Abstract

We present a polarization rotation and coupling scheme that rotates a TE0 mode in a silicon waveguide and simultaneously couples the rotated mode to a hybrid plasmonic (HP0) waveguide mode. Such a polarization rotation can be realized with a partially etched asymmetric hybrid plasmonic waveguide consisting of a silicon strip waveguide, a thin oxide spacer, and a metal cap made from copper, gold, silver or aluminum. Two implementations, one with and one without the tapering of the metal cap are presented, and different taper shapes (linear and exponential) are also analyzed. The devices have large 3 dB conversion bandwidths (over 200 nm at near infrared) and short length (< 5 μm), and achieve a maximum coupling factor of ∼ 78% with a linearly tapered silver metal cap.

© 2015 Optical Society of America

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References

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2014 (2)

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photon. Rev. 8, 394–408 (2014).
[Crossref]

S. Kim and M. Qi, “Copper nanorod array assisted silicon waveguide polarization beam splitter,” Opt. Express 22, 9508–9516 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (4)

J. N. Caspers, M. Alam, and M. Mojahedi, “Compact hybrid plasmonic polarization rotator,” Opt. Lett. 37, 4615–4617 (2012).
[Crossref] [PubMed]

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).
[Crossref]

V. J. Sorger, R. F. Oulton, R.-M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS bulletin 37, 728–738 (2012).
[Crossref]

2011 (1)

J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
[Crossref]

2010 (4)

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

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics the case for plasmonics,” Science 328, 440–441 (2010).
[Crossref] [PubMed]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

2009 (1)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

2008 (2)

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

Z. Wang and D. Dai, “Ultrasmall Si-nanowire-based polarization rotator,” J. Opt. Soc. Am. B 25, 747–753 (2008).
[Crossref]

2007 (1)

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

2006 (2)

B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006).
[Crossref]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron 12, 1678–1687 (2006).
[Crossref]

2005 (2)

1998 (1)

1996 (1)

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[Crossref]

1995 (1)

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

Aitchison, J. S.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photon. Rev. 8, 394–408 (2014).
[Crossref]

J. N. Caspers, J. S. Aitchison, and M. Mojahedi, “Experimental demonstration of an integrated hybrid plasmonic polarization rotator,” Opt. Lett. 38, 4054–4057 (2013).
[Crossref] [PubMed]

M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Supermode propagation in low index medium,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2007), paper JThD112.

Alam, M.

Alam, M. Z.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photon. Rev. 8, 394–408 (2014).
[Crossref]

M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Supermode propagation in low index medium,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2007), paper JThD112.

Atwater, H. A.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

Augendre, E.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

Boltasseva, A.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

Briggs, R. M.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

Brongersma, M. L.

M. L. Brongersma and V. M. Shalaev, “Applied physics the case for plasmonics,” Science 328, 440–441 (2010).
[Crossref] [PubMed]

Brooks, C.

Burgos, S. P.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

Caspers, J. N.

Chen, S.

J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
[Crossref]

Dai, D.

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

de Salvo, B.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Delacour, C.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Deng, H.

Djurišic, A. B.

Drachev, V. P.

Elazar, J. M.

Emani, N. K.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

Emboras, A.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Engheta, N.

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[Crossref] [PubMed]

Espiau de Lamaestre, R.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Fan, L.

Fathpour, S.

Fedeli, J.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Feigenbaum, E.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

Fontaine, M.

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[Crossref]

Genov, D.

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

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

Grandidier, J.

R. M. Briggs, J. Grandidier, S. P. Burgos, E. Feigenbaum, and H. A. Atwater, “Efficient coupling between dielectric-loaded plasmonic and silicon photonic waveguides,” Nano Lett. 10, 4851–4857 (2010).
[Crossref] [PubMed]

Grosse, P.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Haus, H.

Ishii, S.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

Jalali, B.

Jessop, P. E.

Kim, S.

Kwong, D.-L.

J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
[Crossref]

Lanzillotti-Kimura, N. D.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).
[Crossref]

Li, Q.

Lo, G.-Q.

J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
[Crossref]

Ma, R.-M.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).
[Crossref]

V. J. Sorger, R. F. Oulton, R.-M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS bulletin 37, 728–738 (2012).
[Crossref]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

Majewski, M. L.

Meier, J.

M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Supermode propagation in low index medium,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2007), paper JThD112.

Mojahedi, M.

M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photon. Rev. 8, 394–408 (2014).
[Crossref]

J. N. Caspers, J. S. Aitchison, and M. Mojahedi, “Experimental demonstration of an integrated hybrid plasmonic polarization rotator,” Opt. Lett. 38, 4054–4057 (2013).
[Crossref] [PubMed]

J. N. Caspers, M. Alam, and M. Mojahedi, “Compact hybrid plasmonic polarization rotator,” Opt. Lett. 37, 4615–4617 (2012).
[Crossref] [PubMed]

M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Supermode propagation in low index medium,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (Optical Society of America, 2007), paper JThD112.

Naik, G. V.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

Najar, A.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Nambiar, S.

A. Emboras, R. M. Briggs, A. Najar, S. Nambiar, C. Delacour, P. Grosse, E. Augendre, J. Fedeli, B. de Salvo, H. A. Atwater, and R. Espiau de Lamaestre, “Efficient coupler between silicon photonic and metal-insulator-silicon-metal plasmonic waveguides,” Appl. Phys. Lett. 101, 251117 (2012).
[Crossref]

Oulton, R. F.

V. J. Sorger, R. F. Oulton, R.-M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS bulletin 37, 728–738 (2012).
[Crossref]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

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

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, Vol. 3 (Academic, 1998).

Pennings, E. C.

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

Pile, D.

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

Qi, M.

Qiu, M.

Rakic, A. D.

Shalaev, V. M.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
[Crossref]

M. L. Brongersma and V. M. Shalaev, “Applied physics the case for plasmonics,” Science 328, 440–441 (2010).
[Crossref] [PubMed]

Soldano, L. B.

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

Song, Y.

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron 12, 1678–1687 (2006).
[Crossref]

Sorger, V. J.

V. J. Sorger, R. F. Oulton, R.-M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS bulletin 37, 728–738 (2012).
[Crossref]

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R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
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R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nature Photon. 2, 496–500 (2008).
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V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
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P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
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Xuan, Y.

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A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2006).

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[Crossref] [PubMed]

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J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
[Crossref]

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V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1, 17–22 (2012).
[Crossref]

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J. Zhang, S. Zhu, S. Chen, G.-Q. Lo, and D.-L. Kwong, “An ultracompact surface plasmon polariton-effect-based polarization rotator,” IEEE Photon. Technol. Lett. 23, 1606–1608 (2011).
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Figures (10)

Fig. 1
Fig. 1 Schematic of the polarization rotation and coupling (PRC) structure. The inset shows the cross-sectional view and parameters; hSi, hSiO2, and hCu are the heights of Si, SiO2, and Cu, respectively. w1 is the width of both Si and HP waveguides, and w2 is the width of Cu and SiO2 in the PRC region. Si, SiO2, Cu, and PMMA regions are colored in blue, grey, yellow, and cyan respectively. The PMMA cladding is not shown in the 3D illustration.
Fig. 2
Fig. 2 From left to right, normalized mode profiles showing the |E|, at Si waveguide, PRC, and HP waveguide cross-sections. Geometric parameters are set to hSi=230 nm, hSiO2 =50 nm, hCu=100 nm, w1=380 nm, and w2=200 nm.
Fig. 3
Fig. 3 Effective refractive index (neff) of the two lowest modes in PRC (red circle line for PRC0 mode and black square line for PRC1 mode), and corresponding coupling length of Lc (blue line) as a function of w2. Green line represents the neff=2.19, which corresponds to the neff of TE0,Si and HP0 modes.
Fig. 4
Fig. 4 Normalized field plots of real Ex (above) and Ey (below) components at the middle of waveguide width w1 (yz-plane), when w2=180 nm, Lc=4 μm, and λ0=1550 nm.
Fig. 5
Fig. 5 (a) CFHP0 and (b) PCE as a function of Lc for different w2 and tapered PRC: w2 = 160 nm (blue), w2 = 180 nm (green), w2 = 200 nm (red), w2 = 220 nm (cyan), w2 = 240 nm (purple), and tapered PRC (dashed black).
Fig. 6
Fig. 6 Estimated PCE of PRC1 mode as a function of w2, using the Eq. (2). FEM mode calculations are conducted for different w2, and the rotation angle of the optical axis θ is obtained using the Eq. (3). Device length is assumed to be Lc.
Fig. 7
Fig. 7 Schematic of the tapered polarization rotation and coupling (TPRC) structure. Geometric parameters and material composition are same as Fig. 1.
Fig. 8
Fig. 8 Calculated CFHP0 as a function of Lc for the TPRC with different metal caps and w2: Cu (w2 = 380 nm, blue), Ag (w2 = 375 nm, green), Au (w2 = 380 nm, red), and Al (w2 = 355 nm, cyan). The free space wavelength is λ0 = 1550 nm.
Fig. 9
Fig. 9 (a) Taper shapes according to shape function in Eq. (4) and Eq. (5), and (b) Corresponding CFHP0 as a function of Lc for different taper shapes: linear (blue), and exponentials with a = 1 (green) and a = −1 (red).
Fig. 10
Fig. 10 The CFHP0 as a function of the free space wavelength λ0 for the PRC (w2 = 180 μm and Lc = 4 μm, in blue) and the TPRC (Lc = 5 μm, in green).

Equations (5)

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P HP 0 , 2 = 1 2 Re [ ( E FEM × H HP 0 * ) z d s ( E HP 0 × H FEM * ) z d s ( E FEM × H HP 0 * ) z d s ] ,
PCE = sin 2 ( 2 θ ) sin 2 ( π L 2 L c ) ,
R = tan ( θ ) = | H x ( x , y ) | 2 d s | H y ( x , y ) | 2 d s .
Linear taper : z = z z 0 z 1 z 0 f ( z ) = z
Exponential taper : z = z z 0 z 1 z 0 f ( z , a ) = e a z 1 e a 1

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