Abstract

A generalization of the fractional Talbot effect to the case of a tapered gradient-index medium for uniform illumination is considered. A unit cell of the fractional Talbot image contains the superposition of unit cell images of the periodic object.

© 2000 Optical Society of America

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References

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  1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–101 and references therein.
  2. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996) and references therein.
    [CrossRef]
  3. G. S. Agarwall, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
    [CrossRef]
  4. E. Silvestre, P. Andres, J. Ojeda-Castañeda, “Self-imaging in GRIN media,” in Second Iberoamerican Meeting on Optics, D. Malacara-Hernandez, S. E. Acosta Ortiz, R. Rodriguez-Vera, Z. Malacara, A. A. Morales, eds., Proc. SPIE2730, 468–471 (1996).
    [CrossRef]
  5. M. T. Flores-Arias, C. Bao, M. V. Pérez, C. Gómez-Reino, “Talbot effect in a tapered gradient-index medium for nonuniform and uniform illumination,” J. Opt. Soc. Am. A 16, 2439–2446 (1999).
    [CrossRef]

1999

1996

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996) and references therein.
[CrossRef]

1995

G. S. Agarwall, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
[CrossRef]

Agarwall, G. S.

G. S. Agarwall, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
[CrossRef]

Andres, P.

E. Silvestre, P. Andres, J. Ojeda-Castañeda, “Self-imaging in GRIN media,” in Second Iberoamerican Meeting on Optics, D. Malacara-Hernandez, S. E. Acosta Ortiz, R. Rodriguez-Vera, Z. Malacara, A. A. Morales, eds., Proc. SPIE2730, 468–471 (1996).
[CrossRef]

Bao, C.

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996) and references therein.
[CrossRef]

Flores-Arias, M. T.

Gómez-Reino, C.

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996) and references therein.
[CrossRef]

Ojeda-Castañeda, J.

E. Silvestre, P. Andres, J. Ojeda-Castañeda, “Self-imaging in GRIN media,” in Second Iberoamerican Meeting on Optics, D. Malacara-Hernandez, S. E. Acosta Ortiz, R. Rodriguez-Vera, Z. Malacara, A. A. Morales, eds., Proc. SPIE2730, 468–471 (1996).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–101 and references therein.

Pérez, M. V.

Silvestre, E.

E. Silvestre, P. Andres, J. Ojeda-Castañeda, “Self-imaging in GRIN media,” in Second Iberoamerican Meeting on Optics, D. Malacara-Hernandez, S. E. Acosta Ortiz, R. Rodriguez-Vera, Z. Malacara, A. A. Morales, eds., Proc. SPIE2730, 468–471 (1996).
[CrossRef]

J. Mod. Opt.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996) and references therein.
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

G. S. Agarwall, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
[CrossRef]

Other

E. Silvestre, P. Andres, J. Ojeda-Castañeda, “Self-imaging in GRIN media,” in Second Iberoamerican Meeting on Optics, D. Malacara-Hernandez, S. E. Acosta Ortiz, R. Rodriguez-Vera, Z. Malacara, A. A. Morales, eds., Proc. SPIE2730, 468–471 (1996).
[CrossRef]

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 3–101 and references therein.

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Figures (2)

Fig. 1
Fig. 1

Geometry for the evaluation of (a) the complex amplitude distribution in a tapered GRIN medium due to a periodic object located at z = 0 and illuminated by a cylindrical uniform beam and (b) the equivalent optical system for the fractional Talbot effect under divergent uniform illumination.

Fig. 2
Fig. 2

Axial position of self-images versus self-image number for a tapered GRIN medium with a divergent linear taper function. Calculations have been made for n0 = 1.5, g0 = 0.01 mm-1, p = 9 μm, λ = 0.7 μm, L = 1 mm, d=15 mm, and α = 3.

Equations (45)

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n2(x, z)=n02[1-g2(z)x2],
T(x0)=m=-+am exp-i 2πmx0p,
ϕ(x0)=T(x0)ψ0(x0),
ψ0(x0)=1d expi πx02λd
ϕ(x; z)=-+ϕ(x0)K(x, x0; z)dx0,
K(x, x0; z)=n0iλH1(z)1/2×exp(ikn0z)expi kn02H1(z) [x2H˙1(z)+x02H2(z)-2xx0],
ϕ(x; z)=1dF(z) exp(ikn0z)expi kn0F˙(z)2F(z) x2×m=-+am× exp-i 2πmxpF(z)×exp-iπm2 λH1(z)n0 p2F(z),
F() (z)=H()2(z)+H()1(z)n0d.
λH1(zβ/α)n0 p2F(zβ/α)=βα,
βp2αλd02=1d-1zβ/α,
zβ/α=F(zβ/α)d=d+zβ/α,
d02=d2H1(zβ/α)n0 zβ/α,
zβ/α=[F(zβ/α)-1]d.
Mt(zβ/α)=zβ/αd=F(zβ/α).
H1(z)=[g0 g(z)]-1/2 sin0zg(z)d z,
H2(z)=g0g(z)1/2 cos0zg(z)d z,
0zβ/αg(z)d z=tan-1n0 g0dp2βλdα-p2β.
g(z)=g01-z/L,
zβ/α=Lexp1g0 tan-1n0 g0dp2βdαλ-p2β-1.
λ Re[Fg(zβ/α)]H1(zβ/α)n0 p2|Fg(zβ/α)|2=βα,
Fg(zβ/α)=U(0)H1(zβ/α)n0+H2(zβ/α),
Tc(x0)=m=-+δ(x0-mp)=1p m=-+ exp-i 2πmx0p,
ϕc(x; zβ/α)
=1pdF(zβ/α) exp(ikn0zβ/α)expi kn0F˙(zβ/α)2F(zβ/α) x2×m=-+exp-i 2πmxpF(zβ/α)exp-i πm2βα,
ϕc(x; zβ/α)
=F(zβ/α)αd1/2 exp(ikn0zβ/α)expi kn0F˙(zβ/α)2F(zβ/α) x2×m=-+δx-mpF(zβ/α)αA(m; α, β),
x=x+12 pF(zβ/α)eβ,
A(m; α, β)=1α s=1α expi πα 2sm+αeβ2-βs2.
m=-+δx+pF(zβ/α)-mpF(zβ/α)αA(m; α, β)
=n=-+δx-npF(zβ/α)αA(n; α, β).
T(x0)=-+t(η)m=-+δ(η-x0+mp)dη=m=-+t(x0-mp),
ϕ(x; z)=-+t(η)expikn0F˙(z)ηx-ηF(z)2×ϕc[x-ηF(z); z]dη,
ϕc(x; z)=-+K(x, x0; z)ψ0(x0)m=-+δ(x0-mp)dx0.
ϕ(x; zβ/α)=1αdF(zβ/α)1/2 exp(ikn0zβ/α)×expi kn0F˙(zβ/α)2F(zβ/α) x2×m=-+t1F(zβ/α) x-mpF(zβ/α)α×A(m; α, β),
ϕ(x, zβ)=1dF(zβ)1/2exp(ikn0zβ)expi kn0F˙(zβ)2F(zβ) x2×m=-+am exp-i 2πmxpF(zβ),
|A(m; α, β)|=1.
|A(m; α, β)|2=1α s=1αt=1αexpi πα 2(s-t)m+αeβ2-β(s-t)(s+t).
u=s-t,
|A(m; α, β)|2=1α s=1αu=s-αs-1expi πα 2um+αeβ2-β(2s-u)u.
|A(m; α, β)|2=1α u=0α-1 expi πα 2um+αeβ2+βu2×s=1α exp-i 2πβusα,
expi πα 2um+αeβ2-β(2s-u)u
=expi πα 2(u+α)m+αeβ2-β(2s-u-α)(u+α).
s=1α exp-i 2πβusα=αδuo,
|A(m; α, β)|2
=1α u=0α-1 expi πα 2um+αeβ2+βu2αδuo=1.

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