Abstract

The effect of focusing into a biaxially birefringent medium on the light distribution in the focal region of a high-NA optical system is investigated with the Debye approach to vectorial diffraction theory. Attention is limited to media with small birefringence. The electric field in the focal region is the sum of the field of the two polarization eigenmodes of the biaxially birefringent medium. Both modes are generally astigmatically aberrated, are defocused with respect to each other, and have a polarization field that is nonuniform over the pupil. The diffraction integrals are calculated numerically on the basis of an expansion of the field close to focus in terms of partial waves.

© 2004 Optical Society of America

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    [CrossRef]
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2003 (1)

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

2002 (3)

2001 (2)

2000 (1)

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

1999 (2)

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

M. Berry, R. Bhandari, S. Klein, “Black plastic sandwiches demonstrating biaxial optical anisotropy,” Eur. J. Phys. 20, 1–14 (1999).
[CrossRef]

1998 (1)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

1997 (4)

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

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

1996 (3)

1995 (3)

1993 (3)

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum; Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
[CrossRef]

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. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

1989 (1)

1986 (1)

1984 (1)

1967 (1)

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

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

Bakx, J. L.

Berry, M.

M. Berry, R. Bhandari, S. Klein, “Black plastic sandwiches demonstrating biaxial optical anisotropy,” Eur. J. Phys. 20, 1–14 (1999).
[CrossRef]

Bhandari, R.

M. Berry, R. Bhandari, S. Klein, “Black plastic sandwiches demonstrating biaxial optical anisotropy,” Eur. J. Phys. 20, 1–14 (1999).
[CrossRef]

R. Bhandari, “Polarization of light and topological phases,” Phys. Rep. 281, 1–64 (1997).
[CrossRef]

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Booker, G. R.

Borg, H. J.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980).

Braat, J. J. M.

Coene, W. M.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Dekker, M. K.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Dhayalan, V.

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

Dirksen, P.

Dow, J.

Flagello, D. G.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Goodman, T. D.

Ichimura, I.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jain, M.

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

Janssen, A. J. E. M.

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

Kant, R.

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. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

Kasami, Y.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kawakubo, O.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Klein, S.

M. Berry, R. Bhandari, S. Klein, “Black plastic sandwiches demonstrating biaxial optical anisotropy,” Eur. J. Phys. 20, 1–14 (1999).
[CrossRef]

Kuijper, M.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Laczik, Z.

Lee, S.-W.

Ling, H.

Lotsberg, J. K.

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

Mansuripur, M.

Masuhara, S.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Meinders, E. R.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Milster, T.

Nakano, J.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Németh, G.

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).

Osato, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Pfeffer, N.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Richards, B.

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

Rosenbluth, A. E.

Schep, K.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

Sithambaranathan, G.

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

J. J. Stamnes, G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A 18, 3119–3129 (2001).
[CrossRef]

Spruijt, L.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Stallinga, S.

Stamnes, J. J.

J. J. Stamnes, G. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

J. J. Stamnes, G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A 18, 3119–3129 (2001).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab: I. Exact and asymptotic results,” Pure Appl. Opt. 6, 33–52 (1997).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

ter Meulen, J. M.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Török, P.

Ubbens, I. P.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Varga, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Visser, T. D.

Wiersma, S. H.

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

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, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure if 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. (Cambridge U. Press, Cambridge, UK, 1980).

Yasuda, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 Gbytes,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Appl. Opt. (2)

Eur. J. Phys. (1)

M. Berry, R. Bhandari, S. Klein, “Black plastic sandwiches demonstrating biaxial optical anisotropy,” Eur. J. Phys. 20, 1–14 (1999).
[CrossRef]

J. Mod. Opt. (3)

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. I. Spherical aberration, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (14)

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]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
[CrossRef]

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum; Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
[CrossRef]

S. Stallinga, “Axial birefringence in high-NA optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2858 (2001).
[CrossRef]

H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Reference sphere centered at the focal point O. The radius is R and the maximum polar angle, defined by the aperture of the lens, is β. A point Q on the reference sphere corresponds to a plane wave contributing to the field at focus. The unit vector kˆ along the wave vector is parametrized by the polar angle θ and the azimuthal angle ϕ, where θ is always smaller than β. The electric field is a linear combination of the polarization vectors pˆ and sˆ.

Fig. 2
Fig. 2

Parametrization of the wavevector k by the polar and azimuthal angles θ and ϕ, respectively, and of the position vector rp by spherical coordinates rp, θp, and ϕp and by dimensionless cylindrical coordinates u, v, and ϕp.

Fig. 3
Fig. 3

Polarization fields of the two eigenmodes E+ and E- make an angle χ-ϕ with pˆ and sˆ, respectively. The angle between the eigenpolarizations and the Cartesian x and y axes is therefore χ. This angle χ can vary with the pupil point Q.

Fig. 4
Fig. 4

Aberration functions of the (a) extraordinary and (b) ordinary modes, and polarization fields of the (c) extraordinary and (d) ordinary modes as a function of pupil coordinates for a uniaxially birefringent layer with optic axis parallel to the optical axis. The circle in (c) and (d) indicates the pupil rim. We have taken Δn=0.016, Δn=0, n¯=1.58, NA=0.85, d=100 μm, and λ=0.405 μm. The aberration functions are scaled by 2π, and the average defocus has been subtracted. The arrows have been added for the sake of clarity; their direction is arbitrary from a physical point of view.

Fig. 5
Fig. 5

Aberration functions of the (a) extraordinary and (b) ordinary modes, and polarization fields of the (c) extraordinary and (d) ordinary modes as a function of pupil coordinates for a uniaxially birefringent layer with optic axis perpendicular to the optical axis. The circle in the bottom panels indicates the pupil rim. We have taken Δn=-2Δn=0.016, n¯=1.58, NA=0.85, d=100 μm, and λ=0.405 μm. The aberration functions are scaled by 2π, and the average defocus has been subtracted. The arrows have been added for the sake of clarity; their direction is arbitrary from a physical point of view.  

Fig. 6
Fig. 6

Aberration functions of the (a) extraordinary and (b) ordinary modes and polarization fields of the (c) extraordinary and (d) ordinary modes as a function of pupil coordinates for a biaxially birefringent layer. The circle in (c) and (d) indicates the pupil rim. We have taken Δn=0.016, Δn=0.002, n¯=1.58, NA=0.85, d=100 μm, λ=0.4059 μm. The aberration functions are scaled by 2π, and the average defocus has been subtracted. The arrows have been added for the sake of clarity; their direction is arbitrary from a physical point of view. This is indicated by the opposing arrows on the line connecting the zero-retardation points.

Fig. 7
Fig. 7

Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for the case of axial birefringence. Parameters used are Δn=0.016, n¯=1.58, NA=0.85, d=100 μm, and λ=0.405 μm. The intensity distribution is plotted logarithmically to show relatively fine details.

Fig. 8
Fig. 8

Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for the case of lateral birefringence with the entrance polarization perpendicular to the principal axis (ordinary mode). Parameters used are Δn=2Δn=0.016, n¯=1.58, NA=0.85, d=100 μm, λ=0.405 μm. The intensity distribution is plotted logarithmically to show relatively fine details.

Fig. 9
Fig. 9

Intensity distribution in the meridional planes parallel (left) and perpendicular (right) to the linear entrance polarization for the case of lateral birefringence with the entrance polarization parallel to the principal axis (extraordinary mode). Parameters used are Δn=2Δn=0.016, n¯=1.58, NA=0.85, d=100 μm, and λ=0.405 μm. The intensity distribution is plotted logarithmically to show relatively fine details.

Fig. 10
Fig. 10

Intensity distribution in the meridional xz (left) and yz (right) planes for a linear entrance polarization along the x axis (top) and along the y axis (bottom) for the case of biaxial birefringence with Δn=0.016, Δn=0.002 (so n3>n1>n2), γ=0, n¯=1.58, NA=0.85, d=100 μm, and λ=0.405 μm. The intensity distribution is plotted logarithmically to show relatively fine details.

Equations (103)

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kˆ=(sinθcosϕ,sinθsinϕ,cosθ),
ρ=(ρcosϕ, ρsinϕ)
ρ=sinθ/sinβ.
pˆ=(cosθcosϕ,cosθsinϕ, -sinθ),
sˆ=(-sinϕ,cosϕ, 0).
rp=(rpsinθpcosϕp, rpsinθpsinϕp, rpcosθp).
rp=λNA vcosϕp, λNA vsinϕp, nλNA2 u,
Eα(rp)=E0πNij=1,2Fαj(rp)Aj,
Fαj(rp)=1π sin2 βRdΩcosθl=1,2v^lαJljexp(iΦ)=1πRd2ρl=1,2v^lαJljexp(iΦ),
v^1=cosϕpˆ-sinϕsˆ=(cosθ cos2 ϕ+sin2 ϕ, -(1-cosθ)sinϕcosϕ, -sinθcosϕ),
v^2=sinϕpˆ+cosϕsˆ=(-(1-cosθ)sinϕcosϕ, cos2 ϕ+cosθ sin2 ϕ, -sinθsinϕ),
Φ=krp=nkrp(sinθsinθpcos(ϕ-ϕp)+cosθcosθp)=2πvρcos(ϕ-ϕp)-2πuρ21+(1-ρ2sin2 β)1/2+2πusin2 β.
J=1001.
J=B exp(iW)1001,
B=1cosθ=(1-ρ2sin2 β)-1/4.
J=R(ϕ)tp00tsR(-ϕ)=tpcos2 ϕ+tssin2 ϕ(tp-ts)sinϕcosϕ(tp-ts)sinϕcosϕtscos2 ϕ+tpsin2 ϕ.
R(ϕ)=cosϕ-sinϕsinϕcosϕ,
tp=2n1cosθ1n1cosθ2+n2cosθ1,
ts=2n1cosθ1n1cosθ1+n2cosθ2,
Wp=W+ΔW=W+kdΔn sin2 θcosθ,
Ws=W.
Jbir=R(ϕ)exp(iWp)00exp(iWs)R(-ϕ)=exp(iWp)cos2 ϕ+exp(iWs)sin2 ϕ[exp(iWp)-exp(iWs)]sinϕcosϕ[exp(iWp)-exp(iWs)]sinϕcosϕexp(iWs)cos2 ϕ+exp(iWp)sin2 ϕ.
U0=12ε0n¯2π2E02N2,
Uα=U0j,l=1,2FαjFαl*AjAl*=U0μ=0,3IαμMμ,
Iα0=12(|Fα1|2+|Fα2|2),
Iα1=12(|Fα1|2-|Fα2|2),
Iα2=Re{Fα1Fα2*},
Iα3=Im{Fα1Fα2*},
M0= |A1|2+|A2|2 =1,
M1= |A1|2-|A2|2 =cos2εcos2ξ,
M2=2 Re{A1*A2}=cos2εsin2ξ,
M3=2 Im{A1*A2}=sin2ε.
a^1=(cosγ,sinγ, 0),
a^2=(-sinγ,cosγ, 0),
a^3=(0, 0, 1);
n1=n¯/(1-Δn/n¯)1/2,
n2=n¯/(1+Δn/n¯)1/2,
n3=n¯/(1-2Δn/n¯)1/2,
n1=n¯+Δn/2,
n2=n¯-Δn/2,
n3=n¯+Δn.
J=R(χ)exp(iW+)00exp(iW-)R(-χ)=exp(iW+)cos2 χ+exp(iW-)sin2 χ[exp(iW+)-exp(iW-)]sinχcosχ[exp(iW+)-exp(iW-)]sinχcosχexp(iW+)sin2 χ+exp(iW-)cos2 χ.
W±=kd2cosθ [a±(b2+c2)1/2],
cos(2χ-2ϕ)=b/(b2+c2)1/2,
sin(2χ-2ϕ)=c/(b2+c2)1/2,
a=Δnsin2 θ-12Δnsin2 θcos(2γ-2ϕ),
b=Δnsin2 θ+12Δn(1+cos2 θ)cos(2γ-2ϕ),
c=Δncosθsin(2γ-2ϕ),
W+=2πdΔnsin2 θλcosθ,
W-=0,
W+=πdΔnλcosθ [1-2 sin2 θcos2(γ-ϕ)],
W-=-πdΔnλcosθ.
cos(2χ-2ϕ)=cos(2γ-2ϕ)-sin2 θcos2(γ-ϕ)1-sin2 θcos2(γ-ϕ),
sin(2χ-2ϕ)=cosθsin(2γ-2ϕ)1-sin2 θcos2(γ-ϕ).
tan2 θc=2ΔnΔn+2Δn.
kx=nksinθcosϕ,
ky=nksinθsinϕ,
kz=nkcosθ,
pˆ=(cosθcosϕ,cosθsinϕ, -sinθ),
sˆ=(-sinϕ,cosϕ, 0).
1ε0ω2c2 Dα=[k2δαβ-kαkβ]Eβ.
Dα=Dppα+Dssα.
Dα=ε0εαβEβ,
εαβ=cosγ-sinγ0sinγcosγ0001 n12000n22000n32×cosγsinγ0-sinγcosγ0001,
Eα=1ε0 (Dpεαβ-1pβ+Dsεαβ-1sβ).
M11-1/n2M12M21M22-1/n2 DpDs=00,
M11=pαεαβ-1pβ=cos2 θcos2(γ-ϕ)n12+cos2 θsin2(γ-ϕ)n22+sin2 θn32,
M12=pαεαβ-1sβ=-1n22-1n12cosθ×sin(γ-ϕ)cos(γ-ϕ),
M21=sαεαβ-1pβ=-1n22-1n12cosθ×sin(γ-ϕ)cos(γ-ϕ),
M22=sαεαβ-1sβ=sin2(γ-ϕ)n12+cos2(γ-ϕ)n22.
[M11-1/n2][M22-1/n2]-M122=0.
1n±2=M11+M222M11-M2222+M1221/2.
D+α=ε0A+n+2[cos(χ-ϕ)pα+sin(χ-ϕ)sα],
D-α=ε0A-n-2[-sin(χ-ϕ)pα+cos(χ-ϕ)sα],
E+α=A+n+2[cos(χ-ϕ)εαβ-1pβ+sin(χ-ϕ)εαβ-1sβ]=A+[cos(χ-ϕ)pα+sin(χ-ϕ)sα+e+kα],
E-α=A-n-2[-sin(χ-ϕ)εαβ-1pβ+cos(χ-ϕ)εαβ-1sβ]=A-[-sin(χ-ϕ)pα+cos(χ-ϕ)sα+e-kα],
cos(2χ-2ϕ)=-(M11-M22)/2{[(M11-M22)/2]2+M122}1/2,
sin(2χ-2ϕ)=-M12{[(M11-M22)/2]2+M122}1/2.
kair=k(sinθcosϕ,sinθsinϕ,cosθ),
sinθ=n±sinθ±.
k±=k(sinθcosϕ,sinθsinϕ, (n±2-sin2 θ)1/2),
A+A-=T+pT+sT-pT-s ApAs,
M11+M222=1n¯2-1n¯3Δnsin2 θ±-12 Δnsin2 θ±cos(2γ-2ϕ),
M11-M222=-1n¯3Δnsin2 θ±+12 Δn(1+cos2 θ±)cos(2γ-2ϕ),
M12=-1n¯3 [Δncosθ±sin(2γ-2ϕ)].
1n±2=1n¯2-a±(b2+c2)1/2n¯3,
n±=n¯+a±(b2+c2)1/2,
kz±=k(n±2-sin2 θ)1/2=kn ¯cosθ+k2cosθ [a±(b2+c2)1/2].
W±=(kz±-kn ¯cosθ)d,
cos(χ-ϕ)A+-sin(χ-ϕ)A-
=tp(cosϕA1+sinϕA2),
-sin(χ-ϕ)A++cos(χ-ϕ)A-
=ts(-sinϕA1+cosϕA2),
A+A-=R(-χ)JenA1A2,
Fαj(rp)=14π0βdθsinθ02πdϕVαj(θ, ϕ)exp(ikrp),
Vαj(θ, ϕ)=4cosθsin2 βs=1,2v^αsJsj,
krp=nkrpcosζ=nkrp[cosθcosθp+sinθsinθpcos(ϕ-ϕp)],
exp(inkrpcosζ)=l=0il(2l+1)jl(nkrp)Pl(cosζ),
Pl(cosζ)=4π2l+1m=-llYlm*(θ, ϕ)Ylm(θp, ϕp),
exp(ikrp)
=4πl=0m=-llilYlm*(θ, ϕ)Ylm(θp, ϕp)jl(nkrp).
Fαj(rp)=l=0m=-llFαjlmYlm(θp, ϕp)jl(nkrp),
Fαjlm=(-i)l0βdθsinθ02πdϕYlm*(θ, ϕ)Vαj(θ, ϕ).

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