1. Introduction

Photonic band gap (PBG) structures are composed of periodically modulated regions of different refractive indices. They modify propagation properties of electromagnetic waves and forbid their propagation in some frequency intervals in the same way semiconductors affect electrons. Bragg reflectors are in fact 1D PBG structures, where stacks of alternating high- and low-index layers reflect the incident radiation at certain frequencies. Bragg reflectors play a central role in several optoelectronic devices such as filters, modulators, vertical-cavity surface emitting lasers, microcavities, light-emitting diodes and resonant-cavity photodetectors [1, 2, 3].

With the ultimate goal of realization of highly integrated optical components and circuits, the surface plasmon polaritons (SPPs) can be considered as a solution to overcome diffraction limit that restricts down-scaling of photonic devices to a lower-size limit of half the effective wavelength [4]. The SPP field profiles decay exponentially into the neighboring media from their maxima at the interface. In contrast to conventional dielectric components that confine electromagnetic waves to an optically dense core, SPPs are localized at the interface resulting in a subwavelength light-confinement. Low-loss transmission through sharp bends and power dividers has been demonstrated in plasmonic waveguides [5].

Plasmonic PBG components have been used in low threshold SPP-based light-emitting diodes and single-mode surface-plasmon lasers [6, 7, 8]. However, most of the experimentally characterized Bragg reflector structures consist of corrugated metallic strips based on the Insulator-Metal-Insulator (IMI) geometry, which cannot provide true subwavelength light confinement, and therefore, is not suitable for high integration levels [9, 10]. In fact, the Metal-Insulator-Metal (MIM) geometry has been proposed for future on-chip optical communication, which can provide both low-loss propagation and light confinement to lateral dimension of 10% the wavelength [4]. MIMBragg reflectors have been proposed in few numerical studies [11, 12].

In this paper, we present an accurate characteristic impedance forMIMwaveguide structures. Using Finite-Difference Time-Domain (FDTD) simulations, we show that impedance matching can diminish the reflection at discontinuities in MIM based heterostructures. Then, we present a general formulation based on the Transfer Matrix Method (TMM) for MIM Bragg reflectors with subwavelength dielectric layer thickness. Based on the TMM analysis, we investigate the impact of geometric and material parameters on propagation loss and PBG strength.

2. MIM waveguide modeling

In MIM structures [Fig. 1(a) inset], assuming the functional form of exp[i(βz-ωt)] for the field components propagating in the z-direction, the dispersion equation for the fundamental TM mode, where E_y,B_x and B_z are identically zero, is given by [4]

 

Fig. 1. (a) Real part and (b) imaginary part of effective index (neff) variation as a function of wavelength for d=25 nm, 50 nm, 100 nm and 200 nm, for Ag/SiO ₂/Ag (solid line) and Ag/Si/Ag (circled line) waveguides. The refractive indices (n) of SiO ₂ and Si are assumed to be 1.46 and 3.4, respectively. A schematic of MIM waveguide is shown in (a) inset.

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where ε_d and ε_m are dielectric functions of dielectric and metallic materials, respectively. We assume silver for the metallic cladding layers, characterized by the Drude model

where the material-dependent constants ω_p=1.38×10¹⁶ Hz and γ=2.73×10¹³ Hz are the bulk plasma and damping frequencies, respectively, and ε(∞)=3.7 [13]. Propagation constant (β) is usually presented by a dimensionless effective index n_eff=β/k ₀ for the guided modes, where k ₀ is the vacuum wave-vector. Variation of the effective index values with wavelength is depicted in Fig. 1 for different dielectric materials and thicknesses (t). Figure 1 shows that the frequency dependent effective index associated with an MIM waveguide, highly depends on both t and the dielectric material (n). Therefore, a periodic change in core thickness as well as in dielectric material may result in an effective index modulation needed to realize Bragg scattering around a desired frequency.

For an MIM waveguide with a dielectric layer of refractive index n, Z₀ associated with the supported Transverse Magnetic (TM) mode is given as follows

where waveguide width w=1µm is assumed in the y-direction. Note that Eq. (4) is valid since electromagnetic field penetration into the metal exponentially decays from the dielectric-metal interface, and the field profile inside the dielectric region is almost uniform at wavelengths far from the surface plasmon resonance. Equation (4) is the general form of the MIM waveguide characteristic impedance $\left(Z=\frac{\beta d}{\omega {\epsilon }_{0}w}\right)$ derived in [14], which is correct assuming air (n=1).

The structures shown in Fig. 2 are the building blocks for both Thickness-Modulated (ThM) and Index-Modulated (InM) Bragg reflectors, where the thickness and dielectric constant of the middle waveguide is different from those of the left and right sections. Assuming n ₁=2, t ₁=140 nm and n ₂=1, effective index and characteristic impedance at λ=1550 nm become n _eff1,1=2.3090+0.0036i and Z₁=Z₃=30.5 Ω.µm, respectively. Iteratively solving for t ₂, so that Z₂(t ₂)=Z₁, we find t ₂=60 nm, for which n _eff,2=1.3201+0.0033i and Z₂(t ₂)=29.9 Ω.µm. In order to validate Eq. (4), we simulated the structures shown in Fig. 2(a) using FDTD, where impedance matching condition is satisfied using Eq. (4). Propagation of light through the discontinuities suffers negligibly due to the impedance matching. For the structure shown in Fig. 2(b), β×d is the same for all sections, however, the FDTD simulation demonstrates considerable reflection at the discontinuities. Therefore, n ² must be considered in the denominator of Eq. (4) for correct calculation of Z.

 

Fig. 2. FDTD simulation results for a thickness modulated waveguide designed based on (a) our characteristic impedance matching technique provided in Eq. 4 and (b) the characteristic impedance matching technique introduced in [14]. The schematic of both structures with actual physical dimensions, effective indices, and characteristic impedances are shown on top of each FDTD simulation result.

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Fig. 3. (a) Schematic of a generalized Bragg reflector with dielectric index and thickness modulations. (b) Bandgap strength variation as a function of thickness for ThM Bragg reflector [t ₂=t ₁/2, n ₁=n ₂=1.46] and InM Bragg reflector [n ₁=1.46, n ₂=3.4, t ₁=t ₂] from TMM and FDTD simulations. (c) Variation of sin ^-1{abs[(Z₁-Z₂)/(Z₁+Z₂)]} versus t ₂.

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3. MIM Bragg reflector formulation

An MIM Bragg reflector is shown in Fig. 3(a). Note that in ThM and InM Bragg reflectors, n ₁=n ₂,t ₁≠t ₂ and n ₁≠n ₂, t ₁=t ₂, respectively. Field analysis of the discontinuity problems (physical dimensions or material discontinuity) can be simplified by modal analysis technique. Considering the dominant propagating mode inside the waveguide, we can find the electric and magnetic fields, which are correlated together by characteristic impedance defined in Eq. 4. The dominant mode in the MIM waveguide is the fundamental TM mode, which has insignificant E_z component. Therefore, this mode can be considered as a quasi-TEMmode [14]. Electromagnetic wave propagation inside an MIM Bragg reflector can be modeled based on the boundary conditions for transverse fields at the discontinuities resulting in a transfer matrix (A) representing the reflection and transmission at the discontinuities, which is defined as

where X is the position of the discontinuity. l and r refer to the left side and the right side of the discontinuity in the heterostructure, respectively, as shown in Fig. 3(a). The total transmission spectra can be calculated by Π_iA_i. Note that this characteristic impedance formulation is general and is applicable for both ThM and InM Bragg reflectors. Figure 3(b) compares the bandgap strength Δω/ω_b calculated using Eq. 5 and FDTD simulations for two ThM and InM Bragg reflectors, where Δω and ω_b are the FWHM of the reflection peak at the central gap frequency and the central gap frequency, respectively.

The first band gap Δω/ω_b∝sin ^-1[(n _eff,2-n _eff,1)/(n _eff,2+n _eff,1)] in a 1D dielectric photonic crystal depends on the effective index contrast, where n _eff,1 and n _eff,2 are the effective indices associated with the two alternately stacked waveguides and the number of periods of the alternating waveguides (N) is assumed to be infinite [15]. Figure 3(c) demonstrates the variation of sin ^-1[|(Z₁-Z₂)/(Z₁+Z₂)|]. Interestingly, The bandgap strength in the case of the ThM and InM Bragg reflectors increases about 35% and 0%, as t ₂ increases from 40 nm to 200 nm in both Figures 3(b) and (c). Therefore, the MIM characteristic impedance defined in Eq. 4 can be utilized to predict the bandgap strength of MIM Bragg reflectors.

4. Design of MIM Bragg reflectors

To optimize the material and geometric parameters of MIM Bragg reflector structures, the design problem must be formulated to maximize the objectives and meet the constraints while effectively exploiting the design space based on the analysis presented in Section 3. In MIM waveguides decreasing t or increasing n both result in higher effective index value, which can be used to create effective index contrast and wide bandgaps in MIM Bragg reflector, but they also increase the propagation loss inside the structure, as shown in Fig. 1(b). In order to facilitate the design optimization, we have developed a dimensionless figure of merit (FOM), the propagation length over the Bragg reflector length

An important characteristic for Bragg reflectors is to find a structure with acceptable reflectivity and the lowest loss, which can be formally posed as

Note that we did not include bandgap strength in Eq. 6, because based on the application, wide or narrow bandgap may be required. Therefore, the maximum peak reflectivity in the bandgap is more appropriate for the definition of the figure of merit.

Higher refractive index materials result in higher propagation loss inMIMwaveguides. However, the characteristic impedance contrast $\left(C_Z=\mid \frac{Z_1-Z_2}{Z_1+Z_2}\mid \right)$ does not change with n in the case of ThM Bragg reflectors, except at high frequencies, where Real{ε_m(ω)}≈-ε_d. Therefore, in order to increase F for ThM Bragg reflectors, low refractive index dielectric should be used.

 

Fig. 4. FOM variation as a function of number of periods and (a) n ₂ for InM Bragg reflector (n ₁=1.46, t ₁=t ₂=100 nm) and (b) t ₂ for ThM Bragg reflector (n ₁=n ₂=1.46, t ₁=100 nm) for fixed central wavelength of 1550 nm. Transmission (TMM and FDTD) and Reflection (TMM) spectra versus wavelength for optimum (c) InM (n ₂=2.6) and (d) ThM (t ₂=50 nm) Bragg reflectors with N=6.

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Figures 4(a) and (b) demonstrate the variations of F as N and the characteristic impedance contrast change in InM and ThM Bragg reflectors for a fixed central wavelength λ_b=1550 nm. As shown in Figures 4(a) and (b), for a fixed C _Z, the number of periods has an indirect relationship on F due to the increase in the propagation loss. In both ThM and InM Bragg reflectors, the maximum achievable F increases with the characteristic impedance contrast. In the InM Bragg reflector, when the minimum reflection criteria is satisfied, a change in C _Z for a fixed N results in an insignificant change in F. Therefore, the Bragg performance is not degraded with the discrete values of available dielectric constants. However, in the ThM Bragg reflector, F rapidly decreases with decreasing t ₂ (increasing C _Z), which is due to increasing loss in narrower MIM waveguides as depicted in Fig. 1(b). The transmission and reflection spectra of the optimum designs with N=6 and λ_b=1550 nm are shown in Figures 4(c) and (d) for InM (F≈13) and ThM (F≈11) Bragg reflectors, respectively. The two Bragg reflector structures demonstrate almost the same performance. In addition, the TMM calculations based on MIM waveguide characteristic impedance and FDTD simulations are in good agreement.

5. Conclusion

In this paper, we presented a characteristic impedance modeling for plasmonic waveguides with Metal-Insulator-Metal (MIM) configuration. Leveraging the MIM impedance model, we presented a general Transfer Matrix Method model for MIM Bragg reflectors and validated our model against FDTD simulations. The results indicate that both periodically stacked dielectric layers of different thickness or different material can achieve the same performance in terms of propagation loss and minimum transmission.