Abstract

We consider the focusing of two-dimensional scalar and electromagnetic waves through a slit aperture in a perfectly reflecting screen and derive exact solutions that are valid everywhere in the region behind the diffracting slit. Corresponding solutions based on various approximate theories are also presented. Numerical comparisons between exact and approximate results are presented in the special issue on mathematics and modeling in modern optics, J. Opt. Soc. Am. A 15(5), (1998).

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Debye, “Das Verhalten von Licht Wellen in der Nähe eines Brennpunktes oder eine Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
    [CrossRef]
  2. V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I (1919); “Diffraction by a parabolic mirror having arbitrary opening” (1920), papers IV and V, respectively (in Russian).
  3. E. Wolf, “Electromagnetic diffractions in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  4. 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]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).
  6. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  7. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  8. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  9. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
    [CrossRef]
  10. D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Physics Laser Tech. Bull. 5, 1–10 (1966).
  11. A. Yoshida, T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).
  12. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [CrossRef]
  13. R. Kant, “An analytical solution of vector diffraction problems for focusing optical systems,” J. Mod. Opt. 40, 337–347 (1993).
    [CrossRef]
  14. C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves Opt. Acoust. 1, 129–132 (1977).
    [CrossRef]
  15. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberrations, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
    [CrossRef]
  16. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. II. Astigmatism and coma,” J. Mod. Opt. 42, 299–320 (1995).
    [CrossRef]
  17. P. Török, P. Varga, Z. Laczic, 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]
  18. C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
    [CrossRef]
  19. J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
    [CrossRef]
  20. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  21. T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high-aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
    [CrossRef]
  22. T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
    [CrossRef]
  23. T. D. Visser, S. H. Wiersma, “Electromagnetic description of the image formation in confocal fluorescence microscopy,” J. Opt. Soc. Am. A 11, 599–608 (1994).
    [CrossRef]
  24. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  25. V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
    [CrossRef]
  26. G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
    [CrossRef]
  27. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  28. J. J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841–855 (1995). Note that there are misprints in Eqs. (31)–(35) in this reference, implying that y>0 and y<0 in those equations are to be replaced by y<0 and y>0, respectively.
    [CrossRef]
  29. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
    [CrossRef]
  30. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
    [CrossRef]
  31. J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
    [CrossRef]
  32. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).
  33. Equations (13.21a) and (13.21b) of Ref. 6 with P1=1 and Q1=0.
  34. Equation (10.14a) of Ref. 6 with θ1=π/2.
  35. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
    [CrossRef]
  36. Equations (18a) and (18b) of Ref. 28, with y being replaced by z.
  37. Equation (21c) of Ref. 28, with z being replaced by y.
  38. Section 4.3.1 of Ref. 6.
  39. Section 4.3.2 of Ref. 6.
  40. Section 5.2 of Ref. 6.
  41. Section 4.2 of Ref. 6.
  42. Equation (4.16c) of Ref. 6.
  43. Section 5.1 of Ref. 6.
  44. Equation (10.14a) of Ref. 6.

1998 (2)

1997 (1)

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

1996 (1)

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

1995 (4)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. II. Astigmatism and coma,” J. Mod. Opt. 42, 299–320 (1995).
[CrossRef]

J. J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841–855 (1995). Note that there are misprints in Eqs. (31)–(35) in this reference, implying that y>0 and y<0 in those equations are to be replaced by y<0 and y>0, respectively.
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

P. Török, P. Varga, Z. Laczic, 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]

1994 (3)

1993 (2)

R. Kant, “An analytical solution of vector diffraction problems 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 aberrations, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

1992 (1)

1991 (1)

1989 (1)

1984 (1)

1983 (2)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
[CrossRef]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1981 (1)

1977 (1)

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

1974 (1)

A. Yoshida, T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

1967 (1)

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

1966 (1)

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Physics Laser Tech. Bull. 5, 1–10 (1966).

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 diffractions 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 of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1957 (1)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I (1919); “Diffraction by a parabolic mirror having arbitrary opening” (1920), papers IV and V, respectively (in Russian).

1909 (1)

P. Debye, “Das Verhalten von Licht Wellen in der Nähe eines Brennpunktes oder eine Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Asakura, T.

A. Yoshida, T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

Barakat, R.

Bloom, A. L.

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Physics Laser Tech. Bull. 5, 1–10 (1966).

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 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.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

Choudhury, A.

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

Debye, P.

P. Debye, “Das Verhalten von Licht Wellen in der Nähe eines Brennpunktes oder eine Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

Dow, J.

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

Eide, H. A.

Farnell, G. W.

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

Gannaway, J.

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

Hsu, W.

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I (1919); “Diffraction by a parabolic mirror having arbitrary opening” (1920), papers IV and V, respectively (in Russian).

Innes, D. J.

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Physics Laser Tech. Bull. 5, 1–10 (1966).

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. II. Astigmatism and coma,” J. Mod. Opt. 42, 299–320 (1995).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberrations, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

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

Laczic, Z.

Larkin, K. G.

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

Lee, S. W.

Li, Y.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Ling, H.

Mansuripur, M.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[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]

Sheppard, C. J. R.

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

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

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
[CrossRef]

Stamnes, J. J.

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

J. J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841–855 (1995). Note that there are misprints in Eqs. (31)–(35) in this reference, implying that y>0 and y<0 in those equations are to be replaced by y<0 and y>0, respectively.
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
[CrossRef]

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 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]

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]

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

Yoshida, A.

A. Yoshida, T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

Ann. Phys. (Leipzig) (1)

P. Debye, “Das Verhalten von Licht Wellen in der Nähe eines Brennpunktes oder eine Brennlinie,” Ann. Phys. (Leipzig) 30, 755–776 (1909).
[CrossRef]

Can. J. Phys. (1)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

IEE J. Microwaves Opt. Acoust. (1)

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

J. Mod. Opt. (4)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. I. Spherical aberrations, curvature of field, and distortion,” J. Mod. Opt. 40, 2293–2310 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations. II. Astigmatism and coma,” J. Mod. Opt. 42, 299–320 (1995).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[CrossRef]

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

T. D. Visser, S. H. Wiersma, “Electromagnetic description of the image formation in confocal fluorescence microscopy,” J. Opt. Soc. Am. A 11, 599–608 (1994).
[CrossRef]

W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
[CrossRef]

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

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

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

T. D. Visser, 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. Laczic, 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]

Opt. Acta (2)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983); see also Sec. 7.2 of Ref. 6.
[CrossRef]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Optik (Stuttgart) (1)

A. Yoshida, T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

Phys. Rev. B (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]

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

E. Wolf, “Electromagnetic diffractions 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 of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt. (4)

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders,” Pure Appl. Opt. 4, 251–262 (1995).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, “Exact two-dimensional scattering of an arbitrary wave by perfectly reflecting elliptical cylinders, strips and slits,” Pure Appl. Opt. 4, 841–855 (1995). Note that there are misprints in Eqs. (31)–(35) in this reference, implying that y>0 and y<0 in those equations are to be replaced by y<0 and y>0, respectively.
[CrossRef]

Spectra-Physics Laser Tech. Bull. (1)

D. J. Innes, A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Physics Laser Tech. Bull. 5, 1–10 (1966).

Trans. Opt. Inst. Petrograd I (1)

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I (1919); “Diffraction by a parabolic mirror having arbitrary opening” (1920), papers IV and V, respectively (in Russian).

Other (14)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

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

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968).

Equations (13.21a) and (13.21b) of Ref. 6 with P1=1 and Q1=0.

Equation (10.14a) of Ref. 6 with θ1=π/2.

Equations (18a) and (18b) of Ref. 28, with y being replaced by z.

Equation (21c) of Ref. 28, with z being replaced by y.

Section 4.3.1 of Ref. 6.

Section 4.3.2 of Ref. 6.

Section 5.2 of Ref. 6.

Section 4.2 of Ref. 6.

Equation (4.16c) of Ref. 6.

Section 5.1 of Ref. 6.

Equation (10.14a) of Ref. 6.

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 (2)

Fig. 1
Fig. 1

Plane wave with propagation vector k=k(xˆ cos ϕ+zˆ sin ϕ) diffracted through a slit of width 2a in a perfectly reflecting screen in the plane z=0.

Fig. 2
Fig. 2

Converging wave with focal point (0, z1) diffracted through a slit of width 2a in a perfectly reflecting screen in the plane z=0.

Equations (94)

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

uPWi(x, z, ϕ)=exp[ik(x cos ϕ+z sin ϕ)],
uPW,slits=-uPW,stripsc,h,
uPW,slith=-uPW,stripsc,s.
uPW,slits=m=0Fom(s, u, v)Som(s, ϕ),
uPW,slith=m=0Fem(s, u, v)Sem(s, ϕ),
Fpm(s, u, v)=Bpm(s)Hpm(1)(s, u)Spm(s, v),
Bpm(s)=(8π)1/2 imNpmJpm(s, 0)Hpm(1)(s, 0),
Fpm(s, u, v)=Bpm(s)Hpm(1)(s, u)Spm(s, v),
Bpm(s)=(8π)1/2 imNpmJpm(s, 0)Hpm(1)(s,0),
s=(½kd)2=(ka)2
x2=a cosh u cos v,z2=a sinh u sin v.
ui(x, z)=0πA(ϕ)uPWi(x, z, ϕ)dϕ,
A(ϕ)=k2πsin ϕ U(k cos ϕ),
U(kx)=-ui(x, 0)exp(-ikxx)dx.
uslits=0πA(ϕ)uPW,slitsdϕ=m=0Fom(s, u, v)Com,
uslith=0πA(ϕ)uPW,slithdϕ=m=0Fem(s, u, v)Cem,
Cpm(s)=0πA(ϕ)Spm(s, ϕ)dϕ.
Ce2r(s)=k=0De2k(2r)(s)Le2k,
Ce2r+1(s)=k=0De2k+1(2r+1)(s)Le2k+1,
Co2r(s)=k=1Do2k(2r)(s)Lo2k,
Co2r+1(s)=k=0Do2k+1(2r+1)(s)Lo2k+1,
Len=0πA(ϕ)cos(nϕ)dϕ,
Lon=0πA(ϕ)sin(nϕ)dϕ.
ui,d(x, 0)=-12iz1H0(2)(kR1)=kz12iR1H1(2)(kR1),
R1=(x2+z12)1/2
ui,d(x, 0)=1(λz1)1/2z1R13/2 expiπ4-kR1.
Ud(kx)=exp[-i(k2-kx2)1/2z1]if|kx|k0otherwise.
Ad(ϕ)=k2πsin ϕ exp(-ikz1 sin ϕ),
Lend=Lend(kz1),Lond=Lond(kz1),
Lend(w)=½[Lnd+(w)+Lnd-(w)],
Lond(w)=12i[Lnd+(w)-Lnd-(w)],
Lnd±(w)=k2π0π sin t exp[-i(w sin tnt)]dt.
ui,c(x, 0)=12kH0(2)(kR1)1(λz1)1/2z1R11/2×exp[i(π/4-kR1)],
Uc(kx)=Ud(kx)/[1-(kx/k)2]1/2
Ac(ϕ)=Ad(ϕ)/sin ϕ=k2πexp(-ikz1 sin ϕ).
Lpnc=Lpnc(kz1)(p=o,e),
Lenc(w)=½[Lnc+(w)+Lnc-(w)],
Lonc(w)=12i[Lnc+(w)-Lnc-(w)],
Lnc±(w)=k2π0π exp[-i(w sin tnt)]dt.
E=yˆEy.
H=ciωμ×E=ciωμ-xˆ Eyz+zˆ Eyx.
Ei=yˆui(x, z)=yˆ0πA(ϕ)uPWi(x, z, ϕ)dϕ,
Hi=ciωμ-xˆ uiz+zˆ uix,
uix=ik0πA(ϕ)cos ϕ uPWi(x, z, ϕ)dϕ,
uiz=ik0πA(ϕ)sin ϕ uPWi(x, z, ϕ)dϕ.
Eslit=yˆuslits,
Hslit=ciωμ-xˆ uslitsz2+zˆ uslitsx2,
uslitsx2=m=0Bom(s)ComXom,
uslitsz2=m=0Bom(s)ComZom,
Xpm=x2[Hpm(1)(s, u)Spm(s, v)]
=ux2Hpm(1)(s, u)Spm(s, v)+vx2Hpm(1)(s, u)Spm(s, v),
Zpm=z2[Hpm(1)(s, u)Spm(s, v)]
=uz2Hpm(1)(s, u)Spm(s, v)+vz2Hpm(1)(s, u)Spm(s, v).
ux2=vz2=2dsinh u cos vcosh2 u-cos2 v,
uz2=-vx2=2dcosh u sin vcosh2 u-cos2 v.
H=yˆHy,
E=icω×H=icω-xˆ Hyz+zˆ Hyx.
Hi=yˆui(x, z)=yˆ0πA(ϕ)uPWi(x, z, ϕ)dϕ,
Ei=icω-xˆ uiz+zˆ uix,
Hslit=yˆuslith,
Eslit=icω-xˆ uslithz2+zˆ uslithx2,
uslithx2=m=0Bem(s)Cem(s)Xem,
uslithz2=m=0Bem(s)Cem(s)Zem.
uI(x2, z2)=12i-u(x, 0) z2H0(1)(kR2)dx,
R=[(x2-x)2+z22]1/2.
uII(x2, z2)=12i- u(x, z)zz=0H0(1)(kR2)dx.
uK(x2, z2)=½[uI(x2, z2)+uII(x2, z2)].
uId(x2, z2)=k24-aa z1z2R1R2H1(2)(kR1)H1(1)(kR2)dx,
ui,d(x, z)=12izH0(2)[kR1(z)]=-k2iz-z1R1(z)H1(2)[kR1(z)],
R1(z)=[x2+(z-z1)2]1/2,R1(0)=R1.
H1(2)(x)=H0(2)(x)-1xH1(2)(x),
ui,d(x, z)zz=0=-k22iz1R12H0(2)(kR1)A(x),
A(x)=1+1kR1R1z12-2 H1(2)(kR2)H0(2)(kR1).
uIId(x2, z2)=k24-aa z12R12H0(2)(kR1)A(x)H0(1)(kR2)dx.
uIc(x2, z2)=-k24i-aa z2R2H0(2)(kR1)H1(1)(kR2)dx,
uIIc(x2, z2)=k24i-aa z1R1H1(2)(kR1)H0(1)(kR2)dx.
uI(x2, z2)=12π-U(kx)exp[i(kxx2+kzz2)]dkx,
U(kx)=-u(x, 0)exp(-ikxx)dx,
kz=(k2-kx2)1/2,Im kz0.
UKd(kx)=-aag(x)exp[ikf(x)]dx,
g(x)=expi π4(λz1)1/2z1R13/2,
f(x)=-R1+kxkx.
UDd(kx)=exp(-ikzz1)for|kx|k sin θ10otherwise.
uDd(x2, z2)=12π-k sin θ1-k sin θ1×exp{i[kxx+kz(z2-z1)]}dkx,
uDd(x2, z2)=-θ1θ1g(θ)exp[ikr cos(θ-ϕ)]dθ,
g(θ)=cos θλ.
Uc(kx)=kkzUd(kx).
uDc(x2, z2)
=k2π-k sin θ1k sin θ1 exp{i[kxx+kz(z2-z1)]}kzdkx,
uDc(x2, z2)=1λ-θ1θ1 exp[ikr cos(θ-ϕ)]dθ,
Eq=yˆuj(x2, z2)(j=I, II, K;q=c, d),
Hq=ciωμ-xˆ ujq(x2, z2)z2+zˆ ujq(x2, z2)x2.
Hq=yˆujq(x2, z2)(j=I, II, K;q=c, d),
Eq=icω-xˆ ujq(x2, z2)z2+zˆ ujq(x2, z2)x2.

Metrics