Abstract

We study theoretically the possibility of performing temporal differentiation of optical signals using a resonant diffraction grating. We demonstrate that the resonant grating allows the calculation of the first-order derivative of an optical signal envelope in the vicinity of waveguide resonant frequencies in the zeroth transmitted diffraction order. The grating is shown to allow the calculation of the fractional derivative of order 1/2 in the vicinity of Rayleigh–Wood anomalies. Numerical simulations based on the rigorous coupled-wave analysis of Maxwell’s equations demonstrate the high-quality differentiation of optical signals with temporal features in the picosecond range.

© 2011 Optical Society of America

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  1. M. Kulishov and J. Azaña, Opt. Express 15, 6152 (2007).
    [CrossRef] [PubMed]
  2. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, Opt. Express 14, 10699 (2006).
    [CrossRef] [PubMed]
  3. M. Kulishov and J. Azaña, Opt. Lett. 30, 2700 (2005).
    [CrossRef] [PubMed]
  4. R. Slavík, Y. Park, M. Kulishov, and J. Azaña, Opt. Lett. 34, 3116 (2009).
    [CrossRef] [PubMed]
  5. L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. LaRochelle, A. Carballar, and J. Azaña, Opt. Lett. 34, 1792 (2009).
    [CrossRef] [PubMed]
  6. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, Opt. Express 15, 371 (2007).
    [CrossRef] [PubMed]
  7. Y. Park, R. Slavik, and J. Azaña, Opt. Lett. 32, 710 (2007).
    [CrossRef] [PubMed]
  8. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
    [CrossRef]
  9. V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
    [CrossRef]
  10. M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
    [CrossRef]
  11. A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
    [CrossRef]
  12. V. Lomakin and E. Michielssen, IEEE Trans. Antennas Propag. 54, 970 (2006).
    [CrossRef]
  13. R. Magnusson, D. Shin, and Z. S. Liu, Opt. Lett. 23, 612(1998).
    [CrossRef]
  14. T. Tamir and S. Zhang, J. Opt. Soc. Am. A 14, 1607 (1997).
    [CrossRef]
  15. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
    [CrossRef]
  16. W. Nakagawa, R.-C. Tyan, P.-C. Sun, F. Xu, and Y. Fainman, J. Opt. Soc. Am. A 18, 1072 (2001).
    [CrossRef]

2011 (2)

A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
[CrossRef]

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
[CrossRef]

2010 (1)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

2009 (2)

2007 (3)

2006 (2)

2005 (1)

2003 (1)

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
[CrossRef]

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

2001 (1)

1998 (1)

1997 (1)

Akimov, A. B.

A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
[CrossRef]

Azaña, J.

Belotelov, V. I.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

Berger, N. K.

Bezus, E. A.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
[CrossRef]

Boudreau, S.

Bykov, D. A.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

Carballar, A.

Doskolovich, L. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

Fainman, Y.

Fischer, B.

Gippius, N. A.

A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Kalish, A. N.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

Kazanskiy, N. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
[CrossRef]

Kulishov, M.

LaRochelle, S.

Levit, B.

Liu, Z. S.

Lomakin, V.

V. Lomakin and E. Michielssen, IEEE Trans. Antennas Propag. 54, 970 (2006).
[CrossRef]

Magnusson, R.

Michielssen, E.

V. Lomakin and E. Michielssen, IEEE Trans. Antennas Propag. 54, 970 (2006).
[CrossRef]

Morandotti, R.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Nakagawa, W.

Park, Y.

Plant, D. V.

Rivas, L. M.

Sarrazin, M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Shin, D.

Slavik, R.

Slavík, R.

Sun, P.-C.

Tamir, T.

Tikhodeev, S. G.

A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Tyan, R.-C.

Vigneron, J.-P.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Vigoureux, J.-M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Xu, F.

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Zhang, S.

Zvezdin, A. K.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

V. Lomakin and E. Michielssen, IEEE Trans. Antennas Propag. 54, 970 (2006).
[CrossRef]

J. Exp. Theor. Phys. (1)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, J. Exp. Theor. Phys. 110, 816 (2010).
[CrossRef]

J. Exp. Theor. Phys. Lett. (1)

A. B. Akimov, N. A. Gippius, and S. G. Tikhodeev, J. Exp. Theor. Phys. Lett. 93, 427 (2011).
[CrossRef]

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

Microelectron. Eng. (1)

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, Microelectron. Eng. 88, 170 (2011).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. B (2)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, Phys. Rev. B 66, 045102 (2002).
[CrossRef]

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, Phys. Rev. B 67, 085415 (2003).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Incident pulse envelope. (b) Absolute values of the transmitted signal amplitude (solid line) and of the analytically calculated derivative (dotted line). (inset) Geometry of a resonant grating (period d = 1010 nm , ridge height h 1 = 620 nm , ridge width r = 530 nm , layer thickness h 2 = 210 nm , ε 1 = 2.1 , ε 2 = 5.5 ).

Fig. 2
Fig. 2

Absolute values of the transmitted signal amplitude (solid line) and of the analytically calculated fractional derivative (dotted line). (inset) Geometry of a resonant grating (period d = λ 0 / ε 1 1 / 2 , grating height h = 650 nm , ridge width r = 570 nm ).

Equations (11)

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E ( z , t ) = exp ( i k ( ω 0 ) z i ω 0 t ) P ( t z / v g ) = + G ( ω ω 0 ) exp ( i k ( ω ) z i ω t ) d ω ,
P tr ( t ) = + T ( ω + ω 0 ) G ( ω ) exp ( i ω t ) d ω ,
H ( ω ) = T ( ω + ω 0 ) H diff ( ω ) = i ω , | ω | < Ω ,
T ( ω ) a + b ω ω p = a ω ω z ω ω p ,
H ( ω ) = T ( ω + ω 0 ) = a i ω i ω 1 / τ ,
P tr ( t ) = i b n = 1 τ n + 1 d n P ( t ) d t n .
E n = Ω 2 π 0 2 π Ω | τ n + 1 d n P ( t ) d t n | 2 d t = | τ | 2 | τ | n Ω n .
γ = E 1 / n > 1 E n = 1 | τ | Ω 1 .
T 1 ( k z ) = n a n k z n = n a n ω 2 ω 0 2 n = T ( ω ) .
T ( ω ) = n = 0 b n ω ω 0 n .
d 1 / 2 P ( t ) d t 1 / 2 + ω G ( ω ) exp ( i ω t ) d ω .

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