Abstract

Dynamics of femtosecond pulses with the telecom carrier wavelength is investigated numerically in a subwavelength layer of an indium tin oxide (ITO) epsilon-near-zero (ENZ) material with high dispersion and high nonlinearity. Due to the subwavelength thickness of the ITO ENZ material, and the fact that the pulse’s propagation time is shorter than its temporal width, multiple reflections give rise to self-interaction in both spectral and temporal domains, especially at wavelengths longer than at the ENZ point, at which the reflections are significantly stronger. A larger absolute value of the pulse’s chirp strongly affects the self-interaction by redistributing energy between wavelengths, while the sign of the chirp affects the interaction in the temporal domain. It is also found that, when two identical pulses are launched simultaneously from both ends, a subwavelength counterpart of a standing-wave state can be established. It shows robust energy localization in the middle of the sample, in terms of both the spectral and temporal intensity distributions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (4)

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

M. López-Ripa, S. Jarabo, and F. J. Salgado-Remacha, “Near-infrared supercontinuum source by intracavity silica-based highly-nonlinear fiber,” Opt. Lett. 44(8), 2016 (2019).
[Crossref]

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

J. Wu and Q. Li, “Highly efficient self-similar spectral compression of hyperbolic secant pulses enhanced by pre-chirping in nonlinear fibres,” J. Opt. 21(8), 085503 (2019).
[Crossref]

2018 (2)

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

X. Niu, X. Hu, S. Chu, and Q. Gong, “Epsilon-near-zero photonics: a new platform for integrated devices,” Adv. Opt. Mater. 6(10), 1701292 (2018).
[Crossref]

2017 (2)

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

2016 (3)

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref]

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

2015 (4)

S. Campione, I. Brener, and F. Marquier, “Theory of epsilon-near-zero modes in ultrathin films,” Phys. Rev. B 91(12), 121408 (2015).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

2013 (4)

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

T. Zhai and X. Zhang, “Epsilon-near-zero metamaterials for tailoring ultrashort pulses,” Appl. Phys. B 113(2), 185–189 (2013).
[Crossref]

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref]

2012 (1)

2011 (1)

2009 (1)

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

2006 (3)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ϵ-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref]

S.-M. Park, T. Ikegami, and K. Ebihara, “Effects of substrate temperature on the properties of Ga-doped ZnO by pulsed laser deposition,” Thin Solid Films 513(1-2), 90–94 (2006).
[Crossref]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14(17), 7617 (2006).
[Crossref]

2003 (1)

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

2002 (1)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

2001 (1)

1983 (1)

S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
[Crossref]

1900 (1)

P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306(3), 566–613 (1900).
[Crossref]

Abdul Khudus, M. I. M.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013), 5th ed.

Agura, H.

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

Alam, M. Z.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref]

Alù, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Amiri, I. S.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Andrekson, P.

Aoki, T.

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

Azzuhri, S. R.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Banerjee, R.

S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
[Crossref]

Barua, A. K.

S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
[Crossref]

Basu, N.

S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
[Crossref]

Batabyal, A. K.

S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
[Crossref]

Biancalana, F.

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
[Crossref]

Boltasseva, A.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090 (2011).
[Crossref]

Boyd, R. W.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref]

Brener, I.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

S. Campione, I. Brener, and F. Marquier, “Theory of epsilon-near-zero modes in ultrathin films,” Phys. Rev. B 91(12), 121408 (2015).
[Crossref]

Brio, M.

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

Campione, S.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

S. Campione, I. Brener, and F. Marquier, “Theory of epsilon-near-zero modes in ultrathin films,” Phys. Rev. B 91(12), 121408 (2015).
[Crossref]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

Capolino, F.

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

Capretti, A.

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

Caputo, J.-G.

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

Caspani, L.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Chu, S.

X. Niu, X. Hu, S. Chu, and Q. Gong, “Epsilon-near-zero photonics: a new platform for integrated devices,” Adv. Opt. Mater. 6(10), 1701292 (2018).
[Crossref]

Ciattoni, A.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
[Crossref]

Clerici, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Cui, Y.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Dal Negro, L.

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

de Ceglia, D.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

De Leon, I.

M. Z. Alam, I. De Leon, and R. W. Boyd, “Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region,” Science 352(6287), 795–797 (2016).
[Crossref]

DeVault, C.

Di Falco, A.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Dross, F.

Drude, P.

P. Drude, “Zur Elektronentheorie der Metalle,” Ann. Phys. 306(3), 566–613 (1900).
[Crossref]

Ebihara, K.

S.-M. Park, T. Ikegami, and K. Ebihara, “Effects of substrate temperature on the properties of Ga-doped ZnO by pulsed laser deposition,” Thin Solid Films 513(1-2), 90–94 (2006).
[Crossref]

Engheta, N.

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ϵ-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref]

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

Faccio, D.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Falco, A. D.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

Ferrera, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

Finot, C.

Ghimire, S.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Gong, Q.

X. Niu, X. Hu, S. Chu, and Q. Gong, “Epsilon-near-zero photonics: a new platform for integrated devices,” Adv. Opt. Mater. 6(10), 1701292 (2018).
[Crossref]

Greffet, J.-J.

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

Guo, Q.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Gwirtz, K.

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

Hansryd, J.

Hosono, H.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Hu, X.

X. Niu, X. Hu, S. Chu, and Q. Gong, “Epsilon-near-zero photonics: a new platform for integrated devices,” Adv. Opt. Mater. 6(10), 1701292 (2018).
[Crossref]

Hugonin, J.-P.

Ibsen, M.

Ikegami, T.

S.-M. Park, T. Ikegami, and K. Ebihara, “Effects of substrate temperature on the properties of Ga-doped ZnO by pulsed laser deposition,” Thin Solid Films 513(1-2), 90–94 (2006).
[Crossref]

Jarabo, S.

Kaipurath, R. P. M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Keeler, G. A.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

Kelley, K.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Kim, J.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090 (2011).
[Crossref]

Kinsey, N.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

Knudsen, S.

Li, Q.

J. Wu and Q. Li, “Highly efficient self-similar spectral compression of hyperbolic secant pulses enhanced by pre-chirping in nonlinear fibres,” J. Opt. 21(8), 085503 (2019).
[Crossref]

Q. Li, K. Senthilnathan, K. Nakkeeran, and P. K. A. Wai, “Nearly chirp- and pedestal-free pulse compression in nonlinear fiber Bragg gratings,” J. Opt. Soc. Am. B 26(3), 432 (2009).
[Crossref]

Liberal, I.

I. Liberal and N. Engheta, “Near-zero refractive index photonics,” Nat. Photonics 11(3), 149–158 (2017).
[Crossref]

Liu, H.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Liu, J.

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

Liu, S.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

Liu, X.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

López-Ripa, M.

Lu, J.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Luk, T. S.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

Maimistov, A.

M. Brio, J.-G. Caputo, K. Gwirtz, J. Liu, and A. Maimistov, “Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: theoretical and numerical analysis,” Wave Motion 89, 43–56 (2019).
[Crossref]

Manjavacas, A.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Maria, J.-P.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Marini, A.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
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Marquier, F.

S. Campione, I. Brener, and F. Marquier, “Theory of epsilon-near-zero modes in ultrathin films,” Phys. Rev. B 91(12), 121408 (2015).
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S. Vassant, J.-P. Hugonin, F. Marquier, and J.-J. Greffet, “Berreman mode and epsilon near zero mode,” Opt. Express 20(21), 23971 (2012).
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Matsushita, T.

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

Mukasa, K.

Naik, G. V.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090 (2011).
[Crossref]

Nakkeeran, K.

Nasir, N. S.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Niu, X.

X. Niu, X. Hu, S. Chu, and Q. Gong, “Epsilon-near-zero photonics: a new platform for integrated devices,” Adv. Opt. Mater. 6(10), 1701292 (2018).
[Crossref]

Okuda, M.

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

Park, S.-M.

S.-M. Park, T. Ikegami, and K. Ebihara, “Effects of substrate temperature on the properties of Ga-doped ZnO by pulsed laser deposition,” Thin Solid Films 513(1-2), 90–94 (2006).
[Crossref]

Parmigiani, F.

Petropoulos, P.

Pietrzyk, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Prasankumar, R. P.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

Qiu, J.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

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S. Ray, R. Banerjee, N. Basu, A. K. Batabyal, and A. K. Barua, “Properties of tin doped indium oxide thin films prepared by magnetron sputtering,” J. Appl. Phys. 54(6), 3497–3501 (1983).
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Razak, M. Z. A.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Richardson, D. J.

Rizza, C.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
[Crossref]

Roelens, M. A.

Roger, T.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Runnerstrom, E. L.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Salgado-Remacha, F. J.

Salim, M. A. M.

M. A. M. Salim, S. R. Azzuhri, M. I. M. Abdul Khudus, M. Z. A. Razak, N. S. Nasir, and I. S. Amiri, “Generation of dual-wavelength ytterbium-doped fibre laser using a highly nonlinear fibre,” Laser Phys. 28(11), 115107 (2018).
[Crossref]

Scalora, M.

A. Ciattoni, C. Rizza, A. Marini, A. D. Falco, D. Faccio, and M. Scalora, “Enhanced nonlinear effects in pulse propagation through epsilon-near-zero media,” Laser Photonics Rev. 10(3), 517–525 (2016).
[Crossref]

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

A. Ciattoni, A. Marini, C. Rizza, M. Scalora, and F. Biancalana, “Polariton excitation in epsilon-near-zero slabs: transient trapping of slow light,” Phys. Rev. A 87(5), 053853 (2013).
[Crossref]

Senthilnathan, K.

Shalaev, V. M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Di Falco, V. M. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616 (2015).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ϵ-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007).
[Crossref]

Sinclair, M. B.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

Suzuki, A.

H. Agura, A. Suzuki, T. Matsushita, T. Aoki, and M. Okuda, “Low resistivity transparent conducting Al-doped ZnO films prepared by pulsed laser deposition,” Thin Solid Films 445(2), 263–267 (2003).
[Crossref]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

Vassant, S.

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[Crossref]

Vincenti, M. A.

T. S. Luk, D. de Ceglia, S. Liu, G. A. Keeler, R. P. Prasankumar, M. A. Vincenti, M. Scalora, M. B. Sinclair, and S. Campione, “Enhanced third harmonic generation from the epsilon-near-zero modes of ultrathin films,” Appl. Phys. Lett. 106(15), 151103 (2015).
[Crossref]

S. Campione, D. de Ceglia, M. A. Vincenti, M. Scalora, and F. Capolino, “Electric field enhancement in ϵ-near-zero slabs under TM-polarized oblique incidence,” Phys. Rev. B 87(3), 035120 (2013).
[Crossref]

Wai, P. K. A.

Wang, Y.

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

Westlund, M.

Wu, J.

J. Wu and Q. Li, “Highly efficient self-similar spectral compression of hyperbolic secant pulses enhanced by pre-chirping in nonlinear fibres,” J. Opt. 21(8), 085503 (2019).
[Crossref]

Yang, Y.

Y. Yang, J. Lu, A. Manjavacas, T. S. Luk, H. Liu, K. Kelley, J.-P. Maria, E. L. Runnerstrom, M. B. Sinclair, S. Ghimire, and I. Brener, “High-harmonic generation from an epsilon-near-zero material,” Nat. Phys. 15(10), 1022–1026 (2019).
[Crossref]

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Yao, Y.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Ye, Y.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Zhai, T.

T. Zhai and X. Zhang, “Epsilon-near-zero metamaterials for tailoring ultrashort pulses,” Appl. Phys. B 113(2), 185–189 (2013).
[Crossref]

Zhang, S.

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

Zhang, X.

T. Zhai and X. Zhang, “Epsilon-near-zero metamaterials for tailoring ultrashort pulses,” Appl. Phys. B 113(2), 185–189 (2013).
[Crossref]

ACS Photonics (1)

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Comparative study of second-harmonic generation from epsilon-near-zero indium tin oxide and titanium nitride nanolayers excited in the near-infrared spectral range,” ACS Photonics 2(11), 1584–1591 (2015).
[Crossref]

Adv. Mater. (2)

Q. Guo, Y. Cui, Y. Yao, Y. Ye, Y. Yang, X. Liu, S. Zhang, X. Liu, J. Qiu, and H. Hosono, “A solution-processed ultrafast optical switch based on a nanostructured epsilon-near-zero medium,” Adv. Mater. 29(27), 1700754 (2017).
[Crossref]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref]

Adv. Opt. Mater. (1)

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Figures (8)

Fig. 1.
Fig. 1. (a) The variation of the real and imaginary parts (the solid and dashed lines, respectively) of the complex permittivity, as a function of the wavelength. (b) The group-velocity dispersion, $\beta _2$, near $\lambda _\textrm {C} = 1.55$ $\mu$m.
Fig. 2.
Fig. 2. Schematic diagram of (a) a single ultrashort pulse incident onto the ITO ENZ material at the telecom wavelength; (b) the setup of counterpropagating pulses used to establish the QSW (quasi-standing-wave) state.
Fig. 3.
Fig. 3. The transmission (dashed-dotted line), reflection (dashed line), and absorption (solid line) coefficients of the ITO sample under the consideration.
Fig. 4.
Fig. 4. The evolution of the intensity of the electromagnetic field, produced by direct simulations of the Maxwell’s equations [Eq. (6)] in the spectral (a) and temporal (b) domains. Panels (c) and (d) show the same pictures on the logarithmic scale.
Fig. 5.
Fig. 5. The spectra for the propagation distance 0 nm (the dashed line) and 500 nm (the solid line), with the initial chirp in Eq. (5) $C = 0.2$ (a), $0.8$ (b), $-0.2$ (c), $-0.8$ (d). (e)-(h) are the same as in (a)-(d), but for the intensity distribution in the temporal domain.
Fig. 6.
Fig. 6. The spectral intensity distribution with chirps (a) $C = 2$ and (c) $C = -2$ for the propagation distance 0 nm (the dashed line) and 500 nm (the solid line). The temporal intensity distribution with chirps (b) $C = 2$ and (d) $C = -2$.
Fig. 7.
Fig. 7. (a) The spectral intensity distribution corresponding to the QSW (quasi-standing-wave) pattern, created by the interplay of two pulses launched in opposite directions. (b) The same on the logarithmic scale.
Fig. 8.
Fig. 8. Panels (a) and (b) display the temporal-domain counterparts of the spectral image of the QSW pattern shown in Fig. 7.

Equations (7)

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ε ( ω ) = ε r + i ε i = ε b ω p 2 ( ω 2 + Γ 2 ) + i ω p 2 Γ ( ω 2 + Γ 2 ) ω .
[ B C ] = [ cos δ i sin δ / n i n sin δ cos δ ] [ 1 n s ] ,
n r = ε r 2 + ε i 2 + ε r 2 , k = ε i 2 n r  =  n r 2 ε r .
R = ( n s B C n s B + C ) ( n s B C n s B + C ) , T = ( 1 R ) ψ , A = ( 1 R ) ( 1 ψ ) ,
u = P 0 cos ( ω 0 t C t 2 2 T 0 2 ) exp ( t 2 2 T 0 2 ) ,
× H = D t , D = ε 0 ε R E + P L + P NL , × E = B t , B = μ 0 H ,
P NL = 4 c ε 0 2 n 2 3 n 2 E ( r , t ) E ( r , t ) E ( r , t ) ,

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