Abstract

We numerically demonstrated 1 × N multimode interference (MMI) splitters based on dielectric-loaded surface plasmon polariton waveguides (DLSPPW). The dependences of real part of effective index, propagation length and lateral mode profiles of the waveguides on the geometrical parameters are analyzed in detail. The transmission efficiency of the MMI splitter is as much as 60%, with their splitting length being in the range of several micrometers. The performance of the device is found in agreement with results predicted by theoretical MMI self-imaging theory. Such MMI splitter is important for implementation of high density photonic integration and lab-on-a-chip applications.

© 2009 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2009 (4)

2008 (5)

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

A. Polman, “Applied physics. Plasmonics applied,” Science 322(5903), 868–869 (2008).
[CrossRef] [PubMed]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon-polariton waveguides at telecommunication wavelengths: Excitation and characterization,” Appl. Phys. Lett. 92(1), 011124 (2008).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008).
[CrossRef]

2007 (3)

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

2006 (3)

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]

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

2005 (1)

1995 (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Aussenegg, F. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Bolger, P.

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-3-310 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2968 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon-polariton waveguides at telecommunication wavelengths: Excitation and characterization,” Appl. Phys. Lett. 92(1), 011124 (2008).
[CrossRef]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

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]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated Optical Components Utilizing Long-Range Surface Plasmon Polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
[CrossRef]

Brongersma, M. L.

L. Cao and M. L. Brongersma, “Active Plasmonics: Ultrafast Developments,” Nat. Photonics 3(1), 12–13 (2009).
[CrossRef]

Cao, L.

L. Cao and M. L. Brongersma, “Active Plasmonics: Ultrafast Developments,” Nat. Photonics 3(1), 12–13 (2009).
[CrossRef]

Chen, Z.

Dereux, A.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2968 .
[CrossRef] [PubMed]

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-3-310 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon-polariton waveguides at telecommunication wavelengths: Excitation and characterization,” Appl. Phys. Lett. 92(1), 011124 (2008).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

Devaux, E.

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]

Ditlbacher, H.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Drezet, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Ebbesen, T. W.

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]

Hohenau, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Holmgaard, T.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-3-310 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2968 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon-polariton waveguides at telecommunication wavelengths: Excitation and characterization,” Appl. Phys. Lett. 92(1), 011124 (2008).
[CrossRef]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[CrossRef]

Kjaer, K.

Krasavin, A. V.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-3-310 .
[CrossRef] [PubMed]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B 78(4), 045425 (2008).
[CrossRef]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007).
[CrossRef]

Krenn, J. R.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Laluet, J. Y.

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]

Larsen, M. S.

Leitner, A.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Leosson, K.

Markey, L.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34(3), 310–312 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-3-310 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded plasmonic waveguide-ring resonators,” Opt. Express 17(4), 2968–2975 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2968 .
[CrossRef] [PubMed]

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Bend- and splitting loss of dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 16(18), 13585–13592 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13585 .
[CrossRef] [PubMed]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, P. Bolger, and A. V. Zayats, “Efficient excitation of dielectric-loaded surface plasmon-polariton waveguides modes at telecommunication wavelengths,” Phys. Rev. B 78(16), 165431 (2008).
[CrossRef]

T. Holmgaard, S. I. Bozhevolnyi, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon-polariton waveguides at telecommunication wavelengths: Excitation and characterization,” Appl. Phys. Lett. 92(1), 011124 (2008).
[CrossRef]

Nikolajsen, T.

Ozbay, E.

E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006).
[CrossRef] [PubMed]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Polman, A.

A. Polman, “Applied physics. Plasmonics applied,” Science 322(5903), 868–869 (2008).
[CrossRef] [PubMed]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Steinberger, B.

B. Steinberger, A. Hohenau, H. Ditlbacher, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides: Bends and directional couplers,” Appl. Phys. Lett. 91(8), 081111 (2007).
[CrossRef]

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Stepanov, A. L.

B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006).
[CrossRef]

Volkov, V. S.

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]

Zayats, A. V.

T. Holmgaard, Z. Chen, S. I. Bozhevolnyi, L. Markey, A. Dereux, A. V. Krasavin, and A. V. Zayats, “Wavelength selection by dielectric-loaded plasmonic components,” Appl. Phys. Lett. 94(5), 051111 (2009).
[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the 1 × N MMI splitter based on dielectric-loaded surface plasmon polariton waveguides under investigation: t, PMMA layer thickness; W1 , PMMA width in the single-mode input and output waveguides; W and L, PMMA width and length in the multimode region; Wgap , distance between the adjacent output waveguides; Wtaper and Ltaper , hemline width and length of the coupling taper structure, respectively.

Fig. 2
Fig. 2

The dependences of real part of effective index (a) and propagation length (b) of the dielectric-loaded surface plasmon polariton waveguides on the waveguide width. The thickness of PMMA layer is fixed at 600 nm and the incident wavelength is 1550 nm. The inset in the graph (a) shows the index difference of the two lowest-order TM00 and TM01 modes and (b) gives a sketch of the transversal geometry under investigation.

Fig. 3
Fig. 3

Normalized field distributions of the magnitude of the vertical component of the electric field |Ey|, calculated by FEM: (a) shows the fundamental TM00 mode for W = 0.6 μm, (b) (c) (d) depict the two odd and one even modes TM00, TM01 and TM02 for W = 4 μm, respectively. The white lines show the interfaces of PMMA ridge and gold film.

Fig. 4
Fig. 4

Electric field intensity distributions in the xz plane located at 300 nm above the gold film for (a) 4-μm wide straight waveguide, (b) 4-μm wide two-fold splitter and (c) 6-μm wide three-fold splitter with taper structures, respectively. The results are calculated by 3D-FDTD. The outlines of the PMMA layers are shown as solid white lines.

Fig. 5
Fig. 5

Electric field intensity distributions (a) along the dashed white line AB and (b) along the dashed white line CD in Fig. 4(b) and Fig. 4(c), respectively. The mode overlap between the adjacent waveguides is very weak.

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