
[CrossRef]

Kino, G. S.

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopes and Related Imaging Systems (Academic, New York, 1996).

Kolb, A.

Lerner, N.

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Mansuripur, M.

Mason, T. G.

T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Matthews, H. J.

Michel, B.

Milster, T.

Osterberg, H.

Ovryn, B.

S. H. Izen, B. Ovryn, “Imaging spheres with general incident wavefronts using a dipole decomposition,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. Conchello, T. Wilson, T. T. Lu, J. M. Lerner, eds., Proc. SPIE3261, 7–16 (1998).
[CrossRef]

B. Ovryn, J. D. Khaydarov, “Forward scattering particle image velocimetry (FSPIV): application of Mie and imaging theory to measure 3D velocities in microscopic flows using partially coherent illumination and high aperture optics,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 243–254 (1997).
[CrossRef]

Ramsey, J. M.

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[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]

Rosenbluth, A. E.

Schaub, S. A.

Schweitzer, D.

Sedat, J. W.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

Sheppard, C. J. R.

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

C. J. R. Sheppard, 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]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Thamm, E.

Török, P.

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

P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
[CrossRef]

van de Hulst, H. C.

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

van Zanten, J. H.

T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

Varga, P.

P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
[CrossRef]

Weise, W.

Whitten, W. B.

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[CrossRef]

Wilkins, J. E.

Wilson, T.

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

T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

W. Weise, P. Zinin, T. Wilson, A. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
[CrossRef] [PubMed]

C. J. R. Sheppard, 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]

Wirtz, D.

T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

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, 6th ed. (Pergamon, New York, 1980).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Zinin, P.

Appl. Opt. (3)

Biophys. J. (1)

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

J. Colloid Interface Sci. (1)

J. C. Crocker, D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[CrossRef]

J. Mod. Opt. (2)

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

P. Török, S. J. Hewlett, P. Varga, “On the series expansion of high-aperture, vectorial diffraction integrals,” J. Mod. Opt. 44, 493–503 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

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

T. Wilson, R. Juskaitis, P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarization microscopy,” Opt. Commun. 141, 298–313 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

T. G. Mason, J. H. van Zanten, D. Wirtz, “Particle tracking microrheology of complex fluids,” Phys. Rev. Lett. 79, 3282–3285 (1997).
[CrossRef]

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

C. J. R. Sheppard, 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]

Proc. R. Soc. London, Ser. A (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]

Rev. Sci. Instrum. (1)

M. D. Barnes, N. Lerner, W. B. Whitten, J. M. Ramsey, “A CCD based approach to high-precision size and refractive index determination of levitated microdroplets using Fraunhofer diffraction,” Rev. Sci. Instrum. 68, 2287–2290 (1997).
[CrossRef]

Other (10)

T. R. Corle, G. S. Kino, Confocal Scanning Optical Microscopes and Related Imaging Systems (Academic, New York, 1996).

B. Ovryn, J. D. Khaydarov, “Forward scattering particle image velocimetry (FSPIV): application of Mie and imaging theory to measure 3D velocities in microscopic flows using partially coherent illumination and high aperture optics,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J. Conchello, T. Wilson, eds., Proc. SPIE2984, 243–254 (1997).
[CrossRef]

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

S. H. Izen, B. Ovryn, “Imaging spheres with general incident wavefronts using a dipole decomposition,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. Conchello, T. Wilson, T. T. Lu, J. M. Lerner, eds., Proc. SPIE3261, 7–16 (1998).
[CrossRef]

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Cited By

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

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Schematic representing the model geometry. A particle, with index of refraction np and radius a, is in a fluid medium with index nf. The particle’s position along the optical axis may change by an amount δ with respect to a reference distance z1 from the fluid–air interface. The quantities z1 and z2 are determined from the experimental parameters by using z1=zf+δ and z2=f-zf/nf, where zf is measured, f is the focal length, and positive δ indicates a greater distance from plane 1. The electromagnetic fields are successively computed over the four labeled planes.

Fig. 2
Fig. 2

Geometry of the basis vectors u^s, u^s, u^sr, u^i, and u^i.

Fig. 3
Fig. 3

Geometry of the fluid–air interface and the parameters that govern the lens transformation.

Fig. 4
Fig. 4

Propagation of the incident field to the image plane.

Fig. 5
Fig. 5

Experimental and predicted scattering shown as a function of transverse distance on the detector plane at six values of δ : (a) -25.70 μm, (b) -6.10 μm, (c) -2.50 μm, (d) 4.00 μm, (e) +14.9 μm, (f) +22.1 μm.

Fig. 6
Fig. 6

(a) Experimental and (b) predicted scattering shown as a function of particle position along the optical axis and transverse distance on the detector plane.

Fig. 7
Fig. 7

Log plots for lc=85 of the coefficients |al| (○) and |bl| (⋅) for 1llc. The coefficients were evaluated by using the experimental parameters.

Fig. 8
Fig. 8

Ratio of the magnitudes of the radial and transverse components of the scattered field, Esu^r/Es-Esu^r, evaluated at the interface, plotted as a function of sin β. The parameters used correspond to experimental values.

Fig. 9
Fig. 9

Relative error in bl (1-{(-i)l[exp(ikr)/(ikr)]}/[hl(1)(kr)]) plotted as a function of l at the interface along the optical axis. The parameters used correspond to experimental values.

Equations (89)

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

E(x, t)=Re[E(x)exp(-iωt)].
Ei=E0u^x exp ikz,
E=Ei+Es.
u^i(ϕ)=cos ϕ u^x+sin ϕ u^y,
u^i(ϕ)=-sin ϕ u^x+cos ϕ u^y.
Ei=(exp ikz)[(E0 cos ϕ)u^i(ϕ)-(E0 sin ϕ)u^i(ϕ)](exp ikz)[Ei(ϕ)u^i(ϕ)+Ei(ϕ)u^i(ϕ)].
u^s(θ, ϕ)u^θ=cos θ cos ϕu^x+cos θ sin ϕ u^y-sin θ u^z,
u^s(ϕ)u^ϕ=-sin ϕ u^x+cos ϕ u^y,
u^sr(θ, ϕ)u^r=sin θ cos ϕ u^x+sin θ sin ϕ u^y+cos θ u^z,
Es(r, θ, ϕ)=Es(r, θ, ϕ)u^s(θ, ϕ)+Es(r, θ, ϕ)u^s(ϕ)+Esr(r, θ, ϕ)u^sr(θ, ϕ).
Es(r, θ, ϕ)=Es(r, θ, ϕ)u^s(θ, ϕ)+Es(r, θ, ϕ)u^s(ϕ).
Es(r, θ, ϕ)=S(r, θ)Ei(ϕ),
Es(r, θ, ϕ)=S(r, θ)Ei(ϕ),
ρ1=z1 tan θ,
Es1-(ρ1, ϕ)=Es(ρ12+z12,tan-1(ρ1/z1),ϕ).
nf sin θ=sin β.
Es1+(ρ1, ϕ)=Es1+(ρ1, ϕ)u^s(β, ϕ)+Es1+(ρ1, ϕ)u^s(ϕ).
Es1+(ρ1, ϕ)=tEs1-(ρ1, ϕ)=tSEi(ϕ),
Es1+(ρ1, ϕ)=tEs1-(ρ1, ϕ)=tSEi(ϕ),
t=2 cos θnfcos β+nf cos θ,
t=2 cos θnfcos θ+nf cos β.
ρ2=z1 tan θ+z2 tan β.
m(θ, β)=dρ2dρ1=1+nf z2 cos3 θz1 cos3 β.
Es2(ρ2, ϕ)=Es2(ρ2, ϕ)u^s(β, ϕ)+Es2(ρ2, ϕ)u^s(ϕ),
Es2(ρ2, ϕ)=ψs12Es1+=ψs12tSEi(ϕ),
Es2(ρ2, ϕ)=ψs12Es1+=ψs12tSEi(ϕ),
ψs12=1m(θ, β) (exp ik0)[(ρ2-ρ1)2+z22]1/2,
zI=zOfzO-f,
zI sin γs=zO sin β,
ρ3=cos βcos γs ρ2.
Es3(ρ3, ϕ)=Es3(ρ3, ϕ)u^s(π-γs, ϕ)+Es3(ρ3, ϕ)u^s(ϕ),
Es3(ρ3, ϕ)=ψs23Es2(ρ2, ϕ)=ψs23ψs12tSEi(ϕ),
Es3(ρ3, ϕ)=ψs23Es2(ρ2, ϕ)=ψs23ψs12tSEi(ϕ),
ψs23=cos γscos β1/2×exp ik0[zO-(ρ22+zO2)1/2]×exp ik0[zI-(ρ32+zI2)1/2].
Ei1-=E0ψi01u^x,
ψi01=exp ikz1.
Ei1+=tEi1-=E0tψi01u^x,
t=2nf1+nf.
Ei2=ψi12Ei1+=E0ψi12tψi01u^x,
ψi12=exp ik0z2.
Ei2(ρ2, ϕ)=Ei2(ϕ)u^i(ϕ)+Ei2(ϕ)u^i(ϕ),
Ei2(ϕ)=ψi12tψi01Ei(ϕ),
Ei2(ϕ)=ψi12tψi01Ei(ϕ).
γi=sin-1(ρ3/f ).
Ei3(ρ3, ϕ)=Ei3(ρ3, ϕ)u^s(π-γi, ϕ)+Ei3(ρ3, ϕ)u^s(ϕ),
Ei3(ρ3, ϕ)=ψi23Ei2(ϕ)=ψi23ψi12tψi01Ei(ϕ),
Ei3(ρ3, ϕ)=ψi23Ei2(ϕ)=ψi23ψi12tψi01Ei(ϕ),
ψi23=cos γi exp ik0[f-(ρ32+f2)1/2].
E3(ρ3, ϕ)=Es3(ρ3, ϕ)+Ei3(ρ3, ϕ)
=Ei3(ρ3, ϕ)u^s(π-γi, ϕ)+Es3(ρ3, ϕ)u^s(π-γs, ϕ)+[Ei3(ρ3, ϕ)+Es3(ρ3, ϕ)]u^s(ϕ).
E4(ρ4, α)=1λz4 expπiλz4 ρ42×0A02πWμ(ρ3)E3(ρ3, ϕ)expπiλz4 ρ32×exp-πiλz4 ρ4ρ3 cos(ϕ-α)ρ3dϕdρ3,
Wμ(ρ3)=1ρ3μA12 1+cos (ρ3-μA)π(1-μ)AμA<ρ3A.
I(ρ4, α)=|E40(ρ4)|2+|E42(ρ4)|2+2 Re{E40(ρ4)[E42(ρ4)]*cos 2α},
Iunpolarized(ρ4)=12π 02πI(ρ4, α)dα=|E40(ρ4)|2+|E42(ρ4)|2.
Mo1l(j)(r, θ, ϕ)=cos ϕ πl(θ)zl(j)(r)u^θ-sin ϕ τl(θ)zl(j)(r)u^ϕ,
Ne1l(j)(r, θ, ϕ)=cos ϕ l(l+1)sin θ πl(θ) zl(j)(r)r u^r+cos ϕ τl(θ) [rzl(j)(r)]r u^θ-sin ϕ πl(θ) [rzl(j)(r)]r u^ϕ,
πl(θ)=1sin θ Pl1(cos θ),
τl(θ)=ddθ Pl1(cos θ).
Ei(r, θ, ϕ)=E0l=1El[Mo1l(1)(kr, θ, ϕ)-iNe1l(1)(kr, θ, ϕ)],
El=il 2l+1l(l+1).
Es(r, θ, ϕ)=E0l=1El[-blMo1l(3)(kr, θ, ϕ)+ialNe1l(3)(kr, θ, ϕ)],
al=nψl(nka)ψ(ka)-ψl(ka)ψ(nka)nψl(nka)ξ(ka)-ξl(ka)ψ(nka),
bl=ψl(nka)ψ(ka)-nψl(ka)ψ(nka)ψl(nka)ξ(ka)-nξl(ka)ψ(nka).
ψl(r)=rjl(r),
ξl(r)=rhl(1)(r),
Es(r, θ, ϕ)=E0l=1lcEl[-blMo1l(3)(kr, θ, ϕ)+ialNe1l(3)(kr, θ, ϕ)].
hl(1)(kr)(-i)l exp ikrikr 1+i l(l+1)2kr,
[hl(1)](kr)(-i)l exp ikrkr 1+i l(l+1)+12kr,
Es=E0l=1lcEl-blπl(θ)hl(1)(kr)+ialτl(θ) [krhl(1)(kr)]krcos ϕ u^θ+Elblτl(θ)hl(1)(kr)-ialπl(θ) [krhl(1)(kr)]krsin θ u^ϕ.
S(r, θ)=E0l=1lcEl-blπl(θ)hl(1)(kr)+ialτl(θ) [krhl(1)(kr)]kr,
S(r, θ)=E0l=1lcElblπl(θ)hl(1)(kr)-ialπl(θ) [krhl(1)(kr)]kr.
Es3(ρ3, ϕ)=Ts3(ρ3)cos ϕ,
Es3(ρ3, ϕ)=-Ts3(ρ3)sin ϕ,
Ei3(ρ3, ϕ)=Ti3(ρ3)cos ϕ,
Ei3(ρ3, ϕ)=-Ti3(ρ3)sin ϕ.
u^s(π-γ, ϕ)=-cos γ cos ϕ u^x+cos γ sin ϕ u^y-sin γ u^z,
u^s(ϕ)=-sin ϕ u^x+cos ϕ u^y,
E3(ρ3, ϕ)=u^x{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]cos2 ϕ+[Ts3(ρ3)+Ti3(ρ3)]sin2 ϕ}+u^y{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]cos ϕ sin ϕ-[Ts3(ρ3)+Ti3(ρ3)]sin ϕ cos ϕ}+u^z{[-Ts3(ρ3)sin γs-Ti3(ρ3)sin γi]cos ϕ}.
E3(ρ3, ϕ)=u^x(12{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]+[Ts3(ρ3)+Ti3(ρ3)]}+12{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]-[Ts3(ρ3)+Ti3(ρ3)]}cos 2ϕ)+u^y(12{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]-[Ts3(ρ3)+Ti3(ρ3)]}sin 2ϕ)+u^z{[-Ts3(ρ3)sin γs-Ti3(ρ3)sin γi]cos ϕ}.
E3(ρ3, ϕ)=[E30(ρ3)+E32(ρ3)cos 2ϕ]u^x+[E32(ρ3)sin 2ϕ]u^y+[E31(ρ3)cos ϕ]u^z,
E30(ρ3)=12{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]+[Ts3(ρ3)+Ti3(ρ3)]},
E31(ρ3)=-Ts3(ρ3)sin γs-Ti3(ρ3)sin γi,
E32(ρ3)=12{[-Ts3(ρ3)cos γs-Ti3(ρ3)cos γi]-[Ts3(ρ3)+Ti3(ρ3)]}.
Jp(ρ)=i-pπ 0π exp(iρ cos ϕ)cos pϕ dϕ.
E4(ρ4, α)=[E40(ρ4)+E42(ρ4)cos 2α]u^x+[E42(ρ4)sin 2α]u^y+[E41(ρ4)cos α]u^z,
E4p(ρ4)=ip πλz4 expπiλz4 ρ420AWμ(ρ3)×expπiλz4 ρ32Jpπρ4ρ3λz4E3ρ(ρ3)ρ3dρ3.
I(ρ4, α)=|E4x|2+|E4y|2
=|E40(ρ4)+E42(ρ4)cos 2α|2+|E42(ρ4)sin 2α|2
=|E40(ρ4)|2+|E42(ρ4)|2+2 Re{E40(ρ4)[E42(ρ4)]*}cos 2α.

Metrics