Abstract

In this letter we present a technique for the implementation of Nth-order ultrafast temporal differentiators. This technique is based on two oppositely chirped fiber Bragg gratings in which the grating profile maps the spectral response of the Nth-order differentiator. Examples of 1st, 2nd, and 4th order differentiators are designed and numerically simulated.

© 2007 Optical Society of America

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References

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  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]
  2. H. J. A. da Silva and J. J. O'Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526- (1989).
    [CrossRef] [PubMed]
  3. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators, " Opt. Express 14, 10699-10707 (2006).
    [CrossRef] [PubMed]
  4. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, " Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating," Opt. Express 15, 371-381 (2007).
    [CrossRef] [PubMed]
  5. Y. Park, R. Slavik, J. Azaña "Ultrafast all-optical first and higher-order differentiators based on interferometers" Opt. Lett. 32, 710-712 (2007).
    [CrossRef] [PubMed]
  6. M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007).
    [CrossRef]
  7. A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
    [CrossRef]
  8. I. Littler, M. Rochette, and B. Eggleton, "Adjustable bandwidth dispersionless bandpass FBG optical filter," Opt. Express 13, 3397-3407 (2005).
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    [CrossRef]
  10. J. Azaña and L. R. Chen, "Synthesis of temporal optical waveforms by fiber Bragg gratings: a new approach based on space-to-frequency-to-time mapping, " J. Opt. Soc. Am. B 19, 2758-2769 (2002).
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  11. S. Longhi, M. Marano, P. Laporta, O. Svelto, "Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings, J. Opt. Soc. Am. B 19, 2742-2757 (2002).
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  13. J. F. BrennanIII and D. L. LaBrake, "Fabrication of chirped fiber bragg gratings of any desired bandwidth using frequency modulation," US patent 6728444 (April 2004).
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2007 (3)

2006 (1)

2005 (2)

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 (2)

2000 (1)

J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’IEEE J. Quantum Electron. 36, 517-527 (2000).
[CrossRef]

1999 (1)

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

1988 (1)

Azaña, J.

Berger, N. K.

Bovard, B.

Chen, L. R.

Eggleton, B.

Eggleton, B. J.

Fischer, B.

Fu, L.

García-Muñoz, V.

M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007).
[CrossRef]

Jepsen, A. G.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

Johnson, A. E.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

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.

Laporta, P.

Levit, B.

Littler, I.

Littler, I. C. M.

Longhi, S.

Maniloff, E. S.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

Marano, M.

Morandotti, R.

Mossberg, T. W.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

Munroe, M. J.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

Muriel, M. A.

M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007).
[CrossRef]

J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’IEEE J. Quantum Electron. 36, 517-527 (2000).
[CrossRef]

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]

Park, Y.

Plant, D. V.

Preciado, M. A.

M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007).
[CrossRef]

Rochette, M.

Slavik, R.

Slavík, R.

Svelto, O.

Sweetser, J. N.

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

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]

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]

Appl. Opt. (2)

Electron. Lett. (1)

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’IEEE J. Quantum Electron. 36, 517-527 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007).
[CrossRef]

J. Opt. Soc. Am. B (2)

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 (3)

Opt. Lett. (1)

Other (2)

H. J. A. da Silva and J. J. O'Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526- (1989).
[CrossRef] [PubMed]

J. F. BrennanIII and D. L. LaBrake, "Fabrication of chirped fiber bragg gratings of any desired bandwidth using frequency modulation," US patent 6728444 (April 2004).

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Figures (2)

Fig. 1.
Fig. 1.

Architecture of the system. Input signal is processed by two oppositely chirped FBGs, which are connected by an optical circulator.

Fig. 2.
Fig. 2.

Plots (a), (b), and (c) show the phase response of the spectral shaper (dotted), the dispersion compensator (dashed), and the whole system (solid). Plots (d), (e), and (f) show the spectral response corresponding to the spectral shaper (solid) for first, second and third example, and to ideal 1st, 2nd, and 4th order differentiator (dashed), respectively. Plots (g), (h), and (i) show the temporal waveforms of the input pulse (dashed), the output pulse corresponding to the system (solid) for first, second and third example, and the output pulse corresponding to ideal 1st, 2nd, and 4th order differentiator (dotted), respectively.

Equations (14)

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F out ( ω ) = ( j ω ) N F in ( ω )
H N ( ω ) = F out ( ω ) F in ( ω ) = ( j ω ) N
H N , w ( ω ) = H N ( ω ) W ( ω ) = ( j ω ) N W ( ω )
H N , w ( ω ) { ( j ω ) N ω operative band = trans ( ω ) ω transient band 0 ω band of interrest
H syst ( ω ) = H a ( ω ) H b ( ω ) = ( R a ( ω ) R b ( ω ) ) 1 2 exp ( j ( ϕ a + ϕ b ) ) H N , w ( ω )
R a ( ω ) H syst ( ω ) 2 H N , w ( ω ) 2 = ω N W ( ω ) 2
n ( z ) = n av , a ( z ) + Δ n max , a 2 A a ( z ) cos [ 2 π Λ 0 , a z + φ a ( z ) ]
C K , a = 4 n av , a 2 ( c 2 ϕ a ̈ )
L a = ϕ a ̈ c Δ ω g , a ( 2 n av , a )
ϕ a ̈ ( Δ t a ) 2 8 π
A a ( z ) = [ ln ( 1 R a ( ω ) ( ω = sign ( C K , a ) Δ ω g , a L a z ) ) 32 n av , a 2 πω 0 2 ϕ a ̈ Δ n max , a 2 ] 1 2
H syst ( ω ) H 1 , w ( ω ) = H 1 ( ω ) W th ( ω ) = { j ω 2 [ 1 + tanh ( 4 16 ω Δ ω ) ] ω Δ ω 2 0 ω > Δ ω 2
R a ( ω ) = { C R ( ω Δ ω g , a ) [ 1 + tanh ( 4 16 ω Δ ω g , a ) ] } 2
A ( z ) = C A ( ln { 1 C R 2 z L a 2 N [ 1 + tanh ( 4 16 z L a ) ] 2 } ) 1 2

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