Abstract

A simple and general approach for designing practical all-optical (all-fiber) arbitrary-order time differentiators is introduced here for the first time. Specifically, we demonstrate that the N th time derivative of an input optical waveform can be obtained by reflection of this waveform in a single uniform fiber Bragg grating (FBG) incorporating N π-phase shifts properly located along its grating profile. The general design procedure of an arbitrary-order optical time differentiator based on a multiple-phase-shifted FBG is described and numerically demonstrated for up to fourth-order time differentiation. Our simulations show that the proposed approach can provide optical operation bandwidths in the tens-of-GHz regime using readily feasible FBG structures.

© 2007 Optical Society of America

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References

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  1. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., Special issue on "Optical Signal Processing," in J. Lightwave Technol. 24, 2484-2767 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
    [CrossRef]

2007 (2)

2006 (2)

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators," Opt. Express 14, 10699-10707 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

2002 (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

1997 (1)

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1989 (1)

Azaña, J.

Berger, N. K.

Da Silva, H. J. A.

Erdogan, T.

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Fischer, B.

Kam, C.H.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Kulishov, M.

Levit, B.

Li, H.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Li, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Morandotti, R.

Ngo, N. Q.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

O’Reilly, J. J.

Park, Y.

Plant, D. V.

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Raymer, M. G.

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Sheng, Y

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Slavík, R.

Tjin, S. C.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Wang, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Wilcox, R. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Wu, C.

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

Yu, S. F.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Zweiback, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Opt. Commun. (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Other (3)

A. Papoulis, Fourier Integral and its Applications, (McGraw-Hill, New York, 1987).

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

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., Special issue on "Optical Signal Processing," in J. Lightwave Technol. 24, 2484-2767 (2006).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Structures of Bragg grating - based first-order and second-order differentiators

Fig. 2.
Fig. 2.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with two symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal second-order differentiator is also represented (dashed, magenta curve).

Fig. 3.
Fig. 3.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based second-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve is the magnitude of the ideal (analytical) second time derivative of the input Gaussian.

Fig. 4.
Fig. 4.

Relative deviation of the reflected waveform from its ideal shape (second time derivative of the input pulse) as a function of the ratio L2 /L1 . Optimal operation is achieved for L2 /L1 =2.

Fig. 5.
Fig. 5.

Structure of a Bragg grating-based third-order optical differentiator.

Fig. 6.
Fig. 6.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with three π-phase shifts properly located to achieve third-order optical differentiation (parameters given in the text). The solid, cyan curve shows the amplitude reflection spectrum of a FBG with three symmetrically located π-phase shifts designed according to the conditions derived for second-order differentiation. For comparison, the amplitude spectrum of an ideal third-order differentiator is also represented (dashed, magenta curve).

Fig. 7.
Fig. 7.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based third-order differentiator together with the ideal third time derivative (dashed, magenta curve) assuming an input 100-ps Gaussian optical pulse

Fig. 8.
Fig. 8.

Graphical representation of the main design condition (Eq. (15)) of a FBG-based third-order optical differentiator.

Fig. 9.
Fig. 9.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with four symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal fourth-order differentiator is also represented (dashed, magenta curve).

Fig. 10.
Fig. 10.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based fourth-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve represents the magnitude of the ideal (analytical) fourth time derivative of the input Gaussian pulses.

Equations (25)

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[ E A ( z 0 + L ) E B ( z 0 + L ) ] = T ( z 0 , L ) [ E A ( z 0 ) E B ( z 0 ) ] = [ T 11 ( L ) T 12 ( z 0 , L ) T 21 ( z 0 , L ) T 22 ( L ) ] [ E A ( z 0 ) E B ( z 0 ) ] ,
T 11 = T 22 * = [ cosh ( γ L ) + j σ γ sinh ( γ L ) ] exp [ j ( π Λ ) L ] ,
T 12 = T 21 * = j κ γ sinh ( γ L ) exp [ j ( π Λ ) ( 2 z 0 + L ) ] ,
Φ 11 = exp ( j φ 2 ) ;
Φ 22 = exp ( j φ 2 ) ;
Φ 12 = Φ 21 = 0
r = T 21 T 22 = r exp ( j ϕ r )
τ = 1 T 22 = τ exp ( j ϕ τ )
T = T ( L 1 + L 2 , L 1 ) ΦT ( L 1 , L 2 ) ΦT ( 0 , L 1 )
T 21 = [ 2 j κ γ sinh ( γ L 1 ) ( cosh ( γ L 1 ) cosh ( γ L 2 ) σ 2 γ 2 sinh ( γ L 1 ) sinh ( γ L 2 ) ) j κ γ sinh ( γ L 2 ) ( κ 2 + σ 2 γ 2 sinh 2 ( γ L 1 ) + cosh 2 ( γ L 1 ) ) ] exp ( j π Λ ( 2 L 1 + L 2 ) )
T 22 = [ j σ γ sinh ( γ L 2 ) ( κ 2 γ 2 σ 2 γ 2 cosh ( 2 γ L 1 ) j σ γ sinh ( γ L 1 ) sinh ( 2 γ L 1 ) ) cosh ( γ L 2 ) ( cosh ( 2 γ L 1 ) j σ γ sinh ( 2 γ L 1 ) ) ] exp ( j π Λ ( 2 L 1 + L 2 ) )
T 21 = j cosh ( κ L 2 ) ( cosh ( 2 κ L 1 ) sinh ( κ L 2 ) sinh ( 2 κ L 1 ) cosh ( κ L 2 ) cosh ( κ L 1 ) ) + O ( σ 2 )
T 22 = cosh ( κ L 2 ) cosh ( 2 κ L 1 ) + j κ ( cosh ( κ L 2 ) sinh ( 2 κ L 1 ) + sinh ( κ L 2 ) ) σ
+ 1 κ ( cosh ( κ L 2 ) sinh ( 2 κ L 1 ) L 1 + L 2 2 sinh ( κ L 2 ) cosh ( 2 κ L 1 ) + 1 κ sinh ( κ L 2 ) sinh ( 2 κ L 1 ) ) σ 2 + O ( σ 3 )
r = 4 j sinh 2 ( κ L 1 ) sinh ( 2 κ L 1 ) κ 2 cosh 2 ( 2 κ L 1 ) σ 2 + O ( σ 3 )
T = T ( L 1 + 2 L 2 , L 1 ) Φ T ( L 1 + L 2 , L 2 ) Φ T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
T 21 = j κ ( ( cosh ( κ ( L 1 + L 2 ) ) cosh ( κ ( L 1 L 2 ) ) ) 2 + + ( sinh ( κ ( L 1 + L 2 ) ) sinh ( κ ( L 1 L 2 ) ) ) 2 2 sinh 2 ( κ ( L 1 + L 2 ) ) ) σ + O ( σ 3 )
T 22 = j + [ sinh ( 2 κ ( L 2 L 1 ) + 2 sinh ( 2 κ L 1 ) ] σ κ +
+ j [ κ L 1 sinh ( 2 κ L 1 ) + 2 sinh 2 ( κ L 1 ) 2 sinh 2 ( 2 κ L 2 ) + + 4 sinh 2 ( 2 κ L 1 ) sinh 2 ( 2 κ L 2 ) sinh ( 2 κ L 1 ) sinh ( 2 κ L 2 ) ] σ 2 κ 2 + O ( σ 3 )
1 2 cosh ( 2 κ L 1 ) + cosh ( 2 κ ( L 2 L 1 ) = 0
α = L 2 L 1 = 1 + cosh 1 ( 2 cosh ( 2 κ L 1 ) 1 ) 2 κ L 1
T 21 n = 0 ( N 2 ) 1 F 2 n L 1 L ( N 2 ) + 1 σ 2 n + O ( σ N ) for even N
T 21 n = 0 ( N 3 ) 2 F 2 n + 1 L 1 L ( N + 1 ) 2 σ 2 n + 1 + O ( σ N ) for odd N
T = T ( L 1 + 2 L 2 + L 3 , L 1 ) Φ T ( L 1 + L 2 + L 3 , L 2 ) Φ T ( L 1 + L 2 , L 3 ) Φ T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
η = L 3 L 1 = 2 ( α 1 )

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