Abstract

On the basis of the unitary transformation and the Lie algebra decomposition technology widely used in quantum mechanics, we obtain the analytical propagator for light beams propagating through an imperfect gradient-index (GRIN) waveguide. The results show that, unlike in the straight GRIN waveguide widely studied, in an imperfect GRIN waveguide, self-imagining phenomena can result in two new effects: one is a phase shift including a global one and a local one, in which the local one results in a change of direction of the light beam propagating through the imperfect GRIN waveguide; the other is a transverse shift of self-image. The transverse deviation occurring in the imperfect GRIN waveguide is also calculated.

© 2004 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner distribution functions and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  2. G. S. Agarwal, “Talbot effect in a quadratic index medium,” Opt. Commun. 119, 30–32 (1995).
    [CrossRef]
  3. L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
    [CrossRef]
  4. M. T. Flores-Arias, C. Bao, M. V. Perez, 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]
  5. C. Gómez-Reino, M. T. Flores-Arias, M. V. Perez, C. Bao, “Fractional and integer Talbot effect for off-axis illumination and for finite object dimension in tapered GRIN media,” Opt. Commun. 183, 365–376 (2000).
    [CrossRef]
  6. M. T. Flores-Arias, C. R. Fernandez-Pousa, M. V. Perez, 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]
  7. X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
    [CrossRef]
  8. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 79–81.
  9. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  10. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: II. Mode selection in lasers and periodic misaligned complex optical systems,” J. Opt. Soc. Am. A 13, 90–96 (1996).
    [CrossRef]
  11. K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
    [CrossRef]
  12. H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
    [CrossRef]
  13. G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
    [CrossRef] [PubMed]

2003 (1)

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

2001 (1)

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

2000 (2)

C. Gómez-Reino, M. T. Flores-Arias, M. V. Perez, C. Bao, “Fractional and integer Talbot effect for off-axis illumination and for finite object dimension in tapered GRIN media,” Opt. Commun. 183, 365–376 (2000).
[CrossRef]

M. T. Flores-Arias, C. R. Fernandez-Pousa, M. V. Perez, 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]

1999 (1)

1998 (1)

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

1996 (1)

1995 (3)

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

X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
[CrossRef]

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
[CrossRef]

1994 (1)

1986 (1)

G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
[CrossRef] [PubMed]

Agarwal, G. S.

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

Arnaud, J.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 79–81.

Bao, C.

Casperson, L. W.

Chen, M. Z.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Dattoli, G.

G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
[CrossRef] [PubMed]

Fernandez-Pousa, C. R.

Flores-Arias, M. T.

Fu, X.

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Gómez-Reino, C.

Guo, H.

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Hu, W.

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Huang, M. C.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Huang, W. D.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Liñares, J.

X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
[CrossRef]

Liu, T. C.

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Lohmann, A. W.

Lu, Y. Y.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Mendlovic, D.

Montero, C.

X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
[CrossRef]

Ozaktas, H. M.

Peng, Y. Y.

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

Perez, M. V.

Prieto, X.

X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
[CrossRef]

Solimeno, S.

G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
[CrossRef] [PubMed]

Sun, X. M.

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

Tang, H. Q.

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

Torre, A.

G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
[CrossRef] [PubMed]

Tovar, A. A.

Wang, X. W.

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

Wu, L. Q.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Yu, L.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Yu, S.

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Zhu, K. C.

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

Zhu, Z. Z.

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

X. Prieto, C. Montero, J. Liñares, “Three-step diffused surface waveguides for fabricating and designing integrated optical components,” J. Mod. Opt. 42, 2519–2163 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

K. C. Zhu, X. M. Sun, X. W. Wang, H. Q. Tang, Y. Y. Peng, “Paraxial propagation of Gaussian beam through curved transverse parabolic graded-index waveguides,” Opt. Commun. 221, 1–7 (2003).
[CrossRef]

C. Gómez-Reino, M. T. Flores-Arias, M. V. Perez, C. Bao, “Fractional and integer Talbot effect for off-axis illumination and for finite object dimension in tapered GRIN media,” Opt. Commun. 183, 365–376 (2000).
[CrossRef]

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

L. Yu, M. C. Huang, L. Q. Wu, Y. Y. Lu, W. D. Huang, M. Z. Chen, Z. Z. Zhu, “Fractional Fourier transform and the elliptic gradient-index medium,” Opt. Commun. 152, 23–25 (1998).
[CrossRef]

Phys. Rev. A (1)

G. Dattoli, S. Solimeno, A. Torre, “Algebraic time-order techniques and harmonic oscillator with time-dependent frequency,” Phys. Rev. A 34, 2646–2653 (1986).
[CrossRef] [PubMed]

Phys. Rev. E (1)

H. Guo, T. C. Liu, X. Fu, W. Hu, S. Yu, “Beam propagation of x rays in a laser-produced plasma and a modified relation of interferometry in measuring the electron density,” Phys. Rev. E 63, 066401 (2001).
[CrossRef]

Other (1)

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 79–81.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of a sinusoidal-function-type bent GRIN waveguide and the coordinate system. A periodic object is situated at its input end (z=0).

Fig. 2
Fig. 2

(a) Geometrical trajectory -γ(z) described by the center of the irradiance pattern under different parameters, where Q/δ is 3 (circles), 1.5 (squares), 0.6 (stars), and 0.2 (triangles). (b) For comparison, f(z) is also plotted (dashed curve).

Fig. 3
Fig. 3

Irradiance distribution for curved GRIN waveguides (solid curve) and straight GRIN waveguides (dashed curve) at the first (q=1) Talbot image for a plane-wave illumination [w(0)= and R(0)=]. Calculations have been completed for object parameters A0=1/2, A1=1/8, and p=7 µm; for curved GRIN-medium parameters n0=1.5, Q=0.01 mm-1, A=0.5 µm, and δ=5Q; and for illumination parameter λ=0.7 µm.

Equations (40)

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n2(x, z)=n02-ni2[x-f(z)]2,
2x2+2z2+k02n2(x, z)E(x, z)=0,
E(x, z)=ψ(x, z)exp(ik0z),
iψ(x, Z)Z=-122ψ(x, Z)x2+12k2Q2[x-f(kZ)]2ψ(x, Z),
ψ(x, z)=ψI(x, z)exp-i2kQ2zf2(η)dη,
iψI(x, Z)Z=Hˆ0ψI(x, Z),
Hˆ0=-122x2+12k2Q2[x2-2xf(kZ)].
Uˆ=exp[iα(z)+iβ(z)x]expγ(z)x,
ψI(x, Z)=UˆΠ(x, Z),
iΠ(x, Z)Z=HˆΠ(x, Z),
Hˆ=Uˆ+Hˆ0Uˆ-iUˆ+Uˆ/Z=-122x2+dγdz+βkx+12k2Q2x2+dβdz-kQ2[γ+f(z)]x+dαdz-γdβdz+β2+k2Q2γ22k+kQ2γf(z).
dγdz+βk=0,
dβdz-kQ2γ=kQ2f(z),
dαdz-γdβdz+β2+k2Q2γ22k=-kQ2γf(z),
Hˆ=-122x2+12k2Q2x2.
expuxddxf(x)=f(eux)
expuddxf(x)=f(x+u)
expud2dx2f(x)=14πu-exp-(x-y)24uf(y)dy
Π(x2, z)=kQ2πi sin(Qz) -expikQ(x22+x12)2 tan(Qz)-ikQx1x2sin(Qz)E(x1, 0)dx1,
E(x2, z)=exp[i(Ө+βx2)]expγx2Π(x2, z)=exp[i(Ө+βx2)]Π(x2+γ, z)=-K(x2, x1; z)E(x1, 0)dx1,
K(x2, x1; z)=exp[i(Ө)+βx2)]kQ2πi sin(Qz)×expikQ[(x2+γ)2+x12]2 tan(Qz)-ikQ(x2+γ)x1sin(Qz)=exp[i(Ө+βx2)]expγx2×Kf(z)=0(x2, x1; z),
d2γdz2+Q2γ=-Q2f(z),
γ(z)=-Q0zf(u)sin[Q(z-u)]du.
T(x0)=mAm exp-2πimx0p,
E(x0, 0)=T(x0)ψ(x0),
ψ(x0)=[w0/w(0)]1/2 exp[iϕ(0)]ψ[x0, U(0)]
ψ[x0, U(0)]=expiπU(0)x02λ
U(0)=1R(0)+iλπw2(0),ϕ(0)=tan-1λdπw02,
E(x, z)=B(z)mAm exp[-Λm(z)]×expiZRΛm(z)Qtan(Qz)+1R(0),
Λm(z)=1Ω(z)mπp2+πmkQ(x+γ)p sin(Qz),
B(z)=QZRw0/w(0)QZR cos(Qz)+ZR sin(Qz)R(0)+i sin(Qz)1/2exp-(x+γ)2w2(z)+iϕ(0)+Ө+βx+iπ(x+γ)2λR(z),
Ω(z)=1w2(0)1+ZR2Qtan(Qz)+1R(0)2,
w2(z)=w2(0)sin2(Qz)Q2ZR2+cos(Qz)+sin(Qz)QR(0)2,
R(z)=2sin2(Qz)+ZR2Q cos(Qz)+sin(Qz)R(0)2Q1+ZR2R2(0)-Q2ZR2sin(2Qz)+2QZR2R(0) cos(2Qz),
zq±=1Qarctan2qp2Qλ1-2qp2λR(0)±1-2qp2πw2(0)2-1,
zq=1QarctanQqp2λ,
f(z)=A sin(δz),
T(x0)=A0+2A1 cos2πx0p.
γ(z)=QAQ sin(δz)-δ sin(Qz)δ2-Q2forδQ12A[δz cos(δz)-sin(δz)]forδ=Q.
I(x, zq)=|B(zq)|2A02+2A12 exp-2π2p2Ωq×cosh2πkQ(x+γq)pΩq sin(Qzq)+cos2qpkQ(x+γq)sin(Qzq)+4A0A1 exp-π2p2ΩqcoshπkQ(x+γq)pΩq sin(Qzq)×cosqpkQ(x+γq)sin(Qzq),

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