Open Access
Add to BibSonomyAbstract
Enhancements up to four orders of magnitude for electric intensity and three orders of magnitude for magnetic intensity are realized in a novel 2D L-shaped nanocavity. This structure makes full use of the dimension confinement, cavity resonance and tip enhancement to increase the electromagnetic intensity. An impedance matching model is developed to design this cavity by regarding the cavity as a load impedance where electromagnetic fields are maximally enhanced when maximum electromagnetic energy is delivered to the load impedance. Our L-shaped nanocavity promises a variety of useful functionalities in sensing, nonlinear spectroscopy and signal processing.
© 2016 Optical Society of America
1. Introduction
Strong light-matter interactions are of fundamental importance for solar cells [1,2], lasers [3,4], sensors [5, 6], all-optical switches [7, 8] and quantum information devices [9, 10]. Usually, in cavities with high reflectivity mirrors such as metallic mirrors (Fabry-Pérot cavities) and Bragg reflectors (photonic crystal cavities) [7,8], electromagnetic (EM) fields can be greatly enhanced. This kind of enhancement originates from the accumulation of multiple reflections inside the cavity, which reaches maximum when reflections are phase-matched. Due to the diffraction limit, however, further applications of traditional cavities are prohibited [11].
Compared with the cavity enhancement, optical focusing is more intuitive to enhance the EM fields by reducing the cross section of propagating optical modes while keeping the energy flux unchanged. However, the smallest point to which a lens or mirror can focus a beam of light is no less than half the wavelength owing to the diffraction limit. Surface plasmon polaritons (SPPs), excitations propagating at the interface between a metal and a dielectric, enable to guide or control EM fields in a dimension as small as few nanometers [12,13]. The use of SPPs can concentrate EM fields more efficiently and has resulted in a generation of nano-optical devices [11,14].
Nanoantennas, which can be viewed as Fabry-Pérot cavity along the long axis, can confine and enhance EM fields in an area much smaller than the diffraction limit [15]. Besides, the sharp tips of nanoantennas can induce localized plasmon resonance and enhance EM fields more efficiently. Nonetheless, the interaction area between free EM field and the nanoantenna is very small. The energy utilization efficiency of a single nanoantenna is actually very low. Hence, antenna arrays are always required to confine sufficient EM energy.
It is advisable to combine the cavity enhancement, SPPs focusing and tip enhancement to achieve large field enhancement. In this work, we propose a novel L-shaped nanocavity in a metal-insulator-metal (MIM) waveguide for EM field enhancement. In addition, an impedance-matching model is developed to design this cavity.
2. Design methodology
For a simple MIM waveguide with a width of w, the dispersion relation of the fundamental TM mode (Ex is odd, Ey and Hz are even) is expressed as [13]
where ε m and ε d are the permittivities of the metal and the dielectric, respectively. k m and k d denote the propagation constants, which are related to the propagation constant β as , where k 0 is the propagation constant in vacuum. By regarding the MIM waveguide as a transmission line, the characteristic impedance is written as [16] where w is the width of the dielectric and wz is the waveguide thickness in the z direction. According to the transmission line theory [17], the reflection coefficient of the voltage between Z 0 and a load impedance ZL is written asIf Γ is already known, ZL can be deduced from Eq. (3) as
In our model, we adopt closed MIM waveguides to enhance the EM field, which is treated as inductances in the transmission line. The structure and the equivalent transmission line are shown in Figs. 1(a) and 1(b), respectively. The corresponding impedance of the inductance is
where ω is the angular frequency of the EM field. Since Γ can be calculated by full-field simulation tools such as the finite-difference time-domain method (FDTD) or the finite element method (FEM), the inductance L 0 can be extracted by combining Eqs. (4) and (5). In our design, the operating wavelength is 1550 nm, the typical telecommunication wavelength. The dielectric is air and the metal is silver. The dielectric function of silver is characterized by the well-known Drude model as with ε ∞=3.7, ω p=9.1 eV and γ=0.018 eV. For the MIM waveguide with a width of w 0=40 nm, the inductance value is calculated to be L 0 = (1.82−0.030i) × 10−21 H by using the FEM in the commercially available software package COMSOL MULTIPHYSICS. Afterwards, the reflection spectrum is obtained from Eq. (3). The validity of this method can be checked by comparing the reflection coefficients with the FEM simulated results. As demonstrated in Figs. 1(c) and 1(d), the transmission line model (solid lines) matches well with the FEM simulation (open circles), which proves the accuracy of our method.
Fig. 1 (a) Closed MIM waveguide and (b) the equivalent transmission line. The waveguide terminal is regarded as a inductance. Γ is the reflection coefficient of the voltage. Comparison of both the real part (c) and imaginary part (d) of reflection coefficients between the transmission line model (solid lines) and the FEM simulation (open circles).
Considering the waveguide terminal as a load impedance , we can simplify the maximization of EM field enhancement in a cavity as a maximization of power delivery in an impedance network. The EM power inside the cavity is damped via Ohmic heat that is proportional to the EM fields intensity, which ensures that the maximal field enhancement corresponds to the maximal Ohmic damping and hence the minimal reflection.
The load impedance is a complex value. So, in order to match this load, both the real and imaginary parts of should be matched. This implies that a general matching configuration must have at least two degrees of freedom. In this case, a stub is added near an MIM waveguide terminal to form an L-shaped cavity as shown in Fig. 2(a). For simplicity, both the stub and the main trunk line have the same width of 40 nm. The height of the stub l and the distance between the stub and the waveguide terminal d offer two degrees of freedom. It should be noted that the geometry of the cavity can be changed to a tooth-shaped filter by opening the waveguide terminal [18, 19].
Fig. 2 (a) Schematic of the L-shaped nanocavity. (b) The equivalent transmission line of the cavity and (c) the simplified shunt circuit.
The stub are rounded by a radius r in our model, which is very important because charge accumulation is associated with the curvature radius. This radius depends on the grain size of the metal and should not be infinitesimal in reality. The influence of the curvature radius on the EM field enhancement will be covered in detail later. The initial value of r is set as 5 nm.
It is convenient to simplify the L-shaped cavity by an equivalent series network [16] as shown in Fig. 2(b). The network is formed by a series connection of two finite transmission lines with the same characteristic impedances Z 0 terminated by the same inductances L 0. The lengths of the two lines are d and l, respectively. The network scheme is further simplified as the equivalent form shown in Fig. 2(c). These two finite transmission lines are replaced with two effective impedances. As for the stub, the value of the effective impedance ZS 1 is obtained from the transmission-line theory by [17]
where β is the propagation constant of the MIM waveguide. The effective impedance of the waveguide terminal ZS 2 is similar to Eq. (6) by replacing l with d. Now the total input impedance of the whole cavity can be expressed asThe total reflection coefficient of the cavity is then obtained according to Eq. (3). It is clear that the reflection is zero only if Z in and Z 0 are equal.
3. EM field enhancement inside the L-shaped cavity
Now the design of the cavity is simplified as the optimization problem of a bivariate function of l and d. Combing Eqs. (3), (6), and (7), the calculated reflectivity is depicted in Fig. 3(a), where the reflectivity is minimal when l = 268 nm and d = 231 nm. To prove our transmission line model, the reflection spectrum is simulated with the optimized d and l using the FEM as shown in Fig. 3(b) (open circles). The maximal mesh size near the inflection point is one fifth of r to make the corner smooth. We can see that the model computed spectrum (solid line) matches well with the FEM simulated one, which proves the accuracy of our transmission line model. The reflection spectrum reaches the minimum of 3.27 % with the spectral width (FWHM) of 28 nm. The Q factor of this cavity is calculated as λ/FWHM=55.35, where λ is the wavelength at the reflection minimum.
Fig. 3 (a) Reflectivity against d and l. The dashed line corresponds to d + l = 499 nm. (b) Comparison between the reflectivity calculated by the transmission line model (solid line) and the one by the FEM simulation (open circles). (c) Magnetic and (d) electric fields distribution along the central axis of the main trunk line. Positions of the stub and the waveguide terminal are represented by the dashed lines, respectively.
In Figs. 3(c) and 3(d) we present the distributions of |Hz| and |Ey| calculated by FEM (open circles) and the impedance-based transfer matrix method (solid line) [20] along the central axis of the main trunk line. All through the paper, both the electric and magnetic fields are normalized by the incident ones. The three dashed lines represent the positions of the stub and the terminal, respectively. The maximum |Hz| occurs at the waveguide terminal with the enhancement factor of 9.07 while the maximum |Ey| occurs near the right side of the stub with the enhancement factor of 8.47.
The magnetic field distribution |Hz| in the whole cavity is calculated by the FEM as shown in Fig. 4(a). Similar to the distribution in the main trunk line, it has the maximal value at the stub terminal with an enhancement factor of 8.94. As shown in Fig. 4(b), the electric field distribution has the maximal value at the stub with the enhancement factor of 8.95, larger than the value in the main trunk line. More interestingly, there is an extremely sharp electric hot spot at the inflection point, making the total electric field enhancement factor as high as 23.12. Top view of |E| around the inflection point is presented in Fig. 4(c).
Fig. 4 (a) Magnetic and (b) electric field enhancement inside the cavity. (c) Top view of |E| around the inflection point. (d) Relationship between the curvature radius r and the maximum |E| or |Ey| (inset).
From the perspective of cavity resonance, the phase shift upon reflection at the waveguide terminal is φ = arg(Γ) = 0.20 rad and the propagation constant of this MIM waveguide is β = (5.89−0.018i)×106 m−1. Considering the interference condition of a Fabry-Pérot cavity
where m is an integer, we can deduce d + l = 499 nm for m = 1. As shown in Fig. 3(a), the dashed line corresponds to d + l = 499 nm. Reflectivity values are minimal at this length in contrast with other sum lengths d + l. Unlike the impedance matching model, the interference condition is unable to determine parameters for the least reflectivity value.As to the acutal L-shaped cavity, the inflection point at the right side of the stub is the electric antinode where the electric field is maximal. It should be noted that the curvature radius of the inflection point would add a small phase change to the interference condition. If the inflection point of the stub is not rounded, the optimized structural parameters are l = 266 nm and d = 231 nm for the reflection minimum. In this case, the maximal field enhancement varies with simulation mesh size around the inflection point, which actually results from different curvature radii.
The electric hot spot originates from the charge accumulation caused by the interference of the EM fields between the main trunk line and the stub. The reason for this large electric field around the inflection point can be explained by employing the Maxwell equations. We investigate a medium without free charges or currents. For harmonic time dependence (∂/∂t = iω), the Ampère-Maxwell equation is given by
For TM modes, with only Hz,Ex and Ey being nonzero, Eq. (9) can be rewritten in cylindrical coordinates as
where E tip represents the total electric field around the inflection point, e r and e φ are the unit vectors along r and φ directions. Since r is comparably small, the first term in the right-hand side of Eq. (10) is much larger than the second term. We can simplify Eq. (10) by omitting the second term to getAccording to Eq. (11), the electric field is inversely proportional to the curvature radius. This can be proved by checking the relationship between the maximal |E| and the curvature radius r. Considering that the tip enhancement appears on the basis of the cavity resonance, it is also important to take into account the influence of the cavity resonance. As shown in the inset of Fig. 4(d), |Ey| is chosen to represent the cavity resonance. The maximal |Ey| in the central axis of the trunk line versus the curvature radius indicates a quadratic relation. Then the maximal value of the total electrical field |E| can be written in the form
By fitting Eq. (12) into the obtained data, we extract the parameters as a = 11.14, b = 27.45 and c = −1.21. The fitted curve (solid line) matches the FEM simulated data (open circles) well, which proves the inverse relationship given by Eq. (11). From another perspective, since the curvature radius r is much smaller than the wavelength, the electrostatics approximation is valid, where electric field is related to the surface charge density σe near a conductor by E = σe/ε 0. It is a lightning rod effect that small curvature radius results in large surface charge density and electric field.
4. Including a quarter-wave transformer
Though large EM field enhancement is achieved in our L-shaped cavity, it is still of great importance to take into account the coupling efficiency between the free space light and the nanoscaled waveguide. Several devices have been proposed for the waveguide coupling such as the multi-section taper [21], air-gap coupler [22] and quarter-wave transformer [23, 24]. In our design the quarter-wave transformer is chosen for its simplicity.
The whole structure is illustrated in the inset of Fig. 5, where the last segment at the right side is our L-shaped cavity. The first segment at the left side has a width of w 2 = 800 nm. This width is larger than half of 1550 nm, so light can be directly coupled into the MIM waveguide. When the parameters of the quarter-wave transformer are set as w 1 = 280 nm and s =270 nm, the total reflection is less than 1%. Due to the dimension confinement caused by the quarter-wave transformer, both of the electric and magnetic fields can be further enhanced. As shown in Fig. 5(a), the total enhancement factor of |Hz| is as large as 35.25. As to the electric field |E|, the total enhancement factor is 127.22 in Fig. 5(b). The electric field intensity is enhanced by four orders of magnitude in such a simple structure. This enhancement is comparable with other extremely intricate designs [25–27 ]. Furthermore, the electric field enhancement of this cavity is related to the grain size of the metal. Since the roughness of single-crystal Au can be polished to be less than 1 nm [28], our cavity has potential to be further increased by an additional order of magnitude.
Fig. 5 (a) Magnetic and (b) electric field enhancement including a quarter-wave transformer. The inset shows the whole structure.
5. Conclusion
We propose an L-shaped nanocavity for EM field enhancement by adding a stub on a closed MIM waveguide. The working principle of the proposed device is modeled as an impedance matching process by regarding the cavity as a load impedance. Once the cavity is impedance matched, there is a surprisingly high electric hot spot at the inflection point of the cavity besides the large EM enhancement in the trunk lines. The amplitude of this electric hot spot is proved to be inversely proportional to the curvature radius. By introducing a quarter-wave transformer in front of this nanocavity, electromagnetic fields can be coupled directly into the cavity and further be enhanced according to the dimensional confinement. In this case, electric field will be enhanced due to three distinct mechanisms: the first enhancement is the dimension confinement caused by the quarter-wave transformer. The second enhancement is the resonance enhancement in the L-shaped cavity and the third enhancement is the tip enhancement caused by charge accumulation at the inflection point of the cavity. The total magnetic intensity enhancement is three orders of magnitude and the electric intensity enhancement is more than four orders of magnitude in our structure. It should be noted that our 2D model is also valid in 3D case by adding an empirical coefficient in front of the 2D impedance [29]. Hence, our 2D cavity can be successfully converted to a 3D one. This L-shaped cavity, which can be fabricated through electron beam or focused ion beam direct-writing lithography, has the prospect of becoming a key element for highly integrated plasmonic circuits.
Acknowledgments
Yinxiao Xiang thanks Prof. Andrey Iljin, Dr. Yajun Wei and the Class 106 for joyful and helpful discussions. This work was financially supported by the National Basic Research Program of China (2013CB328702), the National Natural Science Foundation of China (11504184, 11374006), the China Postdoctoral Science Foundation (2015M571259), Program for Changjiang Scholars and Innovative Research Team in University (IRT0149) and the 111 Project (B07013).
References and links
1. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef] [PubMed]
2. W. Wang, S. Wu, K. Reinhardt, Y. Lu, and S. Chen, “Broadband light absorption enhancement in thin-film silicon solar cells,” Nano Lett. 10, 2012–2018 (2010). [CrossRef] [PubMed]
3. P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Nat. Photonics 6, 16–24 (2012). [CrossRef]
4. C. Symonds, G. Lheureux, J.-P. Hugonin, J.-J. Greffet, J. Laverdant, G. Brucoli, A. Lemaitre, P. Senellart, and J. Bellessa, “Confined tamm plasmon lasers,” Nano Lett. 13, 3179–3184 (2013). [CrossRef] [PubMed]
5. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3, 654–657 (2009). [CrossRef]
6. M. P. Cecchini, A. Wiener, V. A. Turek, H. Chon, S. Lee, A. P. Ivanov, D. W. McComb, J. Choo, T. Albrecht, S. A. Maier, and J. B. Edel, “Rapid ultrasensitive single particle surface-enhanced raman spectroscopy using metallic nanopores,” Nano Lett. 13, 4602–4609 (2013). [CrossRef] [PubMed]
7. X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonic-bandgap microcavity,” Nat. Photonics 2, 185–189 (2008). [CrossRef]
8. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004). [CrossRef]
9. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006). [CrossRef] [PubMed]
10. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010). [CrossRef] [PubMed]
11. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef] [PubMed]
12. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
13. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
14. D. K. Gramotnev and S. I. Bozhevolnyi, “Nanofocusing of electromagnetic radiation,” Nat. Photonics 8, 13–22 (2014). [CrossRef]
15. P. Mühlschlegel, H.-J. Eisler, O. Martin, B. Hecht, and D. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005). [CrossRef] [PubMed]
16. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]
17. D. M. Pozar, Microwave Engineering (John Wiley and Sons, 2009).
18. X.-S. Lin and X.-G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33, 2874–2876 (2008). [CrossRef] [PubMed]
19. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010). [CrossRef] [PubMed]
20. Y. Xiang, W. Cai, L. Wang, C. Ying, X. Zhang, and J. Xu, “Design methodology for all-optical bistable switches based on a plasmonic resonator sandwiched between dielectric waveguides,” J. Opt. 16, 025003 (2014). [CrossRef]
21. G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express 15, 1211–1221 (2007). [CrossRef] [PubMed]
22. R. A. Wahsheh, Z. Lu, and M. A. Abushagur, “Nanoplasmonic couplers and splitters,” Opt. Express 17, 19033–19040 (2009). [CrossRef]
23. P. Ginzburg and M. Orenstein, “Plasmonic transmission lines: from micro to nano scale with λ/4 impedance matching,” Opt. Express 15, 6762–6767 (2007). [CrossRef] [PubMed]
24. S. Kocabas, G. Veronis, D. Miller, and S. Fan, “Transmission line and equivalent circuit models for plasmonic waveguide components,” IEEE J. Sel. Top. Quantum Electron. 14, 1462–1472 (2008). [CrossRef]
25. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]
26. H. Choo, M.-K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal-insulator-metal gap plasmon waveguide with a three-dimensional linear taper,” Nat. Photonics 6, 838–844 (2012). [CrossRef]
27. Y. Xiang, P. Wang, W. Cai, C.-F. Ying, X. Zhang, and J. Xu, “Plasmonic tamm states: dual enhancement of light inside the plasmonic waveguide,” J. Opt. Soc. Am. B 31, 2769–2772 (2014). [CrossRef]
28. E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface-plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009). [CrossRef] [PubMed]
29. D. Li and E.-P. Li, “Impedance calculation and equivalent circuits for metal–insulator–metal plasmonic waveguide geometries,” Opt. Lett. 38, 3384–3386 (2013). [CrossRef] [PubMed]
