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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16646–16653
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A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement

Daoxin Dai and Sailing He  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16646-16653 (2009)
http://dx.doi.org/10.1364/OE.17.016646


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Abstract

A hybrid plasmonic waveguide with a metal cap on a silicon-on-insulator rib (or slab) is presented. There is a low-index material nano-layer between the Si layer and the metal layer. The field enhancement in the nano-layer provides a nano-scale confinement of the optical field (e.g., 50nm × 5nm) when operates at the optical wavelength λ = 1550nm. The theoretical investigation also shows that the present hybrid plasmonic waveguide has a low loss and consequently a relatively long propagation distance (on the order of several tens of λ).

© 2009 OSA

1. Introduction

In the past years people have presented several three-dimensional structures which can support highly localized fields, e.g. narrow gaps between two metal interfaces [8

8. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]

10

10. F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]

,15

15. L. Liu, Z. H. Han, and S. L. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005). [CrossRef] [PubMed]

,16

16. G. Veronis and S. H. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

] and V-grooves in metals [11

11. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef] [PubMed]

,12

12. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

]. However, it is well known that such a nano-scale optical waveguide has a large loss and the propagation distance is usually at the scale of several micrometers. Recently, a hybrid plasmonic waveguide with a dielectric cylinder above a metal surface has been presented for subwavelength confinement and long propagation distance [17

17. 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]

]. However, it is not easy to fabricate such a waveguide structure due to the cylindrical structure. A rectangular plasmonic waveguide should be more attractive because it is possible to fabricate by using the standard planar lightwave circuit technology. In Ref [18

18. M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009). [CrossRef]

], the authors has given an analyses for the dispersion relation and loss of subwavelength confined mode of several metal-GaAs-gap waveguides, e.g., a GaAs-cylinder with a gap of SiO2 on metal, a rectangular GaAs strip above Ag-substrate, and a Ag-gap-GaAs strip on a SiO2 substrate.

2. Waveguide Structure and Analysis

When the thickness of the SiO2 layer between Si and metal is large (e.g., 0.5 μm), the fundamental mode field is confined well in the Si region and the metal layer almost does not influence the mode field distribution. In this case, the present structure is like a regular SOI nanowire. However, when the SiO2 thickness becomes smaller (e.g., <50 nm), the metal layer will introduce a significant influence on the field distribution of the guided mode. As an example, we choose the geometrical dimensions as follows: w co = 200 nm, h m = 100 nm, h SiO2 = 50 nm, and h Si_rib = h Si = 300 nm. We choose the wavelength λ = 1550 nm and the corresponding refractive indices for all the involved materials as n metal = 0.1453 + 11.3587i (Ag) [17

17. 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]

], n SiO2 = 1.445, and n Si = 3.455. Here we consider an intrinsic Si layer which has a negligible material loss at the window around 1550nm. When a doped Si layer is necessary for some special situations (e.g., when P-/N- contact is introduced), the loss due to the doping could be estimated by using the formula given in Ref [22

22. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]

]. For example, when the doping density N = 1019/cm3, the imaginary part of the refractive index n im = 0.2097, which should be included when estimating the loss of the waveguide. For the case with an intrinsic Si layer (which is usually used for passive optical components), Fig. 2
Fig. 2 The calculated field distribution for the major component Ey(x,y) of the quasi-TM fundamental mode of the present hybrid plasmonic waveguide with w co = 200nm and h SiO2 = 50nm. In this figure, the field distributions Ey(0, y) and Ey(x, 0) are also shown. One sees that the field at 50nm-SiO2 nano-layer is enhanced greatly.
shows the field distribution of the major-component Ey(x, y) for the quasi-TM fundamental mode calculated by using an FEM (finite element mothod)-based mode solver. In order to see the profile more clearly, we also plot the field profles Ey(x, 0) and Ey(0, y).

From the curve of Ey(0, y), one sees that the field at 50nm-SiO2 nano-layer is enhanced greatly. It is well known that there is a similar field enhancement in a low-index region in a pure-dielectric horizontal slot waveguide because of the strong discontinuity of the normal component of the electric field at the high-index-contrast interface [2

2. V. R. Almeida, Q. Xu, C. A. Barrios, M. Lipson, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

,20

20. R. Sun, P. Dong, N. N. Feng, C. Y. Hong, J. Michel, M. Lipson, and L. Kimerling, “Horizontal single and multiple slot waveguides: optical transmission at lambda = 1550 nm,” Opt. Express 15(26), 17967–17972 (2007). [CrossRef] [PubMed]

]. For the present hybrid plasmonic waveguide, the principle is different partially. At the Si-SiO2 interface, there is a strong discontinuity of the normal component of the electric field, which is the same as that in a pure-dielectric horizontal slot waveguide. On the other hand, at the SiO2-metal interface, surface plasmon (SP) wave is excited. The electrical field of the excited SP wave decays exponentially at both sides of the interface and has a peak at the interface. In the thin SiO2 layer, the field distribution could be regarded as the sum of two exponential functions. When the SiO2 layer is very thin (smaller than the evanescent penetration depth), the field at SiO2 layer is enhanced greatly, as shown in Fig. 2. In the following parts, we consider the designs with different thicknesses of SiO2 (h SiO2 = 50 nm, 20 nm, or 5 nm) and the other parameters are chosen as h m = 100 nm and h Si = 300 nm.

The insets in Fig. 3 (b) show the field distribution of the major-component Ey of the electrical fields. From these figures, one sees that there is more optical confinement in the Si layer when the core width increases. This is why the effective index real(n eff) and the propagation distance L prop increases as the core width increases, as shown in Fig. 3 (a) and 3(b), respectively. We also see that the thickness h SiO2 of the SiO2 nano-layer plays an important role for the propagation distance. When choosing a thinner SiO2 layer, the propagation distance becomes smaller. For the case with a relatively large thickness h SiO2 (e.g., 50 nm), more power confined in silicon region. Therefore, when the core width decreases, the power confined in the silicon region will change greatly. This is why the influence of the core width on the propagation distances is significant when the thickness h SiO2 is relatively large. For example, the propagation distance decreases from 432 μm to 76 μm when the core width decreases from 0.5 μm to 50 nm. In contrast, for the case with a very thin SiO2 layer (e.g., 5 nm), the propagation distance is around 50 μm and does not change greatly as the core width decreases. In summary, the calculation results in Fig. 3 (b) show that the present hybrid plasmonic waveguide supports a propagation distance on the order of several tens of wavelength λ, which is useful to develop plasmonic waveguide devices.

One should note that there is a trade-off between the dimension of the plasmon waveguide and its propagation distance. Since it is easy to realize a propagation distance over 104 μm by using a singlemode SOI nanowire when the core width w co > 300 nm, in this paper, we focus on the potential of the present hybrid plasmonic waveguides for a relatively long propagation as well as a nano-scale (<100 nm) optical confinement (which is beyond the ability of conventional pure dielectric optical waveguides, e.g., SOI nanowires).

In the analysis above, the Si layer is etched through. We note that the aspect ratio of such a waveguide will be high when the core width becomes very small (e.g., ~100 nm). This will make the fabrication difficult in some degree. A solution to avoid this problem is using a shallowly-etched Si layer (i.e., h Si_rib<H Si, as shown in Fig. 1). Figure 4
Fig. 4 For the cases of h SiO2 = 5 nm and w co = 100 nm, the real part of the effective refractive index n eff and the propagation distance L prop as the rib height h Si_rib decreases. When the height h Si_rib = 0, the Si part becomes a slab waveguide, in which case the propagation distance is close to 100μm (~60λ) and the fabrication is very easy because the etching becomes shallow.
shows the propagation distance L prop and the real part of the effective index as the Si rib height h Si_rib decreases from 0.3 μm to 0. Here we consider the case with h SiO2 = 5 nm (about λ/300) and the core wdith w co = 100 nm (about λ/15) in this example. From this figure, one sees that the propagation distance increases as the rib height h Si_rib decreases. According to the effective index method, a shallow silicon rib makes an equivalent layer with a larger index. This will make more power confined in silicon layer. Therefore, when the silicon rib decreases until zero, both the propagation distance and the real part of the effective index increase, as shown in Fig. 4.

Figure 6
Fig. 6 The calculated coupling length as the separation between two parallel hybrid plasmonic waveguides. The waveguide parameters are: h Si_rib = 0, h SiO2 = 5nm (~λ/300), and w co = 50nm (~λ/30).
shows the calculated coupling length of two parallel hybrid plasmonic waveguides with the same parameters as those used for Fig. 5 (h Si_rib = 0, h SiO2 = 5 nm, and w co = 50 nm). The coupling length is given by L c = π/(β oβ e), where β e and β o are the propagation constants of the even and odd super-modes of the system of the parallel waveguides (as shown by the inset). The coupling length is almost increases exponentially as the separation D increases, which is similar to the conventional dielectric optical waveguides. When the separation is decreased to 100nm, the coupling length is as small as 2.8μm. This makes it possible to realize a compact directional coupler (which is a basic element for photonic integration circuits).

3. Conclusion

We have studied a Si-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement. The present theoretical investigation has shown that a nano-scale (e.g., 50nm × 5nm) optical confinement is obtained with this hybrid plasmonic waveguide when it operates at 1550nm. At the same time, the low-loss enables the present hybrid plasmonic waveguide to have a relatively long propagation distance (on the order of 100 wavelengths). The fabrication the present hybrid plasmonic waveguide is simple and compatible with the standard processes for SOI wafers. Furthermore, our calculation has also shown that one could use a Si slab (instead of Si rib) under the metal cap (see Fig. 5), in which way the fabrication becomes much simpler and easier. With the present hybrid plasmonic waveguide, it is also possible to realize a low-voltage compact optical modulator when the nano-layer material between the Si layer and the metal layer has a high electro-optical coefficient. In order to connect with pure SOI nanowire when necessary, it is possible to introduce mode transformers in the similar way shown in Ref [24

24. N. N. Feng, R. Sun, J. Michel, and L. C. Kimerling, “Low-loss compact-size slotted waveguide polarization rotator and transformer,” Opt. Lett. 32(15), 2131–2133 (2007). [CrossRef] [PubMed]

]. by consisting of several adiabatic tapers.

Acknowledgement

This project was partially supported Zhejiang Provincial Natural Science Foundation (No. J20081048).

References and links

1.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics Devices Based on Silicon Microfabrication Technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [CrossRef]

2.

V. R. Almeida, Q. Xu, C. A. Barrios, M. Lipson, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

3.

L. Thylén, M. Qiu, and S. Anand, “Photonic crystals--a step towards integrated circuits for photonics,” ChemPhysChem 5(9), 1268–1283 (2004). [CrossRef] [PubMed]

4.

T. Goto, Y. Katagiri, H. Fukuda, H. Shinojima, Y. Nakano, I. Kobayashi, and Y. Mitsuoka, “Propagation loss measurement for surface plasmon-polariton modes at metal waveguides on semiconductor substrates,” Appl. Phys. Lett. 84(6), 852–854 (2004). [CrossRef]

5.

R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express 13(3), 977–984 (2005). [CrossRef] [PubMed]

6.

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef]

7.

B. Wang and G. P. Wang, “Surface plasmon polariton propagation in nanoscale metal gap waveguides,” Opt. Lett. 29(17), 1992–1994 (2004). [CrossRef] [PubMed]

8.

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]

9.

K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides,” Opt. Express 13(1), 256–266 (2005). [CrossRef] [PubMed]

10.

F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]

11.

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef] [PubMed]

12.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]

13.

D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30(10), 1186–1188 (2005). [CrossRef] [PubMed]

14.

S. S. Xiao, L. Liu, and M. Qiu, “Resonator channel drop filters in a plasmon-polaritons metal,” Opt. Express 14(7), 2932–2937 (2006). [CrossRef] [PubMed]

15.

L. Liu, Z. H. Han, and S. L. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005). [CrossRef] [PubMed]

16.

G. Veronis and S. H. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]

17.

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]

18.

M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009). [CrossRef]

19.

D. X. Dai, L. Yang, and S. H. He, “Ultrasmall thermally tunable microring resonator with a submicrometer heater on Si nanowires,” IEEE J. Lightwave Technol. 26(6), 704–709 (2008). [CrossRef]

20.

R. Sun, P. Dong, N. N. Feng, C. Y. Hong, J. Michel, M. Lipson, and L. Kimerling, “Horizontal single and multiple slot waveguides: optical transmission at lambda = 1550 nm,” Opt. Express 15(26), 17967–17972 (2007). [CrossRef] [PubMed]

21.

D. X. Dai, L. Liu, L. Wosinski, and S. He, “Design and fabrication of ultra-small overlapped AWG demultiplexer based on -Si nanowire waveguides,” Electron. Lett. 42(7), 400–402 (2006). [CrossRef]

22.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]

23.

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]

24.

N. N. Feng, R. Sun, J. Michel, and L. C. Kimerling, “Low-loss compact-size slotted waveguide polarization rotator and transformer,” Opt. Lett. 32(15), 2131–2133 (2007). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.6680) Optics at surfaces : Surface plasmons
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Optics at Surfaces

History
Original Manuscript: June 23, 2009
Revised Manuscript: August 20, 2009
Manuscript Accepted: August 31, 2009
Published: September 3, 2009

Citation
Daoxin Dai and Sailing He, "A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement," Opt. Express 17, 16646-16653 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16646


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References

  1. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics Devices Based on Silicon Microfabrication Technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [CrossRef]
  2. V. R. Almeida, Q. Xu, C. A. Barrios, M. Lipson, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]
  3. L. Thylén, M. Qiu, and S. Anand, “Photonic crystals--a step towards integrated circuits for photonics,” ChemPhysChem 5(9), 1268–1283 (2004). [CrossRef] [PubMed]
  4. T. Goto, Y. Katagiri, H. Fukuda, H. Shinojima, Y. Nakano, I. Kobayashi, and Y. Mitsuoka, “Propagation loss measurement for surface plasmon-polariton modes at metal waveguides on semiconductor substrates,” Appl. Phys. Lett. 84(6), 852–854 (2004). [CrossRef]
  5. R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express 13(3), 977–984 (2005). [CrossRef] [PubMed]
  6. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004). [CrossRef]
  7. B. Wang and G. P. Wang, “Surface plasmon polariton propagation in nanoscale metal gap waveguides,” Opt. Lett. 29(17), 1992–1994 (2004). [CrossRef] [PubMed]
  8. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003). [CrossRef]
  9. K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides,” Opt. Express 13(1), 256–266 (2005). [CrossRef] [PubMed]
  10. F. Kusunoki, T. Yotsuya, J. Takahara, and T. Kobayashi, “Propagation properties of guided waves in index-guided two-dimensional optical waveguides,” Appl. Phys. Lett. 86(21), 211101 (2005). [CrossRef]
  11. D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29(10), 1069–1071 (2004). [CrossRef] [PubMed]
  12. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]
  13. D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30(10), 1186–1188 (2005). [CrossRef] [PubMed]
  14. S. S. Xiao, L. Liu, and M. Qiu, “Resonator channel drop filters in a plasmon-polaritons metal,” Opt. Express 14(7), 2932–2937 (2006). [CrossRef] [PubMed]
  15. L. Liu, Z. H. Han, and S. L. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005). [CrossRef] [PubMed]
  16. G. Veronis and S. H. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]
  17. 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]
  18. M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21(6), 362–364 (2009). [CrossRef]
  19. D. X. Dai, L. Yang, and S. H. He, “Ultrasmall thermally tunable microring resonator with a submicrometer heater on Si nanowires,” IEEE J. Lightwave Technol. 26(6), 704–709 (2008). [CrossRef]
  20. R. Sun, P. Dong, N. N. Feng, C. Y. Hong, J. Michel, M. Lipson, and L. Kimerling, “Horizontal single and multiple slot waveguides: optical transmission at lambda = 1550 nm,” Opt. Express 15(26), 17967–17972 (2007). [CrossRef] [PubMed]
  21. D. X. Dai, L. Liu, L. Wosinski, and S. He, “Design and fabrication of ultra-small overlapped AWG demultiplexer based on -Si nanowire waveguides,” Electron. Lett. 42(7), 400–402 (2006). [CrossRef]
  22. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
  23. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31(14), 2133–2135 (2006). [CrossRef] [PubMed]
  24. N. N. Feng, R. Sun, J. Michel, and L. C. Kimerling, “Low-loss compact-size slotted waveguide polarization rotator and transformer,” Opt. Lett. 32(15), 2131–2133 (2007). [CrossRef] [PubMed]

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