Abstract

We demonstrate that an asymmetrical π phase-shifted fiber Bragg grating operated in reflection can provide the required spectral response for implementing an all-optical fractional differentiator. There are different (but equivalent) ways to design it, e.g., by using different gratings lengths and keeping the same index modulation depth at both sides of the π phase shift, or vice versa. Analytical expressions were found relating the fractional differentiator order with the grating parameters. The device shows a good accuracy calculating the fractional time derivatives of the complex field of an arbitrary input optical waveform. The introduced concept is supported by numerical simulations.

© 2009 Optical Society of America

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References

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  1. L. M. Rivas, K. Singh, A. Carballar, and J. Azaña, IEEE Photonics Technol. Lett. 19, 1209 (2007).
    [CrossRef]
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2007 (4)

2006 (1)

2005 (1)

1987 (1)

Azaña, J.

Berger, N.

Carballar, A.

L. M. Rivas, K. Singh, A. Carballar, and J. Azaña, IEEE Photonics Technol. Lett. 19, 1209 (2007).
[CrossRef]

Fischer, B.

Garcia-Muñoz, V.

Krcmarík, D.

Kulishov, M.

Levit, B.

Morandotti, R.

Muriel, M.

Oldham, K. B.

K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Park, Y.

Plant, D.

Preciado, M.

Rivas, L. M.

L. M. Rivas, K. Singh, A. Carballar, and J. Azaña, IEEE Photonics Technol. Lett. 19, 1209 (2007).
[CrossRef]

Sakuda, K.

Singh, K.

L. M. Rivas, K. Singh, A. Carballar, and J. Azaña, IEEE Photonics Technol. Lett. 19, 1209 (2007).
[CrossRef]

Slavík, R.

Spanier, J.

K. B. Oldham and J. Spanier, The Fractional Calculus (Academic, 1974).

Yamada, M.

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Figures (4)

Fig. 1
Fig. 1

Scheme of the proposed setup for implementing a fractional temporal differentiator. Δ n 1 and Δ n 2 are the index modulation depths, whereas L 1 and L 2 are the grating lengths at each side of the π phase shift.

Fig. 2
Fig. 2

Differentiator order n as a function of the relative change of (a) the grating lengths ε L and (b) the index modulation depths ε κ . The scatter points represent the numerical (exact) solution, whereas the solid curve represents the approximate (analytical) solution given by Eq. (7).

Fig. 3
Fig. 3

Ideal (dashed curve) and proposed (solid curve) 0.63th-order differentiator frequency response in (a) amplitude and (b) phase, in baseband frequency.

Fig. 4
Fig. 4

(a) Simulated (solid curve) and ideal (dashed curve) time response of the designed 0.63th-order differentiator. The input signal is also shown (dotted curve); all signals were normalized to unity. (b) Deviation factor D between the ideal (theoretical) and the 0.63th-order differentiator output for different temporal widths T 0 of the input signals (solid curve) and the deviation factor between a n th -order ideal fractional differentiation and the 0.63th-order differentiator output as a function of n (dashed curve).

Equations (7)

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T APS - FBG = T FBG 1 T PS T FBG 2 ,
T 11 i = T 22 * i = [ cosh ( γ i L i ) + j ( Δ β γ i ) sinh ( γ i L i ) ] exp ( j β B L i ) ,
T 12 i = T 21 * i = ( κ i γ i ) sinh ( γ i L i ) exp [ j β B L i ] ,
r APS - FBG r 1 + r 2 ( t 1 ) 2 exp [ j φ ]
θ APS - FBG ( Δ β ) Δ β L 1 + arctan [ ε κ ε L sin ( Δ β L T ) 1 ε κ ε L cos ( Δ β L T ) ] ,
Δ β 0 = 1 L T arccos ( 1 + ε L 2 ε κ 2 ( 2 + ε L ) ε κ ε L ( 3 + ε L ) ) .
n 2 π θ APS - FBG ( Δ β ) Δ β = Δ β 0 ,

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