Abstract

A simple evaluation of the phase change in a diffracted wave, in terms of the Cornu spiral, is presented to complement the well-known intensity change, which is routinely obtained for this elegant graphical construction of the Fresnel integrals. This is, to the best of our knowledge, the first presentation of this evaluation. It is shown that the phase of a wave diffracted by a slit is equal to the slope of the line tangent to the Cornu spiral, shifted by π/4.

© 2010 Optical Society of America

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  1. K. G. Bass, Transp. Forum 1, 47 (1984).
  2. V. M. Castaño, Jpn. J. Appl. Phys. 44, 5009 (2005).
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  3. A.-M. Pendrill, Phys. Educ. 40, 517 (2005).
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  4. S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
    [CrossRef]
  5. E. Hecht, Optics,3rd ed. (Addison-Wesley, 1998).
  6. M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).
  7. D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
    [CrossRef]
  8. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
    [CrossRef]
  9. D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  17. A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).
  18. C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).
  19. D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).
  20. M. V. Papalexandris and D. C. Redding, J. Opt. Soc. Am. A 17, 1763 (2000).
    [CrossRef]

2006 (1)

2005 (2)

V. M. Castaño, Jpn. J. Appl. Phys. 44, 5009 (2005).
[CrossRef]

A.-M. Pendrill, Phys. Educ. 40, 517 (2005).
[CrossRef]

2003 (2)

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

2002 (1)

R. Vincent, Ultramicroscopy 90, 135 (2002).
[CrossRef] [PubMed]

2001 (1)

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

2000 (2)

M. V. Papalexandris and D. C. Redding, J. Opt. Soc. Am. A 17, 1763 (2000).
[CrossRef]

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

1999 (4)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

S. B. Tucker, J. Ojeda-Castañeda, and W. T. Cathey, J. Opt. Soc. Am. A 16, 316 (1999).
[CrossRef]

1998 (1)

E. Hecht, Optics,3rd ed. (Addison-Wesley, 1998).

1996 (2)

1995 (2)

H. Hamam and J. L. de-Bougrenet-de-la-Tocnaye, J. Opt. Soc. Am. A 12, 1920 (1995).
[CrossRef]

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

1994 (1)

1984 (1)

K. G. Bass, Transp. Forum 1, 47 (1984).

Artigas, J. M.

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

Bass, K. G.

K. G. Bass, Transp. Forum 1, 47 (1984).

Bernardo, L. M.

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

Castaño, V. M.

V. M. Castaño, Jpn. J. Appl. Phys. 44, 5009 (2005).
[CrossRef]

Cathey, W. T.

Caulfield, H. J.

Chatila, R.

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

de-Bougrenet-de-la-Tocnaye, J. L.

Deng, Z.-T.

Dorsch, R. G.

Ferreira, C.

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Fleury, S.

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

García, J.

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

J. García, D. Mas, and R. G. Dorsch, Appl. Opt. 35, 7013 (1996).
[CrossRef] [PubMed]

Hamam, H.

Hecht, E.

E. Hecht, Optics,3rd ed. (Addison-Wesley, 1998).

Hernandez, C.

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

Illueca, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

Laumond, J.-P.

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

Lorente, A.

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

J. García, D. Mas, and R. G. Dorsch, Appl. Opt. 35, 7013 (1996).
[CrossRef] [PubMed]

Miret, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

Ojeda-Castañeda, J.

Papalexandris, M. V.

Pellat-Finet, P.

Pendrill, A.-M.

A.-M. Pendrill, Phys. Educ. 40, 517 (2005).
[CrossRef]

Perez, J.

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

Pons, A. M.

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

Redding, D. C.

Schamschula, M.

Schen, F.

Soueres, P.

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

Tucker, S. B.

Vazquez, C.

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

Vincent, R.

R. Vincent, Ultramicroscopy 90, 135 (2002).
[CrossRef] [PubMed]

Wang, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

Appl. Opt. (2)

IEEE Trans. Robot. Automat. (1)

S. Fleury, P. Soueres, J.-P. Laumond, and R. Chatila, IEEE Trans. Robot. Automat. 11, 441 (1995).
[CrossRef]

J. Mod. Opt. (3)

A. M. Pons, A. Lorente, C. Illueca, D. Mas, and J. M. Artigas, J. Mod. Opt. 46, 1043 (1999).

C. Illueca, D. Mas, J. Perez, A. M. Pons, and J. M. Artigas, J. Mod. Opt. 48, 811 (2001).

D. Mas, J. Perez, C. Vazquez, C. Hernandez, and C. Illueca, J. Mod. Opt. 50, 1335 (2003).

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

V. M. Castaño, Jpn. J. Appl. Phys. 44, 5009 (2005).
[CrossRef]

Opt. Commun. (2)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, Opt. Commun. 164, 233 (1999).
[CrossRef]

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, Opt. Commun. 227, 245 (2003).
[CrossRef]

Opt. Eng. (1)

D. Mas, C. Ferreira, J. García, and L. M. Bernardo, Opt. Eng. 39, 1427 (2000).
[CrossRef]

Opt. Lett. (2)

Phys. Educ. (1)

A.-M. Pendrill, Phys. Educ. 40, 517 (2005).
[CrossRef]

Transp. Forum (1)

K. G. Bass, Transp. Forum 1, 47 (1984).

Ultramicroscopy (1)

R. Vincent, Ultramicroscopy 90, 135 (2002).
[CrossRef] [PubMed]

Other (2)

E. Hecht, Optics,3rd ed. (Addison-Wesley, 1998).

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge U. Press, 1999).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the diffraction by a slit.

Fig. 2
Fig. 2

Cornu spiral. The coordinates are given by Eqs. (8, 9).

Equations (15)

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Ψ P = i E 0 λ r 0 ρ 0 z z d z d y exp [ i k ( ρ 0 + r 0 + ( y 2 + z 2 ) ρ 0 + r 0 2 ρ 0 r 0 ) ] ,
u = y [ 2 ( ρ 0 + r 0 ) λ ρ 0 r 0 ] 1 / 2 , v = z [ 2 ( ρ 0 + r 0 ) λ ρ 0 r 0 ] 1 / 2
Ψ P = i E 0 2 ( r 0 + ρ 0 ) exp ( i k ( r 0 + ρ 0 ) ) v v d u d v exp [ i π 2 ( u 2 + v 2 ) ] ,
Ψ P = B ( R + i I ) ,
B = i E 0 2 ( r 0 + ρ 0 ) exp ( i k ( r 0 + ρ 0 ) ) ,
R = v v d u d v cos [ π 2 ( u 2 + v 2 ) ] ,
I = v v d u d v sin [ π 2 ( u 2 + v 2 ) ] .
C ( w ) = 0 w d ζ cos ( π 2 ζ 2 ) ,
S ( w ) = 0 w d ζ sin ( π 2 ζ 2 ) .
R = 2 ( C ( w ) S ( w ) ) ,
I = 2 ( C ( w ) + S ( w ) ) ,
w = 2 ( ρ 0 + r 0 ) λ ρ 0 r 0 z .
k ( r 0 + ρ 0 ) + ϕ ,
tan ( ϕ ) = C ( w ) + S ( w ) C ( w ) S ( w ) = 1 + S ( w ) / C ( w ) 1 S ( w ) / C ( w ) = 1 + tan ( θ ) 1 tan ( θ ) = tan ( θ + π 4 ) ,
ϕ = π 2 w 2 + π 4 modulo π .

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