We present a 2-D plasmonic crystal design with visible band-gap by combining a 2-D photonic crystal with TM band-gap and a silver surface. Simulations show that the presence of the silver surface gives rise to an expanded band-gap. A plasmonic crystal defect cavity with Q ~300 and mode volume ~1.9x10−2 (λ/n) 3 can be formed using our design. The total Q of such a cavity is determined by both the radiative loss of the dielectric component, as well as absorption loss to the metal. We provide design criteria for the optimization of the total Q to allow high radiative or extraction efficiency.
© 2014 Optical Society of America
Sub-wavelength scaled metallic resonant devices, known as plasmonic nano-cavities, can greatly alter dipole radiation characteristics [1–5]. In addition to dramatic concentration of optical fields, the creation of well-defined resonant modes in these cavities should provide both spatial and spectral control over light-matter interactions in a manner similar to what has been demonstrated in high Q photonic cavities . Such selective enhancement allows deterministic observation of phenomena including enhanced spontaneous emission, ultra-low threshold lasing and strong coupling between photonic and electronic states [7–9]. Some of these phenomena have also been demonstrated with plasmonic cavities [3, 10].
However, in the formation of these plasmonic cavities, there is much room for improvement of the cavity geometry and structure to ensure well controlled resonant frequencies and lower cavity loss. In this work, we demonstrate a cavity design which combines a photonic energy gap formed by the periodic arrangement of a dielectric photonic crystal and the highly concentrated fields of the surface plasmon polariton at a metal surface to achieve resonances with both high quality factor (Q) and low mode volume.
The photonic energy gap generated by the periodic variation of refractive index in structures known as photonic crystals has been applied for the design of various photonic devices [11, 12]. The coupling of 1-dimensional dielectric photonic crystals to surface plasmons has been previously demonstrated [13–15].While 2-dimensional photonic crystal rods have been used to modulate the propagation of surface plasmon polaritons , the hybridization of 2-D dielectric photonic crystal modes and surface plasmons has not yet been demonstrated. Moreover, sub-wavelength optical cavities formed by 2-D plasmonic crystals have not been realized.
In this work, we analyze the formation and potential performance of a 2-D plasmonic crystal. In comparison with 1-D plasmonic crystals, such a structure produces a complete band-gap for surface plasmon polaritons and can provide a platform for the design of a wide range of photonic-plasmonic devices, such as plasmonic cavities as well as waveguides or reflectors.
2. Modification of band-structures
The underlying dielectric photonic crystal used in the study is formed by hexagonally arranged high index circular rods embedded within a low index matrix; such a structure has been shown to provide sufficiently large TM band gaps . The particular structure we have studied contains high index dielectric rods with a refractive index of 2.5 embedded in an air matrix. The material parameters are chosen to be close to the refractive index of high index transparent dielectrics such as GaN, TiO2 or diamond. The dielectric rods are arranged into a hexagonal crystalline structure with a lattice constant (a) of 250nm. The radius (r) and height (h) of these rods is 70nm (r = 0.28a) and 250nm respectively. The height was chosen to eliminate higher order guiding modes around the frequency of interest. A schematic drawing of the structure is shown in Fig. 1(a).
Using Lumerical, a commercial finite difference time domain (FDTD) simulator, we calculated the photonic band structure of the dielectric rod array, as shown in Fig. 1(b). Several photonic bands are shown in the simulated spectrum within the frequency range of interest. By analyzing the magnitude of the out-of-plane (z-polarized) electric field component, one can distinguish between TE and TM modes, as labeled in Fig. 1(b). A clear TM band-gap is found from 511THz to 557THz, defined by the two lowest frequency TM bands [labeled as TM1 and TM2 in Fig. 1(b)].
Through further investigation into the electric field distribution of the two TM modes on either side of the fundamental band-gap, one can find signature behavior, similar to what pertains to the formation of an electronic band gap in atomic crystals. Figure 2(c) shows the electric field intensity distribution for TM1, TM2 and TE1 bands near the M point of the photonic crystal. The cross-sectional field distribution of both TM1 and TM2 agrees with that of fundamental TM waveguide modes, and the in-plane field distributions resemble standing waves scattered by the rods. At the band edges, periods of the standing waves equal the lattice plane distances of the photonic crystal. The two-fold degeneracy of the standing wave is then broken by the periodic structures, as the lower frequency band tends to be concentrated in the high index rods and the higher frequency bands stay in air. The difference between the spatial distribution of the two bands results in a difference in response to a proximal metal surface.
As the metal layer (silver, modeled with optical parameters measured by Johnson and Christy ) approaches one side of the photonic crystal (i.e. as the parameter d in Fig. 2(a) decreases), the TM modes begin to hybridize with surface plasmon modes, increasing the effective index of the modes. As a result, the frequencies of all TM modes decrease as the metal approaches the rod array [Fig. 2(b)]. A similar effect has been demonstrated in studies on 1D hybrid plasmonic crystals . This contrasts with the behavior of the TE band, which has a polarization perpendicular to the surface plasmon polariton modes and is therefore not strongly affected by the presence of the metal. Because the electric field in TM1 is more concentrated in the high index rods, it experiences a stronger effective index increase with the presence of the metal layer, and hence the amount of mode shifting in TM1 is larger than in TM2. This effect increases the TM band gap as the metal approaches more closely to the dielectric rods.
A plasmonic crystal is formed when the metal-dielectric spacing is significantly smaller than inverse of the exponential decay constant of the surface plasmon mode (10 - 100 nm in our case, depending on the frequency of the mode). The plasmonic band-gap in the simulated structure with d = 10 nm ranges from 441THz to 516THz (581nm to 680nm in vacuum wavelength). This hybridization between TM modes and surface plasmons can be further explored by comparing the field distribution of the hybrid plasmonic crystal to the original photonic crystal. Although the x-y field distributions shown in Fig. 2(d) remain similar to those without metal in Fig. 2(c), the z field profiles of the TM modes are strongly altered. The hybridization draws the field onto the surface of the metal and concentrates the field in the small gap between the metal and the high index rods, resulting in a highly concentrated optical intensity. Both this highly concentrated field and the expanded band-gaps are beneficial for designing high Q and low mode volume cavities.
3. Cavity design and optimization
To create a cavity from a plasmonic band gap structure, we introduced a defect into the center of the plasmonic crystal. In previous photonic crystal cavities formed of dielectric rods, the defect was often produced by modifying only one lattice site to achieve a small mode volume [17, 19]. In order to create highly concentrated modes in the dielectric-metal gap, the mode needs to be well confined within the high index region. Therefore, we use a linear 3 (L3) defect-like cavity design with a relatively larger high index defect formed by connecting three enlarged dielectric rods in our structure, as shown in Fig. 3(a).
The defect is surrounded by 5 layers of dielectric rods with radius r on each side to form the photonic crystal barriers. The entire structure comprises a 13 by 11 hexagonal rod array. To simulate a more realistic device, we filled the air gap between the metal and photonic crystal with a low index spacer/emitter layer with a refractive index of 1.4, approximately equal to the index of refraction of transparent dielectrics such as SiO2, amorphous alumina, and organic thin films. Adding this spacer layer into the plasmonic crystal shifts the band-gap to 417-483THz (621-719nm in vacuum wavelength). Several electric dipoles polarized perpendicularly to the metal surface are embedded near the center of the low index spacer layer to emulate emitters. The simulated spectrum is shown in Fig. 3(b). For the cavity with rd = 120nm, two distinct cavity modes can be observed at 708 nm (424THz) and 638 nm (470THz) with Q = 207 and 197, respectively. The peak wavelengths of these modes can be shifted by changing rd, and the position of the band-gap can be identified in the spectrum via this mode shifting. As the defect size rd increases from 120nm to 140nm [see Fig. 3(b)], both modes shift toward higher wavelengths and a higher-order resonance can be observed on the short-wavelength side of the spectrum as it shifts into the band-gap. If the defect size is instead decreased, the modes shift to shorter wavelengths, and for rd = 90nm, the higher frequency mode shifts outside of the range of the band-gap and can no longer be observed in the spectrum.
Electric field distributions of the two modes of the cavity with rd = 120 nm are plotted as Fig. 3(c), showing a highly concentrated field within the low index spacer layer. The mode volume of the lower frequency mode at 708nm, with a calculated Q = 207, is estimated to be ~2.4x10−3 μm3, which is roughly 1.9x10−2 (λ/n) 3, where n = 1.4 is the refractive index of the material in which the emitter is embedded. The Q/V ratio of such a cavity is comparable to a cavity with Q ~32000 and V ~1 (λ/n) 3. Thus the periodicity of the 2-D dielectric has helped to define the resonant response of the cavity, while the metal component has produced the highly localized field that results in a high value of Q/V, despite the higher absorption loss to the metal. However, there is still further optimization required for these cavities to adjust the Q of the cavity so that we not only maximize the dipole radiation rate (near-field pattern), but also optimize the far field radiation rate, which is related to the extraction efficiency of the plasmonic cavity.
As has been previously noted for 1-D plasmonic cavities , a distinguishing feature of these metal-dielectric plasmonic cavities is the dominance of two major loss pathways in determining the total cavity Q. The cavity Q can be expressed as:20]. However, in a plasmonic cavity, which has inherently high absorption loss and therefore lower QA, this radiation is essential to produce sufficient external quantum efficiency. Moreover, as we maximize QR by changing the design of the structure, the total cavity Q starts to be dominated by the value of QA, and most of the radiated power is lost to metal absorption, rather than through extraction to the far-field.
In order to explore this further optimization on the enhancement of emission and its effective extraction efficiency, we model the system shown in Fig. 4(a), with a dipole located near the field maximum of the cavity mode (labeled near-field). The dipole z-position is located at the center of the spacer layer and the x-y position is as shown in the plot. Figure 4(b) shows the corresponding power radiation spectrum, normalized to the power radiated from a dipole of the same frequency embedded in a bulk dielectric with a refractive index of 1.4. The Purcell factor, defined as the enhancement of spontaneous emission rate by the cavity, at the resonant wavelength of the cavity mode, can be extracted from the peak value of the power spectrum to be ~410 at 708nm.
To study the far-field extraction efficiency, we simulate the far-field radiated power collected into a 0.9NA objective. The power transmission spectrum is also shown in Fig. 4(b) (labeled as far-field). Like the dipole radiation power, the far-field power is normalized by the power radiated into a bulk dielectric of index n = 1.4. Dividing the far-field transmitted power by the dipole radiation power leads to the extraction efficiency spectrum shown as the blue dotted curve in Fig. 4(b).
To alter the contributions of the power radiation rate and extraction efficiency, we modify the geometry of our 2-D plasmonic cavity in the way indicated in Fig. 4(a). By changing the radii, rT, of the dielectric rods that are closest to the defect (cavity) region (i.e., the two rods indicated by dashed lines in Fig. 4(a)), we can alter the far-field radiation of the resonant modes. The important parameters discussed above, Q, Purcell factor, and far-field extraction efficiency, are strongly correlated to the far-field radiation.
The upper panel of Fig. 4(c) shows the dependence of Q on rT of the dipole power radiation and far-field power radiation of the lower frequency. Q reaches a maximum value of 305 when rT/r = 1.33, which indicates a minimum of the radiation loss. The dipole power radiation, i.e. Purcell factor, follows the same trend as Q since it is in principle proportional to Q. However, the far-field power radiation, which is the product of dipole radiation and extraction efficiency, follows a very different trend. This observation indicates that the extraction efficiency is strongly altered by small changes in the cavity geometry. The lower panel of Fig. 4(c) shows the same dependence of Q and extraction efficiency at the peak of the lower frequency mode on rT/r. As anticipated, the extraction efficiencies show inverse correlation to the Q of the mode.
The inverse correlation of Q and extraction efficiency can be more clearly demonstrated as plotted in Fig. 4(d). We can further analyze this dependence as we express the extraction efficiency (Γext) as:Fig. 4(d), the simulated data show good agreement to this model and the fitting provides us with an internal efficiency of ~60% and a QA of ~430. Note that in this fit we approximate both Γint and QA to be constant while in fact they should both be dependent on the particular mode wavelength and field distribution of the modes. As the wavelength and field distribution do not vary strongly in response to change of rT, this approximation is valid in our case and a linear Q-Γext dependence is observed.
These simulations, and the previous discussion, highlight the additional design and optimization requirements and opportunities for plasmonic cavities that have an inherent associated loss. For example, if the desired outcome were maximum far-field power radiation toward all the upper half-space, then a different optimal condition should be obtained with the change of rT. Maximum far-field power radiation, proportional to cavity Q and extraction efficiency, occurs for QR = QA and Q = 0.5QA. The optimal rT for maximum far-field power radiation in the designed geometry then occurs with a Q~215 as we specify rT to be roughly 1.1r, as demonstrated in Fig. 4(c). Seeking maximum radiation into different collection numerical apertures, or simply requiring the highest QR would result in different optimal structures.
In conclusion, we have designed a 2D plasmonic crystal cavity by integrating a 2D photonic crystal with a proximal metal surface. FDTD simulations indicate that a 2D plasmonic band-gap can be formed by hybridizing a dielectric TM photonic band-gap structure with surface plasmon polariton modes on a metal surface. The formation of the plasmonic band-gap allows us to design a high-Q/V plasmonic cavity with a mode volume of less that 2 x10−2 (λ/n) 3.
Studying radiation from a dipole embedded in such a structure provides insights on the design criteria and new opportunities for tunability of these plasmonic cavities. Our calculations suggest that a Purcell factor greater than 400 is possible, and the spectral tolerance for maintaining high enhancement extends over a few nanometers due to the relatively low Q, compared to dielectric cavities. However, additional design optimization is needed to balance cavity enhancement with high extraction efficiency. Therefore, instead of pursuing high values of radiative Q alone, balancing spontaneous emission enhancement and extraction efficiency is the key to optimizing the performance of these cavities.
We gratefully acknowledge financial support from the NSF Nanoscale Science and Engineering Center under Grant No. NSF/PHY-06-46094, the use of the NSF National Nanotechnology Infrastructure Network facilities at Harvard University’s Center for Nanoscale Systems, the assistance of staff members in Harvard University’s Center for Nanoscale Systems and the use of the high-performance computing computer cluster at Harvard.
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