## Conditions for admittance-matched tunneling through symmetric metal-dielectric stacks |

Optics Express, Vol. 20, Issue S5, pp. A578-A588 (2012)

http://dx.doi.org/10.1364/OE.20.00A578

Acrobat PDF (882 KB)

### 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.

© 2012 OSA

## 1. Introduction and background

1. J. C. Fan and F. J. Bachner, “Transparent heat mirrors for solar-energy applications,” Appl. Opt. **15**(4), 1012–1017 (1976). [CrossRef] [PubMed]

2. G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films **306**(1), 92–99 (1997). [CrossRef]

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

4. X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films **441**(1-2), 200–206 (2003). [CrossRef]

*et al*. [5

5. M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF_{2} photonic band gaps,” Appl. Phys. Lett. **72**(14), 1676–1678 (1998). [CrossRef]

6. 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]

7. 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]

9. 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]

10. M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express **15**(19), 11959–11970 (2007). [CrossRef] [PubMed]

12. E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A **79**(3), 033846 (2009). [CrossRef]

1. J. C. Fan and F. J. Bachner, “Transparent heat mirrors for solar-energy applications,” Appl. Opt. **15**(4), 1012–1017 (1976). [CrossRef] [PubMed]

4. X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films **441**(1-2), 200–206 (2003). [CrossRef]

6. 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]

7. 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]

5. M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF_{2} photonic band gaps,” Appl. Phys. Lett. **72**(14), 1676–1678 (1998). [CrossRef]

11. M. J. Bloemer, G. D’Aguanno, M. Scalora, N. Mattiucci, and D. de Ceglia, “Energy considerations for a superlens based on metal/dielectric multilayers,” Opt. Express **16**(23), 19342–19353 (2008). [CrossRef] [PubMed]

2. G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films **306**(1), 92–99 (1997). [CrossRef]

5. M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF_{2} photonic band gaps,” Appl. Phys. Lett. **72**(14), 1676–1678 (1998). [CrossRef]

11. M. J. Bloemer, G. D’Aguanno, M. Scalora, N. Mattiucci, and D. de Ceglia, “Energy considerations for a superlens based on metal/dielectric multilayers,” Opt. Express **16**(23), 19342–19353 (2008). [CrossRef] [PubMed]

7. 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]

13. S. Feng, J. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

14. P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. **47**(3), 230–239 (1957). [CrossRef]

16. P. W. Baumeister, “Radiant power flow and absorptance in thin films,” Appl. Opt. **8**(2), 423–436 (1969). [CrossRef] [PubMed]

_{2} photonic band gaps,” Appl. Phys. Lett. **72**(14), 1676–1678 (1998). [CrossRef]

17. T. W. Allen and R. G. DeCorby, “Assessing the maximum transmittance of periodic metal-dielectric multi-layers,” J. Opt. Soc. Am. B **28**(10), 2529–2536 (2011). [CrossRef]

*T*) and minimizing reflectance (

*R*) in such structures.

## 2. Admittance matching for minimum effective absorbance of a metal film

14. P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. **47**(3), 230–239 (1957). [CrossRef]

*PT*) that, for a sufficiently thin film, is much greater than the maximum transmittance suggested by the bulk optical absorption coefficient. To achieve

_{MAX}*PT*at a given wavelength (which Berning and Turner termed as ‘inducing transmission’ [14

_{MAX}14. P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. **47**(3), 230–239 (1957). [CrossRef]

*T*=

*PT*occurs when

_{MAX}*R*= 0 for both left and right incidence [17

17. T. W. Allen and R. G. DeCorby, “Assessing the maximum transmittance of periodic metal-dielectric multi-layers,” J. Opt. Soc. Am. B **28**(10), 2529–2536 (2011). [CrossRef]

*N*=

_{m}*n*-i

_{m}*κ*) embedded within an arbitrary assembly of otherwise lossless layers, as shown in Fig. 1(a) . The potential transmittance (

_{m}*PT*=

*T*/(1-

*R*)) of the film depends on the properties (i.e. thickness and index) of the film itself and on the optical admittance presented by the exit assembly (

*Y*=

_{out}*H*/

_{out}*E*), which determines the ratio of the magnetic to electric field at the output interface of the absorbing layer. For a given incident angle and state of polarization,

_{out}*PT*is dependent on the properties of the absorbing film only, and can be calculated using the expressions provided previously [17

_{MAX}17. T. W. Allen and R. G. DeCorby, “Assessing the maximum transmittance of periodic metal-dielectric multi-layers,” J. Opt. Soc. Am. B **28**(10), 2529–2536 (2011). [CrossRef]

*α*

_{min}) for an Ag thin film to the bulk optical absorption coefficient (

*α*= 4π

_{m}*κ*/

_{m}*λ*) for Ag. Optical constants of Ag were modeled using the Lorentz-Drude expressions provided by Rakic

*et al.*[18

18. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. **37**(22), 5271–5283 (1998). [CrossRef] [PubMed]

*PT*is the ratio of power flux at the entrance and exit interface of the absorbing film, it follows that

*PT*= exp(-

_{MAX}*α*

_{min}

*d*), or rearranging:

_{m}*α*

_{min}versus Ag film thickness at a wavelength of 550 nm and for normal incidence. Remarkably, for a 10-nm-thick Ag film the minimum effective absorption coefficient is 2 orders of magnitude lower than the bulk absorption coefficient. Note that a multilayer containing an arbitrary number of 10-nm-thick Ag films can have absorbance embodied by this same

*α*, provided that the Ag films are separated by appropriate dielectric layers to produce an optimal admittance match. Band-limited admittance matching is the reason for the surprisingly high transparency of MD stacks containing many skin depths of metal [5

_{min}_{2} photonic band gaps,” Appl. Phys. Lett. **72**(14), 1676–1678 (1998). [CrossRef]

*N*=

_{m}*n*-i

_{m}*κ*remains valid for describing the optical properties of the thin film. As is well known, very thin metal films can exhibit optical properties that deviate from bulk values, such as a higher effective extinction coefficient arising from electron scattering at grain boundaries. Furthermore, quantum confinement effects cannot be neglected for length scales less than ~10 nm [19

_{m}19. W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express **18**(5), 5124–5134 (2010). [CrossRef] [PubMed]

19. W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express **18**(5), 5124–5134 (2010). [CrossRef] [PubMed]

20. 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 Films **517**(8), 2736–2741 (2009). [CrossRef]

*PT*=

*PT*requires that the exit admittance

_{MAX}*Y*is set to a specific, optimal value

_{out}*Y*=

_{op}*X*+ i

_{op}*Z*. In the following, all admittances are expressed in free-space units (i.e. normalized to the admittance of free space). Closed-form expressions for

_{op}*X*and

_{op}*Z*at normal-incidence are provided by Macleod [15]. Those expressions can be generalized for non-normal incidence as follows:

_{op}*η*and

_{R}*η*are the real and imaginary parts of the ‘tilted’ optical admittance [15] of the metal layer, which is unique for TE and TM polarization:where

_{I}*θ*is the complex angle in the metal layer. For consistency with Macleod [15], the effective phase thickness of the metal layer is expressed:Note that the last expression differs from the notation and convention (

_{m}*μ*=

_{c}*μ*+ i

_{r}*μ*) used for the effective phase thickness of the metal layer in our previous work [17

_{i}**28**(10), 2529–2536 (2011). [CrossRef]

*PT*=

*PT*occurs when

_{MAX}*Y*=

_{out}*Y*, whereas

_{op}*T*=

*PT*(and

_{MAX}*R*= 0) occurs when

*Y*=

_{in}= Y_{out}*Y*. Thus, our goal can be restated as follows: given a periodic DMD multilayer, we seek to identify conditions for attaining

_{op}*Y*=

_{in}= Y_{out}*Y*.

_{op}*X*and

_{op}*Z*versus Ag film thickness, at a wavelength of 550 nm and for normal incidence. For reasons that will be explained in the following section, the ratio

_{op}*Z*/

_{op}*X*is also plotted. As expected [15],

_{op}*Y*tends toward

_{op}*N** =

_{m}*n*+ i

_{m}*κ*in the limit of a thick metal film (for example, the Rakic model predicts

_{m}*N*= 0.1342-i3.1688 for Ag at 550 nm wavelength, as indicated by the dashed lines in the figure). Figure 2(b) plots the same quantities versus the normalized transverse wave vector, for a 30 nm thick Ag film at the same wavelength and for both TE and TM polarization. For a fixed metal film thickness, the optimal admittance (and the ratio

_{m}*Z*/

_{op}*X*) shows relatively little variation versus the transverse wave vector (i.e. angle).

_{op}## 3. Admittance matching conditions for a periodic DMD multilayer

**28**(10), 2529–2536 (2011). [CrossRef]

*λ*,

*n*

_{1},

*d*

_{1},

*N*,

_{m}*d*,

_{m}*n*

_{2}, and

*θ*

_{2}) that maximize transmittance (i.e. that produce

*T*=

*PT*) for the unit cell also correspond to conditions for maximum transmittance through the periodic multilayer based on that unit cell.

_{MAX}*n*

_{1}) on an infinitely thick substrate (with real index

*n*

_{2}) [15]:where

*η*

_{1}and

*η*

_{2}are the tilted optical admittances of the dielectric layer and the ambient medium, respectively. For TE polarization

*η*=

_{i}*n*cos

_{i}*θ*and for TM polarization

_{i}*η*=

_{i}*n*/cos

_{i}*θ*, where

_{i}*n*and

_{i}*θ*are the refractive index and the propagation angle (from Snell’s law) in medium

_{i}*i*. Furthermore,

*δ*

_{1}= (2π/

*λ*)

*n*

_{1}

*d*

_{1}cos

*θ*

_{1}is the phase thickness of the dielectric film. Equating the real and imaginary parts of Eq. (5) to

*X*and

_{op}*Z*, respectively, and after some algebraic manipulation, the following admittance matching equation results:The modifier (+/−) on the cosine term arises because the argument of the arcsine can correspond to an angle in one of two possible quadrants. For a given metal layer (i.e. for a given

_{op}*X*and

_{op}*Z*), the equation predicts that admittance matching (when possible) is dependent on the values of

_{op}*η*

_{1}and

*η*

_{2}only. However, given a solution to Eq. (6), the required thickness for the dielectric layer

*n*

_{1}is fixed by the same set of equations as follows:where

*n*

_{1},

*θ*

_{1}

*,*

_{,m}*η*

_{1}

*and*

_{,m}*η*

_{2}

*are particular values that resulted in a solution to Eq. (6). Although not explicitly indicated in Eq. (7), care must be taken to ensure the angle of the arcsine is taken from the same quadrant that produced the solution to Eq. (6). Also note that Eq. (7) corresponds specifically to the minimum thickness that enables the admittance match. At normal incidence, for example, any value*

_{,m}*d*

_{1}=

*d*

_{1}

*+ q(*

_{,m}*λ*

_{1}/2), where q is an integer and

*λ*

_{1}=

*λ*/

*n*

_{1}, will also produce an admittance match [17

**28**(10), 2529–2536 (2011). [CrossRef]

*d*

_{1}. As illustrated below, Eqs. (6) and (7) can be applied to a variety of tunneling problems, including tunneling of propagating waves (i.e. real angles in both the dielectric and ambient media) and tunneling of evanescent waves (i.e. with

*n*

_{2}>

*n*

_{1}and complex angle

*θ*

_{1}).

*X*and

_{op}*Z*are real numbers by definition, and that

_{op}*η*

_{2}is a real number for all cases considered below (i.e. lossless ambient media with real incident angle). Thus, in all cases for which

*η*

_{1}is purely real (i.e. for non-evanescent wave solutions in dielectric layers

*n*

_{1}), solutions to Eq. (6) are restricted to the following range:For a given metal layer, Eq. (8) places a lower limit on the ratio

*η*

_{1}/

*η*

_{2}, below which solutions to Eq. (6) are not possible. The ratio

*Z*/

_{op}*X*is typically high (see Fig. 2), and diverges for increasing Ag film thickness. Thus, an admittance match is typically reliant on high values of

_{op}*η*

_{1}/

*η*

_{2}, especially for large metal thickness.

### 3.1. Normal incidence in air

*n*

_{2}= 1). For normal incidence, the admittance of a medium in free-space units is simply equal to its refractive index. From the preceding discussion, solutions to Eq. (6) are possible provided (

*n*

_{1}- 1/

*n*

_{1}) > 2

*Z*/

_{op}*X*. Given typical values of

_{op}*X*and

_{op}*Z*for a thin Ag film in the visible range (see Fig. 2), this implies that high values of dielectric index (

_{op}*n*

_{1}> 4) are necessary to achieve a perfect admittance match. Moreover, with fixed

*n*

_{2}and for a given metal layer at a given wavelength, only a specific value of

*n*

_{1}results in a solution to Eq. (6). This observation was made previously [17

**28**(10), 2529–2536 (2011). [CrossRef]

**28**(10), 2529–2536 (2011). [CrossRef]

*T*,

*PT*, and

_{MAX}*R*for a specific admittance-matched case (

*λ*= 550 nm, and assuming a unit cell with

*d*= 25 nm,

_{m}*n*

_{1}= 4.732 and

*d*

_{1}= 17.5 nm) indicated by the symbols in Figs. 4(a) and 4(b).

*T*and

*R*were calculated using a transfer matrix technique, and

*PT*was calculated using previously described expressions [17

_{MAX}**28**(10), 2529–2536 (2011). [CrossRef]

*T*=

*PT*and

_{MAX}*R*= 0) is verified at

*λ*= 550 nm. This perfect admittance match holds, in principle, for a DMD multilayer comprising an arbitrary number of such unit cells. To illustrate this, data for both 1- and 20-period cases are shown.

### 3.2. Admittance-matched tunneling of TE-polarized propagating waves

*n*

_{1}>

*n*

_{2}, such as the air-ambient case in the previous section. From above, admittance matching requires a sufficiently high value of the ratio

*η*

_{1}/

*η*

_{2}. Interestingly, for a given

*n*

_{2}and

*n*

_{1}>

*n*

_{2}, this ratio increases with increasing incident angle (

*θ*

_{2}) for TE polarization, but decreases with increasing incident angle for TM polarization. This implies that when

*n*

_{1}>

*n*

_{2}, solutions to Eq. (6) are possible for off-normal incidence of TE-polarized light only. Tunneling of this kind was described by Hooper

*et al.*[7

**97**(5), 053902 (2006). [CrossRef] [PubMed]

*n*

_{1}= 2.3 and

*n*

_{2}= 1, with results at two different wavelengths plotted in Figs. 5(a) and 5(b). For a given wavelength and Ag film thickness, admittance-matched tunneling of TE-polarized light is predicted at a specific incident angle. To further illustrate, a specific data point from these solutions (

*λ*= 550 nm,

*d*= 25 nm,

_{m}*d*

_{1}= 53.71 nm) was used to generate plots of

*T*and

*R*versus incident angle, shown in Figs. 5(c) and 5(d). As expected, admittance-matched tunneling occurs at the incident angle of 75.01 degrees indicated by the symbol in Fig. 5(a). As for the normal-incidence case, this admittance match extends to a multilayer with arbitrary number of unit cells. The 1- and 10-period cases are shown as examples.

*n*

_{2}; example data is shown for

*n*

_{2}= 1.5 in Fig. 5(a), and could represent a DMD stack symmetrically embedded between glass plates. Note, however, that the solutions lie at even higher incident angles in this case. For propagating waves in an external air medium to access these tunneling angles, coupling prisms [7

**97**(5), 053902 (2006). [CrossRef] [PubMed]

### 3.3. Admittance-matched tunneling of TM-polarized evanescent waves

*et al.*[21

21. 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]

*et al.*[6

6. 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]

*et al.*[22

22. S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express **16**(13), 9942–9950 (2008). [CrossRef] [PubMed]

*et al.*[13

13. S. Feng, J. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

22. S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express **16**(13), 9942–9950 (2008). [CrossRef] [PubMed]

*n*

_{2}>

*n*

_{1}, which might represent a periodic DMD multilayer coupled at its entrance and exit faces by high index prisms. This geometry allows plane waves in the ambient media to couple with evanescent waves in the dielectric layers, thereby enabling a straightforward analysis of the Poynting power flow associated with surface-plasmon-mediated tunneling [13

13. S. Feng, J. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

**28**(10), 2529–2536 (2011). [CrossRef]

*θ*

_{2}>

*θ*= sin

_{C}^{−1}(

*n*

_{1}/

*n*

_{2}), where

*θ*is the critical angle for total internal reflection. Assuming

_{C}*n*

_{1}and

*n*

_{2}are real, then

*η*

_{1}and sin

*δ*

_{1}are both purely imaginary (i.e. when

*θ*

_{2}>

*θ*), so that the admittance expressed by Eq. (5) has the same general form for evanescent waves as it had for the propagating wave cases discussed above. This means that, assuming lossless dielectrics, Eqs. (6) and (7) are valid for tunneling of both evanescent and propagating waves. When

_{C}*n*

_{2}>

*n*

_{1}, however,

*η*

_{1}/

*η*

_{2}increases for TM waves and decreases for TE waves (i.e. as the incident angle is increased). Thus, opposite to the situation described in the previous section, here solutions to Eq. (6) are expected for TM-polarized waves only.

*λ*,

*d*,

_{m}*n*

_{1},

*n*

_{2}). However, typically two solutions were found in the evanescent range for a given Ag film thickness and wavelength, as shown by the representative data in Figs. 6(a) and 6(b). Note that the transverse wave vector is defined by

*k*= (2

_{t}*π*/

*λ*)

*n*

_{2}sin

*θ*

_{2}, and that waves in the

*n*

_{1}layers are evanescent when (

*k*/

_{t}*k*) >

_{0}*n*

_{1}. We assumed

*n*

_{1}= 1.631 and a fictitious coupling medium with

*n*

_{2}= 4, to enable comparison with similar structures studied by Feng

*et al.*[13

**13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

*λ*and

*d*(such as for

_{m}*λ*= 550 nm and

*d*< ~42 nm in Fig. 6(a)), no solutions are found.

_{m}*T*and

*R*were plotted in Figs. 6(c) and 6(d) for a particular admittance-matched case (

*λ*= 500 nm,

*d*= 50 nm). Consistent with the solutions in Figs. 6(a) and 6(b), dielectric thicknesses

_{m}*d*

_{1}= 12.77 nm and 46.07 nm result in a perfect admittance match (

*T*=

*PT*and

_{MAX}*R*= 0) at normalized wave vector values of 2.18 and 3.61, respectively. Other values of

*d*

_{1}result in a partial tunneling peak at a different angle, but without a perfect admittance match. The data for

*d*

_{1}= 20 nm is shown as an example.

*n*

_{2}= 4 are not practical. Figure 7 shows results for a more practical combination of indices,

*n*

_{1}= 1.38 and

*n*

_{2}= 1.515, representing the MgF

_{2}-based tunneling structures studied by Dragila

*et al.*[21

21. 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]

10. M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express **15**(19), 11959–11970 (2007). [CrossRef] [PubMed]

12. E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A **79**(3), 033846 (2009). [CrossRef]

22. S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express **16**(13), 9942–9950 (2008). [CrossRef] [PubMed]

23. E. Ray and R. Lopez, “Numerical design and scattering losses of a one-dimensional metallo-dielectric multilayer with broadband coupling of propagating waves to plasmon modes in the visible range,” J. Opt. Soc. Am. B **28**(7), 1778–1781 (2011). [CrossRef]

## 4. Summary and conclusions

**47**(3), 230–239 (1957). [CrossRef]

16. P. W. Baumeister, “Radiant power flow and absorptance in thin films,” Appl. Opt. **8**(2), 423–436 (1969). [CrossRef] [PubMed]

## Acknowledgment

## References and links

1. | J. C. Fan and F. J. Bachner, “Transparent heat mirrors for solar-energy applications,” Appl. Opt. |

2. | G. Leftheriotis, P. Yianoulis, and D. Patrikios, “Deposition and optical properties of optimized ZnS/Ag/ZnS thin films for energy saving applications,” Thin Solid Films |

3. | C. G. Granqvist, “Transparent conductors for solar energy and energy efficiency: a broad-brush picture,” Int. J. Nanotechnol. |

4. | X. Liu, X. Cai, J. Qiao, J. Mao, and N. Jiang, “The design of ZnS/Ag/ZnS transparent conductive multilayer films,” Thin Solid Films |

5. | M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF |

6. | S. Hayashi, H. Kurokawa, and H. Oga, “Observation of resonant photon tunneling in photonic double barrier structures,” Opt. Rev. |

7. | I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. |

8. | S. Anantha Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. |

9. | Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. |

10. | M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express |

11. | M. J. Bloemer, G. D’Aguanno, M. Scalora, N. Mattiucci, and D. de Ceglia, “Energy considerations for a superlens based on metal/dielectric multilayers,” Opt. Express |

12. | E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A |

13. | S. Feng, J. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express |

14. | P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am. |

15. | H. A. Macleod, |

16. | P. W. Baumeister, “Radiant power flow and absorptance in thin films,” Appl. Opt. |

17. | T. W. Allen and R. G. DeCorby, “Assessing the maximum transmittance of periodic metal-dielectric multi-layers,” J. Opt. Soc. Am. B |

18. | A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. |

19. | W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express |

20. | 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 Films |

21. | R. Dragila, B. Luther-Davies, and S. Vukovic, “High transparency of classically opaque metallic films,” Phys. Rev. Lett. |

22. | S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express |

23. | E. Ray and R. Lopez, “Numerical design and scattering losses of a one-dimensional metallo-dielectric multilayer with broadband coupling of propagating waves to plasmon modes in the visible range,” J. Opt. Soc. Am. B |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(310.7005) Thin films : Transparent conductive coatings

**ToC Category:**

Thin Films

**History**

Original Manuscript: April 30, 2012

Revised Manuscript: June 14, 2012

Manuscript Accepted: June 27, 2012

Published: July 5, 2012

**Citation**

T.W. Allen and R.G. DeCorby, "Conditions for admittance-matched tunneling through symmetric metal-dielectric stacks," Opt. Express **20**, A578-A588 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-S5-A578

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### References

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- M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Opt. Express15(19), 11959–11970 (2007). [CrossRef] [PubMed]
- M. J. Bloemer, G. D’Aguanno, M. Scalora, N. Mattiucci, and D. de Ceglia, “Energy considerations for a superlens based on metal/dielectric multilayers,” Opt. Express16(23), 19342–19353 (2008). [CrossRef] [PubMed]
- E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009). [CrossRef]
- S. Feng, J. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express13(11), 4113–4124 (2005). [CrossRef] [PubMed]
- P. H. Berning and A. F. Turner, “Induced transmission in absorbing films applied to band pass filter design,” J. Opt. Soc. Am.47(3), 230–239 (1957). [CrossRef]
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- A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37(22), 5271–5283 (1998). [CrossRef] [PubMed]
- W. Chen, M. D. Thoreson, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin ultra-smooth and low-loss silver films on a germanium wetting layer,” Opt. Express18(5), 5124–5134 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- S. Tomita, T. Yokoyama, H. Yanagi, B. Wood, J. B. Pendry, M. Fujii, and S. Hayashi, “Resonant photon tunneling via surface plasmon polaritons through one-dimensional metal-dielectric metamaterials,” Opt. Express16(13), 9942–9950 (2008). [CrossRef] [PubMed]
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