Abstract

A general transformation of the optical pulse envelope implemented by a single-resonance diffraction grating is studied. The particular cases considered include optical pulse integration and differentiation implemented by the grating in the Wood anomalies and the fractional integration and differentiation of order 1/2 implemented in the Rayleigh–Wood anomalies. The extraordinary-optical-transmission plasmonic gratings are shown to be well suited for the integration in the transmission. Diffraction gratings to perform the integration and semi-integration of optical pulses with temporal features in the picosecond range are designed. Numerical simulations based on the rigorous coupled-wave analysis of Maxwell’s equations are in good agreement with presented theoretical analysis.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On ability of resonant diffraction gratings to differentiate pulsed optical signal,” J. Exp. Theor. Phys. 114, 724–730 (2012).
[CrossRef]

2011 (3)

A. Akimov, N. Gippius, and S. Tikhodeev, “Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies,” JETP Lett. 93, 427–430 (2011).
[CrossRef]

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36, 3509–3511 (2011).
[CrossRef]

2010 (3)

2009 (2)

2008 (3)

2007 (2)

2006 (2)

Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express 14, 12670–12678 (2006).
[CrossRef]

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antennas Propag. 54, 970–984 (2006).
[CrossRef]

2005 (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[CrossRef]

2003 (1)

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003).
[CrossRef]

2002 (2)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express 10, 840–843 (2002).

2001 (1)

1998 (3)

A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

1996 (2)

1995 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Akimov, A.

A. Akimov, N. Gippius, and S. Tikhodeev, “Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies,” JETP Lett. 93, 427–430 (2011).
[CrossRef]

Azaña, J.

Barbara, A.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100, 066408 (2008).
[CrossRef]

Belotelov, V. I.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, V. A. Kotov, and A. K. Zvezdin, “Giant magneto-optical orientational effect in plasmonic heterostructures,” Opt. Lett. 34, 398–400 (2009).
[CrossRef]

Bendickson, J. M.

Berger, N. K.

Bessonov, V.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).

Brundrett, D. L.

Bykov, D. A.

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On ability of resonant diffraction gratings to differentiate pulsed optical signal,” J. Exp. Theor. Phys. 114, 724–730 (2012).
[CrossRef]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36, 3509–3511 (2011).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, V. A. Kotov, and A. K. Zvezdin, “Giant magneto-optical orientational effect in plasmonic heterostructures,” Opt. Lett. 34, 398–400 (2009).
[CrossRef]

Djurišic, A. B.

Dolgova, T.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Doskolovich, L. L.

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On ability of resonant diffraction gratings to differentiate pulsed optical signal,” J. Exp. Theor. Phys. 114, 724–730 (2012).
[CrossRef]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36, 3509–3511 (2011).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, V. A. Kotov, and A. K. Zvezdin, “Giant magneto-optical orientational effect in plasmonic heterostructures,” Opt. Lett. 34, 398–400 (2009).
[CrossRef]

Ebbesen, T. W.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Elazar, J. M.

Fedyanin, A.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Felbacq, D.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

Fischer, B.

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

Giessen, H.

Gippius, N.

A. Akimov, N. Gippius, and S. Tikhodeev, “Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies,” JETP Lett. 93, 427–430 (2011).
[CrossRef]

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Glytsis, E. N.

Grann, E. B.

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

Guenneau, S.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Janner, D.

Kalish, A. N.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, V. A. Kotov, and A. K. Zvezdin, “Giant magneto-optical orientational effect in plasmonic heterostructures,” Opt. Lett. 34, 398–400 (2009).
[CrossRef]

Kotov, V. A.

Kuhlmey, B.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

Kulishov, M.

Le Perchec, J.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100, 066408 (2008).
[CrossRef]

Levit, B.

Lezec, H. J.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, L.

Li, M.

Lomakin, V.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antennas Propag. 54, 970–984 (2006).
[CrossRef]

López-Ríos, T.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100, 066408 (2008).
[CrossRef]

Majewski, M. L.

Michielssen, E.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antennas Propag. 54, 970–984 (2006).
[CrossRef]

Moharam, M. G.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Muriel, M. A.

Nicolet, A.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, 1962).

Park, Y.

Plant, D. V.

Pommet, D. A.

Preciado, M. A.

Pruneri, V.

Quémerais, P.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100, 066408 (2008).
[CrossRef]

Rakic, A. D.

Renversez, G.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

Sarrazin, M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Sato, A.

Shcherbakov, M.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Slavík, R.

Soifer, V. A.

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On ability of resonant diffraction gratings to differentiate pulsed optical signal,” J. Exp. Theor. Phys. 114, 724–730 (2012).
[CrossRef]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36, 3509–3511 (2011).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Sychev, F.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Thio, T.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Tikhodeev, S.

A. Akimov, N. Gippius, and S. Tikhodeev, “Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies,” JETP Lett. 93, 427–430 (2011).
[CrossRef]

Tikhodeev, S. G.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Opt. Express 18, 7569–7574 (2010).
[CrossRef]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Phys. Rev. B 72, 045138 (2005).
[CrossRef]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Turunen, J.

Vabishchevich, P.

P. Vabishchevich, V. Bessonov, F. Sychev, M. Shcherbakov, T. Dolgova, and A. Fedyanin, “Femtosecond relaxation dynamics of surface plasmon-polaritons in the vicinity of Fano-type resonance,” JETP Lett. 92, 575–579 (2011).
[CrossRef]

Vahimaa, P.

Vallius, T.

Vigneron, J.-P.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Vigoureux, J.-M.

M. Sarrazin, J.-P. Vigneron, and J.-M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003).
[CrossRef]

Weiss, T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd ed. (Pergamon, 1965).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Yao, J.

Zolla, F.

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Zvezdin, A. K.

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, V. A. Kotov, and A. K. Zvezdin, “Giant magneto-optical orientational effect in plasmonic heterostructures,” Opt. Lett. 34, 398–400 (2009).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antennas Propag. 54, 970–984 (2006).
[CrossRef]

J. Exp. Theor. Phys. (2)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: the scattering-matrix method,” J. Exp. Theor. Phys. 110, 816–824 (2010).
[CrossRef]

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “On ability of resonant diffraction gratings to differentiate pulsed optical signal,” J. Exp. Theor. Phys. 114, 724–730 (2012).
[CrossRef]

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A. Akimov, N. Gippius, and S. Tikhodeev, “Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies,” JETP Lett. 93, 427–430 (2011).
[CrossRef]

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H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
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J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Why metallic surfaces with grooves a few nanometers deep and wide may strongly absorb visible light,” Phys. Rev. Lett. 100, 066408 (2008).
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F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations Of Photonic Crystal Fibres (Imperial College, 2005).

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Figures (3)

Fig. 1.
Fig. 1.

Geometry of the metallic diffraction grating.

Fig. 2.
Fig. 2.

(a) Transmission spectrum of the integrating grating (period d=1540nm, height h=80nm, slit width a=440nm) (solid blue line) and the model spectrum with Lorentzian profile (dashed red line). (b) Absolute values of the incident pulse envelope (dash-dotted black line; axis on the right), of the transmitted pulse envelope (solid blue line; axis on the left), and of the model transmitted pulse envelope calculated using Eq. (9) (dashed red line; axis on the left).

Fig. 3.
Fig. 3.

(a) Transmission spectrum of the grating for fractional integration (period d=1550nm, height h=280nm, slit width a=60nm) (solid blue line) and the model spectrum (dashed red line). (b) Absolute values of the transmitted pulse envelope (solid blue line), of the fractional integral of order 1/2 (dash-dotted black line), of the error of the fractional integral calculation [the second term in Eq. (17)] (dotted green line), and of the model transmitted pulse envelope calculated using Eq. (17) (dashed red line).

Equations (26)

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E(z,t)=exp(ik(ω0)ziω0t)Pinc(tz/vg)=F(ωω0)exp(ik(ω)ziωt)dω,
Ptr(t)=T(ω+ω0)F(ω)exp(iωt)dω,
[DRDT]=S[IsupIsub],
S1(ω)[DRDT]=0
1/detS(ωp)=0.
S(ω)A+Bωωp,
T(ω)a+bωωp=aωωzωωp,
H˜int(ω)=bω+i/τ,
Ptr(t)=ib0+Pinc(tT)exp(T/τ)dT=ibtPinc(T)exp((tT)/τ)dT.
H˜diff(ω)=a˜ω1ωτ+i,
|b|×τ1.
|a˜|×|τ|11.
T(ω)=T˜(ξ)a+bξξp=aξξzξξp,
H˜intfr(ω)b1ωξp.
h(t)=12π+beiωtωξpdω=b[1iπtiξp×w(ξpit)]θ(t),
Ptr(t)=Pinc(t)*h(t)=b0+Pinc(tT)[1iπTiξp×w(ξpiT)]dT.
Ptr(t)=bi1Γ(1/2)t(tT)1/2Pinc(T)dTibξptw(ξpi(tT))Pinc(T)dT.
|b|×|ξp|11.
H˜difffr(ω)aωωξp=a˜ω1ωξp11,
|a˜|×|ξp|1.
ΔωRW(2(Imξp)2+(Reξp)2+Imξp)2+(2(Reξp)2+(Imξp)2+Reξp)2.
+eiωtωξdω=ξ+eiωtωξ2dω+i1/2+iωeiωtωξ2dω.
+eiωtωξ2dω=2πieitξ2θ(t).
+iωeiωtωξ2dω=D1/2+eiωtωξ2dω=1Γ(11/2)ddt0t2πieiTξ2θ(t)(tT)1/2dT=2πiΓ(1/2)ddtθ(t)0teiTξ2tTdT=2πiΓ(1/2)θ(t)ddteitξ20teiTξ2TdT=2πiΓ(1/2)θ(t)ddt[eitξ2πiξ1erf(itξ)]=2π[iπt+iξeitξ2erf(itξ)]θ(t).
(ωpωn)S(ωn)=2S(ωn).
ωn+1=ωn+2maxeigS(ωn)maxeigS(ωn).

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