## 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

The emergence of the concept of photonic crystals (PhCs)[1] has inspired a great interest[2]–[6]. Waveguiding in the PhCs can be achieved by forming line defects[4]–[5] or by introducing a chain of coupled high Q cavities proposed by Yariv et al.[7], known as coupled-resonator optical waveguide (CROW). In principle, the CROWcombines the effects from both periodic structures and resonators, i.e. waveguiding and enhanced nonlinearity and group delay. Different from the conventional PhCs, the band structure in the CROWs are created from localized modes through evanescent coupling. This character provides the CROW some promising features over the conventional PhCs, such as smaller group veocity and hence larger enhancement of nonlinear effects[8]. In this paper, we provide a systematic study of a new type CROWstructure, coupled-plasmon- resonant waveguide (CPRW) in one-dimensional metallodielectric (M-D) nanofilms. The results of 1D structures can provide insight into 2D and 3D problems, meanwhile avoiding intensive computational demands of higher dimensionality. From the application standpoint, one-dimensional structures can be easy to fabricate. Metallodielectric structures were studied previously in the context of PhCs[9]–[12] where the Bloch bands of traveling waves were considered, i.e. the field is evanescent only inside the metal layers and the field propagates inside the dielectric layers. This means that the Bloch transmission bands are formed from constructive interference. Here we study a similar type of multilayer structure from the CROWperspective. The electromagnetic fields are evanescent within all metal and dielectric layers along the propagation direction of the Bloch waves, i.e. the direction perpendicular to the interfaces of the layers. In this structure, 100% transmission of the evanescent waves is achieved by means of surface plasmon excitation. The bounded surface plasmon fields provide an enhancement at each interface of the metal/dielectric layers. In this structure, the waveguiding is achieved simutaneously along the interfaces and perpendicular to the interfaces with different guiding mechanisms. In the direction parallel to the interfaces, the electromagnetic fields are guided by surface plasmons whereas perpendicular to the interfaces, the waveguiding is achieved through evanescent coupling. In nano-optics, low-dimensional optical or evanescent waves play an essential role when interacting with materials.

This paper discusses a broad range of features inherent in the type of all-evanescent resonant structure studied here. A related, but more specific discussion in the context of transmission characteristics has been given elsewhere [13], where the focus is on achieving wide-bandwidth and length-independent transmission by means of 1-D M-D multilayer structure. We emphasize that the presentwork and [13] only consider the case where the fields are evanescent everywhere in the M-D structure. This is in contrast to many other studies where the fields in the dielectric layers are non-evanescent. In other words the present work and [13] are photonic in nature due to their periodic multilayer structure, but unlike PhC’s, are nanoplasmonic in that the individual layers of the M-D structure have thicknesses much less than the wavelength. This means that the fields within the M-D structure are all localized to the interfaces. In the presentwork, Sections 2, 3 and 4 consider general properties of evanescent wave Bloch modes, transmission oscillation, and defect modes with the proper conditions for resonant tunneling through the M-D structure. In the last Section 5, we discuss potential applications of this structure.

## 2. Bloch modes of evanescent waves

Surface electromagnetic waves can propagate along the boundary separating two media. Such waves are localized and decay exponentially in the direction perpendicular to the interface. The penetration depth of the surface fields into the adjacent media is governed in part by the optical properties of the media. The surface modes can be excited through attenuated-total-reflection and grating coupling techniques. By solving Maxwell’s equations and matching boundary conditions, the dispersion relation of surface waves bound to a planar interface separating two semi-infinite media of metal (*ε*
_{2},*µ*
_{2}) and dielectric (*ε*
_{1},*µ*
_{1}) is given by

and conditions

need to be satisfied. In this work, we will assume non-permeable media, but we include permeability *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.

To better understand the essential physics of the CPRW, we use a one-dimensional lossless periodic structure consisting of alternating layers of metallic and dielectric materials as shown in Fig. 1. The period is *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

Here *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.

To obtain the Bloch waves for a TM mode electric field **E**=(*E*_{x}
,0,*E*_{z}
), we start with the frequency domain wave equation

where ${k}_{0}=\frac{\omega}{c}$, is the wave number in vacuum. We assume the surface wave propagates along the **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

where *z*_{n}
=*z*-*nd*, and

The field amplitudes **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:

where *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 ${K}_{p}^{2}$
>${k}_{0}^{2}$
*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]‖[16] where the Bloch SPP waves propagate parallel to the surface. In this paper, the Bloch evanescent modes are excited with wavevector perpendicular to the interfaces of the layers and this is transverse to the wavevector of the surface waves. The white areas in Fig. 2 are either stop bands or pass bands of traveling waves where no Bloch evanescent waves exist. The transmission passband frequency increases with increasing plasmon wave number *K*_{p}
. Note that the bandgap disappears at certain values of *K*_{p}
depending on the thickness of the layers.

The effective plasma frequency *ω*_{p}
can be engineered by a combination of material properties and properly designed nanostructures on the metal layers[17], such as subwavelength apertures. The nanostructures as described here provide a coupling mechanism for free electrons at different interfaces to oscillate collectively when resonant with the surface waves. This allows for the control of the Bloch evanescent bands. For example, Fig. 3 shows the Bloch evanescent bands when the effective plasma frequency *ω*_{p}
is tailored to the surface plasmon wave number *K*_{p}
through the condition

where *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.

Dispersion relations always have fundamental interest and Fig. 4 shows the Bloch mode dispersion for different thickness of the layers and two different *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

To further confirm the transmission of the Bloch evanescent waves, we apply a transfer matrix method to calculate the transmission of a plane wave incident upon a M-D structure with a finite number of layers. The plane wave is coupled into and out of the structure by prisms. The incident wave excites the Bloch evanescent mode and this allows 100% transmission through large numbers of metal/dielectric layers. Shown in Fig. 6, for three different wavelengths, is the plane wave transmission coefficient versus the number of M-D periods. The wavelength 430nm is in the stop band, and hence the field decays exponentially inside the structure. The other two wavelengths (375 and 495 nm) are in the pass bands. The transmission coefficients oscillate with periodic waveform over the length of the structure. In fact, the transmission oscillation patterns are repeated indefinitely with increasing the length of the structure. A 100% transmission can be achieved periodically at certain lengths depending on the wavelength, lattice constants, and excitation condition. Even though *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.

To better understand this transmission feature, we analyzed the Bloch wave vectors associated with the transmitted fields. Shown in Fig. 7 is the Bloch wavevector *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

If the symmetry of the CPRW as shown in Fig. 1 is altered by introducing lattice imperfections, the band structure will be modified by new electromagnetic states, or defect modes. Shown in Fig. 8 is the transmission spectrum of a 6-period M-D structure. The top plot shows the transmission of the regular structure without the defects. The bottom plot shows the transmission of the modified structure where the thickness of the third and the fifth metal layers is doubled plus an extra metal layer at the dielectric end of the structure. There are three defect modes inside the stop band, two narrow peaks and one flat-top spectrum of bandwidth 20 THz. Using this type of construction, the number of the narrow peaks equals the number of the defects, but the bandwidth of the broadband mode remains almost unchanged upon increasing the number of the defects. Figure 9 shows the dispersion and the group velocity of the three defect modes. As a reference, the dashed blue curves represent the dispersion of the corresponding infinitely periodic structure without defects. The three solid red lines inside the bandgap are the defect modes with the larger slope line being the dispersion of the broadband mode and the smaller slope lines being the dispersion of the two narrow resonances. The group velocities of the narrow resonances are 100 times smaller than the velocity of light in vacuum. The small-slope dispersion curves may be useful in a light stopping scheme[18].

## 5. Potential device applications

The structure considered in this paper can be used in applications of nanoplasmonic devices. The band diagrams shown in Figs. 2 and 3 describe the transmission spectrum of an infinite periodic structure. In general, for a *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.

Nanoplasmonic electro-optical (EO) devices can be built if the dielectric medium of the multilayer structure is composed of EO materials. Analogous to the concept of electronic FPGA (Field-Programmable Gate Array), the architecture of optical FPGA can be constructed from the combination of the electrodes with EO materials. In such structures, the electrodes not only provide the needed electric field, but also enhance the local field stregth. Meanwhile it offers an opportunity of programmable control. Figure 11 shows the geometry of a tunable nanoplasmonic filter. The prisms provide a means to couple light into and out of the device. The refractive index of the prism must be larger than that of the dielectric material and the coupling angle must be greater than the total internal reflection angle of the prism. Note that if the filter between the prisms is made of metal only, the light cannot be transmitted. On the other hand, if the filter is made of all dielectric material, the light cannot be transimitted as well due to the total internal reflection. However, if the filter between the prisms is made from the metal/dielectric multilayer stacks, the 100% transmission can be achieved. This observation indicates the interesting fact that a transparent composite material can be made from non-transparent materials by alternatively stacking thin layers of different materials. Shown in Fig. 12 is the transmission spectrum of the tunable filter with and without applied voltage. The filter is composed of interleaved 9 silver and 8 nonlinear optical polymer layers. By varying the applided voltage, the transmission spectrum of the filter can be changed due to the electro-optical effect.

## 6. Summary

We have presented a detailed study of the coupled-plasmon-resonant waveguide (CPRW) in one-dimensional all-evanescent metallodielectric structures. The transmission spectra of the electromagnetic fields having frequency within the pass bands exhibits thickness-dependent periodic oscillation or wave-packet features. Under certain resonant excitation conditions, defect modes of bandwidth 20THz can be created inside the stop band. The CPRW structures may find potential applications, such as optical filters and optical delay lines. The structures considered here can be fabricated with existing nanotechnology.

## Acknowledgments

This work is supported by ONR Independent Laboratory In-house Research and Independent Applied Research funds.

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