Abstract

We compare the Fresnel diffraction pattern of a thin circular disk with that of a square obstacle, specifically evaluating the on-axis field strength. Photographs of the diffraction patterns reveal some curious features for the square obstacle. Second, the precise electric and magnetic fields behind a conducting circular disk are evaluated without invoking the Fresnel approximation and contrasted with the rigorous electromagnetic result for a metal sphere. The calculations show that the two cases differ only slightly in the Fresnel region. In the near-field new computational results for the sphere are analyzed.

© 1988 Optical Society of America

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References

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  1. M. E. Hufford, “The Diffraction Ring Pattern in the Shadow of a Circular Object,” Phys. Rev. 7, Ser. 2,545 (1916).
    [CrossRef]
  2. L. K. White, “Positive and Negative Tone Near-Contact Printing of Contact Hole Maskings,” RCA Rev. 43, 391 (1982).
  3. G. Forbes, U. Rochester, Institute of Optics; private communication.
  4. Y. P. Kathuria, G. Herziger, “Annular Apertures Focused in the Fresnel Region,” Proc. Soc. Photo-Opt. Instrum. Eng. 288, 505 (1981).
  5. M. Cagnet, M. Francon, J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin; Atlas of Optical PhenomenaPrentice-Hall, Englewood Cliffs, NJ, 1962), Plate 36.
  6. F. S. Harris, “Light Diffraction Patterns,” Appl. Opt. 3, 909 (1964).
    [CrossRef]
  7. V. J. Meixner, W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen an der volkommen leitenden Kreisscheibe und an der Kreisförmigen Offnung im volkommen leitenden ebenen Schirm,” Ann. Phys. 7, 157 (1950).
    [CrossRef]
  8. C. J. Bouwkamp, “On the Diffraction of Electromagnetic Waves by Small Circular Disks and Holes,” Philips Res. Rep. 5, 401 (1950).
  9. C. Flammer, “The Vector Wave Function Solution of the Diffraction of Electromagnetic Waves by Circular Disks and Apertures. I. Oblate Spheroidal Vector Wave Functions; II. The Diffraction Problems,” J. Appl. Phys. 24, 1218 (1953).
    [CrossRef]
  10. I. Chung, C. L. Andrews, L. F. Libelo, “Near-Field Diffraction on the Axes of Disks,” J. Opt. Soc. Am. 67, 1561 (1977); C. L. Andrews, I. Chung, L. F. Libelo, “Diffraction on the Axes of Disks and Apertures,” J. Opt. Soc. Am. 70, 813 (1980); C. L. Andrews, “Near-Field Diffraction, Experimental and Theoretical,” J. Opt. Soc. Am. 72, 1825 (1982).
    [CrossRef]
  11. G. Mie, “Beiträge zur Optik Trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377 (1908).
    [CrossRef]
  12. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).
  13. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  14. K. Kamiuto, “Near-Field Scattering by a Large Spherical Particle Embedded in a Nonabsorbing Medium,” J. Opt. Soc. Am. 73, 1819 (1983).
    [CrossRef]
  15. F. Slimani, G. Grehan, G. Gouesbet, D. Allano, “Near-Field Lorenz-Mie Theory and Its Application to Microholography,” Appl. Opt. 23, 4140 (1984).
    [CrossRef] [PubMed]
  16. D. S. Benincasa, P. W. Barber, J-Z. Zhang, W-F Hsieh, R. K. Chang, “Spatial Distribution of the Internal and Near-Field Intensities of Large Cylindrical and Spherical Scatterers,” Appl. Opt. 26, 1348 (1987).
    [CrossRef] [PubMed]
  17. C. V. Raman, K. S. Krishnan, “On the Diffraction of Light by Spherical Obstacles,” Proc. Phys. Soc. London 38, 350 (1926).
    [CrossRef]
  18. R. E. English, N. George, “Diffraction from a Circular Aperture: On-Axis Field Strength,” Appl. Opt. 26, 2360 (1987).
    [CrossRef] [PubMed]
  19. V. N. Mahajan, “Axial Irradiance and Optimum Focusing of Laser Beams,” Appl. Opt. 22, 3042 (1983).
    [CrossRef] [PubMed]
  20. M. Abramowitz, I. Stegun, Eds., Handbook of Mathematical Functions (U.S. GPO, Washington, DC, 1972), p. 300, Sec. 7.3.
  21. H. Osterberg, L. W. Smith, “Closed Solutions of Rayleigh’s Diffraction Integral for Axial Points,” J. Opt. Soc. Am. 51, 1050 (1961).
    [CrossRef]
  22. See Ref. 20, p. 445, Sec. 10.3.
  23. J. V. Dave, “Scattering of Electromagnetic Radiation by a Large, Absorbing Sphere,” IBM J. Res. Dev. 13, 302 (1969).
    [CrossRef]
  24. W. D. Ross, “Computation of Bessel Functions in Light Scattering Studies,” Appl. Opt. 11, 1919 (1972).
    [CrossRef] [PubMed]
  25. W. J. Wiscombe, “Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505 (1980).
    [CrossRef] [PubMed]
  26. O. B. Toon, T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20, 3657 (1981).
    [CrossRef] [PubMed]

1987 (2)

1984 (1)

1983 (2)

1982 (1)

L. K. White, “Positive and Negative Tone Near-Contact Printing of Contact Hole Maskings,” RCA Rev. 43, 391 (1982).

1981 (2)

Y. P. Kathuria, G. Herziger, “Annular Apertures Focused in the Fresnel Region,” Proc. Soc. Photo-Opt. Instrum. Eng. 288, 505 (1981).

O. B. Toon, T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20, 3657 (1981).
[CrossRef] [PubMed]

1980 (1)

1977 (1)

1972 (1)

1969 (1)

J. V. Dave, “Scattering of Electromagnetic Radiation by a Large, Absorbing Sphere,” IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

1964 (1)

1961 (1)

1953 (1)

C. Flammer, “The Vector Wave Function Solution of the Diffraction of Electromagnetic Waves by Circular Disks and Apertures. I. Oblate Spheroidal Vector Wave Functions; II. The Diffraction Problems,” J. Appl. Phys. 24, 1218 (1953).
[CrossRef]

1950 (2)

V. J. Meixner, W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen an der volkommen leitenden Kreisscheibe und an der Kreisförmigen Offnung im volkommen leitenden ebenen Schirm,” Ann. Phys. 7, 157 (1950).
[CrossRef]

C. J. Bouwkamp, “On the Diffraction of Electromagnetic Waves by Small Circular Disks and Holes,” Philips Res. Rep. 5, 401 (1950).

1926 (1)

C. V. Raman, K. S. Krishnan, “On the Diffraction of Light by Spherical Obstacles,” Proc. Phys. Soc. London 38, 350 (1926).
[CrossRef]

1916 (1)

M. E. Hufford, “The Diffraction Ring Pattern in the Shadow of a Circular Object,” Phys. Rev. 7, Ser. 2,545 (1916).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Ackerman, T. P.

Allano, D.

Andrejewski, W.

V. J. Meixner, W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen an der volkommen leitenden Kreisscheibe und an der Kreisförmigen Offnung im volkommen leitenden ebenen Schirm,” Ann. Phys. 7, 157 (1950).
[CrossRef]

Andrews, C. L.

Barber, P. W.

Benincasa, D. S.

Bouwkamp, C. J.

C. J. Bouwkamp, “On the Diffraction of Electromagnetic Waves by Small Circular Disks and Holes,” Philips Res. Rep. 5, 401 (1950).

Cagnet, M.

M. Cagnet, M. Francon, J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin; Atlas of Optical PhenomenaPrentice-Hall, Englewood Cliffs, NJ, 1962), Plate 36.

Chang, R. K.

Chung, I.

Dave, J. V.

J. V. Dave, “Scattering of Electromagnetic Radiation by a Large, Absorbing Sphere,” IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

English, R. E.

Flammer, C.

C. Flammer, “The Vector Wave Function Solution of the Diffraction of Electromagnetic Waves by Circular Disks and Apertures. I. Oblate Spheroidal Vector Wave Functions; II. The Diffraction Problems,” J. Appl. Phys. 24, 1218 (1953).
[CrossRef]

Forbes, G.

G. Forbes, U. Rochester, Institute of Optics; private communication.

Francon, M.

M. Cagnet, M. Francon, J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin; Atlas of Optical PhenomenaPrentice-Hall, Englewood Cliffs, NJ, 1962), Plate 36.

George, N.

Gouesbet, G.

Grehan, G.

Harris, F. S.

Herziger, G.

Y. P. Kathuria, G. Herziger, “Annular Apertures Focused in the Fresnel Region,” Proc. Soc. Photo-Opt. Instrum. Eng. 288, 505 (1981).

Hsieh, W-F

Hufford, M. E.

M. E. Hufford, “The Diffraction Ring Pattern in the Shadow of a Circular Object,” Phys. Rev. 7, Ser. 2,545 (1916).
[CrossRef]

Kamiuto, K.

Kathuria, Y. P.

Y. P. Kathuria, G. Herziger, “Annular Apertures Focused in the Fresnel Region,” Proc. Soc. Photo-Opt. Instrum. Eng. 288, 505 (1981).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Krishnan, K. S.

C. V. Raman, K. S. Krishnan, “On the Diffraction of Light by Spherical Obstacles,” Proc. Phys. Soc. London 38, 350 (1926).
[CrossRef]

Libelo, L. F.

Mahajan, V. N.

Meixner, V. J.

V. J. Meixner, W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen an der volkommen leitenden Kreisscheibe und an der Kreisförmigen Offnung im volkommen leitenden ebenen Schirm,” Ann. Phys. 7, 157 (1950).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Osterberg, H.

Raman, C. V.

C. V. Raman, K. S. Krishnan, “On the Diffraction of Light by Spherical Obstacles,” Proc. Phys. Soc. London 38, 350 (1926).
[CrossRef]

Rochester, U.

G. Forbes, U. Rochester, Institute of Optics; private communication.

Ross, W. D.

Slimani, F.

Smith, L. W.

Thrierr, J. C.

M. Cagnet, M. Francon, J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin; Atlas of Optical PhenomenaPrentice-Hall, Englewood Cliffs, NJ, 1962), Plate 36.

Toon, O. B.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).

White, L. K.

L. K. White, “Positive and Negative Tone Near-Contact Printing of Contact Hole Maskings,” RCA Rev. 43, 391 (1982).

Wiscombe, W. J.

Zhang, J-Z.

Ann. Phys. (2)

V. J. Meixner, W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen an der volkommen leitenden Kreisscheibe und an der Kreisförmigen Offnung im volkommen leitenden ebenen Schirm,” Ann. Phys. 7, 157 (1950).
[CrossRef]

G. Mie, “Beiträge zur Optik Trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt. (8)

IBM J. Res. Dev. (1)

J. V. Dave, “Scattering of Electromagnetic Radiation by a Large, Absorbing Sphere,” IBM J. Res. Dev. 13, 302 (1969).
[CrossRef]

J. Appl. Phys. (1)

C. Flammer, “The Vector Wave Function Solution of the Diffraction of Electromagnetic Waves by Circular Disks and Apertures. I. Oblate Spheroidal Vector Wave Functions; II. The Diffraction Problems,” J. Appl. Phys. 24, 1218 (1953).
[CrossRef]

J. Opt. Soc. Am. (3)

Philips Res. Rep. (1)

C. J. Bouwkamp, “On the Diffraction of Electromagnetic Waves by Small Circular Disks and Holes,” Philips Res. Rep. 5, 401 (1950).

Phys. Rev. (1)

M. E. Hufford, “The Diffraction Ring Pattern in the Shadow of a Circular Object,” Phys. Rev. 7, Ser. 2,545 (1916).
[CrossRef]

Proc. Phys. Soc. London (1)

C. V. Raman, K. S. Krishnan, “On the Diffraction of Light by Spherical Obstacles,” Proc. Phys. Soc. London 38, 350 (1926).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

Y. P. Kathuria, G. Herziger, “Annular Apertures Focused in the Fresnel Region,” Proc. Soc. Photo-Opt. Instrum. Eng. 288, 505 (1981).

RCA Rev. (1)

L. K. White, “Positive and Negative Tone Near-Contact Printing of Contact Hole Maskings,” RCA Rev. 43, 391 (1982).

Other (6)

G. Forbes, U. Rochester, Institute of Optics; private communication.

M. Cagnet, M. Francon, J. C. Thrierr, Atlas of Optical Phenomena (Springer-Verlag, Berlin; Atlas of Optical PhenomenaPrentice-Hall, Englewood Cliffs, NJ, 1962), Plate 36.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

See Ref. 20, p. 445, Sec. 10.3.

M. Abramowitz, I. Stegun, Eds., Handbook of Mathematical Functions (U.S. GPO, Washington, DC, 1972), p. 300, Sec. 7.3.

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Figures (6)

Fig. 1
Fig. 1

Diffraction geometry. An infinitely thin, perfectly conducting plane screen containing an aperture A is placed in the x,y plane at z = 0. The electric field is to be evaluated at the position vector r given the tangential components of the electric field at the position vector r′ in the aperture. A normally incident, x-polarized electric field is shown to the left of the aperture.

Fig. 2
Fig. 2

Axial field strength squared, circular disk vs square obstacle. |V0(z)|2 as computed from Eqs. (12) and (13) is plotted for 2a = 1 mm and λ = 0.5 μm for z in the Fresnel region. Note that z is in millimeters. The circular disk curve (—) is a constant value while the square obstacle curve (---) decreases from a value of unity in the far zone to <0.10 at the Fresnel region boundary.

Fig. 3
Fig. 3

Photographs of diffraction patterns behind the circular disk and square obstacle. An expanded and collimated He–Ne laser beam illuminated a thin circular disk; the pattern was recorded on film at z = 100 mm (a). In the same conditions the square obstacle pattern was recorded (b). Increasing the exposure by a factor of 16 brings out some details of the patterns for the circle (c) and especially for the square (d).

Fig. 4
Fig. 4

On-axis Poynting vector, circular disk vs metal sphere. |S| as computed from Eqs. (16) and (19) is plotted vs log(z) for 2a = 1 mm and λ = 0.5 μm. In the figure z is in millimeters. Note that the Poynting vector for the circular disk (—) goes smoothly to zero as z → 0, while the Poynting vector for the metal sphere (---) goes to zero at the edge of the sphere, i.e., z = 0.5 mm.

Fig. 5
Fig. 5

On-axis field strengths, circular disk vs metal sphere. The square modulus of E and H (normalized by ɛ/μ) for the circular disk (—) and the metal sphere (---) are plotted vs log(z). Again z is in millimeters. The fields for the sphere are oscillating with a period of λ/2 so only the envelope of modulation is shown; this envelope is the same for both E and H.

Fig. 6
Fig. 6

Examination of oscillatory behavior of axial fields behind a metal sphere. The square modulus of the fields, |E|2 (—) and |H|2/(ɛ/μ) (---), are shown for (a) z = 0.5 mm, (c) z = 1.0 mm, and (e) z = 10.0 mm. The phase difference between the fields is shown for (b) z = 0.5 mm, (d) z = 1.0 mm, and (f) z = 10.0 mm.

Equations (21)

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E ( r ) = 1 2 π × A n × E ( r ) exp ( i k R ) R d x d y ,
R | r r | = ( x x ) 2 + ( y y ) 2 + z 2 ,
i k H ( r ) = ε / μ × E ( r ) ,
E inc ( r ) = exp ( i k z ) x ; H inc ( r ) = ε / μ exp ( i k z ) y ,
E x ( 0 , 0 , z ) = 1 2 π A E x ( x , y , 0 ) exp ( i k R 0 ) R 0 × ( i k 1 R 0 ) ( z R 0 ) d x d y ,
H y ( 0 , 0 , z ) = ε / μ 2 π i k A { E x ( x , y , 0 ) exp ( i k R 0 ) R 0 × [ z 2 + x 2 R 0 2 ( k 2 3 i k R 0 + 3 R 0 2 ) + 2 R 0 ( i k 1 R 0 ) ] E y ( x , y , 0 ) exp ( i k R 0 ) R 0 [ x y R 0 2 ( k 2 3 i k R 0 + 3 R 0 2 ) ] } d x d y ,
E x ( 0 , 0 , z ) = 1 2 π A E x inc ( x , y , 0 ) exp ( i k R 0 ) R 0 × ( i k 1 R 0 ) ( z R 0 ) d x d y ,
H y ( 0 , 0 , z ) = ε / μ 2 π i k A E x inc ( x , y , 0 ) exp ( i k R 0 ) R 0 × [ z 2 + x 2 R 0 2 ( k 2 3 i k R 0 + 3 R 0 2 ) + 2 R 0 ( i k 1 R 0 ) ] d x d y .
z 3 π 4 λ a 4 ,
A circle = 1 circ ( x 2 + y 2 / a ) ,
A square = 1 rect ( x / a ) rect ( y / a ) .
V 0 ( z ) E x ( 0 , 0 , z ) ,
V 0 circle ( z ) = exp ( i k z ) exp ( i π a 2 / λ z ) ,
V 0 square ( z ) = exp ( i k z ) { 1 + 2 i [ C ( t a ) + i S ( t a ) ] 2 } ,
t a = a 2 λ z ,
E x ( 0 , 0 , z ) = z d exp ( i k d ) ,
H y ( 0 , 0 , z ) = ε / μ 2 [ 1 + z 2 d 2 + a 2 i k d 3 ] exp ( i k d ) ,
S = S | S inc | = z 2 d 2 ( 2 z + a 2 / z 2 d ) z .
E r = 0 , E θ = exp ( i k z ) + 1 k z n = 1 i n + 1 ( 2 n + 1 2 ) × [ a n ζ n ( 1 ) ( k z ) + i b n ζ n ( 1 ) ( k z ) ] , E φ = 0 , H r = 0 , H θ = 0 , H φ = ε / μ { exp ( i k z ) + i k z n = 1 i n + 1 ( 2 n + 1 2 ) × [ a n ζ n ( 1 ) ( k z ) i b n ζ n ( 1 ) ( k z ) ] } ,
a n = ψ n ( k a ) ζ n ( 1 ) ( k a ) ; b n = ψ n ( k a ) ψ n ( 1 ) ( k a ) ,
S = Re { E θ H φ * } ε / μ z .

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