## Abstract

In this paper, we present a low-loss plasmonic Bragg reflector structure with high light-confinement. We show that periodic changes in the dielectric materials of the metal-insulator-metal waveguides can be utilized to design efficient subwavelength Bragg reflectors and micro-cavities. FDTD simulation results of the designed Bragg reflector using realistic material parameters justify that the transfer matrix calculations are adequate for the design purposes.

© 2006 Optical Society of America

## 1. Introduction

Optical devices, which are based on surface plasmon polaritons (SPPs), have recently been considered as solutions to overcome the diffraction in dielectric structures that limits the miniaturization of photonic devices. SPPs are electromagnetic modes coupled to the collective electron oscillations propagating along the interface between dielectric and metallic materials at the optical and near infra red frequency spectrum. In this frequency range, the metal exhibits a negative dielectric function. The electromagnetic field associated with SPPs is bounded along the interface and decreases exponentially in the direction perpendicular to the interface resulting in a subwavelength confinement of the electromagnetic mode [1].

Photonic crystals, periodic arrays of dielectric scatters in homogenous media, affect the properties of photons in the same way a semiconductor affects electrons. Therefore, photons can have band structures, forbidden frequency intervals, known as photonic band gaps (PBG), as well as localized defect states. PBGs and micro-cavities realized by intentionally introducing defects in the periodicity, have various applications, such as sharp bending of light and the control of spontaneous emission and zero-threshold lasing [2, 3].

Periodic gratings introduced on metallic surfaces can lead to photonic band gaps as SPPs propagate along the metallic surfaces. These structures have been used to build SPP mirrors and beam splitters [4, 5]. SPP-based Bragg reflectors (1D photonic crystals) have been previously fabricated by engraving slots into metal strips [6]. Since these corrugated metal strips are based on the insulator-metal-insulator (IMI) geometry, they did not provide the necessary confinement needed for subwavelength photonic devices [7]. This is due to the much longer penetration depth of SPP fields into the dielectric medium compared to that into the metallic medium at the metal-dielectric interface. In addition, the contrast between the effective indices of the alternating layers is small, resulting in small band gaps for the corrugated metal strips [8], as demonstrated by the equation of the first band gap (Δ*ω _{g}*) in a 1D photonic crystal,

where *n*
_{eff,1} and *n*
_{eff,2} are the effective indices associated with the two alternately stacked waveguides, and *ω _{c}* is the central frequency of in the gap [9]. As Shown in Equation 1, small contrast in the effective indices leads to narrow PBGs.

SPP Bragg reflectors were also proposed based on the metal-insulator-metal (MIM) geometry, where different metallic materials are alternately stacked (heterowaveguides) [10]. MIM waveguides (Figure 1) can provide spatial light confinement with lateral dimensions of less than 10% of the free wavelength [11]. Also, they can exhibit zero propagation loss at bends and power splitters with dimensions much smaller than the wavelength of the optical mode [12]. Such MIM heterowaveguide can effectively confine the guided light. However, metallic materials with considerably different dielectric constants are needed in order to realize wide PBGs with high refection and low transmission for the frequencies inside the gaps. The dielectric constants of low loss metals (such as Ag, Au and Cu) are not different enough to produce an effective contrast [13]. This requires using a lossy metal such as aluminum together with a relatively low-loss metal such as silver. The utilization of a lossy material significantly increases the ohmic losses in an already lossy MIM type structure.

In this paper, we present a new MIM-based low-loss Bragg reflector structure with high light confinement. We show that periodic changes in the dielectric materials of the MIM waveguides can be utilized to design effective filtering around the Bragg frequency. In order to avoid the approximation errors in the analysis, silver and aluminum are characterized by the optical constants from [13] and [14], respectively. The fundamental symmetric modes are considered because their field profiles are suitable for end-fire excitations [15] and also there are the dominant modes in terms of the lowest loss.

## 2. MIM Bragg reflector

Assuming that all field components have the functional form of *exp*[*i*(*βz*-*ωt*)], the dispersion equation for the p-polarized symmetric mode in MIM structures can be written as

where *ε _{d}*,

*ε*and

_{m}*t*are dielectric constants of dielectric, metallic materials and the dielectric thickness as shown in Figure 1(a), respectively [11]. The propagation constant (

*β*) is usually represented as a dimensionless effective index

*n*=

_{eff}*β*/

*k*

_{0}for the guided modes, where

*k*

_{0}is the free space wave-vector, and propagation is assumed to be in the

*z*-direction.

Figure 1 displays the variation of the *n _{eff}* values for the slab symmetricmode of MIM waveguides with silver or aluminum as the metal, and air or silicon dioxide (

*SiO*

_{2},

*n*=1.46) as the dielectric. It is shown in Figure 1(a) that the real parts of the associated

*n*values for twoMIM waveguides with the same metallic and different dielectric materials can exhibit better contrast compared to those of the waveguides with different metal and one dielectric. Therefore, when alternately stacked, the resulting frequency gap is wider if we change the dielectric instead of the metal. The propagation length of a propagating field inside a waveguide (

_{eff}*L*=1/

_{p}*Imag*(

*β*)) depends on the imaginary parts of the

*n*values shown in Figure 1(b). It can be seen that the propagation lengths in the MIM waveguides degrade about one order of magnitude when we replace silver with aluminum. In fact, aluminum is not suitable for waveguiding purposes because the imaginary part of the dielectric function is relatively large over the visible and infra red spectrum and it causes significant resistive heating [14, 7]. We can see that lower refractive index dielectric materials result in smaller

_{eff}*Imag*[

*n*] values in plasmon slot waveguides. Therefore, it is necessary to use dielectric materials with low refractive index in order to reduce the loss for the passive device applications.

_{eff}Figure 2 compares the variation of the *n _{eff}* values for MIM waveguides with silver as
the metal, and

*SiO*

_{2}or nanoporous

*SiO*

_{2}(

*PSiO*

_{2},

*n*=1.23) as the dielectric for

*t*=30

*nm*. Nanoporous silicon oxide has been employed as a low-index material in microelectronic and photonic applications [16]. In order to produce a practical Bragg reflector design, we propose to use

*PSiO*

_{2}instead of air as a low refractive index material in MIM waveguides. As depicted in Figures 1(a) and 2(a), the real parts of the

*n*values are always greater for the waveguides with higher index dielectric materials at frequencies less than the surface plasmon frequency (

_{eff}*ω*), which is the frequency where

_{sp}*Real*(

*ε*)=-

_{m}*Real*(

*ε*). In the cases of the

_{d}*Ag-PSiO*

_{2}and

*Ag-SiO*

_{2}waveguides, the surface plasmon wavelengths (

*λ*=2

_{sp}*π*/

*ω*) are 346

_{sp}*nm*and 356

*nm*, respectively [13]. In addition, the

*n*curves become almost horizontal lines at wavelengths longer than about

_{eff}*λ*=1

*µm*for MIM waveguides with silver. Figure 2(a) indicates that we can use alternately stacked

*Ag-SiO*

_{2}and

*Ag-PSiO*

_{2}to realize Bragg reflectors since the associated

*n*curves are well separated.

_{eff}For the Bragg reflector shown in Figure 3(c), alternately stacking *Ag-PSiO*
_{2} and *Ag-SiO*
_{2} waveguides, gives rise to an effective index modulation. For these waveguides with *t*=30 *nm*, *n _{eff}*,1=1.878+0.008

*i*and

*n*

_{eff,2}=2.241+0.009

*i*at

*λ*=1.55

*µm*, respectively. According to the Bragg condition [

*d*

_{1}

*Real*(

*n*

_{eff,1})+

*d*

_{2}

*Real*(

*n*

_{eff,2})=

*nλ*/2], we can realize Bragg scattering around

_{b}*λ*=1.55

*µm*, by choosing

*d*

_{1}=200

*nm*and

*d*

_{2}=180

*nm*. The transmission spectrum of SPPs propagation through this structure can be calculated by the standard transfer matrix method. As shown in Figure 3, there is PBG around

*λ*=1.55

*nm*when a finite number of the periods of the alternating layers (

*N*) is considered. For the plasmonic reflector, as the imaginary parts of the

*n*values of MIM structures increase rapidly with the frequency near

_{eff}*ω*as shown in Figure 2(b), the transmission drops for the wavelengths smaller than about

_{sp}*λ*=1

*µm*. It should be noted that the propagation loss is automatically considered in the results form the transfer matrix method calculations. For example for

*N*=15, at

*λ*=1.55

*µm*and

*λ*=1

*µm*, the output power is about 3

*dB*and 2.2

*dB*less than the input, respectively.

Theoretically Bragg scattering occurs for any number of periods. However, in the cases of the alternating *Ag-PSiO*
_{2} and *Ag-SiO*
_{2} waveguides of dielectric thickness *t*=30*nm*, we found that the minimum N required to realize transmission less than 1% in the gap, is *N*=15. As shown in Figure 3(a), the increase of the number of periods give rise to higher losses.

By introducing a defect in the periodicity, micro-cavities can be formed which trap the incident radiation for applications such as ultra low-threshold lasers and light-emitting diodes [17]. Figure 3(b) shows the transmission spectrum when the 8*th* period is replaced with a single *Ag-PSiO*
_{2} waveguide of 30 *nm* thickness and 390 *nm* length. We can define the cavity quality factor as *Q*=*λ*
_{0}/Δ*λ* where *λ*
_{0} and Δ*λ* are the central resonance wavelength and the full width at half maximum of the SPP defect mode, respectively. This quantity describes the ratio of the energy stored in the cavity at resonance to the energy escaping from the cavity per cycle of oscillation. The quality factor for the cavity in Figure 3(b) is found to be about 145.

We verify the results form the transfer matrix calculations for the Bragg reflector using the finite-difference time-domain (FDTD) method with the perfectly matched layer (PML) as the boundary condition. The structure in Figure 4(a) is simulated choosing the grid sizes in the *x* and *y* directions to be 5 *nm*×2.5 *nm* by a 2D FDTD solver. This structure is excited by a dipole source located in the middle of the feeding waveguide (*Ag-PSiO*
_{2}, *t*=30 *nm*).

The field profiles, |*H _{x}*|

^{2}, associated with the SPP propagation through this reflector are displayed in Figure 4. Figure 4(c) shows that the incident radiation is reflected at the

*λ*=1.55

*µm*, while it propagates through the structure at

*λ*=1

*µm*and

*λ*=1.9

*µm*that are outside the frequency gap [Figures 4(b) and 4(d)]. The transmitted power is in good agreement with the spectrum shown in Figure 3(b). Simulation results for the microcavity described in the previous section is shown in Figure 5. As depicted in this figure, the incident radiation couples into the microcavity structure (390

*nm*long

*Ag-PSiO*

_{2}waveguide).

## 3. Conclusion

We presented a new SPP based low-loss Bragg reflector structure with high light-confinement. We showed that alternately changing of the dielectric materials of the MIM waveguides can be utilized to design effective subwavelength Bragg reflectors and micro-cavities. FDTD simulation results with realistic material parameters were presented which revealed that the transfer matrix approach utilizing the effective indices associated with subwavelength waveguides, is adequate for the design purposes. The reflectors and micro-cavities are expected to have applications in highly integrated photonic circuits and SPP-based devices.

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