Abstract

An optical system consisting of two objective lenses in a confocal arrangement is examined. It is shown that a simple algebraic relation exists between the electric field in the back focal plane of the first objective lens, which focuses the incident light, and the Fourier transform of the electric field in the focal plane of the same lens. The relation holds for high angles. If a thin object is placed in the focal plane it is possible to write the electric field by use of a Fourier transform relation at the exit aperture of the second lens. The theory is generalized for objects that are positioned at oblique angles with respect to the optical axis of the system. This configuration is clearly identical to the setup of a spatially resolving ellipsometer.

© 2000 Optical Society of America

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References

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  1. M. Erman, J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60, 859–873 (1986).
    [CrossRef]
  2. K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).
  3. 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]
  4. P. Török, P. Varga, Z. Laczik, 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]
  5. P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
    [CrossRef]
  6. V. S. Ignatovsky, “Diffraction by a paraboloidal mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).
  7. P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloidal mirrors. I. Theory,” J. Opt. Soc. Am. A (to be published).
  8. 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]
  9. C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
    [CrossRef]
  10. 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]
  11. S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
    [CrossRef]
  12. P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [CrossRef]
  13. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  14. 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]
  15. P. Török, P. D. Higdon, R. Juškaitis, 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]
  16. P. D. Higdon, P. Török, T. Wilson, “Imaging properties of high aperture multiphoton fluorescence scanning microscopes,” J. Microsc. 193, 127–141 (1999).
    [CrossRef]
  17. P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopy,” Opt. Commun. 181, 7–18 (2000).
    [CrossRef]
  18. C. J. R. Sheppard, P. Török, “An electromagnetic theory of imaging in fluorescence microscopy and imaging in polarization fluorescence microscopy,” Bioimag. 5, 205–218 (1997).
    [CrossRef]
  19. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986).
  20. R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1996).
  21. W. Wang, A. T. Friberg, E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947–1953 (1995).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968).
  23. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1985).

2000 (1)

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopy,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

1999 (2)

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

1998 (6)

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

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

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

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

1997 (1)

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

1995 (3)

1986 (1)

M. Erman, J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60, 859–873 (1986).
[CrossRef]

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]

1973 (1)

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

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]

1920 (1)

V. S. Ignatovsky, “Diffraction by a paraboloidal mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

Booker, R.

Erman, M.

M. Erman, J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60, 859–873 (1986).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968).

Higdon, P. D.

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

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

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

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a paraboloidal mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

Jiang, D.

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]

Juškaitis, R.

P. Török, P. D. Higdon, R. Juškaitis, 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]

S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
[CrossRef]

Laczik, Z.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1996).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1985).

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]

Semenko, A. I.

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

Semenko, L. V.

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

Shatalin, S.

S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (1998).
[CrossRef]

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

Sokolov, V. K.

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

Stamnes, J. J.

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]

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

Svitashev, K. K.

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

Tan, J. B.

S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
[CrossRef]

Theeten, J. B.

M. Erman, J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60, 859–873 (1986).
[CrossRef]

Török, P.

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopy,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

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

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

C. J. R. Sheppard, P. Török, “Dependence of Fresnel number on aperture stop position,” J. Opt. Soc. Am. A 15, 3016–3019 (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. D. Higdon, R. Juškaitis, 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. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

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

P. Török, P. Varga, Z. Laczik, 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]

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloidal mirrors. I. Theory,” J. Opt. Soc. Am. A (to be published).

Varga, P.

Wang, W.

Wilson, T.

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

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

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

S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
[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]

Wolf, E.

W. Wang, A. T. Friberg, E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947–1953 (1995).
[CrossRef]

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]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1985).

Bioimag. (1)

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

J. Appl. Phys. (1)

M. Erman, J. B. Theeten, “Spatially resolved ellipsometry,” J. Appl. Phys. 60, 859–873 (1986).
[CrossRef]

J. Microsc. (2)

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

S. Shatalin, R. Juškaitis, J. B. Tan, T. Wilson, “Reflection conoscopy and microellipsometry of isotropic thin film structures,” J. Microsc. 179, 241–252 (1995).
[CrossRef]

J. Mod. Opt. (1)

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

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

Opt. Commun. (5)

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

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

P. Török, P. D. Higdon, R. Juškaitis, 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, “Imaging of small birefringent objects by polarised light conventional and confocal microscopy,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

Opt. Spectrosc. (1)

K. K. Svitashev, A. I. Semenko, L. V. Semenko, V. K. Sokolov, “Ellipsometer based on convergent light beam,” Opt. Spectrosc. 34, 542–544 (1973).

Proc. R. Soc. London Ser. A (2)

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]

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]

Trans. Opt. Inst. (1)

V. S. Ignatovsky, “Diffraction by a paraboloidal mirror with arbitrary aperture,” Trans. Opt. Inst. 1(5), 1–30 (1920).

Other (5)

P. Varga, P. Török, “Focusing of electromagnetic waves by paraboloidal mirrors. I. Theory,” J. Opt. Soc. Am. A (to be published).

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

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. of California Press, Berkeley, Calif., 1996).

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1985).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram showing the notation used for the steady coordinate system.

Fig. 2
Fig. 2

Schematic diagram showing the notation used for the transformed coordinate system.

Equations (23)

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

Ein=cos α expiβxex+sin α expiβyey×exp-iωt-kz,
E1x1, y1, z1; tE1f1s=UsEinz=-f1; t,
Us=|sz|1/2szsx2+sy2σ-2-1-szsxsyσ-2sx-1-szsxsyσ-2szsy2+sx2σ-2sy-sx-sysz,
E-x, y, z; t=-ikf12πΩ1szE1f1s; t×expikxsx+ysy+zszdsxdsy,
E-x, y, 0; t=-ikf12πΩ1szE1f1sx, f1sy; t×expikxsx+ysydsxdsy.
Fkx, ky=expiωt2π-E-x, y, 0; t×exp-ikxx+kyydxdy.
Fkx, ky=if12π expiωt1kzE1f1kxk, f1kyk; t.
Tkx, kyFkx, ky,
Tkx, ky=τxxτxyτxzτyxτyyτyzτzxτzyτzz
E+x, y, z=exp-iωt-Tkx, kyFkx, ky×expikxx+kyy+kzzdkxdky.
Gx, y, z=-gsx, syexpikxsx+ysy+zszdsxdsy
Gx, y, z-2πikzrexpikrrgxr, yr,
E2f2s=-f1f2Tk x2f2, y2f2E1f1f2 x2, f1f2 y2,
E2S=E2f2s=-f1f2TksE1f1s.
Eout =U-1E2,
EoutP2=-f1f2U-1sTsUsEin,
v=Mv,  v=M-1v,
M=cos Φ0-sin Φ010sin Φ0cos Φ
E1f1s=UsEinz=-f1; t,
E-x, y, z; t=-ikf12πΩ1szsE1f1sx, f1sy, f1sz×expikxsx+ysy+zsz×cos Φ-sxsz sin Φdsxdsy.
Fekz, ky=-if12π expiωt1kxcos Φ-kxkz sin Φ×E1f1kxk, f1kyk
E+x, y, z=exp-iωt-Tekx, kyFekx, ky×expi-kxx+kyy+kzzdkxdky.
EoutP2=-f1f2cos Φ+sxsz sin Φ×U-1tTetUtEin,

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