Abstract

Rigorous diffraction theory is applied to compute the electromagnetic fields in the neighborhood of polygonal obstacles of three- and two-dimensional shape, including two-dimensional gratings. For obstacles such as a rectangular depression, a polygonal groove, or an array of grooves present in a metallic substrate, we calculate and measure significant departures from what a scalar diffraction approach would predict as soon as the width of the obstacle becomes smaller than one wavelength. For such widths, it is shown that the apparent depth of a groove can be smaller or larger than the geometrical depth, depending on the polarization of the incident light and on the optical constants of the substrate.

© 1979 Optical Society of America

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References

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  1. J. J. M. Braat and G. Bouwhuis, Appl. Opt. 17, 2022 (1978).
    [CrossRef] [PubMed]
  2. J. G. Dil and B. A. J. Jacobs, Digest of 1977, APS-URSI Symposium in Palo Alto, Ca., U.S.A., p. 44 (unpublished).
  3. A. Sommerfeld, Optics, (Academic, New York, 1964), pp 273–289.
  4. C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
    [CrossRef]
  5. A. T. De Hoop, Proc. Kon. Ned. Akad. Wet. B58, 1401 (1955).
  6. A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conference Publication 13, (Stevenage, Peter Peregrinus Ltd., 1977) Chap. 6.
  7. R. F. Harrington, Field computations by moment methods, (MacMillan, New York, 1968),
  8. W. Maue, Phys. 126, 701 (1949).
    [CrossRef]
  9. R. Petit, C. R. Acad. Sci. (Paris) 264, 1441 (1967).
  10. K. K. Mei and J. G. Van Bladel, IEEE AP-15, 795 (1967).
  11. G. A. Gray and R. E. Kleinman, Proceedings 1977 URSI Symposium on Electromagnetic Wave Theory, 329 (1977) (unpublished).
  12. F. L. Neerhoff, Proc. R. Soc. Lond. A 342, 237 (1975).
    [CrossRef]
  13. J. P. Hugonin and R. Petit, Opt. Commun. 22, 221 (1977).
    [CrossRef]
  14. P. M. Van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55 (1974).
    [CrossRef]
  15. A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975).
    [CrossRef]
  16. J. P. J. Heemskerk and C. H. F. Velzel, Private communication on phase depths measurements, available in print at Philips Research Lab., Eindhoven, The Netherlands.
  17. B. A. J. Jacobs, Appl. Opt. 17, 2001 (1978).
    [CrossRef] [PubMed]
  18. E. G. Loewen, M. Neviere, and D. Maystre, Appl. Opt. 16, 2711 (1977).
    [CrossRef] [PubMed]
  19. T. B. A. Senior, Radio Sci. 10, 911 (1975).
    [CrossRef]

1978 (2)

1977 (2)

1975 (3)

F. L. Neerhoff, Proc. R. Soc. Lond. A 342, 237 (1975).
[CrossRef]

A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975).
[CrossRef]

T. B. A. Senior, Radio Sci. 10, 911 (1975).
[CrossRef]

1974 (1)

P. M. Van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55 (1974).
[CrossRef]

1967 (2)

R. Petit, C. R. Acad. Sci. (Paris) 264, 1441 (1967).

K. K. Mei and J. G. Van Bladel, IEEE AP-15, 795 (1967).

1955 (1)

A. T. De Hoop, Proc. Kon. Ned. Akad. Wet. B58, 1401 (1955).

1954 (1)

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

1949 (1)

W. Maue, Phys. 126, 701 (1949).
[CrossRef]

Borburgh, J. C. M.

P. M. Van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55 (1974).
[CrossRef]

Bouwhuis, G.

Bouwkamp, C. J.

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

Braat, J. J. M.

De Hoop, A. T.

A. T. De Hoop, Proc. Kon. Ned. Akad. Wet. B58, 1401 (1955).

A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conference Publication 13, (Stevenage, Peter Peregrinus Ltd., 1977) Chap. 6.

Dil, J. G.

J. G. Dil and B. A. J. Jacobs, Digest of 1977, APS-URSI Symposium in Palo Alto, Ca., U.S.A., p. 44 (unpublished).

Gray, G. A.

G. A. Gray and R. E. Kleinman, Proceedings 1977 URSI Symposium on Electromagnetic Wave Theory, 329 (1977) (unpublished).

Harrington, R. F.

R. F. Harrington, Field computations by moment methods, (MacMillan, New York, 1968),

Heemskerk, J. P. J.

J. P. J. Heemskerk and C. H. F. Velzel, Private communication on phase depths measurements, available in print at Philips Research Lab., Eindhoven, The Netherlands.

Hugonin, J. P.

J. P. Hugonin and R. Petit, Opt. Commun. 22, 221 (1977).
[CrossRef]

Jacobs, B. A. J.

B. A. J. Jacobs, Appl. Opt. 17, 2001 (1978).
[CrossRef] [PubMed]

J. G. Dil and B. A. J. Jacobs, Digest of 1977, APS-URSI Symposium in Palo Alto, Ca., U.S.A., p. 44 (unpublished).

Kleinman, R. E.

G. A. Gray and R. E. Kleinman, Proceedings 1977 URSI Symposium on Electromagnetic Wave Theory, 329 (1977) (unpublished).

Loewen, E. G.

Maradudin, A. A.

A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975).
[CrossRef]

Maue, W.

W. Maue, Phys. 126, 701 (1949).
[CrossRef]

Maystre, D.

Mei, K. K.

K. K. Mei and J. G. Van Bladel, IEEE AP-15, 795 (1967).

Mills, D. L.

A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975).
[CrossRef]

Neerhoff, F. L.

F. L. Neerhoff, Proc. R. Soc. Lond. A 342, 237 (1975).
[CrossRef]

Neviere, M.

Petit, R.

J. P. Hugonin and R. Petit, Opt. Commun. 22, 221 (1977).
[CrossRef]

R. Petit, C. R. Acad. Sci. (Paris) 264, 1441 (1967).

Senior, T. B. A.

T. B. A. Senior, Radio Sci. 10, 911 (1975).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics, (Academic, New York, 1964), pp 273–289.

Van Bladel, J. G.

K. K. Mei and J. G. Van Bladel, IEEE AP-15, 795 (1967).

Van den Berg, P. M.

P. M. Van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55 (1974).
[CrossRef]

Velzel, C. H. F.

J. P. J. Heemskerk and C. H. F. Velzel, Private communication on phase depths measurements, available in print at Philips Research Lab., Eindhoven, The Netherlands.

Appl. Opt. (3)

Appl. Phys. (1)

P. M. Van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55 (1974).
[CrossRef]

C. R. Acad. Sci. (Paris) (1)

R. Petit, C. R. Acad. Sci. (Paris) 264, 1441 (1967).

IEEE (1)

K. K. Mei and J. G. Van Bladel, IEEE AP-15, 795 (1967).

Opt. Commun. (1)

J. P. Hugonin and R. Petit, Opt. Commun. 22, 221 (1977).
[CrossRef]

Phys. (1)

W. Maue, Phys. 126, 701 (1949).
[CrossRef]

Phys. Rev. B (1)

A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975).
[CrossRef]

Proc. Kon. Ned. Akad. Wet. (1)

A. T. De Hoop, Proc. Kon. Ned. Akad. Wet. B58, 1401 (1955).

Proc. R. Soc. Lond. A (1)

F. L. Neerhoff, Proc. R. Soc. Lond. A 342, 237 (1975).
[CrossRef]

Radio Sci. (1)

T. B. A. Senior, Radio Sci. 10, 911 (1975).
[CrossRef]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, Rep. Prog. Phys. 17, 35 (1954).
[CrossRef]

Other (6)

G. A. Gray and R. E. Kleinman, Proceedings 1977 URSI Symposium on Electromagnetic Wave Theory, 329 (1977) (unpublished).

A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conference Publication 13, (Stevenage, Peter Peregrinus Ltd., 1977) Chap. 6.

R. F. Harrington, Field computations by moment methods, (MacMillan, New York, 1968),

J. G. Dil and B. A. J. Jacobs, Digest of 1977, APS-URSI Symposium in Palo Alto, Ca., U.S.A., p. 44 (unpublished).

A. Sommerfeld, Optics, (Academic, New York, 1964), pp 273–289.

J. P. J. Heemskerk and C. H. F. Velzel, Private communication on phase depths measurements, available in print at Philips Research Lab., Eindhoven, The Netherlands.

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Figures (7)

FIG. 1
FIG. 1

Scattering geometry of the rectangular depression present in a plane metallic substrate.

FIG. 2
FIG. 2

Scattering geometries to test the validity of Eq. (2.10) with Eqs. (2.5)(2.6).

FIG. 3
FIG. 3

Amplitude (a) and phase (b) distributions of the equivalent electric surface currents on the bottom of an aluminum depression of length l, width w, and depth d with l = 1.46 w = 7d = 1.21 λ0. The shaded planes represent the scalar diffraction limit; |je| is plotted in units | k0E(0)/ 0|.

FIG. 4
FIG. 4

Amplitudes and phases of the equivalent electric surface currents on the faces u and w of the scattering geometries of Insert I (dotted), II (light solid), and III (heavy solid). Parallel polarization is represented on the left-hand side, perpendicular polarization on the right-hand side. The scalar diffraction limits are given by the sharp horizontal lines; |je| is plotted in units |k0E(0)/ 0|.

FIG. 5
FIG. 5

Average amplitudes and phases of the equivalent electric surface currents in both polarizations on the bottom of a groove as a function of groove width (a) and of groove depth (b). The scalar limits are given by sharp lines; the amplitude is plotted in units | k0E(0)/ 0|.

FIG. 6
FIG. 6

Experimental setup to measure the phase difference between the zero and first order diffracted by the reflection grating R. T stands for rotating transmission grating, l for lens, B for beam splitter, and D1 and D2 are detectors.

FIG. 7
FIG. 7

Phase difference between zero and first Order diffracted by a titanium reflection grating as a function of wavelength and polarization. The heavy solid and dashed curves are calculated with rigorous diffraction theory, the sharp line gives the scalar diffraction limit, and the circles, two of which have error bars, represent experimental values.

Equations (27)

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J ( r p ) = S d S ( r ) G ( r p | r ) J ( r ) .
J ( r ) [ j m ( r ) j e ( r ) ] [ ν ̂ ( r ) × E ( r ) ν ̂ ( r ) × E ( r ) ] .
G ( r p | r ) exp ( i k 0 | r p r | ) 4 π | r p r | ,
k 0 ω ( 0 μ 0 ) 1 / 2 = 2 π / λ 0 ,
k = k 0 ( n + i k ) .
L S { r p | r } J ( r ) = J ( r p ) J ( 0 ) ( r p ) , r p S ,
L S { r p | r } = [ ν ̂ ( r p ) × p × S d S ( r ) { G ( r p | r ) G 0 ( r p | r ) } i w μ 0 ν ̂ ( r p ) × [ S d S ( r ) { G ( r p | r ) G 0 ( r p | r ) } ν ̂ ( r p ) i w μ 0 × [ S d S ( r ) { k 2 G ( r p | r ) k 0 2 G 0 ( r p | r ) } + p p S d S ( r ) { G ( r p | r ) k 2 G 0 ( r p | r ) } k 0 2 ] + p p d S ( r ) { G ( r p | r ) G 0 ( r p | r ) } ] ν ̂ ( r p ) × p × S d S ( r ) { G ( r p | r ) G 0 ( r p | r ) } . ]
L c { r p | r } [ j m , τ ( r ) j e , y ( r ) ] = [ j m , τ ( r p ) j e , y ( r p ) ] [ j m , τ ( 0 ) ( r p ) j e , y ( 0 ) ( r p ) ] , r p C .
L C { r p | r } [ j m , y ( r ) j e , τ ( r ) ] = [ j m , y ( r p ) j e , τ ( r p ) ] [ j m , y ( 0 ) ( r p ) j e , τ ( 0 ) ( r p ) ] , r p C .
τ ̂ ( r ) y ̂ × ν ̂ ( r ) .
i = 1 N S i ( r i ) = S .
i = 1 N L S i { r p | r i } J ( r i ) = J ( r p ) J ( 0 ) ( r p ) .
L S { r p | r } J ( r ) = J ( r p ) J ( 0 ) ( r p ) , r p S ,
L S d { r p | r } J ( r ) = J ( r p ) J p l ( r p ) , r p S d ,
J p l ( r p ) = L S c { r p | r } J p l ( r ) J p l ( r p ) , r p S d .
L S t { r p | r } [ J ( r ) J p l ( r ) ] 0 , r p S d ,
R m = 1 p one period exp { i 2 φ ( x ) } exp { 2 π i m x / p } d x ,
ψ 10 = arg ( R 0 ) arg ( R 1 ) = π arctan ( c tan ( φ ) 1 2 w / p ) .
I 1 = cos ( Ω p t + ψ 10 ) ,
I 2 = cos ( Ω p t + ψ 10 ) ,
L c { r p | r } = [ C d C ( r ) ν ̂ ( r ) p τ [ G ( r p | r ) G 0 ( r p | r ) ] i w μ 0 C d C ( r ) [ G ( r p | r ) G 0 ( r p | r ) ] y ̂ ν ̂ ( r p ) i w μ 0 × C d C ( r ) τ ̂ ( r ) [ k 2 G ( r p | r ) k 0 2 G 0 ( r p | r ) ] y ̂ ν ̂ ( r p ) × C d C ( r ) ν ̂ ( r ) τ ̂ ( r ) τ [ G ( r p | r ) G 0 ( r p | r ) ] τ ̂ ( r p ) p τ i w μ 0 C d C ( r ) τ ̂ ( r ) τ [ G ( r p | r ) G 0 ( r p | r ) ] + y ̂ ν ̂ ( r p ) × C d C ( r ) τ ̂ ( r ) ν ̂ ( r ) p τ [ G ( r p | r ) G 0 ( r p | r ) ] ]
L c { r p | r } = [ y ̂ ν ̂ ( r p ) × C d C ( r ) ν ̂ ( r ) τ ̂ ( r ) τ [ G ( r p | r ) G 0 ( r p | r ) ] i w μ 0 y ̂ ν ̂ ( r p ) × C d C ( r ) τ ̂ ( r ) { G ( r p | r ) G 0 ( r p | r ) } + y ̂ ν ̂ ( r p ) × C d C ( r ) τ ̂ ( r ) ν ̂ ( r ) p τ [ G ( r p | r ) G 0 ( r p | r ) ] i w μ 0 τ ̂ ( r p ) p τ C d C ( r ) τ ̂ ( r ) τ ( G ( r p | r ) k 2 G 0 ( r p | r ) k 0 2 ) 1 i w μ 0 C d C ( r ) [ k 2 G ( r p | r ) k 0 2 G 0 ( r p | r ) ] C d C ( r ) ν ̂ ( r ) p τ [ G ( r p | r ) G 0 ( r p | r ) ] , ]
τ x x ̂ + z z ̂ .
G ( r p | r ) = d y exp [ i k | r p r | ] 4 π | r p r | = i 4 H 0 ( 1 ) ( k | r p r | ) ,
G ( r p | r ) = m = ( i / 2 k 0 γ m p ) exp [ i k 0 α m ( x p x ) + i k 0 γ m | z p z | ] ,
α m ( 1 / k 0 ) ( k 0 α 0 + 2 π m / p )
γ m ( 1 / k 0 ) ( k 2 k 0 2 α m 2 ) 1 / 2 .