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  • Editor: Bernard Kippelen
  • Vol. 20, Iss. S5 — Sep. 10, 2012
  • pp: A578–A588
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Conditions for admittance-matched tunneling through symmetric metal-dielectric stacks

T.W. Allen and R.G. DeCorby  »View Author Affiliations


Optics Express, Vol. 20, Issue S5, pp. A578-A588 (2012)
http://dx.doi.org/10.1364/OE.20.00A578


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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

Tunneling of electromagnetic radiation through metal films has captured the attention of researchers for a variety of reasons. Since the 1970s [1

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

], for example, dielectric-metal-dielectric (DMD) coatings have been studied in the context of solar control and thermal emittance windows [2

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

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]

], and as transparent electrodes for displays [4

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]

]. In the late 1990s, Bloemer et al. [5

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

] showed that periodic MD multilayers can exhibit wide bands of high transparency, and suggested their application as transparent conductors and radiation shields. Others have explored the analogy between optical tunneling and the quantum mechanical tunneling of electrons over potential barriers [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]

,7

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]

]. More recently, metal-dielectric stacks have been studied in the context of ‘superlenses’ [8

8. 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).

,9

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]

], since they support plasmon-mediated tunneling of TM-polarized evanescent waves. Energy transport through such structures remains an active topic for study [10

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

12

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

].

Symmetric DMD tri-layers have played a central role in the aforementioned research [1

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

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

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

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]

], including antireflection-coated MD stacks [5

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

,11

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]

], which are essentially periodic structures with a DMD unit cell. Much of the knowledge about these structures derives from numerical studies which have shown, for example, that the use of high-index dielectric layers tends to enhance transparency [2

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

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

,11

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]

]. Theoretical treatments, on the other hand, have typically relied on simplifications such as the use of a lossless metal assumption [7

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

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]

]. In our view, it has not been widely recognized that these various bodies of work can be unified within the framework of potential transmittance (PT) theory [14

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]

,15

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

]. For example, PT theory explains [16

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

] the observation that transmittance can be increased by sub-dividing a given thickness of metal into some number of thinner, appropriately spaced metal layers [5

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

,8

8. 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).

].

The goal of our work is to assess the limits of tunneling-based transparency, given a periodic DMD multilayer containing real, absorbing metal layers. In a previous, transfer-matrix-based numerical study [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]

], we used PT theory as a general framework for assessing maximum transmittance. Here, we provide an extended theoretical treatment, and derive a general equation that encapsulates the conditions for admittance-matched tunneling. This approach offers new insight into previously disjointed research on tunneling through MD stacks. Moreover, it directly addresses the important problem of maximizing transmittance (T) and minimizing reflectance (R) in such structures.

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

As first explained by Berning and Turner [14

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]

], the transparency of a thin absorbing film is highly dependent on its boundary conditions. In fact, there is a maximum potential transmittance (PTMAX) that, for a sufficiently thin film, is much greater than the maximum transmittance suggested by the bulk optical absorption coefficient. To achieve PTMAX at a given wavelength (which Berning and Turner termed as ‘inducing transmission’ [14

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]

]), one must design the surrounding media so that the exit optical admittance (viewed from the perspective of the absorbing film) attains a particular value determined by the optical constants and thickness of the film [15

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

]. T = PTMAX occurs when 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]

], which illustrates that inducing transmission is essentially equivalent to admittance matching an absorbing film to the ambient media.

Consider a single absorbing film (with index Nm = nm-iκm) embedded within an arbitrary assembly of otherwise lossless layers, as shown in Fig. 1(a)
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.
. The potential transmittance (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 (Yout = Hout/Eout), 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, PTMAX is dependent on the properties of the absorbing film only, and can be calculated using the expressions provided previously [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]

].

While not typically stated in terms of absorption, one implication of PT theory is that the absorbance of a thin film can be significantly lower than the value predicted by the bulk optical constants. This is especially true for very thin films, but the possibility of reduced absorption (with appropriate admittance matching) extends to surprisingly large film thickness. To illustrate, we compared the minimum effective absorption coefficient (αmin) for an Ag thin film to the bulk optical absorption coefficient (αm = 4πκm/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]

]. Noting that PT is the ratio of power flux at the entrance and exit interface of the absorbing film, it follows that PTMAX = exp(-αmindm), or rearranging:

αmin=ln(PTMAX)/dm.
(1)

As an example, Fig. 1(b) shows a plot of αmin 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 αmin, 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

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

].

Implicit above, and to PT theory in general, is the assumption that the bulk refractive index Nm = nm-iκm 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

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]

]. Nevertheless, carefully deposited Ag films have been shown [19

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

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]

] to exhibit bulk properties for thickness as low as ~10-12 nm. Based on this, we restrict our theoretical analysis to films > 10 nm thick, and assume that bulk optical constants can be applied in this range. In practice, deviation from bulk properties is possible depending on the film deposition technique, and should be considered.

Figure 2(a)
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.
shows a representative plot of Xop and Zop 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 Zop/Xop is also plotted. As expected [15

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

], Yop tends toward Nm* = nm + iκm in the limit of a thick metal film (for example, the Rakic model predicts Nm = 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 Zop/Xop) shows relatively little variation versus the transverse wave vector (i.e. angle).

3. Admittance matching conditions for a periodic DMD multilayer

For the single unit cell, the exit admittance from the perspective of the metal layer is that presented by a single thin film (with real index n1) on an infinitely thick substrate (with real index n2) [15

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

]:
Yout=η2cosδ1+iη1sinδ1cosδ1+i(η2/η1)sinδ1,
(5)
where η1 and η2 are the tilted optical admittances of the dielectric layer and the ambient medium, respectively. For TE polarization ηi = ni cosθi and for TM polarization ηi = ni /cosθi, where ni and θi are the refractive index and the propagation angle (from Snell’s law) in medium i. Furthermore, δ1 = (2π/λ)n1d1cosθ1 is the phase thickness of the dielectric film. Equating the real and imaginary parts of Eq. (5) to Xop and Zop, respectively, and after some algebraic manipulation, the following admittance matching equation results:
Xop=2η2±cos[sin1{2ZopXop(η1/η2η2/η1)}](1η22η12)+(1+η22η12).
(6)
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 Xop and Zop), the equation predicts that admittance matching (when possible) is dependent on the values of η1 and η2 only. However, given a solution to Eq. (6), the required thickness for the dielectric layer n1 is fixed by the same set of equations as follows:
d1,m=(λ4πn1cosθ1,m)sin1{2ZopXop(η1,m/η2,mη2,m/η1,m)},
(7)
where n1, θ1,m, η1,m and η2,m 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 d1 = d1,m + q(λ1/2), where q is an integer and λ1 = λ/n1, will also produce an admittance match [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]

]. In general, the admittance matching conditions are attained for periodically repeating values of d1. 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 n2 > n1 and complex angle θ1).

Note that Xop and Zop are real numbers by definition, and that η2 is a real number for all cases considered below (i.e. lossless ambient media with real incident angle). Thus, in all cases for which η1 is purely real (i.e. for non-evanescent wave solutions in dielectric layers n1), solutions to Eq. (6) are restricted to the following range:
(η1η2η2η1)2ZopXop.
(8)
For a given metal layer, Eq. (8) places a lower limit on the ratio η1/η2, below which solutions to Eq. (6) are not possible. The ratio Zop/Xop is typically high (see Fig. 2), and diverges for increasing Ag film thickness. Thus, an admittance match is typically reliant on high values of η1/η2, especially for large metal thickness.

3.1. Normal incidence in air

We first consider the simplest but practically important case of normal incidence from an ambient air medium (i.e. n2 = 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 (n1 - 1/n1) > 2Zop/Xop. Given typical values of Xop and Zop for a thin Ag film in the visible range (see Fig. 2), this implies that high values of dielectric index (n1 > 4) are necessary to achieve a perfect admittance match. Moreover, with fixed n2 and for a given metal layer at a given wavelength, only a specific value of n1 results in a solution to Eq. (6). This observation was made previously [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]

], but in that case solely on the basis of a transfer-matrix numerical study. Solutions to Eqs. (6) and (7) were obtained as a function of Ag film thickness using a commercial software tool (Matlab), and representative data is shown in Figs. 4(a)
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.
and 4(b).

The results in Fig. 4(a) provide a theoretical underpinning for, and are in exact agreement with, the transfer-matrix results reported previously (see 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.
in [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]

]), other than some numerical noise in the earlier results. Clearly, an impractically large dielectric index is required to facilitate a perfect admittance match for the Ag film at normal incidence. Nevertheless, it is illustrative to consider the implications of the solutions to Eqs. (6) and (7). As an example, Figs. 4(c) and 4(d) show the predicted T, PTMAX, and R for a specific admittance-matched case (λ = 550 nm, and assuming a unit cell with dm = 25 nm, n1 = 4.732 and d1 = 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 PTMAX was calculated using previously described expressions [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]

]. As expected, a perfect admittance match (i.e. T = PTMAX and 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

As an example, solutions to Eqs. (6) and (7) were obtained for fixed indices n1 = 2.3 and n2 = 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, dm = 25 nm, d1 = 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.

To our knowledge, this efficient tunneling of off-normal TE-polarized waves through multi-period DMD stacks has not been studied previously. It might have implications for the realization of polarizing and angularly selective filters. However, it should be noted that the perfect admittance match is dependent on the symmetry of the structures considered, including the assumption of identical input and exit media. The air-ambient case considered would certainly present practical challenges. Solutions exist for higher values of n2; example data is shown for n2 = 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

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]

] or other means of momentum matching would be required.

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

Plasmon-mediated tunneling through symmetric, single-period DMD stacks was originally reported 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]

] and later by Hayashi 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]

]. Analogous tunneling through multi-period DMD stacks was studied experimentally by Tomita 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]

] and theoretically by Feng 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]

], who proposed optical filters and modulators based on the concept. Relevant to the present discussion, the low transmittance reported in [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]

] can be attributed in part to non-optimal admittance matching in their structures.

The existence and the nature of the solutions to Eq. (6) were found to depend on the stack parameters (λ, dm, n1, n2). 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)
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.
and 6(b). Note that the transverse wave vector is defined by kt = (2π/λ)n2sinθ2, and that waves in the n1 layers are evanescent when (kt/k0) > n1. We assumed n1 = 1.631 and a fictitious coupling medium with n2 = 4, to enable comparison with similar structures studied by Feng 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]

]. For some combinations of λ and dm (such as for λ = 550 nm and dm < ~42 nm in Fig. 6(a)), no solutions are found.

The implications of the solutions to Eqs. (6) and (7) are analogous to those for the propagating wave cases above. As an example, T and R were plotted in Figs. 6(c) and 6(d) for a particular admittance-matched case (λ = 500 nm, dm = 50 nm). Consistent with the solutions in Figs. 6(a) and 6(b), dielectric thicknesses d1 = 12.77 nm and 46.07 nm result in a perfect admittance match (T = PTMAX and R = 0) at normalized wave vector values of 2.18 and 3.61, respectively. Other values of d1 result in a partial tunneling peak at a different angle, but without a perfect admittance match. The data for d1 = 20 nm is shown as an example.

From an experimental perspective, coupling prisms with n2 = 4 are not practical. Figure 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).
shows results for a more practical combination of indices, n1 = 1.38 and n2 = 1.515, representing the MgF2-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]

].

As mentioned in the introduction, much of the recent interest in plasmon-mediated tunneling through DMD structures is motivated by their potential to transmit the evanescent fields from an object [8

8. 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).

]. In these superlens applications, there is a need to design MD structures offering both high transmittance and low reflectance, ideally for a wide range of transverse wave vectors and over the entire operational wavelength range of the lens. In particular, significant reflectance of energy back towards the object contributes to image distortion [10

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

]. The model system studied above (a symmetric DMD stack, symmetrically embedded in lossless ambient media) is not expected to capture all pertinent details of the superlens. For example, transfer of energy associated with the evanescent fields of an object necessarily involves optical absorption by the detector (at the image plane) and finite reflection from the lens [12

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

]. Nevertheless, the prism-coupled DMD stack can provide insight regarding the qualitative nature of tunneling through the multilayer superlens [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]

,23

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

Using potential transmittance theory, we derived a general equation describing the conditions for admittance-matched tunneling through periodic DMD multi-layers. We furthermore verified its applicability to tunneling problems involving both propagating and evanescent waves. For normal incidence, the equation predicts that a perfect match occurs only for specific and large values of the dielectric refractive index. For off-normal incidence and assuming fixed dielectric and ambient medium indices, the equation predicts that a perfect admittance match occurs at specific tunneling angles. For TE-polarized light, solutions are found in the propagating wave regime. For TM-polarized light solutions are found in the evanescent-wave regime. To our knowledge, these various classes of tunneling have not previously been unified within a single theoretical framework.

In all cases, matching is predicted to occur at a single wavelength and/or tunneling angle. It seems probable that using two or more dielectrics would enable structures that are matched at more than one wavelength or tunneling angle. The rich body of literature on multi-layer anti-reflection coatings and on induced transmission filters [14

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

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

] might provide substantial guidance in this regard, but such studies are left for future work.

Acknowledgment

The work was supported by the National Sciences and Engineering Research Council of Canada.

References and links

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]

5.

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 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]

8.

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).

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]

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]

12.

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

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]

15.

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

16.

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

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]

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]

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]

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]

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]

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

  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 Films306(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 Films441(1-2), 200–206 (2003). [CrossRef]
  5. M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 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]
  8. 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).
  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. Express15(19), 11959–11970 (2007). [CrossRef] [PubMed]
  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. Express16(23), 19342–19353 (2008). [CrossRef] [PubMed]
  12. E. Fourkal, I. Velchev, and A. Smolyakov, “Energy and information flow in superlensing,” Phys. Rev. A79(3), 033846 (2009). [CrossRef]
  13. 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]
  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]
  15. H. A. Macleod, Thin-Film Optical Filters, Third Edition (Institute of Physics Publishing, 2001).
  16. P. W. Baumeister, “Radiant power flow and absorptance in thin films,” Appl. Opt.8(2), 423–436 (1969). [CrossRef] [PubMed]
  17. T. W. Allen and R. G. DeCorby, “Assessing the maximum transmittance of periodic metal-dielectric multi-layers,” J. Opt. Soc. Am. B28(10), 2529–2536 (2011). [CrossRef]
  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]
  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. Express18(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 Films517(8), 2736–2741 (2009). [CrossRef]
  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]
  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. Express16(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. B28(7), 1778–1781 (2011). [CrossRef]

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