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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 17066–17076
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Active unidirectional propagation of surface plasmons at subwavelength slits

Mehdi Afshari Bavil, Zhiping Zhou, and Qingzhong Deng  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 17066-17076 (2013)
http://dx.doi.org/10.1364/OE.21.017066


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Abstract

Highly efficient, active and compact, unidirectional surface plasmon (SP) propagator composed of double subwavelength slits; filled with organic electro-optic (EO) material is proposed and investigated. By selecting appropriate structure parameters, obtained by solving phase relations between slits, the relative phase of SP generated at the slit exit aperture can be tailored. Simulation results show under normal illumination and external voltage of 8.7 V, SP launching efficiency of 55% and unidirectional SP extinction ratio about 47dB at wavelength of 632.8 nm is achieved. The power consumption of the structure is on the order of 9 fJ/bit which meet the power consumption limitation for optical devices. Moreover, the structure is very compact with effective total length of 1.2 µm and thickness of 0.6 µm.

© 2013 OSA

Introduction

The light diffraction at subwavelength scales hinders the minimization of photonic components. Plasmonics that is based on exploiting coupling between light and collective electronic excitations within conducting materials known as surface plasmons (SP) paves a confident way beyond the diffraction limit for future optical integrated circuits. So far, a large diversity of plasmonic nanodevices has been theoretically proposed and experimentally demonstrated [1

1. L. Gao, L. Tang, F. Hu, R. Guo, X. Wang, and Z. Zhou, “Active metal strip hybrid plasmonic waveguide with low critical material gain,” Opt. Express 20(10), 11487–11495 (2012). [CrossRef] [PubMed]

5

5. C. Lee, K. Lo, and T. Mo, “Electrically switchable Fresnel lens based on a liquid crystal film with a polymer relief pattern,” Jpn. J. Appl. Phys. 46(7A), 4144–4147 (2007). [CrossRef]

]. A key device is a SP generator that may efficiently convert the exciting field into surface plasmon polaritons (SPPs). Unfortunately, conventional generators, such as prism and grating suffer from poor light-SPP coupling or massive size. To increase the light coupling efficiency and minimize the structure, subwavelength structures such as slits or ridges on metal surfaces offering small footprints were currently investigated [6

6. W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8(1), 281–286 (2008). [CrossRef] [PubMed]

9

9. J. Lin, J. P. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340(6130), 331–334 (2013). [CrossRef] [PubMed]

]. However, the direction of the SPPs generated from slits cannot be selected due to structure symmetry. The symmetric SPPs propagation may play as a limiting factor for building the efficient functionalized plasmonic circuits; therefore, the need for directional SP propagation is unavoidable. Over the past few years, numerous passive unidirectional SP generators have been developed [10

10. T. Xu, Y. Zhao, D. Gan, C. Wang, C. Du, and X. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008). [CrossRef]

15

15. J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106–081108 (2010). [CrossRef]

]. Two important figure-of-merits of SP propagators are the launching efficiency η in the desired direction and the extinction ratio r defined as the ratio between the SPPs intensity propagated into the desired direction and the intensity launched into the opposite direction. Meanwhile, tradeoff between launching efficiency, extinction ratio and size is still a big challenge. Most recently, Baron et al. designed a unidirectional SP propagator that composed of eleven subwavelength grooves; an extinction ratio of 38 dB and launching efficiency η larger than 52% was reported but unfortunately, the structure is massive nearly 8 µm [16

16. A. Baron, E. Devaux, J. C. Rodier, J. P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact Antenna for Efficient and Unidirectional Launching and Decoupling of Surface Plasmons,” Nano Lett. 11(10), 4207–4212 (2011). [CrossRef] [PubMed]

].

Moreover, active plasmonic devices with externally controlled characteristic enable us to obtain compact devices with high functionality and low power consumption such as photodetection [17

17. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332(6030), 702–704 (2011). [CrossRef] [PubMed]

], florescence enhancements [18

18. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nature Photonics Lett. 3(11), 654–657 (2009). [CrossRef]

], subwavelength imaging and photolithography [19

19. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef] [PubMed]

] and highly integrated nanophotonic devices [20

20. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007). [CrossRef] [PubMed]

, 21

21. D. Yu. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012). [CrossRef] [PubMed]

]. Alternatively investigating the active unidirectional SP propagator remains a main challenge to design plasmonics architecture than can actively couple light from photonics taper or free space to plasmonics components. A few schemes have been proposed and developed [22

22. Y. M. Liu, S. Palomba, Y. S. Park, T. Zentgraf, X. B. Yin, and X. Zhang, “Compact Magnetic Antennas for Directional Excitation of Surface Plasmons,” Nano Lett. 12(9), 4853–4858 (2012). [CrossRef] [PubMed]

24

24. J. Chen, Z. Li, J. Xiao, and Q. Gong, “Efficient All-Optical Molecule-Plasmon Modulation Based on T-shape Single Slit,” Plasmonics , doi:. [CrossRef]

], Liu et al. by bridging the optical antenna theory and the concept of metamaterials developed an all optically controlled compact 1.2µm unidirectional antenna with extinction ratio of more than 23dB [22

22. Y. M. Liu, S. Palomba, Y. S. Park, T. Zentgraf, X. B. Yin, and X. Zhang, “Compact Magnetic Antennas for Directional Excitation of Surface Plasmons,” Nano Lett. 12(9), 4853–4858 (2012). [CrossRef] [PubMed]

]. Chen et al. reported another structure with coupling efficiency of 64% but the extinction ration in only 16.7 dB [23

23. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly Efficient All-Optical Control of Surface-Plasmon-Polariton Generation Based on a Compact Asymmetric Single Slit,” Nano Lett. 11(7), 2933–2937 (2011). [CrossRef] [PubMed]

]. To our knowledge, there is no reported structure for active directional SP propagator using external voltage. Here phase modulation of the SPPs is achieved by replacing the “dielectric” layers with the electrooptically active materials such as electro-optic organic crystals (4-dimethyl-amino-Nmethyl-4-stilbazolium tosylate (DAST) for our case) [25

25. T. Satoh, Y. Toya, S. Yamamoto, T. Shimura, K. Kuroda, Y. Takahashi, M. Yoshimura, Y. Mori, T. Sasaki, and S. Ashihara, “Generation of mid- to far-infrared ultrashort pulses in 4-dimethylamino-N-methyl-4-stilbazolium tosylate crystal,” J. Opt. Soc. Am. B 27(12), 2507–2511 (2010). [CrossRef]

27

27. B. Ruiz, Z. Yang, V. Gramlich, M. Jazbinseka, and P. Gunter, “Synthesis and crystal structure of a new stilbazolium salt with large second-order optical nonlinearity,” J. Mater. Chem. 16, 2839–2842 (2006). [CrossRef]

] and Indium tin oxide [28

28. S. Franzen, C. Rhodes, M. Cerruti, R. W. Gerber, M. Losego, J. P. Maria, and D. E. Aspnes, “Plasmonic phenomena in indium tin oxide and ITO-Au hybrid films,” Opt. Lett. 34(18), 2867–2869 (2009). [CrossRef] [PubMed]

, 29

29. V. J. Sorger, D. Norberto, L. Kimura, R. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).

]. The large electro-optic coefficient of organic crystals, high speed bandwidth operation, low dispersion, low dielectric constants, and compatible with integrated circuit (IC) technology make them ideal candidates for high frequency operation in optoelectronic applications [26

26. L. Dalton and S. Benight, “Theory-Guided Design of Organic Electro-Optic Materials and Devices,” Polymers 3(4), 1325–1351 (2011). [CrossRef]

, 30

30. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. R. Dalton, A. K. Y. Jen, and A. Scherer, “Optical Modulation and Detection in Slotted Silicon Waveguides,” Opt. Express 13(14), 5216–5226 (2005). [CrossRef] [PubMed]

32

32. W. Geis, R. Sinta, W. Mowers, S. J. Deneault, M. F. Marchant, K. E. Krohn, S. J. Spector, D. R. Calawa, and T. M. Lyszczarz, “Fabrication of crystalline organic waveguides with an exceptionally large electro-optic coefficient,” Appl. Phys. Lett. 84(19), 3729–3731 (2004). [CrossRef]

]. Recently slot waveguide filled with EO polymers to electro-optically control the SPPs propagation for nanophotonics application was reported [26

26. L. Dalton and S. Benight, “Theory-Guided Design of Organic Electro-Optic Materials and Devices,” Polymers 3(4), 1325–1351 (2011). [CrossRef]

, 32

32. W. Geis, R. Sinta, W. Mowers, S. J. Deneault, M. F. Marchant, K. E. Krohn, S. J. Spector, D. R. Calawa, and T. M. Lyszczarz, “Fabrication of crystalline organic waveguides with an exceptionally large electro-optic coefficient,” Appl. Phys. Lett. 84(19), 3729–3731 (2004). [CrossRef]

38

38. X. Mei, X. G. Huang, and T. Jin, “A sub-wavelength Electro-optic Switch Based on Plasmonic T-Shaped Waveguide,” Plasmonics 6(4), 613–618 (2011). [CrossRef]

]. ITO, an In2O3 based material that has been doped with Sn, has been widely used as transparent conducting oxides because of its two chief properties, its electrical conductivity and optical transparency, as well as the ease fabrication procedure to make an electrode. We proposed a novel device composed of two slits perforated on metallic silver and filled with electro-optic organic crystal which is sandwiched between silica and ITO layers as contacts. Through mutual interference of the two SP fields with different relative phases excited at the slit exit apertures, the field intensity along one direction on the metal surface can be enhanced or suppressed. Calculations, performed using 2D finite element method (FEM) demonstrate the proposed device offers better performance than those reported so far [15

15. J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106–081108 (2010). [CrossRef]

18

18. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nature Photonics Lett. 3(11), 654–657 (2009). [CrossRef]

, 21

21. D. Yu. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012). [CrossRef] [PubMed]

24

24. J. Chen, Z. Li, J. Xiao, and Q. Gong, “Efficient All-Optical Molecule-Plasmon Modulation Based on T-shape Single Slit,” Plasmonics , doi:. [CrossRef]

] with launching efficiency η of 55% and a large extinction ratio r ≈47 dB at λ = 632.8 nm. In addition, the device is compact (1.2 μm length) and operates under normal illumination. The geometrical parameters were determined by theoretical calculations.

Device structure and theoretical model

N(z)=13π2(8π2meffh2)3/2(EF+eϕ(z))3/2
(2)

EF and h are the Fermi energy and Planck’s constant, respectively. The electron effective mass, meff, for ITO is 0.35 * me [47

47. F. Neumann, Y. A. Genenko, C. Melzer, S. V. Yampolskii, and H. von Seggern, “Self-consistent analytical solution of a problem of charge-carrier injection at a conductor/insulator interface,” Phys. Rev. B 75(20), 205322 (2007). [CrossRef]

] in terms of the free electron mass me = 9.1*10−31 kg. N0, the carrier density when there is no external voltage, is in the order of 9.25*1026 m−3. The boundary value problem is needed to obtain the potential distribution Ф (z) and then to calculate the induced carrier density N(z) in the ITO layer [46

46. A. Melikyan, N. Lindenmann, S. Walheim, P. M. Leufke, S. Ulrich, J. Ye, P. Vincze, H. Hahn, Th. Schimmel, C. Koos, W. Freude, and J. Leuthold, “Surface plasmon polariton absorption modulator,” Opt. Express 19(9), 8855–8869 (2011). [CrossRef] [PubMed]

]. Once the carrier density is obtained, by using plasma frequency ωp:
ωp2=N(z)e2εm
(3)
then the permittivity of ITO layer can be determined by the Drude formula (Eq. (1)). It has been exhibited the relative change of the free carrier density in the thin ITO layer is estimated to be 1% for a 10 V voltage applied and reaches 9.34*1026 m−3 which cannot affect the permittivity considerably [46

46. A. Melikyan, N. Lindenmann, S. Walheim, P. M. Leufke, S. Ulrich, J. Ye, P. Vincze, H. Hahn, Th. Schimmel, C. Koos, W. Freude, and J. Leuthold, “Surface plasmon polariton absorption modulator,” Opt. Express 19(9), 8855–8869 (2011). [CrossRef] [PubMed]

], since our applied external voltage is lower therefore the change in carrier density and thereby permittivity will be much smaller.

The following advantages for our structure can be specified:

  • Relatively high carrier mobility and the low enough carrier density of ITO, which used as electrodes, result in a small change in real part of the dielectric permittivity under our applied voltage.
  • The narrow low refractive index dielectric layer between metallic layer and ITO provides hybrid surface plasmon waveguides with higher confinement, longer propagation length, and lower lose compare to regular structures.
  • Because of the instructive interference of propagating SPPs at exit of slits, the double slit structure has higher light transmission than single slit.
  • DAST which is compatible with SOI technology has a high electro-optic coefficient, so the energy consumption is very low.
  • The structure is very compact and for desired wavelength or slits, appropriate structure parameters can be determined by applying phase retardation equations.

Nanoslits perforated in metal film with width smaller than the incident wavelength as shown in the inset of Fig. 2
Fig. 2 The effective index dependence on the slit width for incident light of 632.8 nm wavelength. Solid and starred lines represent the real and imaginary parts of the effective index, respectively. The inset shows the sketch of a MDM structure.
provides the necessary momentum for the SP modes excitation and scatters part of incident radiation into a plasmonic channel. Once the SPPs reach the slit exit aperture, some of them are scattered into free-space radiation, while the remainder propagate on the metal-dielectric interface. Because of multi-reflection effect inside the nanoslit that is a direct prove of Fabry-Perot resonator nature of nanoslit, the SPPs wavelength squeezes and effective refractive index for nanoslit can be introduced. The effective refractive index dependence on the dielectric medium and slit width for narrower slits can be approximated as [48

48. S. Bozhevolnyi, Plasmonic Nanoguide and Circuits (Pan Stanford Publishing, 2008), pp.10–20.

]:
Νeffεd+0.5(kgap0k0)2+(kgap0k0)2(εdεm+0.25(kgap0k0)2),kgap0=2εdwεm
(4)
where w is the slit width, εm and εd are the dielectric constants of the metal and dielectric, respectively. Figure 2 depicts the relationship between slit width and its effective index Neff for three dielectric mediums as vacuum, DAST and a dielectric with refractive index of 2.3. Figure 2 also implies that changing the slit width or filling the slit region with different dielectrics can tune the phase retardation of the SP generated at the slit exit aperture.

Simulation result

As mentioned before two important figure-of-merits of SP propagators are the launching efficiency η in the desired direction and the asymmetrical SPP extinction ratio r defined as r = 10 logPd/Pu, where Pd and Pu represent the SPP intensity propagated into the desired direction and the intensity launched into the opposite direction. To calculate these parameters, two line power monitors along the Silica thickness are set at 2µm away from the slits to detect the magnetic field distribution as seen in Fig. 1(b). In all simulation, the results are based on the recorded parameters of these monitors. To optimize the structure parameters, some consideration should be done over layers thickness. A compromise must be made between conductivity and transparency of ITO layer, since increasing the thickness and increasing the concentration of charge carriers will increase the material's conductivity, but decrease its transparency. Simulation results show that ITO and substrate thickness should be 10 nm and 200 nm respectively. In fact, the change in substrate thickness has small influence on output results. The thickness of Silica is should be approximately equals with SPPs penetration depth which is about 30 nm [49

49. W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt. 8(4), S87–S93 (2006). [CrossRef]

].

In the absence of external voltage (OFF state) two slits have identical refractive index; The TM-polarized light source causing a fraction of the energy to be coupled into SPP modes at the Ag/SIO2 interface; propagating equally along left and right interfaces as seen in Fig. 3(a)
Fig. 3 a) Logarithmic magnetic field distribution in OFF state. b) shows the magnetic field intensity along X-direction along metalSilica interface, which in both directions is identical.
and the remaining to be transmitted or reflected. Magnetic field intensity along X-direction for Ag/SIO2 is shown in Fig. 3(b); the magnetic field along left side is larger than right side which attributes to prolonging the DAST to silica layer at left slit. It is worthy to note that most of SPPs concentrate within the hybrid layer.

When an external voltage is applied the refractive index of the DAST with the EO coefficient of dn/dE = 3.41 nm/V will change via this equation:
n=n0+dndE(Vh)
(5)
Where V is the external voltage applied to the structure via the ITO electrodes, h, the distance between two electrodes. As mentioned before the material within a nanoslit obtains an effective refractive index which is higher than its own refractive index. Figure 4
Fig. 4 effective refractive index differences as function of applied voltage obtained by using Eqs. (4) and (5). The inset plot shows the effective refractive index versus applied voltage.
shows the relation between the effective refractive index of DAST for different voltage range from 5 V to 10 V when h fixed at 300 nm, obtained via Eqs. (4) and (5) and noting εd = n2. The inset plot shows the effective refractive index difference for different voltages.

When the voltage is ON, the refractive index of left slit increases. Our goal is to design the structure in a way that propagating SPs, generated at the exit aperture of the each slit, travel along one direction on the metal/Silica interface, which can be achieved by modulating the phase difference of SPPs launched separately from the two slits. This requires that SPs interfere constructively along one direction while destructively along the opposite direction, i.e., the relative phases of SPs at two exit apertures take the forms [8

8. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science 340(6130), 328–330 (2013). [CrossRef] [PubMed]

]:
ϕ1+d2πλSP=ϕ2+2Mπ
(6)
ϕ2+d2πλSP=ϕ1+(2M+1)π
(7)
where φ1 and φ2 are the relative phases of generated SPs at the exit apertures for the left and right slits filled with DAST, respectively. d is the interspacing between two slits and M is an arbitrary integer. Since the relative phase and the effective index have the relation of φ1 = Neff-leftt2π/λ + φ', and φ2 = Neff-rightt2π/λ where t is the film thickness, λ is the wavelength of the incident light, Neff-left and Neff-right are the effective refractive index of the left and right slit when the voltage is ON. The propagating light through OE layer undergoes a phase delay, φ', which can be calculated as [50

50. E. A. Bahaa Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991), Chapter 18, pp. 696–737.

]:
ϕ=2π(NeffleftONNeffleftOFF)hλ
(8)
where h is the distance between two electrodes. Simple manipulation of Eqs. (6), (7) and (8) yields:
ϕ1ϕ2=(NeffleftNeffright)t2πλ+ϕ=π2
(9)
According to Eqs. (5) and (9) higher voltage yields in larger effective refractive index differences and thereby thicker metallic film but shorter interspace distances, vice versa lower voltage yield in lower effective refractive index differences thereby thinner metallic film but longer interspace distances. For an increase in refractive index from 2.2 to 2.3 (lowest increase in the order of 10−1), the required voltage is about 8.7 V which is low voltage compatible with silicon electronics. The width w and wavelength are chosen as 100 nm and 632.8 nm; respectively from Eq. (4) the effective refractive index change is 0.21, after substituting the parameters in Eq. (9) the film thickness obtains as 254 nm. For this thickness, maximum phase retardation will happen between slits. In addition, from Eqs. (6) and (7), the interspacing d can be calculated as d = (4N + 1)λSP/4. The dispersion relation of structure with double ITO layers and metallic film is complicated to be solved analytically. Obtaining the dispersion relation of SPs and thereby the SPPs wavelength is a bottleneck task. Here we use simulation result to determine SP wavelength. According to Eqs. (6) and (7) maximum light intensity occurs at some specific interspace distance which is corresponds to SP wavelength (λsp). The asymmetrical extinction ratio is plotted in Fig. 5
Fig. 5 The asymmetrical extinction ratio as a function of interspace distance. The plot has a periodic nature with the periodicity of SPPs wavelength which has about 445 nm. A big asymmetrical extinction ratio about 47 dB for interspace distance of 1008 nm is achieved
as a function of interspace distance. The periodicity of the plot is equal with SP wavelength, which is about 445 nm. The asymmetrical extinction ratio (r) can be achieved as high as 47 dB (Pd/Pu = 108) for interspace distance of 1008 nm.

For this interspace distance, the structure delivers desirable phase retardation for the incident light and mutual interference of the two SP fields with different relative phases happens. The simulation result reveal that the SP fields generated at the slit exit apertures the magnetic field distribution of silver layer possess a strong spatial distribution orientation.

From Figs. 6(a)
Fig. 6 a) The magnetic field intensity under the metallic layer recorded by two remote monitors. dashed lines are the result of right monitor while solid lines stands for the result of left monitor. b) Light intensity along X-direction, under the metallic layer in both states. c) Logarithmic magnetic field distribution when the voltage takes the value 8.7 V (ON state).
-6(c) it can be seen the field intensity along the left direction on the metal surface was increased while the field intensity along the right direction because of destructive interference was suppressed, as was predicted. The most important achievement is that by selecting appropriate parameters the magnetic field at undesired direction because of destructive interferes was approximately quenched. Another point worth noting in Figs. 6(a) and 6(b) is that the simulated SP intensities along the left side are larger than the intensity calculated for the separate single slit. This increase is attributed to the inter-slit effect, which leads part of SP on illuminated surface couple with SP of remote slit and enhances the SP intensity at the slit exit aperture.

The two parallel electrodes with the EO material sandwiched in between form a capacitor with capacitance of Cm=ε0εEOA/h0.05fF/μm where ε0 is the permittivity of vacuum, εEO = 5.3 is the dielectric constant of EO polymer under applied voltage, A is the area of the ITO covering the top and bottom of EO material, and h is the distance between electrodes [34

34. W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009). [CrossRef] [PubMed]

]. The power consumption can be estimated byP=CmV2f/2 [34

34. W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009). [CrossRef] [PubMed]

], given V = 8.7V and f = 470 THz (wavelength = 632.8 nm), the projected power consumption in the device is on the order of 9 fJ/bit. The energy consumption of optical devices has to be in the order of 10 fJ/bit or lower according to an analysis presented by Miller [51

51. D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97(7), 1166–1185 (2009). [CrossRef]

] which our structure meet this requirements.

If the slits fill with inorganic crystals such as BaTiO2 or polymer electrooptic materials, the required voltage for the same structure will be in the order of 35 V and 100 V, respectively. As a result organic crystals with high Electro-optic coefficient are a good candidate for optoelectronic application.

Conclusions

In summary, for the first time by utilizing electrooptic materials we have designed electrically controlled unidirectional SPPs propagator with higher generation efficiency, extinction ratio and compact size compare to previously reported structures. By choosing proper geometrical parameters to manipulate SPs interference at the exit of slits, higher launching efficiency of 55% and higher extinction ratio of 47 dB at wavelength of 632.8 nm was achieved. The structure is very compact with effective length of 1.2µm that can be integrated easily with other plasmonic components for the application in plasmonic circuitry, such as Bragg grating mirrors to realize selective coupling of SPPs into different ports. Furthermore, it works under low voltage (8.7 V) and low power consumption (9 fJ/bit) which is compatible with silicon electronics. In addition, our proposed unidirectional propagator is achieved for normal incidence that is unavoidable in most applications. Our results may have potential applications in plasmonic integrated circuits and on-chip applications.

References and links:

1.

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6.

W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8(1), 281–286 (2008). [CrossRef] [PubMed]

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Q. Li, T. B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett. 98(25), 251109 (2011).

12.

J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett. 97(4), 041113–041115 (2010). [CrossRef]

13.

F. L. Tejeira, S. G. Rodrigo, L. M. Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3(5), 324–328 (2007). [CrossRef]

14.

Y. K. Wang, X. R. Zhang, H. J. Tang, K. Yang, Y. X. Wang, Y. L. Song, T. H. Wei, and C. H. Wang, “A tunable unidirectional surface plasmon polaritons source,” Opt. Express 17(22), 20457–20464 (2009). [CrossRef] [PubMed]

15.

J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett. 97(8), 081106–081108 (2010). [CrossRef]

16.

A. Baron, E. Devaux, J. C. Rodier, J. P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact Antenna for Efficient and Unidirectional Launching and Decoupling of Surface Plasmons,” Nano Lett. 11(10), 4207–4212 (2011). [CrossRef] [PubMed]

17.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332(6030), 702–704 (2011). [CrossRef] [PubMed]

18.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nature Photonics Lett. 3(11), 654–657 (2009). [CrossRef]

19.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010). [CrossRef] [PubMed]

20.

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007). [CrossRef] [PubMed]

21.

D. Yu. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012). [CrossRef] [PubMed]

22.

Y. M. Liu, S. Palomba, Y. S. Park, T. Zentgraf, X. B. Yin, and X. Zhang, “Compact Magnetic Antennas for Directional Excitation of Surface Plasmons,” Nano Lett. 12(9), 4853–4858 (2012). [CrossRef] [PubMed]

23.

J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly Efficient All-Optical Control of Surface-Plasmon-Polariton Generation Based on a Compact Asymmetric Single Slit,” Nano Lett. 11(7), 2933–2937 (2011). [CrossRef] [PubMed]

24.

J. Chen, Z. Li, J. Xiao, and Q. Gong, “Efficient All-Optical Molecule-Plasmon Modulation Based on T-shape Single Slit,” Plasmonics , doi:. [CrossRef]

25.

T. Satoh, Y. Toya, S. Yamamoto, T. Shimura, K. Kuroda, Y. Takahashi, M. Yoshimura, Y. Mori, T. Sasaki, and S. Ashihara, “Generation of mid- to far-infrared ultrashort pulses in 4-dimethylamino-N-methyl-4-stilbazolium tosylate crystal,” J. Opt. Soc. Am. B 27(12), 2507–2511 (2010). [CrossRef]

26.

L. Dalton and S. Benight, “Theory-Guided Design of Organic Electro-Optic Materials and Devices,” Polymers 3(4), 1325–1351 (2011). [CrossRef]

27.

B. Ruiz, Z. Yang, V. Gramlich, M. Jazbinseka, and P. Gunter, “Synthesis and crystal structure of a new stilbazolium salt with large second-order optical nonlinearity,” J. Mater. Chem. 16, 2839–2842 (2006). [CrossRef]

28.

S. Franzen, C. Rhodes, M. Cerruti, R. W. Gerber, M. Losego, J. P. Maria, and D. E. Aspnes, “Plasmonic phenomena in indium tin oxide and ITO-Au hybrid films,” Opt. Lett. 34(18), 2867–2869 (2009). [CrossRef] [PubMed]

29.

V. J. Sorger, D. Norberto, L. Kimura, R. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).

30.

T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. R. Dalton, A. K. Y. Jen, and A. Scherer, “Optical Modulation and Detection in Slotted Silicon Waveguides,” Opt. Express 13(14), 5216–5226 (2005). [CrossRef] [PubMed]

31.

Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, J. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, “Hybrid polymer / sol-gel waveguide modulators with exceptionally large electrooptic coefficients,” Nat. Photonics 1(3), 180–185 (2007). [CrossRef]

32.

W. Geis, R. Sinta, W. Mowers, S. J. Deneault, M. F. Marchant, K. E. Krohn, S. J. Spector, D. R. Calawa, and T. M. Lyszczarz, “Fabrication of crystalline organic waveguides with an exceptionally large electro-optic coefficient,” Appl. Phys. Lett. 84(19), 3729–3731 (2004). [CrossRef]

33.

M. Xu, F. Li, T. Wang, J. Wu, L. Lu, L. Zhou, and Y. Su, “Design of an electro-optic modulator based on a silicon-plasmonic hybrid phase shifter,” J. Light Wave Tech. 31, 1170–1177 (2013).

34.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009). [CrossRef] [PubMed]

35.

L. Alloatti, D. Korn, R. Palmer, D. Hillerkuss, J. Li, A. Barklund, R. Dinu, J. Wieland, M. Fournier, J. Fedeli, H. Yu, W. Bogaerts, P. Dumon, R. Baets, C. Koos, W. Freude, and J. Leuthold, “42.7 Gbit/s electro-optic modulator in silicon technology,” Opt. Express 19(12), 11841–11851 (2011). [CrossRef] [PubMed]

36.

S. Inoue and S. Yokoyama, “Highly compact organic electro-optic modulator based on nanoscale plasmon metal gap waveguides,” SPIE-OSA-IEEE 7631, 763128 (2009).

37.

H. Nasari and M. S. Abrishamian, “Electrically tunable light focusing via a plasmonic lens,” J. Opt. 14(12), 125002 (2012). [CrossRef]

38.

X. Mei, X. G. Huang, and T. Jin, “A sub-wavelength Electro-optic Switch Based on Plasmonic T-Shaped Waveguide,” Plasmonics 6(4), 613–618 (2011). [CrossRef]

39.

L. R. Dalton, W. H. Steier, B. H. Robinson, C. Zhang, A. Ren, S. Garner, A. Chen, T. Londergan, L. Irwin, B. Carlson, L. Fifield, G. Phelan, C. Kincaid, J. Amenda, and A. Jen, “From molecules to opto-chips: organic electro-optic materials,” J. Mater. Chem. 9(9), 1905–1920 (1999). [CrossRef]

40.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]

41.

D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009). [CrossRef] [PubMed]

42.

Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express 18(12), 13173–13179 (2010). [CrossRef] [PubMed]

43.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

44.

F. Michelotti, L. Dominici, E. Descrovi, N. Danz, and F. Menchini, “Thickness dependence of surface plasmon polariton dispersion in transparent conducting oxide films at 1.55 microm,” Opt. Lett. 34(6), 839–841 (2009). [CrossRef] [PubMed]

45.

B. Chiou and J. Tsai, “Antireflective coating for ITO films deposited on glass substrate,” J. Mater. Sci. Mater. Electron. 10(7), 491–495 (1999). [CrossRef]

46.

A. Melikyan, N. Lindenmann, S. Walheim, P. M. Leufke, S. Ulrich, J. Ye, P. Vincze, H. Hahn, Th. Schimmel, C. Koos, W. Freude, and J. Leuthold, “Surface plasmon polariton absorption modulator,” Opt. Express 19(9), 8855–8869 (2011). [CrossRef] [PubMed]

47.

F. Neumann, Y. A. Genenko, C. Melzer, S. V. Yampolskii, and H. von Seggern, “Self-consistent analytical solution of a problem of charge-carrier injection at a conductor/insulator interface,” Phys. Rev. B 75(20), 205322 (2007). [CrossRef]

48.

S. Bozhevolnyi, Plasmonic Nanoguide and Circuits (Pan Stanford Publishing, 2008), pp.10–20.

49.

W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt. 8(4), S87–S93 (2006). [CrossRef]

50.

E. A. Bahaa Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991), Chapter 18, pp. 696–737.

51.

D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE 97(7), 1166–1185 (2009). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.0250) Optical devices : Optoelectronics
(230.2090) Optical devices : Electro-optical devices
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Integrated Optics

History
Original Manuscript: April 30, 2013
Revised Manuscript: June 15, 2013
Manuscript Accepted: June 28, 2013
Published: July 10, 2013

Citation
Mehdi Afshari Bavil, Zhiping Zhou, and Qingzhong Deng, "Active unidirectional propagation of surface plasmons at subwavelength slits," Opt. Express 21, 17066-17076 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17066


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References

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  3. M. Afshari Bavil, L. Gao, and X. Sun, “A compact nanoplasmonics filter and intersection structure based on utilizing a slot cavity and a Fabry–Perot resonator,” Plasmonics, doi:. [CrossRef]
  4. A. E. Çetin, A. A. Yanik, A. Mertiri, S. Erramilli, O. E. Mustecaplıoglu, and H. Altug, “Field-effect active plasmonics for ultracompact electro-optic switching,” Appl. Phys. Lett.101(12), 121113 (2012). [CrossRef]
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  8. F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, “Near-field interference for the unidirectional excitation of electromagnetic guided modes,” Science340(6130), 328–330 (2013). [CrossRef] [PubMed]
  9. J. Lin, J. P. Mueller, Q. Wang, G. Yuan, N. Antoniou, X. C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science340(6130), 331–334 (2013). [CrossRef] [PubMed]
  10. T. Xu, Y. Zhao, D. Gan, C. Wang, C. Du, and X. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett.92(10), 101501 (2008). [CrossRef]
  11. Q. Li, T. B. Bai, and G. Jin, “Experimental demonstration of tunable directional excitation of surface plasmon polaritons with a subwavelength metallic double slit,” Appl. Phys. Lett.98(25), 251109 (2011).
  12. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Efficient unidirectional generation of surface plasmon polaritons with asymmetric single-nanoslit,” Appl. Phys. Lett.97(4), 041113–041115 (2010). [CrossRef]
  13. F. L. Tejeira, S. G. Rodrigo, L. M. Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys.3(5), 324–328 (2007). [CrossRef]
  14. Y. K. Wang, X. R. Zhang, H. J. Tang, K. Yang, Y. X. Wang, Y. L. Song, T. H. Wei, and C. H. Wang, “A tunable unidirectional surface plasmon polaritons source,” Opt. Express17(22), 20457–20464 (2009). [CrossRef] [PubMed]
  15. J. R. Salgueiro and Y. S. Kivshar, “Nonlinear plasmonic directional couplers,” Appl. Phys. Lett.97(8), 081106–081108 (2010). [CrossRef]
  16. A. Baron, E. Devaux, J. C. Rodier, J. P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact Antenna for Efficient and Unidirectional Launching and Decoupling of Surface Plasmons,” Nano Lett.11(10), 4207–4212 (2011). [CrossRef] [PubMed]
  17. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science332(6030), 702–704 (2011). [CrossRef] [PubMed]
  18. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nature Photonics Lett.3(11), 654–657 (2009). [CrossRef]
  19. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010). [CrossRef] [PubMed]
  20. H. A. Atwater, “The promise of plasmonics,” Sci. Am.296(4), 56–62 (2007). [CrossRef] [PubMed]
  21. D. Yu. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett.12(5), 2459–2463 (2012). [CrossRef] [PubMed]
  22. Y. M. Liu, S. Palomba, Y. S. Park, T. Zentgraf, X. B. Yin, and X. Zhang, “Compact Magnetic Antennas for Directional Excitation of Surface Plasmons,” Nano Lett.12(9), 4853–4858 (2012). [CrossRef] [PubMed]
  23. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Highly Efficient All-Optical Control of Surface-Plasmon-Polariton Generation Based on a Compact Asymmetric Single Slit,” Nano Lett.11(7), 2933–2937 (2011). [CrossRef] [PubMed]
  24. J. Chen, Z. Li, J. Xiao, and Q. Gong, “Efficient All-Optical Molecule-Plasmon Modulation Based on T-shape Single Slit,” Plasmonics, doi:. [CrossRef]
  25. T. Satoh, Y. Toya, S. Yamamoto, T. Shimura, K. Kuroda, Y. Takahashi, M. Yoshimura, Y. Mori, T. Sasaki, and S. Ashihara, “Generation of mid- to far-infrared ultrashort pulses in 4-dimethylamino-N-methyl-4-stilbazolium tosylate crystal,” J. Opt. Soc. Am. B27(12), 2507–2511 (2010). [CrossRef]
  26. L. Dalton and S. Benight, “Theory-Guided Design of Organic Electro-Optic Materials and Devices,” Polymers3(4), 1325–1351 (2011). [CrossRef]
  27. B. Ruiz, Z. Yang, V. Gramlich, M. Jazbinseka, and P. Gunter, “Synthesis and crystal structure of a new stilbazolium salt with large second-order optical nonlinearity,” J. Mater. Chem.16, 2839–2842 (2006). [CrossRef]
  28. S. Franzen, C. Rhodes, M. Cerruti, R. W. Gerber, M. Losego, J. P. Maria, and D. E. Aspnes, “Plasmonic phenomena in indium tin oxide and ITO-Au hybrid films,” Opt. Lett.34(18), 2867–2869 (2009). [CrossRef] [PubMed]
  29. V. J. Sorger, D. Norberto, L. Kimura, R. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics1, 17–22 (2012).
  30. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. A. Sullivan, L. R. Dalton, A. K. Y. Jen, and A. Scherer, “Optical Modulation and Detection in Slotted Silicon Waveguides,” Opt. Express13(14), 5216–5226 (2005). [CrossRef] [PubMed]
  31. Y. Enami, C. T. Derose, D. Mathine, C. Loychik, C. Greenlee, R. A. Norwood, T. D. Kim, J. Luo, Y. Tian, A. K.-Y. Jen, and N. Peyghambarian, “Hybrid polymer / sol-gel waveguide modulators with exceptionally large electrooptic coefficients,” Nat. Photonics1(3), 180–185 (2007). [CrossRef]
  32. W. Geis, R. Sinta, W. Mowers, S. J. Deneault, M. F. Marchant, K. E. Krohn, S. J. Spector, D. R. Calawa, and T. M. Lyszczarz, “Fabrication of crystalline organic waveguides with an exceptionally large electro-optic coefficient,” Appl. Phys. Lett.84(19), 3729–3731 (2004). [CrossRef]
  33. M. Xu, F. Li, T. Wang, J. Wu, L. Lu, L. Zhou, and Y. Su, “Design of an electro-optic modulator based on a silicon-plasmonic hybrid phase shifter,” J. Light Wave Tech.31, 1170–1177 (2013).
  34. W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett.9(12), 4403–4411 (2009). [CrossRef] [PubMed]
  35. L. Alloatti, D. Korn, R. Palmer, D. Hillerkuss, J. Li, A. Barklund, R. Dinu, J. Wieland, M. Fournier, J. Fedeli, H. Yu, W. Bogaerts, P. Dumon, R. Baets, C. Koos, W. Freude, and J. Leuthold, “42.7 Gbit/s electro-optic modulator in silicon technology,” Opt. Express19(12), 11841–11851 (2011). [CrossRef] [PubMed]
  36. S. Inoue and S. Yokoyama, “Highly compact organic electro-optic modulator based on nanoscale plasmon metal gap waveguides,” SPIE-OSA-IEEE7631, 763128 (2009).
  37. H. Nasari and M. S. Abrishamian, “Electrically tunable light focusing via a plasmonic lens,” J. Opt.14(12), 125002 (2012). [CrossRef]
  38. X. Mei, X. G. Huang, and T. Jin, “A sub-wavelength Electro-optic Switch Based on Plasmonic T-Shaped Waveguide,” Plasmonics6(4), 613–618 (2011). [CrossRef]
  39. L. R. Dalton, W. H. Steier, B. H. Robinson, C. Zhang, A. Ren, S. Garner, A. Chen, T. Londergan, L. Irwin, B. Carlson, L. Fifield, G. Phelan, C. Kincaid, J. Amenda, and A. Jen, “From molecules to opto-chips: organic electro-optic materials,” J. Mater. Chem.9(9), 1905–1920 (1999). [CrossRef]
  40. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics2(8), 496–500 (2008). [CrossRef]
  41. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express17(19), 16646–16653 (2009). [CrossRef] [PubMed]
  42. Y. Song, J. Wang, Q. Li, M. Yan, and M. Qiu, “Broadband coupler between silicon waveguide and hybrid plasmonic waveguide,” Opt. Express18(12), 13173–13179 (2010). [CrossRef] [PubMed]
  43. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  44. F. Michelotti, L. Dominici, E. Descrovi, N. Danz, and F. Menchini, “Thickness dependence of surface plasmon polariton dispersion in transparent conducting oxide films at 1.55 microm,” Opt. Lett.34(6), 839–841 (2009). [CrossRef] [PubMed]
  45. B. Chiou and J. Tsai, “Antireflective coating for ITO films deposited on glass substrate,” J. Mater. Sci. Mater. Electron.10(7), 491–495 (1999). [CrossRef]
  46. A. Melikyan, N. Lindenmann, S. Walheim, P. M. Leufke, S. Ulrich, J. Ye, P. Vincze, H. Hahn, Th. Schimmel, C. Koos, W. Freude, and J. Leuthold, “Surface plasmon polariton absorption modulator,” Opt. Express19(9), 8855–8869 (2011). [CrossRef] [PubMed]
  47. F. Neumann, Y. A. Genenko, C. Melzer, S. V. Yampolskii, and H. von Seggern, “Self-consistent analytical solution of a problem of charge-carrier injection at a conductor/insulator interface,” Phys. Rev. B75(20), 205322 (2007). [CrossRef]
  48. S. Bozhevolnyi, Plasmonic Nanoguide and Circuits (Pan Stanford Publishing, 2008), pp.10–20.
  49. W. L. Barnes, “Surface plasmon–polariton length scales: a route to sub-wavelength optics,” J. Opt. A, Pure Appl. Opt.8(4), S87–S93 (2006). [CrossRef]
  50. E. A. Bahaa Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1991), Chapter 18, pp. 696–737.
  51. D. A. B. Miller, “Device requirements for optical interconnects to silicon chips,” Proc. IEEE97(7), 1166–1185 (2009). [CrossRef]

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