A metal/multi-insulator/metal waveguide plasmonic Bragg grating with a large dynamic range of index modulation is investigated analytically and numerically. Theoretical formalism of the dispersion relation for the present and general one-dimensional gratings is developed for TM waves in the vicinity of each stop band. Wide-band and narrow-band designs with their respective FWHM bandwidths of 173.4 nm and < 3.4 nm in the 1550 nm band using a grating length of < 16.0 µm are numerically demonstrated. Time-average power vortexes near the silica-silicon interfaces are revealed in the stop band and are attributed to the contra-flow interaction and simultaneous satisfactions of the Bragg condition for the incident and backward-diffracted waves. An enhanced forward-propagating power is thus shown to occur over certain sections within one period due to the power coupling from the backward-diffracted waves.
© 2010 Optical Society of America
Nanophotonics using surface plasmon polaritons (SPPs) that are light waves coupled to oscillations of free elections in a metal  has been attracting much renewed attention worldwide in last decade. The interest in nanophotonics using SPPs, also called plasmonics, stems from its nanometer-scale confinement that overcomes the lower limit of the lateral mode size of a conventional dielectric waveguide [2, 3]. This makes it possible in realizing optical nanocircuitry and in eventually bringing light into nanoelectronics , all using mature semiconductor manufacturing technologies. The confinement and guidance of electromagnetic energy in the sub-wavelength and even toward sub-100-nm scale structures thus become fundamental yet paramount in further down scaling conventional photonic devices.
Among various plasmonic devices, plasmonic Bragg gratings/reflectors realized in metal heterostructures , insulator-metal-insulator (IMI) [6–8], metal-insulator-metal (MIM) [9–12], or metal-dielectric-air  configurations have been drawing continuing interest in recent years. While ultra narrow band (below 1 nm up to 5 nm) plasmonic Bragg gratings have been experimentally demonstrated using the IMI configuration for long-range SPPs [6, 7], narrow band MIM-based structures seem less studied. In , a 3rd-order waveguide Bragg grating in the MIM configuration was numerically demonstrated where a full-width-at-half-maximum (FWHM) bandwidth of 61 nm was reported with a total grating length of 30.6 µm. More recently, we proposed a metal/multi-insulator/metal (MMIM) waveguide plasmonic grating where design examples with the grating lengths of ≤ 6.8 µm and the FWHM bandwidths of approximately 9 nm in the 1310 nm band were numerically demonstrated .
In this paper, general formalism and transmission characteristics of the MMIM waveguide plasmonic Bragg grating with sinusoidal periodic variations are developed and investigated. The present study starts with the simplification of the two-dimensional (2-D) structure into one-dimensional (1-D) longitudinally-modulated medium where the wave equation may be cast into the canonical form of Hill’s differential equation for transverse magnetic (TM) waves. The dispersion relation and its properties are then derived and discussed in Section 3. The dynamic range of the complex effective permittivity with varying structure parameters is treated at length from which wide-band and narrow-band designs are devised, all using the MMIM configuration. Detailed discussions on the transmission characteristics of the narrow-band case are presented, in particular the power vortexes and the enhancement of the forward-propagating power that are found to occur in the stop band.
2. Structure Description
Figure 1 depicts one unit cell of the proposed MMIM waveguide Bragg grating. It consists of a metal (assumed to be silver) slot waveguide within which a sinusoidally width-modulated silicon (Si) stripe is sandwiched by two lower refractive index regions. The x-dependence of the Si stripe width is described by
where A 0 is the unperturbed Si stripe width, Λ denotes the grating period, and AM = hA 0 with h being the width modulation depth (0 < h < 1). For simplicity, the silver boundaries are conformal to the Si stripe at a fixed gap distance w gap. In between the Si stripe and the silver is the lower index region such as silica. The guided mode supported by the MMIM waveguide is symmetrically excited by a 450 nm-wide Si waveguide followed by a 700 nm-long linearly-tapered MMIM transition section. The gap width associated with the input/output linearly-tapered MMIM transition is fixed at 80 nm for achieving a high transmission efficiency.
3. Formulation of the Electromagnetic Problem
The geometry with sinusoidal periodic variations shown in Fig. 1 may be treated as a cascade of an infinite number of linear MMIM sections of varying widths. By calculating the effective permittivity ε eff (or equivalently effective refractive index N eff) of the fundamental mode associated with every single linear section, the grating geometry may be described completely in terms of the longitudinally-modulated medium,
where is the average value of ε eff(x), δM is the effective permittivity modulation depth, K = 2π/Λ â x is the grating vector, and x = x â x is the position vector with â x being the unit vector in the x direction. The 2-D problem may thus be treated as a 1-D grating (i.e., ∂/∂y = 0 and ∂/∂z = 0) with sinusoidally-stratified dielectric medium and can be analytically described by Hill’s differential equation for TM waves 
where ξ = πx/Λ, f(ξ) = (1−δMcos2ξ)−1/2 I(x),
The I(x) in the expression of f(ξ) denotes the longitudinal field variation associated with the z-directed magnetic field and satisfies the following differential equation
where k 0 is the free-space wave vector. Note that when the modulation is absent, δM = 0 and all the coefficients cn vanish except c 0.
Taking Fourier transform of both sides of Eq. (3) yields the following homogeneous difference equation
where (Λ/π)F(κΛ/π) is the Fourier transform of f(ξ) with κ being the Bloch wave vector. An infinite set of homogeneous difference equations for any κ may then be obtained using the fact that
The general form of the dispersion relation in the vicinity of and κΛ/π = q, where p is a positive integer and q an integer, may be obtained based on the two-mode approximation where a 2×2 truncation in the infinite set of difference equations generated from Eq. (7) is applied. For a non-trivial solution to exist, the determinant of the resultant 2×2 matrix must vanish, yielding
where u = (p−q)/2, v = −(p+q)/2, u, v ∈ ℤ, cp is the Fourier series coefficient given in Eq. (4), and the shorthand notation Dm, m ∈ ℤ is given by
Equation (9) is the required dispersion relation since it is essentially an equation for the Bloch wave vector κ. It is noted that in the derivation of Eq. (9), the propagation in the direction of periodic variation of the relative permittivity is assumed for simplicity. Using the fact that
in the vicinity of κΛ/π=q, Eq. (9) can be further simplified to
Equation (12) exhibits a hyperbolic connection,centered at , between and κΛ/π.
Special cases that may be of primary interest are those when p equals q. They correspond to stop bands along m = 0 unperturbed line in the versus κΛ/π diagram. The dispersion relation in the vicinity of is explicitly given by
and can be further reduced to Eq. (12) with q = p. Physically, Eq. (13) represents the resonant coupling between the plane wave components m = 0 and m = −p. Since the center of the stop band occurs at , the Bragg wavelength λB may be estimated by
provided . It can be further reduced to λB≈2√ Λ/p in the limit as δ 2 M becomes negligibly small.
The general theoretical formalism presented above provides physical insights into not only the present waveguide plasmonic Bragg grating but any grating structure with a 1-D sinusoidal dependence of the index variation in the direction of propagation. From Eq. (12), the normalized bandgap is given by , which is two times the distance between the vertex and the center of the hyperbola in the versus κΛ/π diagram. It is a strong function of the grating period Λ, average effective index , and index modulation depth δM for a fixed operating wavelength. The largest bandgap occurs at since, in general, the first-order Fourier series coefficient has the largest value compared to other higher-order coefficients. The bandgap around for p ≥ 2 becomes smaller with the decreasing higher-order coefficient cp. Therefore, and similar to that of the conventional dielectric grating, the narrow-band characteristic associated with the MMIM configuration can be achieved by increasing Λ, increasing , or minimizing δM so as to weaken the grating coupling strength.
4. Results and Discussions
As the theoretical formalism presented above is based upon the effective permittivity standpoint, the effective permittivity calculation and its dynamic range with varying structure parameters are fundamental aspects that need to be investigated. Since the waveguide plasmonic Bragg grating is symmetrically excited, only the odd vector parity mode [Ex(y) is odd, Hz(y) and Ey(y) are even functions] was considered. Figure 2 illustrates the isometric plots of the real and imaginary parts of the N eff associated with the fundamental odd vector parity mode for an unperturbed linear MMIM waveguide with varying w gap and w Si at a free-space wavelength of λ 0 = 1550 nm. The refractive indices n Si = 3.5 and n silica = 1.46 were assumed and the dielectric function of the silver was taken as a complex fit to the empirical data reported in . The N eff values were calculated based on the rigorous transmission-line network approach in conjunction with the transverse resonance condition .
In general, for a fixed w gap, Re[N eff] increases and the magnitude of Im[N eff] (|Im[N eff]|) decreases with the increase in w Si. This must be the case since the increase in w Si comes with the increasing fractions of the total electromagnetic energy being confined in the Si stripe. On the other hand, for a fixed w Si, the decrease in w gap would lead to larger Re[N eff] and |Im[N eff]|. This is due to increasing fractions of the total electromagnetic energy entering into the metal upon decreasing w gap, resulting in slightly larger Re[N eff] and higher |Im[N eff]| that accounts for the ohmic loss inside the metal. Hence for a fixed w Si, MMIM configuration bears the same characteristic as that of conventional MIM plasmonic waveguide where Re[N eff] and |Im[N eff]| increase with the decreasing dielectric thickness.
The presence of the high-index Si stripe adds another degree of freedom to the index modulation in plasmonic grating designs. While the largest Re[N eff] difference for all fixed w Si values in Fig. 2(a) is merely 0.1874 [2.6735 at (w gap, w Si) = (160,100) to 2.8609 at (w gap, w Si) = (50,100)], geometries with a fixed w gap and varying w Si may offer a larger dynamic range of index modulation of up to 0.7692 [from 2.6735 at (w gap, w Si) = (160,100) to 3.4427 at (w gap, w Si) = (160,400)]. The rate of change of Re[N eff] increases rapidly with w Si ≤ 200 nm while the magnitude of Im[N eff] remains < 5.042×10−5 for w Si ≥ 200 nm.
To fully utilize the large dynamic range of the N eff over the spans of w gap and w Si shown in Fig. 2, both wide-band and narrow-band designs operating in the 1550 nm band are numerically demonstrated. The Bragg wavelength is estimated using Eq. (14) and the N eff calculations shown in Fig. 2, which will be elaborated in detail later for the narrow-band designs. Figure 3 shows the transmission spectra of an MMIM grating with Λ, A 0, AM, and w gap being 950, 250, 90, and 130 nm, respectively, and different numbers of periods. The transmission spectra were obtained based upon 2-D Finite Element Method simulations (Comsol Multiphysics) with the maximum mesh size of 20 nm in the entire computational domain so as to achieve the power convergence to the 4th decimal digit. The refractive indices of Si and silica were assumed constant across the spectrum of interest. For the 9-period case, the stop band is centered at λ 0 = 1552.3 nm and the FWHM bandwidth is 173.4 nm. The pass band ripple becomes smaller as the operating wavelength deviates further away from the stop band.
The narrow-band optical filtering can also be realized using the MMIM configuration. As previously noted in Eq. (12), the bandwidth is proportional to the Fourier series coefficient cn, n = 1,2, ⋯. Accordingly, a higher Bragg order would lead to a narrower stop bandwidth at a sacrifice of a longer grating length for a fixed Bragg wavelength. Thus a trade-off between the Bragg order and the grating period always exists if a narrower bandwidth is desired and the constituent materials are unaltered. With the understanding of the narrow-band characteristics, three design examples for narrow-band optical notch filtering were numerically demonstrated. In these examples, the unperturbed Si width A 0 and the constituent materials are identical to those used in the wide-bandwidth case presented above. All cases have 11 periods and provide a minimum extinction ratio of over 16.08 dB at the center of their respective stop bands. In particular, the minimum transmission in case 1 is smaller than −37.60 dB. Because of the sinusoidal apodization, the largest passband ripple among these cases is reduced to less than 0.1 dB over the wavelength span of 1500 nm to 1600 nm.
The Bragg condition may be evaluated using a staircase approximation to the sinusoidally-varying effective refractive index profile. Table 1 gives the detailed information on an 11-section approximation to case 2 presented in Fig. 4. The average effective refractive index may be calculated using the expression √ ≈ ∑n i=1 Re[N eff, i] · gi, where N eff, i is the effective mode index of the fundamental mode associated with the i-th unperturbed section of length li and gi = li/Λ is the corresponding weighting factor. The √ value in case 2 is then found to be 3.3532. Accordingly, the 6th Bragg order is satisfied with the calculated Bragg wavelength λB = 1570.41 nm, which is close to the simulated value of 1550.4 nm. Decreasing w gap or increasing AM enhances the index modulation. When incorporated with the Λ adjustment, w gap and AM add additional degrees of freedom for the Bragg wavelength design. This is demonstrated in cases 1 and 3 in Fig. 4. It is worth mentioning that the FWHM bandwidth could be fairly insensitive to the variation in the period for both the narrow-band and wide-band designs, although the rate of change is larger for the former case due to its narrow-band nature. Take the case 2 in Fig. 4 as an example, the FWHM bandwidth increases (decreases) approximately 0.5 nm (0.65 nm) for a decrease (an increase) of 50 nm in the period while in the wide-band design, it increases (decreases) by approximately 13 nm (4 nm) for a 50-nm decrease (increase) in the period.
The transmission characteristics of the present waveguide plasmonic Bragg grating may be understood through the investigation of the time-average norm power P norm, where
The P norm distributions inside the 6th unit cell of case 2 in Fig. 4 at λ 0 = 1600 nm are shown in Fig. 5(a) as a representative case. Power interchange between the Si stripe and the silica gap regions, as indicated by the arrows in the figure, is observed and occurs repeatedly throughout the entire grating. A quantitative description of the power interchange may be obtained through the calculation of the x-directed time-average power in the Si stripe and the silica gap regions over the span of one period. Figure 5(b) depicts the normalized x-component of the time-average power Px(x), where
and P in is the input power, at λ 0 = 1600 nm as a function of the propagation distance within the 6th unit cell. The co-flow coupling of Px between the Si stripe and the silica gaps is immediately apparent. Px in the silica gap regions (P x, gap) reaches its maximum at the narrowest Si stripe width where Px in the Si stripe (P x, Si) has the minimum. Note that the P x, gap shown in Fig. 6 is the sum of the normalized x-directed power in the upper and lower gap regions. Although not shown in this paper, the integration of the y-directed time-average Poynting vector over y at each x position within one period vanishes because of the reflection symmetry of the structure with respect to the x axis and the symmetric excitation mentioned earlier. Hence the summation of P x, gaps and P x, Si would effectively represent the normalized total time-average power at each x position and remains above 0.976 inside the 6th cell, which confirms the low-loss nature in the pass band.
The co-flow interaction loses its hold in the stop band where both P x, Si and P x, gaps exhibit forward- and backward-propagating characteristics within each unit cell. Fig. 6 shows the P norm distributions and the normalized x-directed time-average power within the 6th unit cell at the Bragg wavelength of 1550.4 nm. The 8 vortexes near the edges of the Si stripe in Fig. 6(a) correspond to the x positions of the crests and valleys of P x, Si and P x, gaps curves in Fig. 6(b). The co-existence of the positive (forward-propagating) P x, Si (P x, gaps) and the negative (backward-propagating) P x, gaps (P x, Si) at the same x position indicates the occurrence of contra-flow coupling, as has been suggested by the hyperbolic form of the dispersion relation, Eq. (12). On the other hand, a one-to-one correspondence exists between the x positions at which the P x, Si (P x, gaps) undergoes zero-crossing in Fig. 6(b) and the 4 power nulls along the center line of the Si stripe (at the silver-silica interfaces) in Fig. 6(a). The zero crossings shown in Fig. 6(b) are formed due to the cancellation of the forward- and backward-propagating Px at these x positions. Likewise, power nulls found along the x axis in the Si stripe and at the silver-silica interfaces are where the cancellations of x- and y-directed time-average power occur simultaneously.
It is immediately apparent that P x, Si around x = 0.4Λ exceeds unity in Fig. 6(b). The physical interpretation may be made in terms of the behavior of the backward-diffracted waves at the Bragg wavelength. The plane-wave spectrum calculated at the end of the 6th unit cell as a representative case is shown in Fig. 7. At this x position, the plane-wave component with ky = 0 is dominant whereas another two components at ky/(k 0√) = ±0.14324 are also appreciable. The diffracted angles θ associated with these components are 0° and ±8.24° (measured from the −x axis), respectively. In other words, the diffracted wave vectors are either parallel to or nearly parallel to the −x axis and are governed by the Floquet condition. It is worth mentioning that the plane-wave spectrum varies with the x position. Specifically, slight changes in the diffracted angles and the normalized intensity associated with each plane-wave component were observed across one period. This is expected since the interference pattern produced by all the Floquet waves has to be consistent with the sinusoidally-modulated index variation that produces them in the first place.
Consider the backward-diffracted wave with ky = 0 (or equivalently θ = 0°), its wave vector is found to be σ = |β inc−6K| = 1.33969×107 (rad/m), where β inc is the incident wave vector, and the corresponding phase velocity refractive index is Nσ = 3.3057. Using these values and the fact that the grating slant angle ϕ = 0° (seen by the backward-diffracted wave) measured from the −x axis, the backward-diffracted wave is found to satisfy the general Bragg condition
in which λ 0 = 1550.4 nm, Λ = 1405 nm, and m = 5.9914 ≈ 6. Note that the refractive index of the backward-diffracted wave Nσ = 3.3057 is very close to the average effective refractive index of the grating, √ = 3.3069 (obtained at λ 0 = 1550.4 nm and m = 6). This also ensures the satisfaction of the 6th Bragg order. Hence the backward-diffracted wave satisfies the same Bragg condition as does the original incident wave (β inc), producing a forward-diffracted wave that is characterized by σ′ = σ −6K. For other waves with diffracted angles θ≠0°, the general Bragg condition is nearly satisfied because of their small angles of diffraction with respect to the −x axis. The backward-diffracted wave couples its power back to the forward-diffracted wave whose power would consequently add up to that of the original incident wave, thus increasing the normalized forward-propagating power to be over unity. This happens to all unit cells before the 7th. Beyond the 6th unit cell, the normalized P x, Si and P x, silica do not exceed unity due to decreasing forward-propagating power flow. Meanwhile, the total normalized power P x, total does obey the power conservation law and decreases with the increasing propagation distance. Hence the formation of the P norm vortexes seen in the figure is a direct result of simultaneous satisfactions of the same Bragg condition for the incident and backward-diffracted waves in the metal/multi-insulator/metal configuration. As the operating wavelength gradually deviates from λB, the vortexes and P norm nulls gradually fade away since the Bragg condition can no longer be satisfied completely, thus allowing more power transmission inside the Si stripe. The vortexes vanish completely in the pass band.
The theoretical formalism of the dispersion relation in the vicinity of stop bands and the transmission characteristics of metal/multi-insulator/metal waveguide plasmonic Bragg grating have been developed and analyzed. The general dispersion relation near the fundamental and higher-order stop bands is analytically derived based upon the effective permittivity standpoint and the Fourier transform of Hill’s differential equation. The present formulation may also be applicable to any 1-D sinusoidally-stratified medium for TM waves. Physical insights into the design principles are, therefore, obtained and good agreement is found between the calculated and simulated values of the Bragg wavelength.
The present MMIM configuration is shown to inherently have a wide dynamic range of the effective refractive index (up to 0.7692). This enables it to support both wide-band and narrow-band applications. The results show that the FWHM bandwidths of 173.4 nm and < 3.4 nm may be achievable with a grating length of < 16.0 µm, all using the MMIM configuration and the same material system (Si, silica, and silver).
For the narrow-band case, co-flow and contra-flow interactions between the time-average power inside the Si stripe and silica gap regions are shown to occur in the pass band and the stop band, respectively. The time-average power vortexes found near the silica-Si interfaces are formed due both to the contra-flow coupling and the satisfaction of the Bragg condition for the backward-diffracted waves. The backward-diffracted wave then couples its power back to the forward-propagating wave and would in turn enhance the forward-propagating power. This may explain the phenomenon that the normalized forward-propagating power exceeds unity over certain sections within one period.
The author wishes to thank Dr. Shun-Der Wu of ASML Taiwan Ltd. for valuable discussions. This research was supported by Grant NSC-98-2218-E-008-001 from the National Science Council, R.O.C. (Taiwan).
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