## Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes

Optics Express, Vol. 13, Issue 11, pp. 4113-4124 (2005)

http://dx.doi.org/10.1364/OPEX.13.004113

Acrobat PDF (191 KB)

### Abstract

We present a systematic study of mode characteristics of multilayer metal-dielectric (M-D) nanofilm structures. This structure can be described as a coupled-plasmon-resonantwaveguide (CPRW), a special case of coupled-resonator optical waveguide (CROW). Similar to a photonic crystal, the M-D is periodic, but there is a major difference in that the fields are evanescent everywhere in the M-D structure as in a nanoplasmonic structure. The transmission coefficient exhibits periodic oscillation with increasing number of periods. As a result of surface-plasmon-enhanced resonant tunneling, a 100% transmission occurs periodically at certain thicknesses of the M-D structure, depending on the wavelength, lattice constants, and excitation conditions. This structure indicates that a transparent material can be composed from non-transparent materials by alternatively stacking different materials of thin layers. The general properties of the CPRW and resonant tunneling phenomena are discussed.

© 2005 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

2. K. M. HO, C. T. Chan, and C. M. Soukoulis, “Existence of Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. **65**, 3152–3155 (1990). [CrossRef] [PubMed]

6. R. L. Nelson and J.W. Haus, “One-dimensional photonic crystals in reflection geometry for optical applications,” Appl. Phys. Lett. **83**, 1089–1091 (2003). [CrossRef]

4. M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. **84**, 2140–2143 (2000). [CrossRef] [PubMed]

5. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one-and two-dimensional photonic crystals,” Phys. Rev. B **65**, 165208 (2002). [CrossRef]

7. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

8. Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E **69**, 056604 (2004). [CrossRef]

9. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B **50**, 16835–16844 (1994). [CrossRef]

12. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures.” J. Appl. Phys. **83**, 2377–2383 (1998). [CrossRef]

## 2. Bloch modes of evanescent waves

*ε*

_{2},

*µ*

_{2}) and dielectric (

*ε*

_{1},

*µ*

_{1}) is given by

*µ*in our expressions to conveniently extend our current work to left-handed materials that will be discussed in a separate paper. We see that the real parts of ε

_{1}and ε

_{2}must be of opposite sign.

*d*=

*d*

_{1}+

*d*

_{2}where

*d*

_{1}and

*d*

_{2}are the thicknesses of the dielectric and metallic layers. The subscript 1 refers to the dielectric medium with constant permittivity (

*ε*

_{1}) and permeability (

*µ*

_{1}). The subscript 2 refers to the metallic material with constant

*µ*

_{2}, and dispersion is given by

*ω*

_{p}is the effective electron plasma frequency and ε

_{2}(

*ω*) is negative when

*ω*<

*ω*

_{p}. In general, the effective plasma frequency

*ω*

_{p}is a function of electron density and surface structure, such as subwavelength holes or slits. If the conditions in Eq. (2) are satisfied, localized surface modes can be formed at interfaces. In the case of thick layers, there is no coupling between the surface waves bound to each metal-dielectric interface. For thin layers, however, the localized surface wave fields are electromagnetically coupled and this will significantly alter the dispersion relation Eq. (1). The resonant coupling leads to coherent oscillations of the electron plasma between different interfaces.

**E**=(

*E*

_{x},0,

*E*

_{z}), we start with the frequency domain wave equation

**x̂**direction with wave number

*K*

_{p}. The electric field in the

*n*th unit cell is given by

**E**

_{n}(

**r**)=

*𝓔*

_{n}(z)exp(

*iK*

_{p}

*x*), where the evanescent portion of the fields are

*z*

_{n}=

*z*-

*nd*, and

**a**

_{n},

**b**

_{n},

**c**

_{n}, and

**d**

_{n}are vectors in the

**x̂**-

**ẑ**plane. The relationship between these amplitudes can be found by enforcing the boundary conditions and the fact that ∇·

**E**=0 inside each layer. Using the Bloch theorem and after some manipulation, we obtain the dispersion relation:

*K*

_{B}is Bloch wave number. The Bloch wave vector is along the

**ẑ**direction. The existence of the Bloch modes requires that |cos(

*K*

_{B}

*d*)|≤1, i.e.

*K*

_{B}is real. Using this condition and the evanescent condition

*ε*

_{1}

*µ*

_{1}, the Bloch bands of the coupled evanescent fields can be obtained when combined with the material dispersion Eq. (3). We let

*ω*<0.6

*ω*

_{p}and the band structure is composed of a set of surfaces in the three-dimensional space (

*ω*,

*K*

_{p},

*K*

_{B}) formed by the frequency

*ω*, the surface plasma wave number

*K*

_{p}, and the Bloch wave number

*K*

_{B}. Figure 2 shows numerical results of the band structure of the evanescent waves at different layer thicknesses. In Fig. 2, the two blue bands represent the transmission pass bands of the evanesent fields, i.e. Bloch modes of evanescent fields. These types of Bloch modes are a consequence of resonant tunneling of the evanescent waves and possess features of both surface plasmon and Bloch waves. As a characteristic of surface modes, the blue areas in Fig. 2 are below the light line of the dielectric medium. The higher frequency band is truncated at the light line since we consider only modes that are bounded to the metal/dielectric interfaces and are evanescent inside both metal and dielectric media. The Bloch evanescent bands discussed here are different from that of surface-plasmon polaritons (SPP’s) reported previously[14

14. R. D. Meade, K. D. Brommer, A.M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B **44**, 10961–10964 (1991). [CrossRef]

16. M. Kretschmann and A. A. Maradudin, “Band structures of two-dimensional surface-plasmon polaritonic crystals,” Phys. Rev. B **66**, 245408 (2002). [CrossRef]

*K*

_{p}. Note that the bandgap disappears at certain values of

*K*

_{p}depending on the thickness of the layers.

*ω*

_{p}can be engineered by a combination of material properties and properly designed nanostructures on the metal layers[17

17. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science **305**, 847 (2004). [CrossRef] [PubMed]

*ω*

_{p}is tailored to the surface plasmon wave number

*K*

_{p}through the condition

*ε*

_{1}and

*µ*

_{1}refer to the parameters of the dielectric layers in the M-D structure and the material dispersion of the metal is given by Eq. (3). The existence of Bloch evanescent modes allows us to better understand a variety of optical tunneling phenomena and enhanced optical transmission. The Bloch bands of the evanescent fields can also exist in periodic structures with alternating left-handed and right-handed materials. The band structure of the evanescent waves also satisfy the scaling law that is the general criterion of the ordinary Bloch bands. The band gaps depend on the material properties and layer thicknesses.

*K*

_{p}values. As indicated in the plots, when the thickness of the metal layers decreases, the coupling increases and hence the width of the bandgap increases. The bandgap disappears in Fig. 4(d) where

*K*

_{p}=1.36

*π*/

*d*. Figure 5 shows dispersion of the surface plasmon modes at two different values of the Bloch wave number

*K*

_{B}. The red broken line represents the light line of the dielectric medium. The two curves below the light line correspond to the two Bloch evanescent bands in Fig. 2. The higher frequency curve is truncated at the light line.

## 3. Transmission oscillation

*d*≪

*λ*, the cumulative thickness is ≫

*λ*. In spite of the fact that the cumulative thickness ≫λ and that the fields are evanescent everywhere, the photons tunnel through the M-D structure. The wave-packet behavior of the transmission coefficient is the result of the evanescent coupling and the stronger the coupling, the broader the envelope of the oscillation.

*K*

_{B}versus the number of the periods for the three wavelengths. For the stop-band wavelength in Fig. 7(c), the Bloch wavevector decays exponentionally with increasing number of the periods, and eventually it becomes imaginary. For the wavelengths (375 and 495 nm) in the pass bands, the Bloch wavevector oscillates with a large amplitude for the first few periods (not shown), then it converges to oscillate about the

*K*

_{B}of the corresponding infinite periodic structure with a small amplitude as shown in Fig. 7(a) and (b). This explains the oscillation and wave-packet features in the transmission coefficients. The carrier of the oscillation in the transmission coefficient is determined by the

*K*

_{B}of the corresponding infinite structure while the envelope of the oscillation is inversely proportional to the bandwidth of the

*K*

_{B}. Notice that the positions of the nodes in Fig. 7 match the positions of the maximum transmission in Fig. 6. Hence, the 100% transmission happens when the Bloch wavevector matches that of the corresponding infinite periodic structure. The number of the periods

*N*between the peaks satisfies the relation

*K*

_{B}

*Nd*≈2

*πm*where

*m*is an integer.

## 4. Defect modes

18. M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A **71**, 013803 (2005) [CrossRef]

## 5. Potential device applications

*N*-period CPRW structure, there are

*N*-1 transmission resonances inside each pass band due to the

*N*-1 couplings. The bandwidth and center frequency of the resonances change with the length of the structure. This property can be used to construct Dense Wavelength Division Multiplex (DWDM) filters for applications of telecommunications. Shown in Fig. 10 is a DWDM filter designed using reflection mode of the CPRW. With the reflection mode and 140 periods, the filter has a flat-top transmission spectrum of 100 GHz channel spacing in the C-band telecommunications window.

## 6. Summary

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

2. | K. M. HO, C. T. Chan, and C. M. Soukoulis, “Existence of Photonic Gap in Periodic Dielectric Structures,” Phys. Rev. Lett. |

3. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

4. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonic Crystals,” Phys. Rev. Lett. |

5. | S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one-and two-dimensional photonic crystals,” Phys. Rev. B |

6. | R. L. Nelson and J.W. Haus, “One-dimensional photonic crystals in reflection geometry for optical applications,” Appl. Phys. Lett. |

7. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

8. | Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, “Finite-size effect on one-dimensional coupled-resonator optical waveguides,” Phys. Rev. E |

9. | V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B |

10. | M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Metallic photonic band-gap materials.” Phys. Rev. B |

11. | S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B |

12. | M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures.” J. Appl. Phys. |

13. | S. Feng, M. Elson, and P. Overfelt, “Transparent photonic band in metallodielectric nanostructures.” submitted to Phys. Rev. B (2005). |

14. | R. D. Meade, K. D. Brommer, A.M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B |

15. | S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full Photonic Band Gap for Surface Modes in the Visible,” Phys. Rev. Lett. |

16. | M. Kretschmann and A. A. Maradudin, “Band structures of two-dimensional surface-plasmon polaritonic crystals,” Phys. Rev. B |

17. | J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science |

18. | M. F. Yanik and S. Fan, “Stopping and storing light coherently,” Phys. Rev. A |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(160.4760) Materials : Optical properties

(240.0310) Optics at surfaces : Thin films

(240.6680) Optics at surfaces : Surface plasmons

(240.7040) Optics at surfaces : Tunneling

(260.3910) Physical optics : Metal optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 22, 2005

Revised Manuscript: May 16, 2005

Published: May 30, 2005

**Citation**

Simin Feng, J. Elson, and Pamela Overfelt, "Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes," Opt. Express **13**, 4113-4124 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4113

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### References

- E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059�??2062 (1987). [CrossRef] [PubMed]
- K. M. HO, C. T. Chan, and C. M. Soukoulis, �??Existence of Photonic Gap in Periodic Dielectric Structures,�?? Phys. Rev. Lett. 65, 3152�??3155 (1990). [CrossRef] [PubMed]
- J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107�??4121 (1996). [CrossRef]
- M. Bayindir, B. Temelkuran, and E. Ozbay, �??Tight-Binding Description of the Coupled Defect Modes in Three-Dimensional Photonic Crystals,�?? Phys. Rev. Lett. 84, 2140�??2143 (2000). [CrossRef] [PubMed]
- S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, �??Analysis of defect coupling in one-and two-dimensional photonic crystals,�?? Phys. Rev. B 65, 165208 (2002). [CrossRef]
- R. L. Nelson and J.W. Haus, �??One-dimensional photonic crystals in reflection geometry for optical applications,�?? Appl. Phys. Lett. 83, 1089�??1091 (2003). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711�??713 (1999). [CrossRef]
- Y.-H. Ye, J. Ding, D.-Y. Jeong, I. C. Khoo, and Q. M. Zhang, �??Finite-size effect on one-dimensional coupled-resonator optical waveguides,�?? Phys. Rev. E 69, 056604 (2004). [CrossRef]
- V. Kuzmiak, A. A. Maradudin, and F. Pincemin, �??Photonic band structures of two-dimensional systems containing metallic components,�?? Phys. Rev. B 50, 16835�??16844 (1994). [CrossRef]
- M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, �??Metallic photonic band-gap materials.�?? Phys. Rev. B 52, 11744�??11751 (1995). [CrossRef]
- S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, �??Large omnidirectional band gaps in metallodielectric photonic crystals,�?? Phys. Rev. B 54, 11245�??11251 (1996). [CrossRef]
- M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, �??Transparent, metallodielectric, one-dimensional, photonic band-gap structures.�?? J. Appl. Phys. 83, 2377�??2383 (1998). [CrossRef]
- S. Feng, M. Elson, and P. Overfelt, �??Transparent photonic band in metallodielectric nanostructures.�?? Submitted to Phys. Rev. B (2005).
- R. D. Meade, K. D. Brommer, A.M. Rappe, and J. D. Joannopoulos, �??Electromagnetic Bloch waves at the surface of a photonic crystal,�?? Phys. Rev. B 44, 10961�??10964 (1991). [CrossRef]
- S. C. Kitson,W. L. Barnes, and J. R. Sambles, �??Full Photonic Band Gap for Surface Modes in the Visible,�?? Phys. Rev. Lett. 77, 2670�??2673 (1996). [CrossRef] [PubMed]
- M. Kretschmann and A. A. Maradudin, �??Band structures of two-dimensional surface-plasmon polaritonic crystals,�?? Phys. Rev. B 66, 245408 (2002). [CrossRef]
- J. B. Pendry, L. Martín-Moreno, F. J. Garcia-Vidal, �??Mimicking Surface Plasmons with Structured Surfaces,�?? Science 305, 847 (2004). [CrossRef] [PubMed]
- M. F. Yanik and S. Fan, �??Stopping and storing light coherently,�?? Phys. Rev. A 71, 013803 (2005) [CrossRef]

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