Abstract

The analysis of light propagation in a tapered gradient-index (GRIN) medium when the input signal is a binary function is considered with the Walsh–Hadamard analysis. The study of the evolution of the sequency spectrum through the GRIN medium by the Walsh–Hadamard transform confirms the self-imaging effect.

© 2004 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. K. G. Beauchamp, Walsh Functions and Their Applications (Academic, New York, 1975).
  3. H. F. Harmuth, “Applications of Walsh functions in communications,” IEEE Spectrum 6, 82–91 (1969).
    [CrossRef]
  4. S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, New York, 1985).
  5. H. C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970).
  6. K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 3–101 and references therein.
  7. C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
    [CrossRef]
  8. O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
    [CrossRef]
  9. O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
    [CrossRef]
  10. O. Trabocchi, C. Gómez-Reino, “Integer and fractional Talbot effect of Walsh functions in a tapered gradient-index medium,” J. Mod. Opt. (to be published).
  11. N. M. Blachman, “Sinusoids versus Walsh functions,” in Proc. IEEE 62, 346–354 (1974).
    [CrossRef]
  12. C. Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chaps. 1 and 2.

1997 (1)

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

1996 (1)

O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
[CrossRef]

1987 (1)

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

1974 (1)

N. M. Blachman, “Sinusoids versus Walsh functions,” in Proc. IEEE 62, 346–354 (1974).
[CrossRef]

1969 (1)

H. F. Harmuth, “Applications of Walsh functions in communications,” IEEE Spectrum 6, 82–91 (1969).
[CrossRef]

Andrews, H. C.

H. C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970).

Bao, C.

C. Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chaps. 1 and 2.

Beauchamp, K. G.

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, New York, 1975).

Blachman, N. M.

N. M. Blachman, “Sinusoids versus Walsh functions,” in Proc. IEEE 62, 346–354 (1974).
[CrossRef]

Colautti, C.

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
[CrossRef]

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

Garavaglia, M.

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

Gómez-Reino, C.

O. Trabocchi, C. Gómez-Reino, “Integer and fractional Talbot effect of Walsh functions in a tapered gradient-index medium,” J. Mod. Opt. (to be published).

C. Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chaps. 1 and 2.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Harmuth, H. F.

H. F. Harmuth, “Applications of Walsh functions in communications,” IEEE Spectrum 6, 82–91 (1969).
[CrossRef]

Patorski, K.

K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 3–101 and references therein.

Pérez, M. V.

C. Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chaps. 1 and 2.

Ruiz, B.

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

Saavedra, G.

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

Sicre, E. E.

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
[CrossRef]

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

Trabocchi, O.

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
[CrossRef]

O. Trabocchi, C. Gómez-Reino, “Integer and fractional Talbot effect of Walsh functions in a tapered gradient-index medium,” J. Mod. Opt. (to be published).

Tzafestas, S. G.

S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, New York, 1985).

IEEE Spectrum (1)

H. F. Harmuth, “Applications of Walsh functions in communications,” IEEE Spectrum 6, 82–91 (1969).
[CrossRef]

J. Mod. Opt. (2)

C. Colautti, B. Ruiz, E. E. Sicre, M. Garavaglia, “Walsh functions: analysis of their properties under Fresnel diffraction,” J. Mod. Opt. 34, 1385–1391 (1987).
[CrossRef]

O. Trabocchi, C. Colautti, G. Saavedra, E. E. Sicre, “Spatial coherence properties of a multiple aperture system. An analysis based on the Walsh functions,” J. Mod. Opt. 44, 715–729 (1997).
[CrossRef]

Opt. Eng. (1)

O. Trabocchi, C. Colautti, E. E. Sicre, “Diffraction properties of a periodic multiple-aperture system: an approach based on the Walsh functions,” Opt. Eng. 35, 94–101 (1996).
[CrossRef]

Proc. IEEE (1)

N. M. Blachman, “Sinusoids versus Walsh functions,” in Proc. IEEE 62, 346–354 (1974).
[CrossRef]

Other (7)

C. Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chaps. 1 and 2.

O. Trabocchi, C. Gómez-Reino, “Integer and fractional Talbot effect of Walsh functions in a tapered gradient-index medium,” J. Mod. Opt. (to be published).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

K. G. Beauchamp, Walsh Functions and Their Applications (Academic, New York, 1975).

S. G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, New York, 1985).

H. C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970).

K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 3–101 and references therein.

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Figures (6)

Fig. 1
Fig. 1

Geometry of a tapered GRIN medium with a bivalued aperture illuminated by a Gaussian beam.

Fig. 2
Fig. 2

Input signal: (a) central part of the transmittance function W896(x0), (b) the WHT.

Fig. 3
Fig. 3

Evolution of w˜896 in the GRIN medium along the z axis.

Fig. 4
Fig. 4

First positive self-image at z2=0.42 mm: (a) central part of the irradiance distribution, (b) the WHT.

Fig. 5
Fig. 5

First negative self-image at z1=0.19 mm: (a) central part of the irradiance distribution, (b) the WHT.

Fig. 6
Fig. 6

Self-image distances calculated by Eq. (27) coincide with maximum and minimum of Fig. 3.

Equations (29)

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WalN(t),    t[0, ξ],    N=0, 1, 2.
Radr(t),    t[0, ξ],    r=0, 1, 2.
dr=ξ21-r.
Radr(t)=signcos2rπtξ.
Radr(x0)=signcos2rπx0ξ+12.
WalN(x0)=r=pm[Radr(x0)]gr.
f(x0)=M=0w˜MWalM(x0),
w˜M=1ξ-ξ/2ξ/2f(x0)WalM(x0)dx0.
n2(x, z)=n02[1-g2(z)x2],
T(x0)=12[1+WalN(x0)]=WN(x0),
ϕ(x0)=T(x0)ψ0(x0),
ψ0(x0)=w0w(0)1/2exp[iφ(0)]ψ[x0; U(0)]
ψ[x0; U(0)]=expiπU(0)x02λ
U(0)=1R(0)+iλπw2(0),
φ(0)=tan-1λZ0πw02,
ϕ(x; z)=-ξ/2ξ/2ϕ(x0)K(x, x0; z)dx0,
K(x, x0; z)=n0iλH1(z)1/2expi2πλn0z×expiπn0λH1(z)[x2H˙1(z)+x02H2(z)-2xx0],
H1(z)=[g0g(z)]-1/2sin0zg(z)dz,
H2(z)=g0g(z)1/2cos0zg(z)dz,
H1(z)=u(z){g0g(z)[1+u2(z)]}1/2,
H2(z)=g0g(z)[1+u2(z)]1/2,
u(z)=tan0zg(z)dz.
Re[G(zν)]n0|G(zν)|2H1(zν)=νdp2λ,    ν=1, 2,
G(z)=U(0)H1(z)n0+H2(z),
2p1.
T(x0)=12[1+Wal896(x0)]=W896(x0).
zR=πn0w2(0)λ
g(z)=g01+z/L,
zν=Lexp1g0Ltan-1νdp2n0g0Z0Z0λ-νdp2-1.

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