Abstract

Amplitude and phase in the diffraction near fields of small phase objects (bars with rectangular cross sections ≤λ2) are calculated according to the scalar Kirchhoff and Rayleigh–Sommerfeld diffraction theories. Results are compared with 3-cm microwave measurements. The agreement between calculation and measurement depends on the size of the phase objects because interference fields under certain circumstances will satisfy the Kirchhoff boundary conditions of the different diffraction theories.

© 1991 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 8.3.
  2. A. Sommerfeld, Optik, Vol. 4der Vorlesungen über Theoretische Physik, 3rd ed. (Akademische Verlagsgesellschaft, Leipzig, 1960), Sec. 34.
  3. C. L. Andrews, “Diffraction pattern in a circular aperture measured in the microwave region,” J. Appl. Phys. 21, 761–767 (1950).
    [CrossRef]
  4. S. Silver, “Microwave aperture antennas and diffraction theory,”J. Opt. Soc. Am. 52, 131–139 (1962).
    [CrossRef] [PubMed]
  5. G. Toraldo di Francia, “Introduction to the modern theory of electromagnetic diffraction,” Atti Fond. Giorgio Ronchi Contrib. Ist. Naz. Ottica 11, 503–551 (1956).
  6. Rayleigh, The Theory of Sound, 2nd ed. (Dover, New York, 1945), Vol. 2, Sec. 278.
  7. N. Mukunda, “Consistency of Rayleigh’s diffraction formulas with Kirchhoff’s boundary conditions,”J. Opt. Soc. Am. 52, 336–337 (1962).
    [CrossRef]
  8. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,”J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [CrossRef]
  9. S. Ganci, “Equivalence between two consistent formulations of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. A 5, 1626–1628 (1988).
    [CrossRef]
  10. F. Kottler, “Diffraction at a black screen, Pt. 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
    [CrossRef]
  11. J. J. Stamnes, Waves in Focal Regions, Adam Hilger Series on Optics and Optoelectronics (Hilger, Bristol, UK, 1986), pp. 24 ff.
  12. J. C. Heurtley, “Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points,”J. Opt. Soc. Am. 63, 1003–1008 (1973).
    [CrossRef]
  13. M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
    [CrossRef]
  14. E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and the Rayleigh–Sommerfeld theories of diffraction at an aperture,”J. Opt. Soc. Am. 54, 587–594 (1964).
    [CrossRef]
  15. O. Bryngdahl, “Diffraction patterns of small phase objects measured in the microwave region,” Ark. Fys. 16, 69–92 (1959).
  16. S. Ganci, “Maggi–Rubinowicz transformation for phase apertures,” J. Opt. Soc. Am. A 3, 2094–2100 (1986).
    [CrossRef]
  17. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  18. R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968), Sec. 3.7.
  19. C. J. Bouwkamp, “Diffraction theory,” Phys. Soc. Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  20. L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, Reading, Mass., 1972), p. 263.
  21. M. Totzeck, “Gültigkeitsbereich der skalaren Beugungstheorien nach Kirchhoff und Rayleigh–Sommerfeld im Nahfeld kleiner Phasenobjekte. Vergleich von Modellrechnungen mit Mikrowellenmessungen,” Doktor-Thesis, D83 (Technical University of Berlin, Berlin, 1989).
  22. The width of the phase objects determines the amount of difference between the three scalar diffraction theories but not which theory yields the best agreement. For an object width of λall three theories agree quite well with measurements.
  23. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
    [CrossRef]

1988 (1)

1986 (1)

1978 (1)

1973 (1)

1966 (1)

1965 (2)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

F. Kottler, “Diffraction at a black screen, Pt. 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

1964 (1)

1962 (2)

1959 (1)

O. Bryngdahl, “Diffraction patterns of small phase objects measured in the microwave region,” Ark. Fys. 16, 69–92 (1959).

1956 (1)

G. Toraldo di Francia, “Introduction to the modern theory of electromagnetic diffraction,” Atti Fond. Giorgio Ronchi Contrib. Ist. Naz. Ottica 11, 503–551 (1956).

1955 (1)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Phys. Soc. Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

1950 (1)

C. L. Andrews, “Diffraction pattern in a circular aperture measured in the microwave region,” J. Appl. Phys. 21, 761–767 (1950).
[CrossRef]

Andrews, C. L.

C. L. Andrews, “Diffraction pattern in a circular aperture measured in the microwave region,” J. Appl. Phys. 21, 761–767 (1950).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 8.3.

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Phys. Soc. Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Diffraction patterns of small phase objects measured in the microwave region,” Ark. Fys. 16, 69–92 (1959).

Ehrlich, M. J.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Eyges, L.

L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, Reading, Mass., 1972), p. 263.

Feit, M. D.

Fleck, J. A.

Ganci, S.

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968), Sec. 3.7.

Held, G.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Heurtley, J. C.

Kottler, F.

F. Kottler, “Diffraction at a black screen, Pt. 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

Marchand, E. W.

Mukunda, N.

Rayleigh,

Rayleigh, The Theory of Sound, 2nd ed. (Dover, New York, 1945), Vol. 2, Sec. 278.

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

Silver, S.

S. Silver, “Microwave aperture antennas and diffraction theory,”J. Opt. Soc. Am. 52, 131–139 (1962).
[CrossRef] [PubMed]

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optik, Vol. 4der Vorlesungen über Theoretische Physik, 3rd ed. (Akademische Verlagsgesellschaft, Leipzig, 1960), Sec. 34.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions, Adam Hilger Series on Optics and Optoelectronics (Hilger, Bristol, UK, 1986), pp. 24 ff.

Toraldo di Francia, G.

G. Toraldo di Francia, “Introduction to the modern theory of electromagnetic diffraction,” Atti Fond. Giorgio Ronchi Contrib. Ist. Naz. Ottica 11, 503–551 (1956).

Totzeck, M.

M. Totzeck, “Gültigkeitsbereich der skalaren Beugungstheorien nach Kirchhoff und Rayleigh–Sommerfeld im Nahfeld kleiner Phasenobjekte. Vergleich von Modellrechnungen mit Mikrowellenmessungen,” Doktor-Thesis, D83 (Technical University of Berlin, Berlin, 1989).

Wolf, E.

Appl. Opt. (1)

Ark. Fys. (1)

O. Bryngdahl, “Diffraction patterns of small phase objects measured in the microwave region,” Ark. Fys. 16, 69–92 (1959).

Atti Fond. Giorgio Ronchi Contrib. Ist. Naz. Ottica (1)

G. Toraldo di Francia, “Introduction to the modern theory of electromagnetic diffraction,” Atti Fond. Giorgio Ronchi Contrib. Ist. Naz. Ottica 11, 503–551 (1956).

IEEE Trans. Antennas Propag. (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,”IEEE Trans. Antennas Propag. AP-13, 334–341 (1965).
[CrossRef]

J. Appl. Phys. (2)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

C. L. Andrews, “Diffraction pattern in a circular aperture measured in the microwave region,” J. Appl. Phys. 21, 761–767 (1950).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Phys. Soc. Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Phys. Soc. Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Prog. Opt. (1)

F. Kottler, “Diffraction at a black screen, Pt. 1: Kirchhoff’s theory,” Prog. Opt. 4, 281–314 (1965).
[CrossRef]

Other (8)

J. J. Stamnes, Waves in Focal Regions, Adam Hilger Series on Optics and Optoelectronics (Hilger, Bristol, UK, 1986), pp. 24 ff.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 8.3.

A. Sommerfeld, Optik, Vol. 4der Vorlesungen über Theoretische Physik, 3rd ed. (Akademische Verlagsgesellschaft, Leipzig, 1960), Sec. 34.

R. F. Harrington, Field Computation by Moment Methods (MacMillan, New York, 1968), Sec. 3.7.

Rayleigh, The Theory of Sound, 2nd ed. (Dover, New York, 1945), Vol. 2, Sec. 278.

L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, Reading, Mass., 1972), p. 263.

M. Totzeck, “Gültigkeitsbereich der skalaren Beugungstheorien nach Kirchhoff und Rayleigh–Sommerfeld im Nahfeld kleiner Phasenobjekte. Vergleich von Modellrechnungen mit Mikrowellenmessungen,” Doktor-Thesis, D83 (Technical University of Berlin, Berlin, 1989).

The width of the phase objects determines the amount of difference between the three scalar diffraction theories but not which theory yields the best agreement. For an object width of λall three theories agree quite well with measurements.

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Figures (7)

Fig. 1
Fig. 1

Two-dimensional diffraction of an incident cylindrical wave by a slit aperture.

Fig. 2
Fig. 2

Geometry of the phase object in the incident field.

Fig. 3
Fig. 3

Diffraction near field behind a weak and very thin phase object (2b = 0.6λ, d = 0.3λ, n = 1.0165, tan δ = 0.0004, Δφ0 = (n − 1)d 360°/λ 1.9°, z ¯ 0.5λ). Comparison of Kirchhoff and Rayleigh–Sommerfeld I and II (Rayl.-So. I and II) calculations (curves) and microwave measurements (diamonds): (a) amplitude, (b) phase distributions. (The rectangular curve shows the transmittance of the phase object in the limit of geometrical optics.)

Fig. 4
Fig. 4

Comparison of the near fields behind a phase object of approximately λ/2 thickness (2b = 0.6λ, d = 0.6λ, n = 1.0179, Δφ = 3.8°, z ¯ = 0.5λ. See Fig. 3 for definitions.

Fig. 5
Fig. 5

Comparison of the near fields behind a phase approximately λ thickness (2b = 0.6λ, d= 0.9λ, n = 1.0179, Δφ = 5.6° z ¯ =0.5λ). See Fig. 3 for definitions.

Fig. 6
Fig. 6

Comparison of the near fields behind a phase object with the same parameters as in Fig. 3, but at a distance of z ¯ =2.0λ). See Fig. 3 for definitions.

Fig. 7
Fig. 7

Scattered field of a point of a thin phase object (dλ). For the field in the object plane ks is normal to n.

Equations (21)

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U I ( r ) = 1 2 π - - b b U 0 ( r ) n [ exp ( i k r - r ) / r - r ] d x d y .
U II ( r ) = - 1 2 π - - b b n [ U 0 ( r ) ] exp ( i k r - r ) / r - r d x d y .
U K ( r ) = ½ [ U I ( r ) + U II ( r ) ] ,
- exp ( i k r - r ) / r - r d y = i π H 0 ( k ρ - ρ ) ,
U I ( ρ ) = i 2 - b b U 0 ( ρ ) n H 0 ( k ρ - ρ ) d x .
U II ( ρ ) = - i 2 - b b H 0 ( k ρ - ρ ) n U 0 ( ρ ) d x .
U 0 ( x , z ) = A 0 exp ( i k · ρ ) = A 0 exp [ i k ( x sin ξ + z cos ζ ) ] .
U 0 / n = U 0 / z = i k U 0 cos ( ξ ) .
ξ = arcsin [ ( arg U 0 ) k x ] .
n H 0 ( k ρ - ρ ) = - k H 0 ( k ρ - ρ ) k ρ - ρ ( ρ - ρ ) ρ - ρ · n = k H 1 ( k ρ - ρ ) cos ( ϑ ) ,
U ( ρ ) = - b b K ( ρ , ρ ) U 0 ( ρ ) d x ,
K ( ρ , ρ ) = { i k 2 H 1 ( k ρ - ρ ) cos ( ϑ ) Rayleigh - Sommerfeld I k 2 H 0 ( k ρ - ρ ) cos ( ξ ) Rayleigh - Sommerfeld II i k 4 [ H 1 ( k ρ - ρ ) cos ( ϑ ) - i H 0 ( k ρ - ρ ) cos ( ξ ) ] Kirchhoff .
t = exp [ i k ( n - 1 ) d ] .
U ( x , 0 ) = { t U 0 ( x ) if x b U 0 ( x ) if x > b = U 0 ( x ) + { ( t - 1 ) U 0 ( x ) if x b 0 if x > b .
U ( ρ ) = - K ( ρ , ρ ) U 0 ( ρ ) d x + - b b K ( ρ , ρ ) ( t - 1 ) U 0 ( ρ ) d x .
U ( ρ ) = U 0 ( ρ ) + - b b K ( ρ , ρ ) ( t - 1 ) U 0 ( ρ ) d x .
U s / n = i k s U s cos ( ξ s ) = 0.
U s ( ρ ) = i k 2 4 object cross section ( n 2 - 1 ) H 0 ( k ρ - ρ ) U 0 ( ρ ) d x d z .
U s ( ρ ) = i k 2 4 - b b d ( n 2 - 1 ) H 0 ( k ρ - ρ ) U 0 ( ρ ) d x .
i k 2 4 d ( n 2 - 1 ) k 2 { exp [ i k d ( n - 1 ) ] - 1 } = k 2 ( t - 1 ) .
U s ( ρ ) = k 2 - b b ( t - 1 ) H 0 ( k ρ - ρ ) U 0 ( ρ ) d x .

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