Abstract

Interference of light has numerous metrological applications because the optical path difference (OPD) can be varied at will between the interfering waves in the interferometers. We show how one can desirably change the optical path difference in diffraction. This leads to many novel and interesting metrological applications including high-precision measurements of displacement, phase change, refractive index profile, temperature gradient, diffusion coefficient, and coherence parameters, to name only a few. The subject fundamentally differs from interferometry in the sense that in the latter the measurement criterion is the change in intensity or fringe location, while in the former the criterion is the change in the visibility of fringes with an already known intensity profile. The visibility can vary from zero to one as the OPD changes by a half-wave. Therefore, measurements with the accuracy of a few nanometers are quite feasible. Also, the possibility of changing the OPD in diffraction allows us to use Fresnel diffraction in Fourier spectrometry, to enhance or suppress diffracted fields, and to build phase singularities that have many novel and useful applications.

© 2009 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 335.
  3. C. V. Raman and I. R. Rao, “Diffraction of light by a transparent lamina.” Proc. Phys. Soc. London 39, 453-457 (1927).
  4. M. P. Givens and W. L. Goffe, “Application of the Cornu spiral to the semi-transparent half plane,” Am. J. Phys. 34, 248-253 (1966).
    [CrossRef]
  5. R. C. Saust, “Fresnel diffraction at a transparent lamina,” Proc. Phys. Soc. London 64, 105-113 (1950).
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    [CrossRef]
  7. M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).
  8. M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
    [CrossRef]
  9. M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode,” Opt. Commun. 272, 349-361 (2007).
    [CrossRef]
  10. A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. (Bellingham) 46, 128001-7 (2007).
    [CrossRef]
  11. N. Bocher and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
    [CrossRef]
  12. G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
    [CrossRef]
  13. K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
    [CrossRef]
  14. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219-276.
    [CrossRef]
  15. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. K. Knop, “Color pictures using the zero diffraction order of phase grating structures,” Opt. Commun. 18, 298-303 (1976).
    [CrossRef]
  19. O. V. Angelsky, S. G. Hanson, A. P. Maksimyak, and P. P. Maksimyak, “On the feasibility for determining the amplitude zeroes in polychromatic fields,” Opt. Express 13, 4396-4405 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. O. V. Angelsky, P. V. Polyanskii, and S. G. Hanson, “Singular-optical coloring of regularly scattered white light,” Opt. Express 14, 7579-7586 (2006).
    [CrossRef] [PubMed]
  23. M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 42, 2397-2405 (2008).
    [CrossRef]
  24. M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
    [CrossRef]
  25. M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored catastrophes,” Opt. Lett. 33, 1863-1865 (2008).
    [CrossRef] [PubMed]

2008 (3)

2007 (2)

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode,” Opt. Commun. 272, 349-361 (2007).
[CrossRef]

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. (Bellingham) 46, 128001-7 (2007).
[CrossRef]

2006 (1)

2005 (3)

2004 (2)

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
[CrossRef]

2002 (2)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211-1213 (2002).
[CrossRef]

2001 (1)

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

1999 (1)

G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
[CrossRef]

1984 (1)

1976 (2)

K. Knop, “Color pictures using the zero diffraction order of phase grating structures,” Opt. Commun. 18, 298-303 (1976).
[CrossRef]

N. Bocher and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

1966 (1)

M. P. Givens and W. L. Goffe, “Application of the Cornu spiral to the semi-transparent half plane,” Am. J. Phys. 34, 248-253 (1966).
[CrossRef]

1950 (1)

R. C. Saust, “Fresnel diffraction at a transparent lamina,” Proc. Phys. Soc. London 64, 105-113 (1950).

1927 (1)

C. V. Raman and I. R. Rao, “Diffraction of light by a transparent lamina.” Proc. Phys. Soc. London 39, 453-457 (1927).

Amiri, M.

M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored catastrophes,” Opt. Lett. 33, 1863-1865 (2008).
[CrossRef] [PubMed]

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode,” Opt. Commun. 272, 349-361 (2007).
[CrossRef]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

Angelsky, O. V.

Bocher, N.

N. Bocher and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

Born, M.

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 335.

Dashtdar, M.

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 42, 2397-2405 (2008).
[CrossRef]

Ebadi, Z.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Givens, M. P.

M. P. Givens and W. L. Goffe, “Application of the Cornu spiral to the semi-transparent half plane,” Am. J. Phys. 34, 248-253 (1966).
[CrossRef]

Goffe, W. L.

M. P. Givens and W. L. Goffe, “Application of the Cornu spiral to the semi-transparent half plane,” Am. J. Phys. 34, 248-253 (1966).
[CrossRef]

Hanson, S. G.

Jamshidi-Ghaleh, K.

K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
[CrossRef]

Karimi, E.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

Khalesifard, H. R.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

Knop, K.

K. Knop, “Color pictures using the zero diffraction order of phase grating structures,” Opt. Commun. 18, 298-303 (1976).
[CrossRef]

Komiya, A.

G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
[CrossRef]

Maksimyak, A. P.

Maksimyak, P. P.

Mansour, N.

K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
[CrossRef]

Marujama, S.

G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
[CrossRef]

Nahal, A.

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).

Pipman, J.

N. Bocher and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

Polyanskii, P. V.

Ponomarenko, S. A.

Raman, C. V.

C. V. Raman and I. R. Rao, “Diffraction of light by a transparent lamina.” Proc. Phys. Soc. London 39, 453-457 (1927).

Rao, I. R.

C. V. Raman and I. R. Rao, “Diffraction of light by a transparent lamina.” Proc. Phys. Soc. London 39, 453-457 (1927).

Sabatyan, A.

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. (Bellingham) 46, 128001-7 (2007).
[CrossRef]

Sahloll-bai, H.

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

Salehi, M.

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

Saust, R. C.

R. C. Saust, “Fresnel diffraction at a transparent lamina,” Proc. Phys. Soc. London 64, 105-113 (1950).

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219-276.
[CrossRef]

Tavassoly, M. T.

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 42, 2397-2405 (2008).
[CrossRef]

M. Amiri and M. T. Tavassoly, “Spectral anomalies near phase singularities in reflection at Brewster's angle and colored catastrophes,” Opt. Lett. 33, 1863-1865 (2008).
[CrossRef] [PubMed]

M. Dashtdar and M. T. Tavassoly, “Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity,” J. Opt. Soc. Am. A 25, 2509-2517 (2008).
[CrossRef]

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. (Bellingham) 46, 128001-7 (2007).
[CrossRef]

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode,” Opt. Commun. 272, 349-361 (2007).
[CrossRef]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
[CrossRef]

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219-276.
[CrossRef]

Visser, T. D.

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Wolf, E.

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

S. A. Ponomarenko and E. Wolf, “Spectral anomalies in a Fraunhofer diffraction pattern,” Opt. Lett. 27, 1211-1213 (2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 335.

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

Yin, M. C.

Yoshihiro, O.

Zhixiony, G.

G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
[CrossRef]

Am. J. Phys. (1)

M. P. Givens and W. L. Goffe, “Application of the Cornu spiral to the semi-transparent half plane,” Am. J. Phys. 34, 248-253 (1966).
[CrossRef]

Appl. Opt. (1)

Iranian J. Phys. (1)

M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R. Khalesifard, “Fresnel diffraction from step in reflection and transmission,” Iranian J. Phys. 5, 237-246 (2001).

J. Opt. Soc. Am. A (1)

J. Phys. D: Appl. Phys. (3)

N. Bocher and J. Pipman, “A simple method of determining diffusion constants by holographic interferometry,” J. Phys. D: Appl. Phys. 9, 1825-1830 (1976).
[CrossRef]

G. Zhixiony, S. Marujama, and A. Komiya, “Rapid yet accurate measurement of mass diffusion coefficients by phase shifting interferometer,” J. Phys. D: Appl. Phys. 32, 995-999 (1999).
[CrossRef]

K. Jamshidi-Ghaleh, M. T. Tavassoly, and N. Mansour, “Diffusion coefficient measurements of transparent liquid solutions using moiré deflectometery,” J. Phys. D: Appl. Phys. 37, 1-5 (2004).
[CrossRef]

Opt. Commun. (5)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, “Spectral modification by line singularity in Fresnel diffraction from 1D phase step,” Opt. Commun. 255, 23-34 (2005).
[CrossRef]

M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1D and 2D phase steps in reflection and transmission mode,” Opt. Commun. 272, 349-361 (2007).
[CrossRef]

K. Knop, “Color pictures using the zero diffraction order of phase grating structures,” Opt. Commun. 18, 298-303 (1976).
[CrossRef]

M. T. Tavassoly, A. Nahal, and Z. Ebadi, “Image formation in rough surfaces,” Opt. Commun. 238, 252-260 (2004).
[CrossRef]

M. T. Tavassoly and M. Dashtdar, “Height distribution on a rough plane and specularly diffracted light amplitude are Fourier transform pair,” Opt. Commun. 42, 2397-2405 (2008).
[CrossRef]

Opt. Eng. (Bellingham) (1)

A. Sabatyan and M. T. Tavassoly, “Application of Fresnel diffraction to nondestructive measurement of the refractive index of optical fibers,” Opt. Eng. (Bellingham) 46, 128001-7 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Proc. Phys. Soc. London (2)

R. C. Saust, “Fresnel diffraction at a transparent lamina,” Proc. Phys. Soc. London 64, 105-113 (1950).

C. V. Raman and I. R. Rao, “Diffraction of light by a transparent lamina.” Proc. Phys. Soc. London 39, 453-457 (1927).

Other (4)

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 335.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, 2001), pp. 219-276.
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).

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Figures (15)

Fig. 1
Fig. 1

Cylindrical wave Σ striking a 1D phase step of height h. The diffracted intensity at point P is given in the text.

Fig. 2
Fig. 2

Profile of a transparent plate of refractive index N immersed in a liquid of refractive index N N . The 1D phase steps are formed at the edges of the plate.

Fig. 3
Fig. 3

Sketch of a circular phase step that can be built by mounting a circular mirror M 1 and an annular mirror M 2 on two coaxial cylindrical stands C 1 and C 2 . The light reflected from the beam splitter B.S. diffracts from the step formed by the mirrors, and the step height can be varied by displacing mirror M 1 in a vertical direction.

Fig. 4
Fig. 4

(a) A 1D phase step of height h is formed by replacing the mirrors in a Michelson interferometer by two rectangular mirrors in such a way that each mirror intersects the alternative halves of the light beam striking the beam splitter. (b) A 1D phase step is formed by mounting two opaque plates O 1 and O 2 in the arms of a MZI at equal distances from the beam splitter B.S.2 in such a way that the plates obstruct the alternative halves of the beam reflecting from the mirrors M 1 and M 2 . The step height is varied by changing the OPD between the arms of the interferometer.

Fig. 5
Fig. 5

FD patterns of light diffracted from 1D phase steps of different heights formed in a Michelson interferometer arrangement and the corresponding intensity profiles over the patterns. (a) h = λ 8 . (b) h = λ 4 . (c) h = 3 λ 8 .

Fig. 6
Fig. 6

FD patterns and the corresponding intensity profiles of light diffracted from circular phase steps of different heights formed by a MZI, (a) h = 5 λ 24 . (b) h = λ 2 . (c) h = 5 λ 6 .

Fig. 7
Fig. 7

Calculated visibility versus the optical path difference divided by wavelength Δ λ for three central fringes in FD from a 1D phase step.

Fig. 8
Fig. 8

Cornu spirals attributed to a 1D phase step of height h = λ 10 or ϕ = 2 π 5 . The bold face parts of the spirals contribute to the amplitude at point P in Fig. 1 associated with points M and M on the spirals.

Fig. 9
Fig. 9

Scheme of a rectangular cell and a plane parallel plate that is installed inside it to study liquid–liquid diffusion by light diffraction.

Fig. 10
Fig. 10

Diffraction patterns of the light diffracted from the edge of a plane parallel plate immersed in a rectangular cell containing pure water over sugar solution of concentration 10% at different times after the initiation of the diffusion. The established refractive index gradient has appeared as the fringes inclined with respect to the plate edge.

Fig. 11
Fig. 11

Experimental realization of Babinet’s principle. (a), (b) The diffraction patterns and intensity profiles of the light diffracted from a slit of 0.24 mm width and an opaque strip of the same width as the slit. (c) The pattern and intensity profile obtained by superimposing the diffracted fields in (a) and (b) in a MZI. (d), (e) The diffraction patterns and intensity profiles of the light diffracted from two complementary straight edges. (f) The pattern and intensity profile obtained by superimposing the diffraction fields in (d) and (e) in a MZI.

Fig. 12
Fig. 12

Enhancing and suppressing light diffraction. (a) FD pattern of light diffracted from a slit. (b), (c) The patterns obtained by superimposing constructively (enhanced mode) and destructively (suppressed mode) the light diffracted from two similar slits installed in a MZI’s arms. (d) The profiles of the intensity distribution of the corresponding diffraction patterns.

Fig. 13
Fig. 13

Diffraction patterns of the light diffracted from a copper wire of thickness 0.4 mm carrying different electric currents after its original diffracted field had been suppressed by the field diffracted from another similar wire installed in the other arm of a MZI. (a) I = 0.12 A . (b) I = 0.25 A . (c) I = 0.41 A . (d) The pattern obtained by superimposing constructively the diffracted fields from the two wires with no electric current. (e) The profiles of the intensity distributions of the corresponding diffraction patterns.

Fig. 14
Fig. 14

Diffraction patterns of white light diffracted from 1D phase steps of slightly different step heights around h = 140 nm .

Fig. 15
Fig. 15

Spectrum of a light beam incident on a 1D phase step of height λ 0 4 . (b) The normalized spectra of the diffracted lights at two points symmetrical with respect to the step edge ( λ 0 = 560 nm ) .

Equations (6)

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I L = I 0 r L r R [ cos 2 ( ϕ 2 ) + 2 ( C 0 2 + S 0 2 ) sin 2 ( ϕ 2 ) ( C 0 S 0 ) sin ϕ ] + I 0 2 [ ( r L r R ) 2 ( 1 2 + C 0 2 + S 0 2 ) + ( C 0 + S 0 ) ( r L 2 r R 2 ) ] ,
I n = cos 2 ( ϕ 2 ) + 2 ( C 0 2 + S 0 2 ) sin 2 ( ϕ 2 ) ( C 0 S 0 ) sin ϕ ,
I n = A + B cos ϕ C sin ϕ ,
A = 1 2 + C 0 2 + S 0 2 , B = 1 2 ( C 0 2 S 0 2 ) , C = C 0 S 0 .
ϕ = k N h [ n 2 sin 2 θ cos θ ] ,
V = 1 2 ( I maL + I maR ) I miM 1 2 ( I maL + I maR ) + I miM ,

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