Abstract

We propose and demonstrate theoretically both broadband and ultra-compact waveguide mode-order converters and polarization rotators based on asymmetrical nanostructures fabricated in silicon nanowires. A TE0-like to TE1-like mode-order converter with the footprint 0.8 × 5.3μm2 is realized by two cascaded trenches fully etched inside silicon nanowires. Within the wavelength interval between 1420nm and 1620nm, the transmittance of the device is larger than 0.95. The incident TE0-like mode is almost completely converted into TE1-like mode with purity larger than 0.9. Moreover, a polarization rotator or TE0-like to TM0-like mode converter with the footprint 0.4 × 6.4μm2 is realized by two cascaded trenches partially etched on the edges of silicon nanowires. Within the wavelength interval between 1360nm and 1560nm, the transmittance of the device is larger than 0.9. The incident TE0-like mode is almost completely converted into TM0-like mode with purity larger than 0.9. Our designs may lead to a new gateway towards manipulating the spatial modes and polarizations of waveguides in small footprint.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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2017 (1)

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

2016 (3)

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

K. Tan, Y. Huang, G.-Q. Lo, C. Lee, and C. Yu, “Compact highly-efficient polarization splitter and rotator based on 90° bends,” Opt. Express 24(13), 14506–14512 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (4)

2013 (5)

2012 (2)

2011 (1)

2007 (2)

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

W. Bogaerts, D. Taillaert, P. Dumon, D. Van Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express 15(4), 1567–1578 (2007).
[Crossref] [PubMed]

1999 (1)

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60(4), 2610–2618 (1999).
[Crossref]

Agarwal, A. M.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Baehr-Jones, T.

Baets, R.

Barwicz, T.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Bergmen, K.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Bogaerts, W.

Bowers, J. E.

Buhl, L. L.

Calvo, M. L.

Cheben, P.

Chen, C. P.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Chen, D.

Chen, J.

Chen, L.

Chen, Y.-K.

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60(4), 2610–2618 (1999).
[Crossref]

Da Ros, F.

Dadap, J. I.

Dai, D.

Desiatov, B.

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

Ding, Y.

Dong, P.

Driscoll, J. B.

Dumon, P.

Elesin, Y.

Fang, Q.

Fernandez, Í. M.

Frandsen, L. H.

Frellsen, L. F.

Gabrielli, L. H.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Goi, K.

Grote, R. R.

Guan, H.

Han, Z.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

He, Y.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Hochberg, M.

Huang, B.

Huang, Y.

Ippen, E. P.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Janz, S.

Jiang, X.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Kärtner, F. X.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Kim, M.-H.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Kusaka, H.

Kwong, D.-L.

Lapointe, J.

Lee, C.

Levy, U.

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

Li, X.

Li, Z.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Lim, A. E.-J.

Liow, T.-Y.

Lipson, M.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Liu, B.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Liu, W.

Lo, G.-Q.

Loncar, M.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Lu, J.

Lu, M.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

J. B. Driscoll, R. R. Grote, B. Souhan, J. I. Dadap, M. Lu, and R. M. Osgood, “Asymmetric Y junctions in silicon waveguides for on-chip mode-division multiplexing,” Opt. Lett. 38(11), 1854–1856 (2013).
[Crossref] [PubMed]

Luo, L.-W.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Mazurski, N.

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

Mitrovic, M.

Novack, A.

Ogawa, K.

Ohana, D.

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

Oka, A.

Ophir, N.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Ortega-Moñux, A.

Osgood, R. M.

Ou, H.

Overvig, A. C.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Peucheret, C.

Pluk, E.

Poitras, C. B.

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Popovic, M. A.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Qiu, C.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Rakich, P. T.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Ramos, C. A.

Schmid, J. H.

Shi, R.

Shi, Y.

Shrestha, S.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Sigmund, O.

Smith, H. I.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Socci, L.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Soref, R. A.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Souhan, B.

Stein, A.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Streshinsky, M.

Su, Y.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Taillaert, D.

Tan, K.

Tu, X.

Vachon, M.

Van Thourhout, D.

Velasco, A. V.

Vuckovic, J.

Wang, C.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Wang, J.

Wang, L.

Watts, M. R.

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60(4), 2610–2618 (1999).
[Crossref]

Xiao, X.

Xie, A.

Xie, C.

Xiong, Y.

Xu, D.-X.

Xu, J.

Yang, Q.

Ye, W. N.

Yu, C.

Yu, N.

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Yu, Y.

Yvind, K.

Zhang, Y.

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Zhou, L.

APL Photonics (1)

Y. Zhang, Y. He, X. Jiang, B. Liu, C. Qiu, Y. Su, and R. A. Soref, “Ultra-compact and highly efficient silicon polarization splitter and rotator,” APL Photonics 1(9), 091304 (2016).
[Crossref]

Nano Lett. (1)

D. Ohana, B. Desiatov, N. Mazurski, and U. Levy, “Dielectric metasurface as a platform for spatial mode conversion in nanoscale waveguides,” Nano Lett. 16(12), 7956–7961 (2016).
[Crossref] [PubMed]

Nat. Commun. (1)

L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergmen, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 3069 (2014).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

Z. Li, M.-H. Kim, C. Wang, Z. Han, S. Shrestha, A. C. Overvig, M. Lu, A. Stein, A. M. Agarwal, M. Lončar, and N. Yu, “Controlling propagation and coupling of waveguide modes using phase-gradient metasurfaces,” Nat. Nanotechnol. 12(7), 675–683 (2017).
[Crossref] [PubMed]

Nat. Photonics (1)

T. Barwicz, M. R. Watts, M. A. Popović, P. T. Rakich, L. Socci, F. X. Kärtner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007).
[Crossref]

Opt. Express (11)

W. Bogaerts, D. Taillaert, P. Dumon, D. Van Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express 15(4), 1567–1578 (2007).
[Crossref] [PubMed]

Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, and C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express 21(8), 10376–10382 (2013).
[Crossref] [PubMed]

D. Chen, X. Xiao, L. Wang, Y. Yu, W. Liu, and Q. Yang, “Low-loss and fabrication tolerant silicon mode-order converters based on novel compact tapers,” Opt. Express 23(9), 11152–11159 (2015).
[Crossref] [PubMed]

J. Lu and J. Vučković, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
[Crossref] [PubMed]

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Opt. Lett. (5)

Phys. Rev. B (1)

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[Crossref]

Other (1)

A summarization of reported waveguide mode-order converters and on-chip polarization rotators including the design principles and performances could be found in Supplementary information of Reference 21.

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Figures (5)

Figure 1
Figure 1 Eigenmodes of silicon nanowire and slot nanowire. (a)–(c) are propagation constants β of TE0-like and TE1-like modes supported by the 800nm-width (X direction) and 260nm-thickness (Y direction) silicon nanowire ((a), inset) and their X-component electric-field distributions at the wavelength of 1550 nm. (d)-(f) are those of the two TE-like eigenmodes supported by the slot nanowire which is the same nanowire ((a), inset) with a 70nm-width and −30nm off-center (in the X direction) trench etched in the core ((d), inset). The eigen-modes are calculated by using COMSOL MULTIPHYSICS software, the refractive index of Silicon is 3.46.
Fig. 2
Fig. 2 Schematic illustration and theoretical demonstration of the TE0-like to TE1-like mode-order converter. Along the 800nm-width and 260nm-thickness silicon nanowire, the proposed mode-order converter (a) consists of one linear inverse taper trench, one 70nm-width and 30nm-off-center trench (slot nanowire), one 70nm-width and 30nm-off-center trench (mirror-image slot nanowire), and one taper trench. The lengths of the fully etched trenches are 0.3μm,1.3μm,3.4μm and 0.3μm, respectively. (b)A schematic illustration of SMM. (c) Purity of TE1-like mode in the transmitted light through the mode-order converter with varying lengths of slot nanowire (h1) and mirror-slot nanowires (h2). The star marks the point of (h1 = 1.3μm, h2 = 3.4μm) (d) A 3D Lumerical FDTD simulation of the X-component electric-field distributions in the device when a TE0-like mode at the wavelength of 1550 nm incidents from the left side. (e) and (f) are total transmittance and purities of TE0- and TE1-like mode in the transmitted light for the incident TE0-like mode with varying wavelengths.
Fig. 3
Fig. 3 Eigenmodes of rectangular and L-shaped silicon nanowire. (a)–(c) are propagation constant β of the TE0-like and TM0-like modes supported by a 400nm-width and 260nm-thickness silicon nanowire ((a), inset) and their transverse electric-field distribution at the wavelength of 1450 nm. (d)-(f) are those of the two hybrid modes supported by the L-shaped silicon nanowire ((d), inset) which is the same nanowire ((a), inset) with a 140nm-width and 200nm-depth trench etched on the positive X direction. The small arrows in (b), (c), (e), (f) denote the directions of the transverse electric-field vectors, and the different color hues denote the relative amplitudes of the electric fields.
Fig. 4
Fig. 4 Schematic illustration and theoretical demonstration of the TE0-like to TM0-like mode converter (PR). Along the 400nm-width and 260nm-thickness silicon nanowire, the PR (a) consists of one taper trench, a 140nm-width trench etched on the positive X direction (L-shaped nanowire), a 5° ridge nanowire, a 140nm-width inverse trench etched on the negative X direction (mirror-image L-shaped nanowire), and one taper trench. Their lengths are 1.5μm, 0.8μm, 1.6μm, 1.0μm and 1.5μm, respectively. All the trenches and the ridge nanowire are partially etched with depths equal to 200nm. (b) A schematic illustration of SMM. (c) Purity of TM0 like mode in the transmitted light through the mode converter with varying lengths of L-shaped nanowire (h1) and mirror- L-shaped nanowires (h2). The star marks the point of (h1 = 0.8μm, h2 = 1.0μm) (d) A 3D Lumerical FDTD simulation of the X-and Y-component electric-field distributions in the device when a TE0-like mode at the wavelength of 1450 nm incident from the left side. (e) and (f) are total transmittance and purities of TE0- and TM0-like mode in the transmitted light from the PR for the incident TE0-like mode with the varying wavelengths.
Fig. 5
Fig. 5 Schematic illustration and theoretical demonstration of the TE0-like to TE1-like mode-order converter (a) and PR (d) on SOI. For the incident TE0-like mode with varying wavelengths, (b) and (c) are total transmittance and purities of TE0- and TE1-like mode in the transmitted light from the mode-order converter, (e) and (f) are total transmittance and purities of TE0- and TM0-like mode in the transmitted light from the PR.

Equations (5)

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E TE 0 = c 1 E TE R + c 2 E TE L , η E TE 1 = c 1 E TE R c 2 E TE L
σ( e x )[ x y z ]=[ x y z ], σ( e x )[ E x ( x,y,z ) E y ( x,y,z ) E z ( x,y,z ) ]=[ E x ( x,y,z ) E y ( x,y,z ) E z ( x,y,z ) ],
2 c 1 E TE R =2 c 2 σ( e x ) E TE L =σ( e x )( E TE 0 η E TE 1 ).
E TE 0 = d 1 E HM 1 + d 2 E HM 2 , γ E TM 0 = d 1 E HM 1 d 2 E HM 2 ,
2 d 1 E HM 1 =2 d 2 σ( e x ) E HM 2 =σ( e x )( E TE 0 γ E TM 0 ).

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