Abstract

An interpretation of the Talbot effect in a tapered gradient-index medium by number theory as the output/input relationship between the integer and the noninteger difference of position and the slope of rays is presented. Unit cell and transverse magnification for Talbot images are evaluated, and two criteria for angular magnification are defined. The study is particularized to a finite set of diffracted rays.

© 2001 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 XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989), 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. 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]
  4. M. T. Flores-Arias, C. R. Fernández-Pousa, M. V. Pérez, C. Bao, C. Gómez-Reino, “Fractional Talbot effect in a tapered gradient-index medium: unit cell,” J. Opt. Soc. Am. A 17, 1007–1011 (2000).
    [CrossRef]
  5. P. Latimer, R. F. Course, “Talbot effect reinterpreted,” Appl. Opt. 31, 80–89 (1992).
    [CrossRef] [PubMed]
  6. P. Szwaykowski, “Talbot effect reinterpreted: comment,” Appl. Opt. 32, 3466–3467 (1993).
    [CrossRef] [PubMed]
  7. P. Latimer, “Talbot effect reinterpreted: reply to comment,” Appl. Opt. 32, 3468–3469 (1993).
    [CrossRef] [PubMed]
  8. P. Szwaykowski, J. Ojeda-Castañeda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
    [CrossRef]
  9. C. Gómez-Reino, “GRIN optics and its applications in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  10. G. H. Gardy, E. M. Wright, An Introduction to the Theory of Number, 5th ed. (Clarendon, Oxford, UK, 1995), pp. 21, 23, 52, 53.
  11. J. C. Fernando, V. Gregori, Matemática Discreta (Editorial Reverté, Barcelona, Spain, 1994), p. 98.

2000 (1)

1999 (1)

1996 (1)

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

1993 (2)

1992 (2)

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

P. Latimer, R. F. Course, “Talbot effect reinterpreted,” Appl. Opt. 31, 80–89 (1992).
[CrossRef] [PubMed]

1991 (1)

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

Course, R. F.

Fernández-Pousa, C. R.

Fernando, J. C.

J. C. Fernando, V. Gregori, Matemática Discreta (Editorial Reverté, Barcelona, Spain, 1994), p. 98.

Flores-Arias, M. T.

Gardy, G. H.

G. H. Gardy, E. M. Wright, An Introduction to the Theory of Number, 5th ed. (Clarendon, Oxford, UK, 1995), pp. 21, 23, 52, 53.

Gómez-Reino, C.

Gregori, V.

J. C. Fernando, V. Gregori, Matemática Discreta (Editorial Reverté, Barcelona, Spain, 1994), p. 98.

Klein, S.

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

Latimer, P.

Ojeda-Castañeda, J.

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

Patorski, K.

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

Pérez, M. V.

Szwaykowski, P.

P. Szwaykowski, “Talbot effect reinterpreted: comment,” Appl. Opt. 32, 3466–3467 (1993).
[CrossRef] [PubMed]

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

Wright, E. M.

G. H. Gardy, E. M. Wright, An Introduction to the Theory of Number, 5th ed. (Clarendon, Oxford, UK, 1995), pp. 21, 23, 52, 53.

Appl. Opt. (3)

Int. J. Optoelectron. (1)

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

J. Mod. Opt. (1)

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 (2)

Opt. Commun. (1)

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

Other (3)

G. H. Gardy, E. M. Wright, An Introduction to the Theory of Number, 5th ed. (Clarendon, Oxford, UK, 1995), pp. 21, 23, 52, 53.

J. C. Fernando, V. Gregori, Matemática Discreta (Editorial Reverté, Barcelona, Spain, 1994), p. 98.

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

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Figures (7)

Fig. 1
Fig. 1

Geometry for evaluating 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.

Fig. 2
Fig. 2

Integer and fractional Talbot images for β/α=1/2 and ν=1.

Fig. 3
Fig. 3

Ray tracing for (a) image and (b) Fourier transform conditions.

Fig. 4
Fig. 4

Transverse magnification of Talbot images versus self-image number for different values of d. Calculations have been made for n0=1.5, g0=0.01 mm-1, λ=0.7 μm, L=1 mm, and p=9 μm.

Fig. 5
Fig. 5

Ray tracing for (a) the first and (b) the second criterion of angular magnification.

Fig. 6
Fig. 6

Variation of the angular magnification with self-image number for the first and the second criteria. Calculations have been made for the values of Fig. 4 and d=15 mm.

Fig. 7
Fig. 7

Fractional and integer Talbot images in a GRIN medium obtained for five rays. Calculations have been made for the values of Fig. 4.

Equations (102)

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n2(x, z)=n02[1-g2(z)x2],
T(x0)=h=-+δ(x0-hp)=1p h=-+ exp-i 2πhx0p,
x˙0h=x˙0hr+x˙0hm,
x˙0hr=x0hn0d,
x˙0hm=mλn0p.
x(z)x˙(z)=H2(z)H1(z)H˙2(z)H˙1(z)x0hx˙0h,
H1(z)=[g0g(z)]-1/2 sin0zg(z)dz=-[g0g(z)]-1H˙2(z),
H2(z)=g0g(z)1/2 cos0zg(z)dz=g0g(z)H˙1(z),
H1(0)=0,
H˙1(0)=1
H2(0)=1,
H˙2(0)=0,
H˙1(z)H2(z)-H1(z)H˙2(z)=1.
x(z)x˙(z)=H2(z)+H1(z)n0dH1(z)H˙2(z)+H1(z)n0dH˙1(z)x0hx˙0hm
x(·)(z)=pA(·)(z)[h+mξ(·)(z)],
ξ(·)(z)=B(·)(z)pA(·)(z),
A(·)(z)=H(·)2(z)+H(·)1(z)n0d,
B(·)(z)=λH(·)1(z)n0p.
h+mξ(z¯)=hp(z¯)pA(z¯),
ξ(z¯)=β/α,
h1p(z¯)=pA(z¯)[h1+m1ξ(z¯)],
h2p(z¯)=pA(z¯)[h2+m2ξ(z¯)],
ξ(z¯)=h1h2-h2h1m1h2-m2h1,
x(z¯)=pA(z¯)α(hα+mβ),
β/α=ν+β/α,
ξ(zν)=ν
ξ(zβ/α)=β/α
ν=h1-h2m2-m1,
βα=h1-h2m2-m1,
ξ(zim)=0ifβ=0orB(zim)=0,
ξ(zft)=ifα=0orA(zft)=0,
H1(zim)=0,
H2(zft)=-H1(zft)n0d,
x(zβ/α)=pA(zβ/α)α(hα+mβ)=p(zβ/α)u,
u=hα+mβ,
p(zβ/α)=pA(zβ/α)α.
h¯α+m¯β=1,
h˜=uh¯-βt,
m˜=um¯+αt,
1|i-j|α-1.
im¯+αt=jm¯+αt˜;
m¯=α(t-t˜)j-i.
j-i=a1a2,
m¯=r1r2,
r1=αa1,
r2=t-t˜a2.
r1(h¯a1+βr2)=1,
|r1|=1,
|h¯a1+βr2|=1
|a1|=α>1,
|j-i|=|a1| |a2||a1|=α,
t-t˜j-i=r, r  Z
m¯=αr,
α(h¯+rβ)=1,
|α|=1,
um¯+αt=(u+α)m¯+αt˜,
t=t˜+m¯.
x0h=ph=p(uh¯-βt),
x0h˜=ph˜=p[(u+α)h¯-βt˜],
Δx0=x0h˜-x0h=p(hα¯+mβ¯)=p,
p=pA(zβ/α)α(u+1)-pA(zβ/α)αu=pA(zβ/α)α.
x(zν)=pA(zν)[h+νm]=ph(zν),
p=pA(zν),
g(z)=g01+(z/L),
x˙(zβ/α)=x˙(zβ/α; u)+tΔx˙(zβ/α),
x˙(zβ/α; u)=u[pA˙(zβ/α)h¯+m¯B˙(zβ/α)],
Δx˙(zβ/α)=pA˙(zβ/α)[αξ˙(zβ/α-β)],
x˙(zβ/α)-x˙(zβ/α)=(t˜-t)Δx˙(zβ/α)=ΔtΔx˙(zβ/α),
Δx˙0=λdα-p2βn0pd,
Ma(1)=Δx˙(zβ/α)Δx˙0=n0p2dA˙(zβ/α)[αξ˙(zβ/α)-β]λdα-p2β=n0p2dA˙(zβ/α)[ξ˙(zβ/α)-ξ(zβ/α)]λd-p2ξ(zβ/α),
Ma(1)=H2-1(zβ/α),
Ma(1)=H2-1(zν).
Δm˜=(u-u)m¯+α(t˜-t)=1
t˜-t=1-(u-u)m¯α,
Δx=x-x=pA(zβ/α)[h˜+(m˜+1)ξ(zβ/α)-h˜-m˜ξ(zβ/α)]=pA(zβ/α)[h¯(u-u)-β(t˜-t)+ξ(zβ/α)].
Δx=pA(zβ/α){[h¯+ξ(zβ/α)m¯](u-u)}=pA(zβ/α)(u-u)α,
Δx˙β/α=pA˙(zβ/α{h˜+(m˜+1)ξ˙(zβ/α)-[h˜+m˜ξ˙(zβ/α)]}=pA˙(zβ/α)ξ˙(zβ/α)=B˙(zβ/α),
Δx˙0=λn0p.
Ma(2)=Δx˙β/αΔx˙0=H˙1(zβ/α),
Ma(2)=H˙1(zν).
Ma(1)Ma(2)=H˙1(zβ/α)H2(zβ/α)=g(zβ/α)g0,
Ma(1)Ma(2)=1;
Ru={(uh¯-βt, um¯+αt)/tZ},
-M,-M+1 ,, 0,1,2 , M-1,M,
0ru<α.
m(u)=ru mod α=ru+αt,
resmod α(m)=ru.
2M+1=αC+R,0R<α,
0<βα,gcd(α, β)=1,
PM=α=22Mϕ(α),
ϕ(α)=αβ/α(1-1/β),
P2=ϕ(2)+ϕ(3)+ϕ(4)=5.
hα+mβ=0;
h/m=-β/α,
h=-βt,
m=αt,
hα+mβ=1;
h˜=h¯-βt,
m˜=m¯+αt.
hypothesis:αh˜+βm˜=1,
Be´zoutsolution:αh¯+βm¯=1,
α(h˜-h¯)+β(m˜-m¯)=0.

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