Abstract

A laser beam impinging at grazing incidence on a 3-D surface produces a diffraction edge wave whose obliquity factor contains, among other things, the information concerning the local radius of curvature of the surface in the plane of incidence. In this paper we show how to retrieve this curvature radius information from light irradiance measurements of the edge wave and from exact electromagnetic diffraction calculations. This new method of optical metrology gives the curvature radius of a surface in the incident plane and at a given point by a single measurement at that point, while other techniques depend on a scanning or a mapping in the vicinity of the point of interest. Moreover, a measuring system using this diffractional method is very easily implemented and can yield a precision in the 5–10-μm range. Experimental results are presented for six metallic circular cylinders and two metallic spheres.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara-Hernandez, M. V. R. K. Murty, A. Cornejo-Rodriguez, “Bibliography of Various Optical Testing Methods,” Appl. Opt. 14, 1065 (1975).
    [CrossRef]
  2. M. V. R. K. Murty, “Newton, Fizeau and Haidinger Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 1–45.
  3. J. H. Bruning, “Fringe Scanning Interferometry,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409– 437.
  4. J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.
  5. J. R. Varner, “Holographic Contouring Methods,” in Handbook of Optica $lography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 595–600.
  6. H. Takasaki, “Moire Topography,” Jpn. J. Appl. Phys. Suppl. 14-1, 441 (1975).
  7. J. D. Evans, “Method for Approximating the Radius of Curvature of Small Concave Spherical Mirrors Using a He–Ne Laser,” Appl. Opt. 10, 995 (1971).
  8. F. M. Smolka, T. P. Caudell, “Surface Profile Measurement and Angular Deflection Monitoring Using a Scanning Laser Beam: a Noncontact Method,” Appl. Opt. 17, 3284 (1978).
    [CrossRef] [PubMed]
  9. J. B. Keller, “Diffraction by a Convex Cylinder,” IRE Trans. Antennas Propag. AP-4, 312 (1956).
    [CrossRef]
  10. J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. 52, 116 (1962).
    [CrossRef]
  11. J. R. Wait, A. M. Conda, “Diffraction of Electromagnetic Waves by Smooth Obstacles for Grazing Angles,” J. Res. Natl. Bur. Stand. Sect. D 63, 181 (1959).
  12. R. G. Kouyoumjian, “The Geometrical Theory of Diffraction and Its Applications,” in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra, Ed. (Springer, New York, 1975), pp. 165–215.
    [CrossRef]
  13. P. Langlois, “Diffraction of an e.m. Wave by an Absorbing Circular Cylinder,” Opt. Acta 30, 1373 (1983).
    [CrossRef]
  14. P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
    [CrossRef]
  15. P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a 3D Gaussian Laser Beam at Grazing Incidence on a Large Absorbing Circular Cylinder,” J. Opt. Soc. Am. A., (June1985), in press.
    [CrossRef]
  16. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave–Part I,” J. Opt. Soc. Am. 52, 615 (1962).
    [CrossRef]
  17. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave–Part II,” J. Opt. Soc. Am. 52, 626 (1962).
    [CrossRef]
  18. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  19. P. Langlois, M. Cormier, R. Beaulieu, M. Blanchard, “Experimental Evidence of Edge Parameter Effects in Light Diffraction, Using Holography,” J. Opt. Soc. Am. 67, 87 (1977).
    [CrossRef]
  20. T. Takenaka, O. Fukumitsu, “Asymptotic Representation of the Boundary-Diffraction Wave for a Three-Dimensional Gaussian Beam Incident Upon a Kirchhoff Half-Screen,” J. Opt. Soc. Am. 72, 331 (1982).
    [CrossRef]
  21. S. Kozaki, “High Frequency Scattering of a Gaussian Beam by a Conducting Cylinder,” IEEE Trans. Antennas Propag. AP-31, 795 (1983).
    [CrossRef]
  22. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, London, 1965), p. 6.
  23. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970).
  24. L. G. Schulz, “Optical Constants of Silver, Gold, Copper, and Aluminum. I. The Absorption Coefficient k,” J. Opt. Soc. Am. 44, 357 (1954).
    [CrossRef]
  25. L. G. Schulz, F. R. Tangherlini, “Optical Constants of Silver, Gold, Copper, and Aluminum. II. The Index of Refraction n,” J. Opt. Soc. Am. 44, 362 (1954).
    [CrossRef]

1985 (1)

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
[CrossRef]

1983 (2)

P. Langlois, “Diffraction of an e.m. Wave by an Absorbing Circular Cylinder,” Opt. Acta 30, 1373 (1983).
[CrossRef]

S. Kozaki, “High Frequency Scattering of a Gaussian Beam by a Conducting Cylinder,” IEEE Trans. Antennas Propag. AP-31, 795 (1983).
[CrossRef]

1982 (1)

1978 (1)

1977 (1)

1975 (2)

1971 (1)

1962 (3)

1959 (1)

J. R. Wait, A. M. Conda, “Diffraction of Electromagnetic Waves by Smooth Obstacles for Grazing Angles,” J. Res. Natl. Bur. Stand. Sect. D 63, 181 (1959).

1956 (1)

J. B. Keller, “Diffraction by a Convex Cylinder,” IRE Trans. Antennas Propag. AP-4, 312 (1956).
[CrossRef]

1954 (2)

Beaulieu, R.

Blanchard, M.

Boivin, A.

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
[CrossRef]

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a 3D Gaussian Laser Beam at Grazing Incidence on a Large Absorbing Circular Cylinder,” J. Opt. Soc. Am. A., (June1985), in press.
[CrossRef]

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometry,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409– 437.

Caudell, T. P.

Conda, A. M.

J. R. Wait, A. M. Conda, “Diffraction of Electromagnetic Waves by Smooth Obstacles for Grazing Angles,” J. Res. Natl. Bur. Stand. Sect. D 63, 181 (1959).

Cormier, M.

Cornejo-Rodriguez, A.

Evans, J. D.

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, London, 1965), p. 6.

Fukumitsu, O.

Keller, J. B.

J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. 52, 116 (1962).
[CrossRef]

J. B. Keller, “Diffraction by a Convex Cylinder,” IRE Trans. Antennas Propag. AP-4, 312 (1956).
[CrossRef]

Kouyoumjian, R. G.

R. G. Kouyoumjian, “The Geometrical Theory of Diffraction and Its Applications,” in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra, Ed. (Springer, New York, 1975), pp. 165–215.
[CrossRef]

Kozaki, S.

S. Kozaki, “High Frequency Scattering of a Gaussian Beam by a Conducting Cylinder,” IEEE Trans. Antennas Propag. AP-31, 795 (1983).
[CrossRef]

Langlois, P.

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
[CrossRef]

P. Langlois, “Diffraction of an e.m. Wave by an Absorbing Circular Cylinder,” Opt. Acta 30, 1373 (1983).
[CrossRef]

P. Langlois, M. Cormier, R. Beaulieu, M. Blanchard, “Experimental Evidence of Edge Parameter Effects in Light Diffraction, Using Holography,” J. Opt. Soc. Am. 67, 87 (1977).
[CrossRef]

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a 3D Gaussian Laser Beam at Grazing Incidence on a Large Absorbing Circular Cylinder,” J. Opt. Soc. Am. A., (June1985), in press.
[CrossRef]

Lessard, R. A.

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
[CrossRef]

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a 3D Gaussian Laser Beam at Grazing Incidence on a Large Absorbing Circular Cylinder,” J. Opt. Soc. Am. A., (June1985), in press.
[CrossRef]

Malacara-Hernandez, D.

Miyamoto, K.

Murty, M. V. R. K.

D. Malacara-Hernandez, M. V. R. K. Murty, A. Cornejo-Rodriguez, “Bibliography of Various Optical Testing Methods,” Appl. Opt. 14, 1065 (1975).
[CrossRef]

M. V. R. K. Murty, “Newton, Fizeau and Haidinger Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 1–45.

Schulz, L. G.

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Smolka, F. M.

Takasaki, H.

H. Takasaki, “Moire Topography,” Jpn. J. Appl. Phys. Suppl. 14-1, 441 (1975).

Takenaka, T.

Tangherlini, F. R.

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Varner, J. R.

J. R. Varner, “Holographic Contouring Methods,” in Handbook of Optica $lography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 595–600.

Wait, J. R.

J. R. Wait, A. M. Conda, “Diffraction of Electromagnetic Waves by Smooth Obstacles for Grazing Angles,” J. Res. Natl. Bur. Stand. Sect. D 63, 181 (1959).

Wolf, E.

Wyant, J. C.

J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.

Appl. Opt. (3)

Can. J. Phys. (1)

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a Transversal 2-D Gaussian Beam at Normal Incidence on an Absorbing Circular Cylinder,” Can. J. Phys. 63, Feb. (1985). (Special issue on Optics).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Kozaki, “High Frequency Scattering of a Gaussian Beam by a Conducting Cylinder,” IEEE Trans. Antennas Propag. AP-31, 795 (1983).
[CrossRef]

IRE Trans. Antennas Propag. (1)

J. B. Keller, “Diffraction by a Convex Cylinder,” IRE Trans. Antennas Propag. AP-4, 312 (1956).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Res. Natl. Bur. Stand. Sect. D (1)

J. R. Wait, A. M. Conda, “Diffraction of Electromagnetic Waves by Smooth Obstacles for Grazing Angles,” J. Res. Natl. Bur. Stand. Sect. D 63, 181 (1959).

Jpn. J. Appl. Phys. Suppl. (1)

H. Takasaki, “Moire Topography,” Jpn. J. Appl. Phys. Suppl. 14-1, 441 (1975).

Opt. Acta (1)

P. Langlois, “Diffraction of an e.m. Wave by an Absorbing Circular Cylinder,” Opt. Acta 30, 1373 (1983).
[CrossRef]

Other (9)

P. Langlois, A. Boivin, R. A. Lessard, “Electromagnetic Diffraction of a 3D Gaussian Laser Beam at Grazing Incidence on a Large Absorbing Circular Cylinder,” J. Opt. Soc. Am. A., (June1985), in press.
[CrossRef]

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

R. G. Kouyoumjian, “The Geometrical Theory of Diffraction and Its Applications,” in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra, Ed. (Springer, New York, 1975), pp. 165–215.
[CrossRef]

M. V. R. K. Murty, “Newton, Fizeau and Haidinger Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 1–45.

J. H. Bruning, “Fringe Scanning Interferometry,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409– 437.

J. C. Wyant, “Holographic and Moire Techniques,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 381–408.

J. R. Varner, “Holographic Contouring Methods,” in Handbook of Optica $lography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 595–600.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, London, 1965), p. 6.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970).

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 (4)

Fig. 1
Fig. 1

Diffraction of a laser beam at grazing incidence on a circular cylinder. The observation point P is located by means of a cylindrical coordinate system (ρ,φ,z) whose origin is at the tangential point of the laser beam, and the z axis is parallel to the cylinder's axis. The nonobstructed part of the incident beam produces a bright spot on the observation screen and the bright line perpendicular to the z axis is associated with the diffraction edge wave.

Fig. 2
Fig. 2

Graphs of the radius of curvature as a function of the ratio RB(φ) defined in Eqs. (1) and (2) for four different diffraction angles: (a) for radii up to 15 mm; (b) for radii up to 1.5 mm.

Fig. 3
Fig. 3

Orthogonal projections of the diffracting setup in the direction of propagation of the incident laser beam: (a) diffraction by a circular cylinder; (b) diffraction by a sphere.

Fig. 4
Fig. 4

Schematic of the experimental setup.

Tables (1)

Tables Icon

Table I Experimental Results

Equations (18)

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

R B ( φ ) = I B ( φ ) / I B ( φ )
R B ( φ ) = | K p ( + φ : k a , n ) K p ( φ : k a , n ) | 2 ,
K p ( φ ; ka , n ) = exp ( ika φ 3 / 6 ) [ 1 φ + Φ p ( φ ; ka , n ) ] ,
Φ p ( φ ; ka , n ) = 2 M 0 { exp ( iM φ t ) [ V ( t ) q p V ( t ) W 2 ( t ) q p W 2 ( t ) ] + exp ( i π 3 ) exp ( 3 + i 2 M φ t ) × [ V ( t ) q p exp ( i 2 π 3 ) V ( t ) W 1 ( t ) q p exp ( i 2 π 3 ) W 1 ( t ) ] } d t ,
M ( ka 2 ) 1 / 3 ,
q p iM n 2 1 n 1 p ,
V ( t ) π A i ( t ) ,
W 1 ( t ) π [ B i ( t ) i A i ( t ) ] ,
W 2 ( t ) π [ B i ( t ) + i A i ( t ) ] .
a λ ,
w 0 λ ,
a ( w 0 λ ) 1 / 2 w 0 .
d 0 w 0 ( w 0 λ ) 2 .
( λ a ) 2 φ 2 1 ,
ρ a ( a λ ) 1 / 3 .
{ | n | > 1 + ɛ , ɛ ( λ / a ) 2 / 3 ,
ph { n } ( λ | n | a ) 2 / 3 .
R B ¯ ( φ = 15 ° )

Metrics