Abstract

A new treatment of the well-known Sommerfeld solution of the problem of plane-wave diffraction from a perfectly conducting half-plane is reported. We show, in both theory and experiment, that the diffraction field (E-polarization) can be represented as a superposition of real physically existing waves, in contrast to geometrical and boundary waves postulated in Sommerfeld’s representation. Our representation includes two pairs of wave components: one pair propagates along the direction of the incident wave, and the other in a mirror-reflected direction. Each wave pair consists of a plane-wave component with an amplitude half that of the incident wave and a nearly plane-wave component with an infinitely extended edge dislocation. On the basis of the proposed interpretation, all features of the half-plane diffraction are explained.

© 2000 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, UK, 1991).
  2. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).
  3. S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 107–108 (1995).
  4. A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
    [CrossRef]
  5. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. 16(2), 21–48 (1888).
    [CrossRef]
  6. A. Sommerfeld, Optics (Academic, New York, 1954).
  7. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschten Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–258 (1917).
    [CrossRef]
  8. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. (Leipzig) 73, 339–364 (1924).
    [CrossRef]
  9. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 162–164 (1957).
    [CrossRef]
  10. A. Rubinowicz, “Simple derivation of the Miyamoto–Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717–718 (1962).
    [CrossRef]
  11. A. Rubinowicz, Kirchhoff’s Diffraction Theory and Its Interpretation on the Basis of Young’s Views (Polish Academy of Sciences Press, Krakow, 1972).
  12. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [CrossRef]
  13. K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962).
    [CrossRef]
  14. K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave—Part II,” J. Opt. Soc. Am. 52, 626–637 (1962).
    [CrossRef]
  15. E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
    [CrossRef]
  16. J. W. Y. Lit, R. Tramblay, “Boundary-diffraction-wave theory of cascaded-apertures diffraction,” J. Opt. Soc. Am. 59, 559–567 (1969).
    [CrossRef]
  17. T. Suzuki, “Extension of the theory of the boundary diffraction wave to systems with arbitrary aperture-transmittance function,” J. Opt. Soc. Am. 61, 439–445 (1971).
    [CrossRef]
  18. G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
    [CrossRef]
  19. G. Otis, J. L. Lachambre, J. W. Y. Lit, P. Lavigne, “Diffracted waves in the shadow boundary region,” J. Opt. Soc. Am. 67, 551–553 (1977).
    [CrossRef]
  20. P. Langlois, A. Boivin, “Thomas Young’s ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265–274 (1985).
    [CrossRef]
  21. R. W. Wood, Physical Optics (Macmillan, New York, 1934).
  22. S. Ganchi, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370–373 (1989).
    [CrossRef]
  23. Yu. I. Terentiev, “Diffraction of light on a thin flat screen with the straight edge,” Opt. Atm. 2, 1141–1146 (1989) (in Russian).
  24. Yu. I. Terentiev, “New facts confirming the objective character of the Young’s description of the origin of the diffraction on a screen,” Opt. Atm. 2, 1325–1327 (1989) (in Russian).
  25. V. N. Smirnov, S. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spektrosk. 76, 1019–1026 (1994) (in Russian).
  26. P. V. Polyanskii, G. V. Polyanskaya, “On a consequence of the Young–Rubinowicz model of diffraction phenomena in holography,” Opt. Appl. 25, 171–183 (1995).
  27. P. V. Polyanskii, G. V. Polyanskaya, “The Young hologram—a fifth type of hologram,” J. Opt. Technol. 64, 321–330 (1997).
  28. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. Soc. R. London, Sect. A 336, 165–190 (1974).
    [CrossRef]
  29. W. Braunbek, G. Laukien, “Einzelheiten zur Halbebenen-Beugung,” Optik (Stuttgart) 9, 174–179 (1952).
  30. J. D. Barnett, F. S. Harris, “Test of the effect of edge parameters on small-angle Fresnel diffraction of light at a straight edge,” J. Opt. Soc. Am. 52, 637–643 (1962).
    [CrossRef]
  31. I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  32. G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
    [CrossRef] [PubMed]

1997 (1)

P. V. Polyanskii, G. V. Polyanskaya, “The Young hologram—a fifth type of hologram,” J. Opt. Technol. 64, 321–330 (1997).

1995 (4)

P. V. Polyanskii, G. V. Polyanskaya, “On a consequence of the Young–Rubinowicz model of diffraction phenomena in holography,” Opt. Appl. 25, 171–183 (1995).

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 107–108 (1995).

1994 (1)

V. N. Smirnov, S. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spektrosk. 76, 1019–1026 (1994) (in Russian).

1989 (3)

S. Ganchi, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370–373 (1989).
[CrossRef]

Yu. I. Terentiev, “Diffraction of light on a thin flat screen with the straight edge,” Opt. Atm. 2, 1141–1146 (1989) (in Russian).

Yu. I. Terentiev, “New facts confirming the objective character of the Young’s description of the origin of the diffraction on a screen,” Opt. Atm. 2, 1325–1327 (1989) (in Russian).

1985 (1)

P. Langlois, A. Boivin, “Thomas Young’s ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265–274 (1985).
[CrossRef]

1977 (1)

1974 (2)

G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
[CrossRef]

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. Soc. R. London, Sect. A 336, 165–190 (1974).
[CrossRef]

1971 (1)

1969 (1)

1966 (1)

1962 (5)

1957 (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 162–164 (1957).
[CrossRef]

1952 (1)

W. Braunbek, G. Laukien, “Einzelheiten zur Halbebenen-Beugung,” Optik (Stuttgart) 9, 174–179 (1952).

1924 (1)

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. (Leipzig) 73, 339–364 (1924).
[CrossRef]

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschten Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–258 (1917).
[CrossRef]

1896 (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

1888 (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. 16(2), 21–48 (1888).
[CrossRef]

Barnett, J. D.

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. Soc. R. London, Sect. A 336, 165–190 (1974).
[CrossRef]

Boivin, A.

P. Langlois, A. Boivin, “Thomas Young’s ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265–274 (1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

Braunbek, W.

W. Braunbek, G. Laukien, “Einzelheiten zur Halbebenen-Beugung,” Optik (Stuttgart) 9, 174–179 (1952).

Crosignani, B.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Di Porto, P.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Duree, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Ganchi, S.

S. Ganchi, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370–373 (1989).
[CrossRef]

Harris, F. S.

Lachambre, J. L.

Langlois, P.

P. Langlois, A. Boivin, “Thomas Young’s ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265–274 (1985).
[CrossRef]

Laukien, G.

W. Braunbek, G. Laukien, “Einzelheiten zur Halbebenen-Beugung,” Optik (Stuttgart) 9, 174–179 (1952).

Lavigne, P.

Lit, J. W. Y.

Maggi, G. A.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. 16(2), 21–48 (1888).
[CrossRef]

Marchand, E. W.

Miyamoto, K.

Morin, M.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. Soc. R. London, Sect. A 336, 165–190 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

Otis, G.

Polyanskaya, G. V.

P. V. Polyanskii, G. V. Polyanskaya, “The Young hologram—a fifth type of hologram,” J. Opt. Technol. 64, 321–330 (1997).

P. V. Polyanskii, G. V. Polyanskaya, “On a consequence of the Young–Rubinowicz model of diffraction phenomena in holography,” Opt. Appl. 25, 171–183 (1995).

Polyanskii, P. V.

P. V. Polyanskii, G. V. Polyanskaya, “The Young hologram—a fifth type of hologram,” J. Opt. Technol. 64, 321–330 (1997).

P. V. Polyanskii, G. V. Polyanskaya, “On a consequence of the Young–Rubinowicz model of diffraction phenomena in holography,” Opt. Appl. 25, 171–183 (1995).

Rubinowicz, A.

A. Rubinowicz, “Simple derivation of the Miyamoto–Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717–718 (1962).
[CrossRef]

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 162–164 (1957).
[CrossRef]

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. (Leipzig) 73, 339–364 (1924).
[CrossRef]

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschten Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–258 (1917).
[CrossRef]

A. Rubinowicz, Kirchhoff’s Diffraction Theory and Its Interpretation on the Basis of Young’s Views (Polish Academy of Sciences Press, Krakow, 1972).

Salamo, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Segev, M.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Sharp, E.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Smirnov, V. N.

V. N. Smirnov, S. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spektrosk. 76, 1019–1026 (1994) (in Russian).

Sommerfeld, A.

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

A. Sommerfeld, Optics (Academic, New York, 1954).

Soskin, M. S.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Strokovskii, S. A.

V. N. Smirnov, S. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spektrosk. 76, 1019–1026 (1994) (in Russian).

Suzuki, T.

Terentiev, Yu. I.

Yu. I. Terentiev, “New facts confirming the objective character of the Young’s description of the origin of the diffraction on a screen,” Opt. Atm. 2, 1325–1327 (1989) (in Russian).

Yu. I. Terentiev, “Diffraction of light on a thin flat screen with the straight edge,” Opt. Atm. 2, 1141–1146 (1989) (in Russian).

Tramblay, R.

Vasnetsov, M. V.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Wang, S.

S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 107–108 (1995).

Wolf, E.

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1934).

Yariv, A.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Am. J. Phys. (1)

S. Ganchi, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370–373 (1989).
[CrossRef]

Ann. Mat. (1)

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Mat. 16(2), 21–48 (1888).
[CrossRef]

Ann. Phys. (Leipzig) (2)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschten Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–258 (1917).
[CrossRef]

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. (Leipzig) 73, 339–364 (1924).
[CrossRef]

Can. J. Phys. (1)

P. Langlois, A. Boivin, “Thomas Young’s ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265–274 (1985).
[CrossRef]

J. Opt. Soc. Am. (10)

E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
[CrossRef]

J. W. Y. Lit, R. Tramblay, “Boundary-diffraction-wave theory of cascaded-apertures diffraction,” J. Opt. Soc. Am. 59, 559–567 (1969).
[CrossRef]

T. Suzuki, “Extension of the theory of the boundary diffraction wave to systems with arbitrary aperture-transmittance function,” J. Opt. Soc. Am. 61, 439–445 (1971).
[CrossRef]

G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
[CrossRef]

G. Otis, J. L. Lachambre, J. W. Y. Lit, P. Lavigne, “Diffracted waves in the shadow boundary region,” J. Opt. Soc. Am. 67, 551–553 (1977).
[CrossRef]

E. W. Marchand, E. Wolf, “Boundary diffraction wave in the domain of the Rayleigh–Kirchhoff diffraction theory,” J. Opt. Soc. Am. 52, 761–767 (1962).
[CrossRef]

K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962).
[CrossRef]

K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary diffraction wave—Part II,” J. Opt. Soc. Am. 52, 626–637 (1962).
[CrossRef]

J. D. Barnett, F. S. Harris, “Test of the effect of edge parameters on small-angle Fresnel diffraction of light at a straight edge,” J. Opt. Soc. Am. 52, 637–643 (1962).
[CrossRef]

A. Rubinowicz, “Simple derivation of the Miyamoto–Wolf formula for the vector potential associated with a solution of the Helmholtz equation,” J. Opt. Soc. Am. 52, 717–718 (1962).
[CrossRef]

J. Opt. Technol. (1)

P. V. Polyanskii, G. V. Polyanskaya, “The Young hologram—a fifth type of hologram,” J. Opt. Technol. 64, 321–330 (1997).

Math. Ann. (1)

A. Sommerfeld, “Mathematische Theorie der Diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

Nature (London) (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature (London) 180, 162–164 (1957).
[CrossRef]

Opt. Appl. (1)

P. V. Polyanskii, G. V. Polyanskaya, “On a consequence of the Young–Rubinowicz model of diffraction phenomena in holography,” Opt. Appl. 25, 171–183 (1995).

Opt. Atm. (2)

Yu. I. Terentiev, “Diffraction of light on a thin flat screen with the straight edge,” Opt. Atm. 2, 1141–1146 (1989) (in Russian).

Yu. I. Terentiev, “New facts confirming the objective character of the Young’s description of the origin of the diffraction on a screen,” Opt. Atm. 2, 1325–1327 (1989) (in Russian).

Opt. Commun. (1)

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Opt. Spektrosk. (1)

V. N. Smirnov, S. A. Strokovskii, “On diffraction of optical Hermite–Gaussian beams from a diaphragm,” Opt. Spektrosk. 76, 1019–1026 (1994) (in Russian).

Optik (Stuttgart) (2)

S. Wang, “On principles of diffraction,” Optik (Stuttgart) 100, 107–108 (1995).

W. Braunbek, G. Laukien, “Einzelheiten zur Halbebenen-Beugung,” Optik (Stuttgart) 9, 174–179 (1952).

Phys. Rev. Lett. (1)

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[CrossRef] [PubMed]

Proc. Soc. R. London, Sect. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. Soc. R. London, Sect. A 336, 165–190 (1974).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics, Bristol, UK, 1999).

A. Rubinowicz, Kirchhoff’s Diffraction Theory and Its Interpretation on the Basis of Young’s Views (Polish Academy of Sciences Press, Krakow, 1972).

R. W. Wood, Physical Optics (Macmillan, New York, 1934).

A. Sommerfeld, Optics (Academic, New York, 1954).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the half-plane screen diffraction: The screen is placed at the plane y=0, x0; r is the distance between the screen edge and observation point P; k is the wave vector of the incident plane wave. The angle α0 (0α0π) is measured from the X axis to the incident wave-vector direction; the angle θ is measured from the X axis to the radius-vector direction. The Z axis coincides with the screen edge. The mirror-reflected wave direction is also shown.

Fig. 2
Fig. 2

Amplitudes of (a) geometrical and (b) boundary wave components of the diffraction field behind the half-plane screen and (c) their superposition, shown as functions of variable X (positive values of X correspond to the region of geometrical shadow). (d) The wave front of the boundary wave (transverse phase Φ) is also shown, to demonstrate its nonsmoothed shape. Dashed lines are parabolic approximations of the right- and left-hand parts of the wave front.

Fig. 3
Fig. 3

(a) Fast depletion of the backward-propagating wave behind the screen. The intensity is normalized on the incident plane-wave intensity and is plotted as a function of V. (b) Intensities of the co-propagating components in the diffraction field: (curve 1) the EDW, (curve 2) the plane wave, and (curve 3) the resulting wave, plotted as functions of U. (c) Gray-scale distribution of the EDW amplitude within a frame approximately ±4 wavelengths around the screen edge. (d) The wave-front shape of the EDW (curve 1), possessing a half-wavelength step with adjacent oscillations, and the continuous wave front of the diffracted wave (curve 2) are plotted for ky=-10.

Fig. 4
Fig. 4

Amplitude distribution of (a) the forward-propagating EDW and (b) the backward-traveling EDW. Amplitudes are normalized on A0. The negative amplitude of the EDW corresponds to the occurring π shift in phase.

Fig. 5
Fig. 5

Amplitude distribution of (a) the total forward-propagating and (b) the backward-traveling waves. Amplitudes are normalized on A0.

Fig. 6
Fig. 6

Top, amplitude distribution of the total field with a frame ±4 wavelengths around the screen; bottom, central fragment (±1.25-wavelength frame around the screen edge) shown as a contour plot.

Fig. 7
Fig. 7

Experimental setup: (a) Mach–Zehnder interferometer for investigation of the forward propagating EDW; (b) Michelson interferometer for investigation of the backward-traveling EDW. 1, He–Ne laser; 2, telescopic beam expander with pinhole (plane-wave former); 3, 6, beam splitters (50%); 4, variable-density filter; 5, 100% reflecting mirror 7, 100% reflecting mirror mounted on piezoshifter; 8, sharp-edge screen; 9, image plane with a CCD camera.

Fig. 8
Fig. 8

(a), (b) Experimental and (c), (d) theoretical (in gray scale) images of the intensity distribution (a), (c) behind the screen and (b), (d) interference between the diffracted wave and the tilted reference plane wave.

Fig. 9
Fig. 9

(a), (b) Experimental and theoretical (c), (d) images appearing when the plane-wave component is subtracted from the diffraction field behind the screen. (a), (c) Intensity distribution of the isolated EDW; (b), (d), the pattern of interference between the EDW and the tilted reference plane wave. The dashed line shows the position of the edge dislocation.

Equations (8)

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

Ez(P)=A0 exp[-i(π/4)]π{exp[-ikr cos(θ-α0)]}×F(-U)-exp[-ikr cos(θ+α0)]F(-V)},
U=2kr cosθ-α02,
V=2kr cosθ+α02
Ez(P)=Ez(g)+Ez(d),
Ez(g)=A0{exp[-ikr cos(θ-α0)]-exp[-ikr cos(θ+α0)]},0θ<π-α0A0 exp[-ikr cos(θ-α0)],π-α0<θ<π+α00,π+α0<θ2π.
aexp(iμ2)dμ=0exp(iμ2)dμ-0aexp(iμ2)dμ=1+i2π21/2-f(a).
Ez(P)=A012×exp[-ikr cos(θ-α0)]-12exp[-ikr cos(θ+α0)]+1-i2πexp[-ikr cos(θ-α0)]f(U)-1-i2πexp[-ikr cos(θ+α0)]f(V).
Ez(P)=A0 exp(-iky)12+1-i2π0Uexp(iμ2)dμ,

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