Abstract

A generalization of the Talbot effect to the case of a tapered gradient-index medium for nonuniform and uniform illumination is considered. Self-image positions are changed by a function depending on the taper profile, the illumination, and the periodic object. An analogy with the conventional lens-imaging formula for both types of illumination is presented.

© 1999 Optical Society of America

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References

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  1. K. Patorski, Progress in Optics, Vol. XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 3–101 and references therein.
  2. F. J. Clauser, S. Li, “Talbot–von Lau interferometry with cold slow potassium,” Phys. Rev. A 49, 2213–2216 (1996).
    [CrossRef]
  3. C. J. Bordé, Fundamental System in Quantum Optics (Elsevier, Amsterdam, 1992).
  4. G. S. Agarwal, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
    [CrossRef]
  5. P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
    [CrossRef]
  6. 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).
  7. C. Gomez-Reino, “GRIN optics and its applications in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).
  9. R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
    [CrossRef]
  10. C. Gómez-Reino, M. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).
  11. D. Joyeux, Y. Cohen-Sabban, “High magnification self-imaging,” Appl. Opt. 21, 625–627 (1982);“Aberration-free nonparaxial self-imaging,” J. Opt. Soc. Am. 73, 707–719 (1983).
    [CrossRef] [PubMed]

1996 (1)

F. J. Clauser, S. Li, “Talbot–von Lau interferometry with cold slow potassium,” Phys. Rev. A 49, 2213–2216 (1996).
[CrossRef]

1995 (1)

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

1992 (1)

C. Gomez-Reino, “GRIN optics and its applications in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

1991 (1)

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

1983 (1)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

1982 (1)

Agarwal, G. S.

G. S. Agarwal, “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).

Bao, C.

C. Gómez-Reino, M. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).

Bordé, C. J.

C. J. Bordé, Fundamental System in Quantum Optics (Elsevier, Amsterdam, 1992).

Clauser, F. J.

F. J. Clauser, S. Li, “Talbot–von Lau interferometry with cold slow potassium,” Phys. Rev. A 49, 2213–2216 (1996).
[CrossRef]

Cohen-Sabban, Y.

Flores-Arias, M. T.

C. Gómez-Reino, M. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).

Gomez-Reino, C.

C. Gomez-Reino, “GRIN optics and its applications in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Gómez-Reino, C.

C. Gómez-Reino, M. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).

Joyeux, D.

Jozwicki, R.

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

Li, S.

F. J. Clauser, S. Li, “Talbot–von Lau interferometry with cold slow potassium,” Phys. Rev. A 49, 2213–2216 (1996).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).

Ojeda-Castañeda, J.

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

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).

Patorski, K.

K. Patorski, 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. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).

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).

Szwaykowski, P.

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Appl. Opt. (1)

Int. J. Optoelectron. (1)

C. Gomez-Reino, “GRIN optics and its applications in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Opt. Acta (1)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

Opt. Commun. (2)

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

P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Phys. Rev. A (1)

F. J. Clauser, S. Li, “Talbot–von Lau interferometry with cold slow potassium,” Phys. Rev. A 49, 2213–2216 (1996).
[CrossRef]

Other (5)

C. J. Bordé, Fundamental System in Quantum Optics (Elsevier, Amsterdam, 1992).

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).

C. Gómez-Reino, M. T. Flores-Arias, C. Bao, M. V. Pérez, “Talbot effect in tapered GRIN media,” in Third Iberoamerican Optics Meeting and Sixth Latin American Meeting on Optics, Lasers, and Their Applications, A. M. Guzman, ed., Proc. SPIE3572, 242–253 (1999).

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).

K. Patorski, Progress in Optics, Vol. XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), pp. 3–101 and references therein.

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Figures (8)

Fig. 1
Fig. 1

Geometry for the evaluation of the complex amplitude distribution in a tapered GRIN medium due to a periodic object located at z=0 and illuminated by a Gaussian beam.

Fig. 2
Fig. 2

Equivalent optical system for the Talbot effect in a tapered GRIN medium under divergent (a) Gaussian illumination and (b) uniform illumination.

Fig. 3
Fig. 3

Equi-index lines for a divergent linear tapered GRIN medium.

Fig. 4
Fig. 4

Dependence of the self-image distances on L for (a) nonuniform illumination and (b) uniform illumination. Calculations have been made for λ=0.7 μm, p=6 μm, d=10 mm, R(0)=0.65 mm, and w0=7 μm.

Fig. 5
Fig. 5

Dependence of the self-image distances on λ for (a) nonuniform illumination and (b) uniform illumination. Calculations have been made for p=6 μm, L=1 mm, d=10 mm, R(0)=0.65 mm, and w0=7 μm.

Fig. 6
Fig. 6

Dependence of the self-image distances on the curvature radius R(0) and d for (a) nonuniform illumination and (b) uniform illumination, respectively. Calculations have been made for λ=0.7 μm, L=1 mm, p=6 μm, and w0=7 μm.

Fig. 7
Fig. 7

Dependence of the self-image distances on p for (a) nonuniform illumination and (b) uniform illumination. Calculations have been made for λ=0.7 μm, L=1 mm, d=10 mm, R(0)=0.65 mm, and w0=7 μm.

Fig. 8
Fig. 8

Transverse magnification versus self-image number for nonuniform illumination and uniform illumination. Calculations have been made for n0=1.5, g0=0.01 mm-1, λ=0.7 μm, L=1 mm, p=6 μm, d=10 mm, R(0)=0.65 mm, and w0=7 μm.

Equations (74)

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n2(x, z)=n02[1-g2(z)x2],
T(x0)=mam exp-i 2πmx0p,
ϕ(x0)=T(x0)ψ0(x0),
ψ0(x0)=w0w(0)1/2 exp[iφ(0)]ψ[x0,U(0)]
ψ[x0,U(0)]=expi πU(0)x02λ
U(0)=1R(0)+i λπw2(0),
φ(0)=tan-1(λd/πw02),
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)=w0w(0)F(z)1/2 exp[iφ(z)]×expi πU(z)λ x2mam exp-i 2πmpF(z) x×exp-i πλm2H1(z)n0p2F(z),
U(z)=n0F˙(z)F-1(z)=n0 d ln F(z)dz.
F(z)=U(0)H1(z)n0+H2(z),
F˙(z)=U(0)H˙1(z)n0+H˙2(z),
φ(z)=φ(0)+kn0z.
ϕ(x; zq)=w0w(zq)1/2 exp[iφ(zq)]×expi πU(zq)λ x2mam exp-i 2πmp(zq) x,
w(zq)=w(0)H2(zq),
U(zq)=U(0) H˙1(zq)H2(zq),
p(zq)=pH2(zq).
mam exp-πλm2 Im[F(z)]n0p2|F(z)|2 H1(z)
×exp-2πm Im[F(z)]p|F(z)|2 x
×exp-i πλm2 Re[F(z)]n0p2|F(z)|2 H1(z)×exp-i 2πm Re[F(z)]p|F(z)|2 x,
Re[F(z)]=H1(z)n0R(0)+H2(z),
Im[F(z)]=H1(z)zR,
zR=πn0w2(0)λ,
|F(z)|=H1(z)n0R(0)+H2(z)2+H12(z)zR21/2,
w(z)=w(0)|F(z)|.
λ Re[F(zν)]n0p2|F(zν)|2 H1(zν)=2ν
2νp2λ=Re[F(zν)]n0|F(zν)|2 H1(zν),
exp-i 2πmp(zν) x,
p(zν)=pMtg(zν)
Mtg(zν)=|F(zν)|2Re[F(zν)]=w2(zν)w2(0)Re[F(zν)],
2νp2λd02=1R(0)-1zν,
zν=R(0)+zν=|F(zν)|2Re[F(zν)] R(0),
zν=|F(zν)|2-Re[F(zν)]Re[F(zν)] R(0),
d02=H1(zν)n0zν R2(0).
fν=λd022νp2,
zνR(0)=|F(zν)|2Re[F(zν)],
Fpg(zν)=H2(zν)+i H1(zν)zRpg,
zRpg=πn0w02λ.
2νp2λ=H1(zν)H2(zν)n0|Fpg(zν)|2
Mtpg(zν)=|Fpg(zν)|2H2(zν).
Fu(z)=H2(z)+H1(z)n0d.
2νp2λ=dH1(zν)n0dH2(zν)+H1(zν),
2νp2λdu02=1d-1zν,
zν=d+zν=Fu(zν)d,
zν=[Fu(zν)-1]d,
du02=H1(zν)n0zν d2.
Mtu(zν)=zνd=Fu(zν),
2νp2λ=H1(zν)n0H2(zν),
Mpu(zν)=H2(zν).
2νp2λ=zνzνR(0) w(0)w(zν)2,
2νp2λR2(0)=1R(0)-1zν,
2νp2λ=zνdzν,
2νp2λd2=1d-1zν,
H1(z)=[g0g(z)]-1/2 sin0zg(z)dz,
H2(z)=g0g(z)1/2 cos0zg(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.
au2(zν)+bu(zν)+c=0,
a=1g0 2νp2n01n02R2(0)+1zR2-λn0R(0),
b=4νp2R(0)-λ,
c=2νp2n0g0.
u(zν)=n0g0R2(0)zR2λ4νp2[zR2+n02R2(0)]-2λR(0)zR2×1-4νp2R(0)λ-1-4n0νp2zRλ2.
4n0νp2zRλ1orνπ4 w(0)p2
upg(zν)=g0λ(zRpg)24νp2n0 1-1-4n0νp2zRpgλ2,
νπ4 w0p2,
uu(zν)=2νp2n0g0ddλ-2νp2
upu(zν)=2νp2n0g0λ
zν=1g0 tan-12νp2n0g0λ,
g(z)=g01+(z/L),
zν=Lexp1g0L tan-1×n0g0R2(0)zR2λ4νp2[zR2+n02R2(0)]-2λR(0)zR2×1-4νp2R(0)λ-1-4n0νp2zRλ2-1
zν=Lexp1g0L tan-12νp2n0g0dλd-2νp2-1
g(z)=g01-(z/L).

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