Abstract

For decades, nonlinear optics has been used to control the frequency and propagation of light in unique ways enabling a wide range of applications such as ultrafast lasing, sub-wavelength imaging, and novel sensing methods. Through this, a key thread of research in the field has always been the development of new and improved nonlinear materials to empower these applications. Recently, epsilon-near-zero (ENZ) materials have emerged as a potential platform to enhanced nonlinear interactions, bolstered in large part due to the extreme refractive index tuning (Δn∼ 0.1 - 1) of sub-micron thick films that has been demonstrated in literature. Despite this experimental success, the theory has lagged and is needed to guide future experimental efforts. Here, we construct a theoretical framework for the intensity-dependent refractive index of the most popular ENZ materials, heavily doped semiconductors. We demonstrate that the nonlinearity when excited below bandgap, is due to the modification of the effective mass of the electron sea which produces a shift in the plasma frequency. We discuss trends and trade-offs in the optimization of excitation conditions and material choice (such material loss, band structure, and index dispersion), and provide a figure of merit through which the performance of future materials may be evaluated. By illuminating the framework of the nonlinearity, we hope to propel future applications in this growing field.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

S. Saha, B. T. Diroll, J. Shank, Z. Kudyshev, A. Dutta, S. N. Chowdhury, T. S. Luk, S. Campione, R. D. Schaller, V. M. Shalaev, A. Boltasseva, and M. G. Wood, “Broadband, High-Speed, and Large-Amplitude Dynamic Optical Switching with Yttrium-Doped Cadmium Oxide,” Adv. Funct. Mater. 30(7), 1908377 (2020).
[Crossref]

2019 (5)

L. D. Whalley, J. M. Frost, B. J. Morgan, and A. Walsh, “Impact of nonparabolic electronic band structure on the optical and transport properties of photovoltaic materials,” Phys. Rev. B 99(8), 085207 (2019).
[Crossref]

O. Reshef, I. De Leon, M. Z. Alam, and R. W. Boyd, “Nonlinear optical effects in epsilon-near-zero media,” Nat. Rev. Mater. 4(8), 535–551 (2019).
[Crossref]

N. Kinsey, C. DeVault, A. Boltasseva, and V. M. Shalaev, “Near-zero-index materials for photonics,” Nat. Rev. Mater. 4(12), 742–760 (2019).
[Crossref]

N. Kinsey and J. Khurgin, “Nonlinear epsilon-near-zero materials explained: opinion,” Opt. Mater. Express 9(7), 2793 (2019).
[Crossref]

N. Izyumskaya, V. Avrutin, K. Ding, Ü Özgür, H. Morkoç, and H. Fujioka, “Emergence of high quality sputtered III-nitride semiconductors and devices,” Semicond. Sci. Technol. 34(9), 093003 (2019).
[Crossref]

2018 (9)

H. Chen, H. Fu, X. Huang, J. A. Montes, T.-H. Yang, I. Baranowski, and Y. Zhao, “Characterizations of the nonlinear optical properties for (010) and (2¯01) beta-phase gallium oxide,” Opt. Express 26(4), 3938 (2018).
[Crossref]

Q. Gao, E. Li, and A. X. Wang, “Comparative analysis of transparent conductive oxide electro-absorption modulators [Invited],” Opt. Mater. Express 8(9), 2850 (2018).
[Crossref]

I. C. Reines, M. G. Wood, T. S. Luk, D. K. Serkland, and S. Campione, “Compact epsilon-near-zero silicon photonic phase modulators,” Opt. Express 26(17), 21594 (2018).
[Crossref]

E. Carnemolla, L. Caspani, C. DeVault, M. Clerici, S. Vezzoli, V. Bruno, V. Shalaev, D. Faccio, A. Boltasseva, and M. Ferrera, “Degenerate optical nonlinear enhancement in epsilon-near-zero transparent conducting oxides,” Opt. Mater. Express 8(11), 3392–3400 (2018).
[Crossref]

S. Vezzoli, V. Bruno, C. DeVault, T. Roger, V. M. Shalaev, A. Boltasseva, M. Ferrera, M. Clerici, A. Dubietis, and D. Faccio, “Optical time reversal from time-dependent Epsilon-Near-Zero media,” Phys. Rev. Lett. 120(4), 043902 (2018).
[Crossref]

A. Krasnok, M. Tymchenko, and A. Alù, “Nonlinear metasurfaces: a paradigm shift in nonlinear optics,” Mater. Today 21(1), 8–21 (2018).
[Crossref]

A. V. Krasavin, P. Ginzburg, and A. V. Zayats, “Free-electron Optical Nonlinearities in Plasmonic Nanostructures: A Review of the Hydrodynamic Description,” Laser Photonics Rev. 12(1), 1700082 (2018).
[Crossref]

N. C. Panoiu, W. E. I. Sha, D. Y. Lei, and G.-C. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

J. Lian, D. Zhang, R. Hong, P. Qiu, T. Lv, and D. Zhang, “Defect-induced tunable permittivity of epsilon-near-zero in indium tin oxide thin films,” Nanomaterials 8(11), 922 (2018).
[Crossref]

2017 (7)

K. D. Leedy, K. D. Chabak, V. Vasilyev, D. C. Look, J. J. Boeckl, J. L. Brown, S. E. Tetlak, A. J. Green, N. A. Moser, A. Crespo, D. B. Thomson, R. C. Fitch, J. P. McCandless, and G. H. Jessen, “Highly conductive homoepitaxial Si-doped Ga2O3 films on (010) β-Ga2O3 by pulsed laser deposition,” Appl. Phys. Lett. 111(1), 012103 (2017).
[Crossref]

H. Peelaers and C. G. Van De Walle, “Sub-band-gap absorption in Ga2O3,” Appl. Phys. Lett. 111(18), 182104 (2017).
[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8(1), 15829 (2017).
[Crossref]

J. Cimek, N. Liaros, S. Couris, R. Stępień, M. Klimczak, R. Buczyński, A. D. Sontakke, K. Biswas, A. Tarafder, R. Sen, K. Annapurna, and B. Er, “Experimental investigation of the nonlinear refractive index of various soft glasses dedicated for development of nonlinear photonic crystal fibers References and links,” Opt. Mater. Express 7(10), 3471 (2017).
[Crossref]

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

X. Li, M. Pietrzyk, D. Faccio, C. Rizza, A. Ciattoni, and A. Di Falco, “Linear and nonlinear optical behavior of epsilon near zero metamaterials: opportunities and challenges,” Proc. SPIE 10111, 101111O (2017).
[Crossref]

Y. Arakawa, K. Ueno, H. Imabeppu, A. Kobayashi, J. Ohta, and H. Fujioka, “Electrical properties of Si-doped GaN prepared using pulsed sputtering,” Appl. Phys. Lett. 110(4), 042103 (2017).
[Crossref]

2016 (1)

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 (9)

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), 616–622 (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]

A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Enhanced third-harmonic generation in Si-compatible epsilon-near-zero indium tin oxide nanolayers,” Opt. Lett. 40(7), 1500–1503 (2015).
[Crossref]

S. Campione, S. Liu, A. Benz, J. F. Klem, M. B. Sinclair, and I. Brener, “Epsilon-Near-Zero Modes for Tailored Light-Matter Interaction,” Phys. Rev. Appl. 4(4), 044011 (2015).
[Crossref]

Z. Ma, Z. Li, K. Liu, C. Ye, and V. J. Sorger, “Indium-Tin-Oxide for High-performance Electro-optic Modulation,” Nanophotonics 4(1), 198–213 (2015).
[Crossref]

A. Catellani, A. Ruini, and A. Calzolari, “Optoelectronic properties and color chemistry of native point defects in Al:ZnO transparent conductive oxide,” J. Mater. Chem. C 3(32), 8419–8424 (2015).
[Crossref]

N. Kinsey, A. A. Syed, D. Courtwright, C. DeVault, C. E. Bonner, V. I. Gavrilenko, V. M. Shalaev, D. J. Hagan, E. W. Van Stryland, and A. Boltasseva, “Effective third-order nonlinearities in metallic refractory titanium nitride thin films,” Opt. Mater. Express 5(11), 2395 (2015).
[Crossref]

E. Sachet, C. T. Shelton, J. S. Harris, B. E. Gaddy, D. L. Irving, S. Curtarolo, B. F. Donovan, P. E. Hopkins, P. A. Sharma, A. L. Sharma, J. Ihlefeld, S. Franzen, and J.-P. Maria, “Dysprosium-doped cadmium oxide as a gateway material for mid-infrared plasmonics,” Nat. Mater. 14(4), 414–420 (2015).
[Crossref]

F. Zhang, K. Saito, T. Tanaka, M. Nishio, and Q. Guo, “Electrical properties of Si doped Ga2O3 films grown by pulsed laser deposition,” J. Mater. Sci.: Mater. Electron. 26(12), 9624–9629 (2015).
[Crossref]

2014 (1)

X. Liu, J. Park, J. H. Kang, H. Yuan, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Quantification and impact of nonparabolicity of the conduction band of indium tin oxide on its plasmonic properties,” Appl. Phys. Lett. 105(18), 181117 (2014).
[Crossref]

2013 (2)

2012 (3)

M. Girtan, “Comparison of ITO/metal/ITO and ZnO/metal/ZnO characteristics as transparent electrodes for third generation solar cells,” Sol. Energy Mater. Sol. Cells 100, 153–161 (2012).
[Crossref]

H. Y. Liu, V. Avrutin, N. Izyumskaya, Ü Özgür, A. B. Yankovich, A. V. Kvit, P. M. Voyles, and H. Morko, “Electron scattering mechanisms in GZO films grown on a-sapphire substrates by plasma-enhanced molecular beam epitaxy,” J. Appl. Phys. 111(10), 103713 (2012).
[Crossref]

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with Tailored Nonlinear Optical Response,” Nano Lett. 12(2), 673–677 (2012).
[Crossref]

2011 (2)

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2009 (1)

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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), 616–622 (2015).
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V. E. Babicheva, N. Kinsey, G. V. Naik, M. Ferrera, A. V. Lavrinenko, V. M. Shalaev, and A. Boltasseva, “Towards CMOS-compatible nanophotonics: Ultra-compact modulators using alternative plasmonic materials,” Opt. Express 21(22), 27326–27337 (2013).
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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).
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S. Saha, B. T. Diroll, J. Shank, Z. Kudyshev, A. Dutta, S. N. Chowdhury, T. S. Luk, S. Campione, R. D. Schaller, V. M. Shalaev, A. Boltasseva, and M. G. Wood, “Broadband, High-Speed, and Large-Amplitude Dynamic Optical Switching with Yttrium-Doped Cadmium Oxide,” Adv. Funct. Mater. 30(7), 1908377 (2020).
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E. Carnemolla, L. Caspani, C. DeVault, M. Clerici, S. Vezzoli, V. Bruno, V. Shalaev, D. Faccio, A. Boltasseva, and M. Ferrera, “Degenerate optical nonlinear enhancement in epsilon-near-zero transparent conducting oxides,” Opt. Mater. Express 8(11), 3392–3400 (2018).
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M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8(1), 15829 (2017).
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A. Capretti, Y. Wang, N. Engheta, and L. Dal Negro, “Enhanced third-harmonic generation in Si-compatible epsilon-near-zero indium tin oxide nanolayers,” Opt. Lett. 40(7), 1500–1503 (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).
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M. A. Vincenti, D. De Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A: At., Mol., Opt. Phys. 84(6), 063826 (2011).
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O. Reshef, I. De Leon, M. Z. Alam, and R. W. Boyd, “Nonlinear optical effects in epsilon-near-zero media,” Nat. Rev. Mater. 4(8), 535–551 (2019).
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N. Kinsey, C. DeVault, A. Boltasseva, and V. M. Shalaev, “Near-zero-index materials for photonics,” Nat. Rev. Mater. 4(12), 742–760 (2019).
[Crossref]

E. Carnemolla, L. Caspani, C. DeVault, M. Clerici, S. Vezzoli, V. Bruno, V. Shalaev, D. Faccio, A. Boltasseva, and M. Ferrera, “Degenerate optical nonlinear enhancement in epsilon-near-zero transparent conducting oxides,” Opt. Mater. Express 8(11), 3392–3400 (2018).
[Crossref]

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[Crossref]

M. Clerici, N. Kinsey, C. DeVault, J. Kim, E. G. Carnemolla, L. Caspani, A. Shaltout, D. Faccio, V. Shalaev, A. Boltasseva, and M. Ferrera, “Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation,” Nat. Commun. 8(1), 15829 (2017).
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[Crossref]

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[Crossref]

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[Crossref]

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S. Benis, D. J. Hagan, and E. W. Van Stryland, “Enhancement Mechanism of Nonlinear Optical Response of Transparent Conductive Oxides at Epsilon-Near-Zero,” in Conference on Lasers and Electro-Optics (OSA, 2018), p. FF2E.1.

S. Benis, P. Zhao, D. J. Hagan, and E. W. Van Stryland, “Nondegenerate, Transient Nonlinear Refraction of Indium Tin Oxide Excited at Epsilon-Near-Zero,” in Nonlinear Optics (OSA, 2017), p. NW1A.3.

R. Amin, R. Maiti, Z. Ma, M. Miscuglio, H. Dalir, and V. J. Sorger, “An ITO-based Mach-Zehnder Modulator with Lateral MOS-Capacitor on SOI Platform,” in Frontiers in Optics + Laser Science APS/DLS (The Optical Society, 2019), p. JW3A.67.

S. Benis, N. Munera, R. Acuña, D. J. Hagan, and E. W. Van Stryland, “Nonlinear Fresnel coefficients due to giant ultrafast nonlinearities in indium tin oxide (Conference Presentation),” in Ultrafast Phenomena and Nanophotonics XXIIIM. Betz and A. Y. Elezzabi, eds. (SPIE, 2019), Vol. 10916, p. 35.

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Figures (5)

Fig. 1.
Fig. 1. a) Schematic of the intraband nonlinearity in ENZ materials, where the reflectivity (permittivity) of the material is changed through the application of a pump beam. b) The change in permittivity occurs due to a modification of the effective mass of the electron sea as the absorbed pump energy elevates electrons to higher energy, higher mass states. c) The resulting change in effective mass red-shifts the plasma frequency of the material, producing the modulation of permittivity at a fixed frequency.
Fig. 2.
Fig. 2. a) Schematic of the change in momentum for electrons near the Fermi-level under an applied field. b) Energy band dispersion adapted from Ref. [40] and the hyperbolic fit used for calculations compared to a parabolic fit. c) Comparison of the inverse effective mass of electrons for doped zinc oxide films at room temperature versus energy into the conduction band.
Fig. 3.
Fig. 3. a) The change in refractive index due to a ${0.4^{}}(TW/c{m^2})$ (black) and ${0.9^{}}(TW/c{m^2})$ (red) pump at 780 nm. Using a relaxation rate of $\tau = {170^{}}(fs)$ [20] on a Gaussian pulse shape with maximum pump with 100 fs full-width half-maximum. A close fit between experimental [24] and theoretical responses is obtained through a deterministic model. b) The peak $\delta n$ and average effective mass of carriers versus the applied pump intensity for the AZO sample.
Fig. 4.
Fig. 4. Figure of merit for the selected materials as a function of optical mobility. Grey dashed lines represent the range of mobilities achieved by films published in literature. If only a single line, an arrow indicates the range of mobilities.
Fig. 5.
Fig. 5. The change in refractive index of a pumped indium doped tin-oxide film in near-degenerate pump-probe conditions (1300 (nm), 1250 (nm) respectively) as experimentally shown by Benis et. al [23] (black markers) and through our model with a slow light factor (red line).

Tables (2)

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Table 1. Approximate comparison of transparent semiconductors for intensity-dependent refractive index calculations. CdO, GaN, and Ga2O3 mobilities are for lower carrier concentrations ( < 0.5 × 10 20 ( c m 3 ) ). N and m a v g are calculated for all materials to generate an ENZ wavelength of 1325 ( n m ) (See Appendix D).

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Table 2. Hyperbolic fit constants used in Section 5

Equations (31)

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δ k = e F / .
v z = cos θ = v F k z k z 2 + k 2
d v z d k z = d v ( k ) d k z cos θ v ( k ) sin θ d θ d k z = d v ( k ) d k cos 2 θ + v ( k ) k sin 2 θ
δ v z = e i ω [ 2 3 v ( k ) k + 1 3 d v ( k ) d k ] F = e i ω m ( k ) F
m ( k ) = [ 2 3 v ( k ) k + 1 3 d v ( k ) d k ] 1
ε ( ω ) = ε q 2 ε 0 ( ω 2 + i ω Γ ) 1 V l = 1 n c a r 1 m ( k l ) = ε N q 2 ε 0 ( ω 2 + i ω Γ ) m a v g
m a v g = [ 1 N V l = 1 n c a r 1 m ( k l ) ] 1
m a v g = 1 N f ( E , μ , T e ) m ( E ) ρ ( E ) d E
f ( E , μ , T e ) = [ exp ( E μ k B T e ) + 1 ] 1
δ ε ( I ) = + N q 2 ε 0 m a v g ( ω 2 + i ω Γ ) δ m a v g ( I ) m a v g
δ ε | ω p ε δ m a v g ( I ) m a v g
δ ( 1 m ) i = ( k 1 m ( k ) ) i δ k = 1 m ( k F ) δ k i k F = 1 m ( k F ) δ E i v F k F
δ ε q 2 ε 0 ω 2 1 V δ k δ ( 1 m ) i = q 2 ε 0 m ( k F ) ω 2 δ U a b s v F k F
δ ε ( ω p ) = ε δ U a b s N v F k F
δ U a b s = I i ( 1 e α d ) τ / d
δ U a b s = α o n n g τ 2 η 0 F i 2
χ e f f ( 3 ) ( ω p ) = ε α o n n g τ 2 η 0 N v F k F = ε α n τ 2 η 0 N E F
χ e f f ( 3 ) ( ω p ) = χ e f f ( 3 ) ( ω p ) Γ ( ω p ) ω p = ε α n Γ ( ω p ) τ 2 ω p η 0 N E F
n 2 ( ω p ) = ε α o n g τ 2 n ( ω p ) N E F = ε α τ 2 n ( ω p ) N E F
α 2 ( ω p ) 2 η o ω c χ e f f ( 3 ) n 2 ( ω p ) = ε α n Γ ( ω p ) τ c n 2 ( ω p ) N E F
f ( E , μ ( I ) , T e ( I ) ) ρ ( E ) d E = N
f ( E , μ ( I ) , T e ( I ) ) E ρ ( E ) d E f ( E , μ 0 , T 0 ) E ρ ( E ) d E = δ U a b s = A I τ / d
ε ( ω , I ) = ε N q 2 ε 0 m a v g ( I ) 1 ω 2 + i ω Γ
F O M = A p u m p [ 1 m a v g d m d E ] [ 1 N d n p r o b e d m a v g ]
d U a b s = U ( 0 ) U ( d z ) = ( 1 R ) I v g [ 1 e α d z ]
d U a b s = U ( 0 ) U ( d t ) = I i v g [ 1 e α v g d t ]
d U a b s = I i v g α 0 n g v g d t
δ U a b s = I i α 0 n g τ = α o n n g τ 2 η 0 F i 2
m ( E ) = [ 2 3 a 2 2 b 2 ( E + a ) + 1 3 a 4 2 b 2 ( E + a ) 3 ] 1
D O S ( E ) = b 3 π 2 a 3 ( E + a ) 2 a 2 ( E + a )
γ ( t ) = { 1 t < 0 e t / τ e l t 0 .