Abstract

We used the theory of potential transmittance to derive a general expression for reflection-less tunneling through a periodic stack with a dielectric-metal-dielectric unit cell. For normal-incidence from air, the theory shows that only a specific (and typically impractically large) dielectric index can enable a perfect admittance match. For off-normal incidence of TE-polarized light, an admittance match is possible at a specific angle that depends on the index of the ambient and dielectric media and the thickness and index of the metal. For TM-polarized light, admittance matching is possible within the evanescent-wave range (i.e. for tunneling mediated by surface plasmons). The results provide insight for research on transparent metals and superlenses.

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References

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2011 (2)

2010 (1)

2009 (3)

D. Owens, C. Fuentes-Hernandez, and B. Kippelen, “Optical properties of one-dimensional metal-dielectric photonic band-gap structures with low-index dielectrics,” Thin Solid Films517(8), 2736–2741 (2009).
[CrossRef]

E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009).
[CrossRef]

C. G. Granqvist, “Transparent conductors for solar energy and energy efficiency: a broad-brush picture,” Int. J. Nanotechnol.6(9), 785–797 (2009).
[CrossRef]

2008 (2)

2007 (2)

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express15(19), 11959–11970 (2007).
[CrossRef] [PubMed]

2006 (1)

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (2)

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50, 1419–1430 (2003).

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

1999 (1)

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev.6(3), 204–210 (1999).
[CrossRef]

1998 (2)

1997 (1)

G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films306(1), 92–99 (1997).
[CrossRef]

1985 (1)

R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett.55(10), 1117–1120 (1985).
[CrossRef] [PubMed]

1976 (1)

1969 (1)

1957 (1)

Allen, T. W.

Anantha Ramakrishna, S.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50, 1419–1430 (2003).

Bachner, F. J.

Baumeister, P. W.

Berning, P. H.

Bloemer, M. J.

Cai, X.

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

Chen, W.

D’Aguanno, G.

de Ceglia, D.

DeCorby, R. G.

Djurisic, A. B.

Dragila, R.

R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett.55(10), 1117–1120 (1985).
[CrossRef] [PubMed]

Elazar, J. M.

Elson, J.

Fan, J. C.

Feng, S.

Fourkal, E.

E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009).
[CrossRef]

Fuentes-Hernandez, C.

D. Owens, C. Fuentes-Hernandez, and B. Kippelen, “Optical properties of one-dimensional metal-dielectric photonic band-gap structures with low-index dielectrics,” Thin Solid Films517(8), 2736–2741 (2009).
[CrossRef]

Fujii, M.

Granqvist, C. G.

C. G. Granqvist, “Transparent conductors for solar energy and energy efficiency: a broad-brush picture,” Int. J. Nanotechnol.6(9), 785–797 (2009).
[CrossRef]

Hayashi, S.

Hooper, I. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Ishii, S.

Jiang, N.

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

Kildishev, A. V.

Kippelen, B.

D. Owens, C. Fuentes-Hernandez, and B. Kippelen, “Optical properties of one-dimensional metal-dielectric photonic band-gap structures with low-index dielectrics,” Thin Solid Films517(8), 2736–2741 (2009).
[CrossRef]

Kurokawa, H.

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev.6(3), 204–210 (1999).
[CrossRef]

Leftheriotis, G.

G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films306(1), 92–99 (1997).
[CrossRef]

Liu, X.

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

Liu, Z.

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Lopez, R.

Luther-Davies, B.

R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett.55(10), 1117–1120 (1985).
[CrossRef] [PubMed]

Majewski, M. L.

Mao, J.

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

Mattiucci, N.

Oga, H.

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev.6(3), 204–210 (1999).
[CrossRef]

Overfelt, P. L.

Owens, D.

D. Owens, C. Fuentes-Hernandez, and B. Kippelen, “Optical properties of one-dimensional metal-dielectric photonic band-gap structures with low-index dielectrics,” Thin Solid Films517(8), 2736–2741 (2009).
[CrossRef]

Patrikios, D.

G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films306(1), 92–99 (1997).
[CrossRef]

Pendry, J. B.

Preist, T. W.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Psaltis, D.

Qiao, J.

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

Rakic, A. D.

Ray, E.

Sambles, J. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett.97(5), 053902 (2006).
[CrossRef] [PubMed]

Scalora, M.

Shalaev, V. M.

Smolyakov, A.

E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009).
[CrossRef]

Stewart, W. J.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50, 1419–1430 (2003).

Sun, C.

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Thoreson, M. D.

Tomita, S.

Tsang, M.

Turner, A. F.

Velchev, I.

E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009).
[CrossRef]

Vukovic, S.

R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett.55(10), 1117–1120 (1985).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50, 1419–1430 (2003).

Wood, B.

Xiong, Y.

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Yanagi, H.

Yianoulis, P.

G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films306(1), 92–99 (1997).
[CrossRef]

Yokoyama, T.

Zhang, X.

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett.72(14), 1676–1678 (1998).
[CrossRef]

Int. J. Nanotechnol. (1)

C. G. Granqvist, “Transparent conductors for solar energy and energy efficiency: a broad-brush picture,” Int. J. Nanotechnol.6(9), 785–797 (2009).
[CrossRef]

J. Mod. Opt. (1)

S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt.50, 1419–1430 (2003).

J. Opt. Soc. Am. (1)

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

Nano Lett. (1)

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett.7(11), 3360–3365 (2007).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Rev. (1)

S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev.6(3), 204–210 (1999).
[CrossRef]

Phys. Rev. A (1)

E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009).
[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(5), 053902 (2006).
[CrossRef] [PubMed]

R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett.55(10), 1117–1120 (1985).
[CrossRef] [PubMed]

Thin Solid Films (3)

D. Owens, C. Fuentes-Hernandez, and B. Kippelen, “Optical properties of one-dimensional metal-dielectric photonic band-gap structures with low-index dielectrics,” Thin Solid Films517(8), 2736–2741 (2009).
[CrossRef]

X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films441(1-2), 200–206 (2003).
[CrossRef]

G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films306(1), 92–99 (1997).
[CrossRef]

Other (1)

H. A. Macleod, Thin-Film Optical Filters, Third Edition (Institute of Physics Publishing, 2001).

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic showing a multilayer containing one absorbing layer. The optical admittance viewed from the perspective of the absorbing layer, looking towards the incidence and exit directions, is labeled as Yin and Yout, respectively. (b) The optimal (i.e. minimum) optical absorption coefficient, as defined in the text, is plotted versus Ag film thickness at a wavelength of 550 nm and for normal incidence. The bulk absorption coefficient for Ag at the same wavelength is represented by the red dotted line.

Fig. 2
Fig. 2

(a) Components of the exit admittance (Xop and Zop) that produce PT = PTMAX for normal incidence are plotted versus Ag film thickness, at a wavelength of 550 nm. The green curve shows the ratio Zop/Xop. The dashed curves show the real (blue) and imaginary (red) parts of the Ag refractive index at the same wavelength. (b) The same quantities are plotted versus normalized transverse wave vector, for a 30-nm thick Ag film at 550-nm wavelength. The dashed and solid curves correspond to TM and TE polarization, respectively.

Fig. 3
Fig. 3

(a) Geometry for plane wave incidence on a symmetric DMD unit cell embedded between identical incidence and exit media. The admittance viewed from the perspective of the absorbing metal film (Yout) is that for a thin film (n1) on a substrate (n2). (b) Schematic illustration of a periodic MD stack composed of DMD unit cells. Conditions (i.e. n1, d1, Nm, dm, n2, θ2) that admittance match the unit cell also result in an admittance matched multilayer.

Fig. 4
Fig. 4

(a) The dielectric index required for admittance-matched tunneling of normally incident waves through a periodic DMD stack (with Ag metal layers and air ambient) is plotted for three different wavelengths. The symbol indicates the data point used in subsequent examples. (b) The minimum dielectric layers thickness that results in admittance-matched tunneling when combined with the dielectric indices in part (a). (c) Predicted transmittance for a 1-period (blue solid line) and 20-period (blue dashed line) DMD multilayer with dm = 25 nm, n1 = 4.732 and d1 = 17.5 nm (as indicated by the symbols in parts (a) and (b)). The red dashed lines indicate PTMAX for each case. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges at 550 nm, indicating a perfect admittance match for any number of periods.

Fig. 5
Fig. 5

(a) Incident angle that results in admittance-matched tunneling of TE waves is plotted versus Ag film thickness, for n1 = 2.3 and n2 = 1, and at two different wavelengths. The dotted curve plots the tunneling angle for λ = 550 nm, n1 = 2.3, and n2 = 1.5. The symbol indicates the data point used in subsequent examples. (b) The minimum dielectric layers thickness that results in admittance-matched tunneling when combined with the indices and tunneling angles from part (a). (c) Predicted transmittance versus incident angle for a 1-period (blue solid line) and 10-period (blue dashed line) DMD multilayer with n1 = 2.3, n2 = 1, λ = 550 nm, dm = 25 nm, and d1 = 53.7 nm (as indicated by the symbols in parts (a) and (b)). The red dashed lines indicate PTMAX for each case. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges at 75.01 degrees, indicating a perfect admittance match.

Fig. 6
Fig. 6

(a) Normalized transverse wave vectors that result in admittance-matched tunneling of TM waves is plotted versus Ag film thickness, for n1 = 1.631, n2 = 4, and two different wavelengths (λ = 500 nm and λ = 550 nm). The symbols indicate data points used in subsequent examples. (b) The minimum dielectric layers thicknesses that result in admittance-matched tunneling when combined with the indices and tunneling angles from part (a). (c) Predicted transmittance for a 1-period DMD multilayer for λ = 500 nm, dm = 50 nm, n1 = 1.631, n2 = 4, and various d1 as indicated by the labels. Note that d1 values corresponding to the data points indicated by the symbols in part (b) result in admittance-matched tunneling at the transverse wave vector values indicated by the corresponding symbols in part (a). The red dashed line indicates PTMAX. (d) Predicted reflectance for the structures in part (c). Note that the reflectance diverges for the two cases that produce a perfect admittance match.

Fig. 7
Fig. 7

As in Fig. 6, except with n2 = 1.515, n1 = 1.38, and for a 632.8 nm wavelength. (a) Incident angle in external medium (θ2) that produces a perfect admittance match. (b) Dielectric layers thickness that produces admittance matched tunneling when combined with the incidence angles from part (a). (c) Transmittance versus incidence angle for a 1 period DMD structure, with λ = 632.8 nm, dm = 25 nm, and d1 set to the values indicated by the symbols in part (b), verifying a tunneling peak at the corresponding angles from part (a). (d) Reflectance versus incidence angle for the same conditions as in part (c).

Equations (8)

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α min = ln( P T MAX ) / d m .
X op = ( ( η R 2 + η I 2 )( η R sinhβcoshβ+ η I sinαcosα ) ( η R sinhβcoshβ η I sinαcosα ) η R 2 η I 2 ( sin 2 α cosh 2 β+ cos 2 α sinh 2 β ) 2 ( η R sinhβcoshβ η I sinαcosα ) 2 ) 1 2 . Z op = η R η I ( sin 2 α cosh 2 β+ cos 2 α sinh 2 β ) ( η R sinhβcoshβ η I sinαcosα )
η m = η R i η I ={ ( n m i κ m )cos θ m [ TE ] ( n m i κ m ) / cos θ m [TM] ,
δ m =αiβ=( 2π λ )( n m i κ m ) d m cos θ m .
Y out = η 2 cos δ 1 +i η 1 sin δ 1 cos δ 1 +i( η 2 / η 1 )sin δ 1 ,
X op = 2 η 2 ±cos[ sin 1 { 2 Z op X op ( η 1 / η 2 η 2 / η 1 ) } ]( 1 η 2 2 η 1 2 )+( 1+ η 2 2 η 1 2 ) .
d 1,m =( λ 4π n 1 cos θ 1,m ) sin 1 { 2 Z op X op ( η 1,m / η 2,m η 2,m / η 1,m ) },
( η 1 η 2 η 2 η 1 ) 2 Z op X op .

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