Abstract

The theory of fractals has already been applied to many fields in science, such as physics, biology, and chemistry. One of the most commonly used fractals in these applications is the Cantor set. Novel fiber Bragg gratings are proposed that combine the present technology of fiber Bragg gratings with the theory of Cantor sets. The principal goal of this work is to analyze how Cantor sets, applied to gratings, can alter their reflectivity spectra. Specifically, it is observed that, as the order of the Cantor set increases, the bandpass reflectivity spectra of these gratings broaden and evolve into more-complex patterns. Also, self-similarity properties can be observed in the spectra of these gratings. Numerical examples demonstrate variations in the spectra of these structures as the fractal order increases.

© 2000 Optical Society of America

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References

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  1. S. Legoubin, M. Douay, P. Bernage, P. Niay, S. Boj, E. Delevaque, “Free spectral range variations of grating-based Fabry–Perot photowritten in optical fibers,” J. Opt. Soc. Am. A 12, 1687–1694 (1995).
    [CrossRef]
  2. H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
    [CrossRef]
  3. R. Zengerle, O. Leminger, “Phase-shifted Bragg-gratingfilters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
    [CrossRef]
  4. R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).
  5. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, UK, 1990).
  6. M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).
  7. G. Musser, “Practical fractals,” Sci. Am., July1999, p. 23.
  8. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).
  9. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguide via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  10. A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).

1999 (1)

G. Musser, “Practical fractals,” Sci. Am., July1999, p. 23.

1995 (2)

R. Zengerle, O. Leminger, “Phase-shifted Bragg-gratingfilters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

S. Legoubin, M. Douay, P. Bernage, P. Niay, S. Boj, E. Delevaque, “Free spectral range variations of grating-based Fabry–Perot photowritten in optical fibers,” J. Opt. Soc. Am. A 12, 1687–1694 (1995).
[CrossRef]

1992 (1)

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

1987 (1)

Barnsley, M.

M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).

Bernage, P.

Boj, S.

Delevaque, E.

Douay, M.

Falconer, K.

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, UK, 1990).

Haus, H. A.

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

Kalli, K.

A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

Lai, Y.

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

Legoubin, S.

Leminger, O.

R. Zengerle, O. Leminger, “Phase-shifted Bragg-gratingfilters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

Musser, G.

G. Musser, “Practical fractals,” Sci. Am., July1999, p. 23.

Niay, P.

Othonos, A.

A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).

Sakuda, K.

Yamada, M.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

Zengerle, R.

R. Zengerle, O. Leminger, “Phase-shifted Bragg-gratingfilters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

H. A. Haus, Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992).
[CrossRef]

J. Lightwave Technol. (1)

R. Zengerle, O. Leminger, “Phase-shifted Bragg-gratingfilters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Sci. Am. (1)

G. Musser, “Practical fractals,” Sci. Am., July1999, p. 23.

Other (5)

A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984).

A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing (Artech House, Norwood, Mass., 1999).

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Chichester, UK, 1990).

M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988).

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Figures (7)

Fig. 1
Fig. 1

Basic cells of Cantor set gratings. Each subgrating has period Λa, and the grating has a period Λb. The blank areas indicate the presence of a gap (no grating). (a) Cantor set grating cell of order 0, (b) Cantor set grating cell of order 1, (c) Cantor set grating cell of order 2.

Fig. 2
Fig. 2

Cantor set gratings of orders (a) 1 and (b) 2. Wavelength spacing Δλ is 5 nm, and the total length L is 10 mm.

Fig. 3
Fig. 3

Cantor set grating of orders (a) 3 and (b) 4. Wavelength spacing Δλ is 5 nm, and the total length L is 10 mm.

Fig. 4
Fig. 4

Reflectivity peaks for (a) the central band and (b) the first lateral bands as a function of Cantor set order and L. Squares, triangles, and crosses mark plots for orders 1, 2, and 3, respectively. The plot for order 4 is unmarked.

Fig. 5
Fig. 5

Reflectivity peaks for (a) the second and (b) the third lateral bands as a function of Cantor set order and L. Squares, triangles, and crosses mark plots for orders 1, 2, and 3, respectively. The plot for order 4 is unmarked.

Fig. 6
Fig. 6

Main peak reflectivity spectra for gratings of orders (a) 1 and (b) 2.

Fig. 7
Fig. 7

Main peak reflectivity spectra for gratings of orders (a) 3 and (b) 4.

Equations (18)

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nco2(z)=na2+Δna2{f1(z)+f2q(z)[1-f1(z)]},
f1(z)=p=-p02pπexpjp π2sinπp2exp-j 2πpΛa,
f2q(z)=m=-bmqexp-j 2πΛb mz.
bm0=0,
bm1=cos(πm) sin(πm/3)πm ifm0;
b01=1/3ifm=0,
bm2=cos(πm)πm2 cos2πm3sinπm9+sinπm3
ifm0;b02=5/9ifm=0,
bm3=cos(πm)πm2 cos2πm3sinπm9+sinπm3+4 sinπm27cos2πm3cos2πm9
if m0;b03=19/27ifm=0.
dA+dz=K1A-Δna22jπexpj2π2 β¯π-1Λaz+m=-m0bmqexpj2π2 β¯λ-mΛbz-2jπm=-bmqexpj2π2 β¯λ-1Λa-mΛbz,
dA-dz=K1*A+Δna2-2jπexp-j2π2 β¯λ-1Λaz+m=-m0bmqexp-j2π2 β¯λ-mΛbz+2jπm=- bmqexp-j2π2 β¯λ-1Λa-mΛbz,
R=|Cmqsinh(γmqL)|2γmq2cosh2(γmqL)+Δβmq2sinh(γmqL)2,
Cmq=-2λβ¯ Δna2bmq ifm0;
C0q=2λβ¯ Δna2(1.0-b0q),
Δβmq=πβ¯λ-2πΛa-2πmΛb,
γmq=Cmq2-Δβmq221/2.
Λbλ22Δλβ¯.

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