Abstract

Transverse-electric (TE) resonant optical tunneling through an asymmetric, single-barrier potential system consisting of all passive materials in two-dimensional (2-D) glass/silver/TiO2/air configuration is quantified at a silver thickness of 35 nm. Resonant tunneling occurs when the incident condition corresponds to the excitation of a radiation mode. Lasing-like transmission occurring at resonance is carefully qualified in terms of power conservation, resonance condition, and identification of the gain medium equivalent. In particular, effective gain (geff) and threshold gain (gth) coefficients, both of which are strong functions of the forward reflection coefficient at the silver-TiO2 interface, are analytically obtained and the angular span over which geff > gth is further verified rigorously electromagnetically. The results show that the present configuration may be treated as a cascade of the gain equivalent (i.e. the silver film) and the TiO2 resonator that is of Fabry-Perot type, giving rise to negative gth when resonant tunneling occurs. The transmittance spectrum exhibiting a gain-curve-like envelope is shown to be a direct consequence of the competition of the resonator loss at the silver-TiO2 interface and the forward tunneling probability through the silver barrier, all controlled by the effective silver barrier thickness.

© 2014 Optical Society of America

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References

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2014 (1)

2013 (2)

Y.-J. Chang and C.-S. Lai, “Toward maximum transmittance into absorption layers in solar cells: Investigation of lossy-film-induced mismatches between reflectance and transmittance extrema,” Opt. Lett. 38, 3257–3260 (2013).
[Crossref] [PubMed]

A. Q. Jian and X. M. Zhang, “Resonant optical tunneling effect: recent progress in modeling and applications,” IEEE J. Sel. Topics Quantum Electron. 19, 9000310 (2013).
[Crossref]

2011 (2)

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).

2006 (1)

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[Crossref] [PubMed]

2005 (3)

S. Longhi, “Resonant tunneling in frustrated total internal reflection,” Opt. Lett. 30, 2781–2783 (2005).
[Crossref] [PubMed]

N. Yamamoto, K. Akahane, and S.-I. Gozu, “All-optical control of the resonant-photon tunneling effect observed in GaAs/AlGaAs multilayered structures containing quantum dots,” Appl. Phys. Lett. 87, 231119 (2005).
[Crossref]

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

2003 (1)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[Crossref]

1999 (1)

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. 6, 204–210 (1999).
[Crossref]

1997 (1)

1991 (1)

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling,” Physica B 175, 257–262 (1991).
[Crossref]

1985 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Akahane, K.

N. Yamamoto, K. Akahane, and S.-I. Gozu, “All-optical control of the resonant-photon tunneling effect observed in GaAs/AlGaAs multilayered structures containing quantum dots,” Appl. Phys. Lett. 87, 231119 (2005).
[Crossref]

Alù, A.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[Crossref]

Castaldi, G.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

Chan, C. T.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

Chang, Y.-J.

Chiao, R. Y.

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling,” Physica B 175, 257–262 (1991).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cojocaru, E.

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).

Engheta, N.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[Crossref]

Galdi, V.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

Gallina, I.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

Gozu, S.-I.

N. Yamamoto, K. Akahane, and S.-I. Gozu, “All-optical control of the resonant-photon tunneling effect observed in GaAs/AlGaAs multilayered structures containing quantum dots,” Appl. Phys. Lett. 87, 231119 (2005).
[Crossref]

Hayashi, S.

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. 6, 204–210 (1999).
[Crossref]

Hooper, I. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[Crossref] [PubMed]

James, G.

G. James, Advanced Modern Engineering Mathematics, 4th ed. (Pearson, 2011).

Jian, A. Q.

A. Q. Jian and X. M. Zhang, “Resonant optical tunneling effect: recent progress in modeling and applications,” IEEE J. Sel. Topics Quantum Electron. 19, 9000310 (2013).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kurokawa, H.

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. 6, 204–210 (1999).
[Crossref]

Kwiat, P. G.

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling,” Physica B 175, 257–262 (1991).
[Crossref]

Lai, C.-S.

Lee, B.

Lee, K.

Longhi, S.

Oga, H.

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. 6, 204–210 (1999).
[Crossref]

Preist, T. W.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[Crossref] [PubMed]

Sambles, J. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[Crossref] [PubMed]

Sheng, P.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

Steinberg, A. M.

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling,” Physica B 175, 257–262 (1991).
[Crossref]

Wen, W.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

Yamamoto, N.

N. Yamamoto, K. Akahane, and S.-I. Gozu, “All-optical control of the resonant-photon tunneling effect observed in GaAs/AlGaAs multilayered structures containing quantum dots,” Appl. Phys. Lett. 87, 231119 (2005).
[Crossref]

Yeh, P.

Zhang, X. M.

A. Q. Jian and X. M. Zhang, “Resonant optical tunneling effect: recent progress in modeling and applications,” IEEE J. Sel. Topics Quantum Electron. 19, 9000310 (2013).
[Crossref]

Zhou, L.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

Appl. Phys. Lett. (1)

N. Yamamoto, K. Akahane, and S.-I. Gozu, “All-optical control of the resonant-photon tunneling effect observed in GaAs/AlGaAs multilayered structures containing quantum dots,” Appl. Phys. Lett. 87, 231119 (2005).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

A. Q. Jian and X. M. Zhang, “Resonant optical tunneling effect: recent progress in modeling and applications,” IEEE J. Sel. Topics Quantum Electron. 19, 9000310 (2013).
[Crossref]

IEEE Trans. Antennas Propag. (1)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (2)

Opt. Rev. (1)

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. 6, 204–210 (1999).
[Crossref]

Phys. Rev. B (2)

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Phys. Rev. Lett. (2)

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[Crossref] [PubMed]

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[Crossref]

Physica B (1)

R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg, “Analogies between electron and photon tunneling,” Physica B 175, 257–262 (1991).
[Crossref]

Prog. Electromagn. Res. (1)

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).

Other (1)

G. James, Advanced Modern Engineering Mathematics, 4th ed. (Pearson, 2011).

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Figures (9)

Fig. 1
Fig. 1 The potential diagram as the quantum mechanical equivalent of asymmetric metal/multi-insulator configuration under TE wave incidence. The geometry under consideration is shown schematically as the inset with εin = 1.5150802 (glass), εm ≈ −18.375568 − 0.502596 j (silver), tm = 35 nm, εco = (2.892819 − 0.000297 j)2 (TiO2), and εout = 1.0.
Fig. 2
Fig. 2 TE transmittance for the configuration in Fig. 1: (a) as a function of incident angle θ and normalized TiO2 slab thickness tcog with λ g = λ 0 / Re [ ε co ] and (b) with a lossless core (εco = 2.8928192) and an ideal metal having εm = −18.375568.
Fig. 3
Fig. 3 Resonant electric and magnetic fields under unit-amplitude incidence (Ey = 1 V/m) with λ0 = 633 nm: (a) field distributions as a function of x position with interface outlines and FEM-based simulations for Ey (real part) (in V/m) (b) and time-average norm power Pave (in W/m2) (c) in the x–z plane. Also shown in (c) is the Poynting box for power conservation calculations.
Fig. 4
Fig. 4 (a) Designation of reflection coefficients seen by right- and left-traveling waves at different interfaces and (b) reflection coefficient (magnitude) experienced by the right-traveling wave at each interface (λ0 = 633 nm and tco = 386.1 nm). Also schematically depicted in (a) are the optical paths involved in the derivation of the threshold gain coefficient.
Fig. 5
Fig. 5 Right- and left-traveling waves associated with the Ey field under unit-amplitude incidence for (a) θ = 41.14° and (b) θ = 20°.
Fig. 6
Fig. 6 Investigations of gain coefficients: (a) time-average power flows at the front (Pave,1) and rear (Pave,2) facets of the silver barrier and (b) effective gain and threshold gain coefficients with varying angles of incidence.
Fig. 7
Fig. 7 Normalized time-average power flow (normalized to the incident power) in the exit region as a verification to effective gain and threshold gain coefficient calculations given in Fig. 6(b).
Fig. 8
Fig. 8 Transmittance T, reflectance R, and absorptance (in silver barrier) AAg spectra of the 2-D glass/silver/TiO2/air configuration at θ = 41.14°.
Fig. 9
Fig. 9 Magnitude of the reflection coefficients | r 2 + | and | r 3 | at the facets of the TiO2 resonator and the transmittance through glass/silver/TiO2 configuration as a function of the effective silver barrier thickness. The incident angle remains at 41.14°.

Tables (1)

Tables Icon

Table 1 FEM-based numerical results for the power conservation check based on the rigorous complex Poynting theorem under unit-amplitude incidence. The 2-D rectangular Poynting box is set to 1421.1 nm in length and 600 nm in height.

Equations (13)

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[ 1 κ in 1 κ co + j 1 κ m 1 κ out tan ( κ out l ) ] [ tan ( κ m t m ) tan ( κ co t co ) 1 ] j [ 1 κ m 1 κ co + j 1 κ in 1 κ out tan ( κ out l ) ] [ tan ( κ m t m ) + tan ( κ co t co ) ] = 0 ,
Z + Z = δ ,
r 2 + r 3 e j 2 κ co t co = 1 1 + 2 δ κ co / ( ω μ 0 ) [ 2 δ κ co / ( ω μ 0 ) 1 + 2 δ κ co / ( ω μ 0 ) ] ( 1 r 2 + e j 2 κ co d 1 r 3 e j 2 κ co d 2 ) ,
( 1 2 ) S = 2 j ω ( 1 4 μ 0 μ | H | 2 1 4 ε 0 ε | E | 2 ) ( 1 2 ω μ 0 μ | H | 2 + 1 2 ω ε 0 ε | E | 2 ) 1 2 E ( J s + J c ) * ,
1 2 C Re [ S ] n ^ d l = S i S i 1 2 ω ε 0 ε i | E | 2 d A i , i = { m , co } ,
1 2 C Im [ S ] n ^ d l = 2 ω S i S i ( μ 0 μ 4 | H | 2 ε 0 ε i 4 | E | 2 ) d A i , i = { in , m , co , out } ,
α m = k 0 [ ( | ε m | + ε in sin 2 θ ) 2 + ε m 2 ] 1 / 4 cos ϕ 2
ϕ = tan 1 [ ε m | ε m | + ε in sin 2 θ ]
| r 2 | = [ R 2 2 + R 0 , Ag 2 2 ( R 2 R 0 , Ag + X 2 X 0 , Ag ) + X 2 2 + X 0 , Ag 2 R 2 2 + R 0 , Ag 2 + 2 ( R 2 R 0 , Ag + X 2 X 0 , Ag ) + X 2 2 + X 0 , Ag 2 ] 1 / 2 ,
X 2 X 0 , Ag < 0 .
P f = P i ( R 1 + R 2 + R 2 R 3 ) e g eff t m e 2 α m t m e 2 α co ( 2 t co ) ,
g th = 2 α m + 1 t m [ 4 α co t co + ln ( 1 R 1 + R 2 + R 2 R 3 ) ] .
g eff = 1 t m ln ( P ave , 2 P ave , 1 ) .

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